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Quark and gluon helicity flip generalized parton distributions (GPDs) encode the information on the nucleon structure in the transversity sector. In order to build a theoretically consistent phenomenological parametrization for these hadronic matrix element within the framework of the dual parametrization of GPDs (or with the equivalent approach of the SO(3) partial waves (PW) expansion with the Mellin-Barnes integral techniques) we establish the set of combinations of parton helicity flip GPDs suitable for the expansion in the cross channel SO(3) PWs.

10.22323/1.203.0227

1407.6119

e8d6b35fbb49ec8659451710103349d91e52f702

Toward modelization of quark and gluon transversity generalized parton distributions Towards modelization of quark and gluon transversity GPDs 23 Jul 2014 28 April -2 May 2014 B Pire K Semenov-Tian-Shansky L Szymanowski S Wallon K Semenov-Tian-Shansky IFPA, Département AGO CPhT École Polytechnique CNRS 91128PalaiseauFrance National Centre for Nuclear Research (NCBJ) Université de Liège 4000Liège, WarsawBelgium, Poland Université Paris-Sud CNRS 91405OrsayFrance Faculté de Physique UPMC Université Paris 06, 4 place Jussieu75252Paris, WarsawFrance, Poland Toward modelization of quark and gluon transversity generalized parton distributions Towards modelization of quark and gluon transversity GPDs 23 Jul 2014 28 April -2 May 2014XXII. International Workshop on Deep-Inelastic Scattering and Related Subjects, * Speaker. Quark and gluon helicity flip generalized parton distributions (GPDs) encode the information on the nucleon structure in the transversity sector. In order to build a theoretically consistent phenomenological parametrization for these hadronic matrix element within the framework of the dual parametrization of GPDs (or with the equivalent approach of the SO(3) partial waves (PW) expansion with the Mellin-Barnes integral techniques) we establish the set of combinations of parton helicity flip GPDs suitable for the expansion in the cross channel SO(3) PWs. Introduction and preliminaries The transversity partonic structure of hadrons constitutes a longstanding challenge for both theoretical and experimental studies [1]. The appealing feature of the transversity dependent sector is that due to the chiral-odd property of transversity quark distributions it turns out to be possible to clearly separate quark and gluonic contents. In view of the notorious experimental difficulties with accessing quark transversity distributions directly through inclusive dilepton production with transversely polarized beam and target [2], several detour approaches were proposed in the present day literature. Our strategy deals with the description of hard exclusive reactions within the collinear factorization approach in terms of transversity dependent generalized parton distributions (GPDs). The chiral-oddity of the quark helicity flip operator prevents the corresponding GPDs for contributing into photon or meson leptoproduction amplitudes at the leading twist. However, recently there were some attempts to circumvent this restriction [3,4]. In particular, the transverse target spin asymmetries measured by COMPASS in vector meson exclusive leptoproduction [5] have been interpreted [4] as a signal for transversity quark contributions. On the contrary, the gluon sector does not suffer from any selection rules and gluon helicity flip GPDs appear at the leading twist level in amplitudes of various hard exclusive reactions, for instance in the deeply virtual Compton scattering (DVCS) O(α s ) contribution to the leptoproduction of a real photon. This contribution can be separated through a harmonic analysis [6]. Particularly, the (3φ ) modulation of the interference contribution to the unpolarized beam-longitudinally polarized target asymmetry seen in the HERMES data [7] for the DVCS on a nucleon may call for a significant gluon transversity contribution. The practical applications of quark and gluon helicity flip GPD formalism require the construction of flexible phenomenological parametrizations of these non-perturbative objects. Below we review the main points of Ref. [8] in which we established the crossed channel properties of parton helicity flip GPDs that are of direct importance for a theoretically consistent model building in the spirit of the double partial wave expansion of GPDs. The phenomenological side of this study is postponed for future publications. Let us briefly specify our set of conventions. Throughout this paper we employ the usual GPD kinematical notations for the average momentum P = 1 2 (p + p ′ ), t-channel momentum transfer ∆ = p ′ − p and the skewness variable ξ . To the leading twist accuracy, the form factor decomposition of the non-forward nucleon matrix element of the quark tensor operatorÔ q T 1 contracted with the appropriate projector involves 4 invariant functions [9]: valued. Moreover, one may check [9] that H q T ,H q T , E q T are even functions of ξ whileẼ q T is an odd function of ξ . 1 2 dλ 2π e ixP + λ N(p ′ )|Ψ(−λ n/2)iσ +i Ψ(λ n/2)|N(p) = 1 2P +Ū (p ′ ) H q T iσ +i +H q T P + ∆ i − ∆ + P i m 2 +E q T γ + ∆ i − ∆ + γ i 2m +Ẽ q T γ + P i − P + γ i m U (p),(1. Similarly, the parametrization of the nucleon matrix element of the appropriately projected gluon tensor operatorÔ g T to the leading twist accuracy involves four invariant functions [9]: 1 P + dλ 2π e ixP + λ p ′ |SG +i (−λ n/2)G j+ (λ n/2)|p = S 1 2P + P + ∆ j − ∆ + P j 2mP +Ū (p ′ ) H g T iσ +i +H g T P + ∆ i − ∆ + P i m 2 + E g T γ + ∆ i − ∆ + γ i 2m +Ẽ g T γ + P i − P + γ i m U (p), (1.2) where the S symbol stands for the symmetrization in the two transverse spatial indices and removal of the corresponding trace. From the combination of hermiticity and T -invariance they are real valued. H g T , E g T ,H g T are even functions of ξ whileẼ g T is an odd function of ξ . Moreover, the C-invariance demands that H g T , E g T ,H g T andẼ g T are even functions of x. SO(3) partial wave expansion of quark and gluon GPDs with helicity flip The realistic strategy for extracting GPDs from the data relies on employing of phenomenologically motivated GPD representations and simultaneous fitting procedures for the complete set of observable quantities. The clue for building up a valid phenomenological representation for GPDs is provided by implementation of the non-trivial requirements following from the fundamental properties of the underlying quantum field theory. Historically, one of the first parametrizations for GPDs suitable for phenomenological applications -the famous Radyushkin double distribution Ansatz -was based on the double distribution representation for GPDs. It is employed within the extremely popular Vanderhaeghen-Guichon-Guidal (VGG) code [10] for the DVCS observables and saw some success in the description of the available data. The alternative way for building up of a GPD representation resides on the expansion of GPDs over a suitable orthogonal polynomial basis in order to achieve the factorization of certain variable dependence. The appealing possibility is to perform the expansion of GPDs over the conformal PW basis in order to achieve the diagonalization of the leading order evolution operator. Nowadays two main versions of such GPD representations are utilized in phenomenology: the one based on the Mellin-Barnes integral techniques [11] and the one using the idea of the Shuvaev-Noritzsch transformation [12]. It turns out extremely instructive to further expand the conformal moments over the basis of the t-channel SO(3) rotation group partial waves. In the context of the Shuvaev transform techniques the resulting GPD representation is known as the dual parametrization of GPDs [13]. Within the Mellin-Barnes integral techniques [11] this version of the conformal partial wave expansion is referred in the literature as the SO(3) partial wave expansion. Each version of the formalism employs a rather intricate mathematical apparatus, however, as argued in [14], these two approaches turn out to be completely equivalent. Finding out the combinations of GPDs suitable for the t-channel SO(3) partial waves and the choice of the appropriate basis of the orthogonal polynomials represents an important task. For example, for the case of the unpolarized quark and gluon nucleon GPDs this kind of analysis gives rise to the so-called electric and magnetic combinations of GPDs [15]: H E {q,g} = H {q,g} + τE {q,g} ; H M {q,g} = H {q,g} + E {q,g} , where τ ≡ ∆ 2 4m 2 . These combinations are to be expanded respectively in terms of P J (cos θ ) and P ′ J (cos θ ), with P J (cos θ ) standing for the Legendre polynomials and θ referring to the t-channel scattering angle in the NN center-of-mass frame. Following the receipt of sect. 4.2 of ref. [15], in order to identify the combinations of quark helicity flip GPDs suitable for the partial wave expansion in the t-channel partial waves, we consider the form factor decomposition of the N-th Mellin moments of quark and gluon helicity flip GPDs analytically continued to the cross channel (t > 0). Thus we are dealing with the form factor decomposition of the N-th Mellin moments of quark helicity flip NN generalized distribution amplitudes (GDAs). To find which partial waves contribute into the corresponding matrix elements, we compute the spin-tensor structures for spinors of definite (usual) helicity in the NN center-ofmass (CMS) frame using the explicit expressions for the nucleon spinors with definite ordinary helicity. Let us briefly review the main stages of the calculation for the case of quark helicity flip GPDs. We project out the combination of the matrix elements with definite helicity J 3 = ±1 of the corresponding operator: N(p ′ , λ ′ )N(−p, λ )|Ô q +1,++...+ T |0 ± i N(p ′ , λ ′ )N(−p, λ )|Ô q +2,++...+ T |0 ≡ N(p ′ , λ ′ )N(−p, λ )|Ô q +(1±i2),++...+ T |0 . (2.1) The combinations (2.1) possess definite phases depending on the azimuthal angle φ . Now one can decompose (2.1) in the partial waves with total angular momentum J. The θ dependence is governed by the Wigner "small-d" rotation functions d J J 3 ,|λ ′ −λ | . For the case |λ ′ − λ | = 0 (i.e. the aligned configuration of nucleon and antinucleon helicities 2 (N ↑N↑ or N ↓N↓ )) and J 3 = ±1 one has to use d J ±1,0 (θ ) = (±1) 1 J(J + 1) sin θ P ′ J (cos θ ). (2.2) For the case when |λ ′ − λ | = 1 (i.e. the opposite helicity configuration of nucleon and antinucleon: N ↑N↓ or N ↓N↑ ) depending on the operator helicity J 3 = ±1 (2.1) is to be expanded in d J ±1,1 (θ ) = 1 J(J + 1) (1 ± cos θ ) P ′ J (cos θ ) + cos θ P ′′ J (cos θ ) ∓ P ′′ J (cos θ ) . (2.3) After the inverse crossing (2.1) back to the s-channel, within the DVCS kinematics cos θ up to higher twist corrections becomes cos θ → 1 ξ β + O(1/Q 2 ), where β ≡ 1 − 4m 2 t . At this stage we switch to massless hadrons so that we could consider hadron helicities as true quantum numbers thus making simple the crossing relation between the corresponding partial amplitudes (in particular excluding mixing). This implies setting β = 1 (which means systematically neglecting the threshold corrections ∼ 1 − 4m 2 t ). However, up to the very end we keep the non-zero mass within the Dirac spinors to keep the counting of independent tensor structures. Finally, we conclude that the following combinations of quark helicity flip GPDs are to be expanded in P ′ J (1/ξ ): τH q T (x, ξ , ∆ 2 ) − 1 2 E q T (x, ξ , ∆ 2 ); −H q T (x, ξ , ∆ 2 ) + τH q T (x, ξ , ∆ 2 ) − 1 2 E q T (x, ξ , ∆ 2 ). (2.4) while the combinations H q T (x, ξ , ∆ 2 ) + τH q T (x, ξ , ∆ 2 ) ± τẼ q T (x, ξ , ∆ 2 ) (2.5) are to be expanded in P ′ J (1/ξ ) + 1∓ξ ξ P ′′ J (1/ξ ). The gluon case can be considered according to the same pattern (see Sec.3.2 of Ref. [8]) giving rise to the combinations of matrix elements expanded in terms of the Wigner functions d J ±2,0 (θ ), d J ±2,1 (θ ) and d J 0,0 (θ ). The method also allows to work out the set of selection rules for the J PC quantum numbers for the t-channel resonance exchanges contributing into the N-th Mellin moments of quark helicity flip GPDs. Due to the CPT invariance, this kind of J PC matching in the cross channel automatically ensures the T invariance and the correct counting of the independent generalized form factors of the operator matrix element in the direct channel. The selection rules we establish coincide with those worked out with the general method of X. Ji and R. Lebed [16]. The alternative method to work out the set of quark and gluon helicity flip GPDs suitable for the partial wave expansion in the cross-channel partial waves consists in the explicit calculation of the cross channel spin-J resonance contributions into corresponding GPD. The advantage of this method is that it is fully covariant and allows to determine the net resonance exchange contributions into scalar invariant functions H q,g T , E q,g T ,H q,g T ,Ẽ q,g T . Within this approach the nucleon matrix element of the light-cone operatorÔ is represented as an infinite sum of t-channel resonance exchange contributions. Symbolically it can be written in the following form: N(p ′ )|Ô |N(p) ∼ ∑ R J ∑ polarizations of R J 1 ∆ 2 − M 2 R J × N(p ′ )R J (∆)|N(p) V R J NN eff.vertex ⊗ 0|Ô|R J (∆) Fourier Transform of DA of R J , where M R J stand for the resonance masses and ⊗ denotes the convolution in the appropriate Lorentz indices. The resulting on-shell polarization sums for spin-J resonances can be performed with the contracted projectors method (see e.g. Chapter I of [17]). This calculation allows to recover the same combinations of quark and gluon helicity flip GPDs suitable for SO(3) PW expansion in the t-channel. Moreover, as a byproduct, we build up a simple f 2 (1270) meson exchange model for gluon helicity flip GPDs that is similar to b 1 meson exchange model for quark helicity flip GPDs suggested in [18]. The relevant f 2 NN coupling constants can be obtained from low energy NN scattering studies and the gluon helicity flip distribution amplitude normalization constant can be estimated as suggested in [19]. This model allows for the first time to work out the physical normalization of gluon helicity flip GPDs. Conclusions In order to construct a theoretically consistent parametrization of these hadronic matrix elements, we work out the set of combinations of the transversity GPDs suitable for the SO(3) PW expansion in the cross-channel. This universal result will help us to build up a flexible parametrization of these important hadronic non-perturbative quantities, using for instance the approaches based on the conformal PW expansion of GPDs such as the Mellin-Barnes integral or the dual parametrization techniques. We also propose a simple f 2 (1270) meson exchange model for gluon helicity flip T 1) where n denotes the light-cone vector and the Latin index i = 1, 2 refers to the transverse spatial directions. Each of the four invariant functions H depend on the usual GPD variables. Due to hermiticity and time reversal invariance, the four invariant functions are real Here and in eq. (1.2) we omit the Wilson gauge links by sticking to the light-cone gauge A + = 0. Note, that our hadron helicity labeling refers to the t-channel. Obviously, when crossing back to the direct channel the helicity λ is reversed. GPDs that allows to estimate the physical normalization of gluon transversity effects. 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Phys. B 629, 323 (2002) [hep-ph/0112108]. . A Airapetian, HERMES CollaborationNucl. Phys. B. 842265A. Airapetian et al. [HERMES Collaboration], Nucl. Phys. B 842, 265 (2011). . B Pire, K Semenov-Tian-Shansky, L Szymanowski, S Wallon, arXiv:1403.0803Eur. Phys. J. A. 5090hep-phB. Pire, K. Semenov-Tian-Shansky, L. Szymanowski and S. Wallon, Eur. Phys. J. A 50, 90 (2014) [arXiv:1403.0803 [hep-ph]]. . M Diehl, arXiv:hep-ph/0101335Eur. Phys. J. C. 19485M. Diehl, Eur. Phys. J. C 19, 485 (2001) [arXiv:hep-ph/0101335]. . M , arXiv:1003.0307Phys. Lett. B. 689156hep-phM. Guidal, Phys. Lett. B 689, 156 (2010) [arXiv:1003.0307 [hep-ph]]. . D Mueller, A Schafer, hep-ph/0509204Nucl. Phys. B. 7391D. Mueller and A. Schafer, Nucl. Phys. B 739, 1 (2006) [hep-ph/0509204]. . A G Shuvaev, hep-ph/9902318Phys. Rev. 60116005A. G. Shuvaev, Phys. Rev. D60, 116005 (1999), [hep-ph/9902318]; . J D Noritzsch, hep-ph/0004012Phys. Rev. 6254015J. D. Noritzsch, Phys. Rev. D62, 054015 (2000), [hep-ph/0004012]. On 'dual' parametrizations of generalized parton distributions. M V Polyakov, A G Shuvaev, hep-ph/0207153M. V. Polyakov and A. G. Shuvaev, "On 'dual' parametrizations of generalized parton distributions", hep-ph/0207153; . M V Polyakov, K M Semenov-Tian-Shansky, arXiv:0811.2901Eur. Phys. J. A. 40181hep-phM. V. Polyakov and K. M. Semenov-Tian-Shansky, Eur. Phys. J. A 40, 181 (2009) [arXiv:0811.2901 [hep-ph]]. . K M Semenov-Tian-Shansky, arXiv:1001.2711Eur. Phys. J. A. 45217hep-phK. M. Semenov-Tian-Shansky, Eur. Phys. J. A 45, 217 (2010) [arXiv:1001.2711 [hep-ph]]. . D Mueller, M Polyakov, K Semenov-Tian-Shansky, in preparationD. Mueller, M. Polyakov and K. Semenov-Tian-Shansky, in preparation. . M Diehl, hep-ph/0307382Phys. Rept. 38841M. Diehl, Phys. Rept. 388, 41 (2003) [hep-ph/0307382]. . X. -D Ji, R F Lebed, hep-ph/0012160Phys. Rev. D. 6376005X. -D. Ji and R. F. Lebed, Phys. Rev. D 63, 076005 (2001) [hep-ph/0012160]. 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In this paper, we theoretically prove that the Gaussian quantum discord state of optical field can be used to complete continuous variable (CV) quantum key distribution (QKD). The calculation shows that secret key can be distilled with a Gaussian quantum discord state against entangling cloner attack. Secret key rate is increased with the increasing of quantum discord for CV QKD with the Gaussian quantum discord state. Although the calculated results point out that secret key rate using the Gaussian quantum discord state is lower than that using squeezed state and coherent state at the same energy level, we demonstrate that the Gaussian quantum discord, which only involving quantum correlation without the existence of entanglement, may provide a new resource for realizing CV QKD.

10.1007/s11434-014-0193-x

1310.4253

0272dc98e3c06073c12ec53677d6f3feac69670c

Applying Gaussian quantum discord to quantum key distribution 11 Jun 2014 Xiaolong Su Institute of Opto-Electronics State Key Laboratory of Quantum Optics and Quantum Optics Devices Shanxi University 030006TaiyuanPeople's Republic of China Applying Gaussian quantum discord to quantum key distribution 11 Jun 2014arXiv:1310.4253v2 [quant-ph]numbers: 0367Bg0367Lx0365Ud4250Dv In this paper, we theoretically prove that the Gaussian quantum discord state of optical field can be used to complete continuous variable (CV) quantum key distribution (QKD). The calculation shows that secret key can be distilled with a Gaussian quantum discord state against entangling cloner attack. Secret key rate is increased with the increasing of quantum discord for CV QKD with the Gaussian quantum discord state. Although the calculated results point out that secret key rate using the Gaussian quantum discord state is lower than that using squeezed state and coherent state at the same energy level, we demonstrate that the Gaussian quantum discord, which only involving quantum correlation without the existence of entanglement, may provide a new resource for realizing CV QKD. I. INTRODUCTION Quantum correlation, which is measured by quantum discord [1][2][3], is a fundamental resource for quantum information processing tasks. It has been shown that some quantum computational tasks based on a single qubit can be carried out by separable (that is, non-entangled) states that nonetheless carries non-classical correlations [4][5][6]. Recently, quantum discord is extended to twomode Gaussian states [7,8]. A two-mode Gaussian state is entangled with Gaussian quantum discord D > 1, when 0 ≤ D ≤ 1 we have either separable or entangled states. Gaussian quantum discord has been experimentally demonstrated too [9][10][11]. Quantum key distribution (QKD) allows two legitimate parties, Alice and Bob who are linked by a quantum channel and an authenticated classical channel, to establish the secret key only known by themselves. Continuous variables (CV) QKD using Gaussian quantum resource state, such as entangled state, squeezed state and coherent state, as the resource state, along with reconciliation and privacy amplification procedure to distill the secret key [12]. There are two type of QKD schemes, one is called prepare-and-measure scheme, the other is entanglement-based scheme. The equivalence between these two type CV QKD schemes has been proved. QKD with coherent state (squeezed state) has been proved to be equivalent to heterodyning (homodyning) one of the two entangled modes of an Einstein-Podolsky-Rosen (EPR) entangled state [13]. Generally, the entanglementbased QKD model is used to investigate the security of CV QKD. The security of CV QKD scheme has been analyzed [14][15][16], and it has been proved to be unconditionally secure, that is, secure against arbitrary attacks over long distance [17,18]. Recently, a CV QKD scheme with thermal states is also proposed and proved to be secure against collective Gaussian attacks [19]. * Electronic address: [email protected] Very recently, it has been shown that quantum discord can be used as a resource for QKD in general [20]. What we concerned is the role of Gaussian quantum discord in CV QKD. In this paper, we apply a two-mode Gaussian discord state, where only quantum correlation exists and without entanglement, to implement CV QKD. The calculation shows that the secret key can be distilled with the two-mode Gaussian discord state against entangling cloner attack, which is the most important and practical example of collective Gaussian attack. The secret key rate of the QKD scheme with Gaussian discord state is increased with the increasing of the quantum discord. The secret key rates of the CV QKD schemes with the Gaussian discord state, squeezed state and coherent state (no-switching QKD) are compared. Although squeezed state and coherent state offer higher secret key rate than the Gaussian discord state, we demonstrate the Gaussian discord can be used to establish secret key. II. THE GAUSSIAN DISCORD STATE AND QKD SCHEME The QKD scheme with a two-mode Gaussian quantum discord state and entangled state is shown in Fig. 1. Figure 1(a) shows a two-mode Gaussian discord state, as shown in [9], which is prepared by correlated (anticorrelated) displacement of two coherent states in the amplitude (phase) quadrature with a discording noise V . Figure 1(b) shows an EPR entangled state with a variance V E = cosh 2r, where r ∈ [0, ∞) is the squeezing parameter. The amplitude and phase quadratures of an optical modeâ are defined asX a =â +â † and Y a = (â −â † )/i, respectively. The variances of amplitude and phase quadratures for a vacuum (coherent) state are V (X v ) = V (Ŷ v ) = 1. The covariance matrix of the twomode Gaussian quantum resource state in Fig. 1(a) and (b) is given by where I and Z are the Pauli matrices σ = αI γZ γZ βI ,(1)I = 1 0 0 1 , Z = 1 0 0 −1 ,(2)α = β = V D = V + 1, γ = V for the two-mode Gaussian discord state and α = β = V E , γ = V 2 E − 1 for the EPR entangled state, respectively. Quantum discord is defined as the difference between two quantum analogues of classically equivalent expression of the mutual information. The Gaussian quantum discord of a two-mode Gaussian state is given by [8] D AB = f ( I 2 ) − f (ν − ) − f (ν + ) + f ( √ E min ),(3) where f (x) = ( x+1 2 ) log x+1 2 − ( x−1 2 ) log x−1 2 , ν ± = ∆ ± √ ∆ 2 − 4 det σ 2(4) are the symplectic eigenvalues of a two-mode covariance matrix σ = A C C B with det σ as the determinant of covariance matrix and ∆ = det A + det B + 2 det C, and E min =    2I 2 3 +(I2−1)(I4−I1)+2|I3| √ I 2 3 +(I2−1)(I4−I1) (I2−1) 2 a) I1I2−I 2 3 +I4− √ I 4 3 +(I4−I1I2) 2 −2I 2 3 (I4+I1I2) 2I2 b) (5) where a) applies if (I 4 −I 1 I 2 ) 2 ≤ I 2 3 (I 2 +1)(I 1 +I 4 ) and b) applies otherwise. I 1 = det A, I 2 = det B, I 3 = det C, I 4 = det σ are the symplectic invariants. PPT criterion is a necessary and sufficient criterion for entanglement of Gaussian state [21,22]. A Gaussian state is entangled iffν − < 1, whereν − is the smallest symplectic eigenvalue of partial transposed covariance matrix for two-mode Gaussian state, which is given by [23]ν − = ∆ − ∆2 − 4 det σ 2 (6) where∆ = det A + det B − 2 det C. Based on the covariance matrix in eq. (1) for the Gaussian discord state, we calculated the quantum discord and smallest symplectic eigenvalue of PPT criterion, which are shown in Fig. 2. As shown in Fig. 2(a), the quantum discord is increased dramatically with the increasing of input variance V D in the region of V D ∈ [1, 100]. When V D > 100, the quantum discord increased slowly with the increasing of V D . The smallest quantum discord is 0.12 at V D = 1. The quantum discord is always smaller than 1. In Fig. 2(b), the smallest symplectic eigenvalue of partial transposed covariance matrix is always 1, which means that there is no entanglement in the Gaussian discord state. Figure 1(c) shows the CV QKD scheme with a twomode Gaussian state as quantum resource state, which can be the two-mode Gaussian discord state or the EPR entangled state. Alice hold modeâ, and transmitted modeb to Bob over the quantum channel. Here, we consider that Alice and Bob perform homodyne (Hom) or heterodyne (Het) detection on their own beam, which corresponds to the CV QKD scheme with homodyne or heterodyne detection. We assume that Eve perform entangling cloner attack [13], which is the most important and practical example of a collective Gaussian attack [17,[24][25][26], to steal the information. She prepares an ancillary EPR entangled states with variance W , which corresponds to the excess noise δ = W − 1 in [27] and ǫ = (W − 1)(1 − T )/T in [13]. W = 1 means there is no excess noise (δ = 0) in the channel, when W > 1, there is excess noise (δ = W − 1) in the channel. She keeps one modeÊ ′′ and mixed the other modeÊ with the transmitted modeb in the quantum channel by a beam splitter, leading to the output modeÊ ′ . Eve's output modes are stored in a quantum memory and detected collectively at the end of the protocol. Eve's final measurement is opti-mized based on Alice and Bob's classical communication. After communication is completed, Alice and Bob perform reconciliation, error correction [28,29] and privacy amplification [30] to distill final secret key. III. SECURITY OF THE CV QKD SCHEME A. Homodyne detection In the CV QKD scheme with homodyne detection, Alice and Bob perform homodyne detection on their own beams to measure the amplitude or phase quadrature, respectively. For CV QKD with EPR entangled state, homodyning one of the entangled beam is equivalent to the CV QKD with squeezed state. So we will compare the Gaussian discord state QKD with squeezed state QKD in this section. In the following, we use the variable X to represent amplitude or phase quadrature of an optical mode to analyze the secret key without losing the generality. Direct reconciliation. In direct reconciliation, Bob attempts to guess what Alice sent. The secret key rate is given by K DR = I(X A : X B ) − I(X A : E),(7) where I(X A : X B ) = H(X B ) − H(X B |X A ),(8) is the mutual information between Alice and Bob, with H(X B ) = (1/2) log 2 V (X B ) and H(X B |X A ) = (1/2) log 2 V (X B |X A ) being the total and conditional Shannon entropies. Eve's information is I(X A : E) = S(E) − S(E|X A ),(9) where S(·) is the von Neumann entropy. The von Neumann entropy of a Gaussian state ρ can be expressed in terms of its symplectic eigenvalues [31] S(ρ) = n k=1 g(ν k ), with g(ν) = 1 2 (ν + 1) log 2 [ 1 2 (ν + 1)] − 1 2 (ν − 1) log 2 [ 1 2 (ν − 1)], where ν = {ν 1 , . ..ν n } are the symplectic eigenvalues of Gaussian state ρ. The symplectic spectrum ν = {ν 1 , ...ν n } of an arbitrary correlation matrix σ can be calculated by finding the (standard) eigenvalues of the matrix |iΩσ|, where Ω defines the symplectic form and is given by [12] Ω = n k=1 0 1 −1 0 .(11) Here is the direct sum indicating adding matrices on the block diagonal. In Fig. 1(c), the covariance matrix of the two-mode Gaussian state distributed between Alice and Bob in the CV QKD is given by σ AB = V A I γ ′ Z γ ′ Z V B I ,(12) where V A = V E , V B = T V E +(1−T )W , γ ′ = T (V 2 E − 1) for the EPR entangled state and V A = V D , V B = T V D + (1 − T )W , γ ′ = √ T V for the Gaussian discord state, respectively. The conditional variance is defined as [32] V X|Y = V (X) − | XY | 2 /V (Y ) . So Bob's conditional variance in homodyne detection is given by V B|A = V B − γ ′2 V A .(13) The mutual information between Alice and Bob is I Hom (X A : X B ) = 1 2 log 2 [V B /V B|A ] , which is same for the direct and reverse reconciliation. Eve's covariance matrix is made up from the modesÊ ′ andÊ ′′ , which is σ E = e v I ϕZ ϕZ W I ,(14) where e v = (1 − T )V A + T W , ϕ = T (W 2 − 1). In order to obtain S(E|X A ) we need to calculate the symplectic spectrum of the conditional covariance matrix σ E|XA , which represents the covariance matrix of Eve's system where modeâ has been measured by Alice using homodyne detection and is given by [12,33,34] σ E|XA = σ E − (V A ) −1 DΠD T ,(15) where Π = 1 0 0 0 , and D is the matrix describing the quantum correlations between Eve' modes and Alice's mode, which is given by D = X E´XA I X E ′′ X A Z = ζI ηZ ,(17) where ζ = √ 1 − T V A , η = 0. Reverse reconciliation. The 3 dB loss limit on the transmission line in the CV QKD [35] can be beaten with the reverse reconciliation [36,37] or the post-selection [38]. In reverse reconciliation, Alice attempts to guess what was received by Bob rather than Bob guessing what was sent by Alice [36]. Such a reverse reconciliation protocol gives Alice an advantage over a potential eavesdropper Eve. In reverse reconciliation, the secret key rate is K RR = I(X A : X B ) − I(X B : E),(18) where the mutual information between Alice and Bob I(X A : X B ) is same with what obtained above. Eve's information is given by I(X B : E) = S(E) − S(E|X B ).(19) The conditional covariance matrix σ E|XB , which represents the covariance matrix of a system where one of the modes has been measured by homodyne detection (in this case Bob), is given by [12,33,34] σ E|XB = σ E − (V B ) −1 DΠD T .(20) Here D is the matrix describing the quantum correlations between Eve' modes and Bob's mode, which is given by D = X E´XB I X E ′′ X B Z = ζ ′ I η ′ Z ,(21) where Figure 3 shows the secret key rate of the CV QKD scheme with homodyne detection, (a) and (b) are corresponding to the direct and reverse reconciliation, respectively. Solid (black) and Dashed (blue) lines are the secret key rates for the Gaussian discord state with variance V D =40 (typical experimental realistic modulation level [36]) and 1000, respectively. Dotted (red) line is the secret key rate for the squeezed state with variance V E =40. All curves are plotted with excess noise W=1. Comparing the solid and dotted lines in Fig. 3, it is obvious that secret key rate for squeezed state is greater than that for Gaussian discord state at the same energy level in both direct and reverse reconciliation. Comparing solid and dashed lines, we find that the secret key rate is increased with the increasing of the discording noise for the CV QKD with the Gaussian discord state with homodyne detection in both direct and reverse reconciliation. ζ ′ = T (1 − T )(W − V A ), η ′ = (1 − T )(W 2 − 1). B. Heterodyne detection In the CV QKD scheme with heterodyne detection, Alice and Bob perform heterodyne detection to measure the amplitude and phase quadratures of their own beams simultaneously. Since heterodyning one of EPR entangled state is equivalent to QKD with coherent state. In this section, we will compare the Gaussian discord state QKD with no-switching coherent state QKD [39]. In heterodyne detection system, a vacuum modeν is mixed with the optical modeâ (b) on a balanced beamsplitter and the output modes are measured by two homodyne detectors respectively. The amplitude quadrature measured by Alice and Bob areX M A = (X a +X ν )/ √ 2 andX M B = (X B +X ν )/ √ 2,V B|A M = V B − γ ′2 /2 V M A .(22) The mutual information between Alice and Bob are I Het (X A : X B ) = log 2 [V B M /V B M |A M ] , which is same for the direct and reverse reconciliation. Direct reconciliation. In order to obtain S(E|X B ) we need to calculate the symplectic spectrum of the conditional covariance matrix σ E|XA,ŶA , which represents the covariance matrix of a system where two modes has been measured by heterodyne detection (in this case Alice), is given by [12,33,34] σ E|XA,ŶA = σ E − (Λ) −1 D(Ωσ A Ω T + I)D T ,(23) where Λ = det σ A +Trσ A + 1, Ωσ A Ω T + I = σ A + I, and D is given by eq. (18). Reverse reconciliation. The correlation matrix σ E|XB ,ŶB , which represents the covariance matrix of a system where two modes has been measured by heterodyne detection (in this case Bob), is given by [12,33,34] where Λ ′ = det σ B +Trσ B + 1, Ωσ B Ω T + I = σ B + I, and the matrix D is same with eq. (22), which describing the quantum correlations between Eve' modes and Bob's mode. Figure 4 shows the secret key rates for the CV QKD schemes with heterodyne detection, (a) and (b) are for the direct and reverse reconciliation, respectively. Solid (black) and Dashed (blue) lines are the secret key rates for the Gaussian discord state with V D = 40 and 1000, respectively. Dotted (red) line is the secret key rate for the entangled state with V E = 40. All curves are plotted with excess noise W = 1. In Fig. 4(a), comparing solid and dotted lines, we find that secret key can be distilled for the Gaussian discord state at lower transmission efficiency than that for coherent state with heterodyne detection. When T > 0.78, secret key rate for coherent state is still higher than that for the Gaussian discord state with heterodyne detection. In Fig. 4(b), comparing solid and dotted lines, it is obvious that no-switching coherent state QKD offers higher secret key rate and longer transmission distance than that the Gaussian discord state QKD. We also noticed that no secret key can be distilled for the Gaussian discord state at lower trans- mission efficiency (T < 0.55) with reverse reconciliation, which is different from coherent state QKD. Comparing solid and dashed lines in Fig. 4(a) and (b), respectively, we find that secret key rate is increased with increasing of the discording noise for both direct and reverse reconciliation in CV QKD with the Gaussian quantum discord state, which is same with the result of homodyne detection. σ E|XB ,ŶB = σ E − (Λ ′ ) −1 D(Ωσ B Ω T + I)D T ,(24) IV. DEPENDENCE OF SECRET KEY RATE ON QUANTUM DISCORD As shown in Fig. 5, the dependence of secret key rate for the Gaussian discord state on quantum discord are investigated at different transmission efficiency with input variance V D ∈ [1, 1000]. Fig. 5(a) and (b) are the case of direct and reverse reconciliation for homodyne detection, respectively. Fig. 5(c) and (d) are the case of direct and reverse reconciliation for heterodyne detection, respectively. It is obvious that secret key rate is monotonically increased with the increasing of quantum discord. Solid (black), Dashed (red) and Dotted (blue) lines are the secret key rates for the Gaussian discord state with transmission efficiency of 0.75, 0.8 and 0.9, respectively. Dash-dotted (green) line in Fig. 5(b) is the secret key rate for the Gaussian discord state with transmission efficiency of 0.3, which means that secret key can be distilled when T < 0.5 in reverse reconciliation for homodyne detection. Comparing these traces, we find that secret key rate is increased with the increasing of transmission efficiency, which is same with the result in Fig. 3 and 4. Most of the secret key rates start from D AB = 0.12, since 0.12 is the smallest quantum discord with V D = 1 as shown in Fig. 2(a). When T = 0.75 (solid line) in Fig. 5(c), secret key can be distilled when D AB > 0.22. V. CONCLUSION In this paper, by considering CV QKD with a twomode Gaussian discord state, which has only quantum correlation and without entanglement, we show that secret key can be distilled against entangling cloner attack. In CV QKD with the Gaussian discord state,the secret key rate is increased with increasing of quantum discord in both homodyne and heterodyne detection schemes with direct and reverse reconciliation. By comparing the secret key rates of CV QKD schemes with the Gaussian discord state, squeezed state and coherent state, we find that squeezed state and coherent state offer higher secret key rate than the Gaussian discord state at the same energy level for both direct and reverse reconciliation. This is a natural result since Gaussian discord of the Gaussian discord state (0 ≤ D ≤ 1) is smaller than that of EPR entangled state (D > 1). This work provides a possible application of Gaussian quantum discord. FIG. 1 : 1Schematic of the CV QKD scheme with a two-mode Gaussian state. (a): The two-mode Gaussian discord state, AM: amplitude modulator, PM: phase modulator, π: π phase shift. (b): EPR entangled state, (c): the CV QKD scheme. The transmission efficiency of quantum channel is modeled by a beam splitter with transmission T. Eve performs entangling cloner attack, where the variance of the ancillary EPR state is W. Hom: homodyne detection, Het: heterodyne detection. FIG. 2 : 2Quantum discord (a) and smallest symplectic eigenvalue of PPT criterion (b) for the Gaussian discord state. FIG. 3 : 3Secret key rates for the CV QKD schemes with homodyne detection. (a): the direct reconciliation, (b): the reverse reconciliation. Solid (black) and Dashed (blue) lines are the secret key rates for the Gaussian discord state with variance VD = 40 and 1000, respectively. Dotted (red) line is the secret key rate for the entangled state with VE=40. All curves are plotted with excess noise W=1. respectively. The corresponding noise variance measured by Alice and Bob are V M A = (V A + 1)/2 and V M B = (V B + 1)/2, respectively. Bob's conditional variance is given by V B M |A M = (V B|A M + 1)/2, where FIG. 4 : 4Secret key rates for the CV QKD schemes with heterodyne detection. (a): the direct reconciliation, (b): the reverse reconciliation. Solid (black) and Dashed (blue) lines are the secret key rates for the Gaussian discord state with VD=40 and 1000, respectively. Dotted (red) line is the secret key rate for the entangled state with VE=40. All curves are plotted with excess noise W=1. FIG. 5 : 5The dependence of secret key rates on quantum discord for the CV QKD schemes with the Gaussian discord state.(a) and (b): the direct and reverse reconciliation for homodyne detection, respectively. (c) and (d): the direct and reverse reconciliation for heterodyne detection, respectively. 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Most papers implicitly assume competing risks to be induced by residual cohort heterogeneity, i.e. heterogeneity that is not captured by the recorded covariates. Based on this observation we develop a generic statistical description of competing risks that unifies the main schools of thought. Assuming heterogeneity-induced competing risks is much weaker than assuming risk independence. However, we show that it still imposes sufficient constraints to solve the competing risk problem, and derive exact formulae for decontaminated primary risk hazard rates and cause-specific survival functions. The canonical description is in terms of a cohort's covariate-constrained functional distribution of individual hazard rates of all risks. Assuming proportional hazards at the level of individuals leads to a natural parametrisation of this distribution, from which Cox regression, frailty and random effects models, and latent class models can all be recovered in special limits, and which also generates parametrised cumulative incidence functions (the language of Fine and Gray). We demonstrate with synthetic data how the generic method can uncover and map a cohort's substructure, if such substructure exists, and remove heterogeneity-induced false protectivity and false exposure effects. Application to real survival data from the ULSAM study, with prostate cancer as the primary risk, is found to give plausible alternative explanations for previous counter-intuitive inferences.

1307.0199

bf85fd8a0304796e1da6e415ad7d31a6a274664b

Generic solution of the heterogeneity-induced competing risk problem in survival analysis Final version of June 30th 2013 J Van Baardewijk Institute for Mathematical and Molecular Biomedicine King's College London Hodgkin Build-ingSE1 1ULLondonUK H Garmo School of Medicine Cancer Epidemiology Group King's College London Guy's Hospital SE1 9RTLondonUK M Van Hemelrijck School of Medicine Cancer Epidemiology Group King's College London Guy's Hospital SE1 9RTLondonUK L Holmberg School of Medicine Cancer Epidemiology Group King's College London Guy's Hospital SE1 9RTLondonUK Acc Coolen Institute for Mathematical and Molecular Biomedicine King's College London Hodgkin Build-ingSE1 1ULLondonUK London Institute for Mathematical Sciences 35a South StW1K 2XFMayfair, LondonUK Generic solution of the heterogeneity-induced competing risk problem in survival analysis Final version of June 30th 2013 Most papers implicitly assume competing risks to be induced by residual cohort heterogeneity, i.e. heterogeneity that is not captured by the recorded covariates. Based on this observation we develop a generic statistical description of competing risks that unifies the main schools of thought. Assuming heterogeneity-induced competing risks is much weaker than assuming risk independence. However, we show that it still imposes sufficient constraints to solve the competing risk problem, and derive exact formulae for decontaminated primary risk hazard rates and cause-specific survival functions. The canonical description is in terms of a cohort's covariate-constrained functional distribution of individual hazard rates of all risks. Assuming proportional hazards at the level of individuals leads to a natural parametrisation of this distribution, from which Cox regression, frailty and random effects models, and latent class models can all be recovered in special limits, and which also generates parametrised cumulative incidence functions (the language of Fine and Gray). We demonstrate with synthetic data how the generic method can uncover and map a cohort's substructure, if such substructure exists, and remove heterogeneity-induced false protectivity and false exposure effects. Application to real survival data from the ULSAM study, with prostate cancer as the primary risk, is found to give plausible alternative explanations for previous counter-intuitive inferences. Introduction For general introductions to the survival analysis literature we refer to the excellent textbooks (Hougaard, 2001;Klein and Moeschberger, 2003;Ibrahim, Chen and Sinha, 2010;Crowder, 2012). The competing risk problem in survival analysis is the question of how to handle the possible contamination of those characteristics of the primary risk that can be extracted from survival data, such as its hazard rate or its cumulative incidence function, by informative censoring. Decontaminating primary risk characteristics means finding what their values would have been in the hypothetical situation where all non-primary risks were disabled. This is nontrivial, because disabling non-primary risks will not only set their cause-specific hazard rates to zero, but it will generally affect also the hazard rate of the primary risk. If all risks have statistically independent event times censoring is not informative, so there is no problem and many simple methods are available for analysis and regression, such as the survival function estimators of (Kaplan and Meier, 1958) or the proportional hazards method (Cox, 1972). Unfortunately, one usually cannot know beforehand whether the risks in a study are uncorrelated, and there are many cases where the independence assumption is clearly incorrect. Tsiatis' nonidentifiability theorem (Tsiatis, 1975) shows that without additional assumptions it is not possible to infer presence or absence of risk correlations unambiguously from survival data. Unaccounted for risk correlations invalidate the standard interpretations of methods such as (Kaplan and Meier, 1958;Cox, 1972), and can lead to 'false protectivity' effects (DiSerio, 1997) and incorrect inferences (Andersen et al, 2012;Dignam, Zhang and Kocherginsky, 2012). Risk correlations are often fingerprints of residual heterogeneity in cohorts, i.e. of variability in individuals and diseases that is not captured by the covariates. For instance, a primary and a secondary risk (or disease) may share common molecular pathways or be jointly influenced by common environmental or lifestyle factors that were not measured. We would then find that those individuals most likely to be censored by the secondary risk are not random, but would be the ones most likely to report a primary risk event (or vice versa), even if they are indistinguishable in covariate terms. Or, that which we presently regard as a single disease could in fact be a spectrum of distinct diseases, each with their own specific associations with covariates. Many authors have therefore tried to model residual cohort heterogeneity, usually by postulating individual cause-specific hazard rates h i r (t) of the Cox type, but with additional individualised risk multipliers: h i r (t) = λ r (t)e p µ=1 β µ r z µ i +ξ i r(1) Here r and i label, respectively, the different risks and the individuals in our cohort, λ r (t) is a common time-dependent base hazard rate of risk r, (β 1 r , . . . , β p r ) is a vector of regression coefficents for risk r, and (z 1 i , . . . , z p i ) is a vector of covariates of individual i. The'frailty factors' ξ i r are assumed to be sampled from a given parametrised distribution, whose parameters must be estimated from the data. If the frailty factors ξ i r do not depend on the individual's covariates we would speak of 'frailty models', e.g. (Vaupel et al., 1979;Zahl, 1997;Yashin and Iachine, 2005;Gorfine and Hsu, 2010). Frailty models are often regarded as representing the impact of unobserved covariates, see e.g. the discussion in (Keiding, Andersen and Klein, 1997). Models in which the frailty factors depend on the observed covariates, e.g. ξ i r = p µ=1 γ µ ir z µ i , are called 'random effects models', e.g. (Vaida and Xu, 1999;DiSerio, 1997) or the more recent application to breast cancer sub-types (Rosner, 2013). See also the textbooks (Wienke, 2010;Duchateau and Jansen, 2008). If the distribution of frailty factors takes the form of discrete clusters (latent classes), an idea that goes back to (Lazardsfeld, 1950), we obtain the so-called latent class models. See e.g. (Muhten and Masyn, 2005) which combines frailty and random effects with covariate-dependent class sizes as in (Reboussin and Anthony, 2001). Further variations include e.g. frailty factors that are allowed to evolve over time, and models in which the cluster membership of individuals (which represents the heterogeneity) is assumed known. Most frailty and random effects studies, however, quantify only the hazard rates of the primary risk. For instance, in the references above the only exceptions are (Zahl, 1997;DiSerio, 1997;Gorfine and Hsu, 2010). Although one may capture many consequences of cohort heterogeneity (such as time-dependence of regression parameters caused by cohort filtering), without modelling also the non-primary risks it is not possible to deal with the competing risk problem. Moreover, none of the above studies derive explicit formulae for decontaminated primary risk measures such as cause-specific hazard rates or survival functions. The line of work initiated by (Fine and Gray, 1999) does not try to deal with the decontamination question. Instead it focuses on finding parametrisations of the covariate-conditioned cumulative incidence function F 1 (t|z) of the primary risk, for which (Fine and Gray, 1999) propose † F 1 (t|z) = Φ Λ(t)e p µ=1 β µ z µ , Φ(x) = 1 − e −x(2) This is conceptually similar to (Cox, 1972), one just parametrises a diffent quantity: in (Cox, 1972) the covariate-conditioned hazard rate of the primary risk, in (Fine and Gray, 1999) the cumulative incidence function. In particular, (Cox, 1972) and (Fine and Gray, 1999) both model the primary risk profile in the presence of all other risks. The approach of (Fine and Gray, 1999) thus studies risks that compete, but does not address the competing risk problem. The hope is that by parametrising F 1 (t|z) directly, one may capture more heterogeneity-induced effects. The quantity F 1 (t|z) appears more informative than the primary risk hazard rate; it is directly measurable and involves also the crude hazard rates of non-primary risks. However, the latter could have been estimated with Cox regression too. The price paid in (Fine and Gray, 1999) for the advantages of the cumulative incidence function is in parameter estimation: expressing the data likelihood in terms of cumulative incidence functions is much more cumbersome than expressing it in terms of hazard rates. Further developments involve e.g. alternative choices for the function Φ(x) (Fine, 2001;Klein and Andersen, 2005), application to the cumulative incidence of non-primary risks (Jeong and Fine, 2007), and the inclusion of frailty factors (Katsahian and Boudreau, 2011). So we face the unsatisfactory situation of having multiple distinct and diverging approaches to the modeling of heterogeneity and competing risks. Only few actually address the competing risk problem, which requires modelling the hazard rates of all risks and their correlations (not just that of the primary risk), and none lead to explicit formulae for decontamined primary risk measures. In this paper we try to construct a generic statistical description of competing risks and a resolution of the competing risk problem that unifies the various schools of thought described above. Our work is based on the observation that virtually all papers implicitly assume that correlations between competing risks are induced by residual cohort heterogeneity. We show how this simple and transparent assumption leads in a natural way to a formalism with exact formulae for decontaminated primary risk measures, in which Cox regression, frailty models, random effect models, and latent class models are all included as special cases, and which produces transparent parametrisations of the cumulative incidence function (which is the language of Fine and Gray). This paper is organised as follows. In section 2 we define the relevant survival analysis quantities and their relations, and state the competing risk problem in mathematical terms. We then inspect in section 3 the relation between cohort level and individual level statistical descriptions, classify the different levels of risk complexity from the competing risk perspective, and define what we mean by heterogeneity-induced competing risks. We derive the implications of having heterogeneity-induced competing risks, and show how that the canonical description for solving the heterogeneity-induced competing risk problem is in terms of the covariate-conditioned functional distribution W[h 0 , . . . , h R |z] of the individual hazard rates of all risks over the cohort. In section 4 we obtain a generic parametrisation of this distribution, which reduces the mathematical description to a joint covariate-conditioned (functional) distribution M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) over the cohort of individual regression parameters and base hazard rates for all risks, which often simplifies to a covariate-independent form M(β 0 , . . . , β R ; λ 0 , . . . , λ R ). We work out the theory in more detail for a natural family of parametrisations M(β 0 , . . . , β R ; λ 0 , . . . , λ R ). We show how the conventional methods (Cox regression, frailty models, random effect models, latent class models) †The authors of (Fine and Gray, 1999) were critcised for interpreting the quantity λ(t)e p µ=1 β µ z µ , with λ(t) = dΛ(t)/dt, as a hazard rate, since this requires unnatural risk sets. We fail to see the need for this interpretation and regard (2) simply as a parametrisation for the cumulative incidence function. In addition, (Fine and Gray, 1999) also includes time-dependent covariates, but this is not their main point. are recovered in special limits, and derive the parametrised cumulative incidence functions for all risks. The remaining sections are devoted to applications of the formalism to synthetic data, as well as real survival data from the ULSAM longitudinal study (ULSAM, 2013;Grundmark et al., 2011), focusing on prostate cancer as the primary risk. The latter application is found to result in appealing and transparent novel explanations for previously counter-intuitive inferences. We end with a summary of our findings. Definitions and general identities In this section we recall the basic definitions and objectives of survival analysis, and define the competing risk problem in mathematical terms. In doing so we will try to stay as close as possible to the notation conventions and terminology of (Klein and Moeschberger, 2003). Survival probability and crude cause specific hazard rates We assume having a cohort of individuals who are subject to R true risks, labelled by r = 1 . . . R. We use r = 0 to indicate the end-of-trial censoring event, since for the mathematical structure of the theory there is no difference between censoring due to alternative true risks and censoring due to termination of the trial. Most of the mathematical relations of survival analysis can be derived directly from the joint distribution P(t 0 , . . . , t R ) of event times (t 0 , . . . , t R ), where t r ≥ 0 is the time at which risk r triggers a failure event. ‡ From this distribution follow the survival function S(t), i.e. the probability that all events happen later than time t, S(t) = ∞ 0 . . . ∞ 0 dt 0 . . . dt R P(t 0 , . . . , t R ) R r=0 θ(t r − t)(3) and the crude cause-specific hazard rates h r (t), i.e. the probability per unit time that failure r occurs at time t if until that time none of the possible events has yet occurred: h r (t) = 1 S(t) ∞ 0 . . . ∞ 0 dt 0 . . . dt R P(t 0 , . . . , t R )δ(t − t r ) R r =r θ(t r − t)(4) Here we used the delta-distribution δ(x), defined by the identity ∞ −∞ dx δ(x)f (x) = f (0) , and the step function, defined by θ(x > 0) = 1 and θ(x < 0) = 0. It is easy to show that the survival function can be written in terms of the crude hazard rates as S(t) = e − R r=0 t 0 ds hr(s) ,(5) We assume that we can only observe the timing and the risk label of the earliest event. The crude cause-specific hazard rates provide the link between theory and observations, since the probability density P (t, r) to find the earliest event occurring at time t and corresponding to risk r, is given by P (t, r) = h r (t)e − R r =0 t 0 ds h r (s) ,(6) ‡This starting point is not fully general, since it assumes that all risks will ultimately lead to failure. One can include the possibility that events have a finite probability of not happening at any time, by adding for each risk r a binary random variable τr to indicate whether or not the calamity button of risk r is pressed at time tr. The above relations hold irrespective of whether P(t 0 , . . . , t R ) describes a large or a small cohort, or even a single individual, although obviously the values of P(t 0 , . . . , t R ) would be different. However, at some point in our paper we will work simultaneously with both cohort level and individual level descriptions, and it will become necessary to specify with further indices to which we refer. Conditioning on covariate information is trivial in the above picture. For simplicity we assume the covariates to be discrete; in the case of continuous covariates one usually finds the same formulae as those which we derive here, but with integrals instead of sums. Knowing the values z ∈ IR p of p covariates means starting from the distribution P(t 0 , . . . , t R |z) which gives the event time statistics of the sub-cohort of those individuals i that have covariate vector z i = z. It is related to the previous distribution of the full cohort via P(t 0 , . . . , t R ) = z P(t 0 , . . . , t R |z)P (z), where P (z) gives the fraction of the cohort that have covariates z. We then obtain the following covariate-conditioned survival functions and crude cause-specific hazard rates: S(t|z) = ∞ 0 . . . ∞ 0 dt 0 . . . dt R P(t 0 , . . . , t R |z) R r=0 θ(t r − t) (7) h r (t|z) = 1 S(t|z) ∞ 0 . . . ∞ 0 dt 0 . . . dt R P(t 0 , . . . , t R |z)δ(t − t r ) R r =r θ(t r − t)(8) with the usual relation between survival and crude hazard rates, and the usual link to observations: S(t|z) = e − R r =0 t 0 ds h r (s|z) (9) P (t, r|z) = h r (t|z)e − R r =0 t 0 ds h r (s|z)(10) If we study a cohort of N individuals, with coresponding N covariate vectors {z 1 , . . . , z N }, the survival data D usually consist of N samples of event time and event type pairs (t, r): D = {(t 1 , r 1 ), . . . , (t N , r N )}(11) Since the probability density for an individual with covariate vector z to report the pair (t, r) is given by (10), the data likelihood P (D) = N i=1 P (t i , r i |z i ) obeys log P (D) = N i=1 log h ri (t i |z i )e − R r=0 t i 0 dt hr(t|zi) = R r=0 N i=1 δ r,ri log h r (t i |z i ) − N i=1 ti 0 dt h r (t|z i )(12) 2.2. Decontaminated cause-specific risk measures -the competing risk problem The aim of survival analysis is to extract statistical patterns from survival data, that allow us to make risk predictions for new individuals, usually conditioned on knowledge of their covariates. We are often interested in one specific primary risk. Many relevant risk-specific quantities can be calculated once we know the crude hazard rates. For instance, the cause-specific cumulative incidence function F r (t), i.e. the probability that event r has been observed at any time prior to time t, is given by F r (t) = t 0 dt S(t )h r (t )(13) Although F r (t) refers to risk r specifically, it can be heavily influenced by the other risks. For instance, if F r (t) is small, this may be because event r is intrinsically unlikely, or because it has the habit of being preceeded by alternative events r = r. One cannot tell. The problem lies in the difference between events having been observed and events having happened prior to a given time. To obtain decontaminated information on an individual primary risk r one must consider the hypothetical situation where all other risks r = r are disabled. This means replacing § P(t 0 , . . . , t R ) → P(t r ) lim Λ→∞ R r =r δ(t r − Λ)(14) with the the marginal event time distribution P(t r ) = . . . [ s =r dt s ]P(t 0 , . . . , t R ) of risk r. Insertion of (14) into (3,4) gives, as expected, zero values for all non-primary crude hazard rates, but it also affects the value of the primary risk hazard rate. One now finds the following formulae for the decontaminated cause-specific survival function and hazard rate for risk r, indicated with tildes to distinguish them from their crude counterparts: S r (t) = ∞ t dt r P(t r ),h r (t) = − d dt logS r (t)(15) or, in the case of covariate conditioning, S r (t|z) = ∞ t dt r P(t r |z),h r (t|z) = − d dt logS r (t|z)(16) In general one will indeed find thath r (t) = h r (t) andh r (t|z) = h r (t|z). Equations (15,16) tell us that to determine the decontaminated risk measures for the primary risk r we must estimate the marginal distributions P(t r ) or P(t r |z) from survival data. Tsiatis showed (Tsiatis, 1975) that this is impossible without further assumptions. His identifiability theorem states that for every P(t 0 , . . . , t R ) there is an alternative distribution P(t 0 , . . . , t R ) that describes independent times, but such that P and P both generate identical cause-specific hazard rates for all risks: P(t 0 , . . . , t R ) = R r=0 h r (t)e − t 0 ds hr(s)(17) in which {h r (t)} are the cause-specific hazard rates of P(t 0 , . . . , t R ). Hence the only information that can be estimated from survival data alone are the (covariate-conditioned) crude causespecicifc hazard rates. One cannot calculate the distributions P(t 0 , . . . , t R ) or P(t 0 , . . . , t R |z) and their marginals. Without further information or assumptions there is therefore no way to disentangle the different risks and identify the decontaminated cause-specific hazard rates and survival functions. This, in a nutshell, is the competing risk problem. One obvious and simple way out is to assume that all risks are statistically independent, i.e. that P(t 0 , . . . , t R |z) = R r=0 P(t r |z). This solves trivially the competing risk problem, since now one finds immediately that h r (t|z) =h r (t|z) for all r, and S r (t|z) = e − t 0 ds hr(s|z) (18) §A valid critical note that has been made at this step (Prentice et al, 1978) is that one cannot be sure that this hypothesis is appropriate; it may be that correlated risks share biochemical pathways such that they can never be deactivated independently. The independence assumption underlies the clinical use of e.g. Cox's proportional hazards regression (Cox, 1972) and of Kaplan-Meier estimators of the cause-specific survival function (Kaplan and Meier, 1958), which would otherwise be unjustified tools for quantifying cause-specific survival regularities. Assuming risk independence may be acceptable in specific cases. For diseases that share molecular pathways, however, the event times will be strongly correlated. This can lead to false protectivity effects, and the independence assumption may generate nonsensical claims. Heterogeneity-induced competing risks We introduce a different assumption on the nature of the event time correlations of different risks: we assume these to be caused by residual cohort heterogeneity. We will call this heterogeneityinduced competing risks. The assumption is transparent and much weaker than assuming risk independence, but still imposes sufficient constraints to allow us to solve the computing risk problem. Connection between cohort level and individual level descriptions To give a precise definition of heterogeneity-induced competing risks we first need to describe the connection between cohort-level and individual level risk descriptions. The standard survival analysis formalism is built solely on the starting point of a joint event time distributions; it can therefore also be applied directly to risk at the level of individuals. Let N be the number of individuals in the cohort to which the distribution P(t 0 , . . . , t R ) refers, labelled by i = 1 . . . N . We write the joint event time distribution of individual i in this cohort as P i (t 0 , . . . , t R ), and the crude cause-specific hazard rates of individual i as h i r (t). It then follows directly from the general theory that S i (t) = ∞ 0 . . . ∞ 0 dt 0 . . . dt R P i (t 0 , . . . , t R ) R r=0 θ(t r − t) (19) h i r (t) = 1 S i (t) ∞ 0 . . . ∞ 0 dt 0 . . . dt R P i (t 0 , . . . , t R )δ(t − t r ) R r =r θ(t r − t)(20) and P i (t, r) = h i r (t)e − R r =0 t 0 ds h i r (s) ,(21)Here S i (t) = exp[− R r=0 t 0 ds h i r (s)] is the survival function of individual i, and P i (t, r) is the probability that the first event for individual i occurs at time t and corresponds to risk r. When collecting survival data in a cohort, we have the added uncertainty of not knowing which individuals were picked from the population, so the connection between the two levels is simply given by P(t 0 , . . . , t R ) = 1 N N i=1 P i (t 0 , . . . , t R )(22)P(t 0 , . . . , t R |z) = i, zi=z P i (t 0 , . . . , t R ) i, zi=z 1(23) For quantities that depend linearly on the joint event time distribution, the link between cohort level and individual level is a simple averaging over the label i, possibly conditioned on covariates, e.g. S(t) = 1 N N i=1 S i (t), P (t, r) = 1 N N i=1 P i (t, r) (24) S(t|z) = i, zi=z S i (t) i, zi=z 1 , P (t, r|z) = i, zi=z P i (t, r) i, zi=z 1(25) However, quantities such as the crude cause-specific hazard rates depend in a more complicated way on P(t 0 , . . . , t R ), via their conditioning on survival. As a consequence, cohort level cause-specific hazard rates are not direct averages over their individual level counterparts. Instead one finds (see appendix A for details): h r (t) = N i=1 h i r (t)e − R r =0 t 0 ds h i r (s) N i=1 e − R r =0 t 0 ds h i r (s) ,(26)h r (t|z) = i,z i =z h i r (t)e − R r =0 t 0 ds h i r (s) i,z i =z e − R r =0 t 0 ds h i r (s)(27) Heterogeneous cohorts and the different levels of risk complexity We always assume our cohorts to be heterogeneous at the level of covariates. The heterogeneity of concern here is in the relation between covariates and risks. A homogeneous cohort is one in which the relation between covariates and risks is uniform, so that the distribution P i (t 0 , . . . , t R ) can depend on i only via z i . Put differently, there exists a function P(t 0 , . . . , t R |z) such that P i (t 0 , . . . , t R ) = P(t 0 , . . . , t R |z i ) for all i The same is then true for the cause-specific hazard rates: h i r (t) = h r (t|z i ) for all i, in which h r (t|z) is related to P(t 0 , . . . , t R |z) via equations (7,8). It also follows directly from (27) that at cohort level the covariate-conditioned event time distribution is P(t 0 , . . . , t R |z) = P(t 0 , . . . , t R |z), and the covariate-constrained crude hazard rates are h r (t|z) = h r (t|z), as expected. A special property of homogeneous cohorts is that uncorrelated individual level risks, i.e. P i (t 0 , . . . , t R ) = R r=0 P i (t r ), imply uncorrelated covariate-conditioned cohort level risks. This follows from (23): P(t 0 , . . . , t R |z) = i, zi=z R r=0 P i (t r ) i, zi=z 1 = i, zi=z R r=0 P(t r |z i ) i, zi=z 1 = R r=0 P(t r |z) = R r=0 P(t r |z)(29) Heterogeneous cohorts, in contrast, are those where (28) does not hold, i.e. our individuals have further 'hidden' features, not captured by the covariates, that impact upon their risks. In such cohorts one will observe a gradual 'filtering': high-risk individuals will drop out early, causing time dependencies at cohort level that have no counterpart at the level of individuals. For instance, in the simplest case where all individuals have stationary hazard rates, viz. h i r (t) = h i r , one would according to (26,27) still find time dependent crude hazard rates at cohort level. In heterogeneous cohorts it is no longer true that having uncorrelated individual level risks implies having uncorrelated covariate-conditioned cohort level risks. It is a trivial exercise to devise examples where P i (t 0 , . . . , t R ) = R r=0 P i (t r ), but still P(t 0 , . . . , t R |z) = R r=0 P(t r |z). It is clear that risk correlations can be generated at different levels, and that there is a natural hierarchy of cohorts in terms of risk complexity, with implications for the applicability of methods: • LEVEL 1: HOMOGENEOUS COHORT, NO COMPETING RISKS individual: P i (t 0 , . . . , t R ) = R r=0 P(t r |z i ) cohort: P(t 0 , . . . , t R |z) = R r=0 P(t r |z) Although the members of the cohort will be different in their recorded covariates, they are homogeneous in terms of the link between covariates and risk. For each individual, the event times of all risks are statistically independent, and their probabilities are determined fully by the covariates alone. Since there is no residual heterogeneity, there is no competing risk problem; crude and true cause-specific hazard rates and survival functions are identical. • LEVEL 2: HETEROGENEOUS COHORT, NO COMPETING RISKS individual: P i (t 0 , . . . , t R ) = R r=0 P i (t r ) cohort: P(t 0 , . . . , t R |z) = R r=0 P(t r |z) Here for each individual the event times of all risks are still statistically independent, but their susceptibilities are no longer determined by their recorded covariates alone (reflecting e.g. disease sub-groups or the impact of further unobserved covariates). However, this residual heterogeneity does not manifest itself in risk correlations at cohort level. One will therefore observe heterogeneity-induced effects, such as 'cohort filtering', but no competing risks. • LEVEL 3: HETEROGENEITY-INDUCED COMPETING RISKS individual: P i (t 0 , . . . , t R ) = R r=0 P i (t r ) cohort: P(t 0 , . . . , t R |z) = R r=0 P(t r |z) Here for each individual the event times of all risks are statistically independent, but their susceptibilities are not determined by their recorded covariates alone, similar to level 2. However, now this residual cohort heterogeneity leads to risk correlations at cohort level, reflecting e.g. common unobserved risk factors or co-morbitities, and thereby to informative censoring. One will now observe competing risks phenomena, such as false protectivity and false exposure. • LEVEL 4: INDIVIDUAL AND COHORT LEVEL COMPETING RISKS individual: P i (t 0 , . . . , t R ) = R r=0 P i (t r ) cohort: P(t 0 , . . . , t R |z) = R r=0 P(t r |z) This is the most complex situation from a modelling point of view, where both at the level of individuals and at cohort level the event times of different risks are correlated. We will again observe competing risk phenomena, but can no longer say where these are generated. In fact, having correlations amongst non-primary risks is harmless in the context of decontaminating primary risk measures. The only issue is whether there are correlations between primary and nonprimary risks. So we could in principle make a further distinction between having P(t 0 , . . . , t R |z) = R r=0 P(t r |z) and P(t 0 , . . . , t R |z) = P(t r |z)P(t 0 , . . . , t r−1 , t r+1 , . . . , t R |z); the latter property being weaker but still sufficient. Here we will not persue this distinction; it is obvious how the theory should be adapted to accommodate non-primary risk correlations. Levels 1 and 2 are those where the assumption of statistically independent risks, underlying the clinical use of e.g. Cox regression and Kaplan-Meier estimators, is valid. At level 2 there is still no competing risk problem, but the heterogeneity demands parametrisations of crude primary hazard rates at cohort level that are more complex than those used in Cox regression, which is the rationale behind the development of frailty models and random effects models, as well as the latent class models of (Muhten and Masyn, 2005). All these methodologies still only model the cause-specific hazard rate of the primary risk, and therefore cannot handle cohorts beyond complexity level 2. Level 4, which includes homogeneous cohorts with individual level competing risks, represents the most complex scenario, which we will not deal with in this paper. Our focus is on level 3: that of cohorts with heterogeneity-induced competing risks. Here the correlations between cohort level event times have their origin strictly in correlations between disease susceptibilities of individuals, e.g. someone with a high hazard rate for a disease A may also be likely to have a high hazard rate for B, for reasons not explained by the recorded covariates. Most papers on frailty, random effects and latent class models assume implicitly that competing risks are induced by such residual heterogeneity. We now show that the assumption of heterogeneity-induced competing risks leads to a transparent resolution of the competing risk problem. Implications of having heterogeneity-induced competing risks In the case of heterogeneity-induced competing risks we have independent event times at the level of individuals, hence for each individual i we can be sure that P i (t r ) = h i r (t)e − t 0 ds h i r (s)(30) The covariate-conditioned cohort level event time marginals are therefore P r (t r |z) = i,zi=z h i r (t)e − t 0 ds h i r (s) i,zi=z 1(31) and via (16) we can write the decontaminated cause-specific survival function and hazard rate as S r (t|z) = i,zi=z e − t 0 ds h i r (s) i,zi=z 1 (32) h r (t|z) = i,zi=z h i r (t)e − t 0 ds h i r (s) i,zi=z e − t 0 ds h i r (s)(33) Here we used ∞ 0 ds h i r (s) = ∞ for all (i, r), which follows from the assumed normalisation of P i (t 0 , . . . , t R ). Expressions (32,33) are similar but not identical to the formulae (18,27) for the decontaminated cause-specific survival function and the crude covariate-conditioned cause-specific hazard rates which we would have taken to be correct had we assumed all risks to be independent: S r (t|z) = e − t 0 ds hr(s|z) (34) h r (t|z) = i,z i =z h i r (t)e − R r =0 t 0 ds h i r (s) i,z i =z e − R r =0 t 0 ds h i r (s)(35) All this is easily interpreted. In formula (33) the probability that individual i survives until time t is correctly given by the factor exp[− t 0 ds h i r (s)] (which causes the 'cohort filtering'), since no risk other than r is active. In contrast, in (35) all risks contribute to cohort filtering. The formulae (33) and (35) will therefore be non-identical, unless we have risk independence, which in (35) would give rise to an identical factor in numerator and denominator that would drop out. Indeed, the differences between (32,33) and (34,35) quantify the severity of the competing risk problem in the cohort at hand. We also see that in the case of a homogenous cohort, where h i r (t) = h r (t|z i ) for all (r, i), one indeed recoversS r (t|z) = S r (t|z) andh r (t|z) = h r (t|z). Similarly, we can work out the formula that provides the link between the theory and survival data. Inserting (21) into (25) immediately leads us to P (t, r|z) = i, zi=z h i r (t)e − R r =0 t 0 ds h i r (s) i, zi=z 1(36) We conclude that the assumption that competing risks (if present) are of the heterogeneity-induced type allows one to derive relatively simple formulae both for the decontaminated cause-specific quantities of interest and for the likelihood of observing individual survival data. What remains is to identify the minimal level of description required for evaluating these formulae, and to determine how the minimum required information can in practice be estimated from survival data. Canonical level of description for resolving heterogeneity-induced competing risks The canonical level of description is the minimal set of observables in terms of which we can write both the decontaminated risk-specific quantities (32,33) (so that we can calculate what we are interested in), and the data likelihood (36) (so it can be estimated from survival data). In (32,33) we need the covariate-constrained distribution of individual hazard rates for the primary risk. In (36) we need in addition the covariate-constrained distribution of the cumulative rates of non-primary risks. In combination we see that the minimal description would be the functional distribution W[h r , h /r |z] = i,zi=z δ F h r −h i r δ F h /r − r =r h i r i,zi=z 1(37) Here δ F denotes the functional δ-distribution, defined by the functional integral identity {df }δ F [f ]G[f ] = G[f ]| f (t)=0 ∀t≥0(38) W[h r , h /r |z] tells us for each possible choice of the function pair {h r (t), h /r (t)}: which fraction of those individuals in our cohort that have covariates z also have the individual primary hazard rates h i r (t) = h r (t) and the cumulative non-primary hazard rates r =r h i r (t) = h /r (t). In practice it will often be advantageous to relax slightly our requirement of a minimal description. The non-primary risks will usually be mutually very different in their characteristics, so finding an efficient parametrisation of the dependence on r =r h i r (t) in W[h r , h /r |z] will be awkward. A slightly redundant alternative choice, but one that is more easily parametrised, would be W[h 0 , . . . , h R |z] = i,zi=z R r=0 δ F h r −h i r i,zi=z 1(39) It gives the joint functional distribution over the cohort of all R + 1 individual cause-specific hazard rates at all times. The distribution (37) follows from (39) via W[h r , h /r |z] = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] δ F h r −h r δ F h /r − r =r h r(40) For independent risks one would simply find the factorised form W[h 0 , . . . , h R |z] = R r=0 W[h r |z]. If we know (39) we can write the decontaminated risk-specific quantities (32,33) as S r (t|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − t 0 ds hr(s) (41) h r (t|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (t)e − t 0 ds hr(s) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − t 0 ds hr(s)(42) whereas their 'crude' counterparts, which would be reported upon assuming independent risks, are S r (t|z) = e − t 0 ds hr(s|z) (43) h r (t|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (t)e − R r =0 t 0 ds h r (s) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − R r =0 t 0 ds h r (s)(44) Having formulae for the latter is useful for quantifying the impact of competing risks in the cohort, via comparison to (41,42). One can easily confirm that if the primary risk r is not correlated with the non-primary risks (32) and (43) as well as (42) and (44) become pairwise identical, as expected. (i.e. if W[h 0 , . . . , h R |z] = W[h 1 |z]W[h 0 , h 2 , . . . , h R |z]), or if there is just one risk, the formulae The data likelihood (36) acquires the form P (t, r|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (t)e − R r =0 t 0 ds h r (s)(45) An alternative formula for P (t, r|z) follows upon combining (27) with (44). In appendix B we show that the two formulae are indeed identical, as they should. Finally, the covariate-conditioned cause-specific cumulative incidence functions F r (t|z) = t 0 ds P (s, r|z) can be written as F r (t|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] t 0 dt h r (t )e − R r =0 t 0 ds h r (s)(46) The level of description (39) is both sufficient and necessary for handling heterogeneity-induced competing risks, apart from the trivial option to combine the non-primary risks r = r into a single non-primary risk, which would lead to (37). More specifically, one cannot work with the crude cohort-level covariate-conditioned hazard rates alone: whereas the latter can all be calculated from W[h 0 , . . . , h R |z] via (44), the converse is not true. In fact it is easy to show that for any W[h 0 , . . . , h R |z] there exists an alternative distribution W[h 0 , . . . , h R |z] that describes a homogeneous cohort, such that W and W give identical crude cohort-level covariate-conditioned causespecific hazard rates, namely W[h 0 , . . . , h R |z] = R r=0 δ F [h r −h r (z)](47) in which h r (z) is the function of time given in (44). Estimation of W[h 0 , . . . , h R |z] from survival data When there is a limited supply of survival data one must determine the relevant quantities in parametrised form, to avoid overfitting, and estimate the parameters from the data. It is still true that the data likelihood can be expressed in terms of the crude cohort-level covariate-conditioned cause-specific hazard rates, so one cannot extract information on W[h 0 , . . . , h R |z] from survival data that is not contained in {h t (t|z)}. However, even relatively simple and natural parametrisations of W[h 0 , . . . , h R |z] will via (44) correspond to nontrivial crude conditioned hazard rates (with time dependencies caused by cohort filtering), that one would have been very unlikely to propose when parametrising directly at the level of the crude hazard rates. This situation mirrors that of using frailty or latent class models for chorts at complexity level 2. We thus assume W[h 0 , . . . , h R |z] to be a member of a parametrised family of conditioned distributions W[h 0 , . . . , h R |z, θ], in which θ ∈ Ω denotes the vector of parameters and Ω is its value domain. For our cohort of N individuals, with covariate vectors {z 1 , . . . , z N }, the available survival data consist of the N samples of event time and event type pairs, D = {(t 1 , r 1 ), . . . , (t N , r N )}. Since the probability density for an individual with covariate vector z to report the pair (t, r) is given by (45), the data likelihood P (D|θ) = N i=1 P (t i , r i |z i ) given the parameters θ is P (D|θ) = N i=1 {dh 0 . . . dh R } W[h 0 , . . . , h R |z i , θ] h ri (t i )e − R r =0 t i 0 ds h r (s)(48) If we concentrate all the survival data in two empirical distributions, P (t, r|z) = i, zi=z δ(t − t i )δ r,ri i, zi=z 1 ,P (z) = 1 N N i=1 δ z,zi(49) (with the Kronecker delta-symbol, δ ab = 1 if a = b and δ ab = 0 otherwise) we can write the log-likelihood L(θ) = log P (D|θ) of the observed data as L(θ) = N zP (z) R r=0 dtP (t, r|z) log {dh 0 . . . dh R } W[h 0 , . . . , h R |z, θ] × h r (t)e − R r =0 t 0 ds h r (s)(50) This log-likelihood can be interpreted in terms of the dissimilarity of the empirical functionP (t, r|z) and the model predictionP (45): (t, r|z, θ) , i.e. the result of substituting W[h 0 , . . . , h R |z, θ] intoL(θ) N = zP (z) R r=0 dtP (t, r|z) logP (t, r|z) − R r=0 dtP (t, r|z) log P (t, r|z) P (t, r|z, θ)(51) The first (entropic) term is independent of θ and the second term is minus the Kullback-Leibler distance D(P ||P ) (Cover and Thomas, 1991) betweenP and P , hence finding the most probable parameters θ is equivalent to minimizing D(P ||P ). From this starting point one can follow different routes for estimating θ, each with specific advantages and limitations, and each with different computational costs. For instance, in maximum likelihood (ML) estimation one simply uses the valueθ for which the data are most likely, θ ML = argmax θ∈Ω L(θ)(52) In the Bayesian formalism one does not commit oneself to one choice for θ, but one uses the full Bayesian posterior parameter probability P (θ|D). Given a parameter prior P (θ) this would give log P (θ|D) = L(θ) + log P (θ) − log Ω dθ P (θ )e L(θ )(53) Finally, in maximum a posteriori probability (MAP) estimation one uses the valueθ for which the Bayesian posterior parameter probability is maximal, θ MAP = argmax θ∈Ω L(θ) + log P (θ)(54) For sufficiently large data sets the above three estimation methods would all become equivalent, i.e. lim N →∞θMAP = lim N →∞θML and lim N →∞ P (θ|D) = δ(θ − θ ML ). This follows from the property lim N →∞ L(θ)/N = lim N →∞ [L(θ) + log P (θ)]/N = lim N →∞ P (θ|D)/N . There are obviously multiple and more advanced variations on the above parameter estimation protocols. For instance, one could reduce the overfitting danger in the ML method by including Aikake's Information Criterion (AIC) or the Bayesian Information Criterion (BIC). Alternative Bayesian routes involve e.g. hyperparameter estimation, or variational approximations of the posterior parameter distribution to reduce computation costs, or model selection to select good parametrisations W[h 0 , . . . , h R |z, θ]. See e.g. (MacKay, 2003). Parametrisation of W[h 0 , . . . , h R |z] A transparent class of parametrisations for W[h 0 , . . . , h R |z, θ] is obtained by assuming that the proportional hazards assumption of Cox holds at the level of individuals. Here we work out the relevant equations, and show how the resulting theory includes the conventional methods (e.g. Cox regression, frailty models, random effect models, latent class analysis) as special simplified cases. Generic parametrisation We note that for each individual i in our cohort we can always write the individual cause-specific hazard rates in the form h i r (t) = λ i r (t) exp(β 0i r + p µ=1 β µi r z i µ ). The time-dependence for each risk is concentrated in λ i r (t). The parameters β 0i r represent individual risk-specific frailties, which have to be normalised in such a way as to remove the redundancy of the parametrisation, i.e. to eliminate the invariance of the individual hazard rates under the transformation {λ i r (t), β 0i r } → {λ i r (t)e −ζ i r , β 0i r + ζ i r }. According to (39), we can then write W[h 0 , . . . , h R |z] as W[h 0 , . . . , h R |z, M] = dβ 0 . . . dβ R {dλ 0 . . . dλ R } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) × R r=0 δ F h r −λ r e β 0 r + p µ=1 β µ r zµ(55) with the short-hand β r = (β 0 r , . . . , β p r ), and with M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) = i, zi=z R r=0 δ F [λ r −λ i r ]δ(β r −β i r ) i, zi=z 1(56) In the language of the previous subsection we thus have a parametrisation in which θ = M. Note that (55) is still completely general. In particular, it does not yet imply a proportional hazards assumption at the level of individuals unless M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) is independent of z. However, it is a useful representation only if M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) depends in a relatively simple way on the parameters {β 0 , . . . , β R } and the functions {λ 0 , . . . , λ R }. To compactify our notation further we introduce the short-hands β · z = β 0 + p µ=1 β µ z µ and Λ t (t) = t 0 ds λ r (s). Inserting (55) into (50) then gives the data log-likelihood L(M) corresponding to (55): L(M) = N zP (z) R r=0 dtP (t, r|z) log dβ 0 . . . dβ R {dλ 0 , . . . , λ R } × M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) λ r (t) e β r ·z− R r =0 Λ r (t) exp(β r ·z)(57) This is equivalent to L(M) = N i=1 log dβ 0 . . . dβ R {dλ 0 , . . . , λ R } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z i ) ×λ ri (t i ) e β r i ·zi− R r =0 Λ r (ti) exp(β r ·zi)(58) The individual cause-specific hazard rates of all individuals are written in a form reminiscent of (Cox, 1972), but with time-dependent factors and time-independent regression and frailty parameters for the R + 1 risks that are not uniform over the cohort, but distributed according to the distribution M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z), in the spirit of fraily and random effects models. However, here this is done for all risks simultaneously, so the complexities of competing risks and false protectivities are captured by the correlation structure of the joint distribution M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z). All applications and examples in the remainder of this paper are based on the generic parametrisation (55). Given (55) one obtains formulae for the various decontaminated and 'crude' causespecific quantities of interest, which are fully exact as long as M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) is kept general. We write the single-risk marginals of M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) as M(β r ; λ r |z) = r =r dβ r {dλ r } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z)(59) For the decontaminated cause-specific survival functions and hazard rates we then get S r (t|z) = dβ r {dλ r } M(β r ; λ r |z) e − exp(β r ·z)Λr(t) (60) h r (t|z) = dβ r {dλ r } M(β r ; λ r |z) λ r (t)e β r ·z−exp(β r ·z)Λr(t) dβ r {dλ r } M(β r ; λ r |z) e − exp(β r ·z)Λr(t)(61) The crude hazard rates and the data probability become h r (t|z) = (62) dβ 0 . . . dβ R {dλ 0 . . . dλ R } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) λ r (t)e β r ·z− R r =0 exp(β r ·z)Λ r (t) dβ 0 . . . dβ R {dλ 0 . . . dλ R } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) e − R r =0 exp(β r ·z)Λ r (t) P (t, r|z) =(63) dβ 0 . . . dβ R {dλ 0 , . . . , λ R } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z)λ r (t)e β r ·z− R r =0 exp(β r ·z)Λ r (t) and, finally, the covariate-conditioned cumulative cause-specific incidence functions are F r (t|z) = dβ 0 . . . dβ R {dλ 0 . . . dλ R } M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) × t 0 dt λ r (t )e β r ·z− R r =0 exp(β r ·z)Λ r (t )(64) Connection with conventional regression methods Since the parametrisation (55) is generic, all existing regression methods that are compatible with the assumption of heterogeneity-induced competing risks will in principle correspond to specific choices for the covariate-conditioned distribution M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z). We label the primary risk as r = 1. All methods that assume primary and non-primary risks to be independent would have M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) = M(β 1 , λ 1 |z)M(β 0 , β 2 , . . . , β R ; λ 0 , λ 2 , . . . , λ R |z) with some specific choice for the form of M(β 1 , λ 1 |z). Examples from this group are • Cox's proportional hazards regression (Cox, 1972) Here one assumes that there is no variability in the parameters (β 1 , λ 1 ) of the primary risk. Elimination of parameter redundancy then means that β 0 1 is absorbed into λ 1 (t), and we find M(β 1 ; λ 1 |z) = δ F [λ 1 −λ] δ(β 0 1 ) p µ=1 δ(β µ 1 −β µ )(65) Via the maximum likelihood method one can express the base hazard rateλ(t) in terms of the regression coefficients {β µ } (giving Breslow's formula), substitution of which then leads directly to Cox's equations (Cox, 1972). See appendix C for details. • Simple frailty models In simple frailty models, such as (Vaupel et al., 1979;Yashin and Iachine, 2005), the frailty parameters of different risks are assumed to be statistically independent, so the heterogeneity of the cohort that impacts upon the primary risk is concentrated in the random parameter β 0 1 : M(β 1 ; λ 1 |z) = δ F [λ 1 −λ] g(β 0 1 ) p µ=1 δ(β µ 1 −β µ )(66) One usually chooses the frailty distribution g(β 0 1 ) to be of a specific parametrised form that allows one to do various relevant integrals over β 0 1 analytically. See appendix C for details. • Simple random effects models In simple random effects models, such as (Vaida and Xu, 1999), one still assumes the parameters of the primary risk to be independent of the non-primary ones, but now the regression coeficients that couple to the covariates are non-uniform: M(β 1 ; λ 1 |z) = δ F [λ 1 −λ] W (β 1 )(67) One then assumes a specific parametrized form for the distribution W (β 1 ) and estimates its parameters from the data. • Latent class models The latent class models of (Muhten and Masyn, 2005) are recovered upon assuming the cohort to consists of a finite number of discrete sub-cohorts. Each is of the type (65), but with a distinct base hazard rate and distinc regression coefficients. The probabilities w for individuals to belong to each sub-cohort are allowed to depend on their covariates z, as in (Reboussin and Anthony, 2001): M(β 1 ; λ 1 |z) = L =1 w( |z) δ F [λ 1 −λ ] δ(β 0 1 ) p µ=1 δ(β µ 1 −β µ )(68)w( |z) = e α 0 + p µ=1 α µ z µ L =1 e α 0 + p µ=1 α µ z µ(69) The above models all focus on the parameters of the primary risk only, and thereby lose the ability to deal with the competing risk problem. Only few papers try to characterise all risk and their possible parameter interactions simultaneously, such as Zahl (1997) or (DiSerio, 1997), but they do not yet develop their ideas into full systematic regression and/or decontamination protocols. Of course there are multiple variations on the above models. These include versions with time-dependent covariates, and models with non-latent classes in the sense that for each individual i one knows the class label (i) ∈ {1, . . . , L}. It is easy to see how they would fit into the generic formulation. A simple latent class parametrisation for heterogeneity-induced competing risks Any description that includes all risks and their correlations, a prerequisite for decontaminating primary risk measures, will have significantly more parameters than those limited to the primary risk. In view of the overfitting danger it is then vital that one limits the complexity of the chosen parametrisation. The difference between frailty and random effects models is only in whether the risk variability relates to known or unknown covariates, so it seems logical to combine both. If we take the heterogeneity to be discrete, but without the covariate dependence of class probabilities of (68), if we assume the end-of-trial risk not to depend on the covariates, and if we choose the base hazard rates of all risks to be uniform in the cohort, we obtain a model family in which M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) = δ(β 0 )δ F [λ 0 −λ 0 ]M(β 1 , . . . , β R ; λ 1 , . . . , λ R ), with M(β 1 , . . . , β R ; λ 1 , . . . , λ R ) = M(β 1 , . . . , β R ) R r=1 δ F [λ r −λ r ](70)M(β 1 , . . . , β R ) = L =1 w R r=1 δ(β r −β r )(71) Hereβ r = (β 0 r , . . . ,β p r ). See Figure 1 for an illustration of what this parametrisation (70) means in terms of individual cause-specific hazard rates in our cohort. For any choice for the number L of assumed latent classes, the remaining parameters to be estimated from the data are: the causespecific hazard rates {λ r (t)} of all risks, the L class sizes w ∈ [0, 1] (subject to L =1 w = 1), the regression coeficients {β µ r } and frailty parameters {β 0 r } of all risks r = 1 . . . R and all latent classes. The remaining parametrisation invariance is {λ r (t),β 0 r } → {λ r (t)e −ζr ,β 0 r + ζ r } for all , which is removed by definingβ 10 r = 0 for all r. Finding the optimal number L of classes is in principle a simple Bayesian model selection problem. · · · · · · · · · · · · · · · ' & $ % Latent class L fraction: wL for all r > 0: h i r (t) =λr(t)eβ L0 r + p µ=1β Lµ r z µ i Fig. 1. Illustration of the parametrisation (70). All individuals i in the cohort are assumed to have personalised cause-specific hazard rates h i r (t) which for all risks r = 1 . . . R are of the proportional hazards form. The cohort is allowed to be heterogeneous in that it may consist of L sub-cohorts (or 'latent classes'), labelled by = 1 . . . L. Each latent class contains individuals with risk-specific frailtiesβ 0 r and with risk-specific regression parametersβ µ r to capture the impact of covariates. The base hazard ratesλr(t) of the risks are assumed not to vary between individuals. The class membership of the individuals in our data set is not known a priori, but can be inferred a posteriori. The log-likelihood (58) of our survival data is at the core of all parameter estimation procedures. For the multi-risk parametrisation (70) it simplifies to the following expression, with our usual shorthand β r · z = β 0 r + p µ=1 β µ r z µ and with δ ab = 1 − δ ab : L(M) = N i=1 logλ ri (t i ) + N i=1 log L =1 w eβ r i ·zi− R r=0Λ r (ti) exp(β r ·zi) = L 0 (M) + L risks (M)(72) with a first term that includes the (mostly irrelevant) end-of-trial censoring information, and a second term that contains the quantities related to actual risks: L 0 (M) = N i=1 δ 0ri logλ 0 (t i ) − N i=1Λ 0 (t i )(73)L risks (M) = N i=1 δ 0ri logλ ri (t i ) + N i=1 log L =1 w e δ0r iβr i ·zi− R r=1Λ r (ti) exp(β r ·zi)(74) Inserting (70) into our formulae for the decontaminated cause-specific survival function and hazard rates of the true risks r > 0 gives the relatively simple and intuitive expressions S r (t|z) = L =1 w e − exp(β r ·z)Λr(t)(75)h r (t|z) =λ r (t) L =1 w eβ r ·z−exp(β r ·z)Λr(t) L =1 w e − exp(β r ·z)Λr(t)(76) The crude hazard rate and the data probability become h r (t|z) =λ r (t) L =1 w eβ r ·z− R r =1 exp(β r ·z)Λ r (t) L =1 w e − R r =1 exp(β r ·z)Λ r (t)(77)P (t, r|z) =λ r (t)e −Λ0(t) L =1 w eβ r ·z− R r =1 exp(β r ·z)Λ r (t)(78) From the crude cause-specific hazard rates h r (t|z) follow as usual the crude cause-specific survival functions S r (t|z) for r = 1 . . . R, via te relation S r (t|z) = exp[− t 0 ds h r (s|z)]. The cumulative cause-specific incidence functions corresponding to (70) for r = 1 . . . R are F r (t|z) = t 0 dt λ r (t )e −Λ0(t ) L =1 w eβ r ·z− R r =1 exp(β r ·z)Λ r (t )(79) The specific parametrisation (70) has two further useful features: • After the class sizes (w 1 , . . . , w L ) have been inferred, for a chosen (or optimised) value of L, one obtains the effective number L eff of classes via Shannon's information-theoretic entropy S (Cover and Thomas, 1991), which takes into account any class size differences: L eff = e S , S = − L =1 w log w(80) • Since our latent classes are defined in terms of the relation between covariates and risk, one cannot predict class membership for individuals on the basis of covariate information alone. However, after having identified the parameters of our cohort, Bayesian arguments allow us to calculate retrospective class membership probabilities for any individual on which we have the covariates z and survival information (t, r). For each class label , the model (70) gives P (t, r|z, ) =λ r (t) eβ r ·z−Λ0(t)− R r =1 exp(β r ·z)Λ r (t)(81) Hence, using P (t, r, |z) = P (t, r|z, )w and P (t, r|z) = L =1 P (t, r|z, )w , we obtain P ( |t, r, z) = w P (t, r|z, ) L =1 w P (t, r|z, ) = w eβ r ·z− R r =1 exp(β r ·z)Λ r (t) L =1 w eβ r ·z− R r =1 exp(β r ·z)Λ r (t)(82) Formula (82) allows us to assign each individual in our cohort retrospectively to the identified latent classes: the probability that individual i belongs to class is given by P ( |t i , r i , z i ). The effective number of classes (80) can also be a practical tool for identifying the optimal value of L, complementary to Bayesian model selection. A useful application of (82) would be to aid the search for informative new covariates that could increase our ability to predict personalised risk in heterogeneous cohorts. Such missing covariates are expected to be features that impact upon risk and which patients in the same class tend to have in common. Finally, instead of imposing by hand the independence of the end-of-trial risk on covariates (to reduce the number of model parameters), one could also treat the end-of-trial risk as any other risk. Any parameter estimation protocol should then report thatβ µ 0 = 0 for all and all µ = 1 . . . p, which gives a convenient sanity test of numerical imlementations. In addition one sometimes finds that this trivial enlargement of the search space reduces the impact of spurious local minima. A unimodal parametrisation for heterogeneity-induced competing risks For unimodal distributions of individual regression parameters a more appropriate parametrisation would be to replace in (70) the latent class distribution M(β 1 , . . . , β R ) by a Gaussian one: M(β 1 , . . . , β R ; λ 1 , . . . , λ R ) = M(β 1 , . . . , β R ) R r=1 δ F [λ r −λ r ](83)M(β 1 , . . . , β R ) = e − 1 2      β 1 −β 1 . . . β R −β R      ·C −1      β 1 −β 1 . . . β R −β R      (2π) (p+1)R/2 Det 1 2 C(84) For this choice the parameters to be estimated are: the base hazard rates {λ r (t)} of all risks, the location {β 1 , . . . ,β R } of the centre of (84), and the entries of the (p+1)R×(p+1)R covariance matrix C. The corresponding risk-dependent part of the data log-likelihood (58) is L risks (M) = N i=1 δ 0ri logλ ri (t i ) + N i=1 log dβ 1 . . . dβ R M(β 1 , . . . , β R ) × e δ0r i β r i ·zi− R r=1Λ r (ti) exp(β r ·zi)(85) For each risk r and each z the linear combination β r ·z will also be a Gaussian variable. This allows us to simplify the above (p+1)R-dimensional integral to a R-dimensional one, which involves the R×R matrix K(z) with entries K rr (z) = z · C rr z, r, r = 1 . . . R Here C rr denotes the (p + 1) × (p + 1) sub-matrix of C with entries (C rr ) µµ = β µ r β µ r M − β µ r M β µ r M . Averages refer to the dstribution (84). The result is (see Appendix D for details): L risks (M) = N i=1 δ 0ri logλ ri (t i ) + R r=1 N i=1 δ rri β r · z i + 1 2 z i · C rr z i + log Dy e − R r =1Λr (ti) exp[β r ·zi+zi·C rr zi+ R r =1 [K 1 2 (zi)] r r y r ](87) Here y ∈ IR R and K 1 2 (z) is the matrix defined by the property [K 1 2 (z)] 2 = K(z). It is unique because K(z) is non-negative definite and symmetric for any z. Finally, Jenssen's inequality tells us that Dy exp[u(y)] ≥ exp Dy u(y). This allows us after integration over y to obtain an explicit lower bound for L risks (M), which becomes an equality in the absence of heterogeneity (i.e. when C → 0) and which is convenient in numerical calculations: L risks (M) ≥ N i=1 δ 0ri logλ ri (t i ) + R r=1 N i=1 δ rri β r · z i + 1 2 z i · C rr z i − R r =1Λ r (t i )eβ r ·zi+zi·C rr zi+ 1 2 zi·C r r zi(88) Application to synthetic survival data To test our regression method under controlled conditions we apply it first to synthetic data with heterogeneity-induced competing risks, generated from populations of the type (70). Details of the numerical generation of these data are given in Appendix E. Our method is required to uncover and map a cohort's risk and association substructure, if such substructure exists, i.e. report the number and sizes of sub-cohorts and their distinct regression parameters for all risks. It should then use this extracted information to generate correct decontaminated survival curves, and assign individuals retrospectively to their latent classes, with statistically significant accuracy. Cohort substructure and regression parameters We generated numerically event times and event types for three heterogeneous data sets A,B and C. Each set has N = 1600 individuals from L = 2 latent classes of equal size, with at most two real risks, and with end-of-trial censoring at time t = 50. Each indivdual i has three covariates (z 1 i , z 2 i , z 3 i ), drawn randomly and independently from P (z) = (2π) −1/2 e −z 2 /2 . All frailty parameters β 0 r are zero. The base hazard rates of the risks are time-independent:λ 1 (t) = 0.05 (primary risk) andλ 2 (t) = 0.1 (if the secondary risk is enabled). Table 1 shows the further specifications of the data sets, together with the results of performing proportional hazards regression (Cox, 1972), and our generic heterogeneous regression according to the latent class log-likelihood (74) where the MAP protocol was complemented with Aikake's Information Criterion as described in (130). The data sets were constructed such that they have fully identical primary risk characterics. In set A there is heterogeneity but no competing risk. In set B a secondary risk is introduced, which in one of the two classes targets individuals similar to those most sensitive to the primary risk (with respect to the first covariate); here one expects false protectivity effects. In set C a secondary risk is introduced, which in one of the two classes targets individuals similar to those least sensitive to the primary risk (with respect to the first covariate); here one expects false exposure effects. As expected, the proportional hazards regression method (Cox, 1972) fails to report meaningful results, since it aims to describe the relation between covariates and the primary risk in each data set with a single regression vector (β 1 1 , β 2 1 , β 3 1 ). The heterogeneous regression based on (74,130) always reports the correct number of classes (L = 2), and the correct class-specific parameters (within accuracy limits determined by numerical search accuracy and finite sample size). Note that the assigment of class labels to identified classes is in principle arbitrary; see e.g. the regression results for data set B, where the class labelled = 2 is labelled = 1 in the data definition. Decontaminated survival functions The second test of the regression method and its numerical implementation is to verify that for all three data sets A, B and C described in table 5.1 it can extract the correct decontaminated covariateconditioned survival curveS 1 (t|z) for the primary risk, from the survival data alone. The result Table 1. Characteristics of three synthetic data sets A, B and C, all of the form (70) with two equally large latent classes and N = 1600 individuals. All three have identical primary risk parameters; they differ only in characteristics of the secondary risk. We also show the results for each set of Cox's propertional hazards regression, and of our generic regression method based on (74) and (130), with parameters extimated via Maximum A Posteriori likelihood augmented with Aikake's Information Criterion. Error bars in regression parameters are of the order of the last specified decimal. Cox regression cannot cope with heterogeneity and reports non-informative parameters, whereas the generic heterogeneous regression method is able to identify reasonably accurately the cohort's substructure (number and sizes of the classes) and its class-specific regression parameters. (t) (jagged solid curves) and crude survival functions S1(t) (smooth solid curves) of the primary risk, calculated for the upper and lower quartiles (UQ, LQ), and the inter-quartile range (IQ) of covariate 1, for the synthetic data of Table 5.1. Dotted red curve: the true primary risk survival functions (90,91) of the upper/lower quartiles (which are identical) and of the inter-quartile range. The primary risk characteristics of all three data sets are identical. In set A there is only the primary risk; although the cohort is heterogeneous, the estimators quantify the primary risk correctly. In set B the secondary risk is seen to cause 'false protectivity' for the primary risk. In set C it causes 'false exposure'. Risk correlations in data sets B and C clearly invalidate the use of Kaplan-Meier estimators and crude survival functions. Bottom row: decontaminated survival curves for the primary risk (solid black), for the same data and covariate ranges. The decontaminated curvesS1 are seen to be reliable estimates of the quantitative characteristics of the primary risk. should be identical in all three cases, since the data sets differ only in the interference effects of a secondary risk. For the primary risk in table 5.1, the correct expression (75) simplifies tõ S 1 (t|z 1 ) = 1 2 e − t 20 exp(2z1) + 1 2 e − t 20 exp(−2z1)(89) From this we can calculate the true primary risk survival curves for the upper and lower quartiles (UQ, LQ) and for the inter-quartile range (IQ). For the present Gaussian-distributed covariates, with zero average and unit variance, the upper and lower quartile survival curves are identical, due to the symmetryS 1 (t| − z 1 ) =S 1 (t|z 1 ). With the usual short-hand Dz = (2π) −1/2 e −z 2 /2 dz, and with the quartile point z Q defined via ∞ z Q Dz = 1 4 , giving z Q ≈ 0.67449, we obtain from (89) the following (exact) formulae: Here the risks are uncorrelated only for data set A (where there is no secondary risk). As soon as the risks are correlated, in data sets B and C, we see in Figure 2 that the Kaplan-Meier estimators and the crude survival functions no longer predict the true (red) curves. As anticipated from Table 5.1, they both underestimate grossly the primary risk in data set B (where the competing risk filters out high-primary-risk individuals) and overestimate the primary risk in data set C (where the competing risk filters out low-primary-risk individuals). In fact, plotting only the upper and lower quartile curves for S KM 1 or S 1 would suggest a strong overall impact of covariate 1 on the primary risk in data sets B and C, where in reality there is none. In contrast, the decontaminated curvesS 1 calculated from our generic heterogeneous regression protocol (lower three panels of Figure 2) do capture and predict the true survival functions of the primary risk from survival data alone, in spite of the presence of the competing risk. LQ, UQ :S 1 (t|z 1 ∈ [z Q , ∞)) = 2 ∞ z Q Dz e − t 20 exp(2z) + e − t 20 exp(−2z) (90) IQ :S 1 (t|z 1 ∈ [−z Q , z Q ]) = 2 z Q 0 Dz e − t 20 exp(2z) + e − t 20 exp(−2z)(91) Retrospective class identification Finally we illustrate with synthetic data the ability of our methodology to identify the classes of the individuals in a given data set, retrospectively, via (82), after having estimated the generating cohort's structure and parameters. We generate heterogeneous data sets with three risks, zero frailty parameters, and three independently generated (zero average and unit variance) Gaussian covariates. Each set has L = 3 classes, with w 1 = w 2 = w 3 = 1 3 , and the following regression parameter vectors: β 1 1 = ( 1 2 , 1 2 , 1 2 ) + (1, 0, 1) β 2 1 = ( 1 2 , 1 2 , 1 2 ) + (−1, −1, 0) (92) β 3 1 = ( 1 2 , 1 2 , 1 2 ) + (0, 1, −1) For the two non-primary risks r = 2, 3 we set β r = 0 for all . The base hazard rates areλ 1 (t) = 1/10,λ 2 (t) = 1/20 andλ 3 (t) = 1/30. For any value of the parameter the average primary risk regression vector over the cohort is 1 3 (β 1 + β 2 + β 3 ) = ( 1 2 , 1 2 , 1 2 ), with the cohort becoming homogeneous for = 0, and increasingly heterogeneous (i.e. separable) as increases. Formula (82) can visualise the overall class assigment by showing the triplets p i = (p i1 , p i2 , p i3 ) for all i as points in IR 3 , on the simplex defined by the conditions p 1 +p 2 +p 3 = 1 and p 1 , p 2 , p 3 ∈ [0, 1]. The result is shown in Fig. 3 for the synthetic data (92), with N = 9600 and λ = 4. Those individuals i that in reality originate from class 1 are indeed seen to be assigned probability vectors p i close to the point (1, 0, 0), those from class 2 tend to have p i close to the point (0, 1, 0), and those from class 3 have p i close to (0, 0, 1). The three corner points correspond to fully confident assignment. To quantify the quality of the retrospective class identification (82) we can assign each i to its most probable class argmax =1...L p i , and define the fraction f of correctly assigned individuals: f = 1 N N i=1 δ i,argmax =1...L p i(93) where i is the true class label of i. Even if the cohort's parameters were known perfectly, due to the intrinsic stochasticity of event times there will always be a fraction of unlikely events and f will always be less than 1. For the data of Fig. 3 one finds the value f ≈ 0.758, to be compared to the benchmark value f = 1 3 that would be obtained for random assignment to the three classes. One must expect the assigment quality f for the synthetic data (92) to increase monotonically with both the degree of class dissimilarity (as measured by ) and the size N of the data set (due to the improved recovery of the true model parameters). This is borne out by simulation experiments with different choices for ( , N ), the results of which are shown in Fig. 4. On average the number of primary (i.e. informative) events in these sets is about N/2, so for the present example one requires data sets with about 250 primary events or more to have good retrospective overall class allocation. From Fig. 3 one can also conclude that the classification reliability can be increased further by limiting oneself to the subset of patients i that have joint probability vectors p i close to one of the corners (1, 0, 0), (0, 1, 0) or (0, 0, 1) of the simplex (with obvious generalisations to L = 3). , and 929 were consored due to end of trial. We observe that for the above values of K, which controls the number of interpolation points in the base hazard rates, the most probable model to explain the ULSAM data would have two classes. Applications to prostate cancer data Prostate cancer (PC) data are notorious for exhibiting significant competing risk effects. The main reason for this is the fact that the disease tends to occur late in life, when there are an increased number of non-primary events whose incidence could correlate with prostate cancer. Here we analyse data from the ULSAM study (ULSAM, 2013), with N = 2047 individuals of which 208 reported PC as the first event. In this study we limited ourselves to a subset of five typical covariates (the ULSAM data set contains more), in view of CPU constraints; this is not a fundamental limitation, and we expect in the near future to have computer code upgrades that will allow us to include more. We refer again to Appendix E for numerical details. Cohort substructure and regression parameters We compare the outcomes of Cox's propertional hazards regression (Cox, 1972) and our generic heterogeneous regression method, based on (74) and (130), with parameters extimated via Maximum A Posteriori likelihood augmented with Aikake's Information Criterion. In Figure 5 we show the rescaled Aikake scores for different combinations of the discrete parameters L (number of classes in the cohort) and K (which controls the time resolution in the parametrised base hazard rates of all risks). Each score is the result of optimising numerically (via the MAP protocol) the regression and frailty coefficients and base hazard rates, for each risk and all classes, as well as the relative class sizes. For K = 1, 2, 3, 4 the most probable explanation of the ULSAM data involves two distinct classes, marking either residual disease or host heterogeneity not captured by covariates. For larger K values the Aikake score will increase further, but mainly due to the model's increasing ability to capture the base rate of the end-of-trial censoring events, which in the ULSAM data set is sharply peaked in time. Moreover, the score differences for different L values will be increasingly dominated by the Aikake-term, which acts to suppress models with many parameters and eventually Table 2. Regression results for the ULSAM prostate cancer (PC) data set, corresponding to some of the (L, K) combinations in Figure 5. We included five covariates: body mass index (BMI, realvalued), serum selenium level (selen, integer-valued), leisure time physical activity (phys1, discrete levels 0/1/2), work physical activity (phys2, discrete levels 0/1/2), and smoking (smok, discrete levels 0/1/2). We report the results of Cox's propertional hazards method, and of our generic regression method based on (74,130), with parameters extimated via Maximum A Posteriori likelihood augmented with Aikake's Information Criterion. We limit ourselves to K = 3 and K = 4. For K ≤ 2 the time dependence of the base hazard rates cannot be captured, as emphasised by nonnegligible regression coefficients for the end-of-trial risk (not shown here). For K ≥ 5 the growing parameter complexity (relative to the number of data points) pushes us increasingly towards the trivial L = 1 option. Error bars in regression parameters are of the order of the last specified decimal. See the main text for discussion and interpretation of the above data. forces us to accept only the trivial homogeneous explanation L = 1. Note that the score differences in Figure 5 are nonneglible: since the posterior likelihood of models relates to the Aikake score approximately via Prob(L|D) ∝ exp[Ψ(L|D)], a difference of e.g. ∆Ψ/N = 0.001 implies a model likelihood ratio of Prob(L|D)/Prob(L |D) ≈ 11.8. In Table 2 we show the regression outcomes for some of the (L, K) combinations in Figure 5 in more detail, together with the corresponding results from proportional hazards regression (Cox, 1972). As expected, Cox regression and the L = 1 explanation (which for the primary risk differs from the Cox protocol in the definition of the base hazard rate and a simple Bayesian prior for regression coefficients) give the same results. They both report weak effects of the five covariates, some of which are slightly unexpected (e.g. increased PC risk due to leisure time physical activity, weak protective effect of smoking). The apparently more probable L = 2 explanation generated by our generic heterogeneous regression method points consistently to a very different picture. It suggests that the ULSAM cohort should be viewed as consisting of two distinct classes of similar size: one class = 1 with relatively healthy individuals (in terms of both primary and secondary risk), and one class = 2 with rather frail individuals. This overall frailty difference follows from the significantly and consistently different frailty parameters of the two classes, and corresponds (t) (jagged solid curve), crude survival functions S1(t) (upper smooth solid curve) and decontaminated survival functionsS1(t) (lower smooth solid curve) of the primary risk. The crude and decontaminated survival curves are calculated from the model (L, K) = (2, 4); see Table 2. Since the crude and decontaminated curves are different, there is informative censoring in the ULSAM cohort. The true PC risk is predicted to be higher than that which would be found upon assuming risk independence, so here the competing risks act to give false protectivity. to hazard ratios (HR) of class membership in the range 0.01 − 0.02 for primary and secondary risks. The similar sizes of the two classes is consistent with the fact that about half of the ULSAM individuals are censored due to end of trial. In the relatively healthy class, the regression coefficients are now much more pronounced, and especially BMI and smoking are recognised as serious PC risk factors. In the frail class the regression coeficients are weak, and one expects the negative coefficients for e.g. BMI and smoking to reflect reverse causal effects: within this group, having a higher BMI and still being able to smoke may well be an indicator of relative health. One should not make the mistake of concluding that the above two-class explanation of the ULSAM data is for certain the best. There are multiple alternative ways to carry out the regression, e.g. by combining all non-primary risks (including the end-of-trial risk) or by setting the regression coefficients of the end-of-trial risk to zero (instead of leaving them to be determined by optimisation, as a test). In addition one could focus on larger K values, where the resulting complexity increase forces us to reject L > 1 solutions. The only safe conclusion to be drawn is that the new two-class explanation of the ULSAM data is both probabilistically and intuitively plausible. Decontaminated survival curves To assess whether the two-class explanation for the ULSAM data in Table 2 leads to different survival predictions, compared to standard methods, as a consequence of informative censoring by the non-primary risks, we calculate the crude and decontaminated primary risk survival functions, see (75,77), with parameters as estimated from our heterogeneous regression. We show also for comparison the Kaplan-Meier estimators S KM 1 (Kaplan and Meier, 1958) of the primary risk; note, however, that in the presence of competing risks these KM curves can no longer be trusted to estimate real survival curves (as emphasised by our previous synthetic data). The results are shown in Figures (t) (jagged solid curve) and crude survival functions S1(t) (smooth solid curve) of the primary risk, calculated for the three covariate subgroups z5 = 0 (nonsmokers), z5 = 1 (ex-smokers), and z5 = 2 (smokers). Right: Kaplan-Meier curves S KM 1 (t) (jagged solid curve) and decontaminated survival functionsS1(t) (smooth solid curve) of the primary risk, calculated for the same three covariate subgroups. The crude and decontaminated survival curves are calculated from the model (L, K) = (2, 4); see Table 2. and the decontaminated primary risk survival curves, i.e. there are clear competing risk effects. The decontaminated curves are significantly lower, implying a false protectivity effect of the competing risk. This makes sense in view of the date in Table 2, where we can see quite clearly that the regression coefficients of primary and secondary risks are indeed correlated; hence the secondary risk tends to remove from the cohort those individuals that are also more likely to have the primary (PC) events. However, the survival functions still show that smoking is associated with slightly decreased PC risk. The explanation is that the type-1 events (PC) occur prdominantly in the frail sub-class of individuals. Discussion When censoring by non-primary risks is informative, i.e. when non-primary events tend to occur at times that are not statistically independent of the primary event times, most of the commonly used survival analysis methods involving risk-specific quantities are no longer applicable, as they tend to be based on the assumption of risk independence. The observed (crude) cause specific hazard rates are no longer estimators of what these hazard rates would have been if all non-primary risks were disabled. Finding the latter 'decontamined' hazard rates and their associated 'decontaminated' cause-specific survival functions from observed survival data is called the competing risk problem. In this study we have developed a generic statistical description of survival analysis with competing risks that unifies the main schools of thought, such as frailty and random effect models, latent class models, and the Fine and Gray approach. We introduced the concept of heterogeneity-induced competing risks, i.e. informative censoring caused at cohort level, by residual (disease-or patient-) heterogeneity that is not captured by covariates, in populations where only at the level of individuals the different risks are independent. This differentiation of types of competing risks leads in a natural way to a classification of survival cohorts from the viewpoint of informative censoring, into four distinct complexity levels. Assuming heterogeneity-induced competing risks is much weaker than assuming risk independence, yet we demonstrated that it still imposes sufficient constraints to solve the competing risk problem. The canonical statistical description of cohorts with heterogeneityinduced competing risks is in terms of its covariate-constrained functional distribution of individual hazard rates of all risks. We derive exact formulae for decontaminated primary risk hazard rates and cause-specific survival functions, expressed in terms of this distribution. Translating the above formalism into a practical epidemiological tool requires constructing rational and efficient parametrisations of the covariate-conditioned functional distribution of hazard rates. We showed that assuming proportional hazards at the level of individuals leads to a natural family of such parametrisations, from which Cox regression, frailty and random effects models, and latent class models can all be recovered in special limits, and which also generates parametrised cumulative incidence functions (the language of Fine and Gray). Applications of the formalism include a better understanding of the nature and sub-structure of patient cohorts, tools for detecting and quantifying informative censoring, and improved outcome prediction from covariates (via decontaminated survival curves). In addition it may aid the search for informative biomarkers, via retrospective class assignment. We used a simple numerical implementation of the method to analyse synthetic data, which revealed how the generic method can uncover and map a cohort's substructure, if such substructure exists, and can indeed remove heterogeneity-induced false protectivity and false exposure effects. We showed that application to real survival data from the ULSAM study, with prostate cancer (PC) as the primary risk, leads to plausible alternative explanations for previous counter-intuitive inferences (such as a weak protective effect on PC of smoking), in terms of distinct sub-groups of patients with distinct risk factors and overall frailties. Although the statistical formalism in the first part of this study is for now completed, we regard the second part, where we construct parametrisations and numerical implementations, only as a first step. In future studies we will implement full Bayesian sampling (instead of the present MAP+AIC parameter estimation protocol), introduce cause specific base hazard rates that may depend on the class label (which was not implemented here to reduce model complexity), and investigate Gaussian mixture models (which can be seen as the integration of the presently proposed latent class and unimodel Gaussian parametrisations). A. Connection between cohort level and individual level cause-specific hazard rates Here we give the derivation of identities (26) and (27). Starting from (4), we multiply both sides by S(t) and insert P(t 0 , . . . , t R ) = 1 N N i=1 P i (t 0 , . . . , t R ). This gives, via (20): h r (t)S(t) = 1 N N i=1 ∞ 0 . . . ∞ 0 dt 0 . . . dt R P i (t 0 , . . . , t R )δ(t − t r ) R r =r θ(t r − t) = 1 N N i=1 S i (t)h i r (t)(94) Insertion of the identity S(t) = 1 h r (t) = N i=1 S i (t)h i r (t) N i=1 S i (t) = N i=1 h i r (t)e − R r =0 t 0 ds h i r (s) N i=1 e − R r =0 t 0 ds h i r (s)(95) In the case of covariate conditioning we repeat the above steps, but now we start from (8) and we use the sub-cohort distribution P(t 0 , . . . , t R |Z) = [ i, zi=z P i (t 0 , . . . , t R )]/[ i, zi=z 1] instead of P(t 0 , . . . , t R ). We then obtain: h r (t|z)S(t|z) = i, zi=z ∞ 0 . . . ∞ 0 dt 0 . . . dt R P i (t 0 , . . . , t R )δ(t − t r ) R r =r θ(t r − t) i, zi=z 1 = i, zi=z S i (t)h i r (t) i, zi=z 1(96) Insertion of the identity S(t|z) = [ i, zi=z S i (t)]/[ i, zi=z 1] and our formula for S i (t) now leads to the claimed result (27) for the covariate-conditioned case: h r (t|z) = i, zi=z h i r (t)e − R r =0 t 0 ds h i r (s) i, zi=z e − R r =0 t 0 ds h i r (s)(97) This completes the proof of identities (26) and (27). B. Equivalence of formulae for data likelihood in terms of W [h 0 , . . . , h R |z] In section 3 we have derived two routes for expressing the covariate-conditioned data likelihood P (t, r|z) in terms of W[h 0 , . . . , h R |z]. The first (direct) route is via (45): P A (t, r|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (t)e − R r =0 t 0 ds h r (s)(98) The second route is to combine (10) with formula (44) for the crude covariate-conditioned causespecific hazard rates, i.e. use the pair P B (t, r|z) = h r (t|z)e − R r =0 t 0 ds h r (s|z)(99 Although at first sight the two recipes for P (t, r|z) may appear to be different, one can show that they are identical (as they should be). We first note that at t = 0 both expressions agree, since P A (0, r|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (0) (101) P B (0, r|z) = h r (0|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (0) {dh 0 , . . . dh R } W[h 0 , . . . , h R |z] = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (0)(102) Next we show that the ratio of P A (t, r|z) and P B (t, r|z) is time-independent. We note that the numerator of (100) is identical to P A (t, r|z), so we may write d dt P B (t, r|z) P A (t, r|z) = d dt e − R r =0 t 0 ds h r (s|z) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − R r =0 t 0 ds h r (s) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − R r =0 t 0 ds h r (s) −1 × d dt e − R r =0 t 0 ds h r (s|z) − e − R r =0 t 0 ds h r (s|z) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] d dt e − R r =0 t 0 ds h r (s) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − R r =0 t 0 ds h r (s) = e − R r =0 t 0 ds h r (s|z) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − R r =0 t 0 ds h r (s) × R r =0 {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (t)e − R r =0 t 0 ds h r (s) {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − R r =0 t 0 ds h r (s) − h r (t|z) = 0(103) with the last line following directly from (100). Since we know that P B (t, r|z)/P A (t, r|z) = 1 at t = 0, we have now established that P B (t, r|z)/P A (t, r|z) = 1 for all t ≥ 0. C. Connection with standard regression methods Here we show how the equations of some conventional regression methods, that focus only on the hazard rates of the primary risk and assume primary and non-primary risks to be independent, are recovered from our generic formulae. For such models one has M(β 0 , . . . , β R ; λ 0 , . . . , λ R |z) = M(β 1 ; λ 1 |z)M(β 0 , β 2 , . . . , β R ; λ 0 , λ 2 , . . . , λ R |z). Inserting this into (58) gives L(M) = L 1 (M) + terms independent of M(β 1 ; λ 1 |z)(104) with, after some simple manipulations, L 1 (M) = N i=1 δ ri,1 log dβ 1 {dλ 1 } M(β 1 ; λ 1 |z i ) λ 1 (t i ) e β 1 ·zi−Λ1(ti) exp(β 1 ·zi) dβ 1 {dλ 1 } M(β 1 ; λ 1 |z i ) e −Λ1(ti) exp(β 1 ·zi) + N i=1 log dβ 1 {dλ 1 } M(β 1 ; λ 1 |z i ) e −Λ1(ti) exp(β 1 ·zi)(105) The independence of primary and non-primary risks causes (62) to simplify to h 1 (t|z) =h 1 (t|z), and (60,61) give the following decontaminated primary risk survival function and hazard rate: S 1 (t|z) = dβ 1 {dλ 1 } M(β 1 ; λ 1 |z) e − exp(β 1 ·z)Λ1(t) (106) h 1 (t|z) = dβ 1 {dλ 1 } M(β 1 ; λ 1 |z) λ 1 (t)e β 1 ·z−exp(β 1 ·z)Λ1(t) dβ 1 {dλ 1 } M(β 1 ; λ 1 |z) e − exp(β 1 ·z)Λ1(t)(107) Let us work out such formulae for the two most popular methods: • Cox's proportional hazards regression Cox's regression method implies choosing (65) for the distribution of primary risk parameters, viz. M(β 1 ; λ 1 |z) = δ F [λ 1 −λ] δ(β 0 1 ) p µ=1 δ(β µ 1 −β µ ) . Insertion into our formulae for the decontaminated survival function and hazard rate of the primary risk gives Cox's recipes S 1 (t|z) = e − exp( p µ=1β µ z µ )Λ(t) ,h 1 (t|z) =λ(t)e p µ=1β µ z µ(108) The contribution to (105) that contains primary risk parameters is then found to be L 1 (M) = N i=1 δ ri,1 logλ(t i ) + p µ=1β µ µ N i=1 δ ri,1 z µ i − N i=1 e p µ=1β µz µ iΛ (t i )(109) If we calculate from this the maximum likelihood estimate for the base hazard rate, via functional differentiation of (109) with respect toλ(t), we recover Breslow's formulâ λ(t) = N j=1 δ rj ,1 δ(t − t j ) N j=1 e p µ=1β µ z µ j θ(t j − t)(110) Insertion into (109) then gives, after some rewriting, the standard formula for the log-likelihood in terms of the remaining primary risk regression parameters (β 1 , . . . ,β p ) in Cox regression: L 1 (β 1 , . . . ,β p ) = L 1 (0) + N i=1 p µ=1β µ z µ i + log N j=1 θ(t j − t i ) N j=1 e p µ=1β µ z µ j θ(t j − t i )(111) • Frailty models The simple frailty models correspond to M(β 1 ; λ 1 |z) = δ F [λ 1 −λ] g(β 0 1 ) p µ=1 δ(β µ 1 −β µ ) Inserting this into (105) leads to L 1 (M) = N i=1 δ ri,1 logλ(t i ) + p µ=1β µ N i=1 δ ri,1 z µ i + N i=1 δ ri,1 log dβ 0 1 g(β 0 1 ) e β 1 0 −Λ(ti) exp(β 0 1 + p µ=1β µ z µ i ) dβ 0 1 g(β 0 1 ) e −Λ(ti) exp(β 0 1 + p µ=1β µ z µ i ) + N i=1 log dβ 0 1 g(β 0 1 ) e −Λ(ti) exp(β 0 1 + p µ=1β µ z µ i )(112) If for the frailty distribution g(β 0 1 ) one chooses g(β 0 1 ) = α α Γ(α) ∞ 0 dθ θ α−1 e −αθ δ(β 0 1 −log θ)(113) with Γ(z) = ∞ 0 dx x z−1 e −x , we will have the normalisation dβ 0 1 e β 0 1 = 1 (which removes the parametrisation redundancy), and one can do the relevant integrals analytically using dβ 0 1 g(β 0 1 ) e −y exp(β 0 1 ) = α α+y α (114) dβ 0 1 g(β 0 1 ) e β 0 1 −y exp(β 0 1 ) = α α+y α+1(115) Here we used the standard identity Γ(α+1) = αΓ(α) for the gamma function. The formulae (106,107) now givẽ S 1 (t|z) = 1+Λ (t i ) α e p µ=1β µ z µ i −α (116) h 1 (t|z) =λ(t)e p µ=1β µ z µ i 1+Λ (t i ) α e p µ=1β µ z µ i −1(117) For the primary contribution to the data likelihood we obtain L 1 (M) = N i=1 δ ri,1 logλ(t i ) + p µ=1β µ N i=1 δ ri,1 z µ i − α N i=1 log 1+Λ (t i ) α e p µ=1β µ z µ i − N i=1 δ ri,1 log 1+Λ (t i ) α e p µ=1β µ z µ i(118) It is no longer possible to find an analytical expression for the base hazard rate in terms of the other parameters, so here one is limited to either choosing convenient parametrised forms or to numerical maximisation. Since dβ 0 1 e 2β 0 1 = 1+ 1 α we have lim α→∞ g(β 0 1 ) = δ(β 0 1 −1), so for α → ∞ all the above equations must reduce to those corresponding to Cox regression, which indeed they do. The parameter α can also be estimated from the data via Bayesian methods (Ibrahim, Chen and Sinha, 2010). For random effects and latent class models one can recover from our generic formulation the various published results in a similar way. Since these models are more involved, fewer steps can typically be taken analytically, and their authors have to turn to numerical determination of parameters sooner. D. The Gaussian integral over regression parameters Here we derive a simplified expression for the log-likelihood (85), by manipulation of the (p+1)×Rdimensional Gaussian integral I(t, r, z) = dβ 1 . . . dβ R M(β 1 , . . . , β R ) e δ0rβ r ·z− R r =1Λr (t) exp(β r ·z) We first define for r = 1 . . . R new Gaussian variables x r = (β r −β r ) · z. According to the distribution (84) they have zero average and covariance matrix entries x r x r = K rr (z), with the matrix K(z) as defined in (86). Hence we may write, with x = (x 1 , . . . , x R ), I(t, r, z) = e δ0rβ r ·z dx e − 1 2 x·K −1 (z)x (2π) R DetK(z) e δ0rxr− R r =1Λr (t) exp(β r ·z+x r )(120) A final transformation x s = R r =1 [K 1/2 (z)] sr y r + δ 0r K sr (z) then brings us after some simple manipulations to the following expression, with the short-hand Dy = R r=1 [(2π) −1/2 e −y 2 r /2 dy r ]: I(t, r, z) = e δ0r[β r ·z+ 1 2 Krr(z)] × Dy e − R r =1Λr (t) exp[β r ·z+δ0rK rr (z)] exp[ R r =1 [K 1/2 (z)] r r y r ](121) With this result, and recalling the definition (86), we may write the data log-likelihood (85) as L risks (M) = N i=1 δ 0ri logλ ri (t i ) + N i=1 log I(t i , r i , z i ) = N i=1 δ 0ri logλ ri (t i ) + N i=1 δ 0ri β ri · z i + 1 2 z i · C riri z i (122) + N i=1 log Dy e − R r =1Λr (ti) exp[β r ·zi+zi·C r i r zi+ R r =1 [K 1/2 (zi)] r r y r ] As a simple test one confrms that in the limit C → 0, i.e. in the case of vanishing heterogeneity, one recovers from this expression the corresponding log-likelihood formula of the Cox model. E. Numerical details Here we give details of a number of numerical procedures that were used in the applications of our method to synthetic and real survival data, to facilitate the reproduction of our results. • Synthetic data All synthetic data used in this paper were generated as follows. For each individual i and each risk r = 1 . . . R we generate numerically a uniformly distributed random variable u ir ∈ [0, 1], and define a latent event time t ir = −τ ir log u ir . The survival data for i are then given by t i = min r∈{1,...,R} t ir , r i = argmin r∈{1,...,R} t ir These synthetic data will correspond to risks that are independent at the level of individuals: P i (t 1 , . . . , t R ) = R r=1 τ −1 ir e −tr/τir(124) with average latent times τ ir = ∞ 0 dt r t r P i (t r ). Let be the latent class in our cohort to which i belongs. We define the τ ir in terms of the individuals' covariate vector z i ∈ IR p : The individual cause-specific hazard rates of our synthetic data will then be as in the model (70) (see Figure 1), but with time-independent base hazard rates and without frailty terms: τ ir = λ −1 r e − pµ=1βh i r (t) = λ r e p µ=1β µ r z µ i(126) • Parametrisation of base hazard rates In our applications of the models (70,83) to synthetic and real data the base hazard rates λ r (t) and the parameters {w ,β µ r } are estimated by a version of the MAP method (54), i.e. by numerical maximization of the posterior probability P (D|θ)P (θ). This requires the base rates to be parametrised, for which we chose an interpolation method. Given the survival data D = {(t 1 , r 1 ), . . . , (t N , r N )}, which cover a time interval from t min = min i∈{1,...,N } t i to t max = max i∈{1,...,N } t i , we define K +1 equidistant time pointst k , with k = 0 . . . K, t k = t min + k K (t max − t min )(127) To eacht k we assign associated base rate parameters (ξ k0 , . . . , ξ kR ) for all risks r = 0 . . . R, which allows us to define the R + 1 base hazard rates λ r (t) for each r and each t ≥ 0 as smooth Gaussian convolutions, with uniform variation time scale σ = 1 2 (t k+1 −t k ): λ r (t|ξ) = K k=0 ξ kr e − 1 2 (t−t k ) 2 /σ 2 K k=0 e − 1 2 (t−t k ) 2 /σ 2 , σ = t max − t min 2K(128) See also Figure 8. The number K is either chosen or determined by Bayesian model selection. Increasing the value of K allows more irregular functions λ r (t) to be modelled, but it also increases considerably the number of parameters, which slows down the parameter estimation and adds to the danger of overfitting. The integrated rates Λ r (t|ξ) = t 0 ds λ r (s|ξ) are obtained numerically from (128) via 11-point Gaussian Quadrature, see e.g. (Press et al, 1992), applied separately to the n intervals [jt/n, (j+1)t/n] with j = 0 . . . n−1 and n = 20. • Numerical parameter optimisation The parameter optimisations used in the applications of our method to data all follow the MAP (Maximum A Posteriori Probability) protocol described in subsection 3.5, complemented by Aikake's Information Criterion (AIC) to limit overfitting; see e.g. (MacKay, 2003). In combination this implies that the optimal parameters of each of the models (70) and (83) are given in terms of their data log-likelihood L(θ) = log P (D|θ) and parameter prior P (θ) bŷ θ = argmin θ Ψ(θ) (129) Ψ(θ) = n par − L(θ) − log P (θ) Here n par = dim(θ) is the number of parameters of each parametrisation, i.e. parametrisation (70) : n par = RL(p+1) + KR + L − 1 parametrisation (83) : n par = 1 2 (p+1) 2 R 2 + 3 2 (p+1)R + KR We choose flat priors for the parameters {ξ r } of the base hazard rates (since the chosen parametrisation already ensures smoothness), flat priors for the weights W in (70) (i.e. the maximum entropy measure), and unit-variance zero-average Gaussian priors for all other parameters (which for the regression coefficients is justified by our decision to normalise all covariates by linear rescaling to zero average and unit variance over the cohort). The optimumθ is determined numerically via a simple adaptation of the Nelder-Mead or downhill simplex method, see e.g. Press et al (1992), in which we complement simplex iterations by repeated randomizations of decreasing amplitude. Setting ξ kr = ξ r for all k in (128) gives the stationary hazard rates λ r (t|ξ) = ξ r for all t ≥ 0, so a rational initialisation of our search algorithms is to set our parameters such that M(β 1 , . . . , β R ) = R r=1 δ(β r ),λ r (t) = ξ r (∀t ≥ 0) in which the {ξ r } are chosen such as to maximise the data log-likelihoods (74) and (85). For the choice (133) the latter formulae both reduce to L risks = R r=1 log(ξ r ) N i=1 δ rir − ξ r N i=1 t i , from which by simple differentiation we obtain ξ r = N i=1 δ rir / N i=1 t i(134) To break possible symmetries we assign small (Gaussian) random values to the regression coefficients, instead of the strictly zero values in (133), and we repeat the numerical searches for multiple random initialisations to reduce the impact of sub-optimal local minima. This leaves the model selection question, i.e. the selection of the parameters K, which controls the complexity of the base hazard rates, and L (the number of allowed distinct subcohorts in parametrisation (70)). For parametrisation (70) we determine for each choice of (L, K) numerically the value Ψ L,K = Ψ(θ), withθ = argmin θ Ψ(θ)| L,K , via the above procedure. The most probable model then corresponds to (L, K) = argmin L,K>0 Ψ L,K . For (83) we determine for each K > 0 numerically the value Ψ K = Ψ(θ), withθ = argmin θ Ψ(θ)| K . Here the most probable model corresponds to K = argmin K>0 Ψ K . • Calculation of parameter error bars via numerical estimation of curvature The error bar σ i associated with each estimateθ i is the standard deviation of the corresponding marginal of the posterior P (θ|D). The latter can be written in terms of the log-likelihood L(θ) and the function Ψ(θ) (130) that is minimised in the MAP method: σ 2 i = dθ P (θ|D)θ 2 i − dθ P (θ|D)θ i 2 = dθ e −Ψ(θ) θ 2 i dθ e −Ψ(θ) − dθ e −Ψ(θ) θ i dθ e −Ψ(θ) 2(135) Close to the most probable pointθ we may expand Ψ(θ) up to quadratic order in the deviation θ−θ, which implies approximating P (θ|D) by a Gaussian distribution aroundθ, i.e. Ψ(θ) = Ψ(θ) + 1 2 (θ −θ) · C −1 (θ −θ) + . . . and find upon neglecting cubic and higher orders that σ i ≈ C ii . The curvature matrix of the function Ψ(θ) atθ can in principle be calculated analytically. Alternatively, it can be obtained via simple numerical probes of Ψ close to the minimumθ, which is the method used in this paper. We note that for a truly quadratic surface Ψ(θ): Ψ(θ+∆θ) − Ψ(θ) = 1 2 ∆θ · C −1 ∆θ(137) We now define specific probes close toθ and the corresponding responses of the function Ψ: ∆θ i = δ ik : ∆Ψ k = Ψ(θ+∆θ) − Ψ(θ)(138) ∆θ i = (δ ik +δ i ) (k = ) : ∆Ψ k = Ψ(θ+∆θ) − Ψ(θ) For sufficiently small to ensure that the quadratic approximation (137) is good one then finds, after some rearranging of terms, the following simple formulae for the entries of C −1 : (C −1 ) kk = 2 −2 ∆Ψ k , k = : (C −1 ) k = −2 (∆Ψ k −∆Ψ k −∆Ψ ) Numerical matrix inversion then leads us to C and the error bars σ i . We averaged (140) prior to inversion over 10 different choices for , namely λ = 10 −3 ( 1 2 ) λ−1 with λ = 1 . . . 10, to obtain robust estimates. Our covariate normalisation ensures that all relevant parameters are of order one, so for uni-modal posterior distributions P (θ|D) the quadratic appoximation should be acceptable for the chosen values. For multi-modal distributions the error bars σ i will quantify only the local parameter uncertainty associated with the most probable pointθ. Fig. 2 . 2Top row: Kaplan-Meier curves S KM 1 Figure 2 1 ( 21shows the true LQ,UQ and IQ survival curves (90,91) for the data sets A, B and C of table 5.1, together with the decontaminated curvesS 1 in (75), as calculated from application of our heterogeneous regression method (74,130), see bottom row. We show also for comparison the Kaplan-Meier estimators S KMKaplan and Meier, 1958) of the primary risk survival function and the crude primary risk survival functions for the same covariate subsets (LQ,UQ,IQ), see top row. The crude survival functions are calculated via S 1 (t|z) = exp[− t 0 ds h r (s|z)], using the crude hazard rate (77) with parameters as estimated from our heterogeneous regression. As expected, the Kaplan-Meier estimators S KM 1 are estimators of the crude survival functions S 1 . However, both S KM 1 and S 1 are, in turn, only estimators of the true survival functionsS 1 of risk 1, i.e. of (90,91), if the different risks are uncorrelated. Fig. 3 .Fig. 4 . 34gives for each individual i, with covariates z i and outcome values (t i , r i ), the probabilities p i = P ( |t i , r i , z i ) for i to belong to each of the classes = 1 . . . L. For L = 3 one Retrospective assigment to classes of individuals in a heterogeneous synthetic cohort of the type (92), with N = 9600 and = 4. Of the 9600 events, 4968 were primary. Top row: posterior joint probabilities pi = (pi1, pi2, pi3) for each individual to belong to the three classes, calculated from the survival data alone, drawn as points in IR 3 . Black: joint likelihoods pi for all i. Red, green and blue: joint likelihoods pi of the three subsets of 3200 individuals that were generated respectively from the classes = 1, 2, 3. The points of each class indeed tend to be positioned close to the corners (1, 0, 0), (0, 1, 0) and (0, 0, 1) that would correspond to perfect allocation. Histograms (below): distributions of {pi1}, {pi2} and {pi3}, calculated for members of the three classes separately. Perfect assignment corresponds to finding most {p i } for each close to the value 1 only for the row of class , with values close to zero for the other rows. In this example the correctly assigned fraction is f ≈ 0.758 Connected markers: classification quality f (the fraction of correctly classified individuals) for synthetic data sets of the type (92) as a function of the data set size N , for different values of the degree of heterogeneity. Dashed: the value f = 1/3 corresponding to random class assigment. In all sets there are three (zero-average and unit-variance random Gaussian) covariates and three risks. The number of primary events in each set is roughly N/2. Fig. 5 . 5Rescaled Aikake-score as a function of the number of classes L, for K ∈ {1, 2, 3, 4} for the ULSAM prostate cancer data set. This set contains survival data on N = 2047 individuals, of whom 208 reported the primary event (prostate cancer), 910 reported a secondary event (death of other causes) Fig. 6 . 6Kaplan-Meier curves S KM 1 Fig. 7 . 76, where we plotted the survival function estimators for all members of the cohort, and 7, where we plotted the survival function estimators conditioned on the value of the fifth covariate (smoking, for which there aqre three values, viz. 0,1,2). We see a clear difference between the crude Left: Kaplan-Meier curves S KM 1 h i r (s)] for the individual survival functions then leads to the claimed identity (26): Fig. 8 . 8Illustration of the base hazard rate parametrisation via Gaussian interpolation. Circles: the base points (t k , ξ kr ), with k = 0 . . . K. The equi-distant timest k are fixed and given by(127); in this example tmin = 0 and tmax = 3. The K +1 values ξ kr are optimised via MAP (maximum a posteriori data likelihood). Solid lines: the corresponding smooth parametrised hazard rates λ(t|ξ) defined in (128). This parametrisation procedure is applied to the base hazard rates of all risks r = 0 . . . R. )h r (t|z) = {dh 0 . . . dh R } W[h 0 , . . . , h R |z] h r (t)e − R {dh 0 . . . dh R } W[h 0 , . . . , h R |z] e − Rr =0 t 0 ds h r (s) r =0 t 0 ds h r (s) J van Baardewijk et al. AcknowledgementsWe are grateful for fruitful discussions with Shola Agbaje, Salma Ayis, Maria D'Iorio, Niels Keiding, Katherine Lawler and Tony Ng, and for financial support from Prostate Cancer UK and the European Union's FP-7 Programme. Competing risks in epidemiology: possibilities and pitfalls. P K Andersen, R B Geskus, T De Witte, H Putter, International Journal of Epidemiology. 41Andersen, P.K., Geskus, R.B., de Witte, T. and Putter H. (2012). Competing risks in epidemiology: possibilities and pitfalls. International Journal of Epidemiology 41, 861-870. T M Cover, J A Thomas, Elements of Information Theory. 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Gaussian channels with memory and with noiseless feedback have been widely studied in the information theory literature. However, a coding scheme to achieve the feedback capacity is not available. In this paper, a coding scheme is proposed to achieve the feedback capacity for Gaussian channels. The coding scheme essentially implements the celebrated Kalman filter algorithm, and is equivalent to an estimation system over the same channel without feedback. It reveals that the achievable information rate of the feedback communication system can be alternatively given by the decay rate of the Cramer-Rao bound of the associated estimation system. Thus, combined with the control theoretic characterizations of feedback communication (proposed by Elia), this implies that the fundamental limitations in feedback communication, estimation, and control coincide. This leads to a unifying perspective that integrates information, estimation, and control. We also establish the optimality of the Kalman filtering in the sense of information transmission, a supplement to the optimality of Kalman filtering in the sense of information processing proposed by Mitter and Newton. In addition, the proposed coding scheme generalizes the Schalkwijk-Kailath codes and reduces the coding complexity and coding delay. The construction of the coding scheme amounts to solving a finite-dimensional optimization problem. A simplification to the optimal stationary input distribution developed by Yang, Kavcic, and Tatikonda is also obtained. The results are verified in a numerical example.

cs/0512097

712421be56fd56379df15d34f8cbdf9a25fed790

Gaussian Channels with Feedback: Optimality, Fundamental Limitations, and Connections of Communication, Estimation, and Control 26 Dec 2005 Jialing Liu Nicola Elia Gaussian Channels with Feedback: Optimality, Fundamental Limitations, and Connections of Communication, Estimation, and Control 26 Dec 2005arXiv:cs/0512097v1 [cs.IT] Gaussian channels with memory and with noiseless feedback have been widely studied in the information theory literature. However, a coding scheme to achieve the feedback capacity is not available. In this paper, a coding scheme is proposed to achieve the feedback capacity for Gaussian channels. The coding scheme essentially implements the celebrated Kalman filter algorithm, and is equivalent to an estimation system over the same channel without feedback. It reveals that the achievable information rate of the feedback communication system can be alternatively given by the decay rate of the Cramer-Rao bound of the associated estimation system. Thus, combined with the control theoretic characterizations of feedback communication (proposed by Elia), this implies that the fundamental limitations in feedback communication, estimation, and control coincide. This leads to a unifying perspective that integrates information, estimation, and control. We also establish the optimality of the Kalman filtering in the sense of information transmission, a supplement to the optimality of Kalman filtering in the sense of information processing proposed by Mitter and Newton. In addition, the proposed coding scheme generalizes the Schalkwijk-Kailath codes and reduces the coding complexity and coding delay. The construction of the coding scheme amounts to solving a finite-dimensional optimization problem. A simplification to the optimal stationary input distribution developed by Yang, Kavcic, and Tatikonda is also obtained. The results are verified in a numerical example. I. INTRODUCTION Communication systems in which the transmitters have access to noiseless feedback of channel outputs have been widely studied. As one of the most important case, the single-input single-output frequency-selective Gaussian channels with feedback have attracted considerable attention; see [1]- [16] and references therein for the capacity computation and coding scheme design for these channels. In particular, [1], [2] proposed ingenious feedback codes (called the Schalkwijk-Kailath codes, in short the SK codes) for additive white Gaussian noise (AWGN) channels, which achieve the asymptotic feedback capacity (i.e. the infinite-horizon feedback capacity, denoted C ∞ ) and greatly reduce the coding complexity and coding delay. [4], [5], [7] presented the extensions of the SK codes to Gaussian feedback channels with memory and obtained tight capacity bounds. [6] presented a rather general coding structure (called the Cover-Pombra structure, in short the CP structure) to achieve the finite-horizon feedback capacity (denoted C T , where the horizon spans from time epoch 0 to time epoch T ) for Gaussian channels with memory; however, it involves prohibitive computation complexity as the coding length (T + 1) increases. By exploiting the special properties of a moving-average Gaussian channel with feedback, [9] discovered the finite rankness of the innovations in the CP structure, which reduces the computation complexity. [10] reformulated the CP structure along this direction, and obtained an SK-based coding scheme to achieve C T with reduced computation complexity. Also along the line of [9], [15] studied a first-order moving-average Gaussian channel with feedback, found the closed-form expression for C ∞ , and obtained an SK-based coding scheme to achieve C ∞ . [11] provided a thorough study of feedback capacity; extended the notion of directed information proposed in [17] and proved that its supremum is the feedback capacity; reformulated the problem of computing C T as a stochastic control optimization problem; and proposed a dynamic programming based solution. This idea was further explored in [12], which uncovered the Markov property of the optimal input distributions for Gaussian channels with memory and eventually reduced the finite-horizon stochastic control optimization problem to a manageable size. Moreover, under a stationarity conjecture that C ∞ equals the stationary capacity (the maximum information rate over all stationary input distributions, denoted C s ), C ∞ is given by the solution of a finite dimensional optimization problem. This is the first computationally efficient 1 method to calculate C s or C T for general Gaussian channels. The stationary conjecture has been recently confirmed, namely C s = C ∞ , and C ∞ is achievable using a (an asymptotically) stationary input distribution [16]. [3] proposed a view of regarding the optimal communication over an AWGN channel with feedback as a control problem. [13] investigated the problem of tracking unstable sources over a channel and introduced the notion of anytime capacity to capture the fundamental limitations in that problem, which reveals intimate connections between communication and control and brings new insights to feedback communication problems. Furthermore, [14] established the equivalence between feedback communication and feedback stabilization over Gaussian channels with memory, showed that the achievable transmission rate is given by the Bode sensitivity integral of the associated control system, and presented an optimization problem based on robust control to compute lower bounds of C s . [14] also extended the SK codes to achieve these lower bounds, and the coding schemes have an interpretation of tracking unstable sources over Gaussian channels. For Gaussian networks with feedback, tight capacity bounds can be found in [14], [18], [19]. For time-selective fading channels with AWGN and with feedback, an SK-based coding scheme utilizing the channel fading information was constructed in [20] to achieve the ergodic capacity. As we can see, it remains an open problem to build a coding scheme with reasonable complexity to achieve C ∞ for a Gaussian channel with memory; note that no practical codes have been found based on the optimal signalling strategy in [12]. In this paper, we propose a coding scheme for frequencyselective Gaussian channels with output feedback. This coding scheme achieves C ∞ , the asymptotic feedback capacity of the channel; utilizes the Kalman filter algorithm; simplifies the coding processes; and shortens the coding delay. The optimal coding structure is essentially a finite-dimensional linear timeinvariant (FDLTI) system, is also an extension of the SK codes, and leads to a further simplification of the optimal stationary signalling strategy in [12]. The construction of the coding system amounts to solving a finite-dimensional optimization problem. Our solution holds for AWGN channels with intersymbol interference (ISI) where the ISI is model as a stable and minimum-phase FDLTI system; through the equivalence shown in [11], [12], this channel is equivalent to a colored Gaussian channel with rational noise power spectrums and without ISI. Note that the rationalness assumption is widely used and not too restrictive, since any power spectrum can be arbitrarily approximated by rational ones. In deriving our optimal coding design in infinite-horizon, we first present finite-horizon analysis (which is closely related to the CP structure) of the feedback communication problem, and then let the horizon length tend to infinity and obtain our optimal coding design which achieves C ∞ . More specifically, in our finite-horizon analysis, we establish the necessity of the Kalman filter: The Kalman filter is not only a device to provide sufficient statistics (which was shown in [12]), but also a device to ensure the power efficiency and to recover the message optimally. This also leads to a refinement of the CP structure, applicable for generic Gaussian channels. Additionally, the presence of the Kalman filter in our coding scheme reveals the intrinsic connections among feedback communication, estimation, and control. In particular, we show that the feedback communication problem over a Gaussian channel is essentially an optimal estimation problem, and the achievable rate of the feedback communication system is alternatively given by the decay rate of the Cramer-Rao bound (CRB) for the associated estimation system. Invoking the Bode sensitivity characterization of the achievable rate [14], we conclude that the fundamental limitations in feedback communication, estimation, and control coincide. We then extend the horizon to infinity and characterize the steady-state of the feedback communication problem. We finally show that our optimal scheme achieves C ∞ . We also remark that the necessity of the Kalman filter in the optimal coding scheme is not surprising, given various indications of the essential role of Kalman filtering (or minimum mean-squared error (MMSE) estimators; or minimum-energy control, its control theory equivalence; or the sum-product algorithm, its generalization) in optimal communication designs. See e.g. [12], [14], [21]- [24]. The study of the Kalman filter in the feedback communication problem along the line of [24] may shed important insights on optimal communication problems and is under current investigation. One main insight gained in this study is that, the perspective of unifying information, estimation, and control, three fundamental concepts, facilitates our development of the optimal feedback communication design. Though the connections between any two of the three concepts have been investigated or are under investigation, a joint study explicitly addressing all three is not available. Our study provides the first example that the connections among the three can be explored and utilized, to the best of our knowledge. In addition to helping us to achieve the optimality in the feedback communication problem, this new perspective establishes the optimality of the Kalman filtering in the sense of information transmission, a supplement to the optimality of Kalman filtering in the sense of information processing proposed by Mitter and Newton [24]. It also leads to a new formula connecting the mutual information in the feedback communication system and MMSE in the associated estimation problem, a supplement to a fundamental relation between mutual information and MMSE proposed by Guo, Shamai, and Verdu [25]. We anticipate that this new perspective may help us to study more general feedback communication problems in future investigations, such as multiuser feedback communications. This paper is organized as follows. In Section II, we introduce the channel models. The problem formulation is given in Section III, followed by the problem solution, i.e. the optimal coding scheme and the coding theorem. In Section IV, we prove the necessity of the Kalman filter in generating the optimal feedback. In Section V, we provide the connections of the feedback communication problem to an estimation problem and a control problem, and express the maximum achievable rate in terms of estimation theory quantities and control theory quantities. In Section VII, we show that our coding scheme is capacity-achieving. Section VIII provides a numerical example. Finally we conclude the paper and discuss future research directions. Notations: We represent time indices by subscripts, such as y t . We denote by y T the vector {y 0 , y 1 , · · · , y T }, and {y t } the sequence {y t } ∞ t=0 . We assume that the starting time of all processes is 0, consistent with the convention in dynamical systems but different from the information theory literature. We use h(X) for the differential entropy of the random variable X. For a random vector y T , we denote its covariance matrix as K (T ) y . For a stationary process {y t }, we denote its power spectrum as S y (e j2πθ ). We denote T xy (z) as the transfer function from x to y. We denote "defined to be" as ":=". We use (A, B, C, D) to represent system x t+1 = Ax t + Bu t y t = Cx t + Du t .(1) II. CHANNEL MODEL In this section, we briefly describe two Gaussian channel models, namely the colored Gaussian noise channel without ISI and white Gaussian noise channel with ISI. Fig. 1 (a) shows a colored Gaussian noise channel without ISI. At time t, this discrete-time channel is described asỹ A. Colored Gaussian noise channel without ISI t = u t + Z t , for t = 0, 1, · · · ,(2) where u t is the channel input, Z t is the channel noise, andỹ t is the channel output. We make the following assumptions: The colored noise {Z t } is the output of a finite-dimensional stable and minimum-phase linear time-invariant (LTI) system Z(z) driven by a white Gaussian process {N t } of zero mean and unit variance, and Z(z) is at initial rest. For any block size (i.e. coding length) of (T + 1), we may equivalently generate Z T by Z T = Z T N T ,(3) where Z T is a (T + 1) × (T + 1) lower-triangular Toeplitz matrix of the impulse response of Z(z). We may abuse the notation Z for both Z(z) and Z T if no confusion arises. As a consequence, {Z t } is asymptotically stationary. 2 Note that there is no loss of generality in assuming that Z(z) is stable and minimum-phase (cf. Chapter 11, [26]), implying that the initial condition of Z(z) generates no effect on the steady-state. Thus we made the initial rest assumption since we mainly focus on the steady-state characterization. Z N t u t Z t y t (a) N t u t y t Z Z y t F (b) N t H z y t s t s t F G u t (c) B. White Gaussian channel with ISI The above colored Gaussian channel induces a new channel, namely a white Gaussian channel with ISI, under a further assumption that Z(∞) = 0 (i.e. Z is proper but non-strictly proper). More precisely, notice that from (2) and (3), we haveỹ which we identify as a stable and minimum-phase ISI channel with AWGN {N t }, see Fig. 1 (b). Here Z −1 (z) is also at initial rest. For any fixed u T and N T , (a) and (b) generate the same channel output y T . 3 Note that Z −1 T is the matrix inverse of Z T , equal to the lower-triangular Toeplitz matrix of impulse response of Z −1 (z). The initial rest assumption on Z −1 can be imposed in practice equivalently by, first driving the initial condition of the ISI channel to any desired value (known to the receiver) before a transmission, and then removing the response due to that initial condition at the receiver. Such an assumption is also used in [11], [12]. We further assume for simplicity that Z(∞) = 1; for cases where g := Z(∞) = 1, we can normalize Z(z) by scaling it by 1/g. Hence, Z T is a lower triangular Toeplitz matrix with diagonal elements all equal to 1 (and thus is invertible). We can then write the minimal state-space representation of H) is observable, and m is the dimension or order of Z −1 . Let us denote the channel from u to y in Fig. 1(b) as F, where Z −1 as (F, G, H, 1), where F ∈ R m is stable, (F, G) is controllable, (F,y T := Z −1 T u T + N T = Z −1 Tỹ T .(5) The channel F is described in state-space as channel F : s t+1 = F s t + Gu t y t = Hs t + u t + N t ,(6) where s 0 = 0; see Fig. 1 (c). Notice that channel F is not essentially different than the channel from u toỹ, since {y t } and {ỹ t } causally determine each other. We concentrate on the case m ≥ 1; the case that m is 0 (i.e., F is an AWGN channel) was solved in [1], [2]. III. PROBLEM FORMULATION IN STEADY-STATE AND THE SOLUTION Before formulating the steady-state communication problem, we distinguish among the three scenarios: Finite-horizon (i.e. finite coding length), infinite-horizon (i.e. infinite coding length), and steady state. Finite-horizon problems often have time-dependent (i.e. time-varying) and horizon-dependent solutions (similar to finite-horizon Kalman filtering). The horizon-dependence may be removed in the infinitehorizon scenario, and furthermore, the time-dependence may be removed in the steady-state scenario. If we find the (stationary, time-invariant) steady-state solution (which by [16] is also the infinite-horizon solution), we can truncate it and employ the truncation to the practical problem in finite-horizon provided that the horizon is large enough. This truncated solution would greatly simplify the implementation while having a performance sufficiently close to finite-horizon optimality. A. Problem formulation For a Gaussian channel with feedback, the channel input may take the form u t = γ t u t−1 + η t y t−1 + ξ t(7) for any γ t ∈ R 1×t , η t ∈ R 1×t , and zero-mean Gaussian random variable ξ t ∈ R which is independent of u t−1 and y t−1 (cf. [11], [12]). Therefore, the channel inputs are allowed to depend on the channel outputs in a strictly causal manner. Our objective in this paper is to design encoder/decoder to achieve the asymptotic feedback capacity, given by C ∞ := C ∞ (P) := sup {ut} stationary,(7) lim T →∞ 1 T + 1 I(u T → y T ) s.t. P∞:=limT →∞ Eu T ′ u T /(T +1)≤P(8) where P > 0 is the power budget and I(u T → y T ) is the directed information from u T to y T (cf. [11]). For more details about C ∞ , refer to [12], [16] and Section VII-A in this paper. The problem of solving C ∞ may be equivalently formulated as minimizing the average channel input power while keeping the information rate bounded from below, namely for R > 0, P ∞ (R) := inf {ut} stationary,(7) lim T →∞ 1 T + 1 Eu T ′ u T . s.t. limT →∞ I(u T →y T )/(T +1)≥R(9) Therefore P ∞ (R) is the inverse function of C ∞ (P), i.e., C ∞ (P ∞ (R)) = R. Approach: Our approach to solve the steady-state communication problem is to investigate the finitehorizon problem first, and then let the horizon increase to infinity, which leads to a unified treatment of infinite-horizon and finite-horizon. Other approaches not pursued in this paper are also possible, such as applying the idea in [14] to the optimal signalling strategy in [12], though they generate results not as rich as the present approach does. B. The coding scheme The rest of this section presents the solution to the above problem. In this subsection, we introduce an encoder/decoder structure and explain how to choose the parameters to ensure the optimality, and then describe the encoding/decoding process, that is, how we assign the message to be transmitted, and how we recover the message. In the next subsection, we present the coding theorem which states that our encoding/decoding structure with the chosen parameters achieves C ∞ . The proof of the theorem will be developed in Sections IV to VII. The encoder/decoder structure In state-space, the encoder and decoder are described as Encoder:      x t+1 = Ax t r t = Cx t u t = r t −r t(10) and Decoder:             ŝ t+1 = Fŝ t + L 2 e t e t = y t − Hŝ t x t+1 = Ax t + L 1 e t r t = Cx t x 0,t = A −t−1x t+1 ,(11)whereŝ 0 = 0,x 0 = 0, A ∈ R (n+1)×(n+1) , C ∈ R 1×(n+1) , L 1 ∈ R n+1 , and L 2 ∈ R m . We call (n + 1) the encoder dimension, x t the encoder state, andx 0,t the decoder estimate. See Fig. 2 for the block diagram. Observe that −r t is the feedback from the decoder based on the channel output y t−1 , and thus u t depends on y t−1 but not y t . It further follows that −r t = G * t y t for some strictly lower triangular Toeplitz matrix G * t . Here A, C, u t , etc. depend on n, but we do not specify the dependence explicitly to simplify notations. Optimal choice of parameters Fix a desired rate R. Let DI := 2 R and n := m − 1 (recalling that m is the channel dimension), and solve the optimization problem Note that we need to solve (12) twice (one for +DI in A and one for −DI in A), and choose the optimal solution as the one with the smaller objective function value. Then we form the optimal A opt based on a a a opt f , and let (n * + 1) be the number of unstable eigenvalues in A opt , where n * ≥ 0. Now let n := n * , solve (12) again, and obtain a new a a a opt f and Σ opt . Then form A opt , let A * = A opt , Σ * = Σ opt , C * := [1, 0 1×n * ], and form A * ,C * , and D * . Let N t H z y t s t F G u t F A z x t C r t r t A z C e t H z F - x t s t - L L x b x ,t A t x t(12)L * := [L * 1 ′ , L * 2 ′ ] ′ := A * Σ * C * ′ C * Σ * C * ′ + 1 .(14) As we will show, (A * , C * ) is observable, and A * has exactly (n * + 1) unstable eigenvalues. We assign the encoder/decoder parameters to the scheme built in Fig We then drive the initial condition s 0 of channel F to zero. Now we are ready to communicate at a rate R using power P ∞ (R) = D * Σ * D * ′ . 4 Encoding/Decoding process 1) Transmission of analog source: The designed communication system can transmit either an analog source or a digital message. In the former case, we assume that the encoder wishes to convey a Gaussian random vector through the channel and the decoder wishes to learn the random vector, which is a ratedistortion problem (or successive refinement problem, see e.g. [13], [27], [28]). The coding process is as follows. Assume that the to-be-conveyed message W is distributed as N (0, I n * +1 ) (noting that any non-degenerate (n * + 1)-variate Gaussian vector W can be transformed into this form). Assume that the coding length is (T + 1). To encode, let x 0 := W . Then run the system till time epoch T , obtaininĝ x 0,t , t = 0, 1, · · · , T . To decode, letŴ t :=x 0,t for t = 0, 1, · · · , T . The quantities of interest include the squared-error distortion, defined as MSE(Ŵ t ) := E(W −Ŵ t )(W −Ŵ t ) ′ .(16) It will become clear that MSE(Ŵ t ) can be pre-computed before the transmission, and thus the coding length can be determined a priori to ensure a desired distortion level. 2) Transmission of digital message: To transmit digital messages over the communication system, let us first fix ǫ > 0 small enough and the coding length (T + 1) large enough. Let Σ * x := [I n * +1 , 0]Σ * [I n * +1 , 0] ′ .(17) Assume that the matrix (A * ′ ) −T −1 Σ * x (A * ) −T −1 has an eigenvalue decomposition as (A * ′ ) −T −1 Σ * x (A * ) −T −1 = E T Λ T E ′ T ,(18) where E T = [e (1) , · · · , e (n * +1) ] is an orthonormal matrix and Λ T is a positive diagonal matrix. Let σ T,i be the square root of the (i, i)th element of Λ T . Let B ∈ R n * +1 be the unit hypercube spanned by columns of E T , that is, B = n * i=0 α (i) e (i) α (i) ∈ [− 1 2 , 1 2 ], i = 0, · · · , n * .(19) Next we partition the ith side of B into (σ T,i ) −(1−ǫ) segments. This induces a partition of B into M T sub-hypercubes, where M T = n * i=0 (σ T,i ) −(1−ǫ) = det (A * ′ ) −T −1 Σ * (A * ) −T −1 − 1−ǫ 2 .(20) We then map the sub-hypercube centers to a set of M T equally likely messages. The above procedure is known to both the transmitter and receiver a priori. Suppose now we wish to transmit the message represented by the center W . To encode, let x 0 := W . Then run the system till time epoch T . To decode, we mapx 0,T into the closest sub-hypercube center and obtain the decoded messageŴ T . We declare an error ifŴ T = W , and call a (an asymptotic) rate R := lim T →∞ 1 T + 1 log M T(21) achievable if the probability of error P E T vanishes as T tends to infinity. We remark that this coding process is the one used in [14] for Gaussian channels with memory, which was an extension of the SK codes. In fact, the original SK coding scheme can be rewritten in a Kalman filter form, and hence it essentially implements the Kalman filtering algorithm. We also remark that, similar to the analog transmission case, the coding length (T + 1) can be pre-determined. As we have seen, the encoder/decoder design and the encoding/decoding process can be done rather easily. The computation complexity for encoding/decoding grows as O(T + 1). Also interestingly, the encoder may be viewed as a control system, and the decoder may be viewed as an estimation system, as pointed out by Sanjoy Mitter and in [13], [29]. Fig. 2 using n * , A * , C * , L * 1 , and L * 2 . Then under the power constraint Eu 2 ≤ P, C. Coding theorem Theorem 1. Construct the encoder/decoder shown in i) The coding scheme transmits an analog source W ∼ N (0, I n * +1 ) from the encoder to the decoder at rate C ∞ (P), with MSE distortion MSE(Ŵ T ) achieving the optimal asymptotic rate-distortion tradeoff given by R = lim T →∞ 1 2(T + 1) log 1 det MSE(Ŵ T ) .(22) ii) The coding scheme can transmit digital message from the encoder to the decoder at a rate arbitrarily close to C ∞ (P), with P E T decays to zero doubly exponentially. The proof of the theorem will be developed in the subsequent four sections. In Section IV, we consider a general coding structure in finite-horizon which may be viewed as a generalization of our optimal coding structure. We show that this general structure essentially contains a Kalman filter. The presence of the Kalman filter links the feedback communication problem to an estimation problem and a control problem, and hence we rewrite the information rate in terms of estimation theory quantities and control theory quantities; see Section V. Sections IV and V are focused on finite-horizon. In Section VI, we extend the horizon to infinity and characterize the steady-state behavior. Then in Section VII, we show that our optimal encoder/decoder design is actually the solution to the steady-state communication problem. IV. NECESSITY OF KALMAN FILTER FOR OPTIMAL CODING In this section, we consider a finite-horizon coding structure that includes our optimal design in Section III as a special case. This general structure is useful since: 1) searching over all possible parameters in the general structure achieves C ∞ , that is, there is no loss of generality or optimality to focus on this structure only; 2) we can show that to ensure power efficiency (to be explained), the general structure necessarily contains a Kalman filter. The general coding structure is in fact a variation of the CP structure (see Appendix II-D), and hence our Kalman filter characterization leads to a refinement of the CP structure. Fig. 3 illustrates the general coding structure, including the encoder and the feedback generator, a portion of the decoder. Below, we fix the time horizon to be {0, 1, · · · , T } and describe the coding structure. A. A general coding structure N t H z y t s t F G u t F G T A z W x t C r t r t Encoder: The encoder follows the dynamics (10). We assume that the encoder dimension (n + 1) satisfies 0 ≤ n ≤ T , W ∼ N (0, I n+1 ), A ∈ R (n+1)×(n+1) , C ∈ R 1×(n+1) , (A, C) is observable, and none of the eigenvalues of A are on the unit circle or at the locations of the eigenvalues of F . We then let Γ n (A, C) := Γ n := [C ′ , A ′ C ′ , · · · , A n′ C ′ ] ′ Γ(A, C) := Γ := [C ′ , A ′ C ′ , · · · , A T ′ C ′ ] ′ K (T ) r (A, C) := K (T ) r := Er T r T ′ .(23) Therefore, Γ n is the observability matrix for (A, C) and is invertible, Γ has rank (n + 1), r T = ΓW , and K (T ) r = ΓΓ ′ . Feedback generator: The feedback signal −r t is generated through the feedback generator G T , i.e. −r T = G T y T .(24) We assume that G T ∈ R (T +1)×(T +1) is a strictly lower triangular matrix. Clearly, the optimal encoder/decoder can be viewed as a special case of the general structure. Throughout the paper, the above assumptions on the encoder/decoder are always assumed unless otherwise specified. For future use purpose, we compute the channel output as y T = (I − Z −1 T G T ) −1 (Z −1 T r T + N T ).(25) Definition 1. Consider the general coding structure shown in Fig. 3. Define C T,n := C T,n (P) := sup A∈R (n+1)×(n+1) ,C,GT 1 T + 1 I(W ; y T ) s.t. Eu T ′ u T /(T +1)≤P(26) and define its inverse function as P T,n (R). In other words, C T,n is the finite-horizon information capacity for a fixed transmitter dimension n. It holds that C n,n = C n and hence lim n→∞ C n,n = C ∞ (see Lemma 1 and Appendix III-B). Moreover, as we will show, C ∞ can be achieved using this structure. B. The presence of Kalman filter We first compute the mutual information in the general coding structure. Fig. 3. Fix any 0 ≤ n ≤ T , and fix any A, C, and G T . Then it holds that Proposition 1. Consider the general coding structure in I(W ; y T ) = I(r T ; y T ) = I(u T → y T ) = 1 2 log det K (T ) y = 1 2 log det(I + Z −1 T K (T ) r Z −1 ′ T ) = 1 2 log det(I + Z −1 T ΓΓ ′ Z −1 ′ T ),(27) which is independent of G T . Proof: I(W ; y T ) = h(y T ) − h(y T |W ) = h(y T ) − h (I − Z −1 T G T ) −1 (Z −1 T r T + N T )|W (a) = 1 2 log det(2πeK (T ) y ) − h(N T ) (b) = I(u T → y T ) = 1 2 log det K (T ) y = 1 2 log det(I + Z −1 T K (T ) r Z −1 ′ T ),(28) where (a) is due to r T = ΓW , det(AB) = det A det B, and det(I − Z −1 T G T ) −1 = 1; and (b) follows from [14]. Proposition 1 implies that I(W ; y T ) is independent of the feedback generator G T , and dependent only on K (T ) r or equivalently on (A, C). Thus, fixed (A, C) implies fixed information rate, and hence the optimal feedback generator has to be chosen to minimize the average channel input power, which turns out to contain a Kalman filter. Note that the counterpart of this proposition in infinite-horizon was proven in [14]. Now we can define, for a fixed (A, C), the information rate across the channel to be R T (A, C) := I(W ; y T ) T + 1 .(29) The optimal feedback generator for a given (A, C) is found in the next proposition. Proposition 2. Consider the general coding structure in Fig. 3. Fix any 0 ≤ n ≤ T . Then i) P T,n (R) = inf A,C,GT :=G * T (A,C) 1 T + 1 Eu T ′ u T s.t. RT (A,C)≥R (30) where G * T (A, C) is the optimal feedback generator for a given (A, C), defined as G * T (A, C) := arg inf (A,C) fixed,GT 1 T + 1 Eu T ′ u T .(31) ii) The optimal feedback generator G * T (A, C) is given by G * T (A, C) = − G * T (A, C)(I − Z −1 T G * T (A, C)) −1 ,(32) where G * T (A, C) is the strictly causal MMSE estimator (Kalman filter) of r T given the noisy observation y T := Z −1 T r T + N T , i.e., G * T (A, C) := arg inf GT ∈R (T +1)×(T +1) 1 T + 1 E(r T − G Tȳ T )(r T − G Tȳ T ) ′ ,(33) where G T is strictly lower triangular. See Fig. 4 (a) for the associated estimation problem, (b) for the Kalman filter G * T (A, C), and (c) for the optimal feedback generator G * T (A, C). Remark 1. Proposition 2 reveals that, the minimization of channel input power in a feedback communication problem is equivalent to the minimization of MSE in an estimation problem. This equivalence yields a complete characterization (in terms of the Kalman filter) of optimal feedback generator G * T (A, C) for a given (A, C). Since our general coding structure is a variation of the CP structure, this proposition leads to the Kalman filter based characterization of the CP structure and hence is an improvement of the Cover-Pombra formulation; see Appendix II-D. Remark 2. Proposition 2 i) implies that we may reformulate the problem of C T,n (or P T,n ) as a twostep problem: In step 1, we fix (A, C), i.e. fixing the rate, and minimize the input power by searching over G; and in step 2, we search over all possible (A, C) subject to the rate constraint. The role of the feedback generator G for any fixed (A, C) is to minimize the input power. Then ii) solves the optimal feedback generator G * T (A, C) by considering the equivalent optimal estimation problem in Fig. 4 (a) whose solution is the Kalman filter. Notice that the Kalman filter can also give us the optimal estimate of the message W . Hence, the Kalman filter leads to both power efficiency and the best estimate of the message. The power efficiency is ensured by the one-step prediction operation of the Kalman filtering, and the optimal recovery of message is ensured by the smoothing operation of the Kalman filtering; therefore, we obtain the optimality of Kalman filtering in the information transmission sense. We finally note that the necessity of the Kalman filter is not surprising given the previous indications in [2], [5], [11], [13], [24], etc. Proof: i) Notice that for any fixed (A, C), R T (A, C) is fixed. Then from the definition of P T,n (R), we have P T,n (R) = inf A,C,GT 1 T + 1 Eu T ′ u T s.t. RT (A,C)≥R = inf A,C inf (A,C) fixed,GT 1 T + 1 Eu T ′ u T s.t. RT (A,C)≥R .(34) Then i) follows from the definition of G * T (A, C). ii) Note that for the general coding structure, it holds that N t H z F G F A z W x t C r t s t y t G T r t (a) H z F G A z C y t L ,t xt e t s t - r t L ,t b G T (b) G T A, C y t F, L ,t , H, x t s t - - r t e t A, L ,t , C,(c)u T = r T + (−r T ) = r T + G T y T .(35) Then, letting G T := −G T (I − Z −1 T G T ) −1(36)andȳ T := Z −1 T r T + N T , we have G T y T = − G Tȳ T . Therefore, G * T (A, C) = arg inf GT 1 T + 1 E(r T + G T y T )(r T + G T y T ) ′ = arg inf GT 1 T + 1 E(r T − G Tȳ T )(r T − G Tȳ T ) ′ .(37) The last equality implies that the optimal solution G * T is the strictly causal MMSE estimator (with onestep prediction) of r T givenȳ T ; notice that G T is strictly lower triangular. It is well known that such an estimator can be implemented recursively in state-space as a Kalman filter (cf. [30], [31]). Finally, from the relation between G T and G T , we obtain (32). The state-space representation of G * T (A, C) needs only a straightforward computation, as shown in Appendix I. We remark that it is possible to derive a dynamic programming based solution ( [11]) to compute C T,n , and if we further employ the Markov property in [12] and the above Kalman filter based characterization, we would reach a solution with complexity O(T ) for computing C T,n and C T . However, we do not pursue along this line in this paper since it is beyond the main scope of this paper. V. FEEDBACK RATE, CRB, AND BODE INTEGRAL We have shown that in the general coding structure, to ensure power efficiency for a fixed (A, C), we need to design a Kalman-filter based feedback generator. The Kalman filter immediately links the feedback communication problem to estimation and control problems. In this section, we present a unified representation for the general coding structure (with G being chosen as G * (A, C)), its estimation theory counterpart, and its control theory counterpart. Then we will establish connections among the information theory quantities, estimation theory quantities, and control theory quantities. A. Unified representation of feedback coding system, Kalman filter, and minimum-energy control In this subsection, we will present the dynamics for the estimation problem and the general coding structure, then show that they are governed by one set of equations, which may also be viewed as a control system. The estimation system The estimation system in Fig. 4 consists of three parts: the unknown source r T to be estimated or tracked, the channel F (without output feedback), and the estimator which we choose as the Kalman filter G * ; we assume that (A, C) is fixed and known to the estimator. The system is described in state-space as estimation system:                            x t+1 = Ax t r t = Cx t unknown sourcē s t+1 = Fs t + Gr t y t = Hs t + r t + N t channel F x t+1 = Ax t + L 1,t e t r t = Cx t s t+1 = Fŝ t + Gr t + L 2,t e t e t =ȳ t − Hŝ t −r t          Kalman filter G * (A, C)(38)with x 0 = W ,s 0 =ŝ 0 = 0, andx 0 = 0. Here L 1,t ∈ R n+1 and L 2,t ∈ R m are the time-varying Kalman filter gains specified in (43). The general coding structure with the optimal feedback generator The optimal feedback generator for a given (A, C) is solved in (32), see Fig. 4 (c) for its structure. We can then obtain the minimal state-space representation of G * T (A, C), and describe the general coding structure with G * T (A, C) as general coding structure:                                x t+1 = Ax t r t = Cx t u t = r t −r t encoder s t+1 = F s t + Gu t y t = Hs t + u t + N t channel F s t+1 = Fŝ t + L 2,t e t e t = y t − Hŝ t x t+1 = Ax t + L 1,t e t −r t = −Cx t      optimal feedback generator G * (A, C) (39) with x 0 = W , s 0 =ŝ 0 = 0, andx 0 = 0. See Appendix I for the derivation of the minimal state-space representation of G * T (A, C). It can be easily shown that r t ,r t , e t , x t , andx t in (38) and (39) are equal, respectively, and it holds that s t −ŝ t =s t −ŝ t .(40) The unified representation Definex t := x t −x t s t := s t −ŝ t =s t −ŝ t X t := x t s t X 0 := W 0 A := A 0 GC F C := [ C H ] D := [ C 0 ] L t := L 1,t L 2,t .(41) Note that X t is the estimation error for [x ′ t , s ′ t ] ′ . Substituting (41) to (38) and (39), we obtain that both systems become control system:      X t+1 = (A − L t C)X t − L t N t = AX t − L t e t e t = CX t + N t u t = DX t ;(42) see Fig. 5 for its block diagram. It is a control system where we want to minimize the power of u by appropriately choosing L t . This is a minimum energy control problem, which is useful for us to characterize the steady-state solution and it is equivalent to the Kalman filtering problem (see [32]). The signal e t in (42) is called the Kalman filter innovation or innovation 5 , which plays a significant role in Kalman filtering. One fact is that {e t } is a white process, that is, its covariance matrix K (T ) e is a diagonal matrix. Another fact is that e T and y T determine each other causally, and we can easily verify that h(e T ) = h(y T ) and det K (T ) y = det K (T ) e . We remark that (42) is the innovations representation of the Kalman filter (cf. [31]). For each t, the optimal L t is determined as L t := L 1,t L 2,t := AΣ t C ′ K e,t ,(43) where Σ t := EX t X ′ t , K e,t := E(e t ) 2 = CΣ t C ′ + 1, and the error covariance matrix Σ t satisfies the Riccati recursion Σ t+1 = AΣ t A ′ − AΣ t C ′ CΣ t A ′ CΣ t C ′ + 1(44) with initial condition Σ 0 := I n+1 0 0 0 ,(45) This completes the description of the optimal feedback generator for a given (A, C). The meaning of a unified expression for three different systems (38), (39), and (42) is that the first two are actually two different non-minimal realizations of the third. The input-output mappings from N T to e T in the three systems are T -equivalent (see Appendix I-B). Thus we say that the three problems, the optimal estimation problem, the optimal feedback generator problem, and the minimum-energy control problem, are equivalent in the sense that, if any one of the problems is solved, then the other two are solved. Since the estimation problem and the control problem are well studied, the equivalence facilitates our study of the communication problem. Particularly, the formulation (42) yields alternative expressions for the mutual information and average channel input power in the feedback communication problem, as we see in the next subsection. We further illustrate the relation of the estimation system and the communication system in Fig. 6: (b) is obtained from (a) by subtractingr t from the channel input and adding Z −1 Tr t back to the channel output, which does not affect the input, state, and output of G * T . It is clearly seen from the block diagram manipulations that the minimization of channel input power in feedback communication problem becomes the minimization of MSE in the estimation problem. i) B. Mutual information in terms of Fisher information and CRB N t A z W x t C r t y t r t Z T G T (a) N t A z W x t C r t y t r t Z T Z T - y t u t m G T G T A, C (b)I(W ; y T ) = 1 2 log det K (T ) e = 1 2 T t=0 log K e,t = 1 2 T t=0 log(CΣ t C ′ + 1) = 1 2 log det MMSE −1 W,T = 1 2 log det I W,T = 1 2 log det CRB −1 W,T ;(46) ii) P T,n (A, C) = 1 T + 1 T t=0 DΣ t D ′ = 1 T + 1 trace(CMMSE r,T ) = 1 T + 1 T t=0 CA t MMSE W,t A t′ C ′ ,(47) where MMSE W,T is the minimum MSE of W , CMMSE r,T is the causal minimum MSE of r T , I W,T is the Bayesian Fisher information matrix of W for the estimation system (38), and CRB W,T is the Bayesian CRB of W [33]. Remark 3. This proposition connects the mutual information to the innovations process and to the Fisher information, (minimum) MSE, and CRB of the associated estimation problem. As a consequence, the finite-horizon feedback capacity C T,n is then linked to the smallest possible Bayesian CRB, i.e. the smallest possible estimation error covariance, and thus the fundamental limitation in information theory is linked to the fundamental limitation in estimation theory. It is also interesting to notice that the Fisher information, an estimation quantity, indeed has an information theoretic interpretation as its name suggests. Besides, the link between the mutual information and the MMSE provides a supplement to the fundamental relation discovered in [25]; the connections between our result and that in [25] is under current investigation. Proof: i) First we simply notice that h(y T ) = h(e T ), and K e,t = CΣ t C ′ + 1. Next, to find MMSE of W , note that in Fig. 4 (a) ȳ T = Z −1 T ΓW + N T(48) and that W ∼ N (0, I), N T ∼ N (0, I). Thus, by [30] we have MMSE W,t = (I + Γ ′ Z −1 T ′ Z −1 T Γ) −1 ,(49)yielding det MMSE W,t = det(I + Z −1 T ΓΓ ′ Z −1 T ′ ) −1 = det(I + Z −1 T K (T,n) r Z −1 T ′ ) −1 .(50) Besides, from Section 2.4 in [33] we can directly compute the FIM of W to be (I + Γ ′ Z −1 T ′ Z −1 T Γ). Then i) follows from Proposition 1 and (42). ii) Since u t = DX t = Cx t = r t −r t and Ex tx 2 , and then ii) follows. ′ t = A t MMSE W,t A t′ , we have E(u t ) 2 = DΣ t D ′ = CEx tx ′ t C ′ = E(r t −r t ) C. Necessary condition for optimality Before we turn to the infinite-horizon analysis, we show in this subsection that our general coding structure together with the optimal feedback generator satisfies a "necessary condition for optimality" discussed in [15]. The condition says that, the channel input u t needs to be orthogonal to the past channel outputs y t−1 . This is intuitive since to ensure fastest transmission, the transmitter should not transmit any information that the receiver has obtained, thus the transmitter wants to remove any correlation of y t−1 in u t (to this aim, the transmitter has to access the channel outputs through feedback). Proposition 4. In system (39), for any 0 ≤ τ < t, it holds that Eu t y τ = 0. Proof: See Appendix II-E. VI. ASYMPTOTIC BEHAVIOR OF THE SYSTEM By far we have completed our analysis in finite-horizon. We have shown that the optimal design of encoder and decoder must contain a Kalman filter, and connected the feedback communication problem to an estimation problem and a control problem. Below, we consider the steady-state communication problem, by studying the limiting behavior (T going to infinity) of the finite-horizon solution while fixing the encoder dimension to be (n + 1). A. Convergence to steady-state The time-varying Kalman filter in (42) converges to a steady-state, namely (42) is stabilized in closedloop, u t , e t , and y t will converge to steady-state distributions, and Σ t , L t , G * t (A, C), G * t , and K e,t will converge to their steady-state values. That is, asymptotically (42) becomes an LTI system steady-state:      X t+1 = (A − LC)X t − LN t = AX t − Le t e t = CX t + N t u t = DX t ,(51) where L := AΣC ′ K e ,(52) K e = CΣC ′ + 1, and Σ is the unique stabilizing solution to the Riccati equation Σ = AΣA ′ − AΣC ′ CΣA ′ CΣC ′ + 1 .(53) This LTI system is easy to analyze (e.g., it allows transfer function based study) and to implement. For instance, the minimum-energy control (cf. [32]) of an LTI system claims that the transfer function from N to e is an all-pass function in the form of T N e (z) = k i=0 z − a i z − a −1 i (54) where a 0 , · · · , a k are the unstable eigenvalues of A or A (noting that F is stable). Note that this is consistent with the whiteness of innovations process {e t }. The existence of steady-state of the Kalman filter is proven in the following proposition. Notice that (42) is a singular Kalman filter since it has no process noise; the convergence of such a problem was established in [34]. (45), the Riccati recursion (44) generates a sequence {Σ t } that converges to Σ ∞ with rank (n + 1), the unique stabilizing solution to the Riccati equation (53). Proposition 5. Consider the Riccati recursion (44) and the system (42). i) Starting from the initial condition given in ii) The time-varying system (42) converges to the unique steady-state as given in (51). Proof: See Appendix III-A. B. Steady-state quantities Now fix (A, C) and let the horizon T in the general coding structure go to infinity. Let H(e) be the entropy rate of {e t }, DI(A) := k i=0 |a i | be the degree of instability of A, and S(e j2πθ ) be the spectrum of the sensitivity function of system (51) (cf. [14]). Then the limiting result of Proposition 3 is summarized in the next proposition. ii) The average channel input power is given by P ∞,n (A, C) := lim T →∞ 1 T + 1 Eu T u ′ T = DΣD ′ .(56) Remark 4. Proposition 6 links the asymptotic information rate to the entropy rate of the innovations process, to the degree of instability and Bode sensitivity integral ( [14]), to the asymptotic increasing rate of the Fisher information, and to the asymptotic decay rate of MSE and of CRB. Recall that the Bode sensitivity integral is the fundamental limitation of the disturbance rejection (control) problem, and the asymptotic decay rate of CRB is the fundamental limitation of the recursive estimation problem. Hence, the fundamental limitations in feedback communication, control, and estimation coincide. Remark 5. Proposition 6 implies that the presence of stable eigenvalues in A does not affect the rate (see also [14]). Stable eigenvalues do not affect P ∞,n (A, C), either, since the initial condition response associated with the stable eigenvalues can be tracked with zero power (i.e. zero MSE). So, we can achieve C ∞,n by a sequence of purely unstable (A, C), and hence the communication problem is related to the tracking of purely unstable source over a communication channel ( [13], [14]). Proof: Proposition 6 leads to that, the limits of the results in Proposition 3 are well defined. Then R ∞,n (A, C) = lim T →∞ 1 2(T + 1) T t=0 log K e,t = lim T →∞ 1 2 log K e,t = H(e) − 1 2 log 2πe,(57) where the second equality is due to the Cesaro mean (i.e., if a k converges to a, then the average of the first k terms converges to a as k goes to infinity), and the last equality follows from the definition of entropy rate of a Gaussian process (cf. [35]). Now by (54), {e t } has a flat power spectrum with magnitude DI(A) 2 . Then R ∞,n (A, C) = log DI(A). The Bode integral of sensitivity follows from [14]. The other equalities are the direct applications of the Cesaro mean to the results in Proposition 3. VII. ACHIEVABILITY OF C ∞ In this section, we will prove that C ∞,m−1 = C ∞ , leading to the optimality of our encoder/decoder design in Section III in the mutual information sense, and then show that our design achieves C ∞ in the operational sense. A. The optimal Gauss-Markov signalling strategy and a simplification [12] proved that for each input in the form of (7), there exists a Gauss-Markov (GM) input that yields the same directed information and same input power. The GM input takes the form u t = d ′ tss,t + E t ,(58) where d t ∈ R m is a time-varying gain; {E t } is a zero-mean white Gaussian process and E t is independent on N t−1 , u t−1 , and y t−1 ; ands s,t is generated by a Kalman filter (noting that this Kalman filter is different from the Kalman filter obtained in this paper)     s s,t := s t −ŝ s,t s s,t+1 = Fŝ s,t + L s,t e t e t = y t − Hŝ s,t , whereŝ s,0 = 0, L s,t := Q t Σ s,t (H + d ′ t ) ′ + K (t) E G 1 + K (t) E + (H + d ′ t )Σ s,t (H + d ′ t ) ′ ,(60) Q t := F + Gd ′ t , and Σ s,t := Es s,ts ′ s,t is the estimation error covariance of s t , satisfying the Riccati recursion Σ s,t+1 = Q t Σ s,t Q ′ t + K (t) E GG ′ − (Q t Σ s,t (H + d ′ t ) ′ + K (t) E G)(Q t Σ s,t (H + d ′ t ) ′ + K (t) E G) ′ 1 + K (t) E + (H + d ′ t )Σ s,t (H + d ′ t ) ′ .(61) If one lets d t = d and K (t) E = K E for all t, that is, the input {u t } is a stationary process, then the search over all possible d and K E solves C ∞ , that is, C ∞ (P) = max d∈R m ,KE ∈R 1 2 log(1 + K E + (H + d ′ )Σ s (H + d ′ ) ′ )(62) subject to Riccati equation constraint and power constraint Σ s = QΣ s Q ′ + K E GG ′ − (QΣs(H+d ′ ) ′ +KE G)(QΣs(H+d ′ ) ′ +KE G) ′ 1+KE +(H+d ′ )Σs(H+d ′ ) ′ P = d ′ Σ s d + K E .(63) We remark that [12] was focused more on the structure of the optimal input distribution and capacity computation, instead of designing a coding scheme; how to encode/decode a message (rather than using a random coding argument) is not clear from [12]. Now we prove that K E = 0, namely {E t } vanishes in steady-state. 6 This leads to a further simplification of the results in [12]. Proposition 7. For the GM input (58) to achieve C ∞ , it must hold that K E = 0. Proof: See Appendix IV. The vanishing of {E t } in steady-state helps us to show that, our general coding structure shown in Fig. 3 can achieve C ∞ , and the encoder dimension needs not be higher than the channel dimension, namely to achieve C ∞ we need A to have at most m unstable eigenvalues, as we will see in the next subsection. B. Generality of the general coding structure; finite dimensionality of the optimal solution In this subsection, we show that the general coding structure is sufficient to achieve mutual information C ∞ . In other words, if we search over all admissible parameters A, C, G T in the general coding structure, allowing T to increase to infinity and n to increase to (m − 1), then we can obtain C ∞ . Thus, there is no loss of generality and optimality to consider only the general coding structure with encoder dimension no greater than m. Fig. 3. Let C ∞,n := C ∞,n (P) := sup Definition 2. Consider the general coding structure in A∈R (n+1)×(n+1) ,C,G∞ lim T →∞ 1 T + 1 I(W ; y T )(64) subject to P ∞,n := lim T →∞ 1 T + 1 Eu T ′ u T ≤ P.(65) In other words, C ∞,n is the infinite-horizon information capacity for a fixed transmitter dimension. Note that C ∞,n exists and is finite. To see this, note Proposition 6, C ∞,n ≤ C ∞ < ∞, and the fact that The function C ∞,n (P) also induce P ∞,n (R), the "capacity" in terms of minimum input power subject to an information rate constraint. Proposition 8. Consider the general coding structure in Fig. 3. i) C ∞,n is increasing in n; ii) For channel F with order m ≥ 1, C ∞,n = C ∞ for n ≥ m − 1. Proof: See Appendix V-A. This proposition suggests that, to achieve C ∞ , we may first fix the transmitter dimension as (n + 1) and let the dynamical system run to time infinity, obtaining C ∞,n , and then increase n to (m − 1). The finite dimensionality of the optimal solution is important since it guarantees that we can achieve C ∞ without solving an infinite-dimensional optimization problem. C. Achieving C ∞ In this subsection, we prove that our coding scheme achieves C ∞ in the information sense as well as in the operational sense. Proof: See Appendix V-C. Proposition 11. The system constructed in Theorem 1 transmits a digital message W from the transmitter to the receiver at a rate arbitrarily close to C ∞ (P) with P E T decays doubly exponentially. Proof: See Appendix V-D. Note that, the coding length needed for a pre-specified performance level can be pre-determined since Σ * x,T can be solved off-line. Besides, because the probability of error decays doubly exponentially, it leads to much shorter coding length than forward transmission. VIII. NUMERICAL EXAMPLE Here we repeat the numerical example studied in [12]. Consider a third-order channel (i.e. m = 3) with Z −1 := 1 + 0.5z −1 − 0.4z −2 1 + 0.6z −2 − 0.4z −3 .(67) In state-space, Z −1 is described as (F, G, H, 1) where F =   0 −0.6 0.4 1 0 0 0 1 0   G =   1 0 0   H = [ 0.5 −1 0.4 ] .(68) Assume the desired communication rate R is 1 bit per channel use. We first solve (12) with n = m−1 = 2, and find out n * = 1. That is, C ∞ is attained when A has two unstable eigenvalues. Then we solve (12) again with n * = 1, and obtain The asymptotic feedback capacity C∞ and feedforward capacity for channel F with Z −1 = (1 + 0.5z −1 − 0.4z −2 )/(1 + 0.6z −2 − 0.4z −3 ). This yields the optimal power P ∞ = 0.743 (or -1.290 dB). Similar computation generates Figure 7, the curve of C ∞ against SNR or equivalently P. This curve is identical to that in [12]. We then use the obtained A * , C * , and L * to construct the optimal communication scheme. However, we observe that the optimal communication scheme shown in Fig. 2 generates unbounded signals {r t } and {r t } due to the instability of A. This is not desirable for the simulation purpose, though the scheme in the form of Fig. 2 is convenient for the analysis purpose. Here, we propose a modification of the scheme, see Fig. 8. It is easily verify that the system in Fig. 8 is T -equivalent to that in Fig. 2. As we indicate in Fig. 8, the loop including the encoder, the channel, and the feedback link is indeed the control setup, which is stabilized and hence any signal inside is bounded. 6 Note that the encoder now involvesx −1 ; we setx −1 := A −1 W , leading tox 0 = W , the desired value forx 0 . We report the simulation results using the modified communication scheme with the optimal parameters 6 We remark that, in the case of an AWGN channel, the modification coincides with the one studied by Gallager (p. 480, [36]) with minor differences. This modification differs from the more popular feedback communication designs in [1], [2], [14]; notice that, [1] involves exponentially growing bandwidth, [2] involves an exponentially growing parameter α k where α > 1 and k denotes the time index, and [14] generates a feedback signal with exponentially growing power. Thus we consider our modification more feasible for simulation purpose. However, this modification is not yet "practical", mainly because of the strong assumption on the noiseless feedback. A more practical design is under current investigation. given in (69). Fig. 9 (a) shows the convergence ofx 0,t to x 0 , in which x 0 := [−0.2, −0.7] ′ . Fig. 9 (a) also shows the time average of the channel input power, which converges to the optimal power P ∞ = 0.743. To compute the probability of error, we let ǫ = 0.2, i.e., the signalling rate is equal to 0.8C ∞ . We demonstrate that this signalling rate is achieved by showing that the simulated probability of error decays to zero, see Fig. 9 (b). Fig. 9 (b) also plots the theoretic probability of error computed from (137), which is almost identical to the simulated curve. In addition, we see that the probability of error decays rather fast within 28 channel uses. The fast decay implies that the proposed scheme allows shorter coding length and shorter coding delay; here coding delay measures the time steps that one has to wait for the message to be decoded at the receiver with small enough error probability. IX. CONCLUSIONS AND FUTURE WORK We presented a coding scheme to achieve the asymptotic capacity C ∞ for a Gaussian channel with feedback. The scheme is essentially the Kalman filter algorithm, and its construction involves only a finite dimensional optimization problem. We established connections of feedback communication to estimation and control. We have seen that concepts in estimation theory and control theory, such as MMSE, CRB, minimum-energy control, etc., are useful in studying a feedback communication system. We also verified the results by simulations. Our ongoing research includes convexifying the optimization problem (12) to reduce the computation complexity, and finding a more feasible scheme to fight against feedback noise while keeping the feedback signal bounded. In future, we will further explore the connections among information, estimation, and control in more general setups (such as MIMO channels with feedback). APPENDIX I SYSTEMS REPRESENTATIONS AND EQUIVALENCE The concept of system representations and the equivalence between different representations are extensively used in this paper. In this subsection, we briefly introduce system representations and the equivalence. For more thorough treatment, see e.g. [37]- [39]. A. Systems representations Any discrete-time linear system can be represented as a linear mapping (or a linear operator) from its input space to output space; for example, we can describe a single-input single-output (SISO) linear system as y t = M t u t(70) for any t, where M t ∈ R (t+1)×(t+1) is the matrix representation of the linear operator, u t ∈ R t+1 is the stacked input vector consisting of inputs from time 0 to time t, and y t ∈ R t+1 is the stacked output vector consisting of outputs from time 0 to time t. For a (strictly) causal SISO LTI system, M t is a (strictly) lower-triangular Toeplitz matrix formed by the coefficients of the impulse response. Such a system may also be described as the (reduced) transfer function, whose inverse z-transform is the impulse response; by a (reduced) transfer function we mean that its zeros are not at the same location of any pole. A causal SISO LTI system can be realized in state-space as x t+1 = Ax t + Bu t y t = Cx t + Du t ,(71) where x t ∈ R l is the state, u t ∈ R is the input, and y t ∈ R is the output. We call l the dimension or the order of the realization. The state-space representation (71) may be denoted as (A, B, C, D). Note that in the study of input-output relations, it is sometimes convenient to assume that the system is relaxed or at initial rest (i.e. zero input leads to zero output), whereas in the study of state-space, we generally allow x 0 = 0, which is not at initial rest. For multi-input multi-output (MIMO) systems, linear time-varying systems, etc., see [38], [39]. The state-space representation of an causal FDLTI system M(z) is not unique. We call a realization (A, B, C, D) minimal if (A, B) is controllable and (A, C) is observable. All minimal realizations of M(z) have the same dimension, which is the minimum dimension of all possible realizations. All other realizations are called non-minimal. An example We demonstrate here how we can derive a minimal realization of a system. Consider G * T (A, C) in (32) in Section IV, which is given by G * T (A, C) = − G * T (I − Z −1 T G * T ) −1 ,(72) where the state-space representations for G * T (A, C) and Z −1 T are illustrated in Fig. 6 (b) and Fig. 1 (c). Since (72) suggests a feedback connection of G * and Z −1 as shown in Fig. 10, we can write the state-space for G * as                 x t+1 = Ax t + L 1,t e t r t = Cx t s t+1 = Fŝ t + Gr t + L 2,t e t e t =ȳ t − Hŝ t −r t s a,t+1 = F s a,t + Gr t y t = y t + Hs a,t +r t . Then letŝ t :=ŝ t − s a,t , and we have         x t+1 = Ax t + L 1,t e t r t = Cx t s t+1 = Fŝ t + L 2,t e t e t = y t − Hŝ t .(74) It is straightforward to check that this dynamics is controllable and observable, and therefore it is a minimum realization of G * . Fig. 10. G * is a feedback connection of G * and Z −1 . y t r t Z T - y t G T G T B. Equivalence between representations Definition 3. i) Two FDLTI systems represented in state-space are said to be equivalent if they admit a common transfer function (or a common transfer function matrix) and they are both stabilizable and detectable. ii) Fix 0 ≤ T < ∞. Two linear mappings M i,T : R q(T +1) → R p(T +1) , i = 1, 2, both at initial rest, are said to be T -equivalent if for any u T ∈ R q(T +1) , it holds that M 1,T (u T ) = M 2,T (u T ).(75) We note that i) is defined for FDLTI systems, whereas ii) is for general linear systems. i) implies that, the realizations of a transfer function are not necessarily equivalent. However, if we focus on all realizations that do not "hide" any unstable modes, namely all the unstable modes are either controllable from the input or observable from the output, they are equivalent; the converse is also true. ii) concerns about the finite-horizon input-output relations only. Since the states are not specified in ii), it is not readily extended to infinite horizon: Any unstable modes "hidden" from the input and output will grow unboundedly regardless of input and output, which is unwanted. Examples As we mentioned in Section II-B, for any u T and N T , Fig. 1 (a) and (b) generate the same channel outputỹ T . That is, the mappings from (u T , N T ) toỹ T for the two channels are identical, and both are given byỹ T = Z T (Z −1 T u T + N T ).(76) Thus, we say the two channels are T -equivalent. The feedback communication system (39), estimation system (38), and control system (42) are Tequivalent, since for any N T , they generate the same innovations e T . APPENDIX II FINITE-HORIZON: THE FEEDBACK CAPACITY AND THE CP STRUCTURE A. Feedback capacity C T The following definition of feedback capacity is based on [11]. Definition 4. The "operational" or "information" finite-horizon feedback capacity C T , subject to the average channel input power constraint P T := lim T →∞ 1 T + 1 Eu T ′ u T ≤ P,(77) is C T (P) := C T := sup 1 T + 1 I(u T → y T ),(78) where I(u T → y T ) is the directed information from u T to y T , and the supremum is over all possible feedback-dependent input distributions satisfying (77) and in the form u t = γ t u t−1 + η t y t−1 + ξ t (79) for any γ t ∈ R 1×t , η t ∈ R 1×t , and zero-mean Gaussian random variable ξ t ∈ R independent of u t−1 and y t−1 . B. CP structure for colored Gaussian noise channel We briefly review the CP coding structure for the colored Gaussian noise channel specified in Section II-A; see [6], [35] for more details of the CP structure. Let the colored Gaussian noise Z T have covariance matrix K (T ) Z , and u T := B T Z T + v T ,(80) where B T is a (T +1)×(T +1) strictly lower triangular matrix, v T is Gaussian with covariance K (T ) v ≥ 0 and is independent of Z T . 7 This generates channel output y T = (I + B T )Z T + v T .(81) Then the highest rate that the CP structure can achieve in the sense of operational and information is C T,CP (P) = sup 1 T + 1 I(v T ;ỹ T ) = sup 1 2(T + 1) log det K (T ) y det K (T ) Z = sup 1 2(T + 1) log det((I + B T )K (T ) Z (I + B T ) ′ + K (T ) v ) det K (T ) Z ,(82) where the supremum is taken over all admissible K (T ) v and B T satisfying the power constraint P T := 1 T + 1 tr(B T K (T ) Z B ′ T + K (T ) v ) ≤ P.(83) Since the operational capacity definitions in [6] and [11] coincide, we have C T,CP (P) = C T (P). This may also be seen by observing that, any channel input (79) can be rewritten in the form of (80), but since (79) is sufficient to achieve C T , we conclude that (80) is also sufficient to achieve C T . C. CP structure for ISI Gaussian channel By using the equivalence between the colored Gaussian noise channel and the ISI channel F, we can derive the CP coding structure for F, which is obtained from (80) by introducing a new quantity r T as r T := (I + B T ) −1 v T .(84) By Z T = Z T N T andỹ T = Z T y T , we have u T = B T Z T N T + (I + B T )r T y T = Z −1 T (I + B T )Z T N T + Z −1 T (I + B T )r T = Z −1 T (I + B T )(Z T N T + r T ).(85) This implies that, the channel input u T can be represented as Fig. 11. The block diagram of the CP structure for ISI Gaussian channel F. u T = (I + B T ) −1 B T Z T y T + r T ,(86)I B T v T Z T N T y T r T u T I B T B T Z T F which leads to the block diagram in Fig. 11. The capacity C T now takes the form C T (P) = sup 1 2(T + 1) log det K (T ) y = sup 1 2(T + 1) log det Z −1 T (I + B T )(Z T Z ′ T + K (T ) r )(I + B T ) ′ Z −1 T ′ = sup 1 2(T + 1) log det(Z T Z ′ T + K (T ) r )(87) where the supremum is over the power constraint P T := 1 T + 1 tr(B T Z T Z ′ T B ′ T + (I + B T )K (T ) r (I + B T ) ′ ) ≤ P.(88) It is easily seen that the capacity in this form is identical to (82). D. Relation between the CP structure for ISI Gaussian channel and the general coding structure We can establish correspondence relationship between the CP structure for ISI Gaussian channel F in Fig. 11 and the general coding structure for F in Fig. 2. In fact, the general coding structure for F in Fig. 2 was initially motivated by the CP structure for channel F in Fig. 11. For any fixed (K (T ) r , B T ) in the CP structure, define in the general coding structure that G T := (I + B T ) −1 B T Z T A := Γ −1 0 0 I T * * Γ 0 ∈ R (T +1)×(T +1) C := [ 1 0 · · · 0 ] Γ 0 ,(89) where Γ 0 := (K which generates identical channel input u T as (A, C, G T ) does. As a result of the above reasoning, there is a corresponding relation between the CP structure for F and the general coding structure, and the maximum rate over all admissible (K (T ) r , B T ) (namely C T ) equals that over all admissible (A, C, G T ). In other words, we have Lemma 1. C T (P) = C T,T (P). Proof: Note that C T,T is the maximum rate over all admissible (A, C, G T ) with ∈ R (T +1)×(T +1) . This lemma implies that the general coding structure with an extra constraint T = n becomes the CP structure, that is, in the CP structure, the dimension of A is equal to the horizon length. One advantage of considering the general coding structure is that we can allow T = n, which makes it possible to increase the horizon length to infinity without increasing the dimension of A, a crucial step towards the Kalman filtering characterization of the feedback communication problem. Our study on the general coding structure also refines the CP structure. We can now identify more specific structure of the optimal (K (T ) v , B T ). Indeed, we conclude that the CP structure needs to have a Kalman filter inside. We may further determine the optimal form of B T . From (90) and (32), we have that B * T = − G * T (A, C)Z −1 T .(92) Therefore, to achieve C T in the CP structure, it is sufficient to search (K (T ) v , B T ) in the form of K (T ) v := (I − G * T (A, C)Z −1 T )Γ(A, C)Γ(A, C) ′ (I − G * T (A, C)Z −1 T ) ′ B * T := − G * T (A, C)Z −1 T .(93) Additionally, as T tends to infinity, it can be easily shown that {v t } is a stable process in order to achieve C ∞ . E. Proof of Proposition 4: Necessary condition for optimality In this subsection, we show that our general coding structure, in the form of (42), satisfies the necessary condition for optimality as presented in Proposition 4. Since {y t } is interchangeable with the innovations process {e t }, in the sense that they determine each other causally and linearly, it suffices to show that Eu t e τ = 0. Note that u t = DX t = DAX t−1 − DL t−1 e t−1 ,(94) and thus Eu t e t−1 = EDAX t−1 e t−1 − DL t−1 K e,t−1 (a) = EDAX t−1 X ′ t−1 C ′ + EDAX t−1 N t−1 − DAΣ t−1 C ′ = DAΣ t−1 C ′ + 0 − DAΣ t−1 C ′ = 0,(95) where (a) follows from (42) and (43). Similarly we can prove Eu t e τ = 0 for any τ < t − 1. APPENDIX III INFINITE-HORIZON: THE PROPERTIES OF THE GENERAL CODING STRUCTURE A. Proof of Proposition 5: Convergence to steady-state In this subsection, we show that system (42) converges to a steady-state, as given by (51). To this aim, we first transform the Riccati recursion into a new coordinate system, then show that it converges to a limit, and finally prove that the limit is the unique stabilizing solution of the Riccati equation. The convergence to the steady-state follows immediately from the convergence of the Riccati recursion. Consider a coordinate transformation given as A := ΦAΦ −1 := A 0 0 F , C := CΦ −1 , Σ t := ΦΣ t Φ ′ ,(96) where Φ := I n+1 0 −φ I m ,(97) and φ is the unique solution to the Sylvester equation F φ − φA = −GC.(98) Note that the existence and uniqueness of φ is guaranteed by the assumption on A that λ i (−A)+λ j (F ) = 0 for any i and j (see Section IV-A). This transformation transforms A into block-diagonal form with the unstable and stable eigenvalues in different blocks, and transforms the initial condition Σ 0 to Σ 0 := Φ I n+1 0 0 0 Φ ′ = I −φ ′ −φ φφ ′ .(99) Therefore, the convergence of (44) with initial condition Σ 0 is equivalent to the convergence of Σ t+1 = AΣ t A ′ − AΣ t C ′ CΣ t A ′ CΣ t C ′ + 1(100) with initial condition Σ 0 . By [34], Σ t would converge if det 0 0 0 I m − Σ 0 I n+1 0 0 X 22 = 0,(101) where X 22 is a positive semi-definite matrix (whose value does not affect our result here). Since det 0 0 0 I − I −φ ′ −φ φφ ′ I 0 0 X 22 = det −I φ ′ X 22 φ I − φφ ′ X 22 = det(−I) det (I − φφ ′ X 22 + φφ ′ X 22 ) = 0,(102) we conclude that Σ t converges to a limit Σ ∞ . This limit Σ ∞ is a positive semi-definite solution to Σ ∞ = AΣ ∞ A ′ − AΣ ∞ C ′ CΣ ∞ A ′ CΣ ∞ C ′ + 1 .(103) By [31], (103) has a unique stabilizing solution because (A, C) is observable and A does not have any eigenvalues on the unit circle. Therefore, Σ ∞ is this unique stabilizing solution, which can be computed from (103) as (see also [34]) Σ ∞ = Σ 11 0 0 0 (104) where Σ 11 is the positive-definite solution to a reduced order Riccati equation Σ 11 = AΣ 11 A ′ − AΣ 11 (C + Hφ) ′ (C + Hφ)Σ 11 A ′ (C + Hφ)Σ 11 (C + Hφ) ′ + 1 .(105) and has rank (n + 1) (cf. [34]). Thus, Σ t converges to Σ ∞ = Σ 11 Σ 11 φ ′ φΣ 11 φΣ 11 φ ′(106) with rank (n + 1). B. Infinite-horizon feedback capacities If the noise in the colored Gaussian channel forms a (an asymptotic) stationary process, then C T (P) has a finite limit (cf. [15]; the proof utilizes the superadditivity of C T , similar to the case of forward communication capacities studied in [36]), which also has the operational and information meanings. Therefore, we have lim T →∞ C T = C ∞ < ∞,(107) where C ∞ is the operational or information infinite-horizon capacity (cf. [6], [11]). By Lemma 1, the above implies that lim T →∞ C T,T = C ∞ .(108) Note that this does not simply lead to that lim n→∞ lim T →∞ C T,n = C ∞ or C ∞ = C s , since we could not show that the involved limits (including taking the supremum) are interchangeable in this case. APPENDIX IV PROOF OF PROPOSITION 7: K E = 0 In this section, we prove that K E has to be 0 to ensure the optimality in (62). We first derive some properties of the communication system using the stationary GM inputs and the steady-state Kalman filtering. The system dynamics is given by                      u t = d ′s s,t + E t s t+1 = F s t + Gu t y t = Hs t + N t + u t s s,t+1 = s t −ŝ s,t s s,t+1 = Fŝ s,t + L s e t e t = y t − Hŝ s,t = (H + d ′ )s s,t + E t + N t s s,t+1 = Fs s,t + Gu t − L s e t ,(109) whereŝ s,0 = 0 ands s,0 = 0. As before, the Kalman filter innovations {e t } will play an important role. The innovations process is white with variance asymptotically equal to K e = 1 + K E + (H + d ′ )Σ s (H + d ′ ) ′ ,(110) where Σ s := Es ss ′ s . Following the same derivation for Proposition 6, we know that the asymptotic information rate is given by I(E; y) = 1 2 log K e ,(111) which is consistent with the result in [12]. We now invoke the equivalence between the colored Gaussian channel and the ISI channel F, that is, instead of generating y by (109), we generate y by     ỹ t = u t + Z t s c,t+1 = F s c,t + Gỹ t y t = Hs c,t +ỹ t ,(112) where s c,0 = 0. Since Z T = Z T N T , the mapping from (u, N ) to y here is equivalent to that in (109). Therefore, (109) becomes                      u t = d ′s s,t + E t y t = u t + Z t s c,t+1 = F s c,t + Gỹ t y t = Hs c,t +ỹ t s s,t+1 = Fŝ s,t + L s e t e t = y t − Hŝ s,t = (H + d ′ )s s,t + E t + N t s s,t+1 = Fs s,t + Gu t − L s e t ,(113) whereŝ s,0 = 0; see Fig. 12 for the block diagram. Our analysis of this system is facilitated by considering transfer functions. Note that T Eu = S T N u = TZ,(114) where S is the sensitivity, and T := S − 1 is the complimentary sensitivity. (The sensitivity S here should not be confused with the sensitivity in Section V-A.) Then we have u = SE + TZÑ y = S(E + ZN ).(115) Now assume that d and K E form the optimal solution to (62), where K E = 0, for contradiction purpose. We can then compute the corresponding optimal Σ s , L s , S, T, etc. Fix the optimal L s , S, and T. We will show that this leads to: 1) The whiteness of {ỹ t }; 2) L s = G; 3) K E = 0 and hence contradiction. 1) For fixed optimal values of L s , S, and T, suppose that we can have the freedom of choosing the power spectrum of E in (113). Since we have assumed the optimality of a white process {E t }, it must hold that any correlated process {E c,t } does not lead to a larger mutual information than {E t } does. Precisely, assume a stationary correlated process {E c,t } replaces the white process {E t } in (113). Then Since I(E c ;ỹ) = h(ỹ) − h(ỹ|E c ) = h(ỹ) − h(SZN )(117) and h(SZN ) is fixed for fixed S, the above optimization is equivalent to max SE c (e j2πθ ) 1 2 log S S (e j2πθ )S Ec (e j2πθ ) + S S (e j2πθ )S Z (e j2πθ ) dθ, s.t. 1 2 − 1 2 SS(e j2πθ )SE c (e j2πθ )dθ≤P1(119) which we identify as a new forward communication problem, see Fig. 13. In this problem, we want to tune the power spectrum of SE c , the effective channel input, to get the maximum rate. The optimal solution is given by waterfilling, namely, the power spectrum S S (e j2πθ )S Ec (e j2πθ ) needs to waterfill the power spectrum S S (e j2πθ )S Z (e j2πθ ). By optimality of {E t }, K E S S (e j2πθ ) is the waterfilling solution. y Fig. 13. An equivalent forward communication channel. Here SEc is the effective input, SZN is the effective channel noise, andỹ is the output. Since S S (e j2πθ ) = 0 for some θ if and only if S(z) has a zero for that θ on the unit circle, and since S(z) is a finite dimension transfer function with a finite number of zeros, the power spectrum S S (e j2πθ ) cannot have zero amplitude at any interval. This follows that the support of the channel input spectrum K E S S (e j2πθ ) is [−1/2, 1/2]. In waterfilling, if the support of input spectrum is [−1/2, 1/2], then the output spectrum must be flat. This is easily proven by contradiction. Thus, {ỹ t } is a white process. Let us assume that its variance is σ 2 . 2) Note that bothỹ and e have white spectrum, which imposes condition on the choice of L s . The transfer function T ye is illustrated in Fig. 14, where we can see that its structure is a Kalman filter structure. To make e white, it is necessary to choose L s to be the Kalman filter gain (cf. [31]), given by L s := F Σ c H ′ + σ 2 G HΣ c H ′ + σ 2 ,(120) where Σ c is the estimation error covariance matrix and is a nonnegative solution to the Riccati equation Σ c = F Σ c F ′ + σ 2 GG ′ − (F Σ c H ′ + σ 2 G)(F Σ c H ′ + σ 2 G) ′ HΣ c H ′ + σ 2 .(121) Clearly, Σ c = 0 is a solution to the Riccati equation. By [31], it is also the unique nonnegative solution. Hence, we need to choose L s := G. 3) The fact that L s = G leads to reduction of system (113) or equivalently (109). We havẽ In the case that (F − GH) is unstable, the closed-loop of (113) is unstable and cannot transmit information. In the case that (F − GH) is stable, the steady-state of Σ s depends only on (F, G, H) and is independent of the choice of d and K E , and thus (62) s t+1 = (F − GH)s t − GN t Σ s = (F − GH)Σ s (F − GH) ′ + GG ′ .(122) which requires K E = 0. APPENDIX V OPTIMALITY OF THE PROPOSED CODING SCHEME A. Proof of Proposition 8: Finite dimensionality of the optimal scheme i) To show that C ∞,n is non-decreasing as n increases, note that, an encoder (A, C) of dimension (n + 1) can be arbitrarily approximated by a sequence of encoders {(A i , C i )} of dimension (n + 2) in the form of A 0 0 1 , C 1 i ,(125) and therefore the supremum in (64) with encoder dimension (n + 2) is no smaller than the supreme with encoder dimension (n + 1). So C ∞,n is increasing in n. ii) By proposition 6 and the definition for C ∞,m−1 (P), the optimization problem for solving C ∞,m−1 (P) is given by C ∞,m−1 (P) = sup A∈R m×m ,C 1 2 log(CΣC ′ + 1) s.t.Σ=AΣA ′ −AΣC ′ (CΣC ′ +1) −1 CΣA ′ DΣD ′ =P(126) To compare it with C ∞ (P), we rewrite (62) It is then straightforward to verify that 1 2 log(1 + (H + d ′ )Σ s (H + d ′ ) ′ ) = 1 2 log(1 +CΣC ′ ) d ′ Σ s d =DΣD ′ AΣĀ ′ −ĀΣC ′ (CΣC ′ + 1) −1CΣĀ′ =Σ,(128) which yields that C ∞ (P) = sup d∈R m 1 2 log(1 +CΣC ′ ) s.t.Σ=ĀΣĀ ′ −ĀΣC ′ (CΣC ′ +1) −1CΣĀ′ DΣD ′ =P(129) Comparing (129) with (126), we conclude that C ∞,m−1 (P) ≥ C ∞ (P). However, since for each (A, C), the channel input sequence is stationary by the steady-state characterization of the general coding structure, it holds that C ∞,m−1 (P) ≤ C ∞ (P). Therefore, we have C ∞,m−1 (P) = C ∞ (P). Then ii) follows from i) immediately. B. Proof of Proposition 9: Achieving C ∞ in the information sense By Proposition 8, the optimization problem for solving P ∞ (R) in (9) (which is equivalent to solving C ∞ (P)) can be reformulated as [A opt , C opt , Σ opt ] := arg inf A∈R m×m ,C DΣD ′ , s.t.Σ=AΣA ′ −AΣC ′ (CΣC ′ +1) −1 CΣA ′ log DI(A)=R(131) for any desired rate R. Without loss of generality, we may assume that (A, C) is in the observable canonical form, i.e. A := 0 n×1 I n a n a n−1 · · · a 1 C := 1 0 1×n . Observe that det A = a n . Thus, DI(A) = | det A| = |a n | if A does not contain stable eigenvalues, and DI(A) > | det A| = |a n | otherwise. As a consequence, if we search over A with a n fixed to be 2 R or −2 R , we actually enforce DI(A) ≥ 2 R . However, the optimal solution must satisfy DI(A opt ) = 2 R , since otherwise the system has a rate equal to R ∞,m−1 = log DI(A opt ) > R, which would require more power than the case that R ∞,m−1 = R; notice that (131) is a power minimization problem. To summarize, we can remove the constraint log DI(A) = R by letting a n = ±2 R in (132), and the optimal solution A does not contain stable eigenvalues. Furthermore, note that unit-circle eigenvalues do not generate any rate or power and hence can be removed. Thus, if A opt has (n * + 1) unstable eigenvalues, we can solve the optimization problem with A having size (n * + 1) and the obtained optimal solution still achieves C ∞ . C. Proof of Proposition 10: Optimality in the analog transmission The end-to-end distortion is given by MSE(Ŵ t ) = E(W −Ŵ t )(W −Ŵ t ) ′ = E(x 0 −x 0,t )(x 0 −x 0,t ) ′ = E(A −t−1 x t+1 − A −t−1x t+1 )(A −t−1 x t+1 − A −t−1x t+1 ) ′ = EA −t−1x t+1x ′ t+1 A ′−t−1 = A −t−1 Σ x,t+1 A ′−t−1 ,(133) where Σ x,t+1 := [I, 0]Σ t+1 [I, 0] ′ (134) and the expectation is w.r.t. the randomness in W andŴ t . By rate-distortion theory, the above distortion needs an asymptotic rate R satisfying R ≥ lim t→∞ 1 2(t + 1) log P n * +1 det M SE(Ŵ t ) = lim t→∞ 1 2(t + 1) log det A 2t+2 det Σ x,t+1 = log | det A|. (135) From Proposition 9, we know that log | det A * | equals C ∞ and the average channel input power equals P. Because C ∞ is the supremum of asymptotic rate, it follows that the equality in (135) is achieved. Then we see that the proposition holds. D. Proof of Proposition 11: Optimality in digital transmission It is sufficient to show that R ∞,n (A, C) is achievable for any fixed (A, C). To show this, for the fixed (A, C), construct the scheme in Fig. 2 and use G * T , the Kalman-filter based optimal receiver. The closed-loop (42) is stabilized and will converge to its steady-state for large enough T . We can then directly verify that Theorems 4.3 and 4.6 in [14] are applicable to the (steady-state) LTI system. These theorems assert that, if the closed-loop system is stabilized, then we can construct a sequence of codes to reliably (in the sense of vanishing probability of error) transmit the initial conditions associated with the open-loop unstable eigenvalues of A (denoted a 0 , · · · , a k , if any), at a rate R := (1 − ǫ)R ∞,n (A, C)(136) for any ǫ > 0, and in the meantime, P ∞,n (A, C) ≤ P holds. Therefore, we conclude that, for any (A, C), the portion of W that is associated with the unstable eigenvalues of A is transmitted reliably from the transmitter to the receiver at rate arbitrarily close to R ∞,n (A, C). Moreover, we notice that we can achieve C ∞,n by a sequence of purely unstable (A, C) (i.e. k = n), in which the initial condition W is the message being transmitted. This follows that W is transmitted at the capacity rate. In addition, [14] showed that for any choice of x 0 , it holds that P E T = 1 − n i=0 1 − 2Q σ −ǫ T,i 2 ,(137) where σ T,i is the square root of the ith eigenvalue of MSE(x 0,T ), and MSE(x 0,T ) = E(x 0 −x 0,T )(x 0 −x 0,T ) ′ = A −T −1 Σ x,T +1 A ′−T −1 .(138) Note that the expectation is w.r.t. the randomness inx 0,T only, different from (133), and that asymptotically Σ t+1 and hence Σ x,T +1 are independent on the choice of x 0 . It then holds for each i, where λ max (M ) denotes the maximum eigenvalue of M ,σ(M ) denotes the maximum singular value of M , (a) follows from λ(AB) = λ(BA), (b) follows from |λ(A)| ≤σ(A), and (c) is because the maximum singular value is an induced norm. Since Σ x,T +1 converges to steady-state value exponentially, the above implies that, for T large enough, each σ T,i decays to zero exponentially as T increases. Now using the union bound and the Chernoff bound, we have P E T ≤ n i=0 2Q σ −ǫ T,i 2 ≤ n i=0 4 √ 2πσ −ǫ T,i exp − σ −2ǫ T,i 8 ,(140) and hence P E T decreases to zero doubly exponentially since ǫ > 0 and σ T,i decays exponentially. Thus we prove the proposition. Fig. 1 . 1(a) A colored Gaussian noise channel without ISI. (b) The equivalent ISI channel with AWGN. (c) State-space realization of channel F. [a a a opt f , Σ opt ] := arg inf a a af ∈R n DΣD ′ ,s.t. Σ=AΣA ′ −AΣC ′ CΣA ′ /(CΣC ′ +1) Fig. 2 . 2The encoder/decoder structure for F. where A := A 0 GC F , C := [ C H ] , D := [ C 0 ] , A := 0 n×1 I n ±DI a a a f , C := 1 0 1×n . (13) . 2 by letting n := n * , A := A * , C := C * , L 1 := L * 1 , L 2 := L * 2 . Fig. 3 . 3A general coding structure for channel F. Fig. 4 . 4(a) An estimation problem over channel F. (b) The Kalman filter G * T (A, C). (c) The Kalman filter based feedback generator G * T (A, C). Here (A, L1,t, −C, 0) withxt denotes a state-space representation withxt being its state at time t, and x0 being 0; see (41) and (44) for L1,t and L2,t. Fig. 5 . 5The block diagram for the minimum-energy control system. Here the block (A, −L1,t, C, 0) withxt denotes the state-space representation withxt and W being its state at time t and at time 0. Proposition 3 . 3For any fixed 0 ≤ n ≤ T and (A, C), it holds that Fig. 6 . 6Relation between the estimation problem (a) and the communication problem (b). Proposition 6 . 6Consider the general coding structure in Fig. 3. For any n ≥ 0 and (A, C), i) The asymptotic information rate is given by R ∞,n (A, C) C ∞,n (P) = sup A∈R (n+1)×(n+1) ,C,G * (A,C),(65) R ∞,n (A, C). Proposition 9 . 9For the coding scheme described in Theorem 1, R ∞,n * (A * , C * ) = C ∞ (P) and P ∞,n * (A * , C * ) = P.Proof: See Appendix V-B.Proposition 10. The system constructed in Theorem 1 transmits the analog source W ∼ N (0, I) at a rate C ∞ (P), with MSE distortion D(C ∞ (P)), where D(·) is the distortion-rate function. L * = [ −0.506 −0.225 0.573 0.092 −0.327 ] ′ . Fig. 8 . 8The modified feedback communication scheme. Fig. 9 . 9(a) Convergence ofx0,t to x0, and convergence of the average channel input power. (b) Simulated probability of error and theoretic probability of error. and * can be any number. (Note that the case K positive definite can be approached by a sequence of positive definite K (T ) r , and thus it is sufficient to consider only positive definite K (T ) r in establishing the correspondence relation of the two structures.) Then it is easily verified that G T is strictly lower triangular, (A, C) is observable with a nonsingular observability matrix Γ = Γ 0 , and A can have eigenvalues not on the the unit circle and not at the locations of F 's eigenvalues. Therefore, for any given (K (T ) r , B T ), we can find an admissible (A, C, G T ), and it is straightforward to verify that they generate identical channel inputs u T .Conversely, for any fixed admissible (A, C, G T ) with ∈ R (n+1)×(n+1) , we can obtain an admissible (K(T ) r , B T ) as B T := G T Z −1 T (I − G T Z −1 T ) −1 K (T ) r := Γ(A, C)Γ(A, C) ′ , Fig. 12 . 12Block diagram for the communication system using the GM inputs and Kalman filtering, where sc,t is the state for Z −1 with sc,0 = 0, andŝs,t is the state for system (F, Ls, H, 0) withŝs,0 = 0. {E t } yields the maximum achievable rate over all possible {E c,t }, i.e., it solves max Ls,S,T fixed,SE c (e j2πθ ) I(E c ;ỹ). s.t. Eu 2 ≤P Fig. 14 . 14The state-space representation of the transfer function Tye. and (63) in another form, incorporating K E = 0. s Σ s Σ s Σ s . (σ T,i ) 2 ≤ λ max (MSE(x 0,T )) = λ max (A −T −1 Σ x,T +1 A ′−T −1 ) (a) = λ max (A ′−T −1 A −T −1 Σ x,T +1 ) (b) ≤σ(A ′−T −1 A −T −1 Σ x,T +1 ) (c) ≤σ(A ′−T −1 A −T −1 )σ(Σ x,T +1 ) = (σ(A ′ A)) −T −1σ (Σ x,T +1 ) log Sỹ(e j2πθ )dθ.SS(e j2πθ )SE c (e j2πθ )+ST(e j2πθ )SZ (e j2πθ )dθ≤P However, this optimization problem is equivalent to solving, for some P 1 ≥ 0,1 2 − 1 2 s.t. Eu 2 = 1 2 − 1 2 (118) max SE c (e j2πθ ) 1 2 1 2 − 1 2 becomesC ∞ = max Σs fixed,d∈R m ,KE ∈R 1 2 log(1 + K E + (H + d ′ )Σ s (H + d ′ ) ′ ).HΣ s d,s.t. d ′ Σsd≤P−KEs.t. P=d ′ Σsd+KE (123) This is equivalent to max d∈R m ,KE∈R Here we do not mean that their optimization problem is convex. In fact the computation complexity for C f b,T is O(T + 1), and for C f b,∞ the complexity is determined mainly by the channel order, which does not involve prohibitive computation if the channel order is not too high. T = Z T (Z −1 T u T + N T ),(4)2 The difference between a stationarity assumption and an asymptotic stationarity assumption may result from different starting points of the process: If starting from t = −∞, {Zt} is stationary; instead if starting from t = 0 as we are assuming here, {Zt} is asymptotically stationary. They result in exactly the same steady-state analysis of the feedback communication problem. More rigorously, the mappings from (u, N ) toỹ are T -equivalent. For a discussion about systems representations and equivalence between different representations, see Appendix I. We see from(12) that for any channel F, a simple upper bound of the function P∞(R) is given by min{(2 2R − 1)(Z(2 R )) 2 , (2 2R − 1)(Z(−2 R )) 2 }, obtained by using one unstable eigenvalue in A. The innovation defined here is different from the innovation defined in[6] or[12]. KE = 0 was also conjectured and numerically verified by Shaohua Yang (personal communication). This v T is called innovations in[12],[35]; it should not be confused with the Kalman filter innovations in this paper. ACKNOWLEDGEMENTSThe authors would like to thank Anant Sahai, Sekhar Tatikonda, Sanjoy Mitter, Zhengdao Wang, Murti Salapaka, Shaohua Yang, Donatello Materassi, and Young-Han Kim for useful discussion. A coding scheme for additive noise channels with feedback Part I: No bandwidth constraint. J P M Schalkwijk, T Kailath, IEEE Trans. Inform. Theory, IT. 12J.P.M. Schalkwijk and T. Kailath. A coding scheme for additive noise channels with feedback Part I: No bandwidth constraint. IEEE Trans. Inform. Theory, IT-12:172-182, Apr. 1966. A coding scheme for additive noise channels with feedback Part II:Bandlimited signals. J P M Schalkwijk, IEEE Trans. Inform. Theory, IT. 122J.P.M. Schalkwijk. A coding scheme for additive noise channels with feedback Part II:Bandlimited signals. IEEE Trans. Inform. Theory, IT-12(2):183-189, Apr. 1966. Optimum linear transmission of analog data for channels with feedback. J K Omura, IEEE Trans. Inform. Theory. 141J. K. Omura. Optimum linear transmission of analog data for channels with feedback. IEEE Trans. Inform. Theory, 14(1):38-43, Jan. 1968. A general formulation of linear feedback communication systems with solutions. S A Butman, IEEE Trans. Inform. Theory, IT. 15S.A. Butman. A general formulation of linear feedback communication systems with solutions. IEEE Trans. Inform. Theory, IT-15:392-400, 1969. Linear feedback rate bounds for regressive channels. S A Butman, IEEE Trans. Inform. Theory, IT. 22S.A. Butman. Linear feedback rate bounds for regressive channels. IEEE Trans. Inform. Theory, IT-22:363-366, 1976. Gaussian feedback capacity. T M Cover, S Pombra, IEEE Trans. Inform. Theory, IT. 35T.M. Cover and S. Pombra. Gaussian feedback capacity. IEEE Trans. Inform. Theory, IT-35:37-43, 1989. Random coding for additive Gaussian channels with feedback. L H Ozarow, IEEE Trans. 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Let G = (V, E) be a connected finite graph. We investigate two Bogomol'nyi-Prased-Sommerfeld equations on G. We establish necessary and sufficient conditions for the existence and uniqueness of solutions to the two BPS equations.Mathematics Subject Classification (2010) 35A01, 35R02.

2202.09546

42b61110ba1d894349535a9be1ea7cab17b0579e

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO BOGOMOL'NYI-PRASED-SOMMERFELD EQUATIONS ON GRAPHS 19 Feb 2022 Yuanyang Hu EXISTENCE AND UNIQUENESS OF SOLUTIONS TO BOGOMOL'NYI-PRASED-SOMMERFELD EQUATIONS ON GRAPHS 19 Feb 2022BPS equationsfinite graphequation on graphs Let G = (V, E) be a connected finite graph. We investigate two Bogomol'nyi-Prased-Sommerfeld equations on G. We establish necessary and sufficient conditions for the existence and uniqueness of solutions to the two BPS equations.Mathematics Subject Classification (2010) 35A01, 35R02. INTRODUCTION Vortices involve momentous roles in multitudinous fields of theoretical physics comprizing quantum Hall effect, condensed-matter physics, electroweak theory, superconductivity theory, optics, and cosmology. Taubes established the multiple vortex static solutions for the Abelian Higgs model for the first time in [31,32,41]. After that, a large number of work related to vortex equations has been accomplished; see, for example, [4,15,33,35,39] and the references therein. Recently, Lin and Yang pursued a systematic research [22,23] of the multiple vortex equations obtained in [7,8,9,12,27,28,29,30]. They established a great deal of sharp existence and uniqueness theorems. The approach they employ include a priori estimates, monotone iterations, and a degree-theory argument, and constrained minimization. In [40], Yang and Lieb presented a series of sharp existence and uniqueness theorems for the solutions of some non-Abelian vortex equations derived in [5,6], which provide an essential mechanism for linear confinement. Recently, Han [15] established the existence of multipe vortex solutions for the BPS equations derived in [26] from the theory of multi-intersection of D-branes. Chen and Yang [6] proved two sharp existence theorems for the non-Abelian BPS vortex equations arising in the supersymmetric U(1) × SU(N) gauge theory. In this paper, we investigate the existence and uniqueness of the solutions to the two BPS equations in [6,15] on a connected finite graph. During the recent time, the investigation on the equations on graphs has drawn a following and considerable attention from scholars; see, for example, [3,10,11,13,16,17,18,19,20,21,34,37,36] and the references therein. Grigor'yan, Lin and Yang [13] ∆u + ρ he u M he u dx − 1 |M| = 4π N j=1 α q j δ q j − 1 |M| and the relativistic Ablian Chern-Simons equations ∆u 1 = λe u 2 (e u 1 − 1) + 4π k 1 j=1 α j δ p j , ∆u 2 = λe u 1 (e u 2 − 1) + 4π k 2 j=1 β j δ q j on the finite connected graph. Lin and Wu [24] investigated blow-up phenomenons for the nonlinear parabolic equation u t = ∆u + f (u) on a locally finite graph. The paper is organized as follows. In Section 2, we introduce preliminaries and then state our main results. Section 3 is devoted to the proof of Theorem 2.1. In section 4, we give the proof of Theorem 2.2. SETTINGS AND MAIN RESULTS Let G = (V, E) be a connected finite graph, where V denotes the vetex set and E denotes the edge set. Inspired by the work of Huang-Lin-Yau [19], we study two BPS equations on G. One is ∆u j = e u j + l i=1 e u i − (l + 1) + 4π N j s=1 δ p j,s , j = 1, 2, · · · , l (2.1) l is an integer, N j (j = 1, . . . , l) are positive integers, P j,s (j = 1, . . . , l, s = 1, . . . , N j ) are arbitrarily chosen distinct vertices on the graph, and δ p is the Dirac mass at p. The other is Let µ : V → (0, +∞) be a finite measure, and |V |=Vol(V ) = x∈V µ(x) be the volume of V . For each edge xy ∈ E, we assume that its weight w xy > 0 and that w xy = w yx . For any function u : V → R, the Laplacian of u is defined by ∆u i = N j=1 a ij e u j − v 2 + 4π n i s=1 δ p i,s (x), i = 1, . . . , N, (2.2) where a ij = 1 N e 2 2 − g 2 2N + δ ij g 2 2N , i, j = 1, . . . , N,(2.∆u(x) = 1 µ(x) y∼x w yx (u(y) − u(x)),(2.4) where y ∼ x means xy ∈ E. The gradient ∇ of function f is defined by a vector ∇f (x) := [f (y) − f (x)] w xy 2µ(x) y∈V . The associated gradient form reads Γ(u, v)(x) = 1 2µ(x) y∼x w xy (u(y) − u(x))(v(y) − v(x)). (2.5) We denote the length of the gradient of u by |∇u|(x) = Γ(u, u)(x) = 1 2µ(x) y∼x w xy (u(y) − u(x)) 2 1/2 . Denote, for any function u : V → R, an integral of u on V by V udµ = x∈V µ(x)u(x). Our main results can be stated as following: n i < g 2 v 2 8πN |V | + 1 N 1 − 1 N g e 2 n, i = 1, . . . , N. As in [13], we define a sobolev space and a norm by H 1 (V ) = W 1,2 (V ) =    u : V → R : V |∇u| 2 + u 2 dµ < +∞    , and u H 1 (V ) = u W 1,2 (V ) =   V |∇u| 2 + u 2 dµ   1/2 . We next give the following Sobolev embedding and Poincare inequality which will be used later in the paper. Lemma 2.1. ([13, Lemma 5]) Let G = (V, E) be a finite graph. The sobolev space W 1,2 (V ) is precompact. Namely, if u j is bounded in W 1,2 (V ), then there exists some u ∈ W 1,2 (V ) such that up to a subsequence, u j → u in W 1,2 (V ). Lemma 2.2. ([13, Lemma 6]) Let G = (V, E) be a finite graph. For all functions u : V → R with V udµ = 0, there exists some constant C depending only on G such that V u 2 dµ ≤ C V |∇u| 2 dµ. 3. THE PROOF OF THEOREM 2.1 Since V 4π N j s=1 δ p j,s dµ = 4πN j , j = 1, 2, · · · , l, there exists u 0 j such that ∆u 0 j = 4π N j s=1 δ p j,s − 4πN j |V | , x ∈ V, j = 1, 2, · · · , l. (3.1) It is well-known that (u 0 1 , u 0 2 , . . . , u 0 l ) is the unique solution of (3.1) by up to a constant vector C. Set u j = u 0 j + v j , j = 1, 2, · · · , l. Then we know that ∆v j = e u 0 j +v j + l i=1 e u 0 i +v i − (l + 1) + 4πN j |V | , j = 1, · · · , l. (3.2) The following lemma gives a necessary condition for (2.1) to have a solution. Lemma 3.1. If equations (2.1) admits a solution, then max 1≤j≤l {N j } < (l+1) 4π |V |. Proof. Integering (3.2) on V , we deduce that V e u 0 j +v j dµ = |V | − 4πN j + 4π l + 1 l i=1 N i =: K j , j = 1, . . . , l,(3.3) and hence that max 1≤j≤l {N j } < (l+1) 4π |V |. We now finish the proof. Hereafter, we use boldfaced letters to denote column vectors. The following lemma gives a sufficient condition for (2.1) to have a solution. Lemma 3.2. If max 1≤j≤l {N j } < (l+1) 4π |V |, then (2.1) has a solution. Proof. Rewrite (3.2), we have ∆v = AE − Φ, (3.4) where ∆v = (∆v 1 , · · · , ∆v l ) T , E = (e u 0 1 +v 1 , e u 0 2 +v 2 , · · · , e u 0 l +v l ) T , Φ = (l+1− 4πN 1 |V | , · · · , l+ 1 − 4πN l |V | ) and A = (a ij ) l×l =       2 1 1 . . . 1 1 2 1 . . . 1 1 1 2 . . . 1 . . . . . . . . . . . . . . . 1 1 1 . . . 2       . (3.5) In order to find the energy functional of (3.2), we have the following argument. It is easy to see that A = BB T , where B = (b ij ) l×l and b ij =        1+i i , i = j, 1 j(j+1) , i = j, i > j, 0, i = j, i < j. Let q = B −1 v, (3.6) where v = (v 1 , · · · , v l ) T and B −1 is the inverse of matrix B. Then we know that ∆q 1 = √ 2 exp u 0 1 + √ 2q 1 + 1 √ 2 l i=2 exp u 0 i + i−1 k=1 q k k(k + 1) + i + 1 i q i − g 1 , (3.7) ∆q j = j + 1 j exp u 0 j + j−1 k=1 q k (k + 1)k + j + 1 j q j + 1 j(j + 1) l i=j+1 exp u 0 i + i−1 k=1 q k k(k + 1) + i + 1 i q i − g j , j = 1, · · · , l, (3.8) ∆q l = l + 1 l exp u 0 l + l−1 k=1 q k (k + 1)k + l + 1 l q l − g l ,(3.9) where g = (g 1 , · · · , g l ) T = B −1 Φ. Now we define the energy functional I(q) = I (q 1 , . . . , q l ) = V 1 2 l i=1 Γ(q i , q i ) + exp u 0 1 + √ 2q 1 − g 1 q 1 + l i=2 exp u 0 i + i−1 k=1 q k k(k + 1) + i + 1 i q i − g i q i dµ. (3.10) It is easy to check that if I has a critical point, then it is a solution of equations (3.7)-(3.9). We next prove I has a critical point. Define K := (k 1 , . . . , k l ) T . By (3.3), we have K = |V |(B T ) −1 g. For any q ∈ H 1 (V ), we can write q =q +q,(3.11) where Vq dµ = 0 andq = V qdµ |V | . For any q ∈ H 1 (V ) × · · · × H 1 (V ) l , from (3.6), we conclude that I(q) = V 1 2 l i=1 Γ(q i ,q i )dµ + V l i=1 exp (u 0 +v i +v i ) dµ − K Tv = V 1 2 l i=1 Γ(q i ,q i )dµ + V l i=1 exp (u 0 +v i +v i ) dµ − l i=1 K ivi . (3.12) By Jensen's inequality, we obtain exp   V u 0 i +v i +v i dµ |V |   ≤ 1 |V | V exp u 0 i +v i +v i dµ. (3.13) Thus we deduce that V exp u 0 i +v i +v i dµ ≥ |V |exp   1 |V | V u 0 i dµ   ev i =: c i ev i for i = 1, · · · , l. (3.14) By virtue of max 1≤j≤l {N j } < (l+1) 4π |V |, we obtain K i > 0 for i = 1, · · · , l. Thus we deduce that I(q) ≥ V 1 2 l i=1 |∇q i | 2 dµ + l i=1 (c i ev i − K ivi ) ≥ V 1 2 l i=1 |∇q i | 2 dµ + l j=1 K j log c j K j .(3.15) It follows that I is bounded from below in H 1 (V ). Thus we can choose a minimizing sequence {q 1,k , . . . , q l,k } ∞ k=1 of the following minimization problem inf{I(q)|q = (q 1 , . . . , q l ) ∈ H 1 (V ) × · · · × H 1 (V ) l }. In view of lim t→±∞ c i e t − K i t = ±∞ for i = 1, . . . , l. we deduce from (3.15) that {v i,k } ∞ k=1 is a bounded for i = 1, . . . , l. By (3.6), we know that {q i,k } ∞ k=1 is bounded for i = 1, · · · , l. From (3.15), we see that {∇q i,k } ∞ k=1 is bounded in L 2 (V ) for i = 1, · · · , l. From Lemma 2.2, we conclude that {q i,k } ∞ k=1 is bounded in L 2 (V ), i = 1, · · · , l. Thus, {q i,k } ∞ k=1 is bounded in L 2 (V ). Therefore, {q i,k } ∞ k=1 is bounded in H 1 (V ) . Therefore, by Lemma 2.1, there exists q ∞ := (q 1,∞ , . . . , q l,∞ ) ∈ H 1 (V ) × · · · × H 1 (V ) l so that, by passing to a subsequent, q i,k → q i,∞ (3.16) uniformly for x ∈ V as k → +∞ for i = 1, · · · , l. Thus, q ∞ is a critical point of I. Clearly, I is srtictly convex in H 1 (V ). Thus, we know that the solution of equations (3.7)-(3.9) is unique. We now complete the proof. ∆(u − v) = AE(u − v), where E = diag(e ξ 1 , . . . , e ξ l ). We now prove that u 1 ≡ v 1 on V . Let M = max V = (u 1 −v 1 )(x 0 ). We claim that M ≤ 0. Otherwise, M > 0. Thus, we conclude that ∆(u 1 − v 1 )(x 0 ) = [e ξ 1 (u 1 − v 1 )](x 0 ) > 0. (3.17) From (2.4), we obtain ∆(u 1 − v 1 )(x 0 ) ≤ 0. This is impossible. Therefore, we have u 1 ≤ v 1 on V . Thus, we see that u 1 ≡ v 1 on V. (3.18) By a similar argument, we can show that u i ≡ v i on V,(3.19) for i = 2, . . . , l. Therefore, we conclude that u = v on V . The proof is finished. Proof of Theorem 2.1. The desired conclusion follows directly from Lemmas 3.1, 3.2 and 3.3. THE PROOF OF THEOREM 2.2 Set u 0 i be a solution of ∆u 0 i = − 4πn i |V | + 4π n i s=1 δ p i,s (x), i = 1, . . . , N. (4.1) Set u i = u 0 i + U i , i = 1, . . . , N. (4.2) Then (2.2) is transformed into ∆U i = N j=1 a ij e u 0 j +U j − v 2 + 4πn i |V | , i = 1, . . . , N. (4.3) We now write (4.3) in the vector form ∆U =ÂG + F, (4.4) where U = (U 1 , . . . , U N ) T , G = e u 0 1 +U 1 , . . . , e u 0 N +U N T , (4.5) F = 4πn 1 |V | − v 2 N j=1 a 1j , . . . , 4πn N |V | − v 2 N j=1 a N j T =: (f 1 , . . . , f N ) T , (4.6) and = (a ij ) N ×N is a N × N matrix. Applying the Cholesky decomposition theorem, we can write = T T T, (4.7) where T = (t ij ) N ×N is an upper triangular matrix, t 11 = √ a + b, t 12 = t 13 = · · · = t 1N = a t 11 ≡ α 1 > 0, t 22 = (a + b) − α 2 1 , t 23 = t 24 = · · · = t 2N = a − α 2 1 t 22 ≡ α 2 > 0, . . . t (N −1)(N −1) = (a + b) − N −2 i=1 α 2 i , t (N −1)N = a − N −2 i=1 α 2 i t (N −1)(N −1) ≡ α N −1 > 0, t N N = (a + b) − N −1 i=1 α 2 i , a = (1/N) (e 2 /2 − g 2 /2N) and b = g 2 /2N. Set v = (v 1 , . . . , v N ) T , L = T T −1 =: (l ij ) N ×N and v = LU. By (4.4), we have ∆v = T G + LF. We rewrite (4.8) as ∆v i = t ii e u 0 i +t ii v i + i−1 k=1 α k v k +α i N k=1 e u 0 j +t jj v j + j−1 k=1 α k v k + i j=1 l ij f j , i = 1, . . . , N. (4.9) Define the energy functional J(v) = V 1 2 N i=1 Γ(v i , v i ) + N i=1 e u 0 i +t ii v i + i−1 k=1 α k v k + N i=1 i j=1 l ij f j v i dµ. (4.10) The following lemma gives a necessary condition for (2.2) to admit a solution. Lemma 4.1. If (2.2) admits a solution, then n i < g 2 v 2 8πN |V | + 1 N 1 − 1 N g e 2 n, i = 1, . . . , N. (4.11) Proof. Let q i = V e u 0 i +t ii v i + i−1 k=1 α k v k dµ, i = 1, . . . , N (4.12) From (4.9), we get t ii q i + α i N j=i+1 q j = −|V | i j=1 l ij f j =: p i , i = 1, . . . , N (4.13) By (4.13), we deduce that T q = −|V |LF = p, (4.14) where p = (p 1 , . . . , p N ) T and q = (q 1 , . . . , q N ) T . It is easy to check that A −1 = 1 b(Na + b)     (N − 1)a + b −a · · · −a −a (N − 1)a + b · · · −a · · · · · · · · · · · · −a −a · · · (N − 1)a + b     . (4.15) Therefore, we deduce that q i = v 2 |V | + 4πa b(Na + b) N j=1 n i − 4π b n i , i = 1, . . . , N. (4.16) Thus, we have q i = v 2 |V | + 8π 1 g 2 − 1 Ne 2 n − 8πN g 2 n i > 0, i = 1, . . . , N. (4.17) We now complete the proof. We give a sufficient condition for equations (4.3) to have a solution in the following lemma. Proof. For any v = (v 1 , . . . , v N ) ∈ H 1 (V ) × · · · × H 1 (V ) N . By the notation (3.11) and Lemma 4.2. If n i < g 2 v 2 8πN |V | + 1 N 1 − 1 N g e Jensen's inequality, we have V e u 0 i +t ii (v i +v i )+ i−1 k=1 α k (v k +v k ) dµ ≥ |V | exp 1 |V | V u 0 i dµ exp t iivi + i−1 k=1 α kvk =: σ i e t iivi + i−1 k=1 α kvk , i = 1, . . . , N. (4.19) Setw i = t iivi + i−1 k=1 α kvk , i = 1, . . . , N. (4.20) From (4.14), we conclude that J(v) − 1 2 V N i=1 |∇v i | 2 dµ ≥ N i=1 (σ i ew i − q iwi ) ≥ N i=1 q i 1 + ln σ i q i . (4.21) Considering the following minization problem η ≡ inf    J(v) | v ∈ H 1 (V ) × · · · × H 1 (V ) N    . (4.22) Let {(v 1,k , . . . , v N,k )} ∞ k=1 be a minimizing sequence of (4.22), by a similar argument as the proof of Lemma 3.3, we can show that there exists v ∞ := (v 1,∞ , . . . , v N,∞ ) T such that, by passing to a subsequence, v i,k → v i,∞ (4.23) uniformly for x ∈ V as k → +∞ for i = 1, . . . , N. Thus, v ∞ is a critical point of J. It's easy to check that J is strictly convex in H 1 (V ). Thus, we know that the solution of equations (4.9) is unique. The proof is finished. The following lemma gives the uniqueness of solutions to equations (2.2) Proof. Assume that u = (u 1 , . . . , u N ) T and v = (v 1 , . . . , v N ) T are two solutions of the equations (2.2). By mean value theorem, there exists ξ = (ξ 1 , . . . , ξ N ) T such that ∆(u − v) =ÂE(u − v), where E = diag(e ξ 1 , . . . , e ξ N ). We now prove that u 1 ≡ v 1 on V . Let M = max V = (u 1 −v 1 )(x 0 ). We claim that M ≤ 0. Otherwise, M > 0. Thus, we conclude that ∆(u 1 − v 1 )(x 0 ) = [e ξ 1 (u 1 − v 1 )](x 0 ) > 0. (4.24) From (2.4), we obtain ∆(u 1 − v 1 )(x 0 ) ≤ 0. This is impossible. Therefore, we have u 1 ≤ v 1 on V . Thus, we see that u 1 ≡ v 1 on V. (4.25) By a similar argument as above, we can show that u i ≡ v i on V,(4.26) for i = 2, . . . , N, which implies that u ≡ v on V . We now complete the proof. Proof of Theorem 2.2. The desired conclusion follows directly from Lemmas 4.1, 4.2 and 4.3. Next, we give a constrained minization approach to the problem. Denote I i (v) = V e u 0 i +t ii v i + i−1 k=1 α k v k dµ = q i , i = 1, . . . , N,(4.27) We consider the following constrained minization problem γ = inf    J(v) | v ∈ H 1 (V ) × · · · × H 1 (V ) N , I 1 (v) = q 1 , . . . , I N (v) = q N    . (4.28) We now investigate whether the constraints in (4.28) give rise to the so-called "constraints" problem due to the issue of the Lagrange multipliers. For this purpose, let v = (v 1 , . . . , v N ) T be a critical point of J subject to the constraints I i (v) = q N , i = 1, . . . , N. (4.29) Then we can find real numbers σ 1 , . . . , σ N such that d i J = N j=1 σ j d i I j , i = 1, . . . , N,(4.30) where d i (i = 1, . . . , N) denote the Fréchet differention with respect to the i-th arguments, respectively. It follows that, for any z 1 , . . . , z N ∈ H 1 (V ), V Γ(v i , z i ) + t ii e u 0 i +t ii v i + i−1 k=1 α k v k + α i N j=i+1 e u 0 j +t jj v j + j−1 k=1 α k v k + i j=1 l ij f j z i dµ = σ i t ii V e u 0 i +t ii v i + i−1 k=1 α k v k z i dµ + α i N j=i+1 σ j V e u 0 j +t jj v j + j−1 k=1 α k v k z i dµ. (4.31) Taking z 1 , . . . , z N = 1 in (4.31), we deduce that σ i t ii q i + α i N j=i+1 σ j q j = 0, i = 1, . . . , N,(4.32) and hence that ξ N = ξ N −1 = · · · = ξ 1 = 0, (4.33) which reveals that all terms in (4.30) arising from the Lagrange multiplies are automatically absent. Thus, a solution of (4.28) satisfies (4.9). Applying the notation (3.11), we rewrite Thus, we can rewrite (4.10) as J(v) − N i=1 V 1 2 Γ(v i ,v i ) dµ = N i=1 q i − N i=1 p ivi = N i=1 i j=1 p i l ij ln I j (v) + N i=1 q i − p i i j=1 l ij ln q j . (4.36) From (4.14), there exists constant C = C(L, p, q) such that Thus we obtain is bounded in L 2 (V ). By Lemma 2.2, {(∇v 1,k , . . . , ∇v N,k )} ∞ k=1 is bounded in L 2 (V ). Hence, we see that {(v 1,k , . . . , v N,k )} ∞ k=1 is bounded in H 1 (V ). By Lemma 2.1, we deduce that, by passing to a subsequence, v i,k → w i,∞ (4.40) J(v) = 1 2 N i=1 V Γ(v i ,v i ) dµ + N i=1 q i ln I i (v) − C.J(v) − 1 2 N i=1 V |∇v i | 2 dµ ≥ N i=1 q i ln µ i − C. uniformly for x ∈ V as k → +∞ for i = 1, . . . , N. Thus, by the notation (3.11), we know thatw i,∞ satisfies (4.35) for i = 1, . . . , N, and that γ = J(w ∞ ), where w ∞ := (w 1,∞ , . . . , w N,∞ ). Therefore, we know that w ∞ is a solution to the problem (4.28). It follows that w ∞ is a solution to the problem (4.9). 3) n i (i = 1, . . . , N) are positive integers, v, e, g are constants and N is a positive integer. Lemma 3. 3 . 3There exists at most one solution of equations (2.1).Proof. Suppose that u = (u 1 , . . . , u l ) T and v = (v 1 , . . . , v l ) T are two solutions of the equations (2.1). 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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron.AbstractThe free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied in[10]. In [24] the authors proved an upper bound of the constrained free energy using Guerra's interpolation. In this paper, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman-Sims-Starr scheme in [4] and the synchronization mechanism used in the vector spin models in[22]and[23]. We derive a vector version of the Aizenman-Sims-Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda-Guerra identities to prove the matching lower bound.

10.1214/20-ejp431

1806.09772

08b2c5b1e0bf3900cf104741c07c16a76dbf151f

Free energy of multiple systems of spherical spin glasses with constrained overlaps 2020 Justin Ko Free energy of multiple systems of spherical spin glasses with constrained overlaps J. Probab 2528202010.1214/20-EJP431Submitted to EJP on May 2, 2019, final version accepted on February 8, 2020.spin glassesfree energyParisi formulaspherical models E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron.AbstractThe free energy for multiple systems of spherical spin glasses with constrained overlaps was first studied in[10]. In [24] the authors proved an upper bound of the constrained free energy using Guerra's interpolation. In this paper, we prove this upper bound is sharp. Our approach combines the ideas of the Aizenman-Sims-Starr scheme in [4] and the synchronization mechanism used in the vector spin models in[22]and[23]. We derive a vector version of the Aizenman-Sims-Starr scheme for spherical spin glass and use the synchronization property of arrays obeying the overlap-matrix form of the Ghirlanda-Guerra identities to prove the matching lower bound. Introduction In [32], Talagrand proved a formula for the free energy of the spherical mixed even-pspin model originally considered by Crisanti and Sommers in [9]. It was later extended to general mixed p-spin models by Chen in [4]. This formula is the analogue of the classical Parisi formula for the Sherrington-Kirkpatrick model [25,26] proved in [33]. This paper is on the free energy of multiple copies of spherical spin glasses with constrained overlaps introduced in [10,11]. The free energy of this model was studied in [24], where an analogue of the Guerra replica symmetry breaking bound [15] was derived and used in several applications. The goal of this paper is to prove that this upper bound is sharp. There are several motivations for this paper. In [2], spectral gap estimates for generic spherical models were proved under various conditions on the Parisi measure. Our free energy formulas can be used to possibly prove large deviation principles to extend these spectral gap estimates to the larger class of mixed even-p-spin spherical models. Another * University of Toronto, Canada. E-mail: [email protected] application of the free energy formula is possibly proving that chaos in temperature in some full-RSB spherical models cannot be detected at the level of the free energy, as was predicted in [28] and recently proven geometrically in [31]. See [6] for some related results on temperature chaos for spherical models. The main tool that allows us to prove the matching lower bound is the overlap synchronization mechanism developed by Panchenko in [21,22,23] to study multispecies models and models with vectors spins. This mechanism is a consequence of the ultrametric structure of generalized overlaps that satisfy the Ghirlanda-Guerra identities [13,14] which was proved in [19]. Synchronization was used recently in other contexts in [16,8], and in this paper we give another application. Besides this, our proof is based on a variant of the Aizenman-Sims-Starr scheme for spherical models developed in [4]. Lastly, we refer the reader to [18,17,5,3,12,30,7] for other recent work where various aspects of the spherical models have been studied. Model description Fix n ≥ 1. The main goal is to find a formula for the free energy of n constrained copies of spherical spin glasses. The copies are coupled by constraining their overlaps and can possibly exist at different temperatures. We start by introducing the usual spherical spin glass model. Let S N be the sphere in R N of radius √ N and denote the configuration of the jth copy by σ(j) = σ 1 (j), . . . , σ N (j) ∈ S N . (2.1) For p ≥ 2, the p-spin Hamiltonian is denoted by H N,p (σ(j)) = 1 N (p−1)/2 1≤i1,...,ip≤N g i1,...,ip σ i1 (j) · · · σ ip (j), (2.2) where g i1,...,ip are i.i.d. standard Gaussian for all p ≥ 2 and indices (i 1 , . . . , i p ). The corresponding even mixed p-spin Hamiltonian for the jth copy at inverse temperature (β p (j)) p≥2 is denoted by H j N (σ) = p≥2 β p (j)H N,p (σ(j)). (2.3) We assume that the inverse temperatures satisfy p≥2 2 p β 2 p (j) < ∞ for all j ≤ n, so that (2.3) is well-defined, and that β p (j) = 0 for odd p. We now introduce the model for a system of n copies of spherical spin glass. A configuration of n copies can be viewed as vector spins, σ = (σ 1 , . . . , σ N ) ∈ (R n ) N , (2.4) where the vector entries of σ are denoted by σ i = σ i (1), . . . , σ i (n) ∈ R n . (2.5) The configurations σ are restricted to the set S n N = σ ∈ (R N ) n | σ(j) = √ N for all j ≤ n , (2.6) where · is the Euclidean norm on R N . The Hamiltonian of n copies of even mixed p-spin models of spherical spin glasses is denoted by H N (σ) = j≤n H j N (σ). Free energy of multiple systems of spherical spin glasses The upper indices ≥ 1 of the configurations σ index sequences of spin configurations. The Hamiltonian is a Gaussian process indexed by σ ∈ S n N with covariance given by functions of normalized inner products. The inner products, or overlaps, of the configurations of copy σ (j) and σ (j ) is denoted by R j,j , = R j,j , σ (j), σ (j ) = 1 N i≤N σ i (j)σ i (j ). (2.8) The overlaps of vector configurations σ and σ are given by the overlap matrices R , = R(σ , σ ) = R j,j , j,j ≤n = 1 N i≤N σ i ⊗ σ i . (2.9) The overlaps are always normalized by the dimension of the vectors in the inner product. Let x ∈ R n and let A = (A j,j ) j,j ≤n ∈ R n×n . Consider the real valued convex function ξ j,j (x) = p≥2 β p (j)β p (j )x p (2.10) and its matrix valued counterpart ξ(A) = ξ j,j (A j,j ) j,j ≤n = p≥2 (β p ⊗ β p ) A p , (2.11) where ⊗ is the outer product on vectors in R n and is the Hadamard product on n × n matrices. It is easy to check that the mixed p-spin Hamiltonian of the copies (2.3) are centered Gaussian processes with covariance EH j N σ H j N σ = N ξ j,j R j,j , , (2.12) and the Hamiltonian (2.7) is a centered Gaussian process with covariance EH N (σ )H N (σ ) = N Sum(ξ(R , )), (2.13) where the sum of all entries in a matrix is denoted by Sum(A) = j,j ≤n A j,j . (2.14) The limit of the free energy We now define the constrained free energy. Let Q = Q j,j j,j ≤n be a n × n symmetric positive semidefinite matrix with off-diagonals, Q j,j ∈ [−1, 1] for j = j and diagonals Q j,j = 1. Given ε > 0, we denote the set of spins with constrained self overlaps by Q ε N = σ ∈ S n N | R(σ, σ) − Q ∞ ≤ ε , (2.15) where · ∞ is the infinity norm on n × n matrices. For an external field h = h(j) j≤n ∈ R n , we define the free energy as (2.16) where the reference measure λ n N = λ ⊗n N is the product of normalized uniform measures λ N on S N . EJP 25 (2020), paper 28. F ε N (β, Q) = 1 N E log Q ε N exp H N (σ) + j≤n h(j) i≤N σ i (j) dλ n N (σ), Free energy of multiple systems of spherical spin glasses We will prove the limit of (2.16) can be expressed as a Parisi type functional. We begin by introducing some notation. Let Γ n = A | A is a n × n positive-semidefinite matrix , (2.17) denote the space of n × n matrices, and let Π = π : [0, 1] → Γ n | π is left-continuous, π(x 1 ) ≤ π(x 2 ) for x 1 ≤ x 2 (2.18) denote the space of left-continuous monotone paths on Γ n . The notation π(x 1 ) ≤ π(x 2 ) means π(x 2 ) − π(x 1 ) ∈ Γ n . Distances between paths are given by the metric d(π,π) = 1 0 π(x) −π(x) 1 dx, (2.19) where A 1 = j,j |A j,j |. These paths are the functional order parameters of p-spin models with vector spins. Consider a discrete path π ∈ Π connecting 0 and Q, (2.20) This path can be encoded with a sequence of real numbers π(x) = Q k for x k−1 < x ≤ x k for 0 ≤ k ≤ r, π(0) = 0, π(1) = Q.0 = x −1 < x 0 < · · · < x r = 1, (2.21) and a monotone sequence of n × n symmetric positive semi-definite matrices 0 = Q 0 ≤ Q 1 ≤ · · · ≤ Q r = Q. (2.22) Recall definition (2.11), and denote θ(A) = θ j,j (A j,j ) j,j ≤n = A ξ (A) − ξ(A), (2.23) where ξ (A) = (ξ j,j (A j,j )) j,j ≤n is the matrix of entry wise derivatives of ξ. The matrix given by ∆ k = ξ (Q k ) − ξ (Q k−1 ), 1 ≤ k ≤ r, (2.24) is positive semidefinite. This can seen by applying the Schur product theorem to the Hadamard product representation (2.11). Given a symmetric positive definite matrix Λ, for k ≤ r we define recursively Λ r = Λ, Λ k = Λ k+1 − x k ∆ k+1 for 0 ≤ k ≤ r − 1. (2.25) Let | · | be the determinant of n × n matrices and consider the set L := L(π) = {Λ ∈ Γ n | |Λ 0 | > 0}. (2.26) For Λ ∈ L and discrete π ∈ Π, we define the following functional P β,Q (Λ, π) = 1 2 tr(ΛQ) − n − log |Λ| + (Λ −1 0 h, h) + 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | (2.27) − 0≤k≤r−1 x k · Sum θ(Q k+1 ) − θ(Q k ) . (2.28) The dependence on β is through the functions ξ and θ defined in (2.11) and (2.23). The following is the main result: Theorem 2.1. For n ≥ 1 and h ∈ R n , the limit of the free energy at inverse temperature β and constraint Q is given by lim ε→0 lim N →∞ F ε N (β, Q) = inf π,Λ P β,Q (Λ, π). (2.29) The infimum is over Λ ∈ L and discrete paths given by (2.21) and (2.22) over all r ≥ 1. Remark. If det(Q) = 0, we show in Lemma 4.2 that for all fixed β and π, inf Λ P β,Q (Λ, π) = −∞. By concentration of measure, this implies that asymptotically degenerate sequences of vector configurations have exponentially low probability of appearing in the product Gibbs measure in the N → ∞ limit. Remark. Our form of the Parisi functional P β,Q (Λ, π), is missing the 1 2 tr(Λ −1 0 ∆ 1 ) that appears in [24]. This is because we assumed x 0 > 0 in (2.21) while x 0 = 0 in [24]. By applying L'Hôpital's rule and Jacobi's formula, this term can be recovered by observing lim x0→0 1 2x 0 log |Λ 1 | |Λ 1 − x 0 ∆ 1 | = lim x0→0 1 2 |Λ 1 − x 0 ∆ 1 | −1 tr(|Λ 1 − x 0 ∆ 1 |(Λ 1 − x 0 ∆ 1 ) −1 ∆ 1 ) = 1 2 tr(Λ −1 0 ∆ 1 ). Outline of the paper: We begin by using an analogue of Guerra's interpolation to prove the upper bound in Section 3. In Section 4, we prove the sharpness of functionals that appeared in the upper bound using classical large deviations. We begin the proof of the lower bound by using the Poincaré limit to derive an analogue of the Aizenman-Sims-Starr scheme for high dimensional spherical spin glass models in Section 5. In Section 6, we introduce a perturbation of the Hamiltonian that will force the overlaps under the asymptotic Gibbs measure to satisfy the synchronization properties used in the study of vector spin glass models. In Section 7 we combine all the results and finish the proof of the lower bound using standard cavity computations. Upper bound -Guerra's interpolation Remark. Throughout this paper, we will denote by L any constant that depends only on the global parameters of the model such as the number of copies n, and the inverse temperature parameters β. The constant can change even within the same equation. We begin by proving the upper bound of the free energy. Lemma 3.1. For n ≥ 1 and h ∈ R n , lim ε→0 lim sup N →∞ F ε N (β, Q) ≤ inf Λ,π P β,Q (Λ, π). (3.1) A version of this upper bound was proved in Section 2 of [24]. We will provide a different proof using the Ruelle probability cascades and Guerra's interpolation. The main difference is the following proof will hold without the condition that the diagonals of Λ are greater than 1. Consider the sequence of real numbers 0 = x −1 < x 0 < · · · < x r = 1, (3.2) and the sequence of n × n positive semi definite matrices 0 = Q 0 ≤ Q 1 ≤ · · · ≤ Q r = Q. (3.3) Let (v α ) α∈N r be the weights of the Ruelle probability cascades [29] corresponding to the sequence (3.2). For paths α 1 , α 2 ∈ N r , we denote the common vertices by α 1 ∧ α 2 = min 0 ≤ j ≤ r | α 1 1 = α 2 1 , . . . , α 1 j = α 2 j , α 1 j+1 = α 2 j+1 (3.4) and α 1 ∧ α 2 = r if α 1 = α 2 . Consider independent centered Gaussian processes Z(α) = Z j (α) j≤n and Y (α) indexed with α ∈ N r and covariances Cov(Z(α 1 ), Z(α 2 )) = ξ (Q α 1 ∧α 2 ), (3.5) Cov(Y (α 1 ), Y (α 2 )) = Sum θ(Q α 1 ∧α 2 ) . Let Z i (α) be an independent copy of Z(α) also independent of Y (α). A Gaussian interpolation argument will bound the free energy with functions of these Gaussian processes. F ε N (β, Q) ≤ 1 N E log α∈N r v α Q ε N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dλ n N (σ) − 1 N E log α∈N r v α exp √ N Y (α) + Lε. (3.7) Proof. The result follows from Gaussian interpolation. For 0 ≤ t ≤ 1, we define the interpolating Hamiltonian H t (σ, α) = √ tH N (σ) + i≤N j≤n σ i (j) √ 1 − tZ j i (α) + h(j) + √ t √ N Y (α), on S n N × N r . For a given a constraint Q, we define the interpolating free energy function ϕ(t) = 1 N E log α∈N r v α Q ε N exp H t (σ, α) dλ n N (σ). Let · t be the average on Q ε N × N r with respect to the Gibbs measure G(dσ, α) ∝ v α exp H t (σ, α) dλ n N (σ). A straightforward computation shows ϕ (t) = 1 N E ∂ ∂t H t (σ, α) t . By Gaussian integration by parts [20, Lemma 1.1], 1 N E ∂ ∂t H t (σ, α) t = 1 2 E Sum ξ(R 1,1 ) − R 1,1 ξ (Q α 1 ∧α 1 ) + θ(Q α 1 ∧α 1 ) t (3.8) − 1 2 E Sum ξ(R 1,2 ) − R 1,2 ξ (Q α 1 ∧α 2 ) + θ(Q α 1 ∧α 2 ) t . (3.9) We use convexity to bound (3.9). Since β p = 0 for odd p, ξ j,j (x) is a convex function for all j, j ≤ n and therefore lies above all its tangent lines. That is, ξ j,j (a) − aξ j,j (b) + θ j,j (b) ≥ 0 for all a, b ∈ R. which implies, Sum ξ(R 1,2 ) − R 1,2 ξ (Q α 1 ∧α 2 ) + θ(Q α 1 ∧α 2 ) is non-negative.E Sum ξ(R 1,1 ) − ξ(Q α 1 ∧α 1 ) − (R 1,1 − Q α 1 ∧α 1 ) ξ (Q α 1 ∧α 1 ) t . (3.10) The self overlaps are constrained, so R 1,1 − Q α 1 ∧α 1 ∞ ≤ ε. The Lipschitz continuity of ξ implies (3.8) is bounded by Lε, for some constant L that does not depend on N . These bounds on (3.8) and (3.9) imply ϕ (t) ≤ Lε. (3.11) By the mean value theorem, (3.11) gives us the upper bound ϕ(1) ≤ ϕ(0) + Lε, (3.12) where ϕ(1) = F ε N (β, Q) + 1 N E log α∈N r v α exp √ N Y (α), (3.13) ϕ(0) = 1 N E log α∈N r v α Q ε N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dλ n N (σ). (3.14) Rearranging terms finishes the proof of the upper bound. 1 N E log α∈N r v α exp √ N Y (α) = 0≤k≤r−1 x k · Sum θ(Q k+1 ) − θ(Q k ) . (3.15) The term in (3.7) containing Z(α) can be computed similarly after decoupling the constraint on Q ε N using Lagrange multipliers and rotational invariance [24, Lemma 1]. Let ν N be the standard Gaussian measure on R N . We write ω(j) ∈ R N in its polar coordinate form ω(j) = (s j σ(j)), where s j = ω(j) √ N ∈ R + and σ(j) = √ N ω(j) ω(j) ∈ S N . Let γ N denote the law of s j under ν N . By rotational invariance, the law of σ(j) under ν j is λ N , and σ(j) and s j are independent. We express (3.14) in terms of a Gaussian integral. Lemma 3.3. There exists a δ ∈ (0, ε), such that (3.14) is bounded above by 1 N E log α∈N r v α Ω ε,δ N exp i≤N j≤n ω i (j) Z j i (α) + h(j) dν n N (ω) − n log ν N (E δ N ) N + Lδ (3.16) where the δ shell around Q ε N is denoted by (3.17) and the δ neighbourhood of the radial component is denoted by Ω ε,δ N = ω = (s j σ(j)) j≤n ∈ (R N ) n | σ ∈ Q ε N , s j ∈ [ √ 1 − δ, √ 1 + δ] for all j ≤ nE δ N = {x ∈ R N | x ∈ [ (1 − δ)N , (1 + δ)N ]}. Proof. We will use a Gaussian interpolation argument. LetZ j i (α) be an independent copy of Z j i (α). For 0 ≤ t ≤ 1, we define the interpolating Hamiltonian H t (ω, α) = √ t i≤N j≤n σ i (j)Z j i (α) + √ 1 − t i≤N j≤n ω i (j)Z j i (α) + i≤N j≤n σ i (j)h(j), on Ω ε,δ N × N r . The corresponding interpolating free energy function is denoted by ϕ(t) = 1 N E log α∈N r v α Ω ε,δ N exp H t (ω, α) dν n N (ω). Let · t be the average on Ω ε,δ N × N r with respect to the Gibbs measure G(dω, α) ∝ v α exp H t (ω, α) dν n N (ω). By Gaussian integration by parts, ϕ (t) = 1 N E ∂ ∂t H t (ω, α) t = E E ∂H t (ω 1 , α 1 ) ∂t ·H t (ω 1 , α 1 )−E ∂H t (ω 1 , α 1 ) ∂t ·H t (ω 2 , α 2 ) t . Computing the covariances, we get ϕ (t) = 1 2 E Sum R(σ 1 , σ 1 ) ξ (Q α 1 ∧α 1 ) − R(ω 1 , ω 1 ) ξ (Q α 1 ∧α 1 ) t − 1 2 E Sum R(σ 1 , σ 2 ) ξ (Q α 1 ∧α 2 ) − R(ω 1 , ω 2 ) ξ (Q α 1 ∧α 2 ) t . Since ω i (j) = s j σ i (j) and s j ∈ [ √ 1 − δ, √ 1 + δ], we have the bound Sum R(σ 1 , σ 2 ) ξ (Q α 1 ∧α 2 ) − R(ω 1 , ω 2 ) ξ (Q α 1 ∧α 2 ) ≤ δn 2 ξ (1) ∞ . By the triangle inequality, |ϕ (t)| ≤ n 2 δ ξ (1) ∞ = Lδ, resulting in the bound ϕ(1) ≤ ϕ(0) + Lδ. (3.18) The ending term of the interpolation can be simplified using rotational invariance of ν N , ϕ(1) = 1 N E log α∈N r v α Ω ε,δ N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dν n N (ω) = 1 N E log α∈N r v α [ √ 1−δ, √ 1+δ] n Q ε N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dλ n N (σ)dγ n N (s) = 1 N E log α∈N r v α Q ε N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dλ n N (σ) + n log ν N (E δ N ) N . (3.19) Substituting (3.19) into (3.18) gives the bound 1 N E log α∈N r v α Ω ε,δ N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dν n N (ω) (3.20) ≤ 1 N E log α∈N r v α Ω ε,δ N exp i≤N j≤n s j σ i (j)Z j i (α) + σ i (j)h(j) dν n N (ω) − n log ν N (E δ N ) N + Lδ. On the set Ω ε,δ N , the Cauchy-Schwarz inequality implies j≤n i≤N s j σ i (j)h(j) − σ i (j)h(j) ≤ δ √ N j≤n σ(j) · |h(j)| ≤ δLN. Therefore, we can replace σ i (j)h(j) with ω i (j)h(j) in the upper bound of (3.20) and absorb the error into Lδ giving 1 N E log α∈N r v α Ω ε,δ N exp i≤N j≤n ω i (j) Z j i (α) + h(j) dν n N (ω) − n log ν N (E δ N ) N + Lδ, the required upper bound in (3.16). We now explicitly compute the upper bound of (3.16). We denote the subset of R N n constrained by coupling the overlaps with, Ω ε N = ω ∈ (R N ) n | R j,j ω, ω ∈ [Q j,j − ε, Q j,j + ε] for all j, j ≤ n . (3.21) For δ < ε, Ω ε,δ N ⊂Ω 2ε N so (3.16) is bounded above by 1 N E log α∈N r v α Ω2ε N exp i≤N j≤n ω i (j) Z j i (α) + h(j) dν n N (ω) − n log ν N (E δ N ) N + Lε. (3.22) For any ω ∈Ω 2ε N and Λ ∈ L, j,j ≤n Λ j,j Q j,j − 1 N j,j ≤n i≤N Λ j,j ω i (j )ω i (j) 1 ≤ 2ε Λ 1 . Therefore, adding and subtracting 1 2 i≤N ((Λ − I)ω i , ω i ) from the exponent implies (3.22) can be bounded above by 1 N E log α∈N r v α Ω2ε N exp i≤N j≤n ω i (j) Z j i (α) + h(j) − 1 2 i≤N ((Λ − I)ω i , ω i ) dν n N (ω) + 1 2 tr(ΛQ) − n 2 − n log ν N (E δ N ) N + 2ε Λ 1 − Lε. (3.23) SinceΩ 2ε N ⊂ (R N ) n , if we define the function, Y r,i (α) = 1 (2π) n/2 R n exp j≤n ω i (j) Z j i (α) + h(j) − 1 2 j,j ≤n Λ j,j ω i (j)ω i (j ) dω i (3.24) then our upper bound (3.23) can be written as 1 N E log α∈N r v α i≤N Y r,i (α) + 1 2 tr(ΛQ) − n 2 − n log ν N (E δ N ) N + ε Λ 1 − Lε. (3.25) The term containing Y r,i (α) in (3.25) can be computed recursively. Let z k = (z j k ) j≤n be a Gaussian vector with covariance ∆ k defined in (2.24) and let z k be independent for 1 ≤ k ≤ r. For i ≤ M let z k,i be an independent copy of z k . We define the recursion starting with Y r,i = log 1 (2π) n/2 R n exp j≤n ω i (j) 1≤k≤r z j k,i + h(j) − 1 2 j,j ≤n Λ j,j ω i (j)ω i (j ) dω i (3.26) with subsequent values for 0 ≤ k ≤ r − 1 given recursively by Y k,i = 1 x k log E k exp x k Y k+1,i ,(3.27) where E k refers to expectation with respect to the random vector z k+1,i . The z i are i.i.d. so Y 0,i = Y 0,1 for all i ≤ N .Y 0,1 + 1 2 tr(ΛQ) − n 2 − n log ν N (E δ N ) N + ε Λ 1 − Lε. (3.28) In this model, Y 0,1 has a closed form. Starting from the start of the recursion, a direct computation (see equation (2.17) in [24]) shows Here (·, ·) is the scalar product of vectors in R n . The first term is non-random and will Y r,1 = − 1 2 log |Λ| + 1 2 Λ −1 1≤k≤r z k,1 + h , 1≤k≤r z k,1 + h . propagate through the recursion. The second term can be computed recursively using the following result: Lemma 3.4. Let g be a Gaussian vector with covariance C. Then for any y ∈ R n and x ∈ (0, 1], 1 x log E exp x 2 A −1 (y + g ), y + g = 1 2x log |A| |A − xC| + 1 2 (A − xC) −1 y, y . Proof. The one dimensional case was proven in [32, Lemma 3.5]. We will prove the analogous result for R n . The expectation can be computed explicitly as follows, E exp x 2 A −1 (y + g ), y + g = |C| −1 (2π) n 1/2 R n exp x 2 A −1 (y + z ), y + z − 1 2 C −1 z, z dz = |C| −1 (2π) n 1/2 R n exp x 2 (A − xC) −1 y, y − 1 2 (C −1 − xA −1 )(z − By ), (z − By ) dz = |C| −1 |C −1 − xA −1 | 1/2 exp x 2 (A − xC) −1 y, y where the matrix B is given by B = x(C −1 − xA −1 ) −1 A −1 . The conclusion follows immediately if we rewrite the matrices in the normalizing constant as, (C −1 − xA −1 ) = C −1 (A − xC)A −1 , which implies |C −1 − xA −1 | = |C| −1 |A − xC||A| −1 . Using Lemma 3.4 to compute the recursion gives the appropriate closed form. Corollary 3.5. If |Λ 0 | > 0, then Y 0,1 = − 1 2 log |Λ| + 1 2 Λ −1 0 h, h + 1 2 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | . (3.30) Proof. Using Lemma 3.4 to compute the expectation of the second term in (3.29) recursively implies Y r−1,1 = − 1 2 log |Λ| + 1 2x r log |Λ r | |Λ r − x r−1 ∆ r | (3.31) + 1 2 (Λ r − x r−1 ∆ r ) −1 1≤k≤r−1 z k,1 + h , 1≤k≤r−1 z k,1 + h . (3.32) Again, the terms in (3.31) are non-random, so they propagate through the recursion. Computing the terms in (3.32) inductively using repeated applications of Lemma 3.4 implies Y 0,1 = − 1 2 log |Λ| + 1 2 Λ −1 0 h, h + 1 2 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | . (3.33) Free energy of multiple systems of spherical spin glasses Notice ν N concentrates around the sphere of radius √ N in high dimensions, so the Gaussian term will vanish in the limit by the weak law of large numbers. That is, for any δ > 0, lim N →∞ n log ν N (E δ N ) N = lim N →∞ n log(P(| 1 N N i=1 g 2 i − 1| ≤ δ)) N = 0,(3.1 N E log α∈N r v α Q ε N exp i≤N j≤n σ i (j) Z j i (α) + h(j) dλ n N (σ) ≤ 1 2 tr(ΛQ) − n − log |Λ| + (Λ −1 0 h, h) + 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | .F ε N (β, Q) ≤ inf Λ,π P β,Q (Λ, π), (3.36) completing the proof of the upper bound. Sharpness of the upper bound We now prove for every fixed path π, the upper bound (3.35) is asymptotically sharp in the sense that it attains equality after minimizing over Λ. This fact will be used again when a similar functional appears in the proof of the lower bound. The proof of this sharpness for the replica symmetric case can be found in [24,Lemma 4]. We will provide a proof of the general case below. Let π be any fixed discrete monotone path characterized by the sequences (3.2) and f 1 N (π) = 1 N E log α∈N r v α Ω ε,δ N exp i≤N j≤n ω i (j) Z j i (α) + h(j) dν n N (σ). (4.1) We will prove the matching lower bound of (3.35) by decoupling the functional f 1 N (π) from the constraint Q and explicitly computing its value recursively. Lemma 4.1. For all 0 < δ < ε, lim inf N →∞ f 1 N (π) ≥ inf Λ 1 2 tr(ΛQ) − n − log |Λ| + (Λ −1 0 h, h) + 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | . Recall (3.21), the subset of R N n constrained by coupling the overlaps, Ω δ N = ω ∈ (R N ) n | R ω, ω − Q ∞ ≤ δ . (4.2) Clearly, there exists a δ * < ε such that Ω ε,δ N ⊇Ω δ * N , so f 1 N (π) ≥ 1 N E log α∈N r v α Ωδ * N exp i≤N j≤n ω i (j) Z j i (α) + h(j) dν n N (ω). (4.3) We introduce the Lagrange multipliers Λ ∈ L defined in (2.26). Like in the proof of the upper bound, since R(ω, ω) − Q ∞ ≤ δ * for ω ∈Ω δ * N , adding and subtracting the 2 i≤N ((Λ − I)ω i , ω i ) from the exponent implies (4.3) is bounded below by 1 N E log α∈N r v α Ωδ * N exp i≤N j≤n ω i (j) Z j i (α) + h(j) − 1 2 i≤N ((Λ − I)ω i , ω i ) dν n N (ω) + 1 2 tr(ΛQ) − n 2 − δ * Λ 1 . (4.4) We view the quantity on the first line of (4.4) as a function of Λ and the region of integration. In general, we denote this integral over sets V ⊂ (R N ) n by The map V → Φ V (Λ) is monotone and, in particular, Φ V (Λ) ≤ F (Λ). Furthermore, the function F (Λ) does not depend on N , and was computed using the recursion (3.27) giving the closed form in Corollary 3.5, Φ V (Λ) = 1 N E log α∈N r v α V exp i≤N j≤n ω i (j) Z j i (α) + h(j) − 1 2 i≤N ((Λ − I)ω i , ω i ) dν n N (ω)F (Λ) = 1 2 − log |Λ| + Λ −1 0 h, h + 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | . (4.7) We will prove that minimizing over Λ removes the dependence on the constraint Q asymptotically. We start by showing there exists a unique Λ * that minimizes 1 2 tr(ΛQ) + F (Λ) if the lower bound (4.15) is finite. 2. If Q is non-degenerate, then there exists a Λ * ∈ L that minimizes 1 2 tr(ΛQ) + F (Λ) and satisfies ∂ ∂t 1 2 tr (Λ * + tB)Q + F (Λ * + tB) t=0 = 0 (4.9) for any symmetric matrix B. Proof. Consider the eigendecomposition of Λ ∈ L, Λ = U DU T . Using this change of variables and (2.25), we see (4.7) can be rewritten in terms of U and D as, 1 2 tr(ΛQ) + F (Λ) = 1 2 tr DU T QU − log |D| + D − 0≤k<r x k U T ∆ k+1 U −1 (U T h), (U T h) + 0≤k≤r−1 1 x k log |D − k+1≤ <r x U T ∆ +1 U | |D − k≤ <r x U T ∆ +1 U | . (4.10) In this form, the infimum is over positive semidefinite diagonal matrices D and orthogo- nal matrices U such that |D − 0≤ <r x U T ∆ +1 U | > 0, inf Λ 1 2 tr(ΛQ) + F (Λ) = inf D,U 1 2 tr(U DU T Q) + F (U DU T ) . Case (1): Suppose Q is degenerate, i.e. |Q| = 0. There exists an orthogonal matrix U , corresponding to the eigendecomposition of Q, such thatD = U T QU andD 11 = 0. Given this U , we choose diagonal matrix D with diagonal entries large enough such that all the Gershgorin discs of D − k≤ <r x U T ∆ +1 U are contained in the positive real half plane for all k ≤ r − 1. In particular for all k ≤ r − 1, the smallest eigenvalue of D − k≤ <r x U T ∆ +1 U is strictly positive and will remain bounded away from zero if we increase the value of the first diagonal element. That is, there exists a c > 0 such that lim inf D11→∞ λ min D − k≤ <r x U T ∆ +1 U ≥ c > 0 for all k ≤ r − 1. (4.11) We fix all entries D jj for 2 ≤ j ≤ n and show (4.10) diverges to −∞ as we take the first entry D 11 → ∞. For the above choice of D and U , we have tr DU T QU − log |D| = n i=1 D ii (U T QU ) ii − n i=1 log D ii = n i=2 D ii (U T QU ) ii − n i=2 log D ii − log D 11 , which implies lim D11→∞ tr DU T QU − log |D| = −∞. (4.12) We will now show that the remaining terms of (4.10) are finite. Let ν k min denote the smallest eigenvalue of D − k≤ <r x U T ∆ +1 U . By (4.11), we have lim inf D11→∞ ν k min ≥ c for all k ≤ r − 1. Bounding the quadratic form with the largest eigenvalue of the associated matrix implies lim D11→∞ D − 0≤k<r x k U T ∆ k+1 U −1 (U T h), (U T h) ≤ ν 0 min −1 h 2 ≤ c −1 h 2 < ∞. (4.13) The logarithm terms in (4.10) can be bounded by the minimum eigenvalues in a similar manner. It suffices to show an arbitrary term in the sum is bounded, that is, lim D11→∞ 1 x k log |D − k+1≤ <r x U T ∆ +1 U | |D − k≤ <r x U T ∆ +1 U | < ∞. (4.14) If we define the matrices A k : = D − k≤ <r x U T ∆ +1 U and B k := x k U T ∆ k+1 U , then log |D − k+1≤ <r x U T ∆ +1 U | |D − k≤ <r x U T ∆ +1 U | = log |A k + B k | |A k | = log |A −1 k (A k +B k )| = log |I+A −1 k B k |. Bounding this with the largest eigenvalue, we see log |I + A −1 k B k | ≤ n log λ max (I + A −1 k B k ). Using submultiplicativity of the spectral norm and the lower bound on the smallest eigenvalue of A k in (4.11), we have λ max (I +A −1 k B k ) = 1+λ max (A −1 k B k ) ≤ 1+λ max (A −1 k )λ max (B k ) ≤ 1+c −1 λ max (B k ) < ∞, giving the required bound in (4.14). Therefore, for a particular U , we can construct a sequence of diagonal matrices D with arbitrary large first diagonal element such that 1 2 tr(U DU T Q) + F (U DU T ) is unbounded. In particular, we have inf Λ 1 2 tr(ΛQ) + F (Λ) = −∞. Case (2): Consider the case when Q is positive definite. We will prove that (4.7) attains a minimum at some point Λ * ∈ L. By Hölder's inequality, F (Λ) is a convex function of Λ, so any local minimizer is also a global minimizer. We will prove that the minimizer is attained in a compact subset of Γ n under the spectral norm on symmetric matrices Λ 2 = λ max (Λ). Because Q is positive definite, the diagonal elements of U T QU is positive and uniformly bounded away from 0 for all orthogonal matrices U . That is, the first term in (4.10) can be bounded below by tr DU T QU −log |D| = j≤n D jj (U T QU ) jj −log |D jj | ≥ j≤n D jj λ min (Q)−log |D jj | which clearly diverges to ∞ if any diagonal element D jj → ∞. The remaining terms in (4.10) are non-negative, so 1 2 tr(ΛQ) + F (Λ) → ∞ if Λ 2 → ∞. Since Λ − Λ 0 ≥ 0, we also have 1 2 tr(ΛQ) + F (Λ) → ∞ if Λ 0 2 → ∞ by monotonicity. By definition (2.25), we have Λ k+1 = Λ k + x k ∆ k+1 . By submultiplicativity of · 2 , ∆ k+1 2 = Λ k Λ −1 k ∆ k+1 2 ≤ Λ k 2 Λ −1 k ∆ k+1 2 , so the last term in (4.7) can be bounded below by log |Λ k+1 | |Λ k | = log |I + x k Λ −1 k ∆ k+1 | ≥ log(1 + x k Λ −1 k ∆ k+1 2 ) ≥ log(1 + x k Λ k −1 2 ∆ k+1 2 ). Let k * be the smallest index such that ∆ k * +1 = 0, then it is clear the above term diverges as Λ k * 2 → 0. Since the sequence (3.3) is monotone, we have Λ 0 2 = Λ k * 2 . The rest of the terms in (4.7) are bounded or positive for fixed π, so 1 2 tr(ΛQ) + F (Λ) → ∞ if Λ 0 → 0. We have shown, 1 2 tr(ΛQ)+F (Λ) is unbounded if Λ 0 2 → 0 or Λ 0 2 → ∞. Therefore, there exists a 0 < c < C < ∞ such that the minimizer is attained in the compact set L * = {Λ ∈ Γ n | c ≤ Λ 0 2 ≤ C} ⊆ L. The matrix B = 0≤k≤r x k ∆ k is a fixed positive semidefinite matrix, so the map f : Γ n → R defined by f (Λ) = Λ − B 2 = Λ 0 2 is continuous. Therefore, L * = f −1 ([c, C]) is closed and clearly bounded, so it is compact. By the extreme value theorem, there exists a Λ * ∈ L such that 1 2 tr(ΛQ) + F (Λ) attains its minimum at Λ * . Furthermore, our function is convex and Λ * is an interior point of L, so the minimizer Λ * is unique and satisfies the critical point condition, ∂ ∂t 1 2 tr (Λ * + tB)Q + F (Λ * + tB) t=0 = 0, for all symmetric matrices B. Lemma 4.1 is trivially satisfied when the infimum is −∞, so we focus on the nondegenerate case moving forward. We now prove asymptotic sharpness of the upper bound using a standard large deviations calculation to decouple the constraints. Free energy of multiple systems of spherical spin glasses Proof. Consider the partition (R N ) n =Ω δ * N ∪ j,j ≤n V + j,j ∪ j,j ≤n V − j,j where V + j,j = ω | R j,j (ω, ω) ≥ Q j,j + δ * , (4.16) V − j,j = ω | R j,j (ω, ω) ≤ Q j,j − δ * .Φ V (Λ * ) ≤ F (Λ * ) − c.tR j,j (ω, ω) ≤ t(Q j,j − δ * ). Let B be a matrix such that B j,j = B j ,j = 1 and is zero everywhere else. Adding and subtracting 1 2 tN R j,j (ω, ω) and 1 2 tN R j ,j (ω, ω) in the exponent, by symmetry of Q, we Since U (0) = F (Λ * ), the critical point condition (4.9) implies U (0) = −δ * . In particular, there is a t * such that U (t * ) < U (0). Since (4.19) holds for all t > 0, there is a c such that have Φ V (Λ * ) ≤ t(Q j,j − δ * ) + Φ V (Λ * + tB) ≤ t(Q j,j − δ * ) + F (Λ * + tB) = −tδ * − 1 2 tr(Λ * Q) +Φ V (Λ * ) ≤ U (t * ) ≤ U (0) − c = F (Λ * ) − c. Recall the sets (4.16), (4.17), and (4.2) form a partition of (R N ) n . A consequence of the recursion in the Ruelle probability cascades (see equation (118) in the proof of [22,Lemma 7] or the proof of [23, Lemma 6]) implies F (Λ * ) ≤ log(2n 2 + 1) N x 0 + max max V Φ V (Λ * ), ΦΩδ * N (Λ * ) where the maximum over V is over the halfspaces of the form (4.16), (4.17). Our bounds in (4.18) ensures we cannot have F (Λ * ) ≤ log(2n 2 + 1) N x 0 + max V Φ V (Λ * ) for N sufficiently large. Therefore, we must have F (Λ * ) ≤ log(2n 2 + 1) N x 0 + ΦΩδ * N (Λ * ). Taking N → ∞ completes the proof. f 1 N (π) ≥ inf Λ 1 2 tr(ΛQ) − n − log |Λ| + (Λ −1 0 h, h) + 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | . We have shown that Theorem 2.1 is trivially satisfied for degenerate constraint Q just from examining the upper bound. The case for positive definite constraint Q is much harder and will require some preliminary work before attempting the cavity computations. We begin by introducing a variant of the Aizenman-Sims-Starr scheme. The Aizenman-Sims-Starr scheme Before we can complete the cavity computations to prove the lower bound, we first prove an analogue of the Aizenman-Sims-Starr scheme [1] for spherical spin glass models with vector spins. The extension to this model is non-trivial because the uniform measure on the sphere is not a product measure, so the usual proof of the scheme fails. This section follows the proof of the Aizenman-Sims-Starr scheme adapted for spherical models in [4]. The main difference is the Aizenman-Sims-Starr scheme (see Lemma 6.3) will be with respect to a Gaussian reference measure as opposed to the surface measure in [4]. This form was chosen for convenience, because it matches the form of the functional (3.23). To simplify notation, we first prove an analogue of the Aizenman-Sims-Starr scheme with no external field. We will explain how to reintroduce the external field at the end of Section 6. Consider the partition function with h = 0 for a system of size N , We denote spin configurations from the system of size M + N with ρ = (σ, ω) ∈ S n M +N where σ ∈ R N denotes the bulk coordinates and ω ∈ R M denotes the cavity coordinates. Z N (Q, ε) = Q ε N exp H N (σ) dλ n N (σ) We proceed like the traditional Aizenman-Sims-Starr scheme and split the Hamiltonian into the cavity fields [20, Section 3.5] H M +N (σ, ω) d = j≤n H j M,N (σ) + i≤M j≤n ω i (j)Z j i (σ) + r(ρ), (5.3) H N (σ, ω) d = j≤n H j M,N (σ) + √ M j≤n Y j (σ).EZ j i (σ )Z j i (σ ) = δ i,i ξ j,j R j,j , + O M N , (5.6) EY j (σ )Y j (σ ) = θ j,j R j,j , + O M N ,(5.7) and the remainder term r(ρ) has covariance, Er(ρ )r(ρ ) = O M 2 M + N . (5.8) We will prove that we can replace the cavity fields Z(σ) and Y (σ) with centered Gaussian fields z i (σ) and y(σ) taking values in R n indexed by σ ∈ S n N , with covariances We start as usual with the inequality Ez j i (σ )z j i (σ ) = δ i,i ξ j,j R j,j , ,(5.lim inf N →∞ 1 N E log Z N (Q, ε) ≥ 1 M lim inf N →∞ E log Z M +N (Q, ε) − E log Z N (Q, ε) = 1 M lim inf N →∞ E log Z M +N (Q, ε) Z M,N (Q, ε) − E log Z N (Q, ε) Z M,N (Q, ε) . (5.13) The surface measure λ M +N appearing in Z M +N is not a product measure, so the standard proof of the Aizenman-Sims-Starr does not apply after this point. Instead, recall that the δ shell around Q ε M is denoted by Ω ε,δ M = ω = (s j τ (j)) j≤n ∈ (R M ) n | τ ∈ Q ε M , s j ∈ [ √ 1 − δ, √ 1 + δ] for all j ≤ n , (5.14) where s j ∈ R + and τ (j) ∈ S M are the radial and angular components of the polar form of ω(j) (see after (3.15) for the formulas). We will prove the following lower bound of (5.13). 1 M lim inf N →∞ E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (ω) M,N − E log exp √ M y(σ) M,N − Lδ. (5.15) The main difference between the bound (5.15) and the traditional Aizenman-Sims-Starr representation is the Gaussian reference measure appearing in the first cavity field. This measure appears as a consequence of the Poincaré limit, which states that the standard Gaussian measure in R M is the limiting distribution of projected uniform distributions on S N +M as N tends to infinity. Poincaré limit We first explain a method to asymptotically decouple λ M +N into an approximate product measure over the spheres S N × S M . The distribution of the projection of S N +M onto R M under λ N +M converges weakly to the Gaussian distribution ν M on R M in the Poincaré limit [27]. In particular, the distribution of the cavity coordinates under the normalized surface measure will be approximately Gaussian for large N . For large M , ν M will concentrate around S M . We first introduce some notation and state this result in one dimension. For K ≥ 1, we denote the unit sphere in R K with S 1 K and |S 1 K | its surface area. Let A M,N = M j=1 − M + N + 1 − j, M + N + 1 − j ,(5.dν M,N (x) = f M,N (x) dx, where f M,N (x) = b M,N M j=1 1 − x 2 j M + N + 1 − j M +N −j−2 2 , (5.17) with normalizing coefficient b M,N = M j=1 |S 1 M +N −j | |S 1 M +N +1−j | √ M + N + 1 − j .S M +N g(ρ) dλ M +N (ρ) = A M,N S N g(ψ(σ, ω)) dλ N (σ)dν M,N (ω). We will need a multidimensional version of this argument. To simplify notation, for n copies of S N +M , we define a j := a (ω(j)) keeping the dependence on the cavity coordinate ω(j) implicit. Similarly, we define Ψ(σ, ω) = ψ σ(j), ω(j) Proof. We apply Lemma 5.2 to each coordinate j ≤ n. The region of integration is a product set and we are integrating a non-negative function, so we are able to rearrange the order of integration by Fubini's Theorem. We apply Corollary 5.3 to lower bound Z M +N (Q, ε) with an integral over the product set of bulk and cavity coordinates. To simplify notation, we denote the transformed coordinates withρ = (σ,ω) := Ψ(σ, ω) (5.20) wherẽ σ = a j M +1 σ 1 (j), . . . , a j M +1 σ N (j) j≤n andω = a j 1 ω 1 (j), . . . , a j M ω M (j) j≤n (5.21) are the respective transformed bulk and cavity coordinates. For an arbitrary non-negative function g on S n M +N , (5.19) implies Q ε M +N g(ρ) dλ n M +N (ρ) = S n M +N 1 Q ε M +N (ρ)g(ρ) dλ n M +N (ρ) = S n N A n M,N 1 Q ε M +N (σ,ω)g(σ,ω) dν n M,N (ω)dλ n N (σ).1 Q ε M +N (σ,ω) ≥ 1 Q ε N (σ) 1 Ω ε/2,δ M (ω). (5.23) Proof. We will find conditions on δ such that (5.23) holds. Let δ > 0 and take σ ∈ Q ε N and ω ∈ Ω The set Ω ε/2,δ M is bounded, so the corresponding transformed overlaps R(σ,σ) and R(ω,ω), can be approximated by the standard overlaps R(σ, σ) and R(τ , τ ) of configurations σ and τ on the spheres S N and S M respectively. Firstly, since ω(j) 2 < M (1 + δ) for all ω ∈ Ω ε/2,δ M and j ≤ n, we have the relation lim N →∞ N − a j M +1 a j M +1 N = ω(j) 2 + ω(j ) 2 2 − M ≤ M δ. (5.25) Since R j,j (σ,σ) = a j M +1 a j M +1 R j,j (σ, σ), for all N sufficiently large N N + M R(σ,σ) − N N + M R(σ, σ) ∞ ≤ LM δ N + M .(R(ω,ω) − R(ω, ω) ∞ = 0, uniformly on Ω ε/2,δ M . Likewise, on Ω ε/2,δ M , ω(j) − τ (j) 2 = s j τ (j) − τ (j) 2 ≤ δM for all j ≤ n, so R(ω, ω) − R(τ , τ ) ∞ ≤ δ. Therefore, the triangle inequality implies for all ω ∈ Ω ε/2,δ M , M N + M R(ω,ω) − M N + M R(τ , τ ) ∞ ≤ LM δ N + M .R(ρ,ρ) − Q ∞ = N M + N R(σ,σ) − Q + M M + N R(ω,ω) − Q ∞ ≤ N M + N R(σ, σ) − Q + M M + N R(τ , τ ) − Q ∞ + LM δ N + M ≤ N ε N + M + M ε 2(N + M ) + LM δ M + N = ε + M N + M Lδ − ε 2 . Choosing δ ≤ ε 2L , we have R(ρ,ρ) − Q ∞ ≤ ε. In particular, this means Consequently, we are able to decouple the surface measure, which resolves the first major obstacle in the proof of the Aizenman-Sims-Starr representation. (σ,ω) ∈ Q ε N +M ⊇ σ ∈ Q ε N × ω ∈ Ω ε/2, Proof of Lemma 5.1 Using (5.28), we can derive a lower bound for the first term in (5.13). Lemma 5.5. For every ε > 0, there exists a δ ∈ (0, ε) such that lim inf N →∞ E log Z M +N (Q, ε) Z M,N (Q, ε) ≥ lim inf N →∞ E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (ω) M,N − LM δ. (5.29) Proof. We start by splitting the left hand side of (5.29) in three parts E log Z M +N (Q, ε) Step 3: By definition of · M,N , the last term in (5.30) satisfies lim inf N →∞ E log Ω ε/2,δ M J M,N dν n M (ω) Z M,N (Q, ε) = lim inf N →∞ E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (τ ) M,N . Step 4: Combining the inequalities in Step 1, Step 2, and Step 3 with the factorization (5.30) finishes the proof. We now derive a lower bound for the second term appearing in (5.13). Lemma 5.6. We have lim N →∞ −E log Z N (Q, ε) Z M,N (Q, ε) ≥ lim N →∞ −E log exp √ M y(σ) M,N . Proof. The proof by Gaussian interpolation is standard, see for example [4,Lemma 2]. Consider the interpolating Hamiltonian, H t (σ) = j≤n H j M,N (σ) + √ t √ M Y j (σ) + √ 1 − t √ M y j (σ) , Recalling (5.4), consider the corresponding interpolating function, ϕ(t) = E log Q ε N exp H t (σ)dλ N (σ). Differentiating ϕ, we have ϕ (t) = 1 2 E E ∂H t (σ 1 ) ∂t · H t (σ 1 ) − E ∂H t (ρ 1 ) ∂t · H t (σ 2 ) t where · t is the Gibbs average on Q ε N with respect to the Hamiltonian H t (σ). The covariances are given by E ∂H t (σ 1 ) ∂t · H t (σ 2 ) = M j,j ≤n EY j (σ 1 )Y j (σ 2 ) − Ey j (σ 1 )y j (σ 2 ) = O M N , for any σ 1 , σ 2 ∈ Q ε N . Integrating ϕ (t), we have ϕ(1) = ϕ(0) + O M N . Notice (5.4) implies ϕ(1) = Z N (Q, ε). Taking N → ∞, and normalizing both sides by Z M,N (Q, ε) finishes the proof. The proof of Lemma 5.1 is now immediate. Proof of Lemma 5.1. Applying Lemma 5.5 and Lemma 5.6 to (5.13), we have the lower limit of (5.13) is bounded below by 1 M lim inf N →∞ E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (ω) M,N − E log exp √ M y(σ) M,N − Lδ, finishing the proof of Lemma 5.1. Perturbation, Ghirlanda-Guerra identities, and their consequences Using the Aizenman-Sims-Starr scheme, we can approximate the lower bound of the free energy with continuous functionals of the distribution of the overlap array. In particular, we have the terms E log computing the value of the lower bound in the limit, we must first understand the limiting distribution of this overlap. Our main tool is a perturbation of the Gibbs measure that, in the limit, will force the overlaps to satisfy the matrix version of the Ghirlanda-Guerra identities [22,Theorem 3] that in turn imply a powerful synchronization property [22,Theorem 4] in addition to the main consequences of the usual identities [20,Section 3]. These consequences will be summarized at the end of this section. In this section, we introduce this perturbation of the Hamiltonian. We face two main obstacles. Firstly, the usual proof of the Ghirlanda-Guerra identities requires the self-overlaps R(σ, σ) to be constant, which is not immediate in our setting because self-overlaps are only constrained to lie within an ε window Q. Secondly, we need to find a suitable perturbation to give us the matrix version of the Ghirlanda-Guerra identities. Both of these issues are resolved in detail in Section 4 and Section 5 of [22]. They can be adapted to our setting with a few minor modifications. Modified coordinates: We begin by introducing a transformation of the coordinates that was used to control the self overlaps in the vector spin models [22,Section 3]. This transformation will fix the self overlaps allowing us to apply the usual proof of the Ghirlanda-Guerra identities. We use essentially the same change of variables as defined in [22, Section 3] with two main differences. Firstly, since we only need to find a bound for positive definite constraints Q, we do not need to truncate the constraints like in [22]. Secondly, the spins σ i are bounded by a universal constant in the vector spin models, while the individual spins in the spherical models have entries bounded by N . In our setting, we will need to use a slightly different approach to obtain the relevant bounds on the distortion. Let λ min (Q) > 0 denote the smallest eigenvalue of Q. We first state this transformation as it appears in Section 3 of [22]. and, for any R 1 , R 2 such that both R 1 − Q ∞ ≤ ε and R 2 − Q ∞ ≤ ε, A(R 1 ) − A(R 2 ) ∞ ≤ L ε R 1 − R 2 ∞ . (6.2) In the spherical model, we will also need uniform control on A(R) ∞ . Since our constant Q is positive definite, this fact follows as an immediate consequence of (6.1) and (6.2). Free energy of multiple systems of spherical spin glasses Corollary 6.2. If ε ≤ 1, each matrix A(R) constructed in Lemma 6.1, also satisfies the bound A(R) ∞ ≤ L. By norm equivalence, we see (A − I)B 2 ≤ (A − I)B 2 F ≤ L √ ε. Since B is invertible and the · 2 norm is sub-multiplicative we have, A − I 2 = (A − I)BB −1 2 ≤ (A − I)B 2 B −1 2 . Therefore, by norm equivalence, if we assume ε ≤ 1, A − I ∞ ≤ √ n A − I 2 ≤ L √ n √ ε B −1 2 ≤ L √ n B −1 2 , which implies A(Q) is uniformly bounded for all ε ≤ 1. Furthermore, by (6.2), for any A(R) such that R − Q ∞ ≤ ε, we have A(R) − A(Q) ∞ ≤ L ε R − Q ∞ ≤ L. (6.4) Therefore, all matrices A(R) lie within a closed ball around A(Q), which implies that A(R) ∞ is uniformly bounded for all ε ≤ 1. Remark: Corollary 6.2 also holds if we assume is Q is only positive semidefinite. The matrix A(R) has an explicit construction in the proof of Lemma 4 in [22], that only depended on a subset of the eigenvalues of Q. Therefore, there are finitely many possible constructions of A(Q), so we can apply the bound (6.4) to each possible values A(Q) to conclude the uniform bound A(R) ∞ ≤ L. Lemma 6.1 implies there exists a coordinate transform that fixes the self overlaps. For each σ ∈ Q ε N , suppose A σ = A(R(σ, σ)) is chosen as in Lemma 6.1. Denote the modified coordinates byσ = (A σ σ i ) i≤n := A σ σ and observe the corresponding modified overlap satisfies R(σ,σ) = R(A σ σ, A σ σ) = 1 N i≤N (A σ σ i )(A σ σ i ) T = AR(σ, σ)A T = Q. (6.5) The bounds (6.1), (6.2), and (6.3) are used to show the modified overlap matrix is close to the usual overlap. Notice that, R(σ ,σ ) − R(σ , σ ) ∞ ≤ R(σ ,σ ) − R(σ ,σ ) ∞ + R(σ ,σ ) − R(σ , σ ) ∞ . To control the first term, by the Cauchy-Schwarz inequality we have, R(σ ,σ ) − R(σ ,σ ) ∞ ≤ sup j,j ≤n 1 N N i=1 A σ σ i (j)A σ σ i (j ) − σ i (j)A σ σ i (j ) ≤ 1 N sup j,j ≤n (A σ − I)σ (j) A σ σ (j ) ≤ sup j≤n (A σ − I)σ (j) √ N A σ ∞ ≤ A σ ∞ tr(R((A σ − I)σ , (A σ − I)σ )) 1/2 . Using observation (6.5), the bounds (6.1) and (6.3) imply R(σ ,σ ) − R(σ ,σ ) ∞ ≤ Lε 1/4 . A similar computation applied to the second term gives a similar bound, R(σ ,σ ) − R(σ , σ ) ∞ ≤ Lε 1/4 . Therefore, the modified overlap only differs from the overlap by a factor of ε 1/4 , R(σ ,σ ) − R(σ , σ ) ∞ ≤ Lε 1/4 . (6.6) The bounds (6.6) and (6.2) will ensure this change of variables will not affect the limiting values in the perturbed Aizenman-Sims-Starr scheme that we introduce next. Perturbed Hamiltonian: We now define the perturbation that will force the overlaps to satisfy the matrix version of the Ghirlanda-Guerra identities in [22]. This perturbation is identical to the one introduced in Section 5 of [22]. We summarize the key steps below. We denote the family of parameters θ = (p, m, n 1 , . . . , n m , ν 1 , . . . , ν m ). (6.7) For each θ, there exists Gaussian processes h θ (σ) indexed by σ ∈ S n N with mean 0 and covariance C θ , = Cov h θ (σ ), h θ (σ ) = j≤m R p , ν j , ν j nj . (6.8) Furthermore, for ν ∈ [−1, 1] n and σ ∈ S n N , the covariance is bounded by n 2p(n1+···+nm) . We denote the countable set of parameters with Θ = {θ | p ≥ 1, m ≥ 1, n 1 , . . . , n m ≥ 1, ν 1 , . . . , ν m ∈ ([−1, 1] ∩ Q) n }. (6.9) Let j 0 : ([−1, 1] ∩ Q) n → N be a one-to-one function. We denote an enumeration of θ ∈ Θ with j(θ) = p + n 1 + · · · + n m + j 0 (ν 1 ) + · · · + j 0 (ν m ) + 22m. (6.10) Let (u θ ) θ∈Θ be a random sequence of i.i.d. uniform random variables in [1,2]. We define the interpolating Hamiltonian, h N (σ) = θ∈Θ 2 −j(θ) n 2(n1+···+nm) u θ h θ (σ). (6.11) The covariance of this process is bounded by 1, and given explicitly by (6.13) and the corresponding perturbed partition function Cov h N (σ ), h N (σ ) = θ∈Θ 2 −2j(θ) n 4(n1+···+nm) u 2 θ j≤m R p , ν j , ν j nj .Z pert N (Q, ε) = Q ε N exp H pert N (σ) + i≤N j≤n h(j)σ i (j) dλ n N (σ).1 N E log Z N (Q, ε) = lim inf N →∞ 1 N E log Z pert N (Q, ε). (6.15) Perturbed Aizenman-Sims-Starr scheme: The Aizenman-Sims-Starr scheme proved in Section 5 has to be modified slightly to account for the extra perturbation term in the Hamiltonian. Let · pert be the average on Q ε N with respect to the Gibbs measure G pert N (σ) = exp H pert M,N (σ) + i≤N h(j)σ i (j) Z pert M,N (Q, ε) , (6.16) where H pert M,N (σ) = H M,N (σ) + s N h N (σ) and Z pert N (Q, ε) = Q ε N exp H pert M,N (σ) + i≤N j≤n h(j)σ i (j) dλ n N (σ). (6.17) The following modification of Lemma 5.1 will be used in the proof of lower bound. Lemma 6.3. For s N = N γ , h = 0 andσ = (A σ σ i ) i≤N we have lim inf N →∞ 1 N E log Z pert N ≥ 1 M lim inf N →∞ E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (ω) pert − E log exp √ M y(σ) pert − Lδ − Lε 1/4 . (6.18) Proof. Only a small modification needs to be made to adapt the proof of Lemma 5.1 to this setting. We start from the bound in (5.13), lim inf N →∞ 1 N E log Z pert N (Q, ε) ≥ 1 M lim inf N →∞ E log Z pert M +N (Q, ε) − E log Z pert N (Q, ε1 M lim inf N →∞ E log Q ε N Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) + H M,N (σ) + s N +M h N +M (A ρρ ) dν n M (ω) dλ n N (σ) − E log Q ε N exp √ M y(σ) + H M,N (σ) + s N h N (A σ σ) dλ n N (σ) − Lδ,(6.20) after canceling the normalization terms. The perturbation term s N +M h N +M (A ρρ ) needs to appear as s N h N (σ) in the normalization (6.17). We can use an interpolation to show that we can replace perturbation term at the cost of a small error term that vanishes as N → ∞. Consider the interpolating Hamiltonian H t (ρ) = i≤M j≤n ω i (j)z j i (σ) + H M,N (σ) + √ ts N +M h N +M A ρρ + √ 1 − ts N h N (σ), and the interpolating free energy ϕ(t) = E log Q ε N Ω ε/2,δ M exp H t (σ, ω) dν n M,N (ω)dλ n N (σ). Conditionally on u θ , to show that |ϕ (t)| = o(1) after integrating by parts, we will need to control E dH t (ρ 1 ) dt H t (ρ 2 ) = s 2 N +M Eh N +M A ρ 1ρ 1 )h N +M A ρ 2ρ 2 ) − s 2 N Eh N (σ 1 )h N (σ 2 ) = (N + M ) 2γ g(R(A ρ 1ρ 1 , A ρ 2ρ 2 )) − N 2γ g(R(A σ 1 σ 1 , A σ 2 σ 2 ) ) , (6.22) where g is the covariance function of h N given by (6.12). The function g and its derivatives is bounded on compacts uniformly for all parameters u θ . Using (5.24) and (5.25) R(ρ 1 ,ρ 2 ) − R(σ 1 , σ 2 ) ∞ = sup j,j ≤n (a j M +1 (ω 1 )a j M +1 (ω 2 )N − N )R j,j (σ 1 , σ 2 ) M + N − M R j,j (σ 1 , σ 2 ) M + N + M R j,j (ω 1 ,ω 2 ) M + N = O(N −1 ), and therefore, by Lemma 6.1, R(A ρ 1ρ 1 , A ρ 2ρ 2 ) − R(A σ 1 (σ 1 ), A σ 2 (σ 2 )) ∞ = A ρ 1 R(ρ 1 ,ρ 2 )A T ρ 2 − A σ 1 R(σ 1 , σ 2 )A T σ 2 ∞ ≤ L R(ρ 1 ,ρ 2 ) − R(σ 1 , σ 2 ) ∞ ε = O((N ε) −1 ). Using the Taylor series of g(R(A ρ 1ρ 1 , A ρ 2ρ 2 )) around R(A σ 1 σ 1 , A σ 2 σ 2 ), we see (N + M ) 2γ g(R(A ρ 1ρ 1 , A ρ 2ρ 2 )) = (N + M ) 2γ g(R(A σ 1 σ 1 , A σ 2 σ 2 )) + O(N −1−2γ /ε). Since (N + M ) 2γ − N γ = O(N −1−2γ ) we see (6.22) satifies (N + M ) 2γ g(R(A ρ 1ρ 1 , A ρ 2ρ 2 )) − N 2γ g(R(A σ 1 σ 1 , A σ 2 σ 2 )) = O(N −(1−2γ) /ε). The above bound holds uniformly for u θ , so combined with the fact γ < 1 2 , means that we can replace s N +M h N +M (A ρρ ) with s N h N (σ) and the error introduced vanishes as N → ∞. Normalizing both terms in (6.20) by E log Z pert M,N (Q, ε) implies lim inf N →∞ 1 N E log Z pert N ≥ 1 M lim inf N →∞ E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (ω) pert − E log exp √ M y(σ) pert − Lδ. (6.23) When we characterize the limiting distribution of the overlap array, we will require the self overlaps to be constant. Replacing σ with the modified coordinatesσ in the cavity fields achieves this. Starting from (6.23), an interpolation argument will prove that the cavity fields can be replaced with E log Ω ε/2,δ M exp i≤M j≤n ω i (j)z j i (σ) dν n M (ω) pert ,(6.Z j i (σ; t) = √ tz j i (σ) + √ 1 − tz j i (σ), and the corresponding interpolating function ϕ(t) = E log Q ε/2 M exp i≤M j≤n τ i (j)Z j i (σ; t) dλ n M (τ ) pert . LetR , := R(σ ,σ ), a standard integration by parts computation will show |ϕ (t)| ≤ R 1,1 ξ (R 1,1 ) − ξ (R 1,1 ) − R 1,2 ξ (R 1,2 ) − ξ (R 1,2 ) ∞ ≤ n 2 ξ (1) R 1,1 −R 1,1 ∞ + R 1,2 −R 1,2 ∞ ≤ Lε 1/4 since R 1,1 −R 1,1 ∞ ≤ϕ(0) ≥ ϕ(1) − sup t∈[0,1] |ϕ (t)| ≥ ϕ(1) − Lε 1/4 . The bound for (6.25) is similar to above, and is proved using the interpolation Y (σ; t) = √ ty(σ) + √ 1 − ty(σ). Applying the bounds (6.24) and (6.25) to (6.23) finishes the proof. Remark: We assumed h = 0 in the computations above to simplify notation. If h was non-zero, then the lower bound (6.18) The bound (6.27) follows by a simple modification of the above proof. The external field can be decoupled into its cavity and non-cavity coordinates immediately, The first summation appears in the Gibbs average (6.28) and the second summation appears in the cavity field term (6.26). However, the external field in the exponent of (5.28) will appear as i≤N j≤n h(j)a j M +1 σ i (j) + i≤M j≤n h(j)a j i ω i (j) (6.30) Consequences of the perturbation: The lower bound (6.27) is a continuous functional of the distribution of the modified arrays (R(σ ,σ )) , ≥1 under the Gibbs average E(G pert N ) ⊗∞ [23, Lemma 8], so it suffices to study the distribution of the modified array. To this end, we state matrix version of the Ghirlanda-Guerra identities and several of its consequences. These are identical to [22,Section 5] and can now be applied in this setting with no modification. The entries of the overlaps are in [−1, 1], so the probability distributions on finite dimensional subsets of the infinite array are tight. Therefore, by the selection theorem, there exists a subsequence such that all finite dimensional distributions of (R(σ ,σ )) , ≥1 converge weakly. Furthermore, there exists a non-random sequence of parameters (u N θ ) (see [22,Lemma 5] and [20, Lemma 3.3]), possibly changing in N , such that the limiting array, denoted by (R , ) , ≥1 also satisfies a matrix version of the Ghirlanda-Guerra identities. Consider k replica of this limiting array,R k = (R , ) , ≤k , we have: Ef (R k )C 1,k+1 = 1 k Ef (R k )EC 1,2 + 1 k k =2 Ef (R k )C 1, ,(6.31) where C , = ϕ (R p , ν 1 , ν 1 ), . . . , (R p , ν m , ν m ) . (6.32) We have two main consequences of Lemma 6.4. If we take ν i = e i the standard basis vectors in R n , (6.31) implies the traces of the overlap array, denoted by (T , ) , ≥1 = (tr(R , )) , ≥1 , satisfy the usual Ghirlanda-Guerra identities, Ef (T k )g(T 1,k+1 ) = 1 k Ef (T k )Eg(T 1,2 ) + 1 k k =2 Ef (T k )g(T 1, ), g : R → R. (6.33) where T k = (T , ) , ≤k is a sample of k replicas from the array of traces and g is a measurable function. In particular, we are able to apply all the consequences of the standard Ghirlanda-Guerra identities to (T , ) , ≥1 . Furthermore, (6.31) implies a synchronization property for overlap matrices [22,23]: For each k, we can encode the discrete probability measures with a sequences of parameters x −1 = 0 < x 0 < x 1 < . . . < x r = 1 0 = q 0 < q 1 < . . . < q r = n = tr(Q) (7.3) such that µ k (q) = x p for q p ≤ q < q p+1 . (7.4) Let (v α ) α∈N r be the Ruelle probability cascades corresponding to (7.3). Let (α ) ≥1 be an i.i.d. sample from N r according to the weights (v α ) α∈N r , it follows that the array (T k , ) , ≥1 = (q α ∧α ) , ≥1 also converges to (tr(R M , )) , ≥1 by Theorem 2.13 and Theorem 2.17 in [20]. From here, we use the synchronization mechanism to recover a sequence of monotone paths in Π that describes the distribution of the limiting overlap matrix array (R N,M , Lemma 8]. In summary, by choosing a discretization µ k close enough to µ in L 1 , we can find a corresponding discrete pathπ M := π k encoded by the sequences (7.6) such that lim inf N →∞ F ε N (β, Q) ≥ f 1 M (π M ) − f 2 M (π M ) − Lδ − Lε 1/4 . (7.9) The lower bound holds for all M , so we can take a sub-sequential limit as M → ∞. However, we cannot apply Lemma 4.1 to compute the lower bound, because the pathŝ π M may change in M . To resolve this, notice that by monotonicity of the paths,π M →π along some subsequence [23,Section 7]. Furthermore, there exists a discretizationπ ε of π such that d(π,π ε ) ≤ ε 1/4 . This approximation will introduce at most Lε 1/4 error by the Lipschitz continuity of f 1 M (π) and f 2 M (π), so lim inf N →∞ F ε N (β, Q) ≥ lim inf M →∞ f 1 M (π ε ) − f 2 M (π ε ) − Lδ − Lε 1/4 . These paths are now fixed, so we can now compute its limit as M → ∞. Applying Lemma 4.1 to decouple the constraint on Q asymptotically shows lim inf M →∞ f 1 M (π ε ) ≥ inf Λ 1 2 tr(ΛQ) − n − log |Λ| + (Λ −1 0 h, h) + 0≤k≤r−1 1 x k log |Λ k+1 | |Λ k | , where (Λ k ) 0≤k≤r are defined with respect to the sequences (x k ) −1≤k≤r and (Q k ) 0≤k≤r encoded byπ ε . By the recursive computations (3.15), lim M →∞ f 2 M (π ε ) = 0≤k≤r−1 x k · Sum θ(Q k+1 ) − θ(Q k ) . Taking ε → 0 and consequently δ → 0 removes all the error terms, so we conclude lim ε→0 lim inf N →∞ F ε N (β, Q) ≥ inf Λ,π P β,Q (π, Λ). (7.10) Lemma 3. 2 . 2For all N > 0, there exists a constant L such that ( 3 . 329) EJP 25 (2020), paper 28. Page 9/34 http://www.imstat.org/ejp/ Free energy of multiple systems of spherical spin glasses bound in (3.35) holds for all Λ ∈ L. Applying the bounds (3.35) and (3.15) to (3.12) and taking the infimum over all discrete paths encoded by the monotone sequences ( and denote the functional appearing in (3.16) by F (Λ) := Φ R N n (Λ). Lemma 4. 2 . 2Given a positive semi-definite constraint Q: ΛQ) + F (Λ) = −∞. Lemma 4. 3 . 3For any δ * > 0 and positive definite Q, monotone function Φ V defined in(4.5). For Λ * satisfying (4.9), by considering values near this critical point, we will show there exists a constant c > 0 such that for all half-spaces V in (4.16) or(4.17) show this for V = V − j,j for j = j . The proof for the other cases are similar. For all t ≥ 0 and ω ∈ V − j,j , (Λ * + tB)Q) + F (Λ * + tB) =: U (t). corresponding partition function for a system of size M + N , Z M +N (Q, ε) = Q ε M +N exp H N +M (ρ) dλ n M +N (ρ). H M,N (σ) := j≤n H j M,N (σ) is defined like H N (σ) but with normalization (M + N ) −(p−1)/2 . The covariance of this Hamiltonian is given by fields Z(σ) and Y (σ) in (5.3) and (5.4) are centered Gaussian processes with covariances: M,N be the average with respect to the Gibbs measure, G M,N (dσ) = exp H M,N (σ) dλ n N (σ) Z M,N (Q, H M,N (σ) dλ n N (σ). Lemma 5. 1 . 1Let ν M be the standard normal distribution on R M . There exists a constant L such that for any ε > 0 and M ≥ 1, there exists a δ ∈ (0, ε) such that (5.13) is bounded below by pointwise limit of (5.17) converges to the standard normal distribution on R M dν M (x) = f M (x) dx, corresponding map for ψ : S N × A M,N → S M +N given by ψ(σ, ω) = σ 1 a M +1 (ω), . . . , σ N a M +1 (ω), ω 1 a 1 (ω), . . . , ω M a M (ω) for σ ∈ S N and ω ∈ A M,N . The surface measure on S M +N can be decoupled as follows: Lemma 5.2. [4, Lemma 3] Suppose g is a nonnegative function defined on S M +N . Then for ρ = (σ, ω) ∈ S M +N we have j≤n = a j M +1 σ 1 (j), . . . , a j M +1 σ N (j), a j 1 ω 1 (j), . . . , a j M ω M (j) j≤n , to represent the transformation applied coordinate-wise. The following result explains how the surface measure on S M +N decouples asymptotically. Corollary 5.3. Suppose g is a nonnegative function defined on S n M +N . Then for ρ = (σ, ω) . 4 . 4split the integral over Q ε M +N into a product set over Q ε N × Ω For any ε > 0 and N sufficiently large, there exists a δ ∈ (0, ε) such that . The overlaps of the transformed coordinatesρ defined in (5.20) satisfy R(ρ,ρ) = N M + N R(σ,σ) + M M + N R(ω,ω). and (5.27) imply for N sufficiently large 5.4 to (5.22) and taking g(ρ) = exp H M +N (ρ), we have for N sufficiently large E log Z M +N (Q, ε) ≥ E log +N σ,ω dν n M,N (ω)dλ n N (σ). (5.28) Lemma 5.1 are continuous functionals of the distributions of the overlap array (R , ) , ≥1 under the Gibbs measure E(G M,N ) ⊗∞ [23, Theorem 1.3]. Before Lemma 6. 1 . 1[22, Lemma 4] Let ε < λ min (Q). For each positive definite matrix R such that R − Q ∞ ≤ ε, there exists a positive semidefinite matrix A = A(R) such that ARA T = Q.Furthermore, we have the bounds tr (A − I)R(A − I) We first find a bound on A = A(Q). Since Q is positive definite, by the Cholesky decomposition, there exists an invertible matrix B such that Q = BB T . By (6.1), we have tr (A − I)Q(A − I) T = tr (A − I)BB T (A − I) T = (A − I)B 2 F ≤ L √ ε. we denote the sequence s N = N γ . Recall the modified coordinates defined in the previous section denoted withσ = (A σ σ i ) i≤N . We define the perturbed Hamiltonian H pert N (σ) = H N (σ) + s N h N (σ), N → 0, a straightforward Gaussian interpolation argument shows lim inf N →∞ pert is the average on Q ε N with respect to the Gibbs measure with external field,G N (dσ) ∝ exp H pert M,N (σ) + i≤N j≤n h(j)σ i (j) dλ n N (σ). Lemma 6.4. [22, Theorem 3] Given any measurable function ϕ : R m → R and f = f (R k ), the array satisfies the Ghirlanda-Guerra identities 34 ) 34where g 1 , g 2 , . . . are i.i.d. standard normals. The other terms vanish by taking ε → 0, socombining (3.30) and (3.34) with (3.28), gives the bound lim ε→0 lim sup N →∞ is a compact set so lim N →∞ a (ω) = 1 uniformly for all 1 < ≤ M .5.26) Secondly, Ω ε/2,δ M Therefore, lim N →∞ ) .(6.19) Since the s N h N (σ) terms in the perturbed Hamiltonian are independent with all other Gaussian processes, we can leave the s N h N terms untouched by the interpolations in the proof of Lemma 5.1. The exact same computations imply that Lemma 5.1 can be applied in this setting to conclude that (6.19) is bounded below by at the cost of Lε 1/4 error. We only prove (6.24) because the proof of (6.25) is almost identical.Consider the Hamiltonian,24) and E log exp √ M y(σ) pert (6.25) Lε 1/4 by Lemma 6.1 and (6.6). Integrating the quantity above implies http://www.imstat.org/ejp/ ω i (j)z j i (σ) dλ n N (σ).We bound each of the terms in (5.30) separately.Step 1: We show the first term in(We now use Gaussian interpolation to control the term on the right hand side.Recall the Gaussian fields in (5.3) and(5.9). We define the interpolating Hamiltonian, using the corresponding transformed coordinates (5.21), the Gaussian processes in (5.33) are given bybe the corresponding interpolating Hamiltonian. By Gaussian integration by parts,where · t is the average with respect to the Gibbs measure on Q ε N × Ω ε/2,δ M proportional to exp(H t ) with respect to the reference measure λ n N × ν n M,N . We now compute the covariances of the cavity fields in(5.33).For all ≤ M + 1 and j ≤ n, a j (ω) → 1 uniformly on Ω ε/2,δ M by compactness. The leading terms of (5.6) and (5.8) do not grow in N , so by continuity the terms (5.35) and (5.36) in (5.38) vanish in the limit.We now compute the covariances containing H j 1,t . The covariance of (5.5) is order N , so it is not obvious that the differences of the covariances are small. We resolve this by using identity (5.25) and applying the mean value theorem. If we let a j i (ω 1 ) := a i (ω 1 (j)),The mean value theorem impliesfinishing the bound of (5.31).Step 2: We show the second term in (5.30) satisfiesThis proof is identical to the proof of Lemma 5 in[4].The key observationTo simplify notation, let f n M (ω) := j≤n f M (ω(j)) and f n M,N (ω) := j≤n f M,N (ω(j)). almost surely. Recall that Φ is non-decreasing and Lipschitz. This allows us to approximate its distribution with a random measure generated by the Ruelle probability cascades.Jensen's inequality impliesWe begin by characterizing the array (tr(R M , )) , ≥1 consisting of the traces of the limiting array. As a consequence of the generalized Ghirlanda-Guerra identities (6.33), the array of traces also satisfies the usual Ghirlanda-Guerra identities. We denote the distribution of tr(R M 1,2 ) with µ(q) = P tr(R M 1,2 ) ≤ q . We defineand observe (Q k , ) , ≥1 converges to the distribution of (R M , ) , ≥1 because Φ is Lipschitz. It also follows that the discrete path π k (x) = Q k for x k−1 < x ≤ x k for 0 ≤ k ≤ r, π(0) = 0, π(1) = Q.(7.5)induced byIn particular, we have d(π, π k ) → 0 as µ k → µ in L 1 .Recall the Gaussian processes Z j i (α) and Y (α) defined in the (3.5) and (3.6) andconsider the following functionals of the discrete paths associated with the approximating arrays (Q α ∧α ) , ≥1 : The covariances of Z j i (α) and z j i (σ), and Y (α) and y(σ) are given by the same functions of arrays so the difference of the functionals (7.7), (7.8) and the functional appearing in (7.1) can be approximated by the same continuous bounded function of the array[23, . 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We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented triangles, investigated recently in [17] and [18]. Moreover, we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in [17] and[19].

10.1007/s10801-018-0867-6

1706.07693

0012033e267e4cd6de61a734bdffbe420b6a5732

FROM BRAUER GRAPH ALGEBRAS TO BISERIAL WEIGHTED SURFACE ALGEBRAS 10 Apr 2018 Karin Erdmann Andrzej Skowroński FROM BRAUER GRAPH ALGEBRAS TO BISERIAL WEIGHTED SURFACE ALGEBRAS 10 Apr 2018arXiv:1706.07693v3 [math.RT]Brauer graph algebraWeighted surface algebraBiserial weighted surface algebraSymmetric algebraSpecial biserial algebraTame algebraPeriodic algebraQuiver combinatorics 2010 MSC: 05E9916G2016G7020C20 We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented triangles, investigated recently in [17] and [18]. Moreover, we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in [17] and[19]. Introduction and the main results Throughout this paper, K will denote a fixed algebraically closed field. By an algebra we mean an associative, finite-dimensional K-algebra with an identity. For an algebra A, we denote by mod A the category of finite-dimensional right Amodules and by D the standard duality Hom K (−, K) on mod A. An algebra A is called self-injective if A A is an injective module, or equivalently, the projective modules in mod A are injective. Two self-injective algebras A and B are said to be socle equivalent if the quotient algebras A/ soc(A) and B/ soc(B) are isomorphic. Symmetric algebras are an important class of self-injective algebras. An algebra A is symmetric if there exists an associative, non-degenerate, symmetric, K-bilinear form (−, −) : A × A → K. Classical examples of symmetric algebras include in particular, blocks of group algebras of finite groups and Hecke algebras of finite Coxeter groups. In fact, any algebra A is the quotient algebra of its trivial extension algebra T(A) = A ⋉ D(A), which is a symmetric algebra. By general theory, if e is an idempotent of a symmetric algebra A, then the idempotent algebra eAe also is a symmetric algebra. Brauer graph algebras play a prominent role in the representation theory of tame symmetric algebras. Originally, R. Brauer introduced the Brauer tree, which led to the description of blocks of group algebras of finite groups of finite representation type, and they are the basis for their classification up to Morita equivalence [10,25,29], see also [2]. Relaxing the condition on the characteristic of the field, one gets Brauer tree algebras, and these occurred in the Morita equivalence classification of symmetric algebras of Dynkin type A n [22,35]. If one allows arbitrary multiplicities, and also an arbitrary graph instead of just a tree, one obtains Brauer graph algebras. These occurred in the classification of symmetric algebras of Euclidean type A n [7]. It was shown in [36] (see also [37]) that the class of Brauer graph algebras coincides with the class of symmetric special biserial algebras. Symmetric special biserial algebras occurred also in the Gelfand-Ponomarev classification of singular Harish-Chandra modules over the Lorentz group [23], and as well in the context of restricted Lie algebras, or more generally infinitesimal group schemes, [20,21], and in classifications of tame Hecke algebras [3, 4,14]. There are also results on derived equivalence classifications of Brauer graph algebras, and on the connection to Jacobian algebras of quivers with potential, we refer to [1,11,26,31,32,34,37]. We recall the definition of a Brauer graph algebra, following [36], see also [37]. A Brauer graph is a finite connected graph Γ, with at least one edge (possibly with loops and multiple edges) such that for each vertex v of Γ, there is a cyclic ordering of the edges adjacent to v, and there is a multiplicity e(v) which is a positive integer. Given a Brauer graph Γ, one defines the associated Brauer quiver Q Γ as follows: • the vertices Q Γ are the edges of Γ; • there is an arrow i → j in Q Γ if and only if j is the consecutive edge of i in the cyclic ordering of edges adjacent to a vertex v of Γ. In this case we say that the arrow i → j is attached to v. The quiver Q Γ is 2regular (see Section 2). Recall that a quiver is 2-regular if every vertex is the source and target of exactly two arrows. Any 2-regular quiver has a canonical involution (−) on the arrows, namely if α is an arrow theᾱ is the other arrow starting at the same vertex as α. The associated Brauer graph algebra B Γ is a quotient algebra of KQ Γ . The cyclic ordering of the edges adjacent to a vertex v of Γ translates to a cyclic permutation of the arrows in Q Γ , and if α is an arrow in this cycle, we denote vertex v by v(α). Let C α be the product of the arrows in the cycle, in the given order, starting with α, this is an element in KQ Γ . The associated Brauer graph algebra B Γ is defined to be KQ Γ /I Γ , where I Γ is the ideal in the path algebra KQ Γ generated by the elements: (1) all paths αβ of length 2 in Q Γ which are not subpaths of C α , (2) C e(v(α)) α − C e(v(ᾱ)) α , for all arrows α of Q Γ . In [17] and [18] we introduced and studied biserial weighted surface algebras, motivated by tame blocks of group algebras of finite groups. Given a triangulation T of a 2-dimensional real compact surface, with or without boundary, and an orientation #‰ T of triangles in T , there is a natural way to define a quiver Q(S, #‰ T ). We showed that these quivers have an algebraic description: they are precisely what we called triangulation quivers. A triangulation quiver is a pair (Q, f ) where Q is a 2-regular quiver, and f is a permutation of arrows of order 3 such that t(α) = s(f (α)) for each arrow α of Q. A biserial weighted surface algebra B(S, #‰ T , m • ) is then explicitly given by the quiver Q(S, #‰ T ) and relations, depending on a weight function m • , and if described using the triangulation quiver, we get a biserial weighted triangulation algebra B(Q, f, m • ) (see Section 2). Algebras of generalized dihedral type (see [18,Theorem 1]) which contain blocks with dihedral defect groups, turned out to be (up to socle deformation) idempotent algebras of biserial weighted surface algebras, for very specific idempotents. Biserial weighted surface algebras belong to the class of Brauer graph algebras. It is therefore a natural question to ask which other Brauer graph algebras occur as idempotent algebras of biserial weighted surface algebras. This is answered by our first main result. Theorem 1. Let A be a basic, indecomposable, finite-dimensional K-algebra over an algebraically closed field K of dimension at least 2. Then the following statements are equivalent: (i) A is a Brauer graph algebra. (ii) A is isomorphic to the idempotent algebra eBe for a biserial weighted surface algebra B and an idempotent e of B. The main ingredient for this is Theorem 4.1. This gives a canonical construction, which we call * -construction. A byproduct of the proof of Theorem 1 is the following fact. Corollary 2. Let A be a Brauer graph algebra over an algebraically closed field K. Then A is isomorphic to the idempotent algebra eBe of a biserial weighted surface algebra B = B(S, #‰ T , m • ), for a surface S without boundary, a triangulation T of S without self-folded triangles, and an idempotent e of B. Moreover, we can adapt the * -construction to algebras socle equivalent to Brauer graph algebras, and prove an analog for the main part of Theorem 1: Theorem 3. Let A be a symmetric algebra over an algebraically closed field K which is socle equivalent but not isomorphic to a Brauer graph algebra, and assume the Grothendieck group K 0 (A) has rank at least 2. Then (i) char(K) = 2, and (ii) A is isomorphic to an idempotent algebraēBē, whereB is a socle deformed biserial weighted surface algebraB = B(S, #‰ T , m • , b • ). Here S is a surface with boundary, T is a triangulation of S without self-folded triangles, and b • is a border function. Recall that an algebra A is called periodic if it is periodic with respect to action of the syzygy operator Ω A e in the module category mod A e , where A e = A op ⊗ K A is its enveloping algebra. If A is a periodic algebra of period n then all indecomposable non-projective right A-modules are periodic of period dividing n, with respect to the syzygy operator Ω A in mod A. Periodic algebras are self-injective, and have connections with group theory, topology, singularity theory and cluster algebras. In [17] and [19] we introduced and studied weighted surface algebras Λ(S, #‰ T , m • , c • ), which are tame, symmetric, and we showed that they are (with one exception) periodic algebras of period 4. They are defined by the quiver Q(S, #‰ T ) and explicitly given relations, depending on a weight function m • and a parameter function c • (see Section 6). Most biserial weighted surface algebras occur as geometric degenerations of these periodic weighted surface algebras. Our third main result connects Brauer graph algebras with a large class of periodic weighted surface algebras. Theorem 4. Let A be a Brauer graph algebra over an algebraically closed field K. Then A is isomorphic to an idempotent algebra eΛe of a periodic weighted surface algebra Λ = Λ(S, #‰ T , m • , c • ), for a surface S without boundary, a triangulation T of S without self-folded triangles, and an idempotent e of Λ. There are many idempotent algebras of weighted surface algebras which are neither Brauer graph algebras nor periodic algebras. We give an example at the end of Section 6. This paper is organized as follows. In Section 2 we recall basic facts on special biserial algebras and show that Brauer graph algebras, symmetric special biserial algebras, and symmetric algebras associated to weighted biserial quivers are essentially the same. In Section 3 we introduce biserial weighted surface algebras and present their basic properties. In Section 4 we prove Theorem 1. This contains an algorithmic construction which may be of independent interest. Sections 5 and 6 contain the proofs of of Theorems 3 and 4, and related material. In the final Section 7 we present a diagram showing the relations between the main classes of algebras occurring in the paper. For general background on the relevant representation theory we refer to the books [5,13,38,40], and we refer to [13,15] for the representation theory of arbitrary self-injective special biserial algebras. Special biserial algebras A quiver is a quadruple Q = (Q 0 , Q 1 , s, t) consisting of a finite set Q 0 of vertices, a finite set Q 1 of arrows, and two maps s, t : Q 1 → Q 0 which associate to each arrow α ∈ Q 1 its source s(α) ∈ Q 0 and its target t(α) ∈ Q 0 . We denote by KQ the path algebra of Q over K whose underlying K-vector space has as its basis the set of all paths in Q of length ≥ 0, and by R Q the arrow ideal of KQ generated by all paths in Q of length ≥ 1. An ideal I in KQ is said to be admissible if there exists m ≥ 2 such that R m Q ⊆ I ⊆ R 2 Q . If I is an admissible ideal in KQ, then the quotient algebra KQ/I is called a bound quiver algebra, and is a finite-dimensional basic K-algebra. Moreover, KQ/I is indecomposable if and only if Q is connected. Every basic, indecomposable, finite-dimensional K-algebra A has a bound quiver presentation A ∼ = KQ/I, where Q = Q A is the Gabriel quiver of A and I is an admissible ideal in KQ. For a bound quiver algebra A = KQ/I, we denote by e i , i ∈ Q 0 , the associated complete set of pairwise orthogonal primitive idempotents of A. Then the modules S i = e i A/e i rad A (respectively, P i = e i A), i ∈ Q 0 , form a complete family of pairwise non-isomorphic simple modules (respectively, indecomposable projective modules) in mod A. Following [39], an algebra A is said to be special biserial if A is isomorphic to a bound quiver algebra KQ/I, where the bound quiver (Q, I) satisfies the following conditions: (a) each vertex of Q is a source and target of at most two arrows, (b) for any arrow α in Q there are at most one arrow β and at most one arrow γ with αβ / ∈ I and γα / ∈ I. Background on special biserial algebras may be found for example in [8,13,33,39,41]. Perhaps most important is the following, which has been proved by Wald and Waschbüsch in [41] (see also [8,12] for alternative proofs). Proposition 2.1. Every special biserial algebra is tame. If a special biserial algebra is in addition symmetric, there is a more convenient description. We propose the concept of a (weighted ) biserial quiver algebra, which we will now define. Later, in Theorem 2.6 we will show that these algebras are precisely special biserial symmetric algebras. Definition 2.2. A biserial quiver is a pair (Q, f ), where Q = (Q 0 , Q 1 , s, t) is a finite connected quiver and f : Q 1 → Q 1 is a permutation of the arrows of Q satisfying the following conditions: (a) Q is 2-regular, that is every vertex of Q is the source and target of exactly two arrows, (b) for each arrow α ∈ Q 1 we have s(f (α)) = t(α). Let (Q, f ) be a biserial quiver. We obtain another permutation g : Q 1 → Q 1 defined by g(α) = f (α) for any α ∈ Q 1 , so that f (α) and g(α) are the arrows starting at t(α). Let O(α) be the g-orbit of an arrow α, and set n α = n O(α) = |O(α)|. We denote by O(g) the set of all g-orbits in Q 1 . A function m • : O(g) → N * = N \ {0} is said to be a weight function of (Q, f ). We write briefly m α = m O(α) for α ∈ Q 1 . The multiplicity function m • taking only value 1 is said to be trivial. For any arrow α ∈ Q 1 , we single out the oriented cycle B α = αg(α) . . . g nα−1 (α) mα of length m α n α . The triple (Q, f, m • ) is said to be a (weighted ) biserial quiver. The associated biserial quiver algebra B = B(Q, f, m • ) is defined as follows. It is the quotient algebra B(Q, f, m • ) = KQ/J(Q, f, m • ), where J(Q, f, m • ) is the ideal of the path algebra KQ generated by the following elements: (1) αf (α), for all arrows α ∈ Q 1 , (2) B α − Bᾱ, for all arrows α ∈ Q 1 . We assume that Q is not the quiver with one vertex and two loops α andᾱ such that α = B α and Bᾱ =ᾱ are equal in B, that is we exclude the 2-dimensional algebra isomorphic to K[X]/(X 2 ). Assume m α n α = 1, so that α = B α and Bᾱ are equal in B. By the above assumption, Bᾱ lies in the square of the radical of the algebra. Then α is not an arrow in the Gabriel quiver Q B of B, and we call it a virtual loop. The following describes basic properties of (weighted) biserial quiver algebras. Then B is a basic, indecomposable, finite-dimensional symmetric special biserial algebra with dim K B = O∈O(g) m O n 2 O . Proof. It follows from the definition that B is the special biserial bound quiver algebra KQ B /I B , where Q B is obtained from Q by removing all virtual loops, and where I B = J(Q, f, m • ) ∩ KQ B . Let i be a vertex of Q and α,ᾱ the two arrows starting at i. Then the indecomposable projective B-module P i = e i B has a basis given by e i together with all initial proper subwords of B α and Bᾱ, and and B α (= Bᾱ), and hence dim K P i = m α n α + mᾱnᾱ. Note also that the union of these bases gives a basis of B consisting of paths in Q. We deduce that dim K B = O∈O(g) m O n 2 O . As well, the indecomposable projective module P i has simple socle generated by B α (= Bᾱ). We define a symmetrizing K-linear form ϕ : B → K as follows. If u is a path in Q which belongs to the above basis, we set ϕ(u) = 1 if u = B α for an arrow α ∈ Q 1 , and ϕ(u) = 0 otherwise. Then ϕ(ab) = ϕ(ba) for all elements a, b ∈ B and Ker ϕ does not contain any nonzero one-sided ideal of B, and consequently B is a symmetric algebra (see [40, Theorem IV.2.2]). We wish to compare Brauer graph algebras and biserial quiver algebras. For this we start analyzing the combinatorial data. Let Q be a connected 2-regular quiver. We call a permutation g of the arrows of Q admissible if for every arrow α we have t(α) = s(g(α)). That is, the arrows along a cycle of g can be concatenated in Q. The multiplicity function of a Brauer graph Γ taking only value 1 is said to be trivial. Lemma 2.4. There is a bijection between Brauer graphs Γ with trivial multiplicity function and pairs (Q, g) where Q is connected 2-regular, and g is an admissible permutation of the arrows of Q. Proof. (1) Given Γ, we take the quiver Q = Q Γ , as defined in the introduction. (1a) We show that Q Γ is 2-regular. Take an edge i of Γ, it is adjacent to vertices v, w (which may be equal). If v = w then the edge i occurs both in the cyclic ordering around v and of w, so there are two arrows starting at i, and there are two arrows ending at i. If v = w then the edge i occurs twice in the cyclic ordering of edges adjacent to v, so again there are two arrows starting at i and two arrows ending at i. (1b) We define an (admissible) permutation g on the arrows. Given α : i → j, let v be the vertex such that α is attached to v, then there is a unique edge k adjacent to v such that i, j, k are consecutive edges in the ordering around v, and hence a unique arrow β : j → k, also 'attached' to v, and we set g(α) := β. This defines an admissible permutation on the arrows. Writing g as a product of disjoint cycles, gives a bijection between the cycles of g and the vertices of Γ. Namely, let the cycle of g correspond to v if it consists of the arrows attached to v. (2) Suppose we are given a connected 2-regular quiver Q and an admissible permutation g, written as a product of disjoint cycles. Define a graph Γ with vertices the cycles of g, and edges the vertices of Q. Each cycle of g defines a cyclic ordering of the edges adjacent to the vertex corresponding to this cycle. Hence we get a Brauer graph. (3) It is clear that these give a bijection. Remark 2.5. In part (1b) of the above proof, we may have i = j. There are two such cases. If the edge i is adjacent to two distinct vertices of Γ then i is the only edge adjacent to a vertex v and we have g(α) = α. We call α an external loop. Otherwise the edge i is a loop of Γ, and then g(α) = α. In this case the cycle of g passes twice through vertex i of the quiver. We call α an internal loop. The Brauer graph Γ comes with a multiplicity function e defined on the vertices. Given (Q, g), we take the same multiplicity function, defined on the cycles of g, which gives the function m • which we have called a weight function. The permutation g determines the permutation f of the arrows where f (α) = g(α) for any arrow α. Clearly f is also admissible, and f and g determine each other. We have seen that the combinatorial data for B Γ are the same as the combinatorial data for B(Q, f, m • ). Therefore B Γ is in fact equal to B(Q, f, m • ). In the definition of a biserial quiver we focus on (Q, f ), this is motivated by the connection to biserial weighted surface algebras, which we will define later. The following compares various algebras. The equivalence of the statements (i) and (iii) was already obtained by Roggenkamp in [36, Sections 2 and 3] (see also [1,Proposition 1.2] and [37, Theorem 1.1]). We include it, for completeness. Theorem 2.6. Let A be a basic, indecomposable algebra of dimension at least 2, over an algebraically closed field K. The following are equivalent: (i) A is a Brauer graph algebra. (ii) A is isomorphic to an algebra B(Q, f, m • ) where (Q, f, m • ) is a (weighted) biserial quiver. (iii) A is a symmetric special biserial algebra. Proof. As we have just seen, (i) and (ii) are equivalent. The implication (ii) ⇒ (iii) follows from Proposition 2.3. We prove now (iii) ⇒ (ii). Assume that A is a basic symmetric special biserial algebra, let A = KQ A /I where Q A is the Gabriel quiver of A. We will define a (weighted) biserial quiver (Q, f, m • ) and show that A is isomorphic to B(Q, f, m • ). Since A is special biserial, for each vertex i of Q A , we have |s −1 (i)| ≤ 2 and |t −1 (i)| ≤ 2. The algebra A is symmetric, therefore for each vertex i ∈ Q 0 , we have |s −1 (i)| = |t −1 (i)|: Namely, if |s −1 (i)| = 1 then by the special biserial relations, the projective module e i A is uniserial. It is isomorphic to the injective hull of the simple module S i , and hence |t −1 (i)| = 1. If |t −1 (i)| = 1 then by the same reasoning, applied to D( Ae i ) ∼ = e i A it follows that |s −1 (i)| = 1. Let ∆ := {i ∈ (Q A ) 0 | |s −1 (i)| = 1} , to each i ∈ ∆ we adjoin a loop η i at i to the quiver Q A , which then gives a 2-regular quiver. Explicitly, let Q : = (Q 0 , Q 1 , s, t) with Q 0 = (Q A ) 0 and Q 1 is the disjoint union (Q A ) 1 {η i : i ∈ ∆}. We define a permutation f of Q 1 . For each i ∈ ∆, there are unique arrows α i and β i in Q A with t(α i ) = i = s(β i ), and we set f (α i ) = η i and f (η i ) = β i . If α is any arrow of Q A with t(α) not in ∆, we define f (α) to be the unique arrow in (Q A ) 1 with αf (α) ∈ I. With this, (Q, f ) is a biserial quiver. We define now a weight function m • : O(g) → N * , where g =f . For each j ∈ ∆, we have g(η j ) = η j , and we set m O(ηj) = 1. Let α be some arrow of Q A starting at vertex i, and let n α = |O(α)|. Since A is symmetric special biserial, there exists m α ∈ N * such that B α := αg(α) . . . g nα−1 (α) mα is a maximal cyclic path in Q A which does not belong to I, and spans the socle of the indecomposable projective module e i A. The integer m α is constant on the g-orbit of α and we may define m O(α) = m α . It remains to show that by suitable scaling of arrows one obtains the stated relations involving paths B α . Fix a symmetrizing linear form ϕ for A. Fix an orbit of g, say O(ν), there is a non-zero scalar d ν such that for all arrows α in this orbit we have ϕ(B α ) = d ν . We may assume d ν = 1. Namely, we can choose in O(ν) an arrow, α say, and replace it by λα where λ mα = d −1 ν . The cycles are disjoint, and if we do this for each cycle then we have ϕ(B α ) = 1 for all arrows α. Let i be a vertex of Q A with |s −1 (i)| = 2, and let α,ᾱ be the two arrows starting at i. Then there are non-zero scalars c α and cᾱ such that c α B α = cᾱBᾱ in A. Then we have c α = c α ϕ(B α ) = ϕ(c α B α ) = ϕ(cᾱBᾱ) = cᾱϕ(Bᾱ) = cᾱ. Hence we can cancel these scalars and obtain the required relations. With this, there is a canonical isomorphism of K-algebras A = KQ A /I → B(Q, f, m • ). We will from now suppress the word 'weighted', in analogy to the convention for Brauer graph algebras, where the multiplicity function is part of the definition but is not explicitly mentioned. We will study idempotent algebras, and it is important that any idempotent algebra of a special biserial symmetric algebra is again special biserial symmetric. Proposition 2.7. Let A be a symmetric special biserial algebra. Assume e is an idempotent of A which is a sum of some of the e i associated to vertices of Q A . Then eAe also is a symmetric special biserial algebra. Proof. We may assume that A = B(Q, f, m • ) for a weighted biserial quiver (Q, f, m • ) and eAe is indecomposable, and let Q = (Q 0 , Q 1 , s, t). We will show that eAe = (Q,f ,m • ) = KQ/J(Q,f ,m • ) for a weighted biserial quiver (Q,f ,m • ). We definẽ Q 0 to be the set of all vertices i ∈ Q 0 such that e is the sum of the primitive idempotents e i . For each arrow α ∈ Q 1 with s(α) ∈Q 0 , we denote byα the shortest path in Q of the form αg(α) . . . g p (α) with p ∈ {0, 1, . . . , n α − 1} and t(g p (α)) ∈Q 0 . Such a path exists because αg(α) . . . g nα−1 (α) is a cycle around vertex s(α) = t(g nα−1 (α)) inQ 0 . Then we defineQ 1 to be set of pathsα in Q for all arrows α ∈ Q 1 with s(α) ∈Q 0 . Moreover, forα = αg(α) . . . g p (α), we set s(α) = s(α) andt(α) = t(g p (α)). This defines a 2-regular quiverQ = (Q 0 ,Q 1 ,s,t). Further, for each arrowα = αg(α) . . . g p (α) inQ 1 , there is exactly one arrow β = βg(β) . . . g r (β) inQ 1 such thatt(α) = t(g p (α)) = s(β) =s(β) and f (α) = β, and we setf (α) =β. This defines a biserial quiver (Q,f ). Letg be the permutation ofQ 1 associated tof , and O(g) the set ofg-orbits inQ 1 . Then we define the weight functionm • : O(g) → N * of (Q,f ) by settingm O(α) = m O(α) for each arrow α ∈Q 1 . With these, the biserial quiver algebra B(Q,f ,m • ) = KQ/J(Q,f ,m • ) is isomorphic to eAe. We end this section with an example illustrating Theorem 2.6. This also shows that an idempotent algebra of a Brauer graph algebra need not be indecomposable, by taking e = 1 BΓ − e 4 . Example 2.8. Let Γ be the Brauer graph where we take the clockwise ordering of the edges around each vertex. Then B Γ is the symmetric algebra B(Q, f, m • ) with biserial quiver (Q, f ) • a • b • c • p • d1 β α 8 ϕ G G a 6 6 7 ψ o o ξ G G 6 η o o µ G G 4 δ d d ̺ Ð Ð 3 σ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ d d d 5 ν ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ p h h 2 ω ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ γ d d where the f -orbits are (α ω ̺ p ν µ δ β γ d σ), (η ξ), (a ϕ ψ). Then the g-orbits are O(a) = (a), O(d) = (d), O(p) = (p), O(α) = (α γ σ β ω δ), O(̺) = (̺ ν η ψ ϕ ξ µ). The weight function m • : O(g) → N * is as before given by the multiplicity function of the Brauer graph Γ. We note that C α = αγσβωδ and Cᾱ = C β = βωδαγσ, and v(α) = c = v(β). Biserial weighted surface algebras In this section we introduce biserial weighted surface algebras and describe their basic properties. In this paper, by a surface we mean a connected, compact, 2-dimensional real manifold S, orientable or non-orientable, with boundary or without boundary. It is well known that every surface S admits an additional structure of a finite 2dimensional triangular cell complex, and hence a triangulation (by the deep Triangulation Theorem (see for example [9, Section 2.3])). For a positive natural number n, we denote by D n the unit disk in the ndimensional Euclidean space R n , formed by all points of distance ≤ 1 from the origin. Then the boundary ∂D n of D n is the unit sphere S n−1 in R n , formed by all points of distance 1 from the origin. Further, by an n-cell we mean a topological space homeomorphic to the open disk int D n = D n \ ∂D n . In particular, S 0 = ∂D 1 consists of two points. Moreover, we define D 0 = int D 0 to be a point. We refer to [24,Appendix] for some basic topological facts about cell complexes. Let S be a surface. In the paper, by a finite 2-dimensional triangular cell complex structure on S we mean a finite family of continuous maps ϕ n i : D n i → S, with n ∈ {0, 1, 2} and D n i = D n , satisfying the following conditions: (1) Each ϕ n i restricts to a homeomorphism from int D n i to the n-cell e n i = ϕ n i (int D n i ) of S, and these cells are all disjoint and their union is S. = a b • • a, b, c pairwise different a, b different (self-folded triangle) such that every edge of such a triangle in T is either the edge of exactly two triangles, is the self-folded edge, or lies on the boundary. We note that a given surface S admits many finite 2-dimensional triangular cell complex structures, and hence triangulations. We refer to [9,27,28] for general background on surfaces and constructions of surfaces from plane models. Let S be a surface and T a triangulation S. To each triangle ∆ in T we may associate an orientation (1) for any oriented triangle ∆ = (abc) in #‰ T with pairwise different edges a, b, c, we have the cycle a G G b Ö Ö ✌ ✌ ✌ ✌ ✌ ✌ c ✶ ✶ ✶ ✶ ✶ ✶ ,(2) for any self-folded triangle ∆ = (aab) in #‰ T , we have the quiver a 6 6 9 9 b g g , (3) for any boundary edge a in T , we have the loop a 6 6 . Then Q = Q(S, #‰ T ) is a triangulation quiver in the following sense (introduced independently by Ladkani in [30]). A triangulation quiver is a pair (Q, f ), where Q = (Q 0 , Q 1 , s, t) is a finite connected quiver and f : Q 1 → Q 1 is a permutation on the set Q 1 of arrows of Q satisfying the following conditions: (a) every vertex i ∈ Q 0 is the source and target of exactly two arrows in Q 1 , (b) for each arrow α ∈ Q 1 , we have s(f (α)) = t(α), (c) f 3 is the identity on Q 1 . Hence, a triangulation quiver (Q, f ) is a biserial quiver (Q, f ) such that f 3 is the identity. For the quiver Q = Q(S, #‰ T ) of a directed triangulated surface (S, #‰ T ), the permutation f on its set of arrows is defined as follows: (1) a α G G b β Ö Ö ✌ ✌ ✌ ✌ ✌ ✌ c γ ✶ ✶ ✶ ✶ ✶ ✶ f (α) = β, f (β) = γ, f (γ) = α, for an oriented triangle ∆ = (abc) in #‰ T , with pairwise different edges a, b, c, (2) a α 6 6 β 9 9 b γ g g f (α) = β, f (β) = γ, f (γ) = α, for a self-folded triangle ∆ = (aab) in #‰ T , (3) a α 6 6 f (α) = α, for a boundary edge a of T . If (Q, f ) is a triangulation quiver, then the quiver Q is 2-regular. We will consider only the triangulation quivers with at least two vertices. Note that different directed triangulated surfaces (even of different genus) may lead to the same triangulation quiver (see [17,Example 4.4]). Let (Q, f ) be a triangulation quiver. Then we have the involution¯: Q 1 → Q 1 which assigns to an arrow α ∈ Q 1 the arrowᾱ with s(α) = s(ᾱ) and α =ᾱ. Then we obtain another permutation g : Q 1 → Q 1 of the set Q 1 of arrows of Q such that g(α) = f (α) for any α ∈ Q 1 . We denote by O(g) the set of all g-orbits in Q 1 . The following theorem have been established in [17,Theorem 4.11] (see also [18,Example 8.2] for the case with two vertices). Theorem 3.1. Let (Q, f ) be a triangulation quiver with at least two vertices. Then there exists a directed triangulated surface (S, #‰ T ) such that (Q, f ) = (Q(S, #‰ T ), f ). We present now an example of a triangulation quiver which is not of the form (Q(S, #‰ T ), f ), where S is an oriented surface, T a triangulation of S, and #‰ T the orientation of triangles in T induced from the orientation of S. We observe that the coherent gluing of the oriented triangles T 1 , T 2 , T 3 creates the upper part of the above tetrahedron. But then it is not possible to glue the oriented triangle T 4 in a coherent way with T 1 , T 2 , T 3 . Therefore, the triangulation quiver (Q(T , #‰ T ), f ) has the required property. T , m • ) (see [17] and [18]). Biserial weighted surface algebras belong to the class of algebras of generalized dihedral type, which generalize blocks of group algebras with dihedral defect groups. They are introduced and studied in [18]. We end this section by giving two examples of biserial weighted surface algebras. 1 α1 G G β1 G G 2 α2 Ô Ô ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ β2 Ô Ô ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ ✠ 3 α3 ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ β3 ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ ✺ with f -orbits (α 1 α 2 α 3 ) and (β 1 β 2 β 3 ). Then g has only one orbit which is (α 1 β 2 α 3 β 1 α 2 β 3 ), and hence a weight function m • : O(g) → N * is given by a positive integer m. Then the associated biserial weighted surface algebra B(T, #‰ T , m • ) is given by the above quiver and the relations: α 1 α 2 = 0, β 1 β 2 = 0, (α 1 β 2 α 3 β 1 α 2 β 3 ) m = (β 1 α 2 β 3 α 1 β 2 α 3 ) m , α 2 α 3 = 0, β 2 β 3 = 0, (α 2 β 3 α 1 β 2 α 3 β 1 ) m = (β 2 α 3 β 1 α 2 β 3 α 1 ) m , α 3 α 1 = 0, β 3 β 1 = 0, (α 3 β 1 α 2 β 3 α 1 β 2 ) m = (β 3 α 1 β 2 α 3 β 1 α 2 ) m . The triangulation quiver (Q(T, #‰ T ), f ) is called the 'Markov quiver' (see [18] for a motivation). Proof of Theorem 1 To prove the implication (ii) ⇒ (i), let B be a biserial weighted surface algebra. Then by Theorem 3.1 we may assume B = B(Q, f, m • ) where (Q, f ) is a biserial quiver and f 3 is the identity. Then in particular B is a biserial quiver algebra, and by Theorem 2.6, we see that B is a Brauer graph algebra. Now it follows from Theorem 2.6 and Proposition 2.7 that also eBe is a Brauer graph algebra, and (i) holds. We consider the implication (i)⇒ (ii). Assume A is a Brauer graph algebra, by Theorem 2.6 we may assume A = B(Q, f, m • ) where (Q, f ) is a biserial quiver. To obtain (ii), we must find a biserial quiver (Q * , f * ) with (f * ) 3 = 1 such that A = e * B * e * where B * = B(Q * , f * , m * • ) and e * an idempotent of B * . The following shows that this can be done in a canonical way, the construction gives an algorithm. Furthermore, applying the construction twice gives an interesting consequence. Proof. Let Q = (Q 0 , Q 1 , s, t), and let g be the permutation of Q 1 associated to f . We define a triangulation quiver (Q * , f * ) as follows. We take Q * = (Q * 0 , Q * 1 , s * , t * ) with Q * 0 := Q 0 ∪ {x α } α∈Q1 , Q * 1 := {α ′ , α ′′ , ε α } α∈Q1 and s * (α ′ ) = s(α), t * (α ′ ) = x α , s * (α ′′ ) = x α , t * (α ′′ ) = t(α), s * (ε α ) = x f (α) , t * (ε α ) = x α . Moreover, we set f * (α ′′ ) = f (α) ′ , f * (f (α) ′ ) = ε α , f * (ε α ) = α ′′ . We observe that (Q * , f * ) is a triangulation quiver. Let g * be the permutation of Q * 1 associated to f * . We notice that, for any arrow α of Q, we have g * (α ′ ) = α ′′ , g * (α ′′ ) = g(α) ′ , and g * (ε α ) = ε f −1 (α) . For each arrow β ∈ Q * 1 , we denote by O * (β) the g * -orbit of β. Then the g * -orbits in Q * 1 are O * (α ′ ) = α ′ α ′′ g(α) ′ g(α) ′′ . . . g nα−1 (α) ′ g nα−1 (α) ′′ , O * (ε α ) = ε f rα−1 (α) ε f rα−2 (α) . . . ε f (α) ε α , for α ∈ Q 1 , where n α is the length of the g-orbit of α and r α is the length of the f -orbit of α in Q 1 . We define the weight function m * • by m * O * (α ′ ) = m α and m * O * (εα) = 1 for all α ∈ Q 1 . Let B * = B(Q * , f * , m * • ) be the biserial triangulation algebra associated to (Q * , f * , m * • ) and let e * be the sum of the primitive idempotents e * i in B * associated to all vertices i ∈ Q 0 . Using the proof of Proposition 2.7 we see directly that the idempotent algebra e * B * e * is isomorphic to B. It follows also from the definition of f * that Q * has no loops fixed by f * , and (ii) holds. In particular, we conclude that f * (ε α ) = ε α for any arrow α ∈ Q 1 . Hence, the triangulation quiver (Q * * , f * * ) has no loops, and consequently it has also no self-folded triangles, and (iii) follows. Finally, by (i), B * is isomorphic to an idempotent algebraêB * * ê of B * * = B(Q * * , f * * , m * * • ) for the corresponding idempotentê of B * * . Taking e * * = e * ê , we obtain that B is isomorphic to the idempotent algebra e * * B * * e * * , and hence (iv) also holds. We give some illustrations for the * -construction. (1) A loop α in Q fixed by f is replaced in Q * by the subquiver x α εα 6 6 α ′′ C C s(α) α ′ i i with the f * -orbit (α ′ ε α α ′′ ). (2) A subquiver of Q of the form where (α β γ) is an f -orbit, is replaced in Q * by the quiver x β β ′′ 1 1 ❄ ❄ ❄ ❄ ❄ ❄ ❄ εα × × x α α ′′ 9 9 εγ I I a α ′ h h β ′ c c ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ b γ ′ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ x γ γ ′′ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ε β y y with f * -orbits (α ′′ β ′ ε α ), (γ ′′ α ′ ε γ ) and (β ′′ γ ′ ε β ). (3) A subquiver of Q of the form a α G G b β Ô Ô ✠ ✠ ✠ ✠ ✠ ✠ ✠ c γ ✻ ✻ ✻ ✻ ✻ ✻ ✻ and where (α β γ) is an f -orbit, is replaced in Q * by the quiver of the form Remark 4.3. The construction of the triangulation quiver (Q * , f * ) associated to (Q, f ) is canonical, though a quiver with fewer vertices may often be sufficient. In fact, it would be enough to apply the construction only to the arrows in forbits of length different from 1 and 3. An algebra B(Q, f, m • ) may have many presentations as an idempotent algebra of some biserial triangulation algebra, even for a triangulation quiver (Q ′ , f ′ ) with fewer f ′ -orbits than the number of f * -orbits in the triangulation quiver (Q * , f * ) (see Example 4.7). a α ′ G G x α α ′′ G G εγ Ó Ó ✟ ✟ ✟ ✟ ✟ ✟ b β ′ Ó Ó ✟ ✟ ✟ ✟ ✟ ✟ ✟ x γ ε β G G γ ′′ ✻ ✻ ✻ ✻ ✻ ✻ ✻ x β β ′′ Ó Ó ✟ ✟ ✟ ✟ ✟ ✟ εα ✻ ✻ ✻ ✻ ✻ ✻ c γ ′ ✻ ✻ ✻ ✻ ✻ ✻ with f * -orbits (α ′′ β ′ ε α ), (β ′′ γ ′ ε β ), and (γ ′′ α ′ ε γ ). Remark 4.4. The * -construction described in Theorem 4.1 provides a special class of triangulation quivers. Namely, let (Q, f ) be a biserial quiver, g the permutation of Q 1 associated to (Q, f ), and g * the permutation of Q * 1 associated to (Q * , f * ). Then, for every arrow α ∈ Q 1 , we have in Q * 1 the g * -orbit O * (α ′ ) of even length 2|O(α)| and the g * -orbit O * (ε α ) whose length is the length of the f -orbit of α in Q 1 . In particular, all triangulation quivers (Q ′ , f ′ ) having only g ′ -orbits of odd length do not belong to this class of triangulation quivers. For example, it is the case for the tetrahedral quiver considered in Section 6. We refer also to [17,Example 4.9] for an example of triangulation quiver (Q ′′ , f ′′ ) for which all arrows in Q ′′ 1 belong to one g ′′ -orbit of length 18. 1 α ′ Ó Ó ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ β ′ ' ' ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ x α α ′′ g g ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ✟ ε β G G x β εα o o β ′′ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ ✻ and the f * -orbits are (α ′′ β ′ ε α ) and (β ′′ α ′ ε β ). Further, the g * -orbits are O * (α ′ ) = (α ′ α ′′ ), O * (β ′ ) = (β ′ β ′′ ), O * (ε α ) = (ε α ε β ) and the weight function is m * O * (α ′ ) = m, m * O * (β ′ ) = n, m * O * (εα) = 1. We also note that (Q * , f * ) is the triangulation quiver (Q(T, #‰ T ), f ) associated to the torus T with triangulation T and orientation #‰ T of triangles in T as follows Q * , f * ) is x α εα 6 6 α ′′ @ @ 1 α ′ i i β ′ A A x β β ′′ g g ε β h h with f * -orbits (α ′′ α ′ ε α ) and (β ′′ β ′ ε β ). x a x d x p x α x β x γ x δ x ̺ x σ x ω x ν x µ x ξ x η x ψ x ϕ α ′ εσ σ ′′ β ′ ε δ δ ′′ γ ′ ε β β ′′ ω ′ εα α ′′ ̺ ′ εω ω ′′ δ ′ εµ µ ′′ µ ′ εν ν ′′ ψ ′ εϕ ϕ ′′ d ′ εγ γ ′′ σ ′ ε d d ′′ η ′ ε ξ ξ ′′ ξ ′ εη η ′′ ϕ ′ εa a ′′ a ′ ε ψ ψ ′′ p ′ ε̺ ̺ ′′ ν ′ εp p ′′ where the shaded triangles describe the f * -orbits in Q * 1 . Then we have the following g * -orbits in Q * 1 : O * (α ′ ) = (α ′ α ′′ γ ′ γ ′′ σ ′ σ ′′ β ′ β ′′ ω ′ ω ′′ δ ′ δ ′′ ), O * (d ′ ) = (d ′ d ′′ ), O * (µ ′ ) = (µ ′ µ ′′ ̺ ′ ̺ ′′ ν ′ ν ′′ η ′ η ′′ ψ ′ ψ ′′ ϕ ′ ϕ ′′ ξ ′ ξ ′′ ), O * (p ′ ) = (p ′ p ′′ ), O * (ε α ) = (ε α ε σ ε d ε γ ε β ε δ ε µ ε ν ε p ε ̺ ε ω ), O * (a ′ ) = (a ′ a ′′ ), O * (ε a ) = (ε a ε ψ ε ϕ ), O * (ε η ) = (ε η ε ξ ). Moreover, the weight function m * • : O(g * ) → N * is given by m * O * (d ′ ) = m O(d) = e(d), m * O * (α ′ ) = m O(α) = e(c), m * O * (εα) = 1, m * O * (p ′ ) = m O(p) = e(p), m * O * (µ ′ ) = m O(µ) = e(b), m * O * (εη ) = 1, m * O * (a ′ ) = m O(a) = e(a). Finally, e * = e * 1 + e * 2 + e * 3 + e * 4 + e * 5 + e * 6 + e * 7 + e * 8 . We note that (Q * , f * ) has 16 f * -orbits, all of length three. The We finish this section with a combinatorial interpretation of the * -construction in terms of Brauer graphs. 4.8. Barycentric division of Brauer graphs. Let Γ be the Brauer graph so that B Γ = B(Q, f, m • ), then the algebra B(Q * , f * , m * • ) as in the * -construction of Theorem 4.1 is again a Brauer graph algebra, B Γ * say, by Theorem 2.6. The proof of Lemma 2.4 shows how to construct Γ * : Its vertices are in bijection with the cycles of g * . First, each cycle of g is 'augmented', by replacing an arrow α by α ′ , α ′′ , and this gives a cycle of g * , we call a corresponding vertex of Γ * an augmented vertex. Second, any other cycle of g * consists of ε-arrows, and these cycles correspond to f -cycles of Q, as described in Theorem 4.1. Let F (α) be the f -orbit of α in Q, then we write v F (α) for the corresponding vertex of Γ * , then the arrows attached to this vertex are precisely the ε f t (α) . The edges of Γ * are labelled by the vertices of Q * , that is by the vertices of Q together with the set {x α | α ∈ Q 1 }. The cyclic order around an augmented vertex is obtained by replacing i α −→ j by i α ′ −→ x α α ′′ −→ j in Γ * . A vertex v F (α) has attached arrows precisely the ε f t (α) : x f t+1 (α) → x f t (α) . This specifies the edges adjacent, with cyclic order given by the inverse of the f -cycle of α. We may view Γ * as a 'triangular' graph: (1) Assume that |F (α)| = 1. Then x α is the unique edge in Γ * adjacent to v F (α) , and x α is its own successor in the cyclic order of edges in Γ * around v F (α) . Hence we have in Γ * a self-folded triangle v x α v F (α) i • • which corresponds to a subquiver of (Q * , f * ) of the form x α εα 6 6 α ′′ 9 9 i α ′ i i with f * -orbit (α ′′ α ′ ε α ). (2) Assume that |F (α)| ≥ 2, and let β = f (α) starting at vertex j. Let v, w be the vertices in Γ such that α is attached to v and β is attached to w. Then Γ * has edges x α and x β connecting vertices v and w to vertex v F (β) (= v F (α) ). Then x α is the successor of x β in the cyclic order of edges in Γ * around v F (α) . Hence we have in Γ * a triangle • xα ✞ ✞ ✞ ✞ ✞ ✞ ✞ x β ✼ ✼ ✼ ✼ ✼ ✼ ✼ v F (β) • v j • w which corresponds to a subquiver of (Q * , f * ) of the form x α α ′′ ' ' ✼ ✼ ✼ ✼ ✼ ✼ x β εα o o j β ′ g g ✞ ✞ ✞ ✞ ✞ ✞ with f * -orbit (α ′′ β ′ ε α ). The multiplicity function e * of Γ * is given by e * (v) = e(v) for any vertex v of Γ (where e is the multiplicity function for Γ), and e * (v F (α) ) = 1 for any f -orbit F (α). The Brauer graph Γ * can be considered as a barycentric division of the Brauer graph Γ, and has a triangular structure. Namely, every v F (α) is the vertex of |F (α)| triangles in Γ * whose edges opposite to v F (α) are the edges of Γ corresponding to the vertices in Q along F (α). In this way, we obtain an orientable surface S * without boundary, the triangulation T * of S * indexed by the set of edges of Γ, and the orientation # ‰ T * of triangles in T * such that the associated triangulation quiver (Q(S * , # ‰ T * ), f ) is the quiver (Q * , f * ). The triangulated surface (S * , T * ) can be considered as a completion of the Brauer graph Γ to a canonically defined triangulated surface, by a finite number of pyramids whose peaks are the f -orbits and bases are given by the edges of Γ. We also note that the surface S * (without triangulation T * ) can be obtained as follows. We may embed the Brauer graph Γ into a surface S with boundary given by thickening the edges of Γ. The components of the border ∂S of S are given by the 'Green walks' around Γ on S, corresponding to the f -orbits in Q 1 . Then the surface S * is obtained from S by capping all the boundary components of S by the disks D 2 . Remark 4.9. We note that the orientation # ‰ T * of triangles of the triangulation T * of S * described above is not necessarily coherent. Indeed, take the triangulation quiver (Q(T , #‰ T ), f ), with trivial weight function, considered in Example 3.2. O(α) = (α), O(β) = (β σ δ γ), O(ω) = (ω ̺ η). Then the barycentric division Γ * of Γ is the Brauer graph • • • • • • 3 x α 2 x β x δ x η x ω x δ w a b c u v x ̺ 4 1 x γ with u = v F (α) , v = v F (σ) and w = v F (η) . The ordering of the edges around each vertex is clockwise. The multiplicity function of Γ * takes only the value 1. The Brauer graph Γ admits a canonical embedding into the surface S of the form a 3 b 2 c 4 1 • • • obtained from Γ by thickening the edges of Γ, whose border ∂S has three components given by three different 'Green walks' around Γ on S. The triangulated surface (S * , T * ) associated to the Brauer graph Γ * can be viewed as a canonical completion of S to a triangulated surface. Proof of Theorem 3 This theorem describes algebras socle equivalent to Brauer graph algebras. By Theorem 2.6 this is the same as describing algebras socle equivalent to a biserial quiver algebra A = B(Q, f, m • ) where (Q, f ) is a biserial quiver. We show that such algebras can be described using the methods of [18,Section 6]. Then we show that the * -construction for the biserial quiver algebras can be extended. Let (Q, f ) be a biserial quiver. A vertex i ∈ Q 0 is said to be a border vertex of (Q, f ) if there is a loop α at i with f (α) = α. We denote by ∂(Q, f ) the set of all border vertices of (Q, f ), and call it the border of (Q, f ). The terminology is motivated by the connection with surfaces: If (Q, f ) is the triangulation quiver (Q(S, #‰ T ), f ) associated to a directed triangulated surface (S, #‰ T ), then the border vertices of (Q, f ) correspond bijectively to the boundary edges of the triangulation T of S. If (Q, f ) is the biserial quiver associated to a Brauer graph Γ, then the border vertices of (Q, f ) correspond bijectively to the internal loops of Γ (see Section 2). Definition 5.1. Assume (Q, f, m • ) is a biserial quiver with ∂(Q, f ) not empty. A function b • : ∂(Q, f ) → K is said to be a border function of (Q, f ). We have the quotient algebra B(Q, f, m • , b • ) = KQ/J(Q, f, m • , b • ), where J(Q, f, m • , b • ) is the ideal in the path algebra KQ generated by the elements: (1) αf (α), for all arrows α ∈ Q 1 which are not border loops, (2) α 2 − b s(α) B α , for all border loops α ∈ Q 1 , (3) B α − Bᾱ, for all arrows α ∈ Q 1 . We call such an algebra a biserial quiver algebra with border. Note that if b • is the zero function then B(Q, f, m • , b • ) = B(Q, f, m • ). We summarize the basic properties of these algebras. Proof. Part (ii) is clear from the definition and then part (i) follows from Proposition 2.3. For the last part see arguments in the proof of Proposition 6.3 in [18]. The following theorem gives a complete description of symmetric algebras socle equivalent to a biserial quiver algebra. Theorem 5.3. Let A be a basic, indecomposable, symmetric algebra with Grothendieck group K 0 (A) of rank at least 2. Assume that A is socle equivalent to a biserial quiver algebra B(Q, f, m • ). ( 1) If ∂(Q, f ) is empty then A is isomorphic to B(Q, f, m • ). (2) Otherwise A is isomorphic to B(Q, f, m • , b • ) for some border function b • of (Q, f ). Proof. Let B = B(Q, f, m • ) = KQ/J where J = J(Q, f, m • ) . Let A be a symmetric algebra such that A/ soc(A) is isomorphic to B/ soc(B), and we can assume that these are equal, using an isomorphism as identification. In particular, A is of the form KQ/I with the same quiver, and I is an ideal of KQ. We assume A is symmetric, therefore for each i, the module e i A has a 1-dimensional socle which is spanned by some ω i ∈ e i Ae i , and we fix such an element. Then let ϕ be a symmetrizing linear form for A, then ϕ(ω i ) is non-zero. We may assume that ϕ(ω i ) = 1. In the given algebra B we define monomials A α in the arrows by setting B α = A α g −1 (α) when α is not a virtual loop, and then as well B α = αA g(α) . Note that if α is a virtual loop then A α is not defined. With this, the elements A α belong to the socle of B/ soc(B) and hence also to the socle of A/ soc(A). Therefore they cannot lie in the socle of A (because if so then they would be zero in A/ soc(A)). Then 0 = A α J = soc(e i A) where i = s(α). We have that A α J is spanned by A α β, A α γ where β = g −1 (α) and γ = f (g −2 (α)). (I) We may assume that A α β = B α in A: (and hence is equal to B α in KQ): If not, then we have A α β = 0, and then A α γ = 0. We will show that we may interchange β and γ. Since A α γ = 0, in particular g −2 (α)γ = 0 and also t(γ) = i = s(α). Since γ = f (g −2 (α)) we know that g −2 (α)γ belongs to the socle of A. It is non-zero, which implies that A α = g −2 (α) (and m α = 1), and therefore α = g −2 (α), and γ = f (α). We claim that g(α) = α. Namely if we had g(α) = α then both α and f (α) would be loops at vertex i and |Q 0 | = 1, which contradicts our assumption. Hence the cycle of g containing α is (α g(α)), of length two. We claim that also the f -cycle of α (in B) has length two. Namely ifᾱ is the other arrow starting at i and ρ is the other arrow ending at j = t(α) then we must have by the properties of f and g that f (ρ) = β and f (β) =ᾱ. This implies that f (γ) = α and hence f has a cycle (α γ). It follows that there is an algebra isomorphism from B to the biserial quiver algebra B ′ given by the weighted biserial quiver obtained from (Q, f, m • ) by interchanging β and γ (which form a pair of double arrows) and fixing all other arrows of Q. We replace B by B ′ and the claim follows. (II) We show that relation (1) holds in A. Let α be an arrow which is not fixed by f , then αf (α) belongs to the socle of A. We can write αf (α) = a α B α = a α αA g(α) for some a α ∈ K (unless g(α) is a virtual loop). Say i = s(α) then αf (α) = αf (α)e i . (a) If i = t(f (α)) then this is zero, in fact this holds for any choice of α, f (α). (b) Otherwise, we set f (α) ′ := f (α) − a α A g(α) and we replace f (α) by f (α) ′ . (If a cycle of f has a virtual loop then αf (α) and f −1 (α)α are not cyclic paths, so they are zero and do not need adjusting.) These modifications must be iterated. Take a cycle of f , say it has length r, so that r ≥ 2. Assume first this cycle contains an arrow α such that f r−1 (α)α is not a cyclic path. We may start with α and adjust f (α), f 2 (α), . . . , f r−1 (α) as described above. Then f r−1 (α) ′ · α = 0, by (a) above. Otherwise, for any α in the cycle, f r−1 (α)α is cyclic, and then we must have r = 2. We adjust f (α) as described in (b) and have αf (α) ′ = 0 in A, and we must show that as well f (α) ′ α = 0. By the assumption, f (α) ′ α = cω i for some c ∈ K. We have c = ϕ(cω i ) = ϕ(f (α) ′ α) = ϕ(αf (α) ′ ) = ϕ(0) = 0. (III) We show that relation (3) holds in A. For each arrow α ∈ Q 1 , we have B α = c α ω s(α) for some c α ∈ K * . We claim that c σ = c α for any arrow σ in the g-orbit O(α) of α. Indeed, if σ belongs to O(α), then c σ = c σ ϕ ω s(σ) ) = ϕ c σ ω s(σ) ) = ϕ(B σ ) = ϕ(B α ) = ϕ c α ω s(α) ) = c α ϕ ω s(α) ) = c α . Since K is algebraically closed, we may choose d α ∈ K * such that d mαnα α = c −1 α . Replacing now the representative of each arrow α ∈ Q 1 in A by its product with d α , we obtain a new presentation A ∼ = KQ/I ′ such that B α = ω s(α) for any arrow α ∈ Q 1 . This does not change the relations (1) obtained above. Therefore, we may assume that, if i ∈ Q is any vertex, α andᾱ are the arrows in Q with source i, then B α = ω i = Bᾱ in A. (IV) We show that relation (2) holds in A. When the border ∂(Q, f ) of (Q, f ) is empty, there is nothing to do (and A is isomorphic to B). Assume now that ∂(Q, f ) is not empty. Then for any loop α with i = s(α) ∈ ∂(Q, f ), we have α 2 = αf (α) = b i ω i = b i B α for some b i ∈ K. Hence, we have a border function b • : ∂(Q, f ) → K, and A is isomorphic to the algebra B(Q, f, m • , b • ). Recall that a self-injective algebra A is biserial if the radical of any indecomposable non-uniserial projective, left or right, A-module is a sum of two uniserial modules whose intersection is simple. Theorem 3 follows from Theorems 2.6, 3.1, 5.3 and the following relative version of Theorem 4.1 (see Remark 4.3). Proof. The construction of (Q # , f # , m # • ) is analogous to the * -construction in Theorem 4.1. We take the notation as in Theorem 4.1 and in addition we denote by Q b 1 the set of all border loops of the quiver. We define a triangulation quiver (Q # , f # ) as follows. We take Q # = (Q # 0 , Q # 1 , s # , t # ) with Q # 0 := Q 0 ∪ {x α } α∈Q1\Q b 1 , Q # 1 := Q b 1 ∪ {α ′ , α ′′ , ε α } α∈Q1\Q b 1 , s # (β) = s(β) = t(β) = t # (β) for all loops β ∈ Q b 1 , and s # (α ′ ) = s(α), t # (α ′ ) = x α , s # (α ′′ ) = x α , t # (α ′′ ) = t(α), s # (ε α ) = x f (α) , t # (ε α ) = x α , for any arrow α ∈ Q 1 \ Q b 1 . Moreover, we set f # (η) = η for any loop η ∈ Q b 1 , and f # (α ′′ ) = f (α) ′ , f # (f (α) ′ ) = ε α , f # (ε α ) = α ′′ , for any arrow α ∈ Q 1 \Q b 1 . We observe that (Q # , f # ) is a triangulation quiver with ∂(Q # , f # ) = ∂(Q, f ). Let g # be the permutation of Q # 1 associated to f # . For each arrow β in Q # 1 , we denote by O # (β) the g # -orbit of β in Q # 1 . Then the g # -orbits in Q # 1 are O # (η) = η g(η) ′ g(η) ′′ . . . g nη−1 (η) ′ g nη−1 (η) ′′ , for any loop η ∈ Q b 1 , and O # (α ′ ) = α ′ α ′′ g(α) ′ g(α) ′′ . . . g nα−1 (α) ′ g nα−1 (α) ′′ , O # (ε α ) = ε f rα−1 (α) ε f rα−2 (α) . . . ε f (α) ε α , for any arrow α ∈ Q 1 \ Q b 1 (where r α is the length of the f -orbit of α). We define the weight function m # • : O(g # ) → N * by m # O # (η) = m η for any loop η ∈ Q b 1 , and m # O # (α ′ ) = m α and m # O # (εα) = 1 for any arrow α ∈ Q 1 \ Q b 1 . Let B # = B(Q # , f # , m # • ) be the biserial weighted triangulation algebra associated to (Q # , f # , m # • ) and e # the sum of the primitive idempotents e # i in B # associated to the vertices i ∈ Q 0 . Then it follows from the arguments as in the proof of Proposition 2.7 that B is isomorphic to the idempotent algebra e # B # e # . Moreover, let b • be a border function of (Q, f ) and b # • be the induced border function of (Q # , f # ), that is b # i = b i for any border vertex i. Then it follows from the description of g # -orbits in Q # 1 and the definition of the weight function m # • that B(Q, f, m • , b • ) is isomorphic to the idempotent algebra e # B(Q # , f # , m # • , b # • )e # . We give an example to illustrate the #-construction in Theorem 5.4. . It has quiver Q, and to simplify the notation for the relations, we use the notion of B α for an arrow α, as it has appeared throughout, ̺ 2 = b 1 B ρ , B ρ = B α , αβ = 0, η 2 = b 2 B η , B η = B β , βγ = 0, µ 2 = b 3 B µ , B µ = B γ , γσ = 0, ξ 2 = b 4 B ξ , B ξ = B σ , σα = 0. Note that the algebra B(Q, f, m • ) is given by the quiver Q and the above relations such that all b i are zero. It follows from the arguments as in [ x α x γ x β x σ 1 ̺ 2 η 3 µ 4 ξ α ′ α ′′ β ′ β ′′ γ ′ γ ′′ σ ′ σ ′′ εα ε β εγ εσ with f # -orbits (̺), (η), (µ), (ξ), (α ′ ε σ σ ′′ ), (β ′ ε α α ′′ ), (γ ′ ε β β ′′ ), (σ ′ ε γ γ ′′ ). Further, there are two g # -orbits: O # (α ′ ) = (α ′ α ′′ η β ′ β ′′ µ γ ′ γ ′′ ξ σ ′ σ ′′ ̺), O # (ε α ) = (ε α ε σ ε γ ε β ). The weight function m # • takes only value 1, and the border function b # • is b # 1 = b 1 , b # 2 = b 2 , b # 3 = b 3 , b # 4 = b 4 . (a) The relations from vertex 1 are ρ 2 = b 1 B ρ , B ρ = B α ′ . There are analogous relations from each of the vertices 2, 3, 4. (b) The relations from vertex x α are B εσ = B α ′′ , α ′′ β ′ = 0, ε σ σ ′′ = 0. There are analogous relations from each of the vertices x β , x γ , x σ . We observe now that B(Q, f, m • , b • ) is isomorphic to the idempotent algebra e # B(Q # , f # , m # • , b # • )e # where the idempotent e # is the sum of the primitive idempotents at the vertices 1, 2, 3, 4. Moreover, the algebras B(Q, f, m • ) and e # B(Q # , f # , m # • )e # are also isomorphic. Finally, we note that if K has characteristic 2 and b # • = b • is non-zero, then the algebras B(Q # , f # , m # • , b # • ) and B(Q # , f # , m # • ) are not isomorphic. Proof of Theorem 4 We recall the definition of a weighted triangulation algebra. Let (Q, f ) be a triangulation quiver with at least two vertices, and let g, n • and m • be defined as for biserial quiver algebras. The additional datum is a function c • : O(g) → K * = K \ {0} which we call a parameter function of (Q, f ). We write briefly m α = m O(α) and c α = c O(α) for α ∈ Q 1 . The parameter function c • taking only value 1 is said to be trivial. We assume that m α n α ≥ 3 for any arrow α ∈ Q 1 . For any arrow α ∈ Q 1 , define the path A α = αg(α) . . . g nα−1 (α) mα−1 αg(α) . . . g nα−2 (α), if n α ≥ 2, A α = α mα−1 , if n α = 1, in Q of length m α n α − 1. Then we have A α g nα−1 (α) = B α = αg(α) . . . g nα−1 (α) mα of length m α n α . Then, following [17], we define the bound quiver algebra Λ(Q, f, m • , c • ) = KQ/I(Q, f, m • , c • ), where I(Q, f, m • , c • ) is the admissible ideal in the path algebra KQ of Q over K generated by the elements: (1) αf (α) − cᾱAᾱ, for all arrows α ∈ Q 1 , (2) βf (β)g(f (β)), for all arrows β ∈ Q 1 . The algebra Λ := Λ(Q, f, m • , c • ) is called a weighted triangulation algebra of (Q, f ). Moreover, if (Q, f ) = (Q(S, #‰ T ), f ) for a directed triangulated surface (S, #‰ T ), then Λ is called a weighted surface algebra, and if the surface and triangulation is important we denote the algebra by Λ(S, #‰ T , m • , c • ). We note that the Gabriel quiver of Λ is equal to Q, this holds because we assume m α n α ≥ 3 for all arrows α ∈ Q 1 . We have the following proposition (see [17,Proposition 5.8]). (iii) Λ is not isomorphic to a singular tetrahedral algebra. Following [17], a singular tetrahedral algebra is the weighted surface algebra given by a coherent orientation of four triangles of the tetrahedron and the weight and parameter functions taking only value 1. The triangulation quiver of such algebra is the tetrahedral quiver of the form where the shaded triangles denote f -orbits and white triangles denote g-orbits. The following theorem is an essential ingredient for the proof of Theorem 4. Theorem 6.3. Let B = B(Q, f, m • ) be a biserial weighted triangulation algebra where Q has no loops, and Λ * = Λ(Q * , f * , m * • , c * • ) the weighted triangulation algebra associated to the weighted triangulation quiver (Q * , f * , m * • ) and the trivial parameter function c * • of (Q * , f * ). Then the following statements hold: (i) Λ * is a periodic algebra of period 4. (ii) B is isomorphic to the idempotent algebra e * Λ * e * for an idempotent e * of Λ * . Proof. For each arrow ̺ in Q * 1 , we set m * ̺ = m * O(̺) and n * ̺ = |O * (̺)|. We observe first that n * ̺ ≥ 3, and hence m * ̺ n * ̺ ≥ 3, for any arrow ̺ in Q * 1 , and consequently Λ(Q * , f * , m * • , c * • ) is a well defined weighted triangulation algebra. Indeed, it follows from the assumption on Q that the triangulation quiver (Q * , f * ) has neither loops nor self-folded triangles. Moreover, the f -orbits in Q 1 have length 3, and the gorbits in Q 1 are of length at least 2. Then it follows from the proof of Theorem 4.1 that the g * -orbits in Q * 1 are O * (α ′ ) = α ′ α ′′ g(α) ′ g(α) ′′ . . . g nα−1 (α) ′ g nα−1 (α) ′′ , O * (ε α ) = ε f 2 (α) ε f (α) ε α , for all arrows α ∈ Q 1 . Then the required inequalities hold. Further, it follows from Remark 4.4 that (Q * , f * ) is not the tetrahedral quiver. Then, applying Theorem 6.2, we conclude that Λ * is a periodic algebra of period 4. Let e * be the sum of all primitive idempotents in Λ * corresponding to the vertices of Q. We claim that e * Λ * e * is isomorphic to B. Observe that every f -orbit in b γ ′ ε γ α ′′ α ′ β ′′ ε α γ ′′ ε β β ′ The algebra e * Λ * e * has arrows α = α ′ α ′′ , β = β ′ β ′′ , γ = γ ′ γ ′′ , and it follows that in e * Λ * e * we have αβ = α ′ α ′′ β ′ β ′′ = α ′ α ′′ f * (α ′′ )g * f * (α ′′ ) = 0, βγ = β ′ β ′′ γ ′ γ ′′ = β ′ β ′′ f * (β ′′ )g * f * (β ′′ ) = 0, γα = γ ′ γ ′′ α ′ α ′′ = γ ′ γ ′′ f * (γ ′′ )g * f * (γ ′′ ) = 0. Further, let i be a vertex of Q, and let α and σ =ᾱ the two arrows in Q 1 with source i. By the proof of Theorem 4.1, the g * -orbits are O * (α ′ ) = α ′ α ′′ g(α) ′ g(α) ′′ . . . g nα−1 (α) ′ g nα−1 (α) ′′ , O * (σ ′ ) = σ ′ σ ′′ g(σ) ′ g(σ) ′′ . . . g nσ −1 (σ) ′ g nσ−1 (σ) ′′ . Moreover, m * α ′ = m * O * (α ′ ) = m O(α) = m α and m * σ ′ = m * O * (σ ′ ) = m O(σ) = m σ . Hence we have in Q * the cycles B α ′ = α ′ α ′′ g(α) ′ g(α) ′′ . . . g nα−1 (α) ′ g nα−1 (α) ′′ mα , B σ ′ = σ ′ σ ′′ g(σ) ′ g(σ) ′′ . . . g nσ−1 (σ) ′ g nσ−1 (σ) ′′ mσ , and B α ′ = B σ ′ in Λ * (see [17,Lemma 5.3]), and this gives the equality B α = B σ = Bᾱ in e * Λ * e * . Therefore, e * Λ * e * is isomorphic to B. We may now complete the proof of Theorem 4. Let B = B(Q, f, m • ) be a biserial quiver algebra. Then it follows from Theorem 4.1 that B is isomorphic to the idempotent algebra e * * B * * e * * of the biserial triangulation algebra B * * = B(Q * * , f * * , m * * • ) for some idempotent e * * of B * , and Q * * has no loops. Applying now Theorem 6.3 we conclude that B * * is isomorphic to the idempotent algebra eΛe of a periodic weighted triangulation algebra, for an idempotent e of Λ. Since e * * is a summand of e, we have B ∼ = e * * B * * e * * ∼ = e * * (eΛe)e * * = e * * Λe * * . Then Theorem 4 follows from Theorems 2.6 and 3.1. Remark 6.4. Let Λ = Λ(Q, f, m • , c • ) be a weighted triangulation algebra. Then the biserial triangulation algebra B = B(Q, f, m • ) is not an idempotent algebra eΛe of Λ. On the other hand, if Λ is not a tetrahedral algebra, then B is a geometric degeneration of Λ (see [17,Proposition 5.8]). Example 6.5. Let (Q, f ) be the Markov quiver in Example 3.4 and m a positive integer associated to the unique g-orbit (α 1 β 2 α 3 β 1 α 2 β 3 ) in Q 1 . Then the associated weighted triangulation algebra Λ = Λ(Q, f, m • , c • ) with trivial parameter We present now an example of an idempotent algebra of a periodic weighted surface algebra which is neither a Brauer graph algebra nor a weighted surface algebra. ξ 2 = αηβδσµωνγ, ξ 2 α = 0, αβ = ξαηβδσµων, αβδ = 0, νδ = θ̺θ, η 2 = βδσµωνγξα, η 2 β = 0, βγ = ηβδσµωνγξ, βγξ = 0, νδσ = 0, µ 2 = ωνγξαηβδσ, µ 2 ω = 0, γα = δσµωνγξαη, γαη = 0, σω = ̺θ̺, δ̺ = γξαηβδσµω, δ̺θ = 0, ωθ = µωνγξαηβδ, ωθδ = 0, σων = 0, ̺ν = σµωνγξαηβ, ̺νγ = 0, θσ = νγξαηβδσµ, θσµ = 0. Let e = e 1 + e 2 + e 3 + e 4 be the sum of primitive idempotents of Λ at the vertices 1, 2, 3, 4, and B = eΛe the associated idempotent algebra. Then B is given by the quiver ∆ of the form 1 α ξ X X 4 γ d d ■ ■ ■ ■ ■ ■ ϕ G G 3 ψ o o µ z z 2 β X X ✉ ✉ ✉ ✉ ✉ ✉ η X X with the arrows ϕ = δσ and ψ = ων, and the induced relations: ξ 2 = αηβϕµψγ, ξ 2 α = 0, αβ = ξαηβϕµψ, αβδ = 0, ϕψ = 0, η 2 = βϕµψγξα, η 2 β = 0, βγ = ηβϕµψγξ, βγξ = 0, ψϕ = 0, µ 2 = ψγξαηβϕ, µ 2 ψ = 0, γα = ϕµψγξαη, γαη = 0. Then B is not a special biserial algebra, and therefore it is not a Brauer graph algebra. Further, B is not a weighted surface algebra, because we have zero-relations ϕψ = 0 and ψϕ = 0 of length 2. On the other hand, by general theory, the algebra B = eΛe is tame and symmetric. Diagram of algebras The following diagram shows the relations between the main classes of algebras occurring in the paper. , and Λ * * * = B(Q * * * , f * * * , m * * * • , 1), with 1 denoting the trivial parameter function of (Q * * * , f * * * ). Proposition 2 . 3 . 23Let (Q, f, m • ) be a weighted biserial quiver and B = B(Q, f, m • ). ( 2 ) 2For each 2-dimensional cell e 2 i , ϕ 2 i (∂D 2 i ) is the union of k 1-cells and k 0-cells, with k ∈ {2, 3}. (3) For any different 1-cells e 1 i and e 1 j , we have ϕ 1 i (∂D 1 i ) = ϕ 1 j (∂D 1 j ). Then the closures ϕ 2 i (D 2 i ) of all 2-cells e 2 i are called triangles of S, and the closures ϕ 1 i (D 1 i ) of all 1-cells e 1 i are called edges of S. The collection T of all triangles ϕ 2i (D 2 i ) is said to be a triangulation of S. We assume that such a triangulation T of S has at least two different edges, or equivalently, there are at least two different 1-cells in the considered triangular cell complex structure on S. Then T is a finite collection T 1 , . . . , T n of triangles of is self-folded, with the self-folded edge a, and the other edge b. Fix an orientation of each triangle ∆ of T , and denote this choice by #‰ T . Then the pair (S, #‰ T ) is said to be a directed triangulated surface. To each directed triangulated surface (S, #‰ T ) we associate the quiver Q(S, #‰ T ) whose vertices are the edges of T and the arrows are defined as follows: shaded triangles denote the f -orbits, considered in[17, Example 6.7]. Suppose that (Q(T , #‰ T ), f ) = (Q(S, #‰ T ), f ) for a triangulation T of an oriented surface S and the induced orientation #‰ T of triangles in T . Since the triangulation T is given by a finite 2-dimensional triangular cell complex structure on S, we conclude that S is a coherent gluing of four For a triangulation quiver (Q, f ) and a weight function m • : O(g) → N * , the associated weighted biserial quiver algebra B(Q, f, m • ) = KQ/J(Q, f, m • ) is said to be a biserial weighted triangulation algebra. Moreover, if (Q, f ) = (Q(S, #‰ T ), f ) for a directed triangulation surface (S, #‰ T ), then B(Q(S, #‰ T ), f, m • ) is called a biserial weighted surface algebra, and denoted by B(S, #‰ Example 3. 3 . 3Consider the disk D = D 2 with the triangulation T and orientation #‰ T of triangles in T as f -orbits (α β γ), (σ δ ω), (ξ), (η). Then the g-orbits are O(α) = (α ω η σ γ ξ) and O(β) = (β δ). Hence a weight function m • : O(g) → N * is given by two positive integers m O(α) = m and m O(β) = n. Then the associated biserial weighted surface algebra B(D, #‰ T , m • ) is given by the above quiver and the relations: (αωησγξ) m = (ξαωησγ) m , ξ 2 = 0, αβ = 0, σδ = 0, (σγξαωη) m = (ησγξαω) m , η 2 = 0, βγ = 0, δω = 0, (ωησγξα) m = (βδ) n , (γξαωησ) m = (δβ) n , γα = 0, ωσ = 0. Example 3.4. Consider the torus T with the triangulation T and orientation #‰ T of triangles in T as Theorem 4. 1 . 1Let B = B(Q, f, m • ) be a biserial quiver algebra. Then there is a canonically defined weighted triangulation quiver (Q * , f * , m * • ) such that the following statements hold.(i) B is isomorphic to the idempotent algebra e * B * e * of the biserial triangulation algebra B * = B(Q * , f * , m * • ) with respect to a canonically defined idempotent e * of B * . (ii) The triangulation quiver (Q * , f * ) has no loops fixed by f * . (iii) The triangulation quiver (Q * * , f * * ) has no loops and self-folded triangles. (iv) B is isomorphic to the idempotent algebra e * * B * * e * * of the biserial triangulation algebra B * * = B(Q * * , f * * , m * * • ) with respect to a canonically defined idempotent e * * of B * * . Remark 4 . 2 . 42The statement (i) of the above theorem also holds if we replace the canonically defined weight function m * • by a weight functionm * • such that m O * (α ′ ) = m α andm O * (εα) is an arbitrary positive integer, for any arrow α ∈ Q 1 . Example 4. 5 . 5Let Γ be the Brauer tree function e(a) = m and e(b) = n. Then the associated Brauer graph algebra B Γ is the algebra B(Q, f, m • ) associated to the biserial quiver (Q, f, m • ) where Q is of the with f (α) = β, f (β) = α, g(α) = α, g(β) = β, and m O(α) = m, and m O(β) = n. If m = 1, then B Γ is the truncated polynomial algebra K[x]/(x n+1 ). The associated triangulation quiver (Q * , f * ) is of the form e(a) = m for some m ∈ N * . Then the associated Brauer graph algebra B Γ is the algebra B(Q, f, m • ) where the quiver Q is of the f (α) = α, f (β) = β, g(α) = β, g(β) = α, and m O(α) = m. The associated triangulation quiver ( Further, the g * -orbits are O * (α ′ ) = (α ′ α ′′ β ′ β ′′ ), O * (ε α ) = (ε α ), O * (ε β ) = (ε β ), and m * O * (α ′ ) = m, m * O * (εα) = 1, m * O * (ε β ) = 1.Note that (Q * , f * ) is the triangulation quiver (Q(S,#‰T ), f ) associated to the sphere S with triangulation T given by two self- Example 4. 7 . 7Let (Q, f, m • ) be the weighted biserial quiver considered in Example 2.8. Then the triangulation quiver (Q * , f * ) is of the form Brauer graph algebra B Γ = B(Q, f, m • ) is also isomorphic to the idempotent algebra e ′ B ′ e ′ of a biserial triangulation algebra B ′ = B(Q ′ , f ′ , m ′ • ) for the triangulation quiver (Q ′ , f ′ f ′ -orbits described by the shaded triangles (all of length three), a weight function m ′ • of (Q ′ , f ′ ), and where the idempotent e ′ is the sum of the primitive idempotents in B ′ associated to the vertices 1 Proposition 5 . 2 . 52Let (Q, f ) be a biserial quiver such that ∂(Q, f ) is not empty, and letB = B(Q, f, m • , b • ), and B = B(Q, f, m • ) where m • and b • are weight and border functions. Then the following statements hold. (i)B is a basic, indecomposable, finite-dimensional, symmetric, biserial algebra with dim KB = O∈O(g) m O n 2 O . (ii)B is socle equivalent to B. (iii) If K is of characteristic different from 2, thenB is isomorphic to B. Theorem 5 . 4 . 54Let B = B(Q, f, m • ) where Q has at least two vertices, and where the border ∂(Q, f ) is not empty. Then there is a canonically defined weighted triangulation quiver (Q # , f # , m # • ) such that the following statements hold. (i) |∂(Q, f )| = |∂(Q # , f # )|. (ii) B is isomorphic to the idempotent algebra e # B # e # of the biserial weighted triangulation algebra B # = B(Q # , f # , m # • ) with respect to a canonically defined idempotent e # of B # . (iii) For any border function b • of (Q, f ) and the induced border function b # • of (Q # , f # ), the algebras B(Q, f, m • , b • ) and e # B(Q # , f # , m # • , b # • )e # are isomorphic. f -orbits (α β γ σ), (̺), (η), (µ), (ξ). Then the border ∂(Q, f ) of (Q, f ) is the set Q 0 = {1, 2, 3, 4} of all vertices of Q, and ̺, η, µ, ξ are the border loops. Further, g has only one orbit, O(α) = (α η β µ γ ξ σ ̺). We take the weight function m • : O(g) → N * with m O(α) = 1. Moreover, let b • : ∂(Q, f ) → K be a border function. Then we describe the associated algebra B(Q, f, m • , b • ) Proposition 6. 1 . 1Let (Q, f ) be a triangulation quiver, m • and c • weight and parameter functions of (Q, f ).Then Λ = Λ(Q, f, m • , c • ) is a finite-dimensional tame symmetric algebra of dimension O∈O(g) m O n 2 O .We have also the following theorem proved in [17, Theorem 1.2] (see also[6, Proposition 7.1] and[16, Theorem 5.9] for the case of two vertices). Theorem 6. 2 . 2Let Λ = Λ(S, #‰ T , m • , c • ) be a weighted surface algebra over an algebraically closed field K. Then the following statements are equivalent:(i) All simple modules in mod Λ are periodic of period 4. (ii) Λ is a periodic algebra of period 4. Example 6. 6 . 6Let S be a triangle with one puncture, T the triangulation of S f -orbits (ξ), (η), (µ), (α β γ), (δ ̺ ν), (σ ω θ). Hence we have two g-orbits:O(α) = (α η β δ σ µ ω ν γ ξ)andO(̺) = (̺ θ).Take the weight function m • : O(g) → N * given by m O(α) = 1 and m O(̺) = 2. Moreover, let c • : O(g) → K * be the trivial parameter function. Then the associated weighted surface algebra Λ = Λ(S, #‰ T , m • , c • ) is given by the quiver Q(S, #‰ T , ) and the relations: for a weighted biserial quiver algebra B = B(Q, f, m • ), B * = B(Q * , f * , m * • ) 18, Example 6.5] that if K has characteristic 2 and b • is non-zero, then the algebras B(Q, f, m • , b • ) and B(Q, f, m • ) are not isomorphic.The triangulation quiver (Q # , f # ) is of the form1 2 3 4 AcknowledgementsThe results of the paper were partially presented during the Workshop on Brauer Graph Algebras held in Stuttgart in March 2016. 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(Karin Erdmann) Mathematical Institute, Oxford University, ROQ, Oxford OX2 6GG, United Kingdom E-mail address: [email protected] (Andrzej Skowroński) Faculty of Mathematics and Computer Science, Nicolaus Coper- nicus University, Chopina 12/18, 87-100 Toruń, Poland E-mail address: [email protected]

In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs). These functions, referred to as TMDs, are defined as the Fourier transforms of hadronic matrix elements of nonlocal combinations of gluon fields. The nonlocality is bridged by gauge links, which have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. For gluons, the specific correlator with one future and one past pointing gauge link is, in the limit of small x, related to a correlator of a single Wilson loop. We present the parametrization of Wilson loop correlators in terms of Wilson loop TMDs and discuss the relation between these functions and the small-x 'dipole' gluon TMDs. This analysis shows which gluon TMDs are leading or suppressed in the smallx limit. We discuss hadronic targets that are unpolarized, vector polarized (relevant for spin-1/2 and spin-1 hadrons), and tensor polarized (relevant for spin-1 hadrons). The latter are of interest for studies with a future Electron-Ion Collider with polarized deuterons.

10.1007/jhep10(2016)013

1607.01654

89baf024f8a0c3a2e86adfa3343b209baac3e267

Gluon and Wilson loop TMDs for hadrons of spin ≤ 1 15 Sep 2016 Daniël Boer [email protected] Van Swinderen Institute for Particle Physics and Gravity University of Groningen Nijenborgh 4NL-9747 AGGroningenThe Netherlands Sabrina Cotogno [email protected] Department of Physics and Astronomy VU University Amsterdam De Boelelaan 1081NL-1081 HVAmsterdamThe Netherlands Science Park 105NL-1098 XGNikhef, AmsterdamThe Netherlands Tom Van Daal Department of Physics and Astronomy VU University Amsterdam De Boelelaan 1081NL-1081 HVAmsterdamThe Netherlands Science Park 105NL-1098 XGNikhef, AmsterdamThe Netherlands Piet J Mulders [email protected] Department of Physics and Astronomy VU University Amsterdam De Boelelaan 1081NL-1081 HVAmsterdamThe Netherlands Science Park 105NL-1098 XGNikhef, AmsterdamThe Netherlands Andrea Signori [email protected] Department of Physics and Astronomy VU University Amsterdam De Boelelaan 1081NL-1081 HVAmsterdamThe Netherlands Science Park 105NL-1098 XGNikhef, AmsterdamThe Netherlands Ya-Jin Zhou [email protected] Department of Physics and Astronomy VU University Amsterdam De Boelelaan 1081NL-1081 HVAmsterdamThe Netherlands Science Park 105NL-1098 XGNikhef, AmsterdamThe Netherlands School of Physics Key Laboratory of Particle Physics and Particle Irradiation (MOE) Shandong University 250100JinanShandongChina Gluon and Wilson loop TMDs for hadrons of spin ≤ 1 15 Sep 2016Prepared for submission to JHEPQCD PhenomenologySpin and Polarization EffectsWilson loops In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs). These functions, referred to as TMDs, are defined as the Fourier transforms of hadronic matrix elements of nonlocal combinations of gluon fields. The nonlocality is bridged by gauge links, which have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. For gluons, the specific correlator with one future and one past pointing gauge link is, in the limit of small x, related to a correlator of a single Wilson loop. We present the parametrization of Wilson loop correlators in terms of Wilson loop TMDs and discuss the relation between these functions and the small-x 'dipole' gluon TMDs. This analysis shows which gluon TMDs are leading or suppressed in the smallx limit. We discuss hadronic targets that are unpolarized, vector polarized (relevant for spin-1/2 and spin-1 hadrons), and tensor polarized (relevant for spin-1 hadrons). The latter are of interest for studies with a future Electron-Ion Collider with polarized deuterons. Contents 1 Introduction In high energy collisions gluons become more important with increasing energy, due to the decreasing longitudinal momentum fraction x that is typically being probed. This region has for example been studied by experiments at the Hadron-Electron Ring Accelerator (HERA) in inclusive deep inelastic scattering (DIS) and currently by experiments at the Large Hadron Collider (LHC) in proton-proton collisions. In less inclusive processes one can in addition become sensitive to the transverse momentum distribution of gluons. There is a rich variety of gluon transverse momentum dependent (TMD) parton distribution functions (PDFs), or TMDs for short, especially if one includes the polarization of hadrons. At the Relativistic Heavy Ion Collider (RHIC), experiments with spin-polarized protons are conducted and in future experiments, such as at an Electron-Ion Collider (EIC), polarized deuteron beams may be used. For this reason it is useful to parametrize gluon TMD correlators as efficiently and systematically as possible for unpolarized, vector, and tensor polarized hadrons and to consider specifically the small-x region. This is the intention of this paper. In the present work, the starting point for the gluon TMD correlators are Fourier transforms of hadronic matrix elements of field strength tensors connected by Wilson lines or gauge links [1][2][3][4][5][6] that bridge the nonlocality of the field operators, ensuring color gauge invariance. The nonlocality includes transverse directions [1,2], in which case one can consider, besides gluon-gluon correlators, also the matrix element of a single Wilson loop operator, which in this work is referred to as the Wilson loop correlator. The gauge invariant correlators are parametrized in terms of TMDs, depending on the longitudinal momentum fraction x and the transverse momentum k 2 T [7]. Including transverse momentum dependence, i.e. going beyond collinear kinematics, gives rise to a wealth of azimuthal asymmetries. This is particularly true when polarization degrees of freedom of the hadrons involved are considered, giving for instance rise to single spin asymmetries [6,[8][9][10][11]. The parametrizations in terms of TMDs have been extensively studied, especially for the quark case, for different polarizations of hadrons up to and including spin 1 [10,[12][13][14][15][16][17][18]. In the collinear case, the parametrization in terms of PDFs for gluons in tensor polarized spin-1 hadrons has first been considered in refs. [19,20]. A further proliferation of TMDs comes from the structure, i.e. the path dependence, of the gauge links. The gauge links depend on the process and as a consequence they give rise to observable process dependence and thus to a proliferation of TMDs. Since the dependence can be traced to the color flow in the hard scattering process, it is in principle possible to unravel this dependence [6,[21][22][23]. In some cases one may find how different TMDs and processes are related, but in some cases TMDs with different gauge links are not related at all, encoding independent information [24]. Here we limit ourselves to TMDs appearing in those contributions to the cross sections that are leading in inverse powers of the hard scale, referred to as leading twist TMDs. We will not be concerned with higher twist contributions [25] nor with QCD corrections that are of higher order in the strong coupling α s , relevant for the evolution and the large transverse momentum region [26][27][28]. The higher twist contributions would generally involve correlators with more fields. In order to facilitate the study of the evolution of these TMDs we will discuss the transition to impact parameter space, without further studying the evolution itself. We present the parametrizations of the gluon-gluon and Wilson loop TMD correlators in terms of TMDs of definite rank for unpolarized, vector polarized, and tensor polarized hadrons, the latter being considered here for the first time. We also provide a new treatment of the connection between the gluon-gluon correlator at small x and the Wilson loop correlator. This confirms the results of some specific examples that have been discussed in an earlier paper [29]. Parametrizations of gluonic TMD correlators In 2001, Mulders and Rodrigues [7] presented the first parametrization of the gluon-gluon light-front correlator in terms of TMDs considering both unpolarized and vector polarized hadrons. In 2007, a different nomenclature for those TMDs was proposed by Meißner, Metz, and Goeke in ref. [30], in close analogy to the ones for quarks. In this section we extend the analyses of refs. [7,30] by parametrizing both the gluon-gluon and Wilson loop correlators for unpolarized, vector polarized, as well as tensor polarized hadrons. The light-front correlators are expanded in a Lorentz basis of completely symmetric traceless tensors built from the partonic momentum k T (see appendix C.1 for the definitions of the relevant symmetric traceless tensors), and are expressed in terms of TMDs. Furthermore, a more systematic way of naming the various TMDs is introduced, keeping and extending the notation proposed in ref. [30]. We start with outlining the most relevant variables. We denote by P and k the hadron and parton momenta respectively. We parametrize k in terms of the dimensionful vectors P and n, where n is a lightlike vector satisfying n 2 = 0 and P ·n = 1: k µ = xP µ + k µ T + (k·P − xM 2 ) n µ , (2.1) where M is the mass of the hadron. The transverse direction is projected out using the metric tensor in transverse space, g µν T ≡ g µν − P {µ n ν} (curly brackets denote symmetrization of the indices), with nonvanishing elements g 11 T = g 22 T = −1. For a polarized hadron we employ a spin vector S needed to describe vector polarization for any hadron with spin ≥ 1/2 and a symmetric traceless spin tensor T to describe tensor polarization for hadrons with spin ≥ 1 [18,31]. We again parametrize S and T in terms of the dimensionful vectors P and n, 1 S µ = S L P µ M + S µ T − M S L n µ , (2.2) T µν = 1 2 2 3 S LL g µν T + 4 3 S LL P µ P ν M 2 + S {µ LT P ν} M + S µν T T − 4 3 S LL P {µ n ν} − M S {µ LT n ν} + 4 3 M 2 S LL n µ n ν ,(2.3) ensuring the relations P 2 = M 2 , P ·S = 0, P µ T µν = 0. (2.4) For a spin-1/2 hadron only a spin vector is needed to parametrize the density matrix. For a spin-1 hadron also a tensor is required. While the spin vector S for a spin-1 hadron signals a polarized hadron with m = 1 along that direction (in case of its length being one), the spin tensor T corresponds to particular combinations of spin states (see e.g. refs. [18,31]). The spin tensor has five independent parameters, namely S LL , the two components of the transverse vector S LT , and the two independent components of the symmetric traceless transverse tensor S T T . We note that one could reinstate the combination P ·n = P + by replacing everywhere n → n/P ·n. Introducingn ≡ (P − 1 2 M 2 n)/P ·n, such that n·n = 1, one can work with light cone components a + = a·n and a − = a·n. Hence, in the infinite momentum framē n corresponds to the target hadron direction and n to the conjugate direction. They are defined frame independently, however. In order to get the more natural interpretation in the hadron rest frame, one has the covariantly defined time-and spacelike directions, t ≡ P M ,ẑ ≡ P M − M n,(2.5) which become the standard time and spatial z-directions in the hadron rest frame. They are useful since the spin vector and tensor only contain the spacelike combinationẑ: S µ = S Lẑ µ + S µ T , T µν = 1 2 4 3 S LL ẑ µẑν + 1 2 g µν T +ẑ {µ S ν} LT + S µν T T . (2.6) Unpolarized hadrons The gluon-gluon correlator For a color gauge invariant description of gluon correlations in hadrons one can consider the (unintegrated) gluon-gluon correlator as a starting point, Γ [U,U ′ ] µν;ρσ (k; P, n) ≡ d 4 ξ (2π) 4 e ik·ξ P | F µν (0)U [0,ξ] F ρσ (ξ)U ′ [ξ,0] |P ,(2.7) where color summation, a trace in color space (Tr c ), is implicitly assumed. The Wilson lines U [0,ξ] and U ′ [ξ,0] guarantee color gauge invariance. Even though without specifying a process the path integrations could run along arbitrary paths, we have already included a dependence on the lightlike four-vector n, that enters upon consideration of staple-like gauge links running along the light-front (ξ·n = 0) via lightlike ξ·P = ±∞. A possible parametrization of the unintegrated correlator in eq. (2.7), constrained by hermiticity and parity conservation and respecting relations induced by time reversal (see appendix A), is 2 Γ [U,U ′ ] µν;ρσ (k; P, n) = M 2 A 1 ǫ µναβ ǫ ρσ αβ + A 2 P [µ g ν][ρ P σ] + A 3 k [µ g ν][ρ k σ] + (A 4 + iA 5 ) P [µ g ν][ρ k σ] + (A 4 − iA 5 ) k [µ g ν][ρ P σ] + (A 6 /M 2 ) P [µ k ν] P [ρ k σ] + M 4 A ′ 7 n [µ g ν][ρ n σ] + M 2 (A ′ 8 + iA ′ 9 ) P [µ g ν][ρ n σ] + M 2 (A ′ 8 − iA ′ 9 ) n [µ g ν][ρ P σ] + M 2 (A ′ 10 + iA ′ 11 ) k [µ g ν][ρ n σ] + M 2 (A ′ 10 − iA ′ 11 ) n [µ g ν][ρ k σ] + M 2 A ′ 12 P [µ n ν] P [ρ n σ] + M 2 A ′ 13 k [µ n ν] k [ρ n σ] + (A ′ 14 + iA ′ 15 ) P [µ k ν] P [ρ n σ] + (A ′ 14 − iA ′ 15 ) P [µ n ν] P [ρ k σ] + (A ′ 16 + iA ′ 17 ) P [µ k ν] k [ρ n σ] + (A ′ 16 − iA ′ 17 ) k [µ n ν] P [ρ k σ] + M 2 (A ′ 18 + iA ′ 19 ) P [µ n ν] k [ρ n σ] + M 2 (A ′ 18 − iA ′ 19 ) k [µ n ν] P [ρ n σ] , (2.8) where A i = A i (k·n, k·P, k 2 ) and the completely antisymmetric Levi-Civita tensor ǫ µνρσ is fixed by taking ǫ −+12 = 1. Terms with coefficients A 5 , A ′ 9 , A ′ 11 , A ′ 15 , A ′ 17 , A ′ 19 are T -odd, and a prime on the coefficient indicates that the corresponding Lorentz structure includes the four-vector n. As it turns out, these structures do not give rise to any leading twist TMDs (see ref. [32] for the analogous case for quarks). As we are only interested in leading twist functions, we will later on omit the terms containing n from our description of the gluon-gluon correlators in case of polarized hadrons. Integrating eq. (2.7) over k·P , one obtains the TMD (light-front) correlator Γ [U,U ′ ] µν;ρσ (x, k T ; P, n) ≡ dξ·P d 2 ξ T (2π) 3 e ik·ξ P | F µν (0)U [0,ξ] F ρσ (ξ)U ′ [ξ,0] |P ξ·n=0 . (2.9) The relevant correlator showing up in leading terms in the inverse hard scale can be recognized by counting P ∝ Q and n ∝ 1/Q, with Q denoting the hard scale. Suppressing the P and n dependence, which of course is present in the definition of transverse directions and in the paths of the gauge links, the leading (usually referred to as leading twist) correlator is then Γ ij (x, k T ) ≡ Γ [U,U ′ ] ni;nj (x, k T ; P, n). (2.10) Employing constant or symmetric traceless tensors, the light-front correlator is parametrized in terms of leading twist (i.e. twist-2) TMDs of definite rank. For the unpolarized correlator one obtains Γ ij (x, k T ) = x 2 − g ij T f 1 (x, k 2 T ) + k ij T M 2 h ⊥ 1 (x, k 2 T ) ,(2.11) where the expressions of the TMDs in terms of the coefficients A i can be found in appendix B. Throughout this paper, the remaining dependence of TMDs on the gauge link as well as a reference to gluons, such as in f g[U,U ′ ] 1 (x, k 2 T ) , is implicitly assumed, so we often simply write f 1 (x, k 2 T ), etc. We note that integration over k T in eq. (2.11) leads to the collinear correlator Γ ij (x) ≡ d 2 k T Γ ij (x, k T ) = − xg ij T 2 f 1 (x),(2.12) parametrized in terms of a collinear PDF. Integrating over k·n = x shows that this normalization is in agreement with the momentum sum rule for gluons taking the form 0 ≤ 1 0 dx xf 1 (x) ≤ 1, (2.13) which is not saturated because there is also a contribution from quarks. The Wilson loop correlator Again we start with a fully unintegrated correlator, now containing a Wilson loop operator, Γ [loop] 0 (k; P ) ≡ d 4 ξ (2π) 4 e ik·ξ P | U [loop] |P , (2.14) where we implicitly include color tracing. The above quantity is a path-dependent quantity that reduces to the normalization N c P |P upon integration over d 4 k. In certain processes the latter contribution is subtracted, involving the operator U [loop] −I, such as in diffractive scattering [33]. As this subtraction only matters at k = 0, we will not consider it here. To make contact with TMD correlators we can construct the loop from two staple-like paths along n, possibly including additional (color averaged) loops [23] in U [0,ξ] U ′ [ξ,0] , but now without 'parton' fields residing at 0 and ξ, Γ [U,U ′ ] 0 (k; P, n) ≡ d 4 ξ (2π) 4 e ik·ξ P | U [0,ξ] U ′ [ξ,0] |P . (2.15) In the unintegrated amplitude expansion constrained by hermiticity and parity conservation for an unpolarized hadron, just one (T -even) amplitude remains: 3 Γ [U,U ′ ] 0 (k, P, n) = B 1 M 2 , (2.16) with B 1 = B 1 (k·n, k·P, k 2 ). The absence of the 'parton' fields and the structure of the loop on the light-front still allows integration over k·P , and invariance in the ξ·P direction implies a delta function δ(k·n): Γ [U,U ′ ] 0 (x, k T ; P, n) ≡ dξ·P d 2 ξ T (2π) 3 e ik·ξ P | U [0,ξ] U ′ [ξ,0] |P ξ·n=0 = δ(x) Γ [U,U ′ ] 0 (k T ; P, n),(2.17) where the loop correlator integrated over k·P and k·n is given by Γ [U,U ′ ] 0 (k T ; P, n) ≡ d 2 ξ T (2π) 2 e ik T ·ξ T P | U [0,ξ] U ′ [ξ,0] |P ξ·n=0 . (2.18) Note that this correlator allows for azimuthal dependence in k T . In the limit x → 0 we have t ≡ k 2 = k 2 T = −k 2 T (see also appendix B). Bearing in mind the proportionality to the longitudinal extent L of the loop, L ≡ dξ·P = 2π δ(0), the light-front correlator in eq. (2.18) is parametrized in terms of TMDs as follows (we suppress now the dependence on P and n): Γ [U,U ′ ] 0 (k T ) = πL M 2 e(k 2 T ), (2.19) where the expression of the function e in terms of the coefficient B 1 can be found in appendix B. The correlator in eq. (2.18) appears for instance in the dipole cross section ∝ d 2 r ⊥ /(2π) 2 e −ik ⊥ ·r ⊥ Tr U (0)U † (r ⊥ ) /N c , where U is a gauge link running along the light cone from −∞ to +∞, up to endpoints forming a loop (see e.g. ref. [34]). In section 3 we will elaborate on the link between the Wilson loop operator and the gluon-gluon correlator at zero longitudinal momentum (i.e. x = 0). We will consider two specific gauge links, namely a future and a past pointing staple-like gauge link, the simplest ones denoted by [+] and [−] respectively. These two gauge links also make up the rectangular Wilson loop U [ ] ≡ U [+] [0,ξ] U [−] [ξ,0] , consisting of Wilson lines running from −∞ to ∞ along the n direction, at some transverse separation ξ T . This loop can be written as a 'square' of the form O(0)O † (ξ) for a specific nonlocal operator O: 20) which are just the ingredients in the dipole operator [34] now including transverse pieces. U [ ] = U n [−∞,0 T ;∞,0 T ] U T [∞,0 T ;∞,ξ T ] U n [∞,ξ T ;−∞,ξ T ] U T [−∞,ξ T ;∞,0 T ] = U T [−∞,∞ T ;−∞,0 T ] U n [−∞,0 T ;∞,0 T ] U T [∞,0 T ;∞,∞ T ] × U T [−∞,∞ T ;−∞,ξ T ] U n [−∞,ξ T ;∞,ξ T ] U T [∞,ξ T ;∞,∞ T ] † ,(2.From eq. (2.20) it follows that e [ ] (k 2 T ) is positive definite. Similarly, also f [+,−] 1 (x, k 2 T ) is positive definite. Vector polarized hadrons The gluon-gluon correlator Let us now consider vector polarized hadrons. Since here we are only interested in vector polarization (we already discussed the unpolarized case), we would like to single out those terms from the parametrization of the correlator that describe a vector polarized hadron (i.e. terms containing S). To that end, we define ∆Γ µν;ρσ (k; P, S) ≡ 1 2 [Γ µν;ρσ (k; P, S) − Γ µν;ρσ (k; P, −S)] . (2.21) A possible parametrization of this unintegrated correlator that is constrained by hermiticity and parity conservation and respects relations induced by time reversal (see appendix A) is 4 ∆Γ µν;ρσ (k; P, S) = − 2M A 7 ǫ µνρσ k·S + iM A 8 ǫ µνP [ρ S σ] − ǫ ρσP [µ S ν] + iM A 9 ǫ µνS[ρ P σ] − ǫ ρσS[µ P ν] + iM A 10 ǫ µνk[ρ S σ] − ǫ ρσk[µ S ν] + iM A 11 ǫ µνS[ρ k σ] − ǫ ρσS[µ k ν] + iA 12 M ǫ µνP [ρ P σ] − ǫ ρσP [µ P ν] k·S + iA 13 M ǫ µνk[ρ k σ] − ǫ ρσk[µ k ν] k·S + iA 14 M ǫ µνP [ρ k σ] − ǫ ρσP [µ k ν] k·S + iA 15 M ǫ µνk[ρ P σ] − ǫ ρσk[µ P ν] k·S + A 16 + iA 17 M ǫ µνP S k [ρ P σ] + A 16 − iA 17 M ǫ ρσP S k [µ P ν] + A 18 + iA 19 M ǫ µνkS k [ρ P σ] + A 18 − iA 19 M ǫ ρσkS k [µ P ν] + A 20 + iA 21 M ǫ µνkP P [ρ S σ] + A 20 − iA 21 M ǫ ρσkP P [µ S ν] + A 22 + iA 23 M ǫ µνkP k [ρ S σ] + A 22 − iA 23 M ǫ ρσkP k [µ S ν] + A 24 + iA 25 M 3 ǫ µνkP k [ρ P σ] k·S + A 24 − iA 25 M 3 ǫ ρσkP k [µ P ν] k·S,(2.22) where we have employed the notation ǫ abcd ≡ ǫ µνρσ a µ b ν c ρ d σ and square brackets denote antisymmetrization of the indices. The terms with coefficients A 7 , A 16 , A 18 , A 20 , A 22 , A 24 are T -odd, and we note that the ones with coefficients A 8 up to A 15 are slightly different from those in ref. [7]. Employing symmetric traceless tensors in k T , the light-front correlator is parametrized in terms of leading twist (i.e. twist-2), definite rank TMDs as follows (in analogy to eq. (2.10)): ∆Γ ij (x, k T ) = ∆Γ ij L (x, k T ) + ∆Γ ij T (x, k T ), (2.23) where 5 ∆Γ ij L (x, k T ) = x 2 iǫ ij T S L g 1 (x, k 2 T ) + ǫ {i T α k j}α T S L 2M 2 h ⊥ 1L (x, k 2 T ) , (2.24) ∆Γ ij T (x, k T ) = x 2 − g ij T ǫ S T k T T M f ⊥ 1T (x, k 2 T ) + iǫ ij T k T ·S T M g 1T (x, k 2 T ) − ǫ k T {i T S j} T + ǫ S T {i T k j} T 4M h 1 (x, k 2 T ) − ǫ {i T α k j}αS T T 2M 3 h ⊥ 1T (x, k 2 T ) ,(2.h ⊥ 1L , f ⊥ 1T , h 1 , and h ⊥ 1T are T -odd. The only surviving collinear PDF is the rank-0 function g 1 , where we have omitted the index 'L' on g 1 ≡ g 1L . Note that h 1 = h 1T . The function h 1 now corresponds to the function −∆H T in the originally proposed parametrization in ref. [7]. The link to the more traditional parametrization is found by using the identity ǫ {i T α k j}αβ T S T β = ǫ k T {i T k j} T k T ·S T + 1 4 k 2 T S {j T ǫ i}k T T + k {j T ǫ i}S T T . (2.26) We can now recast eq. (2.23) into the more traditional, quite compact form ∆Γ ij (x, k T ) = x 2 g ij T ǫ k T S T T M f ⊥ 1T (x, k 2 T ) + iǫ ij T g 1s (x, k 2 T ) − ǫ k T {i T S j} T + ǫ S T {i T k j} T 4M h 1T (x, k 2 T ) − ǫ k T {i T k j} T 2M 2 h ⊥ 1s (x, k 2 T ) ,(2.27) where we have made use of the shorthand notation (2.28) and likewise for h ⊥ 1s . The functions h 1 and h 1T are related as g 1s (x, k 2 T ) ≡ S L g 1L (x, k 2 T ) + k T ·S T M g 1T (x, k 2 T ),h 1 (x, k 2 T ) ≡ h 1T (x, k 2 T ) + k 2 T 2M 2 h ⊥ 1T (x, k 2 T ). (2.29) The function h 1 is a rank-1 function, h 1T contains both rank-1 and rank-3 pieces, and h ⊥ 1T is a rank-3 function. Note that the function h 1 for gluons is, in spite of similarity in name, quite different from the quark transverse polarization (transversity) function h 1 . The Wilson loop correlator For the same reason as in the case of the gluon-gluon correlator, we define ∆Γ [U,U ′ ] 0 (k; P, S, n) ≡ 1 2 Γ [U,U ′ ] 0 (k; P, S, n) − Γ [U,U ′ ] 0 (k; P, −S, n) . (2.30) A possible parametrization of this unintegrated correlator that is constrained by hermiticity and parity conservation and respects relations induced by time reversal (see appendix A) is ∆Γ [U,U ′ ] 0 (k; P, S, n) = B 2 M 3 ǫ nP kS , (2.31) which is a T -odd term. The loop correlator integrated over k·P and k·n is parametrized in terms of TMDs as follows: ∆Γ [U,U ′ ] 0 (k T ) = πL M 2 ǫ S T k T T M e T (k 2 T ),(2.32) where the expression of the T -odd function e T in terms of the coefficient B 2 can be found in appendix B. Tensor polarized hadrons The gluon-gluon correlator We now include tensor polarization, which is relevant for spin-1 hadrons. Similarly as for the vector polarized case, we define ∆Γ µν;ρσ (k; P, T ) ≡ 1 2 [Γ µν;ρσ (k; P, T ) − Γ µν;ρσ (k; P, −T )] ,(2.33) where we have taken the vector polarization to be zero (i.e. S = 0). A possible parametrization of this unintegrated correlator that is constrained by hermiticity and parity conservation and respects relations induced by time reversal (see appendix A) is ∆Γ µν;ρσ (k; P, T ) = A 26 k [µ T ν][ρ k σ] + A 27 P [µ T ν][ρ P σ] + (A 28 + iA 29 ) k [µ T ν][ρ P σ] + (A 28 − iA 29 ) P [µ T ν][ρ k σ] + A 30 + iA 31 M 2 k α T α[µ k ν] k [ρ P σ] + A 30 − iA 31 M 2 k α T α[ρ k σ] k [µ P ν] + A 32 + iA 33 M 2 k α T α[µ P ν] k [ρ P σ] + A 32 − iA 33 M 2 k α T α[ρ P σ] k [µ P ν] + M 2 A 34 g µ[ρ T σ]ν − g ν[ρ T σ]µ + (A 35 + iA 36 ) k α T α[µ g ν][ρ k σ] + (A 35 − iA 36 ) k α T α[ρ g σ][µ k ν] + (A 37 + iA 38 ) k α T α[µ g ν][ρ P σ] + (A 37 − iA 38 ) k α T α[ρ g σ][µ P ν] + A 39 k α k β T αβ ǫ µνκλ ǫ ρσ κλ + A 40 M 2 k α k β T αβ P [µ g ν][ρ P σ] + A 41 M 2 k α k β T αβ k [µ g ν][ρ k σ] + (A 42 + iA 43 ) M 2 k α k β T αβ P [µ g ν][ρ k σ] + (A 42 − iA 43 ) M 2 k α k β T αβ k [µ g ν][ρ P σ] + A 44 M 4 k α k β T αβ P [µ k ν] P [ρ k σ] ,(2.∆Γ ij (x, k T ) = ∆Γ ij LL (x, k T ) + ∆Γ ij LT (x, k T ) + ∆Γ ij T T (x, k T ), (2.35) where ∆Γ ij LL (x, k T ) = x 2 − g ij T S LL f 1LL (x, k 2 T ) + k ij T S LL M 2 h ⊥ 1LL (x, k 2 T ) , (2.36) ∆Γ ij LT (x, k T ) = x 2 − g ij T k T ·S LT M f 1LT (x, k 2 T ) + iǫ ij T ǫ S LT k T T M g 1LT (x, k 2 T ) + S {i LT k j} T M h 1LT (x, k 2 T ) + k ijα T S LT α M 3 h ⊥ 1LT (x, k 2 T ) , (2.37) ∆Γ ij T T (x, k T ) = x 2 − g ij T k αβ T S T T αβ M 2 f 1T T (x, k 2 T ) + iǫ ij T ǫ β T γ k γα T S T T αβ M 2 g 1T T (x, k 2 T ) + S ij T T h 1T T (x, k 2 T ) + S {i T T α k j}α T M 2 h ⊥ 1T T (x, k 2 T ) + k ijαβ T S T T αβ M 4 h ⊥⊥ 1T T (x, k 2 T ) . (2.38) The expressions of the TMDs in terms of the coefficients A i can be found in appendix B. The functions g 1LT and g 1T T are T -odd. In the collinear case the rank-0 functions f 1LL and h 1T T survive. The former function was also called b 1 in the quark case, and the latter function shows up in the structure function ∆(x, Q 2 ) discussed in ref. [19] and is called ∆ 2 G(x) in ref. [20]. The Wilson loop correlator Similarly to the vector polarized case, we define ∆Γ [U,U ′ ] 0 (k; P, T, n) ≡ 1 2 Γ [U,U ′ ] 0 (k; P, T, n) − Γ [U,U ′ ] 0 (k; P, −T, n) ,(2.39) where we have taken the vector polarization to be zero (i.e. S = 0). A possible parametrization of this unintegrated correlator that is constrained by hermiticity and parity conservation and respects relations induced by time reversal (see appendix A) is ∆Γ [U,U ′ ] 0 (k; P, T, n) = B 3 M 4 k µ k ν T µν + B 4 n µ n ν T µν + B 5 M 2 k µ n ν T µν . (2.40) The loop correlator integrated over k·P and k·n is parametrized in terms of TMDs as follows: [ξ,0] . We will start from the Wilson loop correlator integrated over k·P and k·n given in eq. (2.18). To study its k T dependence, we use the results in eq. (15) of ref. [23] to calculate k i T k j T Γ 0 . Performing one partial integration in 0 and the other in ξ and using the relevant gluonic pole factor C [ ] GG = 4, we obtain ∆Γ [U,U ′ ] 0 (k T ) = πL M 2 S LL e LL (k 2 T ) + k T ·S LT M e LT (k 2 T ) + k αβ T S T T αβ M 2 e T T (k 2 T ) ,(2.k i T k j T Γ [ ] 0 (k T ) = 4 d 2 ξ T (2π) 2 e ik T ·ξ T P | G i T (0) U [+] [0,ξ] G j T (ξ) U [−] [ξ,0] |P ξ·n=0 = dη·P dη ′ ·P d 2 ξ T (2π) 2 e ik T ·ξ T P | F ni (η ′ ) U [+] [η ′ ,η] F nj (η) U [−] [η,η ′ ] |P η ′ ·n=η·n=0, η ′ T =0 T , η T =ξ T = 2πL dξ·P d 2 ξ T (2π) 3 e ik·ξ P | F ni (0) U [+] [0,ξ] F nj (ξ) U [−] [ξ,0] |P ξ·n=k·n=0 = 2πL Γ [+,−] ij (0, k T ),(3.1) which implies that Γ [+,−] ij (0, k T ) = k i T k j T 2πL Γ [ ] 0 (k T ). (3.2) The dependence on k T is in fact in this limit just the dependence on t ≡ k 2 , remaining after the integration over k·n = x and k·P (the mass spectrum of intermediate states). Thus it is appropriate to write the previous equation as Γ [+,−] ij (x, k T ) x→0 −→ k i T k j T 2πL Γ [ ] 0 (k T ) k 2 T =t . (3.3) The above results agree with the result in ref. [34] where in the small-x limit f [+,−] 1 (x, k 2 T ) becomes proportional to the dipole cross section. In ref. [29] that connection was made on the correlator level for the case of a transversely polarized hadron, which corresponds to the above eq. (3.1) and will also be discussed below. For unpolarized hadrons the right-hand side of eq. (3.3) is given by the parametrization in eq. (2.19). It follows that Γ ij U (x, k T ) = x 2 − g ij T f 1 (x, k 2 T ) + k ij T M 2 h ⊥ 1 (x, k 2 T ) x→0 −→ k i T k j T 2M 2 e(k 2 T ) = 1 2 − g ij T k 2 T 2M 2 e(k 2 T ) + k ij T M 2 e(k 2 T ) , (3.4) which implies that lim x→0 xf 1 (x, k 2 T ) = k 2 T 2M 2 lim x→0 xh ⊥ 1 (x, k 2 T ) = k 2 T 2M 2 e(k 2 T ). (3.5) This means that h ⊥ 1 must be maximal [7], i.e. h ⊥ 1 = 2M 2 f 1 /k 2 T , as it is in fact the case in the small-x k T -factorization approach [35] and in the framework of the color glass condensate [36]. This result indicates that the unpolarized dipole gluon distribution grows as 1/x towards small x, apart from subdominant modifications from resummation of large logarithms in 1/x and higher twist effects. For longitudinally polarized hadrons eq. (3.3) implies that g 1 and h ⊥ 1L are less divergent than 1/x in the limit of small x. For g 1 this is in accordance with the fact that in DGLAP and CCFM evolution the splitting kernel lacks the 1/x factor of the kernel of f 1 , see e.g. ref. [37]. Again this does not include resummation of large logarithms in 1/x leading to nonlinear evolution, which may alter the result in the very small x region [38][39][40]. Now let us consider transversely polarized hadrons. The right-hand side of eq. (3.3) is given by the parametrization in eq. (2.32). We find for the symmetric part (symmetric in i, j) of the transversely polarized gluon-gluon correlator ∆Γ ij T sym (x, k T ) = x 2 − g ij T ǫ S T k T T M f ⊥ 1T (x, k 2 T ) − ǫ k T {i T S j} T + ǫ S T {i T k j} T 4M h 1 (x, k 2 T ) − ǫ {i T α k j}αS T T 2M 3 h ⊥ 1T (x, k 2 T ) x→0 −→ k i T k j T 2M 2 ǫ S T k T T M e T (k 2 T ) = 1 2 − g ij T ǫ S T k T T M k 2 T 2M 2 e T (k 2 T ) − ǫ k T {i T S j} T + ǫ S T {i T k j} T 4M k 2 T 2M 2 e T (k 2 T ) + ǫ {i T α k j}αS T T 2M 3 e T (k 2 T ) , (3.6) which implies that lim x→0 xf ⊥ 1T (x, k 2 T ) = lim x→0 xh 1 (x, k 2 T ) = − k 2 T 2M 2 lim x→0 xh ⊥ 1T (x, k 2 T ) = 1 2 lim x→0 xh 1T (x, k 2 T ) = k 2 T 2M 2 e T (k 2 T ), (3.7) in agreement with the leading logarithmic result of ref. [29]. It involves the C-odd operator structure U [ ] − U [ ] † ( see appendix A) surviving in ∆Γ [ ] 0 (k; P, S, n) for a transversely polarized proton, which is the dipole odderon operator [41]. Therefore, this is also referred to as the spin-dependent odderon [42]. The odderon operator O ⊥ 1T as defined in ref. [29] and used in a model calculation in ref. [43] is related to e T by O ⊥ 1T = π e T /(2M 2 ). The odderon in transverse spin asymmetries in elastic scattering has earlier been considered in refs. [44][45][46], but without discussion of its operator structure. As the only nonzero function in the unpolarized case is the even rank function e(k 2 T ), which survives for the C-even Wilson loop operator combination U [ ] +U [ ] † , it also follows that there appears no spin-independent odderon in this formalism, or rather that it is less divergent than 1/x in the limit of small x. This suggests that it will be suppressed in the small-x limit compared to the C-even leading contribution. For spin-1 hadrons eq. (3.3) implies that at small x three tensor polarized TMDs remain, while the rest becomes zero. To be specific, for longitudinal-longitudinal (LL) polarization we find ∆Γ ij LL (x, k T ) = x 2 − g ij T S LL f 1LL (x, k 2 T ) + k ij T S LL M 2 h ⊥ 1LL (x, k 2 T ) x→0 −→ k i T k j T 2M 2 S LL e LL (k 2 T ) = 1 2 − g ij T S LL k 2 T 2M 2 e LL (k 2 T ) + k ij T S LL M 2 e LL (k 2 T ) , (3.8) which implies that lim x→0 xf 1LL (x, k 2 T ) = k 2 T 2M 2 lim x→0 xh ⊥ 1LL (x, k 2 T ) = k 2 T 2M 2 e LL (k 2 T ). (3.9) For the case of longitudinal-transverse (LT ) polarization, we find for the symmetric part of the gluon-gluon correlator ∆Γ ij LT sym (x, k T ) = x 2 − g ij T k T ·S LT M f 1LT (x, k 2 T ) + S {i LT k j} T M h 1LT (x, k 2 T ) + k ijα T S LT α M 3 h ⊥ 1LT (x, k 2 T ) x→0 −→ k i T k j T 2M 2 k T ·S LT M e LT (k 2 T ) = 1 2 − g ij T k T ·S LT M k 2 T 4M 2 e LT (k 2 T ) + S {i LT k j} T M k 2 T 4M 2 e LT (k 2 T ) − k ijα T S LT α M 3 e LT (k 2 T ) , (3.10) which implies that lim x→0 xf 1LT (x, k 2 T ) = lim x→0 xh 1LT (x, k 2 T ) = − k 2 T 4M 2 lim x→0 xh ⊥ 1LT (x, k 2 T ) = k 2 T 4M 2 e LT (k 2 T ). (3.11) For the case of transverse-transverse (T T ) polarization, we find for the symmetric part of the gluon-gluon correlator ∆Γ ij T T sym (x, k T ) = x 2 − g ij T k αβ T S T T αβ M 2 f 1T T (x, k 2 T ) + S ij T T h 1T T (x, k 2 T ) + S {i T T α k j}α T M 2 h ⊥ 1T T (x, k 2 T ) + k ijαβ T S T T αβ M 4 h ⊥⊥ 1T T (x, k 2 T ) x→0 −→ k i T k j T 2M 2 k αβ T S T T αβ M 2 e T T (k 2 T ) = 1 2 − g ij T k αβ T S T T αβ M 2 k 2 T 6M 2 e T T (k 2 T ) + S ij T T k 4 T 4M 4 e T T (k 2 T ) − S {i T T α k j}α T M 2 k 2 T 3M 2 e T T (k 2 T ) + k ijαβ T S T T αβ M 4 e T T (k 2 T ) , (3.12) which implies that lim x→0 xf 1T T (x, k 2 T ) = 2M 2 3k 2 T lim x→0 xh 1T T (x, k 2 T ) = − 1 2 lim x→0 xh ⊥ 1T T (x, k 2 T ) = k 2 T 6M 2 lim x→0 xh ⊥⊥ 1T T (x, k 2 T ) = k 2 T 6M 2 e T T (k 2 T ). (3.13) Summary and discussion We have parametrized the gluon light-front correlators in terms of definite rank TMDs using a basis of symmetric traceless tensors in k T . In table 1 we list the leading twist TMDs (multiplied by x), their rank, and their behavior under time reversal and charge conjugation. The rank-0 functions are the ones that also appear as collinear PDFs. In the last column the x → 0 limit is considered for the functions xf [+,−] ... (x, k 2 T ). We emphasize that the connection to the Wilson loop or dipole TMDs applies only to the TMDs with one future and one past pointing link. Some of these functions are expected to be zero at x = 0, others become equal to the TMDs e ... (k 2 T ) in the Wilson loop operator. Conjecturing that the dependence on k 2 T reflects the analytic behavior in k 2 , it simplifies the picture for the gluon TMDs xf [+,−] ... (x, k 2 T ) at small x, several of them becoming proportional to one another. The C-and T -behavior of the TMDs in the gluon-gluon correlator and those in the Wilson loop correlator correctly match. The functions with a nonvanishing limit are expected to behave as 1/x, or a slightly modified power after resummation of other small-x effects, e.g. with an ln(1/x) behavior. The functions with a vanishing limit are the ones for longitudinally polarized hadrons as well as those linked to circular gluon polarization (the g-type TMDs). We end with a brief discussion of the experimental possibilities to study the gluon TMDs. The unpolarized and vector polarized gluon TMDs could be investigated in processes at RHIC, at the LHC, possibly at a future polarized fixed-target experiment at the LHC called AFTER@LHC [47], and at an EIC [48]. For instance, the unpolarized gluon TMDs could be studied at the LHC and at AFTER@LHC in (pseudo)scalar C-even heavy quarkonium production, such as χ c,b and η b,c in the color-singlet configuration [49,50]. Another option is to consider pseudovector quarkonium such as J/Ψ and Υ, which is predominantly in the color-singlet configuration when produced in gluon fusion together with an additional isolated photon in the final state [51]. The latter is also useful to investigate the QCD evolution of the gluon distributions. The linearly polarized gluon TMDs could be studied by measuring cos(2φ) modulations in processes such as dijet or heavy quark pair production in electron-proton or electron-nucleus collisions [52,53] and in virtual photon-jet pair production in pp or pA collisions [36]. They can also be accessed through heavy quarkonium production in (un)polarized pp collisions [49,50] in association with other gluon TMDs. The most promising processes that directly give access to the gluon Sivers effect are p ↑ p → γ jet X at RHIC and AFTER@LHC [54], p ↑ p → J/Ψ γ X or p ↑ p → J/Ψ J/Ψ X at AFTER@LHC [55], and ep ↑ → e ′ cc X at an EIC [56]. Production of color-singlet heavy quarkonium states [50] and of photon pairs from polarized proton collisions [57] are also valid possibilities. For some of these processes TMD factorization has not been proven yet, neither for general x nor for small x. In order to experimentally probe the functions that remain in the small-x limit, additional processes such as DIS, Drell-Yan, semi-inclusive DIS, or p A → h X offer possibilities. For a discussion and a more detailed list of relevant processes see refs. [29,58,59]. The study of tensor polarized gluon TMDs would be possible at the experiments proposed to investigate polarized deuterons, e.g. at the EIC option put forward at Jefferson Lab (JLEIC) [48,60,61], or at COMPASS [62], although there the region of small x is very limited. Ref. [30] Ref. [7] Rank even even e T T Table 1. An overview of the leading twist gluon TMDs for unpolarized, vector polarized, and tensor polarized hadrons. In the second and third column, the names of the functions in this paper are compared to the ones in refs. [7,30]. In the fourth column we list the rank of the function. Furthermore, we list the properties (even/odd) under time reversal (T ) and charge conjugation (C), see appendix A. In the last column it is indicated to which e-type function the TMD reduces in the limit x → 0. As a shorthand, we use the moment notation f (n) T C Limit x → 0 xf 1 xf 1 xG 0 even even e (1) xh ⊥ 1 xh ⊥ 1 xH ⊥ 2 even even e xg 1 xg 1L −x∆G L 0 even odd 0 xh ⊥ 1L xh ⊥ 1L −x∆H ⊥ L 2 odd even 0 xf ⊥ 1T xf ⊥ 1T −xG T 1 odd odd e (1) T xg 1T xg 1T −x∆G T 1 even even 0 xh 1 xh 1T + xh ⊥(1) 1T −x∆H T 1 odd odd e (1) T xh ⊥ 1T xh ⊥ 1T −x∆H ⊥ T 3 odd odd −e T xf 1LL... (x, k 2 T ) ≡ [k 2 T /(2M 2 )] n f ... (x, k 2 T ). A Constraints on correlators The gluon-gluon and Wilson loop correlators in this paper are constrained by hermiticity and parity (P ). In the parametrizations hermiticity ensures that the functions are real and parity conservation only allows for P -even terms. By simply omitting the spin vector and/or tensor from the expressions above, we obtain the constraints that apply to unpolarized (omitting S and T ), vector polarized (omitting T ), and tensor polarized (omitting S) hadrons. The effect of charge conjugation (C) symmetry for gluons is found by writing down the conjugate correlator involving the conjugate field A c µ = −A † µ = −A µ , such that F µν c = −F µν and U c [0,ξ] = U † [0,ξ] . As for the quark case, charge conjugation does not really give a constraint (hermiticity has already been used), but it enables us to connect partons at negative x (and k T ) to antipartons at positive x (and k T ), which is relevant for sum rules and for the relation to antiprotons. For gluons the C-behavior only becomes important due to the gauge link structure, in particular for the situation with two different gauge links. The correlator for the charge conjugated fields becomes Γ c[U,U ′ ]µν;ρσ (x, k T ) = Γ [U † ,U ′ † ]µν;ρσ (x, k T ), (A.7) which after rewriting and using the hermiticity constraint can also be related to the correlator at negative x (and k T ), Γ c[U,U ′ ]µν;ρσ (x, k T ) = Γ [U ′ † ,U † ]µν;ρσ * (−x, −k T ) = Γ [U ′ † ,U † ]ρσ;µν (−x, −k T ). (A.8) For the TMDs, depending on rank and symmetry in Lorentz indices, the relations become either f [U,U ′ ] 1 (−x, k 2 T ) = −f [U ′ ,U ] 1 (x, k 2 T ), (A.9) or g [U,U ′ ] 1 (−x, k 2 T ) = +g [U ′ ,U ] 1 (x, k 2 T ), (A.10) referred to as C-even and C-odd respectively. The TMDs f 1 , h ⊥ 1 , h ⊥ 1L , g 1T , f 1LL , h ⊥ 1LL , g 1LT , f 1T T , h 1T T h ⊥ 1T T , and h ⊥⊥ 1T T are C-even, whereas g 1 , f ⊥ 1T , h 1 , h ⊥ 1T , f 1LT , h 1LT , h ⊥ 1LT , and g 1T T are C-odd. The C-property is of special interest for the Wilson loop correlator or in general for correlators containing a loop Tr c ( U [0,ξ] U ′ [ξ,0] ) = Tr c (U [0,ξ] U ′ † [0,ξ] ). One has the additional property Γ [U † ,U ′ † ] 0 (k T ) = Γ [U ′ ,U ] 0 (k T ), thus one finds Γ c[U,U ′ ] 0 (k T ) = Γ [U † ,U ′ † ] 0 (k T ) = Γ [U ′ ,U ] 0 (k T ) = Γ [U ′ † ,U † ] 0 (−k T ) = Γ [U,U ′ ] 0 (−k T ). (A.11) Hence in the Wilson loop correlator the C-even and C-odd functions can be directly identified with the even and odd rank functions. For the TMDs in the correlator Γ [ ] (k T ) the functions e, e LL , and e T T are C-even and the functions e T and e LT are C-odd. The C-even and C-odd functions are also the ones that would appear in the correlators Γ [ ] 0 (k T ) ± Γ [ † ] 0 (k T ) /2 respectively. The C-behavior of the TMDs in the gluon-gluon and Wilson loop correlators is consistent with the small-x matching in section 3. B Definitions of TMDs In this appendix the definitions of the various TMDs are given in terms of the coefficients A i and B i that have been introduced in the parametrizations at the level of the unintegrated correlators. B.1 The gluon-gluon correlator Let us denote by Γ(k) the gluon-gluon correlator for any type of polarization, 6 then the light-front correlator is defined as Γ(x, k T ) ≡ dk·P Γ(k) = M 2 2 [dσdτ ] Γ(k), (B.1) where we have introduced the shorthand notation [dσdτ ] ≡ dσdτ δ τ − xσ + x 2 + k 2 T M 2 , (B.2) with the dimensionless invariants σ and τ given by σ ≡ 2k·P M 2 , τ ≡ k 2 M 2 , (B.3) 6 Lorentz indices are omitted for simplicity, since they are not relevant here. spanning regions in remnant mass M 2 R ≡ (P − k) 2 and in the partonic virtuality k 2 . For both of these, the main contribution comes from small (hadronic) values (i.e. σ and τ of order one). The (leading twist) TMDs that occur in the parametrization of the gluon-gluon correlator for the various types of polarization in eqs. (2.11), (2.23), and (2.35), are related to the coefficients A i as follows: xf 1 (x, k 2 T ) ≡ M 2 [dσdτ ] A 2 + 2xA 4 + x 2 A 3 + k 2 T 2M 2 A 6 , (B.4) xh ⊥ 1 (x, k 2 T ) ≡ M 2 [dσdτ ] A 6 , (B.5) xg 1 (x, k 2 T ) ≡ 2M 2 [dσdτ ] A 8 + A 9 + x (A 10 + A 11 ) + σ 2 − x [A 12 + x (A 14 + A 15 ) + x 2 A 13 + k 2 T 2M 2 A 19 + A 23 + σ 2 − x A 25 , (B.6) xh ⊥ 1L (x, k 2 T ) ≡ −2M 2 [dσdτ ] A 18 + A 22 + σ 2 − x A 24 , (B.7) xf ⊥ 1T (x, k 2 T ) ≡ M 2 [dσdτ ] [A 16 − A 20 + x (A 18 − A 22 )] , (B.8) xg 1T (x, k 2 T ) ≡ −M+ A 40 + 2xA 42 + x 2 A 41 + (σ − 3x)A 30 − (σ − 2x) 2 4 − k 2 T 2M 2 A 44 , (B.12) xh ⊥ 1LL (x, k 2 T ) ≡ − 2M 2 3 [dσdτ ] A 26 − A 32 + (σ − 3x)A 30 − (σ − 2x) 2 4 − k 2 T 2M 2 A 44 ,(B.+ k 2 T 2M 2 A 30 + x − σ 2 A 44 , (B.16) xh ⊥ 1LT (x, k 2 T ) ≡ −M 2 [dσdτ ] A 30 + x − σ 2 A 44 , (B.17) xf 1T T (x, k 2 T ) ≡ M 2 2 [dσdτ ] A 40 + 2xA 42 + x 2 A 41 + k 2 T 6M 2 A 44 , (B.18) xg 1T T (x, k 2 T ) ≡ M 2 2 [dσdτ ] (A 33 + xA 31 ) , (B.19) xh 1T T (x, k 2 T ) ≡ − M 2 2 [dσdτ ] A 27 + 2xA 28 + x 2 A 26 + k 2 T M 2 (A 32 + xA 30 ) − k 4 T 4M 4 A 44 , (B.20) xh ⊥ 1T T (x, k 2 T ) ≡ M 2 2 [dσdτ ] A 32 + xA 30 − k 2 T 3M 2 A 44 , (B.21) xh ⊥⊥ 1T T (x, k 2 T ) ≡ M 2 2 [dσdτ ] A 44 . (B.22) B.2 The Wilson loop correlator For the Wilson loop correlator, translation invariance in the ξ·P direction forces k·n = x to be zero and the integration over x is actually naturally the first to be done, even before the integration over k·P . The remaining dependence is on the invariant k 2 , which for vanishing x is just k 2 = k 2 T = −k 2 T . The TMDs in the parametrization of the Wilson loop correlator for the various types of polarization in eqs. (2.19), (2.32), and (2.41) depend on t = k 2 and are related to the coefficients B i as follows: e(k 2 T ) ≡ M 2 2πL dx dσ B 1 , (B.23) e T (k 2 T ) ≡ M 2 2πL dx dσ B 2 , (B.24) e LL (k 2 T ) ≡ − M 2 4πL dx dσ 2B 4 + (σ − 2x)B 5 + (σ − 2x) 2 2 − k 2 T M 2 B 3 , (B.25) e LT (k 2 T ) ≡ − M 2 4πL dx dσ [B 5 + (σ − 2x)B 3 ] , (B.26) e T T (k 2 T ) ≡ M 2 4πL dx dσ B 3 . (B.27) Γ ij LT (x, b T ) = x 2 − iM g ij T b T ·S LTf (1) 1LT (x, b 2 T ) − M ǫ ij T ǫ S LT b T Tg (1) 1LT (x, b 2 T ) + iM S {i LT b j} Th (1) 1LT (x, b 2 T ) − iM 3 b ijα T S LT α 6h ⊥(3) 1LT (x, b 2 T ) , (C.19) Γ ij T T (x, b T ) = x 2 M 2 g ij T b αβ T S T T αβ 2f (2) 1T T (x, b 2 T ) − iM 2 ǫ ij T ǫ β T γ b γα T S T T αβ 2g (2) 1T T (x, b 2 T ) + S ij T Th 1T T (x, b 2 T ) − M 2 S {i T T α b j}α T 2h ⊥(2) 1T T (x, b 2 T ) + M 4 b ijαβ T S T T αβ 24h ⊥⊥(4) 1T T (x, b 2 T ) . . The expressions of the TMDs in terms of the coefficients A i can be found in appendix B. The functions 41) where the expressions of the TMDs in terms of the coefficients B i can be found in appendix B.3 The gluon-gluon correlator at small x In this section we discuss the relation between the gluon-gluon correlator at small x and the Wilson loop correlator. This connection only applies to the gluon-gluon correlator with the staple-like [+] and [−] gauge links. In the Wilson loop correlator those gauge links constitute the rectangular Wilson loop U [ ] ≡ U Time reversal (T ) transformations relate correlators with time-reversed gauge link structures, e.g. time reversal interchanges the staple-like [+] and [−] gauge links. Hence, time reversal invariance is not used as a constraint in the parametrizations of the correlators. For the gluon-gluon correlator the constraints are as follows: Hermiticity: Γ [U,U ′ ] ρσ;µν * (k, P, S, T, n) = Γ [U,U ′ ] µν;ρσ (k, P, S, T, n), (A.1) Parity: Γ [U,U ′ ] µν;ρσ (k, P, S, T, n) = Γ [U,U ′ ] µν;ρσ (k,P , −S,T ,n), (A.2) Time reversal: Γ [U,U ′ ] µν;ρσ * (k, P, S, T, n) = Γ [U T ,U ′T ] have introduced the notationā µ ≡ δ µν a ν andb µν ≡ δ µρ δ νσ b ρσ . Concerning the gauge links, these constraints are based on the properties U † [0,ξ] = U [ξ,0] , U P [0,ξ] = U [0,ξ] , and U T [0,ξ] = U [−0,−ξ]. By omitting the gauge links from the gluon-gluon correlator, the dependence on n is no longer present. Furthermore, the gluon-gluon correlator is antisymmetric in both the pair of indices µ, ν and ρ, σ.For the Wilson loop correlator the constraints read: light-front Wilson loop correlator for a spin-1 hadron is given in b T 2 [dσdτ ] 2A 12 + A 17 + A 21 + 2x (A 14 + A 15 ) + x (A 19 + A 23 ) + 2x 2 A 13 + k 2 xh 1 (x, k 2 T ) ≡ 2M 2 [dσdτ ] A 16 + A 20 + x (A 18 + A 22 ) + dσdτ ] A 27 − 2A 34 + 2xA 28 + x 2 A 26 + 2(σ − 2x) (A 37 + xA 35 ) T M 2 A 26 − A 32T M 2 A 25 , (B.9) k 2 T 2M 2 A 24 , (B.10) xh ⊥ 1T (x, k 2 T ) ≡ 2M 2 [dσdτ ] A 24 , (B.11) xf 1LL (x, k 2 T ) ≡ M 2 3 [+ (σ − 2x) 2 2 A 40 + 2xA 42 + x 2 A 41 − k 2 13 ) 13xf 1LT (x, k 2 T ) ≡ −M 2 [dσdτ ] A 37 + xA 35 + σ 2 − x A 40 + 2xA 42 + x 2 A 41− k 2 T 4M 2 A 30 + x − σ 2 A 44 , (B.14) xg 1LT (x, k 2 T ) ≡ − M 2 2 [dσdτ ] A 29 + x − σ 2 (A 33 + xA 31 ) , (B.15) xh 1LT (x, k 2 T ) ≡ M 2 2 [dσdτ ] A 28 + xA 26 + σ 2 − x (A 32 + xA 30 ) We use the definition of SLL that is used in ref.[18], which differs by a numerical factor from the definition in ref.[31]. Relevant mass dimensions are [Γ] = −2 and [Ai] = −4. Relevant mass dimensions are [Γ0] = −6 and [Bi] = −4. As already mentioned in the previous subsection, we omit gauge links for gluon-gluon correlators in the case of polarized hadrons, hence there is no dependence on the four-vector n (which is the case in eq. (2.8)). Since gauge links will always be present, we will, however, still allow for T -odd terms in the parametrizations. Throughout the paper, momenta indicated in boldface are two-dimensional vectors on the transverse plane rather than four-vectors. We define k µ T = [0, 0, kT ] etc., so that e.g. kT ·ST = −kT ·ST . AcknowledgmentsWe would like to acknowledge useful discussions with Paul Hoyer and Jian Zhou. This research is part of the research program of the "Stichting voor Fundamenteel Onderzoek der Materie (FOM)", which is financially supported by the "Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)" as well as the EU FP7 "Ideas" programme QWORK (contract no. 320389). YZ is financially supported by the China Scholarship Council.C Symmetric traceless tensors and TMDs in b T -spaceC.1 Symmetric traceless tensorsIn this appendix we list the completely symmetric and traceless tensors k i 1 ...in T that are built from the partonic momentum k T . Up to rank n = 4, these are given byProducts of k T can be decomposed into symmetric traceless tensors as follows:(C.7)The symmetric traceless tensor k i 1 ...in T of rank n ≥ 1 only has two independent components. This allows for a decomposition in polar coordinates:in terms of two real numbers |k T | and ϕ.C.2 TMDs in b T -spaceMathematically, TMD factorization decomposes a cross section as a product of functions in b T -space, where b T is Fourier conjugate to the partonic transverse momentum. As a byproduct, TMD evolution is multiplicative in b T -space. For these reasons, it is useful to consider the light-front correlators as a function of b T . We define correlators and functions in b T -space as Fourier transforms of the ones in k T -space:Computing directly eq. (C.9), we can see that the functions entering the parametrizations ofΓ ij (x, b T ) are not the ones in eq. (C.10), but their n-th derivatives with respect to b 2 T , n being the rank of the function in k T -space:where J n (z) is the Bessel function of the first kind of order n, which is defined as J n (z) = 1 2πi n 2π 0 dϕ e inϕ e iz cos ϕ .(C.12)In eq. 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The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they frequently show up perturbed by lowerorder terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that, surprisingly, for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework.

10.1090/mcom/3158

1509.06463

c695b3abaa15b047476a791e2aeb9cbd7eea4797

FINITE ELEMENT EXTERIOR CALCULUS WITH LOWER-ORDER TERMS 25 Feb 2016 Douglas N Arnold And Lizao Li FINITE ELEMENT EXTERIOR CALCULUS WITH LOWER-ORDER TERMS MATHEMATICS OF COMPUTATION 00025 Feb 2016arXiv:1509.06463v2 [math.NA] The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they frequently show up perturbed by lowerorder terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that, surprisingly, for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework. Introduction The vector Laplace equation, and, more generally, the Hodge Laplace equation associated to a complex, arise in many applications. The discretization of such equations is a basic motivation of the Finite Element Exterior Calculus (FEEC) [1,2]. In many applications the equations include variable coefficients and lower-order terms. While the former is included in the standard FEEC framework through weighted inner products, the latter is not, which is the subject of this work. One might expect that lower-order perturbations degrade neither the stability nor the convergence rates of stable Galerkin methods. However, this need not to be true. While stable choices of finite elements for the unperturbed Hodge Laplacian remain stable for the perturbed equation, we find that certain lower order perturbations result in decreased rates of convergence. Other choices of element pairs or perturbations do not lower the convergence rate. The situation is subtle. First, to fix ideas, we consider a simple example taken from magnetohydrodynamics [11,Chapter 3]: given vector fields f, v on a domain Ω ⊂ R 3 , find a vector field B satisfying: curl curl B − curl(v × B) = f, div B = 0, in Ω, B · n = 0, (curl B − v × B) × n = 0, on ∂Ω. Physically, B is the non-dimensionalized magnetic field inside a conductor moving with a velocity field v. The system admits a solution only when div f = 0. In that case, the solution also satisfies the following vector Laplace equation, which is solvable for any data and hence more suitable for discretization: (1.1) − grad div B + curl curl B − curl(v × B) = f. For a mixed method, we introduce σ = curl B − v × B and solve the coupled system: σ − curl B + v × B = 0, curl σ − grad div B = f. A common stable choice of mixed elements, at least when v = 0, seeks B in the space of Nédéléc face elements of the second kind of degree r ≥ 1, and σ in the space of Nédéléc edge elements of the second kind of degree (r + 1) [13,14]. In this case, for the unperturbed problem, that is when v = 0, the convergence for the L 2error in σ is of optimal order O(h r+2 ) if the solution is smooth enough. However, as we show in Section 3 and verify by numerical computation in Section 7, when v does not vanish, the L 2 -convergence for σ is reduced to order O(h r+1 ). A similar phenomenon was observed in mixed methods for the scalar Laplacian by Demlow [6]. But the vector case we study has more surprises. For example, consider the vector Laplacian perturbed by a zeroth-order term: for some real coefficient A, − grad div u + curl curl u + Au = f, in Ω, u × n = 0, div u = 0, on ∂Ω. This problem arises, for example in electromagnetism where u is the electric field and A is the conductivity coefficient. If we use the same mixed finite element method just considered, here with σ = curl u, then the L 2 error in σ is one order suboptimal for a general matrix coefficient A, but optimal if A is a scalar coefficient. We now summarize convergence rates derived from our main abstract theorems applied to the perturbed Hodge Laplace problem, of which the previous vector Laplace problem is an instance. These are important in and directly relevant to many applications. In particular, our results for the vector cases, that is, 1-forms in 2D and 1-and 2-forms in 3D, are new. On a domain Ω in R n , a k-form has n k coefficients and the meaning of the exterior derivative d and the codifferential δ depends on k. For example, when n = 3 and k = 2, we have the case considered before with δd + dδ = − grad div + curl curl. For any 0 ≤ k ≤ n, the k-form Hodge Laplace equation seeks a k-form u satisfying: (1.2) L 0 u := (dδ + δd)u = f, with proper boundary conditions. Our abstract theory applies to equation (1.2) perturbed by general lower order terms: Lu = [(d + l 3 )(δ + l 2 ) + (δ + l 4 )(d + l 1 ) + l 5 ]u = f, with proper boundary conditions. In Section 3, we allow l i to be general linear operators, but here we assume that they are multiplication by smooth coefficient fields. For example, l 1 takes a k-form to (k + 1)-form, thus may be viewed as a n k+1 × n k matrix field. The mixed formulation solves simultaneously for u and for σ = (δ + l 2 )u, which is a (k − 1)-form. On a simplicial triangulation of Ω, for any r ≥ 1, we have four canonical pairs of mixed finite elements for (σ, u): P r+1 Λ k−1 × P r Λ k , P − r+1 Λ k−1 × P r Λ k , P r Λ k−1 × P − r Λ k , P − r Λ k−1 × P − r Λ k . In dimension ≤ 3, all these are classical mixed finite elements. For example, the pair consisting of Nédéléc face elements of the second kind and Nédéléc edge elements of the second kind before is P r+1 Λ 1 × P r Λ 2 . For more, the Periodic Table of the Finite Elements (http://femtable.org/) collects all these elements and their correspondence to classical elements. A main result in FEEC is that the above four pairs lead to stable mixed finite element methods for the unperturbed Hodge Laplace problem. Further, the rates of convergence for the L 2 -errors in σ, dσ, u, and du are optimal, as determined by the approximation properties of the spaces. Thus, for example, if the pair P r+1 Λ k−1 × P r Λ k is used, the order of convergence for the L 2 -error in σ, dσ, u, and du are r + 2, r + 1, r + 1, and r, respectively, while, for P − r Λ k−1 × P − r Λ k all four L 2 -errors converge with order r. For the perturbed system, our discrete stability result (Theorem 3.2) implies that when the perturbed problem is uniquely solvable at the continuous level (which is the generic case, as we will prove), then the mixed discretization of this problem using any of the four stable pairs before is still stable for sufficiently fine discretization. Further, if we have full elliptic regularity (for example on a smooth domain [10]), then our improved error estimate (Theorem 3.4) implies that the L 2 convergence rates for the unperturbed Hodge Laplacian still hold for the perturbed problem with a few exceptions, as summarized in Table 1: P r+1 Λ k−1 × P r Λ k      r + 2, if no l 2 , l 4 , l 5 , r + 1, otherwise.      r + 1, if no l 4 , r, otherwise. r + 1 r P − r+1 Λ k−1 × P r Λ k r + 1      r + 1, if no l 4 , r, otherwise. r + 1 r P r Λ k−1 × P − r Λ k      r + 1, if no l 2 , l 5 , r, otherwise. r r r P − r Λ k−1 × P − r Λ k r r r r For example, in our first example, B is a 2-form and equation (1.1) has an l 2 lower-order term, which leads to reduced L 2 -error rates for σ. In practice, this suggests that the smaller space P − r+1 Λ k should be used for σ = curl B, because the use of the bigger space P r+1 Λ k does not improve the convergence rates for σ or any other quantity. In our second example, the zeroth-order term is a generic l 5 term, hence it also degrades the L 2 -error rate in σ. We observe that some lower-order terms, namely l 1 and l 3 terms, do not degrade the error rates in any of the cases above and that the L 2 -convergence rates for u and du are unaffected by the lowerorder terms. We also note that, most surprisingly, the lower-order term l 4 has the worst effect on the convergence rates of the error in σ, degrading the L 2 -error rates in both σ and dσ, yet σ has no apparent dependence on l 4 . Historically, the effect of lower-order terms on the convergence of finite element methods was first studied by Schatz [18], for the scalar Laplace equation with Lagrange elements. The key tool in that analysis was the Aubin-Nitsche duality argument, which was introduced to prove L 2 -error estimates in both non-mixed [15] and mixed methods [9]. These ideas guide the current techniques. Directly relevant to this work are the studies by Douglas and Roberts [7,8] and Demlow [6] on mixed finite element discretization of the scalar Laplace equation, which is the Hodge Laplace equation for n-forms in n dimensions. The primary mixed finite elements for this problem are the Raviart-Thomas (RT) family P − r Λ n−1 [17] and the BDM family P r Λ n−1 [5,4]. Douglas and Roberts proved the optimal L 2 convergence rates for both variables when the RT family is used. Demlow showed that for BDM elements, even for constant coefficients and smooth solutions, there is degradation of the convergence rate for σ in the problem − div(grad u+ bu)+cu = f while there is no degradation if the same problem is formulated as − div grad u + b · grad u + cu = f . All these classical results on non-mixed and mixed finite elements for scalar Laplace problem with lower-order terms can be read off directly from Table 1. Our approach here gives a uniform derivation of all classical L 2 -estimates for the scalar/vector Laplacian perturbed by lower-order terms under the extended abstract FEEC framework. The rest of this paper is organized as follows. We first briefly review the basic FEEC framework for the unperturbed abstract Hodge Laplacian in Section 2. Then we lay out our extended abstract framework and state our two main discrete result: stability theorem (Theorem 3.2) and improved error estimates (Theorem 3.4) in Section 3. After that, we prove the well-posedness theorems at the continuous level in Section 4. Then we prove the two main discrete results in Section 5 and Section 6 respectively. Finally, in Section 7, we show that the estimates are sharp for the Hodge Laplacian case through numerical examples. Review of the abstract FEEC fraemwork FEEC is an abstract framework for analyzing mixed finite element methods [1,2]. A Hilbert complex (W k , d k ) is a sequence of Hilbert spaces W k and closed denslydefined linear operators d k : W k → W k+1 with closed range satisfying d k+1 •d k = 0. We use ( · , · ) to denote the W -inner product and · to denote the W -norm. Let d * k+1 be the adjoint of d k , V k = D(d k ), and V * k+1 := D(d * k+1 ). From this definition, (d k u, v) = (u, d * k+1 v) , for all u ∈ V k and v ∈ V * k+1 . One important structure is the Hodge decomposition: W k = Z k ⊕ (Z k ) ⊥W = B k ⊕ H k ⊕ (Z k ) ⊥W = B k ⊕ H k ⊕ B * k , where Z k = ker d k , B k = im d k−1 , H k = Z k ∩ (B k ) ⊥ , and B * k = im d * k+1 = (Z k ) ⊥W . We assume the Hilbert complex satisfies the compactness property, that V k ∩ V * k is a compact densely embedded subspace of W . This is true for all the cases we are interested in. For example, the de Rham complex on Lipschitz domains in R n satisfies this property [16]. The abstract Hodge Laplacian is the unbounded operator L k 0 : W k → W k defined by L k 0 = d k−1 d * k +d * k+1 d k with the domain D(L k 0 ) = {u ∈ V k ∩V * k | du ∈ V * k+1 , d * u ∈ V k−1 }. In the following, we drop the index k when it is clear from the context. For example, L 0 = dd * + d * d. It is known that for any f ∈ W k , there exists a unique u ∈ D(L 0 ) such that L 0 u = f mod H, u ⊥ H. Let K 0 : f → u be the solution operator above. It is known that K 0 is self-adjoint and compact as a map W → W . The mixed discretization of the abstract Hodge Laplacian is well-understood. The problem above can be formulated in the mixed weak formulation: given f ∈ W , find (σ, u, p) ∈ V k−1 × V k × H k such that (2.1) (σ, τ ) − (u, dτ ) = 0, ∀τ ∈ V k−1 , (dσ, v) + (du, dv) + (p, v) = (f, v), ∀v ∈ V k , (u, q) = 0, ∀q ∈ H k . For each k, let V k h be a sequence of discrete subspaces of V k indexed by h. V k h is called dense if ∀u ∈ V k , lim h→0 inf v∈V h u − v V = 0. Clearly, density is necessary for V k h to be good approximations of V k . However, it is known that this alone is not sufficient for the Galerkin projection to be a convergent method. The key additional properties are the subcomplex property that dV k h ⊂ V k+1 h and the existence of V -bounded cochain projections, that is, bounded projections π k h : V k → V k h satisfying dπ h = π h d. The main result in FEEC states that the Galerkin projection of system (2.1) using dense discrete subspace admitting bounded cochain projections is stable. More precisely, Theorem 3.8 of [2] states that the bilinear form associated with the system (2.1) B 0 ((σ, u, p), (τ, v, q)) := (σ, τ ) − (u, dτ ) + (dσ, v) + (du, dv) + (p, v) − (u, q), satisfies the inf-sup condition on (V k−1 h × V k h × H k h ) 2 . This implies quasi-optimal convergence rates in the V -norm. Further if the bounded projections can be extended to W -bounded cochain projections, that is, the extension to π k h : W k → V k h exists and π k h W →W are bounded uniformly in h, then we get decoupled error estimates estimates for each variable in the W -norm. To state this precisely, we need some notations. We use a b to express that a ≤ Cb for some generic constant C. Following [2], we let (2.2) δ 0 = (I − π h )K 0 W k →W k , µ 0 = (I − π h )P H W k →W k , η 0 = max j=0,1 { (I − π h )dK 0 W k−j →W k−j+1 , (I − π h )d * K 0 W k+j →W k+j−1 }, α 0 = η 2 0 + δ 0 + µ 0 , where P H is the W -orthogonal projection from W to H ⊂ W . All these quantities converge to 0 as h → 0 due to the compactness and density assumption. For example on smooth or convex polyhedral domains for the de Rham complex, it is known that η 0 = O(h), δ 0 = O(h min(2,r+1) ), µ 0 = O(h r+1 ), where r is the largest degree of complete polynomials in V h . We use the following notation for the best approximation: for w ∈ V k , E(w) := inf v∈V k h w − v . Theorem 3.11 of [2] bounds the W -norm error of each variable in terms of best approximation errors: (2.3) σ − σ h E(σ) + η 0 E(dσ), d(σ − σ h ) E(dσ), u − u h E(u) + (η 2 0 + δ 0 )[E(dσ) + E(p)] + η 0 [E(du) + E(σ)], d(u − u h ) E(du) + η 0 [E(dσ) + E(p)], p − p h E(p) + µ 0 E(dσ). For example, using P r+1 Λ k−1 × P r Λ k and assuming full elliptic regularity, we have E(σ) = O(h r+2 ) and E(dσ) = O(h r+1 ) for any r ≥ 1. Therefore, we get the optimal rates σ − σ h = O(h r+2 ). Similarly, the error rates for all variables can be shown to be optimal for all four canonical FEEC element pairs. Let P h : W → V h be the W -orthogonal projection and K 0h : V k h → V k h be the discrete solution operator K 0h : f → u h . These convergence estimates can be stated using operators (cf. Corollary 3.17 of [2]): (2.4) K 0 − K 0h P h W →W α 0 , dK 0 − dK 0h P h W →W + d * K 0 − d * h K 0h P h W →W η 0 . The FEEC approach not only gives optimal error rates, but also captures the important structures of the Hodge Laplacian. We have the discrete Hodge decomposition: V k h = Z k h ⊕ B * k,h = B k h ⊕ H k h ⊕ B * k,h . The map π k h ensures that Z h ⊂ Z and B h ⊂ B,(2.5) ∀f ∈ W, f = dd * K 0 f + d * dK 0 f + P H f ∈ B ⊕ B * ⊕ H, that is, solving the Hodge Laplacian problem with data f leads to the Hodge decomposition of f . At the discrete level, similarly, we have: (2.6) ∀f ∈ V h , f = dd * h K 0h f + d * h dK 0h f + P H f ∈ B h ⊕ B * h ⊕ H h . This orthogonality is the key ingredient in deriving decoupled W -norm error estimates for each variable above and plays an important role in our analysis as well. Main results We start by identifying W k with its dual and form a Gelfand triple V k ∩ V * k ⊂ W k ⊂ (V k ∩ V * k ) ′ . We extend L 0 = dd * + d * d to an operator V ∩ V * → (V ∩ V * ) ′ which is more suitable for studying perturbations: for all u, v ∈ V k ∩ V * k , L 0 u, v := (d * u, d * v) + (du, dv). Our main operator L : V k ∩ V * k → (V k ∩ V * k ) ′ is obtained by perturbing each abstract differential: for all u, v ∈ V k ∩ V * k , (3.1) Lu, v := ((d * + l 2 )u, (d * + l * 3 )v) + ((d + l 1 )u, dv) + (l 4 du, v) + (l 5 u, v) , where l i : W → W are bounded linear maps between appropriate levels for i = 1, . . . , 5. More succinctly, we write, L = (d + l 3 )(d * + l 2 )u + d * (d + l 1 )u + l 4 du + l 5 u. The grouping is convenient for the mixed form later. We prove in Lemma 4.1 of the next section that this L satisfies a Gårding inequality. Then standard techniques in elliptic PDE theory imply that L is invertible up to some arbitrarily small perturbation to l 5 . Therefore, generically, it is reasonable to assume that L is a bounded isomorphism. Let D be the natural domain on which L maps W k → W k : (3.2) D := {u ∈ V k ∩ V * k | (d * + l 2 )u ∈ V k−1 , (d + l 1 )u ∈ V * k+1 }. Our main perturbed problem is: given f ∈ W k , find u ∈ D such that (3.3) Lu = f. We reformulate it in the mixed form: given f ∈ W k find (σ, u) ∈ V k−1 × V k satisfying (σ, τ ) − (u, dτ ) − (l 2 u, τ ) = 0, ∀τ ∈ V k−1 , (3.4a) ((d + l 3 )σ, v) + ((d + l 1 )u, dv) + (l 4 du, v) + (l 5 u, v) = (f, v), ∀v ∈ V k . (3.4b) The first equation (3.4a) is equivalent to u ∈ V * k and σ = (d * + l 2 )u. The second equation (3.4a) is equivalent to (d + l 1 )u ∈ V * k+1 and d * (d + l 1 )u = f − (d + l 3 )σ − l 4 du − l 5 u. Hence, if (σ, u) solves (3.4), then u solves (3.3) . Therefore, it makes sense to use this mixed formulation to solve our problem. In addition to the assumption that L is a bounded isomorphism, we need more regularity assumptions on l i to ensure that L −1 (W ) ⊂ D so that (3.3) has a solution. One of our main tool for analyzing the discretization is the duality argument, where the dual problem L ′ z = g has to be solved as well. Here L ′ : V k ∩ V * k → (V k ∩ V * k ) ′ is the dual of L. We collection the conditions under which all these continuous problems are well-posed in a theorem: (3.5) (d * + l 2 )L −1 (W ) ⊂ V k−1 , (d * + l * 3 )(L ′ ) −1 (W ) ⊂ V k−1 . Then both the perturbed problem 3.3 and its mixed formulation 3.4 are well-posed. The proof is given in section 4. The regularity assumption is very mild, without which the perturbed problem does not even make sense. The solution operator to the dual problem L ′ z = g will be used frequently, so we give it a name. Let K = (L ′ ) −1 : (V k ∩ V * k ) ′ → V k ∩ V * k . Our first discrete result is the following fundamental theorem on mixed methods for problems perturbed by lower-order terms. W → W : dl * 3 K, (l * 1 d − l * 2 d * )K. Let V k h be a sequence of dense subcomplexes of V k admitting W -bounded cochain projections. Then the Galerkin projection of the mixed system (3.4) using the pair V k−1 h × V k h is stable in the sense that there exist positive constants h 0 , C 0 such that for any h ∈ (0, h 0 ], there exists a unique discrete solution (σ h , u h ) ∈ V k−1 h × V k h satisfying (3.4) for test functions in V k−1 h × V k h and that σ h V + u h V ≤ C 0 f . The proof is nontrivial and is given in Section 5. For the de Rham complex, suppose all the lower-order terms are multiplication by smooth coefficients and the domain has (1 + ǫ)-regularity that Kf H 1+ǫ f L 2 for ǫ > 0. By definition, L ′ = (d + l * 2 )(d * + l * 3 ) + (d * + l * 1 )d + d * l * 4 + l * 5 and K = (L ′ ) −1 , we have (dd * + d * d)K = I − (l * 2 d * + dl * 3 + l * 1 d + d * l * 4 + l * 5 + l * 2 l * 3 )K is bounded L 2 → L 2 . The compactness assumptions are satisfied due to the compact embedding of Sobolev space H s into L 2 for s > 0. This proves the statement on the stability of mixed discretization of Hodge Laplacian in the introduction. It is well-known that stability guarantees optimal error rates in the energy norm: Corollary 3.3. Under the assumptions of Theorem 3.2, if h ≤ h 0 , the unique discrete solution (σ h , u h ) satisfies σ − σ h V + u − u h V (I − π h )σ V + (I − π h )u V . As is common for mixed methods, the energy norm estimate is crude because it couples errors of different variables. For example, for the unperturbed Hodge Laplacian solved with the FEEC pair P r+1 Λ k+1 ×P r Λ k , the convergence rate in σ is in fact h 2 higher than that in du but is lumped together with it. Our next finer discrete result gives the decoupled W -norm estimates for each variable similar to estimates (2.3) for the unperturbed problem. For that, we need more assumptions. To simplify the bookkeeping, we define some approximation quantities: (3.6) δ = max{δ 0 , (I − P h )l * 3 K , (I − P h )(l * 5 − l * 2 l * 3 )K , (I − P h )P B l * 4 K , }, η = max{η 0 , µ 0 , δ, (I − P h )dl * 3 K , (I − P h )(l * 1 d − l * 2 d * )K , }, α = δ + η 2 + µ 0 . where η 0 , δ 0 , µ 0 are defined in equation (2.2) and all the operator norms are in · W →W . As, before, due to the compactness assumptions and density, all δ, η, α → 0 as h → 0. Theorem 3.4. In addition to the assumptions of Theorem 3.2, assume that d(l * 1 d − l * 2 d * + l * 5 − l * 2 l * 3 )K W →Wis bounded, then we have the following improved error estimates: σ − σ h E(σ) + (η + χ 45 √ α)E(dσ) + (χ 24 + χ 3 η + χ 5 √ η)E(u) + (χ 3 α + χ 45 √ α)E(du), d(σ − σ h ) E(dσ) + (χ 3 α + χ 4 + χ 5 η)E(du) + (χ 45 + χ 3 η + χ 2 χ 3 )E(u) + (χ 3 + χ 4 η)E(σ), u − u h E(u) + ηE(du) + ηE(σ) + (α + χ 45 √ α)E(dσ), d(u − u h ) E(du) + ηE(dσ) + χ 1345 E(u) + (χ 3 + χ 145 η)E(σ), where χ i...j denote the presence of lower-order terms. For example, χ 125 = 1 if l 1 = 0 or l 2 = 0 or l 5 = 0, and χ 125 = 0 otherwise. The proof is subtle and is given in Section 6. Corollary 3.5. Suppose a Hodge Laplacian problem satisfies full 2-regularity: Kf H s+2 f H s , for all s ≥ 0, (for example, on a smooth domain), then the error estimates for the discretization using FEEC elements are given by Table 1. Proof. Following the discussion after equation (2.2), we have η 0 = O(h), δ 0 = O(h min(2,r+1) ), µ 0 = O(h r+1 ), where r is the largest degree of complete polynomials in V h . Using the best approximation estimates for FEEC elements, we see that η = O(h) and δ = O(h min(2,r+1) ) as well. Plugging these and the best approximation estimates into Theorem 3.4, we get the rates in Table 1. Well-posedness at the continuous level In this section, we establish well-posedness results for the continuous problem and its mixed formulation. 4.1. Well-posedness for the primal form. First, we prove that the perturbed bounded operator is almost always an isomorphism. Lemma 4.1. Let (W k , d) be a Hilbert complex having the compactness property with domains V k . Let L be defined as in (3.1). Then, L + λI has a bounded inverse for all λ ∈ C except at a discrete subset (so at most countable). Proof. Let M = max i l i W →W and γ = 4M 2 + M + 1/2. Then L + γI is coercive on V k ∩ V * k : Lu, u + γ(u, u) ≥ (1/2)[(u, u) + (d * u, d * u) + (du, du)]. The compactness property ensures that I : V k ∩ V * k ֒→ (V k ∩ V * k ) ′ is compact, which makes I(L + γI) −1 compact on (V k ∩ V * k ) ′ . Spectral theory then implies that I + µI(L + γI) −1 has a bounded inverse for all µ ∈ C except at a discrete subset. Then composing with the bounded isomorphism L + γI on the right proves the claim. In particular, this shows that either L is invertible or L + ǫI is invertible for any small enough nonzero ǫ. Then, we prove the well-posedness of our main problem. Lemma 4.2. Suppose L defined in equation (3.1) is a bounded isomorphism and (d * + l 2 )L −1 (W ) ⊂ V k−1 . Then L −1 (W ) ⊂ D, where D is defined in (3.2). In particular, problem (3.3) has a unique solution. Proof. Since L is already an isomorphism, we only need to show that L −1 (W ) ⊂ D. For any f ∈ W , let u = L −1 f . Then by definition, ((d * +l 2 )u, (d * +l * 3 )v)+((d+l 1 )u, dv)+(l 4 du, v)+(l 5 u, v) = (f, v), ∀v ∈ V k ∩V * k . By assumption, (d * + l 2 )u ∈ V k−1 . Thus, we have, ((d + l 1 )u, dv) = (f − (d + l 3 )(d * + l 2 )u − l 4 du − l 5 u, v), ∀v ∈ V k ∩ V * k . There is no d * v in the above. By the density of V k ∩ V * k in W k , we conclude that the above holds for all v ∈ V k . Hence, (d + l 1 )u ∈ V * k+1 . Thus u ∈ D proves the claim. We then prove a similar result for the dual problem. Let D ′ be the natural domain on which L ′ maps W k → W k : D ′ := {u ∈ V k ∩ V * k | (d * + l * 3 )u ∈ V k−1 , (d + l * 4 )u ∈ V * k+1 }. Using the same argument, we get, Lemma 4.3. Suppose L defined in equation (3.1) is a bounded isomorphism and (d * + l * 3 )(L ′ ) −1 (W ) ⊂ V k−1 . Then (L ′ ) −1 (W ) ⊂ D ′ . 4.2. Well-posedness for the mixed form. We then turn to mixed system (3.4). Its associated bilinear form B : (V k−1 × V k ) 2 → R is: (4.1) B((σ, u), (τ, v)) := (σ, τ ) + (du, dv) + (dσ, v) − (u, dτ ) − (l 2 u, τ ) + (l 3 σ, v) + (l 1 u, dv) + (l 4 du, v) + (l 5 u, v). We call this bilinear form well-posed if and only for any (g, f ) ∈ V ′ k−1 × V ′ k , there exists a unique solution (σ, u) ∈ V k−1 × V k satisfying: B((σ, u), (τ, v)) = g, τ V ′ ×V + f, v V ′ ×V , ∀(τ, v) ∈ V k−1 × V k . From the discussion after equation 3.4, we see that the well-posedness of B implies that L −1 (W ) ∈ D. But it also implies the well-posedness of the dual mixed problem: given any (g, f ) ∈ V ′ k−1 × V ′ k , find (ξ, z) ∈ V k−1 × V k satisfying B((ρ, w), (ξ, z)) = g, ρ V ′ ×V + f, w V ′ ×V , ∀(ρ, w) ∈ V k−1 × V k . A similar argument shows that the well-posedness of B implies (L ′ ) −1 (W ) ∈ D ′ as well. We collect these results in a lemma: Moreover, the converse is also true. Proof. The only if part is clear from the previous lemmas. We only need to show the if part. First, we show that B satisfies a Gårding-like inequality: there exist positive constants a, b, c depending only on l i W →W such that (4.2) B((σ, u), (σ, u + adσ)) ≥ b( σ V + u V ) 2 − c u 2 , ∀(σ, u) ∈ V k−1 × V k . Direct computation using Cauchy-Schwarz inequality shows that there exist constants c 1 , c 2 depending only on l i W →W such that B((σ, u), (σ, u)) ≥ (1/2)( σ 2 + du 2 ) − c 1 u 2 , B((σ, u), (0, dσ)) ≥ (1/2) dσ 2 − c 2 ( σ 2 + du 2 + u 2 ). Multiplying the second inequality by any positive a < 1/(2c 2 ) and adding it to the first inequality, we get the claim. Second, fix any (σ, u) ∈ V k−1 × V k . We solve a dual problem using u as data: let z = cKu and ξ = −(d * + l * 3 )z. By assumption, ξ ∈ V k−1 . Direct computation shows that B((ρ, w), (ξ, z)) = (cu, w), ∀(ρ, w) ∈ V k−1 × V k . Finally, we add (ξ, z) to our choice of test functions in the first step and get B((σ, u), (σ + ξ, u + adσ + z)) ≥ b( σ V + u V ) 2 . Further, from the definition of (ξ, z), we have σ + ξ V + u + adσ + z V ≤ M σ V + u V , where the constant M depends only on a, d(d * + l * 3 )K W →W , and dK W →W . Thus B satisfies the inf-sup condition. Similarly, for any fixed nontrivial (τ, v) ∈ V k−1 × V k , we let u = L −1 v and σ = (d * + l 2 )u. By assumption, σ ∈ V k−1 . Direct computation shows that B((σ, u), (τ, v)) = (v, v) > 0. It is well-known that these two conditions imply that B is well-posed [3]. Given these lemmas, Theorem 3.1 is clearly true. Discrete Stability through new projections In this section, we prove Theorem 3.2. The idea of the proof is similar to the that of the if part of Lemma 4.5. First, due to the subcomplex property, the estimate (4.2) still holds at the discrete level with the same constants a, b, c. Second, fix any (σ, u) ∈ V k−1 h × V k h . We can again solve a dual problem using u as data: let z = cKu and ξ = −(d * + l * 3 )z. Then we have ξ ∈ V k−1 , ξ V + z V u independent of h, and B((ρ, w), (ξ, z)) = (cu, w), ∀(ρ, w) ∈ V k−1 × V k . But we can no longer add (ξ, z) to our choice of test functions because (ξ, z) is not discrete. In the rest of this section, we construct a discrete pair ( ξ h , z h ) ∈ V k−1 h ×V k h such that (5.1) ξ h V + z h V ξ V + z V uniformly in h, and |B((ρ, w), (ξ − ξ h , z − z h ))| ≤ ǫ h ( ρ V + w V )( ξ V + z V ), for all (ρ, w) ∈ V k−1 h × V k h , where ǫ h → 0 as h → 0. Given such a pair, we can add it to our choice of test functions: B((σ, u), (σ + ξ h , u + adσ + z h )) ≥ (b − cǫ h )( σ V + u V ) 2 . Further, there also exists M > 0 bounded uniformly in h, such that σ + ξ h V + u + adσ + z h V ≤ M ( σ V + u V ). Choose a sufficiently small h 0 such that ǫ h0 < b/c. Then for all h < h 0 , the bilinear form B((σ, u), (τ, v)) satisfies the inf-sup condition on V k−1 h × V k h with the inf-sup constant bounded uniformly below by (b − cǫ h0 )/M . Since V k−1 h × V k h is of finite dimension, this establishes the well-posedness. Thus Theorem 3.2 is proved. An obvious choice for (ξ h , z h ) in (5.1) is the elliptic projection given by B((ρ, w), (ξ h , z h )) = B((ρ, w), (ξ, z)), ∀(ρ, w) ∈ V k−1 h × V k h . Then ǫ h = 0. But since we have not proved the well-posedness of B on the discrete level, we neither know a discrete solution exists nor can we show the uniform estimates. The next most obvious choice is obtained using the elliptic projection of the unperturbed problem: B 0 ((ρ, w, p), (ξ h , z h , q h )) = B 0 ((ρ, w), (ξ, z)), ∀(ρ, w, p) ∈ V k−1 h × V k h × H k h . Then we have the existence and uniform bounds. But the second estimate in (5.1) fails. In what follows, we develop two new projection operators to correct the elliptic projection for the unperturbed problem so that both conditions in (5.1) holds. In fact, we do a lot more. Our elaborately chosen (ξ h , z h ) will not only satisfy (5.1), but also have explicit and optimal error rates in quantities like ξ − ξ h , z − z h , and |B((ρ, w), (ξ − ξ h , z − z h )|. This is made precise in Theorem 5.3. This result also form the basis of the improved error estimates later. Moreover, the two new projection operators enjoy many properties making them interesting in their own right. Generalized Canonical Projection. Thus far, we have three projections in FEEC: the orthogonal projection P h , the cochain projection π h which commutes with d but has no orthogonality property, and the elliptic projection K 0h P h L 0 which misses the harmonic part and is only well-defined on the subspace D(L 0 ). Here, we introduce a new projection operator Π h : V k → V k h given by Π h := P Z h + d * h K 0h P h d. In the above and for the rest of this paper, for any subspace X of W , we use the notation P X : W → X for the W -orthogonal projection. Among other properties, this Π h satisfies a commutative property generalizing that of the canonical projection for classical elements like Raviart-Thomas. Π h w w + η 0 dw , (I − Π h )w (I − π h )w + η 0 dw . Moreover, it satisfies "partial orthogonality": for any w, v ∈ V , |((I − Π h )w, v)| ( (I − π h )w + η 0 dw )( (I − π h )v + η 0 dv ) + α 0 dv dw . Proof. The stability of the unperturbed discrete problem (2.1) implies that Π h is uniformly bounded in the V -norm. By subcomplex property, we have P h dv = dv for v ∈ V h . Thus, Π h v = P Z h v + d * h K 0h dv = P Z h v + P B * h v = v, showing that Π h is a projection. We know from equation (2.6) that for the un- perturbed problem dd * h K 0h = P B h . This proves that dΠ h = dd * h K 0h P h d = P B h d. We then prove the first two estimates. Fix any w ∈ V . We split w − Π h w = (P Z − P Z h )w + (P B * w − P B * h Π h w). The second term can be bounded using the error estimates (2.4) for K 0h : P B * w − P B * h Π h w = d * K 0 dw − d * h K 0h P h dw η 0 dw . We then deal with the first term. The subcomplex property ensures Z h ⊂ Z and the cochain property of π h ensures π h Z ⊂ Z h . These two lead to (P Z − P Z h )w (I − π h )P Z w . But (I − π h )P Z w = (I − π h )w − (I − π h )P B * w and P B * = d * K 0 d. Thus, (P Z − P Z h )w (I − π h )w + (I − π h )d * K 0 dw (I − π h )w + η 0 dw . Combining the estimates for the two parts, we get, (I − Π h )w = (P Z − P Z h )w + (P B * w − P B * h Π h w) (I − π h )w + η 0 dw . By triangle inequality, we get Π h w ≤ w + (I − Π h )w w + η 0 dw as well. This proves the first two estimates. Finally, for any w, v ∈ V , we have ((I − Π h )w, v) = ((P Z − P Z h )w, v) + ((P B * − P B * h Π h )w, (I − π h )v) + ((P B * − P B * h Π h )w, π h v) . The first two terms can be bounded as before. The last term is bounded by (d * K 0 dw − d * h K 0h P h dw, π h v) = ((K 0 − K 0h P h )dw, π h dv) α 0 dw dv , where the error estimate (2.4) is used again. This finishes the proof. Modified elliptic projection. We modify the unperturbed elliptic projection slightly to accommodate the harmonic forms. Theorem 5.2. For any z ∈ D(L 0 ), let z h = K 0h P h L 0 z + P H h P H z. Then, z − z h α 0 L 0 z , d(z − z h ) + d * z − d * h z h η 0 L 0 z , P h (d * dz + dd * z) − (d * h dz h + dd * h z h ) ≤ µ 0 L 0 z . Proof. By equation (2.5), we have P H ⊥ z = K 0 L 0 z. Thus we have the splitting z − z h = (P H ⊥ z − P H ⊥ h z h ) + (P H z − P H h z h ) = (K 0 − K 0h P h )L 0 z + (I − P H h )P H z. The first term has been estimated by (2.4). For the second term, since P H h P H = P Z h P H and π h Z ⊂ Z h , we have (I − P H h )P H z W ≤ (I − π h )P H z W ≤ (I − π h )P H W →W z = µ 0 z , which proves the first estimate. The second estimate follows from (2.4) directly. Finally for the last estimate, we use the continuous and discrete Hodge decomposition (2.5) (2.6), we get (d * h d + dd * h )z h = (d * h d + dd * h )K 0h P h L 0 z = (P B h + P B * h )P h L 0 z. Moreover, by definition, P H L 0 z = 0. Thus, P h (d * dz + dd * z − d * h dz h − dd * h z h ) = P h L 0 z − (P B h + P B * h )P h L 0 z = P H h L 0 z. The right-hand side is just p − p 0 for the unperturbed problem with L 0 z as data. By (2.3), P H h L 0 z = (P H h − P H )L 0 z 0 + µ 0 P B L 0 z ≤ µ 0 L 0 z , which proves the last estimate. Projection of the dual solution. We are now ready to construct the discrete pair (ξ h , z h ) satisfying the conditions (5.1) in the proof of the discrete stability theorem. In fact, we prove a stronger result where the first variable ρ is allowed to be in V instead of V h and derive explicit error estimates. Theorem 5.3. Under the assumption of Theorem 3.2, for any g ∈ W k , let z = Kg, ξ = −(d * + l * 3 )z, z h = K 0h P h L 0 z + P H h P H z, and ξ h = −d * h z h − Π h l * 3 z. Then, z − z h α 0 g , d(z − z h ) η 0 g , ξ − ξ h η g . Further, for any (ρ, w) ∈ V k+1 × V k h , we have, |B((ρ, w), (ξ −ξ h , z −z h ))| [η ρ +α dρ +(µ 0 +χ 123 η +χ 5 α) w +χ 4 α dw ] g . Proof. Using the regularity assumption that L 0 K W →W is bounded, the estimate for z − z h and d(z − z h ) follows directly from Theorem 5.2. From the same theorem, for ξ, we have ξ − ξ h ≤ d * z − d * h z h + (I − Π h )l * 3 z η 0 g + (I − Π h )l * 3 Kg . For the second term, using quantities defined in (3.6), we have: (I − Π h )l * 3 Kg (I − π h )l * 3 Kg + η dl * 3 Kg η g . The last estimate is just a direct computation using the error estimates in Theorem 5.2, quantities defined in (3.6), and the Cauchy-Schwarz inequality. For the proof of Theorem 3.2 at the beginning of this section, we get (ξ h , z h ) by applying this theorem to g = cu ∈ V h ⊂ W . We note that (ξ, z) here is the same as the one defined there. We check that condition (5.1) is satisfied. First, z h V = K 0h P h L 0 z + P H h P H z V z V , ξ h V = − d * h K 0h P h L 0 z − Π h l * 3 z V dξ + z V ξ V + z V , where the constants depend only on the stability constant of the continuous and discrete unperturbed problem, which is either independent of h or bounded uniformly in h. Second, as mentioned before, the compactness assumptions and density, α, η, µ 0 → 0 as h → 0. Thus condition (5.1) is verified. This finishes the proof of Theorem 3.2. Proof of Improved Error Estimates In this section, we prove Theorem 3.4. To make the notation more compact, we use e u := u − u h and E u = (I − π h )u. Corresponding quantities for σ are similarly defined. The Galerkin orthogonality equation reads: (e σ , τ ) − (e u , dτ ) − (l 2 e u , τ ) = 0, ∀τ ∈ V k−1 h , (6.1a) ((d + l 3 )e σ , v) + ((d + l 1 )e u , dv) + (l 4 de u , v) + (l 5 e u , v) = 0, ∀v ∈ V k h . (6.1b) 6.1. Preliminary estimates for de σ and de u . Optimal estimates for these two terms can be obtained directly from the error equations (6.1) with carefully chosen test functions. Lemma 6.1. For any (σ, u) solving (3.4) and (σ h , u h ) solving its Galerkin projection, de σ dE σ + χ 3 e σ + χ 4 de u + χ 5 e u . Proof. Restricting the test function space to B h in equation (6.1b) leads to: (6.2) P B h (de σ + l 3 e σ + l 4 de u + l 5 e u ) = 0. Thus, de σ = (I − P B h )de σ + P B h de σ = (I − P B h )de σ − P B h (l 3 e σ + l 4 de u + l 5 e u ). Because π h maps to B to B h , we have (I − P B h )de σ (I − π h )de σ = dE σ proving the claim. Lemma 6.2. For any (σ, u) solving (3.4) and (σ h , u h ) solving its Galerkin projection, de u dE u + η de σ + χ 145 e u + χ 3 e σ . Proof. Let v h = P B * h (π h u − u h ) in the second error equation (6.1b). We have (6.3) (de u , dv h ) = −(de σ , v h ) − [(l 1 e u , dv h ) + (l 3 e σ + l 5 e u , v h )] − (l 4 de u , v h ). By discrete Poincaré inequality v h dv h for v h ∈ B * h , the second term in (6.3) becomes |(l 1 e u , dv h ) + (l 3 e σ + l 5 e u , v h )| (χ 15 e u + χ 3 e σ ) dv h . Because de u = d(u − π h u + π h u − u h ) = dE u + dv h , the last term in (6.3) satisfies |(l 4 de u , v h )| = |(l 4 dE u , v h ) + (l 4 dv h , v h )| ( dE u + v h ) dv h . We finally estimate the first term in (6.3) . Let v = P B * v h . Then d(π h v − v h ) = 0 implies π h v − v h ∈ Z h , so (v h − v) ⊥ (π h v − v h ). Thus, v − v h ≤ (I − π h )v = (I − π h )d * Kdv h η dv h . This implies, |(de σ , v h )| = |(de σ , v − v h )| η de σ dv h . Combining all these estimates and de u ≤ dE u + dv h gives the estimate in the claim. 6.2. Duality lemma. The optimal W -norm estimates for e u and e σ require more work. |(Π h e u , g)| [η e σ + α de σ + (µ 0 + χ 123 η + χ 5 α) e u + (µ 0 η + χ 12345 α) de u ] g . Proof. Given g, let (ξ, z) and (ξ h , z h ) be defined as in Theorem 5.3. We have, (Π h e u , g) = B((e σ , Π h e u ), (ξ, z)) = B((e σ , Π h e u ), (ξ h , z h )) − B((e σ , Π h e u ) , (e ξ , e z )). Galerkin orthogonality (6.1) states that B((e σ , e u ), (τ, v)) = 0 for all discrete (τ, v). Thus, (Π h e u , g) = −B((0, (I − Π h )e u ), (ξ h , z h )) − B((e σ , Π h e u ) , (e ξ , e z )). The second term above can be estimated by the last inequality in Theorem 5.3 and Π h e u e u + η de u , dΠ h e u de u . The result is the following bound: |B((e σ , Π h e u ), (e ξ , e z ))| [η e σ + α de σ + (µ 0 + χ 123 η + χ 5 α) e u + (µ 0 η + χ 4 α) de u ] g . We bound the first term before by splitting it into three parts: |B((0, (I − Π h )e u ), (ξ h , z h ))| ≤ |((I − Π h )e u , dξ h ) + (d(I − Π h )e u , dz h )| + |(l 2 (I − Π h )e u , ξ h ) + (l 1 (I − Π h )e u , dz h ) + (l 5 (I − Π h )e u , z h )| + |(l 4 d(I − Π h )e u , z h )| =: Q 1 + Q 2 + Q 3 . We have Q 1 = 0 because P B h Π h = P B h and dΠ h = P B h d. The second term is: Q 2 ≤ |((I − Π h )e u , l * 2 ξ + l * 1 dz + l * 5 z)| + |((I − Π h )e u , l * 2 e ξ + l * 1 de z + l * 5 e z )|. We note that l * 2 ξ + l * 1 dz + l * 5 z = [−l * 2 d * + l * 1 d + (l * 5 − l * 2 l * 3 )] Kg. The first term above can be bounded by the regularity assumptions and the last estimate in Theorem 5.1. The second term can be bounded using estimates in Theorem 5.3. The result is: Q 2 [(χ 12 η + χ 2 δ + χ 5 α) e u + χ 125 α de u ] g . For Q 3 , we have, Q 3 = |((I − P B h )de u , l * 4 z h )| ≤ |((I − P B h )de u , l * 4 z)| + |((I − P B h )de u , l * 4 e z )|. For the first term ((I − P B h )de u , l * 4 z) = (de u , (P B − P B h )l * 4 z)e σ E σ + (η + χ 45 √ α) de σ + (χ 24 + χ 3 η + χ 5 √ η) e u + (χ 3 α + χ 45 √ α) de u . Proof. Equation (6.1a) implies: (e σ , e σ ) = (e σ , (I − Π h )e σ ) + (e σ , Π h e σ ) = (e σ , (I − Π h )e σ ) + (l 2 e u , Π h e σ ) + (e u , dΠ h e σ ). The first term above is bounded by: |(e σ , (I − Π h )e σ )| ≤ e σ (I − Π h )σ ( E σ + η dE σ ) e σ . The second term is bounded using Π h e σ e σ + η de σ from theorem 5.1, |(l 2 e u , Π h e σ )| χ 2 e u ( e σ + η de σ ) The last term is estimated by the duality lemma. Because P B h Π h = P B h , we have, (e u , dΠ h e σ ) = (e u , P B h de σ ) = (P B h e u , P B h de σ ) = (Π h e u , P B h de σ ). We apply 6.3 with g = P B h de σ and get |(Π h e u , g)| [η e σ + α de σ + (µ 0 + χ 123 η + χ 5 α) e u + (µ 0 η + χ 12345 α) de u ] P B h de σ . From equation (6.2), we have, P B h de σ χ 3 e σ + χ 4 de u + χ 5 e u . Combining all these estimates, we get the estimate in the claim. 6.5. Proof of Improved Error Estimates Theorem. The estimates in Theorem 3.4 are derived from the four preliminary estimates Lemma 6.1, Lemma 6.2, Lemma 6.4, and Lemma 6.5. Using 1 for quantities which are bounded and ǫ for quantities which goes to zero as h → 0, the four preliminary estimates has the following structure:     de σ de u e σ e u         0 1 1 1 ǫ 0 1 1 ǫ ǫ 0 1 ǫ ǫ ǫ 0         de σ de u e σ e u     +     dE σ dE u E σ E u     For example, we can substitute the first line into the second and hide the ǫ de u term in the left-hand side of the second line assuming h is sufficiently small. This, in effect, switches de σ in the second line to dE σ . Due the vanishing diangonal and epsilon lower-triangle of the matrix above, this procedure can be repeated to eliminate all unknown error terms like e u and switch them with known error terms like E u . After this linear algebra exercise, we get the estimates in Theorem 3.4. Numerical examples In this section, we show through numerical examples that the error rates given by Theorem 3.4 in Table 1 are in fact achieved and cannot be improved. In 3D, there are four cases of the Hodge Laplace problems for differential forms of degree 0, 1, 2, 3. The 0-form and 3-form cases lead to the scalar Laplace problem in the non-mixed form and mixed form respectively. The numerical results in these two cases are well-known [18,7,8] and will not be duplicated here. We focus on the 1-form and 2-form case. Let Ω = [0, 1] 3 be the unit cube in R 3 . The 1-form mixed Hodge Laplace problem with natural boundary conditions is: given f ∈ L 2 , find u ∈ D satisfying: (grad +l 3 )(− div +l 2 )u + curl(curl +l 1 )u + l 4 curl u + l 5 u = f, in Ω, u · n = 0, (curl u + l 1 u) × n = 0, on ∂Ω. We choose the following smooth function as the exact solution: At the discrete level, since σ = (− div +l 2 )u is a 0-form where P r Λ 0 = P − r Λ 0 , we only have two element pairs: P r Λ 0 × P r−1 Λ 1 and P r Λ 0 × P − r Λ 1 . On the same domain Ω = [0, 1] 3 as before, the 2-form mixed Hodge Laplace problem with natural boundary conditions is: given f ∈ L 2 , find u ∈ D satisfying: (curl +l 3 )(curl +l 2 )u − grad(div +l 1 )u + l 4 div u + l 5 u. in Ω, u × n = 0, (div +l 1 )u = 0, on ∂Ω. We choose the following smooth function as the exact solution in this case: At the discrete level, we have all four canonical pairs: P r+1 Λ 1 × P r Λ 2 , P − r+1 Λ 1 × P r Λ 2 , P r Λ 1 × P − r Λ 2 , P − r Λ 1 × P − r Λ 2 . In all the numerical experiments, we obtain a quasi-uniform triangulation of size m for the unit cube Ω by first triangulating it uniformly with an m × m × m mesh and then perturbing each interior mesh node randomly within 20% of the mesh size 1/m in all three coordinate directions. All four pairs of the canonical FEEC elements in dimension ≤ 3 of all degrees are supported by the open source finite element package FEniCS [12], in which all our numerical codes are implemented. For example, for the unperturbed 1-form problem, with P 2 Λ 1 × P 1 Λ 2 , we get Clearly the convergence rates for σ and dσ in L 2 are reduced by 1 as predicted. m σ − σ h rate d(σ − σ h ) rate u − u h rate d(u − u h ) There are too many cases for us to list all the detailed results. We instead only summarize the numerical results here. First, the error rates in Table 1 are guaranteed for all FEEC elements for the Hodge Laplace problem. Second, for each case with a reduction in the error rates predicted in Table 1, there is at least one Hodge Laplace problem with a certain form degree which can only converge at that reduced rate. In this sense, the rates in Table 1 is optimal. However, we note that the rates in Table 1 do no represent an upper bound for all possible cases. For example, when the l 5 term is given by multiplication by a smooth scalar coefficient, we do not observe a reduction of convergence rates in the L 2 -error of σ. We also observed that for 1-forms in 3D, an l 2 -term given by multiplication by a generic smooth coefficient do not degrade the L 2 -error rate in σ. The full numerical results along with the python source code used in FEniCS can be found at the companion code repository at https://bitbucket.org/lzlarryli/feeclotexp. We note that due to the randomness involved (random perturbation applied to the mesh), the error numbers will not be exactly the same but very close to what we have listed here. Theorem 3 . 1 ( 31Continuous well-posedness). Let (W, d) be a Hilbert complex with the compactness property. Suppose L defined in equation (3.1) is a bounded isomorphism and Theorem 3. 2 ( 2Discrete stability). Under the assumptions of Theorem 3.1, suppose further that (dd * + d * d)K W →W is bounded and the following operators are compact Lemma 4. 4 . 4Suppose B defined in (4.1) is well-posed and let L be defined as in (3.1). Then L is a bounded isomorphism, L −1 (W ) ⊂ D, and (L ′ ) −1 (W ) ⊂ D ′ . Lemma 4. 5 . 5Suppose L defined in (3.1) is a bounded isomorphism. Then B defined in (4.1) is well-posed if and only if condition (3.5) holds. Theorem 5 . 1 . 51Suppose (W, d) is a Hilbert complex satisfying the compactness property and V k h are dense discrete subcomplexes admitting W -bounded cochain projections. Then Π h is a projection uniformly bounded in the V -norm. Further dΠ h = P B h d. Let η 0 , α 0 be defined as in equation (2.2). Then, for any w ∈ V , Lemma 6. 3 . 3Under the assumptions of Theorem 3.4, for any g ∈ W , πx cos πz cos πx sin πy cos πy sin πz πy sin πz sin πz sin πy sin πz sin πx sin πz sin πx sin πy sin πx sin πx sin πy πx + 3) sin πy sin πz sin πx(cos πy + 2) sin πz sin πx sin πy(cos πz + Table 1 . 1L 2 -error rates for FEEC elements solving Hodge Laplace equationelements σ dσ u du can be bounded using 3.6. The second term can be bounded by |(de u , l * 4 e z )| and Theorem 5.3. The final result is, Q 3 χ 4 α de u g . Combining the all estimates together, we get the estimate in the claim.6.3. Preliminary estimate for e u . Lemma 6.4. Under the assumptions of Theorem 3.4, we have e u E u + η e σ + α de σ + η de u .Proof. Let g = Π h e u in Lemma 6.3 we get an estimate for (Π h e u , Π h e u ). Then,e u ≤ Π h e u + (I − Π h )e u Π h e u + E u + η de u .For sufficiently small h, we hide the e u term on the right in the left-hand side. The result is the estimate in the claim.6.4. Preliminary estimate for e σ . Lemma 6.5. Under the assumptions of Theorem 3.4, Thus, for example, σ − σ h L 2 = 2.766 × 10 −1 on a 2 × 2 × 2 mesh. The rates are computed between the two successive errors. The optimal rates of 3, 2, 2, 1 for σ, dσ, u, du are clear. With an l 4 lower-order perturbation, we get:rate 2 2.766e-01 2.362e+00 2.244e-01 1.441e+00 4 3.940e-02 2.37 8.529e-01 1.24 6.961e-02 1.43 7.434e-01 0.81 8 4.504e-03 3.20 2.312e-01 1.93 1.859e-02 1.95 3.740e-01 1.01 16 5.208e-04 2.98 5.716e-02 1.93 4.659e-03 1.91 1.868e-01 0.96 m σ − σ h rate d(σ − σ h ) rate u − u h rate d(u − u h ) rate 2 4.190e-01 3.522e+00 2.215e-01 1.434e+00 4 1.259e-01 1.44 1.583e+00 0.96 6.773e-02 1.42 7.335e-01 0.81 8 3.689e-02 1.54 7.393e-01 0.96 1.874e-02 1.61 3.749e-01 0.84 16 9.968e-03 1.86 3.595e-01 1.02 4.799e-03 1.93 1.868e-01 0.99 MR 2269741 (2007j:58002) 2. , Finite element exterior calculus: from Hodge theory to numerical stability. Douglas N Arnold, Richard S Falk, Ragnar Winther, 281-354. MR 2594630Bull. Amer. Math. Soc. (N.S.). 15258005Finite element exterior calculus, homological techniques, and applications, Acta NumerDouglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calcu- lus, homological techniques, and applications, Acta Numer. 15 (2006), 1-155. MR 2269741 (2007j:58002) 2. , Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 2, 281-354. MR 2594630 (2011f:58005) Error-bounds for finite element method. Ivo Babuška, 322-333. MR 0288971Numer. Math. 166166Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970), 322-333. MR 0288971 (44 #6166) Mixed finite elements for second order elliptic problems in three variables. Franco Brezzi, Jim Douglas, Jr , Ricardo Durán, Michel Fortin, 237-250. MR 890035Numer. Math. 51265190Franco Brezzi, Jim Douglas, Jr., Ricardo Durán, and Michel Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987), no. 2, 237-250. MR 890035 (88f:65190) Two families of mixed finite elements for second order elliptic problems. Franco Brezzi, Jim Douglas, Jr , L D Marini, 217-235. MR 799685Numer. Math. 47265133Franco Brezzi, Jim Douglas, Jr., and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), no. 2, 217-235. MR 799685 (87g:65133) Suboptimal and optimal convergence in mixed finite element methods. Alan Demlow, electronic). MR 1897944SIAM J. Numer. Anal. 39665216Alan Demlow, Suboptimal and optimal convergence in mixed finite element methods, SIAM J. Numer. Anal. 39 (2002), no. 6, 1938-1953 (electronic). MR 1897944 (2003e:65216) MR 667620 (84b:65111) 8. , Global estimates for mixed methods for second order elliptic equations. Jim Douglas, Jr , Jean E Roberts, 39-52. MR 771029Mat. Apl. Comput. 1165122Math. Comp.Jim Douglas, Jr. and Jean E. Roberts, Mixed finite element methods for second order elliptic problems, Mat. Apl. Comput. 1 (1982), no. 1, 91-103. 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OxfordOxford University Press76205Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre, Mathematical methods for the magnetohydrodynamics of liquid metals, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford, 2006. MR 2289481 (2008i:76205) Automated solution of differential euqations by the finite element method. Lecture Notes in Computational Science and Engineering. Anders Logg, Kent-Andre Mardal, and Garth WellsSpringerAnders Logg, Kent-Andre Mardal, and Garth Wells (eds.), Automated solution of differen- tial euqations by the finite element method, Lecture Notes in Computational Science and Engineering, Springer, Nov 2012. MR 592160 (81k:65125) 14. , A new family of mixed finite elements in R 3. J.-C Nédélec, 57-81. MR 864305Numer. Math. 35365145Numer. Math.J.-C. Nédélec, Mixed finite elements in R 3 , Numer. Math. 35 (1980), no. 3, 315-341. MR 592160 (81k:65125) 14. , A new family of mixed finite elements in R 3 , Numer. 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Let {X k } k≥Z be a stationary sequence. Given p ∈ (2, 3] moments and a mild weak dependence condition, we show a Berry-Esseen theorem with optimal rate n p/2−1 . For p ≥ 4, we also show a convergence rate of n 1/2 in L q -norm, where q ≥ 1. Up to log n factors, we also obtain nonuniform rates for any p > 2. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics. . This reprint differs from the original in pagination and typographic detail. 1 2 M. JIRAK rate of convergence has been addressed under numerous different setups with respect to the metric and underlying structure of the sequence {X k } k∈Z in the literature. Perhaps one of the most important metrics is the Kolmogorov (uniform) metric, given asThe latter has been studied extensively in the literature under many different notions of (weak) dependence for {X k } k∈Z . One general way to measure dependence is in terms of various mixing conditions. In the case of the uniform metric, Bolthausen [6] and Rio[43]showed that it is possible to obtain the rate r n = √ n in(1.3), given certain mixing assumptions and a bounded support of the underlying sequence {X k } k∈Z ; see also [9, 12, 25, 33], among others, for related results and extensions. Under the notion of α-mixing, Tikhomirov [45] obtained r n = n 1/2 /(log n) 2 , provided that E[|X k | 3 ] < ∞ and the mixing coefficient decays exponentially fast; see also [2]. Martingales constitute another important class for the study of (1.3). Some relevant contributions in this context are, for instance, Brown and Heyde [26], Bolthausen [7] and more recently Dedecker et al. [11]. In the special case of functionals of Gaussian or Poissonian sequences, deep results have been obtained by Noudin and Peccati et al.; see, for instance, [37, 38] and [39]. Another stream of significant works focuses on stationary (causal) Bernoullishift processes, given as X k = g k (ε k , ε k−1 , . . .) where {ε k } k∈Z is an i.i.d. sequence. (1.4) The study of (1.3) given the structure in (1.4) has a long history, and dates back to Kac [30] and Postnikov [42]. Ibragimov [28] established a rate of convergence, r n = n 1/2 / √ log n, subject to an exponentially fast decaying weak dependence coefficient. Using the technique of Tikhomirov [45], Götze and Hipp obtained Edgeworth expansions for processes of type (1.4) in a series of works; cf. [19-21]; see also Heinrich [24] and Lahiri [32]. This approach, however, requires the validity of a number of technical conditions. This includes in particular a conditional Crámer-like condition subject to an exponential decay, which is somewhat difficult to verify. In contrast, it turns out that a Berry-Esseen theorem only requires a simple, yet fairly general dependence condition where no exponential decay is required. Indeed, we will see that many popular examples from the literature are within our framework. Unlike previous results in the literature, we also obtain optimal rates for p ∈ (2, 3) given (infinite) weak dependence, which to the best of our knowledge is new (excluding special cases as linear processes). The proofs are based on an m-dependent approximation (m → ∞), which is quite common in the literature. The substantial difference here is the subsequent treatment of the m-dependent sequence. To motivate one of the main ideas of the proofs, . As a dependence measure, we then consider the quantity sup k∈Z X k − X (l,′) k p , p ≥ 1. Dependence conditions of this type are quite general and easy to verify in many cases; cf.[1,48]and the

10.1214/15-aop1017

1606.01617

c7058eff91a624c5d26878ff826728db0dfda502

BERRY-ESSEEN THEOREMS UNDER WEAK DEPENDENCE May 2017. 2016 Moritz Jirak Humboldt Universität zu Berlin BERRY-ESSEEN THEOREMS UNDER WEAK DEPENDENCE The Annals of Probability 443May 2017. 201610.1214/15-AOP1017 Let {X k } k≥Z be a stationary sequence. Given p ∈ (2, 3] moments and a mild weak dependence condition, we show a Berry-Esseen theorem with optimal rate n p/2−1 . For p ≥ 4, we also show a convergence rate of n 1/2 in L q -norm, where q ≥ 1. Up to log n factors, we also obtain nonuniform rates for any p > 2. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics. . This reprint differs from the original in pagination and typographic detail. 1 2 M. JIRAK rate of convergence has been addressed under numerous different setups with respect to the metric and underlying structure of the sequence {X k } k∈Z in the literature. Perhaps one of the most important metrics is the Kolmogorov (uniform) metric, given asThe latter has been studied extensively in the literature under many different notions of (weak) dependence for {X k } k∈Z . One general way to measure dependence is in terms of various mixing conditions. In the case of the uniform metric, Bolthausen [6] and Rio[43]showed that it is possible to obtain the rate r n = √ n in(1.3), given certain mixing assumptions and a bounded support of the underlying sequence {X k } k∈Z ; see also [9, 12, 25, 33], among others, for related results and extensions. Under the notion of α-mixing, Tikhomirov [45] obtained r n = n 1/2 /(log n) 2 , provided that E[|X k | 3 ] < ∞ and the mixing coefficient decays exponentially fast; see also [2]. Martingales constitute another important class for the study of (1.3). Some relevant contributions in this context are, for instance, Brown and Heyde [26], Bolthausen [7] and more recently Dedecker et al. [11]. In the special case of functionals of Gaussian or Poissonian sequences, deep results have been obtained by Noudin and Peccati et al.; see, for instance, [37, 38] and [39]. Another stream of significant works focuses on stationary (causal) Bernoullishift processes, given as X k = g k (ε k , ε k−1 , . . .) where {ε k } k∈Z is an i.i.d. sequence. (1.4) The study of (1.3) given the structure in (1.4) has a long history, and dates back to Kac [30] and Postnikov [42]. Ibragimov [28] established a rate of convergence, r n = n 1/2 / √ log n, subject to an exponentially fast decaying weak dependence coefficient. Using the technique of Tikhomirov [45], Götze and Hipp obtained Edgeworth expansions for processes of type (1.4) in a series of works; cf. [19-21]; see also Heinrich [24] and Lahiri [32]. This approach, however, requires the validity of a number of technical conditions. This includes in particular a conditional Crámer-like condition subject to an exponential decay, which is somewhat difficult to verify. In contrast, it turns out that a Berry-Esseen theorem only requires a simple, yet fairly general dependence condition where no exponential decay is required. Indeed, we will see that many popular examples from the literature are within our framework. Unlike previous results in the literature, we also obtain optimal rates for p ∈ (2, 3) given (infinite) weak dependence, which to the best of our knowledge is new (excluding special cases as linear processes). The proofs are based on an m-dependent approximation (m → ∞), which is quite common in the literature. The substantial difference here is the subsequent treatment of the m-dependent sequence. To motivate one of the main ideas of the proofs, . As a dependence measure, we then consider the quantity sup k∈Z X k − X (l,′) k p , p ≥ 1. Dependence conditions of this type are quite general and easy to verify in many cases; cf.[1,48]and the 1. Introduction. Let {X k } k∈Z be a zero mean process having second moments E[X 2 k ] < ∞. Consider the partial sum S n = n k=1 X k and its normalized variance s 2 n = n −1 Var[S n ]. A very important issue in probability theory and statistics is whether or not the central limit theorem holds, that is, if we have lim n→∞ P S n ≤ x ns 2 n − Φ(x) = 0, (1.1) where Φ(x) denotes the standard normal distribution function. Going one step further, we can ask ourselves about the possible rate of convergence in (1.1), more precisely, if it holds that lim n→∞ d(P Sn/ √ ns 2 n , P Z )r n < ∞ for a sequence r n → ∞, (1.2) where d(·, ·) is a probability metric, Z follows a standard normal distribution and P X denotes the probability measure induced by the random variable X. The rate r n can be considered as a measure of reliability for statistical inference based on S n , and large rates are naturally preferred. The question of let us assume p = 3 for a moment. Given a weakly, m-dependent sequence {X k } k∈Z , one may show via classic arguments that (1.5) provided that E[|X 1 | 3 ] < ∞ and s 2 n > 0. Note, however, since X k is weakly dependent, one finds that ∆ n ≤ C m/nE[|X 1 | 3 ],m −3/2 |E[S 3 m ]| ≤ C √ m . (1.6) Hence if one succeeds in replacing E[|X 1 | 3 ] in (1.5) with (1.6), one obtains the optimal rate r n = √ n. A similar reasoning applies to p ∈ (2, 3). Unfortunately though, setting this idea to work leads to rather intricate problems, and a technique like that of Tikhomirov [45] is not fruitful, inevitably leading to a suboptimal rate. Our approach is based on coupling and conditioning arguments and ideal (Zolotarev) metrics. Interestingly, there is a connection to more recent results of Dedecker et al. [11], who consider different (smoother) probability metrics. We will see that at least some of the problems we encounter may be redirected to these results after some preparation. 2. Main results. Throughout this paper, we will use the following notation: for a random variable X and p ≥ 1, we denote with X p = E[X p ] 1/p the L p norm. Let {ε k } k∈Z be a sequence of independent and identically distributed random variables with values in a measurable space S. Denote the corresponding σ-algebra with E k = σ(ε j , j ≤ k). Given a real-valued stationary sequence {X k } k∈Z , we always assume that X k is adapted to E k for each k ∈ Z. Hence we implicitly assume that X k can be written as in (1.4). For convenience, we write X k = g k (θ k ) with θ k = (ε k , ε k−1 , . . .). The class of processes that fits into this framework is large and contains a variety of functionals of linear and nonlinear processes including ARMA, GARCH and related processes (see, e.g., [18,46,48]), but also examples from dynamic system theory. Some popular examples are given below in Section 3. A nice feature of the representation given in (1.4) is that it allows us to give simple, yet very efficient and general dependence conditions. Following Wu [47], let {ε ′ k } k∈Z be an independent copy of {ε k } k∈Z on the same probability space, and define the "filter" θ examples below. Observe that if the function g = g k does not depend on k, we obtain the simpler version sup k∈Z X k − X (l,′) k p = X l − X ′ l p . (2.2) Note that it is actually not trivial to construct a stationary process {X k } k∈Z that can only be represented as X k = g k (θ k ); that is, a function g independent of k such that X k = g(θ k ) for all k ∈ Z does not exist. We refer to Corollary 2.3 in Feldman and Rudolph [16] for such an example. We will derive all of our results under the following assumptions. Assumption 2.1. Let {X k } k∈Z be stationary such that for some p ≥ 2: (i) X k p < ∞, E[X k ] = 0, (ii) ∞ l=1 l 2 sup k∈Z X k − X (l,′) k p < ∞, (iii) s 2 > 0, where s 2 = k∈Z E[X 0 X k ]. In the sequel, B denotes a varying absolute constant, depending only on p, ∞ l=1 l 2 sup k∈Z X k − X (l,′) k p and s 2 . The following theorem is one of the main results of this paper. sup x∈R P S n / ns 2 n ≤ x − Φ(x) ≤ B n p/2−1 , and hence we may select r n = n p/2−1 . Theorem 2.2 provides optimal convergence rates under mild conditions. In particular, it seems that this is the first time optimal rates are shown to hold under general infinite weak dependence conditions if p ∈ (2, 3). Examples to demonstrate the versatility of the result are given in Section 3. In particular, we consider functions of the dynamical system T x = 2x mod 1 in Example 3.2, a problem which has been studied in the literature for decades. Combining Theorem 2.2 with results of Dedecker and Rio [13], we also obtain optimal results for the L q -norm for martingale differences. Theorem 2.3. Grant Assumption 2.1 for some p ≥ 4, and let s 2 n = n −1 S n 2 2 . If {X k } k∈Z is a martingale difference sequence, then for any q ≥ 1 we have R P S n / ns 2 n ≤ x − Φ(x) q dx ≤ Bn −q/2 . BERRY-ESSEEN THEOREMS UNDER WEAK DEPENDENCE 5 Note that in the case q = 1, the results of Dedecker and Rio [13] are more general. The nonuniform analogue to Theorems 2.2 and 2.3 is given below. Here, we obtain optimality up to logarithmic factors. Theorem 2.4. Grant Assumption 2.1 for some p > 2. Then for any x ∈ R, P S n / ns 2 n ≤ x − Φ(x) ≤ n −(p∧3)/2+1 B(log n) p/2 1 + |x| p , where a ∧ b = min{a, b}. As a particular application of Theorem 2.4, consider f (|S n |/ ns 2 n ) where the function f (·) satisfies f (0) = 0 and ∞ 0 |f ′ (x)| 1 + |x| p dx < ∞ (2.3) for some p > 0, and the derivative f ′ (x) exists for x ∈ (0, ∞). If S n p < ∞, property (2.3) implies the identity E f |S n |/ ns 2 n = ∞ 0 f ′ (x)P |S n |/ ns 2 n ≥ x dx, and we thus obtain the following corollary. E f |S n |/ ns 2 n − R f (|x|) dΦ(x) ≤ Bn −(p∧3)/2+1 (log n) p/2 . As a special case, consider f (|x|) = |x| q , q > 0. We may then use Corollary 2.5 to obtain rates of convergence for moments. Corollary 2.6. Grant Assumption 2.1 for some p > 2. Then for any 0 < q < p, we have S n / ns 2 n q q − R |x| q dΦ(x) ≤ Bn −(p∧3)/2+1 (log n) p/2 . In the special case of i.i.d. sequences and 0 < p < 4, sharp results in this context have been obtained in Hall [23]. It seems that related results for dependent sequences are unknown. Applications and examples. All examples considered here are timehomogenous Bernoulli-shift processes; that is, g = g k does not depend on k, and hence equality (2.2) holds. i } i∈N satisfies ∞ i=0 α 2 i < ∞. If ε k 2 < ∞, then one may show that the linear process Y k = ∞ i=0 α i ε k−i exists and is stationary. Let f be a measurable function such that E[X k ] = 0, where X k = f (Y k ). If f is Hölder continuous with regularity 0 < β ≤ 1, that is, |f (x) − f (y)| ≤ c|x − y| β , then for any p ≥ 1 X k − X ′ k p ≤ cα β k ε 0 p . Hence if ∞ i=0 i 2 |α i | β < ∞Then T U 0 = ∞ j=0 2 −j−1 ζ j , where ζ j are Bernoulli random variables. The flow T k U 0 can then be written as T k U 0 = ∞ j=0 2 −j−1 ζ j+k ; see [28]. The study about the behavior of S n = n k=1 f (T k U 0 ) for appropriate functions f has a very long history and dates back to Kac [30]. Since then, numerous contributions have been made; see, for instance, [4,5,13,14,27,28,31,33,35,36,40,42], to name a few. Here, we consider the following class of functions. Let f be a function defined on the unit interval [0, 1], such that 1 0 f (t) dt = 0, 1 0 |f (t)| p dt < ∞ and (3.1) 1 0 t −1 |log(t)| 2 w p (f, t) dt < ∞, where w p (f, t) denotes the L p ([0, 1], λ) modulos of continuity of f ∈ L p ([0, 1], λ) . This setup is a little more general than in [28]. For x ∈ R + , letf (x) = f (x − ⌊x⌋); that is,f is the one-periodic extension to the positive real line. One then often finds the equivalent formulation S n = n k=1f (2 k U 0 ) in the literature. Consider now the partial sum S n = n k=1f (2 k U 0 ). Ibragimov [28] showed that sup x∈R P S n / ns 2 n ≤ x − Φ(x) ≤ C log n n p/2−1 . (3.2) By alternative methods, Le Borgne and Pène [33], extending Rio [43], managed to remove the logarithmic factor if f ∞ < ∞. A priori, the sequence {T k U 0 } k∈Z does not directly fit into our framework, which, however, can be achieved by a simple time flip. Define the function T n (i) = n − i + 1 for i ∈ {n, n − 1, . . .}, and let ε k = ζ Tn(k) . Then we may write X k = f (T k U 0 ) = f ∞ j=0 ε k−j 2 −j−1 , k ∈ {1, . . . , n}. Note that we have to perform this time flip for every n ∈ N, which, however, has no impact on the applicability of our results. Using the same arguments as in [28], we find that (3.1) implies that for p ∈ (2, 3] ∞ k=1 k 2 X k − X ′ k p < ∞. If s 2 > 0, we see that Assumption 2.1 holds. In particular, an application of Theorem 2.2 gives the rate r n = n p/2−1 , thereby removing the unnecessary log n factor in (3.2) for the whole range p ∈ (2, 3]. In this context, it is useful to work with the transformed block-variables X k = 1 m m−1 l=0 Y mk−l , k ∈ Z, and write X k = g(ε k , ε k−1 ) where ε k = (ζ km , . . . , ζ (k−1)m+1 ) ⊤ ∈ S m ; hence {X k } k∈Z is a two-dependent sequence. This representation ensures that Assumption 2.1(i) and (ii) hold for {X k } k∈Z , independently of the value of m. The drawback of this block-structure is that we loose a factor m, since we have 1 √ nm S n = 1 √ nm n k=1 Y k = 1 n/m n/m k=1 X k , where we assume that n/m ∈ N for simplicity. However, this loss is known in the literature: Theorem 2.2 now yields the commonly observed rate r n = 8 M. JIRAK (n/m) p/2−1 in the context of m-dependent sequences satisfying (3.3); see, for instance, Theorem 2.6 in [9]. In the latter, the rate r n = (n/m) p/2−1 is not immediately obvious, but follows from elementary computations using (3.3). Example 3.4 (Iterated random function). Iterated random functions (cf. [15]) are an important class of processes. Many nonlinear models like ARCH, bilinear and threshold autoregressive models fit into this framework. Let S = R and {X k } k∈Z be defined via the recursion X k = G(X k−1 , ε k ), commonly referred to as iterated random functions; see, for instance, [15]. Let L ε = sup x =y |G(x, ε) − G(y, ε)| |x − y| (3.4) be the Lipschitz coefficient. If L ε p < 1 and G(x 0 , ε) p < ∞ for some x 0 , then X k can be represented as X k = g(ε k , ε k−1 , . . .) for some measurable function g. In addition, we have X k − X ′ k p ≤ Cρ −k where 0 < ρ < 1; (3.5) see [49]. Hence if E[X k ] = 0 and s 2 > 0, Assumption 2.1 holds. As an example, consider the stochastic recursion X k+1 = a k+1 X k + b k+1 , k ∈ Z, where {a k , b k } k∈Z is an i.i.d. sequence. Let ε k = (a k , b k ). If we have, for some p ≥ 2, a k p < 1 and b k p < ∞, E[b k ] = 0, (3.6) then L ε p ≤ a k p < 1, and Assumption 2.1 holds if s 2 > 0. In particular, if a k , b k are independent, then one readily verifies that s 2 = b 0 2 2 1 − a 0 2 2 1 + 2E[a 0 ] 1 − E[a 0 ] , which is strictly positive since |E[a 0 ]| < 1 by Jensen's inequality. Hence if (3.6) holds for p > 2, then Assumption 2.1 holds for p. Analogue conditions can be derived for higher order recursions. Example 3.5 [GARCH(p, q) sequences]. Let S = R. Another very prominent stochastic recursion is the GARCH(p, q) sequence, given through the relations X k = ε k L k where {ε k } k∈Z is a zero mean i.i.d. sequence and L 2 k = µ + α 1 L 2 k−1 + · · · + α p L 2 k−p + β 1 X 2 k−1 + · · · + β q X 2 k−q , with µ, α 1 , . . . , α p , β 1 , . . . , β q ∈ R. We assume that ε k p < ∞ for some p ≥ 2. An important quantity is γ C = r i=1 α i + β i ε 2 i 2 with r = max{p, q}, where we replace possible undefined α i , β i with zero. If γ C < 1, then {X k } k∈Z is stationary; cf. [8]. In particular, it was shown in [3] that {X k } k∈Z may be represented as X k = √ µε k 1 + ∞ n=1 1≤l 1 ,...,ln≤r n i=1 (α l i + β l i ε 2 j−l 1 −···−l i ) 1/2 . Using this representation and the fact that |x − y| p ≤ |x 2 − y 2 | p/2 for x, y ≥ 0, p ≥ 1, one can follow the proof of Theorem 4.2 in [1] to show that X k − X ′ k p ≤ Cρ k where 0 < ρ < 1. Since E[X k ] = E[ε k ] = 0, Assumption 2.1 holds if s 2 > 0. We remark that previous results on ∆ n , in the case of GARCH(p, q) sequences, either require heavy additional assumptions or have suboptimal rates; cf. [27]. Example 3.6 (Volterra processes). In the study of nonlinear processes, Volterra processes are of fundamental importance. Following Berkes et al. [4], we consider X k = ∞ i=1 0≤j 1 <···<j i a k (j 1 , . . . , j i )ε k−j 1 · · · ε k−j i , where S = R and ε k p < ∞ for p ≥ 2, and a k are called the kth Volterra kernel. Let A k,i = k∈{j 1 ,...,j i },0≤j 1 <···<j i |a k (j 1 , . . . , j i )|. Then there exists a constant C such that 4. Proofs. The main approach consists of an m-dependent approximation where m → ∞, followed by characteristic functions and Esseen's inequality. However, here the trouble starts, since we cannot factor the characteristic function as in the classic proof, due to the m-dependence. Tikhomirov [45] uses a chaining-type argument, which is also fruitful for Edgeworth expansions; cf. [19]. However, since this approach inevitably leads to a loss in the rate, this is not an option for Berry-Esseen-type results. In order to circumvent this problem, we first work under an appropriately chosen conditional probability measure P Fm . Unfortunately though, this leads to rather intricate problems, since all involved quantities of interest are then random. We first consider the case of a weakly m-dependent sequence {X k } k∈Z , where m → ∞ as n increases. Note that this is different from Example 3.3. For the general case, we then construct a suitable m-dependent approximating sequence such that the error of approximation is negligible, which is carried out in Section 4.2. The overall proof of Theorem 2.2 is lengthy. Important technical auxiliary results are therefore established separately in Section 4.5. Minor additionally required results are collected in Section 4.6. The proofs of Theorems 2.3 and 2.4 are given in Sections 4.3 and 4.4. To simplify the notation in the proofs, we restrict ourselves to the case of homogeneous Bernoulli shifts, that is, where X k = g(ε k , ε k−1 , . . .), and the function g does not depend on k. This requires substantially fewer indices and notation throughout the proofs, and, in particular, (2.2) holds. The more general nonhomogenous (but still stationary) case follows from straightforward (notational) adaptations. This is because the key ingredient we require for the proof is the Bernoulli-shift structure (1.4) in connection with the summability condition, Assumption 2.1(ii). Whether or not g depends on k is of no relevance in this context. 4.1. m-dependencies. In order to deal with m-dependent sequences, we require some additional notation and definitions. Throughout the remainder of this section, we let X k − X ′ k p ≤ C ∞ i=1 ε 0 i p A k,i . Thus if ∞ k,i=1 k 2 A k,i < ∞X k = f m (ε k , . . . , ε k−m+1 ) for m ∈ N, k ∈ Z, and measurable functions f m : S m → R, where m = m n → ∞ as n increases. We work under the following conditions: Assumption 4.1. Let {X k } k≥Z be such that for some p ≥ 2, uniformly in m: (i) X k p < ∞, E[X k ] = 0, (ii) ∞ k=1 k 2 X k − X ′ k p < ∞, (iii) s 2 m > 0, where s 2 m = k∈Z E[X 0 X k ] = m k=−m E[X 0 X k ]. Observe that this setup is fundamentally different from that considered in Example 3.3. In particular, here we have that Var[S n ] ∼ n. Define the following σ-algebra: F m = σ(ε −m+1 , . . . , ε 0 , ε ′ 1 , . . . , ε ′ m , ε m+1 , . . . , ε 2m , ε ′ 2m+1 , . . .), (4.1) where we recall that {ε k } k∈Z and {ε ′ k } k∈Z are mutually independent, identically distributed random sequences. We write P Fm (·) for the conditional law and E Fm [·] (or E H [·]) for the conditional expectation with respect to F m (or some other filtration H). We introduce S (1) |m = n k=1 X k − E[X k |F m ] and S (2) |m = n k=1 E[X k |F m ], hence S n = n k=1 X k = S (1) |m + S (2) |m . To avoid any notational problems, we put X k = 0 for k / ∈ {1, . . . , n}. Let n = 2(N − 1)m + m ′ , where N, m are chosen such that c 0 m ≤ m ′ ≤ m and c 0 > 0 is an absolute constant, independent of m, n. For 1 ≤ j ≤ N , we construct the block random variables U j = (2j−1)m k=(2j−2)m+1 X k − E[X k |F m ] and R j = 2jm k=(2j−1)m+1 X k − E[X k |F m ], and put Y (1) j = U j + R j , hence S (1) |m = N j=1 Y (1) j . Note that by construction of the blocks, Y (1) j , j = 1, . . . , N are independent random variables under the conditional probability measure P Fm (·), and are identically distributed at least for j = 1, . . . , N − 1 under P . We also put Y (2) 1 = m k=1 E[X k |F m ] and Y (2) j = (j+1)m k=(j−1)m+1 E[X k |F m ] for j = 2, . . . , N . Note that Y (2) j , j = 1, . . . , N is a sequence of independent random variables. The following partial and conditional variances are relevant for the proofs: σ 2 j|m = 1 2m E Fm [(Y (1) j ) 2 ] and σ 2 j = E[σ 2 j|m ], σ 2 |m = 1 n E[(S (1) |m ) 2 |F m ] = 1 N + m ′ /2m N j=1 σ 2 j|m , σ 2 m = E[σ 2 |m ] = 1 N + m ′ /2m N j=1 σ 2 j , σ 2 m = 1 2m m k=1 m l=1 E[X k X l ]. 12 M. JIRAK As we shall see below, these quantities are all closely connected. Note that σ 2 i = σ 2 j for 1 ≤ i, j ≤ N − 1, but σ 2 1 = σ 2 N in general. Moreover, we have the equation 2m σ 2 m = ms 2 m − k∈Z m ∧ |k|E[X 0 X k ]. (4.2) The above relation is important, since Lemma 4.6 yields that under As- sumption 4.1 we have 2 σ 2 m = s 2 m + O(m −1 ). Moreover, Lemma 4.7 gives σ 2 j = σ 2 m + O(m −1 ) for 1 ≤ j ≤ N − 1. We conclude that σ 2 j = s 2 m /2 + O(m −1 ) > 0 for sufficiently large m. (4.3) The same is true for σ 2 N , since m ′ ≥ c 0 m. Summarizing, we see that we do not have any degeneracy problems for the partial variances σ 2 j , 1 ≤ j ≤ N under Assumption 4.1. For the second part S (2) |m , we introduce ς 2 m = n −1 S (2) |m 2 2 . One then readily derives via conditioning arguments that s 2 nm def = n −1 S n 2 2 = n −1 S (1) |m 2 2 + n −1 S (2) |m 2 2 = σ 2 m + ς 2 m . (4.4) We are now ready to give the main result of this section. dition that N = N n = n λ for 0 < λ ≤ p/(2p + 2). Then sup x∈R |P (S n / √ n ≤ x) − Φ(x/s nm )| ≤ c(λ, p)n −p/2+1 , where c(λ, p) > 0 depends on λ, p, ∞ k=1 k 2 X k − X ′ k p and inf m s 2 m > 0. The proof of Theorem 4.2 is based on the following decomposition. Let Z 1 , Z 2 be independent unit Gaussian random variables. Then sup x∈R |P (S n / √ n ≤ x) − Φ(x/s nm )| = sup x∈R |P (S (1) |m ≤ x √ n − S (2) |m ) − P (Z 1 σ m ≤ x − Z 2 ς m )| ≤ A + B + C, where A, B, C are defined as A = sup x∈R |E[P |Fm (S (1) |m / √ n ≤ x − S (2) |m / √ n) − P |Fm (Z 1 σ |m ≤ x − S (2) |m / √ n)]|, B = sup x∈R |E[P |Fm (Z 1 σ |m ≤ x − S (2) |m / √ n) − P |Fm (Z 1 σ m ≤ x − S (2) |m / √ n)]|, C = sup x∈R |P (S (2) |m / √ n ≤ x − Z 1 σ m ) − P (Z 2 ς m ≤ x − Z 1 σ m )|. We will treat the three parts separately, and show that A, B, C ≤ c(λ,p) 3 n −p/2+1 , which proves Theorem 4.2. As a brief overview, the proof consists of the following steps: (a) apply Esseen's smoothing inequality, and factor the resulting characteristic function into a (conditional) product of characteristic functions ϕ j (x) under the conditional probability measure P Fm ; (b) use ideal metrics to control the distance between ϕ j (x) and corresponding Gaussian versions under P Fm ; (c) based on Renyi's representation, control the (conditional) characteristic functions ϕ j (x) under P ; (d) replace conditional variances under the overall probability measure P . One of the main difficulties arises from working under the conditional measure P Fm . For the proof, we require some additional notation. In analogy to the filter θ (l,′) k , we denote with θ (l, * ) k , θ (l, * ) k = (ε k , ε k−1 , . . . , ε ′ k−l , ε ′ k−l−1 , ε ′ k−l−2 , . . .). (4.5) We put θ * k = θ (k, * ) k = (ε k , ε k−1 , . . . , ε ′ 0 , ε ′ −1 , ε ′ −2 , . . .) and X , X ′ k , X * k . This means that we replace every ε ′ k with ε ′′ k at all corresponding places. For k ≥ 0, we also introduce the σ-algebras E ′ k = σ(ε j , j ≤ k and j = 0, ε ′ 0 ) and (4.6) E * k = σ(ε j , 1 ≤ j ≤ k and ε ′ i , i ≤ 0). Similarly, we introduce filtrations E ′′ k and E * * k . Throughout the proofs, we make the following conventions: (1) We do not distinguish between N and N + m ′ /2m since the difference m ′ /2m is not of any particular relevance for the proofs. We use N for both expressions. (2) The abbreviations I, II , III , . . . , for expressions (possible with some additional indices) vary from proof to proof. (3) We use , , (∼) to denote (two-sided) inequalities involving a multiplicative constant. (4) If there is no confusion, we put Y j = (2m) −1/2 Y (1) j for j = 1, . . . , N to lighten the notation, particularly in part A. |m ∈ F m , we obtain that A = sup x∈R |E[P |Fm (S (1) |m / √ n ≤ x − S (2) |m / √ n) − P |Fm (Z 1 σ |m ≤ x − S (2) |m / √ n)]| ≤ E sup y∈R |P |Fm (S (1) |m / √ n ≤ y) − P |Fm (Z 1 σ |m ≤ y)|1(B N ) + 2P (B c N ). Corollary 4.8 yields that P (B c N ) n −p/2 N n −p/2+1 since N ≤ n, and it thus suffices to treat ∆ |m def = sup y∈R |P |Fm (S (1) |m / √ n ≤ y) − P |Fm (Z 1 σ |m ≤ y)|1(B N ). (4.7) Step 1: Berry-Esseen inequality. Denote with ∆ T |m the smoothed version of ∆ |m (cf. [17]) as in the classical approach. Since σ 2 |m ≥ s 2 m /4 > 0 on the set B N by construction, the smoothing inequality (cf. [17], Lemma 1, XVI.3) is applicable, and it thus suffices to treat ∆ T |m . Let ϕ j (x) = E[e ixY j |F m ], and put T = n p/2−1 c T , where c T > 0 will be specified later. Due to the independence of {Y j } 1≤j≤N under P |Fm and since 1(B N ) ≤ 1, it follows that E[|∆ T |m |] ≤ T −T E N j=1 ϕ j (ξ/ √ N ) − N j=1 e −σ 2 j|m ξ 2 /2N |ξ| dξ. (4.8) Put t = ξ/ √ N . Then N j=1 a j − N j=1 b j = N i=1 ( i−1 j=1 b j )(a i −b i )( N j=i+1 a j ), where we use the convention that i−2 j=1 (·) = N j=i+2 (·) = 1 if i − 2 < 1 or i + 2 > N . Hence we have N j=1 ϕ j (t) − N j=1 e −σ 2 j|m t 2 /2 = N i=1 i−1 j=1 ϕ j (t) (ϕ i (t) − e −σ 2 i|m t 2 /2 ) N j=i+1 e −σ 2 j|m t 2 /2 . Note that both {ϕ j (t)} 1≤j≤N and {e −σ 2 j|m t 2 /2 } 1≤j≤N are two-dependent sequences. Since |ϕ j (t)|, e −σ 2 j|m t 2 /2 ≤ 1, it then follows by the triangle inequality, stationarity and "leave one out" that N j=1 ϕ j (t) − N j=1 e −σ 2 j|m t 2 /2 1 ≤ N i=1 i−2 j=1 e −σ 2 j|m t 2 /2 1 ϕ i (t) − e −σ 2 i|m t 2 /2 1 N j=i+2 |ϕ j (t)| 1 ≤ N ϕ 1 (t) − e −σ 2 1|m t 2 /2 1 N −1 j=N/2 |ϕ j (t)| 1 + N N/2−3 j=1 e −σ 2 j|m t 2 /2 1 ϕ 1 (t) − e −σ 2 1|m t 2 /2 1 + N/2−3 j=1 e −σ 2 j|m t 2 /2 1 ϕ N (t) − e −σ 2 N|m t 2 /2 1 = I N (ξ) + II N (ξ) + III N (ξ). We proceed by obtaining upper bounds for I N (ξ), II N (ξ) and III N (ξ). Step 2: Bounding ϕ i (t) − e −σ 2 i|m t 2 /2 1 , i ∈ {1, N }. Let Z i , i ∈ {1 , N } be two zero mean standard Gaussian random variables. Then ϕ i (t) − e −σ 2 i|m t 2 /2 1 ≤ E Fm [cos(tY i ) − cos(tσ i|m Z i )] 1 + E Fm [sin(tY i ) − sin(tσ i|m Z i )] 1 . Due to the very nice analytical properties of sin(y), cos(y), one may reformulate the above in terms of ideal-metrics; cf. [50] and Section 4.5.2. This indeed leads to the desired bound ϕ i (t) − e −σ 2 i|m t 2 /2 1 |t| p m −p/2+1 . (4.9) The precise derivation is carried out in Section 4.5.2 via Lemmas 4.9 and 4.10, and Corollary 4.11. Whether i = 1 or i = N makes no difference. Step 3: Bounding N −1 j=N/2 |ϕ j (t)| 1 : in order to bound N −1 j=N/2 |ϕ j (t)| 1 , we require good enough estimates for |ϕ j (t)| where 0 ≤ t < 1. As already mentioned, we cannot directly follow the classical approach. Instead, we use a refined version based on a conditioning argument. To this end, let us first deal with ϕ j (t). Put Then since F m ∩ σ(−ε −m+1 , . . . , ε 0 ) ⊆ G (l) 1 , we have |ϕ 1 (t)| ≤ E Fm [|E G (l) 1 [e it(2m) −1/2 V (l) 1 (m) ]|]. (4.12) Clearly, this is also valid for ϕ j (t), j = 2, . . . , N , with corresponding G (4.13) and J = {j : N/2 ≤ j ≤ N − 1 and 2 divides j}, and hence J denotes the set of all even numbers between N/2 and N − 1. Then j (x) = E G (l) j [e ix(m−l) −1/2 V (l) j (m) ],N −1 j=N/2 |ϕ j (t)| ≤ j∈J E Fm [|E G (l) j [e it(2m) −1/2 V (l) j (m) ]|] = j∈J E Fm [|ϕ (l) j (x)|], where x = t (m − l)/2m. Note that {V (l) j (m)} j∈J is a sequence of i.i.d. random variables, particularly with respect to P Fm . Hence by independence and Jensen's inequality, it follows from the above that N −1 j=N/2 |ϕ j (t)| 1 ≤ j∈J E Fm [|ϕ (l) j (x)|] 1 (4.14) ≤ j∈J ϕ (l) j (x) 1 = j∈J |ϕ (l) j (x)| 1 . We thus see that it suffices to deal with ϕ (l) j (x). The classical argument uses the estimate ϕ(ξ/ √ σ 2 n) ≤ e −5ξ 2 /18n for ξ 2 /n ≤ c, c > 0 for the characteristic function ϕ. Since in our case ϕ j is random, we cannot use this estimate. Instead, we will use Lemma 4.5, which provides a similar result. In order to apply it, set J = |J | ≥ N/8, H j = 1 √ m − l V (l) j (m) and H j = G (l) j . (4.15) For the applicability of Lemma 4.5, we need to verify that: (i) E H j [H j ] = 0; (ii) there exists a u − > 0 such that P (E |H j [H 2 j ] ≤ u − ) < 1/7, uniformly for j ∈ J ; (iii) H j p ≤ c 1 uniformly for j ∈ J and some c 1 < ∞. Now (i) is true by construction. Claim (ii) is dealt with via Lemma 4.14, which yields that P (E |H j [H 2 j ] ≤ σ 2 m−l ) 1 √ m − l .E G (l) 1 [X k ] = E E l [X k − X (k−l, * ) k ] and E Fm [X k ] = E Fm [X k − X * k ]. (4.17) By stationarity and the triangle and Jensen inequalities, we then have that √ m − l H j p ≤ m k=l+1 X k p + m k=l+1 E E l [X k − X (k−l, * ) k ] p (4.18) + 2 m k=l+1 E Fm [X k − X * k ] p + 2 R 1 p . Using Jensen's inequality and arguing similar to Lemma 4.13, it follows that m k=l+1 E G l [X k − X (k−l, * ) k ] p ≤ m k=l+1 X k − X (k−l, * ) k p ≤ ∞ k=1 k X k − X ′ k p < ∞. Similarly, using also Lemma 4.13 to control R 1 p , we obtain that where x = t (m − l)/2m. It is important to emphasize that both c ϕ,1 , c ϕ,2 do not depend on l, m and are strictly positive. Moreover, we find from (4.16) that l can be chosen freely, as long as m − l is larger than K 0 , which will be important in the next step. Step 4: Bounding and integrating I N (ξ), II N (ξ), III N (ξ). We first treat I N (ξ). Recall that t = ξ/ √ N , hence m k=l+1 E Fm [X k ] p + R 1 p < ∞.|t| p m −p/2+1 |ξ| p n −p/2+1 N −1 . By (4.9), (4.14) and (4.20), it then follows for ξ 2 (m − l) < c ϕ,2 n that I N (ξ) |ξ| p n −p/2+1 (e −c ϕ,1 ξ 2 (m−l)/16m + e − √ N/32 log 8/7 ). (4.21) To make use of this bound, we need to appropriately select l = l(ξ). Recall that N = n λ , 0 < λ ≤ p/(2p + 2) by assumption. Choosing l(ξ) = 1(ξ 2 < n λ c ϕ,2 ) + m − c ϕ,2 n 2ξ 2 ∨ K 0 1(ξ 2 ≥ n λ c ϕ,2 ) and c 2 T < c ϕ,2 /K 0 , we obtain from the above that T −T I N (ξ)/ξ dξ n −p/2+1 . (4.22) In order to treat II N (ξ), let N ′ = N/2 − 3, and B N ′ = {N ′ −1 N ′ j=1 σ 2 j|m ≥ s 2 m /4}. Denote with B c N ′ its complement. Then by Corollary 4.8 (straightforward adaption is necessary) and (4.9), it follows that II N (ξ)1(|ξ| ≤ N ) ≤ N ϕ 1 (t) − e −σ 2 1|m ξ 2 /2 1 N ′ j=1 e −σ 2 j|m ξ 2 /2 1(B N ′ ) 1 + N ϕ 1 (t) − e −σ 2 1|m ξ 2 /2 1 P (B c N ′ ) (4.23) |ξ| p n −p/2+1 e −s 2 m ξ 2 /16 + |ξ| p n −p/2+1 N n −p/2 . Similarly, using ϕ 1 (t) − e −σ 2 1|m ξ 2 /2 1 ≤ 2 one obtains II N (ξ)1(|ξ| > N ) |ξ| p n −p/2+1 e −s 2 m ξ 2 N/16 + n −p/2 N 2 . II N (ξ)/ξ dξ n −p/2+1 |ξ|≤N |ξ| p−1 (e −s 2 m ξ 2 /16 + n −p/2 N ) dξ + N <|ξ|≤T (n −p/2+1 |ξ| p−1 e −s 2 m ξ 2 /16 + n −p/2 N 2 ξ −1 ) dξ (4.25) n −p/2+1 + n −p/2+1 n −p/2 N p+1 + n −p/2 N 2 log T n −p/2+1 , since N = n λ , 0 < λ ≤ p/(2p + 2) by assumption. Similarly, one obtains the same bound for III N (ξ). This completes the proof of part A. Part B. Proof. Let ∆ (2) (x) def = E[P |Fm (Z 1 σ |m ≤ x − S (2) |m / √ n) − P |Fm (Z 1 σ m ≤ x − S (2) |m / √ n)]. Recall that B N = {N −1 N j=1 σ 2 j|m ≥ s 2 m /4} and P (B c N ) n −p/2+1 by Corollary 4.8. Using properties of the Gaussian distribution, it follows that B ≤ sup x∈R |E[∆ (2) (x)1(B N )]| + sup x∈R |E[∆ (2) (x)1(B c N )]| E[|1/σ |m − 1/σ m |1(B N )] + n −p/2+1 . Using (a − b)(a + b) = a 2 − b 2 , Hölders inequality and Lemma 4.7, we obtain that E[|1/σ |m − 1/σ m |1(B N )] σ 2 |m − σ 2 m p/2 n −p/2+1 . Hence we conclude that B n −p/2+1 . Part C. Proof. Due to the independence of Z 1 , Z 2 , we may rewrite C as C = sup x∈R |Φ((x − S (2) |m / √ n)/σ m ) − Φ((x − Z 2 ς m )/σ m )|, where Φ(·) denotes the c.d.f. of a standard normal distribution. This induces a "natural" smoothing. The claim now follows by repeating the same arguments as in part A. Note however, that the present situation is much easier to handle, due to the already smoothed version, and since Y (2) k , k = 1, . . . , N is a sequence of independent random variables. Alternatively, one may also directly appeal to the results in [11]. 4.2. Proof of Theorem 2.2. The proof of Theorem 2.2 mainly consists of constructing a good m-dependent approximation and then verifying the conditions of Theorem 4.2. To this end, set m = cn 3/4 for some c > 0, and note that 1/4 < p/(2p + 2) for p ∈ (2,3]. Let E m k = σ(ε j , k − m + 1 ≤ j ≤ k), and define the approximating sequence as X (≤m) k = E[X k |E m k ] and (4.26) X (>m) k = X k − X (≤m) k = X k − E[X k |E m k ] . We also introduce the corresponding partial sums as |P (S n / √ n ≤ xs n ) − Φ(x)| ≤ A 0 (x, δ) + A 1 (m, n, δ) + max{A 2 (m, n, x, δ) + A 3 (m, n, δ), A 4 (m, n, x, δ) + A 5 (m, n, x, δ)}, where: A 0 (x, δ) = |Φ(x) − Φ(x + δ)|; A 1 (m, n, δ) = P (|S n − S (≤m) n | ≥ δs n √ n); A 2 (m, n, x, δ) = |P (S (≤m) n ≤ (x + δ)s n √ n) − Φ((x + δ)s n /s nm )|; A 3 (m, n, x, δ) = |Φ((x + δ)s n /s nm ) − Φ(x + δ)|; A 4 (m, n, x, δ) = A 2 (m, n, x, −δ) and A 5 (m, n, x, δ) = A 3 (m, n, x, −δ). Proof of Theorem 2.2. As a preparatory result, note that ns 2 n = ns 2 + k∈Z (n ∧ |k|)E[X 0 X k ]. (4.28) Using the same arguments as in Lemma 4.6, it follows that ns n = ns 2 + O(1) > 0. By the properties of Gaussian distribution, sup x∈R |Φ(x/ √ s 2 ) − Φ(x/ √ s n )| n −1 , and we may thus safely interchange s 2 n and s 2 . We first deal with A 1 (m, n, δ). For j ∈ Z, denote with P j (X (>m) k ) the projection operator P j (X (>m) k ) = E[X (>m) k |E j ] − E[X (>m) k |E j−1 ]. (4.29) Proceeding as in the proof of Lemma 3.1 in [29], it follows that for k ≥ 0, P 0 (X (>m) k ) p ≤ 2 min X k − X ′ k p , ∞ l=m X l − X ′ l p . (4.30) An application of Theorem 1 in [48] now yields that n −1/2 S (>m) n p ≤ c(p) ∞ k=1 P 0 (X (>m) k ) p (4.31) for some absolute constant c(p) that only depends on p. By (4.30), it follows that the above is of magnitude ∞ k=L L −2 k 2 X k − X ′ k p + L ∞ k=m m −2 k 2 X k − X ′ k p L −2 + Lm −2 . (4.32) Setting L = m 2/3 , we obtain the bound O(m −4/3 ) = O(n −1 ). We thus conclude from the Markov inequality that P (|S n − S (≤m) n | ≥ δs n √ n) = P (|S (>m) n | ≥ δs n √ n) (δn) −p , hence A 1 (m, n, δ) (δn) −p . (4.33) Note that a much sharper bound can be obtained via moderate deviation arguments (cf. [22]), but the current one is sufficient for our needs, and its deviation requires fewer computations. Next, we deal with A 2 (m, n, x, δ). The aim is to apply Theorem 4.2 to obtain the result. In order to do so, we need to verify Assumption 4.1(i)-(iii) for X X (≤m) k p = E[X k |E m k ] p ≤ X k p < ∞. Hence Assumption 4.1(i) is valid. Case (ii): Note that we may assume k ≤ m, since otherwise (X (≤m) k ) ′ − X (≤m) k = 0, and Assumption 4.1(ii) is trivially true. Put E (m,′) k = σ(ε j , k − m + 1 ≤ j ≤ k, j = 0, ε ′ 0 ). Since E[X k |E m k ] ′ = E[X ′ k |E (m,′) k ], it follows that (X (≤m) k ) ′ − X (≤m) k = E E (m,′) k [X ′ k ] − E E m k [X k ] = E E (m,′) k [X ′ k − X k ] + E E (m,′) k [X k ] − E E m k [X k ] (4.34) = E E (m,′) k [X ′ k − X k ] + E E m k [X ′ k ] − E E m k [X k ] = E E (m,′) k [X ′ k − X k ] + E E m k [X ′ k − X k ]. (4.35) Hence by Jensen's inequality (X (≤m) k ) ′ − X (≤m) k p ≤ 2 X k − X ′ k p , which gives the claim. 22 M. JIRAK Case (iii): We have X (≤m) k = E[X (m, * ) k |E k ]. Then X (>m) k p = E[X k − X (m, * ) k |E k ] p ≤ X k − X (m, * ) k p (4.36) ≤ m −2 ∞ l=m l 2 X l − X ′ l p m −2 . By the Cauchy-Schwarz, triangle and Jensen inequalities, we have |E[X k X 0 ] − E[X (≤m) k X (≤m) 0 ]| ≤ X 0 2 X (>m) k 2 + X k 2 X (>m) 0 2 + X (>m) 0 2 X (>m) k 2 . By (4.36), this is of the magnitude O(m −2 ). We thus conclude that m k=0 E[X k X 0 ] − m k=0 E[X (≤m) k X (≤m) 0 ] m −1 . (4.37) On the other hand, we have k>m E[X k X 0 ] ≤ k>m X 0 2 X * k − X k 2 ≤ 1 m k>m k 2 X k − X ′ k 2 X 0 2 1 m . This yields m k∈Z E[X k X 0 ] − s 2 1 m , (4.38) which gives (iii) for large enough m. Since cn 3/4 , we see that we may apply Theorem 4.2 which yields sup x∈R A 2 (m, n, x, δ) n −p/2+1 . We first consider the case q > 1. Using Theorem 2.2, we have R |∆ n (x)| q dx ≤ ∆ q−1 n R |∆ n (x)| dx n −(q−1)/2 R |∆ n (x)| dx. (4.44) In order to bound R |∆ n (x)| dx, we apply [13], Theorem 3.2, which will give us the bound R |∆ n (x)| dx 1 √ n . (4.45) To this end, we need to verify that k>0 X 2 0 ∨ 1(E[X 2 k − E[X 2 k ]|E 0 ]) 1 + 1 k k i=1 X −i X 0 E[X 2 k − E[X 2 k ]|E 0 ] 1 < ∞ and (4.46) k>0 1 k k i=⌊k/2⌋ |X 0 | ∨ 1E[X i X 2 k − E[X i X 2 k ]|E 0 ] 1 < ∞. Applying the Hölder, Jensen and triangle inequalities, we get X 2 0 ∨ 1(E[X 2 k − E[X 2 k ]|E 0 ]) 1 ≤ X 2 0 ∨ 1 2 X k − X * k 4 X k + X * k 4 k X k − X ′ k 4 . Similarly, with E −i = σ(ε k , k ≤ −i), we obtain that X −i X 0 E[X 2 k − E[X 2 k ]|E 0 ] 1 X −i 4 E[X 0 |E −i ] 4 k X k − X ′ k 4 X i − X * i 4 k X k − X ′ k 4 i X i − X ′ i 4 k X k − X ′ k 4 . M. JIRAK In the same manner, we get that |X 0 | ∨ 1E[X i X 2 k − E[X i X 2 k ]|E 0 ] 1 X i − X * i 4 + X k − X * k 4 i X i − X ′ i 4 + k X k − X ′ k 4 . Combining all three bounds, the validity of (4.46) follows, and hence (4.45). For (4.44), we thus obtain R |∆ n (x)| q dx n −(q−1)/2−1/2 n −q/2 , which completes the proof for q > 1. For q = 1, we may directly refer to [13], Theorem 3.2, using the above bounds. ∆ = sup x∈R |P (Y ≤ x) − Φ(x)|, and assume that Y q < ∞ for q > 0 and 0 ≤ ∆ ≤ e −1/2 . Then |P (Y ≤ x) − Φ(x)| ≤ c(q) ∆(log 1/ ∆) q/2 + λ q 1 + |x| q for all x, where c(q) is a positive constant depending only on q, and λ q = R |x| q dΦ(x) − E[|Y | q ] . Consider first the case where |x| ≤ c 0 √ log n, for c 0 > 0 large enough (see below). Then by the Markov inequality and Lemma 4.12, it follows that P S n 1(|S n | ≤ n) ≤ x ns 2 n − P S n ≤ x ns 2 n n −p/2 . According to a Fuk-Nagaev-type inequality for dependent sequences in [34], Theorem 2, if it holds that However, setting a k = k −1/2−1/(3(p+1)) , an application of the Cauchy-Schwarz inequality yields Then there exist constants c ϕ,1 , c ϕ,2 > 0, only depending on u − , c 1 and p, such that ∞ k=1 (k p/2−1 X k − X ′ k p p ) 1/(p+1) < ∞,∞ k=1 (k p/2−1 X k − X ′ k p p ) 1/(p+1) 2 ≤ ∞ k=1 a 2 k ∞ k=1 a −2 k k (p−2)/(p+1) X k − X ′ k 2p/(p+1) p ∞ k=1 k 2 X k − X ′ k p < ∞ϕ H j (x) = E[exp(ixH j )|H j ].E J j=1 |ϕ H j (x)| e −c ϕ,1 x 2 J + e − √ J/4 log 8/7 for x 2 ≤ c ϕ,2 . Proof. Let (0)) + i(sin(sxH j )) − sin(0))]. Using a Taylor expansion and writing e ix = cos(x) + i sin(x), we obtain that I(s, x) = E H j [H 2 j ((cos(sxH j ) − cosE H j [e ixH j ] = 1 − E H j [H 2 j ]x 2 /2 + x 2 /2 1 0 (1 − s)I(s, x) ds. Using the Lipschitz property of cos(y) and sin(y), it follows that |I(s, x)| ≤ 2E H j [H 2 j |xh| + 2H 2 j 1(|H j | ≥ h)], h > 0. (4.53) For h > 0 we have from the Markov inequality E H j [H 2 j 1(|H j | ≥ h)] ≤ 2 ∞ h xP H j (|H j | ≥ x) dx + h 2 P H j (|H j | ≥ h) ≤ 2h −p+2 ∞ 0 x p−1 P H j (|H j | ≥ x) dx + h 2 P H j (|H j | ≥ h) ≤ 2 + p p h −p+2 E H j [|H j | p ] < h −p+2 E H j [|H j | p ]. We thus conclude from (4.53) that |I(s, x)| ≤ 2E H j [H 2 j ]|xh| + 4h −p+2 E H j [|H j | p ]. This gives us |E H j [e ixH j ] − 1 + E H j [H 2 j ]x 2 /2| (4.54) ≤ E H j [H 2 j ]h|x| 3 + 2h −p+2 x 2 E H j [|H j | p ]. Let I = {1, . . . , J}, and put σ H j = E[H 2 j |H j ] and ρ H j = E[|H j | p |H j ]. Consider ρ H 1,J ≥ ρ H 2,J ≥ · · · ≥ ρ H J,J , where ρ H j,J denotes the jth largest random variable for 1 ≤ j ≤ J . Let E j , j = 1, . . . , J denote i.i.d. unit exponential random variables, and denote with E j,J the jth largest. Further, denote with F ρ j (·) the c.d.f. of ρ H j , j = 1, . . . , J , and with F ρ (·) = min 1≤j≤J F ρ j (·). Using the transformation − log(1 − F ρ j (·)), we thus obtain {ρ H j ≤ x j : 1 ≤ j ≤ J} (4.55) d = {E j ≤ − log(1 − F ρ j (x j )) : 1 ≤ j ≤ J}, x j ∈ R, Next, observe that E J j=1 ϕ H j (x) = E J j=1 ϕ H j (x) (1(A) + 1(A c )) ≤ P (A c ) + E j∈I ϕ H j (x) 1(A) (4.64) ≤ P (A c ) + E j∈I A ϕ H j (x) 1(A) . Moreover, using (4.62) and (4.63) and since |I A | = J/4 on A, it follows that E j∈I A ϕ H j (x) 1(A) ≤ E j∈I A (1 − u(x)) 1(A) ≤ E j∈I A e −u(x) 1(A) ≤ e −u(x)J/8 ≤ e −c ϕ,1 Jx 2 . Hence we conclude from the above and (4.59) that Proof. Since E[X k |E 0 ] = E[X k −X * k |E 0 ], the Cauchy-Schwarz and Jensen inequalities imply ∞ k=0 |E[X 0 X k ]| ≤ X 0 2 ∞ k=0 E[X k |E 0 ] 2 ≤ X 0 2 ∞ k=0 X k − X * k 2 ≤ X 0 2 ∞ k=1 k 2 X k − X ′ k 2 < ∞. The decomposition σ 2 m = s 2 m /2 + O(m −1 ) now follows from (4.2). Claim σ 2 l = s 2 m /2 + O(1) as l → m readily follows from the previous computations. (i) σ 2 j|m − σ 2 j p/2 σ 2 j|m − σ 2 m p/2 + m −1 m −1 for 1 ≤ j ≤ N , 30 M. JIRAK (ii) σ 2 j = σ 2 m + O(m −1 ) for 1 ≤ j ≤ N , (iii) σ 2 |m − σ 2 m p/2 n −1 N 2/p . Proof. We first show (i). Without loss of generality, we may assume j = 1, since m ∼ m ′ . To lighten the notation, we use R 1 = R (1) 1 . We will first establish that σ 2 j|m − σ 2 m p/2 m −1 . We have that 2m(σ 2 1|m − σ 2 m ) = E Fm m k=1 (X ( * * ) k + (X k − X ( * * ) k ) − E Fm [X k ]) + R 1 2 − 2m σ 2 m . By squaring out the first expression, we obtain a sum of square terms and a sum of mixed terms. Let us first treat the mixed terms, which are 2 m k=1 m l=1 E Fm [X ( * * ) k (X l − X ( * * ) l ) + X ( * * ) k E Fm [X l ] + E Fm [X k ](X l − X ( * * ) l )] + 2 m k=1 E Fm [R 1 X ( * * ) k + R 1 (X k − X ( * * ) k ) + R 1 E Fm [X k ]] = I m + II m + III m + IV m + V m + VI m . We will handle all these terms separately. Case I m : We have I m /2 = m l=1 m k=l (· · ·) + m l=1 l−1 k=1 (· · ·) = m l=1 m k=l E Fm [(X l − X ( * * ) l ( * * ) l ) + E Fm [X k ]E Fm [X l ]] + E Fm [R 2 1 ] = 2m σ 2 m + VII m + VIII m + IX m . However, using the results from the previous computations and Lemma 4.13, one readily deduces that VII m p/2 < ∞, VIII m p/2 < ∞, IX m p/2 < ∞. Piecing everything together, we have established that σ 2 j|m − σ 2 m p/2 m −1 . However, from the above arguments one readily deduces that σ 2 j = σ 2 m + O(m −1 ), and hence (i) and (ii) follow. We now treat (iii). Since {Y (1) j } 1≤j≤N is an independent sequence under P Fm , we have Note that since cos(y), sin(y) are bounded in absolute value and are Lipschitz continuous, it follows that up to some finite constant c(α) > 0 we have sin(y), cos(y) ∈ F s for any s > 0. As already mentioned in step 2 of the proof of part A, we will make use of some special ideal-metrics ζ s (Zolotarev metric). For two probability measures P, Q, the metric ζ s is defined as σ 2 |m = N −1 N j=1 σ 2 j|m .n −1 N 2/p .ζ s (P, Q) = sup f (x)(P − Q)(dx) : f ∈ F s . The metric ζ s (P, Q) has the nice property of homogeneity. For random variables X, Y , induced probability measures P cX , P cY and constant c > 0, this means that ζ s (P cX , P cY ) = |c| s ζ s (P X , P Y ). We require some further notation. η 2 j|m = 1 √ 2m E Fm [(Y (1) j ) 2 ]/ σ 2 m = σ 2 j|m / σ 2 m for 1 ≤ j ≤ N − 1, and η 2 m ′ |m = σ 2 m ′ |m / σ 2 m ′ for j = N . As first step toward (4.68), we have the following result. Proof. To lighten the notation, we use Y j = (2m) −1/2 Y We have Y 2 j − S 2 j η 2 j|m = Y 2 j − S 2 j + S 2 j σ −2 m (σ 2 j|m − σ 2 m ) , where we recall that S j and σ j|m are independent. Using the Jensen, triangle and Hölder inequalities and f /2 ∈ F p , it follows that E Fm [(Y 2 j − (S j η j|m ) 2 )(f ′′ (txY j ) − f ′′ (0))] 1 ≤ 2 (Y 2 j − (S j η j|m ) 2 )|txY j | p−2 1 ≤ 2 Y 2 j − (S j η j|m ) 2 p/2 |txY j | p−2 p/(p−2) 2 |txY j | p−2 p/(p−2) × ( Y j − S j p Y j + S j p + S 2 j p/2 σ 2 j|m − σ 2 m p/2 ), where we used that σ m > 0 for large enough m. By Lemmas 4.13 and 4.7, this is of magnitude O(m −1/2 |tx| p−2 ). Hence by adding and subtracting f ′′ (txY j ) and using similar arguments as before, we obtain from the above where we use that S j and η j|m = σ j|m / σ m are independent. This gives the desired result. x 2 I m (x) 1 m −1/2 |x| p + x 2 1 0 (1 − t) E Fm [S 2 j η 2 j|m (f ′′ (txY j ) − f ′′ (txS j ))] 1 dt As next step toward (4.68), we have the following. Proof. To increase the readability, we use the abbreviations σ = σ m and S j = S ( * * ) j|m in the following. The main objective is to transfer the problem to the setup in [11] and apply the corresponding results. To this end, we first perform some necessary preparatory computations. We have that k>l E[X k |E 0 ] p ≤ k>l X k − X * k p ≤ k>l k X k − X ′ k p → 0 (4.72) as l → ∞, hence it follows that m k=0 E[X k |E 0 ] converges in · p . (4.73) Next, note that 2m σ 2 = E E 0 m k=1 X * ε k , ε k−1 , . . . , ε ′ k−l , ε k−l−1 , . . .). ε k , ε k−1 , . . . , ε ′ 0 , ε −1 , . . .) and X Theorem 2 . 2 . 22Grant Assumption 2.1 for some p ∈ (2, 3], and let s Example 3 . 1 ( 31Functions of linear process). Let S = R, and suppose that the sequence {α Example 3. 3 3(m-dependent processes). Consider the zero mean mdependent process Y k = f (ζ k , . . . , ζ k−m+1 ), where m ∈ N and f is a measurable function and {ζ k } k∈Z is i.i.d. and takes values in S. m may depend on n such that n/m → ∞, but we demand in addition and s 2 > 0, then Assumption 2.1 holds.10 M. JIRAK Theorem 4 . 2 . 42Grant Assumption 4.1, and let p ∈ (2, 3]. Assume in ad- ( 5 ) 5We write [as in (4.4)] def = if we make definitions on the fly. . 1 . 1Part A. The proof of part A is divided into four major steps. Some more technical arguments are deferred to Sections 4.5.2 and 4.5.1. Proof. For L > 0, put B L = {L −1 L j=1 σ 2 j|m ≥ s 2 m /4}, and denote with B c L its complement. Since S large enough m−l (say m−l ≥ K 0 > 0) by Lemma 4.6, we may set 0 < u − = s 2 m /8 ≤ σ 2 m−l /2. For showing (iii), it suffices to treat the case j = 1. Note that (for k ≤ m) / 8 8m − l, and hence (iii) follows. We can thus apply Lemma 4.5 with u − = s 2 m /8 and J = |J | ≥ N Lemma 4. 3 . 3For every δ > 0, every m, n ≥ 1 and every x ∈ R, the following estimate holds: we deal with A 3 (m, n, x, δ). Properties of the Gaussian distribution function givesup x∈R A 3 (m, n, x, δ) δ + |s 2 n − s 2 nm |.However, by the Cauchy-Schwarz inequality and (4.32), it follows that |s 2 n − s 2 nm | ≤ n −1 S (>m) thus conclude that sup x∈R A 3 (m, n, x, δ) δ + n −1 . Proof of Theorem 2.3. Recall that ∆ n (x) = P S n ≤ x ns 2 n − Φ(x) and ∆ n = sup x∈R ∆ n (x). 4. 4 . 4Proof of Theorem 2.4. For the proof, we require the following result; cf. [41], Lemma 5.4. Lemma 4 . 4 . 44Let Y be a real-valued random variable. Put − 1 P 12.2 with (4.47) and Lemma 4.4, we see that it suffices to consider λ p with Y = S n 1(|S n | ≤ n). Using again Theorem 2.2 together with (4.47), standard tail bounds for the Gaussian distribution and elementary computations give λ p n −(p∧3)/2+1 (log n) |S n | ≥ x ns 2 n dx. − 1 P 1(4.49) then for large enough c 0 > 0 and x ≥ c 0 √ log n we get P |S n | ≥ |S n | ≥ x ns 2 n dx n −p/2+1 (log n) p/2 log n.(4.51) above conditions, we have the following result. Lemma 4 . 5 . 45Let p > 2, and assume that:(i) E H j [H j ] = 0 uniformly for j = 1, . . . , J , (ii) there exists a u − > 0 such that P (E H j [H 2j ] ≤ u − ) < 1/7 uniformly for j = 1, . . . , J , (iii) E[|H j | p ] ≤ c 1 < ∞ uniformly for j = 1, . . . , J . Lemma 4. 6 . 6Grant Assumption 4.1. Then ∞ k=1 k|E[X 0 X k ]| < ∞ and σ 2 m = s 2 m /2 + O(m −1 ). Moreover, we have σ 2 l = s 2 m /2 + O(1) as l → m. Lemma 4. 7 . 7Grant Assumption 4.1. Then: = {1, 3, 5, . . .} and J = {2, 4, 6, . . .} such that I ∪ J = {1, 2, . . . , N }. Then N Corollary 4. 8 .. 2 . 82Grant Assumption 4.1. Let B = {σ 2 |m ≥ s 2 m /4}. ThenP (B c ) n −p/2 N.Proof. By Markov's inequality, Lemma 4.6 and Lemma 4.7, it follows that for large enough mP (B c ) ≤ P (|σ 2 |m − σ 2 m | ≥ s 2 m /4 − O(m −1 )) s Ideal metrics and applications. The aim of this section is to give a proof for the inequalityϕ i (t) − e −σ 2 i|m t 2 /2 1 |t| p m −p/2+1 , i ∈ {1, N } (4.68)in Corollary 4.11. We will achieve this by employing ideal metrics. Let s > 0. Then we can represent s as s = m + α, where [s] = m denotes the integer part, and 0 < α ≤ 1. Let F s be the class of all real-valued functions f , such that the mth derivative exists and satisfies |f (m) (x) − f (m) (y)| ≤ |x − y| α . (4.69) is independent of F m , and hence σ 2m = E Fm [(S ( * * ) j|m ) 2 ]. Let {Z j }1≤j≤Nbe a sequence of zero mean, standard i.i.d. Gaussian random variables. In addition, let Lemma 4 . 9 . 49Grant Assumption 4.1. Then for f (x) ∈ {cos(x), sin(x)}, η j|m )] 1 m −p/2+1 |x| p if p ∈ (2, 3],where j = 1, . . . , N . in the following. Using Taylor expansion, we havef (y) = f (0) + yf ′ (0) + y 2 f ′′ (t)(f ′′ (ty) − f ′′ (0)) dt. (4.70) Note E Fm [Y j ] = 0 and E Fm [S j η j|m ] = η j|m E[S j ] = 0. Moreover, since σ 2 j|m = η 2 j|m E Fm [S 2 j ] by construction, we obtain from (4.70) that E Fm [f (xY j ) − f (xS j η j|m )] t)E Fm [Y 2 j (f ′′ (txY j ) − f ′′ (0)) − (S j η j|m ) 2 (f ′′ (txS j η j|m ) − f ′′ (0))]/2 dt def = x 2 I m (x). |x| p + m −(p−2)/2 |x| p m −p/2+1 |x| p , Lemma 4 . 10 . 410Grant Assumption 4.1. Then for f (x) ∈ {cos(x), sin(x)}, it holds that E Fm [f (xZ j σ j|m ) − f (xS ( * * ) j|m η j|m )] 1 |x| p m −p/2+1 if p ∈ (2, 3],where j = 1, . . . , N . and s 2 > 0, then Assumption 2.1 holds. Example 3.2 [Sums of the form f (t2 k )]. Consider the measure preserving transformation T x = 2x mod 1 on the probability space ([0, 1], B, λ), with Borel σ-algebra B and Lebesgue measure λ. Let U 0 ∼ Uniform[0, 1]. Proof of main lemmas. 4.5.1. Bounding conditional characteristic functions and variances. Suppose we have a sequence of random variables {H j } 1≤j≤J and a sequence of filtrations {H j } 1≤j≤J , such that both {E H j [H 2 j ]} 1≤j≤J and {E H j [|H j | p ]} 1≤j≤J are independent sequences. Note that this does not necessarily mean that {H j } 1≤j≤J is independent, and indeed this is not the case when we apply Lemma 4.5 in step 4 of the proof of part A. Introduce the conditional characteristic functionby Assumption 2.1. Hence (4.49) holds, and thus (4.50) and (4.51). To com- plete the proof, it remains to treat the case |x| > c 0 √ log n. But in this case, we may directly appeal to (4.50) which gives the result. 4.5. Note that {σ 2 j|m } j∈I is a sequence of independent random variables, and the same is true for {σ 2 j|m } j∈J . Then by Lemma 4.12, it follows thatBERRY-ESSEEN THEOREMS UNDER WEAK DEPENDENCEj=1 σ 2 j|m − σ 2 j p/2 ≤ j∈I σ 2 j|m − σ 2 j p/2 + j∈J σ 2 j|m − σ 2 j p/2 . N j=1 σ 2 j|m − σ 2 j p/2 N 2/p σ 2 j|m − σ 2 j p/2 for p ∈ (2, 3], 33 which by (i) is of the magnitude O(N 2/p m −1 ). Hence we conclude from (4.67) that σ 2 |m − σ 2 m p/2 (l, * ) k = g(θ (l, * ) k ), and in particular, we have X * k = X (k, * ) k . Similarly, let {ε ′′ k } k∈Z be independent copies of {ε k } k∈Z and {ε ′ k } k∈Z . For l ≤ k, we then introduce the quantities X (l,′′) k , X (l, * * ) k , X ′′ k , X ( * * ) k in analogy to X (l,′) k , X (l, * ) k Acknowledgments. I would like to thank the Associate Editor and the anonymous reviewer for a careful reading of the manuscript and the comments and remarks that helped to improve and clarify the presentation. I also thank Istvan Berkes and Wei Biao Wu for stimulating discussions.27which is the well-known Renyi representation; cf.[10,44]. In particular, by the construction of F ρ (·) it follows thatfor 0 ≤ u + < ∞. Let u + H = − log(1 − F ρ (u + )) and u + H (J) = J/2(u + H − log 2). We wish to find a u + such that u + H > log 2. This is implied by F ρ (u + ) > 1/2. We will now construct such an u + . Sinceit follows that c 1 /c 2 ≥ 1 − P (ρ H j < c 2 ). Hence choosing u + = c 2 = 4c 1 , we obtain F ρ j (u + ) ≥ 3/4 and hence F ρ (u + ) ≥ 3/4, which leads to u + H ≥ J/2 log 2. Thus by known properties of exponential order statistics (cf.[10,17]), we havefor sufficiently large J . We thus conclude thatLet us denote this set with A + = {ρ H J/2,J ≤ u + }, and put I +Note that the index set I + A has at least cardinality J/2 given event A + . For the sake of simplicity, let us assume that |I + A | = J/2, which, as is clear from the arguments below, has no impact on our results. Let us introduce Now, similar to as before, let F σ j (·) be the c.d.f. of σ H j , j = 1, . . . , J , and put F σ (·) = max 1≤j≤J F σ j (·), u − H = − log(1 − F σ (u − )) and u − H (J) = J/4 × (log 4/3 − u − H ) for some 0 ≤ u − ≤ u + . We search for a u − > 0 such that u − H < 28 M. JIRAK log 7/6, which is true if max 1≤j≤J F σ j (u − ) < 1/7. However, this is precisely what we demanded in the assumptions. Then proceeding as before, we haveWe thus conclude from (4.57) and the construction of F σ (·) thatCombining (4.56) and (4.58) we obtainAlso note that by the (conditional) Lyapunov inequality, we haveNote that |I A | ≥ J/4 on the event A, and, by the above, we getUsing (4.54), this implies that for every j ∈ I A , we haveHence, if (u + ) p/2 x 2 /2 < 1 and h = x −1/(p−1) , we conclude from the above and the triangle inequality that for j ∈ I A ,Since 0 < u − , u + < ∞ and δ(p) > 0, there exist absolute constants 0 < c ϕ,1 , c ϕ,2 , chosen sufficiently small, such thatthe Cauchy-Schwarz (with respect to E Fm ) and Jensen inequalities thus yieldit follows that II m = 0. Case III m : It follows via the Jensen and triangle inequalities thatThe Cauchy-Schwarz (with respect to E Fm ) and Jensen inequalities then giveCase IV m : Note that XThe Cauchy-Schwarz (with respect to E Fm ) and Jensen inequalitities then yieldCase V m : The Cauchy-Schwarz (with respect to E Fm ) and Jensen inequalities yieldCase VI m : Proceeding as above and using (4.65), we get VI m p/2 < ∞. It thus remains to deal with the squared terms, which areUsing Lemmas 4.9 and 4.10, the triangle inequality gives the following corollary, which proves (4.68).4.6. Some auxiliary lemmas. We will frequently use the following lemma, which is essentially a restatement of Theorem 1 in[48], adapted to our setting.For the sake of completeness, we sate this result in the general, nontimehomogenous but stationary Bernoulli-shift context.BERRY-ESSEEN THEOREMS UNDER WEAK DEPENDENCE37Recall thatLemma 4.13. Grant Assumption 4.1. Then:Proof. Without loss of generally, we assume that j = 1 since m ∼ m ′ . 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(1980). Convergence rate in the central limit theorem for weakly dependent random variables. Teor. Verojatnost. i Primenen. 25 800-818. MR0595140 R S Tsay, Analysis of Financial Time Series. Hoboken, NJ. MR2162112Wiley2nd ed.Tsay, R. S. (2005). Analysis of Financial Time Series, 2nd ed. Wiley, Hoboken, NJ. MR2162112 Nonlinear system theory: Another look at dependence. W B Wu, Proc. Natl. Acad. Sci. USA. Natl. Acad. Sci. USA1022172215electronicWu, W. B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150-14154 (electronic). MR2172215 Strong invariance principles for dependent random variables. W B Wu, 2294-2320. MR2353389Ann. Probab. 35Wu, W. B. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294-2320. MR2353389 Limit theorems for iterated random functions. W B Wu, X Shao, J. Appl. Probab. 412052582Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425-436. 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A notion of unique ergodicity relative to the fixed-point subalgebra is defined for automorphisms of unital C * -algebras. It is proved that the free shift on any reduced amalgamated free product C * -algebra is uniquely ergodic relative to its fixed-point subalgebra, as are autormorphisms of reduced group C * -algebras arising from certain automorphisms of groups. A generalization of Haagerup's inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C * -algebras, is proved.

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UNIQUE ERGODICITY OF FREE SHIFTS AND SOME OTHER AUTOMORPHISMS OF C * -ALGEBRAS 16 Aug 2006 Beatriz Abadie Ken Dykema UNIQUE ERGODICITY OF FREE SHIFTS AND SOME OTHER AUTOMORPHISMS OF C * -ALGEBRAS 16 Aug 2006 A notion of unique ergodicity relative to the fixed-point subalgebra is defined for automorphisms of unital C * -algebras. It is proved that the free shift on any reduced amalgamated free product C * -algebra is uniquely ergodic relative to its fixed-point subalgebra, as are autormorphisms of reduced group C * -algebras arising from certain automorphisms of groups. A generalization of Haagerup's inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C * -algebras, is proved. Introduction Let Ω be a compact Hausdorff space and T a homeomorphism of Ω onto itself. In the terminology of [10], (see also [8] and [2], where slightly different terminology is used), T is called uniquely ergodic if there is a unique Tinvariant Borel probability measure µ on Ω, (with respect to which T is then necessarily ergodic). Oxtoby shows [10, 5.1] that if T is uniquely ergodic, then the ergodic averages 1 n n−1 k=0 f • T k (1) converge uniformly to the constant f dµ, as n → ∞. The homeomorphisms of Ω are in 1-1 correspondence with the automorphisms of the C * -algebra C(Ω) of all continuous, complex-valued functions on Ω and the Borel probability measures on Ω are by Riesz's Theorem in 1-1 correspondence with the states of C(Ω). There is a natural noncommutative version of unique ergodicity. Let A be a unital C * -algebra and let α be an automorphism of A. An α-invariant state of A always exists, and can be found, for example, by taking a weak limit of averages 1 n n−1 k=0 φ • α k(2) of any state φ. We say α is uniquely ergodic if there is a unique α-invariant state of A. It is not difficult to show (based on Oxtoby's argument [10, 5.1]) that α is uniquely ergodic if and only if for every a ∈ A the ergodic averages 1 n n−1 k=0 α k (a)(3) converge in norm to a scalar multiple of the identity as n → ∞ and, in this case, the invariant state evaluated at a is equal to this limit. (A more general result is proved in Theorem 3.2 below.) Our interest in these topics grew out of a question asked by David Kerr [7]: Is the free shift on C * r (F ∞ ) uniquely ergodic? A positive answer to Kerr's question follows from Haagerup's inequality [3]. This argument is described in section 2 below. In considering more general free shift automorphisms, we were motivated to consider a broader notion of unique ergodicity. If A is a unital C * -algebra and α an automorphism of A, consider the fixed-point subalgebra A α = {a ∈ A | α(a) = a}.(4) We say that α is uniquely ergodic relative to its fixed-point subalgebra if every state of A α has a unique α-invariant state extension to A. In the case when A α consists only of scalar multiples of the identity element, this reduces to the usual notion of unique ergodicity. In section 3, we give some alternative characterizations of unique ergodicity relative to the fixed-point subalgebra. It turns out to be equivalent to norm convergence of the ergodic averages (3) as n → ∞ for all a ∈ A. Thus, unique ergodicity relative to the fixed-point subalgebra implies (by taking the limit of the ergodic averages) existence of a unique α-invariant conditional expectation from A onto A α . However (see Question 3.4) we do not know whether the converse direction holds. After seeing that the free shift on C * r (F ∞ ) is uniquely ergodic, it is natural to ask whether free shifts on other reduced free product C * -algebras and even on reduced amalgamated free product C * -algebras are uniquely ergodic relative to their fixed point subalgebras. We give an affirmative answer in Theorem 6.1. A technical result that we use is an extension of Haagerup's inequality to the setting of reduced amalgamated free product C * -algebras. Haagerup's inequality says that the operator norm of an element of C * r (F ∞ ) that is supported on words of length n is no greater than n + 1 times the ℓ 2 -norm. It is a fundamental inequality, and has been generalized in several different directions; see, for example, [4], [5], [1], [6], [12]. One such generalization is [1, 3.3], in the context of reduced free product C * -algebras with amalgamation over the scalars, which applies to all finite linear combinations of words of fixed block length n. A strong generalization, due to Ricard and Xu [12], is in the context of reduced amalgamated free product C * -algebras; they prove bounds on operator norms that apply to all matrices over all finite linear combinations of words of fixed block length n. In Proposition 5.1, we prove a generalization of Haagerup's inequality in the setting of reduced amalgamated free product C * -algebras. Our bound on the operator norm applies only to certain linear combinations of words of block length n, but our bound has a rather nice form. In fact, as Eric Ricard kindly showed us, our Proposition 5.1 follows from the results of Ricard and Xu. However, we nonetheless present our direct proof here, as it is slightly simpler (for being a more specific result). To summarize the contents: section 2 contains the proof of unique ergodicity of the free shift on C * r (F ∞ ); section 3 gives alternative characterizations of unique ergodicity relative to the fixed-point subalgebra, and contains a generalization of the argument from the previous section to groups with property (RD) of Jolissaint; section 4 recalls the construction of the reduced amalgamated free product of C * -algebras; section 5 contains a generalization of Haagerup's inequality to reduced amalgamated free product C * -algebras; section 6 proves that free shifts are uniquely ergodic relative to their fixed-point subalgebras. Acknowledgement. This work was carried out while the first author was visiting the Mathematics Department of Texas A&M University. She would like to thank the members of the department for their warm hospitality. The authors thank Thierry Fack for a helpful comment that lead to Proposition 3.5, and David Kerr and Eric Ricard for helpful conversations. 2. The free shift on C * r (F ∞ ) is uniquely ergodic Here, C * r (F ∞ ) is the reduced group C * -algebra of the free group on infinitely many generators {g i } i∈Z and the free shift is the automorphism α of C * r (F ∞ ) arising from the automorphism of the group that sends g i to g i+1 . The C * -algebra C * r (F ∞ ) is densely spanned by the left translation operators λ h acting on ℓ 2 (F ∞ ), (h ∈ F ∞ ). If h = e is the trivial group element, then λ h is the identity element 1 and α k (λ h ) ≤ (p + 1) 1 n n−1 k=0 α k (λ h ) 2 = p + 1 √ n ,(6) where · 2 refers to the norm of the corresponding element in ℓ 2 (F ∞ ). We conclude that the averages appearing on the left-hand-side of (6) tend to zero as n → ∞, and this proves that the free shift is uniquely ergodic and that its unique invariant state is the canonical tracial state τ defined by τ (λ h ) = 1, h = e 0, h = e.(7) 3. Unique ergodicity relative to the fixed-point subalgebra In this section, we prove certain conditions equivalent to unique ergodicity relative to the fixed-point subalgebra. Observation 3.1. Let A be a C * -algebra and let φ : A → C be a selfadjoint linear functional, namely a bounded linear functional such that φ(a * ) is the complex conjugate of φ(a). Recall (see [11, 3.2.5]) that the Jordan decomposition of φ is the unique pair φ + and φ − of positive linear functionals such that φ = φ + − φ − and φ = φ + + φ − . Suppose α ∈ Aut(A) and φ is α-invariant. Then φ = φ • α = φ + • α − φ − • α and φ = φ + + φ − = φ + • α + φ − • α . By uniqueness, it follows that φ + and φ − are both α-invariant. Recall that a conditional expectation from a C * -algebra A onto a C *subalgebra B is a projection E of norm 1 from A onto B. A classical result of Tomiyama [13] is that such a projection E is automatically completely positive and satisfies the conditional expectation property. Theorem 3.2. Let α be an automorphism of a unital C * -algebra A and let A α be its fixed-point subalgebra as in (4). Then the following five statements are equivalent: (i) Every bounded linear functional on A α has a unique bounded, αinvariant linear extension to A. (ii) Every state of A α has a unique α-invariant state extension to A. (iii) The subspace A α + {a − α(a) | a ∈ A} is dense in A. (iv) For all a ∈ A, the sequence of ergodic averages 1 n n−1 k=0 α k (a)(8) converges in norm as n → ∞. (v) We have A α + {a − α(a) | a ∈ A} = A,(9) where the closure is with respect to the norm topology. These five statements imply the following statement: (vi) There exists a unique α-invariant conditional expectation E from A onto A α . Furthermore, if (i)-(v) hold, then the conditional expectation E in (vi) is given by the formula E(a) = lim n→∞ 1 n n−1 k=0 α k (a).(10) Definition 3.3. We say α is uniquely ergodic relative to its fixed-point subalgebra if the equivalent statements (i)-(v) hold. Proof of Theorem 3.2. (i) =⇒ (ii) is clear. (ii) =⇒ (iii): Suppose, to obtain a contradiction, that (ii) holds but x ∈ A and x / ∈ A α + {a − α(a) | a ∈ A}. (11) By the Hahn-Banach Theorem, there is a bounded linear functional φ : A → C such that φ(x) = 0, φ(A α ) = {0} and φ • α = φ. Taking the real and imaginary parts, we may without loss of generality assume that φ is self-adjoint. Let φ = φ + − φ − be the Jordan decomposition of φ. Then φ + and φ − are α-invariant, by Observation 3.1. Moreover, φ + and φ − agree on A α . Either both restrict to zero on A α , in which case φ ± (1) = 0 and φ ± = 0, or φ ± are nonzero multiples of states on A and by statement (ii), φ + and φ − must agree on all of A. This contradicts φ(x) = 0. (iii) =⇒ (iv): Let a ∈ A and ǫ > 0. Let c ∈ A α and b ∈ A be such that a − (c + b − α(b)) < ǫ.(12) If n ≥ m, then 1 n n−1 k=0 α k (a) − 1 m m−1 k=0 α k (a) (13) < 2ǫ + 1 n n−1 k=0 α k (b − α(b)) − 1 m m−1 k=0 α k (b − α(b))(14)= 2ǫ + 1 n (b − α n (b)) + 1 m (b − α m (b)) (15) ≤ 2ǫ + 4 b m .(16) Taking m large enough, the upper bound (16) can be made < 3ǫ. This shows that the sequence of ergodic averages (8) is Cauchy. (iv) =⇒ (vi)+ (10): Let E be defined by the formula (10). Clearly, E restricts to the identity map on A α . One easily shows E = 1 and E •α = α•E = E. So E is an α-invariant conditional expectation from A onto A α . If E ′ : A → A α is any α-invariant conditional expectation onto A α , then E ′ (a) = 1 n n−1 k=0 E ′ (α k (a)) = E ′ 1 n n−1 k=0 α k (a) .(17) Taking the limit as n → ∞ gives E ′ (a) = E ′ (E(a)) = E(a).(18) (iv)+(vi)+(10) =⇒ (i): Let τ : A α → C be a bounded linear functional. Then τ • E is an α-invariant extension of τ to A. To show uniqueness, suppose φ : A → C is any bounded, α-invariant, linear extension of τ . Then φ(a) = 1 n n−1 k=0 φ(α k (a)) = φ 1 n n−1 k=0 α k (a) .(19) Taking the limit as n → ∞ gives φ(a) = φ(E(a)) = τ (E(a)),(20)so φ = τ • E. We have now proved the equivalence of (i)-(iv), and that these imply (vi) and (10). (i)+(vi) =⇒ (v): Since A = A α + ker E, it will suffice to show ker E ⊆ {a − α(a) | a ∈ A}.(21) Suppose, to obtain a contradiction, x ∈ ker E but x / ∈ {a − α(a) | a ∈ A}. By the Hahn-Banach Theorem, there is a bounded linear functional φ : A → C such that φ(x) = 0 and φ • α = φ. By (i), we must have φ = φ • E, so φ(x) = 0, a contradiction. (v) =⇒ (iii) is clear. Question 3.4. In Theorem 3.2, is (vi) equivalent to (i)-(v)? It was kindly pointed out to us by Thierry Fack that the argument used in section 2 applies more generally. Indeed, as the following proposition shows, the argument carries over to groups with property (RD), as defined by Jolissaint in [5]. Note that by [4] this includes the case of Gromov's hyperbolic groups. Proposition 3.5. Let G be a group with property (RD) for a length function L and let β be an L-preserving automorphism of G such that all orbits of β are either singletons or infinite. Let H = {h ∈ G | β(h) = h}. Then the automorphism α induced by β on C * r (G) is uniquely ergodic relative to its fixed-point subalgebra, which is the canonical copy of C * r (H) in C * r (G). Proof. If g ∈ G is such that β(g) = g, then by [5, Remark 1.2.2] there exist positive numbers C and s such that 1 n n−1 0 α k (λ g ) ≤ C 1 n n−1 0 α k (λ g ) 2,s,L = C √ n (1 + L(g)) s ,(22) and this upper bound approaches zero as n goes to ∞. If β(g) = g, then 1 n n−1 0 α k (λ g ) = λ g(23) for all n. Now one easily sees that condition (iv) of Theorem 3.2 holds and that C * r (H) is the fixed-point subalgebra for α. The construction of reduced amalgamated free product C * -algebras In this section we will review in some detail and thereby set some notation for the reduced amalgamated free product of C * -algebras, which was invented by Voiculescu [14]. We first set some notation concerning a right Hilbert C * -module E over a C * -algebra B (see [9] for a general reference on Hilbert C * -modules). If x ∈ E, then we let |x| = x, x 1/2 ∈ B (24) and the norm of x is defined by x E = |x| B .(25) Let B be a unital C * -algebra, let I be a set with at least two elements and for every i ∈ I let A i be a unital C * -algebra containing a copy of B as a unital C * -subalgebra and having a conditional expectation φ i : A i → B such that for each a i ∈ A i there exists x ∈ A i for which φ i (x * a * i a i x) = 0. We denote by E i = L 2 (A i , φ i ) the right Hilbert C * -module over B obtained by separation and completion of A i with respect to the inner product x, y = φ i (x * y). For a i ∈ A i , we denote byâ i the image of a i in E i under the canonical map. There is a faithful * -representation π i of A i on E i by adjointable operators given by π i (x)(ŷ) = (xy)ˆ,(26) for x, y ∈ A i . We will often omit the reference to π i and write simply av to denote π i (a)(v), for a ∈ A i and v ∈ E i . This inclusion B ⊆ A i yields a copy of B as a complemented Hilbert C * -submodule of E i , and we write E i = B ⊕ E • i and let H i : E i → E • i be the orthogonal projection onto E • i . So, for example, we have H i (â) = (a − φ i (a))ˆ, (a ∈ A i ). Since π i (b) sends E • i into E • i whenever b ∈ B, we regard E • i as equipped with a left B-action via π i . We consider the right Hilbert B-module E = B ⊕ m∈N i 1 ,...,im∈I i j =i j+1 E • i 1 ⊗ B E • i 2 ⊗ B · · · ⊗ B E • im ,(28) where the tensor products are with respect to the right Hilbert B-module structures and the left actions of B described above, and where the summand B in (28) denotes the C * -algebra B with its usual Hilbert C * -module structure over itself. There is a faithful * -representation of A i by adjointable operators on E, which is denoted by a → λ i a and which can be defined by λ i a (b) = φ i (ab) + H i ((ab)ˆ) ∈ B ⊕ E • i , (b ∈ B)(29) and, considering a simple tensor x 1 ⊗ · · · ⊗ x m(30) where m ≥ 1, x j ∈ E • i j , i 1 , . . . , i m ∈ I and i j = i j+1 for all i = 1, . . . , m − 1, by λ i a (x 1 ⊗ · · · ⊗ x m ) =            H i (â) ⊗ x 1 ⊗ · · · ⊗ x m + φ i (a)x 1 ⊗ x 2 ⊗ · · · ⊗ x m , i = i 1 H i (ax 1 ) ⊗ x 2 ⊗ · · · ⊗ x m + (a * )ˆ, x 1 x 2 ⊗ · · · ⊗ x m , i = i 1 .(31) Note that for b ∈ B, λ i b is the same for all i. We will write λ a or simply a instead of λ i a , when no confusion will result. The reduced amalgamated free product C * -algebra (A, φ) = ( * B ) i∈I (A i , φ i )(32) consists of the C * -algebra A generated in L(E) by the set {λ i a : a ∈ A i , i ∈ I} and the conditional expectation φ : A → B defined by φ(a) = a1 B , 1 B , (a ∈ A).(33) Thus, the C * -algebra A is the closed span of B together with the set of all words of the form w = a 1 . . . a n (34) where a i ∈ A • k(i) , k(1), . . . , k(n) ∈ I and k(i) = k(i+ 1) for all i ∈ {1, . . . , n − 1}. 5. Some norm estimates in reduced amalgamated free product C * -algebras The main result of this section is the following norm estimate, which applies to certain linear combinations of words of length n in reduced amalgamated free product C * -algebras. It is a version of the Haagerup inequality. Proposition 5.1. Suppose n ≥ 1 and consider f = k∈K a k,1 a k,2 · · · a k,n ∈ A, where K is a finite subset of I n such that for all k = (k(1), . . . , k(n)) ∈ K we have k(i) = k(i + 1) for all i ∈ {1, . . . , n − 1} and where a k,i ∈ A • k(i) for all k ∈ K and i ∈ {1, . . . , n}. Suppose, furthermore, that if k, k ′ ∈ K and k = k ′ , then k(1) = k ′ (1) and k(n) = k ′ (n). (36) Then f ≤ (2n + 1) k∈K n i=1 a k,i 2 1/2 .(37) Before we get to the proof, we consider some preliminary constructions and results. Let us define some elementary adjointable operators on E, in terms of which we will later describe the action of a word w as in (34) on a tensor x 1 ⊗ · · · ⊗ x m in (30). Notation 5.2. Let P 0 denote the orthogonal projection of E onto the summand B ⊆ E and for m ≥ 1 let P m denote the orthogonal projection of E onto i 1 ,...,im∈I i j =i j+1 E • i 1 ⊗ B E • i 2 ⊗ B · · · ⊗ B E • im . (38) Notation 5.3. For k ∈ I, let Q k denote the orthogonal projection of E onto m≥1 i 1 ,...,im∈I i j =i j+1 i 1 =k E • i 1 ⊗ B E • i 2 ⊗ B · · · ⊗ B E • im .(39) Note that Q k and P m commute. Notation 5.4. Given k ∈ I and y ∈ E • k , let ψ(y) = ψ k (y) ∈ L(E) be given by ψ(y)b = (yb)ˆ∈ E • k , (b ∈ B)(40) and, for x 1 ⊗ · · · ⊗ x m as in (30), ψ(y)(x 1 ⊗ · · · ⊗ x m ) = 0, i 1 = k y ⊗ x 1 ⊗ · · · ⊗ x m , i 1 = k.(41) Therefore, we have ψ(y) = Q k ψ(y)(1 − Q k ),(42)ψ(y) * b = 0, (b ∈ B) (43) ψ(y) * (x 1 ⊗ · · · ⊗ x m ) =      0, i 1 = k y, x 1 , i 1 = k, m = 1 y, x 1 x 2 ⊗ x 3 ⊗ · · · ⊗ x m , i 1 = k, m > 1 (44) ψ(y) * ψ(y) = |y| 2 (1 − Q k ) (45) ψ(y) = y .(46) Notation 5.5. For k ∈ I and a ∈ A k , we let ρ(a) = ρ k (a) ∈ L(E) be defined by ρ(a)b = 0, (b ∈ B)(47) and, for x 1 ⊗ · · · ⊗ x m as in (30), ρ(a)(x 1 ⊗ · · · ⊗ x m ) = (H k (ax 1 )) ⊗ x 2 ⊗ · · · ⊗ x m , i 1 = k 0, i 1 = k.(48) (Recall that H k : E k → E • k is the orthogonal projection.) Therefore, we have ρ(a) = Q k ρ(a)Q k (49) ρ(a) ≤ a .(50) To ease notation, for a ∈ A k we let a † = (a * )ˆ∈ E k .(51) The following lemma describes how a word w = a 1 · · · a n as in (34) can act on a tensor x 1 ⊗ · · · ⊗ x m as in (30). What can happen is: w can first devour some initial string x 1 ⊗ · · · ⊗ x q of the tensor. Then it can either push some more stuff onto the tensor from the left, or it can instead act on the next letter x q+1 and then push some more stuff onto the tensor from the left. This is all that can happen, because neighboring letters in w and neighboring x j in x 1 ⊗ · · · ⊗ x m are constrained to come from different A • k , respectively different E • i . It's not too difficult to see this by considering some examples. We give a more precise statement and a rigorous proof below. Lemma 5.6. Let n ≥ 1 and let k = (k(1), . . . , k(n)) ∈ I n be such that k(i) = k(i + 1) for all i ∈ {1, . . . , n − 1}. Let w = a 1 · · · a n , where a i ∈ A • k(i) for all i ∈ {1, . . . , n}. Let m, r ≥ 0 be integers. (i) If r > m + n or r < |m − n|, then P r wP m = 0. (ii) If r = m + n − 2s with s ∈ {0, 1, . . . , min(m, n)}, then P r wP m = ψ(â 1 )ψ(â 2 ) · · · ψ(â n−s ) · ψ(â † n−s+1 ) * ψ(â † n−s+2 ) * · · · ψ(â † n ) * P m .(52) (iii) If r = m + n − 2s + 1 with s ∈ {1, 2, . . . , min(m, n)}, then P r wP m = ψ(â 1 )ψ(â 2 ) · · · ψ(â n−s ) · ρ(a n−s+1 ) ψ(â † n−s+2 ) * ψ(â † n−s+3 ) * · · · ψ(â † n ) * P m . (53) Proof. The following equation is equivalent to parts (i)-(iii) of the Lemma taken together: wP m = min(m,n) s=0 P n+m−2s ψ(â 1 ) · · · ψ(â n−s ) · ψ(â † n−s+1 ) * · · · ψ(â † n ) * P m + min(m,n) s=1 P n+m−2s+1 ψ(â 1 ) · · · ψ(â n−s )ρ(a n−s+1 ) · ψ(â † n−s+2 ) * · · · ψ(â † n ) * P m .(54) We will prove (54) by induction on n. For n = 1, taking first m ≥ 1 and using the fact that φ k(1) (a 1 ) = 0 together with (31), (41), (44), and (48), we find a 1 P m = ψ(â 1 ) + ρ(a 1 ) + ψ(â † 1 ) * P m (55) = P m+1 ψ(â 1 )P m + P m ρ(a 1 )P m + P m−1 ψ(â † 1 ) * P m ,(56) while in the case m = 0, using (29), (40), (43), and (47), we find a 1 P 0 = ψ(â 1 )P 0 = P 1 ψ(â 1 )P 0 .(57) Thus, (54) is proved in the case n = 1. Now let n ≥ 2 and set w ′ = a 2 a 3 · · · a n . By the induction hypothesis, we have w ′ P m = min(m,n−1) s=0 P n+m−2s−1 ψ(â 2 ) · · · ψ(â n−s ) · ψ(â † n−s+1 ) * · · · ψ(â † n ) * P m (58) + min(m,n−1) s=1 P n+m−2s ψ(â 2 ) · · · ψ(â n−s )ρ(a n−s+1 ) · ψ(â † n−s+2 ) * · · · ψ(â † n ) * P m .(59) Now we multiply both sides of (58) and (59) on the left by a 1 , and use (56) and (57), as needed. For example, from (58) consider a 1 P n+m−2s−1 ψ(â 2 ) · · · ψ(â n−s )ψ(â † n−s+1 ) * · · · ψ(â † n ) * P m .(60) If s < n − 1, then the initial part of (60) is a 1 P n+m−2s−1 ψ(â 2 ) = P n+m−2s ψ(â 1 )P n+m−2s−1 ψ(â 2 ) + P n+m−2s−1 ρ(a 1 )P n+m−2s−1 ψ(â 2 ) + P n+m−2s−2 ψ(â † 1 ) * P n+m−2s−1 ψ(â 2 ) = P n+m−2s ψ(â 1 )P n+m−2s−1 ψ(â 2 ) = P n+m−2s ψ(â 1 )ψ(â 2 ), where we have used ρ(a 1 )P n+m−2s−1 ψ(â 2 ) = ρ(a 1 )Q k(1) P n+m−2s−1 Q k(2) ψ(â 2 ) = 0 (62) ψ(â † 1 ) * P n+m−2s−1 ψ(â 2 ) = ψ(â † 1 ) * Q k(1) P n+m−2s−1 Q k(2) ψ(â 2 ) = 0.(63) If s = n − 1 < m, then the initial part of (60) is a 1 P m−s ψ(â † 2 ) * = P m−s+1 ψ(â 1 )P m−s ψ(â † 2 ) * + P m−s ρ(a 1 )P m−s ψ(â † 2 ) * + P m−s−1 ψ(â † 1 ) * P m−s ψ(â † 2 ) * = P m−s+1 ψ(â 1 )ψ(â † 2 ) * + P m−s ρ(a 1 )ψ(â † 2 ) * + P m−s−1 ψ(â † 1 ) * ψ(â † 2 ) * ,(64) while if s = n − 1 = m, then the initial part of (60) is a 1 P 0 ψ(â † 2 ) * = P 1 ψ(â 1 )P 0 ψ(â † 2 ) * = P 1 ψ(â 1 )ψ(â † 2 ) * .(65) Turning now to (59), we consider a 1 P n+m−2s ψ(â 2 ) · · · ψ(â n−s )ρ(a n−s+1 )ψ(â † n−s+2 ) * · · · ψ(â † n ) * P m . We find that the initial part of (66) is a 1 P n+m−2s ψ(â 2 ) = P n+m−2s+1 ψ(â 1 )ψ(â 2 ), s < n − 1 P m−s+2 ψ(â 1 )ρ(a 2 ), s = n − 1.(67) Taking all of these cases into account, we prove (54). Lemma 5.7. Let f be as in Proposition 5.1. Let m, r be nonnegative integers. Then P r f P m 2 ≤ k∈K n i=1 a k,i 2 .(68) Proof. If r < |m − n| or r > m + n, then by Lemma 5.6(i), we have P r f P m = 0. Case I. Suppose r = m+n−2s for s ∈ {0, 1, . . . , min(m, n)} and with s < n. By Lemma 5.6(ii), we have P r f P m = k∈K ψ(â k,1 ) · · · ψ(â k,n−s )ψ(â † k,n−s+1 ) * · · · ψ(â † k,n ) * P m(69) and (P r f P m ) * (P r f P m ) = = k,k ′ ∈K P m ψ(â k,n ) · · · ψ(â k,n−s+1 )ψ(â † k,n−s ) * · · · ψ(â † k,1 ) * · ψ(â k ′ ,1 ) · · · ψ(â k ′ ,n−s )ψ(â † k ′ ,n−s+1 ) * · · · ψ(â † k ′ ,n ) * P m . By the hypothesis (36), if k = k ′ , then k(1) = k ′ (1) and, consequently, ψ(â † k,1 ) * ψ(â k ′ ,1 ) = ψ(â † k,1 ) * Q k(1) Q k ′ (1) ψ(â k ′ ,1 ) = 0.(71) Therefore, using also (46), we get P r f P m 2 ≤ k∈K n i=1 â k,i 2 ≤ k∈K n i=1 a k,i 2 .(72) Case II. Suppose r = m + n − 2s for s = n ≤ m. Then (69) becomes P r f P m = k∈K ψ(â † k,1 ) * · · · ψ(â † k,n ) * P m(73) and we have (P r f P m )(P r f P m ) * = k,k ′ ∈K ψ(â † k,1 ) * · · · ψ(â † k,n ) * P m ψ(â k ′ ,n ) · · · ψ(â k ′ ,1 ). Again, by the hypothesis (36), if k = k ′ , then k(n) = k ′ (n) and, consequently, ψ(â † k,n ) * P m ψ(â k ′ ,n ) = ψ(â † k,n ) * Q k(1) P m Q k ′ (1) ψ(â k ′ ,n ) = 0.(75) Using again (46), we get (72) also in this case. Case III. Suppose r = m + n − 2s + 1 for s ∈ {1, . . . , min(m, n)}. Then using Lemma 5.6(iii) and proceeding similarly to above, we obtain the estimate P r f P m 2 ≤ k∈K a k,n−s+1 2 1≤i≤n i =n−s+1 â k,i 2 ≤ k∈K n i=1 a k,i 2 .(76) Remark 5.8. The left-hand inequalities in (72) and (76) are better than required in (68). In fact, (72) and (76) seem to be quite close in spirit to the inequality obtained in [1, 3.3], which applied to free products with amalgamation over the scalars. This shows σ(f ) ≤ (2n + 1)γ, which implies (37). Free shifts Let D be a unital C * -algebra, and let E D B : D → B be a conditional expectation onto a unital C * -subalgebra B. For each integer i ∈ Z let (A i , φ i ) be a copy of (D, E D B ). Let (A, φ) = ( * B ) i∈I (A i , φ i ) be the reduced amalgamated free product, and let a → λ i a denote the embedding of A i in the free product algebra A arising from the free product construction, as descibed in section 4. The free-shift automophism α on A is the automorphism of A given by α(λ i a ) = λ i+1 a for all a ∈ A and i ∈ Z. Theorem 6.1. Let α be the free-shift automorphism on the amalgamated free product C * -algebra A as given in (88) above. Then B is the fixedpoint subalgebra for α and α is uniquely ergodic relative to its fixed-point subalgebra. Proof. We will show lim n→∞ 1 n n−1 k=0 α k (a) = φ(a) for every a ∈ A. This will imply that B is the fixed-point subalgebra for α and that condition (iv) of Theorem 3.2 holds. It will suffice to show (89) for all a ∈ B and words a of the form w = a 1 a 2 · · · a p for some p ≥ 1 and a i ∈ A • k(i) , and some k(i) ∈ Z with k(i) = k(i + 1), i = 1, . . . , p − 1. Since B is α invariant, (89) is clear for a ∈ B. So assume a = w as above. Then φ(w) = 0 and n−1 k=0 α k (w) is a finite linear combination of words of length p to which Proposition 5.1 applies, and we have 1 n n−1 k=0 α k (w) ≤ 1 n (2p + 1)n 1/2 p i=1 a i .(90) Thus, we get lim n→∞ 1 n n−1 k=0 α k (w) = 0, as required. n. If h is a nontrivial element of word length p, then byHaagerup' Proof of Proposition 5 . 1 . 51Let σ : B → L(H) be a faithful * -representation of B on a Hilbert space H. Then the internal tensor product H = E ⊗ σ H is a Hilbert space and the * -representationσ : L(E) → L( H) given bỹ σ(a) = a ⊗ id H is faithful.Let v ∈ H. The stable rank of some free product C * -algebras. K J Dykema, U Haagerup, M Rørdam, ibid. 94Duke Math. J. 90213K.J. Dykema, U. Haagerup and M. Rørdam, 'The stable rank of some free product C * -algebras,' Duke Math. J. 90 (1997), 95-121, 'Correction,' ibid. 94 (1998), 213. Strict ergodicity and transformation of the torus. H Furstenberg, American J. Math. 83H. Furstenberg, 'Strict ergodicity and transformation of the torus,' American J. Math. 83 (1961), 573-601. An example of a non nuclear C * -algebra which has the metric approximation property. U Haagerup, Invent. Math. 50U. Haagerup, 'An example of a non nuclear C * -algebra which has the metric approx- imation property,' Invent. Math. 50 (1979), 279-293. Groupes hyperboliques, algèbres d'opérateurs et un théorème de Jolissaint. P De La Harpe, C. R. Acad. Sci. Paris, Série I. 307P. de la Harpe, 'Groupes hyperboliques, algèbres d'opérateurs et un théorème de Jolissaint,' C. R. Acad. Sci. Paris, Série I 307 (1988), 771-774. Rapidly decreasing functions in reduced C * -algebras of groups. P Jolissaint, Trans. Amer. Math. Soc. 317P. Jolissaint, 'Rapidly decreasing functions in reduced C * -algebras of groups,' Trans. Amer. Math. Soc. 317 (1990), 167-196. Strong Haagerup inequalities for free R-diagonal elements. T Kemp, R Speicher, math.OA/0512481preprintT. Kemp and R. Speicher, 'Strong Haagerup inequalities for free R-diagonal ele- ments,' preprint, math.OA/0512481. . D Kerr, private communicationD. Kerr, private communication. La theorie generale de la mesure dans son application a l'etude des systems dynamiques de la mecanique non lineaire. N Kryloff, N Bogoliouboff, Ann. Math. 38N. Kryloff and N. Bogoliouboff, 'La theorie generale de la mesure dans son application a l'etude des systems dynamiques de la mecanique non lineaire,' Ann. Math. 38 (1937), 65-113. A toolkit for operator algebraists. C Lance, C * -Modules Hilbert, London Mathematical Society Lecture Notes Series. 210Cambridge University PressC. Lance, Hilbert C * -modules. A toolkit for operator algebraists, London Mathemat- ical Society Lecture Notes Series, 210, Cambridge University Press, 1995. Ergodic Sets. J C Oxtoby, Bull. Amer. Math. Soc. 58J.C. Oxtoby, 'Ergodic Sets,' Bull. Amer. Math. Soc. 58 (1952), 116-136. G K Pedersen, C * -algebras and their automorphism groups. LondonAcademic PressG.K. Pedersen, C * -algebras and their automorphism groups, Academic Press, London, 1979. Khintchine type inequalities for reduced free products and applications. E Ricard, Q Xu, math.OA/0505302preprintE. Ricard and Q. Xu, 'Khintchine type inequalities for reduced free products and applications,' preprint, math.OA/0505302. On the projection of norm one in W * -algebras. J Tomiyama, Proc. Japan Acad. 33J. Tomiyama, 'On the projection of norm one in W * -algebras,' Proc. Japan Acad. 33 (1957), 608-612. Symmetries of some reduced free product C * -algebras,' in Operator Algebras and Their Connections with Topology and Ergodic Theory. D Voiculescu, Lecture Notes in Mathematics. 1132Springer-VerlagD. Voiculescu, 'Symmetries of some reduced free product C * -algebras,' in Operator Algebras and Their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics, 1132. Springer-Verlag, 1985 pp. 556-588. College Station TX 77843-3368. USA., Permanent address: Centro de Matemáticas, Facultad de Ciencias. Iguá. 4225Dept. of Mathematics, Texas A&M UniversityCP. E-mail address: [email protected]. of Mathematics, Texas A&M University, College Station TX 77843- 3368. USA., Permanent address: Centro de Matemáticas, Facultad de Cien- cias, Iguá 4225, CP 11 400, Montevideo, Uruguay. E-mail address: [email protected] College Station TX 77843-3368, USA. E-mail address: Ken. Dept. of Mathematics, Texas A&M [email protected]. of Mathematics, Texas A&M University, College Station TX 77843- 3368, USA. E-mail address: [email protected]

Working at a φ-factory, KLOE/KLOE-2 has collected a very large and clean sample of η decays, which allow to perform a variety of precision measurements. This report presents the recent results on the η-meson decays from KLOE/KLOE-2 experiments, including the search of the C-violating decay η → γγγ, the determination of the CP asymmetry in η → π + π − e + e − , the precision measurement of the η → π + π − π 0 Dalitz plot distribution, the search for the CPviolating decay η → π + π − and the precision measurement of O(p 6 ) dominated decay η → γγπ 0 .

1904.12034

af8b5a0616fe64466586e4925aeeb06dc4fd67db

Study of η-meson Decays at KLOE/KLOE-2 26 Apr 2019 Xiaolin Kang [email protected] KLOE Collaboration INFN-Laboratori Nazionali di Frascati Via E. Fermi 4000044FrascatiRMItaly Study of η-meson Decays at KLOE/KLOE-2 26 Apr 2019 Working at a φ-factory, KLOE/KLOE-2 has collected a very large and clean sample of η decays, which allow to perform a variety of precision measurements. This report presents the recent results on the η-meson decays from KLOE/KLOE-2 experiments, including the search of the C-violating decay η → γγγ, the determination of the CP asymmetry in η → π + π − e + e − , the precision measurement of the η → π + π − π 0 Dalitz plot distribution, the search for the CPviolating decay η → π + π − and the precision measurement of O(p 6 ) dominated decay η → γγπ 0 . Introduction After being discovered more than 50 years ago, the η and η ′ mesons are still included in the physics program of various experiments, because of their interesting properties. Both η and η ′ are eigenstates of parity P and G, charge conjugation C, and combined CP parity, (I G (J P C ) = 0 + (0 −+ )). Therefore, both decays can serve as laboratories for the testing of conservation or breaking of these discrete symmetries. Moreover, as η is a pseudo-goldstone boson of the strong interaction gauge theory, its strong and electromagnetic decays are forbidden at tree level [1] and are particularly suitable to measure transition form factors, investigate decay dynamics and test different ChPT models. In addition, it is possible to search for new phenomena, such as light dark matter bosons, in rare or forbidden η decays [2,3]. The KLOE/KLOE-2 experiments [4] were operated at DAΦNE, an e + e − collider running at a center of mass energy of ∼ 1020 MeV, the mass of the φ meson, at the INFN Frascati Laboratory. In March 2018 the KLOE-2 experiment completed its data taking campaign and collected 5.5 f b −1 of data. Together with the KLOE experiment, a total data sample of 8 f b −1 has been collected. Copious η mesons are produced via the radiative decay of the φ meson, φ → γη, with a branching ratio of 1.3% [5]. The η mesons can be clearly identified by their recoil against a photon with monochromatic energy of 363 MeV. The KLOE/KLOE-2 data sample corresponds to 3.1 × 10 8 η mesons, which allow the accurate analysis of specific decays and to search for rare or forbidden ones. Search for the forbidden decay η → γγγ The decay η → γγγ is forbidden by charge-conjugation invariance, if the weak interaction is not taken into account. KLOE has performed a search for this decay in the data sample collected in the years 2001 and 2002 with an integrated luminosity of 410 pb −1 and set the upper limit B(η → γγγ) ≤ 1.6 × 10 −5 at 90% confidence level (CL) [6], which is the most stringent result obtained to data. Four isolated photons are required to select the candidate events. A kinematic fit imposing energy-momentum conservation and time-of-flight equal to the velocity of light is performed for the 4γ final states, and the χ 2 value is required to be less than 25. The dominant background is given by the process e + e − → γω, where the initial-state radiation of a hard photon is followed by the decay ω → γπ 0 (γγ). In addition, φ → γπ 0 , γf 0 (π 0 π 0 ) and γa 0 (ηπ 0 ) also can mimic fourphoton events because of the loss of photons, addition of photons from machine background or shower splitting. Those events can be rejected by requiring the invariant mass of any γγ pair to lay out of the π 0 mass range, (90,180)MeV. 8268 events survive the above cuts. In the decay φ → γη, the recoil photon has an energy of 363 MeV in the rest frame of the φ, and it is also the most probable energetic photon (γ hi ) in the signal decay chain. Fig. 1-left shows the monte carlo (MC) simulated energy distribution of the γ hi , E(γ hi ), for the signal. Fig. 1-right shows the E(γ hi ) distribution for the data sample. No peak is observed in the signal region, defined as 350 < E(γ hi ) < 379.75 MeV. To estimate the background, polynomials are used to fit the E(γ hi ) sidebands, (280, 350) and (379.75,481.25) MeV. Neymans construction procedure is used to evaluate the upper limit, which gives the best upper limit to date, 1.6 × 10 −5 @ 90% CL. By analysing the complete KLOE/KLOE-2 data sample, the upper limit is expected to reach 3.6 × 10 −6 @ 90% CL. Figure 1. Distribution of the E(γ hi ) in the φ rest frame for the MC simulated signal (left) and data (right). The shaded interval is the signal region, the dots with error bars are KLOE data, the superimposed curve is the polynomial fit. 3. η → π + π − e + e − By comparing the precision measurement of the branching ratio of η → π + π − e + e − with the predictions from different theoretical approaches, such as Vector Meson Dominance model and ChPT, it is possible to probe the η meson's electromagnetic structure [7]. In addition, a possible CP -violating mechanism, not directly related to the most widely studied flavor changing neutral process, has been proposed for this decay [8]. This mechanism could induce interference between the parity-conserving magnetic amplitudes and the parity-violating electric amplitudes. Such CP -violation effect could be tested by measuring the polarization of the virtual photon and would result in an asymmetry in the angle φ between pions and electrons decay planes in the η rest frame, defined as A φ = π/2 0 dΓ dφ dφ− π π/2 dΓ dφ dφ π/2 0 dΓ dφ dφ+ π π/2 dΓ dφ dφ = N sinφcosφ>0 −N sinφcosφ<0 N sinφcosφ>0 +N sinφcosφ<0 . In the η decay this asymmetry is constrained by experimental [9] and SM [10] upper limits on the CP -violating decay η → π + π − at the level of O(10 −4 ) and O(10 −15 ), respectively. KLOE has measured the φ → π + π − e + e − decay using a 1.73 f b −1 data sample collected at φ meson peak [11]. The main background sources are φ → π + π − π 0 events (with π 0 Dalitz decay) and φ → γη either with η → π + π − π 0 (with π 0 Dalitz decay) or η → π + π − γ (with photon conversion on the beam pipe). Continuum background e + e − → e + e − (γ) events with photon conversions, split tracks or interactions with some material in the region of DAΦNE quadrupoles inside KLOE can also contaminate the signal and are studied using off-peak data taken at √ s = 1 GeV. The π + π − e + e − invariant mass distribution, M ππee , is shown in Fig. 2-left. The background contribution is evaluated by performing a fit to the sidebands of the M ππee invariant mass distribution with the shapes of the two background contributions obtained from MC. For the signal estimate, the η mass region, (535,555) MeV, is considered, counting the event number after background subtraction. The resulting number of signal events, N π + π − e + e − = 1555 ± 52, is used to extract the branching ratio: B(η → π + π − e + e − (γ)) = (26.8 ± 0.9 stat ± 0.7 syst ) × 10 −5 . The decay plane asymmetry A φ has been evaluated for the events in the signal region after background subtraction. The obtained value is A φ = (−0.6 ± 2.6 stat ± 1.8 syst ) × 10 −2 . This is the first measurement of this asymmetry. The distribution of the sinφcosφ variable is show in the right panel of Fig. 2. Figure 2. π + π − e + e − invariant mass spectrum around the η mass (left) and the sinφcosφ distribution in the signal region (right). The dots with error bars are data, the black histograms are the expected distributions obtained by the sum of the MC contribution to signal (dark grey), φ background (light grey) and continuum background (white). Dalitz plot of η → π + π − π 0 The decay η → π + π − π 0 is an isospin-violating process. Electromagnetic contributions to the process are expected to be very small and the decay is induced dominantly by the strong interaction via the u, d quark mass difference [12]. A precise study of this decay can lead to a very accurate measurement of Q 2 = (m 2 s −m 2 )/(m 2 d − m 2 u ), withm = (m d + m u )/2, and thus put a stringent constraint on the light quark masses. The conventional decay amplitude square for η → π + π − π 0 is parametrized as |A(X, Y )| 2 = 1 + aY + bY 2 + cX + dX 2 + eXY + f Y 3 + gX 2 Y + hXY 2 + lX 3 + . . ., with X = √ 3/Q η (T π + − T π − ) and Y = 3T π 0 /Q η − 1. T is the kinetic energy of the pions in η rest frame, Q η = m η − m π + − m π − − m π 0 is the excess energy of the reaction, m η/π are the nominal masses from the Particle Data Group [5]. The Dalitz plot distribution has been recently studied by KLOE [13] with the world's largest signal sample of ∼ 4.7 × 10 6 events, based on 1.6 f b −1 φ data, improving the previous measurement [14]. From a fit to the Dalitz plot density distribution, shown in Fig. 3, we obtained the most precise determinations of the parameters that characterize the decay amplitude. The results, together with a comparison with recent experimental measurements, are summarized in Table 1. This large statistics sample is also sensitive to the g parameter. As expected from C-parity conservation, the odd powers of X (c, e, h and l) are consistent with zero. Figure 3. Dalitz plot for η → π + π − π 0 in the data sample after background subtraction. In addition, we have also checked the C-parity conservation by measuring the left-right, quadrants and sextants charge asymmetries [17], which give the values A LR = (−5.0 ± 4.5 +5.0 −11 ) × 10 −4 , A Q = (+1.8 ± 4.5 +4.8 5. Search for η → π + π − The decay η → π + π − violates both P and CP invariance. In the SM, this decay can proceed only via the CP -violating weak interaction, through mediation by a virtual K 0 S meson, with a branching ratio B(η → π + π − ) ≤ 2 × 10 −27 [10]. By introducing the possible contribution of the CP -violating QCD θ term to this decay, the upper limit may reach the order of 10 −17 [10]. While allowing a CP violation in the extended Higgs sector gives a slightly larger upper limit, a order of 10 −15 [10]. Any detection of larger branching fractions would indicate a new source of CP violation in the strong interaction, beyond any considerable extension of the SM. With the first 350 pb −1 of data, KLOE searched for evidence of this decay, setting the best upper limit to date, B(η → π + π − ) ≤ 1.3 × 10 −5 at 90% CL [9]. Using a data sample of 1.6 f b −1 collected at the φ meson peak, KLOE-2 is updating this result. In this analysis, γπ + π − final states are selected, contaminated by radiative Bhabha events γe + e − , γµ + µ − and ρ(ππ)π with one lost photon. A direction match between the missing momentum of π + π − and the selected γ is performed to reduce π + π − π 0 backgrounds. γe + e − backgrounds can be separated from the γπ + π − signal by the different flight time inside the detector under different mass hypothesis for each charged track. While γµ + µ − events can be rejected using the so-called track-mass variable, computed assuming the φ decays to two particles of identical mass and a photon, shown in the left plot of Fig. 4. The survived candidates are mainly e + e − → (γ)γπ + π − . The π + π − mass spectra, M (π + π − ), is used to look for the presence of η resonance, shown in the right plot of Fig. 4. No peak is observed in the distribution of M (π + π − ) in the vicinity of m η . A Bayesian method is used to obtain the upper limit of the signal number. In the fit to M (π + π − ), the background is described by a third-order polynomial and the signal is described with the dedicated MC simulated shape. The preliminary upper limit on the branching ratio is determined to be 6.3 × 10 −6 at 90% CL. With the whole KLOE and KLOE-2 data, the upper limit is expected to reach 2.7 × 10 −6 @ 90% CL. Figure 4. The track-mass distribution for different decay channels (left) and the π + π − invariant mass spectrum between 500 and 600 MeV (right). The signal shape in arbitrary units. 6. The η → γγπ 0 decay The doubly radiative decay η → γγπ 0 offers a unique window on pure p 6 terms of the Chiral Lagrangian, since there is no O(p 2 ) contribution, and the O(p 4 ) contribution is highly suppressed [18]. The branching ratio for this decay has been measured by several experiments in the past [5]. However, their results show large discrepancies. Using a data sample of 450 pb −1 of data, KLOE obtained the preliminary result B(η → γγπ 0 ) = (8.4 ± 2.7 ± 1.4) × 10 −5 [19], which is around 2σ lower than the predictions from Chiral-loop, LσM and VMD [18,20] and the latest result from Crystall Ball Collaboration at AGS [21] and A2 Collaboration at MAMI [22]. The experimental challenges of measuring this decay is the smallness of the rate for doubly radiative decays and the large rate of background contamination, in particular from η → 3π 0 → 6γ decays with lost or merged photons. A correct estimation of the background is crucial for this analysis. Reanalysis for this decay with more statistics is ongoing at KLOE-2, by performing kinematic fits under different signal or background hypothesis to suppress backgrounds. Furthermore, multi-variable analysis with cluster shape information is under tuning to separate single photon from merged photon. The final measurement is forthcoming. Moreover, this decay also provides a probability for the search of a B boson at the QCD scale, via the decay chain η → γB, B → γπ 0 [2]. Summary The KLOE/KLOE-2 collaboration has achieved various precision measurements on η-meson decays, including the tests of discrete symmetries, studies of the decay dynamics and searches for rare decays using the KLOE data sample. In 2018 KLOE-2 has successfully completed its data taking at the φ-meson peak. The whole KLOE/KLOE-2 data sample amounts now to about 8 f b −1 . Ongoing analyses will produce more precise results in the next years. −2. 3 ) 3× 10 −4 and A S = (−0.4 ± 4.5 +3.1 −3.5 ) × 10 −4 , respectively. All of them are consistent with zero at the level of 10 −4 . Table 1 . 1Recent experimental measurement of the Dalitz plot parameters for η → π + π − π 0 .a b d f g KLOE(16) [13] −1.095 ± 0.004 0.145 ± 0.006 0.081 ± 0.007 0.141 ± 0.011 −0.044 ± 0.016 KLOE(16) [13] −1.104 ± 0.004 0.142 ± 0.006 0.073 ± 0.005 0.154 ± 0.008 - KLOE(08) [14] −1.090 ± 0.020 0.124 ± 0.012 0.057 ± 0.017 0.14 ± 0.02 - WASA [15] −1.144 ± 0.018 0.219 ± 0.051 0.088 ± 0.023 0.115 ± 0.037 - BESIII [16] −1.128 ± 0.017 0.153 ± 0.017 0.085 ± 0.018 0.173 ± 0.035 - AcknowledgmentsWe warmly thank our former KLOE colleagues for the access to the data collected during the KLOE data taking campaign.We thank the DAΦNE team for their efforts in maintaining low background running conditions and their collaboration during all data taking. We want to thank our technical staff: G.F. . B M K Nefkens, J W Price, Phys. Scripta T. 99114B. M. K. Nefkens and J. W. Price, Phys. Scripta T 99, 114 (2002). . S Tulin, Phys. Rev. D. 89114008S. Tulin, Phys. Rev. D 89, 114008 (2014). . M Reece, L T Wang, J. High Energy Phys. 090751M. Reece and L. T. Wang, J. High Energy Phys. 0907, 051 (2009). hep-ex/0002030Proc. 19th International Symposium on Lepton and Photon Interactions at High Energies. S. Bertolucci19th International Symposium on Lepton and Photon Interactions at High EnergiesStanford, USAThe KLOE Detector Technical ProposalThe KLOE Collaboration, The KLOE Detector Technical Proposal, LNF-93/002, 1993; S. Bertolucci (KLOE Collaboration), A Status Report of KLOE at DAΦNE, in Proc. 19th International Symposium on Lepton and Photon Interactions at High Energies, Stanford, USA, 1999, hep-ex/0002030. . M Tanabashi, Particle Data GroupPhys. Rev. D. 9830001M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018). . A Aloisio, KLOE CollaborationPhys. Lett. B. 59149A. Aloisio et al. (KLOE Collaboration), Phys. Lett. B 591, 49 (2004). . L G Landsberg, Phys. Rept. 128301L. G. Landsberg, Phys. Rept. 128, 301 (1985). . D N Gao, Mod. Phys. Lett. A. 171583D. N. Gao, Mod. Phys. Lett. A 17, 1583 (2002). . F Ambrosino, KLOE CollaborationPhys. Lett. B. 606276F. Ambrosino et al. (KLOE Collaboration), Phys. Lett. B 606, 276 (2005). . C Jarlskog, E Shabalin, Phys. Scr. T. 9923C. Jarlskog, E. Shabalin, Phys. Scr. T 99, 23 (2002); . E Shabalin, Phys. Scr. T. 99104E. Shabalin, Phys. Scr. T 99, 104 (2002). . F Ambrosino, KLOE CollaborationPhys. Lett. B. 675283F. Ambrosino et al. (KLOE Collaboration), Phys. Lett. B 675, 283 (2009). . J S Bell, D G Sutherland, Nucl. Phys. B. 4315J. S. Bell and D. G. Sutherland, Nucl. Phys. B 4, 315 (1968); . R Baur, J Kambor, D Wyler, Nucl. Phys. B. 460127R. Baur, J. Kambor and D. Wyler, Nucl. Phys. B 460, 127 (1996); . C Ditsche, B Kubis, U G Meissner, Eur. Phys. J. C. 6083C. Ditsche, B. Kubis, and U. G. Meissner, Eur. Phys. J. C 60, 83 (2009). . A Anastasi, KLOE-2 CollaborationJ. High energy Phys. 160519A. Anastasi et al. (KLOE-2 Collaboration), J. High energy Phys. 1605, 019 (2016). . F Ambrosino, KLOE CollaborationJ. High energy Phys. 08056F. Ambrosino et al. (KLOE Collaboration), J. High energy Phys. 0805, 006 (2008). . P Adlarson, WASA-at-COSY CollaborationPhys. Rev. C. 9045207P. Adlarson et al. (WASA-at-COSY Collaboration), Phys. Rev. C 90, 045207 (2014). . M Ablikim, BESIII CollaborationPhys. Rev. D. 9212014M. Ablikim et al. (BESIII Collaboration), Phys. Rev. D 92, 012014 (2015). . J G Layter, J A Appel, A Kotlewski, W Lee, S Stein, J J Thaler, Phys. Rev. Lett. 29316J. G. Layter, J. A. Appel, A. Kotlewski, W. Lee, S. Stein and J. J. Thaler, Phys. Rev. Lett. 29, 316 (1972). . L Ametller, J Bijnens, A Bramon, F Cornet, Phys. Lett. B. 276185L. Ametller, J. Bijnens, A. Bramon and F. Cornet, Phys. Lett. B 276, 185 (1992). . B , KLOE CollaborationDi Micco, KLOE CollaborationActa Phys. Slov. 56403B. Di Micco et al. (KLOE Collaboration), Acta Phys. Slov. 56, 403 (2006). . R Escribano, S Gonzàlez-Solís, R Jora, E Royo, arXiv:1812.08454hep-phR. Escribano, S. Gonzàlez-Solís, R. Jora and E. Royo, arXiv:1812.08454 [hep-ph]. at AGS). S Prakhov, Crystal Ball CollaborationPhys. Rev. C. 7815206S. Prakhov et al. (Crystal Ball Collaboration at AGS), Phys. Rev. C 78, 015206 (2008). . B M K Nefkens, A2 Collaboration at MAMIPhys. Rev. C. 9025206B. M. K. Nefkens et al. (A2 Collaboration at MAMI), Phys. Rev. C 90, 025206 (2014).

Time-and Angle-resolved photoelectron spectroscopy from surfaces can be used to record the dynamics of electrons and holes in condensed matter on ultrafast time scales. However, ultrafast photoemission experiments using extreme-ultraviolet (XUV) light have previously been limited by either space-charge effects, low photon flux, or limited tuning range. In this article, we describe XUV photoelectron spectroscopy experiments with up to 5 nA of average sample current using a tunable cavity-enhanced high-harmonic source operating at 88 MHz repetition rate. The source delivers >10 11 photons/s in isolated harmonics to the sample over a broad photon energy range from 18 to 37 eV with a spot size of 58 Â 100 lm 2 . From photoelectron spectroscopy data, we place conservative upper limits on the XUV pulse duration and photon energy bandwidth of 93 fs and 65 meV, respectively. The high photocurrent, lack of strong space charge distortions of the photoelectron spectra, and excellent isolation of individual harmonic orders allow us to observe laser-induced modifications of the photoelectron spectra at the 10 À4 level, enabling time-resolved XUV photoemission experiments in a qualitatively new regime.

10.1063/1.5045578

1801.08124

bf50b99f14115229a47a13578f4c7acd935722fd

Ultrafast extreme ultraviolet photoemission without space charge Christopher Corder Stony Brook University 11794-3400Stony BrookNew YorkUSA Peng Zhao Stony Brook University 11794-3400Stony BrookNew YorkUSA Jin Bakalis Stony Brook University 11794-3400Stony BrookNew YorkUSA Xinlong Li Stony Brook University 11794-3400Stony BrookNew YorkUSA Matthew D Kershis Brookhaven National Laboratory 11973UptonNew YorkUSA Amanda R Muraca Stony Brook University 11794-3400Stony BrookNew YorkUSA Michael G White Stony Brook University 11794-3400Stony BrookNew YorkUSA Brookhaven National Laboratory 11973UptonNew YorkUSA Thomas K Allison Stony Brook University 11794-3400Stony BrookNew YorkUSA Ultrafast extreme ultraviolet photoemission without space charge 10.1063/1.5045578(Received 21 June 2018; accepted 16 August 2018; published online 6 September 2018) Time-and Angle-resolved photoelectron spectroscopy from surfaces can be used to record the dynamics of electrons and holes in condensed matter on ultrafast time scales. However, ultrafast photoemission experiments using extreme-ultraviolet (XUV) light have previously been limited by either space-charge effects, low photon flux, or limited tuning range. In this article, we describe XUV photoelectron spectroscopy experiments with up to 5 nA of average sample current using a tunable cavity-enhanced high-harmonic source operating at 88 MHz repetition rate. The source delivers >10 11 photons/s in isolated harmonics to the sample over a broad photon energy range from 18 to 37 eV with a spot size of 58  100 lm 2 . From photoelectron spectroscopy data, we place conservative upper limits on the XUV pulse duration and photon energy bandwidth of 93 fs and 65 meV, respectively. The high photocurrent, lack of strong space charge distortions of the photoelectron spectra, and excellent isolation of individual harmonic orders allow us to observe laser-induced modifications of the photoelectron spectra at the 10 À4 level, enabling time-resolved XUV photoemission experiments in a qualitatively new regime. I. INTRODUCTION Angle-resolved photoelectron spectroscopy (ARPES) using synchrotron radiation has become an essential tool for condensed matter physics and surface science. The high spectral brightness of synchrotron radiation allows photoelectron spectra to be recorded with photocurrents in the nano-Ampere range. These large photocurrents, parsed by sophisticated electron energy analyzers, 1-3 enable detailed studies of the electronic structure of solids and surfaces in energy, momentum, and spin. Tuning the photon energy throughout the extreme ultraviolet (XUV, 10-100 eV) allows experiments to map the energy dispersion relation for momentum perpendicular to the surface (k z ), interpret the contributions of final state effects to the measured energy distribution curves (EDC), and choose between increased surface or bulk sensitivity. 4 Shortly after the development of high-power femtosecond lasers and discovery of highorder harmonic generation (HHG), in which a broad range of laser-harmonics are coherently emitted from a field-ionized medium, 5 HHG was applied to surface photoemission experiments. [6][7][8] Indeed, the range of photon energies typically emitted from HHG driven by Ti:Sapphire lasers in noble gasses is nicely coincident with the range of photon energies used by ARPES beamlines at synchrotrons. In addition to the dramatically reduced cost and footprint compared to a synchrotron source, the HHG pulses had the advantage that they could be orders a) C. Corder and P. Zhao contributed equally to this work. b) [email protected]. 2329-7778/2018/5(5)/054301/16 V C Author(s) 2018. 5, 054301-1 of magnitude shorter than the $100 ps pulse durations of synchrotrons, enabling ultrafast timedomain studies. However, it also became immediately apparent that photoemission experiments using HHG would be drastically limited compared to what is possible at synchrotrons. 8,9 The principle limitation comes from the so-called "vacuum space-charge" effect. 10 Consider a synchrotron experiment illuminating the sample with $10 12 photons/s causing $10 11 electrons/s (16 nA) to be emitted from the surface. The synchrotron photon pulses arrive at $100-1000 MHz repetition rate, so each burst of electrons emitted concurrently from a single pulse contains only $1000-100 electrons. In contrast, due to the high peak powers required to drive the HHG process efficiently, HHG is typically restricted to <100 kHz repetition rates. In order to maintain the same photocurrent, the electrons must then be concentrated by more than 1000 times, to more than 10 6 electrons/pulse. The charging of space at these electron densities distorts the photoelectron spectrum on the eV energy scale, whereas synchrotron beamlines now routinely record photoelectron spectra with meV resolution. 11 Practitioners of timeresolved photoelectron spectroscopy using HHG are then forced to compromise on the applied photon flux, focused spot size, resolution, fidelity of the signal, or some combination thereof. 9,12,13 With the constraint of space charge setting the fundamental limits on the performance of photoemission experiments, this phenomenon has been extensively studied over a wide range of electron kinetic energies and pulse durations, both experimentally and theoretically. 9,10,12,[14][15][16][17][18] For sub-ps pulses and electron kinetic energies in the $5-100 eV range produced from conductive samples, both shifts and broadening of the photoelectron spectra features are observed to scale with linear electron density q N=D, where N is the number of electrons emitted from the sample per pulse and D is the spot size of the light on the sample. Expressed in terms of the average sample current (I sample ) and repetition rate (f rep ), this gives DE s;b ¼ m s;b I sample ef rep D ; (1) where DE s;b is the energy shift (s) or energy broadening (b) of the photoelectron spectrum, e is the charge of the electron, and m s,b are empirical scaling factors. Working in the extremeultraviolet (XUV), space charge effects do not depend strongly on the sample studied or the photon energy since the photoelectric yield is dominated by secondary electrons and the full Brillouin zone of inner valence bands. It is common practice to use the results obtained for simple metals in general considerations of the problem, and reports of the slope parameters m s,b in the literature have varied by a factor of 2. 9,10,12,14,18 Recently, using Cu (001), Pl€ otzing et al. 12 have studied these effects for multiple spot sizes and determined m b ¼ 2.1  10 À6 eV mm and m s ¼ 3.2  10 À6 eV mm with an estimated systematic uncertainty of less than 20%. Figure 1 illustrates the constraints on attainable sample current for a given resolution according to Eq. (1) using m b from Ref. 12. The dashed lines indicate the space charge limits for sub-ps laserbased systems of different repetition rates assuming a 1 mm spot size-large by ARPES standards. Even at the high repetition rate of 100 kHz and the coarse resolution of 100 meV, space charge constraints still limit the sample current to 760 pA. For pump/probe experiments, these data-rate limitations are particularly stark for several reasons. First is that pump excitation adds additional dimensions to the data set. At a minimum, data should be recorded at several pump-probe delays and pump fluences, and there are also the parameters of pump wavelength and polarization. Second is that the signal of interest is inherently small, since only a fraction of the sample's electrons are excited by the pump. Combined, these two factors result in the need for orders of magnitude more data than ground state studies done at synchrotrons, but space charge limits the data rate to be orders of magnitude lower. Experiments have then been almost exclusively restricted to strongly excited samples using absorbed fluences on the order of $1 mJ/cm 2 , [19][20][21] such that laser excitation produces changes to the EDC visible on a linear scale. In addition to probing different physics than can be accessed in the low-fluence regime, 22 at high fluences ultrashort pump pulses also produce many electrons through multiphoton processes which add to the space-charge problem, [23][24][25] and pump-induced space charge can also have a non-trivial dependence on the pump-probe delay, making space charge effects difficult to separate from the dynamics of interest. 26 The inverse dependence of Eq. (1) on f rep has motivated a great deal of work on highpower lasers and HHG source development. Although the highest HHG repetition rates have been reported using high-power frequency combs resonantly enhanced in optical cavities, 33,34 their reliability and suitability for time-resolved photoemission have faced skepticism from several authors. 8,30,35 Instead, there has been great investment in other approaches including HHG from high-power Ti:Sapphire and parametric amplifiers, 35,36 HHG from high power fiber lasers, 37,38 HHG generated within 39 and at the output 40 of thin-disk lasers, HHG from solids, 41,42 and HHG in the near-fields of nanostructures. 43 Despite these intense efforts, HHGbased photoemission comparable to that done with tunable synchrotron radiation has not been realized using any platform. In this article, we demonstrate the application of a tunable cavity-enhanced HHG (CE-HHG) source, specially designed for this purpose, to the difficult problem of XUV photoemission. By performing experiments with high flux at 88 MHz repetition rate, nanoamperes of sample current can be generated from a sub-100 lm laser spot with space charge effects estimated to be less than 10 meV, comparable to synchrotron-based ARPES experiments. 17,32 We observe laser-induced modifications of the EDC at the 10 À4 level in only minutes of integration time, demonstrating the feasibility of time-resolved ARPES measurements from perturbatively excited samples. In Sec. II, we describe critical and unique details of the light source along with its performance. In Sec. III, we demonstrate both static and time-resolved photoelectron spectroscopy measurements free of strong space-charge effects using nA sample currents. In Sec. IV, we compare this work to previous efforts and discuss how the system can be further improved. II. LIGHT SOURCE AND BEAMLINE The experimental setup is shown in Fig. 2. A home-built 80 W, 155 fs frequency comb laser with a repetition rate of 88 MHz and a center wavelength of 1.035 lm (h ¼ 1.2 eV) is passively amplified in a 6 mirror enhancement cavity with a 1% transmission input coupler and (1) for different repetition rates by assuming 1 mm spot size and symbols represent published results applying HHG to surface photoemission. Yellow circles represent results from tunable HHG systems and purple squares represent setups where the photon energy is not tunable in-situ. Symbols with black edges represent space-charge limited spectrometers, and symbols with red edges represent systems that are not yet space-charge limited. For space-charge-limited systems, the (x, y) positions represent the case where space charge broadening and the photon bandwidth add equally in quadrature. See Appendix A for detailed explanation on how symbol placement was calculated based on published results. 12,14,[27][28][29][30][31][32] a finesse of F > 560. We have described the laser in detail previously. 44 The laser is locked to the cavity using a two-point Pound-Drever-Hall lock as described in Refs. 45 and 46. Harmonics are generated at a 24 lm FWHM intracavity focus and reflected from a sapphire wafer placed at Brewster's angle for the resonant 1.035 lm light. Noble gasses are injected to the focus using a fused silica capillary with a 100 lm inside diameter. We optimize the nozzle position by moving it to maximize the photocurrent observed on a stainless steel vacuum photodiode (VPD1). 47 Typical photocurrents from VPD1 are in the range of 100 to 300 nA. When generating harmonics, we also dose each intracavity optic with a mix of ozone and O 2 from a commercial ozone generator to prevent hydrocarbon contamination, allowing continuous operation. Typical intracavity powers for generating harmonics range from 5 to 11 kW, depending on the generating gas and desired harmonic spectrum, corresponding to intracavity peak intensities in the range of 0.6 to 1.3  10 14 W/cm 2 . Intracavity nonlinear effects are observed from both the HHG gas and self-phase modulation in the Brewster plate, dropping the cavity's power enhancement and necessitating careful tuning of servo loop offsets. 46,48,49 For example, the power enhancement drops from 270 at low power to 200 at 7.5 kW of intracavity power when generating harmonics in krypton. The outcoupled harmonics are collimated by a 350 mm focal length toroidal mirror at 3 grazing angle (TM1) that forms the first part of a single off-plane grating pulse-preserving monochromator similar to the design of Frassetto et al. 50 The harmonics strike a motorized grating at a 4 grazing angle and are refocused by a second f ¼ 350 mm toroidal mirror (TM2) at an adjustable slit. For all data presented here, the monochromator grating has 150 grooves/ mm and is blazed for optimum diffraction efficiency for k ¼ 35 nm. With this grating, the monochromator selects an individual harmonic with tolerable pulse broadening but does not narrow the transmitted harmonic bandwidth. The exit slit plane of the monochromator is 1:1 imaged to the sample using another 350 mm focal length toroid at 3 grazing angle (TM3). Mirror TM3 is electrically floated such that the photocurrent of electrons ejected from the mirror surface can be used as a passive XUV intensity monitor. All beamline optics are gold coated and the XUV light is polarized perpendicular to the plane of incidence (s-polarization). We detect the XUV flux exiting the monochromator and delivered to the sample using four separate detectors: an aluminum coated silicon photodiode (PD, Optodiode AXUV100Al), the photocurrent from TM3, the photocurrent from the sample, and the photocurrent from an Al 2 O 3 vacuum photodiode 51 (VPD2) placed at the end of the surface science chamber. Figure 3(a) shows a typical HHG spectrum from xenon gas measured using each of the four detectors as the monochromator grating is rotated. The observed harmonic linewidths in Fig. 3(b) are due to the intentionally small resolving power of the pulse-preserving monochromator, not the intrinsic harmonic linewidth. FIG. 2. CE-HHG source and beamline. High-order harmonics of a resonantly enhanced Yb:fiber frequency comb are generated at the focus of a 6 mirror enhancement cavity and coupled into an XUV beamline. A pulse-preserving monochromator selects one harmonic which is focused on a sample under UHV conditions. BP ¼ Brewster plate , VPD ¼ vacuum photodi- ode, TM ¼ toroidal mirror, PD ¼ XUV photodiode, GJ ¼ gas jet, IC ¼ input coupler. The photon flux can be calculated using the measured photocurrent from all the detectors and literature values for the quantum efficiencies. All of these separate calculations agree within a factor of 2. Since contamination and surface oxidation only cause the quantum efficiency of XUV detectors to decrease, all calculated photon fluxes represent lower limits. In Fig. 3(b), the higher of the two lower limits from the PD or VPD2 are plotted as a function of photon energy and for three different generating gasses: argon, krypton, and xenon. As can be seen in Fig. 3(b), even using a single monochromator grating, by changing the generating gas, a flux of more than 10 11 photons per second is delivered to the sample over a broad tuning range. At lower photon energies, the higher efficiency of generation in Kr and Xe compensates the reduced diffraction efficiency of the grating blazed for 35 eV. Higher fluxes can be obtained at lower photon energies using different gratings. For example, we have observed 7  10 11 photons/s in the 21st harmonic from krypton (h ¼ 25.1 eV) using a 100 groove/mm grating blazed for 55 nm. These fluxes are within one order of magnitude of what is available from many state-of-the-art synchrotron beamlines dedicated to ARPES. 3,11,52 Critically, since at 88 MHz, FIG. 3. (a) An HHG spectrum from xenon gas measured by rotating the monochromator grating while using the four detectors after the exit slit. The photodiode current (black) uses the left y-axis, whereas the photoemission current from three downstream surfaces uses the right y-axis. Note that the harmonic linewidths are not resolved with the pulse-preserving monochromator design. (b) The photon flux delivered to the sample for each harmonic generated with the three gases. The flux has been calibrated using literature values for quantum efficiencies and no corrections for mirror losses have been made. (c) The 27th harmonic from Ar imaged with a Ce:YAG crystal at the sample position. (d) Lineouts through the centroid of (c) fit with Gaussian functions demonstrating 58 lm  100 lm spot size (FWHM). 7  10 11 photons/s corresponds to only 8000 photons/pulse-also comparable to synchrotronsall of this flux is usable for high-resolution photoemission experiments. To measure the XUV spot at the sample, we image the fluorescence from a Ce:YAG scintillator plate placed at the sample position. Figure 3(c) shows the image and Fig. 3(d) shows Gaussian fits in the both the horizontal and vertical along lineouts intersecting the image centroid. The data indicate a clean elliptical beam with a FWHM of 58 lm in the horizontal and 100 lm in the vertical. Also, we measure that approximately 70% of the XUV light can be transmitted through a 100 lm diameter pinhole oriented at 45 to the beam axis. This spot size is again similar to what is used at synchrotron beamlines. 11,52 When comparing to previous HHG results, it is important to note that in our case this small spot size and high flux are actually usable for experiments due to the absence of space-charge effects at 88 MHz repetition rate. A small spot size enables studying spatially inhomogeneous samples (for example, produced by exfoliation 53 ), requires less pump-pulse energy in pump/probe experiments, and is necessary for achieving high angular resolution in ARPES. We evaluated the amplitude noise and long term stability of the system using the PD detector. For the long term stability, we recorded a series of monochromator scans over a 1 h period without any human tuning of the laser alignment or servo loop. Figure 4(a) shows the relative power in the harmonics from Kr with a delivered flux greater than 10 11 photons/ s measured every 3 min. The RMS fluctuations averaged over all the harmonics over this period are 5%. Similar results are obtained for HHG in Ar or Xe as well. On longer timescales, slow drifts in the laser alignment into the cavity and servo-loop offsets require occasional tuning to maintain the flux levels at those shown in Fig. 3(b). It is also important to note that since more than 100 pA of photocurrent is observed from TM3, drifts in the XUV flux can also be normalized using this in-situ monitor, as is commonly done at synchrotrons. At the time of writing, we have run the source on a near-daily basis without venting the vacuum system or performing any alignment of the in-vacuum optics for more than 2 months with stable and reproducible results, enabling the photoelectron spectroscopy experiments discussed in Sec. III. For pump-probe experiments, it is often advantageous to use lock-in detection to extract small signals from large backgrounds. Figure 4(b) shows the amplitude noise (relative intensity noise, RIN) of the 23rd harmonic from Kr measured using the photodiode current amplified with a transimpedance amplifier and recorded with an FFT spectrum analyzer. For frequencies above 400 Hz, the RIN level is below À60 dBc/Hz, which can enable small differences in the photoelectron spectra to be recorded via lock-in detection. At this noise level, EDC or ARPES signals up to 10 6 counts/s/bin can be photoelectron-shot-noise limited with proper correction for drift using the TM3 photocurrent. III. PHOTOEMISSION Photoelectron spectroscopy measurements are performed under ultra-high vacuum conditions in a surface science endstation equipped with a hemispherical electron energy analyzer (VSW HA100). The analyzer is specified to have an angular acceptance of 64 at the input and angle-integrated spectra are recorded from a channeltron detector at the exit as voltages scan the electron kinetic energy. The endstation is also equipped with a sputter gun, a LEED, a quadrupole mass spectrometer, an Al Ka x-ray source, and a sample manipulator that can be cooled and heated between 100 and 1000 K. Also mounted on the sample manipulator are the Ce:YAG scintillator and pinhole mentioned in Sec. II. For all data presented here, the sample is oriented normal to the analyzer axis and 45 to the XUV beam. The electric field vector of the XUV light is in the plane of incidence (p-polarized) and the analyzer axis. Figure 5 shows photoelectron spectra from an Au (111) surface at 100 K temperature obtained using each harmonic between the 7th (h ¼ 8.4 eV) and 33rd (h ¼ 39.5 eV). Each spectrum was acquired with 34 meV steps individually measured with 1 s of integration for a total scan time of $6 min or less. At the d-band peaks, the electron count rates can exceed 1 MHz. These static spectra are in good agreement with those recorded by Kevan et al. 54,55 using tunable synchrotron radiation. The clearly visible dispersion of the d-bands at binding energies between 3 and 7 eV and the large photon energy dependence of the relative amplitudes of the peaks highlight the importance of conducting photoemission experiments with a tunable source. The same final state effects also strongly influence time-resolved photoelectron spectra of excited states and tunability here should be considered no less important, as has been emphasized by previous authors. 56 The resolution of the setup can be determined by analyzing the sharpness of the Fermi edge and is dominated by the energy analyzer. Fermi edge widths as low as 110 meV are measured depending on alignment. The best resolution we have been able to observe in any photoemission experiment using this analyzer is 89 meV using a He I lamp and a Kr gas target. From this data, we can place a conservative upper limit on the single harmonic photon energy bandwidth of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð110 meVÞ 2 À ð89 meVÞ 2 q ¼ 65 meV (details in supplementary material). We have reason to believe that the single harmonic linewidth is lower than this. The single harmonic linewidth from HHG is determined by the duration of time over which harmonics are generated with comparable efficiency and at the same frequency, or the emission window. Usually in HHG systems, the emission window is determined by ionization gating, where phase-matching is lost due to the free-electron plasma generated in field ionization. 5,35,57 The plasma phase shifts on the fundamental pulse gate the harmonic emission when they approach $p/q, where q is the harmonic order. However, in CE-HHG, because the power enhancement relies on the constructive interference of the pulse in the cavity with pulses coming from the laser, time-dependent phase shifts are restricted to be less than 2p=F , 58 roughly one order of magnitude smaller than the scale relevant for ionization gating. Under these conditions, the emission window is determined by the high harmonic dipole, d q (t), and not ionization gating, such that harmonic linewidths significantly narrower than achieved in single-pass HHG systems under their typical operating conditions are expected. Indeed, Mills et al. 27 have reported single harmonic linewidths of 32 meV from a cavity-enhanced HHG source with similar parameters to ours, even starting with shorter fundamental laser pulses. In our setup, more indication that the resolution of the photoelectron spectra reported here is limited by the hemispherical analyzer comes from experiments where we have intentionally lengthened the intracavity IR pulses by detuning the comb/cavity coupling 45 and observed no changes in the EDC. In the absence of ionization gating, increasing the pulse duration increases the emission window. Further investigation of the attainable energy resolution and its potential manipulation will be the subject of future work as we upgrade our electron energy analyzer. Please see the supplementary material for additional discussion and calculations regarding the HHG physics. The color indicates the gas used to generate the harmonic; Ar (black), Kr (red), and Xe (blue). Each EDC is normalized to the photocurrent measured at TM3, and spectra taken with photon energies above 25.1 eV have been enlarged by Â5. Even with the current analyzer-limited resolution, we demonstrate here that the absence of space-charge allows for time-resolved photoemission experiments that are both qualitatively and quantitatively different than what is done with space-charge limited systems. Figure 6 shows two photoelectron spectra near the Fermi edge of the Au (111) on a logarithmic scale, one with and another without a parallel polarized 1.035 lm wavelength laser excitation overlapped in space and time. The spectra were taken with 3 nA of sample current, or approximately 215 electrons/pulse. Consider first the black curve taken with the pump laser off. For a 100 kHz system with our spot size (or even somewhat larger), this sample current would result in broadening and shifting of the Fermi edge on the eV scale instead of the Շ10 meV effects here estimated using Eq. (1). Furthermore, on a logarithmic scale, space charge effects can cause long high energy tails in the photoelectron spectrum 10,21,25 that make it difficult to observe small signals from weakly excited samples. Here, with excellent harmonic isolation from our pulse-preserving monochromator and the absence of space charge effects, a precipitous drop of four orders of magnitude is observed in the EDC at the Fermi edge. Next, consider the red curve taken with the pump laser on. Before discussing the laserinduced features of the spectrum, consider first what is not observed. The photoelectron spectrum is not shifted, broadened, or distorted due to space charge produced by the laser excitation as is commonly observed in pump probe experiments. 21,[23][24][25] This is because the high data rate enables the experiment to be performed under the low pump intensity of 1.3  10 9 W/cm 2 . We measured the sample current from the pump excitation alone and found it to depend strongly on the region of sample probed, as in Ref. 21, but always at least one order of magnitude less than the XUV probe, or less than 22 electrons/pulse produced by the pump. The reflectivity of the gold sample is >97% and the laser-induced features in the EDC are dominated by the now well-known laser-assisted photoelectric effect (LAPE). 21 Briefly, LAPE is a dressing of the ionized electrons by the IR laser field producing sidebands at intervals of the photon energy. 21,[59][60][61] In Fig. 6(a), a sideband of the surface state peak at 24.8 eV is observed 1.2 eV higher at 26 eV. Figure 6(b) shows a measurement of the sideband amplitude at 26.0 eV kinetic energy as a function of the time delay between the IR pump and XUV probe. A cumulative integration time of 7.5 s for each time delay with 10 fs size steps results in a total scan time of 10 min. Within statistical error, identical widths and time-zero positions are observed for cross correlations taken at both higher and lower kinetic energies, further confirming the LAPE mechanism, as hot electrons closer to the Fermi energy would show observable lifetimes. 62,63 LAPE can be used to determine the time resolution of the instrument. A Gaussian fit to the cross-correlation in Fig. 6(b) gives a FWHM of 181 fs. The pump laser pulse duration at the sample position was not independently measured for this experiment, but at the output of the laser the pulse duration was measured to be 165 6 10 fs and optimal compression gives 155 fs. 44 Taking the lowest possible value of the laser pulse duration then gives a conservative upper limit for the XUV pulse duration at the sample of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð181 fsÞ 2 À ð155 fsÞ 2 q ¼ 93 fs. Figure 6(c) shows amplitudes for the sideband at 26 eV kinetic energy for different pump pulse intensities obtained from fits to time-resolved scans as shown in Fig. 6(b), but with a 20 fs time delay step and 5 s integration time. The sideband amplitude is observed to be linear in the laser intensity with a slope of 1.34 6 0.13  10 À12 cm 2 /W, and in excellent agreement with theory (1.3  10 À12 cm 2 /W) for our laser and experimental geometry, as calculated in Appendix B. Even with the modest time of 3.3 min used to acquire each data point in Fig. 6(c), sideband amplitudes as small as 6  10 À4 with an uncertainty of 1  10 À4 are easily observed. We also note that with a multichannel electron analyzer, the delay-dependence for the full energy window of Fig. 6(a) (or larger) could be obtained in parallel with no increase in data acquisition time. IV. CONCLUSIONS Pump-probe experiments in condensed matter physics can broadly be divided into nonperturbative "high-fluence" and perturbative "low-fluence" experiments. 22 High-fluence experiments study photo-induced phase transitions or non-thermally accessible meta-stable states. Low-fluence experiments, where the system is excited as "gently as possible," 22 study phenomena such as exciton dynamics or electron-photon coupling. While ARPES using tunable synchrotron radiation is often the method of choice for studying a material's electrons in their ground states, excited-state XUV ARPES studies have been limited to the high-fluence regime, where optical excitation produces substantial changes to the EDC visible on a linear scale. Currently, to study dynamics in the low-fluence regime, workers in the field turn away from the clarity and fidelity of XUV ARPES and instead pursue other methods that have the required sensitivity but are limited in scope and harder to interpret. For example, low-fluence experiments are typical for the more sensitive techniques of optical spectroscopy, 64 two-photon photoemission, 65 and laser-based ARPES using 6 eV probe light 66 but have been extremely difficult using space-charge limited HHG systems. In this work, we have shown how with high repetition rate, small laser-induced modifications of the photoelectron spectrum can be recorded without space-charge artifacts in reasonable acquisition times. We have observed LAPE with sideband amplitudes in the 10 À3 to 10 À4 range, orders of magnitude smaller than typical. 19,21,67 To our knowledge, these are the smallest LAPE signals observed from a surface, but more importantly they mimic the response of weakly excited samples in the low-fluence regime where only a small fraction (i.e., 10 À4 of the surface state) of the sample's electrons are excited, demonstrating the feasibility of low-fluence excited state XUV ARPES experiments. It is important to note that with the appropriate photoelectron analyzer, the 88 MHz repetition rate of the XUV pulses also does not preclude high-fluence experiments. The photoelectrons are emitted from the sample in discreet bunches which can be either detected individually using time-of-flight (TOF) methods or gated using time-dependent voltages in the analyzer. 68 Thus, if it is desirable to excite the sample at a lower repetition rate for high-fluence experiments or long-lived sample excited states, one can record photoelectrons from both short (fs) and long (ns) pump-probe delays simultaneously via time-resolved detection of the electrons. This has been implemented in synchrotron experiments. 69,70 In this way, the repetition rate of the experiment is only limited by the recovery time of the sample, and one can tune the repetition rate to optimize the data rate for a given experiment. We also note that for excited state studies needing to record only a small energy window, the 88 MHz repetition rate is also ideally suited to state-of-the art TOF photoelectron analyzers, which can achieve orders-of-magnitude improvements over hemispherical analyzers. 1, 68 The trade-offs between sample current, repetition rate, energy window, and resolution are discussed in greater detail in the supplementary material. Whereas space-charge sets a fundamental limit on most HHG-based photoemission instruments, the current time and energy resolution of the system do not represent any inherent limitations of the frequency-comb based methods used here and are instead limited simply by the laser pulse duration and energy analyzer performance. Both of these are straightforward to improve. For example, in our setup, sub-100 fs resolution could be obtained by leaving the XUV probe arm unchanged and implementing nonlinear pulse compression in the pump arm, 71,72 which has no demands on the temporal coherence of the pulse train. CE-HHG can also be performed with shorter driving pulses, 73,74 if desired. Single-grating pulse preserving monochromators have been shown to be compatible with temporal resolutions as small as 8 fs. 75 Returning to Fig. 1, we use the conservative upper limit of 65 meV to compare the performance of our system against previous time-resolved photoemission work. 12,14,[27][28][29][30][31][32] As can be seen, the present work enables a dramatic improvement over space-charge limited systems operating at lower repetition rate. Notable also is the work of Jones and co-workers 27 who have demonstrated to our knowledge the best resolution in HHG-based ARPES using a fixed photonenergy CE-HHG platform based on grating output coupling. 76 However, the grating output coupling method makes dynamic harmonic selection difficult and also introduces larger pulse front tilts than the pulse-preserving monochromator, resulting in larger focused spot sizes and XUV pulse durations. In this work, we have developed a light source that, at the sample, reproduces the parameters of tunable synchrotron radiation critical for the success of data-intensive surface photoemission experiments, but with pulse durations $1000 times shorter, effectively eliminating the data-rate limitations imposed on previous efforts by space charge. Upgrading the electron analyzer to a multi-channel version will allow frames of time-resolved ARPES measurements to be accumulated at rates comparable to or larger than that attained synchrotron beamlines. enabling experiments from perturbatively excited samples and a much fuller exploration of the multidimensional data set. SUPPLEMENTARY MATERIAL See supplementary material for the data used to determine the experimental energy resolution, an estimation of harmonic linewidth in cavity-enhanced HHG, and a discussion of the suitability of this XUV source for use with modern TOF energy analyzers. Both the development of HHG sources and their application to surface photoemission have been very active fields of research. 12,14,[27][28][29][30][31][32]35,77 To our knowledge, this work represents the first tunable HHG source applied to photoemission with nA sample currents capable of achieving sub-100 meV resolution in a spot size less than 1 mm. In Fig. 1, we have attempted to compare this work to previous published efforts applying HHG to sub-ps time-resolved surface photoemission. A time resolution of 1 ps corresponds to a 2 meV Fourier limit to the energy resolution. The following criteria were made in choosing which results to include the following: (1) Only systems with photon energy greater than 10 eV were included. (2) Only systems capable of achieving more than 1 pA of sample current were included. (3) Only systems driven by lasers with sub-ps pulse duration, such that they could, in principle, perform sub-ps time-resolved experiments, were included. (4) Only systems with published reports of being applied to photoemission have been included. For the energy resolution of photoemission experiments, several authors have shown that space charge broadening adds in quadrature with the photon energy bandwidth 10,12 DE ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DE 2 b þ DE 2 h q ; (A1) with DE b being the broadening due to space-charge and DE h being the photon energy bandwidth. The black dashed lines in Fig. 1 For space-charge limited systems (black edges), the (x,y) positions represent the case where the sample current is such that DE b ¼ DE h , and the energy resolution is broadened by a factor of ffiffi ffi 2 p . Since achieving resolution equal to the photon energy bandwidth requires substantially reduced sample current, this represents a practical compromise. The y-ordinate is calculated I sample ¼ ef rep DDE h =m b , with D ¼ 1 mm. The x-ordinate is then ffiffi ffi 2 p times the published photon energy bandwidth. For ARPES, experiments are usually run with spot sizes smaller than 1 mm, 78 such that even lower sample currents than plotted on Fig. 1 are required to maintain energy resolution. Furthermore, we comment that for real-space imaging with photoemission electron microscopy, the space-charge constraints are even more severe. 79,80 APPENDIX B: THEORY FOR LAPE AMPLITUDE The laser-assisted photoelectric effect can be considered the result of dressing of the freeelectron wavefunction. 21,59,61 In the limit of electron kinetic energies much larger than the dressing laser photon energy and ponderomotive energy much less than the photon energy, the amplitude of the nth sideband (A n ) becomes (in atomic units) A n ¼ J 2 n p Á E 0 x 2 ; (B1) where p is the vector momentum of the electron, E 0 is the laser electric field vector amplitude at the surface, and x is the laser frequency. The geometry of the experiment influences observed sideband amplitudes in two ways. First, energy can only be transferred between the laser field and the electron at the surface, such that for metallic surfaces only the component of the electric field normal to the surface contributes to LAPE. 21 Second, due to the dot product in Eq. (B1), only the component of the electric field along the detection direction contributes. For our geometry, with the sample oriented 45 to incident p-polarized beam and electrons detected along the surface normal, these factors are one and the same, and in the limit that the argument of the Bessel function in (B1) is much less than one we have (in SI units) A 1 % 4paIE kin m e hx 4 cos 2 45 ; where a is the fine structure constant, I is the laser intensity ignoring the effects of the surface on the laser electric field, E kin is the kinetic energy of the electron, m e is the mass of the electron, and h is Planck's constant. In the experiment, the laser intensity at which free electrons are generated varies in both space and time due to the finite extent of the XUV and laser beams. The observed sideband amplitude will thus be the space-time average where I peak is the peak intensity of the laser, G laser ðx; y; tÞ is a 3D Gaussian envelope function for the incident laser beam with unit amplitude, and G XUV ðx; y; tÞ is a normalized 3D Gaussian envelope function for the XUV beam. Evaluating the space-time overlap integral with 1/e 2 radii w x;laser ¼ 150 lm; w y;laser ¼ 150 lm; w x;XUV ¼ 49 lm; w y;XUV ¼ 85 lm, and FWHM pulse durations T laser ¼ 165 fs and T XUV ¼ 93 fs, the integral is 0.72 and the theoretical estimate for the observed sideband amplitude is hA 1 i ¼ 1:3  10 À12  I peak ½W=cm 2 . hA 1 i ¼ 2paE FIG. 1 . 1Constraints on sample current and energy resolution due to space charge effects. Dashed lines are created by evaluating Eq. FIG. 4 . 4(a) Normalized XUV flux measured with PD after the monochromator for harmonics 15-25 from Kr over 1 h without human intervention. (b) Relative intensity noise (RIN) of the 23rd harmonic (red), intracavity laser light (green), and Yb:fiber laser (blue), along with the detector noise floor (black dashed). FIG. 5 . 5Static photoelectron spectra of an Au (111) surface taken with harmonics 7 through 33, vertically offset for clarity. FIG. 6 . 6(a) The photoelectron spectrum of the Au (111) Fermi edge taken without (black) and with a 1.035 lm pump pulse (red) at a peak intensity of 1.3  10 9 W/cm 2 . A LAPE sideband of the surface state peak at 24.8 eV is observed at 26 eV. (b) The magnitude of the sideband at a kinetic energy of 26 eV as a function of pump probe time delay. The crosscorrelation has a FWHM of 181 fs. (c) The amplitude of the sideband at 26 eV as a function of pump peak intensity. 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There is an ongoing debate in the literature about whether the present global warming is increasing local and global temperature variability. The central methodological issues of this debate relate to the proper treatment of normalised temperature anomalies and trends in the studied time series which may be difficult to separate from time-evolving fluctuations. Some argue that temperature variability is indeed increasing globally, whereas others conclude it is decreasing or remains practically unchanged. Meanwhile, a consensus appears to emerge that local variability in certain regions (e.g. Western Europe and North America) has indeed been increasing in the past 40 years. Here we investigate the nature of connections between external forcing and climate variability conceptually by using a laboratory-scale minimal model of mid-latitude atmospheric thermal convection subject to continuously decreasing 'equator-to-pole' temperature contrast ΔT, mimicking climate change. The analysis of temperature records from an ensemble of experimental runs ('realisations') all driven by identical time-dependent external forcing reveals that the collective variability of the ensemble and that of individual realisations may be markedly different -a property to be considered when interpreting climate records.To quantify connections between climate change and the temporal variability of a climate index the typical procedure researchers follow is comparing its recently observed fluctuations to those from a base period 1-9 . This approach is inherently built on the naïve assumption of ergodicity, a property that does not apply to far-from-equilibrium processes. In 'climate-like' nonlinear, evolving systems the only way to acquire appropriate expectation values-as 'climate is what you expect, weather is what you get' 10 -would be ensemble averaging over a multitude of parallel realisations of the system's response to the same time-dependent forcing, all obeying the same physical laws and differing only in their initial conditions. It is to be emphasized that differences between the ensemble members represent an inherent property of the problem, internal variability, and cannot only be associated with 'measurement errors' . The ensemble average of the paths of such parallel realisations in the space of essential variables would then trace out a time-evolving, so-called snapshot-or pullback-chaotic attractor 11, 12 . It seems quite appropriate to adapt this approach to the description of any highly nonlinear chaos-like process, like e.g. turbulence.The concept's applicability in climatology has been demonstrated in numerical models ranging from minimal models 12-14 to intermediate complexity GCMs 15 , concluding that the snapshot attractor framework provides the only self-consistent definition of 'climate' from the dynamical systems point of view. Obviously, for the actual Earth system only a single observable realisation exists but experiments in a laboratory characterised by 'climate-like' externally forced dynamics can be repeated multiple times and thus provide a real world test-bed for this approach, whose evaluation has so far been limited to numerical investigations.The tabletop-size rotating, differentially heated annular wave tank we use for this purpose is a widely studied experimental minimal model of the mid-latitude Earth system[16][17][18][19](Fig. 1a, Methods). It captures the two essential components of large-scale atmospheric circulation: lateral ('meridional') temperature difference and rotation. The working fluid (de-ionised water) is located in the annular cavity between two vertically aligned co-axial cylindrical side walls: the one at the center (simulating the North Pole) is cooled, whereas the rim (representing the equator) is heated with computer-controlled thermostats. The tank is mounted on a turntable and rotates around its axis of symmetry. The adjustable parameters (fluid depth, rotation rate, temperature contrast) are set Published: xx xx xxxx OPEN www.nature.com/scientificreports/

10.1038/s41598-017-00319-0

1702.07048

3faf50265dbb570cba17743cb3252a2e31cba115

Temperature fluctuations in a changing climate: an ensemble- based experimental approach OPEN Published: xx xx xxxx Miklós Vincze [email protected] von Kármán Laboratory for Environmental Flows Eötvös University H-1117BudapestHungary MTA-ELTE Theoretical Physics Research Group H-1117BudapestHungary Ion Dan Borcia Department of Aerodynamics and Fluid Mechanics Brandenburg University of Technology Cottbus-Senftenberg D-03046CottbusGermany Uwe Harlander Department of Aerodynamics and Fluid Mechanics Brandenburg University of Technology Cottbus-Senftenberg D-03046CottbusGermany Temperature fluctuations in a changing climate: an ensemble- based experimental approach OPEN Published: xx xx xxxx10.1038/s41598-017-00319-0Received: 28 September 2016 Accepted: 20 February 20171 Scientific RepoRts | 7: 254 |. Correspondence and requests for materials should be addressed to M.V. ( There is an ongoing debate in the literature about whether the present global warming is increasing local and global temperature variability. The central methodological issues of this debate relate to the proper treatment of normalised temperature anomalies and trends in the studied time series which may be difficult to separate from time-evolving fluctuations. Some argue that temperature variability is indeed increasing globally, whereas others conclude it is decreasing or remains practically unchanged. Meanwhile, a consensus appears to emerge that local variability in certain regions (e.g. Western Europe and North America) has indeed been increasing in the past 40 years. Here we investigate the nature of connections between external forcing and climate variability conceptually by using a laboratory-scale minimal model of mid-latitude atmospheric thermal convection subject to continuously decreasing 'equator-to-pole' temperature contrast ΔT, mimicking climate change. The analysis of temperature records from an ensemble of experimental runs ('realisations') all driven by identical time-dependent external forcing reveals that the collective variability of the ensemble and that of individual realisations may be markedly different -a property to be considered when interpreting climate records.To quantify connections between climate change and the temporal variability of a climate index the typical procedure researchers follow is comparing its recently observed fluctuations to those from a base period 1-9 . This approach is inherently built on the naïve assumption of ergodicity, a property that does not apply to far-from-equilibrium processes. In 'climate-like' nonlinear, evolving systems the only way to acquire appropriate expectation values-as 'climate is what you expect, weather is what you get' 10 -would be ensemble averaging over a multitude of parallel realisations of the system's response to the same time-dependent forcing, all obeying the same physical laws and differing only in their initial conditions. It is to be emphasized that differences between the ensemble members represent an inherent property of the problem, internal variability, and cannot only be associated with 'measurement errors' . The ensemble average of the paths of such parallel realisations in the space of essential variables would then trace out a time-evolving, so-called snapshot-or pullback-chaotic attractor 11, 12 . It seems quite appropriate to adapt this approach to the description of any highly nonlinear chaos-like process, like e.g. turbulence.The concept's applicability in climatology has been demonstrated in numerical models ranging from minimal models 12-14 to intermediate complexity GCMs 15 , concluding that the snapshot attractor framework provides the only self-consistent definition of 'climate' from the dynamical systems point of view. Obviously, for the actual Earth system only a single observable realisation exists but experiments in a laboratory characterised by 'climate-like' externally forced dynamics can be repeated multiple times and thus provide a real world test-bed for this approach, whose evaluation has so far been limited to numerical investigations.The tabletop-size rotating, differentially heated annular wave tank we use for this purpose is a widely studied experimental minimal model of the mid-latitude Earth system[16][17][18][19](Fig. 1a, Methods). It captures the two essential components of large-scale atmospheric circulation: lateral ('meridional') temperature difference and rotation. The working fluid (de-ionised water) is located in the annular cavity between two vertically aligned co-axial cylindrical side walls: the one at the center (simulating the North Pole) is cooled, whereas the rim (representing the equator) is heated with computer-controlled thermostats. The tank is mounted on a turntable and rotates around its axis of symmetry. The adjustable parameters (fluid depth, rotation rate, temperature contrast) are set Published: xx xx xxxx OPEN www.nature.com/scientificreports/ to yield approximate dynamical similarity to the terrestrial atmosphere in terms of thermal Rossby number, Ro T , and Taylor number, Ta (Fig. 1c, Methods) 16,17 . We log simultaneously (sampling rate 1 Hz, differential resolution 0.05 K) point-wise local temperature values via five digital co-rotating thermometers, three of which penetrates into the free top surface of the working fluid cavity from above, spaced uniformly along a radius (Fig. 1b). The spatial average, T i (t), of these signals from three different 'latitudes' is used here as a surrogate for 'meridional' mean temperature (index i refers to the i-th ensemble member, i.e. experimental run). Since there is no azimuthal bias in the annulus -as there is, e.g. in the terrestrial atmosphere, due to land-ocean differences, topographical effects, etc. -we would expect the statistical properties of temperature fluctuations to be the same at different azimuths. Thus, it is safe to assume that such a longitudinal average can also be considered a proper surrogate of the global average. The other two identical sensors measure the forcing temperatures at the center (inner cylinder) and in the outer sidewall, whose difference ΔT quantifies the temperature contrast driving the sideways convection. The novelty of our experiments lies in the procedure of intentionally changing the thermal boundary conditions in time, while keeping the rotation rate fixed (so that a 'day' i.e. one revolution of the tank takes P = 3 s). After a 'base period' of ca. 2600 revolutions of constant ΔT the cooling element at the center is turned off. Following this abrupt change in heat flux T i (t) is kept logged for another 3000 revolutions of time, corresponding to a 'global warming' scenario with gradually increasing polar temperatures. We note, that it is generally accepted that the North-South temperature contrast has been decreasing (and will continue to decrease) in the Northern Hemisphere due to climate change as reported, e.g. in the latest assessment report of IPCC 20 . The recent alarming findings 21 about the rapidly melting Arctic also underline the existence of this phenomenon, showing twice as fast warming of the Arctic as that of the global mean. Results Based on our criteria for the external forcing sequence ΔT i (t) to be accepted as 'identical' (Methods) the analysis was restricted to nine experimental runs and 10 000 s of continuous data from each of them with the onset of 'climate change' (hereafter marked as time zero, t = 0) occurring exactly at half time in all cases. The forcing ΔT(t) in each considered realisation (Fig. 2a) follows an exponential decay with characteristic timescale τ = 1085 s for t > 0. The system's response T i (t) in each run, and even their ensemble average 〈T〉(t) shows significant fluctuations ( Fig. 2b and c) due to the geostrophic turbulent flow dominated by irregular cyclonic (warm) and anticyclonic (cold) vortices 18 . Addressing variability in the system we first demonstrate the difference between the 'traditional' measuresbased on single realisations -and the ensemble statistics through the example of standard deviations. We find that the centered running variances (within 501 s long windows) of the residuals of T i (t) following a 4-degree polynomial detrending in the different realisations may exhibit seemingly opposite tendencies (Fig. 3a), and are thus not representative. In the two chosen paths, one reaches the largest variability in the t < 0 base period. Although the Figure 1. Thermal convection in planetary atmospheres and in the laboratory. (a) Schematic diagram of the mid-latitude atmosphere of Earth, illustrating the basic boundary condition with a meridional temperature contrast ΔT between the warm equator (red) and a polar region (blue). The system is rotating at angular velocity Ω. (b) Sketch of the differentially heated rotating annulus with its geometric parameters (a = 4.5 cm, b = 12 cm, d = 4.5 cm) for which the boundary conditions are similar to those of the real atmosphere: warm outer rim (red), cold inner rim (blue). The locations of the three co-rotating thermometers, which were submerged by 0.5 cm into the bulk from above the water surface are also shown (black dots). The average of these signals at each time t yielded 'meridional mean temperature' T(t). (c) Schematic regime diagram for rotating laterally heated systems 16,17 statistical comparison of the t < 0 and t > 0 intervals yields no significant difference, the mean and median indeed are somewhat smaller in the latter case (not shown). Thus -if only this particular record was known -one could speculate that the fluctuations of temperature generally decreased in the 'climate change' phase compared to the base period. The other exemplary case shows just the opposite trend: a slight, statistically insignificant increase in mean variability after t = 0. Meanwhile, in terms of the ensemble variance σ e , i.e. the standard deviation of the nine considered realisations T i (t)(i = 1, …, 9) around 〈T〉(t) at each time instant t (Fig. 3a), the system's real sensitivity to changing ΔT is revealed (Fig. 3b). The mean of σ e (t) shifts significantly by ca. 6.5% from 0.35 to 0.38 °C at t > 0 and, more strikingly, the histogram changes from left-modal (skewness: 0.37) to right-modal (skewness: −0.10) after the initiation of 'climate change' . This result indicates that the paths of the realisations differ from each other more in the presence of nonstationary forcing than in the base period: even if the transition is hardly noticable in the variance patterns of one single realisation its effect on the whole ensemble is apparent. The typical time difference τ c between successive local extrema of the fluctuating temperature records T i (t) serves as a measure of the temporal variability of the 'weather' in the system. The local maxima (minima) indicate the crossing of cyclonic (anticyclonic) eddies at the thermometer locations. We calculate the peak-to-peak time differences for each ensemble members, after a removal of a 5th order polynomial trend and applying a 61-point running mean for smoothing. The statistics of the obtained values of τ c combined from all experimental runs shows a significant shift when comparing the t < 0 and t > 0 periods; the mean increased from 199.2 s to 214.7 s (by around 8%), the median from 183 s to 209 s (by around 14%). This finding is consistent with the theoretical expectations: smaller ΔT yields cyclonic and anticyclonic eddies of smaller size, scaling with the so-called Rossby deformation radius L R , i.e. proportionally to the square root of the imposed temperature gradient 22 : ∝ ∆ L T R . Whereas the drift velocity c of baroclinic eddies is determined by the thermal wind balance and scales as c ∝ ΔT μ , where the exponent μ has been found to be between 0.88 and 1.17 in earlier experiments 19,23 , μ = 1 being the theoretical value. Thus, the crossing timescale is expected to follow a τ c ≈ L R /c ∝ ΔT 0.5−μ dependence, yielding an increasing trend with decreasing ΔT(t) in time. Even in this respect, single-realisation statistics could be misleading: due to the (geostrophic) turbulent nature of the flow the values of τ c exhibit large variance in all cases that can easily suppress the slight trend. Further exploring temporal correlations we apply the method of detrended fluctuation analysis (DFA) 24, 25 , a strandard procedure for measuring the variability of a signal around its local trend in time windows of length n samples as a function of n. DFA4 removes local (cubic) trends, thus it is more suitable for our present purpose than e.g. Fourier transforms, since DFA4 can readily handle nonstationarities (Methods). The DFA4 spectra from all realisations (Fig. 4a) follow the same scaling properties, exhibiting power law-type scaling with two scale breaks. Below t n ≈ 40 s and above t n ≈ 400 s the scaling exponents are δ = 0.87 and 1.1, respectively, implying 1/f noise-like correlated fluctuations. Between these crossover points δ = 2.1 is found, characteristic for geostrophic turbulence 26,27 : it can be shown that if the DFA4 spectra exhibit power-law scaling then the Fourier power spectrum of the time series in the frequency domain, S(ω) also does, following S(ω) ∝ ω −β , where β = 2δ − 1 connects the two exponents 28 , yielding in the present case, β ≈ 3. This is in good agreement with the theoretical result for isotropic geostrophic (two dimensional) turbulence 29 . It is to be noted that the ΔT-dependence of the exponent β has been analyzed via comparing the ensemble-averaged power spectra of different (overlapping) sections of the time series T i (t), but no trend could be established (for more details, we refer to the Supplementary information). Thus, it can be stated that the 'quality' of geostrophic turbulence did not change throughout the 'climate change' period. Concerning the differences between the stationary (t < 0) and 'changing' (t > 0) records (turquoise and orange curves in Fig. 4a, respectively), their fluctuations up to a window size t n * ≈ 160 s are perfectly identical in the statistical sense. This is also apparent from the averages of the two sets of spectra in Fig. 4a (red and black thick curves). On the t n > t n * scale, however, the fluctuations of the 'changing' records are significantly larger. Note, that this timescale is still about an order of magnitude below τ = 1085 s i.e. the characteristic time of the 'climate change' ΔT(t > 0), but is of the same order as the empirical delay time of ∼200 s of the dynamics estimated from the crossover point of the linear temperature trends of 〈T〉(t) in the two periods (Fig. 2d). Also shown are the DFA4 spectra of the ensemble averages 〈T〉(t < 0) (blue line) and 〈T〉(t > 0) (green line) following the same scaling and the same separation of the stationary and 'changing' branches at t n * , as discussed above. Multiplying the fluctuation spectra of the ensemble averages by = N 3 (N = 9 being the sample size of the ensemble) yields perfect match with the average of the single-realisation spectra. This property shows that the fluctuations of different realisations are perfectly uncorrelated on all time scales t n < τ: uncorrelated fluctuations average out following N 1/ , whereas 'ensemble-correlated' fluctuations would remain unaffected by the ensemble averaging. Here the latter are absent; no 'collective variability' can be identified in the ensemble, despite of the identical forcing sequence ΔT(t). Obviously, on the time scale of τ collective behaviour does exist -the trend itself -but such large time windows are not sampled properly and are not evaluated in the spectra. The lack of collective fluctuations on the sub-τ scales highlights the largely nonlinear nature of the system's response to changing forcing. The time-scale-dependence of variability amplification caused by 'climate change' is visualized in Fig. 4b, where the ratios of DFA4 fluctuation spectra in the 'changing' phase relative to the 'base period' of the same run -and those of the ensemble average -are plotted. (Due to the logarithmic vertical axis this practically reflects the differences of the respective graphs in Fig. 4a). Here again it becomes manifest that 'climate change' does not affect the variability on the time scales below t n * from the ensemble average point of view (the average amplification is close to zero), still, one can also easily spot individual realisations with either markedly increased or decreased variability in this spectral band as well. Above t n * all realisations exhibit clearly amplified variability. For the ensemble average it reaches a maximum increase of 47% at around t n * and stays around 20% for the t n > t n * timescales up to t n ≈ 800 s. To determine the statistical significance of the above results, we have carried out Monte Carlo statistical testing using a standard inverse-Fourier surrogate data method 30 (Methods). The null-hypothesis of the testing is that there are no fundamental changes in the dynamics of fluctuations between the t < 0 'base period' and the t > 0 'warming phase' . If this was the case, the fluctuations during the warming would exhibit very similar distribution and spectral properties as in the base period, superimposed onto a warming trend. In order to model this hypothesis, 10 model 'warming' time series were created for each of the 9 ensemble members (i.e. 90 time series in total) using the Fourier amplitude spectra of their corresponding 'base periods' but shuffling their phases. The resulting model series were then superimposed onto a polynomial warming trend, imitating the temporal development of the ensemble average 〈T〉(t > 0), shown in Fig. 2a, to yield a realistic increasing trend (Methods). Comparing the DFA4 spectra of the model series to their respective base periods in the considered timescale range yields the 'amplification factors' shown with turquoise curves in Fig. 5. The green curves corresponding to the actual ensemble members and the thick black curve denoting the ensemble average are repeated from Fig. 4b. The red dashed curve shows the mean of the model results and the dotted curves represent the ±3σ interval. The vertical domain covered by the turquoise curves can be understood as a measure of variability that is due to finite-size effects and the imposed trend itself. It is apparent, however, that the measured ensemble data follow a markedly different distribution, thus the null-hypothesis in the considered timescale-range of t n > t n * can be rejected with a high confidence. Towards the larger timescales, comparable to the typical eddy-crossing times τ c and also to the characteristic time of baroclinic adjustment (as mentioned earlier), a clear increase of fluctuations can be observed, indicating real dynamical differences, not merely statistical artifacts. Discussion The present work provides, to the best of our knowledge, the first results from any laboratory experiment aiming to model the effects of climate change on mid-latitude atmospheric circulation. The authors do not claim that the lessons learned from the presented experimental minimal model could be directly applied or compared to the processes of the Earth system and the ongoing climate change. Perfect hydrodynamic similarity is impossible to achieve, thus the ratios between all of the relevant timescales (corresponding to the rotation, baroclinic adjustment, crossing time of cyclones, the changing of the temperature contrast ΔT) cannot be set to scale properly. Nevertheless, the studied model as a dynamical system does share some important features with the climate system on the conceptual level: both are rotating, turbulent hydrodynamic systems, driven by the incoming differential heat fluxes, a forcing that changes in time. Due to the time-dependence of the forcing, these systems cannot reach an equilibrium state. Therefore, if one intends to survey the variability between the possible outcomes of such a process at any time instant, it is essential to consider a whole ensemble of realisations, subject to the same forcing scenario and differing only in their initial conditions. Despite of the large variability of the ensemble that is due to the nonlinear nature of the processes and the finite length of the studied records, the fluid dynamical interpretation of the observed flow phenomena is relatively straightforward. The system is in the state of well-developed geostrophic turbulence, that yields a power-law scaling in the power spectra of the fluctuations in both the wavenumber-and the frequency domain (Supplementary). The characteristic size of the cyclonic and anticyclonic eddies (corresponding to warm and cold temperature anomalies, respectively) tends to decrease as the 'meridional' temperature gradient drops, in agreement with the theoretical expectations. In parallel, the zonal drift velocities decrease even faster during the process, therefore the characteristic timescale of 'weather change' at a fixed measurement location increases significantly. This timescale is of the same order as the typical response time of the flow to the changes in the forcing (baroclinic adjustment) therefore fluctuations were found to increase markedly in this spectral band. 'One experiment is no experiment' has been the mantra of researchers for ages, but the idea behind the saying has always been the separation of measurement errors from significant signals. Here, however, the fluctuations are just as inherent, fully deterministic and dominant features of the underlying nonlinear processes -just like in the Earth system -as the large-scale trends themselves. The reason for the increasing ensemble variance lies in the system's extreme sensitivity to initial conditions -a ubiquitous property of chaotic, long-range correlated systems. The authors firmly believe that the only proper approach for carrying out laboratory experiments on non-stationary turbulence would be conducting and systematically evaluating, ensembles of runs. In observational climatology this is not a viable option; we have only one Earth. Yet, the present experimental demonstration may help to increase awareness of the fact that a climate-like dynamical system can undergo a transition towards larger variability even without noticeable effects on the temporal fluctuations of one particular realisation. This message applies to the GCM community as well: climate variability information from a single numerical run (e.g. CO 2 doubling scenario) could be misleading as it does not necessarily represent the full complexity of the underlying ensemble dynamics. Methods Non-dimensional parameters, hydrodynamic similarity. In large-scale environmental flows Rossby number Ro ≡ U/(2|Ω|L) -with U being the magnitude of the horizontal flow velocity, L the horizontal extent of the domain and Ω the angular frequency of the planetary rotation -quantifies the characteristic ratio of hydrodynamic acceleration and Coriolis acceleration. In the dynamics of atmospheric convection the thermal boundary conditions and the relationship ρ(T) between the density and temperature of the fluid parcels are of fundamental importance just as well. A convenient nondimensional combination for quantifying all these factors is the thermal Rossby number Ro T (or Hide number), defined as α = ∆ Ω gd T L Ro (2 ) ,(1)T 2 2 where α is the coefficient of volumetric thermal expansion for the fluid, d is the vertical scale, and ΔT is the 'meridional' temperature contrast 22 . For our calculations the annular gapwidth b − a was taken as horizontal scale L for the experiments. Besides Ro T the kinematic viscosity ν of the medium also plays an important role in the dynamics; it introduces a 'viscous cutoff ' that dissipates too weak thermal winds and also damps the baroclinic instability of larger wavenumbers. This effect is parametrised by Taylor number Ta that accounts for the ratio of rotational and viscous effects, and reads as ν = Ω . Ta L d 4(2) 2 5 2 Ro T and Ta are used in tandem to characterize the different dynamical regimes in rotating, thermally driven systems, such as planetary atmospheres and their minimal models in the laboratory (Fig. 1c). Experimental procedures, data selection. For a detailed description of the experimental wave tank and the heating and cooling mechanism we refer to ref. 19. The temperature records were obtained using an ALMEMO temperature sensor array of NiCr sensors with a relative resolution of 0.05 K and 1 Hz sampling rate. The sensors were fixed onto a co-rotating mast above the free surface of the rotating annulus, and penetrated by 0.5 cm into the water surface. The data was transported in real-time via the co-rotating data aquisition module ALMEMO 8590-9, equipped with UHF/Bluetooth antenna. The initial temperature of the working fluid (de-ionised water) was set to 24.5 ± 0.5 °C before each measurement. After switching on the heating thermostats for the differential heating a transient period of 7600 s followed in order to reach quasi-equilibrium dynamics in each experimental run. Only after this period we started to log the data of the 5000 s long 'base period' . The nine experimental runs considered in this work were selected based on the criterion that the forcing time series ΔT(t) of each realisation must not deviate by more than 0.3 °C from the ensemble average 〈ΔT〉 at any time t (two of the original 11 experiments were thus excluded). The thermographic images of Fig. 2c were obtained by an InfraTec VarioCam infrared camera mounted above the set-up, operating in the spectral wavelength range of 7.5-14 μm. These thermograms can be considered to represent surface temperature structures, since the penetration depth of this wavelength range into water is less than a millimeter. The images were taken during an additional experiment following the same forcing sequence, but with the thermometers removed from the working fluid for the sake of visibility. Therefore this run was not a member of the ensemble. Detrended fluctuation analysis. DFAp 24, 28 is a robust and easily implemented analysis of the temporal scaling properties of a fluctuating and non-stationary bounded time series x t . Firstly a summation is applied to yield a cumulated (unbounded) time series X t : ∑ = −〈 〉 = X x x [ ] ,(3)t i t i 1 where 〈x〉 denotes the mean of the time series. Next, the profile is divided into non-overlapping time windows Y j of length n and for each a local least square polynomial fit ξ(p − 1) j of order p − 1 is calculated. Finally, the fluctuation is obtained as the root-mean-square deviation from the trend as where N is the number of n-sized windows of the time series. Note, that care must be taken to the fact that the congruence between N and the length of the time series is often not zero. To preserve the remaining section the applied algorithm repeats the same dividing procedure from the end of x t , thus, practically 2N segments are generated and the applied fluctuations are combined accordingly. We determined the DFAp fluctuation functions with p = 2 … 8 for the time series T i (t) and observed that no significant differences appear between the spectra for p > 4, therefore we limited our presentation of the results for the DFA4 computations only. ∑ ξ =    − −    = F n N Y p ( ) 1 ( ( 1) ) ,(4) Surrogate data for the statistical testing. The surrogate data for the model time series were generated using the method developed by Schreiber and Schmitz and described in ref. 30. The implementation of the algorithm is included in the open source software package TiSeAn 3.0.1 for nonlinear time series analysis 31 whose routine 'surrogates' have been used for the present work. The principle of the method is the following: if the null hypothesis was true, the typical realisations of the process are expected to share the same power spectrum and amplitude distribution, thus such model time series need to be generated. This is carried out iteratively by the following procedure from the prescribed distribution and Fourier spectra of the actual data. } is obtained. In a given iteration step, the shuffled data {x n (i) } is brought to the desired sample power spectrum by taking the Fourier transform of {x n (i) }, replacing the squared amplitudes {S k 2,(i) } by {S k 2 } and then transforming back. The phases of the complex Fourier components are kept. This step enforces the correct spectrum but usually the distribution will be modified. Therefore, in the next step the resulting series is rank-ordered to assume exactly the values taken by {x n }. Then, the spectrum of the resulting {x n (i+1) } will be modified again. These steps have to be repeated several times; at each iteration stage the remaining discrepancy between the obtained and the desired spectra and distributions is checked and the iterations continue until a given accuracy is reached. For finite N a convergence in the strict sense is not expected. Eventually, the transformation towards the correct spectrum will result in a change which is too small to cause a reordering of the values. Thus, after rescaling, the sequence is not changed. For each resulting model series, an increasing (5th order polynomial) warming trend was added. The properties of this warming trend were derived from fitting the polynomial formula to 〈T〉(t) in the 'climate change' period (t > 0). Thus, 90 model time series were obtained -10 for each ensemble member -inheriting the power spectra and the rank-ordering of the original corresponding base period (t < 0) data. Figure 2 . 2Temperature trends and fluctuations in the experiment ensemble. (a) Temperature difference ΔT as measured between the outer and inner cylindrical sidewalls of the annular tank in all runs are shown (green) alongside their ensemble average at each time t (black). (b) Time series T(t) for each experimental realisation, coloured turquoise in the base period and orange in the 'climate change' (t > 0) phase. One exemplary realisation of T(t) is repeated in red and shifted above by + 2.5 °C for better visibility. The ensemble average 〈T〉 (t) of all nine realisations (black solid curve) and the corresponding ensemble standard deviation ± 1σ e range (dotted black lines) is also indicated. (c) Two infrared thermographic snapshots (heat maps) of the surface temperature patterns (obtained in an additional control experiment) during the base period with temperature contrast ΔT = 11 K (left) and toward the end of the 'climate change' period (at t = 4640 s). Orange (blue) areas are warmer (colder) than average. (d) A blow-up of the ensemble average time series 〈T〉 (t) showing linear trend lines fitted to the t < 0 (dashed) and t > 0 (dotted) periods. Their crossing point is found at t = 212 s that serves as an empirical measure of the delay time of the system's response (baroclinic adjustment) to the abrupt change in the forcing at t = 0. Figure 3 . 3Variances and characteristic times of the 'global mean temperature' time series. (a) The collective standard deviation σ e of the ensemble at each time t (turquoise and orange curves) in the two periods (the colour coding is as inFig. 2b). For comparison, time series of the 501-point running standard deviations of two experimental realisations are also shown (black and red curves) calculated after detrending with 4th order polynomials in both cases. (b) Histograms of the ensemble standard deviation σ e for the base period (t < 0, turquoise) and the 'climate change' period (t > 0, orange). (c) Histograms of the peak-to-peak time differences τ c of the fluctuations of T i (t) in the two periods, determined after 61-point running averaging and 5th order polynomial detrending. Here data from all individual realisations T i are combined. The colour coding is as in panels (a) and (b). Figure 4 . 4Detrended fluctuation analysis of the 'global mean temperature' ensemble. the length of the window length in seconds). The spectra from the base period (turquoise curves) and from the 'climate change' period (orange) exhibit indistinguishable behaviour (involving a scale break around = . .e. t n * ≈ 160 s), where the two sets of graphs detach and those corresponding to 'climate change' reach larger fluctuations. The averages of the spectra are also plotted for the base period (black) and the 'changing' period (red). The DFA4 spectra of the ensemble average 〈T〉 (t) in the base period (blue) and the 'changing' period (green) show practically the same behaviour as the corresponding spectral averages. Upward shifting of these ensemble average spectra by ≈ . log-scale graph would yield practically identical curves to the aforementioned averages. (b) Amplification factors of the DFA4 fluctuations of the 'climate change' period with respect to the base period in each individual run (green curves) and in the ensemble average 〈T〉 (t) (thick black curve). Figure 5 . 5Significance testing of the amplification factors of the DFA4 fluctuations. The amplification of the DFA4 fluctuations of the ensemble members and of the ensemble average are repeated from Fig. 4b with the same color coding in the 100 ≤ n ≤ 1000 domain. The turquoise curves indicate the 90 surrogate model time series. The average of the model spectra (red dashed curve) and the upper and lower bounds of the ±3σ intervals (red dotted curves) are also plotted. Scientific RepoRts | 7: 254 | DOI:10.1038/s41598-017-00319-0 First, a sorted list of the values {x n } and the squared amplitudes of the Fourier transform of {x n }, obtained, where N is the number of data points. Then a random shuffle of the data (without replacement) {x n (0) , where ν denotes the kinematic viscosity of the medium (for the precise formulation of these nondimensional parameters see Methods). The main flow regimes are indicated and the approximate positions of three planetary mid-latitude circulations are also shown16 : The vertical arrow represents the trajectory of the dynamics in our experiment during the imposed 'climate change' scenario: Ta ≈ 9.18 × 10 8 stays constant, whereas thermal Rossby number decreases from Ro T ≈ 0.041 to Ro T ≈ 0.013.in terms of thermal Rossby number ∆ ∼ Ω Ro T/ T 2 and Taylor number ν ∼ Ω Ta / 2 2  Ro 1 T Venus ; Ro T Mars ≈ 0.2; Ro T Earth ≈ 0.06. Scientific RepoRts | 7: 254 | DOI:10.1038/s41598-017-00319-0 © The Author(s) 2017 AcknowledgementsThis research was funded by and conducted in the framework of the European High-performence Infrastructures in Turbulence (EuHIT) programme. U.H. and I.D.B. acknowledge financial support from the German Science Foundation (DFG) in the frame of the Research Unit MS-GWaves (HA 2932/8-1). The inspiring and fruitful discussions with Tamás Tél, Imre M. Jánosi and Anna Kohári and the comments from the two anonymous referees are highly acknowledged.Author ContributionsM.V. led the interpretation of results and the writing of the manuscript. I.D.B. and U.H. prepared and oversaw the experimental set-up. All authors reviewed the manuscript.Additional InformationSupplementary information accompanies this paper at doi:10.1038/s41598-017-00319-0Competing Interests: The authors declare that they have no competing interests.Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ Debate heating up over changes in climate variability. L Alexander, S Perkins, Environmental Research Letters. 841001Alexander, L. & Perkins, S. Debate heating up over changes in climate variability. Environmental Research Letters 8, 041001 (2013). A decade of weather extremes. D Coumou, S Rahmstorf, Nature Climate Change. 2Coumou, D. & Rahmstorf, S. A decade of weather extremes. Nature Climate Change 2, 491-496 (2012). Quasi-resonant circulation regimes and hemispheric synchronization of extreme weather in boreal summer. 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Cells can sense and respond to mechanical signals over relatively long distances across fibrous extracellular matrices. Recently proposed models suggest that long-range force transmission can be attributed to the nonlinear elasticity or fibrous nature of collagen matrices, yet the mechanism whereby fibers align remains unknown. Moreover, cell shape and anisotropy of cellular contraction are not considered in existing models, although recent experiments have shown that they play crucial roles. Here, we explore all of the key factors that influence longrange force transmission in cell-populated collagen matrices: alignment of collagen fibers, responses to applied force, strain stiffening properties of the aligned fibers, aspect ratios of the cells, and the polarization of cellular contraction. A constitutive law accounting for mechanically-driven collagen fiber reorientation is proposed. We systematically investigate the range of collagen fiber alignment using both finite element simulations and analytical calculations. Our results show that tension-driven collagen fiber alignment plays a crucial role in force transmission. Small critical stretch for fiber alignment, large fiber stiffness and fiber strainhardening behavior enable long-range interaction. Furthermore, the range of collagen fiber alignment for elliptical cells with polarized contraction is much larger than that for spherical cells with diagonal contraction. A phase diagram showing the range of force transmission as a function of cell shape and polarization and matrix properties is presented. Our results are in good agreement with recent experiments, and highlight the factors that influence long-range force transmission, in particular tension-driven alignment of fibers. Our work has important relevance to biological processes including development, cancer metastasis and wound healing, suggesting conditions whereby cells communicate over long distances. * Corresponding Author: [email protected] Long-range force transmission has significant relevance in normal physiology and pathophysiology over a range of length scales. At the level of single cells, mechanically-based cell-cell communication over long distances regulates patterning, including both tube formation and the detachment of cells from multicellular aggregates(7,9,10). At the tissue level, longrange force transmission may drive the development of tendons, ligaments, and muscle (4); it has the potential to mediate other large-scale architectural rearrangements typical of developmental processes as well (11). Long-distance force transmission between groups of cells, or cells and the matrix, may also mediate tissue-scale rearrangements in pathological settings such as pulmonary fibrosis and liver cirrhosis (12, 13). There are some experimental data implicating it in cancer metastasis(7,14,15), although other work suggests caveats to these findings (16).

10.1016/j.bpj.2014.09.044

1409.6377

b654386888236481cb5f4cb2ba1b353de24e8ee0

Long Range Force Transmission in Fibrous Matrices Enabled by Tension-Driven Alignment of Fibers Hailong Wang Department of Materials Science and Engineering Abhilash Nair Department of Materials Science and Engineering Christopher S Chen Department of Biomedical Engineering Boston University 02215BostonMA Rebecca G Wells Departments of Medicine (GI) and Pathology and Laboratory Medicine University of Pennsylvania 19104PhiladelphiaPA Vivek B Shenoy Department of Materials Science and Engineering Long Range Force Transmission in Fibrous Matrices Enabled by Tension-Driven Alignment of Fibers Cells can sense and respond to mechanical signals over relatively long distances across fibrous extracellular matrices. Recently proposed models suggest that long-range force transmission can be attributed to the nonlinear elasticity or fibrous nature of collagen matrices, yet the mechanism whereby fibers align remains unknown. Moreover, cell shape and anisotropy of cellular contraction are not considered in existing models, although recent experiments have shown that they play crucial roles. Here, we explore all of the key factors that influence longrange force transmission in cell-populated collagen matrices: alignment of collagen fibers, responses to applied force, strain stiffening properties of the aligned fibers, aspect ratios of the cells, and the polarization of cellular contraction. A constitutive law accounting for mechanically-driven collagen fiber reorientation is proposed. We systematically investigate the range of collagen fiber alignment using both finite element simulations and analytical calculations. Our results show that tension-driven collagen fiber alignment plays a crucial role in force transmission. Small critical stretch for fiber alignment, large fiber stiffness and fiber strainhardening behavior enable long-range interaction. Furthermore, the range of collagen fiber alignment for elliptical cells with polarized contraction is much larger than that for spherical cells with diagonal contraction. A phase diagram showing the range of force transmission as a function of cell shape and polarization and matrix properties is presented. Our results are in good agreement with recent experiments, and highlight the factors that influence long-range force transmission, in particular tension-driven alignment of fibers. Our work has important relevance to biological processes including development, cancer metastasis and wound healing, suggesting conditions whereby cells communicate over long distances. * Corresponding Author: [email protected] Long-range force transmission has significant relevance in normal physiology and pathophysiology over a range of length scales. At the level of single cells, mechanically-based cell-cell communication over long distances regulates patterning, including both tube formation and the detachment of cells from multicellular aggregates(7,9,10). At the tissue level, longrange force transmission may drive the development of tendons, ligaments, and muscle (4); it has the potential to mediate other large-scale architectural rearrangements typical of developmental processes as well (11). Long-distance force transmission between groups of cells, or cells and the matrix, may also mediate tissue-scale rearrangements in pathological settings such as pulmonary fibrosis and liver cirrhosis (12, 13). There are some experimental data implicating it in cancer metastasis(7,14,15), although other work suggests caveats to these findings (16). Introduction Cells in fibrous matrices sense and respond to mechanical forces over distances many times their diameter. Although cells cultured on polyacrylamide gels fail to sense substrate stiffness or the presence of other cells beyond a distance of about 20-25 µm (1-3), long-range force sensing (250-1000 µm) between cells in fibrous gels has been appreciated for decades. Stopak and Harris and later Miron-Mendoza et al. placed fibroblast explants into collagen gels and observed collagen realignment parallel to the connecting axes between explants, with translocation of collagen fibrils towards the explants, shortening of the axes, and fibroblast migration across the newly-aligned collagen fibril bridges (4,5). Others have shown that single cells as well as cell colonies are able to align and compact collagen fibers over long distances (6,7) and that these aligned fibers are required for long-range cell-cell interactions (7,8). More recently, Winer et al. showed that single cells in fibrin gels were able to stiffen the gels both locally and globally (9). Previous studies attempting to explain the mechanism of long-range force transmission have implicated applied strain and the presence of a fibrous network (6). While some investigators suggest that the strain-hardening properties of fibrous materials could explain long-range mechanical communication (9,17) more recent evidence (8,18) suggests that the fibrous nature of the ECM, specifically the presence of cross-linked fibers (primarily collagen) is critical in order for force to be transmitted over scales that are 10-20 times the diameters of the cells. Ma et al. used microscopy images to develop finite element models that included fibers that bridge pairs of interacting cells in a collagen matrix (8). They found that including discrete fibers along with a non-linear strain hardening matrix leads to long range transmission of forces, with the fibers carrying most of the loads and non-linear and isotropic matrix mechanics playing a relatively minor role. In other words, the fibrous nature of the collagen matrix, rather than a nonlinear response to force, determined the extent of force transmission. It should be noted that since the fiber distribution in the model of Ma et al. was obtained from experiments, the model cannot predict how an initially random fiber network under strain yields reinforcing fibrous structures in response to forces from contractile cells. Multi-scale finite element models, where discrete fiber networks are employed to determine forces at nodes, have also been used to study force transmission in fibrous gels (18). It has also been observed that the shapes of cells play a crucial role in the transmission of forces. Fabry and coworkers reported that invasive tumor cells are elongated and spindle shaped compared to their non-invasive counterparts and they observed through displacements of beads in the matrices that force transmission is much longer ranged in the former than in the latter case (19). Elongated cells have also been found to be polarized (i.e. the forces they exert are aligned with their long axes). Although these and other studies (20,21) have considered the role of individual cell and matrix elements in force transmission, none have addressed in an integrated way the impact of fiber realignment, the shape of the cells, the anisotropy and the magnitude of the contractile forces and the mechanical properties of fibrous gels on the long-range nature of force transmission. In this work we develop a new non-linear and anisotropic constitutive description of fibrous materials that accounts for the long-range force transmission. We incorporate the fact that these fibrous materials stiffen preferentially along the directions of tensile principal stretches. We start from random and isotropic distributions of fibers, and from there study how mechanical anisotropy evolves as loads are applied. We have developed a finite element implementation of this constitutive law and have used it to study interactions of cells in 3D matrices and on fibrous substrates. In the case of simple cell geometries (spheres, ellipsoids, polarized vs. non polarized), we solve for the stress fields by analytic methods. Thus, we describe here an approach to systematically determine the role of fiber alignment, non-linear elasticity of fibers, cell shape, and polarization of contraction in long-range force transmission. We show that collagen fiber alignment is critical and that anisotropy in cell shape and contraction result in significantly greater collagen alignment and force transmission. A New Constitutive Law for Fibrous Matrices We first developed a new constitutive law to explain the behavior of fibrous matrices and to serve as the foundation for further simulations examining the impact of cells and their contractility on these matrices. To start, we carried out discrete fiber simulations (see Appendix A). We assume that when a fibrous matrix undergoes stretch, there are two families of fibers: the set of fibers that align with the direction of the maximum principal stretch as the material is loaded (fibers colored red in Fig. 1b) and the set of fibers that do not align with the applied load and thus display an isotropic mechanical response. When we plot stress vs. strain for such collagen networks (Fig. 1c), we find that there is a "knee" in the curve representing strain stiffening. This "knee," which according to our simulations requires the presence of the two families of fibers, is in good accord with experimental data (experiment, Ref (22) and Fig. 1d). For strains below a typical threshold (typically 5-10%, depending on collagen concentration and crosslinking), the network shows a nearly isotropic response, without stiffening. Beyond this threshold, there is a transition to a stiffening response that is concomitant with the formation of aligned fibers in the direction of maximum tensile stretch. With increased loading, the numbers of these highly aligned fibers increase, leading to the observed stiffening and to the alignment shown in insets in Fig. 1a and Fig. 1b. Figure 1: (a)-(b) Discrete fiber simulations of a random fiber network before (a) and after (b) shear deformation (50% shear strain). Insets show that the initial random distributions of fibers (a) develop a peak close to the 45° orientation (b), which coincides with the direction of maximum principle stretch. Fibers (with axial strain > 1%) that reorient along the tensile loading axis are colored in red. The white arrow in (b) indicates the direction of principle tensile stretch. (c) The stress-strain curves of collagen I under uniaxial deformation derived experimentally [Ref (22)] (black) are in good accord with those predicted from our constitutive law (red). The "knee" indicates strain stiffening at strains around 10%. The material parameters that provide the best fit to the experimental data are ! = 1.1, = 1 + 1 − 2 ! /(1 − ) ! , ! = 8.5, = 10, ! = 2kPa, = 0.3. (d) Stress-strain curves under uniaxial tension (black) and shear (red) deformations from discrete fiber simulations and from our constitutive law. The material parameters that provide the best fit to the discrete simulations are ! = 1.05, = 0.17, = 1.4, ! = 10kPa, = 0.49. To capture the presence of these two distinct families of aligned and isotropic fibers when developing our constitutive law, we assume that the overall energy density of the collagen network consists of two contributions(23), = ! + ! ! = 2 ( ! − 3) + 2 ( − 1) ! (1) ! = ( ! ) ! !!! Here the first term ! ( ! , ) captures the isotropic response, which we describe using the neo-Hookean hyperelastic model, where and are initial bulk and shear moduli, respectively and ! is the contribution from the aligned fibers. In the above equation, !" = ! / ! is the deformation gradient tensor, where X and x labeled the reference and deformed coordinates respectively and = and = are the right and left Cauchy-Green deformation tensor, respectively. The invariants , and can be defined as (23), = det ( ) = ! ! ! !!! ⨂ = ! ! ⨂ ! !!! (2) where ! , ! , ! are the principle stretches, ! is the first invariant of the deviatoric part of and and are the unit vectors in the principle stretch orientations in the reference state and deformed state, respectively. The functional form ( ! ) is chosen such that the system stiffens only in the direction of tensile principal stretches (beyond a critical value of tensile stretch as observed in experiments and discrete fiber simulations). This is accomplished by decomposing the Cauchy stress (true stress), , into isotropic ( ) and fibrous contributions ( ) (23), = 2 • ( / ) • ! / , = + = − 1 + dev( )/ (3) = 1 ! ! ! ⨂ , ! !!! where is the identity tensor and = / !/! is the left modified Cauchy-Green tensor. The principal components of the filamentous contribution can be obtained from ( ! ) ! = 0, ! < ! ! ( ! − ! ) ! − ! ! − ! !!! + 1 , ! ≤ ! < ! ! ! − ! + 1 + (1 + ! − ! ) !!! − 1 + 1 , ! ≥ !,(4) chosen such that the principal stresses vanish below the critical (tensile) principal stretch, ! and show a stiffened response characterized by the modulus ! and a strain hardening exponent, . In order to ensure that the derivative of the stress-strain curve is continuous near the transition point, a smooth interpolation function is used between ( ! − ! /2, ! + ! /2), where the transition width, ! = 0.25 ! , transition exponent = 5, and we have defined ! = ! − ! /2, ! = ! + ! /2. The functional form ( ! ) which leads to Eq. 4 is provided in Appendix B. Biophysical basis for the constitutive law: The strain energy function and the stresses we propose depend on two parameters (the initial bulk and shear moduli) for the isotropic response and three parameters (the critical stretch, ! , the initial modulus of the fibrous phase, ! and the strain hardening exponent of the fibrous phase, ) for the anisotropic response. We have determined these parameters for collagen networks by comparing the stress-strain curves for uniaxial and shear deformation from discrete network simulations with our constitutive model (Fig. 1d). The biophysical basis that underlies the constitutive law that we have postulated is the presence of two families of fibers, clearly evident from the discrete fiber simulations: the first family (red in Fig. 1) is aligned with the principal axes (shown by the white arrow in Fig. 1) and are in a state of tension while the second family of fibers (black, in compression) provide an isotropic background stress that opposes alignment. The stress at any material point is the sum of the stresses from these two components (Eq. 3). The degree of the interaction between the two families of fibers is determined by the parameter ! / ! -when this ratio is large, the isotropic part provides little resistance to alignment. A systematic study of the range of force transmission as a function of this parameter is given below. With the two families of fibers, our model captures the key features of the response of a collagen network to force, in particular the knee and the subsequent hardening response. Results Having developed a constitutive law, we use it in analytical calculations and finite element simulations to study the impact of the material parameters of the isotropic and fibrous components of the matrix, the shape of cells and the polarization of cell contractile forces on force transmission in fibrous matrices. We have simulated cells on fibrous as well as linear and non-linear substrates to identify the key factors that allow for long range force transmission in fibrous matrices. All simulations were carried out using the finite element package Abaqus (24) by implementing the material model of the new fibrous constitutive law in a user material subroutine (details of the implementation are given in Appendix B). The numerical simulations were performed in a finite deformation setting (i.e. the effect of geometry changes on force balance and rigid body rotations are explicitly taken into account). Force transmission in 3D matrices depends on the fibrous components and the magnitude of the contractile strains To determine the impact of the fibrous component of the matrix on force transmission, we consider matrices that are linearly elastic, hyperelastic (neo-Hookean) and fibrous (characterized by the constitutive law Eq. 4). We consider the case of a spherical cell or contractile explant of radius in a 3D matrix contracting isotropically and inwardly by an amount ! (contractile strain = ! / ) . In our calculations, we apply the boundary condition on the radial displacement (= ! ) at the cell-matrix interface and determine the elastic fields in the matrix by applying both symmetry (or periodic) and fixed (all displacements and rotations vanish) conditions at the top and bottom surfaces of the matrix located at a distance ~ 10 from the center of the cell. In the case of the linearly elastic material, the scaled displacement fields ( / ! ) are independent of the magnitude of the contractile strain, ! , whereas this is not the case for non-linear materials. For both the neo-Hookean and the isotropic response of the fibrous material, the material parameters are chosen such that the Young's moduli and Poisson ratios are the same as that of the linear elastic material at small strains. We find that the displacement fields decay rapidly within a distance on the order of the cell diameter in non-fibrous materials ( Fig. 2a-black, blue and green curves), while the displacement fields are long ranged in the fibrous matrix ( Fig. 2a-red and orange curves). The range of interaction in the fibrous matrix is more than 20× the radius of the cell as evidenced by the fact that the boundary condition (periodic vs. fixed) has an impact on the displacement fields; the cells in this case are able to feel its periodic image since the displacement field does not completely vanish at the boundaries. To gain further insight into the range of elastic fields, we plotted the total force = 4 ! , normalized by the force at the cell-matrix interface, in Fig. 2b. We find that the decay of the total force in strain-hardening hyperelastic matrices is more rapid than in the case of the linearly elastic material, whereas the transmission of force is very long ranged in fibrous matrices. In Appendix C, we have derived a closed form expression for the decay of the force distribution as a function of the material parameters of the fibrous phase. These analytical calculations and the simulations in Fig. 2(c-f) clearly show that the fibrous components and not the isotropic strain-hardening response lead to long-range force transmission. Force transmission in 3D matrices depends on the shape of cells or explants and cell polarization Next, we consider the effect of shape and contraction anisotropy on force transmission in elastic and fibrous matrices. Unlike prior work that focused on the role of shape and cell polarization in linear elastic materials (25,26), here we consider fibrous materials described by the constitutive laws derived in Sec. 2. We model elongated cells as prolate spheroids described by the shape, ( / ) ! +( / ) ! + ( / ) ! = 1. Here and represent the length of the semi-minor and semimajor axes of the prolate spheroid, respectively. The polarization of active forces is modeled by assuming that the contractile strains (determined by molecular motors and regulation of adhesion sites) along the long axis of the spheroid, ! are greater than the strain along the short axis, ! . In order to compare shapes with different aspect ratios, = 1 − / and strain polarizations, = 1 − !!! ! !!! ! /(1 − ! ! ! ! ) , we assume that the volume of the cells prior to ( ! = 4 ! /3 = 4 ! /3) and after contraction ( ! = 4 1 − ! ! 1 − ! ! ) are the same in all cases. Note that = 0 corresponds to a sphere, while ~1 is a highly elongated prolate spheroid. Similarly, = 0 corresponds to isotropic contraction, while = 1 represents a fully polarized cell ( Fig. 3(a-d)). Here is the radius of the sphere as = 0. The above definitions also provide a definition for the size of a cell = !/! !/! , which is the geometric mean of the lengths of the semi-major and semi-minor axes of the elongated cell (which can be considerably shorter than the length of the semi-major axis for a highly elongated cell). The effect of shape and contraction and shape anisotropies on the range of force transmission in fibrous matrices is shown in Fig. 3 (a-d). Here the colored regions represent the extent of the aligned fibrous region, where the fibers are aligned with the tensile principal axis of strain tensor. We find that while shape and contraction anisotropy leads to an increase in the extent of the fibrous region, the effect is significantly amplified when both these factors are present simultaneously. We can understand this by noting that both shape and contraction anisotropy lead to concentration of tensile strains along the long axes of the cells. However, this effect is considerably magnified when the shape is elongated and the cell is polarized; significant concentration of tensile stresses in this case (Fig. 3d) leads to formation of extended regions where fibers are aligned. A heat map of the range of force transmission as a function of these parameters is given in Fig. 3e, for the case where the volume contraction is 1 − ! / ! = 55%. We find that the extent of the fibrous region can be as high as 20× the characteristic size of the fully polarized cells for = 2/3, as has been observed by several groups (8,18). The influence of the magnitude of volume contraction of the cell on the range of force transmission is plotted in Fig. 3f -our simulations show that range of force transmission generally increases with increase in overall contractile strain, although the effect is much more pronounced in elongated and polarized cells on account of the stress concentration effects discussed above. Thus, our analytical calculations and simulations collectively show that, in addition to the fibrous components of the matrix, elongated cells and polarized contraction leads to long-range force transmission. Long-range transmission in 3D matrices varies with the stiffness and strain-hardening exponent of the fibrous component and the critical strain for fiber formation We show in this section that in the material model that we have developed, the relative contributions of the fibrous and isotropic strain-hardening components to the overall mechanical response depends on three parameters: the ratio of the initial elastic moduli of the two components, ! / ! , the strain hardening exponent of the fibrous phase, m, and the critical strain for the onset of the fibrous response. A more pronounced fibrous response is obtained when ! / ! and m are large and when the critical stretch, ! is small (leading to an early transition to the aligned fiber phase). The extent of the aligned fibrous region that surrounds an elongated ( = 2/3) and fully polarized ( = 1) cell is shown in Fig. 4(a-d) as a function of the material parameters that characterize the fibrous phase. The simulations show that the range of force transmission increases with increasing modulus and the strain hardening exponent of the fibrous phase and with decreasing values of the critical strain for transitioning to the fibrous phase. These parameters are determined by the density of fibers, the numbers of crosslinkers per fiber and the porosity of the fibrous gels as discussed in Sec. 2. Cells sense farther into fibrous substrates than into linear and strain hardening substrates Recent work has demonstrated that fibroblasts sense deeper into collagen and fibrin gels (typically > 65 ) than they do into polyacrylamide gels (characteristic sensing distances of < 5 ) (18). In order to determine the characteristics of these gels responsible for characteristic sensing distances, we carried out calculations to determine cell sensing distance as a function of the thickness of gels constrained on one of the sides by a rigid (glass) substrate. Following earlier work (1), we assume the cell is circular and that it contracts radially inwards by pulling on the cell-substrate boundary. We apply displacement boundary conditions to this boundary (radial displacement ( )/ = 0.2 ) and the bottom surface is clamped to the underlying glass substrate. All other surfaces are free of any traction. As in earlier work (1), we find that in both linear elastic and non-linear strain-hardening materials, the sensing distance is close to the radius of the cell, R. Increasing the gel thickness by a factor of 5 from 2/3 to 10/3 has very little impact on the spatial profiles of the displacement fields. On the other hand, cells are able to sense much deeper into fibrous gels as evidenced by the slower decay of the displacement fields of cells on thicker substrates. Our results for the sensing distances (Fig. 5g) show that cells on fibrous gels can sense up to 8× their radii compared to 1.8× the radii on strain-hardening substrates. Cells sense other cells located at distances ~20 times their size in fibrous 3D matrices Interactions between pairs of cells play a key role in cell clustering during morphogenesis as well in pathological processes such as fibrosis, wound healing and metastasis. Based on our results (above) regarding the elastic fields of cells in different types of matrices, it is reasonable to guess that cell-cell interactions are significant when their separations are of the order of twice the sensing distance of a single cell. We verified this hypothesis by explicitly simulating the interactions between two cells in 3D fibrous and non-fibrous matrices as well as on substrates. The clear role of fibrous matrices in mediating cell-cell interactions is shown in Fig. 6 where significant overlap and alignment of strain-fields are observed for pairs of cells located in fibrous matrices at a distance of 10× their size. There is no overlap of strain fields for cells on non-fibrous substrates. Using these simulations, we confirm that cell-cell interactions become significant when cell spacing is twice the sensing distance, which is in agreement with the result in Fig. 6. Color represents the normalized radial displacement (0 − 1) (increasing from blue to red). The geometry and boundary conditions of (b) and (e) are same as those in Fig 2f and Fig 5f, respectively. Our simulations also clearly show the formation of collagen lines observed experimentally between pairs of cells (4, 5); we find that that the alignment of fibers coincides with the line that connects the centers of the two contractile cells both in 3D matrices and on substrates ( Fig. 6c and 6e). Thus, we find that fibrous but not neo-Hookean matrices enable cells to form collagen lines and interact mechanically with other cells at long range. Summary and Discussion In summary, we have developed a new constitutive law for fibrous matrices that predicts the following key cell behaviors: 1. Both shape and contraction anisotropy are important for long-range force transmission. These features of cells lead to stress concentration at the poles, which in turn leads to fiber alignment. Elongated prolate spheroidal cells with polarized contraction are able to sense the mechanical environment over much larger distances than spherical cells exhibiting diagonal contraction. 2. Tension-driven fiber alignment plays a crucial role in mechanosensing: small critical stretch for fiber alignment ( ! ), large fiber stiffness ( ), and fiber strain hardening behavior ( ) enable long-range interactions. Cells in 3D fibrous matrices and cells on 2D fibrous substrates sense rigid boundaries and other cells over relatively long distances compared to cells in and on linear and strainhardening isotropic materials. The range of force transmission increases with increasing contractility for cells in fibrous matrices while increasing contractility of cells cannot lead to enhancement of mechanosensing distances in linear and strain-hardening materials. 4. Cells in 3D fibrous matrices sense rigid boundaries over 10 × their diameters and other cells over 20 × their diameters. Cells on 2D fibrous substrates sense radial rigid boundaries up to 8 × their radii and thicknesses up to 3.5 ×their radii. Sensing distances can be further enhanced by increasing cell elongation, polarization and contractility. These findings are highly relevant biologically. They suggest that the presence of a fibrous matrix, as well as the material properties of that matrix, determine the nature of the mechanical interactions between groups of cells and between cells and boundaries in a range of settings including development, cancer metastases, and wound healing and fibrosis. This is consistent with the experimental observation that increased collagen cross-linking is associated with many of these processes, and suggests that studying the impact of other matrix proteins on fibrous collagen matrices may yield important insights into normal biology and pathology. Similarly, elongated cell shape and polarized cell contractility enhance long-range mechanical interactions; our results are consistent with experimental observations that cells involved in many of these processes are elongated and contractile (and may have undergone an epithelial to mesenchymal transition). Derivation of the constitutive law: The constitutive law for fibrous matrices we have proposed is non-linear with respect to the orientation and the magnitudes of the principal strains. The direction of the stiffened fibrous response coincides with the principal orientations whose principal strains are above a critical threshold. As we show below, these two features are critical to capture the key features of long-range force transmission observed in experiments. In this regard, the detailed form of the constitutive law of the matrix is not crucial as along as it captures the orientational anisotropy and stiffening that naturally arises along the principal directions upon loading. We have verified this idea by studying force transmission in matrices (Fig. 7) with other functional forms of response (Appendix D), but those retain the general features of anisotropic stiffening that coincides with the principal strain orientations. In particular, our constitutive law shares some common features with modified Cauchy-Green deformation tensors (23,27), but there are some crucial differences that are essential to obtaining long range force transmission. In this previously-published work, the collagen network is modeled as a hyperelastic material reinforced by two families of fibers whose orientations depend on the directions of principal stress (Appendix D). Note, however that unlike our formulation, their constitutive laws are based on the invariants of the modified Cauchy-Green deformation tensor. As we show in Appendix E, long-range force transmission cannot be observed when modified Cauchy-Green deformation tensors are used. We have therefore modified the constitutive law where we use the principal stretches, which are the eigenvalues of the Cauchy-Green deformation. We have, however, retained the feature that collagen fibers form only along those directions where the stretches are tensile. In essence, the law previously proposed relies on the deviatoric components of the dyadic ⨂ ( being the principal stretch), which as we show in Appendix E cannot give long-range force transmission since an incompressible material is similar to an isotropic material without tension-driven alignment of collagen fibers (Eq. C10 in Appendix C). Fig. 2f). (c). Contour plot of normalized radial displacement ( )/ ( ) on a fibrous substrate, which is similar to the result in Fig. 5f. Colors (from blue to red) represent the normalized radial displacement (0 − 1). The geometry and boundary conditions for (b) and (c) are same as in Fig. 2f and Fig. 5f, respectively. Sensing of thickness and lateral boundaries by cells on substrates: Our results are consistent with published experimental data on cell sensing distances. Both computational modeling (1, 2) and experimental observations (3,28,29) suggest that cells cultured on polyacrylamide (PA) gels (linear elasticity) cannot sense nearby cells beyond one cell length apart (<40 µμm) (1) and substrate thickness beyond half a cell length away (<20 µμm) (2). In contrast to cells on PA gels, human mesenchymal stem cells (hMSCs) and 3T3 fibroblasts on fibrin gels were shown to sense and respond to mechanical signals up to five cell lengths away (9), consistent with the results shown in Fig. 5f and 5g. Leong et al. studied the role of collagen I gel thickness on the fate of hMSCs and found that the mechanosensing distance for these cells is about 130 µμm, which corresponds to approximately 4.3× cell radii, also in agreement with our work. Recently, Rudnicki et al. designed sloped collagen and fibrin gel cultures to investigate thickness sensing. They found human lung fibroblast (HLF) and 3T3 fibroblast cell areas gradually decrease as gel thickness increases from 0 to 150µμm, with spreading affected on gels as thick as 150 µμm (18). Since the spreading radius in the case of the 150 µμm thick gel is 20 µμm, the mechanosensing distance for substrate thickness is 7.5× cell radii (18). While these multiscale simulations suggest sensing distances of 3.7× cell radii (sensing distance of 50 µμm for a cell radius of 13.4 µμm), our results show that cells sense boundaries up to 3.5 × their radii on fibrous substrates compared to 1.8× their radii on strain-hardening substrates (Fig. 5h). Thus, our work provides a good estimate for sensing distances on fibrous substrates. While most of the experimental work has focused on thickness sensing, recently Mohammadi et al. developed a model system to examine sensing of lateral boundaries in floating thin collagen gels populated with 3T3 fibroblasts (30). They found that cell-induced deformation fields extended to, and were resisted by, the grid boundaries 250 µμm away (30) suggesting a sensing distance for lateral rigid boundaries of about 8 × cell radii. These results are consistent with our calculations in Fig. 5g that show that both lateral and thickness sensing distances are similar in magnitude. Mechanosensing in 3D gels: Our results are consistent with published experimental work on the importance of cell shape, cell contractility, contractile strains, and local fiber alignment on longrange force transmission. Gjorevsk and Nelson examined the strain fields around engineered 3D epithelial tissues in collagen I gels. They found that linear elasticity cannot explain the longranged nature of the strain fields but reported that mechanical heterogeneities caused by stiffening near the poles of elongated contractile epithelial tissues can explain the decay of strain fields (31). Our results clearly show that long-range displacement fields within matrices can be captured by tension-driven local fiber alignment, and that heterogeneities result from the anisotropic shape of the cell domain and the anisotropic contraction of cells (Fig 3). Cell contractility results in reorganization of the ECM to provide contact guidance that facilitates 3D migration and invasion (4,5,32). The fiber alignments observed between nearby cells in 3D matrices (4,5,32) are clearly shown in our FEM simulations (Fig 6c and 6e). Experimental work has shown that treatment of cells to abolish actomyosin contractility leads to dissolution of the collagen lines, in agreement with our results that show that the magnitude of contractile strains play an important role in determining the range of force transmission. Recent experiments on mammary acini in collagen gels show that they can interconnect by forming long collagen lines up to around 10 × acini size (7). Guo et al. find that these lines and interactions are initiated by traction forces created by the cells and not by diffusive factors (10). They also found collagendensity dependent transmission of force up to 10 × cell radii for interacting acini. Our results show that cells in 3D fibrous matrices can sense the radial rigid boundaries up to 10 × their diameters and the other cells up to 20 × their diameters (Figs. 3 and 6), which is in very good agreement with these experiments. Furthermore, Ma et al. suggest that the fibrous nature of the ECM leads to reorganization of the collagen fibers leading to areas of higher fiber density near the cells over relatively long distances (10 cell-diameters) (8). The mechanism whereby this reorganization proceeds (starting from a random network) is discussed in our work. Koch et al. studied the effect of anisotropic cell shape and contractility on the range of force transmission in invasive and non-invasive cancer cells (19). They found that both lung and breast carcinoma cells were significantly elongated compared to the non-invasive cells, which were observed to have rounder shapes. Cell shape anisotropy was accompanied by a larger sensing distance, suggesting that directionality of traction forces is important for cancer cell invasion, consistent with our results (Fig. 3). In sum, we present a new constitutive law that describes the behavior of cells in matrices. All of the parameters for our constitutive law can be obtained either from experiments or from fiber simulations as have been done in Fig. 1. Our findings are relevant to a variety of normal and pathological processes and, importantly, as highlighted in detail above, are consistent with an extensive body of experimental work. We hope that this work will inspire further experiments where the mechanical properties of the ECM are tuned by varying the fiber density and degree of crosslinking to validate our predictions. APPENDIX A: Discrete fiber simulations We developed a finite element based 2D discrete fiber model that captures all aspects of network mechanics including non-affine stiffening, fiber alignment and bending-stretching transitions following our earlier work on crosslinked biopolymer networks (33). The 2D random fiber networks representing collagen gels are created with linear elastic fibers and rigid crosslinks (Fig. 1a). Fibers are uniformly distributed in the computational domain and a crosslink is formed when two fibers intersect. Collagen fibers have diameter in the range of few 100 nanometers to few microns and moduli of few 100 kPa (34)(35)(36). As the persistence length of collagen fibers is in the range of few microns, these fibers are typically modeled as linear elastic. Fibers are modeled using shear flexible Timoshenko beam elements in the finite element package, ABAQUS (24). Collagen gel considered in experiments is converted into a computational network (with equivalent fiber density) using the approach of Stein, Andrew M., et al (37). For the given concentration and volume of the gel, fiber radius is given by = ! ! ! !"# where ! ( ! ) is the volume of the gel, ! (= 1 − 5 / ) is the mass density of collagen, ! = 0.73 / is the specific volume of collagen, ( ) is the radius of the fibers and !"# ( ) is the total length of collagen in the gel. The 3D variables converted into equivalent the 2D ones by transforming quantities per unit volume to quantities per unit area. Fiber radius is assumed to be 250 and from the above relation, the total length of fiber in the gel is calculated for varying collagen concentrations. The fibers have both flexural and stretching rigidities and the crosslinks are assumed to be rigid (38). A parametric study for various collagen concentrations ( 2, 3, 4 5 / ) , simulating simple shear deformation shows good agreement with the experimentally observed strain sweep results (39). Increasing gel concentration reduces the collagen mesh size (distance between two crosslinks) leading to a stiffer response. The reduction in the length of the fiber between the crosslinks affects the bending characteristics and leads to an increase in the initial stiffness and a decrease the knee strain. APPENDIX B: Finite element implementation of the fibrous constitutive law All simulations were performed in a finite deformation setting. The matrices are modeled using 4-node bilinear axisymmetric quadrilateral elements. The axisymmetric constitutive law, the equilibrium condition, !" / ! = 0, and the boundary conditions constitute a well-posed boundary value problem. We implemented the constitutive equation in a user material model in the finite element package ABAQUS (24). The tangent modulus tensor in the material description , the tangent modulus tensor for the convected rate of the Kirchhoff stress , the tangent modulus tensor for the Jaumann rate of the Kirchhoff stress , and the material Jacobin (needed for the user material model) can be expressed as (23,40) !"#$ !" = 4 ! !" !" !"#$ !" = !" !" !" !" !"#$ !" (B1) !"#$ !" = !"#$ !" + !" !" + !" !" !"#$ !" = !"!" !" / Here the second Piola-Kirchhoff stress = / , !"#$ !" = !"#$ ! + !"#$ ! !"#$ ! = 1 2 !" !" + !" !" + !" !" + !" !" − 2 3 !" !" − 2 3 !" !" + 2 9 !" !" !! + 2 − 1 !" !" (B2) = 1 ! ! ! 1 ! ! ! ⨂ ⨂ ⨂ ! !!! + ! ! ! − ! ! ! ! ! − ! ! ( ⨂ ⨂ ⨂ + ⨂ ⨂ ⨂ ) ! !,!!! !!! + ⨂ + ⨂ Here we have adopted the abbreviations ( ⨂ ) !"#$ = !" !" and ( ⨂ ) !"#$ = !" !" . We define ! = 1 ( ! ) ! ! (B3) If λ ! → λ ! , ! ! ! ! ! !! ! ! ! ! ! ! ! !! ! ! gives us 0/0 and must be determined using the limiting conditions (23), lim ! ! →! ! σ ! ! ! − σ ! ! ! ! ! − ! ! = 1 2 σ ! ! ! − σ ! B4 Integrating Eq. 4, the energy function ( ! ) can be expressed as, ( ! ) = 0, ! < ! ! ( ! − ! λt ) ! ( ! − ! ) ! 8(1 + )(2 + ) , ! ≤ ! < ! ! − 1 2 + 3 + ! − ! 1 + + 1 + ! − ! !!! 1 + 2 + + ! 1 + + ( ! − ! )( ! − ! ) 1 + + 4( ! − ! ) ! 2 + 3 + ! , ! ≥ ! (B5) The second derivative of Eq. 4 can be expressed as, ! ( ! ) ! ! = 0, ! < ! ! ! − ! ! − ! ! , ! ≤ ! < ! ! (1 + ! − ! ) ! , ! ≥ ! (B6) Here ! = ! − ! /2, ! = ! + ! /2. APPENDIX C: Analytical linear solution for the spherically symmetric case We further introduce Green-Lagrange strain tensor = ( − )/2. For infinitesimal strains with | | ≪ 1, = 1 + tr( ) = + 2 (C1) ! = (1 + 2 ! ) !/! = 1 + ! Substituting Eq. C1 into Eq. 2 = ! ⨂ ! !!! (C2) The fiber energy function in Eq. 1 can also be expressed as ( ! ) = ( ! ), ( ! ) ! = ( ! ) ! ! ! = ( ! ) ! (C3) Substituting Eq. C3 into Eq.3, we get = ! + ! , = tr + 2 dev , (C4) = ( ! ) ! ⨂ ! !!! For linear bulk and fibrous response (λ ! = 1 and m = 0 in Eq. 4), Eq. C4 can be rewritten as, = ! + ! = ! 3(1 − 2 ) tr( ) + ! 1 + dev( ) (C5) = ! ! !!! ⨂ . For infinitesimal strains, we have the geometric relations, ! = , ! = ! = , = 1, (C6) Here is the radial displacement and the constitutive law Eq. C5 can be rewritten as, ! = ! 1 − 2 1 + 1 − + 2 + ! (C7) ! = ! = ! 1 − 2 1 + ( + ) The condition for mechanical equilibrium !! ! !" + ! ! ( ! − ! ) = 0 can then be written as, 1 + 1 + 1 − 2 1 − ! ! 2 2 + 2 − 2 ! = 0 (C8) The boundary condition is ( ! ) = ! , (∞) = 0 (C9) The solution is ( )/ ! = ( ! / ) ! (C10) ! ( )/ ! ( ! ) = ( ! / ) !!! Here = ! ! ( !!! !!! + 1) and = (!!!)(!!!!) (!!!) ! ! ! ! The strains and stresses can then be expressed as ! = ! ! ( ! ) !!! (C11) ! = ! = − ! ! ( ! ) !!! ! = ! 1 + 1 − 2 [ 1 − − 2 ] + ! ! ! ( ! ) !!! ! = ! = ! (1 + )(1 − 2 ) [ − 1] ! ! ( ! ) !!! In the limit of strong fibrous response, ! / ! ≫ 1, we find that the exponent → 1, whereas for an isotropic material for which ! / ! ≪ 1, we find that → 2. Thus, stresses decay less precipitously, leading to an increased zone of influence in fibrous materials. This result is also consistent with theoretical estimates by Sander (41), who considered a less general case, ! / ! ≫ 1, without including the effect of the Poisson's ratio, . APPENDIX D: Strain energy function with the modified right Cauchy-Green tensor Holzapfel et al. (23,27) developed a constitutive law to describe the mechanical response of arterial tissue with a strain energy function ! = ! ! , + ! = ! ! , + ! ( ! ) !!!,! D1 = 0, ! < 1 ! ! !! ! exp ! ! − 1 ! − 1 , ! ≥ 1, where the first term ! represents the isotropic bulk response of the matrix (same as our model) and the second term ! represents anisotropic stiffening due to two families of reinforcing collagen fibers that evolve during loading. The modified right Cauchy-Green tensor is = / !/! . ! , ! and ! are the modified invariants of , which represent the squares of the stretches along the two families of fibers, ! = tr( ) ! = ! = (D2) where and are the unit vectors along the fibers in the reference configuration. Then, the Cauchy stress has the form, An iterative procedure starting with an arbitrary configuration of the fibers is implemented to find the fiber vectors in the reference and current configurations, and . By considering this constitutive law for the case of spherically-symmetric contractile strain, we show in Appendix E that this constitutive law cannot show long-range transmission of forces. = + = + 2 ! ( ! ) To enable the long range formation in fibrous media, the above strain energy function for collagen fiber alignment can be modified by using a Cauchy-Green deformation tensor instead of a modified Cauchy-Green deformation tensor. Denoting the principal stretches by ! , we retain the functional form of the function, ( ! ) , such that it vanishes when the principal stretches are negative to get ( ! ) = 0, ! < 1 !! 2 !! [exp( !! ( ! ! − 1) ! ) − 1], ! ≥ 1 (D5) ( ! ) ! = 0, ! < 1 2 !! exp( !! ( ! ! − 1) ! )( ! ! − 1) ! , ! ≥ 1 (D6) ! ( ! ) ! ! = 0, ! < 1 2 !! exp( !! ( ! ! − 1) ! )[4 !! ! ! − 8 !! ! ! + (3 + 4 !! ) ! ! − 1], ! ≥ 1 (D7) Here !! and !! are the parameters for initial stiffness and strain-hardening. Note that ! in the original form is replaced with ! . We set χ = (1 + )(1 − 2 ) !! /(1 − ) ! = 0.2 and !! = 500 in our numerical simulations (Fig. 7). APPENDIX E: Analytical solution for the constitutive law with the modified right Cauchy-Green tensor Consider the special case of a spherical cell with isotropic contraction embedded in a fibrous matrix. As in the case of linear analysis in Appendix B, the deviatoric constitutive law in Eq. D3 can be rewritten for infinitesimal strains, = tr( ) + 2 + ! ! dev( ⨂ !!!,! ) (E1) Here the fiber energy function can be express as ! = ! with ! = 1 + 2 ! . For spherical symmetry, the deviatoric strain e ! = ! ! ( ! − ! ) ≥ 0 and e ! = e ! = ! ! ( ! − ! ) ≤ 0, so Eq. E1 can be rewritten as, ! = ! 3(1 − 2 ) ! + 2 ! + 2 3 [ ! 1 + + ! ]( ! − ! ) (E2) σ ! = ! 3(1 − 2 ) ! + 2 ! − Figure 2 : 2Displacement and force profiles in 3D linearly elastic, neo-Hookean and fibrous matrices with a spherical and isotropically contracting cell (radius = R). (a)-(b) Normalized radial displacement ( )/ ( ) and force ( )/ ( ) as functions of the normalized distance / (the boundaries are located at a distance = 100 ) from the center. We have chosen the critical stretch, ! = 1, fibrous modulus = 50 and the strain stiffening parameter = 0 for the fibrous matrix; (c)-(f) Contour plot of normalized radial displacement ( )/ ( ) for fibrous matrices with = 50 and = 30, ! = 2kPa and Poisson's ratio = 0.3 (same as ! and for linear matrices). For neohookean matrices / ! = 1/2(1 + ), / ! = 1/3(1 − 2 ). Figure 3 : 3The influence of shape and contraction anisotropies ( and , respectively) of contractile cells on distance ! / over which forces are transmitted (measured by the extent of aligned fibrous regions in the matrices): (a)-(d) Contour plots of aligned (colored) and isotropic (white) regions for the 4 cases with = 1, 2/3, and = 0, 1. Colors (from blue to red) represent maximum principle stretches (1.04-1.1); (e) Contour plots of ! / as function of shape anisotropy and contraction anisotropy . Colors (from blue to red) represent ! / (4 − 20); (f) Normalized transmission distance ! / vs volume contraction for the 4 cases in (a)-(d). Yellow ellipsoids with red arrows indicate contractile cells with different values of and . Material parameters for the fibrous matrix are ! = 1.04, = 50, = 30. The volume contraction is 55% for all the cases (a-d). The matrix size is 20 × the contractile cell radius ( / = 20) and the symmetry boundary conditions are applied at all boundaries. Figure 4 : 4The influence of material parameters of fibrous matrices on the transmission distance ! / : (a)-(d) Contour plots of aligned (colored) and isotropic (gray) regions for the 4 cases with ! = 1.02 − 1.04, = 10 − 50 and = 0 − 30. Colors represent maximum principle stretch (1.04 − 1.1) (increasing from blue to red). (e) Normalized transmission distance ! / vs volume contraction for the 4 cases in (a)-(d). Shape and contraction anisotropies are = 2/3 and = 1 and the volume contraction is 55% for all the cases. The matrix size is 20 × the contractile cell radius ( / = 20) and the symmetry boundary conditions are applied all boundaries. Figure 5 : 5Mechanosensing distances for contractile cells on linear, neo-Hookean and fibrous substrates with thickness = 2/3 − 10/3 , where R is the radius of the cell: (a)-(f) Contour plots of the normalized radial displacement ( )/ ! ( ! = (R)) with normalized thickness / = 2/3 (a)-(c) and / = 10/3 (d)-(f). (g) Normalized radial displacement ( )/ ! on the substrate surface as a function of the normalized distance / . (h) Normalized force transmission distance ! / as a function of the normalized thickness / (chosen with the criterion that the displacement fields decay by 90%, or ! / ! = 0.1). Circle (black), square (blue) and triangle (red) indicate linear, neo-Hookean and fibrous substrates, respectively. Material parameters for the fibrous matrix are ! = 1.02, = 50, = 30. The substrate radius is 10 × the cell radius ( / = 10) and the bottom boundary is clamped. Figure 6 : 6Interactions of pairs of contractile cells in neo-Hookean and fibrous matrices : (a-b) Contour plots of maximum principle strain in 3D matrices; (c) Vector plots of maximum principle strain (which coincides with the orientation of the collagen lines) in a 3D fibrous matrix; (d-e) Contour plots of maximum principle strain on 3D substrates. Colors (from blue to red) represent maximum principle strain (0.04-0.1). Lengths of red lines represent the magnitude of the maximum principle strain (0.04-1) and their orientations show the directions of fiber alignment. For the fibrous matrices, colored and gray regions represent aligned fibrous and isotropic regions, respectively. We have chosen ! = 1.04, = 50, = 30 for the fibrous material. Figure 7 : 7Force transmission for the material with strain energy function similar to that given in Ref. (23, 27) ( ! , = 0.3, = 0.25, !! = 500). (a) Blue and red curves represent bulk and fibrous contributions to the stress, respectively. (b) Contour plot of normalized radial displacement ( )/ ( ) in fibrous matrices (which is similar to the result in From boundary conditions:! = ! , ∞ = 0, the solution of Eq. E5 is ( )/ ! = ( ! / ) ! (E6) ! ( )/ ! ( ! ) = ( ! / ) !Comparing this with Eq. C 10, we find that the constitutive law of Holzapfel et al.(23,27) does not show long range force transmission. AcknowledgementsWe thank Tom Lubensky for insightful discussions on fibrous constitutive laws. 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Standard two-parameter compressions of the infinite dimensional dark energy model space show crippling limitations even with current SN-Ia data. Firstly they cannot cope with rapid evolutionour best-fit to the latest SN-Ia data shows late and very rapid evolution to w 0 = −2.85. However all of the standard parametrisations (incorrectly) claim that this best-fit is ruled out at more than 2σ primarily because they track it well only at very low redshift, z ≤ 0.2. Further they incorrectly rule out the observationally compatible region w ≪ −1 for z > 1. Secondly the parametrisations give wildly different estimates for the redshift of acceleration, which vary from z acc = 0.14 to z acc = 0.59. Although these failings are largely cured by including higher-order terms (≥ 3 parameters) this results in new degeneracies and opens up large regions of previously ruled-out parameter space. Finally we test the parametrisations against a suite of theoretical quintessence models. The widely used linear expansion in z is generally the worst, with errors of up to 10% at z = 1 and 20% at z ≥ 2. All of this casts serious doubt on the usefulness of the standard two-parameter compressions in the coming era of high-precision dark energy cosmology and emphasises the need for decorrelated compressions with at least three parameters. Subject headings:

10.1086/427023

astro-ph/0407364

fed7ead27c8cb6797c887e4a6094a3109af1128a

THE ESSENCE OF QUINTESSENCE AND THE COST OF COMPRESSION 1 Mar 2007 Draft version May 13, 2019 Bruce A Bassett Pier Stefano Corasaniti Martin Kunz Department of Physics Institute of Cosmology and Gravitation Kyoto University KyotoJapan University of Portsmouth PO1 2EGPortsmouth Astronomy Centre ISCAP Columbia University 10027New YorkNYUSA University of Sussex BN1 9QJBrightonUK THE ESSENCE OF QUINTESSENCE AND THE COST OF COMPRESSION 1 Mar 2007 Draft version May 13, 2019Draft version May 13, 2019 Preprint typeset using L A T E X style emulateapj v. 08/22/09Subject headings: Standard two-parameter compressions of the infinite dimensional dark energy model space show crippling limitations even with current SN-Ia data. Firstly they cannot cope with rapid evolutionour best-fit to the latest SN-Ia data shows late and very rapid evolution to w 0 = −2.85. However all of the standard parametrisations (incorrectly) claim that this best-fit is ruled out at more than 2σ primarily because they track it well only at very low redshift, z ≤ 0.2. Further they incorrectly rule out the observationally compatible region w ≪ −1 for z > 1. Secondly the parametrisations give wildly different estimates for the redshift of acceleration, which vary from z acc = 0.14 to z acc = 0.59. Although these failings are largely cured by including higher-order terms (≥ 3 parameters) this results in new degeneracies and opens up large regions of previously ruled-out parameter space. Finally we test the parametrisations against a suite of theoretical quintessence models. The widely used linear expansion in z is generally the worst, with errors of up to 10% at z = 1 and 20% at z ≥ 2. All of this casts serious doubt on the usefulness of the standard two-parameter compressions in the coming era of high-precision dark energy cosmology and emphasises the need for decorrelated compressions with at least three parameters. Subject headings: INTRODUCTION The issue of dark energy dynamics is perhaps the most pressing today in cosmology. There are claims both for and against dynamics (Bassett et al. 2002, Alam et al. 2003, Jassal et al. 2004. But it is a subject dogged by gauge problems 1 (Jonsson et al. 2004;Virey et al. 2004;Wang and Tegmark 2004). Claims for dynamics (or the lack thereof) can be pure gauge artifacts, mirages induced by the parametrisations that are unrelated to the data. For example, Riess et al. (2004) and Jassal et al. (2004) claim that current SN-Ia data are inconsistent with rapid evolution of dark energy. Such conclusions must always implicitely refer to a finite dimensional subspace of the full dark energy model space, and broadening the class of models studied can (and in this case does) lead to complete reversal of such conclusions. Fig. (2) provides a explicit counterexample. The aim of compression is to summarise, as aggressively as possible, the key features of dark energy properties in a few parameters and to facilitate discrimination between models. We will see that standard expansions 1 Consider two dark energy parametrisations, P 1 and P 2 . We are interested in estimating the true value of an observable O, such as w(z = 0). Under a change in parametrisation the best estimate of O will change to O 2 = O 1 + δO. This is a gauge artifact since the change in O has nothing to do with the real universe. Even worse, since the two parametrisations will usually be rather different we have no guarantee that δO will be small. fail to achieve either of these goals, even when they are extended to higher order. The main result of this work is that compression of the dark energy space into low-dimensional subspaces, while convenient and easy to work with, can give seriously misleading results. Since these results are used in the design of upcoming surveys it is bad news for cosmology in general. If one does not impose the weak energy condition (WEC), w ≥ −1, then the results can border on the completely useless. The rest of this article delimits, as precisely as possible, the quicksands and danger areas in the use of two-parameter compressions. As a first sobering example, consider constraints on w(z) when we do not impose the WEC. One-parameter studies give constraints such as −1.38 < w DE < −0.82 at 2σ (Melchiorri et al. 2003) suggesting that a model with w = −5 at z = 2 would be ruled out at more than 10σ. Instead a little thought makes it clear that if w(z) can vary freely then there is no lower bound on w for z ≥ 1 since this merely changes how fast the already irrelevant and rapidly diminishing dark energy density decreases. If the rapid drop in w occurs at z > 1 this leaves essentially no observable trace (Bassett et al. 2002;Corasaniti et al. 2003). This is clearly reflected in the likelihoods in Fig. (1) which allow for w < −100 at z ∼ 1. How can we hope to cover such possibilities with simple one or two parameter compressions? The dark energy literature overflows with one, two and higher-dimensional compressions of w DE (z), e.g. (Efstathiou 1999;Huterer and Turner 2001;Weller and Albrecht 2002;Bassett et al. 2002;Corasaniti and Copeland 2002;Linder 2003;Jassal et al. 2004). Compressions also exist for ρ DE (z) (Wang and Freese 2004;Wetterich 2004) while decorrelated reconstructions of w DE (z) have been proposed in (Huterer and Starkmann 2003) and (Hu 2004). The most basic parametrisation, namely describing the dark energy with a constant equation of state w = w 0 , is well-known to introduce a severe bias (see for instance Maor et al. 2002, Virey et al. 2004 in parameter estimation. Compressions invoking two parameters which somewhat alleviate this problem have been introduced in (Efstathiou 1999;Linder 2003;Jassal et al. 2004). However, as we will see, these models all struggle to describe rapid evolution. This is not surprising. With two parameters one may fix w at z = 0 and w at high z, but one can do nothing about the time nor the rapidity of the transition between the two extremes. Caldwell and Doran (2004) circumvented this by considering thirteen different one and two-parameter models, some exhibiting rapid transitions. The use of more than two parameters offers the opportunity to test the dark energy evolution with different data sets and consistently account for the dark energy perturbations at all redshifts (Bassett et al. 2002;Corasaniti and Copeland 2002;Corasaniti et al. 2004) but is, of course, more computationally intensive, and care must be taken to accurately capture the dark energy dynamics of rapid transitions; see Appendix C of (Corasaniti et al. 2004). THE PARAMETRISATIONS For our study we consider two distinct classes of compressions. First are standard Taylor expansions of w DE (z) and second is the Kink, a physically-motivated compression. The Taylor expansions are all of the form: w DE (z) = n=0 w n x n (z)(1) where we consider four different choices for the 'expansion' functions, x n (z). Namely: x 0 (z) ≡ 1 ; x n ≡ 0, n ≥ 1 (constant w) (2) x n (z) ≡ z n (redshift) (3) x n (z) ≡ (1 − a) n = z 1 + z n (scale factor) (4) x n (z) ≡ (log(1 + z)) n (logarithmic) (5) To linear order (n ≤ 1) these were first discussed by Huterer and Turner (2001) & Weller and Albrecht (2002), Chevallier and Polarski (2000) and Linder (2003), and Efstathiou (1999) for the redshift, scale-factor and logarithmic expansion functions respectively. Later we will consider their performance at higher order (n ≥ 2). The Kink, on the other hand, is not an expansion. It is a 4-parameter model which accurately captures the behaviour of quintessence (Bassett et al. 2002, Corasaniti and Copeland 2002, Corasaniti et al. 2004). The extra parameters allow us to model very rapid transitions in (1). w DE (z), a freedom we will need: w DE (a) = w 0 + (w m − w 0 ) 1 + e a t ∆ 1 + e (a t −a) ∆ 1 − e (1−a) ∆ 1 − e 1 ∆ ,(6) a is the scale factor, w 0 and w m are the present and matter-dominated values of the dark energy equation of state, a t is the value of the scale factor at the transition from w m to w 0 and ∆ controls the width of the transition. Other formulations of the Kink, with relative merits, are discussed in Appendix A of Corasaniti et al. (2004). There are other parametrisations but these are the most widely used today and lessons learned from these compressions will apply to many others in the literature. CONSTRAINTS FROM SN-IA In this section we investigate whether the different parametrisations above give rise to different best-fits to current type Ia supernova (SN-Ia) data. Recently 6 new SN-Ia at redshift z > 1.25 have been discovered using the Hubble Space Telescope, providing further evidence for a transition from decelerated to accelerated expansion in the past (Riess et al. 2004). In order to be conservative we use only the gold sample of (Riess et al. 2004), containing 157 data points. In our analysis we assume a flat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. The assumption of flatness is required to achieve reasonable error bars (c.f. Kunz and Bassett 2004;Dicus and Repko 2004). Fortunately this is now a data-driven assumption and particularly harmless for this study since we are primarily interested in testing compressions rather than deriving constraints. We will also assume the prior Ω m = 0.27 ± 0.04. This can be justified from CMB data, as shown in Kunz et al. (2003) and Corasaniti et al. (2004), the best-fit values for background FLRW parameters are not affected strongly by dark energy dynamics. The luminosity distance is given by d L (z) = (1 + z) z 0 dz ′ H(z ′ ) ,(7) and H(z) = H 0 [Ω m (1 + z) 3 + (1 − Ω m )f (z)] 1/2 ,(8) where f (z) is the solution to the continuity equation f (z) = exp 3 z 0 1 + w DE (z ′ ) 1 + z ′ dz ′ .(9) Our analysis methods are described in detail in Corasaniti et al. (2004). We use a Markov-Chain Monte Carlo code to find the constraints on the dark energy parameters for each parametrisation. As usual we marginalise analytically over the normalisation of d L which takes care of the Hubble constant as well, leaving Ω m as the only remaining parameter apart from those describing the equation of state. Figure 1 shows the marginalised one-dimensional likelihoods for the parameters of the dark energy compressions. The main point of that figure is that the various parametrisations have similar likelihoods at the same order, but that the likelihoods at different orders are completely different. Here by order we mean the maximum value of n in eq. (1). Hence "linear" or "first-order" (n ≤ 1) refers to the standard two-parameter expansions with only w 0 , w 1 non-zero, while 2nd-order corresponds to n ≤ 2 and has non-zero w 2 . The best-fit values and 1σ errors for the first-order redshift, scale factor and logarithmic parametrisations are given in table 1; these are consistent with those found in (Feng et al. 2004;Gong 2004). We infer the 'maximised' limits on the redshift dependence of the equation of state by computing for a given parametrisation the highest and lowest w DE (z) for the models in the chains with χ 2 < χ 2 min + 4. Models with w(z) outside those limits are therefore expected to be "bad" fits to the SN-Ia data. We plot the result in Figure 2. For the two-parameter models, the limits are very similar to those obtained by marginalising over all other parameters, as is expected since they are nearly Gaussian. We have also checked that they coincide with limits from Gaussian error propagation. We now discuss the constraints derived from the Kink formula, Eq. (6). The best fit to the data has a χ 2 = 172.6 and is characterized by: w 0 = −2.85, w m = −0.41, a t = 0.94 and log(∆) = −1.52, corresponding to a rapidly varying equation of state with a transition from w m to w 0 at z t = 0.1. This best-fit is shown in figure 2 and clearly exits the 2σ limits from the twoparameter compressions, first from below at z ∼ 0 and then from above, at z ∼ 0.2. This graphically illustrates the limitations of the standard parametrisations and shows how they artificially rule out models which should give the strongest signals for dark energy dynamics . Although the Kink formula is by construction everywhere under control, the dependence on its parameters is highly non-linear so the peaks of the marginalised likelihood distributions do not match the values of the best fit model. In fact the peaks of the 1-dimensional marginalised likelihoods are shifted with the respect to the best fit values after marginalising. Hence the marginalised likelihoods do not coincide with the maximized ones, as they would if the likelihood distributions were Gaussian. It is not just one good fit to the data which violates the 2σ limits of all the two-parameter compressions either. For instance, the model with w 0 = −1.46, w m = 0.16, a t = 0.88 and log (∆) = −0.7 has χ 2 = 173.9, while the model with w 0 = −1.11, w m = 6.13, a t = 0.40 and log (∆) = −.98, has χ 2 = 175.9. Both are excellent fits to the data but are supposedly ruled out by the n ≤ 1, linear redshift, scale-factor and logarithmic parametrisations of equations (3-5). The conclusion that rapid evolution of dark energy is ruled out by current data is therefore a 'gauge' artifact. We have shown that rapid variations of the dark energy equation of state are perfectly consistent with, and in fact provide better fits to the Gold sample than do mod-els without rapid transitions. This conclusion remains even after including CMB and large scale structure data (Corasaniti et al. 2004). The pathological behaviour of ruling out models which are very good fits to the data can be rectified by the inclusion of higher order terms, n ≥ 2. Indeed, since the data allow w 1 to be large in all cases, higher order terms in the redshift, scale-factor and Log expansions cannot be neglected. Therefore we have extended our analysis in order to include second order corrections (n ≤ 2) to Eq. (3)-(5). Comparing the yellow likelihoods in Figure 1 with the red ones we see that by the allowed values of w 0 are shifted significantly towards more negative values, consistent with, but broader than the kink confidence interval. Secondly huge values of w 1 ∼ 50 and w 2 ∼ −100 are consistent with the data. This suggests that strong dark energy dynamics is not ruled out and that higher order terms must be taken into account. As mentioned in the introduction, this comes from the fact that w 1 and w 2 are strongly degenerate in all cases and that there is no lower bound on w at z > 1, illustrating the huge effect of imposing the weak energy condition w ≥ −1. In this case we have both a lower bound (from the weak energy condition) and an upper bound (from the data) on w. As the second order parametrisations are described by a parabola in their respective expansion variables, they end up being strongly constrained. On the other hand, we have also considered much higher order (n ≤ 6), and found that severe internal degeneracies lead to finely balanced coefficients. Thus a Taylor expansion in z becomes very unstable at high redshift, and even expansions in 1−a can hardly be called "under control" anywhere since the expansion coefficients become of order 10 3 . Model selection with Information Criteria and the Bayesian Evidence In table 2 we report the values of the χ 2 for the best fit models of each parametrisation. As we can see the fourparameter Kink formula has the lowest χ 2 , followed by the scale-factor, the logarithmic and redshift parametrisation. Accounting for second order terms (n ≤ 2) in the expansions provide fits even better than the best Kink parametrisation which is non-trivial since they have one less parameter, although now the allowed parameter space volume is huge. At this point we can ask how many parameters are actually necessary to describe the dark energy with current SN-Ia data? Following Liddle (2004), we compute the Akaike information criterion (Akaike 1974) AIC = −2 ln L + 2k(10) and Bayesian information criterion (Schwarz 1978) BIC = −2 ln L + k ln N ,(11) where L is the maximum likelihood, k is the number of model parameters and N is the number of data points. We also compute the Bayesian evidence E (Sivia 1996;Saini, Weller and Bridle 2004), both with thermodynamic integration and nested sampling (Skilling 2004). It is worth noting that fully degenerate parameters do not contribute to the Evidence, so that specifically the Kink model is less disfavored than the number of parameters suggests naively. For the same reason we find that E grows very slowly when going to even higher order in the expansion-type parametrisations, although these cases are already disfavored by Bayesian statistics. The preferred parametrisation is ΛCDM -it is indeed remarkable that a model with a single free parameter fits the data so well. WHEN DID ACCELERATION BEGIN? One of the key characteristics of dynamical dark energy is that the redshift at which the universe begins accelerating, z acc , is characteristically different from that in ΛCDM with the same Ω DE today. This is manifest in the CMB as a modified integrated Sachs-Wolfe effect (Bassett et al. 2002;Corasaniti et al. 2003) which is degenerate with reionisation (Corasaniti et al. 2004). The SN-Ia offer the possibility to break this degeneracy and therefore it is crucial to use a parametrisation that can accurately estimate z acc without bias. Using a simple linear expansion of the deceleration parameter q, Riess et al. (2004) estimated z acc = 0.46 ± 0.13. Here we compare the predictions of the various parametrisations for z acc . The 1d likelihoods for z acc are shown in Figure 3 and in table 3 we report the confidence intervals. We have two main points. First, all of the parametrisations predict different best-fit values and 1σ error bars for z acc , ranging from z acc = 0.14 for the redshift expansion to z acc = 0.59 for the scale-factor expansion (see also Dicus and Repko 2004). The logarithmic, constant and Kink parametrisations all have similar best-fits, but the first two have overly narrow error bars relative to the Kink predictions. The largest error bars correspond to the scale-factor expansion, eq. (4). Given the importance of accurately estimating z acc this variance forcefully argues for the need to go beyond twodimensional parametrisations in handling future, highquality data. As a second point it is interesting that the best-fits for z acc for all the parametrisations are lower than in the ΛCDM model. This may be novel evidence for dark energy dynamics. We leave this issue for future work. The marginalised 1d likelihood for zacc as estimated from the same SN-Ia data for each of the various dark energy parametrisations. The wide variance casts doubt on their usefulness. All of them peak at a lower zacc than in the standard ΛCDM, perhaps giving novel evidence for dark energy dynamics. Only the scale-factor expansion, eq. (4) shares significant overlap with ΛCDM, but that is expected since the transition is typically slow in that parametrisation. CONCLUSIONS This Letter shows the limitations of standard one and two-parameter compressions of the infinite dimensional space of dark energy models. We have highlighted the dangers in using constraints derived using these parametrisations, particularly regarding the possibility of rapid evolution in the dark energy, which none of the standard compressions can follow, and in defining allowed regions of parameter space which are completely wrong in the case where the weak energy condition (w > −1) is not imposed. Rapid evolution provides a superlative fit to current SN-Ia data (as measured by χ 2 ), despite claims to the contrary in the literature which were based on two-parameter compressions. Indeed, all of the twoparameter expansions we studied wrongly rule out such rapid evolution at 2σ or more. This is extremely damning evidence, especially since these compressions are typically used in the planning for the next-generation experiments which will provide data of significantly higher quality. In addition, the standard parametrisations also miss the fact that w has no lower-bound at z > 1 if the weak energy condition is not imposed, artifically cuttingout vast swaths of parameter space due to their innate limitations. Further problems occur in estimating the redshift at which the universe began accelerating, z acc . There is a nearly 300% variation in the best-fit for z acc , depending on parametrisation. Interestingly, all the tested parametrisations gave best-fits for z acc below that of ΛCDM, providing unusual cross-parametrisation evidence for dark energy dynamics. Nevertheless, use of the Bayesian information criteria for model selection prefers the cosmological constant over the other models which remains the model to beat. The severe inadequacy of the standard two-parameter expansions lead us to consider higher-order terms (n ≥ 2) with one of more extra parameters, e.g. w 2 . While this brings the rapid evolution models within the allowed region of parameter space it leads to severe degeneracies (see figure 1) which may make the parametrisations impotent for constraining the space of theoretical dark energy models, particularly when w < −1. For completeness we have also compared four standard parametrisations against a test-bed of quintessence models (Appendix) and found that while the kink and scalefactor expansion are excellent, the expansions in z and log(1 + z) can lead to large errors for z ≥ 1. We conclude that confidence intervals inferred from standard two-parameter expansions often do not deserve that name and are typically untrustworthy, even with current data. The wealth and quality of dark energy data we will aquire over the next decade will demand significantly better performance. It is a pleasure to thank Rob Caldwell, Rob Crittenden, Ruth Durrer, Dragan Huterer, Eric Linder, Andrew Liddle, Roy Maartens, Takahiro Tanaka and Jochen Weller for discussions or comments. P.S.C. is grateful to the Michigan Center for Theoretical Physics for hospitality during the "Dark Side of the Universe" workshop where part of this work was done. P.S.C. is supported by Columbia Academic Quality Fund. M. K. is supported by PPARC while BB is supported by a Royal Society/JSPS fellowship. APPENDIX In this appendix we address the problem of how accurately the various parametrisations listed earlier do in capturing the dynamics of standard quintessence models. We proceed by considering a test-bed of four popular quintessence models, which provide a wide range of different behaviours for w(z), with smooth, slow and rapid transitions. Our test-bed suite of models is: the Albrecht-Skordis model (Albrecht and Skordis 2001), V (Q) = M 4 e −αQ [(Q − A) 2 + B] s/2 ,(1) with α = 5, A = 54.5, B = 0.01 and s = 5; the Two Exponential potential (Barreiro, Copeland and Nunes 2000), V (Q) = M 4 (e −αQ + e −βQ ),(2) with slopes α = −9 and β = 2; the inverse power-law potential (Ratra and Peebles 1988), 5), to the different quintessence models as function of zmax for each parametrisation for n ≤ 1. At z ∼ 1 the expansion to first order in z ('redshift') and log(1 + z) ('logarithmic') typically show errors around 10% while the kink and scale-factor parametrisations have errors typically at the 1% level at z ∼ 1. V (Q) = M 4+α /Q α ,(3) with α = 0.6; the Supergravity (SUGRA) inspired model (Brax and Martin 1999), V (Q) = M 4 e −Q 2 /2 /Q α ,(4) with α = 6. For the kink and each two-parameter expansion (n ≤ 1) we compute the best-fit (using an MCMC search) to the different quintessence models in the redshift range 0 < z < z max and plot the average residual quadratic errors as a function of z max , as shown in Figure 4, where Res = 1 N z Nz i=1 (w Q (z i ) − w best par (z i )) 2 ,(5) with N z the total number of equally spaced bins in the range [0, z max ], and we have chosen N z = 1000. Comparing the residuals indicates how well each formula reproduces a given quintessence model as function of the redshift interval. For z max ≥ 0.3, the Kink and scale-factor formulae provide best-fits which are at least one order of magnitude better than the redshift and logarithmic expansions, except perhaps in the Two-Exponential case. Only for z max < 0.3 are the residuals of the redshift and logarithmic formulas of the same order as those of the other parametrisations. This is easy to understand from Fig. (2). The best-fit solution is almost perfectly linear for z < 0.1 but then suddenly deviates. A linear fit will be excellent until z ∼ 0.1, after which it rapidly becomes bad. We do not consider the case of constant w which can at best fit the average value of w(z) and whose errors are expected to be the worst. Fig. 1 . 1-Expansion order is more important than parametrisation. 1-d marginalised likelihoods for the different classes of parametrisations. The narrow curves are the linear likelihoods (constant w DE , n = 0, is dotted, first-order expansions, n ≤ 1, are dashed). The solid lines show the likelihoods at secondorder (n ≤ 2) and are much wider due to degeneracies and the absence of a lower-bound on w for z ≥ 1. The curves correspond to expansions in (a) scale-factor, (b) log and (c) redshift respectively. Finally the 1-d likelihood for wm of the Kink is shown (dot-dashed) and exhibits strongly non-Gaussian wings that extend to very large values of |wm|. For definition of the order of the expansion see eq. Fig. 2 . 2-Parametrisations struggle with rapid evolution. Maximised limits on w DE (z) for the redshift (red dashed line), scale-factor (green dash-dotted), logarithmic (blue dotted) and Kink (solid black line) parametrisations. The best-fit kink solution passes well outside the limits of all the parametrisations (except for the kink) both at z ∼ 0 and at z ∼ 0.2, showing their inability to capture rapid dynamics which leads to their incorrectly ruling it out. Fig. 3 . 3-When did acceleration begin? Fig. 4 . 4-Plot showing the mean quadratic error of the best-fit, eq. ( TABLE 1 1Best fit values and 1σ confidence intervals at linear order (n ≤ 1).w 0 w 1 Redshift −1.30± 0.43 0.52 1.57± 1.58 1.41 Scale-factor −1.48± 0.57 0.64 3.11± 2.98 3.12 Log −1.39± 0.50 0.57 2.25± 2.19 2.15 TABLE 2 Bayesian 2Evidence, BIC, AIC and χ 2 values of best fit for different parametrisations. k is the total number of fitting parameters which include the dark energy density Ω DEmodel k χ 2 BIC AIC − ln(E) ΛCDM 1 177.6 182.7 179.6 93 w=const 2 177.6 187.7 181.6 96 linear 3 174.5 189.7 180.5 99 logarithmic 3 174.2 189.4 180.2 98 scale factor 3 174.0 189.2 180.0 98 quadratic 4 172.1 192.3 180.1 100 logarithmic II 4 172.2 192.4 180.2 100 scale factor II 4 172.3 192.5 180.3 99 Kink 5 172.6 197.9 182.6 96 TABLE 3 Best fit values and 1σ confidence intervals on zacc at linear order (n ≤ 1). 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We aim to tackle the interesting yet challenging problem of generating videos of diverse and natural human motions from prescribed action categories.The key issue lies in the ability to synthesize multiple distinct motion sequences that are realistic in their visual appearances. It is achieved in this paper by a twostep process that maintains internal 3D pose and shape representations, action2motion and motion2video. Ac-tion2motion stochastically generates plausible 3D pose sequences of a prescribed action category, which are processed and rendered by motion2video to form 2D videos. Specifically, the Lie algebraic theory is engaged in representing natural human motions following the physical law of human kinematics; a temporal variational auto-encoder (VAE) is developed that encourages diversity of output motions. Moreover, given an additional input image of a clothed human character, an entire pipeline is proposed to extract his/her 3D detailed shape, and to render in videos the plausible motions from different views. This is realized by improving existing methods to extract 3D human shapes and textures from single 2D images, rigging, animating, and rendering to form 2D videos of human motions. It also necessitates the curation and reannotation of 3D human motion datasets for training purpose. Thorough empirical experiments including ablation study, qualitative and quantitative evaluations manifest the applicability of our approach, and demonstrate its competitiveness in addressing related tasks, where components of our approach are compared favorably to the state-of-thearts.

10.1007/s11263-021-01550-z

2111.06925

61b27367ae5b39719eee517c5fab377b3bb1763c

Action2video: Generating Videos of Human 3D Actions Chuan Guo Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Xinxin Zuo [email protected]. Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Sen Wang Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina · Xinshuang Liu Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Shihao Zou Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Minglun Gong [email protected]. Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Li Cheng [email protected]. Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Chuan Guo Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Xinxin Zuo Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Sen Wang Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Shihao Zou Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Li Cheng Department of Electrical and Computer Engi-neering Xinshuang Liu is with the School of Software, Ts-inghua University University of Alberta 100084BeijingChina Action2video: Generating Videos of Human 3D Actions Received: date / Accepted: dateNoname manuscript No. (will be inserted by the editor) Minglun Gong is with the School of Computer Science, Uni-versity of Guelph.3D human motion generation · video motion synthesis · Lie algebraic human representation · temporal variational autoencoders We aim to tackle the interesting yet challenging problem of generating videos of diverse and natural human motions from prescribed action categories.The key issue lies in the ability to synthesize multiple distinct motion sequences that are realistic in their visual appearances. It is achieved in this paper by a twostep process that maintains internal 3D pose and shape representations, action2motion and motion2video. Ac-tion2motion stochastically generates plausible 3D pose sequences of a prescribed action category, which are processed and rendered by motion2video to form 2D videos. Specifically, the Lie algebraic theory is engaged in representing natural human motions following the physical law of human kinematics; a temporal variational auto-encoder (VAE) is developed that encourages diversity of output motions. Moreover, given an additional input image of a clothed human character, an entire pipeline is proposed to extract his/her 3D detailed shape, and to render in videos the plausible motions from different views. This is realized by improving existing methods to extract 3D human shapes and textures from single 2D images, rigging, animating, and rendering to form 2D videos of human motions. It also necessitates the curation and reannotation of 3D human motion datasets for training purpose. Thorough empirical experiments including ablation study, qualitative and quantitative evaluations manifest the applicability of our approach, and demonstrate its competitiveness in addressing related tasks, where components of our approach are compared favorably to the state-of-thearts. Introduction Human-centric activities always play a key role in our daily life. In recent years, noticeable progresses have been made in video forecasting Gao et al., 2019) and synthesis (Zhu et al., 2020;Tulyakov et al., 2018;Vondrick and Torralba, 2017;Denton and Fergus, 2018). Meanwhile, it remains a substantial challenge in generating realistic videos of diverse and plausible human motions. This is evidenced in many recent video generation efforts Cai et al., 2018;Kim et al., 2019), where the appearances of synthesized human characters are unfortunately either blurring or surreal, and are still far from being photo-realistic; their motions are often distorted and unnatural. These observations stress the importance of properly modeling human body postures & temporal articulations, as well as the surface shapes and textures of the local body parts. It also motivates us to examine the problem of generating videos of human motions based on action categories, the basic ingredient of human behaviors. Due to the complexity of human articulations and pose dynamics, generating human videos is far from being trivial. Existing efforts usually represent human motions in 2D space, which are then rendered pixel-wise to form 2D videos. Moreover, extra information such as Our action2video pipeline generates human full-body motion videos of prescribed actions in two steps: action2motion first generates diverse and natural 3D motions of predefined actions; motion2video proceeds to extract 3D surface shape and texture from an additional 2D input image, and to render 2D videos of the generated motions. an initial 2D pose or a partial/entire motion sequence is usually required, which is practically undesirable. For instance, Yang et al. (2018) produces deterministic sequence of 2D motions, which is followed by synthesizing the appearances frame-by-frame through adversarial training. Action-conditioned 2D human behavior modeling is also studied in Cai et al. (2018), where 2D pose generator and motion generator are trained progressively. Very recently, the efforts of (Weng et al., 2019;Huang et al., 2020) consider the related task of extracting 3D characters from single images, which is then animated to form 3D motions; de Souza et al. (2020) addresses another related task of generating human action videos by composing the human motions and scenes with probabilistic graphical models in 3D game engine;. However, the motions used in both methods are real-life motions that have been made available in prior, instead of being synthesized on the spot. Overall, the existing methods fall short in the following aspects: 1) direct modeling of 2D motions is inherently insufficient to capture the underlying 3D human pose articulations and shape deformations. The absence of 3D geometric information often leads to visual distortions and ambiguities; 2) coordinate locations of body joints are commonly used as the human pose representation, which undesirably entangle the human skeletons and their motion trajectories. Moreover, this creates extra barriers in modeling human kinematics; 3) initial poses often impede the diversity of generated human dynamics. For example, in actions such as warm up and boxing, initial poses crucially influence the formation of the rest sequences; and 4) the popular choice of pixel-to-pixel synthesis among existing efforts on action conditioned video generation has been evidenced incapable of generating detailed and high-resolution views. The aforementioned observations inspire us to consider a two-step pipeline: action2motion generates diverse & natural 3D human motions from prescribed action categories, and motion2video proceeds to extract human character out of an additional input image, to rig, animate, and render to form 2D videos, as illustrated in Fig. 1. In action2motion, we aim at generating diverse motions to traverse the motion space, and to cover various styles of individuals performing the same type of actions; meanwhile, each motion is expected to be visually plausible. This leads to our temporal variational autoencoder (VAE) approach using Lie algebra pose representation. Inspired by the work of Denton and Fergus (2018) in generic video generation, here we leverage the posterior distribution learned from previous poses as a learned prior to gauge the generation of present pose; by tapping into the recurrent neural net (RNN) implementation, this learned prior also encapsulates tem-poral dependencies across consecutive poses. For pose representation, human pose could be characterized as a kinematic tree based on human body kinematics. There are multiple advantages of using Lie algebraic representation over the popular joint-coordinate representation: (i) Lie representation disentangles the skeleton anatomy, temporal dynamics, and scale information; (ii) it faithfully encodes the anatomical constraints of skeletons by following the forward kinematics (Murray et al., 1994); (iii) the dimension of Lie algebraic space corresponds exactly to the degree of freedom (DoF), which is more compact compared to joint-coordinate representation. In practice, the adoption of Lie representation notably mitigates the change-of-length and trembling phenomenons prevailing in joint coordinates representations; it also facilitates the generation of natural, lifelike motions, and simplifies the training process. Furthermore, a global and local movement integration module is used to infer the global pose trajectory from temporal articulations of body parts. This promotes consistence between local shape deformations and global motion trajectory (i.e. direction and velocity), especially when synthesizing locomotion actions such as walking and jumping. It is followed in our pipeline by motion2video, where a 3D character is extracted, rigged, animated according with stochastically generated motions, and rendered to form 2D videos. In fact, animating 3D characters remains an open problem. A common strategy is to extract their 3D shapes and textures from a single input image. Prior efforts such as Weng et al. (2019) align the silhouette and texture of single image to a 3D human shape (e.g. SMPL (Loper et al., 2015)). Due to single input view, nonetheless, they fail to synthesize body textures of unseen views. Recent deep learning methods (Lazova et al., 2019;Saito et al., 2019Saito et al., , 2020Huang et al., 2020;Zheng et al., 2021) shed lights on reliable recovery of 3D surfaces and textures from single images. Meanwhile their results suffer from either lowfidelity, with input image resolution limited to at most 512×512 (Saito et al., 2019;Huang et al., 2020;Zheng et al., 2021), or ill-posed texturing on occluded areas and novel view (Saito et al., 2020). A simple strategy is developed in our work, leading to improved texture mapping in these cases. In summary, our main contributions are three-fold: first, a novel two-step pipeline of action2motion & mo-tion2video is proposed to address the challenging problem of 3D human motion & video generation from action type and single image; second, a dedicated Lie Algebra based VAE framework is developed, capable of producing diverse life-like human motions from prescribed action categories; third, as part of our pipeline, an improved strategy is used in extracting 3D shapes and textures from single images, that is capable of synthesizing visually-appealing texture of unseen views. Moreover, an in-house 3D human motion dataset, Hu-manAct12, has been curated. This paper differs from our preceding effort (Guo et al., 2020) in a number of aspects: -A more general problem of 3D human video generation is considered here, where the task of ac-tion2motion examined in Guo et al. (2020) becomes the first step of our solution pipeline. The motion2video step is entirely new from Guo et al. (2020). -A new local-global movement integration module is proposed, which significantly improves the synthesized 3D locomotion results when comparing to Guo et al. (2020). -A much broader and more thorough discussion is provided comparing to our short version (Guo et al., 2020). It also includes applications to latent interpolation, action transition, outpainting, as well as evaluation of the synthesized motions from coarsevs. fine-grained action categories. Related Work Our focus is to review literature related to generating video of human full-body motions, instead of the more generic theme of video generation (Tulyakov et al., 2018;Denton et al., 2017;Vondrick et al., 2016). Our tally includes the discussion of action video generation (Sec. 2.1), the generation of human motions (Sec. 2.2), motion transfer and rigid body animation (Sec. 2.3). We also review related activities of VAE sequence modeling (Sec. 2.4), skeletal human pose representation (Sec. 2.5), and 3D human motion datasets (Sec. 2.6). Action Video Generation The task of generating human action videos has drawn research attentions very recently. In the work of Cai et al. (2018), 2D human motions are generated from known actions, they are then synthesized into 2D videos frame-by-frame with U-Net (Ronneberger et al., 2015) and a dedicated image discriminator. In Yang et al. (2018), based on an initial 2D pose extracted from a given image, a deterministic sequence of future 2D poses is produced for given action category; this pose sequence are subsequently used to guide video generation via adversarial training. A similar method is considered in Kim et al. (2019), where future 2D poses are instead generated stochastically with variational auto-encoder. These efforts focus on tiny pixel-wise video generation, and human poses are manipulated in 2D image space. A recent work (de Souza et al., 2020) propose to generate 3D human videos directly from 3D game engine using scene composition rules and procedural animation techniques. Our work differs from this work in two folds: 1) de Souza et al. (2020) generate 3D motions by extracting atomic motions from existing motion capture (Mo-Cap) datasets, then stitches these atomic motions into action sequences through predefined rules. For example, a walking animation involves repetitions of swinging a left leg, then swinging a right leg, as well as corresponding pendular arm movements. However, this process is fairly labor-intensive. In our work, diverse 3D actions are automatically produced from a learned generative model end-to-end; 2) de Souza et al. (2020) animate artist-designed 3D avatars (rigid and clothed), while our method generates videos by rigging and animating characters with their 3D shapes and textures extracted from single 2D images. Human Motion Generation In addition to video generation, there are also research efforts focusing on synthesizing human motions, usually in the form of 2D or 3D skeletons, where the input could be of various forms, including but not limited to audio and text. One trendy research direction aims to generate deterministic motion sequences, which is typically realized by RNN models. For example, Tang et al. (2018) and Shlizerman et al. (2018) adopt LSTM models to translate music beats to body motion dynamics. In the efforts of Lin et al. (2018), Ahn et al. (2018), Plappert et al. (2018), and Yamada et al. (2018), human motions are generated from textual descriptions through a encoder-decoder RNN model. Ahuja and Morency (2019) considers a closely related task of constructing a joint embedding space between sentences and human pose sequences. The work of Stoll et al. (2020) engages neural machine translation model with attention mechanism for text-to-sign-pose prediction. Similarly, a recurrent architecture is used in Pavllo et al. (2020) to unfold an input global trajectory to locomotive humanoid movements. To enable the stochasticity of human dynamics, deep generative models are also considered. Habibie et al. (2017) propose a recurrent variational autoencoder model for global trajectory based locomotion generation. Lee et al. (2019) use GANs model to generate diverse movements from music signals. Huang et al. (2021) explore a curriculum training strategy to allow variable sequence lengths. In Cai et al. (2018), a two-stage GAN framework is proposed to generate 2D human motion pro-gressively. To synthesize human motions from scratch, Zhao et al. (2020) and Zhao and Ji (2018) make use of Bayesian inference; the work of Yan et al. (2019) instead considers a combined strategy of graph convolutional networks and GANs. The recent work of Xu et al. (2020) synthesizes novel motions by free combination of style and content codes extracted from existing MoCap library. Motion Transfer and Rigid Body Animation Motion transfer is a traditional topic, aiming to transfer human motions from a source object to target. Recent deep learning based efforts typically consider 2D pixel-wise approaches, where mappings from source and target are based on local pixels or 2D patches. Wang et al. (2018) and Chan et al. (2019), for example, directly learn to map between human poses and appearances of one specific source subject. The aim of (Siarohin et al., 2019;Wang et al., 2019a;Lee et al., 2020;Liu et al., 2019a) is to work toward a more general problem of driving an arbitrary target image with a source 2D pose sequence or videos. This is often realized by establishing connections between the source pose sequence and the target textured shape extracted from an given image, followed by warping the reference image to form the target video frame-by-frame. Although assembling promising results, the mainstream pixel-wise approaches nonetheless possess a number of limitations, including its innate difficulties in dealing with changing views or lifting to 3D motion spaces, as well as the level of complications in producing high-resolution and sharp images. The works of Aberman et al., 2020) also consider a similar task, where motions from the source 3D character are re-targeted to 3D characters with different skeletons (e.g. joint number, bone lengths). Meanwhile, the 3D shapes of these target characters have been artistically designed and well-rigged ahead of time. Meanwhile, it has also been a continuous line of research on rigid body animation of 2D/3D human characters that is especially empowered by advances in computer graphics techniques. Early work such as Zhou et al. (2012) uses a simple pose-retrieval framework, where a segmented garment database indexed by 2D skeleton poses is built for online searching during human image animation. Rigged human models are exploited in later endeavors for articulated object animation. In Hornung et al. (2007), characters extracted from 2D pictures are driven as-rigid-as-possible by external 3D MoCap sequences. At intermediate steps, a 2D mesh with 2D skeleton is constructed for the shape extracted from input image. Weng et al. (2019) further lifts this animation process into 3D space. Specifically, a semi-naked SMPL template is drawn out of 2D images, and deformed to a rigged 3D mesh model with boundary that closely matches to the human silhouette in input image. The recent work of Huang et al. (2020) learns to directly predict a 3D animatable clothed human shape from a single image. VAE in Sequence Modeling Variational autoencoder (Kingma and Welling, 2014) are the encoder-decoder neural nets trained by maximizing the marginal data likelihood with variational methods. It has been widely used in the so-called deep generative models as a powerful learning technique in addressing various learning scenarios, including conditional generation (Sohn et al., 2015), semi-supervised learning (Kingma et al., 2014;Siddharth et al., 2017), controllable generation , few-shot learning (Schonfeld et al., 2019), disentangle representation learning (Ding et al., 2020;Zhu et al., 2020;Higgins et al., 2016) and VAE-GAN architecture (Larsen et al., 2016). To work with sequential data, VAEs are typically plugged in a recurrent network model, e.g. GRU and LSTM. Variational RNN (Chung et al., 2015), a pioneer work, uses vanilla RNN to model temporal dependencies in intermediate time-frames. The RNN output of previous frame is used in generating posterior and prior distributions, as well as the follow-up decoding process. Variational RNN has been particularly favored in speech generation and handwriting character generation. Bowman et al. (2016) and Yang et al. (2017) investigate the LSTM-based VAE for NLP modelling based on a sequence-to-sequence architecture, where the sequence encoder predicts a posterior distribution, from which the sequence decoder samples a latent vector and reconstruct the sequence. More specifically, temporal VAE models has been considered in motion and video generation. Marwah et al. (2017) consider generating videos from textual caption, which is incorporated as semantic attentive vectors and fed to their temporal VAE. In VideoVAE (He et al., 2018), on the other hand, a structured latent unit is devised to model conditional factors including motion category and an initial frame to complete the rest frames. To predict future frames under uncertainty, Denton and Fergus (2018) inspect the use of two separate RNNs to capture temporal dependencies of conditional posterior and prior spaces. Similar network structure is also scrutinized in Wang et al. (2019b), where it is extended to synthesize videos with pre-specified start and end frames. In terms of 3D motion prediction, given a start human pose, Habibie et al. (2017) complete the rest 3D human motion with a LSTM-based VAE model. In Yan et al. (2018), similar model is engaged to learn the transition from observed sequence to future sequence for stochastic motion forecasting. A very recent work by Aliakbarian et al. (2020) adopts VAE and a mix-and-perturbation strategy to statistically predict future motions. Skeletal Human Pose Representation A number of human pose representations have been considered over the years. The most-often used option is the joint-coordinate representation (Han et al., 2017;Hussein et al., 2013) that directly characterizes the human pose by an ordered sequence of 2D/3D joint coordinates. It has a few variants: Wang et al. (2012) consider incorporating the pair-wise relative positions of neighboring joints; meanwhile, only those informative joints are utilized in Chaaraoui et al. (2014). Part-based method is another line of pose representation. Specifically, a human pose is modeled as a ordered list of body parts. For example, in Yacoob and Black (1999), human body is divided into five main parts (i.e. torso and four limbs); pose sequences are then formulated by the displacement and rotations of body parts over time. Alternatively, the work of Müller (2007) models the temporal information using dynamic time warping. Finally, Lie group or axis-angle based representation (Gavrila et al., 1995;Vemulapalli et al., 2014;Huang et al., 2017;Xu et al., 2017;Liu et al., 2019b;Pavllo et al., 2020) characterizes the skeleton as a kinematic tree, with its articulations realized by forward kinematics. 3D Human Motion Datasets CMU MoCap (CMU, 2003) and HDM05 have more than 100,000 3D poses and 2,000 3D motion sequences that are associated with succinct textual descriptions. Unfortunately, the motions are markedly uneven-distributed over action categories. UTKinect-Action (Xia et al., 2012) and MSR-Action3D (Li et al., 2010), on the other hand, have much smaller tally of motion sequences. NTU-RGBD (Liu et al., 2020) is by far the largest human motion dataset, consisting of over 100,000 motions belonging to 120 classes. Nevertheless, the joint positions acquired from Microsoft Kinect-I cameras are notably inaccurate. These observations motivate us instead curating our in-house 3D human action dataset, HumanAct12, as well as revamping the pose annotations of NTU-RGBD. Preliminary Backgrounds Variational Auto-Encoder Variational auto-encoder(VAE) (Kingma and Welling, 2014) consists of an encoder and a decoder, which are normally two separate neural networks. Its goal is to learn a θ-parameterized generative model, p θ (x, z), over data x and latent variables z. Technically, the learning objective is to maximize the likelihood function of x, which could be further formulated as a marginal likelihood with regard to the latent variable z, p θ (x) = z p θ (x|z)p θ (z). Following the variational principle, a φ-parameterized neural network(i.e. encoder), q φ (z|x), is engaged to approximate the unknown posterior distribution p θ (z|x). We thus obtain the the following evidence lower bound (ELBO) to our data likelihood function: log p θ (x) = log z p θ (x|z)p(z) ≥ E q φ (z|x) log p θ (x|z) − D KL (q φ (z|x) p(z)). (1) The first ELBO term encourages the generated samples to be sufficiently close to the real samples; the second term penalizes KL-divergence between the prior and the approximated posterior distribution. Subsequently, the original objective of maximizing the data likelihood over data x becomes that of maximizing over the θand φ-parameterized ELBO function. In Sohn et al. (2015), a follow-up conditional variational auto-encoder (CVAE) framework is conceived by introducing a conditional variable, y, as log p θ (x|y) = log z p θ (x|z, y)p(z|y) ≥ E q φ (z|x,y) log p θ (x|z, y) − D KL (q φ (z|x, y) p(z)). (2) Lie Groups and Lie Algebras In what follows, we provide a succinct introduction of Lie groups and Lie algebra basics. Interested readers may refer to (Murray et al., 1994) for more details. Lie groups. Mathematically, a Lie group is a group as well as a smooth manifold. 3D rotation transformations, also known as the Special Orthogonal group, SO3 = {R ∈ R 3×3 |R R = I, det(R) = +1}, is a classical example of Lie group. Moreover, the product of multiple SO3 groups (i.e. a kinematic chain) is still a Lie group. In other words, for a tree-structured human skeleton model, each of the kinematic chains corresponds to a point in Lie group SO(3) × SO(3) × · · · × SO(3). As a consequence, it is usually far from being trivial in terms of optimization in such a curved space. We instead work in its tangent space, also known as Lie algebra so(3)being a flat space, our familiar linear algebra techniques could work again. Lie algebras. The tangent space of Lie group SO(3) at identity I 3 is referred to as its Lie algebra so(3). Each element of so(3) is in the form of a 3×3 skew-symmetric matrixŴ , aŝ W =   0 −w 3 w 2 w 3 0 −w 1 −w 2 w 1 0   ,(3) which essentially spans a 3-dimensional vector space, w = (w 1 , w 2 , w 3 ) ∈ R 3 . Exponential map. To map from a Lie algebra el-ementŴ ∈ so(3) to a point in the manifold (i.e. Lie group), R ∈ SO(3), an exponential map exp : so(3) → SO(3) is formulated as R = exp (Ŵ ) = I+ sin( w ) w Ŵ + 1 − cos( w ) w 2Ŵ 2 . (4) Here · is a vector norm. Since w is periodically mapped to R, in practice we normally limit w by its norm within the range of [−π, π]. Its inverse map, the logarithm map log(SO(3)): SO(3) → so(3) map be similarly constructed. Our Approach The pipeline of our approach, action2video, consists of two steps: step one (action2motion) synthesizes human pose sequences from a prescribed action category (Sec. 4.1); step two (motion2video) extracts a specific 3D human shape and texture from a reference image to render the generated motions into 2D videos (Sec. 4.2). Step One: Action2Motion Our action2motion framework comprises a temporal VAE (Sec. 4.1.2) with a Lie algebra based representation (Sec. 4.1.1). We also investigate four strategies to decode neural hidden unit to obtain global 3D positions of motions (Sec. 4.1.3 and Sec. 4.1.4). Disentangled Representation with Lie Algebra As shown in Fig. 2, a human pose could be characterized in the form of a kinematic tree that consists of five kinematic chains: main spine and four limbs. Meanwhile, this skeleton model is formed by N oriented edges (i.e. bones) E = {e 1 , . . . , e N } that interconnect N + 1 joints. By incorporating Lie algebraic apparatus, motion of 3D joints could be decomposed into three parts: skeleton anatomical information, motion trajectories, and bone lengths. For each skeletal bone, e n , a local coordinate is attached, with the bone itself being aligned with the xaxis and its starting joint being stuck to the coordinate origin. The relative 3D locations between two consecutive bones could be modeled as a series of 3D rigid transformations. Specifically, given two connected bones e n and e n+1 along a kinematic chain, a joint c = (x, y, z) in the local coordinate of e n amounts to a transformed location c = (x , y , z ) in the local coordinate of e n+1 , by exercising the following transformation c 1 = R n d n 0 1 c 1 .(5) Here, R n ∈ R 3×3 is a rotation matrix, d n = (b n , 0, 0) ∈ R 3 a translation vector along x-axis, and b n the length of bone e n . For a 3D rotation matrix R ∈ SO(3), the associated Lie algebraic vector w ∈ so(3) is an axis-angle vector. For a human skeleton, the exact degree of freedom (DoF) of a axis-angle vector is determined by the rotation orientations of two successive bones, and is up to 3. For example, if two bones are oriented in the same or reverse direction, w is a zero vector with 0 DoF; if one bone only rotates along one axis, then the DoF reduces to 1. Mapping Lie algebra parameters to 3D positions. Now we focus on an articulate object with K kinematic chains; assume the k-th chain have m k joints, with each joint parameterized by a 3-dimensional so(3) vector, w k i , i ∈ {1, 2, . . . , m k }. A human pose is thus represented by composition of Lie algebra vectors of joints/bones on kinematics chains, p Lie = (w 1 1 , . . . , w 1 m1 , . . . , w K 1 , . . . , w K m K ). Now, the 3D position of a joint i in a chain k, J k i , is obtained following a exponential map of the Lie algebraic values, also known as forward kinematics, as J k i =   i−1 j=0 exp(Ŵ k j )   d k i + J k i−1 .(6) Here d k i = (b k i , 0, 0), with b k i representing the bone length of e k i . In addition, forward kinematics typically starts from a root joint whose position J 0 ∈ R 3 , and Lie algebraic valuesŴ 0 stand for the global location and orientation of the entire human body. In our representation, the global location J 0 is independent from the pose. Therefore, given a motion with T successive poses, the sequence (J 0,1 , . . . , J 0,T ) ∈ R 3×T makes up the body motion trajectory, with J 0,t denoting its global location at frame t. Accordingly, the 3D coordinates vector of a body pose, formally denoted as p = (J 1 1 , . . . , J m1 1 , . . . , J 1 K , . . . , J m K K ) could be obtained by the jointwise forward kinematics of a composition of bone lengths, root position, and Lie algebraic vector. For simplicity, we denote this mapping as Γ(p Lie ) : p Lie → p. Overall, a human motion is represented by three parts: -Lie algebra parameters M Lie = p 1 Lie , . . . , p T Lie . -Root trajectory (J 0,1 , . . . , J 0,T ): root trajectory could be represented by either absolute root locations or relative translations between consecutive root locations. The latter works better in our setting. -Bone lengths (b 0 , . . . , b N ): due to the invariant nature of bone lengths of human skeleton, the skeleton bone lengths are acquired from typical real-life human bodies, and are fixed over time. This also reciprocally enables us to generate motions with controllable body scales by manipulating the bone lengths. Conditioned Temporal VAE Consider a real motion or pose sequence M = (p 1 , . . . , p T ). Our temporal VAE aims to maximize the likelihood of the pose sequence M. At time t, a posterior network q φ (z t |p 1:t ) approximates the true posterior distribution conditioned on p 1:t−1 . Then, with sampled latent variables z 1:t and previous states p 1:t−1 , our RNN genera- Fig. 3 Visual diagram of action2motion, the first step in our pipeline. Top row shows the training phase: at time t, the posterior and prior networks take as input a concatenation of three parts -action category a, time counter c t and immediate pose vector (p t or p t−1 ). The generator receives an addition latent vector z t that is sampled from the learned posterior distribution. Afterwards, the 3D joints of current pose is obtained from the decoder of generator through pose decoding module. Bottom row depicts the testing phase: a latent vector is alternatively sampled from the prior distribution, which triggers the aforementioned process in generating 3D pose sequences. tor p θ (p t |p 1:t−1 , z 1:t ) reconstructs the current pose p t . This leads to the following variation lower bound: a. plain log p θ (M) ≥ t E q φ (zt|p1:t) log p θ (p t |p 1:t−1 , z 1:t ) − D KL (q φ (z t |p 1:t ) p(z t )) .(7) Note at time t, our RNN module takes as input the immediate past frame p t−1 and z t . The influence from previous time slices p 1:t−2 and z 1:t−1 lies in the ability of RNN module capturing long-term temporal dependencies. In terms of the prior p(z t ), one option is to consider an identity Normal distribution, N (0, I). This is unsuitable though for the motion generation problem, as the pose variation varies over time. Take running motions as example, the temporal pose variances are typically relatively small, which however could become significantly larger when e.g. the runner makes a Uturn. Inspired by the observation that the variation of present pose is highly correlated to its past timesteps (Denton and Fergus, 2018), we model its prior by a neural network that conditions on its previous steps p 1:t−1 , p ψ (z t |p 1:t−1 ). This leads to a re-formulation of the ELBO objective function log p θ (M) ≥ t E q φ (zt|p1:t) log p θ (p t |p 1:t−1 , z 1:t ) − D KL (q φ (z t |p 1:t ) p ψ (z t |p 1:t−1 )) ,(8) where the distance penalty between prior and posterior distributions further encourages temporal consistency. Architecture of Action2Motion Our action2motion step consists of three main components: posterior network, prior network, and generator, which are shown in Fig. 3. The input vector contains the following parts: the pose vector p t or p t−1 , an onehot vector a to encode action category, and c t ∈ [0, 1], a time-counter to keep record of where we are in the sequence generation progress. As depicted in Fig. 3, during training, a noise vector is sampled from the posterior distribution q φ (z t |·), and fed into the generator, which then produces the final 3D pose prediction by running through the pipeline of encoder E n , GRU unit GRU θ , decoder D n , and pose decoding module. In testing, as the real data p t is not available, z t is instead sampled from the learned prior distribution, p ψ (z t |·). Specifically, our encoder E n and decoder D n are composed of linear fully connected layers with different weights, and updated with the whole network. Moreover, our posterior network (q φ ) and prior network (p ψ ) utilize the same architecture, but with different parameters. They are respectively described as: h t = E n (p t , a, c t ), c t = t T (µ φ (t), σ φ (t)) = GRU φ (h t )(9) and h t−1 = E n (p t−1 , a, c t ), c t = t T (µ ψ (t), σ ψ (t)) = GRU ψ (h t−1 ).(10) Further investigation of the pose decoding module is provided in the following section. Fig. 4 illustrates the four pose decoding variants investigated in our work. The most straightforward and commonly-used approach is Fig. 4(a), where the 3D joint locations are directly and simultaneously regressed from the decoder. It however contains redundant parameters, and does not follow the kinematics law that dictates the 3D articulations of the body skeleton. Alternatively, the Fig. 4(b) variant incorporates Lie algebraic representation, which is the one adopted in our previous work (Guo et al., 2020). The decoder here contains two vectors, skeletal Lie algebraic valuesp t Lie , and global root positionĴ 0,t . The final 3D joints are produced by forward kinematics (see Sec. 4.1.1). Though working well for many motion scenarios, it encounters issues when local body movements and global motions are highly correlated. Take action walk for example, the instantaneous velocity of walking is significantly affected by the movement of legs; independently generating global and local body motions is observed to lead to e.g. sliding-feet phenomenon, as depicted in Fig. 12. Pose Decoding Global and local movement integration. Existing efforts in motion forecasting or generation usually predict only relative body joint positions, this is, relative to the root joint, at the cost of neglecting the global motion all together Yan et al., 2019;Liu et al., 2019b;Xu et al., 2017). In other words, the root joint of human full-body is fixed to coordinate origin during the entire motion sequence. Recently, Adeli et al. (2020) consider global motion by directly enforcing MSE or 2 loss between predicted and ground-truth root joint locations, which is similar to the Fig. 4(a) variant. Intuitively, the transition between two consecutive poses, measured by the displacement of the root joint in the two frames, is highly correlated to the body gesture of these two poses. Consider a person who is walking on a flat ground, his walking pace depends upon how wide his legs span. This inspires us to propose a global and local movement integration unit (GLMI) which, rather than predicting global transition and local joints concurrently, will first generate relative poses, then infer global motion from consecutive local poses, as illustrated in Fig. 4 (c). Herep t Lie is the Lie parameter vector produced by the generator, which is then transformed to 3D joint locationsp o t through forward kinematics; p o t−1 is the offset value of 3D coordinates of previous pose; h o t is a hidden vector containing upstream information. The three vectors are fed into a fully connected layer, MLP, which then produces the velocity (i.e. relative translation)V 0,t at time t. Finally, the 3D global positionp t could be obtained by summation of the three components: root position of previous pose J 0,t−1 , estimated velocityV 0,t , and the current local posep o t . Mathematically, this process is expressed as Fig. 5 Illustration of the motion2video process. Shapes and textures of 3D human characters are extracted from single 2D images, that are rigged, animated with motions generated from the action2motion step, and rendered to produce final videos. p t Lie , h o t = D e (h θ t ) p o t = Γ(p t Lie ) V 0,t = MLP(p o t , p o t−1 , h o t ) p t =p o t + J 0,t−1 +V 0,t .(11) To further capture the temporal dependency of a global trajectory, another version of GLMI is also proposed, with the backbone of MLP replaced by recurrent units, GRU, as presented in Fig. 4(d). Besides, a trajectory alignment loss between the predicted veloci-tiesV 0,t and real velocities V 0,t is also introduced, to encourage accurate velocity estimation. Among these variants, the GLMI-M variant is found to produce the overall best results, and is utilized in our approach by default. Final Objective To summarize, our final objective function becomes L θ,φ,ψ = − T t=1 E q φ (zt|p1:t,a,ct) log p θ (p t |p 1:t−1 , z 1:t , a, c t ) − λ kl D KL (q φ (z t |p 1:t , a, c t ) p ψ (z t |z 1:t−1 , a, c t )) − λ align V 0,t −V 0,t 2 ,(12) where λ kl and λ align are two tuning parameters to tradeoff among reconstruction error L rec , KL-divergence, and trajectory alignment loss. Empirically, a larger λ kl is observed to enhance the quality of generated motions but may decrease their diversity; and vice versa for a smaller λ kl . For the reconstruction error (the first term in Eq. (12)), the per-joint loss suggested in Aksan et al. (2019) is considered, as L rec (p t ,p t ) = N +1 k=1 J k,t −Ĵ k,t 2 .(13) Here N + 1 denotes the number of skeletal joints. In our work, the trajectory alignment loss is only used in the methods of Fig. 4(c) and (d), where the models are trained with the re-parameterization trick of Kingma and Welling (2014). Training Strategy One common issue in sequence modeling is the discrepancy of information exposure during training vs. testing phases. For example, in a RNN model, a ground-truth pose is taken as input to generate next pose in training; while in testing phase, a generated pose is used instead to produce next pose. To mitigate the issue, a mixed training strategy is adopted here, that chooses whether to use (or not to use) teacher forcing by randomly draws from a Bernoulli distribution, V ∼ Bernoulli(p tf ). In particular, teacher forcing is chosen for the entire sequence p 1:T if V is 1, and not if otherwise. As a boundary condition in generating the initial posep 1 , its previous pose input p 0 for the prior network (q ψ ) is a zero vector. In addition, curriculum learning (Bengio et al., 2009) is used in the training phase that is to progressively increase the value of λ kl . Step Two: Motion2Video Recall in step one of our approach, action2motion, diverse motions are generated from prescribed action categories. At this point, a motion is shown as a sequence of 3D skeletal articulations. To produce videos, it remains to settle the full-body shapes and textures of the involved human characters. This is addressed in step two, motion2video, where a specific setup is conceived: a reference person image is presented as input, from which 3D shape and texture of the person are extracted; this is followed by rigging and animating the characters with synthesized motions from the action2motion step, and rendering to generate final 2D videos. Unlike existing motion transfer methods (Chan et al., 2019;Liu et al., 2019a;Wang et al., 2019a) that emphasize in 2D space, our work advocates a fully 3D approach, and we claim our 3D-enabled modelling choice helps to preserve the geometric and appearance aspects in the final video production. our motion2video process that is to be detailed in the following subsections. Human Shape Reconstruction from a Single 2D Image From a single 2D image, a 3D human character is extracted to preserve sufficient geometric and textural details consistent with the input. PIFu (Saito et al., 2019) and PIFuHD (Saito et al., 2020) are the two state-ofthe-art methods on single-image based human shape recovery that have their unique pros and cons. The 3D shapes and textures extracted by both methods are reasonably adhere to their 2D image inputs. Meanwhile, the texture map extracted by PIFu (Saito et al., 2019) has relatively low resolution and accuracy, see e.g. the protruded knee pointed by the red arrow in Fig. 6(b). Although PIFuHD produces high-resolution 3D human geometry construction, notable errors are introduced at the unseen side by the symmetric assumption. As e.g. shown by the red arrows in Fig. 6(c), the frontal human face is also erroneously synthesized at the back side of the 3D character head. Aiming at refining the reconstruction results, our improved variant takes advantage of PIFuHD in better estimating 3D geometry and camera-view appearance, as well as PIFu in better inpainting of texture for the unseen views. Moreover, we also adopt a heuristic in producing smooth transition near the boundary of visible and occluded surface regions, as follows: to detect the stitching boundary, we project the character (facing Z + direction) onto XY plane and match the edge of 2D silhouette with the 3D character; for a point x in the transition region or inside the occluded region O with color c x , its color c x is expected to be close to the color c p x of the corresponding point on PIFu surface; at the same time, c x should also be close to those of its neighbors, N x . This is formulated as the following convex objective function, min x∈O c x − c p x 2 + λ nn 1 |N x | x ∈Nx c x − c x 2 .(14) In practice, the vertex colors c x in O are iteratively updated until a consistent convergence. For transition near the boundaries, only the second term of Eq. (14) is considered. As shown in Fig. 6(d), our result is able to leverage the benefits of of both PIFu and PIFuHD methods, and produces a more natural transition near the boundary regions. Rigging, Animation, and Rendering Fitting SMPL for extracted 3D shape. The SMPL human shape, a generative 3D human representation controlled by pose and shape parameters, is used to facilitate the follow-up rigging and animation process. This requires to fit SMPL as close as possible to the reconstructed 3D human shape that amounts to estimating the pose (θ) and shape (β) parameters by minimizing the following composite objective, L(β, θ) = L surface (β, θ) + λ j L joints (β, θ) + λ r L reg (θ).(15) The joints fitting term L joints enforces the joints location of the SMPL shape to match with the predicted 3D joints from 2D image. Here, the initial 3D joints predic-tionĴ c is obtained by regressing 2D joints from input image with OpenPose (Cao et al., 2021), and by inverse projection into the reconstructed 3D human shape. Denote f (·) a transformation function of specific joint from initial position to current position following skeleton kinematics chain. Denote ρ(·) a differentiable Geman-McClure penalty function (Geman and McClure, 1987), and w the confidence of 2D joint prediction. We have, L joints (β, θ) = i∈|J| ω i ρ f (J(β) i , θ) −Ĵ c,i .(16) Then the surface fitting term L surface is applied to minimize distance between vertex S i of the reconstructed human shape S and its nearest vertex v of the SMPL shape M(β, θ), L surface (β, θ) = i∈|S| min v∈M(β,θ) S i − v 2 .(17) Finally, the pose regularization term L reg (θ) penalizes unusual poses through the learned Gaussian mixture model from CMU dataset (CMU, 2003). Following (Bogo et al., 2016), it is of the form L reg (θ) = − log i (g i N (θ; µ θ,i , Σ θ,i )),(18) where N (θ; µ θ,i , Σ θ,i ) is a Gaussian distribution with its mean µ θ,i and variance Σ θ,i , and g i are weights of mixture Gaussian model. In practice, to minimize the above objective function, during the first two iterations we only consider the joints and the pose regularization constraints for quick convergence; the surface constraint is then incorporated during the rest iterations. 3D model deformation and animation. After obtaining the above optimized SMPL model that closely fits to the reconstructed 3D human mesh model, the SMPL model is used as an anchor to deform the 3D models to new poses. To start with, the vertex-level correspondences between the SMPL surface and the 3D human model are established by nearest neighbor search. In addition, body part information is used to eliminate possible mismatched pairs, especially these around the inter-joint of arms and torso. Specifically, the body parts information of reference image could be obtained using DensePose (Alp Güler et al., 2018), which then are back-projected to the surface of the 3D shape. As SMPL shape has pre-defined body segmentation, this could be utilized to filter out vertex pairs coming from different body parts. Next, we compute a displacement map from the optimized SMPL mesh to their correspondences on the 3D human model, S j = M i (β * , θ * ) + d i→j .(19) where β * and θ * are the optimized shape and pose parameters of the SMPL model. S j and M i (β * , θ * ) are the correspondences and d i→j is the displacement from optimized SMPL model to reconstructed 3D human model. Intuitively, to repose the human shape, we could acquire the target positions S * of shape vertices by applying the displacement map to the reposed SMPL as in Eq.(19). However, this will lead to imperfections due to free-form deformation. Following Zuo et al. (2020), we instead utilize the vertices of S * as control points to deform the 3D human model as rigid as possible, by enforcing a local rigidity constraint. The locally rigid deformation R and the deformed human modelŜ are obtained by minimizing the following objective, L def (R,Ŝ) = i∈|S| j∈Ni k ij (Ŝ i −Ŝ j ) − R i (S i − S j ) 2 + l∈|S| Ŝ l − S * ,l 2 .(20) Here N i is the set of the neighboring vertices of S i ; k ij is the corresponding weights of neighboring vertices. R i is a rotation matrix. The above objective function is optimized by iteratively solving the rotation matrix R and the deformed meshŜ (Sorkine and Alexa, 2007). Rendering. The target 3D shape are deformed and driven by the generated pose sequences frame-by-frame, which are subsequently fed into 3D game engine (Unity3D) to integrate physical conditions such as illuminations and shadows and produce the final videos. Specifically, spot light and directional light are used to illuminate the character from top. Four cameras, fixed at half height of the 3D character, are aimed at the subject to record the front, back, left side and right side views, respectively. Empirical Evaluations A comprehensive set of experiments are conducted to systematically evaluate the performance of our action2video approach, which consists of the two-step pipeline of ac-tion2motion and motion2video. We start by introducing the related datasets, and our implementation details. This is followed by a detailed examination of our ac-tion2motion process at Sec. 5.1, and comparisons for our motion2video with related efforts at Sec.5.2. Finally, Sec. 5.3 provides a holistic evaluation of our full pipeline, action2video. Datasets. Ideally, we expect to work with motion datasets that contain considerable amount of distinct motion clips of various action categories, and with proper 3D pose annotations. In practice, we achieve this by postprocessing existing popular datasets, including reannotating 3D positions of NTU-RGBD (Shahroudy et al., 2016) and action categories of CMU MoCap (CMU, 2003). We also curate an in-house dataset, Human-Act12. In these three datasets, all human poses are uniformly annotated into 3D joints connected into 5 kinematics chains, with pelvis being the root joint. -NTU-RGBD is a large-scale 3D human motion dataset containing nearly one million motion sequences of 120 action types. Its pose annotation (i.e. 3D joint positions) is from MS Kinect readout, which is known unreliable and temporally unstable. In our experiments, the state-of-art video 3D shape estimation method (Kocabas et al., 2020) is employed to re-estimate the 3D poses from video feeds. Note in our scenario, it's sufficient for these poses to appear realistic, and they are not necessarily matched perfectly with the true poses. A subset of 13 distinct actions are further selected in our empirical evaluation, such as cheer up, pick up, salute, consisting of 3,900 motion clips. Each pose is represented by 18 joints (i.e. 17 bones). -CMU MoCap is dataset accurately annotated by motion capture markers, with 2,605 pose sequences. However, the dataset is not originally organized by action types. We identify 8 distinct actions based on their motion captions, including running, walking, climbing, jumping. In the end, 1,088 motions are re-organized by action type, with each skeleton constituting 22 3D joints (i.e. 21 bones). In implementation, these pose sequences are down-sampled from 100 HZ to a frequency of 12 HZ. -HumanAct12 is our in-house dataset that comes with proper annotations. It consists of 1,191 motion clips and 90,099 frames in total, which are categorized into 12 coarse-grained action categories, including e.g. warm up, lift dumbbell, and 34 finegrained action types such as warm up (Leg pressing), lift dumbbell (with right hand). The fine-grained annotations give more specific and dedicated information of the motions. We test our model on both coarse-and fine-grained annotations. Our dataset, HumanAct12, contains more accurate and stable 3D position annotations compared to NTU-RGBD; and has more well-organized action annotations than CMU MoCap. Note each body pose contains 24 joints (i.e. 23 bones). To showcase that our pipeline could work with wide range of applications, input images from myriad sources are considered in our experiments, as displayed in Fig. 7. They include images from the BUFF dataset , People Snapshot dataset (Alldieck et al., 2018), as well internet images, computer-generated (CG) images 1 , and our in-house captured images. BUFF 1 https://renderpeople.com/3d-people/ dataset provides 26 4D human sequences with different cloth styles and performing different actions. We then render 2D images from these human shapes. People Snapshot dataset contains 12 subjects and 24 video sequences with different backgrounds. More examples are provided in the supplementary file. Implementation Details. Our action2video pipeline is mostly implemented by PyTorch. For all encoder layers, the output size is set to 128. One-layer GRU is used for prior network, posterior network and pose decoding module, while generator uses two-layer GRU. The hidden unit size of GRU is 128. And the noise vector z and h o t has the dimension of 30 and 20 respectively. The Adam optimizer is applied for training throughout all experiments, with learning rate of 0.0002, weight decaying of 0.00001, and default parameter values including β 1 = 0.9, β 2 = 0.999. Our model is trained with mini-batch size of 128. To stabilize the training process, teacher forcing rate p tf is set to 0.6. The values of aforementioned hyper-parameters are fixed throughout our empirical experiments across all datasets. Afterwards, we generate motions with length of 60, 100 and 60 on NTU-RGBD, CMU MoCap and Hu-manAct12, respectively. The hyper-parameter λ kl is a trade-off between reconstruction constraints and KLdivergence penalty. During training, the value of λ kl for all datasets are initialized with 0.001 and linearly increased to 0.1, 0.1 and 0.01 at the end for above datasets respectively. During training, the value of λ align is set to 10 throughout these experiments. In motion2video step, to extract 3D shape from single image, λ nn and 10 neighbors are used in Eq. (14) for occluded region. The values of λ j and λ r in Eq. (15) are set to 2.0 and 0.2, respectively. Step 1: Action2motion Thorough evaluations of the action2motion step are carried out in this section. They include both quantitative and qualitative reports of motion generation results, and fine-grained analysis of the locomotion generation module; We also provide demonstrations of specific action2motion applications such as motion interpolation in the latent space, motion transition, and 3D motion outpainting. By default, the action2motion GLMI-M variant is utilized in our approach. Evaluations We start by introducing a tally of evaluation metrics and baseline methods used throughout this section, which is followed by a series of qualitative and quantitative evaluations. Evaluation Metrics. We aim to evaluate the generated motions from the aspects of being natural and diverse. To achieve this, the three metrics in Lee et al. (2019) are adopted in our evaluations: Frechet Inception Distance(FID) to characterize the visually realistic aspect, Diversity and Multimodality to quantify the diverse levels. The action recognition accuracy is additionally used to gauge the similarity between generated motions and real-life motions, as well as the degree of generated motions belonging to the prescribed action. FID is perhaps the most important indicator in our scenario. A lower FID suggests a better result. For multimodality and diversity, a result is claimed better only if its diversity and multimodality scores are closer to their respective values obtained from real motions. To calculate these metrics, we rely on a feature extractor to obtain the high-level features of motions. Since there is no standard implementation of such motion feature extraction, a vanilla RNN action recognition classifier is trained for each dataset; and the final layer of classifier is used as the motion feature extractor. We elaborate these four metrics as below: -Frechet Inception Distance(FID): FID is an effective metric to evaluate the overall quality in motion generation. A large amount (in our case, 3,000) of generated motions and real motions are sampled and then are transformed to two sets of features. For real motion, we sample from test set with replacement. Then, FID is measured by computing the dis-tance between the feature distribution of generated motions and that of the real motions. -Recognition Accuracy: Recognition accuracy is calculated as the accuracy of applying a pre-trained RNN action recognition classifier to the motion of interest. -Diversity: Diversity indicates the variance of the motions across all action types. Specifically, a large set of motions are sampled from all varieties of action types, from which two subsets are randomly sampled with the same size S d . The corresponding sets of motion feature vectors {v 1 , . . . , v S d } and {v 1 , . . . , v S d } are extracted respectively. Then, the diversity of this set of motions is evaluated by Diversity = 1 S d S d i=1 v i − v i 2 ,(21) where S d = 200 is used throughout our experiments. The multimodality is defined as Multimodality = 1 C × S m C c=1 Sm i=1 v c,i − v c,i 2 ,(22) where S m = 20 is used in our experiments. Baseline methods. Since the problem of action2motion, aka action-conditioned 3D human motion generation, is relatively new, there are few existing methods to compare with. We thus adapt the state-of-art methods from related areas to our context, as follows: -CondGRU. Condition GRU is used as a deterministic baseline in our setting, which is also the principal model for audio-to-motion translation in Shlizerman et al. (2018) and text-to-motion generation in (Ahn et al., 2018;Stoll et al., 2020). Here, a small modification of the model is made that the input is the concatenation of condition vector and pose vector at present step and the output is the pose vector for next step. learned to produce input latent vector for pose generator to synthesize pose at each time. By using adversarial training, the entire generated pose sequences are judged by a motion discriminator. We adapt this method for 3D human motion generation through necessary modifications. -Act-MoCoGAN. MoCoGAN (Tulyakov et al., 2018) is a widely used method for both conditional and unconditional video generation. While generating a video, the input noise vector are composed of two parts: one is a shared vector over time, another is a instinct noise vector sampled at each time. These two inputs are expected to map to the stationary content and dynamic motions in videos. In our experiment, to generate 3D human dynamics, we keep the original architecture and replace the video and image discriminators to motion and pose discriminators, respectively. -Dancing2Music. Dancing2Music (Lee et al., 2019) generates 2D dancing motion sequences from audio signals, which consists of two main stages, decomposition and composition. During decomposition, a motion sequence is segmented into short motion snippets, with dance unit VAE (DU-VAE) model being trained to generate these motion snippets given the latent vectors of motion content and an initial frame; during composition, a music-to-movement GAN (MM-GAN) is trained to generate latent vectors of motion snippet contents conditioned on the given music signals. To make a meaningful comparison, the official implementation is adapted by replacing the music signals with action categories. -LatentTransition. Wang et al. (2020) consider a two-stage GAN (Cai et al., 2018), with a Bi-LSTM being employed to produce input latent vectors for pose generation. An additional auxiliary action classifier further ensures the action-awareness of the generative model. -Action2Motion (plain). Oue action2motion variant by adopting the pose decoding module of Fig. 4(a), where the 3D position of joints are directly produced from generator. -Action2Motion (w/ Lie). Our action2motion variant with the pose decoding module of Fig. 4(b), where the Lie algebra parameters and root joint locations are generated independently. -Action2Motion (GLMI-M). Our action2motion variant with the pose decoding module of Fig. 4(c), where both the Lie algebra and GLMI are used, and GLMI is implemented by MLP. -Action2Motion (GLMI-R). Our action2motion variant with the pose decoding module of Fig. 4(d), where both the Lie algebra and GLMI are used, and GLMI is implemented by GRU network instead. Visual comparisons. Fig. 8 provides qualitative comparisons of skeletal motions generated from different methods: given an action category of warm up, two motions of length 60 are sampled, with every 6th frame being displayed. Conditional GRU (Shlizerman et al., 2018) requires as input an initial ground-truth pose to kick-start its generation process. Unfortunately the generated poses often collapse into a cloud of 3D points near the root joint. Two-stage GAN (Cai et al., 2018) produces better results, which however are still perceptually not satisfactory. The skeletal sequence result of Act-MoCoGAN by Tulyakov et al. (2018) is visually the best among these three methods. The generated poses nonetheless often froze to a fixed posture quickly. Dancing2Music (Lee et al., 2019) shows capability of yielding natural poses and motions. Meanwhile, a single such motion usually contains multiple actions, with the motion context deviating from the prescribed action type. For instance, in the left column of Fig. 8, the stick man first performs lift dumbbell (from t = 1 to t = 18), then a shorttime warm up (from t = 24 to t = 36), and finally drifting into drinking. On the other hand, LatentTransition always starts with natural poses, then struggles with proper modeling of long-term motion dependencies, which typically deteriorates to unrecognizable movements. These results are in sharp contrast to that of our four action2motion variants, whose results are in general visually more appealing. Here, the action2motion (plain) variant sometimes generate visual defects noticeable to human eyes. For example, in the left column of Fig. 8, the arm bone lengths of the same individual abnormally vary from t = 1 to t = 24. This is due to the intrinsic 3D-coordinate skeletal representation adopted by the plain variant that does not obey the underlying skeletal kinematics. Skeletal motions generated by the other action2motion variants are typically more faithfully resemble to real-life motions, which we attribute to their adherence to kinematics by their use of Lie group/algebraic skeletal representations. Diversity is another important evaluation criteria. In Fig. 8, motions generated from conditional GRU tends to be visually least appealing; this is followed by those of two-stage GAN and LatentTransition; the results of Act-MoCoGAN often suffers from the mode collapsing issue, with similar results popping up after multiple separate runs; In comparison, Dancing2Music Table 1 Performance evaluation on HumanAct12 benchmark on coarse-grained and fine-grained action categories, respectively. ± indicates 95% confidence interval. ↑ (or ↓) is higher (or lower) the better; → means closer to real motion scores the better. For performance, bold face specifies the best method, with underscore referring to the second best. Methods HumanAct12 ( is capable of producing diverse motions by transiting between different short motion snippets. However, the generated motions could not be faithfully aligned to the prescribed action type; On the contrary, our ac-tion2motion variants are shown to be capable of generating both diverse and consistent motions. Moreover, our action2motion framework is also capable of producing motions from fine-grained action categories, as showcased in Fig. 9. The motions generated by our action2motion (GLMI-M) variant faithfully assemble the subtle characteristics of local motions (e.g. leg pressing and chest expansion), and body parts (e.g. left hand and right hand) from a range of fine-grained action types. Quantitative comparisons. Quantitative evaluations are conducted on a range of datasets. Specifically, Table 1 displays results on our in-house HumanAct12 dataset, where coarse-grained and fine-grained action annotations are both considered; Table 2 presents comparison results on the popular benchmarks of CMU MoCap and NTU-RGBD. Considering the stochastic nature of motion generation, each experiment is repeated 20 times, a statistical confidence interval of 95% is reported in both tables. Note action2motion (GLMI) is however not applicable to the post-processed NTU-RGBD dataset, since the re-estimated pose sequences from videos does not contain global trajectory information. Among the four evaluation metrics in both tables, FID is perhaps the most important indicator, as it evaluates the overall quality of the generated motions. Recognition accuracy quantifies how well a generated motion fits into an action category. Diversity and multimodality (i.e. MModality) are metrics quantifying the diversity aspects of the generated motions. Note the values of FID (or accuracy) is lower (or higher) the better; for Diversity and MModality though the values are as close to the real motion scores the better. From Table 1 and Table 2, we have the following observations. As a deterministic method, conditional GRU fails to generate diverse motions that is essentially an one-tomany mapping problem. GAN models such as two-stage GAN, Act-MoCoGAN and LatentTransition have improved upon conditional GRU in both metrics of FID and recognition accuracy. The considerably high accuracy obtained by Act-MoCoGAN may be attributed to its use of action classifier during training. A sharp drop of FID is observed in Dancing2Music, which however comes at the price of much lower accuracy. Meanwhile, our action2motion clearly outperforms the rest on FID, and the GLMI-M variant consistently excels among the four action2motion variants. The success could be partly attributed to the incorporation of Lie algebraic pose representation. Given substantial performance on FID and perhaps also accuracy scores, the scores of diversity and multimodality are also important indicators for the model capacity of producing diverse motions. Note for diversity and multimodality, the higher values do not necessarily reflect better performance; instead the values are best to be close to those from the real motions, denoted as → in Tables 1 and 2. Act-MoCoGAN generates motions with severely limited diversity. Overall, our action2motion variants, while performing best on FID and accuracy, also maintain a considerable extent of diversity and multimodality. Crowd-sourced Subjective Evaluation. In addition to the aforementioned objective experiments, two user studies are conducted on Amazon Mechanical Turk. The principal criteria used in these two user surveys are the visual perceptual quality of the motion, and the magnitude it is adhere to the intended action categories. Users who possess hit approval rate higher than 97% and 1000 completed hits are considered. The first user study is illustrated in Fig. 10, which compares the first two action2motion variants, ours (plain) and ours (w/ Lie), with baseline methods. Here, same amount (i.e. 36) of motions are generated by different methods. The users are then asked to rank their preferences of these motions evenly sampled over all action categories. Our action2motion variants receive the highest user ratings. Contrarily, conditional RNN, twostage GAN and LatentTransition are the three least performed methods. Dancing2Music and Act-MoCoGAN rank somewhere in-between. More positive feedback is observed in our action2motion plain variant, with 10% motions being graded the first by users. By adopting the Lie algebraic representation, our ours w/ Lie variant further narrows the gap to real motions, with 54% generated motions being secured at the top-2 spots by user ratings. The second user study compares bewteen our two action2motion variants: ours (GLMI) and ours (w/ Lie). As GLMI-M outperforms GLMI-R in most cases, we focus on the evaluation of GLMI-M in this survey. Here the motions are generated following the same protocol conceived in the first study. As shown in table 3, ours with GLMI earns more appreciation from users when compared with ours (w/ Lie), with over a half motion sequences (i.e. 54.4 %) being preferred by users. When comparing to real motions, samples generated by ours (w/ Lie) are slightly inferior to real-life human motions, with 46.2% being preferred. Meanwhile ours (GLMI-M) is almost indistinguishable to the real motions. The results suggest the potentials of applying our algorithm to more interesting VR/AR applications. We further investigate the global displacement aspect of the generated motions. As demonstrated in Fig. 11, motions generated from ours (GLMI) are always more preferred by users than those from ours (w/ Lie) over all these four action categories. In summary, our GLMI-M variant, i.e. ours (GLMI), delivers overall best results among our four action2motion Table 4 Performance evaluation over CMU MoCap dataset on two locomotion action types. ± indicates 95% confidence interval. ↑ (or ↓) is higher (or lower) the better; → means closer to real motion scores the better. For performance, Bold face specifies the best method, with underscore referring to the second best. variants, which are often indistinguishable from real-life human motions. Locomotion Generation Analysis Locomotions (e.g. walking) are the most common activities in our daily life, which typically involve fullbody displacements. Fig. 12 visually compares walking motions produced with vs. without our global local movement integration (GLMI) module. When without, the walking motions appear surreal like ghost haunting on the ground, with arm and leg local movements not tuned to its global motion trajectory. By contrast, our proposed GLMI module significantly mitigates these issues. For example, the waving patterns of left (or right) arm is now synchronized with the right (or left) leg; the local-part moments are also well in agreement with the full-body motion trajectories. variants perform best over all the three metrics. In contrast, ours (plain) attains worst results, which we attribute to the missing modules of Lie algebraic representation and GLMI. Moreover, GLMI-M , i.e. GLMI with MLP implementation, works best in generating Walking motions, while GLMI-R takes the lead in Jump Forward. Interpolation in Latent Space Generative models could be regarded as a function mapping between points in a latent space and those in the real data space. Meanwhile, similar to the concept of well-posed problems, a well-learned generative model is expected to behave smoothly from a small perturbation in the latent space. In other words, when we perform interpolations between two distinct latent codes, their generated motions are supposed to transit smoothly. It is thus of interest to examine how interpolations in the latent space would change the motion generation behaviors of our action2motion. It also demonstrates the model capability in producing non-existent samples. The task is a bit complicated in our situation, as our model generates motion sequences instead of single images. Alternatively, we use the first poses as anchors to perform interpolation between two motions. Specifically, the first poses of two pose sequences are selected. Then, a series of points can be created on the linear path between the latent vectors (i.e. noise vectors) of these two poses. After that, these points are input as initial latent vectors into our model to kick-start the generation of rest poses. Fig. 13 considers lift dumbbell action. Here two pose sequences are deliberately selected from motions generated by action2motion (GLMI-M), where the first poses of the two sequences are a person lifting with the left (and the right) hand, respectively. We have the following observations. 1) As demonstrated in the first column, transition from the left hand pose to the right hand pose is realistic at the first poses, by gradually putting one hand down and lifting another hand up. 2) From each of these initial interpolated poses, a visually natural motion sequence is generated. 3) Interestingly the interpolation leads to the generation of a novel motion, lift dumbbell with both hands. Action Transition To showcase the flexibility of our motion synthesis process, action transition is explored by switching the action categories during sequence generation. Exemplar results are presented in Fig. 14. To our surprise, our ac-tion2motion model is able to produce unseen motions through action transition. In the first row of Fig. 14 (Alldieck et al., 2018) and the bottom one from BUFF dataset . Comparing with the state-of-the-art methods of PaMIR (Zheng et al., 2021), PIFu (Saito et al., 2019) and PIFuHD (Saito et al., 2020), our approach improves upon PIFuHD and PIFu by integrating their otherwise segregated strengths of high-resolution geometry and high-quality texture at novel views. However, all drinking motions in our training set are performed in standing poses. As shown in these examples, the resulting motion sequences are rather realistic and with natural transitions which is well maintained in transitions of not only two actions, but also three actions. This experiment clearly demonstrates the capacity of our approach in synthesizing unseen motions that goes beyond merely memorizing training examples. Motion Outpainting Our method could also serve as a motion outpainting tool: provided the initial few poses, apply our method to complete the rest of the motion sequence. This is realized by simply fixing the beginning poses, and generating the rest. Executing multiple independent runs usually creates distinct yet plausible outcomes. Fig. 15 illustrates such an example. Here black poses denote the fixed initial poses of Walk. This is completed by our model with visually plausible walking motions of distinct velocities and directions. This also suggests the necessity of modeling motion forecasting and generation in a non-deterministic manner. Method Average Rank↓ PIFuHD (Saito et al., 2020) 3.60 PIFu (Saito et al., 2019) 2.36 PaMIR (Zheng et al., 2021) 2.26 Ours 1.77 Table 5 Quantitative comparison of reconstructing 3D human shape & texture from single images. The numbers are averaged user preference ranks, with ↓ meaning the numbers are lower the better. Step 2: Motion2video Side-by-side evaluations are performed in terms of reconstructing 3D human shapes & textures from single images in Sec. 5.2.1, and animation in Sec. 5.2.2. 3D Shape and Texture Reconstruction Here we focus on the evaluation of reconstructing 3D human shape & texture from single images, where the respective part of our approach is compared side-byside with the state-of-the-arts, namely PaMIR (Zheng et al., 2021), PIFu (Saito et al., 2019) and PIFuHD (Saito et al., 2020). PaMIR (Zheng et al., 2021) combines parametric SMPL body model with deep implicit function for robust 3D shape reconstruction. In our comparison, 30 images are obtained from a wide variety of sources, including the BUFF dataset , the People Snapshot dataset (Alldieck et al., 2018), internet images, CG image, and our in-house captured images. Following the network architectures, the input resolution of PaMIR and PIFU is 512 × 512, whereas the input image resolution is 1024 × 1024 for PIFuHD and our approach. Exemplar results of reconstructed textured shapes from single input images are shown in Fig. 16. The shapes and textures extracted by PaMIR and PIFu commonly lack details, and are oftentimes inaccurate. For example, the 3D shape of lady produced by PaMIR is overly slim, together with an smooth face that lack geometric details which is noticeable especially from side-views. PIFuHD is capable of recovering 3D shapes with better facial geometry and in high-resolution, yet the texture is often visually unpleasantly wrong, especially when viewing from the back. In contrast, our method maintains a delicate balance of shape and texture, thus stands at a better position in facilitating the follow-up animation and realistic rendering processes in our pipeline. For quantitative evaluation, user study is further conducted to measure the perceptual quality of the comparison methods. For each input image, 20 Amazon mechanical turk Workers are enrolled to rank their preferences over the shapes reconstructed by their corresponding comparison methods. Table 5 displays the average rank of each method, with more detailed rank distributions presented in Fig. 18. Our method clearly stands out with the most appreciations from users, where almost half (i.e. 51%) results are ranked the first. By contract, PIFuHD is the least preferred one, of which 78% results are placed as least favorable. In-between are PIFu with the second lowest average rank, and PaMIR that receives considerable more positive feedback compared to PIFuHD. Preference Percentage Ours Over 0.843 Ours Over 0.593 Ours Over (Weng et al., 2019) 0.703 Table 6 Crowd-sourced subjective assessment to compare the videos animated with the same image and motion, produced by Ours, Liquid Warping GAN , ARCH and Weng et al. (2019). Motion2Video Animation In Fig. 17, We present two single image animation showcases using our method. 3D shape and texture are predicted from input images, which are driven by two challenging motions, cartwheel, from Adobe Mixamo 2 . As shown, our method could obtain accurate shape and texture predictions from all views, as well as plausible animations with provided motions. In what follows, we elaborate the comparisons between our method and other three state-of-the-art image animation methods Huang et al., 2020;Weng et al., 2019). For quantitative evaluation, we conduct user study on Mechanical Turk which pairs the videos animated with the same image and motion (b) [Weng et al.] (c) Our method from our and comparison method, and request the workers to determine which one that is "more realistic". For each animation, 50 workers with Hit approval rate higher than 97% are enrolled for perceptual assessment. Comparison with Liquid Warping GAN . Liquid Warping GAN is a learning based motion transfer method in pseudo-3D space, where 3D SMPL model estimated from reference video frames are used to re-pose the person in source image. Fig. 19 presents the animated videos by our method (bottom) and Liquid Warping GAN (top), when feeding with the same input image and motion. While successfully modeling the motion dynamics, the individual images obtained by Liquid Warping GAN are very blurring such that the characteristic personal landmarks of face or T-shirt logo are nearly unrecognizable. In contrast, the animation results of our method are of high-resolution and high quality. A user study is performed for quantitative evaluation, based on 22 animations from Liquid Warping GAN and our method covering a variety of input images and motion sequences, including composed of 9 Mixamo motions and 13 motions generated by our action2motion step. As shown in Table 6, 84.3% of our animations are preferred by users. Comparison with ARCH . ARCH Huang et al. (2020) uses a semantic deformation field to produce 3D rigged full-body human avatars from a single image, which is already animatable. However, the implementation and pre-trained model of ARCH has not been released yet. We managed to obtain 3 animated 3d model sequences from the authors with our provided images and Mixamo motions. We render video frames of these 3d model sequences in Unity3D with the same environment setting (e.g. light, camera) as ours. Fig. 19 presents a visual comparison between ARCH's result (middle) and our result (bottom). Though ARCH shows capability of generating reasonable rendering, the person appearance is yet to be realistic. For example, the pants comes with several blue debris; the two feet of the man are in wrong color (black); and the texture of T-shirt is overly bright. A user study is again conducted regarding the 3 animations from ARCH and our method. As given in Table 6, our method earns more preference (i.e. 59.3%) from users. Please refer to the supplementary video for more visual comparisons. Comparison with Weng et al. (2019). the work of Weng et al. (2019) is also closely related to part of our motion2video step, where a 3D character is extracted out of a single image and is further animated to form videos. Their implementation is unfortunately not publicly available, instead we obtain from the authors of Weng et al. (2019) two animated action sequences (i.e. sit and walk) from the two input images provided by us. Note that the motions involved in Weng et al. (2019) are real MoCap motion sequences, while our motions are generated by ourselves. For an easy side-byside visual comparison, we hand pick two of our generated motions that resemble the animations used (Weng et al., 2019). The walk and sit visual results are displayed and compared in Figs. 20 and 21, respectively. When viewing from frontal view, the results of Weng et al. (2019) possess incomplete and distorted errors including the incomplete feet ( Fig. 20(b)), over-slim arms, and torn pants ( Fig. 21(b)), as highlighted by red arrows. These artifacts come from the fact that the textures are directly copied and pasted from the 2D input image, which is inadequate to maintain intact appearance in 3D geometry. In comparison, our results are noticeably better at preserving detailed structure and appearance, e.g. around the feet. When inspecting from the side and back views of the extracted 3D characters that are not directly visible from the input image view, the textured results of Weng et al. (2019) are simply mirrored from the frontal region, as shown in the back side of head and torso -the visual results are thus significantly deteriorated to being funny. In contrast, our results preserve reasonable 3D shape and consistent appearance across multiple views including the frontal view. Moreover, a similar user study is conducted among the two set of generated videos. As in Table 6, our method is 70.3% more preferred over Weng et al. (2019). The Full Action2Video Pipeline This section is devoted to the examination of our full ac-tion2video pipeline. We start by comparing with stateof-the-art 2D-based human video generation results. Further experiments also demonstrate the capacity of our action2video approach in accommodating input images from different sources. Comparison with existing methods. The work of Kim et al. (2019) is state-of-art in generating human motion videos, which is 2D-based and relies on largescale training set of videos. Fig. 22 presents a comparison of their results and ours that share in common similar poses and views. Compared with our results, the frames of Kim et al. (2019) is of low resolution (128x128). Moreover, there are visible lack of details of face, hands & clothes, and unrealistic shape deformations, which we attribute to their innate 2D based limitations. For example, lengths of legs and arms in Kim et al. (2019) of the same lady character vary over time. Moreover, as presented in the middle row of Fig. 22, the exemplar video result generated by engaging Liquid Warping GAN based on the same motion generated by our action2motion step, where edges and facial details are very foggy and fuzzy, when comparing to our results shown at the bottom row. Diverse input image sources. This experiment is to evaluate the flexibility of our action2video pipeline in accommodating input images from varied sources. Fig. 23 presents our action2video results based on BUFF images (e.g. 1st row), People Snapshot images (e.g. 2nd row), Internet images (e.g. 4th row), these captured by our mobile-phone (e.g. 3rd row) as the input images. Overall our approach is able to adapt to these different applications, and to produce videos of visually pleasing quality. More visual results are shown in the supplementary video. Multiple camera views. Fig. 24 displays an exemplar video sequence generated by our approach, that is inspected from four different views. It demonstrates 1) our extracted 3D shape and clothing texture are reasonably realistic when examined in different rendered views, and 2) compared to the popular 2D-based methods, our generated videos are consistent among distinct views. Conclusion and Discussion Conclusion. We propose an action2video approach to tackle the exciting and challenging problem of generating natural and diverse 3D motions & videos of human actions. This is accomplished in this paper by a 2-step pipeline: action2motion focuses on generating 3D human motions, which are then turned into videos by motion2video. Empirical studies demonstrate the effectiveness of our approach. Limitation and Future Work. Our approach performs reasonably well in practice; empirically it out- performs the state-of-the-art methods in many aspects along the full pipeline. On the other hand, we recognize that our training set, primarily the in-house Hu-manAct12 dataset is relatively small, which contains 1,191 motions. For future work, we plan to acquire a larger dataset with broader set of actions, to generate motions and videos from a wider range of human activities including interactions with multiple people, with surroundings and objects, and to improve the reconstructed shape details such as fingers. Furthermore, we would investigate its possible applications such as augmenting data for human-centric tasks (action recognition, pose estimation), and VR/AR. Fig. 1 1Fig. 1 Our action2video pipeline generates human full-body motion videos of prescribed actions in two steps: action2motion first generates diverse and natural 3D motions of predefined actions; motion2video proceeds to extract 3D surface shape and texture from an additional 2D input image, and to render 2D videos of the generated motions. Fig. 2 2An example of human skeleton which consists of 21 joints and 20 body parts. Fig. 4 4Four variants of the pose decoding module conceived in our work: (a) direct generation of 3D joint positions; (b) generation with Lie algebraic representation; and (c)-(d) global and local movement integration (GLMI)-based generation with Lie algebraic representation, implemented by multi-layer perceptron (GLMI-M) or GRU (GLMI-R). Fig. 5 Fig. 6 56A comparison of reconstructing 3D characters from single images by the original methods of PIFu, PIFuHD, and our improved variant. Each 3D reconstruction result is shown in front, side, and back views. Salient errors are pointed by the red arrows. See text for details. Fig. 7 7Input images used in our experiments are from different sources, including (a) BUFF dataset (Zhang et al., 2017), (b) People Snapshot dataset (Alldieck et al., 2018), (c) internet images, (d) CG image, and (e) our in-house captured images. See text for details. -Multimoldality: Different from diversity, multimodality indicates how much the sampled motions vary within each action category. Suppose there are C action types in the set of motion sequences. For the c-th action, two subsets with same size S m are randomly sampled, which are then transformed to two subset of feature vectors {v c,1 , . . . , v c,Sm } and {v c,1 , . . . , v c,Sm }. Fig. 8 8-Two-stage GAN.Cai et al. (2018) propose a twostage GAN method for 2D human motion generation based on action types. In particular, a Wasserstein GAN(Arjovsky et al., 2017) is first trained as the pose generator. After that, the motion generator is Visual comparison of motions generated by the baseline methods and our four action2motion variants. Two warm up motion sequences are sampled for each of the comparison methods. Every 6th frame is shown. See text for details. Best viewed in Adobe Acrobat Reader to activate the animations by clicking the boxed items in the top row. Note each item in the top row is with specific tag and color corresponding to its row of motion sequence displayed below. ( a )Fig. 9 a9Fine grained generation of Lift dumbbell (b) Fine grained generation of Warm up Motion examples of fine-grained action categories generated by our action2motion (GLMI-M). Every 6th frame is shown. (a) Lift dumbbell with (from top to bottom) right hand, left hand, both hand, both hand over head, and both hand over head and squat. (b) Warm up with (from top to bottom) alt chest expansion, chest expansion, wrist circles, left side reach and right side reach. Best viewed in Adobe Acrobat Reader to activate the animations by clicking the boxed items in the top row. Note each item in the top row is with specific tag corresponding to its row of motion sequence displayed below. Fig. 10 Fig. 11 1011Crowd-sourced subjective assessment results of motions generated by comparison methods. For each method, there is a bar of different colors (from red to blue) indicating the percentage of corresponding preference levels (least to most preferred). See text for details.Preference PercentageOurs (GLMI-M) Over Ours (w/ Lie) 0.544 Ours (w/ Lie) Over Real Motions 0.462 Ours (GLMI-M) Over Real Motions 0.501Table 3Crowd-sourced subjective assessment to compare motions sampled from Ours (GLMI-M), Ours (w/ Lie), and real motions. Crowd-sourced subjective assessment to compare generated motions together with their global displacements from Ours (GLMI-M) and Ours (w/ Lie). Fig. 14 14Action transition examples. Every 5th frame is shown. The top three rows show transition between two actions. from top to bottom, they are sit-drink, jump up-lift dumbbell, lift dumbbell-jump up, respectively. The bottom two rows display transition of three actions, which are (from top to bottom) sit-jump up-sit and sit-jump up-lift dumbbell, respectively. Best viewed in Adobe Acrobat Reader to activate the animations by clicking items in the top row. Note each item in the top row is with specific tag corresponding to its row of motion sequence displayed below. Fig. 15 15Examples of motion outpainting of Walking. Provided several initial poses (in black), our method completes the rest motion sequence with multiple plausible outcomes. Best viewed in Adobe Acrobat Reader to see the animations upon clicking. Fig. 17 17Two animation results of our method. Given single images of frontal view of individuals shown on the left, their 3D shapes are reconstructed, 2D videos are obtained, using prescribed off-the shelf motion sequences. The videos produced by our method are visually plausible. Fig. 18 18User preference distributions of reconstructing 3D human shape & texture from single images. Fig. 19 19Comparing our method (bottom) with Liquid Warping GAN (Liu et al., 2019a) (top) and ARCH (Huang et al., 2020) (middle), animated using the same input image and motion sequence. Results are displayed by pairing the corresponding video frames. Fig. 21 21Comparing our method with Weng et al. (2019) by animating sitting motions. Fig. 22 22Visual comparison of three methods: (top) a stateof-the-art 2D-based method Kim et al. (2019), (middle) Liquid Warping GAN (Liu et al., 2019a), and (bottom) ours. Fig. 23 23Exemplar videos produced by our action2video pipeline. Given a reference image and a specific action category, our action2video could extract 3D human shapes & cloth textures, and animate & render into diverse motion videos. For boxing and throw actions, one video are shown, each animated by a different 3D character extracted from a single image; similarly, two distinct videos and four distinct videos are presented for the warm-up action and walking action respectively. Fig. 24 24An generated walking video from the following views: (a) front, (b) right-side, (c) back, and (d) left-side. Table 2 2Performance evaluation on CMU MoCap and NTU-RGBD Dataset. ± indicates 95% confidence interval. As NTU- RGBD dataset does not have global motion trajectory annotations available, our GLMI-M & GLMI-R variants that could not be fairly evaluated here. ↑ (or ↓) is higher (or lower) the better; → means closer to real motion scores the better. For performance, bold face specifies the best method, with underscore referring to the second best. Methods CMU MoCap NTU-RGBD FID↓ Accuracy↑ Diversity→ MModality→ FID↓ Accuracy↑ Diversity→ MModality→ Real motions 0.064 ±.006 0.936 ±.002 6.130 ±.079 2.726 ±.066 0.031 ±.004 0.999 ±.001 7.108 ±.048 2.194 ±.025 Table 4 4quantitatively evaluates the effects of incorporating GLMI module for locomotion generation on CMU MoCap dataset. The same evaluation metrics of Section 5.1.1 are considered here. The number of motion sampling is set to 500. Overall, ours with GLMIFig. 13Examples of motion interpolation in lift dumbbell. Every 6th frame is shown. See text for details. Best viewed in Adobe Acrobat Reader to activate the animations by clicking the boxed items in the top row. Note each item in the top row is with specific tag corresponding to its row of motion sequence displayed below.Click↓ Frame 1 6 12 18 24 30 36 42 48 54 60 Interpolation * - -- + # o ø x , after switching from sit down to drink, the character starts to open the bottle and drink with a sitting pose.Fig. 16 A qualitatively comparison of reconstructing 3D human shape & texture from single image. The input single images are show on the left, where the top image is from People Snapshot datasetPIFuHD Ours PaMIR PIFu Input PaMIR PIFu PIFuHD Ours Input Fig. 20 Comparing our action2video with Weng et al. (2019) by animating walking motions. For each given image on the left, we show the results of Weng et al. (2019) (middle column) and ours (right column) from different views. Weng et al. (2019) fail to build an intact 3D texture model (e.g. incomplete feet), and the appearance of unseen part is distorted. Our method could generate plausible animation from all angles.0 ∘ 90 ∘ 120 ∘ 0 ∘ 90 ∘ 120 ∘ (a) input 180 ∘ (b) [Weng et al.] (c) Our method (a) input www.mixamo.com Acknowledgements This work is partly supported by the NSERC Discovery Grant and the UAHJIC grants. We would like to thank the Amazon MTurk workers for their efforts in user studies, and the anonymous reviewers for helping to improve the presentation of our manuscript. We also want to thank Chungyi Weng, Yuanlu Xu and Zerong Zheng for their great help on reproducing the works of Photo Wake-Up, ARCH and PaMIR. Skeleton-aware networks for deep motion retargeting. 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It is shown that a single molecular magnet placed in a rapidly oscillating magnetic field displays the phenomenon of quenching of tunneling processes. The results open a way to manipulate the quantum states of molecular magnets by means of radiation in the terahertz range. Our analysis separates the time evolution into slow and fast components thereby obtaining an effective theory for the slow dynamics. This effective theory presents quenching of the tunnel effect. In particular, stands out its difference with the so-called coherent destruction of tunneling. We support our prediction with numerical evidence based on an exact solution of the Schrödinger's equation.

10.1016/j.jmmm.2015.08.014

1508.04499

ccf8435d66a9ff13baf4c0294437b0d03fe2a76c

Dynamical quenching of tunneling in molecular magnets María José Santander Departamento de Física Recursos Educativos Quántica SantiagoChile Universidad de Santiago de Chile and CEDENNA Avda. Ecuador 3493SantiagoChile Alvaro S Nunez Departamento de Física Facultad de Ciencias Físicas y Matemáticas Universidad de Chile Casilla 487-3SantiagoChile A Roldán-Molina Instituto de Física Pontificia Universidad Católica de Valparaíso Avenida Universidad 330Curauma, ValparaísoChile Roberto E Troncoso Departamento de Física Centro para el Desarrollo de la Nanociencia y la Nanotecnología CEDENNA 3493, 9170124SantiagoAvda. EcuadorChile Universidad Técnica Federico Santa María 1680ValparaísoAvenida España, Chile Dynamical quenching of tunneling in molecular magnets Tunnel effectTopological QuenchingMagnetization Dynamics It is shown that a single molecular magnet placed in a rapidly oscillating magnetic field displays the phenomenon of quenching of tunneling processes. The results open a way to manipulate the quantum states of molecular magnets by means of radiation in the terahertz range. Our analysis separates the time evolution into slow and fast components thereby obtaining an effective theory for the slow dynamics. This effective theory presents quenching of the tunnel effect. In particular, stands out its difference with the so-called coherent destruction of tunneling. We support our prediction with numerical evidence based on an exact solution of the Schrödinger's equation. Introduction Since a few decades molecular magnetic materials have arisen as a new test ground for several phenomena in quantum behavior of finite size magnetic systems [1,2,3,4,5]. Molecular magnets have attracted attention Email addresses: [email protected] (María José Santander), [email protected] (Alvaro S. Nunez), [email protected] (Roberto E. Troncoso) due to their potential in the implementation of several molecular spintronic devices [6,7,8]. The long spin coherence times displayed turn them into promising candidates in the context of quantum computing, where the molecule spin is used to encode q-bits [10,5,9]. The quantum mechanical degrees of freedom associated with molecular magnets can be manipulated and controlled with great accuracy by the application of external magnetic fields [2,3,4,5]. It has been proposed [10] that suitable manipulation of molecular magnets with time dependent magnetic fields can be used to control the population of their quantum states thereby paving the road toward an implementation of a quantum computation scheme known as the Grover's algorithm. Within the same framework in this work we propose a way to control the quantum mechanical state of a molecular magnet by means of a rapidly varying magnetic field. We will show that radiation within the range of terahertz frequencies can be used to quench the quantum state of a molecular magnet thereby providing a useful tool for potential applications. There is previous work relating radiation in the terahertz range with magnetic properties of materials, for example terahertz radiation has been used to control the spin waves of antiferromagnets [11,12] and in Ref. [13] hybrid magnetic structures have been used to generate radiation in the terahertz range. Our predictions allow an extension of such control into the subject of molecular magnets. We study the dynamics of the magnetization of a molecular magnet exposed to circularly polarized terahertz radiation. In response to such perturbations the system can display a quenching of the tunneling rate between energetically equivalent states giving rise to a trapping of the quantum mechanical state. This effect is analogous to an interesting effect in classical mechanics the trapping of a classical particle that can be achieved by introducing a rapidly oscillating potential V (x, t) = V 0 (x) + V 1 (x, t) [14,15]. Under the action of such forces the slow dynamics of the particle is trapped by an effective potential V eff = V 0 (x) + F 2 /(2mω 2 ) where F is the force associated with the oscillating potential, F = −∂ x V 1 (x, t), ω its frequency and the bar represents an average over an entire cycle of the oscillating force. Based on those ideas it was proposed and demonstrated by Kapitza [15] that a pendulum with a rapidly vibrating point of suspension would be stabilized in the upward position. Once stabilized the pendulum was shown to display small oscillations around its new equilibrium configuration. The main result of this paper is that a similar result holds for the quantum mechanical state associated with the spin of a molecular magnet. In this sense we can say that the prediction corresponds to a Kapitza effect in Hilbert space. Instead of promoting transitions between states, the high frequency radiation traps the state of the spin in a given configuration keeping it from describing tunnel transitions into other configurations. A quantitative statement of this effect is encoded in the tunneling time (the time that it takes for the system to tunnel from one minima to another) which is seen to diverge in certain circumstances. In response to the oscillatory disturbance the tunnel effect is suppressed and the states are frozen in a given configuration. A similar effect of suppression of tunneling has been reported in the literature concerning spins where the effect is attributed to interference of Berry's phases of different paths associated with tunneling. In this context the effect has been dubbed quenching of the tunnel amplitude [26,3]. Effective slow dynamics As is common in the theoretical studies of molecular magnets our study is based on the reduction of the electronic degrees of freedom, to some effective low-energy Hamiltonian, based entirely on localized spin degrees of freedom of the magnetic ions within the molecule [16]. In this context, the total energy has two contributions H = H 0 + H 1 , one arising from the intrinsic anisotropy [2], in the form: H 0 = −DS 2 x + E(S 2 z − S 2 y ).(1) For D > E this Hamiltonian represents a quantum spin with easy axis along the x direction and a hard axis along the z direction. The case E = 0 has been previously studied in Ref. [17]. This Hamiltonian can be used as a model for describing the magnetic degrees of freedom of several single molecule nanomagnets [2]. Among them the most widely used ones are the Mn 12 − ac molecule [5] (with S = 10, D = 0.55 K and E = 0.02 K), the molecular complex Fe 8 molecular magnet [3] (with S = 10, D = 0.29 K and E = 0.05 K) and Ni 4 [23] (with S = 4, D = 0.75 K and vanishingly small E). The spin is perturbed by a circularly polarized time dependent external field [12]: H 1 = −h(cos ωt S x + sin ωt S y )(2) Here ω is the frequency of the oscillation of the magnetic field in the x, y plane and h = gµ B H with H being the amplitude of the oscillating magnetic field and g the gyromagnetic ratio (of order 1 in the examples given). Regarding the geometry we can say that it is dominated by the anisotropy terms in the Hamiltonian. To apply our theory in the experimental setting the incident radiation must be polarized in the plane perpendicular to the hard axis. In a molecular magnet based crystal this can be achieved by selecting the crystal orientation with respect to the incident light. In the remaining parts of this article we will address the problem of how the spin responds to the perturbation in the limit of large ω. In the special case of spin 1/2 all thê ⌦ 1 ⌦ 2h (t) H e↵ Figure 1: Schematics of the proposed arrangement. A rapidly rotating magnetic field, h(t), in the plane induces a slow dynamics characterized by a stationary effective field, H eff . The direction of the effective magnetic field is perpendicular to the plane. The slow dynamics is characterized by two degenerate classical ground statesΩ 1 andΩ 2 . In general, there are oscillations between those states mediated by quantum tunneling. However, for certain values of the parameters the quantum mechanical oscillations between those two states are quenched. anisotropic contributions reduce to the identity. The resulting behavior has the characteristic form of Rabi oscillations [21]. For higher spin the interaction between the magnetic moment and a radiation field is more complex and has been addressed in several references. For instance in Ref. [22] the effect of photon assisted tunneling events was reported in Fe 8 samples irradiated with circularly polarized light. In Ref. [23] microwave spectroscopy was used to reveal quantum superpositions of high spin states (S = 4) in Ni 4 . It is possible to derive a general treatment of a quantum system driven by rapidly varying potentials [24]. In general, the goal is to find an effective equation for the dynamics spanned by the Schrödinger equation: i ∂ t |Ψ = H|Ψ(3) As in the classical case [25], the dynamics of the wave vector can be separated into two components, a fast one that varies in a period of the potential and a slow one that evolves at a slower pace. The method consists in an extension of the method of multiple scales from classical mechanics into quantum mechanics. We separate the slow and fast dynamics, by means of a unitary transformation, and proceed to write an effective theory that involves only the slow variables. The slow dynamics is affected by the rapid motion and is described by an effective Hamiltonian. We start from the time dependent Hamiltonian H and perform a time dependent unitary transformation exp(iF(t)). The basic idea is to absorb the time dependency of the Hamiltonian into the operator F(t) and to obtain an effective time independent Hamiltonian given by: H eff = e iF He −iF + i ∂e iF ∂t e −iF .(4) To find the specific representations of H eff and F we write them as power series in 1/ω: H eff = H (0) eff + 1 ω H (1) eff + 1 ( ω) 2 H (2) eff + · · · (5) F = 1 ω F (1) + 1 ( ω) 2 F (2) + · · ·(6) Expanding Eq. (4) and equating both sides order by order we found recursive relations between F (i) and H (i) eff . The expression for F (i) is found by enforcing that every term in the expansion of the effective Hamiltonian be time independent. A lengthy, but straightforward calculation leads to the following first contributions in the expansion for F: F (1) = −h (sin ωtS x − cos ωtS y )(7) and F (2) = −ih (cos ωt[S x , H 0 ] + sin ωt[S y , H 0 ]) ,(8) while for the Hamiltonian we obtain: H (1) eff = − h 2 2 S z(9) and H (2) eff = h 2 4 ([[S x , H 0 ], S x ] + [[S y , H 0 ], S y ])(10) Collecting the contributions up to second order we obtain an effective Hamiltonian: H eff = H 0 − h 2 2 ω S z + h 2 4 2 ω 2 ([[S x , H 0 ] , S x ] + [[S y , H 0 ] , S y ]) ,(11) In the limit of small DS/ ω, i.e. for sufficiently rapid variations of the magnetic field, the effective Hamiltonian corresponds to the original stationary contribution to the energy plus an effective field pointing in the direction perpendicular to the plane of polarization of the magnetic field. For common molecular magnets such as the ones described above the condition DS/ ω 1 sets the stage in a magnetic field oscillating in the terahertz range [5]. The direction and magnitude of this effective field is independent of H 0 . We can explain the origin of this contribution in terms of the following semi classical argument. The equation of motion of a spin in absence of any anisotropy is simply given by the Landau-Lifshitz equation, d S dt = S × h(t), where h(t) correspond to the oscillating magnetic field. The motion can be understood as a succession of changes each of one is aprecession around the moving field. As shown in Fig. 2 the complicated sequence of small precessions leads to a simple motion, a steady precession around the z axis. This can be associated with an effective field pointing along the same axis. As shown in Eq. (12) this field is generally present regardless of the base Hamiltonian. This is not true for the following terms in the expansion. The second order term depends explicitly upon the stationary Hamiltonian. A fundamental result of this paper follows after a direct calculation of this second order term, it can be easily verified that the contribution to order ω −2 merely shifts the values of the constants D and E with a correction of order h 2 /( ω) 2 . Hereafter we will denote those corrected values byD andẼ respectively. We conclude that the effective theory describes a spin with corrected anisotropy energies exposed to an effective magnetic field in the direction perpendicular to the plane of polarization of the time varying magnetic field. From this result we can infer several properties of the slow dynamics described by the quantum states. For spin 1/2 we can use our result in Eq. (12) in a direct way. Since the Hamiltonian H 0 reduces to the identity (a property of the spin 1/2 operators) the evaluation of the effective Hamiltonian is straightforward. We have a very simple result, namely: which provides a two level system with energy levels equal to ± h 2 4 ω . A system prepared in a mixed state will oscillate [21] between the eigenstates with a frequency determined by this splitting. For greater spins the anisotropy contribution plays an essential role in the dynamics as we will show in the next section. H eff = − h 2 2 ω S z ,(12) Dynamical suppression of quantum tunneling The slow dynamics reduces to the well-known problem of a spin under the action of an external magnetic field, H eff , along the hard axis, whose strength is given by gµ B H eff = h 2 /(2 ω). In this way we see that the resulting Hamiltonian is well studied in several contexts [2]. We can readily infer a number of properties regarding the behavior of the quantum moment by exploiting this analogy. We start by analyzing the classical limit. It is clear that the ground state is doubly degenerate. Using spherical coordinates, Ω = (sin θ cos φ, sin θ sin φ, cos θ), as shown in Fig. 1, we find the two minima located at: cos θ 0 = h 2 4S ω(D +Ẽ)(13) and φ 1,2 = 0, π. The degeneracy of the classical ground states is lifted by quantum fluctuations associated with the tunnel effect between the two minima. The resulting energy splitting ∆ between the two lowest lying states characterizes the tunneling time by the relation T = /∆. A detailed calculation of this splitting has been made and the results are highlighted in Fig. 4. A state initially prepared around one of the minima will oscillate, tunneling across the energy barrier, into the other minima in a characteristic time T . It is possible to find the tunneling time by direct diagonalization of the effective Hamiltonian. The tunnel splitting ∆ oscillates as a function of the effective magnetic field. For certain specific values of the effective field a phenomenon known as the suppression or quenching of the tunnel effect can be observed. For those values of H eff the tunnel splitting vanishes and the tunneling time diverges. The state is, therefore, trapped in a given state. The quenching of the tunneling processes, revealed by the reduction of the energy splitting between the two lowest lying energy states, has an origin that can be traced back to the interference between different tunneling paths. Such behavior is better understood in terms of a semiclassical analysis of the tunneling process as is given by the instanton technique [26,27,28,29]. The origin of the oscillation of the tunneling gap is the interference between the Berry's phases associated with complementary paths that accomplish the reversal. The result is [26] ∆ = ∆ 0 cos(SΩ) where Ω is the solid angle subtended by the complementary paths and ∆ 0 is a monotonous function of h. Whenever SΩ = (2n + 1)π/2 for integer n the splitting vanishes. It is important to note that the dynamical supression of the Rabi oscillations due to the rapidly oscillating magnetic field arises due to the interference between tunneling amplitudes associated with different reversal paths and is, therefore, different in origin from the well-known coherent suppression of tunneling that has been predicted [30] in one dimensional systems. This expression makes explicit the oscillations of the gap as function of the external driving field. However, the most important fact is that it emphasizes their origin in the Berry's phase interference of complementary paths that achieve the tunnel reversal. For this reason, the effect is known as Berry's phase interference quenching of the tunnel effect [26]. This semiclassical argument ought to be contrasted with the exact diagonalization of the Hamiltonian given in Eq. (12) for different values of the magnetic field intensity, h. After a direct numerical diagonalization, it is possible to calculate the energy gap between the two lowest lying states. This gap is shown, for S = 1, S = 3/2, S = 2 and S = 5/2, in the continuous lines in the Fig. 4 where the tunneling times are plot as functions of h 2 /(2 ω). In those figures it is evident the oscillatory behavior displayed by the gap and the quenching associated with its zeroes (indicated by arrows). We conclude that the effective slow dynamics of a spin under a circularly polarized magnetic field will be described by oscillations between two states |Ω 1 and |Ω 2 , where Ω 1,2 correspond to the unit vector with polar angles (θ 0 , φ 1,2 ). Here, the state |Ω corresponds to the spin coherent state oriented along the direction of Ω [16]. Those oscillations correspond to tunneling events. Furthermore, it can be concluded that those oscillations are quenched for certain values of the amplitude of the oscillating magnetic field. In the next section this prediction will be contrasted with exact numerical results. Comparison with exact results We now contrast the predictions made so far with exact numerical results. To that end we solve numerically the time dependent Schrödinger equation Eq. (3). To characterize the slow dynamics generated by the Schrödinger equation we have prepared the state initially in the ket |Ω 1 and let it evolve. We have computed the projection of the state vector |Ψ(t) on the kets |Ω 1 and |Ω 2 with typical behavior shown in Fig. 3 where we plot | Ω 1 |Ψ | 2 and | Ω 2 |Ψ | 2 . We see that the projection on the initial state |Ω 1 starts at its maximum value, one, and then it is reduced to zero. Meanwhile the projection on the second state describes the opposite behavior. We remark that the oscillation in the amplitude is accompanied by a rapidly oscillatory component that corresponds to the fast dynamics. This behavior, reminiscent to that of a two level system, corresponds to the quantum tunneling between the states described by the states |Ω 1 and |Ω 2 . The characteristic time T that takes a transition from one state to the other correspond to a direct quantification of the tunneling time. This time can be inferred directly from the numerical solution. For different values of the field we calculate such tunneling time. The results, for S = 1, S = 3/2, S = 2 and S = 5/2, are displayed in Fig. 4. In the different calculations we have chosen units in which D = 1 and selected a value for E = 0.1D. For the numerical calculations we have selected the driving frequency as ω = 50D/ . The results are in evident agreement with those obtained from the effective theory. This provides strong evidence in favor of the validity of the effective Hamiltonian (Eq. (12)) to describe an important aspect of the complicated dynamics displayed by the system. These exact results show that the tunnel effect is effectively quenched by the time dependent magnetic field for certain values of the field intensity and frequency. This quenching is made manifest by a divergence of the tunneling time associated with quantum transitions between the two degenerate classical ground states. Discussion In this paper we have presented a detailed study of the dynamics described by a quantum spin when exposed to a rapidly varying magnetic field. The problem is analogous to the pendulum, proposed by Kapitza, with a rapidly oscillating suspension point. Just like in the Kapitza's pendulum the motion of the quantum states is decomposed into two components. We have a rapid behavior whose characteristic time is given by the external potential and a slow contribution. The slow dynamics is affected by the rapid motion and is described by an effective Hamiltonian. By performing this separation in a systematic fashion we have proved that the slow dynamics is accounted for by an effective Hamiltonian that has a Zeeman-like contribution in the direction perpendicular to the plane of polarization of the magnetic field. The effective Hamiltonian displays the interesting effect of geometric-phase induced quenching of the tunnel effect that can be used to freeze the quantum states in well defined configurations. Just like the Kapitza's pendulum is trapped in the vertical position, unstable when the driving force is absent, the spin of the molecular magnet is trapped in a given quantum state. In this sense we talk about a Kapitza effect in Hilbert space. This effect provides a tool to control the quantum states by means of high frequency fields. For a typical molecular nanomagnet our analysis sets the desired frequencies in the terahertz range. In addition we have presented numerical evidence, based on a direct solution of the Schrödinger equation, that confirms the predictions of the effective description. Figure 2 : 2Semiclassical origin of the effective magnetic field. (a) the spin precesses around an initial field h 1 . This precession is interrupted when the magnetic field changes into h 2 . Under the new field the precession axes is changed and a new precessional motion is described. A second change into h 3 changes once again the precession axis. The net result of this sequence of changes is an effective precession around the z axis. (b) Exact solution of the Landau-Lifshitz equation for ω = 5h. The exact results match the qualitative argument of panel (a). The results are consistent with a precession around an effective magnetic field pointing in the z axis. Figure 3 : 3Characteristic behavior of the projections | Ω 1 |Ψ | 2 and | Ω 2 |Ψ | 2 as functions of time. The initial state is |Ω 1 and in the slow dynamics regime the system manifest well defined tunneling oscillations between the two classical ground states. The tunneling time, defined as the characteristic time that takes the system from one configuration to the other, is denoted by T . The slow dynamics is dressed by very fast oscillations. Figure 4 : 4Tunneling time is shown for spin S = 1, S = 3/2, S = 2, and S = 5/2 as a function of the intensity of the magnetic field h ef f = h 2 /(2 ω). The tunneling time, obtained directly from the numerical solutions of Eq. (3), is represented by the dots. It is compared with the predictions from the effective theory given by Eq.(12). The effect of quenching of the tunnel effect implies a divergence of the tunneling time as indicated by the arrows. AcknowledgmentThe authors acknowledge support from Fondo Nacional de Desarrollo Científico y Tecnológico Grant No. 1150072; Basal Program Center for Development of Nanoscience and Nanotechnology (CEDENNA); and Anillo de Ciencia y Tecnonología ACT Grant No. 1117; . R Sessoli, Nature. 365141R. Sessoli et al., Nature. 365, 141 (1993). Molecular Nanomagnets. Jacques Villain, Dante Gatteschi, Roberta Sessoli, Oxford University PressJacques Villain, Dante Gatteschi and Roberta Sessoli, Molecular Nano- magnets. Oxford University Press (2006). . W Wernsdorfer, Science. 284133W. Wernsdorfer et al., Science. 284, 133 (1999). . R Sessoli, Angew. Chem. Int. Ed. 42268R. Sessoli, et al. Angew. Chem. Int. Ed., 42, 268 (2003). . Jonathan R Friedman, Myriam P Sarachik, Annu. Rev. Condens. Matter Phys. 1109Jonathan R. Friedman and Myriam P. Sarachik, Annu. Rev. Condens. Matter Phys. 1 109, (2010). . A Rocha, Nature Materials. 4A. Rocha et al., Nature Materials 4, 335 -339 (2005) . L Bogani, W Wernsdorfer, Nature Mater. 7179L. Bogani and W. Wernsdorfer, Nature Mater, 7, 179 (2008). . Stefano Sanvito, Chem. Soc. Rev. 403336Stefano Sanvito, Chem. Soc. Rev.,40, 3336 (2011). . Eugenio Coronado, Arthur J Epsetin, J. Mater. Chem. 191670Eugenio Coronado and Arthur J. Epsetin, J. Mater. Chem., 19, 1670, (2009). . N Michael, Daniel Leuenberger, Loss, Nature. 410789Michael N. Leuenberger and Daniel Loss, Nature 410, 789 (2001). . Tobias Kampfrath, Nature Photonics. 531Tobias Kampfrath, et al. Nature Photonics, 5, 31 (2011). . S Takayoshi, H Aoki, T Oka, Phys. Rev. B. 9085150S. Takayoshi, H. Aoki, and T. Oka, Phys. Rev. B 90, 085150 (2014). . Tobias Kampfrath, Nature Nanotechnology. 8256Tobias Kampfrath, et al. Nature Nanotechnology, 8, 256 (2013). . L D Landau, E M Lifshitz Mechanics, Oxford: PergamonL. D. Landau and E. M. Lifshitz Mechanics, Oxford: Pergamon, (1989). . P L Kapitza Zh, Eksp. Teor. Fiz. 21588P. L. Kapitza Zh. Eksp. Teor. Fiz. 21 588, (1951). Assa-Auerbath, Interacting Electrons and Quantum Magnetism. Springer-Verlag PressAssa-Auerbath., Interacting Electrons and Quantum Magnetism. Springer-Verlag Press (1994). . J L Van Hemmen, H Hey, W F Wreszinski, J. Phys. A: Math. Gen. 306371J. L. van Hemmen, H. Hey and W. F. Wreszinski, J. Phys. A: Math. Gen. 30, 6371 (1997); . J L Van Hemmen, W F Wreszinski, Phys. Rev. B. 571007J. L. van Hemmen and W. F. Wreszinski, Phys. Rev. B 57, 1007 (1998). Jean Zinn-Justin, Path Integrals in Quantum Mechanics. Oxford University PressJean Zinn-Justin., Path Integrals in Quantum Mechanics. Oxford Uni- versity Press (2006). Eugene M Chudnovsky, Macroscopic Quantum Tunneling of the Magnetic Moment. Cambridge University PressEugene M. Chudnovsky et al., Macroscopic Quantum Tunneling of the Magnetic Moment. Cambridge University Press (1998). Quantum Tunneling of Magnetization. Leon Gunther, Nato ASI Series. Leon Gunther et al., Quantum Tunneling of Magnetization. Nato ASI Series (1994). . I I Rabi, Phys. Rev. 51652I.I. Rabi, Phys. Rev. 51, 652 (1937). . L Sorace, Phys. Rev. B. 68220407L. Sorace et al. Phys. Rev. B 68, 220407 (2003). . E Barco, Phys. Rev . Lett. 93157202E. del Barco et al. Phys. Rev . Lett. 93, 157202 (2004). . S Rahav, Phys. Rev. A. 6813820S. Rahav et al. Phys. Rev. A 68, 13820 (2003) . Ali H Nayfeh, Dean T Mook, Nonlinear Oscillations, WileyAli H. Nayfeh, Dean T. Mook, Nonlinear Oscillations, Wiley (2004). . Anupam Garg, Europhys. Lett. 22205Anupam Garg., Europhys. Lett. 22, 205 (1993). . Zhi-De Chen, Phys. Rev. B. 6585313Zhi-De Chen et al., Phys. Rev. B. 65, 085313 (2002). . J M Florez, Alvaro S Nunez, P Vargas, J. Mag. Magn. Mat. 3223623Florez, J. M.; Nunez, Alvaro S.; Vargas, P., J. Mag. Magn. Mat. 322 3623(2010). . J M Florez, P Vargas, Alvaro S Nunez, Physica B-Condensed Matter. 4042791Florez, J. M.; Vargas, P.; Nunez, Alvaro S., Physica B-Condensed Mat- ter 404 2791 (2009). . F Grossmann, T Dittrich, P Jung, P Hnggi, Phys. Rev. Lett. 67516F. Grossmann, T. Dittrich, P. Jung, and P. Hnggi, Phys. Rev. Lett. 67, 516 (1991)

In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of 2π. For two clocks on counterrotating equatorial circular orbits around the Earth, the effect is about 10 −7 seconds per revolution, which is quite large. We introduce a general relativistic definition of the gravitomagnetic clock effect which is valid for arbitrary pairs of orbits. This includes rotations in the same direction and different initial conditions, which are crucial if the effect can be detected with existing satellites or with payloads on nondedicated missions. We also derive the post-Newtonian expansion of the general relativistic expression and calculate the effect for the example of a satellite of a Global Navigation Satellite System compared to a geostationary satellite.

10.1103/physrevd.90.044059

1406.6232

46708a1fa8dca7650985ffde9aad563ba3fe05bb

Generalized gravitomagnetic clock effect 28 Aug 2014 Eva Hackmann Claus Lämmerzahl ZARM University of Bremen Am Fallturm28359BremenGermany Institute for Physics, University Oldenburg ZARM University of Bremen Am Fallturm28359, 26129Bremen, OldenburgGermany and, Germany Generalized gravitomagnetic clock effect 28 Aug 2014 In General Relativity, the rotation of a gravitating body like the Earth influences the motion of orbiting test particles or satellites in a non-Newtonian way. This causes, e.g., a precession of the orbital plane known as the Lense-Thirring effect and a precession of the spin of a gyroscope known as the Schiff effect. Here, we discuss a third effect first introduced by Cohen and Mashhoon called the gravitomagnetic clock effect. It describes the difference in proper time of counterrevolving clocks after a revolution of 2π. For two clocks on counterrotating equatorial circular orbits around the Earth, the effect is about 10 −7 seconds per revolution, which is quite large. We introduce a general relativistic definition of the gravitomagnetic clock effect which is valid for arbitrary pairs of orbits. This includes rotations in the same direction and different initial conditions, which are crucial if the effect can be detected with existing satellites or with payloads on nondedicated missions. We also derive the post-Newtonian expansion of the general relativistic expression and calculate the effect for the example of a satellite of a Global Navigation Satellite System compared to a geostationary satellite. I. INTRODUCTION Within Einstein's General Relativity the rotation of an astronomical object like the Earth has a purely relativistic effect on the motion of orbiting objects which is usually referred to as "frame dragging". Maybe better terminology can be found in analogy to electrodynamics by denoting the effects due to mass currents as "gravitomagnetic". Both terms summarize at least two well-known effects. First, the Lense-Thirring effect calculated in 1918 [1,2], which in the weak field can be interpreted as a precession of the longitude of the ascending node. Solar System measurements of the Lense-Thirring precession were achieved with the LAGEOS mission and will be further improved using the LARES satellite, see e.g. [3]. Second, the Schiff effect calculated in 1960 [4,5], which describes the precession of a gyroscope orbiting a rotating object. This effect was measured by the Gravity Probe B experiment [6,7]. In this paper we are interested in another effect caused by the rotation of the gravitating object, the so-called gravitomagnetic clock effect. It refers to different time measurements of two clocks orbiting a rotating astronomical object, one in the direction of rotation, i.e. on a prograde orbit, and the other against the direction of rotation, i.e. on a retrograde orbit. There are several versions of this clock effect which differ in the details of their definitions [8][9][10]. Here, we discuss what the authors of [8] call the observer-dependent two-clock clock effect. This is the difference of the proper times of two clocks, one on a pro-and the other on a retrograde orbit, * [email protected] † [email protected] after a complete revolution of 2π. Note that the proper time measured by a clock on a geodesic is invariant. The observer enters the discussion through the notion of a "full revolution" which depends on the frame of reference [11]. Alternative definitions of the gravitomagnetic clock effect include e.g. the difference in the proper time of the two clocks after a fixed coordinate time of an observer [9] or the difference in proper time of the two clocks at their meeting point [10]. The gravitomagnetic clock effect considered here was first correctly derived and studied in detail by Cohen and Mashhoon [12] following an idea shortly mentioned in [13,14]. For two counterrevolving clocks on circular orbits of the same radius in the equatorial plane of the Earth Cohen and Mashhoon found that the effect is of the order of 10 −7 seconds per revolution, independent of the radius of the two clocks. Compared to the increasing accuracies of space-based clocks this seems to be quite large. A generalization to the parametrized post-Newtonian formalism also including the effects of the nonspherical shape of the Earth was considered in [15]. Eccentric and inclined orbits were discussed in [9,16], but the requirement of identical initial orbital parameters, apart from the sense of rotation, was so far not removed. A dedicated satellite mission to measure this effect called Gravity Probe C lock was proposed by Gronwald and others [17]. Gravitational and nongravitational error sources for such a mission were also discussed [17][18][19][20][21]. From these analyses, it can be concluded that the most challenging task for a mission to measure the clock effect is not the stability of the orbiting clocks but the precise tracking of the two satellites. This is needed because of the imperfect cancellation of the Keplerian periods of the two clocks, which induces large errors in the measurement. In this paper, we find a fully general relativistic defini-tion of the gravitomagnetic clock effect in Kerr spacetime, which also generalizes the clock effect to two arbitrary geodesics, including rotations in the same direction and different initial orbital parameters (see also [22]). We use for this definition the fundamental frequencies of bound orbits in Kerr spacetime given by Schmidt [23] and elaborated by Fujita and Hikida [24]. This procedure is completely analogous to the definition of the perihelion shift and the Lense-Thirring effect in terms of fundamental frequencies [24,25]. The generalization of the gravitomagnetic clock effect to two arbitrary geodesics would, in principle, allow to use existing satellites for a measurement of the effect as long as they carry stable clocks and can be tracked with sufficient accuracy. We also derive a post-Newtonian expansion of the generalized gravitomagnetic clock effect which can be handled more conveniently than the fully general relativistic expression and should still be sufficiently accurate for orbits in the gravitational field of the Earth. This expression is then applied to a spacecraft of a Global Navigation Satellite System (GNSS) compared to a geostationary satellite. The paper closes with a summary. II. FUNDAMENTAL FREQUENCIES IN KERR SPACETIME We start with a review and an extension of the fundamental frequencies in Kerr spacetime given by Schmidt [23] and by Fujita and Hikida [24]. The Kerr metric in Boyer-Lindquist coordinates reads ds 2 = ∆ r ρ 2 cdt − a sin 2 θdϕ 2 − ρ 2 ∆ r dr 2 − ρ 2 dθ 2 − sin 2 θ ρ 2 (acdt − (r 2 + a 2 )dϕ) 2 ,(1) where ds 2 = c 2 dτ 2 with the proper time τ , ∆ r = r 2 + a 2 − 2M r, ρ 2 = r 2 + a 2 cos 2 θ with M = Gm/c 2 and a = J/(mc), where m is the mass and J > 0 is the angular momentum of the gravitating object. Here G is the gravitational constant and c the speed of light. The geodesic equation in Kerr spacetime can be completely separated due to the existence of four constants of motion. This is the specific energyẼ and the specific angular momentum in the direction of the symmetry axes L z ,Ẽ = g ttṫ + g tϕφ =: c 2 E ,(2)L z = −g ϕϕφ − g tϕṫ =: cL z ,(3) where the dot denotes a derivative with respect to τ . The two remaining constants are the normalization g µνẋ µẋν = ǫc 2 with ǫ = 1 for massive test particles and the Carter constant K [26]. There are some alternative forms of the Carter constant; we use here K such that K = (aE − L z ) 2 for motion in the equatorial plane. With these constants of motion we get the equations of motion in the form [27] dr dλ 2 = R 2 − ∆ r (ǫr 2 + K) =: R ,(4)dθ dλ 2 = K − ǫa 2 cos 2 θ − T 2 sin 2 θ =: Θ ,(5)dϕ dλ = a ∆ r R − T sin 2 θ =: Φ r (r) + Φ θ (θ) ,(6)c dt dλ = r 2 + a 2 ∆ r R − aT =: T r (r) + T θ (θ) . (7) where R = (r 2 + a 2 )E − aL z ,(8)T = aE sin 2 θ − L z .(9) Here λ is the "Mino time" which is connected to the proper time by cdτ = ρ 2 dλ. It is an auxiliary parameter introduced to completely decouple the equations of motions. Note that the equations of motions can be rewritten in a dimensionless form by dividing each by the appropriate power of M and redefininḡ r = r/M ,t = ct/M ,ā = a/M , L z = L z /M ,K = K/M 2 ,λ = λM .(10) In general, the motion of test particles in Kerr spacetime neither has the form of a conic section nor lies in an orbital plane. This is due to a mismatch of the periodicities of the radial and latitudinal motion, which, in general, differ from each other and from 2π [23,24]. Let us consider a bound orbit of a massive test particle (i.e. ǫ = 1) which does not cross a horizon. In this case the radial motion oscillates between the periapsis r p and the apoapsis r a . Also the test particle oscillates around the equatorial plane between two extremal values θ min,max with θ max = π −θ min . The radial and latitudinal periods Λ r,θ are then given by Λ r = 2 rp rp dr R(r) ,(11)Λ θ = 2 θmax θmin dθ Θ(θ) ,(12) i.e.r(λ + Λ r ) =r(λ) and θ(λ + Λ θ ) = θ(λ) for allλ. The conjugate fundamental frequencies are defined as Υ r = 2π/Λ r and Υ θ = 2π/Λ θ . As the ϕ, t, and τ motions are not periodic we have to use a somewhat different approach to define the corresponding fundamental frequencies. We write the coordinate as a part which is linear in λ plus perturbations in r and θ, ϕ(λ) = Υ ϕλ + Φ r osc (r) + Φ θ osc (θ) ,(13)Υ ϕ := Φ r (r) + Φ θ (θ) λ ,(14) where · λ := lim (λ2−λ1)→∞ 1 λ 2 −λ 1 λ 2 λ1 · dλ(15) is an infinite time average with respect toλ. The functions Φ r osc (r) and Φ θ osc (θ) represent oscillatory deviations from the average. They are defined by Φ r osc (r) = Φ r (r)dλ − Φ r (r) λλ ,(16)Φ θ osc (θ) = Φ θ (θ)dλ − Φ θ (θ) λλ(17) and have periods Λ r and Λ θ , respectively. As Υ ϕ containsr-and θ-dependent parts which are periodic functions with respect toλ, the integral (15) can be reduced to an integral over one period. Therefore we find Υ ϕ = 2 Λ r rā rp Φ r (r)dr R(r) + 2 Λ θ θmax θmin Φ θ (θ)dθ Θ(θ) .(18) Analogously, we may define Υ t = 2 Λ r rā rp T r (r)dr R(r) + 2 Λ θ θmax θmin T θ (θ)dθ Θ(θ) ,(19)Υ τ = 2 Λ r rā rpr 2 dr R(r) + 2 Λ θ θmax θmin a 2 cos 2 θdθ Θ(θ) .(20) III. GENERAL DEFINITION OF THE GRAVITOMAGNETIC CLOCK EFFECT The gravitomagnetic clock effect considered here was studied already in 1993 by Cohen and Mashhoon [12]. They showed that two clocks on circular equatorial orbits of the same radius but orbiting in different directions show after a revolution of 2π a time difference of τ + − τ − ≈ 4π J mc 2 ,(21) where τ + is the proper time of the corotating and τ − of the counterrotating clock. Here J is the angular momentum of the Kerr black hole, as before. For satellites orbiting the Earth this yields an effect of the order of 10 −7 s per revolution, which is surprisingly large. The key element is here the measurement after a full revolution of 2π. For measurements after a specific coordinate time or at the meeting point of the clocks the effect is much smaller [9,10]. The result (21) was generalized to spherical orbits with small inclination in [9] and further to orbits with small eccentricity in [16] (their equation (31)), t + − t − ≈ 4π J cos i mc 2 −3 √ 1 − e 2 + + 4 − 2 cos 2 ϕ 0 tan 2 i (1 + e cos(ϕ 0 − g)) 2 ,(22) where i is the inclination measured from the equatorial plane, e is the eccentricity, g is the argument of the pericenter, and ϕ 0 − g is the true anomaly at t = 0. As remarked in [16], at the considered level of approximation the coordinate times used in (22) may be replaced by the proper times. Note that the expression (22) depends on the initial position of the two clocks. Here we introduce a fully general relativistic expression for the gravitomagnetic clock effect based on fundamental frequencies. As outlined in section II the functions ϕ(λ),t(λ), andτ (λ) can be written as a part which is linear inλ plus periodic perturbations. As only the linear parts contribute to the average secular increase of the coordinate, we may use this to define observable quantities like the perihelion shift and the Lense-Thirring effect [24,25]. Analogously, for the gravitomagnetic clock effect we may define a functionτ : ϕ →τ (λ(ϕ)) by using the linearized functionsλ(ϕ) = Υ −1 ϕ ϕ,τ (λ) = Υ τλ , τ (ϕ) := Υ τ Υ −1 ϕ ϕ .(23) In the Newtonian limit, τ (2π) (as well as the corresponding t(2π)) reduces to the Keplerian time of revolution 2π d 3 Gm , where d is the semimajor axis; see (29) below. Assume now two clocks moving along arbitrary geodesics with given periapsides r p,n , apoapsides r a,n , and maximal inclinations θ max,n , n = 1, 2. For each orbit we may calculate the proper time for a revolution of ±2π using (23),τ n (±2π; a), where the sign in front of 2π indicates pro-(+) or retrograde (−) motion and the additional argument indicates the dependence ofτ on the Kerr rotation parameter a > 0. We may also calculate the corresponding value in case the rotation of the central object would vanish,τ n (±2π; 0). To extract the purely gravitomagnetic effect, we define a new observable ∆τ gm :=τ 1 (±2π) + ατ 2 (±2π) ,(24) where the factor of proportionality α is calculated such that the usual gravitoelectric effects just cancel each other. This condition determines α via ∆τ gm = 0 for a = 0, 0 =τ 1 (±2π; 0) + ατ 2 (±2π; 0)(25) and, therefore, α = −τ 1 (±2π; 0) τ 2 (±2π; 0) .(26) The sign in front of 2π has to be chosen for each orbit according to its sense of rotation, i.e. +2π (−2π) for prograde (retrograde) orbits. The actual calculation procedure for ∆τ gm is then as follows: for both clocks, calculate the energies E n , the angular momenta L z,n , and the Carter constants K n , n = 1, 2, by using dr dλ (r p,a ) = 0 in (4) and dθ dλ (θ max ) = 0 in (5). As one may choose E > 0 without loss of generality, for each orbit this gives two solutions, one for a prograde orbit with L z > 0 and one for a retrograde orbit with L z < 0, from which we choose according to their sense of rotation. Note that the values E n , L z,n and K n depend on the rotation parameter a of the central object, E n = E n (a), L z,n = L z,n (a), K n = K n (a). The corresponding values E n (0), L z,n (0), K n (0) can be determined by setting a = 0 in (4) and (5). Then use (18) and (20) (or the corresponding expressions in terms of Jacobian elliptic integrals given in appendix A) to calculatē τ n (±2π; a) andτ n (±2π; 0), which gives α and ∆τ gm . Note that the value of the clock effect (24) depends on the numeration of the two clocks. If we denote by ∆τ (2,1) gm the clock effect with reversed clock labels as compared to ∆τ (1,2) gm we find ∆τ (2,1) gm = −τ 2 (±2π; 0) τ 1 (±2π; 0) ∆τ (1,2) gm .(27) This ambiguity can be removed if ∆τ gm is referred to the Schwarzschild orbit time of the first clock, ∆τ gm τ 1 (±2π; 0) =τ 1 (±2π; a) τ 1 (±2π; 0) −τ 2 (±2π; a) τ 2 (±2π; 0) .(28) The absolute value of this quantity does not change if the two clocks interchange their labels. IV. POST-NEWTONIAN EXPANSION We explore now the definition (24) by deriving an expansion, where we assume that the rotation parameter a and the mass parameter M are small compared to the radii of the clock orbits. This holds for the exterior gravitational field of the Earth where M/r 7 × 10 −10 and a/r 6 × 10 −7 . Let us assume that the orbital parameters r p , r a and θ max are fixed for both orbits. Therefore, to derive the expansion of τ (±2π) for small a we need the expansions of the constants of motions in terms of a, which are given with some additional details in appendix B. We find τ (±2π) ≈ 2π d 3 Gm 1 − 3(1 + e 2 ) 2(1 − e 2 ) M d (29) ± 2π(cos i(3e 2 + 2e + 3) − 2e − 2) (1 − e 2 ) 3 2 a c , where d is the semimajor axis, e is the eccentricity, and i is the inclination, which are defined via r p = d(1 − e), r a = d(1 + e), and θ max = π/2 + i. The sign in (29) has to be chosen according to the sense of rotation, the plus (minus) sign for prograde (retrograde) motion. Consider now the gravitomagnetic clock effect for two clocks on geodesics with identical orbital parameters r p , r a , and θ max , one on a pro-and the other on a retrograde orbit. Here α = −1 and with τ + = τ (+2π) and τ − = τ (−2π) we find which for circular equatorial geodesics (e = 0, i = 0) reduces to the formula (21) derived in [12]. For fixed values of the inclination the effect is visualized in figure 1. At a fixed eccentricity the clock effect (30) vanishes if cos i = 2(1 + e) 3e 2 + 2e + 3 τ + − τ − ≈ 4πJ mc 2 cos i(3e 2 + 2e + 3) − 2e − 2) (1 − e 2 ) 3 2 ,(30) and gets negative for larger inclinations, see also figure 1. Note that the expansion (29) is not valid for polar orbits, see appendix B. However, for nearly polar orbits the ex-pression (29) holds and the clock effect (30) approaches a maximal absolute value, τ − − τ + → 8πJ mc 2 1 + e (1 − e 2 ) 3 2 . (32) Equation (30) differs from the expression (22) in a key aspect: it does not depend on the initial position of the clocks. This is due to the definition via fundamental frequencies, which are given by averages over infinite Mino time. This procedure is completely analogous to the derivation of the Lense-Thirring effect Ω LT in the fully general relativistic setting [24,25], Also, (30) is valid for any 0 ≤ e < 1 and 0 ≤ i < π/2, whereas (22) assumes small eccentricities and inclinations. But also in the limit of small e or small i the expressions (30) and (22) differ. This is, of course, natural as (30) is independent of the initial conditions. Ω LT = Υ ϕ Λ θ − 2π( For two general orbits we derive a post-Newtonian expression of α (see (26)), α ≈ − d 3 2 1 d 3 2 2 − 3d 1 2 1 2d 5 2 2 d 1 (1 + e 2 2 ) 1 − e 2 2 − d 2 (1 + e 2 1 ) 1 − e 2 1 M (34) and from (29) the post-Newtonian expansion of the gravitomagnetic clock effect to lowest order, ∆τ gm ≈ a c s 1 2π(cos i 1 (3e 2 1 + 2e 1 + 3) − 2e 1 − 2) (1 − e 2 1 ) 3 2 − s 2 2πd 3 2 1 (cos i 2 (3e 2 2 + 2e 2 + 3) − 2e 2 − 2) d 3 2 2 (1 − e 2 2 ) 3 2 (35) where the indices indicate the first/second clock. Here s 1 and s 2 are equal to +1 (prograde motion) or −1 (retrograde motion) according to the sense of rotation of the respective orbit. Note that the expressions (35) and (29) diverge for e → 1, which is not surprising as r p = d(1 − e) > r B , where r B is the radius of the central body, requires d → ∞ for e → 1. Therefore, in this limit the orbit time itself diverges and it does not make sense to consider the clock effect. V. APPLICATION TO SATELLITES OF THE GNSS We apply now the post-Newtonian expression (35) for the calculation the gravitomagnetic clock effect to satellites of the Global Navigation Satellite Systems (GNSS), which carry very stable clocks with frequency stabilities of about 10 −14 over ten thousand seconds. For a detection of the effect, the proper times of the satellite clocks after a revolution of 2π have to be communicated to an observer on the ground, who also determines the orbital parameters d, e, and i for each clock. Note that the expression (29) relates these parameters and, therefore, may be used for a consistency check. Using d and e of both clocks we then calculate α by (34) and then the gravitomagnetic clock effect (24). As all GNSS satellites are on nearly circular prograde orbits we need a pair of satellites with different inclinations. The GPS and Galileo satellite systems operate on very similar inclination (55 • and 56 • , respectively) whereas the GLONASS system operates at slightly larger inclination (65 • ), which is still quite close to the GPS and Galileo systems. The Chinese COMPASS system, however, includes geostationary satellites. Therefore, we compare here a geostationary orbit with a Galileo and a GLONASS satellite orbit. For the Galileo satellites we assume an inclination of i Ga = 56 • , an eccentricity of e = 0, and a semimajor axis of d Ga = 29593 km, and for the GLONASS system i GL = 64.8 • , e = 0, and d GL = 25471 km. As orbital parameters of the geostationary satellite we take i Ge = 0, e = 0, and d Ge = 42157 km. We insert this into the expression (35) together with the mass and rotation parameters of the Earth, M For the GLONASS satellite we find ∆τ gm ≈ −9.87 × 10 −8 sec , ∆τ gm τ GL (±2π; 0) ≈ −2.44 × 10 −12 .(38) In principle, the clocks on board the GNSS satellites, including the geostationary COMPASS satellite, should be able to detect this effect. However, we assumed here geodesic motion in a Kerr spacetime in the absence of any disturbing forces. Therefore, a careful analysis of the influence of gravitational and environmental perturbations has to be carried through to judge the measurability of the effect. An analysis of disturbing effects for the situation of identical but counterrevolving clock orbits can be found in [17][18][19][20][21]. Let us only note two major points here. First, the gravitomagnetic clock effect is quite large compared to the sensitivity of the clocks but tiny compared to the measured proper times for a full revolution, see also [21]. If we assume an uncertainty in the semimajor axis of ∆d, then ∆τ /τ ≈ 3/2 ∆d/d, which implies that in the above examples the semimajor axes must be known with an accuracy of about 10 µm. As the gravitomagnetic clock effect accumulates over every revolution this stringent requirement can be relaxed with sufficiently long observation times. Another theoretical possibility to achieve a high accuracy for the value of d would be to use a second clock in each satellite, whose position with respect to the first clock is very well known. If the measured proper time of this second clock, say τ ′ (±2π), is inserted in equation (29) we may calculate its semimajor axis d ′ , assuming e ′ and i ′ for this clock are known to sufficient precision. If d ′ = d + δd, where δd is very well known, we may calculate the semimajor axis d of the first clock which is used to detect the gravitomagnetic effect. Second, the high accuracy for the inclination mentioned in [20] does not apply for the situation considered here, as we assumed different inclinations for the two satellites. If we assume an uncertainty of ∆(cos i), we find for orbits of eccentricity e = 0 that ∆τ ≈ 6π a c ∆(cos i), which implies that the inclinations in the examples above should be known to an accuracy of at least 0.03 degrees. VI. SUMMARY We presented a generalization of the gravitomagnetic clock effect [12] for two clocks moving on arbitrary geodesics in the Kerr spacetime. The definition uses the concept of fundamental frequencies of bound orbits in Kerr spacetime introduced by Schmidt [23] and elaborated by Fujita and Hikida [24] based on a formulation of the geodesic equations in terms of the Mino time [27]. We also derived the post-Newtonian expansion of the effect which yields a more convenient formula and should still be sufficiently accurate for clocks moving in the gravitational field of the Earth. For the example of a GNSSlike satellite orbit compared to a geostationary orbit we found that the effect is of the order of 10 −8 seconds per revolution and relative to the orbit time of order 10 −12 . The novel aspect of this generalized definition of the gravitomagnetic clock effect is that the two clocks may have arbitrary initial conditions and may follow completely different geodesics. This point is crucial if the effect should be tested with existing satellites or with a piggyback payload on another scientific mission. It also enables to consider the effect for astronomical objects. If, for example, two pulsars orbiting Sagittarius A* would be found, the gravitomagnetic clock effect could provide a consistency check of orbital data or of the value of the rotation parameter of the central black hole. Here we considered geodesic motion in the Kerr spacetime, which is, of course, a very idealized situation. For a more realistic treatment it is certainly necessary to consider numerous perturbing effects, both of gravitational and nongravitational origin. Besides the stable clocks needed for a measurement of the gravitomagnetic clock effect a precise tracking of the clocks will be crucial. ACKNOWLEDGMENTS We thank Hansjrg Dittus, Norman Gürlebeck, Sven Herrmann, Bahram Mashhoon, and Volker Perlick for valuable discussions. Financial support from the German Research Foundation (DFG) through the research training group "Models of Gravity" is gratefully acknowledged. We also thank the Collaborative Research Center "Relativistic geodesy and gravimetry with quantum sensors" (geo-Q) for support. Appendix A: Calculation of fundamental frequencies The radial and latitudinal periods Λ r and Λ θ as well as the fundamental frequencies Υ ϕ , Υ t , and Υ τ can be expressed in terms of Jacobian elliptic integrals. These are implemented in several computer algebra systems like Mathematica or Maple and can be calculated easily and quickly. In general, every integral of the form b a Q(z)dz P (z) ,(A1) where Q is a rational function and P is a polynomial of degree three or four, can be expressed in terms of elliptic integrals. If P has only real zeros and a, b are two neighbouring zeros of P , then a substitution of the form z = αnx 2 +β nx 2 +1 can be used to transform the above integral to the form C 1 0 Q(z(x))dx (1 − x 2 )(1 − k 2 x 2 ) ,(A2) where C is a constant and 0 < k < 1. Now Q(z(x)) can be decomposed in partial fraction and the integral can be expressed in terms of complete elliptic integrals of the first, second, and third kind, K(k) = 1 0 dx (1 − x 2 )(1 − k 2 x 2 ) , E(k) = 1 0 (1 − k 2 x 2 )dx (1 − x 2 )(1 − k 2 x 2 ) ,(A3)Π(n, k) = 1 0 dx (1 − nx 2 ) (1 − x 2 )(1 − k 2 x 2 ) . In our case we encounter the polynomials R(r) (see (4)) and with ν = cos 2 θ in (5) dν dλ 2 = 4ā 2 (1 − E 2 )ν 3 + 4(K − (āE −L z ) 2 )ν (A4) + 4(2āE(āE −L z ) −K − ǫā 2 )ν 2 =: Θ ν (ν) . For bound orbits R has four real zerosr 1 <r 2 <r p <r a . All radial integrals can then be transformed to Jacobian elliptic integrals by the substitution r = αnx 2 +β nx 2 +1 with α =r 2 , β =r p , and n = −r a−rp ra−r2 . The radial period is then given by Λ r = 4K(k r ) (1 − E 2 )(r p −r 1 )(r a −r 2 ) , (A5) k 2 r = (r a −r p )(r 2 −r 1 ) (r a −r 2 )(r p −r 1 ) . For bound orbits Θ ν has three real zeros 0 = ν 0 < ν max < 1 < ν 1 . With ν = αnx 2 +β nx 2 +1 where β = ν max , α = ν 1 , and n = − νmax ν1 we find Λ θ = 4K(k θ ) ā 2 (1 − E 2 )ν 1 , k 2 θ = ν max ν 1 .(A6) In the same way we may transform the integrals appearing in the definitions of Υ ϕ (18), Υ t (19), and Υ τ (20). We find Υ ϕ = 1 K(k r ) 1 0 Φ r (x)dx (1 − x 2 )(1 − k 2 r x 2 ) + 1 K(k θ ) 1 0 Φ θ (x)dx (1 − x 2 )(1 − k 2 r x 2 ) , (A7) where Φ r (x) =ā (r p −r 2 ) (h 1 − h 2 ) 2 i=1 (r p − h i ) −1 (−1) i R(h i ) (r 2 − h i )(1 − N r,i x 2 ) +ā R(r 2 ) ∆r 2 , N r,i = (r a −r p )(r 2 − h i ) (r a −r 2 )(r p − h i ) ,(A8) with the horizons h 1,2 = 1 ± √ 1 −ā 2 and Φ θ (x) = T (ν 1 ) ν 1 − 1 +L (ν 1 − ν max ) (ν 1 − 1)(1 − ν max )(1 − N θ x 2 ) , N θ = ν max (1 − ν 1 ) ν 1 (1 − ν max ) .(A9) In terms of Jacobian elliptic integrals this reads Υ ϕ =ā (r p −r 2 ) (h 1 − h 2 ) this expansion we consider them as functions ofā and compare the coefficients in 0 = R(r) − (1 − E 2 )(r a −r)(r −r p )(r −r 2 )(r −r 1 ) , 0 = Θ ν (ν) − 4ā 2 (1 − E 2 )ν(ν max − ν)(ν 1 − ν) ,(B1) taking into account that ν 1 = c 2ā −2 + c 1ā −1 + O(ā 0 ) for some constants c 2 , c 1 . Without loss of generality E can be assumed as positive and we find FIG. 1 . 1The gravitomagnetic clock effect (30) around the Earth. Top: The solid line is for equatorial orbits, the dashed line for inclination i = π/4, and the dotted line indicates the limit values for i → π/2. Bottom: The solid line is for spherical orbits, the dashed line for eccentricity e = 0.2, and the dotted line for e = 0.5. the post-Newtonian expansion (see appendix B for details) reduces to Ω LT ≈ 4π/(is the classical result[1] if it is referred to the Newtonian orbit time 2π d 3 /(Gm). ≈ 4 . 44346×10 −3 m and a c ≈ 1.317×10 −8 sec. For the Galileo satellite (clock 1) compared to the geostationary satellite (clock 2) we find ∆τ gm ≈ −7.54 × 10 −8 sec . (36) Referred to the Schwarzschild orbit time of the Galileo satellite we get ∆τ gm τ Ga (±2π; 0) ≈ −1.49 × 10 −12 . i=1 Analogously we getwhere N r,3 =r a −r p r a −r 2 .(A13)Appendix B: Details of the post-Newtonian expansionTo determine the post-Newtonian expansion of the general gravitomagnetic clock effect (24) we first need an expansion for smallā of the constants of motion E, L z , andK as well as the zerosr 1 ,r 2 , and ν 1 .wherer p =p(1 + e) −1 ,r a =p(1 − e) −1 , and ν max = sin 2 i. Here the upper sign corresponds to prograde and the lower sign to retrograde motion.These expansions must then be inserted in the expressions (A10) and (A11) to derive the post-Schwarzschild expansion of the gravitomagnetic clock effect. To first order inā we getwhere cos i = √ 1 − ν max = sin θ max and, (B12)Note that the expansion (B9) is not valid for polar orbits. 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Satellite gravitational orbital perturbations and the gravitomagnetic clock effect. Int. J. Mod. Phys. D, 10:465, 2001. On the possibility of measuring the gravitomagnetic clock effect in an Earth space-based experiment. L Iorio, H I M Lichtenegger, Class. Quantum Grav. 22119L. Iorio and H.I.M. Lichtenegger. On the possibility of measuring the gravitomagnetic clock effect in an Earth space-based experiment. Class. Quantum Grav., 22:119, 2005. The gravitomagnetic clock effect and its possible observation. H I M Lichtenegger, L Iorio, B Mashhoon, Ann. Phys. (Berlin). 15868H.I.M. Lichtenegger, L. Iorio, and B. Mashhoon. The gravitomagnetic clock effect and its possible observation. Ann. Phys. (Berlin), 15:868, 2006. Is it possible to measure the gravitomagnetic effect with clocks?. E Hackmann, C Lämmerzahl, F Merkle, Proc. of the 64th International Astronautic Congress (IAC). of the 64th International Astronautic Congress (IAC)Beijing, Chinato be publishedE. Hackmann, C.Lämmerzahl, and F.Merkle. Is it pos- sible to measure the gravitomagnetic effect with clocks? In Proc. of the 64th International Astronautic Congress (IAC), Beijing, China, 2013. to be published. Celestial mechanics in Kerr spacetime. W Schmidt, Class. Quantum Grav. 192743W. Schmidt. Celestial mechanics in Kerr spacetime. Class. Quantum Grav., 19:2743, 2002. Analytical solutions of bound timelike geodesic orbits in Kerr spacetime. R Fujita, W Hikida, Class. Quantum Grav. 26135002R. Fujita and W. Hikida. Analytical solutions of bound timelike geodesic orbits in Kerr spacetime. Class. Quan- tum Grav., 26:135002, 2009. Observables for bound orbital motion in axially symmetric space-times. E Hackmann, C Lämmerzahl, Phys. Rev. D. 8544049E. Hackmann and C. Lämmerzahl. Observables for bound orbital motion in axially symmetric space-times. Phys. Rev. D, 85:044049, 2012. Global structure of the Kerr family of gravitational fields. B Carter, Phys. Rev. 1741559B. Carter. 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This paper describes the low temperature nuclear and magnetic structures of La 2 O 2 Se 2 .Fe 2 O by analysis of X-ray and neutron diffraction data. The material has been demonstrated to order antiferromagnetically at low temperatures, with T N = ~90 K a propagation vector of k = (½0½), resulting in a spin arrangement similar to that in FeTe, despite there being no apparent lowering in symmetry of the nuclear structure. a) "AFM1" b) Figure 6 (Colour online) Proposed (AFM1 (a) & AFM6 (b)) and observed (AFM3 (c) & (d)) magnetic structures of La 2 O 2 Se 2 .Fe 2 O; O = red, Se = green, Fe = blue/purple; the blue/purple colour of the Fe 2+ ions relates similarly orientated moments; the magnetic cell in (c) is shaded a) b) J 2b J 1b J 2a J 1a

10.1103/physrevb.81.214433

1005.4600

78141d4a677d9e9c196d31fb87550238f5f565dc

Determination of the Low Temperature Nuclear and Magnetic Structures of La 2 O 2 Se 2 .Fe 2 O David G Free Department of Chemistry Durham University DH1 3LEDurhamUK John S O Evans Department of Chemistry Durham University DH1 3LEDurhamUK Determination of the Low Temperature Nuclear and Magnetic Structures of La 2 O 2 Se 2 .Fe 2 O 1 This paper describes the low temperature nuclear and magnetic structures of La 2 O 2 Se 2 .Fe 2 O by analysis of X-ray and neutron diffraction data. The material has been demonstrated to order antiferromagnetically at low temperatures, with T N = ~90 K a propagation vector of k = (½0½), resulting in a spin arrangement similar to that in FeTe, despite there being no apparent lowering in symmetry of the nuclear structure. a) "AFM1" b) Figure 6 (Colour online) Proposed (AFM1 (a) & AFM6 (b)) and observed (AFM3 (c) & (d)) magnetic structures of La 2 O 2 Se 2 .Fe 2 O; O = red, Se = green, Fe = blue/purple; the blue/purple colour of the Fe 2+ ions relates similarly orientated moments; the magnetic cell in (c) is shaded a) b) J 2b J 1b J 2a J 1a Introduction There has been significant recent interest in mixed anion materials due largely to the discovery of superconductivity in layered oxypnictide systems. 1-3 Work on oxychalcogenides has uncovered materials with interesting magnetic and optical electronic properties, such as LaOCuS, which has been demonstrated to be a transparent p-type semiconductor, emitting blue light on excitation at room temperature. 4 In this paper we describe the low temperature structural and magnetic properties of the mixed anion material, La 2 O 2 Se 2 .Fe 2 O, Figure 1a. 7 Neutron data were collected using HRPD at ISIS over a time of flight window of 10-210 ms (d = 0.2-16.4 Å) from 12-300K, with the sample mounted in a 5 mm vanadium slab can, for a total of 66 µAh. 14 X-ray data were collected over the sample temperature range using a Bruker d8 Advance diffractometer, with a LynxEye silicon strip detector, from 5-120° 2θ with a step size of 0.021° and collection time of 8 s per step; sample temperature was controlled using an Oxford Cryosystems PheniX CCR cryostat. 15 Neutron data were analysed over data ranges of 15-200 ms for each of three neutron banks, and 20-120° 2θ for the X-ray data. Combined X-ray and neutron refinements were performed in GSAS for the 300 and 12 K data collections. [16][17] A total of 84 variables were refined for the 300 K data (2 cell parameters, 2 atom coordinates, 5 isotropic thermal displacement parameters, 45 background parameters (XRD = 12; ND 168° = 12, 90° = 12, 30° = 9), 9 terms for TOF x-axis calibration (3 per ND bank), 3 absorption correction terms (1 per ND bank), a zero error term (XRD), 4 scale factors (1 per data set), 13 profile coefficients (4 XRD, 3 per ND bank). A total of 85 variables were used for the 12 K data (the additional parameter arising from description of the Fe 2+ moment). Variable temperature data shown were collected at HRPD with 6 K intervals for 2.5 µAh, and analysed using two techniques, the SEQGSAS routine in GSAS, and in TOPAS Academic using the local program multitopas. [16][17][18][19] (2) 118.08 (2) La z / c 0.18445(5) 0.18438(3) 0.18407(2) Se z / c 0.09669(9) 0.09624(3) 0.09618(2) La U iso / 100×Å 2 0.68(4) 0.456(8) 0.052(4) Fe U iso / 100×Å 2 1.96(15) 0.785(8) 0.186(4) Se U iso / 100×Å 2 1.06(6) 0.575(9) 0.090(4) O(1) U iso / 100×Å 2 1.0(4) 0.60(1) 0.249(7) O(2) U iso / 100×Å 2 1.3(8) 1.31(3) 0.33(1) Fe M x / µ B - - 2.82(3) R wp / % - 4.d La-O 2.3764(2) La-O-La (1) 105.346(7) d La-Se 3.3102(3) La-O-La Results and Discussion The structure was confirmed at room temperature by combined Rietveld refinement ( Figure 1b) of X-ray and neutron data, and results are given in Table 1. [16][17]20 Refinement of occupancies on individual sites, using a fixed La 3+ occupancy, confirms the expected composition of the material at 300 K (site occupancies refining to In the structurally similar pnictide superconductors, as well as in FeSe and FeTe, structural transitions from tetragonal to orthorhombic or monoclinic symmetry occur at or around T N . 1, 22-28 Despite the use of high resolution neutron (HRPD instrumental resolution ∆d/d ~ 4×10 -4 Å) and X-ray data, we see no evidence of peak splitting at low temperature. We note, however, that there is weak evidence of a discontinuity in the parameter used to describe peak shape strain broadening at T N , which could indicate a subtle lowering of symmetry. The magnitude of the effect is, however, small and corresponds to a change in peak full width half maximum of the (200) reflection of ~4%. There is also weak evidence from the enhanced U 33 Possible magnetic structures were investigated using a Monte-Carlo approach as implemented in SARAh Refine, which interfaces with the GSAS software suite. [16][17]29 For I4/mmm parent symmetry, with a propagation vector of k = (½0½), two independent (2)). Results from this analysis showed good agreement with experimental data can be achieved using basis vector Ψ 1 associated with irreducible representation Γ 2 for Fe(1) and Ψ 2 associated with Γ 3 for Fe(2); 1 a contour plot of χ 2 as a function of these basis vectors can be seen in Figure 5, where lower χ 2 values are shown in blue and higher shown in red. Full refinement of the low temperature model using a combination of powder X-ray and neutron diffraction data, with equated moments on Fe sites, gave a good fit to the experimental data. These results are included in Canting of the moments was also investigated and whilst results revealed a small contribution of the moment along the z axis (0.23(5) µ B ), the refinement statistics were not significantly improved from the original model. Higher quality data would be needed to clarify this. The spin arrangement within a layer is shown in Figure 6c, though we note that the data of Figure 5 show that the alignment of the Fe(2) spins within a layer, relative to those of Fe(1), is not determined. The commensurate spin ordering is in contrast to that reported for Ba 2 F 2 Se 2 .Fe 2 O. 13 between corner-sharing octahedra, J 2 is the edge-sharing ~98˚ (see Table 2 Conclusions In conclusion, neutron diffraction data have shown that the magnetic moments within This arrangement has all interactions frustrated and is contrary to calculations that predicted the AFM1 and AFM6 models to be the most likely arrangements. 8,13 We see no evidence of peak splitting at T N which might reflect a lowering of symmetry, though there is a marked change in the c/a ratio at this temperature. The observation of antiferromagnetism at this temperature is in agreement with susceptibility measurements published by Mayer et al. Figure 1 ( 1materials often crystallise with layered structures, allowing separation of the oxide and chalcogenide / pnictide ions. A review of the structures and properties of different layered oxychalcogenide and oxypnictide materials has been given by Clarke et al.. Colour online) (a) Nuclear structure of La 2 O 2 Se 2 .Fe 2 O and (b) Rietveld refinement of the nuclear model of La 2 O 2 Se 2 .Fe 2 O at 300 K; La = white, O = red, Se = green, Fe = blue. Data shown in (b) are from the HRPD backscattering bank (the observed pattern is in blue, calculated in red and the difference in grey) Figure 2 (Figure 3 ( 23Colour online) Plots showing the effects of temperature on the a (a) and c (b) cell parameter, U 33 values for all sites (c), and the parameter used to describe strain broadening (d); solid lines in (a) and (b) are guides to the eye (see text) On cooling the sample below 90 K, extra peaks became visible in the 90° and 30° banks of HRPD data, at d-spacings of 3.01, 3.27, 3.50, 3.62, 4.44, 5.48 and 6.78 Å, that are not present in the X-ray data (Figure 3a & Figure 4), consistent with magnetic ordering of the Fe 2+ ions. These peaks were indexed using a cell with dimensions 2a × a × 2c, indicating a magnetic propagation vector of k = (½0½). Plots of the nuclear cell parameters are shown in Figure 2a & b. The a parameter shows a continuous contraction (with perhaps a hint of discontinuity at around 90 K), whilst the c parameter shows a marked discontinuity at ~90 K. This is emphasised inFigure 2, where both cells have been fitted using an Einstein model of thermal expansion. Only data above 105 K were used for c, so the equated θ E of 211(4) K is dominated by the a axis data.21 The thermal displacement parameter of the O(2) ion is observed to decrease continuously on cooling, in particular the U 33 parameter. These results can be seen inFigure 2c, where each atom has been modelled anisotropically, and the U 33 values have been plotted as a function of temperature. The high U 33 of O(2), corresponding to displacement above and below the [Fe 2 O] 2+ plane, could indicate a local distortion of Fe-O-Fe bond away from 180°. Colour online) Plots showing the effect of temperature on (a) the evolution of the observed magnetic peaks and (b) the magnitude of the moment on the Fe 2+ ion (b); the pseudo film plot shows the 2.6-4.6 Å (90-160 ms) region of the 90° HRPD bank. Figure 4 ( 4Colour online) Rietveld refinement of (a) the nuclear only and (b) nuclear plus magnetic models of La 2 O 2 Se 2 .Fe 2 O at 12 K; the data shown are the 2.6-4.6 Å region from the 90° HRPD bank, and inset are data from the 3.0-9.0 Å region from the 30° bank. The observed pattern is shown in blue, calculated in red and the difference in grey; magnetic reflections are labelled. Figure 5 ( 5The observation of an antiferromagnetic arrangement of moments supports Mayer's earlier magnetic susceptibility measurements. Colour online) Contour plot of the dependance of χ 2 (z) on the Γ 2 Ψ 1 and Γ 3 Ψ 2 basis vectors (x,y), obtained by mixing of all allowed basis vectors Kabbour et al. have discussed possible ordering patterns within [Fe 2 O] 2+ layers for A 2 F 2 Se 2 .Fe 2 O (A = Ba, Sr) based on DFT calculations; Zhu et al. have performed similar calculations on La 2 O 2 Se 2 .Fe 2 O. 8, 13 There are three important exchange interactions to consider, and we adopt the same labels as Zhu et al.. 2 J 2 ' is a 180˚ Fe-O-Fe interaction ) interaction Fe-Se-Fe, and J 1 is a face-sharing interaction comprising Fe-Se-Fe (~64°), Fe-O-Fe (90°) and potentially direct Fe-Fe exchange; though Fe-Fe distances are ~6% larger than in FeSe 1-x and Fe 1+x Te. 25-28 GGA+U density functional theory calculations predict J 2 ' as antiferromagnetic, J 2 as ferromagnetic and J 1 as antiferromagnetic. Predictions for J 2 ' and = -3.28, J 2 = +0.78 and J 1 = -3.38 for La 2 O 2 Se 2 .Fe 2 O for U = 4.5 eV (negative J's correspond to AF interactions), and predict the magnetic phase diagram for different exchange interaction strengths. For different values of U the ground state is predicted to be either the AFM1 or AFM6 models of Figure 6 (Kabbour's 2F3A and 1A2F models, respectively). In the former case the J 2 ' Fe-O-Fe interactions are frustrated but J 2 and J 1 are satisfied. In the latter case half of the J 1 interactions are frustrated. The observed magnetic structure doesn't correspond to either of the predicted structures and is shown in Figure 6 (c & d). Spins are aligned ferromagnetically along the b direction and antiferromagnetically along a. Note that the relative orientation of spins on the Fe(1) and Fe(2) lattices within a layer is not determined. Along the doubled cell edge Fe-O-Fe interactions are antiferromagnetic as expected, whereas they are ferromagnetic along b. Conversely J 2 interactions are satisfied along the short axis, b, but not along a.Half of the J 1 interactions are similarly frustrated, as seen in the LaOFeAs systems. The magnetic structure is similar to that observed for Fe 1+x Te samples, with x < 0.1(Figure 7a 33 Figure 7 ( 337& b), which have the same k = (½0½) propagation vector and spin arrangement, with the majority spin direction in the ab plane, but predominantly aligned along the short axis.27- 28, 32-33 In this system magnetic ordering is accompanied by a clear phase transition to a monoclinic P2 1 /m cell with a/b = 1.01 and β = 89.2˚, and hence the observed magneticordering can be described by a single irreducible representation. The symmetry lowering leads to exchange interactions differing in different directions, as labelled in Figure 7b, and the region of the magnetic phase diagram in which the AFM3 structure is stable has been discussed by Fang et al. Colour online) Nuclear (a) and magnetic (b) structures of and Fe 1.086 Te; Fe = blue/purple, Te = gold This material, like LaOCuS and the LaOFeAs superconductors, contains layers of edge-sharing La 4 O tetrahedra. These are CuO 2 type and can also be described as a network of face-sharing octahedra, where the transition metal centred octahedron is made up of two axial oxide ions and four equatorial selenide ions. This material, and its oxysulfide analogue, have semiconducting properties and have been described as Mott insulators. 8 A large temperature independent contribution to their magnetic susceptibility with a broad maximum around 100 K, To the best of our knowledge low temperature neutron diffraction experiments to study magnetic order in La 2 O 2 Se 2 .Fe 2 O have not been reported, however, theoretical studies ofseparated from [Fe 2 O] 2+ layers by Se 2-ions, which complete a square antiprismatic coordination of La 3+ . The [Fe 2 O] 2+ transition metal layers are a rare example of the anti- suggests antiferromagnetic ordering at low temperature. Other [M 2 O] containing materials include Na 2 Pn 2 .Ti 2 O and a recently reported family where the Na + ions have been replaced by [A 2 O 2 ] 2+ fluorite layers. 9-11 These materials exhibit anomalous transitions in magnetic susceptibility and electrical resistivity, corresponding to CDW/SDW instabilities; magnetic ordering has not been observed from neutron powder diffraction. Also structurally related to these materials is Na 1.9 Cu 2 Se 2 .Cu 2 O, which contains layers of edge-sharing Cu 4 Se tetrahedra separated from square-planar Cu 2 O layers by Na + ions. 12 the material have suggested two possible structures, denoted AFM1 and AFM6 (see later, Figure 6), depending on the magnitude of U (where U is the Mott-Hubbard interaction energy). 8 Studies on the B 2 F 2 Q 2 .Fe 2 O (B = Sr, Ba; Q = S, Se) family, which also contains square-planar [Fe 2 O] 2+ layers, suggest magnetic ordering occurring at 84 (neutron data) and 95-97 K (susceptibility data) for Ba 2 F 2 Se 2 .Fe 2 O and Sr 2 F 2 Se 2 .Fe 2 O, respectively. 13 Neutron data suggest an incommensurate structure for Ba 2 F 2 Se 2 .Fe 2 O, though no detailed magnetic structures were published. Calculations, again, showed the AFM1 and AFM6 arrangements to be the most stable. La 2 O 2 Se 2 .Fe 2 O for this study was prepared from stoichiometric amounts of La 2 O 3 (Sigma-Aldrich, 99.9%), Fe (Aldrich, 99.9+%) and Se (Alfa-Aesar, 99.999%). The resulting powder was pressed into a 5 mm pellet and placed inside a 7 mm high density alumina crucible. This was sealed in a quartz ampoule under vacuum and heated in a furnace with the following routine: ramp to 600°C at 1°.min -1 and dwell for 12 h, ramp to 800°C at 0.5°.min -1 and dwell for 1 h, ramp to 1000°C at 1°.min -1 and dwell for 12 h. After this the furnace was allowed to cool to room temperature. Analysis of the product by powder X-ray diffraction confirmed that the correct phase had been obtained. This routine was slightly different to that employed by Mayer et al., as single crystals of the material were not required. 7 Table 1 1Results from combined X-ray / neutron Rietveld refinements of La 2 O 2 Se 2 .Fe 2 O at 300 and 12 K, with single crystal values from the literature for comparisonMayer et al. 7 T = 300 K T = 12 K Space group I4/mmm I4/mmm I4/mmm* a / Å 4.0788(2) 4.084466(9) 4.075725(6) c / Å 18.648(2) 18.59798(7) 18.53719(5) V / Å 3 310.24 310.268(2) 307.931(1) * The magnetic contribution to the data was modelled as a separate phase with the magnitude and direction of the moments constrained using the AFM3 model (see later).96 3.95 χ 2 - 1.118 1.903 Table 2 2Bond lengths and angles for La 2 O 2 Se 2 .Fe 2 O at 12 K Inter-atomic distances / Å Bond angles / °d Fe-Fe 2.88198(1) Fe-Se-Fe (1) 64.298(7) d Fe-O 2.03786(1) Fe-Se-Fe (2) 97.62(1) d Fe-Se 2.7080(3) Se-Fe-Se 82.38(1) Table 1 . 1The refined moment of 2.83(3) µ B compares to values of 2.25(8) µ B observed for Fe 1.068 Te at 67 K, 0.36(5) µ B for LaOFeAs at 8 K, and 3.32 µ B for FeO at 77 K. 1, 27, 30 La 2 O 2 Se 2 .Fe 2 O order with an AFM3 arrangement at ~90 K, similar to that in Fe 1.086 Te. 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In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Ampère equation which arise naturally in the study of the conical Kähler-Einstein metric.

10.1007/s00526-019-1498-z

1609.03111

41cc25946030c17bf8e50f975bce13570e50140b

EXPANSION FORMULA FOR COMPLEX MONGE-AMPÈRE EQUATION ALONG CONE SINGULARITIES 11 Sep 2016 Hao Yin Kai Zheng EXPANSION FORMULA FOR COMPLEX MONGE-AMPÈRE EQUATION ALONG CONE SINGULARITIES 11 Sep 2016 In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Ampère equation which arise naturally in the study of the conical Kähler-Einstein metric. Introduction Let M be a closed Kähler manifold and D be a smooth divisor in M . A Kähler metric g is said to be of cone angle 2πβ (with 0 < β < 1) along D if g is smooth away from D and for each p ∈ D, there is a holomorphic coordinates {z 1 , · · · , z n } around p with D = {z 1 = 0} such that g is comparable to the standard cone metric g cone = |z| 2β−2 1 dz 1 dz 1 + dz 2 dz 2 + · · · + dz n dz n . If the metric is Einstein away from the divisor, then it is called conical Kähler-Einstein metric. The study of Kähler cone metrics could be traced back to Tian [11]. Since then, there are researches on uniqueness and existence of conical Kähler-Einstein metrics [6,9]. Recently, in [3], Donaldson introduced a new function space C 2,α β (see Section 2 for the definition) and proved a Schauder estimate of the linear equation, which provides the key analysis tool and stimulates more research work along this line. It is now known that the uniqueness and existence of conical Kähler-Einstein metrics in C 2,α β space, i.e. the Kähler potential of the conical Kähler-Einstein metric lies in this new space, could be proved without knowing the higher order regularity near the divisors (see [7] and references therein). Nevertheless, the expansion formula has been used in [10] to prove a Chern number inequality for some conical Kähler metrics. Therefore, it is interesting to understand it down to earth and more accurately, and this is the main topic we would explore in this paper. For those metrics generated by using Donaldson's Schauder estimate, the Kähler potential (together with its tangential derivatives) lies automatically in the C 2,α β space. Under the angle restriction 0 < β < 1 2 , the higher regularity near the divisor is studied by Brendle [1], meanwhile the higher order Donaldson's spaces are defined and used to improve the regularity of the more general constant scalar curvature Kähler metric with cone singularities in [2,8]. For 0 < β < 1, Jeffres, Mazzeo and Rubinstein [5] proved that the metric is polyhomogeneous, in the sense that ϕ KE (r, θ, Z) ∼ j,k≥0 N j,k l=0 a j,k,l (θ, Z)r j+k/β (log r) l Date: September 13, 2016. 1 where r = |z 1 | β /β, θ = argz 1 and Z = (z 2 , · · · , z n ). We refer the readers to Theorem 2 of [5] for the complete detail of this result. The main result of this paper is to show that by following a method of [12], which is very different from the method in [5], we can prove an expansion with more detailed information. Among other things, we can show that N j,k ≤ max {0, k − 1} and j is even. Briefly speaking, the extra information is obtained by making full use of the fact that the section of the cone (modulo tangential directions) is S 1 . Moreover, we shall see in the proof of the main theorem that the exact form of the expansion relies both on this geometric structure of the singularity and on the particular nonlinear structure of the complex Monge-Ampère equation. It makes a very interesting comparison to note that in [12], the author showed that the expansion of the Ricci flow solution on conical surfaces involves no log term. Since the nature of the regularity problem is local, we start with a local form. Its relation to the conical Kähler-Einstein metric shall be clear in a minute. Consider a solution ϕ to the following singular Monge-Ampère equation (1.1) det(ϕ ij ) = e λϕ+h |z 1 | 2−2β on B 1 ⊂ C n . Here B 1 is the ball of radius 1 centered at the origin. Assume that (S1) h is a smooth function of z = (z 1 , . . . , z n ) ∈ B 1 ⊂ C n ; (S2) ϕ is in C 2,α β space (see Section 2.2 for the definition); (S3) there is some constant c > 1 such that 1 c ω cone ≤ √ −1∂∂ϕ ≤ cω cone , where ω cone = √ −1∂∂ 1 β 2 |z 1 | 2β + n i=2 |z i | 2 . To describe the regularity of the solution ϕ, we need some definitions. Define the polar coordinates (ρ, θ, ξ) by ρ = 1 β |z 1 | β , z 1 = |z 1 | e iθ , ξ = (z 2 , · · · , z n ). Let T log be the set of functions ρ 2j+ k β (log ρ) m cos lθ, ρ 2j+ k β (log ρ) m sin lθ with j, k, m, l satisfying (T1) k, j, l, m = 0, 1, 2, · · · ; (T2) k−l 2 ∈ N ∪ {0}; (T3) m ≤ max {0, k − 1}. The main theorem of this paper is Theorem 1.1. Suppose that ϕ is a solution to (1.1) on B 1 ⊂ C n and that (S1)-(S3) hold. Then for each fixed ξ with |ξ| < 1 2 and q > 0, ϕ has an expansion up to order q in the sense that there exists η in Span(T log ) such that ϕ(ρ, θ, ξ) = η +Õ(q). Here Span(T log ) is the vector space of finite linear combinations of T log andÕ(q) stands for a function of ρ, θ satisfying (ρ∂ ρ ) k1 ∂ k2 θÕ (q) ≤ C(k 1 , k 2 , ξ)ρ q for any k 1 , k 2 ∈ N ∪ {0} . Moreover, any derivatives of ϕ along ξ direction have expansion up to any order in the same sense. Before we discuss the application of Theorem 1.1, we briefly introduce the ideas involved in its proof. It follows from Donaldson's linear theory that the tangential regularity of ϕ is not a problem at all, hence we treat z 2 , · · · , z n as parameters and study a regularity problem on conical surface. For that purpose, we set△ = ∂ 2 ρ + 1 ρ ∂ ρ + 1 β 2 ρ 2 ∂ 2 θ and rewrite (1.1) into the form△ ϕ = RHS where the RHS is an expression involving derivatives of ϕ(see Section 2.3). The first ingredient of the proof is a formal analysis, which we use to decide which functions are needed for the expansion of ϕ. Roughly speaking, we pretend that ϕ and its tangential derivatives are finite linear combinations of functions in T for some set of functions T and compute the RHS above so that the result is the linear combination of some other set of functions T rhs , which may not be identical to T . We look for the smallest Span(T ) such that every function in T rhs lies in the image△(Span(T )). Moreover, we require that Span(T ) contains the terms in the expansion of bounded harmonic functions (see (1) in Lemma 4.16). This is how we obtain T log . The second ingredient is an estimate of ϕ (Theorem 3.1) away from the singular set {z 1 = 0}, which serves as the starting point of the bootstrapping argument in the proof of Theorem 1.1. The idea is that away from the singular set, since the complex Monge-Ampère equation is elliptic and that we have assumed Donaldson's C 2,α β estimate, we should be able to get estimates for higher order derivatives of ϕ, which blow up at some fixed rate near the singular set. The rest of the proof is a bootstrapping argument. Once we know that ϕ and its tangential derivatives have expansion up to a certain order, we can improve this order by at least 1. The details of this argument appear in Section 4.2 and 4.3. To apply Theorem 1.1 to the regularity problem of conical Kähler-Einstein metric, we briefly recall the basic setting. Suppose that M is a compact Kähler manifold with a smooth Kähler form ω 0 and D is a smooth divisor in M , whose corresponding line bundle L has a global holomorphic section s vanishing on D. Assume that (1.2) c 1 (M ) = µ[ω 0 ] + (1 − β)[D] for some µ ∈ R. Here [D] is the cohomology class defined by the closed (1, 1) current defined by the divisor D. By (1.2), there exists an (smooth) hermitian metric h 0 of L such that its curvature form Θ h0 satisfies (1.3) Ric(ω 0 ) = µω 0 + (1 − β)Θ h0 For δ sufficiently small, ω D = ω 0 + δi 2π ∂∂ |s| 2β h0 is a Kähler metric on M \ D and is asymptotically a cone metric along D (see Section 4.3 in [3]). For ψ ∈ C 2,α β (M ) (the Donaldson Hölder space, see Section 2 for the precise definition), the Kähler metric ω ψ = ω D + i 2π ∂∂ψ is called a conical Kähler-Einstein metric if Ric(ω ψ ) = µω ψ + (1 − β)D in the sense of currents, where by abuse of notation, we also use D for the current associated to the divisor D. By the Poincaré-Lelong formula, this is equivalent to (1.4) Ric(ω ψ ) = µω ψ + i 2π ∂∂ |s| 2(1−β) h0 + (1 − β)Θ h0 . Subtracting (1.3) from (1.4) yields (in the sense of currents) ∂∂ log ω n ψ ω n 0 = ∂∂(µψ + µδ |s| 2β h0 ) + ∂∂ |s| 2(1−β) h0 , which implies (1.5) ω n ψ ω n 0 = |s| 2(1−β) h0 e µ(ψ+δ|s| 2β h 0 ) . To reduce the global equation (1.5) into a local one, we take a holomorphic coordinate system {z i } around some p ∈ D such that D (in this neighborhood) is given by {z 1 = 0}. We pick the trivialization of L such that s is given by z 1 and denote the hermitian metric in this trivialization by a real-valued functionh. Moreover, we assume that ω 0 = i 2π ∂∂ψ 0 . Keeping the above notations in mind and setting (1.6) ϕ = ψ 0 + δ |s| 2β h0 + ψ, we obtain (1.7) det ϕ ij = det(ψ 0 ) ij |z 1 | 2(2−β)h1−β e µ(ϕ−ψ0) , which is just equation (1.1) if we take λ = µ and (1.8) h = log det(ψ 0 ) ij ·h 1−β e −µψ0 . Hence, we can apply Theorem 1.1 to get Corollary 1.2. If ω ψ is a conical Kähler-Einstein metric as defined above, then ϕ (hence ψ) has an expansion up to any order as defined in Theorem 1.1. The proof of this corollary is given in Section 4.4. Acknowledgements. The work of H. Yin is supported by NSFC 11471300. The work of K. Zheng has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 703949, and was also partially supported by the Engineering and Physical Sciences Research Council (EPSRC) on a Programme Grant entitled "Singularities of Geometric Partial Differential Equations" reference number EP/K00865X/1. Preliminaries We include in this section a few elementary discussions needed for this paper. First, we define several coordinate systems and explain the notations that we use. Second, we prove a few lemmas on how estimates are translated between different coordinate systems. Finally, we rewrite the complex Monge-Ampère equation for the use of the proof of our main result. Various coordinates. Recall that D is a smooth divisor of the Kähler manifold M . We shall list a few coordinate systems which appear naturally in the study of conical Kähler metric. Please note that some of them are defined in a neighborhood of any point in D and some are defined in a small ball away from D. (1) The holomorphic coordinates z. This is the holomorphic coordinates (z 1 , · · · , z n ) of the underlying complex manifold such that the divisor D is given by z 1 = 0. It is the most natural coordinate system and we have introduced our equation (1.1) using it. Partial derivatives of ϕ with respect to z coordinates are denoted by ϕ i , ϕj, ϕ ij and so on. The roles played by z 1 and z l (l = 2, · · · , n) are very different. To emphasize this, we usually write ξ for any one of z 2 , · · · , z n . Hence, ϕ ξ can be any one of ϕ 2 , · · · , ϕ n and the same convention applies to ϕξ, ϕ ξξ and so on. (2) The polar coordinates (ρ, θ, ξ). If (z 1 , · · · , z n ) is the holomorphic coordinates above, the polar coordinates are defined by ρ = 1 β |z 1 | β , z 1 = |z 1 | e iθ , ξ = (z 2 , · · · , z n ). Here we take ξ as a real vector in R 2n−2 . Partial derivatives are denoted by ∂ ρ , ∂ θ , ∇ ξ , ∇ 2 ξ and so on. (3) The lifted holomorphic coordinates v (or V -coordinate for short). This coordinate system is only defined in small neighborhood away from (but near) D. More precisely, given a point Z 0 = (ρ 0 , θ 0 , ξ 0 ) (in the polar coordinates) with ρ 0 = 0 and θ 0 ∈ [0, 2π), we consider a neighborhood of Z 0 defined by (2.1) Ω = {(ρ, θ, ξ)| ρ 0 /2 < ρ < 2ρ 0 , θ 0 − 0.1 < θ < θ 0 + 0.1, |ξ 0 − ξ| < ρ 0 } . For points in Ω, it makes sense to define v 1 := 1 β z β 1 = 1 β |z 1 | β e iβθ . We write v = (v 1 , ξ) = (v 1 , · · · , v n ) ∈ C n . Note that for any such Z 0 , the lifted holomorphic coordinates are defined for |v − v 0 | ≤ c β ρ 0 for some constant c β < 1 (depending only on β) and v 0 = (ρ 0 e iβθ0 , ξ 0 ). Partial derivatives are denoted by ϕ V,i , ϕ V,j , ϕ V,ij and so on. As before, we use ∇ V,ξ to indicate ∂ vi or ∂v i for i = 2, · · · , n. (4) The scaled lifted holomorphic coordinatesṽ (orṼ -coordinate for short). This is simply a scaling (by ρ 0 ) and translation of v (by v 0 ) so thatṽ is defined in B c β (0) ⊂ C n . Precisely,ṽ = v − v 0 ρ 0 . Partial derivatives are denoted by ϕṼ ,i , ϕṼ ,j , ϕṼ ,ij and so on and ∇Ṽ ,ξ is understood as in V -coordinates. We also define Hölder spaces by using (scaled) lifted holomorphic coordinates. C k,α V is the set of function f : B c β ρ0 (v 0 ) → R that is C k,α in the usual sense (i.e. with respect to the derivative and distance given by the v-coordinates) and the norm · C k,α V is also the same with the usual Hölder norm · C k,α (Bc β ρ 0 (v0)) . Here we use the subscript to emphasize that we are using the V -coordinates around v 0 . Similar convention holds for C k,α V . 2.2. Estimates in various coordinates. There is little doubt that all the above mentioned coordinate systems are important in the study of conical Kähler metrics and that with some efforts, one can switch between them if necessary. In this section, we prove a few lemmas along this line. First, we show how a bound of weighted derivatives (in polar coordinates) can be proved by using the lifted holomorphic coordinates. Lemma 2.1. Suppose that u : B 1 ⊂ C n → R is a function of holomorphic coordinates. The following are almost equivalent in the sense that: (ii) implies (i), while (i) implies (ii) if the 1/2's in (ii) are replaced by 1/4. (i) For any k 1 , k 2 , k 3 ∈ Z + ∪ {0}, there exists constant C(k 1 , k 2 , k 3 ) > 0 such that (ρ∂ ρ ) k1 ∂ k2 θ (ρ∇ ξ ) k3 u ≤ C(k 1 , k 2 , k 3 ) for any ρ ∈ (0, 1/2) and any |ξ| < 1/2. (ii) For any k ∈ Z + ∪{0}, any Z 0 = (ρ 0 , θ 0 , ξ 0 ) with ρ 0 ∈ (0, 1/2) and |ξ 0 | < 1/2, the k-th order derivatives of u on B c β in scaled lifted holomorphic coordinates are bounded by a constant C(k) (independent of Z 0 ). Proof. The proof is based on the following computation. For the ρ and θ part, recall that v 1 = 1 β z β 1 and z 1 = re iθ , which implies ∂ v1 = z 1−β 1 ∂ z1 and ∂v 1 =z 1−β 1 ∂z 1 and ∂ z1 = 1 2 e −iθ ∂ r + e i( 3 2 π−θ) 1 r ∂ θ ∂z 1 = 1 2 e iθ ∂ r + e i(− 3 2 π+θ) 1 r ∂ θ . Putting the above equations together and recalling that ρ = 1 β r β give (2.2) ∂ṽ 1 = ρ 0 2ρ e −iβθ (ρ∂ ρ ) + β −1 e i( 3 2 π−βθ) ∂ θ and (2.3) ∂v 1 = ρ 0 2ρ e iβθ (ρ∂ ρ ) + β −1 e i(− 3 2 π+βθ) ∂ θ . Along the tangent direction, for l = 2, · · · , n, ∂ṽ l = ρ 0 ∂ z l = ρ 0 2ρ (ρ∂ x l − iρ∂ y l ) ∂v l = ρ 0 ∂z l = ρ 0 2ρ (ρ∂ x l + iρ∂ y l ). (2.4) Here x l and y l are the real and imaginary part of z l so that ∇ ξ = (∂ x1 , . . . , ∂ xn , ∂ y1 , · · · , ∂ yn ). If (i) holds, an immediate consequence of (2.2), (2.3) and (2.4) is that (ii) holds for k = 1. For larger k, we observe that any coefficient w in front of ρ∂ ρ , ∂ θ and ρ∇ ξ in (2.2),(2.3) and (2.4) satisfies (ρ∂ ρ ) l1 (∂ θ ) l2 (ρ∇ ξ ) l3 w ≤ C(l 1 , l 2 , l 3 ) when ρ ∈ (ρ 0 /2, 2ρ 0 ) for some constants C(l 1 , l 2 , l 3 ). One may derive from (2.2), (2.3) and (2.4) the formulas for higherṽ-derivatives and with the help of the above observation, we conclude that (i) implies (ii) for any k. To see that (ii) implies (i), we solve from (2.2), (2.3) and (2.4) (2.5) ρ∂ρ = ρ ρ 0 e iβθ ∂ṽ 1 + e −iβθ ∂v 1 (2.6) ∂ θ = βρi ρ 0 e iβθ ∂ṽ 1 − e −iβθ ∂v 1 and for l = 2, · · · , n, ρ∂ x l = ρ ρ 0 (∂ṽ l + ∂v l ) ρ∂ y l = ρ ρ 0 (∂ṽ l − ∂v l ). (2.7) If v is any coefficient in front of ∂ṽ l or ∂v l with l = 1, · · · , n in (2.5), (2.6) and (2.7), then v obviously satisfies (i), hence by what we have just proved, v also satisfies (ii). In particular, anyṽ-derivatives of v in this sameṼ -coordinates around Z 0 are bounded. With this observation, (ii) implies (i) by iterated use of (2.5), (2.6) and (2.7). Corollary 2.2. Suppose that h : B 1 → R is a smooth function in holomorphic coordinates z. Then h satisfies (i) and (ii) in Lemma 2.1. In particular, for each Z 0 = (ρ 0 , θ 0 , ξ 0 ) with ρ 0 ∈ (0, 1/4) and |ξ 0 | < 1/4, if by using theṼ -coordinate around Z 0 we take h as a function defined on B c β , then h C k V (Bc β ) ≤ C(k) independent of Z 0 . Proof. Being a smooth function in z implies that (r∂ r ) k1 (∂ θ ) k2 (∇ ξ ) k3 h ≤ C(k 1 , k 2 , k 3 ). The corollary follows from Lemma 2.1 and the fact that r∂ r = βρ∂ ρ . Next, we recall the Donaldson's C 2,α β norm and show some implications (in the lifted holomorphic coordinates) when it is bounded. With a background conical metric (for example, g cone as given in the introduction), one can define C α β (B) space and C α β (B) norm for functions defined on B, where B stands for some ball in C n centered at the origin. Moreover, there is a subspace C α β,0 (B) of C α β (B) which consists of all u ∈ C α β (B) satisfying u| {z1=0} ≡ 0. A function u defined on B is said to be in (Donaldson's) C 2,α β space if and only if (see Section 2 of [1]) (D1) u ∈ C α β (D2) |z 1 | 1−β ∂ z1 u, |z 1 | 1−β ∂z 1 u ∈ C α β,0 ; (D3) for k = 1, |z 1 | 1−β ∂ 2 z kz1 u and |z 1 | 1−β ∂ 2 z1z k u ∈ C α β,0 ; (D4) for k = 1 and l = 1, |z 1 | 2−2β ∂ 2 z1z1 u, ∂ 2 z kzl u and ∂ 2 z k z l u ∈ C α β . Moreover, u C 2,α β (B) is defined to be the sum of the C α β norms of all the above functions. There is an equivalent way (see [1]) of defining the same space in terms of the polar coordinates: (if we replace (D1-D4) above by) (P1) u ∈ C α β ; (P2) ∂ ρ u, 1 ρ ∂ θ u are in C α β,0 ; (P3) ∇ 2 ξ u, ∂ ρ ∇ ξ u, ρ −1 ∂ θ ∇ ξ u and△u are in C α β where△ = ∂ 2 ρ + 1 ρ ∂ ρ + 1 β 2 ρ 2 ∂ 2 θ . Here is an estimate in the lifted holomorphic coordinates for functions with bounded C 2,α β norm. Lemma 2.3. Suppose that u ∈ C 2,α β (B 1 ). For any Z 0 = (ρ 0 , θ 0 , ξ 0 ) with ρ 0 ∈ (0, 1 2 ) and |ξ 0 | < 1/2, u as a function in the lifted holomorphic coordinates are bounded in C 2,α V satisfying u V,ij C α V (Bc β ρ 0 (v0)) ≤ C u C 2,α β (B1) for some C > 0 independent of Z 0 and u. Proof. First of all, we note that the distance functions involved in the definition of C α V (B c β ρ0 (v 0 )) and C 2,α β (B 1 ) are the same. To see this, we consider two points (ρ 1 , θ 1 , ξ 1 ) and (ρ 2 , θ 2 , ξ 2 ) with ρ 1 , ρ 2 ∈ (ρ 0 /2, 2ρ 0 ) and θ 1 , θ 2 ∈ (θ 0 − 0.1, θ 0 + 0.1). The distance between them with respect to g cone = dρ 2 + ρ 2 β 2 dθ 2 + dξ 2 is the square root of (2.8) |ξ 1 − ξ 2 | 2 + |(ρ 1 cos(βθ 1 ) − ρ 2 cos(βθ 2 ))| 2 + |(ρ 1 sin(βθ 1 ) − ρ 2 sin(βθ 2 ))| 2 . A nice way to compute this distance is to set η i = βθ i for i = 1, 2 and notice that |η 1 − η 2 | < 0.2. On the other hand, the V -coordinates of (ρ 1 , θ 1 , ξ 1 ) and (ρ 2 , θ 2 , ξ 2 ) are (ρ 1 e βθ1 , ξ 1 ) and (ρ 2 e βθ2 , ξ 2 ), so that the Euclidean distance between them in B c β ρ0 (v 0 ) is the same as the square root of (2.8). Next, we study the relation between u V,ij with the quantities in (D1-D4). Since u V,11 = |z 1 | 2−2β u 11 and u V,kl = u kl (k, l = 1), they appear in (D1-D4) directly and hence are in C α β (B 1 ) and (by what we have just proved) in C α V (B c β ρ0 (v 0 )) when restricted to B c β ρ0 (v 0 ). When k = 1, we claim that the C α V (B c β ρ0 (v 0 )) norm of (2.9) u V,1k = z 1 |z 1 | 1−β |z 1 | 1−β u 1k is independent of Z 0 . By (D3) and the first part of the proof, we know the C α V (B c β ρ0 (v 0 )) norm of |z 1 | 1−β u 1k is independent of Z 0 . For any two points V 1 = (ρ 1 e iβθ1 , ξ 1 ) and V 2 = (ρ 2 e iβθ2 , ξ 2 ) in B c β ρ0 (v 0 ), we have |θ 1 − θ 2 | ≤ C |V 1 − V 2 | , which implies that the C α V (B c β ρ0 (v 0 )) norm of z1 |z1| 1−β = e i(1−β)θ is independent of Z 0 so that the claim is proved. The proof of the lemma is finished because the same proof also works for u V,k1 . 2.3. The complex Monge-Ampère equation. The aim of this section is to rewrite the equation (2.10) det(ϕ ij ) = e λϕ+h |z 1 | 2−2β . into a form that will be useful in the proof of Theorem 1.1. More precisely, we want to • put the equation into the polar coordinates; • move anything other than△ϕ (see (P3) above) to the right hand side; • characterize the nonlinear structure of the right hand side in a useful form. It turns out that we need to carry this out not only for ϕ but also for all its tangent derivatives. Multiplying the first row and the first column of ϕ ij by |z 1 | 1−β , we obtain from (2.10) (2.11) det |z 1 | 2(1−β) ϕ 11 |z 1 | 1−β ϕ 1ξ |z 1 | 1−β ϕ ξ1 ϕ ξξ = e λϕ+h . Here ϕ 1ξ is the row vector (ϕ 12 , · · · , ϕ 1n ), ϕ ξ1 is the column vector (ϕ 21 , · · · , ϕ n1 ) and ϕ ξξ is the matrix ϕ ij 2≤i,j≤n . Setting P = |z 1 | 1−β ∂ z1 and recalling that we have defined △ = ∂ 2 ρ + 1 ρ ∂ ρ + 1 β 2 ρ 2 ∂ 2 θ = 4 |z 1 | 2(1−β) ∂ 2 ∂z 1 ∂z 1 , we obtain (2.12) det 1/4△ϕ P ϕξ P ϕ ξ ϕ ξξ = exp(λϕ + h). Finally, let's expand the left hand side of (2.12) by the definition of det to see (A) det ϕ ξξ △ ϕ = (P ϕξ ·P ϕ ξ )#F (ϕ ξξ ) + 4 exp(λϕ + h) Here P ϕξ·P ϕ ξ stands for a term P ϕj ·P ϕ i for i, j = 2, · · · , n, F (ϕ ξξ ) is a polynomial (with constant coefficients) of ϕ ij (i, j = 2, · · · , n) and # means a sum of products of P ϕξ ·P ϕ ξ and F (ϕ ξξ ). The importance of the fact that the P derivatives come in conjugate pairs can be seen from the following lemma. Lemma 2.4. Let f and g be any complex valued functions. P f ·P g is a quadratic polynomial (with constant coefficients) of ∂ ρ f, ∂ ρ g, 1 ρ ∂ θ f and 1 ρ ∂ θ g. Proof. By the definition of P , we have P = 1 2 (cos θ − i sin θ)∂ ρ + β −1 (− sin θ − i cos θ) 1 ρ ∂ θ . The proof of this lemma is just the following direct computation, P f ·P g = 1 4 (cos θ − i sin θ)∂ ρ f + β −1 (− sin θ − i cos θ) 1 ρ ∂ θ f · (cos θ + i sin θ)∂ ρ g + β −1 (− sin θ + i cos θ) 1 ρ ∂ θ g = 1 4 ∂ ρ f ∂ ρ g + β −2 ( 1 ρ ∂ θ f )( 1 ρ ∂ θ g) + iβ −1 ∂ ρ f ( 1 ρ ∂ θ g) − iβ −1 ( 1 ρ ∂ θ f )∂ ρ g . Next, we take tangential derivatives of the equation. By tangential derivative, we mean ∂ z k and ∂z k for k = 2, · · · , n. It turns out that the exact value of k is not important so that we write ∂ ξ and ∂ξ for simplicity. For example, when we write ϕ ξξξ , we mean ∂ z k 1 ∂ z k 2 ∂z k 3 ϕ for any k 1 , k 2 , k 3 = 2, · · · , n. Moreover, we write χ for a finite sequence of ξ and/or ξ and use a subscript of number to denote the length of the sequence. For example, by ϕ χ2 , we mean one of ϕ ξξ , ϕ ξξ , ϕξ ξ , ϕξξ. We shall derive an equation of ϕ χ similar to (A). The key point is to show that the equation will have a similar right hand side as in (A). We claim that if the length of χ is l, then det(ϕ ξξ )△ϕ χ = F (ϕ χ2 , . . . , ϕ χ l+2 )#(P ϕ χa ·P ϕ χ b ) (A χ ) + H(h, · · · , h χ l , ϕ, · · · , ϕ χ l+2 ). Here F and H are smooth functions of their arguments and a, b = 1, · · · , l + 1. Remark 2.5. For the rest of the paper, it is enough to know that F and H are smooth functions of ϕ and h and their tangential derivatives. The claim follows from direct computation and induction. Note that we may use F and H for different smooth functions in different lines below. Although the exact formula can be obtained, it is irrelevant to us. Take ∂ ξ of (A) to get det(ϕ ξξ )△ϕ ξ (2.13) = −∂ ξ det(ϕ ξξ )△ϕ + ∂ ξ (P ϕξ ·P ϕ ξ )#F (ϕ ξξ ) + 4 exp(λϕ + h) . Noticing that • ∂ ξ det(ϕ ξξ ) and ∂ ξ F (ϕ ξξ ) are smooth functions of ϕ ξξ and ϕ ξξξ ; • ∂ ξ (P ϕξ ·P ϕ ξ ) is a sum of P ϕ ξξ ·P ϕ ξ , P ϕξ ·P ϕ ξξ ; • ∂ ξ exp(λϕ + h) is obviously a smooth function of ϕ and h and ∂ ξ ϕ, ∂ ξ h. • as a consequence of (A), △ϕ = det(ϕ ξξ ) −1 P ϕξ ·P ϕ ξ #F + 4 exp(λϕ + h) , we conclude that the right hand side of (2.13) is of the required form in (A χ ) and thus the claim is proved if the length of χ is one. The general case shall follow by a similar computation which we omit. By the assumption of the Theorem 1.1, det(ϕ ξξ ) is Hölder continuous so that we can rewrite (A χ ) as det(ϕ ξξ )(0, ξ)△ϕ χ = F #(P ϕ χa ·P ϕ χ b ) + H (B χ ) + (det(ϕ ξξ )(0, ξ) − det(ϕ ξξ )(ρ, θ, ξ))△ϕ χ . Here det(ϕ ξξ )(0, ξ) is short for det(ϕ ξξ )(0, θ, ξ) since it is independent of θ. Interior estimates for ϕ In this section, we prove higher order estimates for the potential function ϕ away from the singular set. It is quite natural that as one gets closer and closer to the singular set, the estimates become worse and worse. Such a phenomenon usually appears as a weighted derivative estimate. It serves as a starting point of the bootstrapping argument in the proof of the main result in this paper. Theorem 3.1. Let ϕ be a C 2,α β (B 1 ) solution of (2.10) as given in Theorem 1.1. Then given any k 1 , k 2 , k 3 ∈ N ∪ {0}, we have (3.1) (ρ∂ ρ ) k1 (∂ θ ) k2 (∇ ξ ) k3 ϕ ≤ C(k 1 , k 2 , k 3 ), ∀ρ ∈ (0, 1/2) and |ξ| < 1/2, for some constant C(k 1 , k 2 , k 3 ). Remark 3.2. We remark that the same estimate was proved as the Step 1 in Section 4 of [5]. We still include a complete proof here because (1) we can not follow the scaling argument on page 135 of [5] and (2) since we have assumed Donaldson's estimate, i.e. the Hölder continuity of the complex Hessian of ϕ, it looks a bit strange if we use the Evans-Krylov theorem again. By Lemma 2.1, it suffices to show that for any Z 0 = (ρ 0 , θ 0 , ξ 0 ) with ρ 0 = 0, any derivatives of (∇ ξ ) k3 ϕ in scaled lifted holomorphic coordinates around Z 0 are uniformly bounded. Hence in what follows, we have Z 0 fixed and work in thẽ V -coordinates around it. The proof is still a scaling argument. However, we find it necessary to study first the scaling of the complex Hessian of ϕ, which may be and should be regarded as a Kähler metric and which satisfies an elliptic system. By using some well known estimates for this elliptic system, we show that the complex Hessian has the desired weighted estimates, or equivalently, it is reasonably bounded in the scaled lifted holomorphic coordinates (see Lemma 2.1). This is the purpose of Lemma 3.3 and Lemma 3.4, which we prove in the next subsection. 3.1. Metric (complex Hessian) in (scaled) lifted holomorphic coordinates. First, we put (2.10) in the lifted holomorphic coordinates around Z 0 = (ρ 0 , θ 0 , ξ 0 ). For that purpose, we multiply the first column of ϕ ij byz 1−β 1 and the first row by z 1−β 1 and notice that ∂ v1 = z 1−β 1 ∂ z1 and ∂v 1 =z 1−β 1 ∂z 1 to get (3.2) log det(ϕ V,ij ) = λϕ + h. The assumptions in Theorem 1.1 implies (3.3) 1 c δ ij ≤ ϕ V,ij ≤ cδ ij , by (S3) and (3.4) ϕ V,ij C α V (Bc β ρ 0 (v0)) ≤ C,(3.5) (∇ l V,ξ ϕ) V,ij C α V (Bc β ρ 0 (v0)) ≤ C(l). Here l is any natural number and C(l) depends on l but not on Z 0 . Next, we move to work in scaled lifted holomorphic coordinates by setting (3.6)φ(ṽ) = ρ −2 0 ϕ(v 0 +ṽρ 0 ).(3.7) 1 c δ ij ≤φṼ ,ij ≤ cδ ij , (3.8) φṼ ,ij C α V (Bc β ) ≤ C and for l = 1, 2, · · · , (3.9) (∇ lṼ ,ξφ )Ṽ ,ij C 0 (Bc β ) ≤ Cρ l 0 . If we denote g ij :=φṼ ,ij , for i, j = 1, · · · , n then g ij is a good metric with C α control in B c β by (3.7) and (3.8). In order to obtain higher order estimates, we derive an elliptic system satisfied by g ij . By (3.6), taking ∂ṽ i ∂v j of (3.2) and using ∂ k g ij = ∂ i g kj (which is the Kähler condition if we regard g ij as a Kähler metric, or the switching order of derivatives if we regard g ij as the complex Hessian), we get (3.10) △ g g ij − g km g nl ∂g im ∂ṽ n ∂g kj ∂v l = λρ 2 0 g ij + hṼ ,ij , where △ g = g ij ∂ 2 ∂ṽi∂vj . If we neglect the dependence of △ g on g ij itself, this is a quasilinear elliptic system whose nonlinear term is quadratic in the gradient of the unknown. By Corollary 2.2, the only non-homogeneous term in (3.10) satisfies (3.11) hṼ ,ij C k V (Bc β ) ≤ C(k) independent of Z 0 . There is a rich theory about elliptic systems of this kind. In particular, the theorems in Chapter VI of [4] imply that there exists some α ′ > 0 such that for any η > 0 (3.12) g ij C 1,α ′ V (Bc β −η ) ≤ C for some constant C depending on η, α ′ , c in (3.7) and the constant in (3.8). For the reader's convenience, we include a complete proof of this in the Appendix and what we have just used is Lemma A.3 there. With (3.12) and (3.11), the usual Schauder estimates applied to (3.10) give higher order estimates of g ij . That is, Lemma 3.3. There are constants C(k) independent of Z 0 such that g ij C k V (B 3c β /4 ) ≤ C(k). For the proof of Theorem 3.1, we also need estimates for ∇ lṼ ,ξ g ij . Lemma 3.4. For any l ∈ N, there are constants C(k, l) independent of Z 0 such that (3.13) (∇Ṽ ,ξ ) l g ij C k V (B c β /2 ) ≤ C(k, l)ρ l . Proof. Pick any sequence of η l satisfying 0 < η 1 < η 2 < · · · < 1 4 c β . We will prove (3.14) (∇Ṽ ,ξ ) l g ij C k V (B 3c β /4−η l ) ≤ C(k, l)ρ l , which is obviously stronger than (3.13). For the proof below, we allow the constants to depend on this particular choice of η l . Taking ∇Ṽ ,ξ of (3.10) yields △ g (∇Ṽ ,ξ g ij ) + G 1 (g, ∇Ṽ g, ∇ 2 V g)#∇Ṽ ,ξ g + G 2 (g, ∇Ṽ g)#∇Ṽ (∇Ṽ ,ξ g) =(∇Ṽ ,ξ h)Ṽ ,ij , (3.15) which is a linear elliptic system in which we take ∇Ṽ ,ξ g ij as the unknowns. By Lemma 3.3, g ij , G 1 and G 2 are bounded in any C k norm on B 3c β /4 so that the Schauder estimate gives (3.16) ∇Ṽ ,ξ g C k+2,α V (B (3c β /4−η 1 ) ) ≤ C ∇Ṽ ,ξ g C 0 (Bc β ) + ∇Ṽ ,ξ h C k+2,α V (Bc β ) . Since h is a smooth function, we may apply Corollary 2.2 to ∇ V,ξ h to get (3.17) ∇ V,ξ h C k+2,α V (Bc β ) ≤ C which is equivalent to (3.18) ∇Ṽ ,ξ h C k+2,α V (Bc β ) ≤ Cρ 0 . Combining (3.9), (3.18) and (3.16) proves (3.14) in case l = 1. We now prove by induction that for l ≥ 2, (∇Ṽ ,ξ ) l g ij satisfies △ g (∇Ṽ ,ξ ) l g ij + G 1 (g, ∇Ṽ g, ∇ 2 V g)#(∇Ṽ ,ξ ) l g + G 2 (g, ∇Ṽ g)#∇Ṽ ((∇Ṽ ,ξ ) l g) = (∇Ṽ ,ξ ) l h Ṽ ,ij + H l , (3.19) where H l is the sum of terms of the following form G(g, ∇Ṽ g, ∇ 2 V g)(∇Ṽ ) a1 (∇Ṽ ,ξ ) b1 g · · · · · (∇Ṽ ) as (∇Ṽ ,ξ ) bs g. Here 2 ≤ s ≤ l is an integer, for each i = 1, · · · , s, a i takes only three possible values, namely, 0, 1 or 2 and b i is a positive integer strictly smaller than l, satisfying b 1 + · · · + b s = l. Moreover, G, depending on a i and b i , is a smooth function of its arguments. Obviously, when l = 1, H l must be zero and (3.19) is exactly (3.15). Assume that (3.19) holds for l. Direct computation gives △ g (∇Ṽ ,ξ ) l+1 g ij + G 1 (g, ∇Ṽ g, ∇ 2 V g)#(∇Ṽ ,ξ ) l+1 g + G 2 (g, ∇Ṽ g)#∇Ṽ ((∇Ṽ ,ξ ) l+1 g) = (∇Ṽ ,ξ ) l+1 h Ṽ ,ij + ∇Ṽ ,ξ H l + ∇Ṽ ,ξ g ·· #(∇Ṽ ) 2 (∇Ṽ ,ξ ) l g + ∇Ṽ ,ξ G 1 (g, ∇Ṽ g, ∇ 2 V g) #(∇Ṽ ,ξ ) l g + ∇Ṽ ,ξ (G 2 (g, ∇Ṽ g)) #∇Ṽ ((∇Ṽ ,ξ ) l g). (3.20) Here g ·· stands for one of g ij . It follows from the chain rule and the definition of H l that (3.19) holds for l + 1 as well. With (3.19), we can finish the proof of this lemma. Similar to (3.18), we have (3.21) (∇Ṽ ,ξ ) l h C k+2,α V (Bc β ) ≤ Cρ l 0 . Moreover, by induction, if (3.14) is proved for l ′ < l, (3.22) H l C k,α V (Bc β −η l−1 ) ≤ Cρ l 0 . (3.14) for l follows from the Schauder estimate of (3.19) by (3.21), (3.22) and (3.9). With these preparations, we move on to the proof of Theorem 3.1. 3.2. The case k 3 = 0, 1. While we get good control for the complex Hessian of ϕ(ṽ) (see (3.8)), the C 0 norm ofφ is made worse by a multiple of ρ −2 0 , i.e., (3.23) φ(ṽ) C 0 (Bc β ) ≤ Cρ −2 0 and the C 0 norm of the first order derivative ofφ becomes (3.24) ∇Ṽφ(ṽ) C 0 (Bc β ) ≤ Cρ −1 0 . To prove Theorem 3.1 for the case k 3 = 0, 1, we consider the equation satisfied by ∇Ṽφ, ∇Ṽφ C k+2,α V (B c β /4 ) ≤ C ∇Ṽφ C 0 (B c β /2 ) + ∇Ṽ h C k,α V (B c β /2 ) ≤ Cρ −1 0 , where we have used (3.24) and (3.26) in the last inequality above. By (3.6) and (3.27), (3.28) ϕ C k,α V (B c β /4 ) ≤ C for k ≥ 1, which is exactly what we need for the case k 3 = 0 after a further translation to the polar coordinates (see Lemma 2.1). For k 3 = 1, it suffices to notice that (3.27) implies ∇ V,ξ ϕ C k,α V (B c β /4 ) ≤ C for any k ≥ 0. 3.3. The case k 3 > 1. While the basic strategy of the proof in this case is similar, we need to use some extra information provided by the tangential regularity of ϕ: (3.29) (∇Ṽ ,ξ ) k3φ C 0 (Bc β ) ≤ Cρ k3−2 0 , which follows from the boundedness of ∇ k3 V,ξ ϕ and (3.6). By (3.6) again and Lemma 2.1, the proof of Theorem 3.1 for the case k 3 > 1 is reduced to the claim that (3.30) (∇Ṽ ,ξ ) k3φ C l,α V (B c β /4 ) ≤ C(l)ρ k3−2 0 for l ∈ N. The rest of this section is devoted to the proof of (3.30), which is an induction on k 3 . We first notice that (3.30) for the case k 3 = 1 is a special case of (3.27). For k 3 = 2, we take one more ∇Ṽ ,ξ of the following equation (see (3.25)) △ g (∇Ṽ ,ξφ ) = λρ 2 0 ∇Ṽ ,ξφ + ∇Ṽ ,ξ h to get (3.31) △ g (∇ 2 V ,ξφ ) = λρ 2 0 ∇ 2 V ,ξφ + (∇Ṽ ,ξφ )Ṽ ,ij #∇Ṽ ,ξ g ij + ∇ 2 V ,ξ h. The idea is to take this as a linear equation of ∇ 2 V ,ξφ defined on B c β and to apply the Schauder estimate. For this purpose, we check that: (a) the coefficients of △ g are known to be good by Lemma 3.3; (b) by (3.29), the C 0 (B c β ) norm of ∇ 2 V ,ξφ is bounded by a constant independent of ρ 0 , or a constant multiple of ρ k3−2 0 since k 3 = 2; (c) the non-homogeneous term is (by switching the order of derivatives) (∇Ṽ ,ξφ )Ṽ ,ij #∇Ṽ ,ξ g ij + ∇ 2 V ,ξ h = ∇Ṽ ,ξ g ij #∇Ṽ ,ξ g ij + ∇ 2 V ,ξ h,(3.33) ∇ 2 V ,ξ h C k,α V (B c β /2 ) ≤ Cρ 2 0 ≤ C. By (a)-(c) above, applying the Schauder estimate to (3.31) concludes the proof of (3.30) for k 3 = 2. For k 3 = 3, we take one more ∇Ṽ ,ξ to (3.31) and find a similar equation with a more complicated non-homogeneous term. The key point is again that the C k,α V norm of this non-homogeneous term is now bounded by Cρ 0 = Cρ k3−2 0 by the same reason as above so that we can obtain (3.30) (for k 3 = 3) from (3.29) by using the Schauder estimate. We repeat this argument to see that (3.30) for any k 3 (hence Theorem 3.1) holds. Expansion In this section, we prove Theorem 1.1 and Corollary 1.2. Formal consideration. For any q ∈ R, define T = ρ 2j+ k β cos lθ, ρ 2j+ k β sin lθ| j, k, l ∈ N ∪ {0} , k − l 2 ∈ N ∪ {0} . If we write Span(T ) for the set of finite linear combinations of functions in T , it is easy to check that Span(T ) is multiplicatively closed. In [12], the author proved that the Ricci flow solution (on surface) has an expansion using terms in T . Please note that β in this paper is β + 1 in that paper. The key observation there is that if the unknown functions are assumed to have an expansion using functions in T , then by the Ricci flow equation, we know the△ of the unknown function has the same expansion, which is a consequence of the nonlinear structure of the Ricci flow equation and the fact that Span(T ) is multiplicatively closed. Moreover, the fact that Span(T ) is also closed under the application of△ −1 allows us to prove that the unknown function has the desired expansion using terms in T by an induction argument. Before we proceed, let's show how the complex Monge-Ampère equation is more complicated, which forces us to consider terms involving log ρ. Recall the equation of ϕ and ϕ χ det(ϕ ξξ )(0, ξ)△ϕ χ = F #(P ϕ χa ·P ϕ χ b ) + H (B χ ) + (det(ϕ ξξ )(0, ξ) − det(ϕ ξξ )(ρ, θ, ξ))△ϕ χ , where F and H are smooth functions of ϕ and h and their tangent derivatives. Now let's look at the right hand side of (B χ ). Since h and its tangent derivatives are smooth functions of z 1 and z 1 = β 1/β ρ 1 β cos θ + √ −1ρ 1 β sin θ , they should have an expansion in T , which is later proved in Lemma 4.13. If all of ϕ χ 's have expansion involving terms in T , then formally, we expect that F and H also have expansion involving terms in T . The problem is the product P ϕ χa ·P ϕ χ b . Remark 4.1. In the rest of this paper, the functions in T shall frequently appear in the argument. For simplicity, we discuss the cos term only and understand that a minor modification of the proof works for the sin term. By the definition of T and Lemma 2.4, P ϕ χa ·P ϕ χ b is, up to some error, a sum of (4.1) ρ 2j+ k β −2 cos lθ with j + k ≥ 2, k − l 2 ∈ N ∪ {0} . Some terms of (4.1) which may appear in the expansion of the right hand side of (B χ ) cause trouble when we apply△ −1 to the right hand side. To see this, we compute△ ρ σ+2 cos lθ = (∂ 2 ρ + 1 ρ ∂ ρ − l 2 ρ 2 (β) 2 )ρ σ+2 cos lθ (4.2) = ((σ + 2) 2 − l 2 (β) 2 )ρ σ cos lθ, (4.3) which implies that: if ρ k β −2 cos kθ (a special case in (4.1)) appears in the expansion of the right hand side of (B χ ), then we can not find a match in the expansion of the left hand side of (B χ ). This forces us to consider more terms than T . The simple computatioñ △(ρ k β (log ρ) cos kθ) = 2k β ρ k β −2 cos kθ motivates us to include (in the expansion of ϕ and ϕ χ ) ρ k β log ρ cos kθ for k ≥ 2. Here k ≥ 2 is a consequence of the j + k ≥ 2 in (4.1). Concluding the formal discussion above, we define Definition 4.2. T log is defined to be the set of ρ 2j+ k β (log ρ) m cos lθ, ρ 2j+ k β (log ρ) m sin lθ satisfying (1) k, j, l, m = 0, 1, 2, · · · ; (2) k−l 2 ∈ N ∪ {0}; (3) m ≤ max {0, k − 1}. It is trivial to check that Span(T log ) is multiplicatively closed. We shall see in later proofs that (3) plays a subtle role in balancing the solvability of△ and the nonlinear structure of the right hand side of (B χ ). Lemma 4.3. Except the case k = l = m + 1 and j = 0 and the case k = l = j = 0, for any v = ρ 2j+ k β (log ρ) m cos lθ in T log , we have u ∈ Span(T log ) such that △u = ρ −2 v. Proof. The proof relies on the following computatioñ △ρ σ (log ρ) m cos lθ = σ 2 − l 2 β 2 ρ σ−2 (log ρ) m cos lθ (4.4) +2σmρ σ−2 (log ρ) m−1 cos lθ +m(m − 1)ρ σ−2 (log ρ) m−2 cos lθ. (1) If either j = 0 or k = l, we prove the lemma by induction on m as follows. Notice that in this case, we always have 2j + k β > l β because k ≥ l as required in the definition of T log . When m = 0, v = ρ σ cos lθ with σ = 2j + k β = l β . (4.4) implies that△v is a multiple of ρ −2 v, which proves the lemma in this case. Assume that the case m ≤ m 0 for some m 0 < max {0, k − 1} is proved and that m 0 + 1 ≤ max {0, k − 1} such that v = ρ 2j+ k β (log ρ) m0+1 cos lθ is in T log . By taking u 1 = v, (4.4) again implies that△u 1 is a multiple of ρ −2 v up to a linear combination of ρ 2j+ k β −2 (log ρ) m0−1 cos lθ (if m 0 ≥ 1) and ρ 2j+ k β −2 (log ρ) m0 cos lθ. By the induction hypothesis, there exists u 2 , u 3 ∈ Span(T log ) such that the above two terms are△u 2 and△u 3 respectively. The desired u is then the linear combination of u 1 , u 2 and u 3 . (2) For the rest of the proof, we assume j = 0 and k = l = 0. In this case, σ = l β and the first term in the right hand side of (4.4) vanishes. Notice that there is nothing to prove for k = l = 1, so we assume that k ≥ 2 and prove by induction that the lemma holds for m = 0, 1, · · · , k − 2. When m = 0, i.e. v = ρ k β cos kθ, we can take u = β 2k ρ k β (log ρ) cos kθ in Span(T log ), because as a special case of (4.4), △ρ k β (log ρ) cos kθ = 2k β ρ k β −2 cos kθ. Assume that lemma is proved for m ≤ m 0 and that m 0 + 1 ≤ k − 2. Let v = ρ k β (log ρ) m0+1 cos kθ and take u 1 = ρ k β (log ρ) m0+2 cos kθ in T log . By (4.4),△u 1 is a multiple of ρ −2 v up to a multiple of ρ k β −2 (log ρ) m0 cos kθ, which is△u 2 for some u 2 ∈ Span(T log ) by the induction hypothesis. Again, the desired u is a linear combination of u 1 and u 2 . Motivated by this lemma, we define T rhs = ρ −2 T log \ ρ −2 , ρ k β −2 (log ρ) k−1 cos kθ| k ∈ N , where ρ −2 T log is the set of ρ −2 v for each v ∈ T log . A simple observation is that T log ⊂ T rhs . With (4.5), Lemma 4.3 can be formulated as Lemma 4.5. For each v ∈ Span(T rhs ), there is u in Span(T log ) such that △u = v. Besides its connection with Lemma 4.3, the definition of T rhs in (4.5) is important also in the analysis of the structure of the right hand side of (B χ ). Briefly speaking, we will show in Section 4.3 that if ϕ χ has expansion using functions in T log , then the right hand side of (B χ ) has an expansion in T rhs . This explains the subscript in the notation. For that purpose, we shall need the following lemma. Lemma 4.6. (1) If η 1 ∈ Span(T log ) and η 2 ∈ Span(T rhs ), then η 1 ·η 2 ∈ Span(T rhs ). (2) If η 1 and η 2 are in Span(T log ), but neither of them is constant function, then ρ −2 η 1 · η 2 ∈ Span(T rhs ). Proof. (1) Assume without loss of generality that η 1 = ρ 2j1+ k 1 β (log ρ) m1 cos l 1 θ and η 2 = ρ −2+2j2+ k 2 β (log ρ) m2 cos l 2 θ, where k i , j i , m i , l i are subject to the restrictions of Definition 4.2 and Definition 4.4 respectively. Direct computation gives (4.6) ρ 2 η 1 η 2 = 1 2 (Y 1 − Y 2 ), where Y 1 = ρ 2(j1+j2)+ k 1 +k 2 β (log ρ) m1+m2 cos(l 1 + l 2 )θ Y 2 = ρ 2(j1+j2)+ k 1 +k 2 β (log ρ) m1+m2 cos(l 1 − l 2 )θ. (4.7) By the definition of T rhs , it suffices to show that (a) Y 1 and Y 2 are in T log , but (b) neither Y 1 or Y 2 is in (4.8) 1, ρ k β (log ρ) k−1 cos kθ| k ∈ N . The proof of (a) is trivial and is in fact the same as the proof of the claim that Span(T log ) is multiplicatively closed. For (b), we first notice that neither Y 1 or Y 2 can be constant. Otherwise, we must have 2(j 1 + j 2 ) + k 1 + k 2 β = m 1 + m 2 = 0, and hence j 2 = k 2 = m 2 = 0, which is a contradiction to η 2 ∈ T rhs . If there is k ∈ N such that Y 1 = ρ k β (log ρ) k−1 cos kθ, then 2(j 1 + j 2 )β + (k 1 + k 2 ) = k (4.9) m 1 + m 2 = k − 1 (4.10) l 1 + l 2 = k. (4.11) By k 1 ≥ l 1 , k 2 ≥ l 2 and j 1 , j 2 ≥ 0, we know j 1 = j 2 = 0, k 1 = l 1 , k 2 = l 2 . If k 1 = 0, then k 1 = m 1 = l 1 = j 1 = 0, i.e. η 1 = 1, in which case the lemma is trivial and there is nothing to prove. If k 2 = 0, then k 2 = m 2 = l 2 = j 2 = 0 and η 2 = ρ −2 , which is a contradiction to the assumption that η 2 ∈ T rhs . If both k 1 and k 2 are positive, then m 1 ≤ k 1 − 1 and m 2 ≤ k 2 − 1, which contradicts (4.10). In summary, we have proved that Y 1 is not in (4.8). If there is k ∈ N such that Y 2 = ρ k β (log ρ) k−1 cos kθ, then 2(j 1 + j 2 )β + (k 1 + k 2 ) = k (4.12) m 1 + m 2 = k − 1 (4.13) |l 1 − l 2 | = k. (4.14) A similar discussion yields a contradiction. Hence, the proof of (1) is done. (2) Assume η 1 = ρ 2j1+ k 1 β (log ρ) m1 cos l 1 θ and η 2 = ρ 2j2+ k 2 β (log ρ) m2 cos l 2 θ and let Y 1 and Y 2 be defined as in (4.7). Since η 1 · η 2 = 1 2 (Y 1 − Y 2 ), it suffices to prove that (as before) both Y 1 and Y 2 are in T log , but neither is in (4.8). Again, the first assertion is trivial. By our assumption that neither of η 1 and η 2 is constant function, we have (4.15) j 1 β + k 1 > 0 and j 2 β + k 2 > 0. A consequence of this is that Y 1 and Y 2 are not constant function. It remains to exclude the possibility that Y 1 (or Y 2 ) is ρ k β (log ρ) k−1 cos kθ for some k ∈ N. In that case, (4.9) and (4.10) hold as before and imply that (4.16) 2(j 1 + j 2 )β + (k 1 + k 2 ) = m 1 + m 2 + 1. This is a contradiction to m 1 ≤ max {0, k 1 − 1} and m 2 ≤ max {0, k 2 − 1}, unless (at least) one of k 1 and k 2 is zero. If k 1 = m 1 = 0, then (4.15) implies that j 1 > 0, and hence (4.16) implies k 2 < m 2 + 1, which is a contradiction to the definition of T log . 4.2. Finite expansion. Throughout this section, we fix ξ and take ϕ and ϕ χ as functions of ρ and θ alone. We write B r for the set of (ρ, θ) with ρ ∈ (0, r) and θ ∈ S 1 . Definition 4.7. A function u defined in B 1/2 is said to be inÕ(q) for some q ∈ R if and only if there are constants C(k 1 , k 2 ) for all k 1 , k 2 = 0, 1, 2, · · · such that (ρ∂ ρ ) k1 ∂ k2 θ u ≤ C(k 1 , k 2 )ρ q on B 1/2 . Remark 4.8. Sometimes, we abuse the notation by usingÕ(q) to denote a function in it. Theorem 3.1 implies that ϕ and ϕ χ are inÕ(0). In fact, we have Lemma 4.9. Suppose that u is in bothÕ(0) and Donaldson's space C 2,α β . Then there is some q > 1 such that u = c +Õ(q) for some constant c. Proof. Being in C 2,α β implies that ∂u ∂ρ + 1 ρ ∂u ∂θ ≤ Cρ α and that u(0, θ) is a constant independent of θ. By the Newton-Lebnitz formula u(ρ, θ) = u(0, θ) + ρ 0 ∂u ∂ρ (t, θ)dt, we have (4.17) |u(ρ, θ) − u(0, θ)| ≤ Cρ 1+α . For each fixed ρ 0 and θ 0 , set v(ρ, θ) = u(ρρ 0 , θ) − u(0, θ). If v is regarded as a function of (ρ, θ) defined on Q := (ρ, θ)| 1 2 < ρ < 2, θ 0 − 0.1 < θ < θ 0 + 0.1 , we have v C k (Q) ≤ C(k) for any k because u is inÕ(0). On the other hand, (4.17) gives v C 0 (Q) ≤ Cρ 1+α 0 . The usual interpolation yields that (4.18) v C k (Q) ≤ C ′ (k)ρ q 0 for any 1 < q < 1 + α and another sequence of constants C ′ (k) (depending on q). The lemma follows if (4.18) is translated into inequalities of u. Similarly, a function u is said to have an (rhs)-expansion up to order q if and only if there is η ∈ Span(T rhs ) such that the above holds. Obviously, if 2j + k β > q or 2j + k β = q with m = 0, then ρ 2j+ k β (log ρ) m cos lθ ∈Õ(q). Therefore, η in the above definition of expansion is only unique up to terms like these. Similar observation applies to the (rhs)-expansion. Lemma 4.9 and the fact that ϕ and ϕ χ are in C 2,α β imply that ϕ and ϕ χ have expansion up to order q for some q > 1. The proof of the main theorem is a bootstrapping argument starting from this. For the use of the next section, we need the following three lemmas, which explains the reason why we want Span(T log ) to be multiplicatively closed. Lemma 4.12. If u i is inÕ(q i ) for i = 1, 2, then u 1 · u 2 is inÕ(q 1 + q 2 ). Lemma 4.13. Suppose F (x 1 , · · · , x N ) is a smooth function of N variables. Assume that u 1 , · · · , u N have expansions up to order q for some q ≥ 0. Then F (u 1 , · · · , u N ) has an expansion up to order q. The proof of Lemma 4.12 is trivial and the proof of Lemma 4.11 is the combination of the facts that Span(T log ) is multiplicatively closed, that any function in Span(T log ) is inÕ(0) and that for v 1 (v 2 ) inÕ(q 1 )(Õ(q 2 )) respectively, we have v 1 · v 2 ∈Õ(q 1 + q 2 ). All the above mentioned facts can be verified directly by Definition 4.7. Lemma 4.13 is a simple generalization of the Lemma 6.8 in [12], where the case N = 1 is proved. For general N , it suffices to replace the Taylor expansion formula of one variable by the Taylor expansion of multiple variables. We refer the readers to [12] for details of the proof. Lemma 4.14. If ϕ and all of ϕ χ have expansion up to order q for some q > 1, then the right hand side of (B χ ) has a (rhs)-expansion up to order q − 1. then u has an expansion up to order q ′ for any q ′ < q + 2. Before the proof of these two lemmas, we show how they imply Theorem 1.1. Proof of Theorem 1.1 (assuming Lemma 4.14 and 4.15). We have as our starting point that ϕ and ϕ χ are in Donaldson's C 2,α β space. Theorem 3.1 and Lemma 4.9 imply that ϕ and ϕ χ have expansions up to some order q with q > 1. The rest of the proof is easy bootstrapping argument using Lemma 4.14 and 4.15. The proof of Lemma 4.14 depends heavily on the structure of the complex Monge-Ampère equation. Proof of Lemma 4.14. The right hand side of (B χ ) consists of three terms: (a) F (ϕ χ2,...,ϕχ l+2 )#(P ϕ χa ·P ϕ χ b ); (b) H(h, · · · , h χ l , ϕ, · · · , ϕ χ l ); (c) (det(ϕ ξξ )(0, ξ) − det(ϕ ξξ )(ρ, θ, ξ))△ϕ χ . We discuss (b) first. By our assumption in Theorem 1.1, h and its tangent derivatives are smooth in z 1 . Hence, they have expansion up to any order. One can either check this directly by using the definition, or use Lemma 4.13 and notice that z 1 (as a function of (ρ, θ)) has expansion up to any order. By the assumptions about ϕ in Lemma 4.14, Lemma 4.13 again implies that H has expansion up to order q. Since T rhs ⊃ T log , the term (b) has the required expansion. For (c), we notice that there are η 1 , η 2 ∈ Span(T log ) such that det(ϕ ξξ )(0, ξ) − det(ϕ ξξ )(ρ, θ, ξ) = η 1 +Õ(q) and△ ϕ χ = ρ −2 η 2 +Õ(q − 2). Here, the existence of η 1 follows from Lemma 4.13 and the assumptions of the lemma. For η 2 , we first findη 2 satisfying ϕ χ =η 2 +Õ(q) and then check that△ mapsÕ(q) toÕ(q − 2) and that for any v ∈ Span(T log ),△v is in Span(ρ −2 T log ) (see (4.4)). We claim that (4.19) η 1 , η 2 ∈Õ(q) for someq > 1. On one hand, every function (hence η 1 and η 2 ) in Span(T log ) is inÕ(0). On the other hand, it is easy to see that both η 1 and η 2 have no constant term. Note that all the rest of the functions in T log are bounded by Cρq for someq > 1 (depending on β). With (4.19) in mind, we compute (c) as (η 1 +Õ(q))(ρ −2 η 2 +Õ(q − 2)) = ρ −2 η 1 · η 2 + η 1 ·Õ(q − 2) + ρ −2 η 2 ·Õ(q) +Õ(q) ·Õ(q − 2). Given (4.19) and Lemma 4.12, the sum of the last three terms above is inÕ(q − 1). The second part of Lemma 4.6 implies that ρ −2 η 1 · η 2 is in Span(T rhs ) and hence finishes the proof for (c). For the proof for (a), we claim that there exists η 4 ∈ Span(T rhs ) such that (4.20) P ϕ χa ·P ϕ χ b = η 4 +Õ(q − 1). Before we prove the claim, we see how it implies that (a) has the required expansion. By Lemma 4.13, we know that F has expansion up to order q so that F #(P ϕ χa ·P ϕ χ b ) = (η 3 +Õ(q))(η 4 +Õ(q − 1)) for some η 3 ∈ Span(T log ). By the first part of Lemma 4.6, η 3 · η 4 is in Span(T rhs ). The remaining terms, η 4 ·Õ(q), η 3 ·Õ(q − 1) andÕ(q − 1) ·Õ(q) are obviously iñ O(q − 1). The rest of the proof is devoted to the proof of the claim. By our assumption, we may assume that ϕ χa = η a +Õ(q) and ϕ χ b = η b +Õ(q) for η a and η b in Span(T log ). By Lemma 2.4, the right hand side of (4.20) is a quadratic polynomial of ∂ ρ η a +Õ(q − 1), 1 ρ ∂ θ η a +Õ(q − 1) and ∂ ρ η b +Õ(q − 1), 1 ρ ∂ θ η b +Õ(q − 1). By the definition of T log , ∂ ρ η a , ∂ ρ η b , 1 ρ ∂ θ η a and 1 ρ ∂ θ η b are inÕ(0) because among all terms in T log , except the constant term which is killed by the derivative, the lowest order term decays faster than ρ. This implies that the terms like ∂ ρ η a ·Õ(q − 1) is inÕ(q − 1). It remains to show ∂ ρ η a · ∂ ρ η b is in Span(T rhs ). Similar argument works for ∂ ρ η a · 1 ρ ∂ θ η b , 1 ρ ∂ θ η b · ∂ ρ η a and 1 ρ 2 ∂ θ η a · ∂ θ η b . This is a consequence of the second part of Lemma 4.6. In fact, if we setη a = (ρ∂ ρ )η a andη b = (ρ∂ ρ )η b , then ∂ ρ η a · ∂ ρ η b = ρ −2η a ·η b and we can check thatη a andη b are non-constant function in Span(T log ). Next, we move to the proof of Lemma 4.15. Proof of Lemma 4.15. Note that we apply the following argument to every equation (B χ ) simultaneously. By the assumption, the right hand side is given by η +Õ(q) for some η ∈ Span(T rhs ). Lemma 4.5 implies that there exists η ′ in Span(T log ) such that△ η ′ = η. Pretending the constant on the left hand side of (B χ ) is 1, we have (4.21)△(ϕ χ − η ′ ) =Õ(q). To finish the proof of Lemma 4.15, we need Lemma 4.16. (1) (Lemma 6.7 in [12]) Any bounded (△) harmonic function defined on (ρ, θ)| ρ ∈ (0, 1], θ ∈ S 1 has expansion up to any order. (2) (Lemma 6.10 in [12])For each v ∈Õ(q) and q ′ < q + 2, there is u inÕ(q ′ ) with△ u = v. Since this lemma appeared in exactly the same form and is proved in detail in [12], we refer the readers to that paper for the proof. With the above lemma and (4.21), there exists f ∈Õ(q ′ ) for any q ′ < q + 2 satisfying△ (ϕ χ − η ′ ) =△(f ), which implies that ϕ χ − η ′ − f is a bounded harmonic function and hence has expansion up to any order by (1) To apply Theorem 1.1, we need to check (S1)-(S3). Since ψ 0 is the potential function of the smooth Kähler form ω 0 in the holomorphic coordinates {z i }, we know that ψ 0 is smooth and that (ψ 0 ) ij is smooth and positive definite. Moreover, being the metric of the line bundle L,h is also smooth and positive. Hence (S1) follows from (1.8). It follows from the definition of C 2,α β (see (D1)-(D4) in Section 2.2) and β ∈ (0, 1) that smooth functions in holomorphic coordinates are in C 2,α β for some α > 0 depending on β. So ψ 0 ,h ∈ C 2,α β . Moreover, one can check directly that |z 1 | 2β is also in C 2,α β . The best way to see this is to notice that |z 1 | 2β = β 2 ρ 2 and to use the equivalent definition (P1)-(P3). Therefore, |s| 2β h0 =h β |z 1 | 2β is in C 2,α β . Finally, since ψ is in C 2,α β as assumed in the definition of conical Kähler-Einstein metric, so is ϕ, which confirms (S2) (for some α > 0). (S3) follows from the definition of conical Kähler-Einstein metric. Now, we can apply Theorem 1.1 to see that ϕ has the expansion up to any order. To see that this is also true for ψ, it suffices to check that ϕ − ψ = ψ 0 + δ |s| 2β h0 (by (1.6)) has expansion up to any order. Since ψ 0 andh are smooth in holomorphic coordinates and |s| 2β h0 =h β |z 1 | 2β =h β β 2 ρ 2 , it remains to check all smooth functions in holomorphic coordinates have expansions up to any order, which is a consequence of Lemma 4.13 and the fact that z 1 has expansion up to any order. Appendix A. Estimate of some elliptic system In this appendix, we prove some estimates for the following elliptic system defined on B 1 ⊂ C n , (A.1) △ g g ij − g km g nl ∂g im ∂z n ∂g kj ∂zl = λρ 2 0 g ij + hṼ ,ij . When h = 0, this is the equation satisfied by the metric tensor of Kähler-Einstein metric. We always assume (A.2) λρ 2 0 + hṼ ,ij C 0 (B1) ≤ Λ. The methods used here are from the book of Giaquinta [4] and are by now classical. In contrast to the theorems in [4], we prove effective estimates instead of just regularity statements. Remark. Note that in this appendix, the real dimension of the domain is 2n, instead of n. We first bound the L 2 norm of the gradient of g ij . Lemma A.1. Suppose that g ij are some smooth complex-valued functions defined on B 1 ⊂ C n solving (A.1) whose coefficients satisfy (A.2). If for some σ > 0, we have σ −1 g ij ≤ δ ij ≤ σg ij on B 1 , then ∇g ij L 2 (B 3/4 ) ≤ C(σ, Λ, g ij C α (B1) ). Proof. For any point x 0 in B 3/4 and R > 0 to be determined in the proof, let η be some smooth cut-off function supported in B R (x 0 ) with η ≡ 1 in B R/2 (x 0 ) and |∇η| ≤ CR −1 . Multiplying both sides of (A.1) by (g ij − g ij (x 0 ))η 2 and freezing the coefficients of the leading term in (A.1) gives 0 = g kl (x 0 )∂ k∂l g ij (g ij − g ij (x 0 ))η 2 (A.3) −(g kl (x 0 ) − g kl )∂ k∂l g ij (g ij − g ij (x 0 ))η 2 −(g · g · Dg · Dg + λρ 2 0 g + h) · (g ij − g ij (x 0 ))η 2 . Note that we have omitted subscripts in the above computation when they are not essential to the proof. By the Hölder continuity of g ij , we have (A.4) g ij − g ij (x 0 ) ≤ CR α . Integration by parts of (A.3), (A.4) and Young's inequality imply that |Dg| 2 η 2 ≤ C R α |Dg| η |∇η| + R α |Dg| 2 η 2 +C R 2α |Dg| η |∇η| + R α |Dg| 2 η 2 + CR 2n+α ≤ ( 1 2 + CR α ) |Dg| 2 η 2 + CR 2α |∇η| 2 + CR 2n+α . Now we can choose R so small (depending only on σ, α and the Hölder norm of g ij ) that the first term in the right hand side is absorbed by the left hand side to give (A.5) B R/2 (x0) |Dg| 2 dx ≤ CR 2n−2+2α ≤ C. The lemma then follows from the above inequality by covering B 3/4 by balls of radius R. Next, we prove C γ estimate of g ij for any α < γ < 1. Lemma A.2. For g ij as in Lemma A.1 and any α < γ < 1, we have g ij C γ (B 3/4 ) ≤ C(σ, Λ, γ, g ij C α (B1) ). Proof. For any x 0 ∈ B 3/4 and R ≤ 1/4, let v ij be the solution of g kl (x 0 )∂ k∂l v ij = 0 on B R (x 0 ) v ij = g ij on ∂B R (x 0 ). By Theorem 2.1 on page 78 of [4] (applied to Dv ij ), there is a constant depending only on σ such that for 0 < ρ < R, (A.6) Bρ(x0) Dv ij 2 ≤ C( ρ R ) 2n BR(x0) Dv ij 2 . Setting w ij = g ij − v ij , we get (using (A.6)) Bρ(x0) Dg ij 2 (A.7) ≤ Bρ(x0) Dv ij 2 + Bρ(x0) Dw ij 2 ≤ C( ρ R ) 2n BR(x0) Dv ij 2 + C Bρ(x0) Dw ij 2 ≤ C( ρ R ) 2n BR(x0) Dg ij 2 + C BR(x0) Dw ij 2 . Using the equation satisfied by v ij , we may rewrite (A.1) as follows g kl (x 0 )∂ k∂l (g ij − v ij ) = −(g kl − g kl (x 0 ))∂ k∂l g ij + g · g · Dg · Dg + λρ 2 0 g + h. Since w ij vanishes on ∂B R (x 0 ), we can use it as the test function of the above equation to obtain BR(x0) Dw ij 2 ≤ C BR(x0) ∂ k∂l g ij w(g kl − g kl (x 0 )) + |w| |Dg| 2 + |w| ≤ C BR(x0) |Dg| |∇w| g kl − g kl (x 0 ) + |w| |Dg| 2 + |w| . Using Young's inequality, we get (A.8) BR(x0) Dw ij 2 ≤ C BR(x0) |Dg| 2 ( g kl − g kl (x 0 ) 2 + |w|) + |w| . The maximum principle implies that osc BR(x0) v ij ≤ osc BR(x0) g ij , which implies that (A.9) w C 0 (BR(x0)) ≤ osc BR(x0) v ij + osc BR(x0) g ij ≤ CR α . Putting (A.7), (A.8) and (A.9) together yields the following decay estimate Bρ(x0) |Dg| 2 ≤ C ( ρ R ) 2n + R α BR(x0) |Dg| 2 + CR 2n+α . Dividing both sides of the above equation by ρ 2n−2 and setting ρ/R = τ give ρ 2−2n Bρ(x0) |Dg| 2 ≤ C(1 + τ −2n R α ) τ 2 R 2−2n BR(x0) |Dg| 2 + Cτ 2−2n R 2 . By picking τ small so that 2Cτ (2−2γ) = 1 and then R 1 small so that 1 + τ −n R α < 2 for all R < R 1 , we have ρ 2−2n Bρ(x0) |Dg| 2 ≤ τ 2γ R 2−2n BR(x0) |Dg| 2 + C(γ)R 2 for R < R 1 . From here, a routine iteration shows that ρ 2−2n Bρ(x0) |Dg| 2 ≤ Cρ 2γ . Hence, by the Hölder inequality and the equivalence between the Companato space and the Hölder space, we have g ij C γ (B 3/4 ) ≤ C(σ, Λ, γ, g ij C α (B1) ). Finally, we prove the C 1,α estimate. Lemma A.3. For g ij as in Lemma A.1, there exists some α ′ ∈ (0, 1) such that g ij C 1,α ′ (B 1/2 ) ≤ C(σ, Λ, g ij C α (B1) ). Proof. For any x 0 ∈ B 1/2 and R ≤ 1/4, as in the proof of Lemma A.2, let v ij be the solution of g kl (x 0 )∂ k∂l v ij = 0 on B R (x 0 ) v ij = g ij on ∂B R (x 0 ). Again, applying Theorem 2.1 on page 78 of [4] to Dv ij , we get a constant depending only on σ such that (A.10) Bρ(x0) Dv ij − (Dv ij ) x0,ρ 2 ≤ C( ρ R ) 2n+2 BR(x0) Dv ij − (Dv ij ) x0,R 2 . Here (u) x,r means the average of u in the ball B r (x). Next, we set w ij = g ij − v ij on B R (x 0 ). Triangle inequalities and (A.10) imply that Bρ(x0) Dg ij − (Dg ij ) x0,ρ 2 (A.11) ≤ Bρ(x0) Dv ij − (Dv ij ) x0,ρ 2 + Bρ(x0) Dw ij − (Dw ij ) x0,ρ 2 ≤ C( ρ R ) 2n+2 BR(x0) Dv ij − (Dv ij ) x0,R 2 + C Bρ(x0) Dw ij 2 ≤ C( ρ R ) 2n+2 BR(x0) Dg ij − (Dg ij ) x0,R 2 + C BR(x0) Dw ij 2 . Using w ij as the test function of (A.1) as in the proof of Lemma A.2 gives BR(x0) Dw ij 2 ≤ C BR(x0) ∂ k∂l g ij w(g kl − g kl (x 0 )) + |w| |Dg| 2 + |w| ≤ C BR(x0) |Dg| |∇w| g kl − g kl (x 0 ) + |w| |Dg| 2 + |w| . The Young's inequality and the fact that g ij lies in C γ (B 3/4 ) imply that Dg ij − (Dg ij ) x0,ρ 2 ≤ C( ρ R ) 2 R −2n BR(x0) Dg ij − (Dg ij ) x0,ρ 2 +C(R/ρ) 2n R γ ′ . As before, picking τ ∈ (0, 1) with Cτ 2−γ ′ = 1 and setting ρ = τ R, we get ρ −2n Bρ(x0) Dg ij − (Dg ij ) x0,ρ 2 ≤ τ γ ′ R −2n BR(x0) Dg ij − (Dg ij ) x0,ρ 2 +C(τ )R γ ′ . Iteration again implies that Bρ(x0) Dg ij − (Dg ij ) x0,ρ 2 ≤ Cρ 2n+γ ′ , which concludes the proof of the lemma by Theorem 1.2 in Chapter III of [4]. ( 3 . 325) △ g (∇Ṽφ) = λρ 2 0 ∇Ṽφ + ∇Ṽ h, which is obtained by takingṼ -derivative of the Monge-Ampère equation satisfied byφ (see (3.2) and (3.6)) log det(φṼ ,ij ) = λρ 2 0φ + h. By Lemma 3.3 and Corollary 2.2, any C k,α V norm of the metric g is bounded and (3.26) ∇Ṽ h C k,α V (B c β /2 ) ≤ C so that the Schauder estimate applied to ( Definition 4 . 410. A function u is said to have an expansion up to order q if and only if there is η ∈ Span(T log ) such that u = η +Õ(q). Lemma 4 . 11 . 411Suppose that u 1 and u 2 have expansions up to order q. Then so does u 1 · u 2 . 4. 3 . 3The proof of Theorem 1.1. The proof of the main theorem relies on the following two lemmas. Lemma 4 . 15 . 415If v has a (rhs)-expansion up to order q and u is a bounded solution to△ u = v, where the first term is estimated byLemma 3.4 (3.32) ∇Ṽ ,ξ g ij #∇Ṽ ,ξ g ij C k,α V (B c β /2 ) ≤ Cρ 2 0 ≤ C and the second term by Corollary 2.2 (applying to ∇ 2 ξ h) of Lemma 4.16. 4.4. The proof of Corollary 1.2. By (1.7) and (1.8), we know ϕ satisfies (1.1). |Dg| 2 ≤ CR 2n−2+2γin the proof of Lemma A.2. By Lemma A.2, we can choose and fix γ so that (A.12) for some γ ′ > 0 (small). Combining (A.11) and (A.12), we getBR(x0) |Dw| 2 ≤ CR γ BR(x0) |Dg| 2 + CR 2n+γ . Notice that we have proved (see (A.5)) BR(x0) BR(x0) |Dw| 2 ≤ CR 2n+γ ′ ρ −2n Bρ(x0) Ricci flat Kähler metrics with edge singularities, International Mathematics Research Notices. S Brendle, S. Brendle, Ricci flat Kähler metrics with edge singularities, International Mathematics Re- search Notices 2013 (2013), no. 24, 5727-5766. Geodesics in the space of Kähler cone metrics, I. S Calamai, K Zheng, American Journal of Mathematics. 1375S. Calamai and K. Zheng, Geodesics in the space of Kähler cone metrics, I, American Journal of Mathematics 137 (2015), no. 5, 1149-1208. Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications. S Donaldson, S. Donaldson, Kähler metrics with cone singularities along a divisor, Essays in mathematics and its applications, 2012, pp. 49-79. Multiple integrals in the calculus of variations and nonlinear elliptic systems. M Giaquinta, Princeton University PressM. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton University Press, 1983. Kähler-Einstein metrics with edge singularities. T Jeffres, R Mazzeo, Y Rubinstein, Annals of Mathematics. 183T. Jeffres, R. Mazzeo, and Y. Rubinstein, Kähler-Einstein metrics with edge singularities, Annals of Mathematics 183 (2016), 95-176. Uniqueness of Kähler-Einstein cone metrics. T Jeffres, Publicacions matematiques. 442T. Jeffres, Uniqueness of Kähler-Einstein cone metrics, Publicacions matematiques 44 (2000), no. 2, 437-448. L Li, K Zheng, arXiv:1511.02410A continuity method approach to the uniqueness of Kähler-Einstein cone metrics. arXiv preprintL. Li and K. Zheng, A continuity method approach to the uniqueness of Kähler-Einstein cone metrics, arXiv preprint arXiv:1511.02410 (2015). Uniqueness of constant scalar curvature Kähler metrics with cone singularities, I: Reductivity. arXiv:1603.01743arXiv preprint, Uniqueness of constant scalar curvature Kähler metrics with cone singularities, I: Reductivity, arXiv preprint arXiv:1603.01743 (2016). Kähler-Einstein metrics singular along a smooth divisor. R Mazzeo, JournéesÉquations aux dérivées partielles. R. Mazzeo, Kähler-Einstein metrics singular along a smooth divisor, JournéesÉquations aux dérivées partielles (1999), 1-10. The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality. J Song, X Wang, Geom. Topol. 201J. Song and X. Wang, The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality, Geom. Topol. 20 (2016), no. 1, 49-102. Kähler-Einstein metrics on algebraic manifolds. G Tian, Transcendental methods in algebraic geometry. G. Tian, Kähler-Einstein metrics on algebraic manifolds, Transcendental methods in alge- braic geometry, 1996, pp. 143-185. H Yin, arXiv:1605.08836Analysis aspects of Ricci flow on conical surfaces. arXiv preprintH. Yin, Analysis aspects of Ricci flow on conical surfaces, arXiv preprint arXiv:1605.08836 (2016). School of mathematical sciences, university of science and technology of China, Hefei, 230026, China E-mail address. Hao Yin: [email protected] Mathematics Institute. Zheng: [email protected] of WarwickSchool of mathematical sciences, university of science and technology of China, Hefei, 230026, China E-mail address, Hao Yin: [email protected] Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK E-mail address, Kai Zheng: [email protected]

In 1980 J. Powell [Po]proposed that five specific elements sufficed to generate the Goeritz group of any Heegaard splitting of S 3 , extending work of Goeritz [Go] on genus 2 splittings. Here we prove that Powell's conjecture was correct for splittings of genus 3 as well, and discuss a framework for deciding the truth of the conjecture for higher genus splittings.

1804.05909

b1f6dda958a726ac4db0cf889723ec69e68c66bf

POWELL MOVES AND THE GOERITZ GROUP Michael Freedman Martin Scharlemann POWELL MOVES AND THE GOERITZ GROUP In 1980 J. Powell [Po]proposed that five specific elements sufficed to generate the Goeritz group of any Heegaard splitting of S 3 , extending work of Goeritz [Go] on genus 2 splittings. Here we prove that Powell's conjecture was correct for splittings of genus 3 as well, and discuss a framework for deciding the truth of the conjecture for higher genus splittings. Following early work of Goeritz [Go], the genus g Goeritz group of the 3-sphere can be described as the isotopy classes of orientationpreserving homeomorphisms of the 3-sphere that leave the standard genus g Heegaard surface T g invariant. Goeritz identified a finite set of generators for the genus 2 Goeritz group; that work has been recently updated, extended and completed, to give a full picture of the group (see [Sc], [Ak], [Cho]). In 1980 J. Powell [Po] extended Goeritz' set of generators to a set of five elements that he believed would generate the Goeritz group for any fixed higher-genus splitting. Unfortunately, his proof that these suffice contained a serious gap; here we prove that they do suffice for genus 3 splittings, using largely techniques that were unknown in 1980, and introduce methods that could be helpful in deciding the conjecture for higher genus splittings. Powell's actual viewpoint on the Goeritz group, which we will adopt, is framed somewhat differently. Following Johnson-McCullough [JM] (who extend the notion to arbitrary compact orientable manifolds) consider the space of left cosets Diff(S 3 )/ Diff(S 3 , T g ), where Diff(S 3 , T g ) consists of those orientation-preserving diffeomorphisms of S 3 that carry T g to itself. The fundamental group P g of this space projects to the genus g Goeritz group (with kernel π 1 (SO(3)) = Z 2 [Po,p.197]) as follows: A non-trivial element is represented by an isotopy of T g in S 3 that begins with the identity and ends with a diffeomorphism of S 3 that takes T g to itself; this diffeomorphism of the pair (S 3 , T g ) represents an element of the Goeritz group as defined earlier. The advantage of this point of view 1 is that an element of the Goeritz group can be viewed quite vividly: it is a sort of excursion of T g in S 3 that begins and ends with the standard picture of T g ⊂ S 3 . These excursions are what are pictured by Powell in Figure 1. The study of motion groups of three dimensional objects has a deep connection to the emerging study of 3 + 1 − D topological quantum field theory (TQFT) and also to four dimensional questions such as the smooth Schoenflies problem. One may think of P g as a higher dimensional analog of the braid group B n , the motion group of n-points in the plane. In the last 50 years B n has been closely studied with its unitary (2 + 1 − D TQFT) representations becoming the main focus in the last 30 years. With increasing interest in 3 + 1 − D TQFTs, P g and its unitary representations are a natural target for condensed matter theorists. We would like to thank Alex Zupan for helpful conversations; his upcoming [Zu] interleaves the Powell Conjecture with the question of the connectivity of the complexes of reducing spheres that arise from Heegaard splittings of S 3 . Composing Powell generators Let T g ⊂ S 3 be the standard genus g Heegaard surface in S 3 , dividing S 3 into the genus g handlebodies A g and B g . Definition 1.1. Any finite composition of Powell generators, illustrated in Figure 1, will be called a Powell move. around the complementary g-punctured sphere is a Powell move. It follows that there will be no need to keep track of the labels of the standard summands. Furthermore, D ω allows us not to worry about the orientation of the longitude of the 1-handle within each summand. Example: Suppose S 3 = A ∪ T ∪B is a genus g Heegaard in S 3 and {a 1 , ..., a g } and {b 1 , ..., b g } are orthogonal sets of disks in A and B respectively (orthogonal means |a i ∩ b j | = δ i,j ). There is an obvious orientation preserving homeomorphism of S 3 that carries T to T g and each pair ∂a i , ∂b i to the meridian and longitude respectively of one of the standard summands of T g . It follows from the comments above that any two such homeomorphisms are Powell equivalent. More broadly, but in a similar spirit we have: Lemma 1.4. Any braid move of a collection of standard genus 1 summands over their complementary surface is a Powell move. Figure 1 shows an example in the case of two summands. Proof. Let S be the collection of standard genus 1 summands which we are isotoping. From the brief discussion above it suffices to assume that the first Powell standard solid handle H 1 is in S and H 2 is not, and then to show that any braid move of H 1 around the punctured torus T ∩ H 2 is a Powell move. The composition D θ D ω D θ D ω isotopes H 1 around the longitude of H 2 . The generator D ν isotopes H 1 around the meridian of H 2 . But any braid move of a point in a punctured torus is a composition of such circuits around the meridian and around the longitude. Let {c 1 , ..., c g−1 } be the disjoint separating circles on T g shown in Figure 2, with each c i separating the first i standard summands from the last g − i standard summands. Note that each c i bounds a disk in both A and B and so defines a reducing sphere S i for T g . Extension 1: Suppose S 3 = A ∪ T ∪B is a genus g Heegaard surface in S 3 as above. Let g 1 + g 2 = g. Suppose {a 1 , ..., a g 1 } and {b 1 , ..., b g 1 } are orthogonal sets of disks in A and B respectively. There are (many) orientation preserving homeomormorphisms of S 3 that take T to T g , and each ∂a i , ∂b i to the meridian and longitude respectively of one of the first g 1 standard summands of T g . Lemma 1.5. If the Powell Conjecture is true for genus g 2 splittings of S 3 then all such homeomorphisms T → T g are Powell equivalent. Proof. Let h, h : (S 3 , T ) → (S 3 , T g ) be two such homeomorphisms. Let c = h h −1 (c g 1 ) ⊂ T g . Then c, like c g 1 , separates T g into a genus g 1 surface containing the first g 1 handles and a genus g 2 surface disjoint from them. Also c, like c g 1 , bounds disks in both A and B. If we cut off the first g 1 handles from T g then c and c g both bound disks, and so can be isotoped to coincide, in the resulting genus g 2 surface. It follows then from Lemma 1.4 that a Powell move will carry the first g 1 1-handles to themselves and take c to c g 1 . It follows that we may assume that h h −1 (c g 1 ) takes c g 1 (and so the reducing sphere in which it lies) to itself. The lemma then follows by operating separately on the genus g 1 and g 2 Heegaard splittings determined by S g 1 . Definition 1.6. Let M = A ∪ B be a Heegaard splitting of a compact manifold M . A collection {a 1 , ..., a m } of disjoint disks in A is primitive if there is a collection of disjoint disks {b 1 , ..., b m } in B so that for every i, j, |a i ∩ b j | = δ ij . Extension 2: For S 3 = A ∪ T ∪B as above, suppose {a 1 , ..., a g } is a collection of g primitive disks in A. One can define a homeomorphism from T to T g that takes each ∂a i to the meridian of one of the standard summands of T g . Lemma 1.7. All such homeomorphisms are Powell equivalent. Proof. Let {b 1 , ..., b g } and {b 1 , ..., b g } be different sets of orthogonal disks in B. For each 1 ≤ i ≤ g, let β i (resp β i ) denote the arc ∂b i − ∂a i (resp ∂b i − ∂a i ). Special Case: {b 1 , ..., b g } and {b 1 , ..., b g } can be isotoped (leaving the a i invariant) so that the collection of curves {β 1 , ..., β g } and {β 1 , ..., β g } are disjoint. Let P be the planar surface obtained by compressing T along {a 1 , ..., a g }. For each 1 ≤ i ≤ g β i ∪ β i forms a circle c i in P ; if any c i bounds a disk in P then the corresponding disks b i and b i can be isotoped to coincide; the proof now proceeds by induction on the size of the set S of indices for which this is true; we may as well take S = {1, ..., k}; when k = g we are done. Temporarily ignore the boundary components of P that correspond to all a i , i ≤ k and choose a circle c , > k that is then innermost. Then between ∂b and ∂b there are only 1-handle summands whose indices lie in S. Isotope these once around the curve a (a sequence of moves each corresponding to Powell generator D ν ). This isotopy (Powell equivalent to the identity) moves b to b , adding another index to S and completing the proof in this case. The general case is proven via induction on the number of components of b i ∩ b j , i, j ∈ {1, ..., g}. We can assume that all components of intersection are arcs and say that such an arc is outermost in b i , say, if it cuts off a subdisk of b i that contains neither another arc of intersection nor the point b i ∩a i . Choose an arc of intersection γ that is outermost in, say, b i and let b j be the other disk containing γ. Replace the subdisk of b j that is cut off by γ (on the side disjoint from the point b j ∩ a j ) by the outermost subdisk of b i cut off by γ. This converts {b 1 , ..., b g } to a disjoint family (so an equivalent family, per the Special Case) that intersects {b 1 , ..., b g } in fewer arcs, as required. It will later be shown (see Corollary 3.6) that the choice of just a single primitive disk determines the Powell equivalence type. Cautionary Note: Suppose {a 1 , ..., a g 1 } is a non-separating collection of disks in A and c is a simple closed curve c in T that separates T into one component containing the curves {∂a 1 , ..., ∂a g 1 } and the other a genus g 2 surface. Then c automatically bounds a disk in A; if c also bounds a disk in B then there is an orientation preserving homeomorphism h : (S 3 , T, c) → (S 3 , T g , c g 1 ),. The disks bounded by c in A and B divide each into handlebodies, so the two pieces into which T is divided are each Heegaard splittings of the 3-balls in which they lie. Moreover, if the Powell Conjecture is true in genus g 1 and genus g − g 1 , then any two such homeomorphisms are Powell equivalent. But there is no obvious reason why different choices of c will give Powell equivalent homeomorphisms, since not all braid moves on the a i (even if the a i are a primitive set) are Powell moves. For example, if we rotate a 1 around a ∂ reducing disk for B whose boundary lies on the genus g 2 side of c, c will be taken to a curve c that satisfies the same conditions. Yet it is not immediately apparent that such an isotopy of T to itself is a Powell move. Powell-like moves in a more general setting Following Lemma 1.4 and the remarks that precede it we can focus on four types of Goeritz elements, each of which is a Powell move, but less stringently defined than Powell's originals: (1) Let S ⊂ S 3 be a reducing sphere for T g bounding a ball B containing g 1 of the standard genus 1 summands. A standard bubble move is an isotopy of B through some path in T g − B that returns B, B ∩ T to itself, see Figure 3. (3) Let B 1 , B 2 be disjoint bubbles, each containing just a standard genus 1 summand, and let v ⊂ T g − (B 1 ∪ B 2 ) be an arc connecting them. Let B be the reducing ball obtained by attaching the 1-handle regular neighborhood of Figure 1 as Powell's move D η 12 . (4) Again let B 1 , B 2 be disjoint bubbles, each containing just a standard genus 1 summand. For i = 1, 2 let µ i ⊂ B be a meridian disk for T g ∩ B i and i ⊂ A be a longitudinal disk for T g ∩ B i . Let v ⊂ T g be an embedded arc connecting ∂µ 1 to ∂ 2 with the interior of v disjoint from µ 1 , µ 2 , 1 , 2 . Then µ 1 ∪ 2 ∪ v is called a standard eyeglass in T g see Figure 6. The disks µ and are called the lenses of the eyeglass, v is the bridge. The eyeglass defines is a natural automorphism (S 3 , T g ) → (S 3 , T g ) called an eyeglass twist which is supported on a 3-ball regular neighborhood of the eyeglass, see Figure 7. Powell's generator D θ is a standard eyeglass twist, in fact the model for this discussion. Each of these moves can be put in a more general setting. Suppose M is any compact oriented 3-manifold and T is a stabilized Heegaard Then all the definitions above make sense. More generally, if we drop the word 'standard' all the concepts make sense even when specific genus 1 summands have not been designated as standard. In fact, the notion of eyeglass, and so eyeglass twist, can often apply even when the splitting M = A ∪ T B is not stabilized. v to B 1 , B 2 . A standard switch is the homeomorphism (B, B ∩ T g ) → (B, B ∩ T g ) shown in Definition 2.1. An eyeglass is the union of two disks, a , b ( the lenses ) with an arc v (the bridge) connecting their boundaries. Suppose an eyeglass η is embedded in M so that the 1-skeleton of η (called the frame) lies in T , one lens is properly embedded in A, and the other lens is properly embedded in B. The embedded η defines a natural automorphism (M, T ) → (M, T ), as illustrated in Figure 7, called an eyeglass twist. Remark: It will be useful later to note that a ∂-compression of one of the lenses of an eyeglass η into T breaks the eyeglass twist around η into a composition of two eyeglass twists η 1 η 2 , where in each η i the ∂-compressed disk is replaced by one of the disks that is the result of the ∂-compression. See Figure 8. Weak reduction Recall ( [CG]) that a Heegaard splitting M = A ∪ T B is weakly reducible if there is an essential disk in A and one in B so that their boundaries are disjoint in T . Since the two lenses in an eyeglass are disjoint, it follows that only weakly reducible splittings can support nontrivial eyeglass twists. In this section we consider families of weakly reducing disks in Heegaard splittings of S 3 and describe conditions which guarantee that an eyeglass twist is a Powell move. Of course the Powell Conjecture would assert that any eyeglass twist is a Powell move. Definition 3.1. Suppose a non-empty collection {a 1 , ..., a m } of disks embedded in A is disjoint from a non-empty collection {b 1 , ..., b n } of disks in B. Then the pair of collections is called weakly reducing. If the complement in T of their boundaries {∂a 1 , ..., ∂a m } ∪ {∂b 1 , ..., ∂b n } consists of planar surfaces in T then the pair of disk collections is complete. If the complement is a single surface then they are called non-separating. Suppose {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g } is a complete non-separating weaklyreducing pair of disk collections for a genus g splitting S 3 = A ∪ T B. Let c ⊂ T be a simple closed curve that is disjoint from the boundaries of the two sets and separates one set from the other. Then c bounds a disk a c in A and a disk b c in B. The union a c ∪ b c is a sphere, dividing S 3 into two balls, and T intersects each ball in a Heegaard surface. Waldhausen's theorem applied to each then shows that there is an orientation preserving homeomorphism (S 3 , T, c) → (S 3 , T g , c g 1 ). Moreover, if the Powell Conjecture is true for genus g 1 and for genus g − g 1 splittings of S 3 then all such homeomorphisms are Powell equivalent. Consider the standard genus g Heegaard splitting S 3 = A ∪ Tg B, in which we denote the meridians of the standard summands by {a 1 , ..., a g } and the longitudes by {b 1 , ..., b g }. In particular |a i ∩ b j | = |∂a i ∩ ∂b j | = 1, 1 ≤ i, j ≤ g. The curve c g 1 ⊂ T g separates T g into two components, T A containing {∂a 1 , ..., ∂a g 1 } and T B containing {∂b g 1 +1 , ..., ∂b g }. Let P A (resp P B ) be the planar surface T A − {a 1 , ..., a g 1 } (resp T B − {b g 1 +1 , ..., b g }) and let P be the combined planar surface P A ∪ c P b . Lemma 3.2. Suppose η is an eyeglass in T g whose lenses consists of some a i , i ≤ g 1 and some b j , j ≥ g 1 + 1 and whose bridge v intersects c g 1 exactly once. Then an eyeglass twist along η is a Powell move. Proof. Observe that, after perhaps some conjugating Powell moves as described at the start of Section 1, Powell's generator D θ can be viewed as an eyeglass twist around the same lenses and a bridge v ⊂ P which also intersects c g 1 once. Now use Lemma 1.4 on the 1 handles in T B to slide them around T A so that afterwards v ∩T A coincides with v ∩T A ⊂ P A . Then symmetrically slide the 1-handles of T A around T B until the entire v = v ⊂ P . Lemma 3.3. Suppose η is an eyeglass in T g whose lenses consist of a disk a ⊂ A with ∂a ⊂ P A and a disk b ⊂ B with ∂b ⊂ P B . Suppose further that the bridge v intersects c g 1 exactly once. Then an eyeglass twist along η is a Powell move. Proof. As in the proof of Lemma 3.2 we can assume that the bridge v lies in P = P A ∪ P B . Then ∂a is coplanar with a number of boundary components of P A (informally, ∂a bounds a disk containing copies of some of the a i ). Noting the remark following Definition 2.1, we can view the twist around η as a composition of twists around each of these components of P A . Do the same for ∂b. The result follows now from Lemma 3.2. Lemma 3.4. Suppose η is an eyeglass in T g whose lenses consist of a disk a ⊂ A with ∂a ⊂ T A and a disk b ⊂ B with ∂b ⊂ T B . Suppose further that the bridge v intersects c g 1 exactly once. Then an eyeglass twist along η is a Powell move. Proof. As before, we may assume that v lies in P . The proof is by induction on the number of arc components of a ∩ (a 1 ∪ ... ∪ a g 1 ) and b ∩ (b g 1 +1 ∪ ... ∪ b g ) . If there are none, the result follows from Lemma 3.3. Otherwise choose an arc of intersection that is outermost in, say, a i . Use the arc to ∂-compress a, breaking up the twist around η into the twist around two eyeglass curves as illustrated in Figure 8. (If one of the new A lenses is inessential then v could have been extended past the ∂-compressing disk, replacing the arc of intersection by a new point of intersection of a i with v. This can be removed as usual.) The result now follows by induction. (But, cautionary note, one of the new eyeglasses has a bridge that intersects a i . Fortunately, having v disjoint from {a 1 , ..., a g 1 } is not part of our hypothesis but was achieved by argument. This condition is needed, else the boundary compression used in the argumentwould intersect v.) Here is an application: Proposition 3.5. Suppose b ⊂ B is a disk orthogonal to a 1 . Then there is a Powell move that leaves a 1 unchanged and carries b 1 to b. Proof. We can assume ∂b intersects a bicollar neighborhood Y ⊂ T g of ∂a 1 in an arc parallel to but disjoint from ∂b 1 ∩ Y . Special Case: b and b 1 are disjoint. Band ∂b to ∂b 1 together using one of the two bands they cut off from the bicollar Y , creating a new disk b + . Push the interior of this band into B to properly embed b + in B. Consider a 1 and b + as lenses of an eyeglass η whose bridge v is one of the small arcs (∂b ∩ Y ) − ∂a 1 . It is easy to see ( Figure 9) that an appropriate eyeglass twist of η will move b 1 to b. The eyeglass visibly satisfies the criterion required by Lemma 3.4 : Let c be the boundary of a regular neighborhood of ∂a 1 ∪b 1 . Then c bounds a punctured torus in T g (in fact the first standard summand) that contains a 1 , is disjoint from b + , and intersects v in a single point. See Figure 10. The general case now follows much as in the proof of Lemma 1.7, via induction on the number of components of b ∩ b 1 : Suppose b is a disk intersecting b 1 in n arcs and the proposition is true for disks that intersect b 1 in fewer than n arcs. Let γ be an arc of b ∩ b 1 that is outermost in b (with reference to the point b ∩ a 1 ) and let b o ⊂ b be the disk γ cuts off from b. Replace the disk in b 1 cut off by γ with b o to get a disk b orthogonal to a 1 that is disjoint from b 1 and intersects b in fewer than n arcs. By the special case just proven there is a Powell move ρ that carries b to b 1 and so carries b to a disk b intersecting ρ(b ) = b 1 in fewer than n arcs. By inductive assumption, there is a Powell move ρ carrying b" to b 1 . Then ρ ρ(b) = ρ (b ) = b 1 as required. Corollary 3.6. Suppose the Powell conjecture is true for genus g − 1 splittings of S 3 and T ⊂ S 3 is a genus g splitting. Then the choice of a single primitive disk a ⊂ A (or primitive b ⊂ B) defines a Powell equivalence class of homeomorphism (S 3 , T ) → (S 3 , T g ). Proof. Since a is primitive, there is a disk b ⊂ B whose boundary intersects the boundary of a in a single point. Choose a homeomorphism h : T → T g which carries the pair (a, b) to the pair (a 1 , b 1 ). By Proposition 3.5 the Powell equivalence class of h does not depend on b. The proof then follows by applying the inductive assumption to the genus g − 1 side of the reducing sphere S 1 for T g . Returning now to the general discussion, we drop the assumption that the a i and b j are primitive, but otherwise maintain the notation above. Suppose {a 1 , ..., a g 1 } and {b g 1 +1 , ..., b g } are a (not necessarily primitive) pair of non-separating weakly-reducing disk collections for T ⊂ S 3 . Let c ⊂ T be a simple closed curve that separates the two sets and η is an eyeglass in T whose lenses consists of a disk a ⊂ A with ∂a ⊂ T A and a disk b ⊂ B with ∂b ⊂ T B . Lemma 3.7. Suppose that the bridge v for η intersects c exactly once. Then an eyeglass twist along η does not change the Powell equivalence class of any homeomorphism h : (S 3 , T, c) → (S 3 , T g , c g 1 ). Proof. First note that there is such a homeomorphism: the hypothesis guarantees that c bounds a disk in both A and B and so is part of a reducing sphere for T . Then Waldhausen's theorem applied to both T A and T B provides a homeomorphism h. (This homeomorphism is unique up to Powell equivalence if the Powell Conjecture is true for lower genus splittings. We do not need that inductive assumption here.) Now apply Lemma 3.4. Theorem 3.8. Suppose {a 1 , ..., a g 1 } and {b g 1 +1 , ..., b g } are a (not necessarily primitive) pair of non-separating weakly-reducing disk collections for T ⊂ S 3 . Then • there is a homeomorphism h : T → T g so that {h(a 1 ), ..., h(a g 1 )} lie on the side of the separating sphere S g 1 that contains c 1 and {h(b g 1 +1 ), ..., h(b g )} lie on the other side and • If the Powell Conjecture is true for genus g 1 and genus g − g 1 splittings, any two such homeomorphisms are Powell equivalent. Proof. Choose any simple closed curve c ⊂ T that lies between the collections {∂a 1 , ..., ∂a g 1 } and {∂b g 1 +1 , ..., ∂b g }. As noted in the proof of Lemma 3.7 there is a homeomorphism h : (S 3 , T, c) → (S 3 , T g , c g 1 ) as required, and its Powell equivalence class depends only on the choice of c. The goal then is to show that the Powell equivalence class does not even depend on c. As before, let P be the connected planar surface T − {a 1 , ..., a g 1 ∪ b g 1 +1 , ..., b g }, so that c ⊂ P separates the 2g 1 components of ∂P corresponding to {∂a 1 , ..., ∂a g 1 } from the 2(g−g 1 ) components corresponding to {∂b g 1 +1 , ..., ∂b g }. Call the former component P A and the latter P B . Suppose c ⊂ P is another such simple closed curve. Picturing P as a 2g punctured sphere, a braid automorphism will move c to c. So we need only show that any braid automorphism ρ that moves the boundary components ∂P ∩P A back to themselves (and so the boundary components of ∂P ∩P B back to themselves) does not change the Powell equivalence class of h. It is a classic result that the "mixed" braid group B g 1 ,g 2 on the gpunctured sphere P A ∪ P B can be generated by a set of g 1 − 1 halftwists in P A (call this subgroup B a ), g 2 − 1, half-twists in P B (call this subgroup B b ) and a single full twist σ along a chosen arc γ connecting a specific component a 0 of ∂P A to a specific component b 0 of ∂P B . This is an eyeglass twist with lenses a 0 and b 0 and bridge γ. Choose γ to be an arc crossing c once. Clearly the subgroups B a and B b commute. The proof that ρ ∈ B g 1 ,g 2 is Powell equivalent to one that lies in B a × B b proceeds by induction on the number n σ , of occurrences of σ in ρ when expressed as a product of these generators. If σ does not appear, then c is preserved by the braid. With no loss the initial segment of ρ can be written αβσ, where α ∈ B a and β ∈ B b . That is, ρ = αβσω, where ω ∈ B g 1 ,g 2 has one less occurence of σ. Then ρ = (αβσβ −1 α −1 )αβω. But it is easy to see that (αβσβ −1 α −1 ) is an eyeglass twist along an eyeglass whose bridge is αβ(γ), an arc that still crosses c once. The proof then follows from Lemma 3.7. To state the implication of Theorem 3.8 a little less formally: Corollary 3.9. If the Powell Conjecture is true for genus g 1 and genus g − g 1 , then a pair {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g } of non-separating weaklyreducing disk collections for T ⊂ S 3 determines a Powell equivalence class of homeomorphisms h : T → T g . In fact, only one set of disks is needed, so long as we know it has at least one complementary set: Theorem 3.10. Suppose the collection {b g 1 +1 , ..., b g } of compressing disks in B can be extended to two possibly different non-separating complete collections {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g } and {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g } of weakly reducing disks for T . If the Powell Conjecture is true for genus g 1 and genus g − g 1 , then the Powell equivalance classes determined by each extension via Theorem 3.8 are the same. Proof. Let H ⊂ S 3 be the genus g 1 handlebody obtained from A by attaching 2-handles along {b g 1 +1 , ..., b g }. Special Case: {a 1 , ..., a g 1 } and {a 1 , ..., a g 1 } are disjoint. We first claim that, in this case, we can proceed from one family to the other by a sequence of substitutions of a single disk at a time. This is obvious (indeed there is nothing to prove) if each a j is parallel to one of the a i , so we induct on the number of a j that are not parallel to any a i . With no loss of generality, say a 1 is not parallel to any a i . Since the set {a 1 , ..., a g 1 } is non-separating in H, H − {a 1 , ..., a g 1 } is a ball containing the disjoint collection of properly embedded disks {a 1 , ..., a g 1 }. Each a i gives rise to a pair of disks (called twins) on the boundary of the ball. Since H − {a 1 , ..., a g 1 } is connected, there is at least one a i , say a 1 , which is parallel to no a j in H and whose twins in the boundary of the ball lie on opposite sides of a 1 . Then replacing a 1 by a 1 changes only a single disk in {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g } and provides a non-separating complete collection {a 1 , a 2 , ..., a g 1 }, {b g 1 +1 , ..., b g } with more disks parallel in A to disks in {a 1 , ..., a g 1 }, completing the inductive step. So, following the claim, we may as well assume that a i = a i for all 2 ≤ i ≤ g 1 . Let W be the solid torus obtained from H by deleting each a i = a i , 2 ≤ i ≤ g 1 . The remaining disks a 1 , a 1 are disjoint meridian disks of W , dividing it into two cylindrical components W ± , each topologically a ball. Now consider the g−g 1 properly embedded arcs {β g 1 , ..., β g } in W , each arc β j dual to the disk b j ⊂ B. If, say, W − contains none of the β j , then all the β j lie in W + . Then there is a curve c ⊂ ∂A ∩ W + that separates {∂a 1 , ∂a 1 , ..., ∂a g 1 } from {∂b g 1 +1 , ..., ∂b g }. Then in S 3 , c bounds disks a c ⊂ A and b c ⊂ B, and so provides a reducing sphere a c ∪ b c for the splitting T that determines, as shown in Theorem 3.8, the same Powell equivalence class for the pair of weakly compressing collections {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g } as it does for {a 1 , a 2 ..., a g 1 }, {b g 1 +1 , ..., b g }, completing the argument in this case. (In effect, {a 1 , ..., a g 1 } and {a 1 , ..., a g 1 } are merely different choices of complete collections of ∂reducing disks for A in the side of the sphere a c ∪ b c on which they lie). The remaining possibility in this special case is that some of the β j lie in each of W ± . There is a curve c ⊂ T that separates {∂a 1 , ..., ∂a g 1 } from {∂b g 1 +1 , ..., ∂b g } and intersects ∂a 1 in two points. (So c bounds a disk a c ⊂ A intersecting the disk a 1 in a single arc γ.) Via Theorem 3.8 this determines a Powell equivalence class for the pair of weakly compressing collections {a 1 , ..., a g 1 }, {b g 1 +1 , ..., b g }. The union of a component of a c − γ and a component of a 1 − γ cuts off a bubble containing all the arcs β j lying in, say, W − . A bubble move around a longitude of ∂W will push a 1 into a new position so that all the β j lie in W + , where the previous argument applies. The bubble itself may not be standard in the given Powell equivalence class but, invoking the inductive assumption that the Powell Conjecture is true for genus g − g 1 , the summand contained in the bubble may be made standard without changing the Powell equivalence class. Then the bubble move itself does not change the Powell equivalence class, completing the proof in this special case. General Case The general case now proceeds classically, by induction on |{a 1 , ..., a g 1 } ∩ {a 1 , ..., a g 1 }|, the number of arcs in which the two systems of A disks intersect. Consider an outermost arc of intersection in a 1 , say, cutting off an outermost disk D ⊂ a 1 . . With no loss assume the arc also lies in a 1 . The correct choice of subdisk D ⊂ a 1 , when attached to D along the arc of intersection will give a disk a D ⊂ A so that {a D , a 2 , ..., a g 1 } also satisfies the hypotheses of the theorem. Since a D is disjoint from a 1 it follows from the special case above that {a D , a 2 , ..., a g 1 } and {a 1 , a 2 , ..., a g 1 } determine the same Powell equivalence class. Since |a D ∩ a 1 | < |a 1 ∩ a 1 | (and no other pairs of disks have an increased number of intersection arcs) the inductive hypothesis implies that {a D , a 2 , ..., a g 1 } and {a 1 , a 2 , ..., a g 1 } determine the same Powell equivalence class. Towards a proof of the Powell Conjecture 4.1. The philosophy. For the purposes of this section, let T 0 denote a copy of the standard genus g ≥ 2 Heegaard surface T g in S 3 . Here is the philosophy behind our (only partially successful) strategy to prove the Powell Conjecture. Start with an element of the Goeritz group, represented by a path τ θ : S 3 → S 3 , 0 ≤ θ ≤ 2π in Diff(S 3 ) that starts out as the identity and has τ 2π (T 0 ) = T 0 . For brevity denote each surface τ θ (T 0 ) ⊂ S 3 by T θ . We would like to find such a representative τ so that for some 0 = θ 0 < θ 1 < θ 2 < .... < θ n < 2π (1) for each θ / ∈ {θ i , 1 ≤ i ≤ n} we can extract information from the surface T θ ⊂ S 3 sufficient to determine a Powell equivalence class of trivializations h θ : (S 3 , T θ ) → (S 3 , T g ). (2) ensure that the information is unchanged throughout each interval in [0, 2π] − {θ i , 1 ≤ i ≤ n} so that h θ τ θ : (S 3 , T 0 ) → (S 3 , T g ) is a well-defined Powell equivalence throughout each interval (3) for each 1 ≤ i ≤ n ensure that the information for θ i − gives the same Powell equivalence class as the information for θ i + so that h θ τ θ : (S 3 , T 0 ) → (S 3 , T g ) is a well-defined Powell equivalence class as we move from one interval to the next. It would follow then that for all 0 ≤ θ ≤ 2π, h θ τ θ : (S 3 , T 0 ) → (S 3 , T g ) are Powell equivalent. In particular h 0 τ 0 is Powell equivalent to h 2π τ 2π . But since the Powell equivalence class of h θ is determined entirely by the surface T θ , and T 2π = T 0 , it follows that we may take h 2π = h 0 so that τ 0 (the identity) is Powell equivalent to τ 2π , as required. Although this is the philosophy, the outcome of our argument is not so neat. Sadly, the information we will be able to extract does not rise to the level (as exhibited, say, in Corollary 3.9 above) that is sufficient to determine Powell equivalence class, at least as far as we have been able to determine. (But it does suffice for the case of genus 3, see Section 5 below.) 4.2. The complex 2C(T ). First we describe the topological information that we will extract. Thinking of C(T ) as the curve complex of T = T 0 , define a 1-complex 2C(T ) as follows: the vertices are disjoint ordered pairs of simple closed curves in T (so each corresponds to an edge in C(T ), with an orientation). The 1-simplices in 2C(T ) are of two types: pairs of pairs ((a, b), (a, b )) with the property that the curves {a, b, b } are pairwise disjoint and pairs of pairs ((a, b), (a , b)) where the curves (a, a , b) are pairwise disjoint. (So we can think of each edge as a 2-simplex in C(T ) in which two of the edges have been oriented to share a head or a tail.) We speak of "shuffling" between (a, b) and (a, b ) and between (a, b) and (a , b). Now let T θ = τ θ (T 0 ), θ ∈ S 1 be a parameterized Heegaard surface in S 3 representing an element of the Goeritz group, as described above. By a circle γ of weak reductions (cwr) representing T θ we mean an edge path γ t , 0 ≤ t ≤ 2π in 2C(T ) so that • each vertex on γ t is a pair (a, b) where a compresses in A and b compresses in B, • γ 2π = τ 2π (γ 0 ), where τ 2π acts via the natural action of M CG(T ) on 2C(T ). We will show that any such parameterization τ θ can be deformed into one that somewhat naturally presents a cwr representing τ θ . Specifically, after the deformation, there will be successive values θ 1 , θ 2 , ...., θ n ∈ (0, 2π) so that for each θ / ∈ {θ i , 1 ≤ i ≤ n} there is a pair of weakly reducing disks (a θ , b θ ) associated to the topological surface T θ ⊂ S 3 so that • the isotopy class of the pair (a θ , b θ ) is unchanged throughout each interval in S 1 − {θ i , 1 ≤ i ≤ n}, • for each 1 ≤ i ≤ n the pair (a θ i + , b θ i + ) differs from the pair (a θ i − , b θ i − ) by a shuffle Technically, the cwr is then the pull-back of this sequence of disk pairs under the parameterization (S 3 , T g ) → (S 3 , T θ ) that defines T θ . Before we show how to construct the cwr, we describe how a proof of the following conjecture would then lead to a proof of the Powell Conjecture. Conjecture 4.1. There is a method of associating to any vertex (a, b) in 2C(T ) for which a bounds a disk in A and b bounds a disk in B, a Powell equivalence class of homeomorphisms h (a,b) (S 3 , T ) → (S 3 , T g ) with these properties: • The method is topological. That is, for any homeomorphism σ : (S 3 , T ) → (S 3 , T ) and vertex (a, b) ∈ 2C(T ), h (σ(a),σ(b)) σ = h (a,b) and • the Powell equivalence class associated to the ends of any edge in 2C(T ) are the same. Combining Conjecture 4.1 with the construction that precedes it, we get a sequence of homeomorphisms h (a 0 ,b 0 ) , h (a 1 ,b 1 ) , ..., h (an,bn) : T → T g which all have the same Powell equivalence class and for which (a n , b n ) = (τ 2π (a 0 ), τ 2π (b 0 )). This implies that h (a 0 ,b 0 ) τ 2π is Powell equivalent to h (τ 2π (a 0 ),τ 2π (b 0 )) τ 2π = h (a 0 ,b 0 ) so τ 2π is Powell equivalent to the identity, as required. 4.3. The Rieck background and its refinement. Recall the classical sweep-out technology applicable to any Heegaard splitting of a closed 3-manifold M = A ∪ T B (see [RS]): Pick a spine in each handlebody A, B, that is a 1-complex X in A (say) whose complement is homeomorphic to T × (0, 1), with T × {1} corresponding to ∂A. This gives rise to a mapping cylinder structure on A, A ∼ = ∂A × [0, 1]/(a × 0) ≡ f (a), some f : ∂A → X. In its classical application, the mapping cylinder structures on A and B can be combined to parameterize the entire complement of the spines in M as T × (−1, 1) describing how most of M is swept-out by copies of T . In the case that M = S 3 there is another natural sweep-out (actually the genus 0 version of the sweep-out just described). Viewing S 3 as the standard 3-sphere in 4-space, a height function in R 4 describes a sweep-out of S 3 from south-pole to north-pole. That is, we can view S 3 also as S 2 × [−1, 1] with S 2 × {0} crushed to the south pole and S 2 × {1} crushed to the north pole. In [R] (based on arguments in [RS]) Rieck proves Waldhausen's theorem by comparing these two "sweep-outs" of S 3 by surfaces, one parameter s ∈ [−1, 1] for the sweep out by spheres S and one parameter t ∈ [−1, 1] for the sweep-out by a genus g ≥ 2 Heegaard surface T . A "graphic" Γ of this (t, s)-square is analyzed to find a weak reduction of T , that is a pair of compressing disks, one in A and the other in B, whose boundaries are disjoint in T . Then [CG] implies that the Heegaard splitting is reducible, and that finishes the proof by induction. The graphic consists of open regions R i where S s and T t intersect transversely, edges or "walls" where the two have a tangency, and cusp points where two types of tangencies cancel. As argued in [RS] only domain walls corresponding to saddle tangencies need to be tracked. Cusps and tangencies of index 2 or 0 can be erased as they amount only to births/deaths of inessential simple closed curves of intersection in S s ∩T t . The most interesting event which occurs are transverse crossings of saddle walls; at this point two independent saddle tangencies occur. It will be useful to very briefly review Rieck's analysis of this graphic: Each region R i ⊂ I × I\Γ is labeled by I or E: I if every component of S s ∩ T t , (t, s) ∈ R i is inessential in T , E otherwise. Call an I region green, labeled I g , if T t ∩ ( s ≥s S s ) ⊂ T t \disks, that is most of T t lies north of S s . Similarly we call an I region yellow, labeled I y , if T t ∩ s <s S s ) ⊂ T t \ disks, that is most of T t lies south of S s . [R, Lemma/Definition 3.4] asserts that every I-region is labeled exactly once as I g or I y . Furthermore, [R,Proposition 3.3] states that no two I regions bearing different color labels can touch, even at a corner. (This is where requiring genus g ≥ 2 comes in; see for example Lemma A.4 in the Appendix.) It is clear that for s sufficiently near -1, (t, s) must lie in an I g region (since most of T t lies north of S s ) and that for s sufficiently near +1, (t, s) must lie in an I y region. It follows that between and I g and I y region there is an entire strip of I × I, running from the side t = −1 to the side t = 1, in which each region is labelled E. Moreover, when t is near −1, T t is close to the spine of A, so the essential curves bound disks in A and when t is near +1, T t is close to the spine of B, so the essential curves bound disks in B. It follows that there must be some region (or two adjacent regions) in such a strip in which both sorts of disks occur, and this provides a weak reduction. Rieck's result can be refined. In the Appendix A we show that the lowest of the strips that appear in Rieck's argument is actually monotonic in t. By this we mean that there is a function r : [−1, 1] → [−1, 1] whose graph lies in the 1-skeleton of the reduced graphic and, for small , {(t, r(t) + )}|t ∈ [−1, 1]} lies entirely inside the lowest strip. To put it another way, the graph of r + in I × I is transverse to the reduced graphic and, for any (t, s) lying in a region that the graph intersects, among the circles in S s ∩ T t , there is at least one circle that is essential in T t . This fact has no real importance in Rieck's argument, but its analogue in our context will be quite useful (though not mathematically essential). 4.4. Adding the parameter θ. The fundamental idea now will be to add a third parameter to Rieck's proof, namely the parameter θ. We first need to show that the choice of spines in A and B above is unimportant, even in the general context of a closed manifold M = A ∪ B. One can think of a spine for A (or B) as a 1-complex X ⊂ int A together with a dual cell structure on A: a proper disk associated with (and normal to) each 1-simplex of X and a 3-ball containing each vertex of X. But such a presentation of the spine involves no choice in the sense that its space of parameters is contractible. The argument for this has two parts: It follows from [Mc] that the choice of disks defining the spine is unimportant since the disk space of A is contractible. (Here the disk space is the simplicial complex whose n-cells are n + 1 pairwise disjoint properly embedded disks, which together divide A into balls.) Once these disks are chosen, Hatcher has shown that the exact placement of the disks in A, and then the ensuing parameterization of the complementary 3-balls, also involves no choice. Putting together these results it follows that once a Heegaard surface T ⊂ M is determined, M is foliated by a family of surfaces M = t∈[−1,1] T t which degenerate at t ∈ {−1, +1}. Here T 0 is of course the original Heegaard surface T = ∂A = ∂B. Moreover, this foliation is canonical, up to choices coming from a weakly contractible parameter space. The disk complex is contractible and the corresponding function spaces whose simplicial realization is the disk complex is therefore weakly contractible. Now add the circular parameter θ ∈ [0, 2π]/0 ≡ 2π and consider the family of Heegaard surfaces T θ ⊂ S 3 . Choose spines for A 0 and B 0 , propagate them along θ by isotopy until at 2π − we see that the initial and final spines are not matching up. However using just the π 1 -aspect of "canonical" above we may isotope the spines to match at 2π = 0. This yields a θ-parameter family of singular foliations T t,θ , (t, θ) ∈ [−1, 1] × [0, 2π]/0 ≡ 2π := I × S 1 . Theorem 4.2. Let T θ be a loop of genus g Heegaard surfaces, g ≥ 2 as described above. There is a circle γ of weak reductions (cwr) representing T θ . Proof. We study the 3-parameter family T t,θ ∩ S s , (t, s, θ) ∈ [−1, 1] × [−1, 1] × [0, 2π]/0 ≡ 2π := [−1, 1] × [−1, 1] × S 1 . These intersections may be regarded as the level set at height = s of a 2-parameter family of functions on T . All told this means we are looking for a 3-parameter family of germs of smooth functions (R 2 , 0) → (R, 0). According to Thom's Jet-transversality Theorem any such family can be perturbed a generic one with local simulation types as listed below. In these dimensions the generic local germs 2 [HW] are represented by: 1. f (x, y) = x codim = 0 2a. f (x, y) = x 2 + y 2 source / sink codim = 1 b. f (x, y) = x 2 − y 2 saddle codim = 1 3a. f (x, y) = x 3 + y 2 birth/death of (1, 2)-handle pair codim = 2 b. f (x, y) = x 3 − y 2 birth/death of (1, 2)-handle pair codim = 2 4a. f (x, y) = x 4 + y 2 dovetail singularity codim = 3 b. f (x, y) = x 4 − y 2 dovetail singularity codim = 3 ( ) These are the germs. We also may invoke genericity to ensure that where a single function on T restricts to singular germs on several different points of T , the local unfoldings are transverse. These is no difficulty making this precise as the stratification of the unfoldings obey the two Whitney conditions [W] are are in fact piecewise smooth submanifolds of the parameter space. So what does the singular set of a graphic in [−1, 1] × [−1, 1] × S 1 look like? It consists of 2D-manifold sheets of two types -saddle and sink -meeting along birth/death 1D strata, which are allowed to have dovetail singularities at isolated points. The 1D and 2D strata may pass transversely through each other at points and the 2D strata may have transverse 1-manifolds of intersection and isolated standard triple points. The picture we must study is much simpler. Just as in [RS], the 2D sheets labeled by source/sink tangencies may be discarded forming the reduced graphic, as crossing these walls only changes the intersection T ∩S by inessential simple closed curves. The remaining saddle-labeled walls (it does not matter for connectivities, but for convenience take their closures) divide [−1, 1] × [−1, 1] × S 1 into complementary open regions {R i }, each of which has a constant topological pattern T ∩ S, up to birth and death of inessential sccs. Thus the 3D reduced graphic G 3 is a straightforward generalization of the 2D one: 2-manifold sheets (perhaps with borders) crossing in double curves and triple points. 4.5. The graphic on an annular surface. After having defined the 3D reduced graphic, we now replace it with a 2D graphic by restricting it to the sub-surface Σ ⊂ [−1, 1] × [−1, 1] × S 1 , where Σ is the graph of the function described in Proposition A.5. Its structure as a graph provides Σ with a natural diffeomorphism to an annulus [−1, 1] × S 1 , parameterized by (t, θ), and that is how we will view it. This structure as a graph allows us, when thinking of the intersection T t,θ ∩ S s that corresponds to a point in Σ, to take the sphere as fixed, so henceforth we drop the subscript s. (When there is little risk of confusion, we will also drop the subscript on T .) Each region of the graphic in Σ corresponds to a transverse intersection of T t,θ with S in which some of the curves of intersection are essential in T t,θ and bound disks in A t,θ or B t,θ , perhaps both. For regions near t = −1, all such disks lie in A t,θ (since T is near the spine of A t,θ , which we may take transverse to the height function s) and we label these regions A. Similarly, near t = +1, all such disks lie in B t,θ and we label such regions B. Let us establish the following labeling and "artistic" convention for the general region R of our annular graphic. If T t,θ ∩ S contains a scc a essential in T t,θ and compressing in the handlebody A t,θ label the region containing (t, θ) by A. (Do this regardless of whether T t,θ ∩ S contains an essential scc compressing in B t,θ ). If T t,θ ∩ S does not contain any essential scc compressing in A t,θ , but does contain a scc compressing in B, label that Region B. Artistic convention: If labels A and B alternate around a saddle wall double point render the boundary between A-regions and B-regions to be a 1-manifold favoring A. We label this 1-manifold M , it separates A regions from B regions. We recall both this labeling and rendering convention with the phrase: "favor A." Since near one boundary component of the annulus Σ the regions are all labelled A and at the other they are all labelled B, there is at least one component of M that is essential in Σ. Such a component will be homotopic in Σ, fixing a point, to a core curve {pt} × S 1 ⊂ [−1, 1] × S 1 . Here then is the plan for the proof of Theorem 4.2. We will show The crux is to understand the intersection locus where two saddle walls cross, what the four possible resolutions of the two saddles look like, and which curves among these resolution could be labeled a or b. (Again a curve is labeled a (resp b) if it is essential in T and compresses in A (resp B).) The singular pattern ξ associated with a saddle wall crossing is present simultaneously in the sphere S and the surface T . Thus the pattern ξ as a 4-valent graph is the same in S and T . Furthermore the tangential information is also preserved so at a saddle tangency we know which legs are opposite (this is not quite the information of a cyclic ordering; it might be called a "dihedral ordering" as the information is a coset of S(4)/D(4)). But interestingly, it is not true that the pairs (N S (ξ), ξ) ⊂ S and (N T (ξ), ξ) ⊂ T are necessarily homeomorphic. (Here N S and N T denote neighborhood in S and T respectively.) The important example is Figure 12; it is a torus of revolution intersected with a sloping plane to produce two circles of intersection. In the plane they have one positive and one negative intersection, whereas in the torus they have two positive intersections. So, overall, on T we have 5 cases to consider: 1, 2, 3, 4, and 5 above. Figure 15 displays the four local resolutions within N (ξ) of ξ in each of the 5 cases. Most of M ⊂ Σ runs along arcs in the graphic, representing saddle walls for which one side is labelled A and the other is labelled B. A neighborhood of the corresponding saddle is a 3-punctured sphere and crossing the saddle wall splits one curve into two: c → c Π c . We start with a simple lemma. Proof. Consider capping off two boundaries of the 3-punctured sphere. Arrows indicate sccs that must intersect Of course the third curve may be inessential in T , so crossing from a region labelled A to one labelled B may represent two curves bounding essential disks in A fusing at the saddle into a curve inessential in T , leaving other unrelated curves bounding disks in B. (In this case the two fused curves at A are isotopic in T .) Or passing through could represent a curve that bounds a disk in A fusing with a curve that is essential in both A and B to create a curve bounding a disk in B, or a similar fission. In each case there is a unique essential intersection curve a bounding a disk in A associated with the saddle wall and it is disjoint from every essential intersection curve bounding a disk in B. To expedite the discussion we introduce: Definition 4.4. Some conventions: • We call a scc a (resp b) if it is essential in T and compresses in A (resp B). • An edge-path in 2C(T ) in which each vertex is of type (a, b) (as just defined) is an admissible path. • Near a saddle wall crossing (swc) point p we call a scc of type a or b local if it is a scc of one of four resolutions of ξ P . Otherwise we call it far. With these conventions, we have just observed that any saddle wall that is part of M is associated to a 'cloud' of vertices in 2C(T ) of the form (a, b), with a uniquely defined, and b any curve of that type coming from the adjacent graphic region labelled B. Any two vertices in such a cloud are connected via the relation (a, b) → (a, b ) since a, b, b are all pairwise disjoint. Recalling our artistic rendering convention for defining the 1-manifold M , we now consider how these clouds of vertices in 2C(T ) are related as we pass from one saddle wall to another, either through or around a corner of a vertex in the graphic representing a saddle wall crossing. Lemma 4.5. For all cases (i) A A B B , (ii) A B B A , (iii) A A A B , and (iv) A B B B there is an admissible path in 2C(T ) from some vertex in one cloud to some vertex in the other. Proof. Since each pair of vertices in a cloud are connected by an edge in 2C(T ), and such an edge is obviously an admissible path, it suffices to replace both occurences of 'some' with 'every' in its two occurences in the lemma. Case (i) A A B B : If any b (or any b ) is far it will be unaffected by passing through any saddle wall, so pick b = b = far. Moreover, passing through the saddle wall between the regions A and A will change the a curve to a disjoint a . Then (a, b) → (a , b) = (a , b ) is an admissible path from one cloud to the other. If b and b are both local there is a subtlety. In all cases at most one diagonal pair of quadrants contain intersecting curves (see Figure 15); call this a dangerous diagonal. But the appropriate choice of a twostep process avoids the dangerous diagonal, so all curves are disjoint and the path is admissible: (a, b) (a, b ) (a , b) or (a , b ) Case (ii) B A B A : It suffices to consider one arc of M . Regardless of whether b is local or far leave it fixed. The only problematic possibility for a transition (a, b) → (a , b) is if a and a are intersecting curves from a dangerous diagonal. In this context case (5) cannot occur since there a and a would have intersection number = 1. But where it is possible, cases (2), (3), and (4), the topology is the same: a and a are crossing non-boundary-parallel sccs on a 4-punctured sphere: Figure 16. But an innermost circle argument reduces the geometry to precisely the case pictured in Figure 16, where we see that all four boundary curves also compress in A. By Lemma 4.3 at most two of these boundary curves are inessential in T ; let ∂ be a boundary curve that is essential in T . Then (a, b) → (∂, b) → (a , b) is an admissible path from one cloud to the other. (Note that the intermediate vertex (∂, b) is not in either cloud.) Call this the lantern trick. Case (iii) A A A B : This is similar to case (ii). Regardless of whether b is local or far, leave it fixed. The only problematic case in passing from (a, b) to (a , b) is if a and a are intersecting sccs from a dangerous diagonal. As in case (ii), case (5) cannot occur. Cases (2) Case (5): This case cannot occur: By intersection number, the three B regions cannot contain a dangerous diagonal pair, so A must be one of the dangerous diagonal pair of quadrants. Then the other quadrant in that pair must be B . The other two quadrants then are represented by the boundary curves in the twice-punctured torus shown in Figure 15, one compressible in A and the other in B. In order to avoid a label A in one of these quadrants, the former curve must be inessential in T so the twice-punctured torus is actually only once-punctured. But in that case, the label B (or B ) for this quadrant implies that there is a (far) b curve, contradicting assumption. We now complete the proof of Theorem 4.2 as described in the plan above. Pick an essential component of the 1-manifold M in Σ and a saddle wall edge in it. Beginning at that edge and moving around the component of M , construct an admissible path through 2C(T ) as described above, until one returns to the original edge. Since the component is homotopic in I × I × S 1 , rel its beginning and ending point, to a simple loop {pt} × S 1 which represents τ , the cwr given by the admissible path also represents τ . The genus 3 case We are now in a position to prove the Powell Conjecture for genus 3 splittings of S 3 , by verifying Conjecture 4.1 in this case. We need a bit of terminology. Suppose H is a genus 3 handlebody and D ⊂ H is a separating disk. Then D divides H into a solid torus H D and a genus 2 handlebody; the meridian disk D of H D is welldefined up to proper isotopy in H and is non-separating in H; call D the surrogate of D in H. Lemma 5.1. Suppose S 3 = A ∪ T B is a genus 3 splitting, and disks a ⊂ A, b ⊂ B are a weakly reducing pair. Then at least one of the following 4 disks is primitive: • a, if a is non-separating in A, • the surrogate of a if a is separating in A • b, if b is non-separating in B • the surrogate of b if b is separating in B Proof. Case 1: a is separating. One possibility in this case is that ∂b lies on the boundary of the solid torus A a ⊂ A. In that case, if b is non-separating, then ∂b must be a longitude of A a , verifying that the surrogate of a (and incidentally also b) is primitive. If b is separating, then ∂b must be parallel to ∂a in T , so together a and b constitute a reducing sphere for T , cutting off a genus 1 Heegaard splitting whose splitting surface is ∂A a . It follows again that the surrogate of a (i. e. the meridian of A a ) is primitive (and incidentally that the surrogate of b is also primitive). The other possibility is that ∂b is essential in the punctured genus 2 component of T − ∂a. This implies that the surrogate of b (if b is separating) is disjoint from the surrogate of b, so we may as well jump immediately to Case 2. Case 2: Both a and b are non-separating. Let T ⊂ S 3 be the torus obtained by simultaneously compressing T along a and b. T bounds a solid torus W on one side or the other, say W lies on the side in which b lies. The surface T obtained from T by compressing only along a, when pushed into W , describes a genus 2 Heegaard splitting W = A ∪ T B in which ∂W = ∂ − B and b is the unique boundaryreducing disk for the compression body B . It follows that b is primitive in T , hence in T . Following Lemma 5.1 and Corollary 3.6 there is an obvious candidate for assigning a Powell equivalence class of homeomorphisms to the pair (a, b): if a (or its surrogate) is primitive, assign the class of homeomorphisms that carries that disk to the disk a 1 in the standard splitting A ∪ T 3 B. If b (or its surrogate) is primitive, assign the class of homeomorphisms that carries that disk to b 3 in the standard splitting. It is only necessary to check that if two of the disks listed in Lemma 5.1 are primitive, the assignments we have made coincide. So suppose both a (or its surrogate) and b (or its surrogate) are primitive. If both a and b are non-separating (so surrogates are not involved) and therefore primitive, a straightforward outermost arc argument will find a disk b ⊥ ⊂ B that is orthogonal to a and disjoint from b. Similarly, one can then find a disk a ⊥ ⊂ A that is dual to b and is disjoint from both a and b ⊥ . There is then an obvious homeomorphism (S 3 , T ) → (S 3 , T 3 ) (well-defined up to the 2-strand braid group of the torus) that takes the four disks a, b ⊥ , b, a ⊥ to, respectively a 1 , b 1 , b 3 , a 3 . The Powell class of this homeomorphism then coincides both with that determined by a and that determined by b. There remains the possibility that one or both primitive disks are surrogates of a or b and intersect, in which case they are orthogonal. There is then a homeomorphism (S 3 , T ) → (S 3 , T 3 ) that carries one disk to a 1 and the other one to b 1 . But a further Powell move (an exchange) carries the pair to a 3 and b 3 . The first homeomorphism is in the Powell equivalence class determined by a or its conjugate and the second by b or its conjugate. Since there is a Powell move taking one to the other, the equivalence classes coincide. The second and final step is to show that any two vertices in 2C(T ) that are connected by an edge determine the same Powell equivalence class. Given the simple rule above for assigning a Powell equivalence class, this will follow immediately from the following lemma. Lemma 5.2. For the genus 3 Heegaard splitting S 3 = A ∪ T B, suppose a ⊂ A, b 1 , b 2 ⊂ B are three pairwise disjoint essential disks . Then either • a is primitive • a is separating and its surrogate is primitive • b 1 and b 2 are parallel in B or • b 1 is the surrogate of b 2 or vice versa. The symmetric statement is true for three pairwise disjoint essential disks (a 1 , a 2 , b) Proof. Case 1: a is separating. If the boundary of either b i lies on the same side of a as a then the disk is either orthogonal to a or the disk together with a is a reducing sphere cutting off a genus 1 summand with a as meridian. In any case, a is primitive and we are done. So we henceforth assume that the boundary of both b i lie on the genus 2 side of T − ∂a, and neither is parallel to ∂a. If, together, b 1 and b 2 separate T then the component containing ∂a is either a torus in which a is a meridian, and so is primitive, or a genus 2 surface. In the latter case, the other component must be attached either by a single disk that cuts off a torus with the other disk as meridian (so one of b 1 , b 2 is surrogate to the other) or attached by two disks, one copy of each b i , that cut off a sphere, in which case b 1 and b 2 are parallel in B. Case 2: a is non-separating. One of the components of B − (b 1 ∪ b 2 ) contains ∂a and has connected boundary; attaching a 2-handle to that component along a creates a 3-manifold N that will still be connected. Also ∂N is still connected, for if it were disconnected, with boundary components ∂ 1 N, ∂ 2 N , each containing a copy of a, then there would be a path through that component of B − (b 1 ∪ b 2 ) from ∂ 1 N to ∂ 2 N . But there is already a path from ∂ 1 N to ∂ 2 N in T − a, since a is nonseparating. Together the paths would give a circle that intersects the surface ∂ 1 N in a single point, which is impossible in S 3 . So let γ ⊂ ∂N be a path from one copy of a in ∂N to the other. Tubing the copies of a together along the boundary of a neighborhood of γ gives a separating disk a + in A, disjoint from the b i , and a is the surrogate for a + . Now apply case 1 to the three disks a + , b 1 , b 2 . Corollary 5.3. The Powell Conjecture is true for genus 3 splittings. Proof. Suppose two vertices in 2C(T ) are connected by an edge. With no loss of generality, the two vertices are (a, b 1 ) and (a, b 2 ) with the three disks a, b 1 , b 2 pairwise disjoint. In each case given by Lemma 5.2 the procedure described for assigning a Powell equivalence class assigns the same equivalence class to both cases. Concluding remarks The methods here seem to hold great promise for the general case, genus g ≥ 4. But there are many difficulties: Presented with a pair of weakly reducing disks a ⊂ A, b ⊂ B, the methods of Casson-Gordon ( [CG]) describe how to proceed to exhibit a complete non-separating weakly reducing collection of disks a 1 , ..., a g 1 , b g 1 +1 , ..., b g for T . But many choices are involved in creating such a collection, and it is not clear that different choices will lead, via Theorem 3.8, to the same Powell equivalence class of homeomorophisms to T g , even under strong inductive assumptions. To begin with, the first step in the Casson-Gordon recipe, following the choice of a, b, is to use the pair to maximally weakly reduce the splitting. But this involves choices: for example, one could choose first to expand a to a maximal collection of compressing disks in A that are disjoint from b. (Or one could do the reverse!) The resulting compression body lying in A may itself be well-defined, via the arguments of [Bo], but there are many possible families of compressing disks for the compression body, and, to apply Theorem 3.8, a specific collection of compressing disks in A is required. The next step in [CG] is to maximally compress the surface T A b just defined towards the side that contains B. The same difficulty arises: the actual compression disks into the B side that one uses involves a choice. On the positive side, the resulting surface T BA b , which is well-defined, can be shown to bound a handlebody M BA b in which T A b is a Heegaard surface, providing perhaps an inductive foothold. Furthermore, Powell-like moves (bubble-moves, eyeglass twists, etc) in the Heegaard surface of T A b ⊂ M BA b can be 'shadowed' by similar moves in the original splitting surface T . But again choice is involved in how to shadow the moves. Also, even once a primitive disk is chosen for the splitting T A b ⊂ M BA b (inductively we might assume that this choice is unique up to Powell-like moves), such a disk does not pick out a unique primitive disk in our original surface T . All along one could hope to show that different choices made lead to Powell equivalent trivializations, but we have failed to find such arguments. Perhaps a good intermediate question is this: Conjecture 6.1. Any eyeglass twist on T g is a Powell move. Of course any eyeglass twist is a Goeritz element, so this conjecture is weaker than the Powell Conjecture. An affirmative solution would bode well for an affirmative solution of the Powell Conjecture as well. A weaker but perhaps still challenging conjecture is this: Conjecture 6.2. Any eyeglass twist on T g is Goeritz conjugate to a Powell move. That is, given an eyeglass twist ρ : (S 3 , T g ) → (S 3 , T g ) there is a homeomorphism h : (S 3 , T g ) → (S 3 , T g ) so that h −1 ρh : (S 3 , T g ) → (S 3 , T g ) is a Powell move. Observe the following cautionary example about eyeglass twists: In the argument of Section 4.5 a crucial weakly reducing pair that may well appear can be viewed as follows: One of the pair, b ⊂ B say, is an innermost disk of B ∩ S for some level sphere S; b lies inside another disk D that is also bounded by a curve of S ∩ T , a curve to which b is tangent. Finally, the complement of b in D lies entirely in A and becomes the disk a when the point of tangency is resolved. See Figure 17. b a ∂D Figure 17. Now imagine rolling the disk b around the inside of D so that the point of tangency circumnavigates ∂D. It is easy to see that this is a Goeritz move and only a bit harder to see that it is an eyeglass twist of b around a. Yet, the methods of Section 4.5 would provide the same pair of disks a, b throughout the Goeritz move. This suggests that, as it stands, Section 4.5 is not in itself powerful enough to resolve even the weaker Conjecture 6.1. Other tools may be needed to solve the problem for at least some types of eyeglass twists. In that spirit, we conclude with one rather limited class of eyeglass twists which we are able to show are Powell moves. These are described using the notation that precedes Lemma 3.7. Definition 6.3. An eyeglass η ⊂ T satisfying the following conditions will be called a short eyeglass. • There is a complete non-separating pair of weakly reducing disk collections {a 1 , ..., a g 1 } and {b g 1 +1 , ..., b g } for T ⊂ S 3 . • There is a simple closed curve c ⊂ T that separates the two sets {∂a 1 , ..., ∂a g 1 } and {∂b g 1 +1 , ..., ∂b g } in T . • The lens a ⊂ A of η has ∂a ⊂ T A . • The lens b ⊂ B of η has ∂b ⊂ T B . • The interior of the bridge v ⊂ η intersects at least one of T A or T B entirely in P A or P B . Proposition 6.4. Suppose η ⊂ T is a short eyeglass as above, with c ⊂ T the separating curve. Then an eyeglass twist along η does not change the Powell equivalence class of any homeomorphism h : (S 3 , T, c) → (S 3 , T g , c g 1 ). The central difference between this lemma and Lemma 3.7 is that we no longer require the bridge v to intersect c in exactly one point; on the other hand, we do require that the bridge of the eyeglass be disjoint either from {∂a 1 , ..., ∂a g 1 } or from {∂b g 1 +1 , ..., ∂b g } Proof. The proof is by induction on |v ∩c|; the case |v ∩c| = 1 is Lemma 3.7. Suppose with no loss of generality that v ∩T A ⊂ P A and that |v ∩c| > 1. Consider the arcs of v ∩ P A that are not incident to ∂a. Each such divides P A into two sub-planar surfaces, one of which cuts off a subarc of c intersecting v in an even number of points. By picking an outermost such arc we may find a properly embedded segment v A ⊂ v that cuts off a planar surface whose interior is disjoint from v. The union of v A and the subarc c A ⊂ c it cuts off is a simple closed curve a c which bounds a disk in A. Create an eyeglass η , one of whose lenses is a c , the other is b ⊂ B and whose bridge intersects c in at least two fewer points. By induction, a twist around η does not change the Powell equivalence class of h, and it vacates entirely everything in P A that lies between v A and c A , putting it on the other side of v A . Then v A can be isotoped across c, reducing |v ∩ c|. Proof. The proof (not the underlying mathematics) is made a bit more complicated because we we do not know when G r is trivial. Pick any z 0 ∈ Z and any > 0. Let r = r(z 0 ). south of the sphere S s . We then have a function r : [−1, 1] → [−1, 1] with the property that for values of s < r(t), none of the first homology of T t,(−∞,s] persists into all of T t , but for any values of s > r(t) some does. It follows that the topology of T t,(−∞s] changes as s rises through r(t), so the graph of r(t) must lie in the 1-skeleton of the reduced graphic. We can say more: For a generic value of t, the point (t, r(t)) will lie in an edge of the reduced graphic, so at that point there is a saddle tangency of T t with S r(t) . That is, T t,(−∞,r(t)+ ] is obtained from T t,(−∞,r(t)− ] by attaching a 1-handle. At a non-generic value of t, the point (t, r(t)) will be a vertex in the graphic. In that case, one can get from the region just below (t, r(t)) to the region just above by passing through two edges in the graphic, so T t,(−∞,r(t)+ ] is obtained from T t,(−∞,r(t)− ] by attaching two 1-handles. By definition of r(t), T t,(−∞,r(t)− ] corresponds to a subsurface of T t whose first homology is trivial in all of T t . In particular, T t,(−∞,r(t)+ ] contains no non-separating simple closed curves i. e., T t,(−∞,r(t)+ ] is planar. We now invoke an easy lemma: Lemma A.4. Suppose P is a compact possibly disconnected planar surface and P + is the surface obtained from P by attaching no more than three 1-handles. Then either genus P + ≤ 1 or some boundary component of P is non-separating in P + . Proof. If any of the 1-handles has its ends attached to different components of P , it is wasted: The resulting surface P is still planar, and the new boundary component created is non-separating in P + if and only if at least one of the boundary components to which it is attached is non-separating in P + . So we could juat replace P with P in the argument, but now with one fewer handle to attach. So we may as well assume that all 1-handles are attached to the same component of P and so we can take P to be connected. In that case, if any 1-handle has exactly one end on any component of ∂P then that component becomes non-separating, finishing the argument. So we may as well assume that no 1-handle has a single end on an original boundary component of P . Now imagine attaching the 1-handles sequentially, say h 1 , h 2 , h 3 . The first handle h 1 must have both ends on the same component of ∂P so the result is still a planar surface P , with exactly two components of ∂P not appearing as components of ∂P . If, after attaching h 2 , the surface is still planar, attaching h 3 can raise the genus to at most 1, completing the proof. So we need to have the attachment of h 2 raise the genus, so it must connect the two new boundary components of P . The resulting surface P now has genus 1 and only a single boundary component not among the components of ∂P . It follows that both ends of h 3 must lie on the same component of ∂P , either the new one or one of the original components of ∂P . In any case, attaching h 3 does not raise the genus any further, completing the proof. Following Lemma A.4, if every boundary component of T t,(−∞,r(t)+ ] is inessential in T t then genus(T t ) ≤ 1. So under the hypothesis that genus(T ) ≥ 2 the function r(t) + for small enough will be the function we seek: its graph in [−1, 1] × [−1, 1] will lie entirely in the regions labelled E. The argument is readily enlarged to include the third parameter θ. That is, Proposition A.5. In the parameter space [−1, 1] × [−1, 1] × S 1 as described in Section 4.3, there is a function r : I × S 1 → I so that the graph (t, r(t, θ) + , θ) ⊂ [−1, 1] × [−1, 1] × S 1 of r + is transverse to the graphic and intersects only those regions of the graphic in which S s ∩ T t,θ contains circles that are essential in T t,θ . Proof. The only additional case to consider is that in which (t, r(t, θ), θ) is a (codimension 3) vertex in the graphic. But in that case, the region just below the vertex can be connected to the region just above the vertex by passing through three saddle "walls". Thus Lemma A.4 is still applicable, now with n = 3. Figure 1 . 1Powell's proposed generators (from [Po]) Figure 2 . 2Figure 2. Figure 3 . 3Standard bubble move (2) Let B be a bubble containing just a single standard genus 1 summand. A standard flip is the homeomorphism (B, B ∩T ) → (B, B ∩ T ) shown in Figure 1 as Powell's move D ω . Figure 4 . 4Standard flip Figure 5 . 5Standard switch Figure 6 . 6Standard eyeglass Figure 7 . 7Eyeglass twist splitting, with some genus 1 summands of T designated as standard. Figure 8 . 8Figure 8. Figure 9. Figure 10. •Figure 11 . 11Each component of M in Σ naturally describes an edge-path in 2C(T ) through vertices (a, b) in which the first curve bounds an essential disk in A and the second an essential disk in B • For any component of M that is essential in Σ the associated edge-path is a cwr representing τ . Figure 12 .Figure 13 .Figure 14 . 121314For our analysis we need to list all possible pairs (N (ξ), ξ) up to homeomorphism. Let us start by enumerating the possibilities on S.These four possible ξs can yield neighborhood pairs on T as shown inFigure 14. (The small circles in the figures represent boundary components of the neighborhood, typically essential circles in T . Lemma 4 . 3 43(two of three). If crossing a saddle wall transforms c → c Π c and if two of the three curves compress in A (resp B) then so does the third. Figure 15 . 15Figure 15. , (3), and (4) are also handled as in case (iii) by the lantern trick. Case (iv) B A B B : If any b, b , b is far, select it and keep it constant, and keep a constant as well. Then the corresponding vertex in 2C(T ) lies in both clouds Now assume all b, b , b are local. Case (1): If a, b, b , b are all disjoint, then (a, b) → (a, b ) is an admissible path. Case (2), (3), (4): The only non-disjoint situation is when b and b are intersecting curves from a dangerous diagonal pair. Again use the lantern trick. The Z 2 distinction (unimportant for our purposes) is best illustrated by Powell's generator D η inFigure 1. It has order g when viewed as a diffeomorphism (S 3 , T g ) → (S 3 , T g ) but is of order 2g when viewed as an isotopy of S 3 to itself. All other germ types such as hyperbolic umbilic, f (x, y) = x 3 + y 3 , have codimensions ≥ 4 and need not be considered. Notice that G r may or may not be trivial. For an example where G r is trivial, let X be the closure of an -neighborhood of the Polish circle C in the plane, and f (x) be the distance from x to C. Then r = 0 and G r = H 1 (C) = 0. Appendix A. Refining theRieck argumentWe begin with a well-known result. Suppose X is a compact topological space and f : X → R is continuous. Then f has a unique minimal value r ∈ R. Now let Z be a topological space, R X denote the space of continuous functions from X to R with the compact-open topology and consider a function f : Z → R X . Denote the image of any z ∈ Z by f z : X → R. Then f gives rise to a function r : Z → R by defining r(z) = min{f z (x)|x ∈ X}.Proposition A.1. If f is continuous then so is r.Proof. The standard proof is this: Since X is compact, the compactopen topology coincides with the topology of uniform convergence, in which the conclusion is obvious. There is also a proof directly from the definition of compact-open topology that is analogous to (but much easier than) the proof of Proposition A.2 below.Notice that the function r could equivalently (but more obscurely) be defined this way: r(z) is the least r ∈ R so that the image ofViewed in this way, the proposition generalizes. In particular, we will be interested in an analogous statement for H 1 .For this version, assume also that H 1 (X) is non-trivial. Then given a continuous function f : X → R and a value y ∈ R, let G y ⊂ H 1 (X) be the image under the inclusion-induced map of H 1 (f −1 (−∞, y]) → H 1 (X). There are obvious properties:• for y below the minimum of f (X), G y is trivial, • y ≤ y =⇒ G y ⊂ G y , and • for y above the maximum of f (X), G y = H 1 (X) = 0.So it makes sense in this situation to define a valueWith this definition then, in analogy to the discussion above, a function f : Z → R X gives rise to a function r : Z → R by setting r(z) to be the value r just defined for the function f z . That is,Since f is continuous, so is r.Claim 1:There is an open neighborhood N of z 0 so that z ∈ N =⇒ r(z) < r + 2 .Proof of Claim 1: First observe that f −1 z 0 (−∞, r + ] is closed in X, hence compact, so {z ∈ Z|f z (f −1 z 0 (−∞, r + ]) ⊂ (−∞, r + 2 )} is an open neighborhood N of z 0 . Put another way, for z ∈ N ,is non-trivial, and this factors through H 1 (f −1 z (−∞, r + 2 )) → H 1 (X), so the image of the latter is non-trivial. Hence r(z) < r + 2 as required.Claim 2:There is an open neighborhood M of z 0 so that z ∈ M =⇒ r(z) > r − 2 .Proof of Claim 2: Paralleling the argument in Claim 1, there is an open neighborhood M of z 0 so that for z ∈ M ,→ H 1 (X) has trivial image; it follows that H 1 (f −1 z (−∞, r − 2 ]) → H 1 (X) has trivial image, so r(z) > r − 2 as required.Hence for z ∈ M ∩N , |r(z)−r(z 0 )| < 2 , showing that r is continuous at z 0 . Now return to the setting of Section 4.3: T is a genus g ≥ 2 Heegaard surface in S 3 and T t , t ∈ [−1, 1] describes a sweep-out of S 3 = A ∪ T B from a spine of A to a spine of B. Similarly S s , s ∈ [−1, 1] describes the sweep-out of S 3 by 2-spheres, from the south pole no the north pole of S 3 . The intersection patterns of the two surfaces give rise to a reduced graphic Γ ⊂ [−1, 1] × [−1, 1] for which a complementary region R is labelled E if and only if for each (t, s) ∈ R, T t ∩ S s contains a circle that is essential in T t . Proposition A.3. There is a function r : [−1, 1] → [−1, 1] whose graph {(t, r(t))} ⊂ [−1, 1] × [−1, 1] is transverse to Γ and each region that the graph intersects is labelled E.Proof. Apply Proposition A.2 to X = T and Z = [−1, 1] by taking for f t the height function s restricted to T t ⊂ S 3 . Denote by T t,(−∞s] the subsurface of T t that lies in ∪ s ≤s S s ., that is, in the part of S 3 that is A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. E Akbas, Pacific J. Math. 236E. Akbas, A presentation for the automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Pacific J. Math. 236 (2008) 201-222. Braids, links and mapping class groups. J Birman, Annals of Mathematics Studies. 82Princeton University PressJ. Birman, Braids, links and mapping class groups, Annals of Mathematics Studies 82. Princeton University Press, 1974. Cobordism of automorphisms of surfaces. F Bonahon, Ann. Sci. cole Norm. Sup. 16270F. Bonahon, Cobordism of automorphisms of surfaces. Ann. Sci. cole Norm. Sup. 16 (1983) 237?270. Reducing Heegaard splittings. A Casson, C Mca, Gordon, Topology and its applications. 27A. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology and its applications, 27 (1987), 275-283. Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two. Cho, Proc. Amer. Math. Soc. 13611131123Cho, Homeomorphisms of the 3-sphere that preserve a Heegaard splitting of genus two, Proc. Amer. Math. Soc. 136 (2008) 11131123 Die Abbildungen der Brezelfläche und der Volbrezel vom Gesschlect 2. L Goeritz, Abh. Math. Sem. Univ. Hamburg. 9L. Goeritz, Die Abbildungen der Brezelfläche und der Volbrezel vom Gess- chlect 2, Abh. Math. Sem. Univ. Hamburg 9 (1933) 244-259. . A , A.. Homeomorphisms of sufficiently large P 2 -irreducible 3-manifolds. Hatcher, Topology. 11Hatcher, Homeomorphisms of sufficiently large P 2 -irreducible 3- manifolds, Topology 11 (1976) 343-347. . A , A.. . A Hatcher, Proof Of The Smale Conjecture, Annals of Mathematics. 117Hatcher, A proof of the Smale Conjecture, Annals of Mathematics, 117 (1983) 553-607. . A Hatcher, J Wagoner, Pseudo-isotopies of compact manifolds. 6AstérisqueA.. Hatcher and J. Wagoner, Pseudo-isotopies of compact manifolds, Astérisque, 6 (1973) 1-275. The space of Heegaard splittings. J Johnson, D Mccullough, J. Reine Angew. Math. 679J. Johnson and D. McCullough, The space of Heegaard splittings, J. Reine Angew. Math. , 679 (2013), 155-179. Virtually geometrically finite mapping class groups of 3-manifolds. D Mccullough, J. Differential Geom. 33D. McCullough. Virtually geometrically finite mapping class groups of 3- manifolds, J. Differential Geom. , 33 (1991) 1-65. Homeomorphisms of S 3 leaving a Heegaard surface invariant. J Powell, Trans. Amer. Math. Soc. 257J. Powell, Homeomorphisms of S 3 leaving a Heegaard surface invariant, Trans. Amer. Math. Soc. 257 (1980) 193-216. A proof of Waldhausen's uniqueness of splittings of S3 (after Rubinstein and Scharlemann). Y Rieck, Workshop on Heegaard Splittings. Coventry12Geom. Topol. Publ.Y. Rieck, A proof of Waldhausen's uniqueness of splittings of S3 (after Ru- binstein and Scharlemann). Workshop on Heegaard Splittings, 277-284, Geom. Topol. Monogr., 12, Geom. Topol. Publ., Coventry, 2007. Comparing Heegaard splittings of non-Haken 3-manifolds. H Rubinstein, M Hyam, Scharlemann, Topology. 35H. Rubinstein, Hyam and M. Scharlemann, Comparing Heegaard splittings of non-Haken 3-manifolds. Topology 35 (1996), 1005-1026. Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting. M Scharlemann, Bol. Soc. Mat. Mexicana bf. 10M. Scharlemann, Automorphisms of the 3-sphere that preserve a genus two Heegaard splitting, Bol. Soc. Mat. Mexicana bf 10 (2004) 503-514. H Whitney, Local properties of analytic varieties. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse. Princeton, N. J.Princeton Univ. PressH. Whitney, Local properties of analytic varieties. Differential and Combi- natorial Topology (A Symposium in Honor of Marston Morse) pp. 205-244, Princeton Univ. Press, Princeton, N. J., 1965. The Powell Conjecture and sphere complexes. A Zupan, to appearA. Zupan, The Powell Conjecture and sphere complexes, to appear. 93106-3080 USA E-mail address: [email protected], and, Mathematics Department. Santa Barbara, CA; Santa Barbara, CAMicrosoft Station Q, University of California ; University of CaliforniaMicrosoft Station Q, University of California, Santa Barbara, CA 93106-6105, and, Mathematics Department, University of California, Santa Barbara, CA 93106-3080 USA E-mail address: [email protected] 93106-3080 USA E-mail address: mgscharl@math. Santa Barbara, CAMartin Scharlemann, Mathematics Department, University of Californiaucsb.eduMartin Scharlemann, Mathematics Department, University of Cali- fornia, Santa Barbara, CA 93106-3080 USA E-mail address: [email protected]

It has been recently demonstrated that the 125 GeV Higgs boson can mediate a long-range force between TeV-scale particles, that can impact considerably their annihilation due to the Sommerfeld effect, and hence the density of thermal relic dark matter. In the presence of long-range interactions, the formation and decay of particle-antiparticle bound states can also deplete dark matter significantly. We consider the Higgs boson as mediator in the formation of bound states, and compute the effect on the dark matter abundance. To this end, we consider a simplified model in which dark matter co-annihilates with coloured particles that have a sizeable coupling to the Higgs. The Higgs-mediated force affects the dark matter depletion via bound state formation in several ways. It enhances the capture cross-sections due to the attraction it mediates between the incoming particles, it increases the binding energy of the bound states, hence rendering their ionisation inefficient sooner in the early universe, and for large enough couplings, it can overcome the gluon repulsion of certain colour representations and give rise to additional bound states. Because it alters the momentum exchange in the bound states, the Higgs-mediated force also affects the gluon-mediated potential via the running of the strong coupling. We comment on the experimental implications and conclude that the Higgs-mediated potential must be taken into account when circumscribing the viable parameter space of related models.

10.1007/jhep04(2019)130

1901.10030

90a48fee8339cac4a17bfb062208854ede709097

Prepared for submission to JHEP Higgs-mediated bound states in dark-matter models 28 Jan 2019 Julia Harz [email protected] Physik Department T70 Technische Universität München James-Franck-Straße85748Garch-ingGermany Laboratoire de Physique Théorique et Hautes Energies Sorbonne Université CNRS LPTHE F-75252ParisFrance Sorbonne Université Institut Lagrange de Paris (ILP) 98 bis Boulevard Arago75014ParisFrance Kalliopi Petraki [email protected] Laboratoire de Physique Théorique et Hautes Energies Sorbonne Université CNRS LPTHE F-75252ParisFrance Nikhef Science Park 1051098 XGAmsterdamThe Netherlands Prepared for submission to JHEP Higgs-mediated bound states in dark-matter models 28 Jan 2019 It has been recently demonstrated that the 125 GeV Higgs boson can mediate a long-range force between TeV-scale particles, that can impact considerably their annihilation due to the Sommerfeld effect, and hence the density of thermal relic dark matter. In the presence of long-range interactions, the formation and decay of particle-antiparticle bound states can also deplete dark matter significantly. We consider the Higgs boson as mediator in the formation of bound states, and compute the effect on the dark matter abundance. To this end, we consider a simplified model in which dark matter co-annihilates with coloured particles that have a sizeable coupling to the Higgs. The Higgs-mediated force affects the dark matter depletion via bound state formation in several ways. It enhances the capture cross-sections due to the attraction it mediates between the incoming particles, it increases the binding energy of the bound states, hence rendering their ionisation inefficient sooner in the early universe, and for large enough couplings, it can overcome the gluon repulsion of certain colour representations and give rise to additional bound states. Because it alters the momentum exchange in the bound states, the Higgs-mediated force also affects the gluon-mediated potential via the running of the strong coupling. We comment on the experimental implications and conclude that the Higgs-mediated potential must be taken into account when circumscribing the viable parameter space of related models. Contents Introduction The nature of dark matter (DM) is one of the biggest open questions in modern particle physics. An intriguing possibility is that the Higgs boson participates in the physics of the dark sector. The discovery of the Higgs and the measurement of its mass impel the thorough investigation of its implications for various DM scenarios. One of the most generic scenarios for the cosmological production of DM asserts that DM arose from the thermal freeze-out of non-relativistic particles that were previously in equilibrium with the Standard Model (SM) plasma in the early universe. This possibility necessitates a sizeable coupling between the SM and DM, and is thus being probed by a variety of direct, indirect and collider experiments. The increasingly more stringent constraints have pushed the viable DM mass range in this class of models toward and beyond the TeV regime, and have given impetus to co-annihilation scenarios [1][2][3][4][5], which provide more flexibility in reproducing the observed DM density via freeze-out. If the 125 GeV Higgs couples to a (multi-)TeV dark sector, then the difference between the two mass scales may give rise to long-range effects. Indeed, in models where DM or its co-annihilating partners couple to a much lighter force mediator, the resulting longrange force distorts the wavepackets of the interacting pairs, giving rise to the well-known Sommerfeld effect [6,7] that alters their annihilation rate [8,9]. Long-range interactions imply also the existence of bound states; the formation and decay of particle-antiparticle bound states opens an additional two-step annihilation channel that can deplete the DM abundance [10,11] and contribute to the DM indirect detection signals [12][13][14][15][16]. While such effects emanating from hidden-sector mediators, the electroweak gauge bosons, or the gluons in co-annihilation scenarios, have been considered , only recently the longrange effect of the Higgs was pointed out [40]. The reasoning for disregarding the latter appears to be twofold: the Higgs boson was considered too heavy to yield a long-range force, and the coupling of the Higgs to DM was assumed to be smaller than (or at best comparable to) the SM gauge couplings. However, in co-annihilation scenarios, the mass of DM or its co-annihilating partners can be in the multi-TeV regime and their coupling to the Higgs can be sizeable, such that the range of the Higgs-mediated interaction becomes comparable to or exceeds the Bohr radius of the interacting pair of particles. Then, the interaction manifests as long-range. Reference [40] demonstrated the significant impact of the Higgs-generated potential on the annihilation cross-sections due to the Sommerfeld effect, and consequently on the DM density. In the present paper, we investigate the impact of the Higgs-mediated force on the existence and formation of bound states, and the associated effect on the DM abundance. For definiteness, we shall consider the simplified model of ref. [40], where DM is assumed to co-annihilate with a coloured scalar that transforms under the fundamental of SU (3) c and possesses a sizeable coupling to the Higgs. This scenario is encountered in the Minimal Supersymmetric SM (MSSM) -with the coloured particle being typically the lightest stop eigenstate -and constitutes one of the last refuges of neutralino DM; as such, it has received a lot of attention recently [2,41,42], with the effect of radiative bound-state formation (BSF) due to gluon exchange computed in [37]. We shall focus on the DM mass range (0. 5 -5) TeV. The lower bound of this interval largely ensures viability against current experimental constraints, while the upper bound implies that DM freezes-out after the electroweak symmetry breaking (EWSB), when the neutral component of the Higgs doublet becomes the mediator that couples to the DM co-annihilating partners. Importantly, this mass interval encompasses the range probed at the LHC and the range of highest interest for the resolution of the hierarchy problem. In fact, retaining the supersymmetry-breaking scale relatively low such that the hierarchy problem is better addressed, and reproducing the measured Higgs mass at the same time, is possible if the coupling of the stops to the Higgs is sizeable. Altogether, this scenario contains the necessary ingredients for the Higgs-mediated force to have an important effect. Similar features are present also in various Higgs-portal models (see e.g. [3,43]), for which we expect our findings to have important implications. In particular, the impact of the Higgs-mediated potential on the DM abundance affects the prediction of the DM couplings and mass. This in turn changes the interpretation of the experimental results and affects the viability of specific scenarios. The viability of models with long-range interactions with respect to indirect detection constraints is rather sensitive to the predicted DM mass, since the expected γ-ray spectrum from annihilations inside galaxies exhibits sharp parametric resonances with varying DM mass. Moreover, thermal-relic DM models are probed at precision collider experiments. The measurement of the DM abundance to an unprecedented precision, Ωh 2 = 0.120 ± 0.001, by the Planck satellite [44] yields a powerful constraint that can meaningfully complement those from colliders and indirect detection. This however requires that the theoretical computation of the relic density is also sufficiently accurate. During the last couple of years, important progress has been made along this direction. While state-of-the-art numerical tools [45][46][47] have employed so far mainly leading order calculations, it has been demonstrated that higher order corrections to the annihilation processes can impact the DM density up to 20% [48][49][50][51]. The Sommerfeld effect and BSF due to the SM gauge bosons can have an even greater impact [37]. As already shown in [40] and we shall see again in the following, the Higgs-mediated force can have a comparable or larger effect that far exceeds the experimental precision in the measurement of the DM density. The paper is organized as follows. In section 2, we describe the simplified model, first introduced in [40], and review the calculation of the DM relic density, including the formation and decay of bound states. We then discuss the scattering and bound states in a mixed Coulomb and Yukawa potential in section 3. In section 4, we compute the BSF cross-sections. Since several different momentum scales enter the calculation, in section 4.4 we discuss the running of the strong coupling and explore its effect on the cross-sections. In section 5, we present our results. We discuss the impact of the Higgs boson on the existence and formation of bound states, and on the effective annihilation cross-section, before demonstrating its effect on the predicted DM density. We conclude in section 6. For easy reference, we summarize the notation used throughout the paper in table 1. s, [8] in annihilation and BSF vertices (singlet, octet) α ann s and α BSF s, [1] , α BSF s, [8] Gluon-generated Coulomb potential coupling in the scattering states (singlet, octet) Description α S g,[1] ≡ (4/3)α S s , α S g,[8] ≡ −α S s /6 in the bound states (singlet, octet) α B g,[1] ≡ (4/3)α B s,[1] , α B g,[8] ≡ −α B s,[8] /6 Bohr momentum of the bound states (R =1,8) κ [R] ≡ µ(α B g,[R] + α h ) Expectationd h ≡ µα h /m h λ [R] ≡ α B g,[R] /α h ζ h ≡ α h /v rel ζ S g,[R] ≡ α S g,[R] /v rel ζ B g,[R] ≡ α B g,[R] /v rel ζ S,[R] ≡ (α h + α S g,[R] )/v rel ζ B,[R] ≡ (α h + α B g,[R] )/v rel Binding energy of n m bound state E n = −γ 2 n (λ, d h ) × κ 2 /(2µ) Kinetic energy of scattering state in CM frame E k = k 2 /(2µ) Wavefunction of n m bound state ψ n m (r) = κ 3/2 χ n (κr) κr Y m (Ω r ) Wavefunction of scattering state φ k (r) = ∞ =0 (2 + 1) χ |k|, (kr) kr P (k ·r) Dimensionless radial space coordinates for bound and scattering states x B ≡ κr and x S ≡ kr Dimensionless time parameters for freeze-out x ≡ m χ /T,x ≡ m X /T 2 Simplified model and relic density Simplified model We assume that DM is a Majorana fermion χ with mass m χ , that co-annihilates with a complex scalar X with mass m X . X is a triplet under SU (3) c and couples to a real scalar h of mass m h = 125 GeV, that aims to resemble the SM Higgs boson. Both χ and X are odd under a Z 2 symmetry, and assumed to be the lightest (LP) and next-to-lightest particles (NLP). The Lagrangian is [40] δL = (D µ,ij X j ) † (D µ ij X j ) − m 2 X X † j X j + 1 2 (∂ µ h)(∂ µ h) − 1 2 m 2 h h 2 − g h m X h X † j X j , (2.1) where D µ,ij = δ ij ∂ µ + ig s G a µ T a ij is the covariant derivative with G a µ being the gluon fields and T a the corresponding generators. We denote the LP-NLP relative mass splitting as ∆ ≡ (m X − m χ )/m χ ,(2.2) and define the total and the reduced mass of an XX † pair, M = 2m X and µ = m X /2. (2.3) We also define the couplings that will appear in the non-relativistic potential in section 2.3 α s ≡ g 2 s /(4π), (2.4) α h ≡ g 2 h /(16π). (2.5) Let us briefly comment on the origin of the trilinear term h|X| 2 . In the corresponding dimensionful coupling, we have factored out m X for convenience. This factorization does not imply that the coupling originates solely from the quartic term |H| 2 |X| 2 ⊃ |H 0 | 2 |X| 2 , with H and H 0 being the Higgs doublet and its neutral component, whose radial excitation after the EWSB is h. The h|X| 2 term may be also sourced by a gauge-invariant trilinear coupling HX † 1 X 2 , where X 1 and X 2 have different SU (2) L × U (1) Y charges and mix after the EWSB to generate the mass eigenstate X. This dynamics is in fact encountered in the MSSM, where X 1 and X 2 would be the two stop fields that couple to the Higgs via a SUSY-breaking A term. Moreover, the X, X 1 , X 2 fields -being scalars -may also have SU (2) L × U (1) Y invariant mass terms. In order to remain agnostic about the underlying UV complete model, we shall treat g h and m X as independent parameters that are not delimited by one another, or by the Higgs vacuum expectation value, v H 246 GeV. Here, we are interested in the long-range effect of the Higgs, which emanates from the h|X| 2 coupling. Thus for simplicity, we do not include quartic terms in the scalar potential, or the couplings of the Higgs to the SM particles. Boltzmann equation for the relic density If ∆ 1, the DM abundance is determined not only by the χ − χ annihilations, but also by the χ − X, χ − X † and X − X † (co)annihilation processes. Following [52], we defineỸ to be the sum over the yields of all (co)annihilating particles, Figure 1. The gluon and Higgs exchange are the leading order contributions to the 2-particleirreducible (2PI) kernel that gives rise to the long-range interaction between X and X † . Y ≡ Y χ + Y X + Y X † = Y χ + 2Y X , (2.6) 2PI = g + h where Y i is the ratio of the number density n i over the entropy density of the universe s ≡ (2π 2 /45) g * S T 3 . We denote by g * S and g * the entropy and energy degrees of freedom respectively, and define g 1/2 * ,eff ≡ g * S √ g * 1 + T 3g * S dg * S dT . (2.7) The evolution of the DM density is described by the Boltzmann equation dỸ dx = − π 45 m Pl m χ g 1/2 * ,eff x 2 σ eff v rel (Ỹ 2 −Ỹ 2 eq ). (2.8) As usual, we have defined the dimensionless time parameter x ≡ m χ /T . The yields in equilibrium of χ, X and X † are (g χ = 2, g X = 3) Y eq χ = 90 (2π) 7/2 g χ g * S x 3/2 e −x , (2.9a) Y eq X = Y eq X † = 90 (2π) 7/2 g X g * S [(1 + ∆)x] 3/2 e −(1+∆)x . (2.9b) While σ eff v rel in general denotes the effective, thermally averaged cross-section that includes all annihilation and co-annihilation processes, each weighted by the number densities of the interacting species, we assume for the purpose of our study that the NLP self-annihilation is the dominant contribution. Then, the effective cross-section becomes σ eff v rel = 2Y eq X Y eq X † σ XX † v rel Ỹ 2 eq = σ XX † v rel 2g 2 X (1 + ∆) 3 e −2x ∆ g χ + 2g X (1 + ∆) 3/2 e −x ∆ 2 , (2.10) where σ XX † v rel ≡ σ ann v rel + σ BSF v rel eff (2.11) includes all processes that deplete the XX † pairs. It comprises not only of the direct annihilation channels, but also of the BSF processes, whose cross-sections must be weighted by the fraction of bound states that decay rather than being ionised, as we discuss below. Non-relativistic potential In the presence of a long-range interaction between two particles, the asymptotic states are not well approximated by plane waves. In order to determine their properties, we must resum the two-particle-irreducible (2PI) diagrams (see e.g. [11]). In our model, the X, X † pairs interact via gluon and Higgs exchange, both of which may result in a sizeable longrange effect [40]. In the non-relativistic and weak coupling regime, the resummation of the gluon and Higgs exchange diagrams shown in fig. 1 amounts to solving the Schrödinger equation with the mixed Coulomb and Yukawa potential V (r) = − α g r − α h r e −m h r , (2.12) where α h has been defined in eq. (2.5) and derived in ref. [15]. The coupling α g is related to α s defined in eq. (2.4) but depends also on the colour representation of the XX † state. The product of a colour triplet and an anti-triplet decomposes into a singlet and an octet state, 3 ⊗3 = 1 ⊕ 8, such that the coupling α g amounts to [53] α g, [1] ≡ α s, [1] × C F = (4/3)α s, [1] , (2.13a) α g,[8] ≡ α s,[8] × (C F − C A /2) = −α s,[8] /6, (2.13b) for the singlet and the octet configurations, where C F = 4/3 and C A = 3 are the quadratic Casimir invariants of the fundamental and adjoint representations of SU (3). In eqs. (2.13), we have differentiated between the strong coupling in the singlet and octet states, α s, [1] and α s, [8] respectively, to accommodate the possibly different momentum transfer along the 2PI diagrams, which affects the value of α s due its running. For the massive mediator h, the interaction manifests as long-range if the range of the potential m −1 h is comparable to or larger than the corresponding Bohr radius (µα h ) −1 . To parametrise the range of the interaction we use the dimensionless ratio d h ≡ µα h /m h . (2.14) The Sommerfeld effect and BSF become significant when the average momentum transfer between the scattering particles, µv rel , is comparable to or lower than the Bohr momentum, µα. To compare the relevant scales in our set-up, we introduce the parameters ζ g,[R] ≡ α g,[R] /v rel , (2.15a) ζ h ≡ α h /v rel , (2.15b) where in eq. (2.15a), R = 1, 8 denotes the colour representation. Because the average momentum transfer along the 2PI diagrams is different in the scattering and the bound states, we shall also discern between ζ S g, [R] and ζ B g, [R] , for which we evaluate the strong coupling at the respective scale, as we shall see in the following (cf. section 4.4 for a summary). However, since the running of α h depends on the underlying UV model, we neglect it in the following. For convenience, we define ζ S,[R] ≡ ζ S g,[R] + ζ h = (α S g,[R] + α h )/v rel , (2.16a) ζ B,[R] ≡ ζ B g,[R] + ζ h = (α B g,[R] + α h )/v rel . (2.16b) Finally, for the bound states, it will be useful to define the parameter λ [R] ≡ α B g,[R] /α h = ζ B g,[R] /ζ h . (2.17) The dimensionless variables defined in eqs. (2.14) to (2.17) suffice to parametrise the pieces of the annihilation and BSF cross-sections that need to be evaluated numerically. Generically, the Sommerfeld effect and the capture into bound states are expected to be significant for |ζ g | O(1) and ζ h , d h O(1), for the Coulomb and Yukawa potentials, respectively. However, in ref. [40] it was shown that the Coulomb potential affects how long-range the Yukawa potential manifests. In section 3, we revisit this point and determine the conditions for the existence of bound states and their properties. Thermal effects. The coupling of the mediators -the gluons and the Higgs -to the relativistic SM plasma modifies the zero-temperature potential (2.12) in several ways. On one hand, it generates thermal contributions to the mediator masses that screen the longrange effect. On the other hand, it gives rise to frequent scatterings between X, X † and the relativistic plasma that allow for non-radiative dissipation of energy from an XX † pair, and thus ensure equilibrium between bound and scattering states. In fact, the XX † spectral function morphs into a continuum that encompasses energies both above and below 2m X [27,28]. These thermal effects have been considered in DM freeze-out computations using linear response theory, first in refs. [29][30][31], where radiative capture processes were neglected, and subsequently in ref. [32] which incorporated BSF via gluon emission. Recently, ref. [54] derived ab initio a formalism for the description of DM long-range interactions in a plasma background, starting from non-equilibrium quantum field theory. In our analysis, we neglect thermal effects. In the scattering states, the screening of the gluon-mediated force due to the thermal gluon masses is negligible, because of the fairly large average velocity of the X, X † particles during DM freeze-out, which ensures that the average momentum transfer in an XX † pair is larger than the thermal gluon mass. The radiative BSF processes we consider are sufficiently rapid to bring bound states in equilibrium at early times. In equilibrium, and at temperatures much higher than the binding energy, the DM depletion via bound-state decay becomes independent of the BSF rate, whether this is due to radiative and/or scattering processes; we return to this point in section 2.5. More importantly, we find that the impact of the bound states on the DM relic density occurs mostly at later times, when the plasma temperature approaches or falls below the binding energy. In this regime, the thermal corrections become less important, and the radiative capture is typically the dominant BSF process. Annihilation The dominant contributions to the direct annihilation of XX † pairs arise from the XX † → gg and XX † → hh channels, shown in fig. 2. We neglect the p-wave suppressed contributions XX † → qq, gh. (Similarly we neglect the p-wave suppressed contribution of the s-channel diagram of XX † → gg). To leading order, the colour-averaged perturbative cross-sections are [40] (σv rel ) perturb XX † →gg = 14 27 π(α ann s ) 2 m 2 X , (2.18a) (σv rel ) perturb XX † →hh = 4πα 2 h 3m 2 X (1 − m 2 h /m 2 X ) 1/2 [1 − m 2 h /(2m 2 X )] 2 , (2.18b) where α ann s ≡ α s (Q = m X ) arises from the gluon emission vertices in the hard annihilation process and must be evaluated at momentum transfer Q = m X . Both the colour-singlet and Figure 2. Tree-level diagrams contributing to the X − X † annihilation into gluons and into Higgs bosons. The u-channel diagram that match the corresponding t-channel diagram (first diagram) is not shown. The s-channel annihilation into gluons (third ) yields a p-wave contribution such that we neglect this process in our computations. octet scattering states contribute in eq. (2.18a), while only the singlet can annihilate into hh. Due to their different gluon-mediated potential the two states are affected differently by the Sommerfeld effect. The full cross-sections are [40] X X † g g X X † g g X X † g g X X † h hX X † 2PI · · · 2PI(σv rel ) XX † →gg = (σv rel ) perturb XX † →gg × 2 7 S 0,[1] + 5 7 S 0,[8] , (2.19a) (σv rel ) XX † →hh = (σv rel ) perturb XX † →hh × S 0,[1] , (2.19b) where S 0, [1] and S 0, [8] indicate the s-wave Sommerfeld factors for the colour-singlet and octet states respectively. For the mixed Coulomb and Yukawa potential (2.12), the s-wave Sommerfeld factor depends on the parameters defined in eqs. (2.14) and (2.15), i.e. S 0 = S 0 (ζ S g , ζ h , d h ), as we discuss in section 3.1. Thus, taking eqs. (2.13) into account, [8] denotes the strong coupling evaluated at the average momentum transfer in the scattering states, Q = µv rel , which is independent of the colour representation. The thermally-averaged total annihilation cross-section is S 0,[1] = S 0 4α S s 3v rel , α h v rel , µα h m h , (2.20a) S 0,[8] = S 0 − α S s 6v rel , α h v rel , µα h m h . (2.20b) Here, α S s ≡ α s (Q = µv rel ) = α S s,[1] = α S s,σ ann v rel = µ 2πT 3/2 d 3 v rel exp − µv 2 rel 2T (σv rel ) XX † →gg + (σv rel ) XX † →hh . (2.21) Bound-state formation, ionisation and decay Formation. Attractive long-range interactions imply the existence of bound states that may form from the scattering states via dissipation of energy. In the non-relativistic regime, Figure 4. X X † 2PI · · · 2PI 2PI · · · 2PI B g C The amplitude for the radiative capture into bound states consists of the (nonperturbative) initial and final state wavefunctions, and the perturbative 5-point function that includes the radiative vertices. The wavefunctions (cf. section 3) are determined by the resummation of the 2-particle-irreducible (2PI) diagrams. The 2PI kernel is shown in fig. 1. the energy available to be dissipated is the difference between the initial and final states, i.e. the kinetic energy of the scattering state in the centre-of-momentum (CM) frame C = + +E k = k 2 /(2µ) = µv 2 rel /2 minus the binding energy of the bound state E n = −γ 2 n × κ 2 /(2µ), where κ ≡ µ(α B g + α h ) is the generalised Bohr momentum of the bound state and γ n parametrises the departure of E n from its Coulomb limit d h → ∞ (cf. section 3.2). For capture into the ground state, the dissipated energy is ω E k − E 10 = µ 2 [v 2 rel + (α B g + α h ) 2 γ 2 10 ]. (2.22) The capture into bound states via emission of an ultrasoft gluon is depicted in figs. 4 and 5. The decompositions 3 ⊗3 = 1 ⊕ 8 and 8 ⊗ 8 = 1 S + 8 A + 8 S + 10 A + 10 A + 27 S imply that the allowed transitions are (X + X † ) [8] → B(XX † ) [1] + g [8] , (2.23a) (X + X † ) [1] → B(XX † ) [8] + g [8] [1 S ] , (2.23b) (X + X † ) [8] → B(XX † ) [8] + g [8] [8 S ] or [8 A ] . (2.23c) In (2.23b) and (2.23c), the indices outside the curly brackets denote the colour representation of the entire final state (bound state plus gluon). When considering the gluon potential solely, only the colour-singlet XX † combination supports a bound state, thus (2.23a) is the sole capture process via gluon emission. However, if the coupling to the Higgs is sufficiently strong, the Higgs exchange can overcome the gluon-mediated repulsion in the octet state and give rise to octet bound states. Then, the processes (2.23b) and (2.23c) may be possible. In section 3.2, we determine the condition for the existence of bound states in a mixed repulsive Coulomb plus attractive Yukawa potential, and in section 4.4 we adapt this condition to the colour-octet states in our model, taking into account the α s running. If the energy (2.22) available to be dissipated exceeds the Higgs mass, then bound states may form also via Higgs emission, according to (X + X † ) [1] → B(XX † ) [1] + h, (2.24a) (X + X † ) [8] → B(XX † ) [8] + h. (2.24b) However, the capture of particle-antiparticle pairs via scalar emission is subject to cancellations that suppress the processes (2.24) by higher orders in α s and α h with respect to (2.19) and (2.23) [15, eqs. (4.10)]. Although other couplings in the scalar potential, such as the quartic h 2 |X| 2 term, can enhance and even dominate the cross-sections for the processes (2.24) [55], these cross-sections remain subdominant to the direct annihilation and/or BSF via gluon emission. In the following, we thus neglect BSF via Higgs emission. The thermally-averaged BSF cross-section is σ BSF v rel = µ 2πT 3/2 d 3 v rel exp − µv 2 rel 2T [1 + f g (ω)] σ BSF v rel , (2.25) where f g (ω) = 1/(e ω/T − 1) is the gluon occupation number, with ω being the energy of the emitted gluon, given by eq. (2.22). The factor 1+f g (ω) accounts for the Bose enhancement due to the final-state gluon, and is necessary to ensure the detailed balance between the bound-state formation and ionisation processes at T ω, which encompasses a significant temperature range that is relevant to the DM freeze-out [10]. 1 Once bound states form, they may either be ionised back into their constituents by the ambient radiation, or decay into radiation, as we now discuss. Ionisation. The ionisation cross-section is related to the BSF cross-section via the Milne relation (cf. e.g. [37, appendix D]), σ ion,[R] = g 2 X g g g [R] µ 2 v 2 rel ω 2 [R] R S σ [R S ]→[R] BSF ,(2.26) where R S , R denote the colour representations of the scattering and bound XX † states, with g [1] = 1 and g [8] = 8 being the colour-singlet and octet degrees of freedom respectively, and g g = 8 being the gluon degrees of freedom. In eq. (2.26), we have made explicit the dependence of the gluon energy (2.22) on the colour representation of the bound state. Note that for the ionisation cross-section of the colour-octet bound states -if they existwe must include the contributions from both processes (2.23b) and (2.23c), i.e. sum over R S = 1, 8 as indicated in eq. (2.26). The ionisation rate of a bound state is then Γ ion,[R] = g g ∞ ω min [R] dω [R] 2π 2 ω 2 [R] e ω [R] /T − 1 σ ion,[R] , 1 As is standard, we have omitted the Bose enhancement factor in the thermal averaging of the annihilation cross-sections (2.21), since the energy of the annihilation products, E = mX, far exceeds the temperature of the relativistic bath at the temperature range relevant for freeze-out, T mX/20. where ω min [R] is the minimum energy required for ionisation, i.e. the binding energy, recovered from eq. (2.22) for v rel = 0. Using eqs. (2.22) and (2.26), we arrive at Γ ion,[R] = 9µ 3 2π 2 g [R] ∞ 0 dv rel v 2 rel exp µ[(α B g,[R] + α h ) 2 γ 2 1,0 + v 2 rel ] 2T − 1 R S σ [R S ]→[R] BSF v rel . (2.27) Note that in eq. (2.27), γ 1,0 also depends on R even though it is not explicitly indicated. From eqs. (2.25) and (2.27), we can explicitly verify the principle of detailed balance, (n eq X ) 2 R S σ [R S ]→[R] BSF v rel = n eq B,[R] Γ ion,[R] ,(2.28) where n eq X and n eq B, [R] are the equilibrium densities of X and of XX † bound states of colour representation [R], respectively. Note that eq. (2.28) is more general than our derivation here may suggest; it holds true independently of what processes -radiative or scattering -may contribute to the capture into and the ionisation of bound states. It is clear from (2.27) and (2.28), that Γ ion decreases exponentially when T drops below the binding energy and most particles in the plasma do not have enough energy to dissociate the bound states. Decay. The decay rate of the = 0 bound states is related to the perturbative s-wave annihilation cross-section times relative velocity as (see e.g. [11]) Γ dec,[R] = (σ s−wave ann,[R] v rel ) |ψ [R] n00 (0)| 2 , (2.29) where ψ [R] n00 (0) indicates the n-th level s-wave bound-state wavefunction of the colour representation R, evaluated at the origin, and will be discussed in section 3.2. The cross-section σ s−wave ann, [R] v rel corresponds to the colour configuration of the bound state and should be averaged over the bound-state colour degrees of freedom only, rather than those of an unbound XX † pair. Using the perturbative annihilation cross-sections (2.18) and taking into account the colour decomposition in eq. (2.19a), we obtain for the ground states Γ dec,[1] = 9 4π(α ann s ) 2 27m 2 X + 4πα 2 h 3m 2 X (1 − m 2 h /m 2 X ) 1/2 [1 − m 2 h /(2m 2 X )] 2 |ψ [1] 100 (0)| 2 , (2.30a) Γ dec,[8] = 9 8 10π(α ann s ) 2 27m 2 X |ψ [8] 100 (0)| 2 . (2.30b) Effective bound-state formation cross-section. The effect of unstable bound states on the DM relic abundance is described by a system of coupled Boltzmann equations for the bound and unbound particles that describe the interplay between bound-state formation, ionisation and decay [10]. Only the bound states that decay into radiation before being ionised contribute to the depletion of the DM abundance. It is thus possible to describe the impact of bound states using a single Boltzmann equation for the unbound particles and an effective BSF cross-section that incorporates the branching ratio of the bound states that decay rather than being ionised. For our model, this is [8] . σ BSF v rel eff = σ [8]→[1] BSF v rel × Γ dec,[1] Γ dec,[1] + Γ ion,[1] + σ [1]→[8] BSF v rel + σ [8]→[8] BSF v rel × Γ dec,[8] Γ dec,[8] + Γ ion, (2.31) The decay becomes faster than ionisation at temperatures comparable to or smaller than the binding energy, when most particles in the thermal bath do not possess enough energy to disassociate the bound states and Γ ion decreases exponentially with decreasing temperature. The effective BSF cross-section (2.31) together with the thermally averaged annihilation cross-sections (2.19) yield the total effective annihilation cross-section (2.11) that determines the DM relic density according to the Boltzmann eq. (2.8). Ionisation equilibrium. If the BSF and ionisation cross-sections are sufficiently large to set Γ ion Γ dec at high temperatures, then the effective BSF cross-section (2.31) becomes independent of the actual BSF cross-section during that time. Instead, it is proportional to the annihilation cross-section, which determines the bound-state decay rate. Indeed, using the detailed balance eq. (2.28) and the decay rate (2.29), we find σ BSF v rel eff large T R S ,R σ [R S ]→[R] BSF v rel Γ dec,[R] Γ ion,[R] = R n eq B,[R] (n eq X ) 2 Γ dec,[R] R g [R] g 2 X exp |E [R] 10 |/T (πm X T ) 3/2 (σ s−wave ann,[R] v rel ) |ψ [R] 100 (0)| 2 ,(2.S bound,[R] |ψ [R] 100 (0)| 2 /(πm X T ) 3/2 . (2.33) To assess the importance of eq. (2.33), it is informative to consider a simple attractive Coulomb potential of strength α, for which |ψ 1,0,0 (0)| 2 = (µα) 3 /π. Then, eq. (2.33) yields π −5/2 [m X α 2 /(4T )] 3/2 1, where the square bracket contains the binding-energy-totemperature ratio, |E 10 |/T . This contribution can be compared with the thermally averaged Sommerfeld factor from the scattering states, which in this regime is 1+(4π) 1/2 [m X α 2 /(4T )] 1/2 . Clearly, the enhancement due to the bound states is subdominant to that due to the scattering states. Including excited states in eq. (2.32) would somewhat increase the contribution of the bound levels, but not change this conclusion. The depletion of DM via BSF becomes more significant as the temperature of the universe lowers and approaches the binding energy [10]; this is in part manifested by the exponential factor in eq. (2.32). In this regime, the bound-state decay becomes comparable to or faster than ionisation, which is now exponentially suppressed, [cf. eq. (2.27)], and σ BSF v rel eff rapidly saturates to σ BSF v rel [cf. eq. (2.31)]. It is important to note that if σ BSF is comparable to exceeds σ ann , then it is possible that the DM depletion via BSF contributes significantly to the effective annihilation rate even starting from temperatures that are larger than the binding energy by a factor of a few (cf. refs. [10,37] and section 5). Wavefunctions and overlap integrals Scattering states The non-relativistic potential due to gluon and Higgs exchange for the scattering states is V S (r) = − α S g r − α h r e −m h r , (3.1) where we will consider both positive and negative α S g , but only α h 0. Wavefunctions. The scattering states are described by a wavefunction φ k (r), parametrised by the continuous quantum number k = µv rel ,(3.2) where k and v rel are the expectation values of the momenta of the interacting particles in the CM frame, and of their relative velocity. The strong coupling α S s that determines α S g in eq. (3.1) is evaluated at Q = |k|. The scattering state wavefunction obeys the Schrödinger equation − ∇ 2 2µ + V S (r) φ k (r) = E k φ k (r), (3.3) with E k ≡ k 2 2µ = µv 2 rel 2 > 0. (3.4) They are normalised according to d 3 r φ * k (r) φ k (r) = (2π) 3 δ 3 (k − k ). (3.5) To solve the Schrödinger equation numerically, we perform the separation of variables φ k (r) = ∞ =0 (2 + 1) χ |k|, (kr) kr P (k ·r),(3.6) and introduce the dimensionless radial space coordinate x S ≡ kr,(3.7) such that the radial Schrödinger equation reads χ |k|, (x S ) + 1 − ( + 1) x 2 S + 2 x S ζ S g + ζ h e −(ζ h /d h ) x S χ |k|, (x S ) = 0,(3.8) where we have used the parameters ζ S g ≡ α S g /v rel , ζ h ≡ α h /v rel and d h ≡ µα h /m h defined in eqs. (2.14) and (2.15). At x S → 0, and for > 0, the second term of eq. (3.8) is dominated by the centrifugal contribution. In this region, the two independent solutions of eq. (3.8) scale as x +1 S (regular) and x − S (irregular). We are interested in the regular solutions, which imply the boundary condition lim x→0 χ |k|, (x S ) = ( + 1) lim x S →0 [χ |k|, (x S )/x S ] . (3.9) The condition (3.9) will be valid also for = 0. To fully specify the wavefunction χ |k|, , we shall also use the asymptotic behaviour at x S → ∞. At large x S , the wavefunction χ |k|, behaves as (see e.g. ref. [56, chapter 7]) χ |k|, (x S ) x S →∞ −→ 1 2i e i(x S +δ ) − e −i(x S − π) , where the phase shifts δ depend on ζ S g , ζ h and d h . This implies that χ |k|, (x S ) 2 + χ |k|, (x S − π/2) 2 = 1 . (3.10) Coulomb limit. In the limit d h → ∞, the Schrödinger eq. (3.8) depends only on ζ S = ζ S g + ζ h , and can be solved analytically, χ C |k|, (x S ) = S C 0 (ζ S ) 2 (2 + 1)! Γ (1 + − iζ S ) Γ (1 − iζ S ) x 1+ S e −ix S 1 F 1 (1 + + iζ S ; 2 + 2; 2ix S ), (3.11a) where 1 F 1 is the confluent hypergeometric function of the first kind, and S C 0 (ζ) ≡ 2πζ 1 − e −2πζ . (3.11b) The sum over the modes in eq. (3.6) can be expressed in closed form such that the total scattering-state wave function in the Coulomb limit reads φ C k (r) = lim d h →∞ φ k (r) = S C 0 (ζ S ) 1 F 1 [iζ S ; 1; i(kr − k · r)] e ik·r . (3.11c) Note that the limit d h → 0 (pure gluon-mediated potential) can be also obtained from the above with the substitution ζ S → ζ S g . Sommerfeld factor S 0 for s-wave annihilation. This is (see e.g. [57]) S 0 (ζ g , ζ h , d h ) = |φ k (r = 0)| 2 = lim x S →0 χ |k|,0 (x S ) x S 2 . (3.12) In the Coulomb limits d h → 0 and d h → ∞, the Sommerfeld factor reduces to lim d h →0 S 0 (ζ S g , ζ h , d h ) = S C 0 (ζ S g ), (3.13) lim d h →∞ S 0 (ζ S g , ζ h , d h ) = S C 0 (ζ S g + ζ h ),(3.14) with S C 0 (ζ) given in eq. (3.11b). The features of S 0 and the impact of Higgs enhancement on the relic density have been discussed in detail in ref. [40] (see in particular figs. 2 and 3 therein). Bound states The non-relativistic potential for the bound states, due to gluon and Higgs exchange, is V B (r) = − α B g r − α h r e −m h r . (3.15) Note the difference in α g with respect to eq. (3.1). While α S g in the scattering potential is to be evaluated at the scale Q = µv rel , α B g in the bound state potential is to be taken at the Bohr momentum scale, as will be discussed below. 16) where {n m} are the standard principal and angular momentum quantum numbers, and the wavefunctions are normalised as Wavefunctions. The Schrödinger equation for bound states is − ∇ 2 2µ + V B (r) ψ n m (r) = E n ψ n m (r) ,(3.d 3 r ψ * n m (r) ψ n m (r) = δ nn δ δ mm . (3.17) We define the generalised Bohr momentum κ ≡ µ(α B g + α h ),(3.18) such that the discrete binding energy levels of the system are E n ≡ −γ 2 n × κ 2 2µ = − 1 2 µ (α B g + α h ) 2 γ 2 n < 0,(3.19) where the factor γ n is determined numerically (see below) and parametrises the departure from the Coulomb limit d h → ∞ where γ n = 1/n. We perform the separation of variables ψ n m (r) = κ 3/2 χ n (κr) κr Y m (Ω r ) . (3.20) Defining the dimensionless space coordinate x B ≡ κr,(3.21) we find that χ n is normalised as ∞ 0 dx B |χ n (x B )| 2 = 1,(3.22) and obeys the radial Schrödinger equation χ n (x B )+ −γ 2 n − ( + 1) x 2 B + 2 (1 + λ)x B λ + exp − x B (1 + λ)d h χ n (x B ) = 0 . (3.23) We recall that the dimensionless parameters λ ≡ α B g /α h and d h ≡ µα h /m h have been defined in eqs. (2.14) and (2.17). It follows that the bound-state wavefunctions χ n and the (normalised) energy eigenvalues γ n depend on λ and d h only. Assuming α h > 0, the necessary condition κ > 0 for bound states to exist implies λ > −1. This condition is necessary but not sufficient, as we shall see below. As in the case of the scattering states, we are interested in the regular solutions of eq. (3.23), which scale as lim x B →0 χ n (x B ) = ( + 1) lim x B →0 [χ n (x B )/x B ]. (3.24) The discrete spectrum of eigenvalues γ n = γ n (λ, d h ) is then determined by requiring that χ n vanish at infinity, lim x B →∞ χ n (x B ) = 0. (3.25) We present some numerical results in fig. 6 and discuss them next. Coulomb limits. In the left panel of fig. 6, the two plateaus indicate the two Coulomb limits, d h → 0 (pure gluon-mediated potential) and d h → ∞ (Coulomb limit of the Higgsmediated potential), which allow for analytic solutions of eq. (3.23). • In the limit d h → ∞, the spectrum of eigenvalues and the radial wavefunction are γ C n = lim d h →∞ γ n = 1/n , (3.26) χ C n (x B ) = 1 n (n − − 1)! (n + )! 1/2 e −x B /n (2x B /n) +1 L (2 +1) n− −1 (2x B /n),(3.27) where L a n are the generalised Laguerre polynomials of degree n, and we assume the normalisation condition • In the limit d h → 0, bound states exist only for λ > 0 (α B g > 0). In this case, the binding energy takes the Coulomb value for the gluon-generated potential only, lim d h →0 E n = 1 n 2 κ 2 g 2µ ,(3.28) where κ g ≡ µα B g . This implies lim d h →0 γ n = λ n(1 + λ) . (3.29) Existence condition. If the Coulomb component of the potential is not attractive, λ 0 (equivalently, α B g 0), the Yukawa potential has to be sufficiently long-range for bound states to exist, i.e. d h has to be sufficiently large. In the right panel of fig. 6, we show the critical value d h,crit (λ) such that for d h d h,crit (λ) at least one bound level exists. We find that in the range −0.96 λ 0, d h,crit can be fitted by the analytical expression d h,crit (λ) d 0 exp (−λ) 0.59 (1 + λ) 1.95 ,(3. Overlap integrals In order to evaluate the amplitudes for the BSF processes of interest, we need the following overlap integrals of the scattering and bound state wavefunctions [37] J k,{n m} (b) ≡ d 3 p (2π) 3 pφ k (p + b)ψ * n m (p) = i d 3 r φ k (r) [∇ψ * n m (r)] e −ibr , (3.31a) Y k,{n m} ≡ 8πµα B s d 3 p (2π) 3 d 3 q (2π) 3 q − p (q − p) 4φ [R S ] k (q)ψ * n m (p) = −iµα B s d 3 r ψ * n m (r) φ k (r)r, (3.31b) where we used the following Fourier transforms For the numerical evaluation of the integrals (3.31), we express them in terms of the radial wavefunctions χ k, and χ n introduced in sections 3.1 and 3.2. The integrals J k,{n m} have been previously analysed in ref. [15, appendix B]. The leading order contribution to J k,{n m} (b) with b ∝ P g is independent of b [11,15] such that in the following we denote J k,{n m} = J k,{n m} (b = 0). Adapting the result to our notation for capture into the ground state, {n m} = {100}, we arrive at 2 φ k (q) = d 3 r φ k (r) e −iq r , φ k (r) = d 3 q (2π) 3φ k (q) e iq r ,(3.J k,{100} −ik κ k 4π κ ∞ 0 dx B χ * 1,0 (x B ) x B − dχ * 1,0 (x B ) dx B χ |k|,1 (x B /ζ B ), (3.33a) where we recall from eq. (2.16b) that ζ B = κ/k. Similarly, the integral (3.31b) becomes Y k,{100} = −i µα B s k √ 4πκ ∞ =0 (2 + 1) ∞ 0 dx B χ * 1,0 (x B ) χ |k|, (x B /ζ B ) dΩr P (k ·r). The angular integral above is (see e.g. ref. [15, eq. (B.7b) with b → 0]) dΩr P (k ·r) = δ ,1 (4π/3)k. Thus, we arrive at the final expression Y k,{100} = −ik µα B s k 4π κ ∞ 0 dx B χ * 1,0 (x B ) χ |k|,1 (x B /ζ B ). (3.33b) We shall use eqs. (3.33) to evaluate the BSF cross-sections in section 4. Coulomb limit. In the limit d h → ∞, the scattering and bound state wavefunctions, (3.11c) and (3.27) are φ C k (r) = S C 0 (ζ S ) 1 F 1 [iζ S ; 1; i(kr − k · r)] e ik·r , (3.34a) ψ C {100} (r) = κ 3 π e −κr , (3.34b) where ζ S ≡ ζ S g + ζ h [cf. eq. (2.16a)], S C 0 (ζ S ) ≡ 2πζ S /(1 − e −2πζ S ) [cf. eq. (3. 11b)], and 1 F 1 is the confluent hypergeometric function of the first kind. Following refs. [11,15,37,58], we derive the Coulomb limit of the overlap integrals using the identity [59] d 3 r e i(k−b)·r−κr 4πr 1 F 1 [iζ S , 1, i(kr − k · r)] = [b 2 + (κ − ik) 2 ] −iζ S [(k − b) 2 + κ 2 ] 1−iζ S ≡ f k,b (κJ C k,{100} (b) = κ 16πκ 3 S C 0 (ζ S ) [∇ b f k,b (κ)], (3.36a) Y C k,{100} = µα B s 16πκ 3 S C 0 (ζ S ) [∇ b f k,b (κ)] b=0 , (3.36b) where [∇ b f k,b (κ)] b=0 =k 2(1 − i ζ S ) k 3 exp [−2ζ S arccot(ζ B )] (1 + ζ 2 B ) 2 , (3.36c) and we used again κ/k = ζ B . As noted before, it suffices to evaluate J k,{100} (b) at b = 0 [11,15] such that Y k,{100} and J k,{100} are related as Y C k,{100} J C k,{100} = α B s α h + α B g . (3.37) with the squared amplitude of J C k,{100} being |J C k,{100} | 2 = 2 6 π κ S C 0 (ζ S ) (1 + ζ 2 S ) ζ 6 B exp [−4ζ S arccot(ζ B )] (1 + ζ 2 B ) 4 . (3.38) In section 4.3, we shall use eqs. (3.37) and (3.38) to obtain analytical expressions for the BSF cross-sections in the Coulomb limit. Bound-state formation cross-sections Having outlined how to evaluate the scattering and bound state wavefunctions, we have the tools at hand to calculate the cross-sections for the BSF processes discussed in section 2.5. Amplitudes The radiative capture via gluon emission has been computed in ref. [37] for general groups and representations, in terms of the overlap integrals (3.31). Adapting the results to our model, we find the following colour-averaged squared amplitudes: (X + X † ) [8] → B(XX † ) [1] + g : 1 3 2 M [8]→[1] k→{n m} 2 = 2 7 πα BSF s M 2 3 3 µ × J [8,1] k,{n m} + 3 2 Y [8,1] k,{n m} 2 , (4.1a) (X + X † ) [1] → B(XX † ) [8] + g : 1 3 2 M [1]→[8] k→{n m} 2 = 2 7 πα BSF s M 2 3 3 µ × J [1,8] k,{n m} − 3 2 Y [1,8] k,{n m} 2 , (4.1b) (X + X † ) [8] → B(XX † ) [8] + g : 1 3 2 M [8]→[8] k→{n m} 2 = 2 7 πα BSF s M 2 3 3 µ × 5 2 J [8,8] k,{n m} 2 , (4.1c) where α BSF s ≡ α s (Q = |P g |) is the strong coupling evaluated at momentum transfer equal to the momentum of the emitted gluon; this factor arises from the gluon emission vertices in the diagrams of fig. 5. |P g | is found from energy-momentum conservation to be |P g | = E k − E n = (µ/2)[(α h +α B g ) 2 γ 2 n +v 2 rel ]. We recall that in eqs. (4.1), the superscripts on the overlap integrals denote, in order, the scattering-state and bound-state colour representations. Cross-sections for capture into the ground state The differential cross-section for the radiative capture into bound states is v rel dσ k→{n m} dΩ = |P g | 64π 2 M 2 µ |M k→{n m} | 2 − |P g · M k→{n m} | 2 . (4.2) For capture into the ground state, the leading-order amplitude is M k→{100} ∝ k, as seen from the expressions (3.33) for the overlap integrals. Thus, carrying out the angular integration, we arrive at σ k→{100} v rel = (1 + ζ 2 B γ 2 1,0 )v 2 rel 48πM 2 |M k→{100} | 2 ,(4.3) where we recall that ζ B ≡ (α h + α B g )/v rel [cf. eq. (2.16b)]. Using the amplitudes (4.1) and the overlap integrals (3.33), we find from eq. (4.3) the coloured-averaged radiative capture cross-sections to be [8] k,1 σ k→{100} v rel = πα BSF s (α h + α B g ) m 2 X 2 7 3 4 × (1 + ζ 2 B γ 2 1,0 ) H, (4.4) with H [8]→[1] ≡ ∞ 0 dx B χ x B ζ B, [1] χ [1] 1,0 (x B ) x B − dχ [1] 1,0 (x B ) dx B + 3 2 α B s, [1] α h + α B g, [1] χ [1] 1,0 (x B ) * 2 , (4.4a) [1] k,1 H [1]→[8] ≡ ∞ 0 dx B χ x B ζ B, [8] χ [8] 1,0 (x B ) x B − dχ [8] 1,0 (x B ) dx B − 3 2 α B s, [8] α h + α B g, [8] χ [8] 1,0 (x B ) * 2 , (4.4b) [8] k,1 H [8]→[8] ≡ 5 2 ∞ 0 dx B χ x B ζ B, [8] χ [8] 1,0 (x B ) x B − dχ [8] 1,0 (x B ) dx B * 2 . (4.4c) We use eqs. (4.4) to evaluate numerically the BSF cross-sections. Coulomb limit In the limit d h → ∞, we may use the analytical expressions (3.37) and (3.38) for the overlap integrals. We find the colour-averaged BSF cross-sections to be σ C BSF v rel = πα BSF s (α h + α B g ) m 2 X 2 11 3 4 f c × S C BSF (ζ S , ζ B ), (4.5a) where f c is a numerical factor that depends on the transition, f c =                1 + 3 2 α B s,[1] α h + 4α B s,[1] /3 2 , [8] → [1], 1 − 3 2 α B s, [8] α h − α B s, [8] and S C BSF (ζ S , ζ B ) ≡ 2πζ S 1 − e −2πζ S (1 + ζ 2 S ) ζ 4 B exp [−4 ζ S arccot(ζ B )] (1 + ζ 2 B ) 3 . (4.5c) We emphasise that, even when not explicitly denoted above, the couplings α BSF s , α B s , α B g , α S g , and therefore also ζ S ≡ (α S g + α h )/v rel and ζ B ≡ (α B g + α h )/v rel , depend on the colour representations of the initial and final states, and are thus different for every transition. The function S C BSF (ζ S , ζ B ) encapsulates all the velocity dependence of σ C BSF v rel . The first two factors in eq. (4.5c) arise solely from the scattering state wavefunction and correspond to the Sommerfeld effect on p-wave processes in the Coulomb limit, [8] , ζ B, [1] (4.7) S C 1 (ζ S ) = [2πζ S /(1 − e −2πζ S )](1 + ζ 2 Sm 2 X × S C BSF (ζ S, The running of α s As it has been pointed out throughout this paper, the various vertices where the strong coupling appears in the annihilation and BSF diagrams are characterised by different momentum transfers Q. 3 Since the strong coupling is sensitive to the momentum transfer, α s = α s (Q), it is important to carefully account for its running. An overview of the various scales is given in table 2. 4 In the occasions that the momentum transfer itself depends on the strong coupling, Q = Q(α s ), we solve numerically the following equation forα, α s (Q(α)) =α. (4.8) The effect of the running of α s is depicted in figs. 7 to 10, with related discussion in the captions. A few remarks are in order here. Scattering states. The average momentum transfer Q = µv rel , implies that α S s becomes suppressed for large m X . Consequently, for large m X , the attraction due to the Higgs can significantly ameliorate or even overcome the repulsion due to the gluons in the colour-octet state. Due to the large multiplicity of the latter, (cf. eq. (2.19a)), this enhances significantly the total annihilation rate, even for moderate values of α h . 5 This is depicted in fig. 7. Bound states. Because the momentum transfer depends on the strong coupling, we solve numerically eq. (4.8). Generally, α h increases the average momentum transfer, thus suppressing α B s , as shown in fig. 8. Overall though, the Bohr momentum κ ≡ µ(α B g + α h ) increases with increasing α h . If α h is sufficiently strong, it overcomes the gluon-mediated repulsion in the octet states, and gives rise to bound levels. In the octet ground state, Q(α s ) = µ(α h − α s /6) × γ 1,0 − α s 6α h , d h , (4.9) where we recall that d h = µα h /m h , and the function γ 1,0 (λ, d h ) has been determined numerically and is depicted in fig. 6. For this momentum transfer, we see graphically in the left panel of fig. 9 that eq. (4.8) has a solution when the coloured lines of fixed α h intersect the horizontal black line. Let us examine the scaling of α s (Q(α))/α withα. • For sufficiently smallα (α 6α h ), the momentum transfer (4.9) is independent of α, i.e. Q µα h × γ 1,0 (0, d h ). Since α s ∼ 0.1 for the m X values of interest, the ratio α s (Q(α))/α is simply 0.1/α, i.e. starts from values 1 and decreases withα. 3 In fact, the smallness of the momentum transfer along the mediators in the ladder diagrams is responsible for the emergence of the non-perturbative phenomena, the Sommerfeld effect and bound states. 4 An improved treatment would incorporate the running of the coupling in the Schrödinger equations for the scattering and bound states, by setting Q = 1/r. The approximations of table 2 are sufficient for our purposes. We refer to [60][61][62] for the computation of the αs running in NRQCD and pNRQCD. 5 This holds provided that v rel is not too low. See related discussion in section 5.1. • At largerα (but stillα < 6α h ), this scaling changes. The factor γ 1,0 tends to 0, i.e. d h approaches d h, crit (λ = −α/(6α h )) [cf. section 3.2 and fig. 6]. This, in turn, drives Q to zero and consequently α s (Q) to infinity. • Between these two limits, α s (Q(α))/α reaches the value 1 if α h is sufficiently large. The minimum value of the Higgs coupling, α h, crit , required for the octet bound states to exist is shown in the right panel of fig. 9. Gluon-emission vertices in BSF. The momentum transfer -which depends here as well on the strong coupling -is the softer scale that enters our calculations, and thus yields the largest values of α s (for fixed values of the other parameters). Since α h increases the binding energy, it suppresses α BSF s . This is seen in fig. 10. Vertices µ 2 v 2 rel + α h − α B s,[8] 6 2 γ 2 n − α B s, [8] 6α h , d h Table 2. The momentum transfer Q at which the strong coupling α s (Q) is evaluated. For the bound states, the functions γ n (λ, d h ) are computed numerically (cf. section 3.2 and fig. 6). We recall that λ [R] ≡ α g,[R] /α h with R = 1, 8, and αh=0.05 Upper left: α S s evaluated at the average momentum exchange in the scattering state, Q = µv rel ; for the colour-singlet state α S g, [1] = (4/3)α S s , while for the colour-octet state α S g, [8] = −α S s /6 within the indicative velocity range 0.2 < v rel < 0.4 that is typical during the DM freeze-out. For comparison, we show α ann s occurring in the perturbative annihilation processes taken at a momentum transfer Q = m X (red diamonds). d h = µα h /m h . ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆x  =2 0 x  = 5 0 x  =2 0 x  = 5 0 Upper right: The parameter ζ g = α S g /v rel that determines the Sommerfeld effect due to gluon exchange only, for the singlet and octet configurations. Lower left: The coloured bands show the thermally-averaged Sommerfeld factor for s-wave annihilation processes,S 0 , evaluated within the indicative range 20 <x ≡ m X /T < 50, during which the DM relic density is mostly determined. The Higgs exchange enhances the attraction / reduces the repulsion between the scattering-state particles (blue bands: α h = 0.05) with respect to gluon-only exchange (gray bands: α h = 0). The thin solid and dotted lines mark the Coulomb limit, m h → 0, which becomes a good approximation for large enough m X . Lower right: The thermally-averaged total Sommerfeld factor for annihilation into gluons. The Higgs exchange leads to a significant enhancement (blue band: α h = 0.05, purple band: α h = 0.2) with respect to gluon-only exchange (gray band: α h = 0). Figure 8. ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆α h =0.1 α h =0.2 α h =0.3 The strong couplings α B s, [1] and α B s, [8] and the corresponding α B g, [1] = 4α B s, [1] /3 and α B g, [8] = −α B s, [8] /6 that determine the colour-singlet (left) and the colour-octet (right) bound-state wavefunctions. The coupling to the Higgs, α h , increases the average momentum transfer within the bound states, thereby suppressing α s . While the effect is modest in case of the colour singlet, a large enough α h is required for the existence of the octet bound state. Figure 10. The strong coupling α BSF s, [1] and α BSF s, [8] at the gluon emission vertex during the radiative capture of a colour-singlet (left panel ) or colour-octet (right panel ) bound state. The emitted gluon carries away the binding energy of the bound state plus the kinetic energy of the scattering state (see table 2) that depends on the relative velocity v rel . As before, we show α BSF s, [1] and α BSF s, [8] in the typical velocity range 0.2 v rel 0.4 during freeze-out. As v rel decreases, the momentum transfer at the vertex drops and α BSF s increases. In the right panel, the cut-off of the bands for low m X reflects the fact that colour-octet bound states exist only for large enough m X and α h (cf. fig. 9). Colour-octet bound states Binding energy [GeV] Colour-octet bound states Figure 11. The binding energy of the singlet and octet bound states, B [1] and B [8] , for various values of the coupling to the Higgs, α h (solid lines). The dotted red lines show the approximate temperature of freeze-out, T FO ≈ m X /30. α h = 0 . 2 α h = 0 . 3 mX 30 α h = 0 .1 The long-range effect of the Higgs Scattering states, bound states, annihilation and BSF rates Based on the computations of previous sections, we now present and summarise the impact of the Higgs-mediated force on the properties of the scattering and bound states, and on the various cross-sections of interest. Scattering states and Sommerfeld effect. The Higgs-mediated potential enhances the attraction in the colour-singlet state, and suppresses or overcomes the repulsion in the octet state. Notably, the effect of the Higgs potential is influenced by the presence of the gluon-mediated potential, as was already shown in ref. [40]. In particular: • In the singlet state, the Higgs enhancement becomes sizeable -and potentially reaches its Coulomb limit for a given coupling strength -for lower masses (more generally, for lower d h ≡ µα h /m h ) than in the absence of the attractive gluon-mediated force [40, fig. 2]. • Conversely, in the octet state, a larger d h is required for the Higgs to affect the longrange interaction. However, the gluon-mediated repulsion is suppressed by a colour factor with respect to the full strength of the strong coupling, α g, [8] = −α s, [8] /6, suggesting that even a modest α h can potentially counteract it. This holds so long as v rel is not too low; at low v rel , the gluon-induced repulsion becomes exponential [cf. the Sommerfeld factor in the Coulomb limit (3.11b) with ζ = ζ S g, [8] −1], and cannot be surmounted by a finite-range attractive force (cf. ref. [40, fig. 3, left panel]). However, the DM freeze-out occurs while v rel is fairly large and the gluon-induced repulsion is milder; this implies that the Higgs-mediated force has a significant effect. Tighter bound states. The Higgs attraction increases the absolute value of the binding energies, as shown in fig. 11. This renders the bound-state dissociation processes inefficient earlier, when the DM density is greater, thus enhancing the efficacy of BSF in depleting DM. In fig. 11, we compare the binding energy with the typical temperature around DM freeze-out, T FO ≈ m X /30. If the binding energy equals or exceeds T FO , then the dissociation of bound states is inhibited already at freeze-out, and the efficiency of BSF in depleting DM is maximal. While this occurs only for very large values of α h , we emphasise that the effective BSF rate can become comparable to or exceed the annihilation rate even at temperatures that are a factor of a few larger than the binding energy, even if it is not maximal (cf. fig. 13). Additional bound states. As already discussed in section 4.4 and shown in fig. 9, for sufficiently large masses and couplings, the Higgs-mediated attractive force implies the existence of colour-octet bound states. However, these bound states are considerably looser than the singlet states, as seen in fig. 11, and have a limited effect on the DM density (except perhaps for α h 0.3). Impact on the strong coupling. The Higgs-mediated force increases the momentum transfer in the bound states and in the gluon-emission vertices of the radiative capture processes. Because of the running of the strong coupling, this suppresses the corresponding values of α s , as discussed in section 4.4 and depicted in figs. 7, 8 and 10. However, the Bohr momentum and the binding energy increase with α h , as seen in fig. 11. Cross-sections. In fig. 12, we illustrate the velocity dependence of the BSF crosssections, and compare them with the direct annihilation processes. • For large v rel , the BSF cross-sections are rather suppressed and subdominant to the annihilation. This is due to the small overlap of the wavefunctions: the average momentum in the scattering states, k = µv rel , is much greater than that in the bound states, which in the Coulomb approximation is the Bohr momentum, κ = µ(α B g +α h ). In this regime, the BSF cross-sections scale as σ BSF v rel ∝ (κ/k) 4 ≈ [(α B g + α h )/v rel ] 4 . • At lower velocities, when k ∼ κ or v rel ∼ α B g + α h , the overlap of the wavefunctions is nearly maximal (although the precise value of v rel at which this occurs depends also on the Bohr momentum of the scattering state, hence on α S g +α h ). In this regime, the cross-section for capture into the tightest bound state (colour singlet) dominates over annihilation. Note that since α s ∼ 0.1, this velocity range is relevant for freeze-out. • At v rel α S g + α h , the Sommerfeld effect becomes important. Because of the gluon-mediated repulsion in the octet scattering states, the [8] → [1] and [8] → [8] capture processes become suppressed with decreasing v rel . The coupling to the Higgs ameliorates this suppression, and large α h can even make these BSF cross-sections increase temporarily as the velocity drops; this can be seen in the right panel of fig. 12. However, at sufficiently low v rel , the exponential Coulomb repulsion cannot be overcome by the finite-range Higgs-mediated attraction. On the other hand, the [1] → [8] cross-section increases steadily with decreasing v rel , due to the gluon and Higgs-mediated attractive forces in the colour-singlet scattering state. Similarly, the annihilation cross-section becomes dominated by the coloursinglet contribution and increases with decreasing v rel . Note that in fig. 12, we show the various cross-sections normalized to the perturbative annihilation cross-section, such that the direct dependence on m X cancels out. Nevertheless, m X affects the scale of α s and thus has indirect impact on the various lines. Let us now focus on the effect of the Higgs. As expected, the annihilation cross-section increases with larger α h . This is both due to the XX † → hh channel, and because the Higgs-mediated attractive potential enhances the annihilation cross-sections for all final states. In part because of the increase in the (perturbative) annihilation cross-section, the relative strength of BSF appears to diminish with increasing α h . The suppression of α BSF s at larger α h due to the running of the strong coupling (cf. fig. 10) contributes to this trend. However, the larger Bohr momentum of the bound states implies that BSF peaks at a larger v rel . This is very important for the DM depletion via BSF, as we discuss next. Thermally averaged cross-sections. In fig. 13, we depict the impact of the Higgs on the thermally averaged annihilation and BSF rates. The features of σ ann v rel follow from the discussion above. For BSF, we present both the actual and the effective thermally averaged cross-sections, σ BSF v rel and σ BSF v rel eff . Several comments are in order. • Despite the suppression of σ BSF v rel at large v rel discussed above, σ BSF v rel appears to be large even at high T ; this is due to the Bose-enhancement factor [cf. eq. (2.25)]. • Because of the rapid ionisation processes, σ BSF v rel eff is suppressed at high T , and in fact becomes independent of σ BSF v rel , as discussed in section 2.5. However, σ BSF v rel eff increases as T drops. For capture from an octet state, σ BSF v rel eff peaks at T ∼ binding energy. The reason is three-fold: (i) This condition is the thermally-averaged equivalent of k ∼ κ, which maximises the overlap of the bound and scattering-state wavefunctions, as discussed above. (ii) At and below this temperature, the ionisation rate becomes exponentially suppressed, therefore σ BSF v rel eff saturates to σ BSF v rel . (iii) At even lower temperatures, the gluon repulsion in the octet state suppresses σ [8]→ [1] BSF v rel and σ [8]→ [8] BSF v rel . • At least for 0.02 α h 0.1, σ [8]→[1] BSF v rel eff overcomes σ ann v rel , even before it saturates to σ [8]→ [1] BSF v rel , i.e. at temperatures higher than the binding energy of the singlet bound state. Overall, BSF yields a sizeable contribution to the total effective annihilation cross-section σ XX † v rel eff for the range of couplings considered. Dark matter relic density We now turn to the impact of the Higgs-mediated force on the relic density. We perform the calculation as described in section 2, using the BSF cross-sections of section 4. In the left panels of figs. 14 and 15, we show, for different values of α h , the NLP-LP mass difference ∆m versus the DM mass m χ that give rise to the observed DM abundance. The width of the bands reflects the uncertainty in the measurement of the DM abundance by Planck. We compare different computations that incorporate combinations of the various nonperturbative effects discussed in this work: the Sommerfeld effect on the direct annihilation processes and BSF, due to the gluon and Higgs mediated potentials. We present the impact of these effect on the relic density in the right panels of figs. 14 and 15. For α h = 0.02, the Higgs enhancement of the direct annihilation gives only a moderate correction on the predicted relic density. However, the full calculation that includes annihilation and BSF with gluon and Higgs exchange, changes the predicted mass gap by up to 10 GeV, with respect to considering gluon exchange only. Given that the usual tools for computing the relic density include a perturbative calculation only, their estimation can be off by a factor of 2.5 − 8.5 depending on the DM mass, even for a small value of α h . With increasing α h , the effect of BSF with gluon exchange only appears to become less significant. This is because it has remained unaffected by α h , while the direct annihilation into two Higgs bosons has become very rapid. However, including the Higgs-mediated potential gives rise to a very sizeable effect. This is true when considering direct annihilation only, and even more so when including BSF. We point out indicatively that for α h = 0.1 and α h = 0.2, ∆m is predicted to be larger by up to 40 GeV and 70 GeV respectively. The above clearly demonstrate that in coloured co-annihilation scenarios, both the Sommerfeld effect and BSF, as well as both the gluon-exchange and Higgs-exchange potentials must be considered in order to obtain a reliable estimation of the relic density. Caveats The present work along with refs. [37,40] aimed to demonstrate the long-range effect of the Higgs boson and the impact of BSF on the DM abundance. In order to not obfuscate these goals, various issues were not addressed, but would need to be properly considered in a comprehensive analysis. We briefly mention the main ones. Our simplified model does not specify the LP-NLP interactions that are hypothesised to establish equilibrium between the two in the early universe. Departure from equilibrium during the NLP freeze-out would imply that the NLP annihilation (direct or via BSF) depletes DM less efficiently than implicitly assumed by the set of eqs. (2.6) to (2.11). This in turn would shift the contours of figs. 14 and 15 to lower values of ∆m and m χ . The LP-NLP interactions, along with their mass difference, determine also the NLP decay rate. The latter must be smaller than the decay rate of the NLP-NLP bound states in order for BSF to deplete DM. This potentially implies that BSF cannot affect the DM density if the LP-NLP mass difference exceeds a threshold that depends on the strength of the LP-NLP interactions. Such a condition can be specified within complete models, e.g. MSSM scenarios, where the couplings of the UV theory determine on the one hand the NLP decay rate, and on the other hand the entirety of the annihilation processes that control the bound-state decay rate. Moreover, these couplings determine the value of α h . Including all significant annihilation channels in the analysis of course affects also the total effective annihilation rate -both directly, and via the bound-state decay rate that determines the efficiency of BSF in depleting DM. This could have a considerable impact on the results shown in figs. 14 and 15, and therefore on the interpretation of the experimental constraints. For very large α h , the radiative capture via Higgs emission that was neglected here, may also become significant, as noted in section 2.5. Ultimately, the precise computation of the DM density requires taking into account the thermal effects that were briefly discussed in section 2.3. This could be potentially done using the formalism of ref. [54], which can account for the transition of the BSF processes from in-to out-of equilibrium. However, this formalism must be first extended to encompass radiative processes that involve ultrasoft modes, which are currently neglected. Finally we note that sizeable couplings of the coloured particles to the Higgs endanger the stability of the SU (3) c symmetric vacuum. The determination of vacuum stability requires a dedicated study for any specific scenario under consideration. However, it has also been argued that this kind of dynamics could imply the emergence of a new phase in the MSSM, where the standard treatment of the vacuum stability does not apply [63]. Ω / Ω DM Figure 14. Left panels: The mass difference ∆m versus the DM mass m χ that reproduce the observed DM density when taking into account: perturbative annihilation only (grey dashed ), annihilation with Sommerfeld effect due to gluon exchange only (grey dotted ), annihilation with Sommerfeld effect due to gluon and Higgs exchange (green), annihilation with Sommerfeld effect and bound state formation with gluon exchange only (blue), annihilation with Sommerfeld effect and bound state formation with gluon and Higgs exchange (red ). The width of the bands corresponds to the 3σ experimental uncertainty on the DM abundance, using the Planck 2018 results. Right panels: For the values of ∆m predicted by the full calculation (red bands on the left panels), we show the ratio of the relic density from each partial calculation to the observed DM abundance. The colour coding is the same as on the left panels. We present the results for α h = 0.02 (up) and α h = 0.05 (down). Conclusions While our searches for DM coupled to the electroweak gauge interactions have returned null results until now, it is plausible that the Higgs constitutes the portal to the dark sector. The discovery of the Higgs boson and the measurement of its properties in collider experiments urge the comprehensive assessment of its implications for DM. If TeV-scale DM or its co-annihilating partners couple to the 125 GeV Higgs, then this interaction may have a sizeable long-range effect. In recent work, we showed that the DM relic abundance in coloured co-annihilation scenarios is affected significantly by the enhancement of the direct annihilation processes due to the Higgs-mediated force [40], and by the formation and decay of bound states due to gluon exchange [37]. Here we considered the effect of the Higgs-generated potential on the formation of bound states, and demonstrated the impact on the DM abundance. With respect to available tools that include perturbative calculations only, we have found that the predicted DM density may differ by up to almost one order of magnitude. While we focused on a particular class of models, we expect that the long-range effect of the Higgs boson is important in a variety of scenarios where the Higgs constitutes the portal to the dark sector. Altogether, in the set-up we considered, we have shown that the Higgs-mediated force affects the DM density in a variety of ways that exhibit some salient features. In summary: • The attractive interaction in the scattering states due to the Higgs exchange enhances both the direct annihilation cross-sections, as well as the capture into bound states. • The Higgs-mediated potential increases the binding energy of the colour-singlet bound states. This implies that the capture rate becomes maximal at larger velocities, and consequently at higher temperatures in the early universe. Moreover, the bound-state dissociation becomes insignificant starting at higher temperatures. As a result, BSF begins to deplete DM efficiently at earlier times, when the DM density is higher, and therefore has a larger impact on the relic abundance. • The Higgs counteracts the gluon-mediated repulsion in the octet states. For large enough mass of the interacting particles and coupling to the Higgs, colour-octet bound states exist. Their formation and decay contributes to the DM depletion rate in the early universe, even if only modestly due to their small binding energy. • The Higgs exchange increases the momentum transfer in the bound states and at the gluon emission vertex in the capture process, hence driving the strong coupling to smaller values. This somewhat quells the strength of the BSF processes at large α h . • The interference of the (gluon-generated) Coulomb potential and the (Higgs-generated) Yukawa potential influences the long-range effect of the latter, both in the scattering and the bound states. The enhancement of the effective DM annihilation rate due to the Higgs-mediated force, implies that the LP-NLP mass difference and/or the DM mass must be larger than previously predicted, in order for DM to attain the observed density via freeze-out. This is particularly important for collider searches, since a larger mass gap yields harder jets that are easier to probe. Similarly, the increased values of the predicted DM masswhich can lie even beyond the range considered here -strengthen the motivation for indirect searches in the multi-TeV regime. However, the accurate interpretation of the experimental constraints necessitates that the effects discussed here are considered within complete models, where various relevant technicalities can be treated properly, as discussed in section 5.3. Figure 3 . 3The annihilation processes are influenced by the Sommerfeld effect, a non-perturbative phenomenon that corresponds diagrammatically to the resummation of the infinite ladder of 2particle-irreducible (2PI) diagrams. The 2PI kernel for our model is shown infig. 1and includes both possible long-range interactions: the gluon and the Higgs exchange (left). The black blob represents the perturbative part of the various annihilation channels, shown infig. 2. Figure 5 . 5Leading order contributions to the radiative part of the capture into bound states via gluon emission (cf. section 4). Figure 6 . 6Left: γ 1,0 (λ, d h ) parametrises the ground-state binding energy,E 1,0 = −γ 2 1,0 κ 2 /(2µ), where κ ≡ µ(α B g + α h ).The two dotted black lines connect the points for which the binding energy is 50% and 90% between the two Coulomb limits, d h → 0 and d h → ∞. Comparing with a simple attractive Yukawa potential (λ = 0, red line), the superposition of an attractive (λ > 0) or repulsive (λ < 0) Coulomb component results in the Yukawa potential having a long-range effect for smaller and larger d h respectively. This is analogous to the features exhibited by the scattering states (see the s-wave Sommerfeld factor for a mixed Coulomb and Yukawa potential in ref.[40, fig. 2]).Right:The critical values d h,crit above which bound states exist in the mixed repulsive Coulomb and attractive Yukawa potential. The points are the numerical results, while the line shows the analytical fit(3.30). m (z)dz = [Γ (n + a + 1)/n!] δ n,m . accuracy of better than 3%. The value d 0 = d h,crit (λ = 0) 0.84 corresponds to a Yukawa potential only and is in agreement with the results of ref. [15, appendix A2]. 32a) ψ n m (q) = d 3 r ψ n m (r) e −iq r , ψ n m (r) = d 3 q (2π) 3ψ n m (q) e iq r . (3.32b) Note that the integrals (3.31) depend on the representations of the scattering and bound states. For simplicity we leave this dependence implicit in this section, but will denote it explicitly in section 4, as J [R S ,R B ] k,{n m} and Y [R S ,R B ] k,{n m} , where R S and R B stand for the scattering-state and bound-state colour representations respectively. Figure 7 . 7The effect of the running of α s on the scattering states and the Sommerfeld enhancement/suppression of the annihilation processes. Figure 9 . 9A colour-octet bound state exists only if the coupling α h that gives rise to the attractive Yukawa potential is sufficiently strong to overcome the repulsive gluon potential.Left: The equation α s (Q(α)) =α has a solution when the coloured lines of fixed α h intersect the horizontal black line. See text for discussion.Right:The minimum coupling α h, crit for the octet bound states to exist as a function of m X . Figure 12 . 12σv rel for annihilation into gluons and Higgs bosons, XX † → gg, hh (blue lines), and for the capture into the ground states via gluon emission, for the different colour transitions: [8] → [1] (red lines), [1] → [8] (light green lines), and [8] → [8] (dark green lines). All σv rel are normalised to the perturbative value for s-wave annihilation, σ 0 ≡ σ pert XX † →gg v rel + σ pert XX † →hh v rel . Lines of different dashing correspond to different values for α h , as denoted in the legends. Left: For m X = 1 TeV and the values of α h considered, only colour-singlet bound states exist. Right: For m X = 4 TeV and larger values of α h , there exist also colour-octet bound states. Figure 13 . 13Comparison of thermally averaged cross-sections vs. the time parameterx = m X /T . Left: We fix m X = 1 TeV, and consider different α h as denoted on the plot. The lines of different dashing denote annihilation (dashed), bound state formation [8] → [1] (wide-spaced dotted ), effective bound state formation [8] → [1] weighted by the fraction that decay rather than being ionized (densely-spaced dotted ), and total effective cross-section that depletes DM (solid line = dashed + densely-spaced dotted ). For the m X and α h considered, only colour-singlet bound states exist. Right: The values m X = 4 TeV and α h = 0.2 allow also for colour-octet bound states. We show the cross-sections for annihilation (dashed) and effective bound state formation for different colour transitions, [8] → [1] (dotted ), [1] → [8] (double-dot dashed ), [8] → [8] (dot dashed ). Figure 15 . 15As in fig. 14, but for α h = 0.1 and α h = 0.2. value of relative velocity of interacting particles in the scattering state v rel , v rel = |v rel |Expectation value of momentum in the scattering state in the CM frame k ≡ µv rel , k ≡ |k| Dimensionless variables that parametrise the long-range effects and the parts of the annihilation and BSF cross-sections that are evaluated numerically (colour representation R =1,8) Table 1 . 1Summary of notation. For the running of the strong coupling, see table 2. 32 ) 32where we included the ground-state contributions only, and recall that E[R] 10 are their binding energies. Equation(2.32) implies that at T |E[R] 10 |, the bound states contribute to the Sommerfeld factor of the colour configuration [R], by ). The factors inside the square brackets in eq. (4.5c) arise from the convolution of the scattering and bound state wavefunctions with the radiative vertices.Comparing eqs. (4.4) and (4.5), yields S C BSF (ζ S , ζ B ) = [(1 + ζ 2 B )/(16f c )] lim d h →∞ H. Two limiting cases of eqs. (4.5) are of particular interest: • For α h α B s , only colour-singlet bound states exist, hence the transition [8] → [1] is the only possible, with f c = 289/64. The corresponding cross-section is σ [8]→[1] BSF v rel = 2 7 17 2 3 5 πα BSF s,[1] α B s,[1] mX =4 TeV, αh = 0.2 〈σannvrel〉mX =1 TeV 〈σann vrel〉 〈σ BSF [8]→[1] vrel〉 〈σ BSF [8]→[1] vrel〉eff 〈σ XX † vrel〉eff 10 10 2 10 3 10 4 10 -24 10 -23 10 -22 10 -21 m X / T σ v rel [cm 3 /s] αh = 0 .0 2 αh = 0 .1 αh = 0 .2 〈σ BSF [8]→[1] vrel〉eff 〈σ BSF [1]→[8] vrel〉eff 〈σ BSF [8]→[8] vrel〉eff 〈σ XX † vrel〉eff Note that in ref.[15], a factor of (−i) has been missed in going from eq. (B.8b) to eq. (B.9b). 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We prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations with monotone drifts, which in particular contains a class of SDEs with reflection in a convex domain.

1104.5096

a9206710aaed9e0c8712ee5a6de743e3d961eacc

LARGE DEVIATIONS FOR MULTI-VALUED STOCHASTIC DIFFERENTIAL EQUATIONS * 27 Apr 2011 Jiagang Ren School of Mathematics and Computational Science Sun Yat-Sen University 510275GuangzhouGuangdongP.R.China Siyan Xu School of Mathematics and statistic Huazhong University of Science and Technology 430074WuhanHubeiP.R.China Xicheng Zhang School of Mathematics and statistic Huazhong University of Science and Technology 430074WuhanHubeiP.R.China School of Mathematics and Statistics The University of New South Wales 2052SydneyAustralia LARGE DEVIATIONS FOR MULTI-VALUED STOCHASTIC DIFFERENTIAL EQUATIONS * 27 Apr 2011arXiv:1104.5096v1 [math.PR] We prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations with monotone drifts, which in particular contains a class of SDEs with reflection in a convex domain. Introduction Consider the following multivalued stochastic differential equation (MSDE in short): dX(t) ∈ b(X(t))dt + σ(X(t))dW (t) − A(X(t))dt, X(0) = x ∈ D(A),(1) where A is a multivalued maximal monotone operator, which will be described below, W (t) = {W k (t), t 0, k ∈ N} is a sequence of independent standard Brownian motions on a filtered probability space (Ω, F , P ; (F t ) t 0 ), b : R m → R m and σ : R m → R m × l 2 are two continuous functions, l 2 stands for the Hilbert space of square summable sequences of real numbers. This type of MSDE was first studied by Cépa in [5,6]. He proved that if b and σ are Lipschitz continuous, then there exists a unique pair of processes (X(t), K(t)) such that X(t) = x + t 0 b(X(s))ds + t 0 σ(X(s))dW (s) − K(t), where K(t) is a process of finite variation (see Definition 2.3 below for more details). Recently, Zhang [14] extended Cépa's result to the infinite dimensional case, and relaxed the Lipschitz assumption on b to the monotone case. It should be noted that when A is the subdifferential of the indicator function of a convex subset of R m , the above MSDE is the same as the usual SDE with reflecting boundary in a convex domain (cf. [1,9]). Moreover, since the subdifferential of any lower semicontinuous convex function is a maximal monotone operator, Cépa's result can also be used to deal with the SDE with discontinuous coefficients. It is well known that there are many literatures to investigate the SDEs with reflecting boundary since the solutions of a large class of PDEs with Neumann boundary and mixed boundary conditions can be represented by the solution of such SDEs (cf. [1]). We now consider the following small perturbation of Eq.(1): dX ǫ (t) ∈ b(X ǫ (t))dt + √ ǫσ(X ǫ (t))dW (t) − A(X ǫ (t))dt, X ǫ (0) = x ∈ D(A), ǫ ∈ (0, 1].(2) The solution of this equation is denoted by (X ǫ (t, x), K ǫ (t, x)). We want to establish the large deviation principle of the law of X ǫ (t, x) in the space S := C([0, T ] × D(A); D(A)), namely, the asymptotic estimates of probabilities P (X ǫ ∈ Γ), where Γ ∈ B(S). In [1], Anderson and Orey considered the same small random perturbation for the dynamical system with reflecting boundary in smooth domain, and obtained the Freidlin-Wentzell's large deviation estimates in C([0, T ]; D(A)). They assumed that the coefficients are bounded and Lipschitz continuous, and the diffusion coefficient is non-degenerate. Using the contraction principle, Cépa [5] only considered the large deviation principle of one dimensional case based on an explicit construction of the solution (cf. [13]). The multi-dimensional case is still open. Compared with the usual SDE, i.e., A = 0, most of the difficulties come from the presence of the process of finite variation, K(t). One only knows that t → K(t) is continuous, and could not prove any further regularity such as Hölder continuity. Therefore, the classical method of time discretization is almost inapplicable (cf. [7]). Our method is based on the recently well developed weak convergence approach due to Dupuis and Ellis [8] (see also [2,3]). This method has been proved to be very effective for various systems (cf. [11, 15, 4, 16, 10, 12, etc.]). In the situation considered in the present paper, however, since we cannot prove the following uniform estimate as in [11]: for any p 2 and s, t ∈ [0, T ], x, y ∈ D(A) sup ǫ∈(0,1) E|X ǫ (t, x) − X ǫ (s, y)| 2p C(|t − s| p + |x − y| 2p ), we cannot obtain the tightness of the laws of X ǫ (t, x) in S. Some technical difficulties for verifying the conditions (LD) 1 and (LD) 2 below need to be overcome. In Section 2, we recall some well known facts about the MSDE and a criterion for Laplace principle. In Section 3, we present our main result and give a detailed proof. Throughout the paper, C with or without indexes will denote different constants (depending on the indexes) whose values are not important. Preliminaries We first give some notions and notations about multivalued operators. Let 2 R m be the set of all subsets of R m . A map A : R m → 2 R m is called a multivalued operator. Given such a multivalued operator A, define: D(A) := {x ∈ R m : A(x) = ∅}, Im(A) := ∪ x∈D(A) A(x), Gr(A) := {(x, y) ∈ R 2m : x ∈ R m , y ∈ A(x)}. We recall the following definitions. Definition 2.1. (1) A multivalued operator A is called monotone if y 1 − y 2 , x 1 − x 2 R m 0, ∀(x 1 , y 1 ), (x 2 , y 2 ) ∈ Gr(A). 2 (2) A monotone operator A is called maximal monotone if for each (x, y) ∈ Gr(A), y − y ′ , x − x ′ R m 0, ∀(x ′ , y ′ ) ∈ Gr(AI O (x) = 0, if x ∈ O, +∞, if x / ∈ O. The subdifferential of I O is given by ∂I O (x) = {y ∈ R m : y, x − z R m 0, ∀z ∈ O} =    ∅, if x / ∈ O, {0}, if x ∈ Int(O), Λ x , if x ∈ ∂O, where Int(O) is the interior of O and Λ x is the exterior normal cone at x. One can check that ∂I O is a multivalued maximal monotone operator in the sense of Definition 2.1. We now give the precise definition of the solution to Eq.(1). Definition 2.3. A pair of continuous and (F t )-adapted processes (X, K) is called a so- lution of Eq.(1) if (i) X(0) = x, and for all t 0, X(t) ∈ D(A) a.s.; (ii) K(0) = 0 a.s. and K is of finite variation; (iii) dX(t) = b(X(t))dt + σ(X(t))dW (t) − dK(t), 0 t < ∞, a.s.; (iv) for any continuous and (F t )-adapted processes (α, β) with (α(t), β(t)) ∈ Gr(A), ∀t ∈ [0, +∞), the measure X(t) − α(t), dK(t) − β(t)dt 0 a.s.. We now recall an abstract criterion for Laplace principle, which is equivalent to the large deviation principle (cf. [3,4,15]). It is well known that there exists a Hilbert space so that l 2 ⊂ U is Hilbert-Schmidt with embedding operator J and {W k (t), k ∈ N} is a Brownian motion with values in U, whose covariance operator is given by Q = J • J * . For example, one can take U as the completion of l 2 with respect to the norm generated by the scalar product h, h ′ U := ∞ k=1 h k h ′ k k 2 1 2 , h, h ′ ∈ l 2 . For a Polish space B, we denote by B(B) its Borel σ-field, and by C T (B) the continuous function space from [0, T ] to B, which is endowed with the uniform distance so that C T (B) is still a Polish space. Define ℓ 2 T := h = · 0ḣ (s)ds :ḣ ∈ L 2 (0, T ; l 2 )(3) with the norm h ℓ 2 T := T 0 ḣ (s) 2 l 2 ds 1/2 , where the dot denotes the generalized derivative. Let µ be the law of the Brownian motion W in C T (U). Then (ℓ 2 T , C T (U), µ) forms an abstract Wiener space. For T, N > 0, set D N := {h ∈ ℓ 2 T : h ℓ 2 T N} and A T N := h : [0, T ] → l 2 is a continuous and (F t )-adapted process, and for almost all ω, h(·, ω) ∈ D N .(4) We equip D N with the weak convergence topology in ℓ 2 T . Then D N is metrizable as a compact Polish space. Let S be a Polish space. A function I : S → [0, ∞] is given. Definition 2.4. The function I is called a rate function if for every a < ∞, the set {f ∈ S : I(f ) a} is compact in S. Let {Z ǫ : C T (U) → S, ǫ ∈ (0, 1)} be a family of measurable mappings. Assume that there is a measurable map Z 0 : ℓ 2 T → S such that (LD) 1 For any N > 0, if a family {h ǫ , ǫ ∈ (0, 1)} ⊂ A T N (as random variables in D N ) converges in distribution to h ∈ A T N , then for some subsequence ǫ k , Z ǫ k · + hǫ k (·) √ ǫ k converges in distribution to Z 0 (h) in S. (LD) 2 For any N > 0, if {h n , n ∈ N} ⊂ D N weakly converges to h ∈ ℓ 2 T , then for some subsequence h n k , Z 0 (h n k ) converges to Z 0 (h) in S. For each f ∈ S, define I(f ) := 1 2 inf {h∈ℓ 2 T : f =Z 0 (h)} h 2 ℓ 2 T ,(6) where inf ∅ = ∞ by convention. Then under (LD) 2 , I(f ) is a rate function. We recall the following result due to [3] (see also [17,Theorem 4.4]). Theorem 2.5. Under (LD) 1 and (LD) 2 , {Z ǫ , ǫ ∈ (0, 1)} satisfies the Laplace principle with the rate function I(f ) given by (6). More precisely, for each real bounded continuous function g on S: lim ǫ→0 ǫ log E µ exp − g(Z ǫ ) ǫ = − inf f ∈S {g(f ) + I(f )}.(7) In particular, the family of {Z ǫ , ǫ ∈ (0, 1)} satisfies the large deviation principle in (S, B(S)) with the rate function I(f ). More precisely, let ν ǫ be the law of Z ǫ in (S, B(S)), then for any B ∈ B(S) − inf f ∈B o I(f ) lim inf ǫ→0 ǫ log ν ǫ (B) lim sup ǫ→0 ǫ log ν ǫ (B) − inf f ∈B I(f ), where the closure and the interior are taken in S, and I(f ) is defined by (6). Main Result and Proof We assume that (H1) A is a maximal monotone operator with non-empty interior, i.e., Int(D(A)) = ∅; (H2) σ and b are continuous functions and satisfy that for some C σ , C b > 0 and all x, y ∈ R m σ(x) − σ(y) L 2 (l 2 ;R m ) C σ |x − y|, x − y, b(x) − b(y) R m C b |x − y| 2 , where L 2 (l 2 ; R m ) denotes the Hilbert-Schmidt space and | · | denotes the norm in R m , and for some C ′ b > 0 and n ∈ N |b(x)| C ′ b (1 + |x| n ). It is well known that under (H1) and (H2), there exists a unique solution (X ǫ , K ǫ ) to Eq.(2) in the sense of Definition 2.3 (cf. [14]). Our main result is stated as follows: I(f ) := 1 2 inf {h∈ℓ 2 T : f =X h } h 2 ℓ 2 T ,(8) where X h (t, x) solves the following equation: dX h (t) ∈ b(X h (t))dt + σ(X h (t))ḣ(t)dt − A(X h (t))dt, X h (0) = x. For proving this result, by Theorem 2.5, the main task is to verify (LD) 1 and (LD) 2 with S := C([0, T ] × D(A); D(A)), Z ǫ = X ǫ , Z 0 (h) = X h . This will be done in Lemmas 3.7 and 3.8 below. Let h ǫ ∈ A T N converge almost surely to h ∈ A T N as random variables in ℓ 2 T , and (X ǫ,hǫ , K ǫ,hǫ ) solve the following control equation: X ǫ,hǫ (t) = x + t 0 b(X ǫ,hǫ (s))ds + t 0 σ(X ǫ,hǫ (s))ḣ ǫ (s)ds + √ ǫ t 0 σ(X ǫ,hǫ (s))dW (s) − K ǫ,hǫ (t),(9) which can be solved by Girsanov's theorem, and (X h , K h ) solve the following deterministic equation: X h (t) = x + t 0 b(X h (s))ds + t 0 σ(X h (s))ḣ(s)ds − K h (t).(10) Let |K| s t denote the total variation of K on [s, t]. We recall the following result due to Cépa [6] (see also [14, Moreover, for any pairs of (X, K) and (X,K) with the property (iv) of Definition 2.3 X(t) −X(t), dK(t) − dK(t) R m 0.(12) Using this property, we first prove the following uniform estimates. 5 Lemma 3.3. For any p 1, there exists C p,T,N > 0 such that for any ǫ ∈ (0, 1) and x, y ∈ D(A) E sup t∈[0,T ] |X ǫ,hǫ (t, x) − X ǫ,hǫ (t, y)| 2p C p,T,N |x − y| 2p .(13) Proof. Set Z ǫ (t) := X ǫ,hǫ (t, x) − X ǫ,hǫ (t, y) and Λ(s) := σ(X ǫ,hǫ (s, x)) − σ(X ǫ,hǫ (s, y)). By Itô's formula, (H2) and (12), we have for any p 1 |Z ǫ (t)| 2p = |Z ǫ (0)| 2p + 2p t 0 |Z ǫ (s)| 2p−2 Z ǫ (s), b(X ǫ,hǫ (s, x)) − b(X ǫ,hǫ (s, y)) R m ds +2p t 0 |Z ǫ (s)| 2p−2 Z ǫ (s), Λ(s)ḣ ǫ (s) R m ds +2p √ ǫ t 0 |Z ǫ (s)| 2p−2 Z ǫ (s), Λ(s)dW (s) R m −2p t 0 |Z ǫ (s)| 2p−2 Z ǫ (s), dK ǫ,hǫ (s, x) − dK ǫ,hǫ (s, y) R m −p t 0 |Z ǫ (s)| 2p−2 ( Λ(s) 2 + 2(p − 1) Z ǫ (s), Λ(s)Λ * (s)Z ǫ (s) R m /|Z ǫ (s)| 2 )ds |Z ǫ (0)| 2p + C t 0 |Z ǫ (s)| 2p ds + 2p t 0 |Z ǫ (s)| 2p · ḣ ǫ (s) l 2 ds +2p √ ǫ t 0 |Z ǫ (s)| 2p−2 Z ǫ (s), Λ(s)dW (s) R m . Set g(t) := E sup s∈[0,t] |Z ǫ (s)| 2p . By BDG's inequality and Young's inequality, we have for any δ > 0 E sup t ′ ∈[0,t] t ′ 0 |Z ǫ (s)| 2p−2 Z ǫ (s), Λ(s)dW (s) R m CE t 0 |Z ǫ (s)| 4p−4 Λ(s) * Z ǫ (s) 2 l 2 ds 1/2 CE sup s∈[0,t] |Z ǫ (s)| 2p t 0 |Z ǫ (s)| 2p ds 1/2 δ · g(t) + C δ t 0 E|Z ǫ (s)| 2p ds.(14) Similarly, we have E t 0 |Z ǫ (s)| 2p · ḣ ǫ (s) l 2 ds NE t 0 |Z ǫ (s)| 4p ds 1/2 δ · g(t) + C δ,N t 0 E|Z ǫ (s)| 2p ds.(15) Letting δ = 1/4 in (14) and (15) and combining the above calculations, we get g(t) g(0) + 1 2 g(t) + C t 0 E|Z ǫ (s)| 2p ds 2g(0) + 2C t 0 g(s)ds, which gives the desired estimate by Gronwall's inequality. E sup t∈[0,T ] |X ǫ,hǫ (t, x)| 2p + E|K ǫ,hǫ (·, x)| 0 T C p,T,N,x .(16) Proof. First of all, as in the proof of Lemma 3.3 we can prove that E sup t∈[0,T ] |X ǫ,hǫ (t, x)| 2p C p,T,N,x .(17) Let a ∈ Int(D(A)) be as in Proposition 3.2. By Itô's formula, (11) and (H2), we have 1 2 |X ǫ,hǫ (t) − a| 2 = 1 2 |x − a| 2 + t 0 X ǫ,hǫ (s) − a, b(X ǫ,hǫ (s)) R m ds + t 0 X ǫ,hǫ (s) − a, σ(X ǫ,hǫ (s))ḣ ǫ (u) R m ds + √ ǫ t 0 X ǫ,hǫ (s) − a, σ(X ǫ,hǫ (s))dW (s) R m − t 0 X ǫ,hǫ (s) − a, dK ǫ,hǫ (s) R m + ǫ 2 t 0 σ(X ǫ,hǫ (s)) 2 L 2 (l 2 ;R m ) ds 1 2 |x − a| 2 + C b t 0 |X ǫ,hǫ (s) − a| 2 ds + t 0 X ǫ,hǫ (s) − a, b(a) R m ds +N t 0 |σ(X ǫ,hǫ (s)) * (X ǫ,hǫ (s) − a)| 2 ds 1/2 + √ ǫ t 0 X ǫ,hǫ (s) − a, σ(X ǫ,hǫ (s))dW (u) R m + µγt −γ|K ǫ,hǫ | 0 t + µ t 0 |X ǫ,hǫ (s) − a|ds + C 2 σ ǫ 2 t 0 (|X ǫ,hǫ (s)| + σ(0)) 2 ds. The desired estimate now follows by (17). σ(X h (s, x))(ḣ ǫ (s) −ḣ(s))ds. The following lemma is easy by Ascoli-Arzela's lemma. Lemma 3.5. w ǫ (·, x) converges a.s. to zero in C([0, T ], D(A)). We now prove the following key lemma. Lemma 3.6. X ǫ,hǫ defined by (9) converges in probability to X h defined by (10) in S. Proof. Set v ǫ (t) := v ǫ (t, x) := X ǫ,hǫ (t, x) − X h (t, x). Then v ǫ (t) = K ǫ,hǫ (t) − K h (t) + t 0 (b(X ǫ,hǫ (s)) − b(X h (s)))ds + t 0 (σ(X ǫ,hǫ (s))ḣ ǫ (s) − σ(X h (s))ḣ(s))ds + √ ǫ t 0 σ(X ǫ,hǫ (s))dW (s). By Itô's formula, we have |v ǫ (t)| 2 = 2 t 0 v ǫ (s), dK ǫ,hǫ (s) − dK h (s) R m +2 t 0 v ǫ (s), b(X ǫ,hǫ (s)) − b(X h (s)) R m ds +2 t 0 v ǫ (s), (σ(X ǫ,hǫ (s)) − σ(X h (s)))ḣ ǫ (s) R m ds +2 t 0 v ǫ (s), σ(X h (s))(ḣ ǫ (s) −ḣ(s)) R m ds +2 √ ǫ t 0 v ǫ (s), σ(X ǫ,hǫ (s))dW (s) R m +ǫ t 0 σ(X ǫ,hǫ (s)) 2 L 2 (l 2 ;R m ) ds =: I ǫ 1 (t) + I ǫ 2 (t) + I ǫ 3 (t) + I ǫ 4 (t) + I ǫ 5 (t) + I ǫ 6 (t) . It is clear that by (12) I ǫ 1 (t) 0 and I ǫ 2 (t) 2C b t 0 |v ǫ (s)| 2 ds. By BDG's inequality and (H2) we also have E sup t∈[0,T ] |I ǫ 5 (t)| + E sup t∈[0,T ] |I ǫ 6 (t)| C √ ǫ. As estimating (15) we have E sup s∈[0,t] |I ǫ 3 (s)| 1 2 E sup s∈[0,t] |v ǫ (s)| 2 + C t 0 E|v ǫ (s)| 2 ds. 8 Set g(t) := E sup s∈[0,t] |v ǫ (s)| 2 . Then we have g(t) 1 2 g(t) + C √ ǫ + E sup s∈[0,t] |I ǫ 4 (s)| + C t 0 E|v ǫ (s)| 2 ds, which implies that g(t) C √ ǫ + 2E sup s∈[0,T ] |I ǫ 4 (s)| + C t 0 g(s)ds. By Gronwall's inequality we get E sup s∈[0,T ] |v ǫ (s)| 2 C √ ǫ + CE sup s∈[0,T ] |I ǫ 4 (s)| .(19) We now deal with the hard term I ǫ 4 . By Itô's formula again, we have For any x ∈ R m , let x δ denote the left-lower corner point in δZ m so that 1 2 I ǫ 4 (t) = v ǫ (t), w ǫ (t) R m − t 0 w ǫ (s)d(K ǫ,hǫ (s) − K h (s)) − t 0 w ǫ (s)(b(X ǫ,hǫ (s)) − b(X h (s)))ds − t 0 w ǫ (s)(σ(X ǫ,hǫ (s))ḣ ǫ (s) − σ(X h (s))ḣ(s))ds − √ ǫ t 0 w ǫ (s)σ(X ǫ,|x − x δ | δ. Noting that ξ n,ǫ 2ξ δ n,ǫ + 2 sup x∈D(A),|x| n sup t∈[0,T ] |v ǫ (t, x) − v ǫ (t, x δ )| 2 , we have for any β > 0 and some α > 0 P (ξ n,ǫ > 4β) P (ξ δ n,ǫ + sup |v ǫ (t, x) − v ǫ (t, x δ )| 2 /β P (ξ δ n,ǫ > β) + Cδ α /β, where the last step is due to Lemma 3.3 and Kolmogorov's criterion. First letting δ be small enough, then ǫ go to zero, we then obtain lim ǫ→0 P (ξ n,ǫ > 4β) = 0. which yields the desired result. Proof. Let h ǫ be a sequence in A T N converge to h in distribution. Since D N is compact and the law of W is tight, {h ǫ , W } is tight in D N × C T (U) by the definition of tightness. Without loss of generality, we assume the law of {h ǫ , W } weakly converges to µ. Then the law of h is just µ(·, C T (U)). Indeed, for any bounded continuous function g on D N , we have E(g(h)) = lim n→∞ E(g(h ǫ )) = D N g(h)µ(dh, C T (U)). By Skorohod's representation theorem, there are (Ω,P ) and {h ǫ ,W ǫ } and {h,W } such that (1) (h ǫ ,W ǫ ) a.s. converges to (h,W ); (2) (h ǫ ,W ǫ ) has the same law as (h ǫ , W ); (3) The law of {h,W } is µ, and the law of h is the same ash. Using Lemma 3.6, we get Φ( 1 √ ǫ · 0ḣ ǫ ds +W ǫ ) → Xh, in probability, where Φ is the strong solution functional(cf. [9]). From this, we derive Φ( 1 √ ǫ · 0ḣ ǫ ds + W ) → X h , in distribution. Thus, (LD) 1 holds. Similar to the proof of Lemma 3.6, one can easily verify that Thus, by Lemmas 3.7, 3.8 and Theorem 2.5, we have proved Theorem 3.1. Theorem 3. 1 . 1Assume that (H1) and (H2) hold. Then the family of {X ǫ (t, x), ǫ ∈ (0, 1)} satisfies the large deviation principle in S := C([0, T ] × D(A); D(A)) with the rate function given by Proposition 3 . 2 . 32Under (H1), there exist a ∈ R m , γ > 0, µ 0 such that for any pair of (X, K) with the property (iv) of Definition 2.3 and all 0 s < t T t s X(r) − a, dK(r) R m γ|K| s t − µ t s |X(r) − a|dr − γµ(t − s). Lemma 3. 4 . 4For any p 1 and x ∈ D(A), there exists C p,T,N,x > 0 such that for any ǫ ∈ [0, 1) |v ǫ (s)| 2 = 0. Thus, we have proven that for all x ∈ D(A) sup t∈[0,T ] |v ǫ (t, x)| 2 → 0, in probability. We now strengthen it by Lemma 3.3 to ξ n,ǫ := sup t∈[0,T ],x∈D(A),|x| n |v ǫ (t, x)| 2 → 0, in probability. Set D δ n := D(A) ∩ {x ∈ R m : |x| n} ∩ δZ m , where δ > 0 and δZ m denotes the grid in R m with edge length δ. It is clear that there are only finite many points in D δ n . Hence ξ δ n,ǫ := sup t∈[0,T ],x∈D δ n |v ǫ (t, x)| 2 → 0, in probability. x∈D(A),|x| n sup t∈[0,T ] |v ǫ (t, x) − v ǫ (t, x δ )| 2 2β) ǫ (t, x) − v ǫ (t, x δ )| Lemma 3. 8 . 8(LD) 2 holds. ) . )Examples 2.2. Suppose that O is a closed convex subset of R m , and I O is the indicator function of O, i.e, |v ǫ (t)| R .Moreover, by BDG's inequality we havehǫ (s))dW (s) =: I ǫ 41 (t) + I ǫ 42 (t) + I ǫ 43 (t) + I ǫ 44 (t) + I ǫ 45 (t). For any δ > 0 and R > 0, we have P sup t∈[0,T ] |I ǫ 41 (t)| δ = P sup t∈[0,T ] |I ǫ 41 (t)| δ; sup t∈[0,T ] |v ǫ (t)| < R +P sup t∈[0,T ] |I ǫ 41 (t)| δ; sup t∈[0,T ] |v ǫ (t)| R P sup t∈[0,T ] |w ǫ (t)| δ/R + P sup t∈[0,T ] By Lemma 3.5 and (16) we know lim ǫ→0 P sup t∈[0,T ] |I ǫ 41 (t)| δ = 0. Noting that sup t∈[0,T ] |I ǫ 42 (t)| sup s∈[0,T ] |w ǫ (s)| · (|K ǫ,hǫ (·)| 0 T + |K h (·)| 0 T ), as above, we also have sup t∈[0,T ] |I ǫ 42 (t)| → 0 in probability. Similarly, we have sup t∈[0,T ] |I ǫ 43 (t)| + sup t∈[0,T ] |I ǫ 44 (t)| → 0 in probability. √ ǫE sup t∈[0,T ] |I ǫ 45 (t)| C √ ǫ. Combining the above calculations, we get sup t∈[0,T ] |I ǫ 4 (t)| → 0 in probability. It is easy to see by (16) that sup ǫ∈(0,1) E sup t∈[0,T ] |I ǫ 4 (t)| 2 < +∞. Hence lim ǫ→0 E sup t∈[0,T ] |I ǫ 4 (t)| = 0. Substituting this into (19) we obtain lim ǫ→0 E sup s∈[0,T ] Small Random Perturbation of Dynamical Systems with Refecting Boundary. R F Anderson, S Orey, Nagoya Math. J. 60Anderson, R.F. and Orey, S.: Small Random Perturbation of Dynamical Systems with Refecting Boundary. Nagoya Math. J., Vol.60, 189-216(1976). A variational representation for certain functionals of Brownian motion. M Boué, P Dupuis, Ann. of Prob. 264Boué, M. and Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. of Prob., Vol. 26, No.4, 1641-1659(1998). A variational representation for positive functionals of infinite dimensional Brownian motion. A Budhiraja, P Dupuis, Acta Univ. Wratislav. No. 201Probab. Math. Statist.Budhiraja, A. and Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Statist. 20, no.1, Acta Univ. Wratislav. No. 2246, 39-61(2000). Large deviations for infinite dimensional stochastic dynamical systems. A Budhiraja, P Dupuis, V Maroulas, to appear in Ann. of Prob.Budhiraja, A., Dupuis, P. and Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems, to appear in Ann. of Prob.. Équations différentielles stochasticcques multivoques. E Cépa, Lect. Notes in Math. Sém. Prob. XXIX. SpringerCépa, E.:Équations différentielles stochasticcques multivoques. Lect. Notes in Math. Sém. Prob. XXIX, Springer, Berlin, 86-107(1995). Probleme De Skorohod Multivoque. E Cépa, Ann. of Prob. 262Cépa, E.: Probleme De Skorohod Multivoque. Ann. of Prob., Vol.26, No.2, 500-532(1998). Large deviations. J D Deuschel, D W Stroock, Academic PressBoston, New YorkDeuschel, J.D. and Stroock, D.W.: Large deviations. Academic Press, Boston, New York, 1988. A Weak Convergence Approach to the Theory of Large Deviations. P Dupuis, R S Ellis, WileyNew-YorkDupuis, P. and Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New-York, 1997. N Ikeda, S Watanabe, Stochastic Differential Equations and Diffusion Processes. Kodansha, Tokyo/North-Holland, Amsterdam2nd ed.Ikeda, N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes, 2nd ed., Kodansha, Tokyo/North-Holland, Amsterdam, 1989. Large deviations for stochastic evolution equations with small multiplcative noise. W Liu, PreprintLiu, W.: Large deviations for stochastic evolution equations with small multiplcative noise. Preprint. Freidlin-Wentzell's large deviations for homeomorphism flows of non-Lipschitz SDEs. J Ren, X Zhang, Bull. Sci. Math. 2 Serie. 129Ren, J. and Zhang, X.: Freidlin-Wentzell's large deviations for homeomorphism flows of non- Lipschitz SDEs. Bull. Sci. Math. 2 Serie, Vol 129/8 pp 643-655(2005). Freidlin-Wentzell's Large Deviations for Stochastic Evolution Equations. J Ren, X Zhang, J. Func. Anal. 254Ren, J. and Zhang, X.: Freidlin-Wentzell's Large Deviations for Stochastic Evolution Equations. J. Func. Anal., Vol.254, 3148-3172(2008). Explicit solutions for multivalued stochastic differential equations. S Xu, Statist. Probab. Lett. 78Xu, S.: Explicit solutions for multivalued stochastic differential equations, Statist. Probab. Lett. 78, 2281-2292(2008). Skorohod problem and multivalued stochastic evolution equations in Banach spaces. X Zhang, Bull. Sci. Math. 1312Zhang, X.: Skorohod problem and multivalued stochastic evolution equations in Banach spaces. Bull. Sci. Math.131(2),175-217(2007). Euler Schemes and Large Deviations for Stochastic Volterra Equations with Singular Kernels. X Zhang, Journal of Differential Equation. 224Zhang, X.: Euler Schemes and Large Deviations for Stochastic Volterra Equations with Singular Kernels. Journal of Differential Equation, Vol. 224, 2226-2250(2008). X Zhang, Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations. PreprintZhang, X.: Stochastic Volterra Equations in Banach Spaces and Stochastic Partial Differential Equations. Preprint. A variational representation for random functionals on abstract Wiener spaces. X Zhang, PreprintZhang, X.: A variational representation for random functionals on abstract Wiener spaces. Preprint.

We show that a group admits a non-zero homogeneous quasimorphism if and only if it admits a certain type of action on a poset. Our proof is based on a construction of quasimorphisms which generalizes Poincaré-Ghys' construction of the classical translation number quasimorphism. We then develop a correspondence between quasimorphisms and actions on posets, which allows us to translate properties of orders into properties of quasimorphisms and vice versa. Concerning examples we obtain new realizations of the Rademacher quasimorphism, certain Brooks type quasimorphisms, the Dehornoy floor quasimorphism as well as Guichardet-Wigner quasimorphisms on simple Hermitian Lie groups of tube type. The latter we relate to Kaneyuki causal structures on Shilov boundaries, following an idea by Clerc and Koufany. As applications we characterize those quasimorphisms which arise from circle actions, and subgroups of Hermitian Lie groups with vanishing Guichardet-Wigner quasimorphisms.

1106.6307

038a96c3d0d549ff3e30bef4e3309e46410edbbb

QUASI-TOTAL ORDERS AND TRANSLATION NUMBERS Gabi Ben Simon And Tobias Hartnick QUASI-TOTAL ORDERS AND TRANSLATION NUMBERS We show that a group admits a non-zero homogeneous quasimorphism if and only if it admits a certain type of action on a poset. Our proof is based on a construction of quasimorphisms which generalizes Poincaré-Ghys' construction of the classical translation number quasimorphism. We then develop a correspondence between quasimorphisms and actions on posets, which allows us to translate properties of orders into properties of quasimorphisms and vice versa. Concerning examples we obtain new realizations of the Rademacher quasimorphism, certain Brooks type quasimorphisms, the Dehornoy floor quasimorphism as well as Guichardet-Wigner quasimorphisms on simple Hermitian Lie groups of tube type. The latter we relate to Kaneyuki causal structures on Shilov boundaries, following an idea by Clerc and Koufany. As applications we characterize those quasimorphisms which arise from circle actions, and subgroups of Hermitian Lie groups with vanishing Guichardet-Wigner quasimorphisms. This article continues our investigation of the relation between biinvariant partial orders on groups and homogeneous quasimorphisms initiated in [2]. There we observed that every homogeneous quasimorphism arises as a multiple of the growth function (in the sense of Eliashberg and Polterovich [7]) of a bi-invariant partial order, but left open the converse question, which asks for a characterization of those bi-invariant partial orders that give rise to quasimorphisms via their growth functions. In the present article we introduce a special class of bi-invariant partial orders on groups, which we call quasi-total orders. We then establish the following results: (a) Growth functions of quasi-total orders are homogeneous quasimorphism (see Theorem 1.2). (b) Conversely, every quasimorphism arises as the growth function of some quasi-total order (up to a multiplicative constant, see Theorem 1.2 and Proposition 1.7). This quasi-total order is not unique in general. (c) Special classes of quasi-total orders give rise to special classes of quasimorphisms. Concerning (c) we will actually prove the following more specific statements: (c1) We describe, which quasi-total orders correspond to homomorphisms (see Proposition 1.3). (c2) We describe, which quasi-total orders correspond to pullbacks of the classical rotation number via some action on the circle (see Proposition 1.9); here our treatment is inspired by [15]. (c3) We also discuss how topological assumptions on the order in question influence the behaviour of the corresponding quasimorphism. Here the concrete consequences are more technical to state; see Proposition 1.10 below. Our approach was originally motivated from our study of quasimorphisms on (finite-dimensional) simple Lie groups. In this context we obtain notably a new interpretation of results of Clerc and Koufany [5], see Theorem 1.11. However, the methods developed in this article apply far beyond this case. For example, our construction includes the construction from [15] as a special case and also provides new constructions of the Rademacher quasimorphism on PSL 2 (Z) and the Brooks quasimorphisms on free groups. Since quasi-total orders are induced by certain actions on posets (to be described below) we obtain from (a) and (b) above a complete characterization of groups admitting a nonzero homogeneous quasimorphism in terms of actions on posets. Unlike existing cohomological characterizations our characterization does not require any local compactness or second countability assumptions on the underlying group, and hence seems particular amenable to the study of infinite-dimensional groups. We now summarize the results of this article in the same order in which they appear in the body of the text. Before we can start we have to recall some basic definitions: Given a group G, a partial order ≤ on G is called bi-invariant if g ≤ h implies both gk ≤ hk and kg ≤ kh for all g, h, k ∈ G. Equivalently, the associated order semigroup G + := {g ∈ G | g ≥ e} is a conjugation-invariant monoid satisfying the pointedness condition G + ∩ (G + ) −1 = {e}. In this case we call the pair (G, ≤) a partially bi-ordered group and refer to the elements of 1 G ++ := {g ∈ G + \ {e} | ∀h ∈ G∃n ∈ N 0 : g n ≥ h} as dominants of the ordered group (G, ≤). We will always assume that (G, ≤) is admissible meaning that G ++ = ∅. In this case the relative growth function γ : G ++ × G → R, (g, h) → γ(g, h) 1 Here and in the sequel we distinguish the set N := {1, 2, . . . } of positive integers and the set of non-negative integers N 0 := N ∪ {0}. given by γ(g, h) := lim n→∞ inf{p ∈ Z | g p ≥ h n } n contains valuable numerical information on the order. (This function was introduced in [7].) Fixing g ∈ G ++ we obtain a growth function γ g : G → R, γ g (h) := γ(g, h). This function is always homogeneous, i.e. satisfies γ g (h n ) = nγ g (h) for all n ∈ N; here we ask whether it happens to be a quasimorphism, i.e. whether it satisfies D(γ g ) := sup h,k∈G |γ g (hk) − γ g (h) − γ g (k)| < ∞. If this is the case, we refer to the constant D(γ g ) as the defect of γ g . With this terminology understood we can now state the problem to be solved in this article: Problem 1. Describe a class C of bi-invariants partial orders on groups such that (i) the growth functions of any order ≤∈ C are nonzero homogeneous quasimorphisms; (ii) every nonzero homogeneous quasimorphism arises as the growth function of some ≤∈ C (say, up to a positive multiple). We now aim to describe a class of orders which solves Problem 1. We start by observing that bi-invariant orders on G arise from (effective) G-actions on posets (not necessarily order-preserving). Indeed, if G acts effectively on a poset (X, ) then we obtain a bi-invariant partial order ≤ on G by setting g ≤ h :⇔ ∀k ∈ G ∀x ∈ X : (kg).x (kh).x. We refer to ≤ as the order induced from the G-action on (X, ). A priori this order may be trivial, but it will be non-trivial in the cases we discuss below. Since every bi-invariant partial order is induced from itself (via the left action of G on itself), specifying a class of bi-invariant partial orders is equivalent to specifying a class of effective G-posets. Given a half-space filtration {H n } n∈Z of X and elements a, b ∈ X we denote by h(a) := sup{n ∈ Z | a ∈ H n } the height of a and by h(a, b) := h(a) − h(b) the relative height of a over b with respect to {H n }. A triple (X, , {H n } n∈Z ) is called a half-space order of width bounded by w := w(X, , {H n }) if (X, ) is a poset, {H n } is a halfspace filtration of X and ∀a, b ∈ X : h(a, b) ≥ w ⇒ a b. In this case we say that a group G acts on (X, , {H n } n∈Z ) by quasiautomorphisms of defect bounded by d if ∀g ∈ G, a, b ∈ X : |h(ga, gb) − h(a, b)| ≤ d. If G moreover acts effectively, then the induced order ≤ on G is called a quasi-total order of defect bounded by d provided the action is unbounded in the sense that ∃g ∈ G, a ∈ X : lim n→±∞ h(g n .a) = ±∞. (1.1) Note that the G-action on X does not necessarily preserve the order; however the above assumptions guarantee that the order is quasipreserved in some sense. Using the above terminology we can now state our solution to Problem 1: Theorem 1.2. The class C of quasi-total orders solves Problem 1, i.e. (i) the growth functions of any quasi-total order ≤ are nonzero homogeneous quasimorphisms; (ii) every nonzero homogeneous quasimorphism on a given group G arises as the growth function of some quasi-total order on that group (up to a positive multiple). In this article we will describe the following subclasses of the class of quasi-total orders more closely (definitions will be given below; other classes of examples will be discussed in subsequent work): • the standard halfspace order on the real line, corresponding to the classical translation number; • quasi-total orders induced from planar group embeddings; examples include the Rademacher quasimorphism and various Brooks type quasimorphisms; • quasi-total orders induced from quasi-total triples; this largest class includes the following subclasses: bi-invariant admissible total orders (which lead to homomorphisms as growth functions); orders induced from total triples of the form (G, , T ); these are closely related to actions on the circle; orders induced from smooth quasi-total orders. These are related to causal structures on manifolds; examples include Guichardet-Wigner quasimorphisms on Hermitian Lie groups of tube type. We now turn to each of this classes individually: The simplest example of a half-space order is given by (R, ≤, {[n, ∞)}), where ≤ denotes the usual total order on R. If Homeo + (S 1 ) denotes the group of orientation-preserving homeomorphisms of the circle and Homeo + Z (R) its universal covering group, then the Homeo + (S 1 )-action on the circle lifts to an action of Homeo + Z (R) on the real line. This action turns out to be by quasi-automorphisms with respect to (R, ≤, {[n, ∞)}), hence gives rise to a quasi-total order, and thereby to a quasimorphism T on Homeo + Z (R). This quasimorphism can be traced all the way back to the work of Poincaré [21,22], where a R/Z-valued continuous map on Homeo + (S 1 ) is constructed. Namely, it turns out that this famous rotation number lifts to a real-valued map on the universal covering group Homeo + Z (R) of Homeo + (S 1 ). This lift, which is sometimes called translation number, is precisely our quasimorphism T . The observation that the translation number is a quasimorphism appears first in Ghys' fundamental work on group actions on the circle [9,10,11,1], where it is related to the universal bounded Euler class. Our proof of Part (i) of Theorem 1.2 is based on a far-reaching abstraction of these classical arguments. In particular, we will construct for every half-space order a corresponding generalized translation number (see Subsection 2.1), which is then shown to coincide (up to a multiple) with the growth functions of the corresponding quasi-total order (see Subsection 2.2). While (R, ≤, {[n, ∞)}) is the most classical examples of a half-space order, it is special in many respects, for example because of the totality of ≤. A more typical example of a half-space order is obtained from the real plane R 2 by defining an order by (x, y) ≺ (x , y ) :⇔ x < x and halfspaces H n by H n := {(x, y) ∈ R 2 | x ≥ n}. More generally, if X ⊂ R 2 is any subset, which intersects all H n \ H n+1 non-trivially, then (X, | X×X , H n ∩ X) is half-space ordered. For example, the group PSL 2 (Z) = Z/2Z * Z/3Z can be embedded into the plane by continuing the pattern given in Figure 1. It is easy to see that the left-action of PSL 2 (Z) on its planar image is by quasiautomorphisms with respect to the resulting half-space order; we thus obtain a quasimorphism on PSL 2 (Z) just from drawing the picture! This quasimorphism happens to be the famous Rademacher quasimorphism; see the discussion in Example 3.11 below. A different class of examples arises from bi-invariant total orders; however, these do not provide interesting quasimorphisms (see Subsection 4): Proposition 1.3. Every admissible bi-invariant total order on a group G is quasi-total; its associated growth functions are homomorphisms. A more interesting class of examples can be obtained by weakening the condition of totality just a little bit: ∈ Aut(X, ) is called dominant if ∀a, b ∈ X ∃n ∈ N : T n a b. Given a poset (X, ) and a dominant automorphism T we call the triple (X, , T ) a quasi-total triple if ∃N (X) ∈ N ∀a, b ∈ X ∃0 k N (X) : (a T k b ∨ b T k a). The quasi-total triple (X, , T ) is called complete if ∀a ∈ X : a T a. An order-preserving action of G on (X, ) is called dominating if it commutes with T and satisfies ∃g ∈ G ∃x ∈ X ∃n ∈ N : g.x T n .x. (1.2) The link between quasi-total triples and quasi-total orders will be established in Subsection 3.1 below. Proposition 1.5. Let (X, , T ) be a complete quasi-total triple, x 0 ∈ X a basepoint and H n := {x ∈ X | x T n .x 0 }. Then (X, , {H n }) is a half-space order. Moreover, if G acts dominatingly and effectively by automorphisms on (X, , T ), then it acts unboundedly and by quasi-automorphisms on (X, , {H n }). In particular, the order induced by on G is quasi-total. Definition 1.6. A quasi-total order ≤ is called special if it is induced from an effective, dominating action of G on a quasi-total triple. This quasi-total triple is then said to realize ≤ (or its growths functions). We will establish the following strengthening of the existence part of Theorem 1.2 in Subsection 3.2 below: Proposition 1.7. Every nonzero quasimorphisms arises as the multiple of a growth function of a special quasi-total order. In fact, it arises as the multiple of a growth functions of an order induced from a quasi-total triple of the form (G, , T ) via the left action of G on itself. Proposition 1.7 will be established by constructing for every given quasimorphism f an explicit quasi-total triple called the tautological realization, which realizes f . This will lead us in Subsection 3.5 to the following characterization result: Corollary 1.8. A group G admits a non-zero homogeneous quasimorphism if and only if if acts dominatingly on some quasi-total triple (X, , T ). Here we do not assume that the action is effective, nor that the triple is complete. We have seen above that we may restrict attention to orders induced from quasi-total triples of the form (G, , T ). It is not possible in general to refine into a total left order on G. Indeed, the corresponding class of quasimorphisms is rather special: Proposition 1.9. Assume that the order ≤ on G is induced from a quasi-total triple (G, , T ) with (G, ) total. Then there exists a homomorphism ϕ : G → Homeo + Z (R) such that the growth functions of ≤ are proportional to the pullback of the translation number via ϕ. See Section 4 for a circle of related ideas. (Note that the proposition concerns bi-invariant partial orders on G induced from total left-orders on itself; these are not to be confused with bi-invariant total orders on G, which we discussed in Proposition 1.3 above.) While we have seen that every quasimorphism can be realized using its tautological realization, there are many reasons to look for nontautological realizations of known quasimorphisms. Firstly, these appear often more naturally; more importantly, additional properties of a realization can often be used to establish corresponding properties of the quasimorphism. In order to illustrate this principle, we discuss the case of globally hyperbolic quasi-total triples. Here a quasi-total triple (X, ≤, T ) is called globally hyperbolic if X comes equipped with a topology such that the finite order intervals [x, y] := {z ∈ X | x ≤ z ≤ y} are compact and the infinite order intervals [x, ∞) := {z ∈ X | x ≤ z}, (−∞, x] := {z ∈ X | z ≤ x}. are closed. Let us call a subset B ⊂ X bounded if its closure is compact. Then we have (see Theorem 2.7): Proposition 1.10. Let (X, ≤, {H n }) be a halfspace order such that (X, ≤) is globally hyperbolic. Let H be a group acting effectively and unboundedly on (X, ≤, {H n }) and denote by T X : H → R the associated translation number. Then the following are equivalent for a subgroup G < H: (i) (T X )| G ≡ 0. (ii) Every G-orbit in X is bounded. (iii) There exists a bounded G-orbit in X. The proposition motivates the question whether a given quasimorphism arises as the growth function associated with a globally hyperbolic quasi-total triple. We study this question in the context of quasimorphisms on simple Lie groups. More precisely, let D be an irreducible bounded symmetric domain with Shilov boundaryŠ and let G be the identity component of the isometry group of D with respect to the Bergman metric. Then any infinite covering of G admits a (unique up to multiples) homogeneous quasimorphism, called Guichardet-Wigner quasimorphism [12,3]. In general, we do not know whether this quasimorphism can be realized using a globally hyperbolic quasi-total triple. However, if the domain D happens to be of tube type, then by a result of Kaneyuki [16] there is a unique (up to inversion) G-invariant causal structure onŠ, which gives rise to a partial order on the universal coveringŘ ofŠ. The fact that this order is closely related to the Guichardet-Wigner quasimorphism was first observed by Clerc and Koufany [5]. In the language of the present paper their results can be reinterpreted as follows: Denote by T the unique deck transformation of the coveringŘ →Š which is non-decreasing with respect to . Also denote byǦ the unique cyclic covering of G which acts transitively and effectively onŘ. Then we have: Theorem 1.11. The triple (Ř, , T ) is a globally hyperbolic quasi-total triple, which induces a quasi-total order ≤ onǦ. The growth functions of ≤ are proportional to the Guichardet-Wigner quasimorphism onǦ. Theorem 1.11 will be proved in Corollary 5.14 below. We remark that the order used in the present article is the closure of the order considered in [5]. For this closed order we establish global hyperbolicity in Corollary 5.15. The combination of Theorem 1.11 and Proposition 1.10 yields the following result: Proof. We first show that the f a are mutually at bounded distance: |f a (g) − f b (g)| = |h(ga) − h(a) − h(gb) + h(b)| = |h(ga) − h(gb) − (h(a) − h(b))| = |h(ga, gb) − h(a, b)| < d. Now let us use this fact to show that they are quasimorphisms: |f a (gk) − f a (g) − f a (k)| ≤ |f a (gk) − f ka (g) − f a (k)| + d = |h(gka) − h(a) − h(gka) + h(ka) − h(ka) + h(a)| + d = d. The assumption that the action of G on X is unbounded implies immediately that each of the functions f a is unbounded. Then standard properties of homogeneization (see e.g. [4]) yield the following results 2.2. Translation numbers of growth functions. Throughout this subsection let (X, , {H n }) denote a fixed half-space order of width w and let G be a group acting by quasi-automorphisms of defect d on (X, , {H n }). We denote by T X := T (X, ,{Hn}) the translation number associated with this action and by ≤ the induced order on G. Our goal is to establish the following result: Theorem 2.4. The growth functions of ≤ are multiples of T X , in particular, they are nonzero homogeneous quasimorphisms. For the proof we need to recall some basic results and concepts from [2]. Given a partially bi-ordered group (G, ≤) with order semigroup G + and a homogeneous quasimorphism f : G → R, we say that f sandwiches G + if for some C > 0 {g ∈ G | f (g) ≥ C} ⊂ G + . (2.1) (In this case automatically G + ⊂ {g ∈ G | f (g) ≥ 0},Proposition 2.5. Suppose that (G, ≤) is a partially bi-ordered group and that f : G → R is a non-zero homogeneous quasi-morphism. If f sandwiches ≤, then ≤ is admissible and for all g ∈ G ++ , h ∈ G we have γ(g, h) = f (h) f (g) . Thus Theorem 2.4 is reduced to establishing the following proposition: Proposition 2.6. With notation as above, the quasimorphism T X sandwiches the partial order ≤. Proof. The quasimorphisms f a are mutually at uniformly bounded distance d, hence at distance d from T X,{Hn} . Now assume T X (g) > w+2d. Then for all k ∈ G, x ∈ X we have h(kg.x, k.x) ≥ h(g.x, x) − d ≥ f x (g) − d ≥ T X (g) − 2d > w, hence kg.x k.x by definition of w. This in turn implies g ≥ e. This finished the proof of Theorem 2.4 and thereby establishes Part (i) of Theorem 1.2. Global hyperbolicity. It is well-known that the classical translation number contains valuable information about orbits of subgroups of Homeo + Z (R) on the real line and circle. For example, the question whether a subgroup H < Homeo + Z (R) has a bounded orbit in R can be decided from the translation number. Indeed, such a bounded orbit exists if and only if T R | H ≡ 0; in this case, in fact all orbits are bounded. In order to obtain similar results for other types of quasimorphisms additional topological assumptions are necessary; concerning the existence of bounded orbits, global hyperbolicity is the key property. Indeed, we have the following result, which was stated as Proposition 1.10 from the introduction: Theorem 2.7. Let (X, ≤, {H n }) be a halfspace order such that (X, ≤) is globally hyperbolic. Let H be a group acting effectively and unboundedly on (X, ≤, {H n }) and denote by T X : H → R the associated translation number. Then the following are equivalent for a subgroup G < H: (i) (T X )| G ≡ 0. (ii) Every G-orbit in X is bounded. (iii) There exists a bounded G-orbit in X. Before we turn to the proof we observe that global hyperbolicity can be characterized in terms of halfspaces: Lemma 2.8. Let (X, ≤, {H n }) be a halfspace order; then (X, ) is hyperbolic if and only if order intervals are closed and the sets H n \ H n+1 are compact. Proof. Denote by w the width of (X, ≤, {H n }). Given n ∈ N let x + n ∈ H n+w+1 and x − n ∈ H n−w−1 \ H n−w . Then H n \ H n+1 ⊂ [x − n , x + n ] , hence global hyperbolicity implies compactness of the sets H n \ H n+1 . For the converse observe that if x ∈ H n , y ∈ H m , then [x, y] ⊂ m+w+1 k=n−w−1 H k \ H k+1 Proof of Theorem 2.7. (i)⇒(iii): We claim that (i) implies that there are no g ∈ G, n ∈ N, x ∈ X satisfying h(g n x, x) ≥ 2d. Indeed, otherwise, we had for all m ∈ N the inequaliy h(g nm .x, g n(m−1) .x) ≥ h(g n x, x) − d ≥ d, whence inductively h(g nm .x, x) ≥ h(g nm .x, g n(m−1)x ) + h(g n(m−1) .x, x) ≥ md, which leads to h(g nm .x, x) nm ≥ d n , and thus T X (g) ≥ d n by passing to the limit m → ∞. This contradiction shows that ∀g ∈ G, n ∈ N, x ∈ X : h(g n x, x) ≤ 2d. Applying the same argument to the reverse order, we can strengthen this to ∀g ∈ G, n ∈ N, x ∈ X : |h(g n x, x)| ≤ 2d. This implies that each orbit is contained in a finite number of strips of the form H n \ H n+1 , hence bounded by the lemma. (ii)⇒(iii): obvious. (iii)⇒(i): Assume that G.x is compact and let g ∈ G. Consider the sequence x n := g n .x. We claim that there exist n − , n + (possibly depending on g and x) such that {x n } ⊂ H n + \ H n − . Observe first that the claim implies that h(g n x, x) is bounded, whence T X (g) = 0; it thus remains to establish the claim. Assume that the claim fails; replacing the order by its reverse if necessary we may assume that h(g n x, x) is not bounded above. We thus find a subsequence n k such that for every y ∈ X there exists k(y) such that for all k > k(y) we have g n k x y. By compactness of G.x there exists an accumulation point x ∞ of x n and since order intervals are closed we have x ∞ ≥ y for all y ∈ X. However, a halfspace order does not admit a maximum. Constructions of quasi-total orders 3.1. Complete quasi-total triples. Throughout this subsection, let (X, , T ) be a complete quasi-total triple. We choose a basepoint x 0 ∈ X and define halfspaces H n for n ∈ Z by H n := [T n .x 0 , ∞). Since T is dominating, these form indeed a half-space filtration for X. Let us describe the height function of (X, {H n }) in terms of T and x 0 : Since the order is complete, there exists an absolute constant C(X) such that for all a, b ∈ X we have (a T C(X) b) ∨ (b T C(X) a). Assume h(a) = n. Then a T n .x 0 and a T n+k .x 0 for k > 0, whence a T n+(2C(X)+1) .x 0 . Thus we have established: Lemma 3.1. If h(a) = n, then T n .x 0 a T n+(2C(X)+1) .x 0 . This property determines h up to a bounded error. Moreover, we can deduce: Lemma 3.2. The triple (X, , {H n }) is a half-space order. Proof. Assume h(a, b) > 2C(X) + 1, and let n := h(a), m := h(b). By the lemma we then have a T n .x 0 T m+(2C(X)+1) .x 0 b. Now assume G acts by automorphisms on the triple (X, , T ). Then we have: Proposition 3.3. For all g ∈ G, |h(ga, gb) − h(a, b)| < 4C(X) + 2. Proof. Assume h(a) = n and h(b) = m. Then we have h(a, b) = n − m, and T n .x 0 a T n+(2C(X)+1) .x 0 , T m .x 0 b T m+(2C(X)+1) .x 0 , and consequently T n .gx 0 ga T n+(2C(X)+1) .gx 0 , T m .gx 0 gb T m+(2C(X)+1) .gx 0 . Now we find k ∈ Z such that T k x 0 g.x 0 T k+(2C(X)+1) .x 0 ; inserting this into the previous set of inequalities we obtain T n+k .x 0 ga T n+k+(4C(X)+2) .x 0 , T m+k .x 0 gb T m+(4C(X)+2) .x 0 . We deduce that n+k ≤ h(ga) ≤ n+k+(4C(X)+2), m+k ≤ h(gb) ≤ m+k+(4C(X)+2), and hence (n − m) − (4C(X) + 2) ≤ h(ga) − h(gb) ≤ (n − m) + (4C(X) + 2), which is to say |h(ga, gb) − h(a, b)| ≤ 4C(X) + 2. Corollary 3.4. Assume G acts by automorphisms on the quasi-total triple (X, , T ). Then G acts by quasi-automorphisms on the associated half-space order (X, , {H n }). We will denote the translation number of the triple (X, , {H n }) by T (X, ,T ) := T (X, ,{Hn}) . Let us show that this translation number does not depend on the choice of basepoint x 0 : For this we define the relative T -height of a, b ∈ X by the formula h T : X 2 → Z, h T (a, b) := inf{m ∈ Z | T m b ≥ a}. Then unravelling definitions yields |h T (a, b) − h(a, b)| < 2C(X) + 2. Thus we obtain: Proposition 3.5. The translation number T (X, ,T ) is given by the formula T (X, ,T ) (g) = lim n→∞ h T (g n a, a) n ; in particular, it is independent of the choice of basepoint x 0 . Tautological realization. We now turn to a proof of Proposition 1.7, whose statement we repeat here for ease of reference: Proposition 3.6. Every nonzero quasimorphisms arises as the multiple of a growth functions of an order induced from a quasi-total triple of the from (G, , T ) via the left action of G on itself. Proof. Let G be a group and f a nonzero (hence unbounded) homogeneous quasimorphism on G. Set x ≺ f y :⇔ ∀g ∈ G : f (gx) < f (gy), (3.1) and let h ∈ G be an element with f (h) >. Denote by ρ h the rightmultiplication by h. Then (G, f , ρ h ) is a quasi-total triple. In view of Proposition 2.5 it remains to show that f sandwiches the partial order (i) (Lexicographic products) Let (X 0 , 0 , T 0 ) be a complete quasi-total triples and (X i , i ) i∈N be a family of arbitrary posets. On X := ∞ i=0 X i define the lexicographic ordering by ≤ f induced by (G, f , ρ h ) on G. For this let g ∈ G with f (g) > D(f ); then for all x ∈ G we have ∀h ∈ G : f (ghx) ≥ f (g) + f (hx) − D(f ) > f (hx),(x i ) ≺ (y i ) :⇔ ∃j ∈ N 0 : (∀i < j : x i = y i ) ∧ (x j ≺ j y j ) and define T : X → X by T (x i ) := (T 0 x 0 , x 1 , x 2 , . . . ). Then (X, , T ) is a quasi-total triple. (ii) (Subtriples) Let (X, , T ) be a complete quasi-total triple. Let Y ⊂ X be a subset and suppose there exists S ∈ Aut(Y, ) such that for all y ∈ Y we have Sy T y. Then (Y, , S) is quasi-total. (iii) (Refinement) Let (X, , T ) be a quasi-total triple and be a refinement of ; then (X, , T ) is a quasi-total triple. These constructions, trivial as they may seem, immediately give rise to a large supply of interesting quasimorphisms. We illustrate this in the following example: Example 3.8. Let (X, , T ) = (R, ≤, x → x + 1) be the standard quasi-total triple, which realizes the classical translation number T R , i.e. the lift of Poincaré's rotation number. Then for any set X (which we consider as a trivial poset (X, =)) we obtain a new quasi-total triple (C X := R × X, , T ) as the lexicographic product. Explicitly, we have (λ, x) (µ, y) :⇔ λ < µ and T (λ, x) := (λ + 1, x). This quasi-total triple induces a quasimorphism T C X = T (C X , ,T ) on G := Aut(C X , , T ), hence on any subgroup of G. The reader might have the impression that these quasimorphisms are a trivial variation of T R , but this is not the case. For concreteness, let X := S 1 ; then it is easy to see that T C X cannot be the pullback of T R via any embedding G → Homeo + Z (R). Indeed, such an embedding does not exist, since Homeo + Z (R) is torsion-free. The notion of a half-space order is even more flexible. The following trivial observation is important: Lemma 3.9. Let (X, , {H n }) be a halfspace order and Y ⊂ X a subset such that Y ∩ (H n \ H n+1 ) = ∅ for all n. Then (Y, | Y ×Y , {H n ∩ Y }) is a halfspace order. In the next section we will apply the following special case: Corollary 3.10. Let Y be a set. Then an embedding ι : Y → R 2 induces a halfspace order on Y by setting y 1 ≺ y 2 :⇔ ι(y 1 ) = (x 1 , z 1 ), ι(y 2 ) = (x 2 , z 2 ), x 1 < x 2 and H n := {y ∈ Y | ι(y) = (x, z), x ≥ n}, provided H n \ H n+1 = ∅ for all n ∈ Z. 3.4. Planar embeddings. In view of the last corollary, a good strategy to construct quasimorphisms on groups is making the group act on a subset of the plane. One way to do so is to embed the group itself into the plane. If the embedding ι : G → R 2 is chosen in such a way that the left action of G on itself induces an unbounded action on ι(G) by quasi-automorphisms, then we obtain a quasi-total order, hence a quasimorphism. We provide two examples where this strategy works: Example 3.11 (Rademacher quasimorphism). Let G = PSL 2 (Z) = Z/2Z * Z/3Z . Denote by S a generator of Z/2Z and by R a generator of Z/3Z, so that G = S, R | S 2 , R 3 . We observe that the translations T 1 := SR and T 2 := SR 2 are of infinite order in G and generate a free semigroup G 0 in G. Every element of G can be written uniquely as either w, Sw, wS or SwS, where w ∈ G 0 . Now the Rademacher quasimorphism f on G can be described as follows (see [1] and also [23,Cor. 4.3]): Given g ∈ G, let w be the element in G 0 such that g ∈ {w, Sw, wS, SwS}. Then f (g) is the number of T 1 s in w minus the number of T 2 s in w. An embedding of G into the plane is depicted in Figure 1. (For better readability we have actually drawn a piece of the Caley graph of G with respect to the generating set {S, R, R 2 }; note however, that by continuing the pattern we will not obtain an embedding of the Cayley graph, but only of G, since edges will intersect already at the next step.) We claim that the action of G on this embedding is by quasi-automorphisms. Indeed, one immediately reduces to showing that G 0 acts by quasi-automorphisms. However, in the above embedding of the Cayley graph, T 1 acts by increasing the x-coordinate by 1, while T 2 acts by decreasing the x-coordinate by 1, whence G 0 even preserves the relative height function. To see that the resulting quasimorphism is indeed the Rademacher quasimorphism, just observe that every g ∈ G with f (g) > 5 maps every point in the Cayley graph to the right and consequently the induced order on G is sandwiched by the Rademacher quasimorphism. Example 3.12 (Brooks quasimorphism). We construct an embedding of the free group G on two generators a and b into the plane. We first label the intersection of the lattice Z 2 with the first and third quadrant as in Figure 2, thereby embedding a subset of G into the plane; in a second step we will extend this embedding to a planar embedding of whole group G by growing hair. To explain this procedure, let w be a vertex in the graph in Figure 2 with the property that at least one of its four neighbours in the Cayley graph of G with respect to {a ±1 , b ±1 } does not yet appear. We will then add the missing neighbour(s) and some further vertices by the following rules: • Assume that the a-neighbour wa is missing; this can only happen if the last letter of w is either a or b −1 . We then add the vertices wa, wab, waba, wabab etc. to the graph. Where precisely we place the first new vertex wa depends on the last letter of w: If it is an a, then we place wa above w (at a height that has not yet been taken); if it is a b −1 we place it two to the right of w at a yet available height. Once wa has been placed, we add wab one to the right of wa at the same height, waba one to the right of wab at the same height etc. For w = a this is depicted in Figure 3. • Similar rules apply to the other types of missing neighbours: If the b-neighbour of w is missing, then the last letter of w is either b or a −1 . In both cases we place wb above w and add wba, wbab, wbaba, etc. to the right. If the a −1 -neighbour of w is missing, then the last letter of w is b or a −1 . In both cases we place wb above w and add wa −1 , After applying this procedure once, every vertex in the original embedding has four neighbours, but the newly added vertices have only two neighbours each; we thus continue by growing hair to them according to the same rules. Repeating this procedure ad infinitum we finally obtain an embedding of G into the plane. Similarly as in the last example it can be checked that the action of G on this embedding is unbounded by quasi-automorphisms using the following key observation: If a word w ∈ G contains ab, respectively b −1 a −1 as a subword n + , respectively n − -times, then the action of w is at uniformly bounded height-distance from a translation by 2(n + −n − ). This fact can be used to show not only that the action is by quasi-automorphisms, but also that the quasimorphism corresponding to the embedding is given (up to a multiple) by the Brooks quasimorphism associated with the word ab, which assigns to w as above the difference n + − n − . Many other Brooks type quasimorphism admit similar realizations. wa −1 b −1 , wa −1 b −1 a −1 , 3.5. Incomplete quasi-total triples. In our construction of quasitotal orders from quasi-total triples we have always assumed that the quasi-total triples in question were complete and the corresponding Gactions were effective. Let us point out that these assumptions can be weakened. We first consider completeness of a quasi-total triple (X, , T ). Let us first observe that T is automatically fixed point-free and non-decreasing, i.e. ∀a ∈ X ∀m ∈ N : T m a a. In general, however, a T need not be strictly increasing. This defect can be repaired as follows: Define a new partial order T by setting a T b :⇔ ∃k ≥ 0 : T k a b. Then T is strictly increasing with respect to T , hence (X, T , T ) is a complete quasi-total triple. We refer to (X, T , T ) as the completion of (X, , T ). The following simple observation explains why the passage from an incomplete to a complete quasi-total triple does not affect the corresponding quasimorphisms. Proposition 3.13. A quasi-total triple (X, ≤, T ) and its completion (X, ≤ T , T ) define the same height function on X, hence give rise to the same translation number. Proof. This follows from T (a, b). h ≤ T ,T (a, b) = inf{m ∈ Z | ∃k ∈ N 0 : T m−k b ≥ a} = inf{m ∈ Z | T m b ≥ a} = h ≤, Note that if G acts dominatingly on a quasi-total triple, then it also acts dominatingly on its completion. By our definition a quasi-total order is induced by an effective dominating G-action. Effectiveness is required to make sure that the induced relation on G is indeed a partial order. However, if G acts dominatingly, but not necessarily effectively on some quasi-total triple (X, , T ), then we obtain an effective dominating diagonal action of G on the quasi-total order (G × X, ≤, T ), where (g, x) < (g , x ) iff x < x and T (g, x) := (g, T x). Combining these two observations we obtain: Corollary 3.14. If a group acts dominatingly on some quasi-total triple (X, , T ), then it acs dominatingly and effectively on some complete quasi-total triple (X , , T ). Combining this with Theorem 1.2 and Proposition 1.7 we obtain: Corollary 3.15. A group G admits a non-zero homogeneous quasimorphism if and only if if acts dominatingly on some quasi-total triple (X, , T ). 3.6. Admissible bi-invariant total orders. In this subsection we will provide a proof of Proposition 1.3. We start by observing: Lemma 3.16. Let ≤ be an admissible bi-invariant total order on a group G. Then ≤ is a special quasi-total order. Proof. Choose h ∈ G ++ and set (X, , T ) := (G, ≤, ρ h ), where ρ h denotes right-multiplication by h. Then it is easy to check that (X, , T ) is a quasi-total triple and that the induced quasi-total order coincides with ≤. It thus remains to establish the following result: Proposition 3.17. Let ≤ be an admissible bi-invariant total order on a group G. Then the growth functions of ≤ are homomorphisms. For the proof we remind the reader of the following simple fact: Lemma 3.18. Let G be a group and ≤ be a bi-invariant partial order on G. Then for all f 1 , f 2 , g 1 , g 2 ∈ Γ we have f 1 ≥ g 1 , f 2 ≥ g 2 ⇒ f 1 f 2 ≥ g 1 g 2 . Proof of Proposition 3.17. We fix g ∈ G ++ and show that γ g is a homomorphism. For this let a, b ∈ G. We may assume without loss of generality that ab ≤ ba. From bi-invariance we then obtain for all w 1 , w 2 ∈ G the inequality w 1 abw 2 ≤ w 1 baw 2 . (3.2) We claim that this implies that for every n ∈ N, (3.3) a n b n ≤ (ab) n ≤ b n a n . Indeed, using (3.2) repeatedly we obtain (ab) n = abab · · · · · ab ≥ a 2 bbab · ... · ab ≥ a 2 babb · ... · ab ≥ a 3 b 3 ab · ... · ab ≥ a n b n , and the other inequality is proved similarly. If we abbreviate γ n (g, h) := inf{p ∈ Z | g p ≥ h n } (h ∈ G), then we obtain γ 1 (g, a n b n ) ≤ γ n (g, ab) ≤ γ 1 (g, b n a n ). On the other hand totality of ≤ yields for every n ∈ N, g γn(g,a)−1 ≤ a n ≤ g γn(g,a) , g γn(g,b)−1 ≤ b n ≤ g γn (g,b) and thus by Lemma 3.18 g γn(g,a)+γn(g,b)−2 ≤ a n b n ≤ g γn(g,a)+γn(g,b)+2 , g γn(g,a)+γn(g,b)−2 ≤ b n a n ≤ g γn(g,a)+γn(g,b)+2 . We deduce that |γ 1 (g, a n b n ) − γ n (g, a) − γ n (g, b)| ≤ 2 |γ 1 (g, b n a n ) − γ n (g, a) − γ n (g, b)| ≤ 2. Combining this with (3.4) we obtain γ n (g, a) + γ n (g, b) − 2 ≤ γ n (g, ab) ≤ γ n (g, a) + γ n (g, b) + 2 Dividing by n and passing to the limit n → ∞ we get γ(g, a) + γ(g, b) ≤ γ(g, ab) ≤ γ(g, a) + γ(g, b). This shows that γ g is a homomorphism. The condition of admitting a bi-invariant total order is rather restrictive. We refer the reader to [19] and the references therein for various characterizations and properties of totally bi-orderable groups. Here we just consider two classes of examples given by free groups and pure braid groups respectively. We refer the reader to [17, Sec. 7.2] for background on their bi-invariant total orders. Example 3.19. Consider first the case of a free group F n on n free generators S = {x 1 , . . . , x n }. The natural total order x 1 < · · · < x n on S then induces a total bi-invariant order on F n via Magnus expansion, see [17,Prop. 7.11]. By Proposition 3.17 the associated growth functions on F n are homomorphisms, and it is easy to see that up to normalization they are given by the counting homomorphism µ xn associated with x n , i.e. if w = s 1 · · · s m with s i ∈ S ∪ S −1 then µ xn (w) = #{i | s i = x n } − #{j | s j = x −1 n }. Example 3.20. Following the notation in [17] we now denote by P n the pure braid group on n strands (i.e. the kernel of the natural surjection of the braid group B n onto the symmetric group on n letters) and by A ij , 1 ≤ i < j ≤ n its canonical generators. We also denote by U n the free group on generators A j,n , 1 ≤ j < n and equip it with the bi-invariant total order induced from the order A 1,n < · · · < A n−1,n on generators. Then we can define inductively a bi-invariant total order on P n by demanding that the morphisms in the short exact sequence 1 → U n → P n → P n−1 → 1 are order-preserving and the order on P 2 ∼ = Z is the standard one [17, p. 281]. Again the associated growth functions are easy to compute; they coincide up to normalization with the iterated projection π n : P n → P n−1 → · · · → P 2 ∼ = Z ⊂ R. To describe π n in terms of generators and relations, observe that the counting homomorphism µ A 12 on the free group F on generators A ij , 1 ≤ i < j ≤ n descends to a homomorphism P n , and this homomorphism coincides with π n . Thus π n counts the occurences of A 12 in a given word in the pure braid group. 4. Total triples and circular quasimorphisms 4.1. Total triples. A very special class of examples of quasi-total triples (X, , T ) is given by totally ordered spaces (X, ) together with a dominating automorphism T . We then say that (X, T ) is a total triple. In this situation the theory simplifies considerably. For instance, the height function admits the following simpler description: Proposition 4.1. Let (X, , T ) denote a complete total triple and let a, b ∈ X. Then h T (a, b) is the unique integer such that T h T (a,b)−1 .b ≺ a T h T (a,b) .b. Now let us specialize further to the case where X coincides with G. In this case is a left-invariant order on G and we have a distinguished basepoint given by a = e. Given g ∈ G define n := n(g) to be the unique integer satisfying T n−1 .e g T n .e; Then, as a special case of the last proposition we see that the function g → n(g) is at bounded distance from the translation number T G, ,T associated with (G, , T ). From this description we see in particular that our construction generalizes a construction of Ito [15]: Corollary 4.2. Let G be a group, a left-invariant total order on G, x ∈ G and ρ x (g) := gx. Assume that (G, , ρ x ) is a total triple. Then the translation number T G, ,ρx is the homogeneization of the quasimorphism ρ G x, constructed in [15]. A particular example seems worth mentioning at this point: Example 4.3. Let B n be again the n-string braid group and denote by σ 1 , . . . , σ n−1 its canonical (Artin) generators. There is a canonical left-ordering on B n , which is described e.g. in [6] and sometimes called the Dehornoy order. If we choose x := ((σ 1 · · · σ n−1 )(σ 1 σ 2 )σ 1 ) 2 , then x is central in B n and (B n , , ρ x ) is a total triple. Combining the last corollary with [15, Example 1], we see that the translation number T Bn, ,ρx is the homogeneization of the Dehornoy floor quasimorphism. In [15] it is always assumed that T = ρ x for some x ∈ G. If G is assumed countable, then this is not a serious restriction: Lemma 4.4. Let G be a countable group and (G, , T ) be a total triple with a dominating G-action. Then there exists a supergroup G 1 of G, a total order 1 on G 1 and an element x ∈ G 1 with the following properties: (i) 1 is a left-invariant, total order on G 1 and 1 | G = . (ii) x ∈ Z(G 1 ) and x is dominant for 1 . (iii) (G 1 , 1 , ρ x ) is a total triple with a dominating G 1 -action. Moreover, G 1 is isomorphic to a quotient of G × Z and the embeddings of G into G × Z and G 1 are compatible. Proof. Let G 1 be the subgroup of Aut(G, , T) generated by G and T and set x := T . Here G acts on itself by left-multiplication. Since G and T commute, this group is a quotient of G×Z and x is central. Note that G 1 acts on G preserving . To define 1 choose an enumeration {g i } i∈N of G with g 1 = e; then define that g 1 h if and only if (gg i ) (hg i ) with respect to the lexicographical order on G N . Since G 1 acts effectively on G, this defines a total order and x is dominant, since T is dominant. Also, 1 is G 1 -invariant, since is. Finally, let g, h ∈ G be distinct; then either g ≺ h or g h. In the former case we have g.g 1 ≺ h.g 1 (since g 1 = e) and thus g ≺ 1 h, while in the second case we have g 1 h. This shows that 1 restricts to on G. Thus in studying total triples (G, , T ) over a countable group G we may focus on the case, where T = ρ x for a central dominant x ∈ G. 4.2. From circular quasimorphisms to total triples. In this subsection we study quasimorphisms which arise from lifts of actions on the circle: Definition 4.5. Let G be a group. A nonzero homogeneous quasimorphism f on G is called circular if there exists an injective homomorphism ϕ : G → Homeo + Z (R) such that f = ϕ * T R . It turns out that circular quasimorphisms are closely related to total triples. The precise relation is somewhat technical, and we offer three different (essentially equivalent) formulations: Proposition 4.6. Let G be a group, and f be a circular homogeneous quasimorphism on G. (i) There exists a left-invariant total order on G such that the growth functions of the order induced from via the left-action of G on itself are multiples of f . (ii) There exists a quasi-total triple (G, 0 , T ) realizing f with the property that 0 can be refined into a left-invariant total order on G. (iii) Assume that f is unbounded on the center of G. Then there exists a total triple (G, , T ) realizing f . Proof. (i) We first recall [19] that every enumeration {q n } of Q defines a total order on H := Homeo + Z (R) by setting g h if and only if (gq n ) ≤ (hq n ) with respect to the lexicographic ordering on R N . Indeed, this follows from the fact that every h ∈ H is uniquely determined by its restriction to Q. We fix such an enumeration and the corresponding ordering once and for all. By construction, is left-invariant. Denote by ≤ H the bi-invariant order on H induced by . Then ≤ H is sandwiched by T R . Indeed, assume T R (h) > 10. Then for all q ∈ R we have h.q > q, whence (hq n ) (eq n ) and thus h ≥ H e. Now assume f : G → R is circular and nonzero, say f = ϕ * T R for some injection ϕ : G → H. For notation's sake let us assume that G is a subgroup of H and ϕ the inclusion. Then the restriction | G defines a left-invariant total order on G. Let ≤ be the order on G induced by | G . From the fact that T R sandwiches ≤ we deduce that f sandwiches f * ≤ H ; since ≤ is a refinement of f * ≤ H , it also sandwiches ≤. (ii) Argue as in (i), but define 0 to be the bi-invariant order induced by and choose T to be right multiplication by some element g ∈ G with ϕ(g) > 10. (iii) Construct as in (i) and choose T to be multiplication by a central element x with f (x) > 10D(f ) + 5. For countable groups we will establish a partial converse to Proposition 4.6 in Theorem 4.7 below. 4.3. From total triples to circular quasimorphisms. The goal of this section is to establish the following partial converse of Proposition 4.6: Theorem 4.7. Let G be a countable group and (G, , T ) be a total triple with a dominating G-action. Denote by ≤ the induced biinvariant order on G. Then the growth functions of ≤ are nonzero circular quasimorphisms. We will first establish the theorem under the additional hypothesis that T = ρ x for some central dominant x ∈ G. In a second step we will then reduce the general case to this case by means of Lemma 4.4. The first step of the proof uses crucially the notion of a dynamical realization [19]: Definition 4.8. Let be a left-invariant total order on G. A dynamical realization of is a pair (ϕ, t) consisting of an injective homomorphism ϕ : G → Homeo + (R) and a ϕ-equivariant embedding t : G → R such that t(e) = 0, inf g∈G t(g) = −∞, sup g∈G t(g) = ∞ and f ≺ g ⇔ ϕ(f ).0 < ϕ(g).0. (4.1) A dynamical realization of is special if ϕ(G) centralizes the transla- tion T : x → x + 1; it is called adapted to x ∈ G if ϕ(x) = T . The following is well-known: Since ϕ(x) is contained in the centralizer of ϕ(G) and ϕ is injective we must have x ∈ Z(G). Also, given any g ∈ G we find n ∈ N with ϕ(g).0 < n = ϕ(x n ).0. We deduce that g x n , which shows that x is a dominant. Thus the conditions are necessary. On the other hand, assume that x ∈ Z(G) is dominant. Then every element in G may be written uniquely as g = g 0 x n with n ∈ Z and e g 0 x. Now define the embedding t as follows: Set t(e) = 0, t(x) = 1 and let {g k } k∈N be an enumeration of the order interval [e, x] with g 1 = e, g 2 = x. Inductively assume t(g 1 ), . . . , t(g i−1 ) have been defined. Then there exists g m , g M such that g m < g i < g M and ]g m , g M [∩{g 1 , . . . , g i−1 } = ∅. We then define t(g i ) := (t(g m ) + t(g M ))/2. Now extend the map t : [e, x] → R to all of G by the formula t(g 0 x n ) = n + t(g 0 ). The action of G on t(G) given by ϕ(g)t(h) := t(gh) extends continuously to the closure of t(G) and can be extended to an action of homeomorphisms on R in a standard way, see [19]. We have ϕ(x).t(g) = t(g + 1), hence ϕ(x) = T on t(G). From the construction of the extension in loc. cit. we deduce ϕ(x) = T on all of R. We now fix a total triple of the form (G, , ρ x ) with x ∈ Z(G) dominant and a dynamical realization (ϕ, t) of adapted to x. As before, we denote by ≤ the order induced by on G. We recall that ≤ is admissible and that its growth functions are multiples of T (G, ,ρx) (g). We now aim to describe these growth functions in terms of the homomorphism ϕ : G → Homeo + Z (R). To this end we observe that ϕ allows us to pullback the classical translation number T R to a homogeneous quasimorphism ϕ * T R on G. The following was observed in [15]: Proposition 4.10 (Ito). Let G be a countable group, a left-invariant total order on G and x ∈ G a central dominant. Let (ϕ, t) be a dynamical realization of adapted to x. Then T (G, ,ρx) − ϕ * T R : G → R is a homomorphism. Proof. By the proof of [15,Theorem 3] the pullback of the bounded Euler class −e b := dT R under ϕ in real bounded cohomology is the class represented by the differential of the quasimorphism denoted ρ G x, in [15]. (In fact, this is even true for the corresponding integral bounded cohomology classes, but we do not need this stronger statement here.) Since T G, ,ρx is at bounded distance from ρ G x, by Corollary 4.2, we deduce that the differential of f := T G, ,ρx − ϕ * T R represents the trivial class in H 2 b (G; R). Now f is both homogeneous and cohomologically trivial, hence a homomorphism. We will strengthen this as follows: Lemma 4.11. Let G be a countable group, a left-invariant total order on G and x ∈ G a central dominant. Let (ϕ, t) be a dynamical realization of adapted to x. Then T (G, ,ρx) = ϕ * T R . Proof. Since both quasimorphisms are homogeneous it suffices to show that they are at bounded distance. For this we may replace T R by the function g → ϕ(g).0 and T (G, ,ρx) (g) by h T (g, e), since those are at bounded distance from the original functions. Now choose n so that ϕ(x) n−1 .0 = n − 1 < ϕ(g).0 ≤ n = ϕ(x) n .0. This implies both |ϕ(g).0 − n| < 1 and x n−1 ≺ g x n+1 , the latter by (4.1). We may rewrite the last chain of inequalities by ρ(x) n−1 .e = x n−1 ≺ g ≺ x n+1 = ρ(x) n+1 .e. From this we deduce that |h T (g, e)−n| < 2, whence |ϕ(g).0−h T (g, e)| < 3. Now we can deduce the theorem: Proof of Theorem 4.7. Let (G, , T ) be any total triple with a dominating G-action. We then construct the extended triple (G 1 , 1 , ρ x ) as in Lemma 4.4 and denote by ≤ 1 the order induced by 1 on G 1 . We then choose a dynamical realization (ϕ 1 , t 1 ) of 1 adapted to x and deduce from Theorem 1.2 and Lemma 4.11 that ≤ 1 is sandwiched by T G 1 , 1 ,ρx = ϕ * 1 T R . We thus find a constant C such that for g ∈ G 1 with T R (ϕ 1 (g)) > C we have ∀x ∈ G 1 : g.x 1 x. (4.2) Denote by ≤ the bi-invariant order induced by on G and by ϕ the composition of the inclusion G → G 1 with ϕ 1 . We then claim that ≤ is sandwiched by ϕ * T R . Indeed, assume g ∈ G satisfies ϕ * T R (g) > C; then (4.2) holds, and in particular ∀x ∈ G : g.x 1 x. But since 1 | G = , this shows that g ≥ e, which yields the desired sandwiching result. For quasimorphisms which are unbounded on the center of G we have obtained a complete characterization of circularity: Corollary 4.12. Let f : G → R be a quasimorphism, which is unbounded on the center of G. Then the following are equivalent: (i) f is circular. (ii) f can be realized by a total triple (G, , T ). For quasimorphism, which are not unbounded on the center of G the situation is slightly more technical, as witnessed by the more complicated formulation of Proposition 4.6 for such quasimorphisms. 5. Smooth quasi-total triples from causal coverings 5.1. Causal coverings. In this section we study quasi-total triples induced by smooth partial orders on manifolds. The notion of smooth partial orders that we use here is discussed in the appendix. From now on we will denote by ( M , C) a causal manifold in the sense of Definition A.6 and by the associated partial order. We also denote by G( M , C) the associated automorphism group (see Definition A.4). The main problem of this section can then be formulated as follows: Problem 2. Given a causal manifold ( M , C), is there an automorphism T ∈ G( M , C), which turns ( M , , T ) into a quasi-total triple? Remark 5.1. For the purpose of this subsection we could as well consider a weakly causal manifold in the sense of Definition A.6 and study the associated strict causality s instead of . We would then ask for an automorphism T of ( M , C) turning ( M , s , T )) into a quasitotal triple. All results of this subsection remain valid in this setting; the difference between s and will only become important when we discuss global hyperbolicity in Subsection 5.3 below. We now fix a causal manifold ( M , C). We oberve that if T as in Problem 2 exists, then it has to be of infinite order. Hence, fix T ∈ Aut( M , C) of infinte order and assume moreover that the group Γ ∼ = Z generated Since M is totally acausal there exists a closed causal loop γ p(a),p(b) at p(a) through p(b). We can lift this loop to a curve γ a,b with initial point a; the result is a causal curve through a and some T -translate of b. Thereby we find integers l(a, b) ∈ Z with a T l(a,b) b. Since M does not contain causal loops, we may assume l(a, a) > 0 for some given basepoint a upon possibly replacing T by its inverse. We claim that this implies l(b, b) > 0 for all b. Indeed, suppose otherwise, say b T k b with k > 0. We then find m > 0 with T −mk a b T mk a, T −mk b b T mk b, hence b T mk (T −mk a) = a and b T −mk (T mk a) = a. This yields a = b and l(a, a) < 0, which is a contradiction. We see in particular that we can choose l(a, b) positive by adding a suitable multiple of l(b, b). This implies that T is dominant. Thus replacing T by T −1 if necessary we will assume from now on that T is the unique dominant generator of Γ. We see from the proof of the last proposition that for any pair a, b ∈ M there exists n(a, b) := min{l(a, b), l(b, a)} ∈ N such that a T n (a,b) b or b T n(a,b) a. However, the number n is in general not uniformly bounded. Equivalently, ( M , , T ) is a quasi-total triple. In the next section we will provide two different criteria which guarantee this property. Before, let us give some elementary examples of quasi-total causal coverings. Firstly, the classical translation number T R is associated with the causal covering R → S 1 . The following example can be considered as a smooth twisting of this trivial example; as in Example 3.8 it is easy to argue that this sort of twisting produces fundamentally different quasimorphisms. Example 5.5. Let M := R×] − 1, 1[ be a strip of bounded diameter with basepoint x 0 := (0, 0) and let C ⊂ R 2 be a closed regular cone which contains the positive x-axis in its interior. Then the translation invariant cone field on R 2 modelled on C restricts to a conal structure C on M , and the conal manifold ( M , C) is in fact causal, since every nonconstant causal curve is strictly monotone in the x-coordinate. Since the cone C contains the positive x-axis in its interior we find x ± ∈ R such that x ± ∈ R × {±1} ∩ C. Choose x ± minimal with this property and set x 0 := 2 max{x + , x − }. Similar twists can also be defined for the examples from Lie groups as discussed below. 5.2. Criteria for quasi-totality. Before we can discuss further examples, we need to develop criteria whoch guarantee quasi-totality. Throughout this section we fix a causal covering M → M and denote by 0 → Γ →Ǧ → G → 1 the associated central extension of automorphism groups. The easiest way to guarantee quasi-totality is to demand enough transitivity of G on M . Definition 5.6. An action of a group G on a space X is almost 2transitive it there exists a G-orbit X (2) ⊂ X 2 with the property that ∀x, y ∈ X ∃z ∈ X : {(x, z), (z, y)} ⊂ X (2) . In this case we call X almost 2-homogeneous and write x y to indicate that (x, y) ∈ X (2) . Then we obtain: Note that the almost 2-transitivity of G on M implies automatically that M is totally acausal. We prepare the proof of Theorem 5.7 by the following lemma: Proof. Let a ∈ M be some basepoint and x := p(a). Then we find z ∈ M with z x and a closed causal loop γ x : [0, 1] → M be a closed causal loop at x with z = γ x (1/2). Letγ a be the lift of γ x with initial point a. Thenγ a (0) γ a (1); we thus find N 0 > 0 such that γ a (1) = T N 0γ a (0). If we define c :=γ a (1/2), then a c T N 0 a. Now let d ∈ M , w := p(d) and assume w z. Then we find g 0 ∈ G 0 with g 0 .x = w and g 0 .z = z. Thus γ w = g 0 .γ x is a closed causal loop at w through z. Now letγ d be a lift of γ w with initial point d. Let g ∈ G be a lift of g 0 ; then g maps a to a point in the fiber of d, and modifying g by a deck transformation if necessary we can assume g.a = d. Then g.γ a is a lift of γ w with initial point d, henceγ d = g.γ a by uniqueness. Now we havê γ d (1) = gγ a (1) = gT N 0γ a (0) = T N 0 gγ a (0) = T N 0γ d (0). Now since γ w (1/2) = z we find l ∈ Z such thatγ d (1/2) = T l c. We thus have established d T l c T N 0 d (5.1) under the assumption p(d) z. Now consider the case of an arbitrary b ∈ M and let y := p(b). We then find d ∈ M such that w := p(d) satisfies y w z. Then (5.1) holds for some l ∈ Z. Moreover, we find h 0 ∈ G 0 with h 0 .(x, z) = (y, w). Define γ y := h 0 .γ x and denote byγ b the lift of γ y with initial point b. By the same argument as before we then shoŵ γ b (1) = T N 0γ b (0). Now γ y (1/2) = w, so we find l ∈ Z withγ b (1/2) = T l d. Thus Proof of Theorem 5.7. Let c and N be as in the lemma. Given a, b ∈ M we find l, l ∈ Z such that b T l c T N b and a T l c T N a. We may assume w.l.o.g. that l ≥ l. Now for all k ≥ 1 we have The almost 2-transitivity condition of Theorem 5.7 is rather strong and not always easy to check in practice. We are thus looking for an alternative condition that ensures quasi-totality. One consequence of quasi-totality is that T k .x T l −l b T l c T N a T kN a, x for all x and a uniformly bounded k. Here we shall assume the slightly stronger condition Let m ij be integers such that b i ≤ T m ij a j and set N := max m ij . Then x ≤ T N y for all x, y ∈ g j U , hence x ≤ T N y for all x ∈ H − , y ∈ H + . Now let x, y ∈ M be arbitrary. We distinguish three cases: T.x x, • If one of them is in H − and the other is contained in H + , then y T N y or y T N x. • If none of them is in H + , apply T until the first of them is. We may assume T k x ∈ H + and T k−1 y ∈ H + , hence T k−1 y ∈ H − . Then T k−1 y ≤ T k+N x, hence y ≤ T N +1 x. • If none of them is in H − we argue dually. We thus obtain x T N +1 y or y T N +1 x in all possible cases. A criterion for global hyperbolicity. We have seen in the last section how compactness of the base manifold can be used to obtain quasi-totality of a given causal covering. This sort of compactness assumption also has implications to global hyperbolicity, which we briefly want to outline here. More precisely, we will establish the folliowing: Theorem 5.10. Let p : M → M be a causal covering and assume that M is compact. Then the partial order on M and its completion T are globally hyperbolic. While up to this point we could have worked with the strict causality s instead of the closed causality , closedness of is clearly necessary for Theorem 5.10 to hold. Concerning the proof of Theorem 5.10 we first observe that the order intervals of are closed by construction; since [a, b] T = N ( M )−1 k=0 N ( M )−1 l=0 [T k a, T −l b] , we see that also T is closed. It thus remains only to show that finite order intervals of T are bounded. From now on, all order intervals will be with respect to T . Our starting point is the following trivial observation: Lemma 5.11. Let a ∈ M and N ∈ N. Then for all x ∈ M there exists b 0 ∈ p −1 (x) such that p −1 (x) ∩ [a, T N a] ⊂ {b 0 , T b 0 , . . . , T N b 0 }. Proof. Let x ∈ M and b ∈ F x := p −1 (x). Consider E b := {k ∈ Z |, T k b T a}. We claim that E b has a minimal element. Indeed, since T is dominating we have T l a T b for some l ∈ Z and hence a T T −l b. Now if T k b T a, then k ≥ −l, for otherwise T k b T a T T −l b, hence T k+l b T b and thus 0 > k + l ≥ 0. Now let k ∈ E b be minimal and b 0 := T k b. Then T k b 0 ≥ a implies k ≥ 0, while T k b 0 ≤ T N a implies k ≤ N . Thus p −1 (x) ∩ [a, T N a] ⊂ {b 0 , . . . , T N b 0 }. Now the key to the proof of Theorem 5.10 is the following lemma: F x ∩ [a, T N a] ⊂ {b 0 , T b 0 , . . . , T N b 0 } for some b 0 ∈ F x . In particular, we find k 1 ∈ N with 0 ≤ k ≤ N and c = T k 1 b 0 . On the other hand we have we have M a ⊂ [a, T N a] ∩ F x ∩ [a, T N a] = ∅ in view of (5.5). We thus find k 2 ∈ N with 0 ≤ k ≤ N and T k 2 b 0 ∈ M a . Then c = T k 1 b 0 = T k 1 −k 2 T k 2 b 0 ∈ T k 1 −k 2 M a . Since −N ≤ k 1 − k 2 ≤ N we obtain c ∈ N n=−N T n M a , and since c ∈ [a, T N a] was chosen arbitrarily, we obtain the desired boundedness result. 5.4. Examples from Lie groups. In this section we explain how the Clerc-Koufany construction of the Guichardet-Wigner quasimorphisms on a simply-connected simple Hermitian Lie groups of tube type [5] can be reinterpreted in the language of the present paper. Let G be an adjoint simple Lie group with maximal compact subgroup K. Then G is called Hermitian if the associated symmetric space G/K admits a G-invariant complex structure J, and of tube type if (G/K, J) is biholomorphic to a complex tube. From now on G will always denote an adjoint simple Hermitian Lie group of tube type. We then find a Euclidean Jordan algebra V such that G/K can be identified with the unit ball D in V C with respect to the spectral norm. We will fix such an identification once and for all. The action of G on D extends continuously to the Shilov boundary ofŠ. SinceŠ is a generalized flag manifold, we obtain a notion of transversality onŠ from the associated Bruhat decomposition. It then follows from the abstract theory of generalized flag manifolds that G acts almost 2-transitively onŠ (see e.g. [25,Lemma 3.30]). If e V denotes the unit element of the Jordan algebra V , then e V ∈Š and the setŠ e V of points inŠ transverse to e V is Zariski open inŠ. The Cayley transform of V C identifiesŠ e V and hence T −e VŠ with V . Thus the (closed) cone of squares in V gives rise to a closed cone Ω ⊂ T −e VŠ . By a result of Kaneyuki [16] there exists a unique G-invariant causal structure C onŠ with C −e V = Ω. The universal covering (Ř, C) of the causal manifold (Š, C) is described in [5]. Namely, it turns out that π 1 (Š) ∼ = Z, so that p :Ř →Š is an infinite cyclic covering. The universal covering G of G acts transitively onŘ; the kernel of this action can be identified with π 1 (G) tors . Thuš G := G/π 1 (G) tors acts transitively and effectively onŘ. Now we claim: Proposition 5.13. The covering p :Ř →Š is a complete quasi-total causal covering. Proof. Identify the tangent space of T −e VŠ with V and choose an inner product ·, · on V such that Ω is a symmetric cone with respect to ·, · [8]. Since e V is contained in the interior of Ω it follows from the self-duality of the latter that ∃ > 0 ∀x ∈ Ω : x, e V ≥ · v .1-form α oň S with α −e V (v) = x, e V , v ∈ T −e VŠ . Since K/M is symmetric, this form is closed. It then follows from (5.6) that α is a uniformly positive 1-form in the sense of Definition A.7. This implies that the pullback β := p * α is a uniformly positive 1-form onŘ. In particular,Ř is causal by Proposition A.8. SinceŠ is a flag variety, the action of G onŠ is almost 2-transitive; just takeŠ (2) to be the set of transverse pairs inŠ (see e.g. [25]). This almost 2-transitivity implies immediately thatŠ is totally acausal, whence p :Ř →Š is a causal covering; in view of Theorem 5.7 it also implies that this causal covering is quasi-total. It remains to show that this covering is total, i.e. T x x for all x ∈Ř. For this it suffices to construct a causal curve joining x and T x; this is established in [5]. In view of the pioneering work in [16] we refer to the partial order oň R as the Kaneyuki order. It was established in [16], that G(Š,Č) = G unless G ∼ = P SL 2 (R). Thus let us assume G ∼ = P SL 2 (R) from now on. Then the central extension associated with the causal covering p :Ř →Š is precisely 0 → Z →Ǧ → G → {e}. We thus obtain a non-trivial quasimorphism TŘ on the simple Lie group G. It follows from the classification of such quasimorphisms in [24] that TŘ is necessarily a multiple of the Guichardet-Wigner quasimorphism onǦ [12]. We have thus proved: Corollary 5.14. The growth functions of the order ≤ onǦ induced from the Kaneyuki order onŘ are multiples of the Guichardet-Wigner quasimorphism. Guichardet-Wigner quasimorphisms are very well understood; see [3] for an explicit formula. The observation that Guichardet-Wigner quasimorphisms are related to the causal structure on the corresponding Shilov boundaries was first made in [5] (see also [4] for an English introduction to their work). However, their precise formulation of this phenomenon is different from ours. Corollary 5.14 allows us to characterize subgroups ofǦ with vanishing Guichardet-Wigner quasimorphism. Indeed, as as special case of Theorem 5.10 we obtain: Appendix A. Partial orders on causal manifold A.1. Causalities on conal manifolds. In various branches of mathematics and physics cone fields in the tangent bundle of a manifold are used to define a causality (i.e. a reflexive and transiive relation) on the manifold itself. The precise definitions of such causalities, however, differ widely in the literature; it thus seem worthwhile to elabarote a bit on the definitions we use in the body of text. In the present paper we are mainly interested in invariant cone fields on homogeneous spaces of finite-dimensional Lie groups, and our definitions are adapted to work well in this context. We refer the reader to [14,13,18] for sources with a point of view similar to ours. A convex, R >0 -invariant closed subset Ω of a vector space V will be called a wedge. A wedge is called a closed cone if it is pointed, i.e. Ω ∩ (−Ω) = {0}. It is called regular if its interior is non-empty. Given a closed regular cone Ω ⊂ V we denote by G(Ω) the group G(Ω) := {g ∈ GL(V ) | gΩ = Ω}. Then we define: Definition A.1. Let M be a d-dimensional manifold and Ω ⊂ R d a closed regular cone. Then a causal structure on M is a principal G(Ω)bundle P → M together with an isomorphism ι : P × G(Ω) V → T M (i.e. a reduction of the structure group of T M from GL d (R) to G(Ω).) The associated fiber bundle C := ι(P × G(Ω) Ω) of P with fiber Ω is called the cone field of the causal structure P . We then refer to the pair (M, C) as a conal manifold. We warn the reader that the term causal manifold is traditionally reserved for a conal manifold with additional properties, see the definition below. We also remark that [14] uses a more general definition of causal structure, but the present definition is sufficient for our purposes. For us it will be important that cone fields can be lifted along coverings: Lemma A.2. Let (M, C) be a conal manifold and M its universal covering. Then there exists a unique cone field C on M such that π 1 (M ) acts by causal diffeomorphisms on ( M , C). Conversely, every π 1 (M )invariant cone field descends to M . Proof. The only way to define a causal structure with the desired property is to set C x = (dp M ) −1 C p M (x) , where dp M : T x M → T p M (x) M is the derivative of the universal covering projection. This defines indeed a causal structure on M , since the triviality condition is local. The second statement is obvious. Given a manifold M and real numbers a < b we call a curve γ : [a, b] → M piecewise smooth if it is continuous and there exists real numbers a = a 0 < a 1 < · · · < a n = b such that γ| (a j ,a j+1 ) is of class C ∞ for j = 0, . . . , n − 1. Now we define: Since the concatenation of piecewise smooth curves is piecewise smooth, the strict causality is indeed a causality. It may or may not be antisymmetric, and it may or may not be smooth. Antisymmetry can sometimes be obtained by passing to a suitable covering. Obtaining a closed causality is difficult in general. Indeed, the closure of s in M × M need no longer be transitive. Fortunately, in homogeneous examples this kind of pathology hardly occurs. To make this precise we define: Definition A.4. Let (M, C) be a conal manifold. A diffeomorphism ϕ of M is called causal with respect to C if dϕ(C m ) = C ϕ(m) for all m ∈ M . The group of all causal diffeomorphisms of (M, C) is denoted G (M, C). A group action G × M → M is causal if G acts by causal diffeomorphisms. In this case the causal structure C is called G-invariant. The conal manifold (M, C) is called uniformly homogeneous if G(M, C) acts transitively on M and every x ∈ M has an open neighbourhood U such that for all x n ∈ U with x n → x there exists a sequence g n ∈ G(M, C) such that g n x n = x and g n → e in the compact-open topology. Note that if G is a finite-dimensional Lie group and H is a closed subgroup, then every G-invariant cone field on G/H is uniformly homogeneous; indeed, this follows from the existence of local sections of the principal bundle G → G/H. This case is actually all we need. In any case in all our examples will be a well-defined causality. We follow [13,14,26,20,18] in calling , rather than s the causality associated with the conal manifold (M, C). A.2. Causal manifolds and positive 1-forms. In this section let (M, C) be a conal manifold. We denote by s the strict causality on M and by its closure. Then we define: Definition A.6. The conal manifold (M, C) is called causal if is a partial order and totally acausal if = M × M . It is called weakly causal if s is a partial order. Note that for (M, C) to be causal we demand in particular that is transitive. We now provide a sufficient condition for M which guarantees causality. To this end we define: Definition A.7. Let (M, C) be a conal manifold. A closed 1-form α ∈ Ω 1 (M ) is called uniformly positive with respect to C if there exists a Riemannian metric on M and > 0 such that for all x ∈ M and all v ∈ C x we have α x (v) ≥ · v . Uniformly positive 1-forms are a special case of positive 1-forms as introduced in [13] refining ideas from [26,20]. Positive 1-forms were introduced to prove antisymmetry of the strict causality on certain 1connected manifolds. Uniformly positive 1-forms play a similar role for the closed causality: Proposition A.8. Let (M, C) be a simply-connected uniformly homogeneous conal manifold admitting a uniformly positive 1-form α. Then (M, C) is causal. Proof. Since M is uniformly homogeneous, is transitive, and it remains to show that it is antisymmetric. Let x, y ∈ M be distinct points and assume x y x. By definition this means that there exist sequences x n → x, x n → x, y n → y, y n → y inŘ such that x n s y n , y n s x n . Let G := G(M, C) and observe that since (M, C) is uniformly homogeneous there exist sequence g n → e, g n → e in G such that y n = g n y, y n = g n y. Now define a n := g −1 n x n , b n := (g n ) −1 y n . Then a n s y s b n , a n → x, b n → x. where L(c) denotes the length of c. Fix a neighbourhood U of x not containing y in its closure and set δ := 2 d(U, y). Choose n 0 such that a n , b n ∈ U for all n ≥ n 0 . By shrinking U if necessary we may assume that U is geodesically convex and relatively compact. Now let c n be a causal curve from a n to b n through y; then L(c n ) ≥ δ/ and thus cn β ≥ L(c n ) · ≥ δ > 0. Now denote by c n a geodesic joining a n to b n in U (parametrized by arclength) and by (c n ) * the same curve with the opposite orientation. Then the concatenation c n #(c n ) * is a closed loop in M ; since M is simply-connected this loop bounds a disc D and thus Now α is bounded on the compact set U , hence their exists C > 0 such that c n α ≤ C · L(c n ) = C · d(a n , b n ). We have thus established for all n > n 0 the inequality 0 < δ ≤ C · d(a n , b n ). Since d(a n , b n ) → 0, this is a contradiction. Definition 1. 1 . 1Let X be a set. A family of subsets {H n } n∈Z of X is called a half-space filtration of X if (H1) H n+1 H n , n ∈ Z. (H2) H n = ∅, H n = X. Figure 1 . 1A planar embedding of PSL 2 (Z) corresponding to the Rademacher quasimorphism Definition 1.4. Let (X, ) be a poset and denote by Aut(X, ) the group of order-preserving permutations of X. Then an element T Corollary 1 . 12 . 112Let H <Ǧ be a subgroup. Then the Guichardet-Wigner quasimorphism vanishes along H if and only if some (hence any) H-orbit inŘ is bounded. Acknowledgement: We would like to thank Uri Bader, Marc Burger, Danny Calegari,Étienne Ghys, Pascal Rolli and Mark Sapir for comments and remarks related to the present article. We also thank Andreas Leiser and Tobias Struble for help with the pictures.The authors also acknowledge the hospitality of IHES, Bures-sur-Yvette, and Hausdorff Institute, Bonn. The first named author is grateful to Departement Mathematik of ETH Zürich and in particular Dietmar Salamon and Paul Biran for the support during this academic year. The second named author was supported by SNF grants PP002-102765 and 200021-127016. 2. Foundations of quasi-total orders 2.1. The translation number of a quasi-total order. Throughout this subsection let (X, , {H n }) denote a fixed half-space order of width w and let G be a group acting by quasi-automorphisms of defect d on (X, , {H n }) (see Definition 1.1). Proposition 2.1. The functions {f a : G → X} a∈X given by f a (g) := h(ga, a) are mutually equivalent quasimorphisms of defect ≤ d. In fact, their mutual distances are uniformly bounded by d. Corollary 2 . 2 .. 3 . 223There exists a nonzero homogeneous quasimorphism T (X, ,{Hn}) : G → R of defect ≤ 2d such that for all a ∈ X, T (X, ,{Hn}) The quasimorphism T (X, ,{Hn}) : G → R is called the translation number associated with the action of G on (X, , {H n }). hence gx f x and thus g ≥ f e, finishing the proof Note that in view of the remarks following Theorem 2.4 we have now completed the proof of Theorem 1.2.3.3. Elementary constructions. The notion of a quasi-total triple is closed under various elementary constructions. The following three persistence properties are immediate from the definition: Proposition 3.7. Figure 2 . 2A planar embedding of a subset of the free group Figure 3 . 3Growing two hairs at a etc. to the left. Finally, if the b −1 -neighbour of w is missing, then the last letter of w is either a or b −1 and we place wb −1 two steps left below, respectively straight below w accordingly. We then add wb −1 a −1 , wb −1 a −1 b −1 etc. to the left. (See again Figure 3.) Proposition 4 . 9 . 49Let G be a countable group and a left-invariant total order on G. a dynamical realization. (ii) There exists a special dynamical realization of adapted to x ∈ G if and only if x is both central and dominant for . Proof. (i) see [19, Prop. 2.1]. (ii) Assume that such a realization exists. by T in G( M , C) acts properly discontinuously on M ; then M := Γ\ M is a manifold and p : M → M is a covering projection. We will denote byǦ the centralizer of T (hence Γ) in G( M , C). Then G :=Ǧ/Γ acts on M andǦ is a central extension of G by Γ. We refer to the central extension 0 → Γ →Ǧ → G → 1 as the central extension associated with the covering p : M → M . Definition 5.2. The covering p : M → M is called a causal covering if M is totally acausal (in the sense of Definition A.6). Lemma 5.3. Assume that p : M → M is a causal covering, Then either T or T −1 is dominant. Proof. Denote by p : M → M the covering projection and let a, b ∈ M . Definition 5. 4 . 4A causal covering M → M is called quasi-total if the number n(a, b) is uniformly bounded. Let T be the translation along the x-axis by x 0 , i.e. T (x, y) := (x+x 0 , y) and let M := M / T . Then M ∼ = S 1 ×] − 1, 1[ and M → M is a quasitotal causal covering. Theorem 5. 7 . 7Let p : M → M be a causal covering. If G :=Ǧ/Γ acts almost 2-transitively on M , then p is quasi-total. Lemma 5 . 8 . 58In the situation of Theorem 5.7 there exists a constant N ∈ N, depending only on M , and a point c ∈ M such that for all b ∈ M there exists l ∈ Z such that b T l c T N b. l +l c T 2N 0 bWe may thus choose N := 2N 0 . Now we deduce: hence T l −l−kN b a. Now choosing k appropriately we can ensure that −N l − l − kN N . Thus ∀a, b ∈ M ∃k ∈ {−N, . . . , N } : (a T k b) ∨ (b T k a). Now (1.2) follows for N ( M ) := 2N . e. completeness of the triple ( M , , T ). We then call the covering p : M → M a complete causal covering. This terminology understood we have the following useful criterion: Theorem 5.9. Assume that M is compact. Then any complete causal covering p : M → M is quasi-total.Proof. We first claim that there exists point a, b ∈ M such that a x b for all x ∈ U . Indeed, choosing a and b close enough we can ensure that exp(Int( C a )) and exp(Int(− C b )) have open intersection. By compactness of M there exists finally many elements g 1 , . . . , g l such that M = H − ∪ H + . For j = 1, . . . , l we set a j := g j a, b j := g j b. Lemma 5 . 12 . 512For every x 0 ∈ M there exists a bounded subset M x 0 ⊂ M such that p(M x 0 ) = M and x x 0 for all x ∈ M x 0 . Now let b ∈ M . We then find N ∈ N such that N ≥ N 0 and [a, b] ⊂ [a, T N a]. It remains to show that [a, T N a] is bounded. We claim that [a, T N a] ⊂ N n=−N T n M a , which implies the desired boundedness. Indeed, let c ∈ [a, T N a] and let x := p(c). We consider the fiber F x := p −1 x. By Lemma 5.11 we have with the compact symmetric space K/M , where M denotes the stabilizer of −e V . Since the stabilizer action of M preserves both e V and the inner product, there exists a K-invariant Corollary 5 . 15 . 515The Kaneyuki order is globally hyperbolic. Combining this observation with Theorem 2.7 we deduce: Corollary 5.16. Let H <Ǧ be a subgroup. Then the following are equivalent: (i) The Guichardet-Wigner quasimorphism vanishes on H. (ii) H has a bounded orbit inŘ. (iii) Every H-orbit inŘ is bounded. Definition A. 3 . 3Let (M, C) be a conal manifold. A piecewise smooth curve γ : [a, b] → M is called C-causal ifγ(t) ∈ C γ(t) for all but finitely many t ∈ [a, b]. The relation s on M obtained by setting x s y if there exists a causal curve γ : [a, b] → M with γ(a) = x and γ(b) = y, is called the strict causality of (M, C). Lemma A. 5 ( 5Hilgert-Olafsson). Let (M, C) be a uniformly homogeneous conal manifold. Then the closure of s in M × M is a causality. Moreover, the order intervals of are the closures of the order intervals of s . Proof. The assumption of uniform homogeneity guarantees that the proof of [14, Prop. 2.2.4] carries over. Denote by d the metric induced by the Riemannian metric, for which α is uniformly positive. Then for any causal curve c : [a, b] (t) (ċ(t)) ≥ L(c) · , Proof. Let a ∈ M be some basepoint. Then there exists an open subset U ⊂ M with the following properties:• There exists a compact neighbourhood V of U , which is evenly covered under p;Indeed, we can take a sufficiently small open subset of exp(Int(C a )).Since M is compact and weakly homogeneous there exist g 1 , . . . , g r ∈ G(M, C) such thatThis is clearly bounded (since the closure of each V j is compact) and covers M . It remains to show that x x j for every x ∈ V j ; this however follows easily by lifting the curve γ p(x) to V j with basepoint x j ; the resulting lift is then a causal curve joining x j with x. Now we can easily deduce the theorem:Proof of Theorem 5.10. Let a ∈ M . We use the lemma to obtain a bounded subset M a of M such that p(M a ) = M and x a for all x ∈ M a . In particular we have x T a for all x ∈ M a . We claim that there exists some N 0 ∈ N such thatIndeed, suppose otherwise. Then there exists a sequence x n ∈ M a with x n ≥ T n a. Since M a is bounded there exists a subsequence n k such that x n k → x ∈ M . Now for every l there exists k 0 ∈ N such that for every k ≥ k 0 we have n k ≥ l and thus x n k T T l a. This implies that x n k T l a for some l ∈ {l − N, · · · , l} for some fixed constant N . By passing to another subsequence we can thus ensure that x n km T l a for k m ≥ k 0 . Since is closed this yields x T l a and thus x T T l−N a. Since l was arbitrary, this contradicts the fact that T is dominating. This contradiction establishes (5.5). Cocycles d'Euler et de Maslov. J Barge, Ghys, Math. Ann. 2942J. Barge andÉ. Ghys. Cocycles d'Euler et de Maslov. Math. Ann., 294(2):235- 265, 1992. Reconstructing quasimorphisms from associated partial orders. G , Ben Simon, T Hartnick, Comm. Math. Helv. to appearG. Ben Simon and T. Hartnick. Reconstructing quasimorphisms from associ- ated partial orders. Comm. Math. Helv., to appear. Surface group representations with maximal Toledo invariant. 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In this paper a deterministic preprocessing algorithm is presented, whose output can be given as input to most state-of-the-art epipolar geometry estimation algorithms, improving their results considerably. They are now able to succeed on hard cases for which they failed before. The algorithm consists of three steps, whose scope changes from local to global. In the local step it extracts from a pair of images local features (e.g. SIFT). Similar features from each image are clustered and the clusters are matched yielding a large number of putative matches. In the second step pairs of spatially close features (called 2keypoints) are matched and ranked by a classifier. The 2keypoint matches with the highest ranks are selected. In the global step, from each two 2keypoint matches a fundamental matrix is computed. As quite a few of the matrices are generated from correct matches they are used to rank the putative matches found in the first step. For each match the number of fundamental matrices, for which it approximately satisfies the epipolar constraint, is calculated. This set of matches is combined with the putative matches generated by standard methods and their probabilities to be correct are estimated by a classifier. These are then given as input to state-of-the-art epipolar geometry estimation algorithms such as BEEM, BLOGS and USAC yielding much better results than the original algorithms. This was shown in extensive

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A General Preprocessing Method for Improved Performance of Epipolar Geometry Estimation Algorithms Maria Kushnir [email protected] Ilan Shimshoni [email protected] Maria Kushnir Ilan Shimshoni Department of Information Systems Department of Information Systems University of Haifa 31905HaifaIsrael University of Haifa 31905HaifaIsrael A General Preprocessing Method for Improved Performance of Epipolar Geometry Estimation Algorithms Received: date / Accepted: dateIJCV manuscript No. (will be inserted by the editor) testing performed on almost 900 image pairs from six publicly available datasets.Fundamental matrix · epipolar geometry estimation · local features · SIFT In this paper a deterministic preprocessing algorithm is presented, whose output can be given as input to most state-of-the-art epipolar geometry estimation algorithms, improving their results considerably. They are now able to succeed on hard cases for which they failed before. The algorithm consists of three steps, whose scope changes from local to global. In the local step it extracts from a pair of images local features (e.g. SIFT). Similar features from each image are clustered and the clusters are matched yielding a large number of putative matches. In the second step pairs of spatially close features (called 2keypoints) are matched and ranked by a classifier. The 2keypoint matches with the highest ranks are selected. In the global step, from each two 2keypoint matches a fundamental matrix is computed. As quite a few of the matrices are generated from correct matches they are used to rank the putative matches found in the first step. For each match the number of fundamental matrices, for which it approximately satisfies the epipolar constraint, is calculated. This set of matches is combined with the putative matches generated by standard methods and their probabilities to be correct are estimated by a classifier. These are then given as input to state-of-the-art epipolar geometry estimation algorithms such as BEEM, BLOGS and USAC yielding much better results than the original algorithms. This was shown in extensive Introduction Epipolar geometry estimation from image pairs with partial scene overlap is a basic problem in computer vision. It is used as a component of many important applications such as vision based robot navigation, structure from motion (SfM) and other multiple view geometry applications. This problem has attracted considerable interest in the computer vision community, interest which continues till this day. Most of the successful algorithms are based on an initial step, in which local features are detected in both images. For each detected feature a local descriptor is computed. These features are then matched based on their local descriptors. For each putative match a prior probability or score is estimated. These putative matches and scores are given as input to the algorithm (Chum et al, 2003;Chum and Matas, 2005;Brahmachari and Sarkar, 2013b;Goshen and Shimshoni, 2008;Raguram et al, 2013a;Tordoff and Murray, 2002). Even though successful algorithms have been proposed to address this problem, it still remains an active field of research. This is because there are several reasons why input given to these algorithms may be challenging. As pointed out for example by Lowe (2004) and tested extensively by Mikolajczyk et al (2005), as the angle between the viewing directions increases, the appearance of local descriptors changes, making them hard to match. Thus, wide baseline images are hard inputs for the algorithms. Urban scenes are also challenging for such algorithms. In such scenes features such as for example windows are repeated several times. In such cases it is hard for the local matching algorithm to match the window in the first image to its corresponding window in the second image. In both these types of cases the percentage of correct matches (inliers) from the set of putative matches is low. When the probabilities are taken into account, the problem is that the percentage of correct matches with high prior probabilities is low. In these cases, even state-of-the-art algorithms tend to fail. For that reason, in this paper, instead of trying to propose a new epipolar geometry estimation algorithm, we present a preprocessing step which is given as input two images and returns a set of putative matches with their associated probabilities. Our method was extensively tested on almost 900 image pairs from different datasets: ZuBuD dataset (Shao et al, 2003), BLOGS dataset (Brahmachari and Sarkar, 2013a), USAC dataset (Raguram et al, 2013b) and Open1, Open2 and Urban datasets (Goldman et al, 2014). Our results are much better than those obtained by the standard initial steps of state-of-the-art algorithms. Consequently, when our output is given as input to them (BEEM (Goshen and Shimshoni, 2008), BLOGS (Brahmachari and Sarkar, 2013b) and USAC (Raguram et al, 2013a) in this paper) they outperform the same algorithms operating on their regular input. Our output is general and can be incorporated within many other algorithms such as Chum et al (2003); Chum and Matas (2005); Tordoff and Murray (2002). The algorithm starts with standard techniques of detecting local features and extracting putative correspondences from them. Using this input we propose a new concept consisting of three steps, running from local to global. In the local step features are clustered together in each image. Clusters with similar features from the first image are matched to clusters of features from the second image and vice versa. The result of this step is a large set of putative matches, most of which are incorrect. In the second step we match pairs of spatially close features (2keypoints) in the first image, to corresponding pairs of features (found in the first step) from the second image. For each 2keypoint match a short descriptor is generated, characterizing the quality of the match. Using a classifier we trained on data from several image pairs, for each 2keypoint match the probability of being correct is estimated. The highest K 2kp 2keypoint matches are chosen. Here already the percentage of correct 2keypoint matches is much higher then was recovered in the first step. In the global step, the 2keypoint matches are used to generate a large number of possible fundamental matrices. For each putative match from the first step we calculate the number of fundamental matrices it supports. Finally we combine putative matches generated by standard methods with those found by our method, and estimate their probabilities to be correct, using a simple classifier. The correlation between these probabilities and the ground truth inlier-outlier labels is much higher. As a result, when we submit the putative matches and the computed probabilities as input to algorithms from the guided RANSAC family, much better results are obtained on challenging datasets. For example, the performance of all three algorithms when run on the Open2 dataset (Goldman et al, 2014) increased considerably. The number of image pairs they succeeded on increased by between 62% and 239% relative to the original performance of those algorithms. This demonstrates the fact that our algorithm improves the quality of the input significantly, resulting in better results of the basic algorithm. Similar improvement was obtained when the algorithm was run as a preprocessing step of USAC on the Urban dataset (Goldman et al, 2014). The paper continues as follows. In Section 2, we review related work concentrating mainly on how the quality of the input affects the performance of the algorithm. Section 3 presents the overview of our method, while the details are given in the next section. Experimental results are presented in Section 5. We conclude in Section 6. Related work In reviewing related work we will concentrate on how the quality of the input effects the algorithm's performance and not on the various components of the algorithms. We will first consider PROSAC (Chum and Matas, 2005) and USAC (Raguram et al, 2013a). The algorithm is given as input a set of putative matches ordered by a score or prior probability. Under this general framework the set of putative matches can be ordered for example using the distance ratio d r method introduced by Lowe (2004). The models are generated in an order consistent with the order of the matches used to generate them. Once a model is generated it is verified using the Statistical Probability Ratio Test (SPRT). The putative matches are tested until the SPRT reaches a decision on whether the model is correct or not. Thus, when the beginning of the list (matches with high scores) is contaminated by a large number of outliers, the number of required iterations increases considerably. When the number of iterations of the algorithm is limited, this also increases the probability of failure. On the other hand, a list consisting of a large number of matches does not effect the running time, since the SPRT process usually reaches a decision quite early in the verification procedure. In algorithms from the Guided RANSAC family (Tordoff and Murray, 2002;Goshen and Shimshoni, 2008) the subset of matches used for model generation is chosen according to their probability. Thus, the performance of the algorithm is similar to that of PROSAC. When the list contains a large number of outliers with high probabilities, the chances of the algorithm to fail are high. A similar behavior occurs in BLOGS (Brahmachari and Sarkar, 2013b). There also, in the global search step, a model is computed from a minimal subset of matches according to a score. Thus, the probability of finding a model consisting only of inliers depends on the quality of the prior scores. Specifically in BLOGS, a new method for putative match ranking was introduced, and their scores are referred to as similarity weights {t k }. Thus, for all these algorithms, if we can assign more accurate probabilities to the putative matches, the algorithms performance should improve considerably. This is exactly the goal of the algorithm we suggest here. We would also like to review two other algorithms which address the problem of matching images containing scenes with repeated structures. In that case the initial stage of the algorithms mentioned above will fail to match a feature belonging to repeated structures to its correct match in the second image, since it will not be able to choose the correct candidate. Thus, this feature will be discarded. In Generalized RANSAC (Zhang and Kosecka, 2006), all possible matches of the feature to similar (normalized cross correlation above a certain threshold) features in the second image are generated but are given low probabilities. On the list of putative matches guided RANSAC is run. Thus, in the case when there are not enough non-repeating inliers in the list with high probabilities, the algorithm might fail. In our previous work (Kushnir and Shimshoni, 2014), a special algorithm was developed to deal with buildings with repeated features. There also, all possible matches of the feature to similar features in the second image are generated. The algorithm assumes that in both images a planar facade is visible. The algorithm tends to fail when this assumption is not satisfied. In this work we propose a method which can successfully deal with general scenes, including the case of repeated structures. Algorithm outline The goal of the algorithm is to generate a set of putative feature matches between the two images, where each match is accompanied by a prior probability (or score) that the match is correct. The higher the quality of this set, the more probable that algorithms from the guided RANSAC family (Chum and Matas, 2005;Raguram et al, 2013a;Brahmachari and Sarkar, 2013b;Goshen and Shimshoni, 2008) will succeed to estimate the epipolar geometry. In this section we will present an overview of the algorithm. The details will be given in the next section. The algorithm, given in pseudo-code in Algorithm 1, is described as follows: Algorithm 1 A General Preprocessing Method for Improved Performance of Epipolar Geometry Estimation 1: Input: images I 1 and I 2 2: Extract SIFT features from I 1 and I 2 3: Find standard putative correspondences {X L } and associate to them distance ratios {d r } 4: Find standard putative correspondences {X B } and associate to them similarity weights {t k } 5: Cluster SIFT features from each image based on descriptor similarity yielding clusters of features 6: Estimate relative roll angle α exp 7: for α ∈ [α exp , 0 • ] do 8: Match clusters from the two images, yielding cluster pairs 9: Generate putative correspondences {X} from the members of the matched clusters 10: Generate all 2keypoints: a pair of features from a main feature point and another feature point which is close to it in the image 11: Match 2keypoints from the first image to the 2keypoints from the second image 12: Use a classifier to assign probabilities to 2keypoint matches 13: Select the top K 2kp of 2keypoint matches 14: Estimate a candidate fundamental matrix from each two matched 2keypoints, yielding K 2kp (K 2kp − 1)/2 matrices 15: For each putative match from {X} count how many candidate fundamental matrices (sf m) support it 16: Assign each putative match from {X} {X L } {X B } a probability that it is correct 17: Use these putative matches and their associated probabilities as input to one of the algorithms from the guided RANSAC family to yield a fundamental matrix and its support 18: end for 19: Choose from the two fundamental matrices, the one with maximal support 20: return The fundamental matrix and the list of its inliers The algorithm is given as input two images I 1 and I 2 . The first step of the algorithm (described in Section 4.1) is to detect features (SIFT in our case) in each image. Those features are used to generate three groups of putative correspondences. Following the standard method introduced by Lowe (2004), we find putative correspondences {X L } and as-sociate to them distance ratios {d r }. The distance ratio is used to assign a prior probability for the correctness of the match (described in Section 4.1). Following the scheme introduced by Brahmachari and Sarkar (2013b), we find putative correspondences {X B } and associate to them similarity weights {t k }. To resolve problems which occur in image pairs which are hard to match, such as scenes which include repeating elements, we cluster detected features (described in Section 4.2). Thus, repeated features or features with very similar descriptors are clustered together. Non-repeating features will belong to clusters of size one. Then each cluster from the first image is matched to the most similar cluster in the second image and vice versa. The result of this step is a large number of putative correspondences {X} (described in Section 4.3). A vast majority of them however, are incorrect. An example of how clustering similar features can help in the case of a scene which includes repeating elements, is shown in Figure 1. For the feature point marked by a Red circle in the upper image Lowe (2004) finds no match and Brahmachari and Sarkar (2013b) find the feature point marked by a Red circle in the bottom image, which is incorrect. When using clustering, as we suggest, the feature point in the upper image belongs to a cluster of size one, while in the bottom image two feature points are clustered together, marked by Green points. Only in the case of clustering, a correct match is generated, namely the correct match is a member of {X}, but not of {X L } or {X B }. In addition, in order to overcome the problem of features looking similar to rotated features (such as window corners), for the clustering step only, the orientation of all the SIFT features in the image is fixed in one specific direction. Thus, for example different corners of a window will not be clustered together. In order to determine this direction, we propose to find a rough relative roll angle α, between the two images, from the differences of SIFT orientations in {X L } and {X B } and denote it α exp . Using this approximation and the fact, that many images are taken with zero roll angle as a prior, all the following steps of our algorithm are repeated twice, once for α = α exp and again for α = 0 • . In order to overcome the problem that the majority of the matches in {X} are incorrect we estimate their probabilities to be inliers. This is done in two steps: In the first step, which is described in Section 4.4, local information is used. We create a pair of features from a main feature point and another feature point which is close to it in the image. This pair of features is called a 2keypoint. These two features are matched to corresponding features in the second image which are also close to each other and belong to matching clusters. An illustration of a 2keypoint match is shown in Figure 2. The decision to work with 2keypoints is a compromise between two contradictory preferences: on the one hand any combination of features contains more information than a single keypoint, which can be used to detect inliers more accurately. In general, the larger the number of features in the combination, the higher the probability that the matched combination is correct. On the other hand, since the probability for feature detection is low, the probability for detecting a large number of features in a combination is even lower. Thus, relying on the minimal subset of features is preferable due to the difficulties in detecting large combinations of features. From the set of 2keypoint matches we would like to choose a subset, which have a high prior probability to be correctly matched. In order to accomplish this, each 2keypoint match is characterized by a short descriptor. The descriptor consists of measures of geometric similarity between the two 2keypoints and a count of the Fig. 2 An example of the advantage in using 2keypoints. Images (a) FLH00010 and (b) FLH00016 taken from the Open2 dataset. (c) Zoom-in of (a). (d) Zoom-in of (b). A correct 2keypoint match generated by our method and ranked in the fifth place. When single keypoint matches are used both Blue and Red matches are ranked much lower. (a) (b) (c) (d) number of possible matches between each 2keypoint in one of the images to 2keypoints in the other image. As the interdependencies between these characteristics are complex, a classifier is trained to learn the probability that the 2keypoint is correctly matched. At test time each 2keypoint match is assigned a probability and the top K 2kp (100 in our application) 2keypoint matches are selected. An example of how the generation and ranking of 2keypoint matches can help in dealing with low ranking matches, is shown in Figure 2. In that example if only single keypoints are used, standard techniques (Lowe, 2004) and (Brahmachari and Sarkar, 2013b) would order the Blue match in the 55th and 16th places respectively, whereas the Red match would be placed in a 161th place or would not be generated at all. On the other hand, when 2keypoint matches are used, the correct 2keypoint is ranked in the fifth place. In the second step, described in Section 4.5, global information is used. Up until now the analysis we performed has been local in nature. We first matched single features and then pairs of close features. In order to be able to assign more accurate probabilities to the matches, the epipolar geometry constraint, which is global in nature, comes into play. In order to generate rough estimations of the fundamental matrix we borrow an idea from the BEEM algorithm (Goshen and Shimshoni, 2008), where it is estimated from only two matches. In our case from each two matched 2keypoints a candidate fundamental matrix is estimated, yielding K 2kp (K 2kp − 1)/2 matrices. As a result of the ranking of the 2keypoints, quite a few of them are generated from inlier matches. The problem is that even in this case they are quite inaccurate. Each of them is supported (i.e., the Sampson distance computed from the matrix and the match is below a certain threshold) by a subset of the inliers and quite a few outliers. Instead of returning the matrix with the largest support, we exploit these matrices in a different way. For each putative match from {X} we count how many candidate fundamental matrices (sf m) support it. This number is a strong indication of the probability that this putative match is inlier. Finally we combine the three groups of putative correspondences {X}, {X L } and {X B }, in order to achieve a set of putative feature matches between the two images, where each match is accompanied by a prior probability that the match is correct (described in Section 4.6). For that purpose we construct a keypoint match descriptor, denoted kpmd, and train a classifier on it. The descriptor consists of the local measures of similarity, namely {d r } and {t k } and the global measure {sf m}. At test time each putative feature match is assigned a probability. As was already mentioned, all the previous steps of our algorithm are repeated twice, once for α = α exp and again for α = 0 • . In order to proceed we run one of the algorithms from the guided RANSAC family twice, once for each set of putative matches, and choose the one with maximal support. The output of the entire process is a fundamental matrix along with its inlier set. Algorithm details We will now delve into the details of the various components of the algorithm. Extraction of putative matches with their distance ratios and similarity weights The algorithm is given as input two images. As a first step we apply feature detection on both images. In general, any feature detector which returns the location, scale, and orientation can be used (e.g.: MSER (Matas et al, 2002), BRISK (Leutenegger et al, 2011), ORB (Rublee et al, 2011), SURF (Bay et al, 2006), SIFT (Lowe, 2004)). In our case we use the implementation of SIFT by Vedaldi and Fulkerson (2008). Following the standard method introduced by Lowe (2004), we find putative correspondences {X L } based on descriptor similarity. The best candidate match for each keypoint in the first image, is found by identifying its nearest neighbor in the second image. The nearest neighbor is defined as the keypoint with maximal normalized cross-correlation from the given descriptor vector. The probability that a match is correct can be determined by taking a distance ratio d r = cos −1 (m k ) cos −1 (m k2 ) , where m k is the similarity to the closest neighbor and m k2 is the second highest similarity in the second image. All matches for which the distance ratio is greater than a certain threshold (in our case 0.9) are rejected. This choice of threshold distance ratio is relatively high (there are many works where 0.85 or even 0.8 are used) and many more matches are kept. This is done since we rely on the next steps of our method to deal with them correctly (See for example Figure 2.). In addition we follow the scheme introduced by Brahmachari and Sarkar (2013b), which is a different way to define putative correspondences and weights. We find putative correspondences {X B } that exhibit the highest similarity measure in both images. This means that putative correspondence x k = (u k , v k ) is a member of {X B } if for a keypoint in the first image u k its nearest neighbor in the second image is v k , and for the keypoint in the second image v k its nearest neighbor in the first image is u k . With each such putative match pair, they associate a confidence measure which is referred to as the similarity weight. They define the similarity weight t k for the correspondence x k as t k = 1 − exp −m k 2 1 − m k1 m k 1 − m k2 m k , where m k1 is the second highest similarity in the first image and m k2 is the second highest similarity in the second image as mentioned above. While the third term of the t k , i.e. 1 − m k 2 m k can be interpreted as another version of d r , the two other terms are new. The second term, i.e. 1 − m k 1 m k is a symmetric complimentary of the third one, and it emphasizes that there should be no difference between the treatment of first and second image. The first term, i.e. (1 − exp −m k ) 2 is a similarity based component. While Lowe in his work did not use the absolute distance/similarity as a measure of similarity, in BLOGS the contribution of the absolute similarity exists. Feature clustering Using the former putative matches as is, is sometimes insufficient due to the following two problems which occur in challenging image pairs. In scenes which include repeating elements (such as for example windows of buildings), the matching process is unable to match the repeated features correctly. In image pairs with wide baselines, the descriptors of the matching features are quite dissimilar and will receive quite low matching scores. We will now deal with the first problem. The second problem will be addressed in Section 4.4. In each image, the features recovered from it in Section 4.1, are clustered based on descriptor similarity. In our algorithm we use agglomerative clustering. The merging of clusters stops when the similarity measure between the closest clusters is below a certain threshold (normalized cross-correlation below 0.85). The result of this process is a set of clusters of features. Nonrepeating features yield clusters of size one. Each cluster is represented by the median descriptor of its members. Fig. 3 An example of keypoint clustering with and without fixed orientation. Image object0008.view04 was taken from the ZuBuD dataset. Black circles: clustering with fixed SIFT orientation. Red points: clustering without fixed orientation. In the latter case three different types of corners are clustered together. In order to overcome the problem, which is common in buildings, of features looking similar to rotated features (such as window corners), for the clustering step only, the orientation of all the SIFT features in the image is fixed in one specific direction. Thus, for example different corners of a window will not be clustered together. An example of keypoints clustering with and without fixing the orientation of all the SIFT features is presented in Figure 3. The Red points show a cluster without fixed orientation. In that case three dif- ferent orientations of the window corner are clustered together. The Black circles are features clustered together, when fixing the SIFT orientation. All of them are upper left corners of a window. In general, clustering without fixed orientation of all the SIFT features in the image, leads not only to larger clusters which can be handled by our method, but to systematic errors when matching features from those clusters. This will be further explained later on. (a) (b) (c) (d) (e) (f) The approach, of defining one specific orientation for all the SIFT features in the image, has been extensively used in the literature in the frame of upright SIFT, and in this work we generalize this idea to any orientation. It is true, that there are many applications such as vision based robot navigation and structure from motion, where all the images are taken with a zero roll angle, which justifies the upright SIFT assumption in all the images. However, since we do not limit our approach to any specific application, we propose to find a rough approximation of the relative roll angle α, between the two images, from the existing data. For that purpose we calculate the difference of SIFT orientations in each putative match found in Section 4.1, and build a kernel density estimation of those angle differences. Although the transformation between the two images is perspec-tive and not affine, the maximal peak α exp of this function can be used as a rough approximation for α. Two examples of α extraction are shown in Figure 4. The red arrows were added to indicate the upright direction. In the first row the image pair, taken with a relative roll of α = 78 • , along with its kernel density estimation of the angle difference is presented. The maximal peak of this kernel density is precisely α exp = 78 • . In the second row an image pair, taken with zero relative roll is shown. The maximal peak of its kernel density estimation is located at α exp = −3 • which is quite a good approximation for α. Using this approximation and the fact that many images are taken with zero roll angle as a prior, we proceed as follows. When running the algorithm, the first image is processed one time using an upright SIFT, while the second image is processed in two orientations [α exp , 0 • ]. Therefore all the following steps of our algorithm, described in Sections 4.3-4.6 are repeated twice, once for each orientation. Generation of all keypoint matches After both of the images have been processed as described above, the next step is to generate pairs of possible matches between features from the two images. The main problem we have to overcome is when a real cluster is segmented into several clusters. We try to deal with this problem as follows. Each cluster from the first image is matched to the closest cluster from the second image using the normalized cross-correlation between the cluster representatives. This process is repeated when the roles of the images are switched. Thus, if a real cluster was over-segmented in one image but not in the other we can still match the clusters from the two images correctly. If there is over-segmentation in both images, not all possible matches will be found. Due to this problem we can not use the distance ratio method suggested by Lowe (2004). Because if the distances to the closest cluster and the second closest cluster are similar, we can not distinguish between oversegmentation and when both clusters in the second image equally far from the cluster in the first image and should not be matched at all. Therefore the closest cluster is always chosen. is segmented into two smaller (Green and Yellow) clusters in (a). In this example due to cluster matching in both directions all the correct matches were found. An example of the result of the clustering process is shown in Figure 5. The large Red cluster in the second image is segmented into two smaller (Green and Yellow) clusters in the first image. The Red cluster was matched to the Yellow cluster when clusters from the second image are matched to clusters in the first image. Because the matching is also done from clusters in the first image to the closest cluster in the second image, the Green cluster is also matched to the Red cluster. In this example all of the correct matches were found together with many incorrect matches. Each pair of clusters which is matched yields a set of putative feature matches from the members of the two clusters. The result of this step is a large number of possible matches most of which are obviously incorrect. We will now refer to the systematic errors mentioned above when explaining the clustering without a fixed orientation of all the SIFT features in the image. As was already shown, when the orientation of all the SIFT features is not kept fixed, features looking similar to rotated features (such as window corners) will be clustered together. In that case, when generating putative feature matches, there will be matches of the same feature (left upper corner in both images for example) and there also will be matches of the rotated features (left upper corner in the first image matched to right lower corner in the second image for example). In the next steps of the algorithm, these matches of the rotated features will all vote together supporting each other and will lead to systematic errors. We therefore chose to keep the orientation of all the SIFT features fixed during the clustering in Section 4.2, to prevent such failures. Generation and ranking of 2keypoint matches Most of the feature pairs generated in the previous stage are incorrect and therefore in order to be able to use them for epipolar geometry estimation prior probabilities have to be assigned to them. This will be done in two steps: a step which uses only local information which will be described here and a step which uses global information described in the next section. Recall that each SIFT feature p has besides a descriptor also a scale s(p) and an orientation angle α(p). These values will be used in our analysis to make it scale and orientation invariant. For each feature point p we add a neighboring feature n. The distance between the features in terms of the scale of p is denoted d = |p−n|/s(p). The angle between the vector connecting p to n with respect to α(p) is denoted θ. This pair of features will be termed a 2keypoint. A 2keypoint {p 1 , n 1 } in the first image is matched to a 2keypoint {p 2 , n 2 } in the second image. Naturally, p 1 and p 2 , and n 1 and n 2 , have to be putative matches. This set of four features is illustrated in Figure 6. We suggest three methods to choose the neighboring pairs {n 1 , n 2 } close to {p 1 , p 2 }. The first method simply takes n i , i = 1, 2 from the K 1 closest features around p i . The second method chooses n i from all the features within a certain distance K 2 s(p i ) in pixels from p i . This parameter is given in units of scale in order to be scale invariant. Finally, the third method chooses n i from the K 3 closest features which belong to the same cluster as p i . Experimentally we found that optimal values are achieved for K 1 = 5, K 2 = 5, and K 3 = 1. In order to estimate the probability that a 2keypoint match consists only of inliers, we have to take into account quite a few factors. Due to the interdependencies between these factors and their effects on the estimated probability, we construct a 2keypoint match descriptor, denoted 2kpmd, and train a classifier on it. The descriptor consists of the following fields: 2kpmd = [N 1 ; N 2 ; dist r ; angle d ; cluster t ; min d ]. The definitions of the fields are as follows: N 1 and N 2 are the number of 2keypoint matches that the 2keypoints {p 1 , n 1 } and {p 2 , n 2 } belong to respectively. The smaller the values, the higher the probability that the 2keypoint match consists of inliers. dist r = min(d 1 /d 2 , d 2 /d 1 ) is the ratio of the distance between p 1 and n 1 in the first image in terms of s(p 1 ) to the distance between p 2 and n 2 in the second image in terms of s(p 2 ). This measure is scale invariant and its value should be close to one. The value angle d = ang dif f (θ 1 , θ 2 ) measures the difference between the angles associated with the two 2keypoints and should be close to zero. The field cluster t is equal one if p i and n i belong to the same cluster and zero otherwise. Finally, the distance between the point and its neighbor also affects the probability that the 2keypoint match is correct. The further the points are, the lower the probability is. We therefore define min d = min(d 1 , d 2 ). In order to train the classifier we chose six image pairs from the ZuBuD dataset. From them four were cases that state-of-the-art algorithms were able to match, while the other two were more challenging. For these image pairs we manually found the ground truth matches and trained on the data a C4.5 decision tree classifier (Quinlan, 1993). This classifier returns for each descriptor the probability that the 2keypoint match consists of inliers. The training set consists of 31352 2keypoint matches of which 4102 are inliers and the rest are outliers. The quality of the classifier was estimated using a 10-fold cross-validation procedure. When choosing a classifier, we tried several options such as random forest (Breiman, 2001), SVM (Vapnik, 1995) and others. The C4.5 decision tree classifier was selected, as the one which not only classifies correctly 91.6% of the 2keypoint matches, but also gives a maximal precision (the proportion of positive results that are true positive) of 73.8%, which is the most important parameter as will be now explained. The 2keypoint matches are then sorted by probability and the highest K 2kp (K 2kp = 100 in our implementation) are chosen. Thus, what is most important is that from the top K 2kp a fair amount of them should be inliers (precision). This is evident from the results on the training set shown in Figure 7. The cumulative precision of the classifier is shown as a function (on a log scale) of the number of 2keypoint matches ordered by the probability returned by the classifier. Dashed curves represent the inlier percentages from all the 2keypoint matches, which would be correct if no classifier existed. Consider for example the hardest case of the image pair (obj066view2,obj066view5). From the top ranked 100 2keypoint matches, 41% were inliers, while their percentage from all the 2keypoint matches was only 3.84%. Global ranking of matches Even though we could use the 2keypoint matches found in the previous step as the input for epipolar geometry estimation, better results can be obtained by exploiting global information. In BEEM (Goshen and Shimshoni, 2008) a method was proposed to generate a rough estimate of the fundamental matrix using only two pairs of matches instead of 7 or 8. This is done by using the similarity transformation between the regions around the corresponding features, to generate three additional matches for each "real" match. The resulting estimated fundamental matrix is quite inaccurate but can be used as a basis for local optimization (Chum et al, 2003), yielding good results. In our case we use two 2keypoint matches (four matched points) as the input for estimating the fundamental matrix. One 2keypoint match could not be used since all the points from each image are too close to each other to generate a meaningful result. For each of the K 2kp (K 2kp − 1)/2 pairs of 2keypoint matches a fundamental matrix F is generated. All the putative matches generated in Section 4.3 are checked to see whether they support F or not. Instead of taking the fundamental matrix with the largest support as the result of our algorithm, we suggest here a method to exploit all the generated fundamental matrices. Since we assume that many of them were generated from inlier 2keypoint matches they are therefore rough estimates of the required solution. Thus, we measure the support of the putative matches. The larger the number of fundamental matrices which support the match (sf m), the higher the probability that the match is correct. An example of the spatial distribution of the inlier matches is shown in Figure 8. Since the fundamental matrices generated from inliers are quite inaccurate, only a small number of matches which lie close to each other, are supported by a large number of fundamental matrices (marked in Blue). The other matches with lower support are distributed around this group in an irregular manner. In Figure 9 we compare results obtained by exploiting global information, to the ones obtained using only the 2keypoint matches found in Section 4.4. For that, we present the cumulative precision (inliers fraction) as a function of the number of putative matches, ordered based on their associated probabilities. The solid curves show the results obtained by exploiting global information, namely cumulative precisions as a function of number of keypoint matches ordered by their Fig. 8 An example of most supported inliers. Image ob-ject0092.view02 was taken from the ZuBuD dataset. Matches supported by more than 600 sf ms are plotted in Blue, matches with more than 200 sf ms in Green, and the rest in Red. sf ms. The dashed curves show the results found in Section 4.4, namely the cumulative precision as a function of the number of keypoint matches ordered by the probability returned by the 2keypoint match classifier. Each color represents a different image pair. As can be seen in the graph, better results are obtained by exploiting global information, in addition to using the 2keypoint match ranking computed in Section 4.4. Con-sider for example the hardest case of the image pair (FLH00010,FLH00016). From the top ranked 100 putative matches ordered by their sf ms, 11% were inliers, while if they were ordered by the probability returned by the 2keypoint match classifier only 5% would be inliers. The result of this step is the set of putative matches {X} described in Section 4.3 and their sf ms {sf m}. Combining all the data As was stated earlier, the goal of the algorithm is to generate a set of putative feature matches between the two images, where each match is accompanied by a prior probability (or score) that the match is correct. For that purpose, in Sections 4.2-4.5 we presented a three step algorithm, running from local to global and generating putative matches and their sf ms. In addition, in Section 4.1, we calculated putative match pairs {X L } and {X B }, which have a large intersection with {X}, along with their distance ratios {d r } and/or a similarity weights {t k }, based on the local features only. In order to incorporate those local scores in our method, we constructed a keypoint match descriptor, denoted kpmd, and trained a classifier on it. The descriptor consists of the following fields: kpmd = [sf m; d r ; t k ]. The definitions of the fields are as follows: sf m is the number of fundamental matrices which support the match, calculated in Section 4.5. d r and t k are the distance ratio and the similarity weight described in Section 4.1 respectively. For those putative matches that miss d r or t k , we attribute ones for d r , and zeros for t k . For those putative matches in {X L } and {X B } that miss the sf m we attribute zeros. In general, the smaller the value of d r and the higher the values of sf m and t k , the higher is the probability that the putative feature match is an inlier. The general idea behind this step is to improve the performance on challenging image pairs, while not harming the performance on easy ones. For that purpose the classifier should operate correctly under different scenarios. On the one hand, when an image pair is challenging, putative match pairs {X L } and {X B } are insufficient and it should rely on {X} and their sf ms. On the other hand, for easy image pairs {X} might be misleading, while relying on {X L } and {X B } works. Since there is no way to know a-priori with which scenario we are dealing with, the classifier should highly rank both: putative matches with high sf ms and missing (or low) t k and d r , and match pairs with missing sf ms but with high t k and/or low d r values. Using the training set described above, we trained a C4.5 decision tree classifier (Quinlan, 1993). This classifier returns for each descriptor the probability that the putative feature match is an inlier. The training set consists of 14255 feature matches from which 1399 are inliers and the rest are outliers. Here again a 10-fold cross-validation procedure was run. The resulting classifier correctly classifies 94.9% of the feature matches. The result of this step is a set of putative matches and their associated probabilities. Epipolar geometry estimation As was already mentioned in Section 4.2, the steps of our algorithm described in Sections 4.3-4.6 are repeated twice, once for each orientation. Therefore at this stage there are actually two sets of putative matches and their associated probabilities. To finalize the process we run an algorithm from the guided RANSAC family twice, once for each set of putative matches, yielding two fundamental matrices. The one with maximal support (the larger number of inliers) is chosen. Experiments Our method is a preprocessing step for state-of-the-art algorithms for epipolar geometry estimation. Therefore, in order to evaluate it, we compared the performance of three known algorithms BEEM, BLOGS and USAC with and without our method. In all the three cases we used the original implementations including all algorithm parameters, as proposed by their authors, available on the Internet. We ran experiments with the same parameters on all the results included in this work. These parameters were automatically selected to produce optimal results. Test Data To demonstrate the generality of our method, we used almost 900 image pairs from six separate publicly available sources for test data. Each image pair except those from the "USAC dataset" came with a small set of ground truth correspondences, which are different from the SIFT features used to estimate the epipolar geometry. These correspondences were used by the authors in their performance evaluation. The mean of roots of their Sampson distances served as our quantitative performance measure. The lower the value, the closer the proposed solution is to the ground truth. ZuBuD dataset (Shao et al, 2003): The dataset contains 1005 color images of 201 buildings (5 images per building) from Zurich, taken from different viewpoints and under different illumination conditions, yielding 2010 image pairs. In (Kushnir and Shimshoni, 2014) two subsets of it were used: the "ZuBuD1 set" of 139 challenging image pairs (two of which we used for training as mentioned in Section 4 and the rest for test) and the "ZuBuD2 set" of relatively easy image pairs. This way we can check the performance of the algorithm on both hard and easy cases. BLOGS dataset (Brahmachari and Sarkar, 2013a): The BLOGS dataset consists of 20 image pairs, some of which have very wide baselines, scale changes, rotations and occlusions. USAC dataset (Raguram et al, 2013b): In the USAC dataset there are 11 image pairs. Since image pairs from this dataset come without control points, we manually marked 16 correspondences for each image pair, serving as the ground truth. Since the "BLOGS dataset" and the "USAC dataset" are quite small, in our experiments we merged them into a single dataset. Open1, Open2 and Urban datasets (Goldman et al, 2014): These three datasets that were collected at different locations include 246, 224 and 108 image pairs respectively. They were used for testing the SOREPP algorithm (Goldman et al, 2015). The datasets present challenging scenarios with wide baseline images, small overlapping regions, scale changes, and nondescript objects that make feature matching difficult. Under these conditions the inlier fractions are often less than 10%. Qualitative results We will start this discussion with a presentation of qualitative results on the three image pairs already mentioned in Figures 1,2,5 and 9. Figure 10 shows the cumulative precision (inliers fraction) as a function of the number of putative matches, ordered based on their associated probabilities (on a log scale). The results of our method are drawn using solid curves. The putative matches ranking, based on the distance ratio proposed by Lowe and used as an initial step of many state-of-theart algorithms such as USAC and BEEM, are drawn using dotted curves. The putative matches ranking, based on similarity weights introduced in BLOGS, is drawn using dashed curves. Different colors represent the different image pairs. One can easily see that our ranking method outperforms the standard ones. In Table 1 we present numeric comparisons of our method on these three examples to these two common techniques for ranking putative matches. The number of matches we generate is ten times larger than the other methods. Even so, as our ranking is much better correlated with the probability to be an inlier, as was already shown in Figure 10, the performance is not hurt. Numerically speaking, we recover more inliers in the top 10 and top 100 ranked matches. This should be translated into improved performance of the subsequent registration process. To check that we ran BEEM, BLOGS and USAC with and without our preprocessing method and report on their success which will be defined in the next section. The first example (object0076.view02,object0076.view04) is an easy case and all the algorithms with or without the preprocessing step succeed on it. The second image pair (FLH00010,FLH00016) is so challenging, that although the number of inliers was increased by our method, it remains unsolved in all the cases. The last image pair (GEO00029,GEO00038) is a typical example of our contribution. Without our method only BEEM found a correct fundamental matrix, whereas when our preprocessing step is used, all three algorithms succeed. General performance To evaluate the general performance of our method, we present a comparison between BEEM, BLOGS and USAC with and without our preprocessing step on the previously mentioned datasets. We use ground truth correspondences and the quantitative performance measure, mentioned in Section 5.1. For every algorithm on each image pair we check this performance measure and consider it as a success when it is smaller than a threshold. In Figure 11 we present the number of correct epipolar geometry estimations on each set of image pairs as a function of the threshold. Although our method is deterministic, algorithms for epipolar geometry estimation are not. Therefore for the sake of proper comparison, we show an average over 5 executions of the algorithms. The error bars represent one standard deviation. On the "ZuBuD2 set" and the "BLOGS+USAC dataset" all of the checked algorithms perform extremely well. The performance after our preprocessing is similar. The results for the other datasets are dramatically lower for all the checked algorithms. This indicates that many of the image pairs are challeng-ing. The significant improvement due to our preprocessing method can be seen in all the checked algorithms. For example, for the "Urban dataset" and the "Open2 dataset" our preprocessing step improved the performance by a factor of two or three for USAC. In Table 2 we present numeric comparisons of the general performance with and without our preprocessing step. For each algorithm we report its results with and without our step, followed by our contribution for this algorithm. Our contribution is the percentage change, computed as follows: result with our step − result without our step result without our step , where results with and without our step are defined as the number of successful image pairs, with performance measure smaller than 10 pixels. From all the checked cases there is a negligible degradation due to our method, of one or two out of 137 image pairs, in ZuBuD2 dataset for BLOGS and USAC. In all other verified cases, our preprocessing step improves performance of all the checked algorithms. In general we can summarize that our preprocessing algorithm yields better results in hard cases and does not degrade on the easy ones. Another issue worth mentioning is run time. Our step is a preprocessing step for any epipolar geometry algorithm. As such, the run time can not be shorter when our method is used. Moreover, as described in Section 4.7, it requires to run the algorithm from the guided RANSAC family twice, once for each set of putative matches. Therefore, the run time with our method is expected to be at least doubled. In Figure 12 we show its time overhead. We defined it as: run time with our step run time without our step , and chose to present it as a function of our contribution, discussed previously. Each point shows one al-gorithm on one of the datasets. There are three algorithms and six datasets, resulting in 18 points. It appears that there is a negative correlation between the time overhead of our method and its contribution, which can be explained as follows. Our method takes almost constant time regardless of the difficulty of the image pair. Epipolar geometry estimation algorithms, on the other hand run much faster on easy image pairs. Therefore, the larger our contribution, the easier it was for USAC/BEEM/BLOGS to finish running. Consider the extreme example of running USAC on the "Open2 dataset". The standard algorithm succeeded on only 27 image pairs, while with our preprocessing step 91 successes were registered (contribution of 239%). This increase in performance was achieved at a factor of 2.7 in running times. As a result, due to the time overhead, we would recommend to apply our method only on challenging image pairs or when the standard method failed. Fig. 12 Time overhead of our method. There is a negative correlation between time overhead of our method and its contribution. Analysis Our method is a combination of several steps. One could naturally ask whether all of these steps are necessary and what are their contributions. To answer those questions we tested several variations of the algorithm, which skip parts of our algorithm or replace them with standard ones. In Figure 13 we chose to present this analysis on the "Open1 dataset" while running USAC. Results on different datasets with other algorithms yielded qualitatively similar results. The Blue curve is the result of the USAC algorithm without our contribution. The Green curve is the result Fig. 13 Analysis of different parts of our method. Major contribution can be attributed both to the 2keypoint matches generation and ranking and to the global ranking of matches. when the 2keypoint match ranking is used as the input for epipolar geometry estimation. As a result we obtain better results than for the original USAC. This indicates that 2keypoint generation is a strong component of our method. The Cyan curve is a result of altering our method. We generate K 2kp (K 2kp − 1)/2 fundamental matrices, but instead of exploiting all of them, as it is done in our algorithm, we present here the fundamental matrix with the largest support as it is usually done. This shows really bad results, mostly because each fundamental matrix is calculated from four pairs of matches instead of 7 or 8, giving quite an inaccurate estimation. We believe that local optimization could yield better results, but this is beyond the scope of this work. The Red curve is based on the putative matches {X} and their {sf m}. Using the training set described above we converted the sf m into a probability measure and used it as the input for epipolar geometry estimation. The resulting Red curve exhibits improved performance with respect to the 2keypoint match ranking. Therefore, this step also yields an important contribution to our method. The Black curve shows the result of our entire method as is, without any changes. Those results are similar to those based on {X} and their {sf m}. This is not surprising, recalling the reasoning behind combining all the measures. As it was already mentioned, the intent was to improve performance on challenging image pairs, while not harming the performance on easy ones. Therefore, on challenging datasets such as the "Open1 dataset", we expect our method to rely mostly on {X} and their {sf m} and have a minor contribution from {X L } and {X B }. This is exactly what is exhibited by the similarity between the Black and Red curves. To verify the contribution of this step, we analyzed the performance of the algorithms on the easier "BLOGS+USAC dataset". We found that relying on putative matches {X} alone, on average degraded the performance relative to our entire method for BLOGS/BEEM/USAC by 3.9%, 3% and 7.4% respectively. Since combining all the measures is relatively cheap, it is still recommended especially in the easy cases, even though its contribution is small. Concluding this section we can state that every step of our algorithm is necessary and that our good results can mostly be attributed to the 2keypoint matches generation and ranking and to the global ranking of matches. Conclusions In this paper we presented a general deterministic preprocessing step for epipolar geometry estimation algorithms. It generates a set of putative feature matches between the two images, accompanied by prior probabilities that each match is correct. The algorithm was tested on almost 900 image pairs from six publicly available datasets. We showed experimentally that the results obtained by state-of-the-art algorithms which use the output of our algorithm outperform the same algorithms which uses the standard input. In general we can summarize that our preprocessing algorithm yields better results in hard cases and does not degrade on the easy ones. This method is general and we believe that it can be used as the initial step of all guided RANSAC algorithms improving their performance. Fig. 11 Performance comparison between several algorithms with and without our preprocessing step on the standard datasets. Solid Red curves: BLOGS. Dashed Red curves: our method followed by BLOGS. Solid Green curves: BEEM. Dashed Green curves: our method followed by BEEM. Solid Blue curves: USAC. Dashed Blue curves: our method followed by USAC. Fig. 1 1An example of why using the standard putative matches as is, is sometimes insufficient, while clustering might help. Images GEO00038 and GEO00029 were taken from the Urban dataset. The feature point marked by a Red circle in the upper image is not matched at all by Lowe or mistakenly matched by BLOGS to a feature point marked by a Red circle in the bottom image. When using clustering, for feature point marked by the Red circle in the upper image we generate two putative matches marked by Green points in the bottom image, one of which is correct. Fig. 4 4An example of a relative roll angle α estimation. Images (a) corridor1 and (b) corridor2 were taken from the BLOGS dataset. The images in (a) and (b) are taken with relative roll of α = 78 • . (c) The kernel density estimation of the angle difference for (a) and (b) with the maximal peak at α exp = 78 • . Images (d) IMG0047 and (e) IMG0106 were taken from the Open1 dataset. The images in (d) and (e) are taken with zero relative roll. (f) The kernel density estimation of the angle difference for (d) and (e) with the maximal peak at α exp = −3 • . Fig. 5 5An example of keypoint clustering and matching. Images (a) object0076.view02 and (b) object0076.view04 were taken from the ZuBuD dataset. The large Red cluster in (b) Fig. 6 6An example of 2keypoints and the parameters used for their matching. Images (a) object0041.view01 and (b) ob-ject0041.view04 were taken from the ZuBuD dataset. (c) Zoom-in of (a). (d) Zoom-in of (b). Green and Magenta points show the 2keypoints generated by the first method. Blue and Red points show a 2keypoint generated by the third method. Fig. 7 72keypoint match classifier performances on the training set. Solid curves represent results of the classifier. Dashed curves show results without the classifier (the inliers percentages from all the 2keypoint matches). When using the classifier, there are many more inliers in the top K 2kp = 100 2keypoint matches. Fig. 9 92keypoint match classifier performance vs. global ranking of matches performance. Red: images object0076.view02 and object0076.view04 were taken from the ZuBuD dataset, shown in Figure 5. Green: images FLH00010 and FLH00016 were taken from the Open2 dataset, shown in Figure 2. Blue: images GEO00029 and GEO00038 were taken from the Urban dataset, shown in Figure 1. Solid curves show the results of exploiting global information, while the dashed curves show the results found in Section 4.4. Better results are obtained by exploiting global information. Fig. 10 A 10comparison between different methods of ranking. Red: images object0076.view02 and object0076.view04 were taken from the ZuBuD dataset, shown inFigure 5. Green: images FLH00010 and FLH00016 were taken from the Open2 dataset, shown inFigure 2. Blue: images GEO00029 and GEO00038 were taken from the Urban dataset, shown inFigure 1. Solid curves are results of our method. Doted curves are based on distance ratio proposed by Lowe. Dashed curves are based on similarity weights introduced in BLOGS. Our ranking method outperforms the standard ones. Table 1 A 1numeric comparison of our method, on several examples, to two common techniques for ranking putative matches. Numeric comparison of general performance with and without our preprocessing step on the standard datasets. Number of successful image pairs, with performance measure smaller than 10 pixels is reported.ZuBuD1 ZuBuD2 BLOGS + USAC DB Urban Open1 Open2 Number of image pairs 137 137 31 108 246 224 BLOGS 69.4 135 24.4 35.8 80.2 52 Our method followed by BLOGS 104.6 133.8 26.8 54.2 119.2 90 Our contribution for BLOGS 50.7% -0.9% 9.8% 51.4% 48.6% 73.1% BEEM 96 134.2 26.2 46.6 104.8 76.6 Our method followed by BEEM 107 134.2 27.2 64 132.4 124.2 Our contribution for BEEM 11.5% 0 3.8% 37.3% 26.3% 62.1% USAC 66 133.8 24.4 23.6 77.6 27 Our method followed by USAC 95.6 132.2 26.2 51 124.8 91.6 Our contribution for USAC 44.8% -1.2% 7.4% 116% 60.8% 239% Table 2 SURF: Speeded up robust features. H Bay, T Tuytelaars, Van Gool, ECCV. Bay H, Tuytelaars T, Van Gool L (2006) SURF: Speeded up robust features. In: ECCV, pp 404-417 . A Brahmachari, S Sarkar, BLOGS datasetBrahmachari A, Sarkar S (2013a) BLOGS dataset. http://www.cse.usf.edu/%7Esarkar/BLOGS/Epipolar- Matching/BLOGS.html Hop-diffusion monte carlo for epipolar geometry estimation between very wide-baseline images. A Brahmachari, S Sarkar, PAMI. 353Brahmachari A, Sarkar S (2013b) Hop-diffusion monte carlo for epipolar geometry estimation between very wide-baseline images. PAMI 35(3):755-762 Random forests. L Breiman, Machine Learning. 451Breiman L (2001) Random forests. Machine Learning 45(1):5-32 Matching with PROSAC-progressive sample consensus. O Chum, J Matas, CVPR. Chum O, Matas J (2005) Matching with PROSAC-pro- gressive sample consensus. In: CVPR, pp I:220-226 Locally optimized RANSAC. O Chum, J Matas, J Kittler, Patt. Recog. Chum O, Matas J, Kittler J (2003) Locally optimized RANSAC. In: Patt. Recog., pp 236-243 . Y Goldman, E Rivlin, I Shimshoni, SOREPP datasetGoldman Y, Rivlin E, Shimshoni I (2014) SOREPP dataset. Robust epipolar geometry estimation using noisy pose priors. Submitted to Computer Vision and Image Understanding Goshen L, Shimshoni I (2008) Balanced exploration and exploitation model search for efficient epipolar geometry estimation. / Sorepp/ Goldman, Y Rivlin, E Shimshoni, I , PAMI. 307http://is.haifa.ac.il/%7ishimshoni/SOREPP/ Goldman Y, Rivlin E, Shimshoni I (2015) Robust epipo- lar geometry estimation using noisy pose priors. Sub- mitted to Computer Vision and Image Understand- ing Goshen L, Shimshoni I (2008) Balanced exploration and exploitation model search for efficient epipolar geom- etry estimation. PAMI 30(7):1230-1242 Epipolar geometry estimation for urban scenes with repetitive structures. M Kushnir, I Shimshoni, PAMI. 3612Kushnir M, Shimshoni I (2014) Epipolar geometry es- timation for urban scenes with repetitive structures. PAMI 36(12):2381-2395 BRISK: Binary robust invariant scalable keypoints. S Leutenegger, M Chli, R Y Siegwart, ICCV. Leutenegger S, Chli M, Siegwart RY (2011) BRISK: Binary robust invariant scalable keypoints. In: ICCV, pp 2548-2555 Distinctive image features from scaleinvariant keypoints. D Lowe, IJCV. 602Lowe D (2004) Distinctive image features from scale- invariant keypoints. IJCV 60(2):91-110 Robust wide baseline stereo from maximally stable extremal regions. J Matas, O Chum, M Urban, T Pajdla, BMVC. Matas J, Chum O, Urban M, Pajdla T (2002) Robust wide baseline stereo from maximally stable extremal regions. In: BMVC, pp 384-393 A comparison of affine region detectors. K Mikolajczyk, T Tuytelaars, C Schmid, A Zisserman, J Matas, F Schaffalitzky, T Kadir, L Van Gool, IJCV. 651-2Mikolajczyk K, Tuytelaars T, Schmid C, Zisserman A, Matas J, Schaffalitzky F, Kadir T, Van Gool L (2005) A comparison of affine region detectors. IJCV 65(1- 2):43-72 C4.5: Programs for Machine Learning. J Quinlan, Morgan Kaufmann Publishers IncQuinlan J (1993) C4.5: Programs for Machine Learning. Morgan Kaufmann Publishers Inc. USAC: a universal framework for random sample consensus. R Raguram, O Chum, M Pollefeys, J Matas, J M Frahm, PAMI. 358Raguram R, Chum O, Pollefeys M, Matas J, Frahm JM (2013a) USAC: a universal framework for random sample consensus. PAMI 35(8):2022-2038 . R Raguram, O Chum, M Pollefeys, J Matas, J M Frahm, USAC datasetRaguram R, Chum O, Pollefeys M, Matas J, Frahm JM (2013b) USAC dataset. http://www.cs.unc.edu/%7Erraguram/usac/ ORB: An efficient alternative to sift or surf. E Rublee, V Rabaud, K Konolige, G R Bradski, ICCV. Rublee E, Rabaud V, Konolige K, Bradski GR (2011) ORB: An efficient alternative to sift or surf. In: ICCV, pp 2564-2571 ZuBuD -Zurich Buildings Database for image based recognition. H Shao, T Svoboda, L Van Gool, 260CVL, Swiss Federal Institute of TechnologyTechnical ReportShao H, Svoboda T, Van Gool L (2003) ZuBuD -Zurich Buildings Database for image based recognition. In: Technical Report 260, CVL, Swiss Federal Institute of Technology Guided sampling and consensus for motion estimation. B Tordoff, D Murray, ECCV. Tordoff B, Murray D (2002) Guided sampling and con- sensus for motion estimation. In: ECCV, pp 82-98 The Nature of Statistical Learning Theory. V N Vapnik, SpringerVapnik VN (1995) The Nature of Statistical Learning Theory. Springer VLFeat: An open and portable library of computer vision algorithms. A Vedaldi, B Fulkerson, Vedaldi A, Fulkerson B (2008) VLFeat: An open and portable library of computer vision algorithms. http: //www.vlfeat.org/ Generalized RANSAC framework for relaxed correspondence problems. W Zhang, J Kosecka, Zhang W, Kosecka J (2006) Generalized RANSAC framework for relaxed correspondence problems. In: 3DPVT, pp 854-860

A maximal subgroup of the Mathieu group M 24 arises as the combined holomorphic symplectic automorphism group of all Kummer surfaces whose Kähler class is induced from the underlying complex torus. As a subgroup of M 24 , this group is the stabilizer group of an octad in the Golay code. To meaningfully combine the symmetry groups of distinct Kummer surfaces, we introduce the concepts of Niemeier markings and overarching maps between pairs of Kummer surfaces. The latter induce a prescription for symmetry-surfing the moduli space, while the former can be seen as a first step towards constructing a vertex algebra that governs the elliptic genus of K3 in an M 24 -compatible fashion. We thus argue that a geometric approach from K3 to Mathieu Moonshine may bear fruit.

10.1090/pspum/090/01522

1303.2931

26739db44473f72e3c675cec9cf9154d3d6dde1a

Symmetry-surfing the moduli space of Kummer K3s 20 Sep 2013 September 24, 2013 Anne Taormina Katrin Wendland Mathematics Institute University of Freiburg D-79104FreiburgGermany Department of Mathematical Sciences Centre for Particle Theory Durham University DH1 3LEDurhamU.K Symmetry-surfing the moduli space of Kummer K3s 20 Sep 2013 September 24, 2013 A maximal subgroup of the Mathieu group M 24 arises as the combined holomorphic symplectic automorphism group of all Kummer surfaces whose Kähler class is induced from the underlying complex torus. As a subgroup of M 24 , this group is the stabilizer group of an octad in the Golay code. To meaningfully combine the symmetry groups of distinct Kummer surfaces, we introduce the concepts of Niemeier markings and overarching maps between pairs of Kummer surfaces. The latter induce a prescription for symmetry-surfing the moduli space, while the former can be seen as a first step towards constructing a vertex algebra that governs the elliptic genus of K3 in an M 24 -compatible fashion. We thus argue that a geometric approach from K3 to Mathieu Moonshine may bear fruit. Introduction This work is motivated by several mysteries related to the Mathieu Moonshine phenomenon. Central to this phenomenon is the elliptic genus of K3, which encodes topological data on K3 surfaces and at the same time is expected to organise a selection of states in N = (4, 4) superconformal field theories (SCFTs) on K3 into representations of the Mathieu group M 24 . The existence of the relevant representations follows from Gannon's result [Gan12], which in turn builds on the work of Cheng, Gaberdiel-Hohenegger-Volpato and Eguchi-Hikami [Che10, GHV10b, GHV10a,EH11]. The precise construction of those representations in terms of conformal field theory data, however, has been completely elusive so far, since the detailed nature of the states governing the elliptic genus has not been pinned down. Indeed, the elliptic genus is a topological invariant generalizing the genera of multiplicative sequences that were introduced by F. Hirzebruch [Hir66]. It can be viewed as the regularized index of a U(1)-equivariant Dirac operator on the loop space of K3 [AKMW87,Wit87]. It also arises from the supertrace over the subsector of Ramond-Ramond states of every superconformal field theory on K3, and hence it counts states with signs [EOTY89,Kap05]. That the net contribution should yield a well-defined representation of any group, let alone of M 24 , is mysterious. However, from the properties of twining and twisted-twining genera it has been argued that one should actually expect this representation to be realized in terms of a vertex algebra X [GPRV12]. We share that view, although not the recent claim by some experts exclusively expecting holomorphic vertex algebras in this context, and casting doubts on whether K3 surfaces bear any key to the Mathieu Moonshine Mysteries [Gan12,GPV13]. In fact, we argue that the resolution of certain aspects of Mathieu Moonshine might benefit from deepening our understanding of the implications of Mukai's work [Muk88], and from building on the insights offered by Kondo [Kon98]. Of course, Mukai has proved in [Muk88] that every holomorphic symplectic symmetry group of a K3 surface is a subgroup of the group M 24 . But he also proved that all these symmetry groups are smaller than M 24 by orders of magnitude. In fact, all of them are subgroups of M 23 . In [TW11] we advertised the idea that presumably, M 24 could be obtained by combining the holomorphic symplectic symmetry groups of distinct K3 surfaces at different points of the moduli space. As a test bed, we proved the existence of an overarching map Θ which allows to combine the holomorphic symplectic symmetry groups of two special, distinct Kummer surfaces in terms of their induced actions on the Niemeier lattice N of type A 24 1 . We also proved that this combined action on N yields the largest possible group that can arise by means of such an overarching map. This group is (Z 2 ) 4 ⋊ A 7 , which we therefore called the overarching finite symmetry group of Kummer surfaces. It contains as proper subgroups all holomorphic symplectic symmetry groups of Kummer surfaces which are equipped with the dual Kähler class induced from the underlying torus. In this note, in Section 1 we briefly recall the Kummer construction and gather the information appearing in [TW11] that is useful for the present work. In Section 2, we introduce the concept of Niemeier markings and generalize the ideas summarized above by showing that the technique introduced for two specific examples of Kummer surfaces in [TW11], namely the tetrahedral and the square Kummer K3, generalizes to other pairs of Kummer surfaces. As an application of this technique, Section 3 constructs three overarching maps for three pairs of Kummer surfaces with maximal symmetry. Section 4 shows that for any pair of Kummer K3s, one can find representatives in the smooth universal cover of the moduli space of hyperkähler structures such that there exists an overarching map analogous to the one constructed in [TW11]. Moreover, there always exists a continuous path between the two representatives of our Kummer surfaces, such that Θ is compatible with all holomorphic symplectic symmetries along the path. This is the idea of symmetry-surfing the moduli space, alluded to in the title of the present paper. Our surfing procedure allows us to combine the action of all holomorphic symplectic sym-metry groups of Kummer surfaces with induced dual Kähler class by means of their induced actions on the lattice N. In fact, this action is independent of all choices of overarching maps. We also prove in Section 4 that the combined action of all these groups is given by a faithful representation of (Z 2 ) 4 ⋊ A 8 on N. The subgroup (Z 2 ) 4 ⋊ A 7 , i.e. the overarching finite symmetry group of Kummer surfaces, is the stabilizer subgroup of (Z 2 ) 4 ⋊ A 8 for one root in the Niemeier lattice N, just as the subgroup M 23 of the Mathieu group M 24 is the stabilizer subgroup of M 24 , which naturally acts on N, for one root in N. We view this as evidence that the Mathieu Moonshine phenomenon is tied to the largest Mathieu group M 24 rather than M 23 , as also argued by Gannon [Gan12]. In Section 5, we highlight the relevance of our geometric approach, and in particular of the Niemeier markings, in the quest for a vertex algebra that governs the elliptic genus of K3 at lowest order. To this effect, we establish a link between our work on Kummer surfaces and a special class of N = (4, 4) SCFTs at central charge c = c = 6, namely Z 2orbifolds of toroidal conformal field theories 1 . This necessitates a transition from geometry to superconformal theory language, which we describe in Appendix A. The upshot is that our surfing idea is natural: the symmetry groups act on the twisted ground states of the Z 2orbifold conformal field theories, and that action completely determines these symmetries. The twisted ground states can be viewed as a stable part of the Hilbert space when one surfs between Z 2 -orbifolds. As such the twisted ground states collect the various symmetry groups just like the Niemeier lattice does by means of our Niemeier markings. In passing we explain how the very idea of constructing a vertex algebra from the field content of SCFTs on K3, which simultaneously governs the elliptic genus and symmetries, motivates why we restrict our attention to symmetry groups that are induced from geometric symmetries in some geometric interpretation, that is, to subgroups of M 24 . Kummer surfaces and quaternions An interesting class of K3 surfaces is obtained through the Kummer construction, which amounts to taking a Z 2 -orbifold of any complex torus T of dimension 2, and minimally resolving the singularities that arise from the orbifold procedure. More specifically, let T = T (Λ) = C 2 /Λ with Λ ⊂ C 2 denote a lattice of rank 4 over Z, and with generators λ i , i ∈ {1, . . . , 4}. The group Z 2 acts naturally on C 2 by (z 1 , z 2 ) → (−z 1 , −z 2 ) and thereby on T (Λ). Using Euclidean coordinates x = (x 1 , x 2 , x 3 , x 4 ), where z 1 = x 1 + ix 2 and z 2 = x 3 + ix 4 , points on the quotient T (Λ)/Z 2 are identified according to x ∼ x + 4 i=1 n i λ i , n i ∈ Z, x ∼ − x. Hence T (Λ)/Z 2 has 16 singularities of type A 1 , located at the fixed points of the Z 2action. These fixed points are conveniently labelled by the hypercube F 4 2 ∼ = 1 2 Λ/Λ, where F 2 = {0, 1} is the finite field with two elements, as F a := 1 2 4 i=1 a i λ i ∈ T (Λ)/Z 2 , a = (a 1 , a 2 , a 3 , a 4 ) ∈ F 4 2 . (1.1) Definition 1.1 The complex surface X Λ obtained by minimally resolving the 16 singularities of T (Λ)/Z 2 is a K3 surface (see e.g. [Nik75]) called a Kummer surface 2 . According to the above definition, the Kummer surface X Λ carries the complex structure induced from the universal cover C 2 of T . It may also be equipped with a Kähler structure 3 , and this is natural if one is interested in the description of finite groups of symplectic automorphisms of Kummer surfaces. We specify such a Kähler structure by choosing a so-called dual Kähler class ω, that is, a homology class which is Poincaré dual to a Kähler class. Indeed, first recall the following: Definition 1.2 Consider a K3 surface X. A map f : X −→ X of finite order is called a symplectic automorphism if and only if f is biholomorphic and it induces the identity map on H 2,0 (X, C). If ω is a dual Kähler class on X and the induced map f * : H * (X, R) −→ H * (X, R) leaves ω invariant, then f is a holomorphic symplectic automorphism with respect to ω. When a dual Kähler class ω on X has been specified, then the group of holomorphic symplectic automorphisms of X with respect to ω is called the symmetry group of X. As an application of the Throughout this work, we focus on Kummer surfaces X Λ,ω 0 , by which we mean that as Kähler structure on X Λ we choose the one induced from the standard Kähler structure of the torus T (Λ) inherited from the Euclidean metric on its universal cover C 2 . Here, ω 0 denotes the corresponding dual Kähler class on X Λ . This restricts the symmetry groups of Kummer surfaces that can be obtained, but is sufficient to argue for the existence of a combined symmetry group (Z 2 ) 4 ⋊ A 8 in Section 4. The generic structure of the symmetry group G of the Kummer surface X Λ,ω 0 is a semidirect product G = G t ⋊G T (see, for example, [TW11,Prop. 3.3.4]). The normal subgroup G t ∼ = (Z 2 ) 4 of G is the so-called translational automorphism group which is induced from the shifts by half lattice vectors 1 2 λ, λ ∈ Λ, on the underlying torus T = T (Λ). The group G T is the normalizer of G t in G. It is the group of symmetries of the Kummer surface induced by the holomorphic symplectic automorphisms of the torus T fixing 0 ∈ C 2 /Λ = T . That is, G T ∼ = G ′ T /Z 2 , where G ′ T is the group of linear holomorphic symplectic 2 We denote by π : T X the corresponding rational map of degree 2, and by π * : H * (T, Z) −→ H * (X, Z) the induced map on homology. 3 For most parts of our work, the Kähler class is degenerate in the sense that it corresponds to an orbifold limit of Kähler metrics. automorphisms of T . These groups and their possible actions on a torus T have been classified by Fujiki [Fuj88], who proves that G ′ T is isomorphic to a subgroup of one of the following groups: the cyclic groups Z 4 , Z 6 , the binary dihedral groups O and D of order 8 and 12, and the binary tetrahedral group T . This actually implies that the symmetry group G is a subgroup of (Z 2 ) 4 ⋊ A 6 , where A 6 is the alternating group on six elements. Moreover 4 , T acts only on the so-called tetrahedral torus, while D acts only on the socalled triangular torus. O can act on the square torus or on the tetrahedral torus, where it is realized as a subgroup of T . Finally the action of the cyclic groups Z 4 and Z 6 agrees with that of a cyclic subgroup of O, D or T , possibly on a torus that does not enjoy the full dihedral or tetrahedral symmetry. In summary, the maximal groups that can occur are O, D and T . By definition, any element of G must leave the complex structure and the dual Kähler class ω 0 of the Kummer surface X Λ,ω 0 invariant. Hence in terms of real local coordinates x = (x 1 , x 2 , x 3 , x 4 ) as above and with respect to standard real coordinate vector fields e 1 , . . . , e 4 , using the notations of [TW11, Section 3], G must preserve each of the following 2-cycles in H 2 (X Λ,ω 0 , R), Ω 1 = e 1 ∨ e 3 − e 2 ∨ e 4 , Ω 2 = e 1 ∨ e 4 + e 2 ∨ e 3 and ω 0 = e 1 ∨ e 2 + e 3 ∨ e 4 . (1.2) Equivalently, every symmetry group G must preserve the hyperkähler structure which is specified by the nowhere vanishing holomorphic 2-form and the Kähler class on X Λ,ω 0 . We can work with local holomorphic coordinates (z 1 , z 2 ) that are induced from the underlying torus. The invariant classes hence are given by dz 1 ∧ dz 2 , and 1 2i ( dz 1 ∧ dz 1 + dz 2 ∧ dz 2 ). Moreover, G T ∼ = G ′ T /Z 2 where G ′ T acts linearly. In other words, G ′ T is a finite subgroup of SU(2). Once a group G ′ T ⊂ SU(2) preserving the lattice Λ has been identified such that Z 2 ⊂ G ′ T , then G T ∼ = G ′ T /Z 2 acts faithfully on the Kummer surface X Λ,ω 0 . It is not surprising that quaternions provide an elegant framework to describe the groups G T ∼ = G ′ T /Z 2 we are interested in when symmetry-surfing [Fuj88,Bri98]. Indeed, we recall a formalism taken from [Bri98] which is tailored to recover the maximal groups G ′ T classified by Fujiki, i.e. G ′ T ∼ = O, D, T . It moreover provides a unified description of the lattice Λ for each torus on which one of these groups can act as automorphism group. In fact, each lattice Λ is given in terms of unit quaternion generators, and the automorphisms act by quaternionic left multiplication. The link between the skew field of quaternions H and lattices Λ ⊂ R 4 is through the natural isomorphism R 4 −→ H, q = (q 0 , q 1 , q 2 , q 3 ) −→ q 0 + q 1 i + q 2 j − q 3 k, (1.3) with H = {q = q 0 + q 1 i + q 2 j + q 3 k | q µ ∈ R, µ ∈ {0, . . . , 3}}. The unit quaternions form a group which is isomorphic to SU(2), and under the identification (1.3) its regular representation on R 4 ∼ = C 2 is realized by left multiplication on H ∼ = R 4 . One immediately checks that with this faithful representation, every unit quaternion leaves the standard holomorphic two-form dz 1 ∧ dz 2 and Kähler class 1 2i (dz 1 ∧ dz 1 + dz 2 ∧ dz 2 ) on R 4 ∼ = C 2 invariant. Hence this identification allows us to realize each of our groups G ′ T in terms of a finite group of unit quaternions. Assume now that Λ ⊂ R 4 ∼ = H is a lattice of rank 4 which carries the faithful action of an automorphism group G ′ T ⊂ SU(2), where G ′ T is one of the maximal groups O, D, T from Fujiki's classification. By the properties of these maximal groups, we can assume without loss of generality that G ′ T has generators a, b, c that are represented by unit quaternions of the formâ = cos( π m ) − i sin( π r ) + j cos( π n ), b = j, (1.4) c = cos( π n ) + j cos( π m ) + k sin( π r ), with the constraint cos 2 ( π m )+cos 2 ( π n ) = cos 2 ( π r ), where the numbers m, n, r ∈ Z determine the group G ′ T [Cox74]. Moreover, for the lattice Λ ⊂ R 4 ∼ = H we can choose the unit quaternion generators 1,â,b,ĉ. Hence in terms of R 4 , we let λ 1 = (1, 0, 0, 0), λ 2 = cos( π m ), − sin( π r ), cos( π n ), 0 , λ 3 = (0, 0, 1, 0), λ 4 = cos( π n ), 0, cos( π m ), − sin( π r ) , be the generators of Λ. We now summarise the data needed for symmetry-surfing the moduli space of Kummer surfaces. We describe the three maximal symmetry groups G T ∼ = G ′ T /Z 2 of Kummer surfaces induced by the holomorphic symplectic automorphisms of some torus T = T (Λ) fixing 0 ∈ C 2 /Λ = T , along with the possible lattices Λ: 1. Dihedral group D 2 ∼ = O/Z 2 ∼ = Z 2 × Z 2 Take the lattice Λ to be Λ 0 := span Z {1,â = i,b = j,ĉ = k}, with {â,b,ĉ} generating the quaternionic group G ′ T ∼ = Q 8 of order 8. It is immediate that Q 8 is the automorphism group of Λ 0 , which is the lattice yielding the square Kummer surface X 0 in [TW11]. There, an equivalent description of the generators of the binary dihedral group O was given by α 1 : (z 1 , z 2 ) −→ (iz 1 , −iz 2 ), α 2 : (z 1 , z 2 ) −→ (−z 2 , z 1 ), (1.5) both of which are of order 4. Alternating group A 4 ∼ = T /Z 2 The lattice Λ may be generated by {1,â = cos( π 3 ) − i sin( 5π 4 ) + j cos( π 3 ),b = j,ĉ = cos( π 3 ) + j cos( π 3 ) + k sin( 5π 4 )}, hence the four lattice vectors that generate Λ may be chosen as λ 1 = (1, 0, 0, 0), λ 2 = ( 1 2 , 1 √ 2 , 1 2 , 0), λ 3 = (0, 0, 1, 0) and λ 4 = ( 1 2 , 0, 1 2 , 1 √ 2 ). One shows that the orbit of λ 1 under the group G ′ T = T yields 24 unit lattice vectors. This lattice is isometric to the lattice Λ 1 := Λ D 4 used in [TW11] to construct the tetrahedral Kummer surface X 1 = X D 4 from the torus T (Λ D 4 ). We will use this Kummer surface in what follows, hence we recall the generators of Λ D 4 : λ 1 = (1, 0, 0, 0), λ 2 = (0, 1, 0, 0), λ 3 = (0, 0, 1, 0), λ 4 = 1 2 (1, 1, 1, 1). (1.6) Generators of the binary tetrahedral group T may be taken to be γ 1 : (z 1 , z 2 ) −→ (iz 1 , −iz 2 ), γ 2 : (z 1 , z 2 ) −→ (−z 2 , z 1 ), (1.7) γ 3 : (z 1 , z 2 ) −→ i+1 2 (i(z 1 − z 2 ), −(z 1 + z 2 )). These generators satisfy the relations γ 4 1 = γ 4 2 = 1 and γ 3 3 = 1 . Note that the minimum number of generators for the group T is 2, and indeed, one has γ 2 = γ 2 1 γ 3 γ 1 (γ 3 ) −1 . Permutation group S 3 ∼ = D/Z 2 Take the lattice Λ 2 generated by {1,â = − cos( π 3 ) + i sin( π 3 ),b = j,ĉ = −j cos( π 3 ) − k sin( π 3 ) }, hence the four lattice vectors that generate Λ 2 may be chosen as λ 1 = (1, 0, 0, 0), λ 2 = (− 1 2 , √ 3 2 , 0, 0), λ 3 = (0, 0, 1, 0), λ 4 = (0, 0, − 1 2 , √ 3 2 ). (1.8) The orbit of λ 1 under the binary dihedral group G ′ T ∼ = D yields 12 unit vectors in Λ 2 . The Kummer surface obtained from T (Λ 2 ) is the triangular Kummer surface X 2 . The generators of D have order 3 and 4, respectively, and they are given by β 1 : (z 1 , z 2 ) −→ (ζz 1 , ζ −1 z 2 ), β 2 : (z 1 , z 2 ) −→ (−z 2 , z 1 ), (1.9) where ζ := e 2πi/3 . Overarching maps and Niemeier markings The description of symmetries of K3 surfaces is most efficient in terms of lattices. To this end, recall that the geometric action of a symmetry group G of a K3 surface X is fully captured by its action on the lattice L G = (L G ) ⊥ ∩H * (X, Z), where L G := H * (X, Z) G . This follows from the Torelli theorem (see the discussion of Def. 1.2) and the very definition of L G as the sublattice of H * (X, Z) on which G acts trivially. On the other hand, if X Λ,ω 0 is a Kummer surface with its induced dual Kähler class, then the induced action of G on the Kummer lattice Π ⊂ H * (X, Z) bears all information about the action of G (see [TW11,Prop. 3 .3.3]): Proposition 2.1 Consider a Kummer surface X Λ,ω 0 with its induced dual Kähler class. Let Π ⊂ H * (X, Z) denote the Kummer lattice, that is, the smallest primitive sublattice of the integral K3 homology which contains the 16 classes E a , a ∈ F 4 2 , that are obtained from blowing up the fixed points F a of the Z 2 -action on the underlying torus (1.1). Then every symmetry of X induces a permutation of the E a . This permutation is given by an affine linear transformation of the labels a ∈ F 4 2 , which in turn uniquely determines the symmetry. In the case of Kummer surfaces we thus have two competing lattices Π and L G which conveniently encode the action of the symmetry group G of X Λ,ω 0 . In [TW11] we argue that neither does L G contain the rank 16 Kummer lattice, nor does, in general, the Kummer lattice contain L G . Instead, combining the two, in [TW11, Prop. 3.3.6] we introduce the lattice M G , which is generated by L G and Π along with the vector υ 0 − υ, where υ 0 , υ are generators of H 0 (X, Z) and H 4 (X, Z) with 5 υ 0 , υ = 1. We argue that in the Kummer case we can generalize and improve some extremely useful techniques introduced by Kondo [Kon98] to this enlarged lattice M G . Indeed, we prove that this lattice allows a primitive embedding into the Niemeier lattice N(−1) with root lattice A 24 1 [TW11, Thm. 3.3.7], where the decoration (−1) indicates that the roots of N(−1) have length square −2. This embedding allows us to view the symmetry group G as a group of lattice automorphisms of N(−1): the action of G on N(−1) is defined such that the embedding ι G : In what follows, we use the notations and conventions of [TW11] throughout. In particular, we fix the Kummer lattice Π within the abstract lattice H * (X, Z) as well as its image under ι G in N(−1) for every Kummer surface, independently of the parameters of the underlying torus. More precisely, we fix a unique marking for all our Kummer surfaces, that is, an explicit isometry of the lattice H * (X, Z) with a standard even, unimodular lattice of signature (4, 20). As is explained in [TW11, Sect. 2.2], the Kummer construction induces a natural such marking, which in particular fixes the position of Π within the lattice H * (X, Z). In this setting, among the data specifying each Kummer surface we have to include the choice of generators λ 1 , . . . , λ 4 ∈ R 4 for the lattice Λ of the underlying torus T = T (Λ). Note that the choice of such a fixed marking amounts to the transition to a smooth universal cover of the moduli space of hyperkähler structures on K3. Similarly to Π ⊂ H * (X, Z), we also fix the position of Π(−1) := ι G (Π) in N(−1) such that Π is common to all Kummer surfaces. To do so, in [TW11, (2.14)] we construct a bijection I : I \ O 9 −→ F 4 2 between the 16 elements of the set I := {1, 2, . . . , 24} that do not belong to our choice of reference octad O 9 := {3, 5, 6, 9, 15, 19, 23, 24} from the Golay code and the vertices of the hypercube F 4 2 . In [TW11, Prop. 2.3.4] we prove that the Q-linear extension of ι G (E a ) := f I −1 ( a) yields an isometry between Π and Π(−1), where {f n , n ∈ I}, denotes a root basis of the root lattice A 24 1 in N(−1). Thus we have fixed the position of Π within N for all Kummer surfaces, similarly to fixing the position of Π within the abstract lattice H * (X, Z). This motivates the Definition 2.2 With notations as above, for a Kummer surface X Λ,ω 0 with symmetry group G, an isometric embedding ι G : M G ֒→ N(−1) is G-equivariant,M G ֒→ N(−1) such that ι G (E a ) = f I −1 ( a) for all a ∈ F 4 2 is called a Niemeier marking. By the above, every Kummer surface X allows a Niemeier marking [TW11, Prop. 4.1.1]. In general, the embedding ι G is not uniquely determined. However, the action of G on N, which is induced by the requirement that ι G is G-equivariant, is independent of all choices: indeed, ι G (E a ) = f I −1 ( a) ∀ a ∈ F 4 2 fixes the action of G on the lattice Π ⊂ N, and by the arguments presented in the discussion of [TW11, Cor. 3.3.8] this already uniquely determines the action of G on all of N. In particular, consider the translational symmetry group G t ∼ = (Z 2 ) 4 discussed in Section 1. Its action on the roots f n , n ∈ I, of N, which is common to all Kummer surfaces, is generated by the following permutations [TW11, Prop. 4.1.1]: [Muk88] that the symmetry group of every K3 surface is isomorphic to a subgroup of one of eleven subgroups of the Mathieu group M 24 , the largest one of which has 960 elements. Hence symmetry groups of K3 surfaces are by orders of magnitude smaller than the group M 24 , whose appearance one expects from Mathieu Moonshine. Therefore, in [TW11] we propose to use Niemeier markings to combine the symmetry groups of distinct Kummer surfaces by means of their actions on the Niemeier lattice N. To underpin this idea by lattice identifications, we propose to extend a given Niemeier marking ι G to a linear bijection Θ : H * (X, Z) −→ N(−1), which restricts to an isometry on the largest possible sublattice of H * (X, Z). More precisely, we propose to construct a map Θ which induces Niemeier markings of all K3 surfaces along a smooth path in the smooth universal cover of the moduli space of hyperkähler structures on K3. If this path connects two distinct Kummer K3s X A and X B , then we call Θ AB an overarching map for X A and X B . This is the key to exhibit an overarching symmetry in the moduli space of Kummer K3s. We say that an overaching map Θ AB for Kummer surfaces X A and X B allows us to surf from one of the corresponding Kummer surfaces to the other in moduli space. For two Kummer surfaces X A , X B with complex and Kähler structures induced from the underlying torus and with symmetry groups G A , G B , respectively, we will argue below that the following holds: under appropriate additional assumptions, one can construct an overarching map Θ AB which restricts to a Niemeier marking, that is to an isometric G k -equivariant embedding ι G k : M G k ֒→ N(−1), for both k = A and k = B, just like the map Θ constructed in [TW11] for the tetrahedral Kummer surface X 1 = X D 4 and the square Kummer surface X 0 . That Θ restricts to the desired Niemeier markings is sufficient to ensure that Θ AB is an overarching map according to the above definition. Indeed, we can always find a path in the smooth universal cover of the moduli space which connects X A and X B , such that all intermediate points of the path are Kummer surfaces with the minimal symmetry group G = G t ∼ = (Z 2 ) 4 . The group G t is compatible with Θ AB by construction. See [TW11, Thm. 4.4.2] for an example -one solely needs to ensure that G t := (Z 2 ) 4 :            ι 1 = (1,span C {Ω 1 , Ω 2 , ω 0 } ⊥ ∩ π * H 2 (T, Z) = {0} along the path. To determine sufficient conditions on the existence of Θ AB , first note that by the above, see also [TW11,Thm. 3.3.7], the lattices M G k share the Kummer lattice Π and the vector υ 0 − υ. By the Definition 2.2 of Niemeier markings ι G : M G ֒→ N(−1), we require Θ AB (E a ) = f I −1 ( a) for all a ∈ F 4 2 . As mentioned above, G k -equivariance of ι G k then already fixes the action of G k on N. An overarching map Θ AB hence only exists if there is an index n 0 ∈ O 9 , such that f n 0 is invariant under the action of both groups G A , G B , such that Θ AB (υ 0 − υ) = f n 0 is consistent with G k -equivariance. For the complementary lattices K G k := ((π * (H 2 (T, Z)) (G k ) T ) ⊥ ∩ π * H 2 (T, Z) for k ∈ {A, B} introduced in [TW11, Thm. 3.3.7], choose bases I ⊥ i k ,k , i k ∈ {1, . . . , N k }, where N k ≤ 3 by construction. If all the vectors I ⊥ 1,A , . . . , I ⊥ N A ,A , I ⊥ 1,B , . . . , I ⊥ N B ,B are linearly independent, then we claim that under one final assumption we can find an overarching map Θ AB for X A and X B as desired 6 . Indeed, first recall from [TW11,(4.3)] that there is a choice between two complementary sets of four labels for the six sets Q ij ⊂ I which specify the image of 7 1 2 π * λ ij under the map Θ AB , up to lattice vectors in N(−1). Choose these quadruplets of labels such that for each Q ij , n 0 ∈ Q ij . Analogously to [TW11, Prop. 4.2.5] this defines a map I through I(π * λ ij ) := Q ij and I(λ+λ ′ ) := I(λ)+I(λ ′ ) by symmetric differences of sets. Since isometric embeddings ι G k : M G k ֒→ N(−1) exist by [TW11, Prop. 4.1.1], we can now find appropriate candidates Θ AB (I ⊥ i k ,k ) ∈ N(−1) such that Θ AB restricts to an isometry on both lattices K G k . Indeed, up to contributions of the form 2∆ with ∆ ∈ N(−1), each Θ AB (I ⊥ i k ,k ) is a linear combination of roots f j with j ∈ I(I ⊥ i k ,k ). Under the final assumption that all the Θ AB (I ⊥ i k ,k ) constructed in this manner are linearly independent, clearly Θ AB can be extended to an overarching map as desired. All our assumptions hold true in two of the three cases for which we shall construct overarching maps and exhibit overarching symmetries in Section 3 below. In one case, the vectors I ⊥ 1,A , . . . , I ⊥ N A ,A , I ⊥ 1,B , . . . , I ⊥ N B ,B fail to be linearly independent. However, the linear dependence results from a repetition of vectors, I ⊥ a,A = I ⊥ b,B , so by listing every vector only once, linear independence is achieved, and the argument goes through as above. This technique allows us to find overarching maps between any two Kummer surfaces, as we shall see in the next two sections. More precisely, for any pair of Kummer surfaces we can find representatives X A and X B in the smooth universal cover of the moduli space of hyperkähler structures, such that an overarching map for X A and X B exists. Hence we can surf between any two points in moduli space. Construction of overarching maps In Section 1, we have identified three distinct Kummer surfaces X k , k ∈ {0, 1, 2}, whose associated tori T = C 2 /Λ have maximal symmetry. In order to explore the overarching symmetry for the moduli space of Kummer surfaces by surfing from X 0 to X 1 and X 2 , and from X 1 to X 2 , we apply the recipe given in Section 2 to construct three overarching maps Θ kℓ , 0 ≤ k < ℓ ≤ 2, that yield overarching symmetry groups for the three pairs of Kummer surfaces (X k , X ℓ ). As was explained in Section 2, the construction of an overarching map requires the existence of a root f n 0 ∈ N(−1), n 0 ∈ O 9 , that is invariant under the action of G k and G ℓ . In the cases of interest to us here, the value of n 0 varies from map to map, but we carefully note down all possible choices, since this will be crucial in the subsequent section. We first summarize the construction of the overarching map Θ 01 valid for the square and tetrahedral Kummer surfaces, which appeared with some additional details in [TW11]. Then we proceed to the construction of the other two maps, Θ 02 and Θ 12 , which are new. This exercise paves the way to Section 4, where we argue that one can combine various overarching groups and obtain an action of a maximal subgroup (Z 2 ) 4 ⋊ A 8 of M 24 on the Niemeier lattice N(−1), overarching the entire Kummer moduli space. Overarching the square and tetrahedral Kummer K3s The full symmetry group of the square Kummer surface X 0 is the group G 0 := (Z 2 ) 4 ⋊ (Z 2 × Z 2 ) of order 64, while that of the tetrahedral Kummer surface X 1 := X D 4 is the group G 1 := (Z 2 ) 4 ⋊ A 4 of order 192. By the discussion in the previous section, there exist Niemeier markings ι G k , k ∈ {0, 1}, which allow the definition of induced actions of the groups G k on the Niemeier lattice N(−1), independently of all choices. Indeed, for the respective generators listed at the end of Section 1, according to [TW11,Sects. 4 (3. 2) The construction of the map Θ 01 requires that one root f n 0 with n 0 ∈ O 9 is invariant under G 0 and G 1 . One checks that indeed n 0 := 5 is the only label in O 9 which is fixed by both groups. According to [TW11, (4.9),(4.21)], the generators of the rank 3 lattices K G 1 and K G 0 are I ⊥ 1,1 = π * λ 14 + π * λ 24 − π * λ 23 , I ⊥ 1,0 = π * λ 14 − π * λ 23 , I ⊥ 2,1 = π * λ 13 + π * λ 24 + π * λ 34 , I ⊥ 2,0 = π * λ 13 + π * λ 24 , I ⊥ 3,1 = −π * λ 12 + π * λ 14 + π * λ 34 , I ⊥ 3,0 = π * λ 34 − π * λ 12 . (3.5) Our choice of images of the generators (3.3) under Θ 01 must ensure that Θ 01 restricts to an isometry on both lattices K G k . Therefore, note that the quadratic form on K G 1 with respect to the basis I ⊥ i,1 , i ∈ {1, 2, 3}, and that on K G 0 with respect to the basis I ⊥ i,0 , i ∈ {1, 2, 3}, are K G 1 :   −4 −2 −2 −2 −4 −2 −2 −2 −4   , K G 0 :   −4 0 0 0 −4 0 0 0 −4   (3.6) according to [TW11,(4.20)] and [TW11,(4.27)]. Then the following gives linearly independent candidates for the Θ 01 (I ⊥ i k ,k ) ∈ N(−1) as desired: Θ 01 : I ⊥ 1,1 −→ f 19 − f 15 , I ⊥ 2,1 −→ f 9 − f 15 , I ⊥ 3,1 −→ f 24 − f 15 , I ⊥ 1,0 −→ f 24 − f 23 , I ⊥ 2,0 −→ f 19 − f 6 , I ⊥ 3,0 −→ f 9 − f 3 . (3.7) Equivalently, Θ 01 :                      π * λ 12 −→ 2q 12 = f 3 + f 6 − f 15 − f 19 , π * λ 34 −→ 2q 34 = f 6 + f 9 − f 15 − f 19 , π * λ 13 −→ 2q 13 = −f 6 + f 15 − f 23 + f 24 , π * λ 24 −→ 2q 24 = −f 15 + f 19 + f 23 − f 24 , π * λ 14 −→ 2q 14 = f 3 − f 9 − f 15 + f 24 , π * λ 23 −→ 2q 23 = f 3 − f 9 − f 15 + f 23 . (3.8) On the Kummer lattice Π, we set Θ 01 (E a ) = f I −1 ( a) , as always. Finally, a consistent choice for the images of υ, υ 0 is Θ 01 : υ 0 −→ 1 2 (f 3 + f 5 + f 6 + f 9 − f 15 − f 19 − f 23 − f 24 ) , υ −→ 1 2 (f 3 − f 5 + f 6 + f 9 − f 15 − f 19 − f 23 − f 24 ) . This completes the construction of the map Θ 01 which is compatible with the symmetry groups of the square (G 0 ) and tetrahedral (G 1 ) Kummer surfaces. Viewed as a linear bijection Θ 01 : H * (X 0 , Z) −→ N(−1), its restriction Θ 01 | M G 0 yields a G 0 -equivariant and isometric embedding of M G 0 in N(−1). Viewed instead as a linear bijection Θ 01 : H * (X 1 , Z) −→ N(−1), its restriction Θ 01 | M G 1 yields a G 1 -equivariant and isometric embedding of M G 1 in N(−1). This property of the overarching map Θ 01 gives us ground to argue that there is an overarching symmetry group for the square and tetrahedral Kummer surfaces, whose action is encoded in the same Niemeier lattice N(−1) through the generators (2.1) of the translational symmetry group G t common to all Kummer surfaces, in addition to the generators (3.1) and (3.2). The group generated this way is a copy of (Z 2 ) 4 ⋊ A 7 ⊂ M 24 . Overarching the square and the triangular Kummer K3s The full symmetry group of the triangular Kummer surface X 2 is the group G 2 := (Z 2 ) 4 ⋊S 3 of order 96, see (1.8) and (1.9). Independently of the choice of Niemeier marking, the induced action of G 2 on the Niemeier lattice is generated by (3.9) The construction of an overarching map Θ 02 for X 0 and X 2 requires a root f n 0 with n 0 ∈ O 9 which is invariant under G 0 and G 2 . From (3.1) and (3.9) we observe that α 2 = β 2 and that n 0 = 15 is the only label in O 9 which is invariant under both groups. (G 2 ) T := S 3 : β 1 = (2, To calculate the generators of the lattice K G 2 following the techniques explained in [TW11], we first need to determine generators of the lattice (π * H 2 (T, Z)) (G 2 ) T . With the basis λ 1 , . . . , λ 4 for the triangular lattice given in (1.8), we obtain primitive generators of that lattice as π * λ 13 − π * λ 24 , π * λ 13 + π * λ 23 + π * λ 14 , π * λ 12 + π * λ 34 , and hence the orthogonal complement K G 2 of (π * H 2 (T, Z)) (G 2 ) T in π * H 2 (T, Z) is generated by the lattice vectors I ⊥ 1,2 := π * λ 12 − π * λ 34 , I ⊥ 2,2 := π * λ 13 + π * λ 23 + π * λ 24 , I ⊥ 3,2 := π * λ 14 − π * λ 23 . Hence the map I described in Section 2 is as in (3.5) for I ⊥ i,0 , i ∈ {1, 2, 3}, and furthermore, I(I ⊥ 1,2 ) = {3, 9}, I(I ⊥ 2,2 ) = {5, 24}, I(I ⊥ 3,2 ) = {23, 24} . We now need to choose the images in N(−1) of the generators (3.10) under Θ 02 such that Θ 02 restricts to an isometry on the lattices K G 0 and K G 2 . To do so, note that the quadratic form on K G 0 with respect to the basis I ⊥ i,0 , i ∈ {1, 2, 3}, and that of K G 2 with respect to the basis I ⊥ i,1 , i ∈ {1, 2, 3}, by (3.6) and (3.10) are K G 0 :   −4 0 0 0 −4 0 0 0 −4   , K G 2 :   −4 0 0 0 −4 2 0 2 −4   . (3.12) Moreover, we have I ⊥ 1,0 = I ⊥ 3,2 and I ⊥ 3,0 = I ⊥ 1,2 , such that we can find candidates for Θ 02 (I ⊥ i k ,k ) ∈ N as desired: Θ 02 : I ⊥ 1,0 −→ f 24 − f 23 , I ⊥ 2,0 −→ f 6 − f 19 , I ⊥ 3,0 −→ f 3 − f 9 , I ⊥ 1,2 −→ f 3 − f 9 , I ⊥ 2,2 −→ f 5 − f 24 , I ⊥ 3,2 −→ f 24 − f 23 . (3.13) For example, we can choose the following map in order to induce (3.13): Θ 02 :                      π * λ 12 −→ 2q 12 = f 5 + f 9 − f 23 − f 24 , π * λ 34 −→ 2q 34 = f 3 + f 5 − f 23 − f 24 , π * λ 13 −→ 2q 13 = f 3 + f 5 − f 9 − f 19 , π * λ 24 −→ 2q 24 = −f 3 − f 5 + f 6 + f 9 , π * λ 14 −→ 2q 14 = f 5 − f 6 + f 19 − f 23 , π * λ 23 −→ 2q 23 = f 5 − f 6 + f 19 − f 24 . (3.14) On the Kummer lattice Π, we set Θ 02 (E a ) = f I −1 ( a) , as before. Finally, a consistent choice for the images of υ, υ 0 is Θ 02 : υ 0 −→ 1 2 (f 3 + f 5 + f 6 − f 9 + f 15 − f 19 − f 23 − f 24 ) , υ −→ 1 2 (f 3 + f 5 + f 6 − f 9 − f 15 − f 19 − f 23 − f 24 ) . (3.15) This completes the construction of the overarching map Θ 02 for the square and the triangular Kummer surfaces. Again, the overarching map Θ 02 leads to an overarching symmetry group, whose action is encoded in the same Niemeier lattice N(−1) through the generators (2.1) of the translational symmetry group G t common to all Kummer surfaces, in addition to the generators (3.1) and (3.9). The resulting group is a copy of (Z 2 ) 4 ⋊ D ⊂ M 24 , where D denotes the binary dihedral group of order 12, as before. Overarching the tetrahedral and triangular Kummer K3s The construction of an overarching map Θ 12 for X 1 and X 2 requires a root f n 0 with n 0 ∈ O 9 which is invariant under G 1 and G 2 , whose generators are given in (3.9) and (3.2). The only label in O 9 which is invariant under both these groups is n 0 = 6. The generators of the rank 3 lattice K G 1 are given in (3.3), and those of the lattice K G 2 by (3.10). From [TW11,(4.3)] we read that n 0 = 6 ∈ Q ij implies Hence the map I described in Section 2 is Q 12 = {5,I(I ⊥ 1,1 ) = {15, 19}, I(I ⊥ 2,1 ) = {9, 15}, I(I ⊥ 3,1 ) = {15, 24}, I(I ⊥ 1,2 ) = {3, 9}, I(I ⊥ 2,2 ) = {5, 24}, I(I ⊥ 3,2 ) = {23, 24}. We now need to choose the images in N of the generators I ⊥ i,1 , i ∈ {1, 2, 3} and I ⊥ i,2 , i ∈ {1, 2, 3} under Θ 12 such that Θ 12 restricts to an isometry on the lattices K G 1 and K G 2 . Given the quadratic form (3.6) for K G 1 and (3.12) for K G 2 , the following gives linearly independent candidates for Θ 12 (I ⊥ i k ,k ) ∈ N: Θ 12 : I ⊥ 1,1 −→ f 19 − f 15 , I ⊥ 2,1 −→ f 9 − f 15 , I ⊥ 3,1 −→ f 24 − f 15 , I ⊥ 1,2 −→ f 3 − f 9 , I ⊥ 2,2 −→ f 5 − f 24 , I ⊥ 3,2 −→ f 24 − f 23 . (3.17) Equivalently, Θ 12 :                      π * λ 12 −→ 2q 12 + 2f 3 − 2f 15 = −f 5 − f 9 + f 23 + f 24 + 2f 3 − 2f 15 , π * λ 34 −→ 2q 34 − 2f 15 = f 3 − f 5 + f 23 + f 24 − 2f 15 , π * λ 13 −→ 2q 13 + 2f 15 − 2f 23 = −f 3 + f 5 + f 9 − f 19 + 2f 15 − 2f 23 , π * λ 24 −→ 2q 24 = −f 15 + f 19 + f 23 − f 24 , π * λ 14 −→ 2q 14 = f 3 − f 9 − f 15 + f 24 , π * λ 23 −→ 2q 23 = f 3 − f 9 − f 15 + f 23 . (3.18) On the Kummer lattice Π, we set Θ 12 (E a ) = f I −1 ( a) . Finally, a consistent choice for the images of υ, υ 0 is Θ 12 : υ 0 −→ 1 2 (f 3 + f 5 + f 6 − f 9 − f 15 − f 19 + f 23 − f 24 ) , υ −→ 1 2 (f 3 + f 5 − f 6 − f 9 − f 15 − f 19 + f 23 − f 24 ) . ( 3.19) This completes the construction of the overarching map Θ 12 which is compatible with the symmetry groups of the tetrahedral (G 1 ) and triangular (G 2 ) Kummer surfaces. Hence there is an overarching symmetry group for the tetrahedral and the triangular Kummer surfaces, whose action is encoded in the same Niemeier lattice N(−1) through the generators (2.1) of the translational symmetry group G t common to all Kummer surfaces, in addition to the generators (3.2) and (3.9). The group thus generated is a copy of (Z 2 ) 4 ⋊ A 7 ⊂ M 24 . 4 Overarching the moduli space of Kummer K3s by (Z 2 ) 4 ⋊ A 8 In this section we argue that our surfing procedure allows us to surf between any two points of the moduli space of Kummer K3s. More precisely, for any two Kummer surfaces with induced dual Kähler class, we can find representatives in the smooth universal cover of the moduli space of hyperkähler structures, such that an overarching map between the two representatives exists. This allows us to combine all symmetry groups of such Kummer surfaces to a larger, overarching group. To see this, let us first consider an arbitrary Kummer surface X Λ,ω 0 with induced dual Kähler class, and let G denote its symmetry group. According to our discussion in Section 1, G = (Z 2 ) 4 ⋊ ( G ′ T /Z 2 ), where G ′ T ⊂ SU(2) is the linear automorphism group of the lattice Λ. Moreover, G ′ T is a subgroup of one of the three maximal linear automorphism groups of complex tori, the binary tetrahedral group T or one of the dihedral groups D, O of order 12 and 8. Let G ′ T = O, T or D, such that G ′ T ⊂ G ′ T , and let Λ = Λ 0 , Λ 1 or Λ 2 denote the corresponding choice of lattice from Section 1 which has G ′ T as its linear automorphism group. Fujiki's classification [Fuj88] implies that we can choose G ′ T and Λ in such a way that there is a smooth deformation of Λ into Λ, call it Λ t with t ∈ [0, 1] and Λ 0 = Λ, Λ 1 = Λ, such that the linear automorphism group of each Λ t with t = 0 is G ′ T . The quaternionic language introduced in Section 1 is particularly useful to check this. For example, if G ′ T = Z 4 , then by Fujiki's results we can choose coordinates such that the action of this group on C 2 is generated by our symmetry α 1 of (1.5), and we can choose G ′ T = O with Λ = Λ 0 the lattice of the square torus. One finds generators λ t 1 , . . . , λ t 4 for the lattices Λ t as desired such that λ t 2 = α 1 ( λ t 1 ) and λ t 3 = α 1 ( λ t 4 ) for every t ∈ [0, 1]. This deformation argument implies that by use of our fixed marking, the invariant sublattices of the integral torus homology, L G ′ T = H * (T, Z) G ′ T and L G ′ T = H * (T, Z) G ′ T , obey L G ′ T ⊂ L G ′ T . Hence for the symmetry group G of X Λ,ω 0 and for the lattices that yield our Niemeier markings we have M G ⊂ M G , see Def. 2.2 and the discussion preceding it. From this it follows that one can find a representative of X Λ,ω 0 in the smooth universal cover of the moduli space of hyperkähler structures such that every Niemeier marking ι G : M G ֒→ N(−1) of the maximally symmetric Kummer surface X Λ,ω 0 restricts to a Niemeier marking ι G := ι G|M G of the Kummer surface X Λ,ω 0 . Hence any overarching map Θ for the maximally symmetric Kummer surface X Λ,ω 0 and any other Kummer K3 X also allows to surf from X Λ,ω 0 to X. Now consider two distinct Kummer surfaces X A and X B with their induced dual Kähler classes. By the above, we can choose maximally symmetric Kummer surfaces X A and X B from the square, the tetrahedral and the triangular Kummer surfaces, such that the following holds: there are representatives of X A and X B in the smooth universal cover of the moduli space of hyperkähler structures such that any Niemeier marking of X A restricts to a Niemeier marking of X A , and analogously for X B and X B . Then by the above, the overarching map Θ AB for X A and X B which was constructed 8 in Section 3 also overarches X A and X B . In other words, we can surf from X A to X B . We conclude that by means of our overarching maps we can surf the entire moduli space of hyperkähler structures of Kummer surfaces. In particular, we can combine the actions of all symmetry groups of Kummer surfaces with induced dual Kähler class by means of their action on the Niemeier lattice N. Recall from Section 2 that by construction, every overarching map Θ AB between Kummer surfaces X A and X B with symmetry groups G A and G B assigns a fixed root Θ AB (υ 0 − υ) = f n 0 ∈ N(−1) to the root υ 0 − υ ∈ H * (X, Z), where n 0 ∈ O 9 is a label in our reference octad from the Golay code 9 . This root f n 0 is fixed under the induced actions of both G A and G B . For the overarching group G AB obtained from G A and G B , which by construction is a subgroup of the stabilizer group (Z 2 ) 4 ⋊ A 8 of the octad O 9 , this implies that G AB additionally fixes one label n 0 ∈ O 9 . Hence G AB is a subgroup of (Z 2 ) 4 ⋊ A 7 , the group which we call the overarching symmetry group of Kummer K3s. In Section 3 we have seen that for two pairs of distinct Kummer surfaces with maximal symmetry, the overarching group yields G AB = (Z 2 ) 4 ⋊ A 7 . The third pair has overarching group (Z 2 ) 4 ⋊ D. Moreover, in each case there exists precisely one label in O 9 which is fixed by both G A and G B . This label, however, is different for each of the three pairs of Kummer K3s with maximal symmetry. It follows that the combined symmetry group for all Kummer K3s with induced dual Kähler class is (Z 2 ) 4 ⋊ A 8 . Interpretation and outlook Let us now explain how our construction fits into the quest for the expected representation of M 24 on a vertex algebra which governs the elliptic genus of K3. As mentioned in the Introduction, the elliptic genus arises from the contribution to the partition function of any superconformal field theory on K3 which counts states in the Ramond-Ramond sector with signs according to fermion numbers. This part of the partition function is modular invariant on its own, inducing the well-known modularity properties of the elliptic genus. The very construction of the elliptic genus, in addition, amounts to a projection onto those states which are Ramond ground states on the antiholomorphic side. The usual rules for fermion numbers imply that the OPE between any two fields in the Ramond sector yields contributions from the Neveu-Schwarz sector only. Hence the expected vertex algebra can certainly not arise in the Ramond-Ramond sector. Of course we can spectral flow the relevant fields into the Neveu-Schwarz sector, where (prior to all projections) they indeed form a closed vertex algebra 10 X . Note that the choice of a spectral flow requires the choice of a holomorphic and an antiholomorphic U(1)-current within the superconformal algebra of our SCFT. For definiteness, we use the spectral flow which maps Ramond-Ramond ground states to (chiral, chiral) states. The resulting vertex algebra X certainly governs the elliptic genus. Its space of states contains the states underlying the well-known (chiral, chiral) algebra X of Lerche-Vafa-Warner [LVW89], which accounts for the contributions to the lowest order terms of the elliptic genus. In Appendix A we describe the (chiral, chiral) algebra X (see (A.1)) more concretely in the context relevant to this work, namely in Z 2 -orbifold conformal field theories C = T /Z 2 on K3, where T denotes the underlying toroidal theory. As expanded upon in Appendix A, the very truncation to the (chiral, chiral) algebra X makes X completely independent of all moduli. In principle, this is a desired effect when aiming at constructing a vertex algebra which governs the elliptic genus, since the latter is independent of all moduli. However, from the action of a linear map on X (generated by the fields in (A.1), independently of all moduli), it is not clear whether or not it is a symmetry, while the Mathieu Moonshine phenomenon dictates that we consider symmetries of some underlying vertex algebra. We shall come back to this 'bottom up' discussion further down, but we first take a closer look at the 'top-down' approach, and consider the action of symmetries of C on the (chiral, chiral) algebra X generated by (A.1). We impose a number of rather severe assumptions on such symmetries, in order to ensure that they descend to symmetries of a candidate vertex algebra that governs the elliptic genus. As mentioned in the Introduction, this graded vertex algebra at leading order is the (chiral,chiral) algebra X . Following [LVW89] we identify X with the cohomology of a K3 surface X. Associated to every Calabi-Yau manifold Y , there is the chiral de Rham complex [MSV99] which furnishes a sheaf of vertex algebras governing the elliptic genus of Y and containing the usual de Rham complex of Y at leading order [BL00,Bor01]. We thus find it natural 11 to restrict our attention to symmetries of C that descend to the chiral de Rham complex of X. To this end, we assume that our SCFT C comes with a choice of generators of the N = (4, 4) superconformal algebra, which in particular fixes the U(1)-currents and a preferred N = (2, 2) subalgebra. As mentioned above, this is already necessary when we choose the spectral flow to X . Recall that the choice of U(1)-currents amounts to the choice of a complex structure in any geometric interpretation of C [AM94]. We furthermore use the notion advertised by [GPRV12], which requires symmetries to fix the superconformal algebra of C pointwise. 12 To identify X with the cohomology of a K3 surface X, we need to perform a large volume limit [Wit82,LVW89]. More generally, according to [Kap05], the space of states singled out by the elliptic genus is mapped to the appropriate cohomology of the chiral de Rham complex of X only in the large volume limit. In order to perform such a large volume limit, 10 Here and in the following, we loosely refer to the space of fields which create states in the Neveu-Schwarz sector, equipped with the OPE, as a "vertex algebra", which however is not a holomorphic VOA. 11 By [BL00,FS07], the CFT orbifold procedure descends to the chiral de Rham complex; this should be the source for the behavior of the twining genera in Mathieu Moonshine, at least for those symmetries that are induced from geometric ones. 12 This, for instance, excludes equivalences of SCFTs induced by mirror symmetry, which acts as an outer automorphism on the superconformal algebra. we need to choose a geometric interpretation of C. Summarising, in view of constructing a vertex algebra from the fields in C, such that X governs the leading order terms of the elliptic genus, we restrict our attention to symmetries that are induced from geometric symmetries. This justifies why so far, in our work, we have searched for explanations of Mathieu Moonshine phenomena within the context of geometric symmetries only. As a further potential justification for this restriction recall the notion of 'exceptional' symmetry groups of sigma models on K3, that is, symmetry groups of such SCFTs which are not realizable as subgroups of M 24 , obtained from the classification in [GHV10a]. According to [GV12], in many cases 'exceptional' symmetry is linked to certain quantum symmetries which as we shall argue cannot be induced from any classical geometric symmetries. Indeed, these symmetries in [GV12] are characterized by the property that they generate a group G, such that orbifolding the K3 model by G yields a toroidal SCFT. We remark that there is no geometric counterpart of such an orbifold construction, which would have to yield a complex four-torus as an orbifold of a K3 surface. Indeed, the odd cohomology of a complex four-torus cannot be restored by blowing up quotient singularities in an orbifold by a symplectic automorphism group of a K3 surface. However, this is only a potential justification for our restriction to geometric symmetries since, according to [GV12], 'exceptional' symmetry groups also occur in a few cases where to date it is not known whether or not such purely non-geometric quantum symmetries are responsible for the 'exceptionality' of the symmetry group. Although the group M 24 itself contains elements that can never act in terms of a geometric symmetry on K3, we are optimistic that every element of M 24 can be obtained as a composition of 'geometric' symmetries. We wish to emphasize that it is immediately clear that the (chiral, chiral) algebra X cannot carry a representation of M 24 . Indeed, (A.2) is the basis of a four-dimensional subspace of the 24-dimensional space X which is invariant under all symmetries that are of interest here, but by the known properties of representations of M 24 , this group can only act trivially on the remaining 20-dimensional space. Hence a vertex algebra which governs the massless leading order terms of the elliptic genus, and which at the same time carries the expected representation of M 24 , must be related to X by some nontrivial map. The Niemeier markings and the overarching maps which were constructed in [TW11] should be viewed as a first approach towards constructing such a map. This claim is based on the observation that, from a geometric viewpoint, the introduction of Niemeier markings is necessary to combine symmetry groups of Kummer surfaces into larger groups. Indeed, it follows from Mukai's results that for any finite group G of lattice automorphisms of H * (X, Z) that is not a subgroup of one of the eleven maximal groups listed in [Muk88], the lattice L G := (H * (X, Z) G ) ⊥ ∩ H * (X, Z) is indefinite and thus violates the signature requirements for symmetries of K3 surfaces. Therefore, we never expected M 24 to act on H * (X, Z) either. It would be interesting to see if the massive sector of the elliptic genus is also subject to a 'no-go theorem' when working in the framework of Z 2 -orbifold CFTs on K3. A priori, the situation could be different, as the original Mathieu Moonshine observation [EOT11] states that in the elliptic genus, the multiplicities of massive characters of the N = 4 superconformal algebra yield dimensions of representations of M 24 . In a forthcoming work [TW13] we present evidence in favour of our expectation that the massive fields which contribute to the elliptic genus are related to a representation of M 24 in a much more immediate fashion. We now return to the 'bottom-up' approach, and investigate more closely the action of symmetry groups on the vertex algebra X , to explain in terms of CFT data how our Niemeier markings and overarching maps are relevant in the context of SCFTs on K3. To this end note that the entire group SL(2, C) acts naturally on the truncated vertex algebra C ⊗ X of (A.1), preserving U(1)-charges. However, a given element of SL(2, C) may not have an extension to a symmetry of the full SCFT C. Whether or not this is the case cannot be determined from the action on the fields listed in (A.1). Indeed, this depends on the moduli of C, but the vertex algebra X has lost its dependence on all moduli due to the truncation, as described earlier. However, as we explain in Appendix A, one may introduce the analog X Z of the lattice of integral homology in the vector space X , and use its structure to determine whether or not an element of SL(2, C) acts as a symmetry of C. By the above, we are only interested in symmetry groups G that are induced by geometric symmetries, and in line with our work so far, we restrict our attention to those that are induced 13 from the underlying toroidal CFT T . By definition, a symmetry of a SCFT must be compatible with all OPEs in that theory. In particular, the standardized OPE (A.4) must be preserved. Following the arguments presented in Appendix A, this implies that each of our symmetry groups G acts as a group of lattice automorphisms on X Z , such that this lattice of fields in our SCFT contains a sublattice X Z G which bears all relevant information about the G-action on our SCFT. This lattice can be identified with the lattice M G which is central to our construction, in that our Niemeier markings isometrically replicate it as a sublattice of the Niemeier lattice N(−1). This allows a more elegant description of G as a subgroup of M 24 , and it enables us to combine the symmetry groups from distinct K3 theories to a larger, overarching group. In other words, our Niemeier marking describes precisely the action of geometric symmetry groups on the vertex algebra which governs the elliptic genus to leading order terms. This justifies the relevance of our construction in the context of our quest to unravel some of the mysteries of the Mathieu Moonshine phenomenon. The picture that we offer here shows how the beautiful interplay between geometry and conformal field theory may yield some keys to the Mathieu Moonshine Mysteries. Such an interplay is expected. On the one hand, the elliptic genus is a purely geometric quantity. On the other hand, this quantity also appears in the context of SCFTs on K3, where its decomposition into N = 4 characters is natural. Notably, it is only after decomposing the elliptic genus into N = 4 characters that one observes the Mathieu Moonshine phenomenon [EOT11]. We expect that order by order, the elliptic genus dictates the construction of representations of M 24 on appropriately truncated vertex algebras arising from SCFTs on K3. In other words, the very representations of M 24 that are observed in the elliptic genus are intrinsic to these SCFTs. The reason why the emerging group is M 24 is still unclear, but we expect it to be rooted in the structure of these SCFTs, where geometry dictates the symmetries which can act on these representations. By symmetry-surfing the moduli space of SCFTs on K3, we expect that the natural representations of geometric symmetry groups on these vertex algebras combine to the action of M 24 . Our construction of overarching maps in [TW11] should be viewed as a very first step towards finding such vertex algebras for the leading order terms of the elliptic genus. In the present work, we show that our overarching maps indeed allow us to combine all relevant symmetry groups, as long as we restrict to Z 2 -orbifold conformal field theories on K3 and their geometric interpretations on Kummer K3s, and to symmetries that are induced geometrically from the underlying toroidal theories. Indeed, since one can easily associate a vertex algebra to the Niemeier lattice N, one could claim that we have solved the problem of constructing a vertex algebra that furnishes the expected symmetries. However, of course we pay dearly since this vertex algebra does not govern the leading order terms of the elliptic genus in any obvious way. Still our approach paves the way to defining the desired vertex algebra. As we have explained above, we expect vertex algebras associated with all remaining orders of the elliptic genus to relate directly to the respective representations of M 24 , and we present evidence in favour of this expectation in [TW13]. while the antiholomorphic counterparts are denoted χ 1 ± (z), χ 2 ± (z). The corresponding holomorphic -antiholomorphic combinations are more appropriate for our purposes, ξ 1 := 1 2 χ 1 + + χ 1 + , ξ 2 := 1 2i χ 1 + − χ 1 + , ξ 3 := 1 2 χ 2 + + χ 2 + , ξ 4 := 1 2i χ 2 + − χ 2 + . Moreover, in every Z 2 -orbifold conformal field theory C = T /Z 2 on K3, there is a 16dimensional space of twisted ground states, generated by fields T a in the Ramond-Ramond sector, where the label a ∈ F 4 2 refers to the fixed point F a as in (1.1) at which the respective field is localized. For ease of notation we denote by T a , a ∈ F 4 2 , the (chiral, chiral) fields which the T a flow to under our choice of spectral flow. Then the following is a list of 24 fields which generate the (chiral, chiral) algebra in every theory C = T /Z 2 on K3: ξ 1 ξ 2 ξ 3 ξ 4 , ξ i ξ j (1 ≤ i < j ≤ 4), 1 ; T a ( a ∈ F 4 2 ), (A.1) where 1 denotes the vacuum field, and where we may restrict our attention to the real vector space X generated by these 24 fields. After truncation of the OPE to chiral primaries [LVW89], the fields listed in (A.1) form a closed vertex algebra X over R. Note that this very truncation makes the vertex algebra completely independent of all moduli. We remark that the real and imaginary parts 14 of the four fields with U(1)-charges (2, 2), (2, 0), (0, 2), (0, 0) in (A.1), ξ 1 ξ 2 ξ 3 ξ 4 , ξ 1 ξ 3 − ξ 2 ξ 4 , ξ 1 ξ 4 + ξ 2 ξ 3 , 1 , (A.2) remain invariant under every symmetry of C. These fields are naturally identified with the cycles π * υ T , Ω 1 , Ω 2 , π * υ T 0 ∈ π * H * (T, R) on K3, with Ω 1 , Ω 2 as in (1.2) and υ T , υ T 0 generators of H 4 (T, Z) and H 0 (T, Z) such that υ T , υ T 0 = 1. The invariance of Ω 1 , Ω 2 under symmetries means that in a given geometric interpretation, one restricts attention to symplectic automorphisms (see [TW11] for further details). In the description of the moduli space of SCFTs on a K3 surface X of [AM94], our SCFT C is specified by the relative position of a positive definite fourplane in H * (X, R) with respect to H * (X, Z). The twoforms Ω 1 , Ω 2 generate a two-dimensional subspace of that fourplane, while the choice of υ T and υ T 0 amounts to the choice of a geometric interpretation of the toroidal theory T which induces a natural geometric interpretation of its Z 2 -orbifold C (see [NW01,Wen01]). Here, the four fields in (A.2) are the real and imaginary parts of the images of the four charged Ramond-Ramond ground states under our choice of spectral flow. These four Ramond-Ramond ground states also furnish a fourplane that can be used to describe the moduli space of superconformal field theories on K3 [NW01]. Note however that the fourplane of [AM94] is not the one generated by the four vectors in (A.2). The vector space X can be identified with the real K3 homology H * (X, R), where the ξ i ξ j with 1 ≤ i < j ≤ 4 are mapped to our generators e i ∨ e j of π * H 2 (T, R), and the T a are in 1 : 1 correspondence with the cycles E a that arise from the minimal resolution of T /Z 2 (see [NW03] for the subtleties in this identification, due to the B-field that is induced by orbifolding). One may, in addition, introduce the analog of the lattice of integral homology for the vector space X , thereby recovering the dependence on the moduli. To appreciate this, note that before truncation the OPE between twist fields T b and T b ′ with b, b ′ ∈ F 4 This leaves us with the lattice Y Z generated over Z by Ψ 1 Ψ 2 Ψ 3 Ψ 4 , Ψ i Ψ j , (1 ≤ i < j ≤ 4), 1 , which is the analog of the lattice π * H * (T, Z) ⊂ H * (X, R). Using the twist fields T a , a ∈ F 4 2 , as additional generators that correspond to the vectors E a in the Kummer lattice, and then performing the usual gluing procedure, one obtains a lattice X Z which can be identified with H * (X, Z) ⊂ H * (X, R). In particular, the relative position of X Z with respect to the basis (A.1) of X determines the respective point in the moduli space. For the SCFT associated with the square Kummer surface 18 , we can choose the eight fields ξ 1 ξ 2 ξ 3 ξ 4 , ξ i ξ j (1 ≤ i < j ≤ 4), 1 as generators of the lattice Y Z . Now note that each symmetry of a Kummer surface X Λ,ω 0 as studied in our work induces a symmetry of a SCFT C = T /Z 2 , with T a toroidal theory 19 associated with the torus R 4 /Λ. By construction, our geometric symmetry groups G enjoy an induced action as groups of lattice automorphisms on the lattice X Z . By definition, the symmetries of a CFT are compatible with all OPEs, hence they must in particular leave the standardized OPEs (A.4) invariant. Since our symmetries are induced by geometric symmetries of the toroidal theory T , they act linearly on the J k (z) and they permute the fields ±W a (z, z). It follows that such symmetries act as lattice automorphisms on the lattice generated by the J k (z). The same thus holds for the lattice generated by their superpartners Ψ k (z) and for the lattice Y Z mentioned above. Since our symmetries also permute the twist fields ± T a amongst each other in a manner compatible with gluing, altogether it follows that they must act as automorphisms of the lattice X Z . By the above, the vector space X can be identified with the K3 homology, and X Z can be identified with the integral homology. In particular, the lattice X Z possesses a sublattice X Z G which can be identified with the lattice M G that is so crucial to our construction, see Def. 2.2. The action of G on X Z G bears all relevant information about the G-action on our SCFT. Our construction hence realizes the very representation of G on X in terms of the action of a subgroup G of M 24 on the Niemeier lattice N. In other words, our Niemeier marking describes precisely the action of the relevant symmetry groups on the (chiral, chiral) algebra. Certainly from the description of the moduli space of SCFTs in terms of cohomological data [AM94,NW01], we are lead to expect that the role of the (chiral, chiral) algebra X along with the lattice X Z in its underlying vector space generalizes to arbitrary K3 theories. and G acts trivially on the orthogonal complement of ι G (M G ) in N(−1). Since the automorphism group of N(−1), up to reflections in the roots of N(−1), is the Mathieu group M 24 , this conveniently realizes every symmetry group G of a Kummer K3 as a subgroup of M 24 . TW11, (4.3)] we read that n 0 = 5 ∈ Q ij implies Q 12 = {3, 6, 15, 19}, Q 13 = {6, 15, 23, 24}, Q 14 = {3, 9, 15, 24}, Q 34 = {6, 9, 15, 19}, Q 24 = {15, 19, 23, 24}, Q 23 = {3, 9, 15, 23}. (3.4) Hence the map I described in Section 2 is I(I ⊥ 1,1 ) = {15, 19}, I(I ⊥ 2,1 ) = {9, 15}, I(I ⊥ 3,1 ) = {15, 24}, I(I ⊥ 1,0 ) = {23, 24}, I(I ⊥ 2,0 ) = {6, 19}, I(I ⊥ 3,0 ) = {3, 9}. we read that n 0 = 15 ∈ Q ij implies Q 12 = {5, 9, 23, 24}, Q 13 = {3, 5, 9, 19}, Q 14 = {5, 6, 19, 23}, Q 34 = {3, 5, 23, 24}, Q 24 = {3, 5, 6, 9}, Q 23 = {5, 6, 19, 24}. (3.11) Torelli theorem for K3 surfaces, the discussion of holomorphic symplectic automorphisms f of a K3 surface X can be entirely rephrased in terms of the induced lattice automorphisms f * of the full integral homology lattice H * (X, Z) (these and other results on geometry and symmetries of Kummer K3s are standard; for a summary, see e.g. [TW11, Thm. 3.2.2]). Then (see [TW11, Prop. 3.2.4] for a proof), Proposition 1.3 Consider a K3 surface X, and denote by G a group of symplectic automorphisms of X. Then G is finite if and only if X possesses a dual Kähler class which is invariant under G. 11)(2, 22)(4, 20)(7, 12)(8, 17)(10, 18)(13, 21)(14, 16), ι 2 = (1, 13)(2, 12)(4, 14)(7, 22)(8, 10)(11, 21)(16, 20)(17, 18), Now recall Mukai's seminal resultι 3 = (1, 14)(2, 17)(4, 13)(7, 10)(8, 22)(11, 16)(12, 18)(20, 21), ι 4 = (1, 17)(2, 14)(4, 12)(7, 20)(8, 11)(10, 21)(13, 18)(16, 22). (2.1) To avoid clumsy terminology, we simply refer to those SCFTs C on K3 which are obtained by the standard Z 2 -orbifold procedure from a toroidal theory as "Z 2 -orbifolds". See the end of this section, items 1.-3., for the precise definitions of the relevant lattices and group actions. On H * (X, Z), we use the standard quadratic form which is induced by the intersection form. We will see below that the assumption of linear independence can be relaxed, but for simplicity of exposition we first consider this case.7 Recall that for T = T (Λ), λ ij := λ i ∨ λ j ∈ H 2 (T, Z) denotes the integral two-cycle specified by the lattice vectors λ i , λ j ∈ Λ. If X A = X B , then there is nothing left to be shown. 9 This fixed label n 0 is responsible for the fact that each G k is a subgroup of M 23 , as we emphasized in[TW11]. This includes the symmetries induced by half lattice shifts in the underlying toroidal theory T . Here and in the following, for a field η ∈ C ⊗ X with η = η 1 + iη 2 and η 1 , η 2 ∈ X , we call η 1 , η 2 the real and the imaginary part of η. Here, we identify R 4 ∼ = (R 4 ) * by means of the standard Euclidean scalar product.16 From[LVW89], we obtain an immediate identification with cohomology, which however is equivalent to homology by Poincaré duality.17 For open strings, one can view χ k + and its antiholomorphic partner χ k + as complex conjugates, where the left and right modes combine into standing waves. In this language, we are simply reviewing the emergence of charge lattices for D-branes. with vanishing B-field on the underlying toroidal theory 19 This leaves a choice of the B-field B T in the toroidal theory T , which must be invariant under our symmetry; of course B T = 0 is always admissible. AcknowledgementsWe thank Matthias Gaberdiel and Roberto Volpato for very helpful discussions. We also thank theHeilbronnA Transition to superconformal field theory Throughout our work, we use homological data to describe geometric symmetries of K3 surfaces. This is natural, since the techniques are well-established in algebraic geometry, but also since the well-known properties of (chiral, chiral) algebras[Wit82,LVW89]recover (co)homological data from sigma model interpretations of SCFTs. This is particularly straightforward for the Z 2 -orbifold conformal field theories which are relevant to our investigations. Since our work is motivated by Mathieu Moonshine [EOT11], which is rooted in conformal field theory, and since the role of the integral (co)homology in (chiral, chiral) algebras seems not so well established, we gather in this appendix the tools needed to make a smooth transition to superconformal field theory.We first need to fix some notations. Every toroidal conformal field theory T possesses two free Dirac fermions on the holomorphic side, which we denote by χ 1 + (z), χ 2 + (z). The complex conjugate fields are denoted χ 1 − (z), χ 2 − (z), such that2}, 2 yields, to leading order, a primary field W b− b ′ (z, z) which does depend on the moduli. This is best measured by means of the OPE between the free bosonic superpartners of the Dirac fermions ξ 1 , . . . , ξ 4 and W b− b ′ (z, z). For convenience of notation, we introduce real, holomorphic U(1)-currents j 1 (z), . . . , j 4 (z), which arise as the superpartners of the real and the imaginary parts of 2χ 1 + (z), 2χ 2 + (z), respectively, and note that the relevant OPE then has the forma l λ l k for a = (a 1 , . . . , a 4 ) ∈ F 4 2 .Here, λ l 1 , . . . , λ l 4 are the Euclidean coordinates of generators λ 1 , . . . , λ 4 of a rank 4 lattice Λ ⊂ R 4 , if the underlying toroidal SCFT T has a geometric interpretation on the torus T = R 4 /Λ, where we identify R 4 with C 2 as usual. We observe that in the truncation procedure yielding the (chiral, chiral) algebra X of (A.1), the moduli-dependent fields W a (z, z) are projected to zero, and therefore the dependence on the moduli disappears from X . However, one may introduce new fieldswhere µ l 1 , . . . , µ l 4 are the Euclidean coordinates of the basis µ 1 , . . . , µ 4 dual 15 to λ 1 , . . . , λ 4 , such that the OPEs with the fields W a (w, w) take the standardized "integral" formThe fermionic superpartners Ψ 1 (z), . . . , Ψ 4 (z) of the new fields J 1 (z), . . . , J 4 (z) and their antiholomorphic counterparts Ψ 1 (z), . . . , Ψ 4 (z) yield a lattice with generatorsover Z. However, to determine a lattice which plays the role of the integral homology of the Kummer surface X, one needs to recall that the identification 16 of X with H * (X, R) rests on the correspondence χ k + ↔ dz k , , χ k + ↔ dz k for k ∈ {1, 2}, with local holomorphic coordinates z 1 , z 2 on X. 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In this talk I am going to present some of recent phenomenological applications of the ITMD [1, 2] formalism extended to account for Sudakov form factor. In particular production of dijets in the future Electron Ion Collider is discussed.

10.22323/1.398.0412

2111.02286

e5d5a084af976865bb459feba219e772a7dad908

Small-x Improved Transverse Momentum Dependent Factorization and some of its recent applications *** The European Physical Society Conference on High Energy Physics (EPS-HEP2021), *** *** 26-30 July 2021 *** *** Online conference, jointly organized by Universität Hamburg and the research center DESY *** Krzysztof Kutak [email protected] Institute of Nuclear Physics Polish Academy of Sciences Radzikowskiego 15231-342Krakow Small-x Improved Transverse Momentum Dependent Factorization and some of its recent applications *** The European Physical Society Conference on High Energy Physics (EPS-HEP2021), *** *** 26-30 July 2021 *** *** Online conference, jointly organized by Universität Hamburg and the research center DESY *** In this talk I am going to present some of recent phenomenological applications of the ITMD [1, 2] formalism extended to account for Sudakov form factor. In particular production of dijets in the future Electron Ion Collider is discussed. Factorization for forward dijet production High energy collisions of protons and heavy nuclei at the Large Hadron Collider (LHC) provide a unique tool to probe dense systems of quarks and gluons. Furthermore recently approved Electron Ion Collider [3] will allow for precision tomographic study of structure of hadrons ranging from proton to lead. In particular interesting are processes where jets or particles are produced in the forward direction with respect to the probe which can be proton or electron. Kinematically, such final states have large rapidities and therefore they trigger events in which the partons from the nucleus carry rather small longitudinal momentum fraction x. This kinematic setup is perfectly suited to investigate the phenomenon of gluon saturation, which is expected to occur at some value of x to prevent violation of the unitarity bound (for a review of this subject see Ref. [4]). The behavior of dense systems of partons when x becomes small is predicted by Quantum Chromodynamics (QCD) and leads to non-linear evolution equations known as B-JIMWLK [5,7] equations (for review see [8,9]), which can be derived within the Color Glass Condensate (CGC) theory. In CGC, the calculation of forward jet production relies on factorization, where the dense target according to theoretical results is described with nonlinear BK-JIMWLK equations. The description of multi-jet production is rather complicated [10]. A novel approach to such processes was initiated in Ref. [1]. The framework is known as the small-x Improved Transverse Momentum Dependent (ITMD) factorization. The ITMD formula accounts for: • complete kinematics of the scattering process with off-shell gluons, • gauge invariant definitions of the TMD gluon densities, • gauge invariant expressions for the off-shell hard matrix elements, • it also recovers the high energy factorization (aka k T -factorization) [11,12] in the limit of large off-shellness of the initial-state gluon from the nucleus and small-x TMD factorization [13] in correlation limit i.e. when momenta of jets are much larger that momentum of offshell gluon. Recently, the ITMD factorization has been proved [14]. Steps in further extension of the formalism to three and more jets were undertaken in Ref. [17,16]. While in the paper [18] the ITMD was generalized to account for masses of produced hadrons as well as longitudinally polarized gluons (see also [19] for discussion of the role of transverse gauge links). Furthermore in the papers [20,21] the ITMD was shown to agree very well with full CGS results (accounting on top of kinematical twist for genuine twist effects) in the region dominated by hard jets i.e. k T , p T > Q s While the original ITMD formula, as well as the works studying the jet correlation limit within CGC, include gluon saturation effects, they do not account for all contributions proportional to logarithms of the hard scale set by the large transverse momenta of jets -the so-called Sudakov logarithms. It has been shown in Refs. [22,23] that inclusion of Sudakov logarithms is necessary in order to describe the LHC jet data at small x but yet before the saturation regime. In the low x domain, the resummation leading to the Sudakov logarithms has been developed in [24,25] see also [26]. In the paper [28], it has been shown for the first time, that the interplay of saturation effects and the resummation of the Sudakov logarithms is essential to describe the small-x forward-forward dijet data. In this contribution we present two results that demonstrate relevance and importance of both effects i.e. nonlinearity accounting for saturation and Sudakov effects accounting for emission of soft gluons. • The first process under consideration is the inclusive dijet production p P p + A (P A ) → j 1 (p 1 ) + j 2 (p 2 ) + X , (1.1) • The second one is the dijet production in Deep Inelastic Scattering e P p + A (P A ) → j 1 (p 1 ) + j 2 (p 2 ) + X ,(1.2) where A can be either the lead nucleus, or a proton. To describe the former of above processes, the hybrid approach has been used where one assumes that the proton p is a dilute projectile, whose partons are collinear to the beam and carry momenta p = x p P p . The hadron A is probed at a dense state. The jets j 1 and j 2 originate from hard partons produced in a collision of the probe a with a gluon belonging to the dense system A. This gluon is off-shell, with momentum k = x A P A + k T and k 2 = −| k T | 2 . The ITMD factorization formula (schematically) reads dσ pA→ j 1 j 2 +X = ∑ a,c,d x p f a/p x p , µ ⊗ 2 ∑ i=1 K (i) ag * →cd (q T , k T ; µ) ⊗ Φ (i) ag→cd (x A , k T , µ) ,(1. 3) The distributions f a/p are the collinear PDFs corresponding to the large-x gluons and quarks in the projectile. The functions K (i) ag * →cd are the hard matrix elements constructed from gauge-invariant off-shell amplitudes [27]. The quantities Φ (i) ag→cd are the TMD gluon distributions introduced in Ref. [1] and parametrise a dense state of the nucleus or the proton in terms of small-x gluons. Figure 2: Azimuthal correlations between the total transverse momentum of the dijet system and the transverse momentum of the scattered electron at EIC in two frames: the LAB frame (left), the Breit frame (right). The calculation has been done within the ITMD framework using KATIE Monte Carlo [31] with the Weizsäcker-Williams gluon distribution obtained from the KS fit to HERA data. Similarly the ITMD formula for e+A collision reads (once the contribution from longitudinally polarized gluons is neglected) dσ eh→e +2 j+X = 1 4xP e ·P h F (3) gg (x, k T , µ) ⊗ |M eg * →e +2 j | 2 , (1.4) where F(3) gg (x, k T , µ) is so called WW gluon density which has the interpretation of gluon number density. To apply the ITMD formula one needs to construct the ITMD densities. One can use their operator definition and calculate their x, k T dependence using B-JIMWLK or using mean field approximation and express the ITMD distributions in terms of convolutions of dipole gluon density which is a solution of BK equation. We followed the second option and constructed the ITMD distributions from distributions given by the KS gluon density [30,2]. As it has been argued above the necessary additional element in order to provide realistic cross section is the Sudakov form factor. For the dijet production at the LHC we used DGLAP based Sudakov form factor [28]. On fig. 1 we show normalized cross sections as functions of ∆φ in p-p and p-Pb collisions. The three panels correspond to three different cuts on the transverse momenta of the two leading jets. The points with error bars represent experimental data from Ref. [29]. The main results for p-Pb collisions are represented by blue solid lines in Fig. 1. The visible broadening comes from the interplay of the non-linear evolution of the initial state and the Sudakov resummation. Similarly in the fig. 2 shows result for predictions for di-jet production at EIC obtained in [32] for the cross section as a function of the angle between electron and dijet system. In this case we used Sudakov obtained in [24]. A control result based on a calculation that neglects the Sudakov form factor is presented there as well. The comparison of the two results clearly shows that while saturation effects are mild, the Sudakov effects are fairly large. This feature is clearly visible in both the LAB and the Breit frame. Conclusions In this contribution the ITMD factorization framework was presented. In particular I discussed its phenomenological applications to LHC and EIC physics. Furthermore it is clear that one needs to account for additional contributions coming from soft gluons and leading to Sudakov form factor. The combined contribution allows to describe shape of the LHC forward-forward dijet spectra. Using this knowledge we provide predictions for the EIC dijet crosssection. Figure 1 : 1Broadening of azimuthal decorrelations in p-Pb collisions vs p-p collisions for different sets of cuts imposed on the jets' transverse momenta. The plots show normalized cross sections as functions of the azimuthal distance between the two leading jets, ∆φ . The points show the experimental data[29] for p-p and p-Pb, where the p-Pb data were shifted by a pedestal, so that the values in the bin ∆φ ∼ π are the same. Theoretical calculations are represented by the histograms with uncertainty bands coming from varying the scale by factors 1/2 and 2. AcknowledgmentsThe research is founded by Polish National Science Center grant no. DEC-2017/27/B/ST2/01985. . P Kotko, K Kutak, C Marquet, E Petreska, S Sapeta, A Van Hameren, 10.1007/JHEP09(2015)106arXiv:1503.03421JHEP. 09106hep-phP. Kotko, K. Kutak, C. Marquet, E. 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Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to[DS16,DS17], this paper emphasizes the role of centers in controlling properties of valuation rings in prime characteristic.

10.2140/ant.2021.15.2485

1707.01649

09bea99c9b60f9b4f294a39f2ace9c5850aaefbe

FROBENIUS SPLITTING OF VALUATION RINGS AND F -SINGULARITIES OF CENTERS 2 Mar 2020 Rankeya Datta FROBENIUS SPLITTING OF VALUATION RINGS AND F -SINGULARITIES OF CENTERS 2 Mar 2020 Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring of an Abhyankar valuation of a function field over a perfect ground field of prime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficiently well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In contrast to[DS16,DS17], this paper emphasizes the role of centers in controlling properties of valuation rings in prime characteristic. Introduction One of the goals of this paper is to prove the following result. Theorem A. Let K be a finitely generated field extension of a perfect field k of prime characteristic. Then the valuation ring of any Abhyankar valuation of K/k is Frobenius split. For a valuation ν of K/k, if Γ ν is the value group and (R ν , m ν , κ ν ) the valuation ring, then ν is Abhyankar if dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /k = tr. deg K/k. Abhyankar valuations, often also called quasimonomial valuations in characteristic 0, extend the class of valuations associated to prime divisors on normal models, since dim Q (Q ⊗ Z Γ ν ) = 1 and tr. deg κ ν /k = tr. deg K/k − 1 for divisorial valuations. Nevertheless, non-divisorial Abhyankar valuations arise naturally in geometry (see [Spi90,ELS03,FJ04,FJ05,JM12,Mus12,Tem13,RS14,Tei14,Pay14,Blu18] for some applications), and they possess many of the good properties of their Noetherian counterparts. For example, the value group of any Abhyankar valuation is a free abelian group of finite rank, and its residue field is a finitely generated extension of the ground field. Divisorial valuation rings over perfect fields of prime characteristic are Frobenius split. Indeed, when a Noetherian local ring R is F -finite, that is when the Frobenius (p-th power) endomorphism F : R → R is a finite ring map, a famous result of Kunz shows that regularity of R is characterized by R being free over its p-th power subring R p [Kun69, Theorem 2.1]. Therefore F -finite regular rings, and consequently divisorial valuation rings over perfect fields, are Frobenius split (the same conclusion can be drawn using the Direct Summand Theorem). Thus Theorem A extends a well-known fact about divisorial valuation rings to a class of valuation rings that behaves the most like divisorial ones. A key ingredient in our proof of Theorem A is the following result of Knaf and Kuhlmann which says that one can locally monomialize finite subsets of Abhyankar valuation rings in any characteristic. Theorem 1. [KK05, Theorem 1] Let K be a finitely generated field extension of a field k of any characteristic, and let ν be an Abhyankar valuation of K/k such that the residue field κ ν is separable over k. Then for any finite set Z ⊆ R ν , R ν is centered on a regular local ring (A, m A , κ A ) essentially of finite type over k with fraction field K satisfying the following properties: (1) The Krull dimension of A equals d := dim Q (Q ⊗ Z Γ ν ). (2) Z ⊆ A, and there exists a regular system of parameters {x 1 , . . . , x d } of A such that every z ∈ Z admits a factorization z = ux a 1 1 . . . x a d d , for some u ∈ A × , and a i ∈ N ∪ {0}. When the ground field k is perfect, any Abhyankar valuation ν admits a local monomialization in the sense of Theorem 1 because κ ν is automatically separable over k. Theorem 1 allows us to additionally choose a center of ν on a regular local k-algebra A such that A has a regular system of parameters {x 1 , . . . , x d } whose valuations freely generate Γ ν and the residue field of A coincides with κ ν (Lemma 3.0.1). Our strategy then is to identify a suitable Frobenius splitting of A that lifts to a Frobenius splitting of R ν . A proof of Theorem A was announced in [DS16,Theorem 5.1], where Frobenius splitting was deduced as a consequence of the incorrect assertion that Abhyankar valuation rings are Ffinite. On the contrary, [DS17, Theorem 0.1] shows that finiteness of Frobenius for valuation rings of function fields (a function field is a finitely generated extension of a base field) is equivalent to the associated valuation being divisorial, and so, Abhyankar valuations rings that are not F -finite are abundant. The current paper rectifies the error in [DS16]. Said differently, ν is Abhyankar precisely when the extension of valuations ν/ν p is defectless, where ν p denotes the restriction of ν to the subfield K p . We refer the reader to [FV11,Page 281] for the definition of defect of an extension of valuations. The second goal of this paper is to generalize the characterization of (1.0.0.1) to valuations, not necessarily of function fields, that admit a Noetherian local center. When a valuation ν of an arbitrary field K is centered on a Noetherian local domain (R, m R , κ R ) such that Frac(R) = K, one has the following beautiful inequality established by Abhyankar [Abh56, Theorem 1]: dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /κ R ≤ dim R. (1.0.0.2) When equality holds in (1.0.0.2), ν behaves like an Abhyankar valuation of a function field. For example, the value group Γ ν is then again a free abelian group of finite rank, and the residue field κ ν is finitely generated over κ R . However, whether a valuation of a function field is Abhyankar is intrinsic to the valuation, while equality in (1.0.0.2) with respect to a center depends, unsurprisingly, on the center as well (see Example 4.0.1 for an illustration). Bearing this difference in mind, we call a Noetherian center R an Abhyankar center of ν, if ν satisfies equality in (1.0.0.2) with respect to R. In practice one is often interested in centers satisfying additional restrictions. For example, in the local uniformization problem for valuations of function fields, one seeks centers that are regular and essentially of finite type over a ground field. Although satisfying equality in (1.0.0.2) is not intrinsic to a valuation, the property of possessing Abhyankar centers from a more restrictive class of local rings may become independent of the center. For example, when K/k is a function field and C is the collection of local rings that are essentially of finite type over k with fraction field K, then a valuation ν admits an Abhyankar center from the collection C precisely when ν is an Abhyankar valuation of K/k. Moreover, then all centers of ν from C are Abhyankar centers (see Section 4). In other words, the property of possessing Abhyankar centers that are essentially of finite type over k is intrinsic to valuations of function fields over k. Our investigation reveals that even when the fraction field of a valuation is not a function field, one can find a reasonably broad class of Noetherian local rings such that the property of admitting an Abhyankar center from this class is independent of the choice of the center. More precisely, we show the following: Theorem 4.0.3. Let (R, m R , κ R ) be a Noetherian F -finite local domain of characteristic p > 0 and fraction field K. Suppose ν is a non-trivial valuation of K centered on R with value group Γ ν and valuation ring (V, m ν , κ ν ). Then R is an Abhyankar center of ν if and only if [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. Since the identity [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] does not depend on the center R, Theorem 4.0.3 implies that possessing F -finite Abhyankar centers is intrinsic to a valuation. Although imposing finiteness of Frobenius on Noetherian centers appears to be a strong restriction, the class of such rings include any center one is likely to encounter is geometric applications. For example, any generically F -finite excellent domain of prime characteristic is also F -finite. Thus Theorem 4.0.3 implies that the property of a valuation possessing generically F -finite, excellent Abhyankar centers is independent of the choice of such centers. We can draw interesting conclusions from Theorem 4.0.3 even for valuations of function fields. First note that it generalizes (1.0.0.1) (see Remark 4.0.8(i)). Furthermore, if K is a function field over a perfect field k of prime characteristic, then Theorem 4.0.3 and (1.0.0.1) imply that any valuation of K/k that possesses an excellent Abhyankar center is an Abhyankar valuation of K/k (Corollary 4.0.5). Admitting excellent Abhyankar centers is a priori much weaker than admitting Abhyankar centers that are essentially of finite type over k, and so Corollary 4.0.5 is not obvious. In fact, the corresponding statement is false when the ground field k has characteristic 0 -one can easily construct a non-Abhyankar valuation of K/C which has an Abhyankar center that is an excellent local domain (see Remark 4.0.8(vi) or [ELS03, Example 1(iv)]). Theorem 4.0.3 does not claim that if a valuation ν of an F -finite field K satisfies [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] , then ν is centered on an excellent local domain; the latter assertion is false even when K is not perfect. Indeed, using the theory of F -singularities of valuations developed in [DS16,DS17], one can show that if the valuation ring of ν is F -finite, then the identity [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] is always satisfied (Section 5, (3)(ii)). However, we prove in this paper that a non-Noetherian F -finite valuation ring cannot be centered on any excellent local domain of its fraction field (Section 5, (8)), and explicitly construct such a valuation ring of K = L(X), where L is a perfect field admitting a non-trivial valuation (Example 5.0.5). The valuation ring of Example 5.0.5 does not contain the ground field L, and so its existence does not contradict local uniformization in positive characteristic. On the contrary, this paper and the author's prior work with Karen Smith [DS17, Theorem 0.1] shows it is impossible to construct non-Noetherian F -finite valuation rings of function fields when the valuation rings contain the ground field (see also Remark 5.0.6). Thus, pathologies such as Example 5.0.5 cease to exist for valuations trivial on the ground field of a function field. In the final section of this paper (Section 5), we summarize all known results on Fsingularities of valuation rings for the convenience of the reader, mainly drawing from the paper [DS16] and the accompanying corrigendum [DS17]. Results are grouped according to the type of F -singularity they characterize, which we hope will make it easier for the reader to navigate [DS16,DS17]. The summary is not just limited to a recollection of old results; many new results are also proved. For example, we show that when ν/ν p is totally unramified or has maximal defect, that is, when [Γ ν : pΓ ν ][κ ν : κ p ν ] = 1 , then the valuation ring of ν cannot be Frobenius split (Section 5, (14)). To put this in context, Theorem A implies, in contrast, that if ν/ν p is defectless for a valuation ν of a function field over a perfect field, then its valuation ring is always Frobenius split. Thus, Frobenius splitting of valuation rings is well-understood when the defect of ν/ν p is one of two possible extremes, but seems to be mysterious otherwise. We also show that if V is valuation ring of Krull dimension 1 whose fraction field is spherically complete with respect to the multiplicative norm induced by the valuation, then V is Frobenius split (Section 5, (15)). This result should be viewed as a generalization of the well-known fact that a complete discrete valuation ring of prime characteristic is Frobenius split. Acknowledgments: The question of Frobenius splitting of valuation rings arose in conversations of Zsolt Patakfalvi, Karl Schwede and Karen Smith. While Frobenius splitting of Abhyankar valuations was suspected in my prior work with Karen, the idea of using local uniformization took shape while I was visiting University of Utah. I thank Karl Schwede, Raymond Heitmann, Linquan Ma and Anurag Singh for many fruitful conversations during my stay in Utah, and Karl for the invitation. I am also grateful to Karen, Linquan and Raymond Heitmann for helpful comments on a draft of this paper, and to Eric Canton, Takumi Murayama and Matthew Stevenson for allowing me to include (Section 5, (15)) that we realized while working on a different project. Further thanks go to Franz-Viktor Kuhlmann, whose question on the relationship between defect and Abhyankar valuations inspired the material of Section 4, and to Steven Dale Cutkosky for helpful conversations. My work was supported by department and summer fellowships at University of Michigan and NSF Grant DMS #1501625. Background and conventions 2.1. Valuations. All valuations are written additively, and local rings are not necessarily Noetherian. We say a valuation ν with valuation ring V is centered on a local ring A, or A is a center of ν (or V ) if A ⊆ V , and the maximal ideal of V contracts to the maximal ideal of A. It is always assumed that the fraction field of a center coincides with the fraction field of the valuation ring. Let K be a finitely generated field extension of a field k, that is K is a function field over k. A valuation ν of K/k (this means ν is trivial on k) with valuation ring (R ν , m ν , κ ν ) and value group Γ ν satisfies the fundamental inequality dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /k ≤ tr. deg K/k. (2.1.0.1) If equality holds in the above inequality, then ν is called an Abhyankar valuation or a quasi-monomial valuation of K/k, and the associated valuation ring of ν is called an Abhyankar valuation ring of K/k. For Abhyankar valuations it is well-known [Bou89, VI, §10.3, Corollary 1] that Γ ν is a free abelian group of finite rank equal to d := dim Q (Γ ν ⊗ Z Q), and the residue field κ ν is a finitely generated field extension of k of transcendence degree n − d, where n := tr. deg K/k. If the ground field k has prime characteristic p and [k : k p ] < ∞, this implies [K : K p ] = [k : k p ]p n and [κ ν : κ p ν ] = [k : k p ]p n−d . An important fact, used implicitly throughout the paper, is that if x, y ∈ K × are non-zero elements such that x + y = 0 and ν(x) = ν(y), then ν(x + y) = inf{ν(x), ν(y)}. A discrete valuation ring (abbreviated DVR) is a Noetherian valuation ring which is not a field. Equivalently, a DVR is a dimension 1 regular local ring. 2.2. Frobenius. Let R be a ring of prime characteristic. We have the (absolute) Frobenius map F : R → R, which maps an element r ∈ R to its p-th power. The target copy of R is usually considered as an R-module by restriction of scalars via F , and is then denoted F * R. In other words, if r ∈ R and x ∈ F * R, then r · x = r p x. For an ideal I of R, by I [p e ] we mean the ideal generated by the p e -th powers of elements of I. Thus I [p e ] is the expansion of I under F e , the e-th iterate of Frobenius. Quite remarkably, the Frobenius map can detect when a Noetherian ring is regular, and the foundational result in the theory of F -singularities is the following: Theorem 2.2.1 ([Kun69], Theorem 2.1). Let R be a Noetherian ring of prime characteristic. Then R is regular if and only if F : R → F * R is a flat ring homomorphism. If Frobenius is a finite map and R is regular local, the above theorem implies that F * R is a free R-module. The freeness of F * R will be important when proving Theorem A. In geometry, finiteness of Frobenius is a mild restriction. For instance, Frobenius is a finite map for the localization of any finitely generated algebra over a perfect field of prime characteristic, and also for any complete local ring whose residue field k satisfies [k : k p ] < ∞. A ring for which Frobenius is finite is called F -finite. Kunz's theorem shows that when R is regular local and F -finite, F * R has many free Rsummands. For an arbitrary ring R, if F * R has at least one free R-summand, we say R is Frobenius split. More formally, R is Frobenius split if F : R → F * R has a left inverse, called a Frobenius splitting, in the category of R-modules. When R is reduced, Frobenius is an isomorphism onto its image R p , and the existence of a Frobenius splitting is equivalent to the existence of an R p -linear map R → R p that maps 1 → 1. Weakening the notion of Frobenius splitting leads to F -purity. We say that R is F -pure when Frobenius is a pure map of R-modules. This means for any R-module M , the induced map F ⊗ id M : M → F * R ⊗ R M is injective. Regular rings are F -pure because Frobenius is flat hence faithfully flat for such rings, and faithful flatness implies purity. Also Frobenius splitting clearly implies F -purity, but the converse is false. For example, any non-excellent DVR of a function field over a perfect ground field is F-pure but not Frobenius split. See (1) and (10) in Section 5 for further discussion, and Example 4.0.1 for a non-excellent DVR of a function field. Proof of Theorem A Unless otherwise specified, throughout this section we assume that K is a finitely generated field extension of an F -finite ground field k, and ν is an Abhyankar valuation of K/k whose residue field κ ν is separable over k. We will prove more generally under these assumptions that the valuation ring R ν is Frobenius split. Also, we let d := dim Q (Q ⊗ Z Γ ν ) and n := tr. deg K/k. The goal is to choose a regular local center of ν satisfying some nice properties, and then extend a Frobenius splitting of this center to a Frobenius splitting of R ν . The center we seek is given by the following lemma. Lemma 3.0.1. Let ν be an Abhyankar valuation as in Theorem 1. Then there exists a regular local ring (A, m A , κ A ) which is essentially of finite type over k with fraction field K satisfying the following properties: (1) R ν is centered on A, and κ A ֒→ κ ν is an isomorphism. (2) A has Krull dimension d, and there exist a regular system of parameters {x 1 , . . . , x d } of A such that {ν(x 1 ), . . . , ν(x d )} freely generates the value group Γ ν . Proof. Choose r 1 , . . . , r d , s 1 , . . . , s t ∈ R ν such that {ν(r 1 ), . . . , ν(r d )} freely generates the value group Γ ν , and the images of s 1 , . . . , s t in κ ν generate the latter over k. Taking Z := {r 1 , . . . , r d , s 1 , . . . , s t }, by local monomialization (Theorem 1) there exists a regular local ring (A, m A , κ A ) essentially of finite type over k with fraction field K such that R ν dominates A, A has dimension d, Z ⊆ A, and there exists a regular system of parameters {x 1 , . . . , x d } of A such that every z ∈ Z admits a factorization z = ux a 1 1 . . . x a d d , for some u ∈ A × , and a i ∈ N ∪ {0}. In particular, each ν(r i ) is Z-linear combination of ν(x 1 ), . . . , ν(x d ), which implies that {ν(x 1 ), . . . , ν(x d )} also freely generates Γ ν . Moreover, our choice of s 1 , . . . , s t implies that κ A = κ ν . Remark 3.0.2. For a valuation ν of K/k, the existence of a center which is an essentially of finite type k-algebra of Krull dimension equal to dim Q (Q ⊗ Z Γ ν ) implies that ν is Abhyankar. Thus, only Abhyankar valuations admit a center as in Lemma 3.0.1. From now on A will denote a choice of a regular local center of ν that satisfies Lemma 3.0.1, and {x 1 , . . . , x d } a regular system of parameters of A whose valuations freely generate Γ ν . Observe that A is F -finite since it is essentially of finite type over an F -finite field. Then Theorem 2.2.1 implies that A is free over its p-th power subring A p of rank equal to [K : K p ] = [k : k p ]p n . For f := [κ ν : κ p ν ] = [k : k p ]p n−d , if we choose 1 = y 1 , y 2 , . . . , y f ∈ A, such that the images of y i in κ A = κ ν form a basis of κ ν over κ p ν , then B := {y j x β 1 1 . . . x β d d : 1 ≤ j ≤ f, 0 ≤ β i ≤ p − 1}, is a free basis of A over A p . Note the elements y j are units in A. With respect to the basis B, A has a natural Frobenius splitting η B : A → A p , given by mapping 1 = y 1 x 0 1 . . . x 0 d → 1, and all the other basis elements to 0. Extending η B uniquely to a K p -linear map η B : K → K p of the fraction fields, we will show that the restriction of η B to R ν yields a Frobenius splitting of R ν , or in other words, η B | Rν maps into R p ν . Claim 3.0.3. For any a ∈ A, either η B (a) = 0 or ν(η B (a)) ≥ ν(a). Theorem A follows easily from the claim using the following general observation. Lemma 3.0.4. Let ν be a valuation of a field K of characteristic p > 0 with valuation ring R ν , and A a subring of R ν such that Frac(A) = K. Suppose ϕ : A → A p e is an A p e -linear map, for some e ≥ 1. Consider the following: (i) For all a ∈ A, ϕ(a) = 0 or ν(ϕ(a)) ≥ ν(a). (ii) For all a, b ∈ A such that ν(a) ≥ ν(b), if ϕ(ab p e −1 ) = 0, then ν(ϕ(ab p e −1 )) ≥ ν(b p e ). (iii) ϕ extends to an R p e ν -linear map R ν → R p e ν . (iv) ϕ extends uniquely to an R p e ν -linear map R ν → R p e ν . Then (ii), (iii) and (iv) are equivalent, and (i) ⇒ (ii). Moreover, if ϕ is a Frobenius splitting of A satisfying (i) or (ii), then ϕ extends to a Frobenius splitting of R ν . Proof. For the final assertion on Frobenius splitting, note that the extension of a Frobenius splitting remains a Frobenius splitting since 1 → 1 in the extension. (i) ⇒ (ii): If ϕ(ab p e −1 ) = 0, we have ν(ϕ(ab p e −1 )) ≥ ν(ab p e −1 ) ≥ ν(b p e ), where the first inequality follows from (i), and the second inequality follows from ν(a) ≥ ν(b). (ii) ⇒ (iii): Extending ϕ to a K p e -linear map ϕ : K → K p , it suffices to show that ϕ| Rν maps into R p e ν . Let r ∈ R ν be a non-zero element. Since K is the fraction field of A and R ν , one can express r as a fraction a/b, for non-zero a, b ∈ A. Note ν(a) ≥ ν(b). Then ϕ(r) = ϕ a b = 1 b p e ϕ(ab p e −1 ). (3.0.4.1) If ϕ(ab p e −1 ) = 0, then ϕ(r) = 0, and r maps into R p e ν . Otherwise by assumption, ν(ϕ(ab p e −1 )) ≥ ν(b p e ), and so, ν( ϕ(r)) = ν(ϕ(ab p e −1 )) − ν(b p e ) ≥ 0, that is ϕ(r) is an element of K p e ∩ R ν = R p e ν . (iii) ⇒ (iv) : Since A and R ν have the same fraction field, any extension of ϕ to R ν is obtained as a restriction to R ν of the unique extension of ϕ to a K p e -linear map ϕ : K → K p e , and so is also unique. See (3.0.4.1) above for a concrete description of how ϕ extends to R ν . To finish the proof of the lemma, it suffices to show (iv) ⇒ (ii). But this also follows easily from (3.0.4.1). Proof of Claim 3.0.3. Recall that B = {y j x β 1 1 . . . x β d d : 1 ≤ j ≤ f, 0 ≤ β i ≤ p − 1} is a basis of A over A p , where the x i and y j are chosen such that {ν(x 1 ), . . . , ν(x d )} freely generates the value group Γ ν , and the images of 1 = y 1 , y 2 , . . . , y f in κ ν form a basis of κ ν over κ p ν . The A p -linear Frobenius splitting η B is given by η B f j=1 0≤β i ≤p−1 c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d = c p 1,0,0,...,0 . Thus, we need to show that either c p 1,0,0,...,0 = 0 or ν(c p 1,0,0,...,0 ) ≥ ν f j=1 0≤β i ≤p−1 c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d . Assuming without loss of generality that f j=1 0≤β i ≤p−1 c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d = 0, we will prove the stronger fact that ν f j=1 0≤β i ≤p−1 c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d = inf{ν(c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d ) : c p j,β 1 ,...,β d = 0}. (3.0.4.2) For two non-zero terms c p j,α 1 ,...,α d y j x α 1 1 . . . x α d d and c p k,β 1 ,...,β d y k x β 1 1 . . . x β d d in the above sum, ν(c p j,α 1 ,...,α d y j x α 1 1 . . . x α d d ) = ν(c p k,β 1 ,...,β d y k x β 1 1 . . . x β d d ) (3.0.4.3) if and only if pν(c j,α 1 ,...,α d ) + α 1 ν(x 1 ) + · · · + α d ν(x d ) = pν(c k,β 1 ,...,β d ) + β 1 ν(x 1 ) + · · · + β d ν(x d ). (3.0.4.4) By Z-linear independence of ν(x 1 ), . . . , ν(x d ), for all i = 1, . . . , d, we get p|(α i − β i ). Since 0 ≤ α i , β i ≤ p − 1, this means that α i = β i for all i, and moreover, then ν(c p j,α 1 ,...,α d ) = ν(c p k,β 1 ,...,β d ). Thus, (3.0.4.3) holds precisely when ν(c p j,α 1 ,...,α d ) = ν(c p k,β 1 ,...,β d ) and α i = β i , for all i = 1, . . . , d. For ease of notation, let us use α as a shorthand for α 1 , . . . , α d , and x α for x α 1 1 . . . x α d d . Then for a fixed non-zero term c p j 1 ,α y j 1 x α , consider the set {c p j 1 ,α y j 1 x α , c p j 2 ,α y j 2 x α , . . . , c p j i ,α y j i x α } of all non-zero terms of f j=1 0≤β i ≤p−1 c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d having the same valuation as c p j 1 ,α y j 1 x α . In particular, by the above reasoning we also have ν(c p j 1 ,α ) = ν(c p j 2 ,α ) = · · · = ν(c p j i ,α ). Adding these terms of equal valuation, in the valuation ring R ν one can write c p j 1 ,α y j 1 x α + c p j 2 ,α y j 2 x α + · · · + c p j i ,α y j i x α = y j 1 + c j 2 ,α c j 1 ,α p y j 2 + · · · + c j i ,α c j 1 ,α p y j i c p j 1 ,α x α , where y j 1 + c j 2 ,α c j 1 ,α p y j 2 + · · · + c j i ,α c j 1 ,α p y j i is a unit in R ν by the κ p ν -linear independence of the images of y j 1 , . . . , y j i in κ ν and the fact that (c j 2 ,α /c j 1 ,α ) p , . . . , (c j i ,α /c j 1 ,α ) p are units in R p ν . Thus, the valuation of the sum c p j 1 ,α y j 1 x α + · · · + c p j i ,α y j i x α equals the valuation of any of its terms. Now rewriting f j=1 0≤β i ≤p−1 c p j,β 1 ,...,β d y j x β 1 1 . . . x β d d by collecting non-zero terms having the same valuation, (3.0.4.2), hence also the claim, follows. Examples 3.0.5. (a) A valuation ring of a function field of a curve over an F -finite ground field is always Frobenius split. Indeed, such a valuation ring is always centered on some normal affine model of dimension 1 of the function field, and so is an F -finite DVR. (b) For a positive integer n, consider Z ⊕n with the lexicographical order. That is, if {e 1 , . . . , e n } denotes the standard basis of Z ⊕n , then e 1 > e 2 > · · · > e n . There exists a unique valuation ν lex on F p (x 1 , . . . , x n )/F p such that for all i ∈ {1, . . . , n}, ν lex (x i ) = e i . The valuation ν lex is clearly Abhyakar since dim Q (Q ⊗ Z Z ⊕n ) = n, which coincides with the transcendence degree of F p (x 1 , . . . , x n )/F p . One can also show that the valuation ring R ν lex has Krull dimension n and residue field F p . The valuation is centered on the regular local ring F p [x 1 , . . . , x n ] (x 1 ,...,xn) such that the valuations of the obvious regular system of parameters freely generate Z ⊕n and the residue field coincides with the residue field of ν lex . Then a Frobenius splitting of R ν lex → R p ν lex is obtained by extending the canonical splitting on F p [x 1 , . . . , x n ] (x 1 ,...,xn) with respect to the basis {x β 1 1 . . . x β n : 0 ≤ β i ≤ p − 1}. This splitting of F p [x 1 , . . . , x n ] (x 1 ,...,xn) maps x α 1 1 . . . x αn n → x α 1 1 . . . x αn n if p|α i for all i, 0 otherwise. (c) Let Γ = Z ⊕ Zπ ⊂ R. There exists a valuation ν of F p (x, y, z)/F p given by ν(x) = ν(y) = 1, ν(z) = π. Then dim Q (Q ⊗ Z Γ) = 2, and the transcendence degree of the residue field κ ν /F p is at least 1 since the image of y/x in the residue field is transcendental over F p . Therefore the fundamental inequality (2.1.0.1) implies that ν is Abhyankar. Although the valuation ν is centered on the regular local ring F p [x, y, z] (x,y,z) , no regular system of parameters can freely generate the value group because the center has dimension 3, whereas the value group is free of rank 2. However, blowing up the origin in A 3 Fp , we see that ν is now centered on the regular local ring F p [x, y/x, z/x] (x,z/x) , and the valuations of the regular system of paramaters {x, z/x} freely generate Γ. Furthermore, the residue field of F p [x, y/x, z/x] (x,z/x) can be checked to coincide with the residue field of the valuation ring. Relabelling y/x and z/x as u, w respectively, a Frobenius splitting on R ν is obtained by extending the Frobenius splitting of F p [x, u, w] (x,w) given by the same rule as in (a) with respect to the transcendental elements x, u, w over F p . Valuations centered on Noetherian, local domains The proof of [DS16, Theorem 5.1] shows that a valuation ν of an F -finite function field K/k of characteristic p is Abhyankar precisely when [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. (4.0.0.1) If ν p denotes the restriction of ν to the subfield K p of K, then the value group of ν p is easily verified to be pΓ ν , and the residue field κ ν p can be identified with κ p ν . Thus, [Γ ν : pΓ ν ] is the ramification index, and [κ ν : κ p ν ] the residue degree of the extension of valuations ν/ν p [DS16, Remark 4.3.3]. Note ν is the unqiue extension of ν p to K since K is a purely inseparable extension of K p . In terms of the theory of extensions of valuations, (4.0.0.1) can be reinterpreted as saying that a necessary and sufficient condition for ν to be Abhyankar is for the unique extension of valuations ν/ν p to be defectless (see [FV11,Page 281] for the definition of defect). There is a natural generalization of the notion of an Abhyankar valuation for valuations of arbitrary fields. The goal of this section is to introduce this more general notion, and investigate to what extent such valuations can be characterized in terms of the defect of ν/ν p . We fix some notation. Let K denote a field of characteristic p > 0 (not necessarily a function field), and ν a valuation of K with valuation ring (V, m ν , κ ν ) centered on a Noetherian local ring (R, m R , κ R ) such that dim(R) < ∞. Recall that centers, by convention, always have the same fraction field as the valuation ring. Let Γ ν be the value group of ν. Abhyankar greatly generalized (2.1.0.1) in [Abh56, Theorem 1], establishing that in the above setup, dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /κ R ≤ dim(R). (4.0.0.2) Moreover, he showed that if equality holds in the above inequality, then Γ ν is a free abelian group and κ ν is a finitely generated extension of κ R . When K/k is a function field, there is a close relationship between Abhyankar valuations of K/k, and those valuations of K/k that admit a Noetherian center with respect to which equality holds in (4.0.0.2). Indeed, if equality holds in (4.0.0.2) for an arbitrary valuation ν of K/k with respect to a center R which is essentially of finite type over k, then ν is an Abhyankar valuation of K/k. To see this, let n = tr. deg K/k. Then tr. deg κ R /k = n − dim(R), because R is essentially of finite type over k with fraction field K, and so tr. deg κ ν /k = tr. deg κ ν /κ R + n − dim(R). This implies dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /k = (dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /κ R ) + n − dim(R) = dim(R) + n − dim(R) = tr. deg K/k. Conversely, a similar reasoning shows that if ν is an Abhyankar valuation of K/k, then ν satisfies equality in (4.0.0.2) with respect to any center which is essentially of finite type over k. However, despite the similarity between (4.0.0.2) and (2.1.0.1), whether a valuation satisfies equality in (4.0.0.2) is not an intrinsic property of the valuation, but also depends on the center R. In contrast, the property of being an Abhyankar valuation is intrinsic to valuations of function fields. To better illustrate this difference, we construct a valuation of F p (X, Y ) with two different Noetherian centers such that equality in (4.0.0.2) is satisfied with respect to one center, but not the other. In our example we work over a base field of characteristic p > 0, but the construction goes through when the ground field has characteristic 0. by mapping X → t and Y → p(t), where p(t) ∈ F p [[t]] such that {t, p(t)} are algebraically independent over F p . Such a power-series exists because F p ((t)) is uncountable, but F p (t) is countable. Moreover, multiplying p(t) by t, we may even assume that t|p(t). Then we get a new valuation ν on F p (X, Y ), given by the composition ν := F p (X, Y ) × i − → F p ((t)) × νt − → Z. The corresponding valuation ring V is a DVR with maximal ideal generated by X. Since ν(X) = ν t (t), ν(Y ) = ν t (p(t)) ≥ 1, (p(t) was scaled so that t|p(t)), we see that ν is centered on F p [X, Y ] (X,Y ) . Furthermore, ν is also trivially centered on its own valuation ring. As F p [[t]] dominates V and has residue field F p , κ ν = F p . Clearly ν satisfies equality in (4.0.0.2) with respect to its valuation ring as a center (this is true more generally for any discrete valuation), but not with respect to the center F p [X, Y ] (X,Y ) . Note that ν is not an Abhyankar valuation of F p (X, Y )/F p , since dim Q (Q ⊗ Z Z) + tr. deg κ ν /F p = 1 = tr. deg F p (X, Y )/F p . Moreover, the valuation ring of ν is not an F -finite DVR. This follows from results stated in the next section, but we include the justification here. Indeed, since the maximal ideal m of V is principal, by [Section 5, (3)(i)] dim κ p ν (V /m [p] ) = p[κ ν : κ p ν ] = p = [F p (X, Y ) : F p (X, Y ) p ] , and so V is not F -finite by [Section 5, (2)]. It turns out that V is also not excellent [Section 5, (10)]. Given the example above, we make the following definition. Definition 4.0.2. Let ν be a valuation centered on a Noetherian local domain R. We say R is an Abhyankar center of ν if dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /κ R = dim(R). To summarize our observations, the property of being an Abhyankar valuation of a function field is intrinsic to a valuation, while whether ν admits Abhyankar centers depends on the centers. However, if additional restrictions are imposed on the class of centers (for instance, if we require centers to be essentially of finite type over k), then the property of possessing these more restrictive Abhyankar centers becomes intrinsic to ν. Theorem 4.0.3 has some interesting consequences that we illustrate first. Corollary 4.0.4. Let ν be a valuation of a field K of characteristic p > 0. If ν admits a Noetherian, F -finite center which is Abhyankar, then any other Noetherian, F -finite center of ν is also an Abhyankar center of ν. In other words, the property of possessing Noetherian, F -finite, Abhyankar centers is intrinsic to a valuation. Corollary 4.0.5. Let ν be a valuation of a function field K/k over a perfect field k of characteristic p > 0 (it suffices for k to be F -finite). The following are equivalent: (1) ν is an Abyankar valuation of K/k. (2) ν admits an Abhyankar center which is an excellent local ring. Proof of Corollary 4.0.5. For (1) ⇒ (2), any center of the Abhyankar valuation ν which is essentially of finite type over k, hence also excellent, is an Abhyankar center of ν. For the converse, let R be an excellent, Abhyankar center of ν. As K is F -finite, R is also Ffinite. This follows from the fact that since R p is excellent (it is isomorphic to R), its integral closure S in K is module finite over R p , because K is a finite extension of K p . But R is an R p -submodule of S, and submodules of finitely generated modules over Noetherian rings are finitely generated. So R is also module finite over R p , that is, R is F -finite. Thus ν satisfies the identity [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] by Theorem 4.0.3, and since K/k is a function field, (4.0.0.1) implies that ν is an Abhyankar valuation of K/k. We will prove Theorem 4.0.3 by first developing a connection between the inequality dim Q (Q ⊗ Z Γ ν ) + tr. deg κ ν /κ R ≤ dim(R) and the quantities [Γ ν : pΓ ν ] and [κ ν : κ p ν ]. This will also shed light on precisely where F -finiteness is used in the proof the theorem. In order to achieve the above goal, we recall some general facts about torsion-free abelian groups and F -finite fields. Lemma 4.0.6. Let p be a prime number, K an F -finite field of characteristic p, and Γ a torsion-free abelian group such that dim Q (Q ⊗ Z Γ) is finite. We have the following: (1) If L is an algebraic extension of K, then [L : L p ] ≤ [K : K p ], with equality if K ⊆ L is a finite extension. In particular, L is then also F -finite. (2) If L is field extension of K of transcendence degree t, then [L : L p ] ≤ p t [K : K p ], with equality if L is finitely generated over K. (3) If s = dim Q (Q ⊗ Z Γ), then [Γ : pΓ] ≤ p s , with equality if Γ is finitely generated. Indication of proof of Lemma 4.0.6. (2) clearly follows from (1), and (3) follows from [DS16, Lemma 5.5], with equality obviously holding when Γ is finitely generated, since Γ is then free. We prove (1) here, which is a minor generalization of [DS16, Lemma 5.8]. To show [L : L p ] = [K : K p ] when K ⊆ L is finite is easy (see [DS16, Section 4.6]). So suppose K ⊆ L is algebraic, and [K : K p ] < ∞. It suffices to show that if a 1 , . . . , a n ∈ L are linearly independent over L p , then n ≤ [K : K p ]. Let L := K(a 1 , . . . , a n ). Since L is algebraic over K, L is a finite extension K, and so by what we already established, [ L : L p ] = [K : K p ]. On the other hand, since a 1 , . . . , a n are linearly independent over L p , and L p ⊆ L p , it follows that a 1 , . . . , a n are also linearly independent over L p . Thus, n ≤ [ L : L p ] = [K : K p ], as desired. Using the previous lemma, we can now relate the ramification index (i.e. [Γ ν : pΓ ν ]) and residue degree (i.e. [κ ν : κ p ν ]) of the extension of valuations ν/ν p to (4.0.0.2): Proposition 4.0.7. Let ν be a valuation of a field K of characteristic p > 0 with valuation ring (V, m ν , κ ν ), centered on Noetherian local domain (R, m R , κ R ) such that [κ R : κ p R ] < ∞. We have the following: (1) [Γ ν : pΓ ν ][κ ν : κ p ν ] ≤ p dim(R) [κ R : κ p R ]. (2) R is an Abhyankar center of ν if and only if [Γ ν : pΓ ν ][κ ν : κ p ν ] = p dim(R) [κ R : κ p R ]. Proof of Proposition 4.0.7. Throughout the proof, let s := dim Q (Q ⊗ Z Γ ν ) and t := tr. deg κ ν /κ R . (1) Abhyankar's inequality (4.0.0.2) implies s + t ≤ dim(R). In particular, s and t are both finite. Using Lemma 4.0.6(3), we get [Γ ν : pΓ ν ] ≤ p s . On the other hand, since κ R is F -finite by hypothesis, and κ ν has transcendence degree t over κ R , Lemma 4.0.6(2) shows [κ ν : κ p ν ] ≤ p t [κ R : κ p R ]. Thus, [Γ ν : pΓ ν ][κ ν : κ p ν ] ≤ p s+t [κ R : κ p R ] ≤ p dim(R) [κ R : κ p R ]. (4.0.7.1) (2) Suppose R is an Abhyankar center of ν, that is, s + t = dim(R). By [Abh56, Theorem 1], Γ ν is a free abelian group of rank s, and κ ν is a finitely generated field extension of κ R of transcendence degree t. Again using Lemma 4.0.6, we get [Γ ν : pΓ ν ] = p s and [κ ν : κ p ν ] = p t [κ R : κ p R ], and so [Γ ν : pΓ ν ][κ ν : κ p ν ] = p s+t [κ R : κ p R ] = p dim(R) [κ R : κ p R ], proving the forward implication. Conversely, if [Γ ν : pΓ ν ][κ ν : κ p ν ] = p dim(R) [κ R : κ p R ] then p dim(R) [κ R : κ p R ] = [Γ ν : pΓ ν ][κ ν : κ p ν ] ≤ p s+t [κ R : κ p R ] ≤ p dim(R) [κ R : κ p R ] , where the inequalities follow from (4.0.7.1). Thus, dim(R) = s + t, which by definition means that R is an Abhyankar center of ν. (4.0.7.2) This is a well-known result that is implicit in the proof of [Kun76, Proposition 2.1]. However, since (4.0.7.2) is crucial for our proof, we briefly indicate how it is established. In [Kun76, Proposition 2.1], Kunz uses the analogue of Noetherian normalization for complete rings to show that when R is F -finite, then for any minimal prime ideal P of the m R -adic completion R, [K : K p ] = p dim( R/P) [κ R : κ p R ] . This shows that dim( R/P) is independent of P, or in other words that R is equidimensional. However, since P is minimal, we then have dim( R/P) = dim( R) = dim(R), which confirms (4.0.7.2). A careful analysis of the proof of [Kun76, Proposition 2.1] reveals that (4.0.8.1) holds for any Noetherian, local domain R such that Ω 1 R/Z is a finitely generated R-module and the completion R is reduced, that is, if R is analytically unramified. We note that when R is F -finite, it satisfies both these properties. Indeed, since R is a finitely generated R p -module, (v) For a Noetherian domain R with F -finite fraction field K, the following are all equivalent: Ω 1 R/Z = Ω 1 R/R p(a) R is F -finite. (b) R is excellent. (c) R is a Japanese/N-2 ring. (d) The integral closure of R p in K is a finite R p -module. We already saw (b) ⇒ (c) ⇒ (d) ⇒ (a) in the proof of Corollary 4.0.5. The hard part is to show (a) ⇒ (b), which follows from [Kun76, Theorem 2.5]. As a consequence, in Theorem 4.0.3, the F -finiteness assumption on centers can be replaced by excellence, provided we assume the ambient field K is F -finite. Hence when the fraction field of a valuation is F -finite, the property of admitting excellent Abhyankar centers is intrinsic to the valuation. In particular, this is true for valuations of function fields over perfect ground fields of prime characteristic. (vi) The analogue of (v) is false for valuations of function fields over ground fields of characteristic 0, that is, whether a valuation admits an excellent Abhyankar center depends on the excellent center. For instance, any DVR of characteristic 0 is automatically excellent [Sta17, Tag 07QW], and by imitating the construction of Example 4.0.1 using the fields C(X, Y ) and C((t)) instead, one can show that there exists a discrete valuation ν of C(X, Y )/C centered on C[X, Y ] (X,Y ) such that the latter is not an Abhyankar center of ν (see [ELS03, Example 1(iv)] for more details). However, ν is also trivially centered on its own valuation ring which is an excellent, Abhyankar center of ν. The same example also shows that Corollary 4.0.5 is false over ground fields of characteristic 0. This remark and (v) indicate that excellent rings in characteristic p > 0 behave very differently from excellent rings in characteristic 0, and that the notion of excellence in prime characteristic is more restrictive than in characteristic 0. Summary of F -singularities for valuation rings We summarize all known results on F-singularities of valuation rings, grouping them according to the type of F-singularity they characterize. While most results are proved in [DS16] and the erratum [DS17], some new results also appear below (with complete proofs). For a valuation ring (V, m, κ) of a field K of characteristic p > 0 with associated valuation ν, singularities of V defined using the Frobenius map are intimately related to properties of the extension of valuations ν/ν p , where ν p denotes the restriction of ν to the subfield K p of K. Recall that the valuation ring of ν p is V p , and the residue field of ν p can be identified with κ p . Furthermore, ν is the unique extension of ν p to K (up to equivalence of valuations), and V is the integral closure of V p in K. In what follows Γ or Γ ν will always denote the value group of a valuation (ring), and κ or κ ν its residue field. Flatness of Frobenius and F-purity: (1) [DS16, Theorem 3.1] The Frobenius endomorphism on any valuation ring of prime characteristic is always faithfully flat. Hence a valuation ring of prime characteristic is F -pure, and so close to being Frobenius split. F -finiteness in general: Let (V, m, κ) be a valuation ring of a field K, with associated valuation ν. A necessary condition for V to be F -finite is that [K : K p ] < ∞, that is, K is F -finite. So we implicitly assume in our discussion of F -finiteness of V that K is F -finite to begin with. Note Ffiniteness of K also implies [κ : κ p ] < ∞, that is, the residue field is always F -finite. This follows by observing that [κ : κ p ] is the residue degree of the extension of valuations ν/ν p and then using [Bou89, VI, §8.1, Lemma 2]. (2) The following are equivalent: (a) V is F -finite. (b) V is a free V p -module of rank [K : K p ]. (c) dim κ p (V /m [p] ) = [K : K p ]. Proof of (2). The equivalence of (a) and (b) is shown in [DS16, Theorem 4.1.1]. Although used in the proof of [DS17, Erratum, revised Corollary 4.3.2], the equivalence of (c) to (a) and (b) is not explicitly stated in [DS16,DS17]. Thus, we include a complete proof here. Let n := [K : K p ]. We show (b) and (c) are equivalent. Suppose V is a free module of rank n over the subring V p , which is a valuation ring of K p . If η is the maximal ideal of V p , we see that V /m [p] ∼ = V ⊗ V p V p /η is a free κ p = V p /η-module of rank n, which proves (b) ⇒ (c). For the converse, suppose dim κ p (V /m [p] ) = [K : K p ] = n. Choose x 1 , . . . , x n ∈ V such that the images of x i in V /m [p] form a κ p -basis, and let L := V p x 1 + · · · + V p x n . Note L is a finitely generated, torsion free V p -module, hence free over V p since finitely generated torsion-free modules over valuation rings are free. To prove (b), it suffices to show that L = V. The rank of L equals dim κ p L/ηL, and it is easy to see that the images of x 1 , . . . , x n in L/ηL form a κ p -basis of L/ηL. Thus, L is a free V p -module of rank n, and so {x 1 , . . . , x n } is a V p -basis of L. Observe that {x 1 , . . . , x n } is also linearly independent over K p , and since [K : K p ] = n, this means that {x 1 , . . . , x n } is a K p -basis of K. Let s ∈ V be a non-zero element, and r 1 , . . . , r n ∈ K p such that s = r 1 x 1 + · · · + r n x n . Clearly V = L, if we can prove that all the r i are elements of V p . By renumbering the x i , and using the fact that V p is a valuation ring of K p , we may assume without loss of generality that r 1 = 0 and r i r −1 1 ∈ V p , for all i ≥ 2. If r 1 ∈ V p , then all the r i are already in V p . If not, then r −1 1 is an element of the maximal ideal, η, of V p . Thus, r −1 1 s = x 1 + r 2 r −1 1 x 2 + · · · + r n r −1 1 x n , which contradicts κ p -linear independence of the images of x 1 , . . . , x n in V /ηV = V /m [p] . Hence all the r i are elements of V p , and we are done. (3) (i) [DS17, Erratum, Lemma 2.2] For (V, m, κ) and K as above, dim κ p (V /m [p] ) = [κ : κ p ] if m is not finitely generated, p[κ : κ p ] if m is finitely generated. Sketch of proof of (i). The assertion follows from the short exact sequence of κ p -vector spaces Proof of (iv 0 → m/m [p] → V /m [p] → κ → 0,κ p ] = dim κ p (V /m [p] ), where we use (3)(i) for the final equality. So V is F -finite by (2). Remark 5.0.3. The proof of (iv) shows more generally that if V is a valuation ring with principal maximal ideal m and such that [Γ : pΓ] = p, then [Γ : pΓ][κ : κ p ] = [K : K p ] implies V is F -finite. F -finiteness and finite field extensions: (4) [DS17, Section 3] Let K ⊆ L be a finite extension of F -finite fields. Let ν be a valuation on K and w an extension of ν to L. Then the valuation ring of ν is F -finite if and only if the valuation ring of w is F -finite. F -finiteness in function fields: Let K be a finitely generated field extension of an F-finite field k. Sketch of proof of (6). The non-trivial implication is that when V is F -finite, its valuation ν is divisorial. However, (3)(ii)(a) and (5) imply that ν is Abhyankar, and so has a finitely generated value group. Using (3)(ii)(d) one then concludes V is a DVR. Now a classical result of Zariski shows that any Abhyankar DVR is divisorial [SZ60, VI, §14, Theorem 31]. Remarks 5.0.4. (i) [DS16, Theorem 5.1] erroneously states that an Abhyankar valuation ring of K/k is F-finite. The justification for why this is wrong is given in Remark 5.0.2. For a counter-example, the valuation ring V of the lexicographical valuation on F p (X, Y )/F p with value group Z ⊕2 (ordered lexicographically) is Abhyankar (see Example 3.0.5(a)), but not F -finite because dim κ p (V /m [p] ) = p[κ : κ p ] = p[F p : F p ] = p = [F p (X, Y ) : F p (X, Y ) p ]. Here the first equality holds by (3)(i) because the maximal ideal of V is principal, with Y being a generator. The second equality holds because the residue field of V is F p . (ii) When not in a function field, it is easy to construct non-Noetherian F -finite valuation rings. ] is a non-Noetherian, F -finite valuation ring of its fraction field F p ((t 1/p ∞ )). More generally, a non-trivial valuation ring of any perfect field of prime characteristic is not Noetherian, but F -finite because Frobenius is an isomorphism for such a ring. Rings of prime characteristic for which Frobenius is an isomorphism are called perfect rings. Such rings have been extensively investigated of late since finding applications in Scholze's work on perfectoid spaces (see [Sch12], [BS16]). While perfect rings are trivially F -finite, there exist non-Noetherian, F -finite valuation rings that are not perfect. Suppose L is a perfect field of prime characteristic equipped with a non-trivial valuation ν with value group Γ ν . For instance L can be a perfectoid field, or the algebraic closure of a field which has non-trivial valuations. Then the residue field κ ν of the associated valuation ring is also perfect. Now consider the group Γ ′ := Γ ν ⊕ Z ordered lexicographically, and the field L(X), where X is an indeterminate. There exists a unique extension w of the valuation ν to L(X) with value group Γ ′ such that for any polynomial f = n i=0 a i X i in L[X], we have w(f ) = inf{(ν(a i ), i) : i = 0, . . . , n}. The residue field κ w of w equals the residue field κ ν [Bou89, VI, §10.1, Proposition 1], hence is also perfect. Also, Γ ′ has a smallest element > 0 in the lex order, namely (0, 1). Thus, if (R w , m w ) is the valuation ring of w, the maximal ideal m w is principal, and in fact generated by X. Using (3)(i), we see that dim κ p w (R w /m [p] w ) = p[κ w : κ p w ] = p = [L(X) : L(X) p ]. (5.0.4.1) Then R w is F -finite by (2), not Noetherian because Γ ′ = Γ ν ⊕ Z has rational rank at least 2, and not perfect because the field L(X) is not perfect. (iii) Curiously, if instead of taking Γ ′ = Γ ν ⊕ Z ordered lexicographically we take Γ ′ = Z ⊕ Γ ν ordered lexicographically in the above construction, the resulting extension w of ν to L(X) (with obvious modifications to the definition of w) does not have an F -finite valuation ring R w . Indeed, then the maximal ideal of R w is not finitely generated, while the residue field κ w still coincides with κ ν , which is perfect. Thus dim κ p w (R w /m [p] w ) = [κ w : κ p w ] = 1 = [L(X) : L(X) p ]. F -finiteness and valuations centered on Noetherian domains: Using Theorem 4.0.3, one can generalize (6) to a non function field setting as follows. (7) Let ν be a non-trivial valuation on K centered on an F -finite, Noetherian local domain R. Then the valuation ring V of ν is F -finite if and only if V is a DVR and R is an Abhyankar center of ν. Proof of (7). For the forward implication, if V is F-finite, then [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] holds automatically (see (3)(ii)), and then Theorem 4.0.3 implies that R is an Abhyankar center of ν. In particular, Γ ν is a non-trivial finitely generated abelian group, and so (3)(ii)(d) ⇒ V is a DVR. This proves the forward implication. Conversely, if R is an Abhyankar center of ν, then Theorem 4.0.3 again implies [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. Since V is also a DVR by hypothesis, it is F -finite by (3)(iv). The above result has the following interesting consequence that we would like to highlight separately. (8) Suppose ν is a valuation of an F -finite field K with valuation ring V that satisfies either of the following conditions: (a) V is F -finite, but not Noetherian. (b) dim(V ) > s, where [K : K p ] = p s . Then ν is not centered on any excellent local domain whose fraction field is K. Proof of (8). Since K is F -finite, a Noetherian domain with fraction field K is excellent if and only if it is F -finite (see Remark 4.0.8(v)). Thus, it suffices to show that ν is not centered on any F -finite, Noetherian local domain if it satisfies (a) or (b). Suppose ν satisfies (a). As V is not Noetherian, (7) implies that ν cannot be centered on any F -finite, Noetherian local domain with fraction field K. If R is an F -finite, Noetherian local ring with fraction field K, then recall that we have the identity p dim(R) [κ R : κ p R ] = [K : K p ]. In particular, this means dim(R) ≤ s, where s is as above. Now if ν is centered on R, then Abhyankar's inequality (4.0.0.2) shows in particular that dim Q (Q ⊗ Z Γ ν ) ≤ dim(R) ≤ s. However, it is well-known that the Krull dimension of V is at most dim Q (Q ⊗ Z Γ ν ) [Bou89, § 10.2, Corollary to Proposition 3]. Then dim(V ) ≤ s, which contradicts the hypothesis of (b). Hence ν cannot be centered on any F -finite, Noetherian domain with fraction field K. Example 5.0.5. Let w be the valuation of L(X) (where L is a perfect field) constructed in Remark 5.0.4(ii). The valuation ring R w satisfies conditions (a) and (b) of (8). We have already observed that R w satisfies (a). To see that R w satisfies (b), note that the value group of w has a proper, non-trivial isolated/convex subgroup. Thus R w has Krull dimension at least 2 [Bou89, §4.5], while [L(X) : L(X) p ] = p. Although R w is a valuation ring of a function field, it does not contain the ground field L. So even though w/w p is defectless, this example does not contradict (5), or the problem of local uniformization in prime characteristic. Remark 5.0.6. If K/k is an F -finite function field, and ν is a valuation of K/k with valuation ring V , then (5) shows that V cannot satisfy (8)(a), while (2.1.0.1) shows that V cannot satisfy (8)(b). Thus, the pathologies of (8) do not arise for valuations of function fields that are trivial on the ground field. Frobenius splitting: (9) [DS16, Corollary 4.1.2] Any F -finite valuation ring is Frobenius split. This follows from (2) because F * V is then a free V -module. Remark 5.0.7. (9) is a special case of a general phenomenon of valuation rings which is independent of the characteristic of the ring. Recall, that a ring R is called a splinter if any module finite ring extension ϕ : R → S has an R-linear left inverse. For example, from recent work of André [And18] (see also [Bha18,HM18]) and earlier work of Hochster [Hoc73], it follows that any regular ring is a splinter. We want to show that valuation rings are also splinters. So let V be a valuation ring (of any characteristic), and ϕ : V → S a module finite ring extension. Choose a prime ideal P of S that lies over the zero ideal of V . Then the composition V ϕ − → S ։ S/P is also a module finite ring extension, and it suffices to show this composite extension splits. Thus we may assume S is a domain, which makes S a finitely generated torsionfree V module. But finitely generated torsion free modules over valuation rings are free. Nakayama's lemma implies there exists a basis of S over V containing 1, and then one can easily construct many splittings of ϕ with respect to such a basis. (10) [DS16, Corollary 4.2.2] The following are equivalent for a Noetherian valuation ring (V, m, κ) with F-finite fraction field K: • V is Frobenius split. • V is F -finite. • V is excellent. • dim κ p (V /m p ) = [K : K p ]. • [Γ : pΓ][κ : κ p ] = [K : K p ]. (11) Any DVR that admits a Noetherian, F -finite, Abhyankar center is Frobenius split. Proof of 11. V is F -finite by (7), hence Frobenius split by (9). (12) Any complete DVR of prime characteristic is Frobenius split. Proof of 12. Let V be a complete DVR of prime characteristic with residue field κ. Since V is equicharacteristic, by Cohen's Structure Theorem for complete rings, We want to show that when ν/ν p is totally unramified and K is not perfect, the valuation ring V of ν cannot be Frobenius split 1 . Note that (5.0.7.1) implies Γ ν = pΓ ν and κ ν = κ p ν , that is, the value group is p-divisible and the residue field is perfect. The p-divisibility of Γ ν shows that V ∼ = κ[[t]],m = m [p] . Then any Frobenius splitting ϕ : V → V p maps the maximal ideal m of V into the maximal ideal of V p , thereby inducing a Frobenius splitting of residue fields ϕ : κ ν → κ p ν . However, κ ν is perfect, so that ϕ is just the identity map. Since K is not perfect, ϕ has a non-trivial kernel, that is, some non-zero x ∈ V gets mapped to 0. By p-divisibility, one can write x = uy p , for a unit u in V , and y = 0. Then 0 = ϕ(x) = y p ϕ(u), which shows that ϕ(u) = 0. But this contradicts injectivity of ϕ, proving that no Frobenius splitting of V exists. Example 5.0.8. Valuations ν such that ν/ν p is totally unramified can occur even in function fields. For example, one can construct a Q-valued valuation of F p (X, Y )/F p (see [Vaq06,Example 8]). We claim that ν/ν p is then totally unramified. Indeed, since Q is not a finitely generated abelian group, ν is not an Abhyankar valuation of F p (X, Y )/F p . It follows by Abhyankar's inequality (2.1.0.1) that if κ ν is the residue field of ν, then tr. deg κ ν /F p = 0, that is, κ ν is an algebraic extension of F p . Consequently, κ ν is a perfect field, and so [κ ν : κ p ν ] = 1. Furthermore, Q is p-divisible. This shows that the equality in (5.0.7.1) holds, that is, ν/ν p is totally unramified. (15) Let V be a valuation ring of Krull dimension 1 and fraction field K. Since the value group of V is an ordered subgroup of R, the induced valuation ν on K gives a multiplicative norm | · | ν : K → R ≥0 via the assignment |x| ν = e −ν(x) for x ∈ K − {0} and |0| ν = 0. The norm | · | ν satisfies the multiplicative analogue of the axioms of a valuation: (a) |xy| ν = |x| ν |y| ν (b) |x + y| ν ≤ max{|x| ν , |y| ν }. In other words, | · | ν is an ultrametric on K. Observe that the valuation ring V of K is then the subring of elements x ∈ K such that |x| ν ≤ 1. Given x ∈ K and r ∈ R ≥0 , the closed ball B(x, r) of radius r centered at x is defined the usual way. A consequence of | · | ν being ultrametric is that if r > 0, then any closed ball is also open. We say that the field K is spherically complete with respect to the norm |·| ν if any nested sequence of non-empty closed balls B 1 ⊃ B 2 ⊃ B 3 ⊃ . . . has a non-empty intersection. Note that here we do not require the centers of the B i to be the same, in which case the assertion that the intersection is non-empty is clear. If K is spherically complete with respect to | · | ν then K is also complete with respect to this norm. One can view the notion of a spherically complete field as a generalization of a field equipped with a complete discrete valuation, for a field of the latter type is always spherically complete [GPS10, Theorem 1.2.13] We now claim that if V has prime characteristic p > 0 and (K, | · | ν ) is spherically complete with respect to |·| ν , then V is Frobenius split 2 . This result may be viewed as a generalization of Frobenius splitting of complete discrete valuation rings (12). Let | · | ν p be the corresponding norm on the subfield K p induced by ν p . Then (K p , | · | ν p ) is also spherically complete. Viewing K as a vector space over K p , one can verify that the norm | · | ν on K gives K the structure of a normed space over K p (i.e. the norm on K is compatible with that on K p ). Since K p is spherically complete, the analytic form of the Hahn-Banach theorem over spherically complete fields [GPS10, Theorem 4.1.1] shows that the identity map id K p : K p → K p extends to a K p -linear map f : K → K p such that for all x ∈ K, |f (x)| ν p ≤ |x| ν . Since the identity map sends 1 → 1, it follows that f also sends 1 → 1, that is, f is a Frobenius splitting of K p . On the other hand, for any x ∈ V , |f (x)| ν p ≤ |x| ν ≤ 1, that is, if x ∈ V , then f (x) ∈ K p ∩ V = V p . Thus, restricting f to V gives a Frobenius splitting of V . F -regularity: Two notions of F -regularity were introduced in [DS16], generalizing strong F -regularity to a non-Noetherian and non F -finite setting-split F -regularity [DS16, Definition 6.6.1] and F -pure regularity [DS16, Definiton 6.1.1]. Split F -regularity just drops the F -finite and Noetherian hypotheses from the definition of strong F -regularity, while F -pure regularity replaces splitting of certain maps by purity and seems to be the better notion for rings that are not F -finite. Split F -regularity ⇒ F -pure regularity, but the converse is false. Indeed, any non-excellent DVR with an F -finite fraction field will be F -pure regular but not split F -regular. In particular, the DVR of Example 4.0.1 is not excellent, so not split F -regular. (16) [DS16, Thm 6.5.1 and Cor. 6.5.4] For a valuation ring V of prime characteristic, V is F -pure regular if and only if it is Noetherian. (17) [DS16, Corollary 6.6.3] Let V be a valuation ring whose fraction field is F -finite. The following are equivalent (see also (10)): • V is split F -regular. • V is Noetherian and F -finite. • V is excellent. • V is Noetherian and Frobenius split. • If V is a valuation ring of a function field, then V is divisorial. Remark 5.0.9. (16) and (17) indicate that F -regularity is perhaps a useful notion of singularity only for Noetherian rings. Open Questions: Just as is the case in geometry, our results indicate that Frobenius splitting is the most mysterious F-singularity for valuation rings with many basic open questions. The proof of Frobenius splitting of Abhyankar valuation rings of function fields over Ffinite ground fields uses the local monomialization result of Knaf and Kuhlmann (Theorem 1), hence also the hypothesis that the residue field of the valuation ring is separable over the ground field. However, it is probably the case that any Abhyankar valuation ring of an F -finite function field is Frobenius split, and this will from our proof if one can remove the separability hypothesis from Theorem 1. Moreover, a natural question is if one can generalize our result on Frobenius splitting of Abhyankar valuations to valuations, not necessarily of function fields, that admit F -finite, Noetherian, Abhyankar centers satisfying 'mild' singularities such as Fregularity. For example, (7) shows that a discrete valuation admitting a Noetherian, F -finite, Abhyankar center is Frobenius split. F -singularities of a valuation ring V are intimately related to basic properties of the corresponding extension of valuations ν/ν p . For example, (5) shows that at least for function fields over perfect ground fields, when ν/ν p is defectless, ν is Abhyankar and its valuation ring is Frobenius split. On the other hand, when ν/ν p has maximal defect, that is when ν/ν p is totally unramified, (14) shows that V cannot be Frobenius split unless it is a perfect ring. However, Frobenius splitting of V remains mysterious when the defect of ν/ν p is not one two possible extremes. For instance, suppose K/k is an F -finite function field. Is there a non-Abhyankar valuation of K/k whose valuation ring is Frobenius split? We can use (17) to conclude that such valuations, if they exist, cannot be discrete. On the other hand, Example 5.0.5 shows that if we relax the condition that the valuation is trivial on the ground field, then there exists a valuation w of K, not trivial on k, whose valuation ring is Frobenius split. Moreover, w has the feature that it is not centered on any excellent domain whose fraction field is K. However, even in this example, the extension w/w p is defectless. Furthermore, in light of (15) it is interesting to ask when a (complete) non-Archimedean and non-perfect field of prime characteristic has a Frobenius split valuation ring. Note that by completing the valuation of Example 5.0.8 it follows that such valuation rings are not always Frobenius split. There are interesting open questions pertaining to Frobenius splitting even for Noetherian valuation rings. As far as we know, it is not known if every excellent DVR of prime characteristic is Frobenius split. (10) (resp. (12)) provides an affirmative answer when the fraction field of a DVR is F -finite (resp. when the DVR is complete). At the same time it is worth recalling that the Frobenius map is always pure for any valuation ring of prime characteristic by (1), and purity seems to be a better notion than splitting for rings that are not F -finite. On the other hand, the proof of [DS16, Theorem 5.1] does establish that a valuation ν of an F -finite function field K/k is Abhyankar if and only if [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. (1.0.0.1) Example 4.0.1. (see also[DS16, Example 4.0.5]) Consider the laurent series field F p ((t)) with the canonical t-adic valuation, ν t , whose corresponding valuation ring is the DVR F p [[t]]. Choose an embedding of fields i : F p (X, Y ) ֒→ F p ((t)) The interplay between Abhyankar valuations and valuations admitting Abhyankar centers raises the natural question: does (4.0.0.1) have an analogue for valuations of fields that are not necessarily function fields? Moreover, can the feature of possessing Abhyankar centers become intrinsic to a valuation if we restrict the class of admissible centers? The next result provides an affirmative answer for a broad class of Noetherian centers. . Let (R, m R , κ R ) be a Noetherian, F -finite local domain of characteristic p > 0 and fraction field K. Suppose ν is a non-trivial valuation of K centered on R with value group Γ ν and valuation ring (V, m ν , κ ν ). Then R is an Abhyankar center of ν if and only if [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. Proof of Corollary 4.0.4. The proof follows easily from Theorem 4.0.3 using the observation that the identity [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] is independent of the choice of a center. . Suppose R is a Noetherian, F -finite, local domain with fraction field K. Then [κ R : κ p R ] and [K : K p ] < ∞. In particular, R satisfies the hypotheses of Proposition 4.0.7, and so Theorem 4.0.3 follows if we can show that [K : K p ] = p dim(R) [κ R : κ p R ]. ) A key point in the proof of Theorem 4.0.3 is that when (R, m, κ) is an F -finite, Noetherian, local domain of characteristic p > 0 and fraction field K, then [K : K p ] = p dim(R) [κ R : κ p R ]. (4.0.8.1) is then a finitely generated R-module, and R is reduced by[Kun69, Lemma 2.4].Theorem 4.0.3, hence Corollary 4.0.4, clearly hold more generally for the class of Noetherian centers of any F -finite field that satisfy (4.0.8.1). While such centers are quite common, it is not difficult to construct generically F -finite Noetherian local, domains that do not satisfy (4.0.8.1). For instance, (4.0.8.1) fails for the non F -finite DVR of F p (X, Y ) constructed in Example 4.0.1. In particular, since regular local rings are analytically unramified, our observations imply that the module of absolute Kähler differentials of the DVR of Example 4.0.1 must not be finitely generated.(ii) Theorem 4.0.3 generalizes (4.0.0.1). Indeed, if K/k is an F -finite function field, then any valuation ν of K/k admits an F -finite, Noetherian center. For instance, by the valuative criterion of properness, ν is centered on a proper k-variety with function field K, and the local ring of this variety at the center is F -finite. Since we observed that ν is an Abhyankar valuation if and only if any center of ν which is essentially of finite type over k is an Abhyankar center, it follows by Theorem 4.0.3 that ν is Abhyankar precisely when [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. ( iii) One can reinterpret Theorem 4.0.3 as saying that an F -finite, Noetherian center R of ν is an Abhyankar center of ν if and only if the extension of valuations ν/ν p is defectless.(iv) When [K : K p ] < ∞, the condition [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ] does not imply that the valuation ν admits a Noetherian, F -finite center. See Example 5.0.5 for a counterexample, which shows that counter-examples can be constructed even for valuations of fields that are not perfect. with the additional observations that when m is not finitely generated, m [p] = m [DS17, Lemma 2.1], and when m is finitely generated, m is principal, so that dim κ p (m/m [p] ) = (p − 1)[κ : κ p ].(ii) For an F-finite, valuation ring V with value group Γ = 0: (a) [K : K p ] = [Γ : pΓ][κ : κ p ] = dim κ p (V /m [p] ) ([DS17, corrected Thm 4.3.1] and (2)). (b) The value group Γ satisfies [Γ : pΓ] = 1 or [Γ : pΓ] = p. (c) If the maximal ideal of V is not finitely generated, then Γ is p-divisible. (d) If Γ is finitely generated, then V is a DVR. Remark 5.0.2. The error in [DS16] arose from the incorrect assertion that [Γ : pΓ][κ : κ p ] = [K : K p ] ⇒ V is F-finite (although the assertion is true when V is a DVR by (3)(iv) below). (iii) If [K : K p ] = [κ : κ p ], then V is F-finite [DS17, revised Corollary 4.3.2]. (iv) If V is a DVR, then V is F -finite if and only if [Γ : pΓ][κ : κ p ] = [K : K p ]. ( 5 ) 5[DS17, Theorem 1.1] A valuation ν of K/k is Abhyankar if and only if [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. See also Theorem 4.0.3 for a generalization of the above result. (6) [DS17, Erratum, Theorem 0.1] For a non-trivial valuation ν of K/k, the associated valuation ring V is F -finite if and only if ν is divisorial. For example, the perfection F p [[t 1/p ∞ ]] := e∈N F p [[t 1/p e ]] of the power series ring F p [[t] as rings, and clearly κ p [[t p ]] is a direct summand of κ[[t]], even when [κ : κ p ] is not finite.(13) [Section 3] For an Abhyankar valuation ring V of an F -finite function field K/k, if the residue field κ is separable over k, then V is Frobenius split. In particular Abhyankar valuation rings over perfect ground fields of prime characteristic are always Frobenius split.(14) We have seen in (5) that Abhyankar valuations in F -finite function fields are characterized as those valuations ν such that ν/ν p is defectless, that is, [Γ ν : pΓ ν ][κ ν : κ p ν ] = [K : K p ]. At the opposite extreme is a valuation ν such that the extension ν/ν p is totally unramified, which means that [Γ ν : pΓ ν ][κ ν : κ p ν ] = 1.(5.0.7.1) The author thanks Ray Heitmann for showing him a proof of (14) in the special case of Q-valuations of Fp(X, Y ). Our argument above is a generalization of Heitmann's argument to all totally unramified extensions ν/ν p . Eric Canton, Takumi Murayama, Matthew Stevenson and the author realized (15) while working on a different project. He is grateful to them for allowing him to reproduce the result in this paper. E-mail address: [email protected] On the valuations centered in a local domain. S Abhyankar, Amer. J. Math. 782S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), no. 2, pp. 321-348. La conjecture du facteur direct. Y André, Publ. Math. Inst. Hautes Études Sci. 1271Y. André, La conjecture du facteur direct, Publ. Math. Inst. Hautes Études Sci. 127(1) (2018), pp. 71-93. On the direct summand conjecture and its derived variant. Inventiones Math. 2122, On the direct summand conjecture and its derived variant, Inventiones Math. 212(2) (2018), pp. 297-317. Existence of valuations with smallest normalized volume. H Blum, Compos. Math. 154H. Blum, Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), pp. 820-849. N Bourbaki, Commutative algebra. Berlin HeidelbergSpringer-VerlagN. Bourbaki, Commutative algebra, Chapters 1-7, Springer-Verlag Berlin Heidelberg, 1989. Projectivity of the Witt vector affine Grassmannian. B Bhatt, P Scholze, Inventiones Math. 2092B. Bhatt and P. Scholze, Projectivity of the Witt vector affine Grassmannian, Inventiones Math. 209 (2016), no. 2, pp. 329-423. Frobenius and valuation rings. R Datta, K E Smith, Algebra Number Theory. 105R. Datta and K.E. Smith, Frobenius and valuation rings, Algebra Number Theory 10 (2016), no. 5, pp. 1057-1090. Correction to the article "Frobenius and valuation rings. Algebra Number Theory. 114, Correction to the article "Frobenius and valuation rings", Algebra Number Theory 11 (2017), no. 4, pp. 1003-1007. Uniform approximation of Abhyankar valuation ideals in smooth function fields. L Ein, R Lazarsfeld, K E Smith, Amer. J. Math. 1252L. Ein and R. Lazarsfeld and K.E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), no. 2, pp. 409-440. The Valuative Tree. C Favre, M Jonsson, Lecture notes in Mathematics. 1853Springer-VerlagC. Favre and M. Jonsson, The Valuative Tree, Lecture notes in Mathematics, vol. 1853, Springer- Verlag Berlin Heidelberg, 2004. Valuations and multiplier ideals. J. Amer. Math. Soc. 183, Valuations and multiplier ideals, J. Amer. Math. Soc. 18 (2005), no. 3, pp. 655-684. The defect, Commutative Algebra-Noetherian and Non-Noetherian Perspectives. F-V Kuhlmann, SpringerNew YorkF-V. Kuhlmann, The defect, Commutative Algebra-Noetherian and Non-Noetherian Perspec- tives, Springer New York, pp. 277-318, 2011. Locally convex spaces over non-Archimedean valued fields, Cambridge studies in advanced mathematics. C Perez-Garcia, W H Schikhof, Cambridge University Press119C. Perez-Garcia and W.H. Schikhof, Locally convex spaces over non-Archimedean valued fields, Cambridge studies in advanced mathematics, vol. 119, Cambridge University Press, 2010. Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic. R Heitmann, L Ma, Algebra Number Theory. 127R. Heitmann and L. Ma, Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic, Algebra Number Theory 12(7), pp. 1659-1674. Contracted ideals from integral extensions of regular rings. M Hochster, Nagoya Math. J. 51M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), pp. 25-43. Valuations and asymptotic invariants for sequences of ideals. M Jonsson, M Mustaţă, Ann. Inst. Fourier (Grenoble). 626M. Jonsson and M. Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, pp. 2145-2209. Abhyankar places admit local uniformization in any characteristic. H Knaf, F-V Kuhlmann, Ann. Sci. École Norm. Sup. 46SérieH. Knaf and F-V. Kuhlmann, Abhyankar places admit local uniformization in any characteristic, Ann. Sci. École Norm. Sup., Série 4 38 (2005), no. 6, pp. 833-846. Characterizations of regular local rings of characteristic p. E Kunz, Amer. J. Math. 913E. Kunz, Characterizations of regular local rings of characteristic p, Amer. J. Math. 91 (1969), no. 3, pp. 772-784. On Noetherian rings of characteristic p. Amer. J. Math. 984, On Noetherian rings of characteristic p, Amer. J. Math. 98 (1976), no. 4, pp. 999-1013. M Mustaţă, IMPANGA lecture notes on log canonical thresholds, Contributions to algebraic geometry: Impanga ecture notes. M. Mustaţă, IMPANGA lecture notes on log canonical thresholds, Contributions to algebraic geometry: Impanga ecture notes, EMS (2012), pp. 407-442. Topology of non-Archimedean analytic spaces and relations to complex algebraic geometry. S Payne, Bull. Amer. Math. Soc. 522S. Payne, Topology of non-Archimedean analytic spaces and relations to complex algebraic geometry, Bull. Amer. Math. Soc. 52 (2014), no. 2, pp. 223-247. The analogue of Izumi's theorem for Abhyankar valuations. G Rond, M Spivakovsky, J. Lon. Math. Soc. 903G. Rond and M. Spivakovsky, The analogue of Izumi's theorem for Abhyankar valuations, J. Lon. Math. Soc. 90 (2014), no. 3, pp. 725-740. Perfectoid Spaces. P Scholze, Publ. Math. Inst. Hautes Études Sci. 1161P. Scholze, Perfectoid Spaces, Publ. Math. Inst. Hautes Études Sci. 116 (2012), Issue 1, pp. 245-313. Valuations in function fields of surfaces. M Spivakovsky, Amer. J. Math. 1121M. Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), no. 1, pp. 107-156. The Stacks Project Authors, Stacks Project. The Stacks Project Authors, Stacks Project, http://stacks.math.columbia.edu, 2017. . P Samuel, O Zariski, Commutative Algebra. IISpringer-VerlagP. Samuel and O. Zariski, Commutative Algebra, Vol II, Springer-Verlag Berlin Heidelberg, 1960. Valuation Theory in Interaction. B Teissier, Overweight deformations of affine toric varieties and local uniformization. ZürichEMS Publishing HouseB. Teissier, Overweight deformations of affine toric varieties and local uniformization , vol. Valuation Theory in Interaction, EMS Series of Congress Reports, pp. 474-565, EMS Publishing House, Zürich, 2014. Inseparable local uniformization. M Temkin, J. Algebra. 3731M. Temkin, Inseparable local uniformization, J. Algebra 373 (2013), no. 1, pp. 65-119. Singularity theory and its applications. M Vaquié, Adv. Stud. Pure Math. 43Math. Soc. JapanValuations and local uniformizationM. Vaquié, Valuations and local uniformization, Singularity theory and its applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo (2006), pp. 477-527.

The 2017 Fake News Challenge Stage 1 (FNC-1) shared task addressed a stance classification task as a crucial first step towards detecting fake news. To date, there is no in-depth analysis paper to critically discuss FNC-1's experimental setup, reproduce the results, and draw conclusions for next-generation stance classification methods. In this paper, we provide such an in-depth analysis for the three top-performing systems. We first find that FNC-1's proposed evaluation metric favors the majority class, which can be easily classified, and thus overestimates the true discriminative power of the methods. Therefore, we propose a new F1-based metric yielding a changed system ranking. Next, we compare the features and architectures used, which leads to a novel feature-rich stacked LSTM model that performs on par with the best systems, but is superior in predicting minority classes. To understand the methods' ability to generalize, we derive a new dataset and perform both in-domain and cross-domain experiments. Our qualitative and quantitative study helps interpreting the original FNC-1 scores and understand which features help improving performance and why. Our new dataset and all source code used during the reproduction study are publicly available for future research 1

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A Retrospective Analysis of the Fake News Challenge Stance Detection Task August 20-26. 2018 Andreas Hanselowski AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn Avinesh Pvs AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn Benjamin Schiller AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn Felix Caspelherr AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn Debanjan Chaudhuri AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn Christian M Meyer AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn Iryna Gurevych AIPHES † Computer Science Department Research Training Group Technische Universität Darmstadt ‡ Smart Data Analytics University of Bonn A Retrospective Analysis of the Fake News Challenge Stance Detection Task Proceedings of the 27th International Conference on Computational Linguistics the 27th International Conference on Computational LinguisticsSanta Fe, New Mexico, USA1859August 20-26. 2018https://www.aiphes.tu-darmstadt.de The 2017 Fake News Challenge Stage 1 (FNC-1) shared task addressed a stance classification task as a crucial first step towards detecting fake news. To date, there is no in-depth analysis paper to critically discuss FNC-1's experimental setup, reproduce the results, and draw conclusions for next-generation stance classification methods. In this paper, we provide such an in-depth analysis for the three top-performing systems. We first find that FNC-1's proposed evaluation metric favors the majority class, which can be easily classified, and thus overestimates the true discriminative power of the methods. Therefore, we propose a new F1-based metric yielding a changed system ranking. Next, we compare the features and architectures used, which leads to a novel feature-rich stacked LSTM model that performs on par with the best systems, but is superior in predicting minority classes. To understand the methods' ability to generalize, we derive a new dataset and perform both in-domain and cross-domain experiments. Our qualitative and quantitative study helps interpreting the original FNC-1 scores and understand which features help improving performance and why. Our new dataset and all source code used during the reproduction study are publicly available for future research 1 Introduction Recently, Pomerleau and Rao (2017) organized the first Fake News Challenge 2 (FNC-1) in order to foster the development of AI technology to automatically detect fake news. The challenge received much attention in the NLP community: 50 teams from both academia and industry participated. The goal of the FNC-1 challenge is to determine the perspective (or stance) of a news article relative to a given headline. An article's stance can either agree or disagree with the headline, discuss the same topic, or it is completely unrelated. Table 1 shows four example documents illustrating these classes. Stance detection is a crucial building block for a variety of tasks, such as analyzing online debates (Walker et al., 2012;Sridhar et al., 2015;Somasundaran and Wiebe, 2010), determining the veracity of rumors on twitter (Lukasik et al., 2016;Derczynski et al., 2017), or understanding the argumentative structure of persuasive essays (Stab and Gurevych, 2017). While stance detection has been previously focused on individual sentences or phrases, the systems participating in FNC-1 have to detect the stance of an entire document, which raises many new challenges. Although the disagreeing article of Table 1 clearly leans against the headline's claim, the fourth sentence would agree to it if considered in isolation. To properly learn from a scientific shared task, there are typically overview and analysis papers that compare the architectures, features, and results of the participating systems. To date, there is, however, no such paper for FNC-1, which is why we conduct a reproduction study of the top three participating systems. Our goal is to independently verify the results reported in the challenge, which is an important asset in empirical research, to critically assess the experimental setup of FNC-1, and to learn building Headline: Hundreds of Palestinians flee floods in Gaza as Israel opens dams Agree (AGR) GAZA CITY (Ma'an) -Hundreds of Palestinians were evacuated from their homes Sunday morning after Israeli authorities opened a number of dams near the border, flooding the Gaza Valley in the wake of a recent severe winter storm. The Gaza Ministry of Interior said in a statement that civil defense services and teams from the Ministry of Public Works had evacuated more than 80 families from both sides of the Gaza Valley (Wadi Gaza) after their homes flooded as water levels reached more than three meters [..] Discuss (DSC) Palestinian officials say hundreds of Gazans were forced to evacuate after Israel opened the gates of several dams on the border with the Gaza Strip, and flooded at least 80 households. Israel has denied the claim as "entirely false". [..] Disagree (DSG) Israel has rejected allegations by government officials in the Gaza strip that authorities were responsible for released storm waters flooding parts of the besieged area. "The claim is entirely false, and southern Israel does not have any dams," said a statement from the Coordinator of Government Activities in the Territories (COGAT). "Due to the recent rain, streams were flooded throughout the region with no connection to actions taken by the State of Israel." At least 80 Palestinian families have been evacuated after water levels in the Gaza Valley (Wadi Gaza) rose to almost three meters. [..] Unrelated (UNR) Apple is continuing to experience 'Hairgate' problems but they may just be a publicity stunt [..] Table 1: Headline and text snippets from document bodies with respective stances from the FNC dataset better methods by understanding their merits and drawbacks. Based on our analysis of the shared task data, we first propose a new evaluation metric for FNC-1 and related document-level stance detection tasks, which is less affected by highly imbalanced datasets. To understand the headroom of the stateof-the-art performance, we additionally estimate the upper bound for this task. In a feature ablation study, we then identify which features contribute to solving the stance detection task. On the basis of our analysis, we combine ideas from previous systems and propose a novel architecture that performs on par with the state-of-the-art systems, but is better able to correctly classify difficult cases. Since generalizability is crucial for the method's future impact, we finally introduce a new evaluation dataset and evaluate how well the FNC-1 models generalize to unseen data from a different domain. In addition to in-domain experiments, we also conduct cross-domain experiments in order to analyze the transfer potential of a method. Related Work Previous works in stance detection mostly considered target-specific stance prediction, whereby the stance of a text entity with respect to a topic or a named entity is determined. Target-specific stance prediction has been performed for tweets (Mohammad et al., 2016;Augenstein et al., 2016;Zarrella and Marsh, 2016) and online debates (Walker et al., 2012;Somasundaran and Wiebe, 2010;Sridhar et al., 2015). Such target-specific approaches are based on structural (Walker et al., 2012), linguistic and lexical features (Somasundaran and Wiebe, 2010) and they jointly model disagreement only and collective stance using probabilistic soft logic (Sridhar et al., 2015) or neural models (Zarrella and Marsh, 2016;Du et al., 2017) with conditional encoding (Augenstein et al., 2016). Stance prediction in tweets (Mohammad et al., 2016;Augenstein et al., 2016;Du et al., 2017) and in online debates (Hasan and Ng, 2013) is different from that of stance detection in a news article, which -while similar -is concerned with stance detection of a news article relative to a statement in natural language. To the best of our knowledge, there is yet no overview or analysis paper on FNC-1 similar to the shared task on detecting stance in twitter (Mohammad et al., 2016;Derczynski et al., 2017;Taulé et al., 2017). To demonstrate the best scientific practices and achieve research transparency, we close this gap by systematically reviewing the top-ranked systems at FNC-1. The FNC-1 stance detection task is inspired by Ferreira and Vlachos (2016), who classify the stance of a single sentence of a news headline towards a specific claim. In FNC-1, however, the task is documentlevel stance detection, which requires the classification of an entire news article relative to a headline. The top performing system in FNC-1 is called SOLAT in the SWEN (Sean et al., 2017) by Talos Intelligence (henceforth: Talos). They use a combination of deep convolutional neural networks and gradient-boosted decision trees with lexical features. Team Athene (Hanselowski et al., 2017) (Riedel et al., 2017) were placed third using a multi-layer perceptron with bag-of-words features. Additionally, recently published work on FNC-1 use a two-step logistic regression based classifier (Bourgonje et al., 2017) and a stacked ensemble of five classifiers (Thorne et al., 2017) which achieve 9th and 11th places respectively. Although multiple systems 3 participated at FNC-1, we focus on the top three systems in this paper, due to the availability of source code and our goal of analyzing what contributes most to good performance. In the remaining paper, we introduce and analyze these three systems in detail. 3 Reproduction of the Fake News Challenge FNC-1 In this section, we take a closer look at the challenge. We briefly discuss the task and dataset of FNC-1, describe the three top-ranked systems and reproduce their results. FNC-1 task and dataset. The task in FNC-1 is learning a classifier f : (d, h) → s that predicts one of four stance labels s ∈ S = {AGR, DSG, DSC, UNR} for a document d with regard to a headline h. If headline and document cover different topics, the stance is s = UNR (unrelated). Otherwise, s is AGR if d agrees and DSG if d disagrees with h. If h and d merely discuss the same topic, but d does not take a definite position, s will be DSC. To evaluate the challenge, the organizers provide a dataset 4 of 300 topics. The topics are represented by claims with 5-20 news article documents each. The dataset is derived from the Emergent project (Silverman, 2017) which addressed rumor debunking. In the project, each news article document was summarized into a headline that reflects the stance of the whole document. Other than for rumor debunking, the FNC-1 organizers match each document with every summarized headline and then label the (d, h) pair with one of the four stance labels S. To generate the unrelated class UNR, headlines and documents belonging to different topics are randomly matched. Document-headline pairs of 200 topics are reserved for training, the remaining document-headline pairs of 100 topics for testing. Topics, headlines, and documents are therefore not shared between the two data splits. To prevent teams from using any unfair means by deriving the labels for the test set from the publicly available Emergent data, the organizers additionally created 266 instances. Table 2 shows the corpus size and label distribution. Participating systems. For our reproduction study, we consider FNC-1's three top-ranked systems. Talos Intelligence's SOLAT in the SWEN team (Sean et al., 2017) won the FNC-1 using their weighted average model (TalosComb) of a deep convolutional neural network (TalosCNN) and a gradient-boosted decision trees model (TalosTree). TalosCNN uses pre-trained word2vec embeddings 5 passed through several convolutional layers followed by three fully-connected and a final softmax layer for classification. TalosTree is based on word count, TF-IDF, sentiment, and singular-value decomposition features in combination with the word2vec embeddings. Team Athene (Hanselowski et al., 2017) was ranked second. They propose a multi-layer perceptron (MLP) inspired by the work of Davis and Proctor (2017). They extend the original model structure to six hidden and a softmax layer and they incorporate multiple hand-engineered features: unigrams, the cosine similarity of word embeddings of nouns and verbs between headline and document tokens, and topic models based on non-negative matrix factorization, latent Dirichlet allocation, and latent semantic indexing in addition to the baseline features provided by the FNC-1 organizers. Depending on the feature type, they either form separate feature vectors for document and headline, or a joint feature vector. Table 3: FNC, F 1 m, and class-wise F 1 scores for the analyzed models on in-domain experiments The UCL Machine Reading (UCLMR) team propose an MLP as well, but use only a single hidden layer (Riedel et al., 2017). Their system was ranked third. As features, they use term frequency vectors of unigrams of the 5,000 most frequent words for the headlines and the documents. Additionally, they compute the cosine similarity between the TF-IDF vectors of the headline and document. The resulting term frequency feature vectors of headline and document are concatenated along with the cosine similarity of the two TF-IDF vectors. Reproduction. Following the instructions from the GitHub repositories of the three teams, 6 we could successfully reproduce the results reported in the competition without significant deviations. Table 3 shows these results in the FNC column of the FNC-FNC setup, which means that the models were trained and tested on the FNC dataset. Since Talos use a combination of two models, we have also included the results of TalosCNN and TalosTree. A first interesting finding is that TalosTree even outperforms the combined model, since the CNN component performed poorly. To understand the merits and drawbacks of the systems, we analyze the performance metrics and the features used, as discussed in the following sections. Performance evaluation In this section, we critically assess the FNC-1 evaluation methodology and we determine a human upper bound for this task in order to identify the room for improvement for the document-level stance detection task. Evaluation metrics. The FNC-1 organizers propose the hierarchical evaluation metric FNC, which first awards .25 points if a document is correctly classified as related (i.e., s ∈ {AGR, DSG, DSC}) or UNR to a given headline. If it is related, .75 additional points are assigned if the model correctly classifies the document-headline pair as AGR, DSG, or DSC. The goal of this weighting schema is to balance out the large number of unrelated instances. Nevertheless, the metric fails to take into account the highly imbalanced class distribution of the three related classes AGR, DSG, and DSC illustrated in Table 2. Thus, models, which perform well on the majority class and poorly on the minority classes are favored. Since it is not difficult to separate related from unrelated instances (the best systems reach about F 1 = .99 for the UNR class), a classifier that just randomly predicts one of the three related classes would already achieve a high FNC score. A classifier that always predicts DSC for the related documents even reaches FNC = .833, which is even higher than the top-ranked system. We therefore argue that the FNC metric is not appropriate for validating the document-level stance detection task. Instead, we propose the class-wise and the macro-averaged F 1 scores (F 1 m) as a new metric for this task that is not affected by the large size of the majority class. The class-wise F 1 scores are the harmonic means of the precisions and recalls of the four classes, which are then averaged to the F 1 m metric. The naïve approach of perfectly classifying UNR and always predicting DSC for the related classes, would achieve only F 1 m = .444, which is clearly different from the proposed systems. By averaging over the individual classes' scores, F 1 m is also applicable to other datasets, which have a different class distribution than the FNC-1 dataset. While the averaged F 1 m objectively reflects the quality of the prediction rather than the class distribution, we can also analyze which classes cannot be properly predicted yet. As the scores in Table 3 indicate, the performance of the three top-ranked systems reach only about F 1 m = .6. Our analysis reveals that TalosCNN does not predict the DSC class yielding an F 1 score of zero. Also the overall performance of this model is low and according to the FNC metric, TalosTree would even outperform TalosComb. In contrast, TalosTree returns almost no predictions for the DSG class, although it performs exceptionally well in terms of FNC. This is because it often predicts the majority class DSC for the related documents. Since there are only few DSG instances in the dataset, the overall performance of this model appears high. Considering the FNC-1 results according to our proposed F 1 m metric, the ranking of the three systems changes: The TalosComb and TalosTree systems are slightly outperformed by UCLMR and clearly outperformed by the Athene system. This is because the two Talos models benefit from the FNC metric definition, favoring the prediction of the majority classes UNR and DSC. On smaller classes, such as DSG, they perform much worse than Athene and UCLMR. Using F 1 m as a metric, the Athene system would be ranked first, as it outperforms UCLMR by 2.1 percentage points. In addition to that, Athene also works best on the DSG, DSC, and UNR class. Human upper bound. In addition to the issues with the evaluation metric, there is also no upper bound reported for the FNC-1 data, although this will help estimating the headroom of the proposed systems with regard to human performance. Therefore, we ask five human raters to manually label 200 instances. The raters reach an overall inter-annotator agreement of Fleiss' κ = .686 (Fleiss, 1971), which is substantial and allows drawing tentative conclusions (Artstein and Poesio, 2008). However, when ignoring the UNR class, the inter-annotator agreement dramatically drops to κ = .218. This indicates that differentiating between the three related classes AGR, DSG, and DSC is difficult even for humans. On the basis of the annotation, we also determine the most probable stance labels according to MACE (Hovy et al., 2013), and compare them to the ground truth from the Emergent project. The agreement of the labels in this case is better, reaching a Fleiss' κ of .807 overall and .552 for the three related classes. The MACE-based most probable label allows us to compute the human upper bound as F 1 m = .754, which we include in Table 9 along with the upper bound per class F1 scores UNR = .997 AGR = .588, DSG =.667, and DSC = .765. Analysis of models and features In this section, we first perform an error analysis in order to be able to find out what the three best performing models are learning and in which cases they fail. In order to address the identified drawbacks, we conduct a systematic feature analysis and derive an alternative model based on our findings. Error analysis Our error analysis for the three analyzed systems shows that the models fail in the following cases: (1) If there is lexical overlap between the headline and the document, the models classify the instance as one of the related classes, even in cases in which the two are unrelated. (2) If the document-headline pair is related, but only contains synonyms rather than the same tokens, the model often misclassifies the case as UNR. (3) If keywords like reports, said, or allegedly are detected, the systems classify the pair as DSC. (4) The DSG class is especially difficult to determine, as only few lexical indicators (e.g., false, hoax, fake) are available as features. The disagreement is often expressed in complex terms which demands more sophisticated machine learning techniques. For example: "If the bizarre story about. . . sounded outlandish, that's because it was". In appendix A.2, we illustrate these errors with concrete examples. The analysis shows that the models exploit the similarity between the headline and the document in terms of lexical overlap. Lexical cue words, such as reports, said, false, hoax play an important role in classification. However, the systems fail when semantic relations between words need to be taken into account, complex negation instances are encountered, or the understanding of propositional content in general is required. This is not surprising since the three models are based on n-grams, bag-of-words, topic models and lexicon-based features instead of capturing the semantics of the text. In this section, we test these features systematically and we propose new features and a new architecture for FNC-1. Feature analysis Throughout our feature analysis, we use the Athene model, which performed best in terms of F 1 m and allows a large number of experiments due to its fast computation. All tests are performed on the FNC-1 development set with 10-fold cross-validation. In the remaining section, we first discuss and evaluate the performance of each feature individually and then conduct an ablation test for groups of similar features. Detailed feature descriptions are included in the supplementary material (section A.1). Figure 1 shows the system performance of the individual features discussed below. FNC-1 baseline features. The FNC-1 organizers provide a gradient-boosting baseline using the cooccurrence (COOC) of word and character n-grams in the headline and the document as well as two lexicon-based features, which count the number of refuting (REFU) and polarity (POLA) words based on small word lists. Figure 1 indicates that COOC performs well, whereas both lexicon-based features are on par with the majority vote baseline. Challenge features. The three analyzed FNC-1 systems rely on combinations of the following features: Bag-of-words (BoW) unigram features, topic model features based on non-negative matrix factorization (NMF-300, NMF-cos) (Lin, 2007), Latent Dirichlet Allocation (LDA-cos) (Blei et al., 2001), Latent Semantic Indexing (LSI-300) , two lexicon-based features using NRC Hashtag Sentiment (NRC-Lex) and Sentiment140 (Sent140) (Mohammad et al., 2013), and word similarity features which measure the cosine similarity of pre-trained word2vec embeddings of nouns and verbs in the headlines and the documents (WSim). The topic models use 300 topics. Besides the concatenated topic vectors, we also consider the cosine similarity between the topics of document and headline (NMF-cos, LDA-cos). The BoW features perform best in terms of F 1 m. While LSI-300, NMF-300 and NMF-cos topic models yield high scores, LDA-cos and WSim fall behind. Novel features. We also analyze a number of novel features for the FNC-1 task which have not been used in the challenge. Bag-of-character 3-grams (BoC) represent subword information. They show promising results in our setup. The structural features (STRUC) include the average word lengths of the headline and the document, the number of paragraphs in the document and their average lengths. The low performance of this feature indicates that the structure of the headline and the documents is not indicative of their stance. Furthermore, we test readability features (READ) which estimate the complexity of a text. Less complex texts could be indicative of deficiently written fake news. We tried the following metrics for headline and document as a concatenated feature vector: SMOG grade (Mc Laughlin, 1969), Flesch-Kincaid grade level and Flesch reading ease (Kincaid et al., 1975), Gunning fog index (Štajner et al., 2012), Coleman-Liau index (Mari and Ta Lin, 1975), automated readability index (Senter and Smith, 1967), LIX and RIX (Jonathan, 1983), McAlpine EFLAW Readability Score (McAlpine, 1997), and Strain Index (Solomon, 2006). However, in the present problem setting these features show only a low performance. The same is true for the lexical diversity (LexDiv) metrics, type-token ratio, and the measure of textual diversity (MTLD) (McCarthy, 2005). We finally analyze the performance of features based on the following lexicons: MPQA (Wilson et al., 2005), MaxDiff , and EmoLex (Mohammad and Turney, 2010). These features are based on the sentiment, polarity, and emotion expressed by headlines and documents, which might be good indicators of an author's opinion. However, our results show that these lexicon-based features are not helpful. Even though the considered lexicons are important for fake-news detection (Shu et al. (2017), Horne and Adali (2017)), for stance detection, the properties captured by the lexicon-based features are not very useful. Feature ablation test. We first remove all features that are more than 10% below the FNC-1 baseline, since they mostly predict the majority class and thus harm the F 1 m score. In Figure 1, we mark these features with an asterisk (*). To quantify their contribution, we perform an ablation test across three groups of related features: (1) BoW and BoC (BoW/C), (2) LSI-topic, NMF-topic, NMF-cos, LDA-cos (Topic), and (3) NRC-POS and WSim (Oth). Table 4 show the results of our ablation test. The BoW and BoC features have the biggest impact on the performance. While the topic models yield further improvements, the NRC-POS and WSim features are not helpful. Hence, we suggest BoW, BoC, and the four topic model based features as the most promising feature set. We evaluate this feature set on the FNC-1 test dataset. The results are included in the featMLP row of Table 3 for the FNC-FNC setting. Although featMLP with the revised feature selection outperforms the best performing FNC-1 system Athene in terms of F 1 m and FNC score, the margin is not significant. Similar to the three FNC-1 systems, we observe a .2 performance drop between the development and test dataset. This is most likely because of the 100 new topics in the test dataset, which have not been seen during training. Thus, the evaluation on the test set can be considered as an out-of-domain prediction. Model analysis In order to increase the overall performance, we conduct additional experiments with an ensemble of the three models featMLP, TalosComb, and UCLMR using hard voting. However, we could not significantly improve the results. Since all models struggle with the DSG class, we have applied different under-and over-sampling techniques to balance the class distribution, but also this technique did not yield improved results. In the error analysis, we observed that the feature-based systems lack semantic understanding. Therefore, we combine a feature-based system with a model that is better able to capture the semantics based Figure 1). Headline: NHL expansion ahead? No, says Gary Bettman Article body: It wasn't very long ago that NHL commissioner Gary Bettman was treating talk of expansion as though he was being asked if he'd like an epidemic of Ebola. But recently the nature of the rhetoric has changed so much that the question is becoming not if, but when. ... Table 5: A correctly classified DSG instance by the stackLSTM on word embeddings and sequential encoding. Sequential processing of information is important in order to get the meaning of the whole sentence, e.g. "It wasn't long ago that Gary Bettman was ready to expand NHL." VS. "It was long ago that Gary Bettman wasn't ready to expand NHL." In Figure 2, we introduce this stackLSTM model, which combines the best feature set found in the ablation test with a stacked long short-term memory (LSTM) network (Hermans and Schrauwen, 2013). We use 50-dimensional GloVe word embeddings 7 (Pennington et al., 2014) in order to generate sequences of word vectors of a headline-document pair. For this, we concatenate a maximum of 100 tokens of the headline and the document. These embedded word sequences v 1 , v 2 , . . . , v n are fed through two stacked LSTMs with a hidden state size of 100 with a dropout of 0.2 each. The last hidden state of the second LSTM is concatenated with the feature set and fed into a 3-layer neural network with 600 neurons each. Finally, we add a dense layer with four neurons and softmax activation function in order to retrieve the class probabilities. Table 3 shows the performance of stackLSTM for the FNC-FNC setup. Our model outperforms all other methods in terms of F 1 m. The difference to Athene and featMLP is, however, not significant. An important advantage of stackLSTM is its improved performance for the DSG class, which is the most difficult one to predict due to the low number of instances. This means that stackLSTM correctly classifies a larger number of complex negation instances. The difference on this difficult DSG class between stackLSTM and all other methods is statistically significant (using Student's t-test). The model predicts more often for the DSG class and gets more of these examples correct without compromising the overall performance. One challenging DSG example, which was correctly classified by the stackLSTM, is given in Table 5. Analysis of the generalizability of the models To test the robustness of the models (i.e. how well they generalize to new datasets), we introduce novel test data for document-level stance detection based on the Argument Reasoning Comprehension (ARC) task proposed by Habernal et al. (2018). In this section, we describe the dataset, analyze the models' performance, and perform cross-domain experiments. ARC dataset. Habernal et al. (2018) manually select 188 debate topics with popular questions from the user debate section of the New York Times. For each topic, they collect user posts, which are highly ranked by other users, and create two claims representing two opposing views on the topic. Then, they Table 7: Example of the original ARC dataset and the generated instance to align with FNC dataset ask crowd workers to decide whether a user post supports either of the two opposing claims or does not express a stance at all. This Argument Reasoning Comprehension (ARC) dataset consists of typical controversial topics from the news domain, such as immigration, schooling issues, or international affairs. While this is similar to the FNC-1 dataset, there are significant differences, as a user post is typically a multi-sentence statement representing one viewpoint on the topic. In contrast, the news articles of FNC-1 are longer and usually provide more balanced and detailed perspective on an issue. To allow using the ARC data for our FNC stance detection setup, we consider each user post as a document and randomly select one of the two claims as the headline. We label the claim-document pair as AGR if the claim has also been chosen by the workers, as DSG if the workers chose the opposite claim, and as DSC if the workers selected neither claim. Table 7 shows an example of our revised ARC corpus structure. In order to generate the unrelated instances, we randomly match the user posts with claims, but avoid that a user post is assigned to a claim from the same topic. Table 6 provides basic corpus statistics. For training and testing, we split the corpus into 80% training/validation set and 20% testing set. As for the FNC-1 corpus, we have also determined a human upper bound for the ARC dataset. Five subjects annotate 200 samples using the four classes. Even though the overall Fleiss' κ = .614 is slightly lower compared to the FNC-1 corpus, the agreement for the three related classes AGR, DSG, and DSC is higher (κ = .383) than for FNC-1. The human upper bound based on MACE is F 1 m = .773. Table 9 contains also the class-wise F 1 scores. In-domain experiments ARC-ARC: The in-domain results for the ARC corpus listed in Table 8 show that the overall performance of all models decreases. Since the models have been constructed to perform well on the FNC-1 dataset, this is not surprising. Nevertheless, for the ARC corpus, the models are better able to distinguish between AGR and DSG instances. We assume this is because the number of DSG instances is substantially larger and is similar to the number AGR instances. The classification of the DSC instances, on the other hand, turns out to be more challenging on the ARC corpus. This is because even if a user post is related to the claim, it often does not explicitly refer to it. With TalosComb being best, the Talos models were better able to generalize to the new data. Even though the stackLSTM is again better on the more difficult minority class (in this case DSC), the structure and features of TalosComb seem to be more appropriate for this problem setting. Cross-domain experiments: In the cross-domain setting we train on the training data of one corpus and evaluate on the test data of the other corpus. The experiments in Table 9 show that the performance of the models is substantially better than the majority vote baseline. We therefore conclude that the two problem settings are related and exhibit a common structure. The results suggest that TalosComb is best able to learn from the ARC corpus, as it is also superior in the ARC-FNC setting. The stackLSTM, on the Discussion and conclusion In this paper, we conducted a thorough analysis of the Fake News Challenge stage one stance detection task. Although this is common for shared tasks, there is yet no analysis or reproduction study of this task, which is why we close this gap. Given that the challenge has attracted much attention in the NLP community with 50 participating teams, a detailed analysis is valuable as it provides insights into the problem setting and lessons learned for upcoming competitions. In our investigation, we evaluated the performance of the three top-scoring systems, critically assessed the experimental setup, and performed a detailed feature analysis, in which we identify high-performing features for the task yielding a new model. We conducted an error analysis and found that the models mostly rely on the lexical overlap for classification. To assess how well the models generalize to a similar problem setting, we experimented with a second, newly derived corpus. We also propose a new evaluation metric based on F 1 scores, since the challenge's metric is highly affected by the imbalanced class distribution of the test data. Using this evaluation setup, the ranking of the top three systems changes. Based on our analysis, we conclude that the investigated stance detection problem is challenging, since the best performing features are not yet able to resolve the difficult cases. Thus, more sophisticated machine learning techniques are needed, which have a deeper semantic understanding, and are able to determine the stance on the basis of propositional content instead of relying on lexical features. Lexical features As lexical features we implement the type-token-ratio (TTR) and the measure of textual lexical diversity (MTLD) (McCarthy, 2005) for the document, and only type-token-ratio for the headline, since MTLD needs at least 50 tokens to be valid. Also, the baseline feature word overlap belongs to this group. It divides the cardinality of the intersection of unique words in headline and document by the cardinality of the union of unique words in headline and document. POS features The POS features amongst others include counters for nouns, personal pronouns, verbs and verbs in past tense, adverbs, nouns and proper nouns, cardinal numbers, punctuations, the ratio of quoted words, and also the frequency of the three least common words in the text. The headline feature also contains a value for the percentage of stop words and the number of verb phrases, which showed good results in the work of Horne and Adali (2017). For the word-similarity feature, [which are mainly based on Ferreira and Vlachos (2016) we calculated average word embeddings (pre-trained word2vec model 8 ) for all verbs (retrieved with Stanford Core NLP toolkit 9 ) of headline/document separately. The cosine similarity between the averaged embeddings of headline and document is taken as a feature, as well as the hungarian distance between verbs of headline and document based on the paraphrase database 10 . The same computation is done for all nouns of headline and document. Additionally the average sentiment of the headline and the average sentiment of the document is used as a feature. A count of negating words of the headline and the document is added to the feature vector as well as the distance from the negated word to the root of the sentence. The number of average words per sentence of headline and document is another feature. The aforementioned features are improved by only selecting a predefined number of sentences of document and headline. Therefore the sentences are ordered by TF-IDF score. Structural features The structural features contain the average word length of the headline and document, and the number of paragraphs and average paragraph length of the document. A.2 Misclassified examples identified in the error analysis Example 1. (ground truth: "unrelated", system predicts: "agree") Headline: CNN: Doctor Took Mid-Surgery Selfie with Unconscious Joan Rivers Document: "A TEENAGER woke up during brain surgery to ask doctors how it was going. Iga Jasica, 19, was having an op to remove a tumour at when the anaesthetic wore off and she struck up a conversation with the medics still working on her." Example 2. (ground truth: "agree", system predicts: "unrelated") Headline: Three Boobs Are Most Likely Two Boobs and a Lie Document: The woman who claimed she had a third breast has been proved a hoax. Example 3. (ground truth: "disagree", system predicts: "discuss") Headline: Woman pays 20,000 for third breast to make herself LESS attractive to men Document: The woman who reported that she added a third breast was most likely lying. Example 4. (ground truth: "disagree", system predicts: "agree") Figure 1 : 1Performance of the system based on individual features Figure 2 : 2Model Architecture of the feature-rich stackLSTM won the second place withDataset headlines documents tokens instances AGR DSG DSC UNR FNC-1 2,587 2,587 372 75,385 7.4% 2.0% 17.7% 72.8% Table 2 : 2Corpus statistics and label distribution for the FNC-1 datasetan ensemble of five multi-layer perceptrons (MLP) with six hidden layers each and handcrafted features. For the prediction they used hard voting. Finally, UCL Machine Reading (UCLMR) Table 4 : 4Results of the feature ablation test. Baseline FNC-1 uses gradient boosting classifier with all FNC-1 baseline features. * states that only the preselected features are used (see Table 6 : 6Corpus statistics and label distribution for the ARC datasetExample from the original ARC dataset Topic Do same-sex colleges play an important role in education, or are they outdated? User post Only 40 women's colleges are left in the U.S. And, while there are a variety of opinions on their value, to the women who have attended ... them, they have been ... tremendously valuable. ... Claims 1. Same-sex colleges are outdated 2. Same-sex colleges are still relevant Label Same-sex colleges are still relevant Generated instance in alignment with the FNC problem setting Stance Headline Document AGR Same-sex colleges are still relevant Only 40 women's colleges are left in the U.S. ... Table 8 : 8SystemsARC-ARC FNC F 1 m AGR DSG DSC UNR FNC, F 1 m, and class-wise F 1 scores for the analyzed models on in-domain experiments FNC F 1 m AGR DSG DSC UNR FNC F 1 m AGR DSG DSC UNRMajority vote .430 .214 0.0 0.0 0.0 .857 TalosComb .725 .573 .593 .598 .160 .944 Athene .680 .548 .516 .482 .190 .933 UCLMR .667 .519 .517 .503 .121 .932 featMLP .690 .526 .526 .506 .144 .934 stackLSTM .685 .524 .451 .518 .194 .935 Upper bound .796 .773 .710 .857 .571 .954 Systems FNC-ARC ARC-FNC Majority vote .430 .214 0.0 0.0 0.0 .857 .394 .210 0.0 0.0 0.0 .839 TalosComb .584 .365 .336 0.0 .195 .929 .607 .388 .279 .183 .113 .977 Athene .523 .340 .340 .244 .138 .894 .548 .321 .277 .097 .028 .882 UCLMR .557 .358 .271 .064 .201 .896 .482 .288 .234 .109 .080 .728 featMLP .586 .389 .321 .159 .171 .906 .585 .351 .322 .111 .033 .939 stackLSTM .591 .401 .321 .191 .182 .910 .613 .373 .343 .116 .082 .950 Upper bound .796 .773 .710 .857 .571 .954 .859 .754 .588 .667 .765 .997 Table 9 : 9FNC, F 1 m and class-wise F 1 scores(F 1 m) based on cross-domain experiments other hand, yields best results when trained on the FNC corpus as the FNC-ARC setting suggests. e.g., http://web.stanford.edu/class/cs224n/reports.html 4 https://github.com/FakeNewsChallenge/fnc-1-baseline 5 https://code.google.com/archive/p/word2vec/ https://github.com/Cisco-Talos/fnc-1; https://github.com/hanselowski/athene_system; https://github.com/uclmr/fakenewschallenge http://nlp.stanford.edu/data/glove.twitter.27B.zip https://code.google.com/archive/p/word2vec/ 9 https://stanfordnlp.github.io/CoreNLP/ 10 http://www.cis.upenn.edu/ ccb/ppdb/ AcknowledgementsThis work has been supported by the German Research Foundation as part of the Research Training Group "Adaptive Preparation of Information from Heterogeneous Sources" (AIPHES) at the Technische Universität Darmstadt under grant No. GRK 1994/1. 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Glove: Global vectors for word representation. Jeffrey Pennington, Richard Socher, Christopher D Manning, Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)Doha, QatarJeffrey Pennington, Richard Socher, and Christopher D. Manning. 2014. Glove: Global vectors for word representation. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP). Doha, Qatar, pages 1532-1543. The Fake News Challenge: Exploring how artificial intelligence technologies could be leveraged to combat fake news. Dean Pomerleau, Delip Rao, Dean Pomerleau and Delip Rao. 2017. The Fake News Challenge: Exploring how artificial intelligence technologies could be leveraged to combat fake news. http://www.fakenewschallenge.org/. Accessed: 2017-10-20. A simple but tough-to-beat baseline for the fake news challenge stance detection task. 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Baird Sean, Sibley Doug, Pan Yuxi, Talos Targets Disinformation with Fake News Challenge Victory. Baird Sean, Sibley Doug, and Pan Yuxi. 2017. Talos Targets Disinformation with Fake News Challenge Victory. http://blog.talosintelligence.com/2017/06/talos-fake-news-challenge. html. Accessed: 2017-12-02. Automated readability index. R J Senter, Edgar A Smith, AMRL-TR-66- 220University of CincinnatiTechnical ReportR.J. Senter and Edgar A. Smith. 1967. Automated readability index. Technical Report AMRL-TR-66- 220, University of Cincinnati. Fake news detection on social media: A data mining perspective. Kai Shu, Amy Sliva, Suhang Wang, Jiliang Tang, Huan Liu, ACM SIGKDD Explorations Newsletter. 191Kai Shu, Amy Sliva, Suhang Wang, Jiliang Tang, and Huan Liu. 2017. Fake news detection on social media: A data mining perspective. ACM SIGKDD Explorations Newsletter 19(1):22-36. Emergent: A real-time rumor tracker. Craig Silverman, Craig Silverman. 2017. Emergent: A real-time rumor tracker. 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Dhanya Sridhar, James R Foulds, Bert Huang, Lise Getoor, Marilyn A Walker, Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing of the Asian Federation of Natural Language Processing. the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing of the Asian Federation of Natural Language ProcessingBeijing, ChinaDhanya Sridhar, James R. Foulds, Bert Huang, Lise Getoor, and Marilyn A. Walker. 2015. Joint models of disagreement and stance in online debate. In Proceedings of the 53rd Annual Meeting of the Associ- ation for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing of the Asian Federation of Natural Language Processing (ACL/IJCNLP). Beijing, China, pages 116-125. Parsing argumentation structures in persuasive essays. Christian Stab, Iryna Gurevych, Computational Linguistics. 433Christian Stab and Iryna Gurevych. 2017. Parsing argumentation structures in persuasive essays. Com- putational Linguistics 43(3):619-659. What can readability measures really tell us about text complexity. Sanja Štajner, Richard Evans, Constantin Orȃsan, Ruslan Mitkov, Proceedings of the LREC 2012 Workshop on Natural Language Processing for Improving Textual Accessibility (NLP4ITA). the LREC 2012 Workshop on Natural Language Processing for Improving Textual Accessibility (NLP4ITA)Istanbul, TurkeySanja Štajner, Richard Evans, Constantin Orȃsan, and Ruslan Mitkov. 2012. What can readability mea- sures really tell us about text complexity. In Proceedings of the LREC 2012 Workshop on Natural Language Processing for Improving Textual Accessibility (NLP4ITA). Istanbul, Turkey, pages 14-21. Overview of the Task on Stance and Gender Detection in Tweets on Catalan Independence. Mariona Taulé, Maria Antònia Martí, Francisco M Rangel Pardo, Paolo Rosso, Cristina Bosco, Viviana Patti, Proceedings of the Second Workshop on Evaluation of Human Language Technologies for Iberian Languages (IberEval) co-located with 33th Conference of the Spanish Society for Natural Language Processing (SEPLN). the Second Workshop on Evaluation of Human Language Technologies for Iberian Languages (IberEval) co-located with 33th Conference of the Spanish Society for Natural Language Processing (SEPLN)Murcia, SpainMariona Taulé, Maria Antònia Martí, Francisco M. Rangel Pardo, Paolo Rosso, Cristina Bosco, and Viviana Patti. 2017. Overview of the Task on Stance and Gender Detection in Tweets on Catalan Inde- pendence. In Proceedings of the Second Workshop on Evaluation of Human Language Technologies for Iberian Languages (IberEval) co-located with 33th Conference of the Spanish Society for Natural Language Processing (SEPLN). Murcia, Spain, pages 157-177. Fake news stance detection using stacked ensemble of classifiers. James Thorne, Mingjie Chen, Giorgos Myrianthous, Jiashu Pu, Xiaoxuan Wang, Andreas Vlachos, Proceedings of the EMNLP 2017 Workshop 'Natural Language Processing meets Journalism. the EMNLP 2017 Workshop 'Natural Language Processing meets JournalismCopenhagen, DenmarkJames Thorne, Mingjie Chen, Giorgos Myrianthous, Jiashu Pu, Xiaoxuan Wang, and Andreas Vlachos. 2017. Fake news stance detection using stacked ensemble of classifiers. In Proceedings of the EMNLP 2017 Workshop 'Natural Language Processing meets Journalism'. Copenhagen, Denmark, pages 80- 83. Stance classification using dialogic properties of persuasion. Marilyn A Walker, Pranav Anand, Robert Abbott, Ricky Grant, Proceedings of the 2012 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (NAACL/HLT). the 2012 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies (NAACL/HLT)Marilyn A. Walker, Pranav Anand, Robert Abbott, and Ricky Grant. 2012. Stance classification using dialogic properties of persuasion. In Proceedings of the 2012 Conference of the North American Chap- ter of the Association for Computational Linguistics: Human Language Technologies (NAACL/HLT). . Q C Montréal, Canada , Montréal, QC, Canada, pages 592-596. Recognizing contextual polarity in phraselevel sentiment analysis. Theresa Wilson, Janyce Wiebe, Paul Hoffmann, Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing (HLT/EMNLP). the Conference on Human Language Technology and Empirical Methods in Natural Language Processing (HLT/EMNLP)Vancouver, BC, CanadaTheresa Wilson, Janyce Wiebe, and Paul Hoffmann. 2005. Recognizing contextual polarity in phrase- level sentiment analysis. In Proceedings of the Conference on Human Language Technology and Empirical Methods in Natural Language Processing (HLT/EMNLP). Vancouver, BC, Canada, pages 347-354. MITRE at SemEval-2016 Task 6: Transfer Learning for Stance Detection. Guido Zarrella, Amy Marsh, Proceedings of the 10th International Workshop on Semantic Evaluation (SemEval-2016). the 10th International Workshop on Semantic Evaluation (SemEval-2016)San Diego, CA, USAGuido Zarrella and Amy Marsh. 2016. MITRE at SemEval-2016 Task 6: Transfer Learning for Stance Detection. In Proceedings of the 10th International Workshop on Semantic Evaluation (SemEval- 2016). San Diego, CA, USA, pages 458-463. NRC-Canada-2014: Recent Improvements in the Sentiment Analysis of Tweets. Xiaodan Zhu, Svetlana Kiritchenko, Saif M Mohammad, Proceedings of the 8th International Workshop on Semantic Evaluation (SemEval). the 8th International Workshop on Semantic Evaluation (SemEval)Dublin, IrelandXiaodan Zhu, Svetlana Kiritchenko, and Saif M. Mohammad. 2014. NRC-Canada-2014: Recent Im- provements in the Sentiment Analysis of Tweets. In Proceedings of the 8th International Workshop on Semantic Evaluation (SemEval). Dublin, Ireland, pages 443-447. . A Supplemental Material A.1 Features: Detailed description. A Supplemental Material A.1 Features: Detailed description no") until the next punctuation mark appears. For the bag-of-characters (BoC) 3-grams are chosen with 5,000 tokens vocabulary, too. For the BoW/BoC feature we use the TF to extract the vocabulary and to build the feature vectors of headline and document. The resulting TF vectors of headline and document get concatenated afterwards. Feature co-occurrence (FNC-1 baseline feature) counts how many times word 1-/2-/4-grams. BoW/BoC features We use bag-of-words (BoW) 1-and 2-grams with 5,000 tokens vocabulary for the headline as well as the document. For the BoW feature. Das and Chenwe add a negation tag "_NEG" as prefix to every word between special negation keywords (e.g. "not. and stop words of the headline appear in the first 100. first 255 characters of the document, and how often they appear in the document overallBoW/BoC features We use bag-of-words (BoW) 1-and 2-grams with 5,000 tokens vocabulary for the headline as well as the document. For the BoW feature, based on a technique by Das and Chen (2007), we add a negation tag "_NEG" as prefix to every word between special negation keywords (e.g. "not", "never", "no") until the next punctuation mark appears. For the bag-of-characters (BoC) 3-grams are chosen with 5,000 tokens vocabulary, too. For the BoW/BoC feature we use the TF to extract the vocabulary and to build the feature vectors of headline and document. The resulting TF vectors of headline and document get concatenated afterwards. Feature co-occurrence (FNC-1 baseline feature) counts how many times word 1-/2-/4-grams, character 2-/4-/8-/16-grams, and stop words of the headline appear in the first 100, first 255 characters of the document, and how often they appear in the document overall. and latent Dirichlet allocation (LDA) (Blei et al., 2001) to create topic models out of which we create independent features. For each topic model, we extract 300 topics out of the headline and document texts. Afterwards, we compute the similarity of headlines and bodies to the found topics separately and either concatenate the feature vectors. ( Deerwester, Topic models We use non-negative matrix factorization (NMF) (Lin, 2007), latent semantic indexing (LSI). NMF, LSI) or calculate the cosine distance between them as a single valued feature (NMF, LDATopic models We use non-negative matrix factorization (NMF) (Lin, 2007), latent semantic indexing (LSI) (Deerwester et al., 1990), and latent Dirichlet allocation (LDA) (Blei et al., 2001) to create topic models out of which we create independent features. For each topic model, we extract 300 topics out of the headline and document texts. Afterwards, we compute the similarity of headlines and bodies to the found topics separately and either concatenate the feature vectors (NMF, LSI) or calculate the cosine distance between them as a single valued feature (NMF, LDA). For all its words, it holds up to eight emotions (anger, fear, anticipation, trust, surprise, sadness, joy, disgust), based on the context they frequently appear in. For headline and document respectively, the emotions for all words are counted as a feature vector. The resulting vectors for headline and document are then concatenated. Lastly, the baseline features polarity words and refuting words are added. Kiritchenko, Lexicon-based features These features are based on the NRC Hashtag Sentiment and Sentiment140 lexicon. Mohammad et al.Mohammad and TurneyFirst, we count how many words with positive, negative, and without polarity are found in the text. Two features sum up the positive and negative polarity values of the words in the texts and another two features are set by finding the word with the maximum positive and negative polarity value in the text. Finally, the last word in the text with negative or positive polarity is taken as a feature. The first one counts refuting words (e.g. "fake", "hoax"), divides the sum by two, and takes the remainder as a feature signaling the polarity of headline or document. The latter one sets a binary feature for each refuting word (e.g. "fraud", "deny") appearing in the headline or documentLexicon-based features These features are based on the NRC Hashtag Sentiment and Sentiment140 lexicon (Kiritchenko et al., 2014; Mohammad et al., 2013; Zhu et al., 2014), as well as for the MPQA lexicon (Wilson et al., 2005) and MaxDiff Twitter lexicon (Rosenthal et al., 2015; Kiritchenko et al., 2014). All named lexicons hold values that signal the sentiment/polarity for each word. The features are computed separately for headline and document, and constructed as proposed by Mohammad et al. (2013): First, we count how many words with positive, negative, and without polarity are found in the text. Two features sum up the positive and negative polarity values of the words in the texts and another two features are set by finding the word with the maximum positive and negative polarity value in the text. Finally, the last word in the text with negative or positive polarity is taken as a feature. Since the MaxDiff Twitter lexicon also contains 2-grams, we decide to take them into account as well, whereas for the other lexicons only 1-grams incorporated. Additionally, we base features on the EmoLex lexicon (Mohammad and Turney, 2010, 2013). For all its words, it holds up to eight emotions (anger, fear, anticipation, trust, surprise, sadness, joy, disgust), based on the context they frequently appear in. For headline and document respectively, the emotions for all words are counted as a feature vector. The resulting vectors for headline and document are then concatenated. Lastly, the baseline features polarity words and refuting words are added. The first one counts refuting words (e.g. "fake", "hoax"), divides the sum by two, and takes the remainder as a feature signaling the polarity of headline or document. The latter one sets a binary feature for each refuting word (e.g. "fraud", "deny") appearing in the headline or document. Flesch-Kincaid grade level, Flesch reading ease, and Gunning fog index. Štajner, Readability features We measure the readability of headline and document with SMOG grade (only document). Coleman-Liau index (Mari and Ta LinMcAlpine; Strain Index (SolomonSenter and SmithThe SMOG grade is only valid if a text has at least 30 sentences. and thus is only implemented for the bodiesReadability features We measure the readability of headline and document with SMOG grade (only document), Flesch-Kincaid grade level, Flesch reading ease, and Gunning fog index (Štajner et al., 2012), Coleman-Liau index (Mari and Ta Lin, 1975), automated readability index (Senter and Smith, 1967), LIX and RIX (Jonathan, 1983), McAlpine EFLAW Readability Score (McAlpine, 1997), Strain Index (Solomon, 2006). The SMOG grade is only valid if a text has at least 30 sen- tences, and thus is only implemented for the bodies. Sick Selfie EXPOSED -Last Photo Of Comic Icon, When She Was Under Anesthesia Document: If the bizarre story about Joan Rivers' doctor pausing to take a "selfie" in the operating room minutes before the 81-year-old comedienne went into cardiac arrest on August 29 sounded outlandish. Headline: Disgusting! Joan Rivers Doc Gwen Korovin's. that's because it wasHeadline: Disgusting! Joan Rivers Doc Gwen Korovin's Sick Selfie EXPOSED -Last Photo Of Comic Icon, When She Was Under Anesthesia Document: If the bizarre story about Joan Rivers' doctor pausing to take a "selfie" in the operating room minutes before the 81-year-old comedienne went into cardiac arrest on August 29 sounded outlandish, that's because it was.

Structure learning of Bayesian networks has always been a challenging problem. Nowadays, massive-size networks with thousands or more of nodes but fewer samples frequently appear in many areas. We develop a divide-and-conquer framework, called partition-estimationfusion (PEF), for structure learning of such big networks. The proposed method first partitions nodes into clusters, then learns a subgraph on each cluster of nodes, and finally fuses all learned subgraphs into one Bayesian network. The PEF method is designed in a flexible way so that any structure learning method may be used in the second step to learn a subgraph structure as either a DAG or a CPDAG. In the clustering step, we adapt the hierarchical clustering method to automatically choose a proper number of clusters. In the fusion step, we propose a novel hybrid method that sequentially add edges between subgraphs. Extensive numerical experiments demonstrate the competitive performance of our PEF method, in terms of both speed and accuracy compared to existing methods. Our method can improve the accuracy of structure learning by 20% or more, while reducing running time up to two orders-of-magnitude.

1904.10900

5dbfbfc000a513ebb0ac3ee486238e406fff9126

Learning Big Gaussian Bayesian Networks: Partition, Estimation and Fusion Jiaying Gu Department of Statistics University of California Los Angeles Qing Zhou Department of Statistics University of California Los Angeles Learning Big Gaussian Bayesian Networks: Partition, Estimation and Fusion Bayesian networkdirected acyclic graphdivide-and-conquerstructure learning Structure learning of Bayesian networks has always been a challenging problem. Nowadays, massive-size networks with thousands or more of nodes but fewer samples frequently appear in many areas. We develop a divide-and-conquer framework, called partition-estimationfusion (PEF), for structure learning of such big networks. The proposed method first partitions nodes into clusters, then learns a subgraph on each cluster of nodes, and finally fuses all learned subgraphs into one Bayesian network. The PEF method is designed in a flexible way so that any structure learning method may be used in the second step to learn a subgraph structure as either a DAG or a CPDAG. In the clustering step, we adapt the hierarchical clustering method to automatically choose a proper number of clusters. In the fusion step, we propose a novel hybrid method that sequentially add edges between subgraphs. Extensive numerical experiments demonstrate the competitive performance of our PEF method, in terms of both speed and accuracy compared to existing methods. Our method can improve the accuracy of structure learning by 20% or more, while reducing running time up to two orders-of-magnitude. Introduction The structure of a Bayesian network for p random variables X 1 , . . . , X p is represented by a directed acyclic graph (DAG) G = (V, E). The node set V = {1, . . . , p} represents the set of random variables, and E = {(j, i) ∈ V × V : j → i} is the edge set, where j → i is a directed edge in G. Let Π G i = {j ∈ V : (j, i) ∈ E} denote the parent set of node i. The joint probability density function f of (X 1 , . . . , X p ) can be factorized according to the structure of G: f (x 1 , . . . , x p ) = p i=1 f (x i |π i ),(1) where f (x i |π i ) is the conditional probability density (CPD) of X i given Π G i = π i . Hereafter, we may use X i and the node i interchangeably. The problem of structure learning of Bayesian networks from data has been an active research area due to its wide applications in machine learning, statistical modeling, and causal inference (Spirtes et al. 1993;Pearl 2000). There are a few different approaches to this problem. The first one is the constraint-based approach, which determines the existence of edges by a sequence of conditional independence tests. The PC algorithm (Spirtes and Glymour 1991) and its further developments (Tsamardinos et al. 2003;Kalisch and Bühlmann 2007;Colombo and Maathuis 2014) are typical examples of constrained-based methods. The second category is so-called score-based learning, which searches for a graphical structure that optimizes a certain scoring function, such as early works in Heckerman et al. (1995); Geiger and Heckerman (1994); Chickering (2002b); Chickering and Meek (2002). Recently, fast algorithms have been developed to handle large and high-dimensional datasets (Fu and Zhou 2013;Xiang and Kim 2013;Aragam and Zhou 2015;Ramsey et al. 2017;Zheng et al. 2018;Yuan et al. 2019). In addition, there are also hybrid methods that combine the above two approaches. These methods first restrict the search space using a constraint-based method, and then learn the DAG structure by optimizing a score over the restricted search space (Tsamardinos et al. 2006;Gámez et al. 2011;Gasse et al. 2012). Despite these great efforts, structure learning of Bayesian networks remains challenging, especially for datasets with a large number of variables. The DAG space grows super-exponentially in the number of nodes p (Robinson 1977), and learning Bayesian networks has been shown to be an NP-hard problem in general (Chickering et al. 2004). Nowadays, it is common to generate and collect data from thousands of variables or more. As p increases, however, many of the aforementioned methods slow down dramatically and become much less accurate, making them incompetent for large datasets. This motivates our development of a divide-and-conquer method that can learn massive-size Bayesian networks efficiently and accurately. Our method consists of three steps, Partition, Estimation and Fusion (PEF for short): 1. P-step: Partition the p nodes into clusters based on a modified hierarchical clustering algorithm. 2. E-step: Apply an existing structure learning algorithm to estimate a subgraph on each cluster of nodes. 3. F-step: Develop a new hybrid method to merge the estimated subgraphs into a full DAG on all nodes. Note that the number of nodes in a cluster is usually much smaller than p. This greatly speeds up structure learning in the estimation step, as most algorithms scale at least as O(p k ) for some k ≥ 2, e.g. Kalisch and Bühlmann (2007). Moreover, this step can be parallelized in an obvious way, leading to further improvement in computational efficiency. The hybrid method in the fusion step first uses statistical tests to generate a candidate set of node pairs between estimated subgraphs, and then maximizes a modified BIC score by adding between-subgraph edges and updating within-subgraph edges. Since our conditional independence tests are performed based on the structure of subgraphs, the number of tests needed for our method is substantially smaller than a constraint-based method on a p-node problem. Our method is designed with maximum flexibility. The user can apply any structure learning algorithm in the second step as long as it outputs a PDAG (partially directed acyclic graphs), including DAGs and CPDAGs (completed PDAGs) as special cases. Our PEF method works very well on Bayesian networks with a block structure to some degree, having relatively weak connections between subgraphs. It is quite common for a large network to show such a block structure, due to the underlying heterogeneity among the nodes (Chin et al. 2015;Decelle et al. 2011;Abbe et al. 2016). From extensive numerical comparisons with existing methods, we find that the PEF method can significantly improve the accuracy of structure learning of Bayesian networks, while reducing computing time substantially, up to two orders-of-magnitude for big graphs. The remaining of this paper is organized as follows. Section 2 contains a necessary background review for our method. Section 3 describes the partition and the estimation steps of the PEF method, while Section 4 develops the fusion step in detail. Section 5 provides numerical results of our method on real networks in comparison to other DAG learning algorithms. Section 6 summarizes this work with a discussion of future directions. Some technical details are deferred to an Appendix. Review of Bayesian networks In this section, we briefly review some concepts about Bayesian networks that are most relevant to our method. The joint distribution P that factorizes according to the DAG structure of a Bayesian network as in (1) satisfies so-called Markov properties (Lauritzen 1996). Let X, Y ∈ V and Z ⊆ V \ {X, Y }. If Z d-separates X from Y in DAG G, then the random variables X and Y are conditionally independent given Z. Using D G (X; Y |Z) to denote d-separation in G and I P (X; Y |Z) for conditional independence in P , the above (global) Markov property says that D G (X; Y |Z) ⇒ I P (X; Y |Z). Faithfulness Note that the implication in a Markov property goes only in one direction. To estimate the structure of a DAG, we need to infer edges from conditional independence statements learned from data, which often requires the faithfulness assumption (Spirtes et al. 1993) to build up the equivalence between the two. Definition 1 (Faithfulness). Suppose G is a DAG equipped with a joint probability distribution P . Then G and P are faithful to each other if and only if I P (X; Y |Z) ⇔ D G (X; Y |Z) for any X, Y ∈ V and Z ⊆ V \ {X, Y }. If (G, P ) satisfies the faithfulness assumption, we can use conditional independence (CI) test to infer d-separation in G. Theorem 1 provides a useful criterion to determine the existence of an edge using CI tests. Theorem 1 (Spirtes et al. (1993)). Suppose (G, P ) satisfies the faithfulness assumption. Then there is no edge between a pair of nodes X, Y ∈ V if and only if there exists a subset Z ⊆ V \ {X, Y } such that I P (X; Y |Z). Consequently, faithfulness is commonly assumed in the development of many structure learning algorithms, especially constraint-based and hybrid methods, such as the PC algorithm and the MMHC algorithm (Spirtes et al. 1993;Tsamardinos et al. 2006). Markov equivalence Multiple DAGs may imply the same set of d-separations, and thus encode the same set of CI statements, if they are Markov equivalent: Definition 2 (Markov equivalence). Two DAGs G and G on the same set of nodes V are Markov equivalent if D G (X; Y |Z) ⇔ D G (X; Y |Z) for any X, Y ∈ V and Z ⊆ V \ {X, Y }. As shown by Verma and Pearl (1990), two DAGs are Markov equivalent if and only if they have the same skeletons and the same v-structures. A v-structure is a triplet {i, j, k} ⊆ V of the form i → k ← j, where i and j are not adjacent, and the node k is called an uncovered collider. DAGs that are Markov equivalent form an equivalence class in the space of DAGs. A Markov equivalence class can be uniquely represented by a CPDAG (Chickering 2002a). The CPDAG for an equivalence class is defined as the PDAG consisting of directed edges for all compelled edges and undirected edges for all reversible edges in the equivalence class. Since DAGs in the same equivalence class cannot be distinguished from observational data, some structure learning algorithms (Chickering and Meek 2002;Spirtes et al. 1993) output a CPDAG, instead of a particular DAG in the equivalence class. Thus, depending on which structure learning algorithm is used, the estimation step of our PEF method may output a DAG, a CPDAG, or in general a PDAG, from each cluster of nodes. The fusion step will merge these PDAGs into a full DAG as the final estimate. Gaussian Bayesian networks In this paper, we focus on Gaussian Bayesian networks for continuous data, in which the conditional distributions are specified by a linear structural equation model, X j = i∈Π G j β ij X i + ε j , j = 1, . . . , p,(2) where ε j ∼ N (0, σ 2 j ) and β ij = 0 if and only if i ∈ Π G j . Let B = (β ij ) p×p be an edge coefficient matrix, which can be regarded as a weighted adjacency matrix for the DAG G, with β ij being the weight for the edge i → j. Put Ω = diag(σ 2 1 , . . . , σ 2 p ) as a p × p diagonal matrix of error variances. Then the joint distribution of (X 1 , . . . , X p ) defined by (2) (B, Σ) = p j=1 − n 2 log(σ 2 j ) − 1 2σ 2 j x j − xB j 2 ,(3) which forms the basis for score-based learning, subject to certain regularization or constraint on model complexity, e.g. the total number of edges in the DAG. For Gaussian random variables, conditional independence is equivalent to zero partial correlation, which is completely determined by the covariance matrix Σ. Consequently, in constraint-based methods, CI tests are performed based on sample partial correlations; see Appendix A.1 for a brief description. Partition and estimation In this section, we describe the first two steps of our PEF method. Besides some modifications to meet our specific needs in learning large networks, these two steps follow quite standard methods in clustering and structure learning. We will devote the entire Section 4 to the fusion step. Partition As we have mentioned in the introduction, the first step (P-step) of our method is to partition nodes into clusters. Each node is associated with a data column x j ∈ R n for j = 1, . . . , p. Let C i , i = 1, . . . , k, be the k clusters generated by the P-step, and S i = |C i | the size of the ith cluster. Accordingly, the underlying DAG G is cut into k subgraphs. Let s w be the number of edges of G within a subgraph, and s b the number of edges between subgraphs. In other words, s b is the number of edges in the partition-cut with respect to the k clusters, which may be recovered later by the fusion step of our algorithm. In general, we want to control s b to a small value so that our recovery of the DAG structure will be more accurate. On the other hand, we wish that k is quite large and the cluster size is as uniform as possible across the k clusters, which will lead to maximum savings in computing time for parallel learning of subgraphs in the E-step. To meet these specific needs for our problem, we propose a modified hierarchical clustering with average linkage that automatically chooses the number of clusters k. Define the distance between two nodes i and j by d(i, j) = 1 − |r ij | ∈ [0, 1],(4) where r ij = cor(x i , x j ) is the correlation between x i and x j for i, j = 1, . . . , p. Hartigan (1981) suggests that one should only consider clusters with at least 5% of the data points, which will be referred to as "big clusters" hereafter. Following this suggestion, we require the minimum cluster size be 0.05p. As a result, there will be at most 20 clusters. Let k max ≤ 20 be the maximum number of clusters specified by the user. For h = 0, 1, . . . , p−1, let C h be the set of clusters formed at the hth step of the hierarchical clustering that proceeds in a bottom-up manner ( Figure 1). In particular, C 0 = {{1}, {2}, ..., {p}} consists of p singleton clusters and C p−1 = {{1, ..., p}} is just one cluster of all p nodes. Let k i be the number of big clusters in C i . We choose k = min k max , max 0≤i≤p−1 k i ,(5) which is the maximum number of big clusters subject to the user-specified k max . Let be the highest level on the dendrogram with k big clusters, i.e. = argmax 0≤i≤p−1 {i : k i = k}.(6) Note that two big clusters will be merged at the next level ( + 1) by the hierarchical clustering. Figure 1 shows an example of k and on a dendrogram. Relabel the clusters in C so that S 1 ≥ S 2 ... ≥ S p− , where S i = |C i |. Then the first k clusters are big clusters of interest. We assign the remaining small clusters to the k big clusters by recursively merging two closest clusters if at least one of them is a small cluster. An outline of our clustering algorithm is shown in Algorithm 1. Note that in Line 6, the distance Algorithm 1 Modified hierarchical clustering 1: Hierarchical clustering given the dissimilarity matrix D = (d(i, j)) p×p . 2: Generate the dendrogram T D of the hierarchical clustering. 3: Choose k by (5) and by (6). d(C i , C j ) := min {d(X, Y ) : X ∈ C i , Y ∈ C j },4: Relabel clusters in C ← C so that S 1 ≥ ... ≥ S p− . 5: while |C| > k do 6: (i * , j * ) ← argmin (i,j) {d(C i , C j ) : i < j and j > k}. 7: C i * ← C i * ∪ C j * , C ← C \ {C j * }. 8: end while 9: Return C = {C 1 , C 2 , ..., C k }. Estimation In the estimation step (E-step) we learn the structure of each subgraph individually. Under our PEF framework, this estimation step acts like a blackbox, and the user may use any structure learning algorithm to estimate the subgraphs without knowing its technical details. The output of this step is in general k PDAGs. Note that both DAGs and CPDAGs are special cases of PDAGs. In this work, we choose the CCDr algorithm (Aragam and Zhou 2015) in the R package sparsebn (Aragam et al. 2019) and the PC algorithm in the R package pcalg (Kalisch et al. 2012) as examples for the E-step. CCDr is a score-based method that outputs a DAG, while the PC algorithm is constraint-based and outputs a CPDAG or PDAG. As such, we can illustrate the performance of the PEF method with different approaches, score-based versus constraintbased, to structure learning. We use the CCDr algorithm for two reasons: 1) It has competitive performance in terms of accuracy for structure learning of DAGs on high-dimensional data, which is our focus. 2) The way it is formulated and coded enables CCDr to learn quite large graphs, allowing for manageable comparisons with PEF in terms of running time. When the time complexity of a structure learning method grows faster than O(p 2 ), the running time of learning small subgraphs in the E-step will be much shorter than estimating the full DAG as a whole. Furthermore, we can easily distribute the estimation step. Suppose in the partition step we have divided nodes into k clusters C 1 , . . . , C k , and the running time for learning a PDAG on C i is t i . Learning k subgraphs on k cores in parallel will reduce the time for the E-step to max{t i , i = 1, . . . , k}, which is usually determined by the size of the largest cluster. According to our discussion in the previous subsection, there will be at most 20 clusters, and computing resource with 20 cores is very common nowadays. As supported by our numerical experiments, we can save majority of the computing time with the E-step. Fusion The fusion step (F-step) is a novel hybrid method developed to add edges between estimated subgraphs from the E-step and to learn the full DAG structure. It proceeds in two stages. First, we generate a candidate edge set A to restrict our search space. By using a sequence of statistical tests, we identify a set A * of candidate edges between subgraphs. Then the candidate edge set A consists of A * and all edges learned in each subgraph from the E-step. Second, we use a modified BIC score to learn the DAG structure by sequentially updating the edges in the set A. The final output of our PEF method is a DAG. Candidate edge set Recall that Theorem 1 provides a justification for using conditional independence tests to infer edges of a DAG. In light of this result, we develop a method to produce a set A * of candidate edges between the subgraphs estimated from the E-step. Let G m = (V m , E m ), m = 1, . . . , k, denote these subgraphs and z(i) ∈ {1, . . . , k} the cluster label of node i. In general, the subgraphs G m are PDAGs. We define the neighbors of a node i in the subgraph G z(i) as N i (z(i)) = {j ∈ V z(i) : j → i ∈ E z(i) or (i, j) ∈ E z(i) }, where j → i denotes a directed edge and (i, j) an undirected one. By Theorem 1, it is sufficient to find any subset of nodes Z such that X i and X j are conditionally independent given Z to conclude that there is no edge between i and j. Unfortunately, for our problem size it is impractical to search all possible subsets. To save calculation, we use the correlationρ ij = cor(R i ,R j ), whereR i is the residual in projecting X i onto its neighbors N i (z(i)) in G z(i) , to filter out unlikely between-subgraph edges. More specifically, we produce an initial candidate setà * = {(i, j) : z(i) = z(j) andρ ij = 0 is rejected at significance levelα},(7) which will be refined further to define A * . Proposition 2 shows that, under certain conditions, A * will include all between-subgraph edges if the test againstρ ij = 0 is perfect. Its proof can be found in Appendix A.2. Proposition 2. Suppose the joint Gaussian distribution of (X 1 , . . . , X p ) defined by (2) is faith- ful to the DAG G. Let R i·A = X i − E(X i | X A ) be the residual after regressing X i onto X A :=(X k ) k∈A . If there is an edge X i → X j in G, then R i·A and R j·B are correlated for any disjoint A, B ⊆ V \ {i, j} as long as R i·A is independent of X A given X B . Since R i·A is the residual after projecting X i onto X A , by definition it is always independent of X A . So the conclusion of the above proposition holds if R i·A and X A do not become dependent after conditioning on X B . Our rule (7) could produce false positive statements: X i and X j may become independent conditioning on other subsets. Therefore, we develop a sequential way to screenà * and define the final candidate edge set A * between subgraphs, described in Algorithm 2 Line 9 to Line 14: We go through each node pair (i, j) ∈à * and run conditional independence test given the union of their updated neighbors, N i (z(i)) ∪ N j (z(j)) ∪ P ij , where P ij = {k : (k, i) ∈ A * or (k, j) ∈ A * }(8) is the set of neighbors of i or j in the current candidate set A * between subgraphs. Algorithm 2 Find candidate edge set A 1: Input data matrix x and estimated subgraphs G 1 , ..., G k . 2: Setà * = ∅. 3: for all pairs (i, j) such that z(i) = z(j) do 4: ifρ ij = 0 is rejected at levelα then 5:à * ←à * ∪ (i, j). 6: end if 7: end for 8: Set A * = ∅. 9: for all (i, j) ∈à * do 10: Let Z = N i (z(i)) ∪ N j (z(j)) ∪ P ij , where P ij is defined in (8). 11: if I P (X i ; X j |Z) is rejected at level α then 12: A * ← A * ∪ (i, j). 13: end if 14: end for 15: Return A = A * ∪ SK(G). Remark 1. In Algorithm 2, node pairs are added to A * sequentially (Line 12) and thus, the result depends on the order we go throughà * . In our implementation, we sort the node pairs iñ A * in the ascending order of their p-values in testing againstρ ij = 0 (Line 4). In this way, node pairs that are more significant will have a higher priority to be included in the set A * . Similarly, we also sort the node pairs in A * according to their p-values calculated in Line 11. Breaking a full DAG into subgraphs not only might introduce false negatives, i.e. the cut edges between two subgraphs, but also it could result in false positive edges within a subgraph. Suppose two non-adjacent nodes i and j share a common parent k in the full DAG, but the P-step has put k into a different cluster than i and j. Then in the subgraph containing i and j, there will be an edge (i, j) in the estimated skeleton since they are not independent without conditioning on k. By cutting some of the edges in the P-step, we have changed the structure of a subgraph, and therefore a structure learning algorithm in the E-step may not recover the true subgraph in the original DAG. To fix this problem, we will revisit all edges learned from the E-step and correct the subgraph structures based on the new edges added between subgraphs. Let G = (V, E) be the PDAG consisting of disconnected subgraphs learned from the E-step and SK(G) be the (undirected) edge set in the skeleton of G, i.e., SK(G) = {(i, j) : (i, j) ∈ E or i → j ∈ E}. Our candidate edge set A is formed by attaching SK(G) to the end of A * (Line 15 in Algorithm 2). The edges of our final output DAG will be restricted to a subset of A. The complete algorithm for finding the candidate edge set A is summarized in Algorithm 2. Learning full DAG structure The last stage in the fusion step is to determine, for each node pair (i, j) ∈ A, whether there is an edge and its orientation if an edge does exist. This is done by minimizing a modified BIC score, called the risk inflation criterion (RIC) (Foster and George 1994), over the candidate edge set in a sequential manner. The RIC score has two components, a log-likelihood part to measure how good a graph G fits the data and a regularization term to enforce sparsity: RIC(G) = −2 ( B, Ω | G) + λd(G),(9) where (· | G) is the log-likelihood (3) evaluated at the MLE ( B, Ω) given the DAG G, d(G) is the number of edges, and λ = 2 log p. We use this score when the number of nodes is large with p > √ n. When p ≤ √ n, we switch back to the regular BIC score, i.e. λ = log n. For each (i, j) ∈ A, we need to compare three models: M 0 : no edge between i and j, M 1 : i is a parent of j,(10) M 2 : j is a parent of i, while holding other edges in G fixed. Since the RIC score (9) is decomposable, this comparison reduces to comparing the score difference for the involved child nodes (see Appendix A.3). If there is a true edge between the two nodes i and j, both M 1 and M 2 will have a lower RIC score than M 0 in the large sample limit, due to the nonzero partial correlation between X i and X j given any other set of variables. Thus, we will add an edge between (i, j) if and only if max {RIC(M 1 ), RIC(M 2 )} < RIC(M 0 ),(11) where RIC(M ) is the RIC score for model M . If criterion (11) is met, we will further decide the edge orientation. To enforce acyclicity, if the edge i → j (or j → i) induces a directed cycle, we add j → i (or i → j). If neither direction induces a directed cycle, we choose the model with a smaller RIC following a default tie-breaking rule. See Appendix A.3 for more technical details. The full fusion step is shown in Algorithm 3, which cycles through A iteratively until the structure of G does not change. Denote by N i (G) the neighbors of node i in the current G. At any iteration, if I P (X i ; X j |N i (G) ∪ N j (G)) according to the conditional independence test (Line 8), we will remove the pair (i, j) from A permanently. This rule is again justified by Theorem 1 under faithfulness. In order to reduce the number of false positive edges for large p, the significance level for all tests, includingα and α in Algorithm 2, is set to 0.001 in our implementation. Algorithm 3 Fuse subgraphs 1: Input data matrix x and estimated subgraphs G 1 , ..., G k . 2: Run Algorithm 2 to generate candidate edge set A. if I P (X i ; X j |N i (G) ∪ N j (G)) then 9: A ← A \ {(i, j)}. Numerical experiments In this section, we test our PEF method on Gaussian data generated from real networks. We choose to use two different structure learning algorithms in the E-step, the CCDr algorithm (Aragam and Zhou 2015) which estimates a DAG and the PC algorithm which outputs a PDAG. We will call these two implementations PEF-CCDr and PEF-PC hereafter. Accordingly, we compare the results from the PEF methods with those from the CCDr and the PC algorithms applied on the whole data, which will demonstrate the advantages of our divide-and-conquer strategy in learning large networks. Data generation All network structures were downloaded from the repository of the R package bnlearn (Scutari (195,338,602,592,1125,1397). In order to generate large DAGs, we replicate each network k times and randomly add some edges between copies of the network. For easy reference, define Net(k, c) to be the DAG composed of k replicates of Net with c · ks sub edges added between subgraphs, where c ≥ 0 is a constant and Net is one of the above six networks. Let s w = ks sub be the number of within-subgraph edges and s b be the number of betweensubgraph edges. Then c = s b /s w is the ratio between the numbers of the two types of edges. In total, ten networks were generated by this scheme. ii. Mixed networks: We combined networks PATHFINDER, ANDES, DIABETES, PIGS, LINK to build a DAG with k = 5 different subgraphs. Similar to scheme (i), we randomly added s b = cs w edges between subgraphs for c ∈ {0, 0.1}. We refer to these two networks as Mix(5, c). iii. MUNIN(k, 0) for k = 1, . . . , 10: MUNIN is the largest network available on the bnlearn repository. We did not add any edges between the subgraphs. So the number of edges for each DAG generated here was s 0 = s w = ks sub . Data sets from the above DAGs were generated according to the linear structural equation model in (2). We drew β ij uniformly from [−1, −0.5] ∪ [0.5, 1] if (i, j) ∈ E and set β ij = 0 otherwise. The error variances σ 2 j , j = 1, ..., p were chosen so that all data columns had the same standard deviation. The number of observations for all simulated data sets were set to n = 1, 000. For each network generated by schemes (i) and (ii), we simulated 10 data sets. Networks in scheme (iii) were mainly used to test the limit of structure learning algorithms, so only 5 data sets were generated from each DAG. Accuracy metrics We propose a few metrics to evaluate the accuracy of PDAGs learned by structure learning algorithms. As DAGs can be regarded as a special class of PDAGs, metrics defined here can be used to assess the quality of estimated DAGs as well. Since we are using observational data, structure learning algorithms may not determine all edge orientations due to Markov equivalence (Definition 2). In our assessment, we take v-structures and compelled edges into account in the following definitions of accuracy metrics: -T, the number of edges in the true graph. -P, the number of predicted edges by a structure learning algorithm. -E, the number of expected edges for which the true graph and the estimated graph coincide. We define an estimated directed edge to be expected if it meets either of the following two criteria: (1) This edge is in the true DAG with the correct orientation; (2) The edge coincides after converting the estimated DAG and the true DAG to CPDAGs. An estimated undirected edge is considered expected if it satisfies condition 2. -R, the number of reversed edges. This is the number of predicted edges in the true skeleton, excluding expected edges. -FP = P − E − R, the number of false positive edges. -SHD = R + M + FP, the structural Hamming distance between the estimated and the true graphs, where M = T − E − R is the number of missing edges. -JI = E/(T + P − E), the Jaccard index, i.e. the ratio of the number of common edges over the size of the union of the edge sets of two graphs. In particular, SHD and JI are overall accuracy metrics. Small SHD and high JI indicate high accuracy in structure learning. Comparison with the CCDr algorithm We will first show the improvement in speed of our PEF-CCDr method compared to the CCDr algorithm. Then we will show that the PEF-CCDr method actually improves the accuracy of the CCDr algorithm. For all the experiments, we ran CCDr provided in the R package sparsebn. The CCDr algorithm outputs a solution path with an increasing number of edges. In order to From Figure 2(a) we see that when the number of subgraphs stayed the same and the size of the sub-graphs became larger, the running time of PEF-CCDr increased monotonically. The scalability of the E-step depends on the CCDr algorithm. Therefore, the running time of the E-step of our PEF-CCDr method increased with the size of the subgraphs, in a similar pattern as the CCDr algorithm did. As reported in Table 1, for the largest network MUNIN(5, 0) included in Figure 2(a), PEF-CCDr was 37 times faster than CCDr. From the lower panel of Table 1 as well as Figure 2(b), we see that as k increased, improvement of our PEF method in speed became more substantial. The number of clusters our PEF-CCDr method identified (k) is shown in Table 1. The PEF-CCDr method identified the correct number of subgraphs for k ≥ 3, and therefore the running time of the E-step stayed comparable to the running time of CCDr on a single MUNIN network (around 1 minute). When Note: p is the number of nodes, T is the total running time, P, E, and F are the running times for the P-step, the E-step, and the F-step, respectively.k is the average number of estimated clusters in the P-step. r T is the ratio of total running time of CCDr over that of PEF-CCDr. the number of subgraphs k = 3, PEF-CCDr was 11 times faster than CCDr, and when k = 9, it was 105 times faster. When k was increased to 10, our device for running the tests, MacBook Pro with 3.1 GHz Intel Core i7 processor, ran out of memory for the CCDr algorithm. Our PEF-CCDr method, on the other hand, took only 5.94 minutes to run MUNIN(10,0). This example shows the huge advantage of our PEF-CCDr method in terms of computational efficiency for learning big Bayesian networks. Accuracy comparison Next, we compare the accuracy between the PEF-CCDr method and the CCDr algorithm. Table 2 reports the summary of accuracy for the two methods on ten networks generated in the first two schemes (Section 5.1). In this and subsequent tables, Net(5) refers to either Net(5,0) or Net(5, 0.1) with the value of c implicitly given by (s 0 , s b ). We see from Table 2 that for all cases the SHD of PEF-CCDr was much smaller than CCDr, and the JI was higher than CCDr. For PATHFINDER(5) with p = 545 < n = 1000, the advantage of our PEF-CCDr method was not as obvious as the rest of the big networks. For all other networks where p > n, the number of expected edges of our PEF-CCDr method increased more than 15% compared to CCDr in most of the cases, while the reversed edges and the false positives decreased more than 20%. The overall metrics SHD decreased more than 20% and the JI increased over 35% for all cases. The number of edges between subgraphs, s b in Table 2, did not show any real impact on the accuracy of PEF-CCDr. This is because when we fuse DAGs from clusters we also correct their structures learned in the E-step. Therefore, even if we cut some edges in the P-step, which may alter the subDAG structures, we can still correct them in the F-step. Therefore, our PEF-CCDr method has some tolerance for errors in the first two steps. Even if the full DAG does not have a clear cluster structure, in which case many edges will be cut in the P-step, PEF-CCDr can still recover a reasonable amount of these edges. This is demonstrated by the comparable performance of our method on DAGs with a different s b in the table. On the other hand, performance of PEF-CCDr clearly depends on the structure learning algorithm plugged in the E-step. In the final F-step, we only remove or flip within-subgraph edges, so the missing edges within any subgraph introduced in the E-step will never be added back in the fusion step. In addition, the learned subgraph structure may also affect our choice for the candidate set A (Section 4.1) and thus the final accuracy. Recovery rate of the fusion step To examine the role of the fusion step, we compare DAGs learned by our full PEF method with DAGs learned from the first two steps only, i.e. the partition step and the estimation step. We call the latter PE-CCDr. Table 3 reports the comparison between these two methods. The rows for PEF-CCDr report the percentage of change in each accuracy metric relative to PE-CCDr. Using ANDES(5, 0.1) with s b = 169 as an example, PEF-CCDr predicted 63% more expected edges with the SHD 55% smaller than that of PE-CCDr. It is clear from the results that the fusion step always improved the structure of an estimated DAG with increased E, JI and decreased R, FP, SHD. The results in Table 3 show that as s b increased, the fusion step recovered an increasing number of expected edges. The number of expected edges recovered by the fusion step can reach 60% of that recovered in the first two steps, such as for ANDES(5, 0.1), DIABETES(5, 0.1) and LINK(5, 0.1). In addition, we see that our fusion step not only recovered expected edges, but also was able to remove reversed and false positive edges. Across different cases, the F-step reduced 30% to 60% FPs and 10% to 60% Rs, which substantially improved the structure learning accuracy. All these observed improvements in accuracy demonstrate the critical role of the fusion step. Not only does it add edges cut by the P-step back to the full DAG, but also gets rid of false positive edges produced by the E-step. This suggests that the fusion step can largely correct the mistakes made by the first two steps, and thus our PEF method may handle networks with a moderate number of between-subgraph edges, relaxing the assumption on a block structure of the true DAG to some degree. Comparison with the PC algorithm In this section, we test our PEF framework with the PC algorithm used for the E-step, which we call PEF-PC. The PC algorithm is a well-known constraint-based method that outputs a PDAG in general. This will complement our comparison with CCDr in the previous subsection, which estimates a DAG via a score-based approach. In our experiments, we used the PC algorithm in the pcalg package (Kalisch et al. 2012). An important tuning parameter of PC is the significance level α for conditional independence tests, (See Table 1 for the definitions of T, P, E, F, and r T .) which controls the sparsity of an estimated graph: The smaller the α, the sparser the estimated graph and the faster the algorithm. With the default setting α = 0.05, PC took too long, more than 24 hours, to learn some DAGs like the PATHFINDER(5) networks. Furthermore, for highdimensional data, a big α usually results in too many false positive edges in the graph learned by the PC algorithm. In order to make an informative comparison, we set α = 10 −4 so that the PC algorithm can produce quite accurate PDAGs within a reasonable amount of time. Another tuning parameter is the maximal size (m.max) of the conditioning sets that are considered in a conditional independence test. The default value of this parameter is infinity, but with this default value, it took up to 6 hours to run PC on a single data set. Thus, in our experiment, we limited this value to 3. We also tried increasing m.max to 5, and got similar results with slightly lower accuracy but much longer running time. The same data for the comparisons in Tables 1 and 2 were used in this experiment as well. The parameter choices for PC in our E-step were the same as those for running PC on full data. Table 4 compares the running time between PC and PEF-PC with paralleling the E-step. As described above, we did fine tuning on the parameters of PC to improve its speed, and consequently, the algorithm ran very fast on these data sets. Even though, PEF-PC was usually 2 to 8 times faster. The running time improvement here was not as substantial as that for the CCDr algorithm, probably because of the different ways these two algorithms scale with the graph size. Next, we compared the estimation accuracy between PC and PEF-PC. To confirm the effect of the fusion step in our method, we also compared with PE-PC, which only includes the first two steps of our PEF framework. Table 5 reports the detailed accuracy metrics. Similar to the results in the comparison with CCDr, we observe significant improvement in accuracy of PEF-PC over PC. For all the networks tested, the Jaccard index of PEF-PC was much higher and the SHD of PEF-PC was much lower than PC. Consistent with Table 3, the fusion step of PEF-PC substantially improved the results from the P-step and the E-step, by recovering more expected edges and correcting many reversed edges. Take the ANDES(5) network with s b = 0 as an example. Our PEF method found 30% more expected edges, while reducing reversed edges by more than 80%, compared to the other two competitors. Remark 2. Comparing the results in this section with those in Section 5.3, it appears that the PEF-PC method outperformed the PEF-CCDr method in terms of accuracy for most of the networks except PATHFINDER(5). This is because the PC algorithm had higher accuracy than the CCDr algorithm on these data. Such differences match our expectation that performance of the PEF framework will depend on the algorithm used in the E-step. On the other hand, our PEF framework showed substantial advantages over both algorithms, demonstrating the robustness of our divide-and-conquer strategy regardless of the performance of the DAG learning algorithm used in the E-step. Discussion We have developed a divide-and-conquer framework for structure learning of massive Bayesian networks from continuous data. The key novel step in our method is the fusion step, which merges the subgraphs learned from subsets of nodes partitioned by a modified clustering algorithm. Our numerical results suggest that this fusion step can correct and fix the DAG structure damaged by the partition step, so that the overall accuracy of the PEF method is seen to be much higher than the structure learning algorithm used in the estimation step. We also observed quite significant boost in speed, ranging from a few folds to orders-of-magnitude. There are certain limitations of our current design and implementation of the PEF method. First of all, in the partition step, we need to calculate and store the dissimilarity matrix for all pairs of nodes. When the number of nodes p is really large, this becomes memory-intensive. A promising potential solution to this issue is to borrow ideas from the subsample clustering method for big data (Marchetti and Zhou 2016). We may subsample a small fraction of the nodes for clustering, and then assign the remaining large number of nodes based on the clustering of the subsample, which can be implemented in a sequential way. For the fusion step, our current implementation takes as input the correlation matrix of the data columns. Again, when the number of nodes is too big, we may implement the algorithm to calculate correlations whenever needed, instead of pre-computing all correlations. Our current fusion step was implemented At a conceptual level, it seems straightforward to generalize the PEF method to discrete Bayesian networks. For discrete data, one can still use our clustering method, with a suitable similarity measure, for the partition step, and plug in an appropriate structure learning algorithm in the estimation step. As for the fusion step, the conditional independence test is no longer for zero partial correlations, instead we may use the G 2 test for discrete data as in the PC algorithm. Finally we may substitute linear regression with the multinomial logistic regression as used in Gu et al. (2019) for BIC-based edge selection (Section 4.2). This is left as future work. where R i·A ⊥ X A (independence). Similarly, regressing X j onto X A∪B∪{i} , we arrive at X j = k∈A∪B β kj X k + β ij X i + ε j , (A.2) with ε j ⊥ X A∪B∪{i} and thus ε j ⊥ R i·A . Plugging (A.1) into (A.2) to eliminate X i , we have X j = k∈A∪B γ kj X k + β ij R i·A + ε j , (A.3) for some γ kj 's after rearranging terms in the summation. Denote by R i·A·B the residual of regressing R i·A onto X B . Since R i·A ⊥ X A | X B by assumption, the coefficient β ij = E(R j·B R i·A·B ) E(R i·A·B ) 2 = E(R j·B R i·A ) E(R i·A·B ) 2 = cov(R j·B , R i·A ) var(R i·A·B ) , where the second equality is due to R j·B ⊥ E(R i·A | X B ). By Theorem 1, β ij = 0, because otherwise I P (X j ; X i |X A∪B ), and thus cov(R j·B , R i·A ) = 0. The proof is complete. A.3 RIC for model selection Recall we want to compare three models, M 0 , M 1 , M 2 , defined in (10). Suppose the current DAG is G, which has no edge between i and j. Now consider the following two linear models X i =β ji X j + k∈Π G i β ki X k + ε i , (A.4) X j =β ij X i + k∈Π G j β kj X k + ε j . (A.5) Then, M 0 is equivalent to β ij = β ji = 0, M 1 equivalent to β ij = 0 and β ji = 0, and M 2 equivalent to β ij = 0 and β ji = 0. Note that when undirected edges exist, we consider all neighbors as the parents. In order to choose from the three models, we calculate their RIC scores. In our implementation, we find least-squares estimates (LSEs) of the regression coefficients for (A.4) and (A.5). Let ji be the log-likelihood evaluated at the LSE under the linear model (A.4), and 0i the loglikelihood under (A.4) when β ji = 0. Similarly, ij denotes the log-likelihood at the LSE for the linear model (A.5), and 0j the log-likelihood when β ij = 0. Since the structure of G is identical except for the node pair (i, j), these four likelihood scores are sufficient for comparing M 0 , M 1 and M 2 . Let (M i ) be the log-likelihood of M i for i = 0, 1, 2. Then we have (M 0 ) = 0i + 0j , (M 1 ) = 0i + ij , and (M 2 ) = ji + 0j . Thus, the RIC selection criterion (11) is equivalent to 2 min{ ij − 0j , ji − 0i } > λ, where λ is the penalty parameter in (9). The motivation for this criterion is to add an edge between i and j only when i ⊥ j|Π G i and i ⊥ j|Π G j (Theorem 1). is a multivariate Gaussian distribution N p (0, Σ) with covariance matrix Σ = (I − B) −T Ω(I − B) −1 , where I denotes the identity matrix. Suppose we have observed an iid sample of size n, x = [x 1 | . . . |x p ] ∈ R n×p , from a Gaussian Bayesian network parameterized by (B, Ω). Let B j be the jth column of B. Then the loglikelihood under this model is mainly for speed purpose. Figure 1 : 1Example for determining k and . Shown is the upper portion of a dendrogram. Red clusters are big clusters with more than 0.05p nodes, and the grey ones are small clusters. The level is marked by the red box, and in this case k = 3. 3 : 3Initialize G to be the PDAG consisting of G 1 , . . . , G k . 4: for all (i, j) ∈ A do 5:if i, j are adjacent in G then RIC max = max (RIC(M 1 ), RIC(M 2 )).12: if RIC max < RIC(M 0 ) then 13: if adding edge i → j induces a cycle then 14: add j → i to G 15: else if adding edge j → i induces a cycle then 16: add i → j to G 22: end for 23: Repeat 4 to 22 until the structure of G does not change and return G. 2010, 2017). The networks used in this work are: PATHFINDER, ANDES, DIABETES, PIGS, LINK, and MUNIN, with the number of nodes p = (135, 223, 413, 441, 724, 1041) and the number of edges s sub = For example, ANDES(5, 0.1) refers to a network constructed by 5 copies of the ANDES network with s b = 0.1s w edges added between the 5 sub-networks. Denote the number of true edges in a DAG by s 0 = s w + s b . We have three network generation schemes: i. Net(5, c) for c ∈ {0, 0.1} and Net∈ {PATHFINDER, ANDES, DIABETES, PIGS, LINK}. Figure 2 :Figure 2 22Log 10 running time for different size of DAGs. The line with -C-is for CCDr and the line with -P-for PEF-CCDr. enforce sparsity, we simply chose the DAG along the solution path with around 1.5p edges, and stopped running CCDr when the number of estimated edges on the path became greater than 2p by setting edge.threshold = 2p. Note that we used exactly the same settings for CCDr applied to learn the full graph and in the E-step of the PEF method. reports the log 10 running times of the two algorithms.Figure 2(a) illustrates how the two methods scaled when the size of the subgraphs increased, tested on the networks Net(5, 0) for Net ∈ {PATHFINDER, ANDES, DIABETES, LINK, MUNIN}. Figure 2(b) illustrates how the two methods scaled when the number of subgraphs increased, using the networks MUNIN(k, 0)for k = 1, ..., 10.Table 1reports the total running time (T) of CCDr and PEF-CCDr, as well as the running time of each step (P, E, F) of PEF-CCDr for all 22 networks (Section 5.1). For the E-step in our PEF-CCDr method, we report the time for parallel estimation of multiple subgraphs. Table 1 : 1Timing comparison (in minutes) between CCDr and PEF-CCDrCCDr PEF-CCDr Network p T T P E Fk r T PATHFINDER(5, 0) 545 0.24 0.10 0.01 0.01 0.08 5.0 2.40 PATHFINDER(5, 0.1) 545 0.23 0.10 0.01 0.01 0.08 5.0 2.30 ANDES(5, 0) 1115 0.93 0.24 0.02 0.02 0.20 9.5 3.88 ANDES(5, 0.1) 1115 0.59 0.38 0.02 0.02 0.34 8.4 1.55 MIX(5, 0) 1910 4.65 0.46 0.06 0.08 0.32 8.6 10.11 MIX(5, 0.1) 1910 2.51 0.67 0.06 0.16 0.45 7.2 3.75 DIABETES(5, 0) 2065 7.38 0.53 0.06 0.09 0.38 8.1 13.92 DIABETES(5, 0.1) 2065 4.73 0.60 0.05 0.08 0.47 8.2 7.88 PIGS(5, 0) 2205 10.84 0.64 0.07 0.16 0.41 5.8 16.94 PIGS(5, 0.1) 2205 6.60 0.74 0.07 0.14 0.53 6.2 8.92 LINK(5, 0) 3620 9.19 1.17 0.17 0.16 0.84 8.1 7.85 LINK(5, 0.1) 3620 9.90 1.59 0.16 0.17 1.26 9.2 6.23 MUNIN(1, 0) 1041 0.93 0.48 0.02 0.20 0.27 7.0 1.94 MUNIN(2, 0) 2082 7.42 1.22 0.07 0.79 0.36 4.2 6.08 MUNIN(3, 0) 3123 22.32 2.03 0.12 1.04 0.87 3.0 11.00 MUNIN(4, 0) 4164 56.42 2.21 0.22 1.13 0.86 4.0 25.53 MUNIN(5, 0) 5205 114.85 3.11 0.34 1.21 1.57 5.0 36.93 MUNIN(6, 0) 6246 204.93 3.18 0.46 1.28 1.44 6.0 64.44 MUNIN(7, 0) 7287 311.59 3.71 0.64 1.28 1.79 7.0 83.99 MUNIN(8, 0) 8328 440.02 4.42 0.82 1.26 2.33 8.0 99.55 MUNIN(9, 0) 9369 542.56 5.15 1.04 1.33 2.78 9.0 105.35 MUNIN(10, 0) 10410 NA 5.94 1.32 1.39 3.23 10.0 NA Table 2 : 2Accuracy comparison between CCDr and PEF-CCDr (s 0 , s b ) Method P E R FP SHD JI PATHFINDER(5), p = 545 (975, 0) CCDr 823.0 252.7 149.6 420.7 1143.0 0.164 PEF-CCDr 660.4 254.7 122.4 283.3 1003.6 0.186 (1073, 98) CCDr 838.0 329.5 123.4 385.1 1128.6 0.209 PEF-CCDr 768.5 361.2 119.1 288.2 1000.0 0.245 ANDES(5), p = 1115 (1690, 0) CCDr 1586.0 931.4 447.0 207.6 966.2 0.397 PEF-CCDr 1563.0 1187.7 224.3 151.0 653.3 0.576 (1859, 169) CCDr 1721.8 1051.6 452.1 218.1 1025.5 0.416 PEF-CCDr 1766.1 1406.1 186.6 173.4 626.3 0.634 DIABETES(5), p = 2065 (3010, 0) CCDr 3166.3 1327.3 1067.9 771.1 2453.8 0.274 PEF-CCDr 2779.8 1580.2 779.1 420.5 1850.3 0.376 (3311, 301) CCDr 3069.6 1499.4 978.9 591.3 2402.9 0.307 PEF-CCDr 3202.7 2010.1 702.4 490.2 1791.1 0.447 PIGS(5), p = 2205 (2960, 0) CCDr 3285.6 1677.4 832.0 776.2 2058.8 0.367 PEF-CCDr 2809.9 1933.5 541.6 334.8 1361.3 0.504 (3256, 296) CCDr 3262.5 1874.0 800.8 587.7 1969.7 0.404 PEF-CCDr 3182.1 2308.3 489.9 383.9 1331.6 0.559 LINK(5), p = 3620 (5625, 0) CCDr 5329.4 2640.6 1421.7 1267.1 4251.5 0.318 PEF-CCDr 5021.9 3211.4 972.6 837.9 3251.5 0.432 (6188, 563) CCDr 5799.6 3096.9 1436.5 1266.2 4357.3 0.348 PEF-CCDr 5849.9 4018.3 878.1 953.5 3123.2 0.501 Mix(5), p = 1910 (2852, 0) CCDr 2893.1 1423.4 766.4 703.3 2131.9 0.329 PEF-CCDr 2620.2 1685.9 526.2 408.1 1574.2 0.446 (3138, 286) CCDr 2923.5 1564.6 766.0 592.9 2166.3 0.348 PEF-CCDr 2965.3 2005.0 497.5 462.8 1595.8 0.489 Table 3 : 3Accuracy comparison between PE-CCDr and PEF-CCDr(s 0 , s b ) Method P E R FP SHD JI PATHFINDER(5), p = 545 (975, 0) PE-CCDr 818.4 250.6 146.4 421.4 1145.8 0.163 PEF-CCDr(%) −19 2 −16 −33 −12 14 (1073, 98) PE-CCDr 830.0 275.0 134.3 420.7 1218.7 0.169 PEF-CCDr(%) −7 31 −11 −31 −18 45 ANDES(5), p = 1115 (1690, 0) PE-CCDr 1652.0 873.3 456.4 322.3 1139.0 0.354 PEF-CCDr(%) −5 36 −51 −53 −43 63 (1859, 169) PE-CCDr 1691.8 860.5 452.3 379.0 1377.5 0.320 PEF-CCDr(%) 4 63 −59 −54 −55 98 DIABETES(5), p = 2065 (3010, 0) PE-CCDr 3119.5 1281.3 1050.6 787.6 2516.3 0.264 PEF-CCDr(%) −11 23 −26 −47 −26 42 (3311, 301) PE-CCDr 3058.5 1256.6 994.6 807.3 2861.7 0.246 PEF-CCDr(%) 5 60 −29 −39 −37 82 PIGS(5), p = 2205 (2960, 0) PE-CCDr 3290.1 1632.7 834.7 822.7 2150.0 0.354 PEF-CCDr(%) −15 18 −35 −59 −37 42 (3256, 296) PE-CCDr 3312.9 1574.7 825.4 912.8 2594.1 0.315 PEF-CCDr(%) −4 47 −41 −58 −49 77 LINK(5), p = 3620 (5625, 0) PE-CCDr 5432.8 2514.9 1432.3 1485.6 4595.7 0.294 PEF-CCDr(%) −8 28 −32 −44 −29 47 (6188, 563) PE-CCDr 5471.0 2422.9 1459.3 1588.8 5353.9 0.262 PEF-CCDr(%) 7 66 −40 −40 −42 91 Mix(5), p = 1910 (2852, 0) PE-CCDr 2848.1 1330.9 765.6 751.6 2272.7 0.305 PEF-CCDr(%) −8 27 −31 −46 −31 46 (3138, 286) PE-CCDr 2880.4 1313.1 779.1 788.2 2613.1 0.279 PEF-CCDr(%) 3 53 −36 −41 −39 75 Table 4 : 4Timing comparison (in minutes) between PC and PEF-PC PC PEF-PC Network p T T P E F r T PATHFINDER(5, 0) 545 3.50 1.89 0.01 1.85 0.03 1.85 PATHFINDER(5, 0.1) 545 3.54 1.64 0.01 1.54 0.09 2.16 ANDES(5, 0) 1115 0.52 0.32 0.01 0.04 0.27 1.63 ANDES(5, 0.1) 1115 0.59 0.39 0.02 0.04 0.33 1.51 DIABETES(5, 0) 2065 2.67 0.57 0.06 0.22 0.29 4.68 DIABETES(5, 0.1) 2065 2.82 0.67 0.05 0.18 0.44 4.21 PIGS(5, 0) 2205 4.33 1.01 0.07 0.62 0.32 4.29 PIGS(5, 0.1) 2205 4.87 0.96 0.07 0.51 0.38 5.07 LINK(5, 0) 3620 8.37 1.12 0.16 0.35 0.61 7.47 LINK(5, 0.1) 3620 9.00 1.36 0.16 0.36 0.84 6.62 MIX(5, 0) 1910 3.05 1.82 0.06 0.93 0.83 1.68 MIX(5, 0.1) 1910 3.70 1.75 0.06 1.08 0.61 2.11 Table 5 : 5Accuracy comparison between PC, PE-PC and PEF-PC with the Rcpp package Armadillo. If we code it in pure C++, the speed of the fusion step may be further improved.(s 0 , s b ) Method P E R FP SHD JI PATHFINDER(5), p = 545 (975, 0) PC 438.0 154.6 190.1 93.3 913.7 0.123 PE-PC 436.9 154.6 189.4 92.9 913.3 0.123 PEF-PC 434.9 255.8 85.0 94.1 813.3 0.222 (1073, 98) PC 521.9 205.1 218.4 98.4 966.3 0.148 PE-PC 492.0 176.1 192.5 123.4 1020.3 0.127 PEF-PC 660.8 342.8 99.8 218.2 948.4 0.247 ANDES(5), p = 1115 (1690, 0) PC 1483.0 1143.8 318.3 20.9 567.1 0.564 PE-PC 1398.4 1050.6 307.1 40.7 680.1 0.516 PEF-PC 1520.7 1423.1 49.2 48.4 315.3 0.796 (1859, 169) PC 1635.4 1230.7 383.1 21.6 649.9 0.544 PE-PC 1387.8 993.7 333.5 60.6 925.9 0.441 PEF-PC 1714.3 1589.1 48.0 77.2 347.1 0.801 DIABETES(5), p = 2065 (3010, 0) PC 2563.0 1875.9 669.6 17.5 1151.6 0.507 PE-PC 2506.2 1807.5 653.5 45.2 1247.7 0.487 PEF-PC 2601.1 2192.0 353.1 56.0 874.0 0.641 (3311, 301) PC 2850.5 2071.7 750.2 28.6 1267.9 0.507 PE-PC 2466.4 1676.9 679.3 110.2 1744.3 0.409 PEF-PC 3009.5 2533.1 324.4 152.0 929.9 0.669 PIGS(5), p = 2205 (2960, 0) PC 2556.6 1881.6 638.5 36.5 1114.9 0.519 PE-PC 2497.2 1797.9 649.9 49.4 1211.5 0.492 PEF-PC 2586.6 2256.0 263.2 67.4 771.4 0.686 (3256, 296) PC 2859.6 2116.5 707.5 35.6 1175.1 0.530 PE-PC 2525.8 1696.3 695.7 133.8 1693.5 0.415 PEF-PC 2989.4 2613.3 228.3 147.8 790.5 0.720 LINK(5), p = 3620 (5625, 0) PC 4752.8 3480.3 1115.8 156.7 2301.4 0.505 PE-PC 4510.0 3182.4 1098.7 228.9 2671.5 0.458 PEF-PC 4734.4 4372.8 218.9 142.7 1394.9 0.730 (6188, 563) PC 5244.1 3818.8 1304.5 120.8 2490.0 0.502 PE-PC 4361.3 2861.3 1199.1 300.9 3627.6 0.372 PEF-PC 5434.0 4924.0 228.4 281.6 1545.6 0.735 Mix(5), p = 1910 (2852, 0) PC 2376.6 1709.4 606.7 60.5 1203.1 0.486 PE-PC 2251.5 1564.3 588.6 98.6 1386.3 0.442 PEF-PC 2409.1 2116.7 201.3 91.1 826.4 0.673 (3138, 286) PC 2621.5 1875.5 679.2 66.8 1329.3 0.483 PE-PC 2280.3 1514.4 617.0 148.9 1772.5 0.388 PEF-PC 2766.3 2380.2 202.8 183.3 941.1 0.676 A AppendixA.1 Partial correlationThe partial correlation between X and Y given Z, ρ XY ·Z , can be calculated using their covariance matrix. 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I Tsamardinos, L E Brown, Aliferis , C F , Machine Learning. 65Tsamardinos, I., Brown, L. E., and Aliferis, C. F. (2006), "The max-min hill-climbing Bayesian network structure learning algorithm," Machine Learning, 65, 31-78. Equivalence and synthesis of causal models. T Verma, J Pearl, Sixth Annual Conference on Uncertainty in Artificial Intelligence. Verma, T. and Pearl, J. (1990), "Equivalence and synthesis of causal models." in Sixth Annual Conference on Uncertainty in Artificial Intelligence, pp. 220-227. A* Lasso for learning a sparse Bayesian network structure for continuous variables. J Xiang, S Kim, Advances in Neural Information Processing Systems. Xiang, J. and Kim, S. (2013), "A* Lasso for learning a sparse Bayesian network structure for continuous variables," in Advances in Neural Information Processing Systems, pp. 2418-2426. Constrained likelihood for reconstructing a directed acyclic Gaussian graph. Y Yuan, X Shen, W Pan, Wang , Z , Biometrika. 106Yuan, Y., Shen, X., Pan, W., and Wang, Z. (2019), "Constrained likelihood for reconstructing a directed acyclic Gaussian graph," Biometrika, 106, 109-125. DAGs with NO TEARS: Continuous Optimization for Structure Learning. X Zheng, B Aragam, P K Ravikumar, E P Xing, Advances in Neural Information Processing Systems. Zheng, X., Aragam, B., Ravikumar, P. K., and Xing, E. P. (2018), "DAGs with NO TEARS: Continuous Optimization for Structure Learning," in Advances in Neural Information Pro- cessing Systems, pp. 9472-9483.

Thiès, Senegal Proposition 4. Let E be an elliptic curve defined over a field k, and let m = 0 be an integer. The m-torsion group of E, denoted by E[m], has the following structure: • E[m] (Z/mZ) 2 if the characteristic of k does not divide m; • If p > 0 is the characteristic of k, then E[p i ] Z/p i Z for any i ≥ 0, or {O} for any i ≥ 0.

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Mathematics of Isogeny Based Cryptography May 10 -23, 2017 11 Nov 2017 Luca De Feo Université de Versailles & Inria Saclay Mathematics of Isogeny Based Cryptography May 10 -23, 2017 11 Nov 2017, Thiès, Senegal Introduction Thiès, Senegal Proposition 4. Let E be an elliptic curve defined over a field k, and let m = 0 be an integer. The m-torsion group of E, denoted by E[m], has the following structure: • E[m] (Z/mZ) 2 if the characteristic of k does not divide m; • If p > 0 is the characteristic of k, then E[p i ] Z/p i Z for any i ≥ 0, or {O} for any i ≥ 0. Introduction These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thiès, Senegal. They try to provide a guide for Masters' students to get through the vast literature on elliptic curves, without getting lost on their way to learning isogeny based cryptography. They are by no means a reference text on the theory of elliptic curves, nor on cryptography; students are encouraged to complement these notes with some of the books recommended in the bibliography. The presentation is divided in three parts, roughly corresponding to the three lectures given. In an effort to keep the reader interested, each part alternates between the fundamental theory of elliptic curves, and applications in cryptography. We often prefer to have the main ideas flow smoothly, rather than having a rigorous presentation as one would have in a more classical book. The reader will excuse us for the inaccuracies and the omissions. Isogeny Based Cryptography is a very young field, that has only begun in the 2000s. It has its roots in Elliptic Curve Cryptography (ECC), a somewhat older branch of public-key cryptography that was started in the 1980s, when Miller and Koblitz first suggested to use elliptic curves inside the Diffie-Hellman key exchange protocol (see Section 4). ECC only started to gain traction in the 1990s, after Schoof's algorithm made it possible to easily find elliptic curves of large prime order. It is nowadays a staple in public-key cryptography. The 2000s have seen two major innovations in ECC: the rise of Pairing Based Cryptography (PBC), epitomized by Joux' one-round tripartite Diffie-Hellman key exchange, and the advent of Isogeny based cryptography, initiated by the works of Couveignes, Teske and Rostovtsev & Stolbunov. While PBC has attracted most of the attention during the first decade, thanks to its revolutionary applications, isogeny based cryptography has stayed mostly discrete during this time. It is only in the second half of the 2010 that the attention has partly shifted to isogenies. The main reason for this is the sudden realization by the cryptographic community of the very possibly near arrival of a general purpose quantum computer. While the capabilities of such futuristic machine would render all of ECC and PBC suddenly worthless, isogeny based cryptography seems to resist much better to the cryptanalytic powers of the quantum computer. In these notes, after a review of the general theory of elliptic curves and isogenies, we will present the most important isogeny based systems, and their cryptographic properties. L A T E X source code available at https://github.com/defeo/ema2017/. Elliptic curves and cryptography Throughout this part we let k be a field, and we denote byk its algebraic closure. We review the basic theory of elliptic curves, and two classic applications in cryptography. The interested reader will find more details on elliptic curves in [66], and on their use in cryptography in [41,31]. Elliptic curves Elliptic curves are projective curves of genus 1 having a specified base point. Projective space initially appeared through the process of adding points at infinity, as a method to understand the geometry of projections (also known as perspective in classical painting). In modern terms, we define projective space as the collection of all lines in affine space passing through the origin. Definition 1 (Projective space). The projective space of dimension n, denoted by P n or P n (k), is the set of all (n + 1)-tuples (x 0 , . . . , x n ) ∈k n+1 such that (x 0 , . . . , x n ) = (0, . . . , 0), taken modulo the equivalence relation if and only if there exists λ ∈k such that x i = λ i y i for all i. The equivalence class of a projective point (x 0 , . . . , x n ) is customarily denoted by (x 0 : · · · : x n ). The set of the k-rational points, denoted by P n (k), is defined as P n (k) = {(x 0 : · · · : x n ) ∈ P n | x i ∈ k for all i} . By fixing arbitrarily the coordinate x n = 0, we define a projective space of dimension n − 1, which we call the space at infinity; its points are called points at infinity. From now on we suppose that the field k has characteristic different from 2 and 3. This has the merit of greatly simplifying the representation of an elliptic curve. For a general definition, see [66,Chap. III]. Definition 2 (Weierstrass equation). An elliptic curve defined over k is the locus in P 2 (k) of an equation Y 2 Z = X 3 + aXZ 2 + bZ 3 ,(1) with a, b ∈ k and 4a 3 + 27b 2 = 0. The point (0 : 1 : 0) is the only point on the line Z = 0; it is called the point at infinity of the curve. It is customary to write Eq. (1) in affine form. By defining the coordinates x = X/Z and y = Y /Z, we equivalently define the elliptic curve as the locus of the equation Now, since any elliptic curve is defined by a cubic equation, Bezout's theorem tells us that any line in P 2 intersects the curve in exactly three points, taken with multiplicity. We define a group law by requiring that three co-linear points sum to zero. Definition 3. Let E : y 2 = x 3 + ax + b be an elliptic curve. Let P 1 = (x 1 , y 1 ) and P 2 = (x 2 , y 2 ) be two points on E different from the point at infinity, then we define a composition law ⊕ on E as follows: y 2 = x 3 + ax + b, • P ⊕ O = O ⊕ P = P for any point P ∈ E; • If x 1 = x 2 and y 1 = −y 2 , then P 1 ⊕ P 2 = O; if P = Q, then the point (P 1 ⊕ P 2 ) = (x 3 , y 3 ) is defined by • Otherwise set λ = y2−y1 x2−x1 if P = Q,x 3 = λ 2 − x 1 − x 2 , y 3 = −λx 3 − y 1 + λx 1 . It can be shown that the above law defines an Abelian group, thus we will simply write + for ⊕. The n-th scalar multiple of a point P will be denoted by [n]P . When E is defined over k, the subgroup of its rational points over k is customarily denoted E(k). Figure 1 shows a graphical depiction of the group law on an elliptic curve defined over R. We now turn to the group structure of elliptic curves. The torsion part is easily characterized. Proof. See [66,Coro. 6.4]. For the characteristic 0 case see also next part. For curves defined over a field of positive characteristic p, the case E[p] Z/pZ is called ordinary, while the case E[p] {O} is called supersingular. The free part of the group is much harder to characterize. We have some partial results for elliptic curves over number fields. Theorem 5 (Mordell-Weil). Let k be a number field, the group E(k) is finitely generated. However the exact determination of the rank of E(k) is somewhat elusive: we have algorithms to compute the rank of most elliptic curves over number fields; however, an exact formula for such rank is the object of the Birch and Swinnerton-Dyer conjecture, one of the Clay Millenium Prize Problems. Maps between elliptic curves Finally, we focus on maps between elliptic curves. We are mostly interested in maps that preserve both facets of elliptic curves: as projective varieties, and as groups. We first look into invertible algebraic maps, that is linear changes of coordinates that preserve the Weierstrass form of the equation. Because linear maps preserve lines, it is immediate that they also preserve the group law. It is easily verified that the only such maps take the form (x, y) → (u 2 x , u 3 y ) for some u ∈k, thus defining an isomorphism between the curve y 2 = x 3 + au 4 x + bu 6 and the curve (y ) 2 = (x ) 3 + ax + b. Isomorphism classes are traditionally encoded by an invariant, which origins can be tracked back to complex analysis. Proposition 6 (j-invariant). Let E : y 2 = x 3 + ax + b be an elliptic curve, and define the j-invariant of E as j(E) = 1728 4a 3 4a 3 + 27b 2 . Two curves are isomorphic over the algebraic closurek if and only if they have the same jinvariant. Note that if two curves defined over k are isomorphic overk, they are so over an extension of k of degree dividing 6. An isomorphism between two elliptic curves defined over k, that is itself not defined over k is called a twist. Any curve has a quadratic twist, unique up to isomorphism, obtained by taking u / ∈ k such that u 2 ∈ k. The two curves of j-invariant 0 and 1728 also have cubic, sextic and quartic twists. A surjective group morphism, not necessarily invertible, between two elliptic curves is called an isogeny. It turns out that isogenies are algebraic maps as well. Theorem 7. Let E, E be two elliptic curves, and let φ : E → E be a map between them. The following conditions are equivalent: 1. φ is a surjective group morphism, 2. φ is a group morphism with finite kernel, 3. φ is a non-constant algebraic map of projective varieties sending the point at infinity of E onto the point at infinity of E . Proof. See [66,III,Th. 4.8]. Two curves are called isogenous if there exists an isogeny between them. We shall see in the next part that this is an equivalence relation. Isogenies from a curve to itself are called endomorphisms. The prototypical endomorphism is the multiplication-by-m endomorphism defined by [m] : P → [m]P. Its kernel is exactly the m-th torsion subgroup E[m]. For most elliptic curves, this is the end of the story: the only endomorphisms are the scalar multiplications. We shall however see some non-trivial endomorphisms soon. Elliptic curves over finite fields From now on we let E be an elliptic curve defined over a finite field k with q elements. Obviously, the group of k-rational points is finite, thus the algebraic group E(k) only contains torsion elements, and we have already characterized precisely the structure of the torsion part of E. Curves over finite fields always have a special endomorphism. Definition 8 (Frobenius endomorphism). Let E be an elliptic curve defined over a field with q elements, its Frobenius endomorphism, denoted by π, is the map that sends (X : Y : Z) → (X q : Y q : Z q ). Proposition 9. Let π be the Frobenius endomorphism of E. Then: • ker π = {O}; • ker(π − 1) = E(k). Corollary 10 (Hasse's theorem). Let E be an elliptic curve defined over a finite field k with q elements, then |#E(k) − q − 1| ≤ 2 √ q. Proof. See [66, V, Th. 1.1]. It turns out that the cardinality of E over its base field k determines its cardinality over any finite extension of it. This is a special case of a special case of the famous Weil's conjectures, proven by Weil himself in 1949 for Abelian varieties, and more generally by Deligne in 1973. Definition 11. Let V be a projective variety defined over a finite field F q , its zeta function is the power series Z(V /F q ; T ) = exp ∞ n=1 #V (F q n ) T n n . Theorem 12. Let E be an elliptic curve defined over a finite field F q , and let #E(F q ) = q+1−a. Then Z(E/F q ; T ) = 1 − aT + qT 2 (1 − T )(1 − qT ) . Proof. See [66, V, Th. 2.4]. We conclude with a theorem that links the isogenies between two elliptic curves with their Frobenius endomorphisms. Theorem 13 (Sato-Tate). Two elliptic curves E, E defined over a finite field k are isogenous over k if and only if #E(k) = #E (k). Application: Diffie-Hellman key exhange Elliptic curves are largely present in modern technology thanks to their applications in cryptography. The simplest of these application is the Diffie-Hellman key exchange, a cryptographic protocol by which two parties communicating over a public channel can agree on a common secret string unknown to any other party listening on the same channel. The original protocol was invented in the 1970s by Whitfield Diffie and Martin Hellman [24], and constitutes the first practical example of public key cryptography. The two communicating parties are customarily called Alice and Bob, and the listening third party is represented by the character Eve (for eavesdropper ). To set up the protocol, Alice and Bob agree on a set of public parameters: • A large enough prime number p, such that p − 1 has a large enough prime factor; • A multiplicative generator g ∈ Z/pZ. Then, Alice and Bob perform the following steps: 1. Each chooses a secret integer in the interval ]0, p − 1[; call a Alice's secret and b Bob's secret. 2. They respectively compute A = g a and B = g b . 3. They exchange A and B over the public channel. 4. They respectively compute the shared secret B a = A b = g ab . The protocol can be easily generalized by replacing the multiplicative group (Z/pZ) × with any other cyclic group G = g . From Eve's point of view, she is given the knowledge of the group G, the generator g, and Alice's and Bob's public data A, B ∈ G; her goal is to recover the shared secret g ab . This is mathematically possible, but not necessarily easy from a computational point of view. Definition 14 (Discrete logarithm). Let G be a cyclic group generated by an element g. For any element A ∈ G, we define the discrete logarithm of A in base g, denoted log g (A), as the unique integer in the interval [0, #G[ such that g log g (A) = A. It is evident that if Eve can compute discrete logarithms in G efficiently, then she can also efficiently compute the shared secret; the converse is not known to be true in general, but it is widely believed to be. Thus, the strength of the Diffie-Hellman protocol is entirely dependent on the hardness of the discrete logarithm problem in the group G. We know algorithms to compute discrete logarithms in a generic group G that require O( √ q) computational steps (see [41]), where q is the largest prime divisor of #G; we also know that these algorithms are optimal for abstract cyclic groups. For this reason, G is usually chosen so that the largest prime divisor q has size at least log 2 q ≈ 256. However, the proof of optimally does not exclude the existence of better algorithms for specific groups G. And indeed, algorithms of complexity better than O( √ #G) are known for the case G = (Z/pZ) × [41], thus requiring parameters of considerably larger size to guarantee cryptographic strength. On the contrary, no algorithms better than the generic ones are known when G is a subgroup of E(k), where E is an elliptic curve defined over a finite field k. This has led Miller [53] and Koblitz [43] to suggest, in the 1980s, to replace (Z/pZ) × in the Diffie-Helman protocol by the group of rational points of an elliptic curve of (almost) prime order over a finite field. The resulting protocol is summarized in Figure 2. Public parameters Finite field F p , with log 2 p ≈ 256, In the 1980s H. Lenstra [48] introduced an algorithm for factoring that has become known as the Elliptic Curve Method (ECM). Its complexity is between Pollard's and Coppersmith's algorithms in terms of number of operations; at the same time it only requires a constant amount of memory, and is very easy to parallelize. For these reasons, ECM is typically used to factor integers having medium sized prime factors. Elliptic curve E/F p , such that #E(F p ) is prime, A generator P of E(F p ). Alice Bob Pick random secret 0 < a < #E(F p ) 0 < b < #E(F p ) Compute public data A = [a]P B = [b]P Exchange data A −→ ←− B Compute shared secret S = [a]B S = [b]A From now on we suppose that N = pq is an integer which factorization we wish to compute, where p and q are distinct primes. Without loss of generality, we can suppose that p < q. Lenstra's idea has its roots in an earlier method for factoring special integers, also due to Pollard. Pollard's (p − 1) factoring method is especially suited for integers N = pq such that p − 1 only has small prime factors. It is based on the isomorphism ρ : Z/N Z → Z/pZ × Z/qZ, x → (x mod p, x mod q) given by the Chinese remainder theorem. The algorithm is detailed in Figure 3a. It works by guessing a multiple e of p − 1, then taking a random element x ∈ (Z/N Z) × , to deduce a random element y in 1 ⊕ (Z/qZ) × . If the guessed exponent e was correct, and if y = 1, the gcd of y − 1 with N yields a non-trivial factor. The p − 1 method is very effective when the bound B is small, but its complexity grows exponentially with B. For this reason it is only usable when p − 1 has small prime factors, a constraint that is very unlikely to be satisfied by random primes. Lenstra's ECM algorithm is a straightforward generalization of the p − 1 method, where the multiplicative groups (Z/pZ) × and (Z/qZ) × are replaced by the groups of points E(F p ) and E(F q ) of an elliptic curve defined over Q. Now, the requirement is that #E(F p ) only has small prime factors. This condition is also extremely rare, but now we have the freedom to try the method many times by changing the elliptic curve. The algorithm is summarized in Figure 3b. It features two remarkable subtleties. First, it would feel natural to pick a random elliptic curve E : y 2 = x 3 + ax + b by picking random a and b, however taking a point on such curve would then require computing a square root modulo N , Input: An integer N = pq, a bound B on the largest prime factor of p − 1; Output: (p, q) or FAIL. 1. Set e = r prime <B r log r √ N ; 2. Pick a random 1 < x < N ; 3. Compute y = x e mod N ; 4. Compute q = gcd(y − 1, N ); 5. if q = 1, N then 6. return N/q , q ; 7. else a problem that is known to be has hard as factoring N . For this reason, the algorithm starts by taking a random point, and then deduces the equation of E from it. Secondly, all computations on coordinates happen in the projective plane over Z/N Z; however, properly speaking, projective space cannot be defined over non-integral rings. Implicitly, E(Z/N Z) is defined as the product group E(F p ) ⊕ E(F q ), and any attempt at inverting a non-invertible in Z/N Z will result in a factorization of N . Exercices Exercice I.1. Prove Proposition 6. Exercice I.4. Using Proposition 12, devise an algorithm to effectively compute #E(F q n ) given #E(F q ). Exercice I.5. Implement the ECDH key exchange in the language of your choice. Exercice I.6 (Pohlig-Hellman algorithm). Let G be a cyclic group of order N = pq, generated by an element g. Show how to solve discrete logarithms in G by computing two separate discrete logarithms in the subgroups g p and g q . Exercice I.7. Implement the ECM factorization method in the language of your choice. Part II Isogenies and applications 6 Elliptic curves over C Definition 15 (Complex lattice). A complex lattice Λ is a discrete subgroup of C that contains an R-basis. Explicitly, a complex lattice is generated by a basis (ω 1 , ω 2 ), such that ω 1 = λω 2 for any λ ∈ R, as Λ = ω 1 Z + ω 2 Z. Up to exchanging ω 1 and ω 2 , we can assume that Im(ω 1 /ω 2 ) > 0; we then say that the basis has positive orientation. A positively oriented basis is obviously not unique, though. Proposition 16. Let Λ be a complex lattice, and let (ω 1 , ω 2 ) be a positively oriented basis, then any other positively oriented basis (ω 1 , ω 2 ) is of the form ω 1 = aω 1 + bω 2 ,ω 1 = cω 1 + dω 2 , for some matrix a b c d ∈ SL 2 (Z). Proof. See [67, I, Lem. 2.4]. Definition 17 (Complex torus). Let Λ be a complex lattice, the quotient C/Λ is called a complex torus. A convex set of class representatives of C/Λ is called a fundamental parallelogram. Figure 4 shows a complex lattice generated by a (positively oriented) basis (ω 1 , ω 2 ), together with a fundamental parallelogram for C/(ω 1 , ω 2 ). The additive group structure of C carries over to C/Λ, and can be graphically represented as operations on points inside a fundamental parallelogram. This is illustrated in Figure 5. a b a + b a [3]a Definition 18 (Homothetic lattices). Two complex lattices Λ and Λ are said to be homothetic if there is a complex number α ∈ C × such that Λ = αΛ . Geometrically, applying a homothety to a lattice corresponds to zooms and rotations around the origin. We are only interested in complex tori up to homothety; to classify them, we introduce the Eisenstein series of weight 2k, defined as G 2k (Λ) = ω∈Λ\{0} ω −2k . It is customary to set g 2 (Λ) = 60G 4 (Λ), g 3 (Λ) = 140G 6 (Λ); when Λ is clear from the context, we simply write g 2 and g 3 . Theorem 19 (Modular j-invariant). The modular j-invariant is the function on complex lattices defined by j(Λ) = 1728 g 2 (Λ) 3 g 2 (Λ) 3 − 27g 3 (Λ) 2 . Two lattices are homothetic if and only if they have the same modular j-invariant. It is no chance that the invariants classifying elliptic curves and complex tori look very similar. Indeed, we can prove that the two are in one-to-one correspondence. Definition 20 (Weierstrass ℘ function). Let Λ be a complex lattice, the Weierstrass ℘ function associated to Λ is the series ℘(z; Λ) = 1 z 2 + ω∈Λ\{0} 1 (z − ω) 2 − 1 ω 2 . Theorem 21. The Weierestrass function ℘(z; Λ) has the following properties: 1. It is an elliptic function for Λ, i.e. ℘(z) = ℘(z + ω) for all z ∈ C and ω ∈ Λ. Its Laurent series around z = 0 is ℘(z) = 1 z 2 + ∞ k=1 (2k + 1)G 2k+2 z 2k . 3. It satisfies the differential equation ℘ (z) 2 = 4℘(z) 3 − g 2 ℘(z) − g 3 for all z / ∈ Λ. 4. The curve E : y 2 = 4x 3 − g 2 x − g 3 is an elliptic curve over C. The map C/Λ → E(C), 0 → (0 : 1 : 0), z → (℘(z) : ℘ (z) : 1) is an isomorphism of Riemann surfaces and a group morphism. By comparing the two definitions for the j-invariants, we see that j(Λ) = j(E). So, for any homotety class of complex tori, we have a corresponding isomorphism class of elliptic curves. The converse is also true. Theorem 22 (Uniformization theorem). Let a, b ∈ C be such that 4a 3 + 27b 2 = 0, then there is a unique complex lattice Λ such that g 2 (Λ) = −4a and g 3 (Λ) = −4b. Proof. See [67,I,Coro. 4.3]. Using the correspondence between elliptic curves and complex tori, we now have a new perspective on their group structure. Looking at complex tori, it becomes immediately evident why the torsion part has rank 2, i.e. why E[m] (Z/mZ) 2 . This is illustrated in Figure 6a; in the picture wee see two lattices Λ and Λ , generated respectively by the black and the red dots. The multiplication-by-m map corresponds then to [m] : C/Λ → C/Λ , z → z mod Λ ; and we verify that it is an endomorphism because Λ and Λ are homothetic. [3] and is generated by a. The kernel of the dual isogenyφ is generated by the vector b in Λ . Within this new perspective, isogenies are a mild generalization of scalar multiplications. Whenever two lattices Λ, Λ verify αΛ ⊂ Λ , there is a well defined map b z φ(z) (b) Isogeny from C/Λ (black dots) to C/Λ (red dots) defined by φ(z) = z mod Λ . The kernel of φ is contained in (C/Λ)φ α : C/Λ → C/Λ , z → αz mod Λ that is holomorphic and also a group morphism. One example of such maps is given in Figure 6a: there, α = 1 and the red lattice strictly contains the black one; the map is simply defined as reduction modulo Λ . It turns out that these maps are exactly the isogenies of the corresponding elliptic curves. Theorem 23. Let E, E be elliptic curves over C, with corresponding lattices Λ, Λ . There is a bijection between the group of isogenies from E to E and the group of maps φ α for all α such that Λ ⊂ αΛ . Proof. See [66,VI,Th. 4.1]. Looking again at Figure 6a, we see that there is a second isogenyφ from Λ to Λ/3, which kernel is generated by b ∈ Λ . The compositionφ • φ is an endomorphism of C/Λ, up to the homothety sending Λ/3 to Λ, and we verify that it corresponds to the multiplication-by-3 map. In this example, the kernels of both φ andφ contain 3 elements, and we say that φ andφ have degree 3. Although not immediately evident from the picture, this same construction can be applied to any isogeny. The isogenyφ is called the dual of φ. Dual isogenies exist not only in characteristic 0, but for any base field. We finish this section by summarizing the most important algebraic properties of isogenies; we start with a technical definition. Definition 24 (Degree, separability). Let φ : E → E be an isogeny defined over a field k, and let k(E), k(E ) be the function fields of E, E . By composing φ with the functions of k(E ), we obtain a subfield of k(E) that we denote by φ * (k(E )). 1. The degree of φ is defined as deg φ = [k(E) : φ * (k(E ))]; it is always finite. 2. φ is said to be separable, inseparable, or purely inseparable if the extension of function fields is. 3. If φ is separable, then deg φ = # ker φ. 4. If φ is purely inseparable, then deg φ is a power of the characteristic of k. 5. Any isogeny can be decomposed as a product of a separable and a purely inseparable isogeny. Proof. See [66, II, Th. 2.4]. In practice, most of the time we will be considering separable isogenies, and we can take deg φ = # ker φ as the definition of the degree. Notice that in this case deg φ is the size of any fiber of φ. Separable isogenies are completely determined by their kernel, as the following proposition shows. Proposition 25. Let E be an elliptic curve, and let G be a finite subgroup of E. There are a unique elliptic curve E , and a unique separable isogeny φ, such that ker φ = G and φ : E → E . Proof. See [66, Prop. III, 4.12]. The proposition justifies introducing the notation E/G for the image curve E . We conclude with a fundamental theorem on isogenies. Theorem 26 (Dual isogeny). Let φ : E → E be an isogeny of degree m. There is a unique isogenyφ : E → E such thatφ • φ = [m] E , φ •φ = [m] E . φ is called the dual isogeny of φ; it has the following properties: 1.φ is defined over k if and only if φ is; 2. ψ • φ =φ •ψ for any isogeny ψ : E → E ; 3. ψ + φ =ψ +φ for any isogeny ψ : E → E ; 4. deg φ = degφ; 5.φ = φ. The endomorphism ring We have already defined an endomorphism as an isogeny from a curve to itself. If we add the multiplication-by-0 to it, the set of all endomorphisms of E form a ring under the operations of addition and composition, denoted by End(E). We have already seen that the multiplication-by-m is a different endomorphism for any integer m, thus Z ⊂ End(E). For the case of finite fields, we have also learned about the Frobenius endomorphism π; so certainly Z[π] ⊂ End(E) in this case. We shall now give a complete characterization of the endomorphism ring for any field. Definition 27 (Order). Let K be a finitely generated Q-algebra. An order O ⊂ K is a subring of K that is a finitely generated Z-module of maximal dimension. The prototypical example of order is the ring of integers O K of a number field K, i.e., the ring of all elements of K such that their monic minimal polynomial has coefficients in Z. It turns out that O K is the maximal order of K, i.e., it contains any other order of K. Definition 28 (Quaternion algebra). A quaternion algebra is an algebra of the form K = Q + αQ + βQ + αβQ, where the generators satisfy the relations α 2 , β 2 ∈ Q, α 2 < 0, β 2 < 0, βα = −αβ. Theorem 29 (Deuring). Let E be an elliptic curve defined over a field k of characteristic p. The ring End(E) is isomorphic to one of the following: • Z, only if p = 0; • An order O in a quadratic imaginary field (a number field of the form Q[ √ −D] for some D > 0); in this case we say that E has complex multiplication by O; • Only if p > 0, a maximal order in the quaternion algebra ramified at p and ∞; in this case we say that E is supersingular. Proof. See [66, III, Coro. 9.4] and [4]. In positive characteristic, a curve that is not supersingular is called ordinary; it necessarily has complex multiplication. We focus again on the finite field case; we have already seen that Z[π] ⊂ End(E). Now, Hasse's theorem can be made more precise as follows. Theorem 30. Let E be an elliptic curve defined over a finite field. Its Frobenius endomorphism π satisfies a quadratic equation π 2 − tπ + q = 0, for some |t| ≤ 2 √ q. Proof. See [66, V, Th. 2.3.1]. The coefficient t in the equation is called the trace of π. By replacing π = 1 in the equation, we immediately obtain the cardinality of E as #E = q + 1 − t. Now, if we let D π = t 2 − 4q < 0, we verify that π ∈ Q[ √ D π ] ; so, at least in the ordinary case, we can affirm that Z[π] ⊂ End(E) ⊂ O K , where K = Q[ √ D π ] is called the endomorphism algebra of E. The structure of the orders of K is very simple in this case. Proposition 31. Let K be a quadratic number field, and let O K be its ring of integers. Any order O ⊂ K can be written as O = Z + f O K for an integer f , called the conductor of O. If d K is the discriminant of K, the discriminant of O is f 2 d K . If O, O are two orders of discriminants f, f , then O ⊂ O if and only if f |f . In our case, we can write D π = f 2 d K , with d K squarefree. Then, any order Z[π] ⊂ O ⊂ O K has conductor dividing f . 15 Before going more in depth into the study of the endomorphism ring, let us pause for a while on a simpler problem. Hasse's theorem relates the cardinality of a curve defined over a finite field with the trace of its Frobenius endomorphism. However, it does not give us an algorithm to compute either. The first efficient algorithm to compute the trace of π was proposed by Schoof in the 1980s [63]. The idea is very simple: compute the value of t π mod for many small primes , and then reconstruct the trace using the Chinese remainder theorem. To compute t π mod , Schoof's algorithm formally constructs the group E[ ], takes a generic point P ∈ E[ ], and then runs a search for the integer t such that π([t]P ) = [q]P + π 2 (P ). The formal computation must be carried out by computing modulo a polynomial that vanishes on the whole E[ ]; the smallest such polynomial is provided by the division polynomial ψ . Definition 32 (Division polynomial). Let E : y 2 = x 3 + ax + b be an elliptic curve, the division polynomials ψ m are defined by the initial values ψ 1 = 1, ψ 2 = 2y 2 , ψ 3 = 3x 4 + 6ax 2 + 12bx − a 2 , ψ 4 = (2x 6 + 10ax 4 + 40bx 3 − 10a 2 x 2 − 8abx − 2a 3 − 16b 2 )2y 2 , and by the recurrence ψ 2m+1 = ψ m+2 ψ 3 m − ψ m−1 ψ 3 m+1 for m ≥ 2, ψ 2 ψ 2m = (ψ m+2 ψ 2 m−1 − ψ m−2 ψ 2 m+1 )ψ m for m ≥ 3. The m-th division polynomial ψ m vanishes on E[m]; the multiplication-by-m map can be written as [m]P = φ m (P ) ψ m (P ) 2 , ω m (P ) ψ m (P ) 3 for any point P = O, where φ m and ω m are defined as φ m = xψ 2 m − ψ m+1 ψ m−1 , ω m = ψ 2 m−1 ψ m+2 + ψ m−2 ψ 2 m+1 . Schoof's algorithm runs in time polynomial in log #E(k), however it is quite slow in practice. Among the major advances that have enabled the use of elliptic curves in cryptography are the optimizations of Schoof's algorithm due to Atkin and Elkies [1,2,25,64,26]. Both improvements use a better understanding of the action of π on E[ ]. Assume that is different from the characteristic, we have already seen that E[ ] is a group of rank two. Hence, π acts on E[ ] like a matrix M in GL 2 (Z/ Z), and its characteristic polynomial is exactly χ(X) = X 2 − t π X + q mod . Now we have three possibilities: • χ splits modulo , as χ(X) = (X − λ)(X − µ), with λ = µ; we call this the Elkies case. • χ does not split modulo ; we call this the Atkin case; • χ is a square modulo . The SEA algorithm, treats each of these cases in a slightly different way; for simplicity, we will only sketch the Elkies case. In this case, there exists a basis P, Q for E[ ] onto which π acts as a matrix M = λ 0 0 µ . Each of the two eigenspaces of M is the kernel of an isogeny of degree from E to another curve E . If we can determine the curve corresponding to, e.g., P , then we can compute the isogeny φ : E → E/ P , and use it to formally represent the point P . Then, λ is recovered by solving the equation [λ]P = π(P ), and from it we recover t π = λ + q/λ mod . Elkies' method is very similar to Schoof's original way of computing t π , however it is considerably more efficient thanks to the degree of the extension rings involved. Indeed, in Schoof's algorithm a generic point of E[ ] is represented modulo the division polynomial ψ , which has degree ( 2 − 1)/2. In Elkies' algorithm, instead, the formal representation of P only requires working modulo a polynomial of degree ≈ . The other cases have similar complexity gains. For a more detailed overview, we address the reader to [64,49,26,70]. Isogeny graphs We now look at the graph structure that isogenies create on the set of j-invariants defined over a finite field. We start with an easy generalization of the Sato-Tate theorem 13. Theorem 33 (Sato-Tate). Two elliptic curves E, E defined over a finite field are isogenous if and only if their endomorphism algebras End(E) ⊗ Q and End(E ) ⊗ Q are isomorphic. An equivalence class of isogenous elliptic curves is called an isogeny class. In particular, we see that it is impossible for an isogeny class to contain both ordinary and supersingular curves. When we restrict to isogenies of a prescribed degree , we say that two curves are -isogenous; by the dual isogeny theorem, this too is an equivalence relation. Remark that if E is -isogenous to E , and if E is isomorphic to E , then by composition E and E are also -isogenous. At this stage, we are only interested in elliptic curves up to isomorphism, i.e., j-invariants. Accordingly, we say that two j-invariants are isogenous whenever their corresponding curves are. Definition 34 (Isogeny graph). An isogeny graph is a (multi)-graph which nodes are the jinvariants of isogenous curves, and which edges are isogenies between them. The dual isogeny theorem guarantees that for every isogeny E → E there is a corresponding isogeny E → E of the same degree. For this reason, isogeny graphs are usually drawn undirected. Figure 7 shows a typical example of isogeny graph, where we restrict to isogenies of degree 3. The classification of isogeny graphs was initiated by Mestre [52], Pizer [59,60] and Kohel [44]; further algorithmic treatment of graphs of ordinary curves, and the now famous name of isogeny volcanoes was subsequently given by Fouquet and Morain [29]. We start with some generalities. Proposition 35. Let E : y 2 = x 3 + ax + b be an elliptic curve defined over a finite field k of characteristic p, and let = p be a prime. 1. There are + 1 distinct isogenies of degree with domain E defined over the algebraic closurek. 2. There are 0, 1, 2 or + 1 isogenies of degree with domain E defined over k. End(E) O K Z[π]3. If E is ordinary, there is a unique separable isogeny of degree p with domain E; there are none if E is supersingular. 4. The map (x, y) → (x p , y p ) is a purely inseparable isogeny of degree p from E to E (p) : y 2 = x 3 + a p x + b p . There are many differences between the structure of isogeny graphs of ordinary curves and those of supersingular ones. We focus here on the ordinary case, and we leave the supersingular one for the last part. Observe that vertical isogenies can only exist for primes that divide the conductor of Z[π], so the horizontal case is the generic one. Like we did for the SEA algorithm we can further distinguish three cases, depending on the value of the Legendre symbol D , i.e., depending on whether π splits (Elkies case), is inert (Atkin case), or ramifies modulo . All possible cases are encoded in the following proposition. Proposition 37. Let E be an elliptic curve over a finite field k. Let O be its endomorphism ring, f its conductor, D its discriminant, π the Frobenius endormphism, f π the conductor of Z[π]. Let be a prime different from the characteristic of k, then the types of degree isogenies with domain E are as follows: • If |f and (f π /f ), there is one ascending isogeny; • If |f and |(f π /f ), there is one ascending isogeny and descending ones; • If f and (f π /f ), there are 1 + D horizontal isogenies, where D represents the Legendre symbol; • If f there are 1 + D horizontal isogenies, plus − D descending isogenies only if |(f π /f ). Proof. See [44,Prop. 21]. Putting the pieces together, we see that graphs of ordinary curves have a very rigid structure: a cycle of horizontal isogenies (Elkies case), possibly reduced to one point (Atkin case), or to two points (ramified case); and a tree of descending isogenies of height v (f π ) (the -adic valuation of the conductor of π). Such graphs are called isogeny volcanoes for obvious reasons (have a look at Figure 7). The action of π on E[ ], or more generally on E[ k ] for k large enough, can be used to determine even more precisely which isogenies are ascending, descending or horizontal. We will not give details here, but see [54,55,37,21]. Application: computing irreducible polynomials In the applications seen in the first part, we have followed an old mantra: whenever an algorithm relies solely on the properties of the multiplicative group F * q , it can be generalized by replacing F * q with the group of points of an elliptic curve over F q (or, eventually, a higher dimensional Abelian variety). Typically, the generalization adds some complexity to the computation, but comes with the advantage of having more freedom in the choice of the group size and structure. We now present another instance of the same mantra, that is particularly remarkable in our opinion: to the best of our knowledge, it is the first algorithm where replacing F * q with E(F q ) required some non-trivial work with isogenies. Constructing irreducible polynomials of arbitrary degree over a finite field F q is a classical problem. A classical solution consists in picking polynomials at random, and applying an irreducibility test, until an irreducible one is found. This solution is not satisfactory for at least two reasons: it is not deterministic, and has average complexity quadratic both in the degree of the polynomial and in log q. For a few special cases, we have well known irreducible polynomials. For example, when d divides q − 1, there exist α ∈ F q such that X d − α is irreducible. Such an α can be computed using Hilbert's theorem 90, or -more pragmatically, and assuming that the factorization of q − 1 is known-by taking a random element and testing that it has no d-th root in F q . It is evident that this algorithm relies on the fact that the multiplicative group F * q is cyclic of order q − 1. At this point our mantra suggests that we replace α with a point P ∈ E(F q ) that has no -divisor in E(F q ), for some well chosen curve E. The obvious advantage is that we now require |#E(F q ), thus we are no longer limited to |(q − 1); however, what irreducible polynomial shall we take? Intuition would suggest that we take the polynomial defining the -divisors of P ; however we know that the map [ ] has degree 2 , thus the resulting polynomial would have degree too large, and it would not even be irreducible. This idea was first developed by Couveignes and Lercier [17] and then slightly generalized in [20]. Their answer to the question is to decompose the map [ ] as a composition of isogenieŝ φ • φ, and then take the (irreducible) polynomial vanishing on the fiber φ −1 (P ). More precisely, let F q be a finite field, and let (q −1) be odd and such that q +1+2 √ q. Then there exists a curve E which cardinality #E(F q ) is divisible by . The hypothesis (q − 1) guarantees that G = E[ ] ∩ E(F q ) is cyclic (see Exercice II.8). Let φ be the degree isogeny of kernel G, and let E be its image curve. Let P be a point in E (F q ) \ [ ]E (F q ), Couveignes and Lercier show that φ −1 (P ) is an irreducible fiber, i.e., that the polynomial f (X) = Q∈φ −1 (P ) (X − x(Q)) is irreducible over F q . To effectively compute the polynomial f , we need one last technical ingredient: a way to compute a representation of the isogeny φ as a rational function. This is given to us by the famous Vélu's formulas [76]. Proposition 38 (Vélu's formulas). Let E : y 2 = x 3 + ax + b be an elliptic curve defined over a field k, and let G ⊂ E(k) be a finite subgroup. The separable isogeny φ : E → E/G, of kernel G, can be written as φ(P ) =   x(P ) + Q∈G\{O} x(P + Q) − x(Q), y(P ) + Q∈G\{O} y(P + Q) − y(Q)   ; and the curve E/G has equation y 2 = x 3 + a x + b , where a = a − 5 Q∈G\{O} (3x(Q) 2 + a), b = b − 7 Q∈G\{O} (5x(Q) 3 + 3ax(Q) + b). Proof. See [19, §8.2]. Corollary 39. Let E and G be as above. Let h(X) = Q∈G\{O} (X − x(Q)). Then the isogeny φ can be expressed as φ(X, Y ) = g(X) h(X) , y g(x) h(x) , where g(X) is defined by g(X) h(X) = dX − p 1 − (3X 2 + a) h (X) h(X) − 2(X 3 + aX + b) h (X) h(X) , with p 1 the trace of h(X) and d its degree. Proof. See [19, §8.2]. The Couveignes-Lercier algorithm is summarized in Figure 8. What is most interesting, is the fact that it can be immediately generalized to computing irreducible polynomials of degree e , by iterating the construction. Looking at the specific parameters, it is apparent that is an Elkies prime for E (i.e., D = 1), and that each isogeny φ i is horizontal, thus their composition eventually forms a cycle, the crater of a volcano. Input: A finite field F q , a prime power e such that (q − 1) and q; Output: An irreducible polynomial of degree e . 1. Take random curves E 0 , until one with |#E 0 is found; 2. Factor #E 0 ; 3. for 1 ≤ i ≤ e do 4. Use Vélu's formulas to compute a degree isogeny φ i : E i−1 → E i ; 5. end for 6. Take random points P ∈ E i (F q ) until one not in [ ]E i (F q ) is found; 7. return The polynomial vanishing on the abscissas of φ −1 i • · · · • φ −1 1 (P ). Exercice II.5. Prove that the dual of a horizontal isogeny is horizontal, and that the dual of a descending isogeny is ascending. E 0 E 1 E 2 E 3 E 4φ0 Exercice II.6. Prove that the height of a volcano of -isogenies is v (f π ), the -adic valuation of the Frobenius endomorphism. Exercice II.7. Let X 2 − tX − q be the minimal polynomial of π, and suppose that it splits as (X − λ)(X − µ) in Z (the ring of -adic integers). Prove that the volcano of isogenies has height v (λ − µ). Exercice II.8. Prove that E[ ] ⊂ E(F q ) implies |(q − 1). Part III Cryptography from isogeny graphs 11 Expander graphs When we talk about Isogeny Based Cryptography, as a topic distinct from Elliptic Curve Cryptography, we usually mean algorithms and protocols that rely fundamentally on the structure of large isogeny graphs. The cryptographically interesting properties of these graphs are usually tied to their expansion properties. We recall some basic concepts of graph theory; for simplicity, we will restrict to undirected graphs. An undirected graph G is a pair (V, E) where V is a finite set of vertices and E ⊂ V × V is a set of unordered pairs called edges. Two vertices v, v are said to be connected by an edge if {v, v } ∈ E. The neighbors of a vertex v are the vertices of V connected to it by an edge. A path between two vertices v, v is a sequence of vertices v → v 1 → · · · → v such that each vertex is connected to the next by an edge. The distance between two vertices is the length of the shortest path between them; if there is no such path, the vertices are said to be at infinite distance. A graph is called connected if any two vertices have a path connecting them; it is called disconnected otherwise. The diameter of a connected graph is the largest of all distances between its vertices. The degree of a vertex is the number of edges pointing to (or from) it; a graph where every edge has degree k is called k-regular. The adjacency matrix of a graph G with vertex set V = {v 1 , . . . , v n } and edge set E, is the n × n matrix where the (i, j)-th entry is 1 if there is an edge between v i and v j , and 0 otherwise. Because our graphs are undirected, the adjacency matrix is symmetric, thus it has n real eigenvalues λ 1 ≥ · · · ≥ λ n . It is convenient to identify functions on V with vectors in R n , and therefore also think of the adjacency matrix as a self-adjoint operator on L 2 (V ). Then can we immediately bound the eigenvalues of G. Proposition 40. If G is a k-regular graph, then its largest and smallest eigenvalues λ 1 , λ n satisfy k = λ 1 ≥ λ n ≥ −k. Proof. See [72, Lem. 2]. Definition 41 (Expander graph). Let ε > 0 and k ≥ 1. A k-regular graph is called a (one-sided) ε-expander if λ 2 ≤ (1 − ε)k; and a two-sided ε-expander if it also satisfies λ n ≥ −(1 − ε)k. A sequence G i = (V i , E i ) of k-regular graphs with #V i → ∞ is said to be a one-sided (resp. two-sided) expander family if there is an ε > 0 such that G i is a one-sided (resp. two-sided) ε-expander for all sufficiently large i. Theorem 42 (Ramanujan graph). Let k ≥ 1, and let G i be a sequence of k-regular graphs. Then max(|λ 2 |, |λ n |) ≥ 2 √ k − 1 − o(1), as n → ∞. A graph such that |λ i | ≤ 2 √ k − 1 for any λ i except λ 1 is called a Ramanujan graph. The spectral definition of expansion is very practical to work with, but gives very little intuition on the topological properties of the graph. Edge expansion quantifies how well subsets of vertices are connected to the whole graph, or, said otherwise, how far the graph is from being disconnected. Definition 43 (Edge expansion). Let F ⊂ V be a subset of the vertices of G. The boundary of F , denoted by ∂F ⊂ E, is the subset of the edges of G that go from F to V \ F . The edge expansion ratio of G, denoted by h(G) is the quantity h(G) = min F ⊂V, #F ≤#V /2 #∂F #F . Note that h(G) = 0 if and only if G is disconnected. Edge expansion is strongly tied to spectral expansion, as the following theorem shows. Theorem 44 (Discrete Cheeger inequality). Let G be a k-regular one-sided ε-expander, then ε 2 k ≤ h(G) ≤ √ 2εk. Expander families of graphs have many applications in theoretical computer science, thanks to their pseudo-randomness properties: they are useful to construct pseudo-random number generators, error-correcting codes, probabilistic checkable proofs, and, most interesting to us, cryptographic primitives. Qualitatively, we can describe them as having short diameter and rapidly mixing walks. Proposition 45. Let G be a k-regular one sided ε-expander. for any vertex v and any radius r > 0, let B(v, r) be the ball of vertices at distance at most r from v. Then, there is a constant c > 0, depending only on k and ε, such that #B(v, r) ≥ min((1 + c) r , #V ). In particular, this shows that the diameter of an expander is bounded by O(log n), where the constant depends only on k and ε. A random walk of length i is a path v 1 → · · · → v i , defined by the random process that selects v i uniformly at random among the neighbors of v i−1 . Loosely speaking, the next proposition says that, in an expander graph, random walks of length close to its diameter terminate on any vertex with probability close to uniform. Proposition 46 (Mixing theorem). Let G = (V, E) be a k-regular two-sided ε-expander. Let F ⊂ V be any subset of the vertices of G, and let v be any vertex in V . Then a random walk of length at least log #F 1/2 /2#V log(1 − ε) starting from v will land in F with probability at least #F/2#V . Proof. See [39]. The length in the previous proposition is also called the mixing length of the expander graph. We conclude this section with two results on expansion in graphs of isogenies. 3. The graph of supersingular curves inF p with -isogenies is connected, + 1 regular, and has the Ramanujan property. Proof. See [66, V, Th. 4.1], [59,60], [8]. Theorem 48 (Graphs of horizontal isogenies are expanders). Let F q be a finite field and let O ⊂ Q[ √ −D] be an order in a quadratic imaginary field. Let G be the graph which vertices are elliptic curves over F q with complex multiplication by O, and which edges are (horizontal) isogenies of prime degree bounded by (log q) 2+δ for some fixed δ > 0. Assume that G is nonempty. Then, under the generalized Riemann hypothesis, G is a regular graph and there exists an ε, independent of O and q, such that G is a one-sided ε-expander. Proof. See [39]. Isogeny graphs in cryptanalysis Besides the applications to point counting mentioned in the previous part, the first application of isogenies in cryptography has been to study the difficulty of the discrete logarithm problem in elliptic curves. One can state several computational problems related to isogenies, both easy and hard ones. Here are some examples. Problem 1 (Isogeny computation). Given an elliptic curve E with Frobenius endomorphism π, and a subgroup G ⊂ E such that π(G) = G, compute the rational fractions and the image curve of the separable isogeny φ of kernel G. Vélu's formulas (Proposition 38) give a solution to this problem inÕ(#G) operations over the field of definition of E. This is nearly optimal, given that the output has size O(#G). However in some special instances, e.g., when φ is a composition of many small degree isogenies, the rational fractions may be represented more compactly, and the cost may become only logarithmic in #G. Problem 2 (Explicit isogeny). Given two elliptic curves E, E over a finite field, isogenous of known degree d, find an isogeny φ : E → E of degree d. Remark that, up to automorphisms, the isogeny φ is typically unique. Elkies was the first to formulate the problem and give an algorithm [25,26] with complexity O(d 3 ) in general, and O(d) in the special context of the SEA algorithm [7,50]. Alternate algorithms, with complexity O(d 2 ) in general, are due to Couveignes and others [13,14,15,23,21]. Problem 3 (Isogeny path). Given two elliptic curves E, E over a finite field k, such that #E = #E , find an isogeny φ : E → E of smooth degree. This problem, and variations thereof, is the one that occurs most in isogeny based cryptography. It is a notoriously difficult problem, for which only algorithms exponential in log #E are known in general. A general strategy to tackle it is by a meet in the middle random walk [30]: 24 E E weak curve strong curve E Figure 9: The meet in the middle attack in weak isogeny classes. choose an expander graph G containing both E and E , and start a random walk from each curve. By the birthday paradox, the two walks are expected to meet after roughly O( √ #G) steps; when a collision is detected, the composition of the walks yields the desired isogeny. The meet in the middle strategy was notoriously used to extend the power of the GHS attack on elliptic curves defined over extension fields of composite degree [35,32]. Without going into the details of the GHS attacks, one of its remarkable properties is that only a small fraction of a given isogeny class is vulnerable to it. Finding an isogeny from an immune curve to a weak curve allows the attacker to map the discrete logarithm problem from one to the other. The average size of an isogeny class of ordinary elliptic curves is O( √ #E), thus the meet in the middle strategy yields an O(#E 1/4 ) attack on any curve in the class: better than a generic attack on the discrete logarithm problem. The attack is pictured in Figure 9. Similar ideas have been used to construct key escrow systems [73], and to prove random reducibility of discrete logarithms inside some isogeny classes [39]. Provably secure hash functions The next application of isogeny graphs is constructing provably secure hash functions. The mixing properties of expander graphs make them very good pseudo-random generators. For the very same reason, they can also be used to define hash functions. The Charles-Goren-Lauter (CGL) construction [8] chooses an arbitrary start vertex j 0 in an expander graph, then takes a random walk (without backtracking) according to the string to be hashed, and outputs the arrival vertex. To fix notation, let's assume that the graph is 3-regular, then the value to be hashed is encoded as a binary string. At each step one bit is read from the string, and its value is used to choose an edge from the current vertex to the next one, avoiding the one edge that goes back. The way an edge is chosen according to the read bit need only be deterministic, but can be otherwise arbitrary (e.g., determined by some lexicographic ordering). The process is pictured in Figure 10. For the process to be a good pseudo-random function, the walks need to be longer than the mixing length of the graph. However this is not enough to guarantee a cryptographically strong hash function. Indeed the two main properties of cryptographic hash functions, translate in this setting as the following computational problems. Problem 4 (Preimage resistance). Given a vertex j in the graph, find a path from the start vertex j 0 to j. Problem 5 (Collision resistance). Find a non-trivial loop (i.e., one that does not track backwards) from j 0 to itself. Charles, Goren and Lauter suggested two types of expander graphs to be used in their constructions. One is based on Cayley graphs, and was broken shortly afterwards [75,58]. The second one is based on graphs of supersingular curves. In this context, the preimage finding problem is an instance of the isogeny path problem, while the collision finding problem is equivalent to computing a non-trivial endomorphism of the start curve j 0 . In this sense, the CGL hash function on expander graphs has provable security, meaning that its cryptographic strength can be provably reduced to well defined mathematical problems thought to be hard. Nevertheless, the CGL hash function has failed to attract the interest of practitioners. For one, it is considerably slower than popular hash functions such as those standardized by NIST. More worryingly, some weaknesses have recently been highlighted [45,57], that could potentially lead to backdoors in standardized parameters. Post-quantum key exchange We come to the last, more powerful constructions based on isogeny graphs. We present here two key exchange protocols, similar in spirit to the Diffie-Hellman protocol discussed in the first part. Both protocols are significantly less efficient than ECDH, however they are relevant because of their conjectured quantum security. In recent years, the case has been made that cryptographic standards must be amended, in view of the potential threat of general purpose quantum computers becoming available. It is well known, indeed, that Shor's algorithm [65] would solve the factorization and the discrete logarithm problems in polynomial time on a quantum computer, thus sealing the fate of RSA, ECDH, and any other protocol based on them. For this reason, the cryptographic community is actively seeking cryptographic primitives that would not break in polynomial time on quantum computers. Both protocols are based on random walks in an isogeny graph. The two participants, Alice and Bob, start from the same common curve E 0 , and take a (secret) random walk to some curves E A , E B . After publishing their respective curves, Alice starts a new walk from E B , while Bob starts from E A . By repeating the "same" secret steps, they both eventually arrive on a shared secret curve E S , only known to them. While the idea may seem simple, its realization is far from easy. Indeed, as opposed to the hash function case, we cannot be content with an arbitrary labeling of the graph edges. We must instead use the algebraic properties of the isogeny graphs to ensure that Alice and Bob's walks "commute". Hard homogeneous spaces The first protocol originates in a preprint by Couveignes [16], but was only later put into practice and popularized by Rostovtsev and Stolbunov [62,68]. It uses random walks in graphs of ordinary curves with horizontal isogenies; in this sense, it is a direct application of Theorem 48. The protocol can be viewed as a special instance of a general construction on Schreier graphs, a generalization of Cayley graphs. Definition 49 (Schreier graph). Let G be a group acting freely on a set X, in the sense that there is a map G × X → X (σ, x) → σ · x such that σ · x = x if and only if σ = 1, and σ · (τ · x) = (στ ) · x, for all σ, τ ∈ G and x ∈ X. Let S ⊂ G be a symmetric subset, i.e. one not containing 1 and closed under inversion. The Schreier g 1 x → x 2 x → x 3 x → x 5 graph of (S, X) is the graph which vertices are the elements of X, and such that x, x ∈ X are connected by an edge if and only if σ · x = x for some σ ∈ S. Because of the constraints on the group action and the set S, Schreier graphs are undirected and regular, and they usually make good expanders (see exercise III.2). Note that Cayley graphs are the Schreier graphs of the (left) action of a group on itself. As an example, take a cyclic group G of order n, then (Z/nZ) × acts naturally on G by the law σ · g = g σ for any g ∈ G and σ ∈ (Z/nZ) × . This action is not free on G, but it is so on the subset P of all generators of G; we can thus build the Schreier graph (S, P ), where S is a symmetric subset that generates (Z/nZ) × . An example of such graph for the case n = 13 is shown in Figure 11, where the set S ⊂ (Z/13Z) × has been chosen to contain 2, 3, 5 and their inverses. By slightly generalizing Couveignes' work [16], we will now show how to construct a key exchange protocol based on this family of Schreier graphs. We will restrict to cyclic groups of prime order p, and we will have the cryptosystem security grow exponentially in log p. Let G = g be a cyclic group of order p; let D ⊂ (Z/pZ) × be a generating set such that σ ∈ D implies σ −1 / ∈ D; and let S = D ∪ D −1 . We call directed route a sequence of elements of D. A directed route ρ ∈ D * , together with a starting vertex g ∈ G, defines a walk in the Schreier graph (S, G) by starting in g, and successively taking the edges corresponding to the labels in ρ. If ρ is a directed route, and g ∈ G, we write ρ(g) for vertex where the walk defined by ρ and g ends. We can now define a key exchange protocol where the secrets are random directed routes, and the public data are vertices of the Schreier graph. The protocol is summarized in Figure 12. A graphical example of this protocol with p = 13 and D = {2, 3, 5} is given in Figure 13. To understand why it works, observe that if ρ is a route of length m ρ = (σ 1 , . . . , σ m ), then ρ(g) = exp g σ i for any g ∈ G. Hence, the order of the steps in a route does not matter: what counts is only how many times each element of D appears in ρ. We immediately realize that this protocol is 27 Public parameters A group G of prime order p, A generating set D ⊂ (Z/pZ) × such that σ ∈ D ⇒ σ −1 / ∈ D, A generator g of G. Alice Bob Pick random secret Figure 12: Key exchange protocol based on random walks in a Schreier graph. Figure 13: Example of key exchange on the Schreier graph of Figure 11. Alice's route is represented by continuous lines, Bob's route by dashed lines. On the left, Bob computes the shared secret starting from Alice's public data. On the right, Alice does the analogous computation. ρ A ∈ D * ρ B ∈ D * Compute public data g A = ρ A (g) g B = ρ B (g) Exchange data g A −→ ←− g B Compute shared secret g AB = ρ A (g B ) g AB = ρ B (g A )g g A g AB g g B g AB nothing else than the classical Diffie-Hellman protocol on the group G, presented in a twisted way. 1 For this protocol to have the same security as the original Diffie-Helman, we need the public keys g A , g B to be (almost) uniformly distributed. Hence, we shall require that the graph is an expander, and that walks are longer than the mixing length; i.e., that D generates (Z/pZ) × , and that walks have length ∼ log p. Since a secret route is simply defined by the number of times each element of D is present, we shall also need #D ∼ log p/ loglog p in order to have a large enough key space. If we respect all these constraints, we end up with a protocol that is essentially equivalent to the original Diffie-Hellman, only less efficient. It is now an easy exercise to generalize to other Schreier graphs. To see how this applies to isogeny graphs, we must take a step back, and define some more objects related to elliptic curves. A fractional ideal is called principal if it is of the form xO for some x ∈ K. Note that the ideals of O are exactly the fractional ideals contained in O; however, from now on we will simply call ideals the fraction ideals, and we will use the name integral ideal for ordinary ones. An ideal I is said to be invertible if there is another ideal J such that IJ = O. Invertible ideals form an Abelian group, written multiplicatively, under the operation IJ = {xy | x ∈ I, y ∈ J}. It is easily verified that O is the neutral element of the group, and that principal ideals form a subgroup of it. The class group is a fundamental object in the study of number fields and their Galois theory. What is relevant to us, is the fact that the elements of Cl(O) are represented by horizontal isogenies, a fact that is developed in the theory of complex multiplication. We only take here a small peek at the theory; see [47,67,18] for a detailed account. Definition 52 (a-torsion). Let E be an elliptic curve defined over a finite field F q . Let O be the endomorphism ring of E, and let a ⊂ O be an integral invertible ideal of norm coprime to q. We define the a-torsion subgroup of E as E[a] = {P ∈ E | α(P ) = 0 for all α ∈ a}. Given an ideal a ⊂ O as above, it is natural to define the (separable) isogeny φ a : E → E a , where E a = E/E[a]. This definition can be readily extended to inseparable isogenies. Since a is invertible, we can show that End Assume that Ell q (O) is non-empty, then the class group Cl(O) acts freely and transitively on it; i.e., there is a map Cl(O) × Ell q (O) → Ell q (O) (a, E) → a · E such that a · (b · E) = (ab) · E for all a, b ∈ Cl(O) and E ∈ Ell q (O), and such that for any E, E ∈ Ell q (O) there is a unique a ∈ Cl(O) such that E = a · E. A set that is acted upon freely and transitively by a group G, is also called a principal homogeneous space or a torsor for G. An immediate consequence of the theorem above is that the torsor Ell q (O) has cardinality equal to the class number h(O). Following on from the connection between isogenies and ideals, suppose that that O splits into prime ideals as O = ll. Set S = {l,l}, then the Schreier graph of (S, Ell q (O)) is exactly the graph of horizontal -isogenies on Ell q (O). More generally, if we let S ⊂ Cl(O) be a symmetric subset, its Schreier graph is a graph of horizontal isogenies, and it is an expander if and only if S generates Cl(O). Based on this observation, we can now give a key exchange protocol based on random walks in graphs of horizontal isogenies. The general idea was already present in Couveignes' work [16], but it was Rostovtsev and Stolbunov who proposed to use isogeny computations to effectively implement the protocol [62,68]. The protocol implicitly uses the set Ell q (O) of elliptic curves over F q with complex multiplication by some order O; however it never explicitly computes O. Instead, it determines parameters in the following order: 1. A large enough finite field F q ; 2. A curve E defined over F q ; 3. The Frobenius discriminant D π = t 2 π − 4q of E is computed through point counting, and it is verified that it contains a large enough prime factor; 4. A set L = { 1 , . . . , m } of primes that split in Z[π], i.e., such that Dπ i = 1; For each prime i , the factorization π 2 − t π π + q = (π − λ i )(π − µ i ) mod i is computed, and one of the roots, say λ i , is chosen arbitrarily as positive direction. The condition on the i 's guarantees that each graph of i -isogenies on Ell q (O) is 2-regular. The choice of a positive direction allows us to orient the graph, by associating to λ i the isogeny with kernel E[ i ] ∩ ker(π − λ i ). The key exchange now proceeds like the ordinary Diffie-Hellman protocol: 1. Alice chooses a random walk made of steps in L along the positive direction; denote the walk by ρ A ∈ L * , and denote by E A = ρ A (E) the curve where the walk terminates. Note that E A only depends on how many times each i appears in ρ A , and not on their order. 2. Bob does the same, choosing a random walk ρ B and computing E B = ρ B (E). Alice and Bob exchange E A and E B . 4. Alice computes the shared secret ρ A (E B ). Bob computes the shared secret ρ B (E A ). The actual computations are carried out by solving explicit isogeny problems (see Problem 2), in much the same way they are done in the Elkies case of the SEA algorithm (see Section 8). The protocol is summarized in Figure 14. We conclude this section with a discussion on the security of the Rostovtsev-Stolbunov protocol. All the protocol's security rests on the isogeny path problem: given E and E A , find an isogeny φ : E → E A of smooth order. To be safe against exhaustive search and meet in the middle attacks as seen in Section 12, the set Ell q (O) must be large. On average # Ell q (O) ∼ √ q, thus we shall take log 2 q ≈ 512 for a security level of at most 128 bits. However, some isogeny classes are much smaller than average, this is why we also need check that D π has a large prime factor. Furthermore, for the public and private curves to be (almost) uniformly distributed in Ell q (O), we need the isogeny graph to be connected; equivalently, we need the ideals ( i , π−λ i ) to generate Public parameters An elliptic curve E over a finite field F q , D π , the discriminant of the Frobenius endomorphism of E, Cl(O). Theorem 48 ensures this is the case if #L ∼ (log q) 2 , however it is usually sufficient to take a much smaller set in practice. It is not enough to have an expander: we also need the random walks to be longer than the mixing length, that is ∼ log q. And, since the key space grows exponentially with #L, rather than with the walk length, we shall also ask that #L ∼ log q/ loglog q. When all conditions are met, the best known attack against this cryptosystem is the meet in the middle strategy, which runs in O( 4 √ q) steps. However, the real case for this system is made by looking at attacks performed on a quantum computer. It is well known that Shor's algorithm [65] breaks the Diffie-Hellman cryptosystem in polynomial time on a quantum computer, and thus it also breaks the protocol of Figure 12. More generally, Shor's algorithm can solve the (generalized) discrete logarithm problem in any Abelian group, and in particular in Cl(O). However, in the Rostovtsev-Stolbunov protocol, the attacker only sees E, E A and E B . Since there is no canonical way to map the curves to elements of Cl(O), it is not enough to be able to solve discrete logarithms in it. Childs, Jao and Soukharev [9] have shown how to adapt quantum algorithms by Regev [61] and Kuperberg [46] to solve the ordinary isogeny path problem in subexponential time. Although their attack does not qualify as a total break, it makes the Rostovtsev-Stolbunov protocol even less practical. Indeed, the protocol is already very slow, mainly due to the relatively large size of the isogeny degree set L. If parameter sizes must be further enlarged to protect against quantum attacks, it seems plausible that the Rostovtsev-Stolbunov protocol may never be used in practice. A set of primes L = { 1 , . . . , m } such that Dπ i = 1, A Frobenius eigenvalue λ i for each i , Alice Bob Pick random secret ρ A ∈ L * ρ B ∈ L * Compute public data E A = ρ A (E) E B = ρ B (E) Exchange data E A −→ ←− E B Compute shared secret E AB = ρ A (E B ) E AB = ρ B (E A ) Supersingular Isogeny Diffie-Hellman We finally come to the last cryptographic construction from isogeny graphs. Compared to the ordinary case, graphs of supersingular isogenies have two attractive features for constructing key exchange protocols. First, one isogeny degree is sufficient to obtain an expander graph; by choosing one small prime degree, we have the opportunity to construct more efficient protocols. Second, there is no action of an Abelian group, such as Cl(O), on them; it thus seems harder to use quantum computers to speed up the supersingular isogeny path problem. But the absence of a group action also makes it impossible to directly generalize the Rostovtsev-Stolbunov protocol to supersingular graphs. It turns out, however, that there is an algebraic structure acting on supersingular graphs. We have seen that, if E is a supersingular curve defined over F p or F p 2 , its endomorphism ring is isomorphic to an order in the quaternion algebra Q p,∞ ramified at p and at infinity. There is more: supersingular curves are in correspondence with the maximal orders of Q p,∞ , and their left ideals act on the graph like isogenies. It would be rather Figure 15: Supersingular isogeny graphs of degree 2 (left, blue) and 3 (right, red) on F 97 2 . technical to go into the details of the theory of quaternion algebras and their maximal orders; instead, we describe the key exchange protocol using only the language of isogenies, with the caveat that its security can only be properly evaluated by also looking at its quaternion counterpart. ker α = A ⊂ E[ e A A ] ker β = B ⊂ E[ e B B ] ker α = β(A) ker β = α(B) E E/ A E/ B E/ A, B α α β β The interested reader will find more details on quaternion algebras in [77,59,60,44,4,45]. The key idea of the Supersingular Isogeny Diffie-Hellman protocol (SIDH), first proposed in [38], is to let Alice and Bob take random walks in two distinct isogeny graphs on the same vertex set. In practice, we choose a large enough prime p, and two small primes A and B . The vertex set is going to consist of the supersingular j-invariants defined over F p 2 , Alice's graph is going to be made of A -isogenies, while Bob is going to use B -isogenies. Figure 15 shows a toy example of such graphs, where p = 97, A = 2 and B = 3. Even this, though, is not sufficient to define a key exchange protocol, because there is no canonical way of labeling the edges of these graphs. We shall introduce, then, a very ad hoc construction leveraging the group structure of elliptic curves. Recall that a separable isogeny is uniquely defined by its kernel, and that in this case deg φ = # ker φ. More precisely, a walk of length e A in the A -isogeny graph corresponds to a kernel of size B . This is illustrated in Figure 16. At this point, we would like to define a protocol where Alice and Bob choose random cyclic subgroups A and B in some large enough torsion groups, and exchange enough information to both compute E/ A, B (up to isomorphism), without revealing their respective secrets. We are faced with two difficulties, though: 1. The points of A (or B ) may not be rational. Indeed, in general they may be defined over a field extension of degree as large as e A A , thus requiring an exponential amount of information to be explicitly represented. Figure 16 shows no way by which Alice and Bob could compute E/ A, B without revealing their secrets to each other. The diagram in We will solve both problems by carefully controlling the group structure of our supersingular curves. This is something that is very hard to do in the ordinary case, but totally elementary in the supersingular one, as the following proposition shows. Theorem 54 (Group structure of supersingular curves). Let p be a prime, and let E be a supersingular curve defined over a finite field F q with q = p m elements. Let t be the trace of the Frobenius endomorphism of E/k, then one of the following is true: • m is odd and t = 0, or p = 2 and t 2 = 2q, or p = 3 and t 2 = 3q; • m is even and t 2 = 4q, or t 2 = q, and j(E) = 0, and E is not isomorphic to y 2 = x 3 ± 1, or t 2 = 0, and j(E) = 1728, and E is not isomorphic to y 2 = x 3 ± x. The group structure of E(F q ) is one of the following: • If t 2 = q, 2q, 3q, then E(F q ) is cyclic; • If t = 0, then E(F q ) is either cyclic, or isomorphic to Z/ q+1 2 Z ⊕ Z/2Z; • If t = ∓2 √ q, then E(F q ) (Z/( √ q ± 1)Z) 2 . Proof. See [77,51]. Of all the cases, the only one we are concerned with is q = p 2 , and E(F q ) (Z/(p ± 1)Z) 2 . The second problem is solved by a very peculiar trick, which sets SIDH apart from other isogeny based protocols. The idea is to let Alice and Bob publish some additional information to help each other compute the shared secret. Let us summarize what are the quantities known to Alice and Bob. To set up the cryptosystem, they have publicly agreed on a prime p and a supersingular curve E such that Figure 17: Supersingular Isogeny Diffie-Hellman key exchange protocol. E(F p 2 ) (Z/ e A A Z) 2 ⊕ (Z/ e B B Z) 2 ⊕ (Z/f Z) 2 . A e B B f ∓ 1, A supersingular elliptic curve E over F p 2 of order (p ± 1) 2 , A basis P A , Q A of E[ e A A ], A basis P B , Q B of E[ e B B ], Alice Bob Pick random secret A = [m A ]P A + [n A ]Q A B = [m B ]P B + [n B ]Q B Compute secret isogeny α : E → E A = E/ A β : E → E B = E/ B Exchange data E A , α(P B ), α(Q B ) −→ ←− E B , β(P A ), β(Q A ) Compute shared secret E/ A, B = E B / β(A) E/ A, B = E A / α(B) It will be convenient to also fix public bases of their respective torsion groups: E[ e A A ] = P A , Q A , E[ e B B ] = P B , Q B . To start the protocol, they choose random secret subgroups A = [m A ]P A + [n A ]Q A ⊂ E[ e A A ], B = [m B ]P B + [n B ]Q B ⊂ E[ e B B ], of respective orders e A A , e B B , and compute the secret isogenies α : E → E/ A , β : E → E/ B . They respectively publish E A = E/ A and E B = E/ B . Now, to compute the shared secret E/ A, B , Alice needs to compute the isogeny α : E/ B → E/ A, B , which kernel is generated by β(A). We see that the kernel of α depends on both secrets, thus Alice cannot compute it without Bob's assistance. The trick here is for Bob to publish the values β(P A ) and β(Q A ): they do not require the knowledge of Alice's secret, and it is conjectured that they do not give any advantage in computing E/ A, B to an attacker. From Bob's published values, Alice can compute β(A) as [m A ]β(P A ) + [n A ]β(Q A ), and complete the protocol. Bob performs the analogous computation, with the help of Alice. The protocol is summarized in Figure 17, and schematized in Figure 18. We end with a discussion on parameter sizes. ) storage) as follows: tabulate all possible walks of length e A /2 starting from E, then iterate over the walks of length e A /2 starting from E A , until a collision Figure 18: Schematics of SIDH key exchange. Quantities only known to Alice are drawn in blue, quantities only known to Bob in red. α(P B ) α(Q B ) E/ B β(P A ) β(Q A ) E/ A α(B) E/ A, B E/ B β(A) α β β α α(B) β(A) is found. The same collision can also be found with O( e A /3 A ) queries to a quantum oracle, using a quantum algorithm due to Tani [71]. Because the isogeny walks are shorter than the diameter, we expect to find only one collision, and that is precisely Alice's secret isogeny. It turns out these are the best known attacks against SIDH, even taking into account the additional information passed by Alice and Bob. Hence, taking log 2 p = n offers a classical security of ∼ n/4 bits, and a quantum security of ∼ n/6 qubits. In conclusion, to obtain a 128qubit and 192-bit secure system, we would have to find a 768-bit prime of the for p = e A A e B B f ±1, with e A log 2 A ∼ e B log 2 B ∼ 384. In practice, we usually take A = 2 and B = 3 for efficiency reasons, and an example of one such prime is p = 2 387 3 242 − 1. Further topics in isogeny based cryptography We conclude these notes with a brief overview of the current research topics in isogeny based cryptography. We only focus on constructions derived from supersingular isogenies, as they currently are the most promising ones. Efficient implementation of SIDH What makes SIDH interesting is its relatively good efficiency, especially when compared with other isogeny based protocols. However, several optimizations are required in order to achieve a compact and fast implementation, competitive with other post-quantum key-exchange candidates. In short, one must optimize each of these levels: • The arithmetic of F p benefits from the special form of p, especially for primes of the form p = 2 a 3 b − 1, as explained in [12,42,6]; • The arithmetic of F p 2 benefits from the fact that −1 is not a square in F p , whenever p = −1 mod 2; • The arithmetic of elliptic curves benefits from using Montgomery models, and optimized formulas for doublings, triplings, scalar multiplications and isogenies [22,12,10,27]; • Field inversions can be avoided using projective coordinates and projectivized curve equations [12]; • The full computation and evaluation of the secret isogeny from a generator of its kernel must be performed using a quasi-linear algorithm first described in [22]. Undoubtedly, the latter is the most novel and surprising of the optimizations. For lack of space, we do not describe any of them here, and we primarily address the interested reader to [22] and [12]. By putting together all the optimizations mentioned above, the SIDH scheme can be made relatively practical, as shown in [22,12], although one or two orders of magnitude slower than other post-quantum competitors. Where SIDH really excels, is in its very short key sizes, actually the shortest among post-quantum candidates, at the time of writing. This key size can be shrunk even more through key compression techniques [3,11]. However, the size of the isogeny graph in SIDH is much larger than the size of the key space, it is thus, in principle, possible to make even shorter keys; how to do this efficiently is still an open question. Security of SIDH We can formally state the security of SIDH as a hardness assumption on a problem called SSDDH. As mentioned previously, the best known algorithms for SSDDH have exponential complexity, even on a quantum computer. Assuming SSDDH is hard, we can formally prove the security of the key exchange against passive adversaries, i.e., those adversaries who can see all messages sent between Alice and Bob, but who do not modify them. We address the interested reader to [22] for the technical details. It is apparent that SSDDH is a very special instance of the isogeny path problem; it is thus conceivable that specially crafted algorithms could break SIDH without solving the generic isogeny path problem. As an illustration, consider the following problem. Problem 7. Let E, A , B , e A , e B , P A , Q A , P B , Q B be the parameters of an SIDH protocol. Let A ∈ E be a point of order e A A , and let φ : E → E/ A . Given E/ A , φ(P B ) and φ(Q B ) compute φ(R) for an arbitrary point R ∈ E of order e A A . It is easy to verify that solving this problem immediately reveals the secret A . Indeed, φ(R) is an element of kerφ, from which we can recoverφ and φ, and thus A . An efficient solution to this problem completely breaks SIDH, without doing anything for the generic isogeny path problem. 2 And indeed, although the security of SSDDH is still unblemished at the time of writing, several polynomial-time attacks have appeared against variations of SIDH. The interested reader will find more details in the following references: • A key-recovery attack against a static key version of SIDH, where Alice uses a long term secret isogeny [33]; • Key-recovery attacks in various leakage models [33,36,74]; • Key recovery attacks against some unbalanced variants of SIDH [56]. Finally, it is worth mentioning that there is a quantum subexponential attack [5] in the case where both E and E/ A are defined over F p . Other protocols Key exchange is not the only public-key protocol that can be derived from isogeny graphs. It is easy, for example, to derive a public-key encryption protocol similar to El Gamal from either the Rostovtsev-Stolbunov protocol or SIDH. We illustrate the second: • Alice's secret key is an isogeny α : E → E/ A ; her public key contains E/ A and the evaluation of α on Bob's basis P B , Q B . • To encrypt a message m, Bob chooses a random β : E → E/ B , and computes the shared secret E/ A, B , which he converts to a binary string s (e.g., by hashing the j invariant of E/ A, B ); he sends to Alice the message (E/ B , β(P A ), β(Q A ), m ⊕ s). • To decrypt, Alice uses E/ B , β(P A ), β(Q A ) to compute the shared secret E/ A, B , which she converts to s, and finally she unmasks m ⊕ s. In [22], it is proven that this protocol is IND-CPA secure under the SSDDH assumption. Achieving IND-CCA security is harder, as the attack against static keys in [33] shows, however it is possible to apply a generic transformation to obtain an IND-CCA secure key encapsulation mechanism. One may expect that digital signatures would also generalize easily to the isogeny setting, but both Schnorr signatures and ECDSA rely on the existence of a group law on the public data, something that is missing both in the ordinary and in the supersingular case. To our rescue, comes a zero-knowledge protocol based on the same construction shown in Figure 16. In this protocol, Alice's secret key is an isogeny α : E → E/ A ; her public key is the curve E/ A , together with a description of the action of α on E[ e B B ], as in SIDH. To prove knowledge of α to Bob, she takes a random subgroup B ⊂ E[ e B B ], computes a commutative diagram as in Figure 16, and sends to Bob the curves E/ B and E/ A, B . To verify that Alice knows the secret, Bob asks her one of two questions at random: After receiving Alice's answer, he accepts only if the points do define isogenies between the curves E, E/ A , E/ B , E/ A, B as expected. The protocol is summarized in Figure 19. Intuitively, if Alice respects the protocol, she always succeeds in convincing Bob. If she cheats, she only has one chance out of two of guessing Bob's challenge and succeed in tricking him. Thus, by iterating the protocol a sufficient number of times, a cheater's chance of success can be made arbitrarily small at exponential pace. The protocol is zero-knowledge because revealing B and α(B) does not reveal anything that Bob does not already know. Revealing β(A) is trickier, and we need to make one more security assumption, named Decisional Supersingular Product (DSSP), to prove zero knowledge. In [22] it is proven that this protocol is secure and zero-knowledge under the SSDDH 3 and DSSP assumptions. 4 Using a generic construction, such as the Fiat-Shamir heuristic [28], it is possible to derive a signature scheme from the zero-knowledge protocol above. Alternative signature schemes based on the same construction, with different desirable properties, are presented in [34,78]. However, all these protocols suffer from the high cost of having to iterate hundreds of times the basic building block of Figure 19. Obtaining an efficient signature scheme from isogeny assumptions is still an open problem. More protocols can be obtained by slightly generalizing the SIDH construction. If we allow the prime to be of the form p = e A A e B B e C C ± 1, we can construct a commutative cube in the same way the square of Figure 16 was constructed. Using primes of this form, Sun, Tian and Wang have proposed a strong designated verifier signature scheme [69]. Adding one more prime D in the mix, Jao and Soukharev have proposed undeniable signatures [40]. The drawback of all these schemes is that, as we add more torsion subgroups to the base curve, the size of the primes grows, making the schemes less and less practical. In general, isogeny graphs are much less flexible than the classical discrete logarithm problem. Many of the protocols that have been built on discrete logarithms fail to be ported to isogeny based cryptography. Devising new post-quantum protocols, retaining some of the desirable properties of classical ones, is a very active area of research in isogeny based cryptography. (x 0 0, . . . , x n ) ∼ (y 0 , . . . , y n ) Figure 1 : 1plus the point at infinity O = (0 : 1 : 0).In characteristic different from 2 and 3, we can show that any projective curve of genus 1 with a distinguished point O is isomorphic to a Weierstrass equation by sending O onto the point at infinity (0 An elliptic curve defined over R, and the geometric representation of its group law. Figure 2 : 2The Diffie-Hellman protocol over elliptic curves 5 Application: Elliptic curve factoring method A second popular use of elliptic curves in technology is for factoring large integers, a problem that also occurs frequently in cryptography.The earliest method for factoring integers was already known to the ancient Greeks: the sieve of Eratosthenes finds all primes up to a given bound by crossing composite numbers out in atable. Applying the Eratosthenes' sieve up to √ N finds all prime factors of a composite number N . Examples of modern algorithms used for factoring are Pollard's Rho algorithm and Coppersmith's Number Field Sieve (NFS). An integer N = pq, a bound B; Output: (p, q) or FAIL.1. Pick random integers a, X, Y in [0, N [; 2. Compute b = Y 2 − X 3 − aX mod N ; 3. Define the elliptic curve E : y 2 = x 3 − ax − b.4. Define the point P = (X : Y : 1) ∈ E(Z/N Z). 5. Set e = r prime <B r log r √ N ; 6. Compute Q = [e]P = (X : Y : Z ); 7. Compute q = gcd(Z , N ); 8. if q = 1, Figure 3 : 3The (p − 1) and ECM factorization algorithms Exercice I. 2 . 2Determine all the possible automorphisms of elliptic curves. Exercice I.3. Prove Proposition 9. Figure 4 : 4A complex lattice (black dots) and its associated complex torus (grayed fundamental domain). Figure 5 : 5Addition (left) and scalar multiplication (right) of points in a complex torus C/Λ. Proof. See [67, I, Th. 4.1]. 3-torsion group on a complex torus (red points), with two generators a and b, and action of the multiplication-by-3 map (blue dots). Figure 6 : 6Maps between complex tori. Figure 7 : 7A volcano of 3-isogenies (ordinary elliptic curves, Elkies case), and the corresponding tower of orders inside the endomorphism algebra. Proposition 36 ( 36Horizontal and vertical isogenies). Let φ : E → E be an isogeny of prime degree , and let O, O be the orders corresponding to E, E . Then, either O ⊂ O or O ⊂ O, and one of the following is true: • O = O , in this case φ is said to horizontal; • [O : O] = , in this case φ is said to be ascending; • [O : O ] = , in this case φ is said to be descending.Proof. See[44, Prop. 21]. Figure 8 : 8Couveignes-Lercier algorithm to compute irreducible polynomials, and structure of the computed isogeny cycle. Exercices Exercice II.1. Prove Lemma 16. Exercice II.2. Prove that y divides the m-th division polynomial ψ m if and only if m is even, and that no division polynomial is divisible by y 2 . Exercice II.3. Using the Sato-Tate theorem 33, prove that two curves are isogenous if and only if they have the same number of points. Exercice II.4. Prove Propostion 35. Theorem 47 ( 47Supersingular graphs are Ramanujan). Let p, be distinct primes, then 1. All supersingular j-invariants of curves inF p are defined in F p p = 11 mod 12 isomorphism classes of supersingular elliptic curves overF p ; Figure 10 : 10Hashing the string 010101 using an expander graph Figure 11 : 11Schreier graph of the generators of a group of order 13 under the action of S = {2, 3, 5, 2 −1 , 3 −1 , 5 −1 } ⊂ (Z/13Z) × . Definition 50 ( 50Fractional ideal). Let O be an order in a number field K. A fractional ideal of O is a non-zero subgroup I ⊂ K such that • xI ⊂ I for all x ∈ O, and • there exists a non-zero x ∈ O such that xI ⊂ O. Proposition 51 ( 51Ideal class group). Let O be an order in a number field K. Let I(O) be its group of invertible ideals, and P(O) the subgroup of principal ideals. The (ideal) class group of O is the quotient Cl(O) = I(O)/P(O).It is a finite Abelian group. Its order, denoted by h(O), is called the class number of O. (E) End(E a ) O, that E a only depends on the class of a in Cl(O), and that the map (a, E) → E a defines a group action of Cl(O) on the set of elliptic curves with complex multiplication by O. Theorem 53. Let F q be a finite field, and let O ⊂ Q[ √ −D] be an order in a quadratic imaginary field. Denote by Ell q (O) the set of elliptic curves defined over F q with complex multiplication by O. Figure 14 : 14Rostovtsev-Stolbunov key exchange protocol based on random walks in an isogeny graph. Figure 16 : 16Commutative isogeny diagram constructed from Alice's and Bob's secrets. Quantities known to Alice are drawn in blue, those known to Bob are drawn in red. Problem 6 ( 6Supersingular DecisionDiffie-Hellman). Let E, A , B , e A , e B , P A , Q A , P B , Q B be the parameters of an SIDH protocol.Given a tuple sampled with probability 1/2 from one of the following two distributions:1. (E/ A , φ(P B ), φ(Q B ), E/ B , ψ(P A ), ψ(Q A ), E/ A, B ), where • A ∈ E is a uniformly random point of order e A A , • B ∈ Eis a uniformly random point of order e B B , • φ : E → E/ A is the isogeny of kernel A , and • ψ : E → E/ B is the isogeny of kernel B ; 2. (E/ A , φ(P B ), φ(Q B ), E/ B , ψ(P A ), ψ(Q A ), E/ C ), where A, B, φ, ψ are as above, and where C ∈ E is a uniformly random point of order e A A e B B ; determine from which distribution the tuple is sampled. •Figure 19 : 19either reveal the point B and its image α(B), • or reveal the point β(A).ParametersPrimes A , B , and a prime p = elliptic curve E over F p 2 of order (p ± 1) 2 ,A basis P B , Q B of E[ e B B ]. Secret key An isogeny α : E → E/ A of degree e A A .Public keyThe curve E/ A , the images α(P B ), α(Q B ).Alice Bob Pick random B ∈ E[ e B B ] of order e B BCompute masking isogenyβ : E → E/ B Commit (E/ B , E/ A, B ) −→ Challenge ←− b ∈ {0, 1} Reveal if b = 0, send (B, α(B))if b = 1, send β(A) −→ Supersingular Isogeny Zero-Knowledge Identification protocol. This work is licensed under a Creative Commons "Attribution-NonCommercial 4.0 International" license. arXiv:1711.04062v1 [cs.CR] 11 Nov 201711 Expander graphs 22 12 Isogeny graphs in cryptanalysis 24 13 Provably secure hash functions 25 14 Post-quantum key exchange 26 15 Further topics in isogeny based cryptography 35 Part I e A A ; and this kernel is cyclic if and only if the walk does not backtrack. Hence, Alice choosing a secret walk of length e A is equivalent to her choosing a secret cyclic subgroup A ⊂ E[ e A A ]. If we let Alice choose one such subgroup, and Bob choose similarly a secret B ⊂ E[ e B B ], then there is a well defined subgroup A + B = A, B , defining an isogeny to E/ A, B . Since we have taken care to choose A = B , the group A, B is cyclic of order e AA e B Since we have full control on p, we can choose it so that E(F q ) contains two large subgroups E[ e A A ] and E[ e B B ] of coprime order. Hence, once e A A and e B B are fixed, we look for a prime of the form p = e AA e B B f ∓ 1, where f is a small cofactor. In practice, such primes are abundant, and we can easily take f = 1. This solves the first problem: E(F q ) now contains e A −1 A ( A + 1) cyclic subgroups of order e A A , each defining a distinct isogeny; hence, a single point A ∈ E(F q ) is enough to represent an isogeny walk of length e A . It is clear that the key space of SIDH depends on the size of the subgroups E[ e A A ] and E[ e B B ], hence we must take e A A ∼ e B B so to make attacks equally hard against Alice or Bob's public data. However this puts serious constraints on the isogeny walks performed in SIDH. Indeed, we have seen that the size of the supersingular isogeny graph is O(p), whereas the size of Alice's (or Bob's) public key space is only O( √ p). Said otherwise, Alice and Bob take random walks much shorter than the diameter of the graph. At the moment, it is not clear how this affects the security of the protocol.To choose an appropriate size for p, we start by looking at attacks that only use the jinvariants published by Alice and Bob. Given curves E and E A , connected by an isogeny of degree e A A , an easy variation on the meet-in-the-middle paradigm finds the secret isogeny in O(e A /2 A ) steps (and O( e A /2 A A minor difference lies in the fact that this protocol avoids non-primitive elements of G, whereas the classical Diffie-Hellman protocol may well use public keys belonging to a subgroup of G. The converse reduction is not evident either: given an oracle solving the isogeny path problem, how can we break SIDH? A partial answer is given in[45,33], where it is shown that, knowing the endomorphism rings of E and E/ A , an attacker can solve the isogeny path problem, and then break SIDH, in polynomial time. Actually, a weaker assumption named CSSI.4 The paper[22] also hints at a variant of the zero-knowledge protocol where Bob challenges Alice to open one out of three commitments, namely one of B, α(B), β(A). 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TaAs is a prime example of a topological semimetal with two types of Weyl nodes, W1 and W2, whose bulk signatures have proven elusive. We apply Landau level spectroscopy to crystals with multiple facets and identify-among other low-energy excitations between parabolic bands-the response of a cone extending over a wide energy range. Comparison with density functional theory studies allows us to associate this conical band with nearly isotropic W2 nodes. In contrast, W1 cones, which are more anisotropic and less extended in energy, appear to be buried too deep beneath the Fermi level. They cannot be accessed directly. Instead, the excitations in their vicinity give rise to an optical response typical of a narrow-gap semiconductor rather than a Weyl semimetal.

10.1103/physrevb.105.l081114

2111.11039

6b91b8dc6c1d65bfe6ecf84b1056c104dc2b634e

Addressing Shape and Extent of Weyl cones in TaAs by Landau level spectroscopy D Santos-Cottin Department of Physics University of Fribourg 1700FribourgSwitzerland J Wyzula LNCMI CNRS-UGA-UPS-INSA 25, avenue des MartyrsF-38042GrenobleFrance F Le Mardelé Department of Physics University of Fribourg 1700FribourgSwitzerland I Crassee LNCMI CNRS-UGA-UPS-INSA 25, avenue des MartyrsF-38042GrenobleFrance E Martino Department of Physics University of Fribourg 1700FribourgSwitzerland IPHYS EPFL CH-1015LausanneSwitzerland G Eguchi Institute of Solid State Physics Vienna University of Technology Wiedner Hauptstrasse 8-101040ViennaAustria Z Rukelj Department of Physics University of Fribourg 1700FribourgSwitzerland Department of Physics Faculty of Science University of Zagreb Bijenička 32HR-10000ZagrebCroatia M Novak Department of Physics Faculty of Science University of Zagreb Bijenička 32HR-10000ZagrebCroatia M Orlita LNCMI CNRS-UGA-UPS-INSA 25, avenue des MartyrsF-38042GrenobleFrance Institute of Physics Charles University CZ-12116Prague, PragueCzech Republic Ana Akrap Department of Physics University of Fribourg 1700FribourgSwitzerland Addressing Shape and Extent of Weyl cones in TaAs by Landau level spectroscopy (Dated: November 23, 2021) TaAs is a prime example of a topological semimetal with two types of Weyl nodes, W1 and W2, whose bulk signatures have proven elusive. We apply Landau level spectroscopy to crystals with multiple facets and identify-among other low-energy excitations between parabolic bands-the response of a cone extending over a wide energy range. Comparison with density functional theory studies allows us to associate this conical band with nearly isotropic W2 nodes. In contrast, W1 cones, which are more anisotropic and less extended in energy, appear to be buried too deep beneath the Fermi level. They cannot be accessed directly. Instead, the excitations in their vicinity give rise to an optical response typical of a narrow-gap semiconductor rather than a Weyl semimetal. Topological systems offer glimpses of relativistic-like physics within a crystal. The two most famous manifestations are Klein tunnelling in graphene [1], and the chiral anomaly [2] in Weyl semimetals. These are materials whose broken inversion symmetry leads to pairs of Weyl nodes in the reciprocal space. The elementary excitations near those nodes are chiral Weyl quasiparticles. In TaAs-the first discovered and the most prominent Weyl semimetal-24 such nodes are sprinkled around the Brillouin zone [3,4]. Among them, 8 are strongly anisotropic W 1 nodes, and 16 are fairly isotropic W 2 nodes [5]. The large number and two different kinds of Weyl nodes have complicated our understanding of the underlying Weyl physics. TaAs was proposed as a platform to benefit from the special properties of Weyl fermions [6][7][8]. Yet, most basic questions remain unanswered: How can one directly access the linearly dispersing Weyl cones? What is their energy range? When do their excitations dominate over other collective excitations? We address the Weyl cones of TaAs through Landau level and infrared spectroscopy, measuring several samples with multiple crystalline facets. Our key finding is that only one kind of Weyl cones in TaAs significantly contributes to its optical response. These cones are isotropic and large, extending over more than 250 meV. Comparison with density functional theory (DFT) studies allows us to recognize them as W 2 cones. In contrast, the W 1 cones are buried too deep below the Fermi energy to be optically accessible. Regardless, the electronic bands in their vicinity strongly contribute to the optical and magneto-optical response, because of their profound anisotropy, mirrored in a large joint density of states. This contribution dominates when the magnetic field is oriented along the [001] axis, effectively masking excitations in the W 2 cones. We grew single crystals of TaAs exposing large mirrorlike surfaces in different crystalline planes: (001), (101), (112). The samples were characterized through magnetotransport and quantum oscillations [9]. Infrared spectra were then taken using FTIR Vertex 70v spectrometer and employing an in situ coating technique [10]. Magnetooptical spectra were measured at 2 K, in fields up to 34 T. A single sample with large (001), (101) and (112) planes allowed us to confirm that all the measurements on different samples are reproducible and consistent [9]. The tetragonal symmetry of TaAs leads to a high number of Weyl points, shown in Fig. 1. Eight W 1 nodes with k c = 0 sit near the time-reversal invariant momentum point Σ of the first Brillouin zone. Sixteen W 2 nodes lurk further away from high symmetry points, with k c = 0. The connecting lines between all pairs of Weyl nodes are parallel with the (001) plane. The crystal unit cell is 4 times longer along the c axis than along the a axis. Such an elongated shape creates a steep angle of ∼ 74 • between the (001) and (101) planes, and an angle of ∼ 67 • between the planes (001) and (112). The (001) plane of TaAs is most commonly addressed in the literature. In contrast, other directions are much less explored. Figure 1a schematically shows the shape of W 1 and W 2 cones in the (001) plane, after the DFT calculations [11]. A cut through the same Weyl cones is shown for the (101) plane in Fig. 1b. Evidently, the W 2 cone is isotropic, but the W 1 cone becomes rather flat in the (101) plane. This flat energy dispersion of the W 1 cone along the c axis leads to an increased density of states (DOS). One may expect the same for the electronic states surrounding these cones. Topological semimetals can be understood in fine detail using Landau level (LL) spectroscopy [12][13][14][15][16][17][18]. Their small Fermi pockets and high carrier mobility enable us arXiv:2111.11039v1 [cond-mat.mes-hall] 22 Nov 2021 to reach the quantum regime and observe the inter-LL transitions, letting us map the band structure. Figures 2a-b and 2e-f show the colorplots of relative magnetoreflectance and its energy derivative, for (101) and (001) facet, respectively. The observed inter-LL resonances are narrow, so we associate the positions of maxima in R B /R 0 spectra directly with the transition energies. The spectra were confirmed on different samples [9], with similar carrier concentrations to the literature [5,19]. Magneto-optical spectra are exceptionally rich. Far below 1 T, TaAs reaches its quantum regime and the inter-LL transitions become apparent. Figure 2a-c shows a striking series of inter-LL transitions along the (101) facet. The spectra are dominated by a series of sublinear lines -a nearly √ B dependence of interband excitations on magnetic field. This indicates a strongly nonparabolic dispersion, as expected for a conical dispersion stemming from Weyl nodes [16,20]. TaAs is a multiband system with a complex magnetooptical response, therefore we cannot analyze the experimental data using simple effective Hamiltonians [13][14][15][16][17]. Instead, to understand our spectra, we adopt a semiclassical approach [21]. Lifshitz-Onsager formula says that in a magnetic field B, the area in momentum space encircled during cyclotron motion of an electron with crystal momentum k n equals S n = πk 2 n = (eB/h)(n + γ). Here n is an integer enumerating individual cyclotron orbits, in fact Landau levels. The parameter γ describes the Berry phase of the explored band. In the semiclassical approach, we may assume that the optical excitations conserve the momentum of electrons which are promoted between the electronic bands. Each interband inter-LL transition, observed at a given B and energy ω, then measures the distance between bands at a given momentum k n = (n + γ)2eB/ . In our case, rather than a Berry phase, γ represents an additional tuning parameter, compensating for the fact that momentum is no longer a good quantum number in quantizing magnetic fields. In reality, the excited electrons change their LL index, or the cyclotron orbit. For a simple system with full rotational symmetry around the direction of B, the selection rules n → n ± 1 apply in the Faraday configuration. The described procedure allows us to deduce the joint profile, E c − E v , of the involved conduction and valence bands. Assigning each excitation a particular orbital integer n, we obtain a fairly smooth profile (Fig. 2d), leading us to conclude that a well-defined conical band extends over more than 250 meV. There is no deviation from linearity up to highest experimentally probed energies. Assuming a full electron-hole symmetry, the slope gives a velocity of v = (2.0 ± 0.2) × 10 5 m/s. This simple result of our analysis is surprising. We observe only one type of Weyl cones, while the other one does not manifest in our data. The occupation effect-Pauli blocking-is the most likely explanation. We may also conclude that the observed conical band is nearly isotropic. Otherwise, the pairs of Weyl cones oriented along a and b crystallographic axes would have to contribute differently to the magneto-optical response when B is applied along the [101] direction, see Brillouin zone in Fig. 1. A weak anisotropy of the traced conical bands may explain the appearance of additional, fairly weak lines in our data (gray points in Fig. 2c). The analysis of the data collected on the (112)-oriented facet-43 • away from the (101) plane-also indicates a single conical band with a nearly identical velocity parameter [9]. DFT studies [4,11] show that W 2 cones should be isotropic. In contrast, W 1 cones have a flat dispersion along the tetragonal c axis, and their character is closer to quasi-twodimensional rather than three-dimensional. Therefore, the conical feature we observe is W 2 . In stark contrast, the (001) data in Fig. 2e-h shows over 50 discernible lines, where linear-in-B transitions completely dominate the response. Clearly, in the (001) plane we are primarily looking at excitations between parabolic-like bands. No trace is seen of √ B-like transitions, which would be primarily expected in a 3D Weyl semimetal. We may identify at least four separate series Fig. 2g. The oscillator strengths of these transitions are considerably stronger than for the excitations observed on other facets [9], effectively masking the √ B-like transitions. At the same time, these linear-in-B excitations are completely missing from the response of other facets. This apparent anisotropy allows us to establish that these excitations originate in the vicinity of the W 1 cones-around the Σ point-where the band structure becomes significantly flatter in the tetragonal c direction [11]. Let us now analyze the response of the (001) facet in greater detail. Repeating the semiclassical approach [22], we first identify which transitions belong together. The energy of each inter-LL transition is then plotted as a function of k n , after assigning each relevant transition an index n. We recognize up to four families of inter-LL transitions on the (001) facet. The most pronounced series of lines extrapolate to 15 meV in the limit of vanishing magnetic fields. More than 15 transitions collapse onto the same line in the energy versus k n plot in Fig. 2h. From this "collapsed" plot, representing joint bands, we see that the underlying nearly parabolic bands have a gap of ∆ ∼ 15 meV. A vertical line at any high field in Fig. 2g intersects a series of equidistant excitations. Their spacing, amounting to 15-20 meV at 30 T, gives the effective mass of these carriers, m * ≈ 0.4m e . The lowest transition, 0→1, emerges at 6-8 T, and is marked in dark red in Fig. 2g. This value gives us the size of the relevant Fermi pocket. In quantum oscillations, we have a frequency of 7-7.5 T, a size of the W 1 pocket which corresponds well to previously reported values [5,9,19]. Such a small pocket size would imply that W 1 is in the chiral regime. Yet, the response in the (001) plane appears to be dictated by a narrow-gap parabolic band. This striking absence of any observable Weyl cone signatures in the (001) orientation is an important result. It tells us that the W 1 cones are inaccessible for optical excitations, despite a very low carrier concentration. TaAs is closely related to TaP, whose (001) plane shows peculiar down-dispersing inter-LL transitions [16]. Those transitions are related to the band inversion and the linear dispersion from W 1 cones. In TaAs, nothing similar can be seen: no lines shift to lower energies as the field increases. This is another strong indication that the Fermi level in TaAs is not placed within the W 1 cones, and the system is above the topological separatrix. Finally, on the (001) facet of TaAs we see no evidence of nonparabolic behavior at high energies, contrary to TaP [16]. In TaAs, the W 1 nodes do not seem to play a role, and the conical dispersion from W 2 points becomes remarkably evident once we move away from the (001) plane. Fig. 2g contains several unusual low-energy lines shown in black. These lines are only seen in high fields. The lowest line begins at 25 T, crosses 7 meV at 30 T, and extrapolates to zero energy at zero field. Assuming this is a cyclotron resonance (CR), its associated effective mass is m * H = 0.5m e . Such heavy carriers in TaAs may be the trivial holes seen in quantum oscillations [5,9]. The CR line onset at 25 T corresponds to the heavy-hole pocket quantum limit. Three roughly equidistant lines appear above the lowest line. One may speculate that these are CR harmonics appearing due to strongly anisotropic band dispersion, similar to bulk graphite with electrons in the vicinity of a Lifshitz transition [23]. The trivial hole pocket in TaAs is indeed anisotropic and warped [5]. One pronounced yet unassigned inter-LL transition in the (001) plane is the line indicated in blue in Fig. 2g. This line does not naturally belong with the straight lines emanating from 15 meV. Instead, it extrapolates linearly into zero energy at zero field, but flattens significantly above 20 T. Such response may represent CR absorption of massive electrons in one of partly occupied, strongly non-parabolic bands. For both (101) and (001) facets, LL spectroscopy paints a full and complex picture of low-energy excitations in TaAs. This knowledge helps us better understand its zero-field infrared spectra. Optical conductivity primarily reflects the joint density of states, and thereby becomes strongly sensitive to the out-of-plane dispersion. Figure 3 shows the zero-field optical spectra of TaAs for (101) and (001) planes. The low-temperature reflectivity shows sharp plasma edges at very low energies (15-20 meV), in line with low carrier concentration and small scattering rate 1/τ , which are necessary conditions to observe the LL quantization in fields below 1 T (Fig. 2a,e). Through Kramers-Kronig transformations, we obtain the complex optical conductivity,σ = σ 1 + iσ 2 . Its real part σ 1 (ω) is shown in Fig. 3c-d. The large carrier mobility is reflected in a narrow Drude response, with 1/τ ∼ 1 meV. Insets in Fig. 3 shows that 1/τ follows a quadratic temperature dependence. Most relevant to the Weyl node physics are the interband excitations, which in TaAs set in at remarkably low energies. The 10 K absorption onset is at 15 meV for the (001) facet, in agreement with the gap from Fig. 2h. Interestingly, σ 1 (ω) has sharper features for the (001) facet, Fig. 3d, than for the (101) facet, Fig. 3c. This is caused by the flat dispersion of the bands near Σ point [11], enhancing their optical response. Contrary to initial reports [24,25], we see several sharp kinks in σ 1 (ω). Besides the absorption onset kink at 15 meV, the peaks at 70 and 128 meV can be linked to the zero-field extrapolation of several lines in Fig. 2h. These may be van Hove singularities that could originate anywhere in the Brillouin zone, but because of the DOS effect, their source is most likely in the bands near the Σ point. All those fine features are superimposed upon a broad background. We find that there is no simple meaning of the slopes in σ 1 (ω), nor a link to Weyl cone effective velocities. The W 1 pocket, which dominates the (001) plane response, is not in its chiral limit. In the (101) plane, Fig 3c, σ 1 (ω) has a smoother profile, with a narrow Drude component and a broad maximum at 120 meV. We expect a contribution of W 2 nodes in this orientation, although it is difficult to say what this contribution should look like. It seems abundantly clear that the W 2 contribution is not simply a straight line from 0 to the maximum cone extent. In conclusion, we found a strong difference in the optical and magneto-optical response of TaAs for different orientations, specifically planes (001) and (101). This anisotropic response is caused by the anisotropy of bands giving rise to the W 1 nodes. Crucially, we found that W 1 nodes are not optically accessible even in low-carrierdensity crystals of TaAs. The magneto-optical response of the (001) facet, most often addressed experimentally, does not show any direct signatures of Weyl physics. Instead, optical experiments can only probe the neighboring parabolic bands. 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A survey of the mSUGRA/CMSSM parameter space is presented. The viable regions of the parameter space which satisfy standard experimental constraints are identified and discussed. These constraints include a 124 − 127 GeV mass for the lightest CP-even Higgs and the correct relic density for cold dark matter (CDM). The superpartner spectra corresponding to these regions fall within the well-known hyperbolic branch (HB) and are found to possess sub-TeV neutralinos and charginos, with mixed Bino/Higgsino LSP's with 200 − 800 GeV masses. In addition, the models possess ∼ 3 − 4 TeV gluino masses and heavy squarks and sleptons with masses mq, ml > mg.Spectra with a Higgs mass m h ∼ = 125 GeV and a relic density 0.105 ≤ Ω χ 0 h 2 ≤ 0.123 are found to require EWFT at around the one-percent level, while those spectra with a much lower relic density require EWFT of only a few percent. Moreover, the SI neutralino-proton direct detection crosssections are found to be below or within the XENON100 2σ limit and should be experimentally accessible now or in the near future. Finally, it is pointed-out that the supersymmetry breaking soft terms corresponding to these regions of the mSUGRA/CMSSM parameter space (m 0 ∝ m 1/2 with m 2 0 >> m 2 1/2 and A 0 = −m 1/2 ) may be obtained from general flux-induced soft terms in Type IIB flux compactifications with D3 branes. PACS numbers:

10.1142/s0217751x13500619

1302.4394

678abc619480d713593d79a4da198de702e28f9f

SUSY into Darkness: Heavy Scalars in the CMSSM 22 Apr 2013 Van E Mayes Department of Chemistry The University of Texas at Tyler 75799TylerTX SUSY into Darkness: Heavy Scalars in the CMSSM 22 Apr 2013arXiv:1302.4394v3 [hep-ph]PACS numbers: A survey of the mSUGRA/CMSSM parameter space is presented. The viable regions of the parameter space which satisfy standard experimental constraints are identified and discussed. These constraints include a 124 − 127 GeV mass for the lightest CP-even Higgs and the correct relic density for cold dark matter (CDM). The superpartner spectra corresponding to these regions fall within the well-known hyperbolic branch (HB) and are found to possess sub-TeV neutralinos and charginos, with mixed Bino/Higgsino LSP's with 200 − 800 GeV masses. In addition, the models possess ∼ 3 − 4 TeV gluino masses and heavy squarks and sleptons with masses mq, ml > mg.Spectra with a Higgs mass m h ∼ = 125 GeV and a relic density 0.105 ≤ Ω χ 0 h 2 ≤ 0.123 are found to require EWFT at around the one-percent level, while those spectra with a much lower relic density require EWFT of only a few percent. Moreover, the SI neutralino-proton direct detection crosssections are found to be below or within the XENON100 2σ limit and should be experimentally accessible now or in the near future. Finally, it is pointed-out that the supersymmetry breaking soft terms corresponding to these regions of the mSUGRA/CMSSM parameter space (m 0 ∝ m 1/2 with m 2 0 >> m 2 1/2 and A 0 = −m 1/2 ) may be obtained from general flux-induced soft terms in Type IIB flux compactifications with D3 branes. PACS numbers: I. INTRODUCTION The recent discovery of a Higgs-like particle with a mass in the range 124 − 127 GeV is perhaps the single greatest development in high-energy physics in recent memory [1,2]. If this particle is indeed the Higgs scalar, it not only represents the final piece of the Standard Model (SM), but can potentially open a window into the world beyond the SM. However, an important question that must be answered is the problem of how such an elementary scalar remains so light against quantum corrections, an issue known as the hierarchy problem. An elegant solution to the hierarchy problem is supersymmetry (SUSY). One of the best motivated and most studied extensions of the Standard Model (SM) is the incorporaton of SUSY into the Minimal Supersymmetric Standard Model (MSSM). However, nature itself is not so elegant since the superpartners have not been observed with the same masses as their SM counterparts, and so SUSY must be a broken symmetry. Although the exact mechanism and scale at which SUSY is broken in nature should it exist is not known, simple calculations suggest that the masses of the superpartners should have O(1 TeV) masses if SUSY solves the hierarchy problem without requiring any fine-tuning. Moreover, it can be shown that there is an upper bound on the Higgs mass in the MSSM, m h 130 GeV [3], which is in nice accord with the Higgs-like resonance observed at the LHC. Moreover, in addition to providing a solution to the hierarchy problem, SUSY with R-parity imposed can provide a natural candidate for dark matter [4][5][6]. Finally, the apparent convergence of the gauge couplings when extrapolated to high energies is more precise when SUSY is incorporated compared to the non-SUSY SM, consistent with the idea of Grand Unification [7,8]. Despite these many attractive features of SUSY, data from the the Large Hadron Collider (LHC) has been infringing upon this rosy scenario as of late. In particular, the LHC has thus far failed to find any new particles beyond the SM. Indeed, direct searches for squarks and gluinos are pushing the mass limits for these particles into the TeV range [9][10][11][12][13]. Furthermore, to obtain a ∼ 125 GeV Higgs mass in the MSSM requires large radiative corrections involving the top/stop sector, requiring large stop squark masses O(TeV) and/or large values of tanβ. In spite of this, reports of the demise of SUSY are greatly exaggerated. Indeed, in some extended models it is possible to obtain a 125 GeV Higgs while maintaining a light spectrum of superpartners [14,15]. Perhaps the most-studied framework for supersymmetry breaking is minimal supergravity (mSUGRA), or equivalently the Constrained MSSM (CMSSM) [16][17][18]. However, to obtain a sufficiently large Higgs mass in mSUGRA/CMSSM seemingly requires heavy squarks and sleptons which generically spoils the naturalness in which the hierarchy problem is solved by introducing some amount of electroweak fine-tuning (EWFT). One possible exception to this is the hyperbolic branch (HB)/focus point (FP) region of the mSUGRA/CMSSM parameter space characterized by large m 0 in comparison to m 1/2 where the amount of required EWFT is minimized in respect to the full parameter space [19][20][21][22][23][24]. Several different groups have recently reassessed the status of mSUGRA/CMSSM in light of the ∼ 125 GeV Higgs discovery [25][26][27][28][29][30][31][32][33][34][35][36][37][38] (see [39] for a similiar analysis in the context of anomaly mediation). It is generally agreed that the mSUGRA/CMSSM parameter space is being squeezed by this discovery and pushed into regions which require a degree of fine-tuning. In [40], a study of the parameter space in regards to fine-tuning was performed and it was concluded that there are no regions where the Higgs is sufficiently heavy and where the relic density may satisfy the WMAP constraint that do not require large fine-tuning. In the following, scans of the mSUGRA/CMSSM parameter space have been performed. Viable regions of the parameter space, which appear to fall within the HB region of the CMSSM parameter space, are identified. In contrast to what was found [40], these regions do not seem to require excessive EWFT. The superpartner spectra corresponding to these regions will be found to possess sub-TeV neutralinos and charginos, with mixed Bino/Higgsino LSP's with 200 − 800 GeV masses. In addition, the models will be shown to possess ∼ 3−4 TeV gluino masses and heavy squarks and sleptons with masses mq, ml > mg. Spectra with a Higgs mass m h ≥ 125 GeV and a relic density 0.105 ≤ Ω χ 0 h 2 ≤ 0.123 are found to require EWFT at around the one-percent level, while those spectra with a relic density much lower require EWFT of only a few percent. Moreover, the spin-independent neutralino-proton cross-sections for direct detection of dark matter for these spectra are below the XENON100 limit [41,42] and should be experimentally accessible in the near future. Finally, it is pointed-out that the supersymmetry breaking soft terms corresponding to these regions of the mSUGRA/CMSSM parameter space (m 0 ∝ m 1/2 with m 2 0 >> m 2 1/2 and A 0 = −m 1/2 ) may be obtained from general flux-induced soft terms in Type IIB flux compactifications with D3 branes. II. PARAMETER SPACE The most studied model of supersymmetry breaking is minimal supergravity (mSUGRA), which arises from adopting the simplest ansatz for the Kähler metric, treating all chiral superfields symmetrically. In this framework, N = 1 supergravity is broken in a hidden sector which is communicated to the observable sector through gravitational interactions. Such models are characterized by the following parameters: a universal scalar mass m 0 , a universal gaugino mass m 1/2 , the Higgsino mixing µ-parameter, the Higgs bilinear Bparameter, a universal trilinear coupling A 0 , and tan β. One then determines the B and |µ| parameters by the minimization of the Higgs potential triggering REWSB [43,44] and leave tan β as a free parameter, while µ is determined by the requirement of REWSB. However, we do take µ > 0 as suggested by the results of g µ − 2 for the muon. In analyzing the resulting data, we consider the following experimental constraints: 1. The WMAP 9-year 2 − σ preferred range [50] for the cold dark matter density, 0.105 [1] The first results from the Planck experiment [51], with a slightly larger value for the dark matter density Ω c h 2 = 0.1199 ± 0.0027, appeared shortly after the first version of the paper was produced. Using the Planck result rather than the WMAP bounds results in a slight shifts in the parameter spaces shown in Figs. [1][2][3][4], but does not alter the fundamental conclusions of this paper. 3. The process B 0 s → µ + µ − which has recently been observed to be in the range In the following, we will not require that the anomalous magnetic moment of the muon [55], 4.7 × 10 −10 ≤ a µ ≈ 52.7 × 10 −10 , is solved by contributions from supersymmetric particles as the spectra that will be studied may only make a small contribution. ≤ Ω χ o h 2 ≤ 02×10 −9 < BF (B 0 s → µ + µ − ) < 4.7 × 10 −9 by LHCb [54]. Furtheremore, there are large hadronic contributions to this anomaly that require delicate m 0 = 6510, A 0 = −1560, tanβ = 30, µ > 0 and m t = 173.1 GeV. Here Ω χ o h 2 = 0.103, σ SI p−χ 0 = 3.575 × 10 −8 pb, ∆ EW = 111.4, Br(B s → µ + µ − ) = 3 .05 × 10 −9 , a µ = 0.2575 × 10 −10 , and Br(b → sγ) = 3.2 × 10 −4 . h 0 H 0 A 0 H ± g χ ± 1 χ ± 2 χ 0 1 χ 0 2 125χ 0 3 χ 0 4 t 1 t 2 u R / c R u L / c L b 1 b 2 728.7 1337 4516 5657 7026 6997 5666 6481 subtractions with large uncertainties [56]. d R / s R d L / s L τ 1 τ 2 ν τ e R / µ R e L / µ L ν e / ν In order to generate superpartner and Higgs spectra, we shall work within the mSUGRA/CMMSM framework. Here we will generate a set of soft terms for the mSUGRA/CMSSM parameter space. We take the top quark mass to be m t = 173.2 ± 0.9 GeV. We vary m 0 and m 1/2 each in increments of 10 GeV between 500 − 7500 GeV for each scan. In addition, we fix A 0 = −m 1/2 as this relation is typical of soft terms induced Here Ω χ o h 2 = 0.113, σ SI p−χ 0 = 3.089 × 10 −8 pb, ∆ EW = 162.9, Br(B s → µ + µ − ) = 3.06 × 10 −9 , a µ = 0.1920 × 10 −10 , and Higgs mass may be also obtained. Please note that although these plots seem to indicate that these spectra lie along a continuous band, they are actually interspersed with spectra Br(b → sγ) = 3.2 × 10 −4 . h 0 H 0 A 0 H ± g χ ± 1 χ ± 2 χd R / s R d L / s L τ 1 τ 2 ν τ e R / µ R e L / µ L ν e / ν µ where SuSpect is not able to converge to a solution. Sample spectra with m h = 125.2 GeV and Ω χ 0 h 2 = 0.103 are shown in Table I satisfies 124 GeV m h 127 GeV and for which the relic density satisifes the WMAP constraint, the gluino mass is in the range 3 − 4 TeV. Thus, these spectra result in the 'Higgsino World' scenario [57]. Due to the heavy masses for the gluino and squarks in these models, it would be very difficult to observe any superpartners at the LHC if the spectrum of superpartners falls into these regions of the parameter space. However, the prospects for observing superpartners at a linear collider or at a higher-energy hadron collider appear to be more promising. Similar results are obtained for tanβ = 20 and tanβ = 40 as can be seen in Fig. 2. While these spectra may not create an observable signal at the LHC, the relic neutralinoproton SI cross-sections for dark matter direct detection are currently being probed by the XENON100 experiment. Plots of the SI neutralino-proton cross-sections vs. neutralino mass are shown in Fig. 3 and Fig. 4. As can be seen from these plots, regions of the parameter space of the parameter space should either be excluded in the next update, or they should see a clear signal. In particular, the XENON1T experiment [58] should be able to completely probe this parameter space. However, it should be noted that the dark matter constraint on this parameter may only be imposed if R-parity is conserved. Thus, even if the XENON1T experiment reports negative results, the supersymmetry parameter space would still be viable if R-parity violation is allowed. III. FINE-TUNING One of the strongest reasons for introducing low-scale SUSY is to solve the hierarchy problem. The parameter space which has been found does this, however it is an important question whether or not this is accomplished naturally without reintroducing any fine-tuning (the little hierarchy problem). Ordinarily, such spectra with large scalar masses would generically be considered fine-tuned. This is not necessarily true for those spectra which fall in the HB region of the parameter space, such as those falling upon the red and black bands of ∆ EW ≡ max(C i )/(M 2 Z /2), (3.1) where C µ ≡ |−µ 2 |, C Hu ≡ −m 2 Hu tan 2 β/(tan 2 β − 1) , and C H d ≡ −m 2 H d /(tan 2 β − 1) . The percent-level of EWFT is then given by ∆ −1 EW . It should be noted that for most of the parameter space explored in this analysis C µ is dominant, and so generally we have ∆ EW = |−µ 2 | /(M 2 Z /2). In [40], it is argued that ∆ EW only provides a measure of the minimum amount of fine-tuning in regards to the electroweak scale and provides no information about the high scale physics involved in a particular model of SUSY breaking. In order to provide a measure of how fine-tuned a particular model is given knowledge of how SUSY is broken at a high energy scale, a parameter called ∆ HS was introduced which is analogous to ∆ EW [40]. For most of the parameter space this parameter is given by ∆ HS = m 2 0 + µ 2 (M 2 Z /2) = ∆ EW + m 2 0 (M 2 Z /2) . (3.2) As we can see, for regions of the mSUGRA/CMSSM parameter space with large scalar masses, ∆ HS is very large even for those cases where ∆ EW is small such as in the HB regions. This simply reflects the fact that, although a particular SUSY spectrum may be completely natural and solve the hierarchy problem without any fine-tuning, obtaining this spectrum within the mSUGRA/CMSSM framework of SUSY breaking requires large cancellations which only happens for specific sets of soft-terms rather than the general parameter space. In Fig. 5, a contour plot of ∆ EW vs. m h is shown for the parameter space satisfying Ω χ 0 h 2 ≤ 0.123. The different colored regions of the plot denote different ranges of Ω χ 0 h 2 . From this plot, it can be seen that the amount of EWFT appears to be proportional to m 2 Hu − m 2 H d as might be expected from the Higgs potential. In addition, the amount of EWFT seems to increase linearly with relic density. For spectra with a relic density 0.105 ≤ Ω χ 0 h 2 ≤ 0.123 and a Higgs mass m h = 124 GeV, ∆ EW ≈ 60 and thus requires minimum EWFT at about the two-percent level. On the other hand, spectra with the same relic density and a Higgs mass m h = 125 GeV requires minimum EWFT at the percent level. Conversley, spectra with a very low relic density can have a 125 GeV Higgs mass and only require EWFT at the five-percent level. However, in this case the neutralino LSP can only provide a small component of the cold dark matter (see [59] for a similar study prior to the discovery Higgs-like resonance). IV. FLUX-INDUCED SOFT TERMS ON D3 BRANES From the analysis of the previous sections, it can be seen that there are spectra within the mSUGRA/CMSSM parameter space which may solve the hierarchy problem while only requiring EWFT of a percent or greater. However, as discussed in the last section it does require a large amount of fine-tuning to obtain these spectra within the specific framework of supersymmetry breaking, mSUGRA/CMSSM. Within this framework, it is rather unnatural to have a universal scalar mass which is so much larger than the universal gaugino mass, m 2 0 >> m 2 1/2 , as large cancellations are required to obtain a light Higgs mass. Clearly, it would be desirable to have a specific model of supersymmetry breaking for which large scalar masses in comparison to the gaugino masses arise naturally. Over the past decade, there has been much progress in constructing realistic models in Type I and Type II string compactifications [60,61]. In these models, the SM fields are localized within the world-volume of D-branes embedded in a closed 10-d closed string background. Physical observables such as gauge and Yukawa couplings are dependent upon the moduli of compactification, which must be stabilized in order to have a true vaccum. It has been shown that in Type IIB compactifications non-trivial backgrounds of NSNS and RR 3-form field strength fluxes generically fix the VEVs of the dilaton and all complex structure moduli. Besides fixing the VEVs of the moduli fields, these fluxes may also induce SUSY-breaking soft terms. As shown in [62], for the most general combination involving both imaginary selfdual (ISD) and imaginary anti-selfdual (IASD) fluxes, the soft terms on D3 branes take the form neutralino is a mixture of Bino and Higgsino. It would be difficult for these spectra to produce an observable signal at the LHC. However, the prospects for their observation at a linear collider are much more promising. m 2 0 = |m1/2| 2 3 [1 − tanθ cos(δ + β)] , (4.1) A ijk = −m 1/2 h ijk . [2] However, there is one problem with this scenario. It is known that only ISD fluxes solve the equations of motion. Thus, it is only possible to stabilize the moduli with ISD fluxes. With only D3 branes and including both ISD and IASD fluxes, one may have soft terms of the desired form, but one would require other nonperturbative effects to stabilize the moduli. While these spectra may not create an observable signal at the LHC, the relic neutralinoproton SI cross-sections for dark matter direct detection are currently being probed by the XENON100 experiment. At present, regions of the parameter space with a Higgs mass m h 124 GeV and a relic density in the range 0.095 Ω χ 0 h 2 0.125 have been excluded by the upper limit on the proton-neutralino SI cross-section from XENON100. However, regions of the parameter space with m h 124 GeV and a relic neutralino density at or below the WMAP limit are still viable. These regions of the parameter space should either be excluded in the next update, or they should see a discernable signal. We have also investigated the question of fine-tuning with respect to both the electroweak scale and the high scale of supersymmetry breaking. We have found that the spectra with a large enough Higgs mass 124 GeV m h 127 GeV and the correct relic density 0.109 Ω χ 0 h 2 0.123 require at least a one-percent EWFT, while spectra satifying the Higgs constraint but which possess a low relic density are fine-tuned at the five-percent level. As these spectra fall into regions of the parameter space where m 2 0 ≫ m 2 Z , these spectra are highly fine-tuned with respect to the high scale, at least within the context of mSUGRA/CMSSM. Finally, we have discussed the inducement of SUSY-breaking soft-terms from supergravity fluxes which appear in Type IIB string compactifications. Such fluxes may be utilized in regards to the moduli stabilization problem of string theory compactifications. We have pointed out that for a general combination of ISD and IASD fluxes with D3 branes, the soft terms may have exactly the same form as those which give rise to the viable parameter space we have investigated in the paper, namely m 0 ∝ m 1/2 with m 0 ≫ m 1/2 and A 0 = −m 1/2 . Thus, in contrast to mSUGRA/CMSSM, having large m 0 compared to m 1/2 can potentially arise naturally within the context of Type IIB flux compactifications, in constrast to the situation with mSUGRA/CMSSM. VI. ACKNOWLEDGMENTS The author would like to thank The University of Texas at Tyler for providing resources which allowed the production of this paper to be possible, as well as D. V. Nanopoulos and J. Maxin for a critical reading of the manuscript. . 123 . 123We consider two cases, one where a neutralino LSP is the dominant component of the dark matter and another where it makes up a subdominant component such that 0 ≤ Ω χ o h 2 ≤ 0.123. 2. The experimental limits on the Flavor Changing Neutral Current (FCNC) process, b → sγ. The results from the Heavy Flavor Averaging Group (HFAG) [52], in addition to the BABAR, Belle, and CLEO results, are: Br(b → sγ) = (355 ± 24 +9 −10 ± 3) × 10 −6 . There is also a more recent estimate [53] of Br(b → sγ) = (3.15 ± 0.23) × 10 −4 . For our analysis, we use the limits 2.86 × 10 −4 ≤ Br(b → sγ) ≤ 4.18 × 10 −4 , where experimental and theoretical errors are added in quadrature. FIG. 1 : 1The mSUGRA m 1/2 vs. m 0 plane with A 0 = −m 1/2 , µ > 0, tanβ = 30, and m t = 173 GeV. The region shaded in black indicates a relic density 0.105 Ω χ 0 h 2 0.123, the region shaded in red indicates Ω χ 0 h 2 0.123, while the region shaded in green has a charged LSP. The black contour lines indicate the lightest CP-even Higgs mass. 4 . 4The lightest CP-even Higgs mass in the range 124 GeV m h 127 GeV as observed by the ATLAS and CMS experiments at the LHC [1, 2]. FIG. 2 : 2The mSUGRA m 1/2 vs. m 0 plane for tanβ = 20 and tanβ = 40 with A 0 = −m 1/2 , µ > 0, and m t = 173.1 GeV. The region shaded in black indicates a relic density 0.102 Ω χ 0 h 2 0.123, while the region shaded in red indicates Ω χ 0 h 2 0.123. The black contour lines indicate the lightest CP-even Higgs mass. II: Low energy supersymmetric particles and their masses (in GeV) for m 1/2 = 1910, m 0 = 7460, A 0 = −1910, tanβ = 30, µ > 0, and m t = 173.1 GeV. in Type IIB string compactifications. Scans are made for different values of tanβ, while µ is determined by the requirement of radiative electroweak symmetry breaking (EWSB). In addition to imposing experimental constraints, the spectra are filtered from the final data set if the iterative procedure employed by SuSpect does not converge to a reliable solution. A contour plot of the m 1/2 vs. m 0 plane for tanβ = 30 is shown in Fig. 1. Regions satisfying different constraints are as indicated on the figure. Here, we can see that there is a linear band shaded in black where the lightest neutralino relic density satisfies 0.105 Ω χ 0 h 2 0.123 which sits inside a broader linear band shaded in red where the relic density statisfies Ω χ 0 h 2 0.123. These bands lie along the HB branch of the mSUGRA/CMSSM parameter space. The values of the lightest CP-even Higgs mass are indicated on the plot by the black contours lines. It can be seen from this plot that there are regions of this parameter space where the relic density is in the range 0.105 Ω χ 0 h 2 0.123 and where the desired FIG. 3 : 3The spin-independent (SI) neutralino-proton direct detection cross-sections vs. neutralino mass for regions of the parameter space where Ω χ 0 h 2 ≤ 0.123. The region shaded in black indicates 0.105 Ω χ 0 h 2 0.123. The upper limit on the cross-section obtained from the XENON100 experiment is shown in blue with the ±2σ bounds shown as dashed curves, while the red dashed curved indicates the future reach of the XENON1T experiment. FIG. 4 : 4The spin-independent (SI) neutralino-proton direct detection cross-sections vs. neutralino mass for regions of the parameter space where Ω χ 0 h 2 ≤ 0.123. The region shaded in yellow-green a Higgs mass m h ≥ 124 GeV. The upper limit on the cross-section obtained from the XENON100 experiment is shown in blue with the ±2σ bounds shown as dashed curves, while the red dashed curve indicates the reach of the XENON1T experiment. FIG. 5 : 5with a Higgs mass m h 124 GeV and a relic density in the range 0.105 Ω χ 0 h 2 0.123 have been excluded by the upper limit on the proton-neutralino SI cross-section from XENON100.However, regions of the parameter space with m h 124 GeV and a relic neutralino density at or below the WMAP limit are still viable, at least within the 2σ range. These regions Contour plot of the ∆ EW vs. m h for the parameter space with Ω χ 0 h 2 ≤ 0.15. The different colors denote different ranges for the neutralino relic density, Ω χ 0 h 2 . The areas covered in green indicate regions of the parameter space with a relic density which falls into the WMAP preferred range. Fig. 1 . 1The amount of fine-tuning with respect to the electroweak scale (EWFT) is typically signified by the fine-tuning parameter For real flux backgrounds (δ = β = 0 mod 2π), and tanθ = 0 the flux-induced soft terms take the dilaton-dominated form of no-scale supergravity. In addition, for tanθ >> −1 one finds that m 0 ∝ m 1/2 with m 0 >> m 1/2 and A = −m 1/2 , which is exactly the form of the soft terms required to match the viable region of the parameter space found in the analysis of the previous sections. Thus, if the MSSM is built on D3 branes in Type IIB string theory with a combination of ISD and IASD fluxes in the background, then it is possible to induce soft terms of the form studied in this paper. In such a model of SUSY-breaking, large scalar masses with m 0 >> m 1/2 are then completely natural in constrast to the situation with mSUGRA/CMSSM where there is no a priori correlation between m 0 and m 1/2 . V. CONCLUSION We have surveyed the mSUGRA/CMSSM parameter space for tanβ = 20, tanβ = 30, and tanβ = 40 with the restriction A 0 = −m 1/2 . We have found that there are viable areas of the parameter space where the lightest CP-even Higgs mass is in the range 124 GeV m h 127 GeV, the relic neutralino density is below the WMAP constraint, Ω χ 0 h 2 ≤ 0.123, and standard experimental constraints are satisfied. These areas of the parameter space appear to lie along the HB regions of the mSUGRA/CMSSM parameter space. The corresponding spectra features neutralinos and charginos with sub-TeV masses, gluino masses in the range 3 − 4 TeV, and heavy squarks and sleptons with masses greater than 4 TeV. The lightest TABLE I : ILow energy supersymmetric particles and their masses (in GeV) for m 1/2 = 1560, TABLE , and with m h = 125.8 GeV and Ω χ 0 h 2 = 0.113 is shown inTable II. As is typical for spectra in the HB region of the parameter space, the lightest neutralino has a large Higgsino component while the squarks and sleptons all have masses greater than the gluino mass. 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We study radiative corrections to massless quantum electrodynamics modified by two dimension-five LV interactionsΨγ µ b ν F µν Ψ andΨγ µ b νF µν Ψ in the framework of effective field theories. All divergent one-particle-irreducible Feynman diagrams are calculated at one-loop order and several related issues are discussed. It is found that massless quantum electrodynamics modified by the interactionΨγ µ b ν F µν Ψ alone is one-loop renormalizable and the result can be understood on the grounds of symmetry. In this context the one-loop Lorentz-violating beta function is derived and the corresponding running coefficients are obtained.

10.1140/epjc/s10052-014-2875-6

1312.1505

037e4e1dc6f2542340a1fd05e977de88249ef906

Electrodynamics Modified by Some Dimension-five Lorentz Violating Interactions: Radiative Corrections 23 Apr 2014 Shan-Quan Lan Department of Physics Nanchang University 330031China Feng Wu [email protected] Department of Physics Nanchang University 330031China Electrodynamics Modified by Some Dimension-five Lorentz Violating Interactions: Radiative Corrections 23 Apr 2014(Dated: April 24, 2014)* Electronic address: 2 We study radiative corrections to massless quantum electrodynamics modified by two dimension-five LV interactionsΨγ µ b ν F µν Ψ andΨγ µ b νF µν Ψ in the framework of effective field theories. All divergent one-particle-irreducible Feynman diagrams are calculated at one-loop order and several related issues are discussed. It is found that massless quantum electrodynamics modified by the interactionΨγ µ b ν F µν Ψ alone is one-loop renormalizable and the result can be understood on the grounds of symmetry. In this context the one-loop Lorentz-violating beta function is derived and the corresponding running coefficients are obtained. I. INTRODUCTION Despite its success to account for nearly all phenomena with precision down to subatomic scales, the standard model of particle physics is incomplete and leaves several issues unsolved. Beyond the standard model, exploring the possible new physics involving a violation of Lorentz symmetry is an interesting and extensively studied subject in recent years. In particular, Colladay and Kostelecký [1] have systematically constructed Lorentz-violating (LV) terms of renormalizable dimensions and many related contents have been intensely investigated [2][3][4][5][6][7][8][9]. Nevertheless, the fact that no significant departure from Lorentz invariance has been observed in precision tests raises a subtle "Lorentz fine-tuning problem" [10] in this context. One possible resolution is that the currently unknown underlying theory prohibits the generation of the renormalizable LV operators at low energies. Probing this scenario at highenergy scales would be interesting but lies beyond the scope of this paper. However, this conception raises the interest in studying the LV terms of nonrenormalizable dimensions. Studies in the literature of nonrenormalizable LV operators are relatively scanty. In the framework of effective field theories, a nonrenormalizable theory treated as a low energy effective field theory, valid up to some mass scale M of new physics, might still be sensible and reliable predictions could be made from it. At low energies, effects due to nonrenormalizable terms are suppressed by inverse powers of M . This power suppression makes nonrenormalizable operators "safer" than the renormalizable ones. A few investigations have been carried out in this direction [11][12][13][14][15]. In this work, we focus on quantum electrodynamics (QED) modified by two dimensionfive LV interactionsΨγ µ b ν F µν Ψ andΨγ µ b νF µν Ψ, whereF µν ≡ 1 2 µναβ F αβ is the dual electromagnetic tensor and the fixed vector background b µ is assumed to be the only source that induces the Lorentz symmetry breaking. Several issues related to these two LV terms have been studied and non-trivial results are obtained [12][13][14][15]. In particular, it is found that with the operatorΨγ µ b νF µν Ψ, a charged spinor possesses a spin-independent magnetic dipole moment density, along with the usual one associated with its spin. Also, the degeneracy of the hydrogen energy spectrum is shown to be completely removed by the CP -even operator Ψγ i b jF ij Ψ. The LV operatorΨγ µ b ν F µν Ψ takes no part in determining the atomic energy spectrum. For more details, see Ref. [15]. From the field-theoretic point of view, it is interesting to study the quantum corrections of an effective theory containing nonrenormalizable LV terms. A general feature of a nonrenormalizable theory is that one would not be able to reabsorb all the ultraviolet (UV) divergent quantum corrections into the coupling constants in the original Lagrangian, and new counterterms permitted by symmetry are needed at each order of perturbative calculations. So far, even in the simplified case where massless QED is modified by the two non-minimal LV operators mentioned above, a comprehensive study of one-loop radiative corrections to this model is still lacking. Some one-loop calculations of the photon self-energy amplitude have been performed 1 [14]. The goal of this work is to fill this gap by determining all the divergent one-loop corrections and identify the higher dimensional counterterms that should be added to the Lagrangian at the beginning so that the theory is consistent at one-loop order. The rest of the paper is organized into three parts. In Sec. II, using the Feynman rules for massless QED modified by two non-minimal LV interactionsΨγ µ b ν F µν Ψ andΨγ µ b νF µν Ψ, the superficial degree of freedom of a general Feynman diagram is determined. We then compute all divergent radiative corrections to the Lagrangian at one-loop order and find out all new counterterms required in order to render the corrections finite. Some related issues are discussed along the way. In Sec. III, based on the results of Sec. II, we investigate the special case where massless QED is modified by only one LV operatorΨγ µ b ν F µν Ψ and argue the renormalizability of the theory in this context. The one-loop beta function for the LV coefficients b α is derived and then used to solve for the running LV coefficients. Our conclusions are given in the final section. II. ONE-LOOP CORRECTIONS We start with the LV model defined by the Lagrangian density as follows: L = − 1 4 F µν F µν +Ψ(iD / − γ µ b ν F µν − γ µ b νF µν )Ψ,(1) where the gauge covariant derivative takes the form D µ = ∂ µ + ieA µ with e being the gauge coupling constant determining the strength of the electromagnetic interaction. The mass dimension of the fixed vector background b µ and b µ is −1. Here b µ and b µ are chosen differently since we have absorbed possible coupling constants of the two LV terms into the fixed vector background. Following the standard procedure, perturbative analysis begins with gauge fixing. Feynman rules for the fermion and photon propagators are the usual ones. With the introduction of the LV terms in the Lagrangian (1), the fermion-photon vertex is given by V µ (q) = −ieγ µ + b · qγ µ − b µ / q − µ αβν b α γ β q ν(2) where q µ is the photon momentum pointing into the vertex. By naive power counting, the superficial degree of divergence D of a Feynman diagram is D = 4 − N γ − 3 2 N e + V,(3) where N γ is the number of external photon legs, N e is the number of external fermion legs, and V is the number of fermion-photon vertices. At one-loop order, we only need to consider the part of the diagram that is either zeroth or first order in coefficients of Lorentz violation. It would be inconsistent to include terms that are nonlinear in LV coefficients without also considering multiloop diagrams which could contribute at the same order [6]. Hereafter, whenever we refer to one-loop, we mean the part of one-loop that is at most linear in LV coefficients. Notice that although the Lagrangian (1) violates CP T , it preserves C parity. Therefore, the conventional Furry theorem holds and any vacuum expectation value of an odd number of currents vanishes. The one-loop four-point photon amplitude, in spite of having positive superficial degree of divergence from (3), is finite because of the requirement of gauge invariance. In summary, at one-loop order, there are four divergent one-particle-irreducible amplitudes, as shown in Fig. 1. For the remainder of this section, we will calculate all the divergent one-loop corrections. In order to extract the UV singularities, we adopt dimensional regularization to evaluate the integrals in d = 4 − dimensional spacetime. The applicability of standard dimensional regularization techniques in LV theories is discussed in [6]. A. Photon self-energy Applying the Feynman rules, an expression corresponding to the one-loop photon selfenergy iΠ µν (q) (Fig. 1a) is iΠ µν (q) = (−1) d d k (2π) d tr V µ (q) i / k V ν (−q) i / k + / q .(4) After manipulating the Dirac matrices, we have iΠ µν (q) = −ed d d k (2π) d 1 k 2 (k + q) 2 {e[k µ (k + q) ν − g µν k · (k + q) + k ν (k + q) µ ] +i ν αβσ b α q σ [k µ (k + q) β − g µβ k · (k + q) + k β (k + q) µ ] −i µ αβρ b α q ρ [k β (k + q) ν − g βν k · (k + q) + k ν (k + q) β ]}.(5) integration variable, and performing the momentum integral, we obtain iΠ µν (q) = ie (4π) d/2 2d Γ(2 − d 2 )(Γ( d 2 )) 2 Γ(d) (−q 2 ) d 2 −2 [e(q µ q ν − g µν q 2 ) − 2ig να g µβ αβσρ b ρ q σ q 2 ] = ie 2 6π 2 (q µ q ν − g µν q 2 ) − ie 3π 2 g να g µβ αβσρ b ρ q σ q 2 + finite part.(6) Some comments regarding this result are in order. First of all, current conservation guarantees that the result (6) obeys the Ward-Takahashi identity. This can be seen by dotting the photon momentum q µ into the amplitude (6), which gives zero. Second, while the divergent term proportional to i(q µ q ν −g µν q 2 ) can be renormalized by the usual QED counterterm proportional to F µν F µν , the second term in (6) shows that the one-loop correction to the photon-photon correlation function due to the LV operatorΨγ µ b νF µν Ψ generates a new type of divergence which cannot be absorbed in the original Lagrangian (1). It is straightforward to show that the new counterterm needed in order to cancel this divergence is of the form b α F µν ∂ µFαν . This is the leading higher derivative term allowed by symmetries. Finally, our result shows that the LV operatorΨγ µ b ν F µν Ψ does not contribute to photon self-energy at this order. B. Fermion self-energy The one-loop diagram contributing to the fermion self-energy is shown in Fig. 1b. This contribution, denoted by −iΣ(p), in Feynman gauge is given by − iΣ(p) = d d k (2π) d V µ (p − k) i/ k k 2 V ν (k − p) −ig µν (p − k) 2 .(7) By direct evaluation, we have − iΣ(p) = − e (4π) d 2 1 0 dx{Γ(2 − d 2 ) x(x − 1)p 2 d 2 −2 [ie(2 − d)x / p + 2(1 − x)x / p µαβν b α p ν γ µ γ β ] +Γ(1 − d 2 )(x(x − 1)p 2 ) d 2 −1 µαβν b α γ µ γ β γ ν } = e (Γ (d/2)) 2 (4π) d 2 (−p 2 ) d 2 −2 2ie Γ(2 − d 2 ) Γ(d − 1) / p − Γ(1 − d 2 ) Γ(d) αβµν b α ((2 − d) / pγ µ γ β p ν − p 2 γ µ γ β γ ν ) = ie 2 8π 2 / p − e 48π 2 αβµν b α (2p ν γ µ γ β / p + p 2 γ µ γ β γ ν ) + finite part = ie 2 8π 2 / p + ie 24π 2 (5p 2 / bγ 5 − 2(p · b) / pγ 5 ) + finite part.(8) In the last step, we have used the identity γ µ γ ν γ λ = η µν γ λ + η νλ γ µ − η µλ γ ν − i σµνλ γ σ γ 5 .(9) Note that Eq. (9) can be applied since the dimension dependence of γ 5 will not affect the results of simple poles in and we are only interested in the divergent terms of one-loop corrections in this paper. The first term in (8) is the usual QED correction. The second and third divergent terms indicate that new counterterms of the formΨ / bγ 5 ∂ 2 Ψ andΨb · ∂ / ∂γ 5 Ψ are needed. These two terms are not gauge invariant. Later, we will show that by combining all new counterterms, we can rewrite the set of counterterms in terms of gauge invariant operators. C. Three-point fermion-photon vertex Now we turn our attention to the vertex corrections. The calculation follows the same steps as for the self-energy diagrams. The one-loop contribution to the three-point vertex ( Fig. 1c), computed in Feynman gauge, is Γ ρ (p , p) = d d k (2π) d −ig µν k 2 V µ (k) i / p − / k V ρ (q) i / p − / k V ν (−k),(10) where p µ = p µ + k µ . After combining denominators by introducing Feynman parameters and shifting to a new loop momentum variable l, we have Γ ρ (p , p) = −2e 2 1 0 dxdydzδ(x + y + z − 1) d d l (2π) d 1 (l 2 + y(1 − y)p 2 + z(1 − z)p 2 − 2yzp · p) 3 { +eγ µ ( / p − / l − y / p − z / p)γ ρ ( / p − / l − y / p − z / p)γ µ −i( / l + y / p + z / p)( / p − / l − y / p − z / p)γ ρ ( / p − / l − y / p − z / p) / b +iγ µ ( / p − / l − y / p − z / p)[(b · p − b · p)γ ρ − b ρ ( / p − / p)]( / p − / l − y / p − z / p)γ µ +i / b ( / p − / l − y / p − z / p)γ ρ ( / p − / l − y / p − z / p)( / l + y / p + z / p) +2i µαβν b α (l + yp + zp) ν γ µ ( / p − / l − y / p − z / p)γ ρ ( / p − − / l − y / p − z / p)γ β −i ρ αβν b α q ν γ µ ( / p − / l − y / p − z / p)γ β ( / p − / l − y / p − z / p)γ µ }.(11) Then, a direct evaluation for the divergent contribution yields (12) Again, the identity (9) has been used in obtaining this result. Γ ρ (p , p) = −ie 2 (4π) 2 {2eγ ρ − 2i µαβν b α (γ µ γ ν γ ρ ( 1 3 / p − 1 6 / p )γ β + γ µ ( 1 3 / p − 1 6 / p)γ ρ γ ν γ β +( 1 3 p + 1 3 p) ν γ µ γ ρ γ β ) + 2i ρ µαβ b α (p − p) β γ µ } + finite part = −ie 3 8π 2 γ ρ − ie 2 24π 2 (5 / bγ 5 (p + p) ρ − b · (p + p)γ ρ γ 5 − b ρ ( / p + / p)γ 5 ) + finite part. The first term in (12) is the usual divergent vertex correction in QED. The other divergent terms in (12) reveal that the LV operatorsΨγ µ b ν F µν Ψ andΨγ µ b νF µν Ψ receive no divergent radiative corrections at one-loop order. Instead, LV counterterms of the formΨ{A µ , ∂ µ } / bγ 5 Ψ andΨb (µ γ ν) {∂ µ , A ν }γ 5 Ψ are required to absorb the new divergences. D. Four-point fermion-photon vertex The radiative corrections to the four-point fermion-photon amplitude in usual QED is finite. However, in the modified LV model (1), this is no longer true. At one-loop order, the four-point fermion-photon vertex receives a correction from the diagram Fig. 1d. In Feynman gauge, the diagram reads Γ µν (p, q 1 , q 2 ) = d d k (2π) d −ig ρσ k 2 V ρ (k) i / p − / k V µ (q 2 ) i / p − / k + / q 1 V ν (q 1 ) i / p − / k V σ (−k),(13) where p µ = p µ + q µ 1 + q µ 2 . Note that due to the symmetry of external photon legs, one only needs to consider the part of this diagram that is symmetric under µ ↔ ν. Defining a shifted momentum l ≡ k − (x + y)p − (y + z)p, one can show that the divergent contribution comes from the following integrals: − 6ie 3 d d l (2π) d 1 0 dxdydzdw δ(x + y + z + w − 1) (l 2 − ∆) 4 αβρσ b α l σ (γ β / l γ µ / l γ ν / l γ ρ − γ ρ / l γ µ / l γ ν / l γ β ),(14) where ∆ = ((x + y)p + (y + z)p) 2 − xp 2 − y(p + q 1 ) 2 − zp 2 . After a straightforward evaluation of the integrals, we arrive at Γ (µν) (p, q 1 , q 2 ) = ie 3 24π 2 (5g µν / bγ 5 − b (µ γ ν) γ 5 ) + finite part. Thus, two countertermsΨA 2 / bγ 5 Ψ andΨb · A / Aγ 5 Ψ are required to cancel these divergences. It is well-known in usual QED that gauge symmetry guarantees the equality of the coefficient of i / p in (8) with that of −ieγ ρ in (12). Here the fact that the coefficient of p 2 / bγ 5 in (8), that of −e / bγ 5 (p + p) ρ in (12), and that of e 2 g µν / bγ 5 in (15) are equal and the coefficient of −2b · p / pγ 5 in (8), that of e(b · (p + p)γ ρ γ 5 + b ρ ( / p + / p)γ 5 ) in (12), and that of −e 2 b (µ γ ν) γ 5 in (15) are equal is due to the same reason. Therefore, it is easy to show that all new counterterms needed to absorb these divergences can be combined into two gauge invariant operatorsΨD 2 / bγ 5 Ψ andΨb µ D (µ D ν) γ ν γ 5 Ψ, as it should be. In summary, starting with massless QED modified by two non-minimal LV interactions Ψγ µ b ν F µν Ψ andΨγ µ b νF µν Ψ, we have computed all UV divergent one-loop corrections and found out that three additional higher-derivative LV operators b α F µν ∂ µFαν ,ΨD 2 / bγ 5 Ψ and Ψb µ D (µ D ν) γ ν γ 5 Ψ should be included in the input Lagrangian in order to render quantum corrections finite and keep the predictiveness of the theory at one-loop order. III. b µ = 0 CASE From the results of straightforward calculations in the previous section, an interesting consequence is that other than the usual QED divergences, all new divergent corrections are induced by the LV operatorΨγ µ b νF µν Ψ. The other operatorΨγ µ b ν F µν Ψ does not contribute to one-loop divergences. The reason for this is that among all the gauge invariant, CP Tviolating and C-preserving operators that are linear in a fixed vector background, operator Ψγ µ b ν F µν Ψ is unique in the sense that none of the other operators has the same P and T transformation properties asΨγ µ b ν F µν Ψ has. More explicitly, b 0 term preserves P parity (and thus violates T parity) and b i terms preserve T parity (and thus violate P parity). It follows thatΨγ µ b ν F µν Ψ cannot mix with other dimension-five operators by quantum corrections. Then, in the special case where b µ = 0 in (1), the results of our analysis in the previous section show that all divergent corrections are the usual QED ones, which can be removed by the renormalization constants and interaction parameters in the original Lagrangian. Hence the theory governed by the Lagrangian L = − 1 4 F µν F µν +Ψ(iD / − γ µ b ν F µν )Ψ,(16) although containg a dimension-five interaction, is one-loop renormalizable. In this circumstance, given the results of the usual QED one-loop divergences, it is straightforward to determine the renormalization constants Z A,Ψ,e,b , which relate the bare fields, the bare coupling constant, and the bare LV coefficients to the renormalized ones by Ψ B = Z Ψ Ψ, A µ B = Z A A µ , e B = Z e e, b α B = (Z b ) α µ b µ .(17) The results are: Z Ψ = 1 − e 2 8π 2 , Z A = 1 − e 2 6π 2 , Z e = 1 + e 2 12π 2 , (Z b ) α µ b µ = b α + 5e 2 24π 2 b α .(18) From these renormalization constants, the beta function β b governing the one-loop running of the LV coefficients b α is found to be (β b ) α = 5 24 e 2 π 2 b α .(19) Solving the renormalization group equation, the one-loop running of the LV coefficients b α is given by b α (µ) = 1 − e 2 (µ 0 ) 6π 2 ln µ µ 0 − 5 4 b α (µ 0 ).(20) This result indicates that the LV coupling b α (µ) becomes weaker at low energies. Notice that this running is slow despite the fact that the mass dimension of b α is negative. In [6], based on the running behaviors of the coefficients associated with LV operators of mass dimension four or less, it is conjectured that there should be a rapid running for LV coefficients with negative mass dimension. However, this is not the case for the theory (16). In fact, Eq. (20) tells us that b α (M P l ) 1.08 b α (M W ),(21) where M P l and M W are, respectively, the Planck and electroweak scales. This modest running behavior is due to the fact that setting b α = 0 enhances the spacetime symmetry group of the specific model (16), which admits the background vector b α as an invariant tensor, from SO (3), SO(2, 1) or SIM(2) (depending on if b α is timelike, spacelike, or lightlike 2 , respectively) to the conformal group SO (4,2). In this paper, we have computed all UV divergent one-particle-irreducible Feynman diagrams in the LV theory (1) at one-loop order. The divergent corrections to the photon self-energy are given in Eq. (6), and those to the fermion self-energy are given in Eq. (8). The divergent corrections to the three-point and four-point fermion-photon vertices are given in Eq. (12) and Eq. (15), respectively. Our results indicate that other than the usual QED divergences, all new divergent corrections are due to the LV operatorΨγ µ b νF µν Ψ. Three additional higher-derivative LV operators b α F µν ∂ µFαν ,ΨD 2 / bγ 5 Ψ andΨb µ D (µ D ν) γ ν γ 5 Ψ are found to be required in order to keep the predictiveness of the theory at one-loop order. We have also shown the one-loop renormalizability of massless QED modified by the operatorΨγ µ b ν F µν Ψ, despite the negative mass dimension of the vector background b α . In this circumstance the one-loop beta function for the LV coefficients b α is determined and solved for the running coefficients. We argue that the slow running of b α (µ) between the electroweak and Planck scales described by Eq. (20) can be understood on the grounds of symmetry. 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We present a form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. It can also be viewed as a generalization of the exact milestoning method, or form of NEUS. We prove convergence of the method under certain assumptions, with expressions for the convergence rate in terms of the process's behavior within each stratum and large scale behavior between strata. We show that the algorithm has a unique fixed point which corresponds to the invariant measure of the process without stratification. We will show how the speeds of two versions of the new algorithm, one with an extra eigenvalue problem step and one without, relate to the mixing rate of a discrete process on the strata, and the mixing probability of the process being sampled within each stratum. The eigenvalue problem version also relates to local and global perturbation results of discrete Markov chains, such as those given by Van Koten, Weare et. al. arXiv:2111.05838v2 [math.PR]

2111.05838

122ce2daee3d2d8ebde3df80385e7487142da497

Convergence of Stratified MCMC Sampling of Non-Reversible Dynamics February 21, 2022 Gabriel Earle Jonathan Mattingly Departments of Mathematics and of Statistical Science Duke University Department of Mathematics University of Massachusetts-Amherst Convergence of Stratified MCMC Sampling of Non-Reversible Dynamics February 21, 2022 We present a form of stratified MCMC algorithm built with non-reversible stochastic dynamics in mind. It can also be viewed as a generalization of the exact milestoning method, or form of NEUS. We prove convergence of the method under certain assumptions, with expressions for the convergence rate in terms of the process's behavior within each stratum and large scale behavior between strata. We show that the algorithm has a unique fixed point which corresponds to the invariant measure of the process without stratification. We will show how the speeds of two versions of the new algorithm, one with an extra eigenvalue problem step and one without, relate to the mixing rate of a discrete process on the strata, and the mixing probability of the process being sampled within each stratum. The eigenvalue problem version also relates to local and global perturbation results of discrete Markov chains, such as those given by Van Koten, Weare et. al. arXiv:2111.05838v2 [math.PR] Introduction Markov Chain Monte Carlo (MCMC) is an often used method to produce samples from a distribution, when a Markov kernel converging to that distribution is known. Stratification of MCMC methods is a well-studied form of rare event sampling. Cases of interest include systems where regimes of low probability have outsize importance, or systems with multiple regimes of high probability but rare transitions between them. In such cases, the sample space can be broken up into smaller parts, called strata, and a Markov Chain can be run within each. The results are then combined in some way, allowing for an estimate of a distribution on the whole space which can be obtained more quickly than via a simple MCMC method. There are many advantages of dividing the pace into strata. Including the fact that many of the computations can be run in parallel and computational resolution and effort can be concentrated in regions of interest. In this article, we present and prove convergence results for a specific stratified MCMC scheme close to those in [4]. The algorithm, which we call the "injection measure method", constructs an estimate of the invariant measure of a Markov chain, and is built specifically with non-reversible Markov chains in mind. It can also be viewed as a version of Non-Equilibrium Umbrella Sampling (NEUS), as in [4], or as an extension of the exact milestoning method, as detailed in [1]. Non-reversible Markov chains sampling problems typically arise in two settings. First, when one samples an invariant measure which does not satisfy detailed balance; and hence, has a nontrivial flux through the system in its stationary state. Such stationary states are often referred to as non-equilibrium steady-states. Second, when one samples the non-reversible Markov chain can be obtained by adding time to the dynamics as one of the state-variables. Because of direction of time ensures that the system has a non trivial flux. As illustrated in [4], the resulting space-time dynamics can be used to study out-of-equilibrium transitions rates and other transient phenomenon. See [4] for more details. The key object of the method, and our analysis of it, is a collection of distributions on the strata known as injection measures. These estimate how particles following the Markov chain are likely to be distributed on the step when they enter one stratum from another. Also associated to each stratum is a corresponding weight, estimating how likely particles are to enter the stratum, relative to the other strata. If the injection measures and weights were known exactly, they could be used to compute estimates of the invariant measure within each stratum, and patch those estimates together with the correct weights. In practice, the injection measures and weights will most likely not be known, and so they must themselves be estimated. We propose estimating them iteratively, finding new injection measures via trajectories started from the current ones. This iteration is the main step of our formulation and those discussed in [4]. The corresponding weights can be calculated in two ways, leading to two versions of the injection measure method. In the first, the weights are found by applying a transition matrix determined by the measures to the previous weights. We call this form of the method the basic version. In the other, the principle eigenvector of the matrix is found, and its entries are taken to be the new weights. This is called the eigenvector version. The eigenvector version performs strictly better in our numerical experiments, but we as yet only have a proof of local convergence for it, whereas we prove global convergence of the basic version. In order to prove our main results, we need some assumptions about the behavior of the Markov chain being sampled and how it interacts with the chosen strata. In addition to some standard regularity assumptions, we will need to assumptions on two types of behavior. The first is a "microscopic" assumption, governing how particles following the chain move within a stratum. In effect, our assumption will be that any two particles starting somewhere in the same stratum have some chance of being coupled at or before the time of exit. We will also need a "macroscopic" assumption, about how the chain moves overall mass between strata. We will have several forms of this, depending on the version of the algorithm and precision of the theorem we wish to prove. However, each form of the assumption roughly states that the chain moves mass between the various strata at a suitable rate, or with enough regularity. The algorithm's overall convergence speed can then be expressed in terms of the "microscopic" coupling probability and the "macroscopic" rate. That is the substance of our main theorems. Specifically, our first result shows that, under the assumptions, the algorithm has a unique fixed point, and the injection measures given by the fixed point are correspond to the original invariant measure. Next, we prove that, if c is the microscopic coupling probability and λ is the macroscopic rate of convergence, then the algorithm converges to the fixed point in total variation at a rate that roughly looks like O(max( √ 1 − c, λ)). We also show that, for a constant r reflecting the sensitivity of the weights to the entries of a macroscopic transition matrix, the eigenvector version converges, for sufficiently good starting estimates, at a rate approaching 1 − c 1 + r . More precisely, we will show that, if G is the matrix of transition rates between strata for the dynamics in equilibrium, then r bounds the relative sensitivities of the invariant distribution of G to small changes in the entries of G. The structure of this article will be as follows. In Section 2, we give the basic notation needed for our results, and state the two versions of the injection measure method explicitly. Section 3, we outline the assumptions needed and state the main theorems we can now prove under them. In Section 4, we give some results of numerical simulations of the method on a simple example system, and relate them to some of our theoretical results. In Section 5, we give the proofs of our theorems in detail. The Injection Measure Method Intuition Behind the Algorithm To motivate the steps of the method, consider a space broken up into J subsets, or strata, and a process X n j , following a kernel P and confined to the j-th stratum. If we wish to sample the invariant distribution restricted to that stratum, then the question becomes what to do if a step drawn according to P would have X n j leave the stratum. One option would be to simply "bounce it back" from the boundary, which would mean setting X n+1 j = X n j . We could then keep track of how many attempted exits occur between the strata and use that to form a matrix G on the set of strata, or {1, . . . , J}. The empirical distributions of the X n j can then be combined, with weights given by the invariant distribution of G. The above scheme works under a key assumption: that the kernel P is reversible. The reason this is needed is that "bouncing a particle back" when it tries to leave a stratum is like saying that another particle comes to replace it at the exact location it left from. The assumption that this occurs for many particles in equilibrium is exactly the assumption that detailed balance holds, at least along the strata boundaries. But we are precisely interested in the non-reversible case. Therefore, we must have a better idea of how, and if, a particle leaving a stratum is replaced. Thus we propose building an injection measure for each stratum, which captures how particles entering that stratum are distributed. In addition, we need weights that capture how many particles enter each stratum, relative to the other strata. Of course, we cannot expect these to reflect how particles entering a stratum are in equilibrium, at first. Therefore, we will use the starting injection measures to calculate the distributions of particles leaving each stratum, called exit measures. Then, for each j, we look at the distribution of all particles leaving other strata, via their exit measures, and entering stratum j. We use this to form a new injection measure and weight for the j-th stratum, for all j. Thus we have an iterative procedure, building new injection measures and weights from old ones. We wish to show that the injection measures thus defined converge over time to some fixed point, called the equilibrium injection measures, and that they give us an estimate of the invariant measure of P . We call the method thus outlined the basic version of our algorithm. There is one further step we can add to the method. Just as in the reversible case, we can make a transition matrix G based on how many particles leave each stratum into each other one, or how much mass the k-th exit measure gives ot the j-th stratum. Then we can replace the old weights on each iteration by the invariant vector of G. We call the method, with this added step, the eigenvector version of the injection measure method. We will be able to show, under certain assumptions, an improved bound for the rate of convergence for this version, with the drawback that we have not yet shown that it converges for any starting injection measures. That is, our theorems only show that it converges to the correct distribution locally. Setting and Notation Suppose we are interested in sampling from the invariant distribution π of a Markov kernel P, defined on a state space A ⊂ R d . The simplest version of the injection measure method of stratified sampling is to break A into subsets, or strata, A1, . . . AJ which partion the space. One then runs the Markov chain P , starting from a measure ν 0 j concentrated with in each strata Aj, until it exist. The algorithm calculates new starting measures {ν 1 j }j biased on the exists from the collection of {Aj}j. This process, which is iterated until the measure {ν n j }j converge also provides a collection of weights {aj} and occupation measure πj in each Aj so that the desired target measure π = ajπj. It the above setting is clear which of the strata Aj contain the process at any given times since they are disjoint. However, we are also interested in the case where there is overlap between the strata, and the point at which a particle is declared to have left one stratum and entered another is possibly random. Though we will consider a more general formulation, for the moment consider the following illustrative setting. Consider the following setup: we have strata A1, . . . , AJ covering A as before, but the strata are no longer disjoint. We also assume we have a partition of unity ψ1, . . . , ψJ whose suport coincides with the A1, . . . , AJ . Namely supp(ψj) = Aj and for all x ∈ A, ψj(x) ≥ 0 and j ψj(x) = 1 . When a particle enters Ajat X0 ∼ νj, a value κ is chosen from some distribution η on [0, 1]. The particle then moves according to P , with its position at time n being Xn ∼ P (Xn−1, · ). When ψj(Xn) < κ, the particle is declared to have exited Aj. The index of the stratum it exits into is chosen from 1, . . . j−1, j+1, . . . J, with probabilities proportional to by ψ1(Xn), . . . , ψj−1(Xn), ψj+1(Xn), . . . , ψJ (Xn). We want our results to cover this more general setting, because it is often computationally useful. In particular, one does not need to insure that the A1, . . . , AJ are disjoint which can be computationally intensive. Additionally, the "softening" of the exit boundary by introducing random exit times seems to soften any artifacts from the strata edges. We can fit this more general case, into our original setting by consider the following augmented space A = {(x, k) : x ∈ A k , 1 ≤ k ≤ J} with strata given by A j = {(x, j) : x ∈ Aj}. The A j are then a partition of A even when the Aj are only a cover of A (namely, not disjoint). We then extend the Markov dynamics of P to A in the stratified setting by defining a collection of kernels {P κ : κ ∈ [0, 1]} on A as follows. If the initial state is (X0, j0) then the new state (X1, j1) is constructed as follows: X1 ∼ P (X0, · ) and j1 = j0 if ψj 0 (X1) ≥ κ. If ψj 0 (X1) < κ then j1 is chosen randomly according the probabilities P (j1 = k) ∝ ψ k (X1) for k = j0 as described above. Then, when a particle is started in A j via the injection measure, a value of κ is drawn according to some probability measure η on [0, 1] , and the particle moves by P κ until it leaves A j . In light of this construction, we can recast this more setting into the initial frame work of disjoint intervals. Since all of the kernels P κ have the same action on x, i.e. the x-marginal of P κ ((x, j), ·) is independent of κ, and the x-marginal of the invariant distribution of each kernel is π. In particular, the kernel P (x, ·) =´[ 0,1] P κ (x, ·)η(dκ) acts the same on x as P . For the remainder of this article, we will use this last setting. We will drop the superscript of A , P κ , etc. and simply assume that the strata Aj are a partition of A, and that we have a family of Markov kernels Pκ on A. We will also require that P (x, ·) =´[ 0,1] Pκ(x, ·)η(dκ), so integrating the kernels over κ gives us back the kernel of the original process we wanted to sample. We will show that, in this setting, the injection measure method gives a way of approximating the invariant distribution of P . In fact, our results apply to any family of kernels Pκ to an collection of kernels Pκ which we will make more explicit later. However, our primary interest is the specific choice of kernels outlined above. Remark 2.1. Note that, if κ is always chosen to be 1, so that there is only one kernel P1, then we are back in the original setting, so our results here cover all the cases in which we are interested. Since the Aj are now assumed to be disjoint, we can define the index of a point in A as follows: Idx(x) = j if x ∈ Aj. Now, we are interested in the point at which the sample process exits a stratum, so, for any X0 = x ∈ A, define the exit time starting from to be τ = inf n ≥ 1 : Idx(Xn) = Idx(X0) where κ is first chosen according to η and then {Xn} n≥0 moves according to Pκ. Note that, since Idx(X0) is possibly random, we define the exit time starting from n = 1, so that an exit does not occur at time 0 even if Idx(X0) = Idx(x). We will write Px and Ex to be the respectively the probability and expected value when the process starts from the initial condition x. Similarly we will write Pν and Eν to be probability and expected value when the initial condition is distributed as a the probability measure ν. We can now define the main object of our algorithm, the exit kernel Q defined on A by Q(x, ·) = Px(Xτ ∈ · ). Note that Q does not depend on κ, because the step where κ is chosen is included in its definition. Given any injection measure ν with ν(Aj) > 0 for all j, we define the associated weights aj(ν) and stratified injection measure νj by aj = ν(Aj) and νj(·) = ν( · ∩ Aj) aj . Note that ν = j ajνj. Next we define transition matrix G by Gij = Pν j (Xτ ∈ Ai) = νjQ(Ai). Finally, for each j, define the exit measure from Aj by ξj(·) = Pν j (Xτ ∈ ·) = νjQ(·). Note that Gij = ξj(Ai). We will denote the above quantities associated with the associated with the equilibrium injection measure ν * by a * j , ν * j , G * and ξ * j . Notice that without loss of generality, we can assume that a * j > 0 for all j. Also, observe that , G * is the matrix which the index process in equilibrium follows. In equilibrium, this is truly a Markov provess on the index space since ν * = ν * Q. In this framework, the basic version of our algorithm proceeds by 1 ν n+1 = ν n Q, and we will want to show that ν n → ν * as n → ∞. Analyzing the stratified processes will require understanding their transition kernels. To this end, define, for each j ∈ {1, ..., J} and x ∈ Aj, the restricted kernel Pκ,j(x, ·) = Pκ(x, · ∩ Aj) Pκ(x, Aj) We will assume throughout that Pκ(x, Aj) > 0 if x ∈ Aj. Define the corresponding un-normalized kernel Pκ,j(x, ·) = Pκ(x, · ∩ Aj) and finally the un-normalized kernel restricted to leaving Aj : Pκ,j(x, ·) = Pκ(x, · ∩ A c j ). Denote the quasi-stationary distribution (QSD) of Pκ on Aj byνκ,j. That is,νκ,j is the unique invariant distribution ofPκ,j. The kernelsPj,Pj, andPj are defined by integrating the associated kernel over κ ∼ η, just as P was defined from Pκ. The last notation we need to introduce has to do with the relationship between injection measures and the invariant measure on the whole space (and approximations of it), which is what we originally wanted to sample from. Given an injection measure νj on a stratum Aj, define the corresponding occupation measure, and it's normalization, by µj(B) = Eν j τx−1 k=0 1 (X k ∈B) and πj(B) = 1 Eτν j Eν j   τν j −1 k=0 1 (X k ∈B)   , for B ⊂ A. Here Eτν j means the expected value of the exit time from Aj for a particle started at νj. If we also have weights aj for the strata, then we can define the total occupation measure on A: µ = j ajµj and π = 1 µ(A) j ajµj = 1 µ(A) µ. (As before, π * j , π * , etc. are defined analogously). Throughout this article, · will mean · T V unless otherwise stated. a, a * will mean the vectors (a1, ..., aJ ), (a * 1 , ..., a * J ), and similarly for other vectors and matrices. We will also denote the weights given by a total injection measure as a(ν) = (a1(ν), ..., aJ (ν)) = (ν(A1), ..., ν(AJ )). Statement of the Algorithm We are now ready to state how the injection measure method proceeds formally. The idea behind the algorithm is to start with some injection measures and weights, and on each iteration, calculate the exit measures given by them, then combine those into new exit measures and weights. The eigenvector version adds a step in which the starting weights are replaced by the eigenvector weights of the transition matrix given by the injection measures. For reference, we state the precise form of the injection measure method in Algorithm 1 Main Results Assumptions Roughly, in order to prove our convergence theorems, we need to first assume that the process is wellbehaved both within each stratum, and in terms of how it moves mass between the strata. Our strategy will then be to show that the exit process moves the weights closer to the true weights, at which point the behavior within each stratum lets us show that coupling can occur. The first two assumptions below can be thought of as non-degeneracy and regularity conditions on P and the strata. The first says, in effect, that P is well behaved and has a unique equilibrium, and that it can move mass into each strata from at least one of the others. The second is a Lyapunov-type condition we can use to control the growth of the exit measure for a well-behaved injection measure, even when some strata are unbounded. for j ← 1 to J do 6: for i ← 1 to M do 7: X n j,i (0) ← Random as ν n j 8: κ n j,i ← Random as η 9: while k < τ do 10: X n j,i (k + 1) ← Random as P κ n j,i (X n j,i (k), · ) 11: µ n j,i ← τ −1 k=0 δ X n j,i (k) 12: µ n j ← 1 M M i=1 µ n j,i 13: for j, k ← 1 to J do 14: G n kj ←| {i : Idx(X n k,i (τ )) = j} | 15: Normalize G to be a probablity transition matrix 16: if Basic Version then 17: for j ← 1 to J do 18: a n+1 j ← k a n k G n kj a n+1 ← a n G n 19: ν n+1 j ← 1 a n+1 j k a n k i:X n k,i (τ )∈Aj δ X n k,i (τ ) New injection measures 20: ν n+1 ← k a n+1 k ν n+1 k and µ n+1 ← k a n+1 k µ n+1 k 21: if Eigenvector Version then 22: z n ← Normalize solution of z n G n = z n z n = (z n 1 , . . . , z n J ) 23: ν n+1 j ← 1 z n j k z n k i:X n k,i (τ )∈Aj δ X n k,i (τ ) New injection measures 24: ν n+1 ← j z n j ν n+1 j and µ n+1 ← k z n+1 k µ n+1 k 25: Return : µ N Approximate of π and P (x, ·) =´[ 0,1] Pκ(x, ·)η(dκ). Assumption (A1). There exists a continuous function V : A → [0, ∞), a compact set K ⊂ A, and b, K > 0, γ ∈ (0, 1) such that 2 for all x ∈ A, κ ∈ [0, 1], PκV (x) ≤ γV (x) + b1K and if x ∈ K and Idx(x) = j, then Exτj ≤ K. Next is our primary assumption on the system's "microscopic" behavior, i.e. it's behavior inside a stratum. Intuitively, it states that any particle starting from any point in a stratum has a chance of coupling with the quasi-stationary distribution (QSD) of Pκ in that stratum before or at the time of exit. Therefore, it's exit distribution will look like, with some probability, what it would have been had the particle been injected according to the QSD. Assumption (A2). For all j and κ ∈ [0, 1], there exists a unique QSDνκ,j of Pκ in Aj. Letνj = [0,1]ν κ,j η(dκ), where η is the distribution κ is drawn from at the start of a trajectory. Then there exists a constant 0 < c < 1 such that, for any j and ν 0 j , ξ 0 j ≥ c ·ξj whereξj is the exit measure from Aj started fromνj, via the kernel Q. That is,ξj =νjQ. Next, we need an assumption on the "macroscopic" behavior, i.e. how mass moves between strata. (Assumption A0 ensures some movement between strata, but not does not ensure that it is global or give quantitative information.) We have three versions of this assumption, and a convergence theorem that holds under each.c The first form says that after enough exits, a particle has a probability, bounded from below, of being in any of the strata. i.e. the exit process explores the space after enough time. The second choice of assumption says that the transition matrix in equilibrium, G * , is geometrically ergodic, with mixing rate λ. This will allow us the prove a convergence result for the basic version with a more precise estimate of the convergence rate, combining the microscopic rate c and macroscopic rate λ. Note that the above assumptions could fail due to the index of a particle on the n-th exit being periodic. For example, if there are only two strata, then the index will keep flipping between one stratum and the other between exits. If this is the only problem keeping the assumptions from holding, it can be remedied by introducing laziness to the exit kernel. That is, Q can be replaced by pQ + (1 − p)I, where I is the identity kernel, and the periodicity will be removed. The last form of the macroscopic assumption pertains to the eigenvector version, and is borrowed from the perturbation results in [14]. Instead of giving the mixing rate of G * , we assume that the invariant distribution vector G * has bounded sensitivity to perturbations in its entries. We will then prove a convergence theorem for the eigenvector version, this time combining c and the sensitivity constants. We state all three macroscopic assumptions below. Assumption (B1). There exists u > 0 and m ≥ 1 such that, for any j, if X0 ∼νj and Xn is the position of the n-th exit from one stratum to another, then Law(Idx(Xn)) ≥ ua * for all n ≥ m. Assumption (B2). There exists m ≥ 1 and λ * ∈ (0, 1) such that, for n ≥ m and any two probability vectors a, b on {1, ..., J}, a(G * ) n+1 − b(G * ) n+1 ≤ λ a(G * ) n − b(G * ) n . Assumption (B3). There exists θ ik > 0 for i, k ∈ {1, ..., J}, depending only on G * such that, if G is an irreducible transition matrix on {1, ..., J} such that, if G, G * ≥ cG 3 and z, z * are the invariant measures of G, G * , resp., then for all j, sup j log(zj) − log(z * j ) ≤ i =k θ i,k | G ik − G * ik | Main Theorems We are now ready to state our main theorems, as well as give a brief strategy for proving them. The detailed proofs will be reserved for section 5. The first theorem states in effect that the equilibrium injection measure exists uniquely, and that the corresponding occupation measure is the original measure we wanted to sample. Our strategy for proving this involves manipulating the sub-stochastic kernelsPκ,j andPκ,j, and how they relate the injection and occupation measures. It is essentially a Poisson equation argument at heart: µj solves a Poisson-like equation in terms of νj, and combining these gives an equation for µ that reduces to µ(I − P ) = 0 in the equilibrium case. Theorem 3.1. Suppose that (A0) and (A1) hold. Then there exists a unique probability measure on A, ν * such that ν * V < ∞ and ν * Q = ν * . Furthermore,for the coresponding occupation measure, µ * = µ * P, where P (x, ·) =´[ 0,1] Pκ(x, ·)η(dκ). Our second main theorem establishes convergence of the basic version after enough steps are performed. We use a fairly straightforward coupling argument to prove it. The idea is that assumption (B1) states that two particles following the exit process Q can eventually be in the same stratum, at which point (A1) says that they have a chance of being coupled after the next iteration of Q. This establishes geometric ergodicity of Q. 1 − c 2 u. In particular, ν k(m+1) − ν * ≤ (1 − c 2 u) k for k ≥ 1. Next, we have a theorem giving a more precise rate of geometric convergence in the long term. To prove it, we will need to construct a new metric which balances the total variation distance between ν n and the true ν * , and the difference between their strata weights. The idea is that, even if no coupling is possible on a given step, the weights will still get closer to the truth, which will allow coupling with ν * to occur at some future time. Since the metric we show contraction in is equivalent to TV-distance, we get the desired result. Theorem 3.3. Suppose that (A0)-(A2) and (B1)-(B2) hold. For any ν 0 such that ν 0 V < ∞, there exists qn, ∀n ≥ 0, such that ν n+1 − ν * ≤ qn ν n − ν * and as n → ∞, qn → q := inf β∈(0,1)   inf 0<α< βc S(1−c) max 1 − βc + αS(1 − c), 1 + (1 − β)αλ 1 + (1 − β)α   (1) where S = 1 1−λ . The expression we get for the limiting rate is complex, but has a meaningful interpretation. Note that it takes the form of a maximum of a rate in terms of c and one in terms of λ. Therefore, it suggests that, of the macro-and microscopic rates, whichever is slower acts as a sort of bottleneck. It can also be shown straightforwardly that if 1 − c = 0, then the limiting rate is λ, and if λ = 0, then the limiting rate is order 1 2 in 1 − c. So we can conjecture that, to first order, it behaves like max( √ 1 − c, λ). Our final theorem establishes geometric convergence of the eigenvector version, for starting guess ν 0 sufficiently close to ν * . The final rate involves the microscopic coupling parameter c and a constant r ∞ relating to how sensitive G * 's eigenvector is to perturbations. In this way, it is like an analogue to the previous theorem, with λ replaced by r ∞ . It also does not have the bottleneck form, suggesting that if λ is a slow rate, then the eigenvector version may be faster than the basic version, which is to be expected. r ∞ = 2(1 − c) sup j (a * j ) sup i,k e θ ik − 1 a * i E = exp 2(1 − c) i sup k =i (θ ik ) Also suppose that ν 0 − ν * < 1 + r ∞ E 2(1 − c) sup i =k θ ik a * i 1 q − 1 , where q = 1 − c r ∞ E+1 . Then there exists pn, ∀n ≥ 0, such that ν n+1 − ν * ≤ pn ν n − ν * and p n → 1 − c r ∞ + 1 . Numerical Simulations We now turn to analyzing the results of numerical simulations of our algorithms. We focus mainly on the eigenvector version here, as in all our simulations its convergence is strictly faster than the basic version. but does not immediately suggest how to prove this. We include the code used to generate our results in the following git repository: https://gitlab.com/gabeearle/julia-code-stratified-2021 We test our method on the two-dimensional Maier-Stein system, as outlined in [11], with parameters used in [8]. This is a relatively simple low-dimensional system, but one which displays the non-reversibility which our methods are suited for. Furthermore, with the chosen parameters, the system displays both increased non-gradient effects and a double-well type invariant distribution. For reference, we state the version of the Maier-Stein system we use here. The dynamics evolve according to the following SDE: du = (u − u 3 − βuv 2 )dt + √ dW 1 t dv = −(1 + u 2 )vdt + √ dW 2 t where β, > 0 and W 1 , W 2 are independent Brownian motions. For β = 1, this system is non-reversible. For β > 4, it displays additional non-gradient behavior and unusual minimum action paths, as detailed in [8]. We use the parameters β = 10 and = 0.01. Fig. 1 Shows an approximate 2D histogram plot of the invariant distribution of this system, computed via a discretization of the dynamics. In Fig. 2, we show several ways to choose the strata. One is a subdivision of the space into 3 vertical ellipses, another into 5 smaller ones. The next is a version in which the 5 strata are rotated, so that they do not perfectly line up with the axes of the system. Finally, we also study the case where the space is divided into 6 circular strata, with varying sizes and more than two strata overlapping at once. We can begin by comparing the 1-dimensional projections of both estimates to get a sense of how well the algorithm approximates the double-well structure of the system. Fig. 3 shows the un-stratified and stratified (with 3000 exits) estimates of the u-marginal after 30 iterations. As we can see, they agree quite well, demonstrating the accuracy of our method. In Fig. 4 we plot, for each of the strata setups, the final total injection measure, i.e. our approximation of ν n , where n = 30 is our final iteration number.Note that in each case, in injection measure has Figure 1: A 2D histogram of the invariant measure for the Maier-Stein system. This approximation of the true density was generated by an (un-stratified) Runge-Kutta Method, with noise, run for 10 8 steps, and averaged over two runs. The u-marginal density of this distribution will be used as our benchmark for the true density when calculating the error of the stratified method in u. Computational time is measured as the total number of steps of time h that each particle has taken, up to the current iteration. Note that, for the two 5-strata setups, a dramatic increase in accuracy occurs somewhere between 1000 and 3000 exits run per iteration. converged to a density roughly along the boundaries of the ellipses, illustrating that the algorithm can be used to estimate this density as well as the invariant distribution of the system. We can also examine how much the number of exits per iteration affects performance. Fig. 5 shows a log-log plot of the error in the u-marginal of the invariant distribution vs. time, for a range of exits the algorithm was run with and for each setup. The error is approximately computed by taking the TVdistance between the histogram from the un-stratified run shown in Fig. 3, and the histogram computed from the stratified run, averaged up to up to iteration n. The time spent is computed as the total number of steps of the SDE taken in all strata up to iteration n. The results, are shown in Fig. 5. We can see that, in the 3-strata setting, increasing the number of exits (hence the number of sample from each exit measure) per iteration does not seem to affect the accuracy per computational time greatly. The runs of the algorithm with more exits do in fact decrease the error more quickly on each iteration, but the iterations also take longer, and in the end the effects balance out, and all three runs have similar trend lines of error vs. time. For both 5-strata setups, this behavior changes dramatically. The version of the algorithm run with 3000 exits still takes longer to perform, but also sees error decrease much more rapidly. The reason for such an increase in accuracy is not immediately clear, and could be a subject of future work. The interesting phenomenon is that there appears to be some threshold, in terms of the work put in on each iteration, at which the compounding error becomes much less of an issue. A possible hint as to why the accuracy threshold occurs can be found by examining how much the injection measure changes for each strata setup and each choice of the number of exits. We measure fluctuations in the total injection measure similarly to the error in the invariant measure: by taking the Figure 6: Fluctuation of the injection measure vs. time for the strata setups. The fluctuation is computed as the TV distance between the v-marginal of the total injection measure on one iteration and on the next. Computational time is computed as in Fig. 6. Corresponding to the jump in accuracy in the 5-strata setups is a lower and more regular fluctuation in the injection measure. TV-distance between a histogram of the approximate ν marginal in the v-direction on one iteration, and the same histogram on the next iteration. The results are shown, for all the setups, in Fig. 6. First, we can see that the total variation change in the injection measure from one iteration to the next remains quite small for each case. However, corresponding to the large drops in error fore the 5-strata setups are cases where, when going from 1000 to 3000 exits, the fluctuation not only decreases but seems to behave periodically. What this means for the algorithm's deeper behavior is not immediately clear, but it could be related to the accuracy threshold observed in Fig. 5. In Fig. 7, we plot the number of points in the injection measure of each stratum, for each setup and some choices of the number of exits per iteration. Unsurprisingly, we see in the 5-strata case that having more exits results in the more points in each injection measure. More specifically, we can see a large difference in how many points are in the "smallest" of the strata. Perhaps the accuracy threshold has some connection to there being enough exits that the least common strata still gets enough points to approximate ν n j reasonably well. In that case, an alternate version of the algorithm could be devised, in which more exits are sampled from the strata neighboring the "smallest" ones, as that is where it is most important to get samples. This would have the advantage of avoiding unnecessary computational work in the strata which naturally get many points injected already. We can already see from Fig. 7 that one of the main advantages of stratified sampling holds here: that the different strata have different sizes, in terms of how likely particles are to enter them, but we put in similar amounts of work in them, by always drawing the same number of points from each injection measure. We can derive some insight into how the error behaves by decomposing into the error in the weights or in each of the strata. In doing so, we can see how the error in the weights relates to the overall error, Figure 7: Injection measures sizes vs. iteration number for the strata setups. The injection measure size is the total number of points that exited into a given stratum on the current iteration. The figure illustrated that the size of each measure converges to some value and then fluctuates around it. Note that some strata have significantly more points injected than others, but the same number of injections are drawn and run to exit on an iteration for each, illustrating that similar amounts of work are done in each strata despite their different sizes. and in each stratum. In Fig. 8, We first plot the "occupation weights" of the strata for the 5 vertical strata setup, that is, their weight as a fraction of the total mass in the invariant measure. We then plot the error in the occupation measure for each stratum. For this, we use as a benchmark the occupation measure computed from an un-stratified run, where the measure in each stratum is formed from all points visited in that stratum over the run. The error is then taken to be the total variation distance between a histogram of the stratified run's occupation measure and this benchmark. Finally, in Fig. 8 we also show in error in the injection weights over time and the overall total variation error over time. Comparing the results for stratified runs with different numbers of exits per iteration, we see that the overall error behaves almost identically to the error in strata 2 and 3. This is not surprising, as those are the strata with by far the highest occupation weight. Since they dominate the mass of the invariant measure, most of the total error will be composed of the error in them. We can see that the error in the dominant strata sees a dramatic drop somewhere between 1000 and 3000 exits, suggesting that the same behavior we saw before in the total error is caused by the drop occurring in strata 2 and 3. Another observation to note is that the error in the weights also behaves very similarly to the error in the dominant strata. A sharp increase in accuracy in those occupation measures when going from 1000 to 3000 exits corresponds to a similar rise in accuracy in the injection weights. Recall that the injection weights at iteration n are given by the eigenvector of the transition matrix G n , which itself is a function of the injection measures and the exit kernel. The occupation and injection measures are also directly related to each other. The behavior in the weights being similar to that of strata 2 and 3 suggests that, for this choice of strata, the weights are especially sensitive to perturbations in those strata. Therefore, it seems that the weights are especially sensitive to perturbations in the corresponding entries in the G matrix. Hence, in this case, the strata which are dominate the invariant measure's mass and those to which the eigenvector weights are most sensitive are the same. On the other hand, for a different choice of strata, we see slightly different behavior. In Fig. 9, we plot the occupation weights, error in occupation measures and weights, and total error in the same way as before, but for the 6 circular strata setup. As before, we can see that the total error behaves similarly to that of the highest weight strata, this time strata 5 and 6. However, the injection weight error behaves more like the occupation error in the previous 4 strata, suggesting that the transition matrix G for this setup is more sensitive to the entries corresponding to transitions from those strata. So as before, some strata are more important to the overall error and some are more important than the weights, but this time the two do not line up. Strata with small occupation weight have a large influence on the accuracy of the eigenvector weights. This is important to note, as it illustrates one of the main motivations for our method: that strata which are visited rarely may still have outsize influence on something of interest. Proofs Existence and Uniqueness of the Equilibrium Injection Measure ν * The first result we must prove is that the fixed point of Q exists, and has the appropriate distribution. We will show this by manipulating the sub-stochastic kernels derived from Pκ, and using the ways they relate the injection, exit and occupation measures. First, however, we must establish some regularity. To this end, our first lemma states that, if a given injection measure has controlled growth with respect to the Lyapunov function in assumption (A1), then the corresponding exit and occupation measures exist, and also have controlled growth. Lemma 5.1. Suppose that (A1) holds. Let νj be a probability measure on Aj, with exit time, exit measure and occupation measure τj, ξj and µj, such that νjV < ∞. Then Eν j τj < ∞, ξj and µjare well defined, and ξjV < ∞, µjV < ∞. Furthermore, if π is the invariant distribution of Pκ for any κ ∈ [0, 1], then πV < ∞ and π |A j (I −Pj) < ∞. Figure 8: Occupation weights, individual strata errors, weights error and total error, for 5 vertical strata setup. The error in individual occupation measures is calculated as the TV distance between the u-histogram generated by an un-stratified run and the u-histogram for the occupation measure from the stratified algorithm, with the same bins as for the overall error. The total error behaves similarly to the error in the largest-weight strata, as does the weight error. Figure 9: Occupation weights, individual strata errors, weights error and total error, for the 6 circular strata setup. Errors are calculated the same way as in Fig. 10. In this case, the total error still behaves similarly to the error in the largest strata, but the weight error is more similar to that of the smaller strata, showing that low-weight strata can have significant importance. Proof. Fix κ ∈ [0, 1]. Let B ⊂ K c where K is defined in A1, and let τB = inf{n : Xn / ∈ B} Where X0 = x and Xn follows Pκ. Define V * B = inf{V (x) : x ∈ B} Following ideas from [13,12], we define the following process: Mn = V (Xn) V * B γ n − b n−1 k=0 1 (X k ∈K) γ k+1 with the filtration Fn = σ(Xn : k ≤ n). Then, using the condition of (A1) to bound E [V (Xn+1) | Fn] = P V (Xn), we can simplify E [Mn+1 | Fn] = E [V (Xn+1) | Fn] V * B γ n+1 − bE n−1 k=0 1 (X k ∈K) γ k+1 | Fn − bE 1 (Xn∈K) γ n+1 | Fn to show that E [Mn+1 | Fn] ≤ V (Xn) V * B γ n − b n−1 k=0 1 (X k ∈K) γ k+1 = Mn. Furthermore, if E | Mn |< ∞, then EV (Xn) < ∞ because the term b n−1 k=0 1 (X k ∈K) γ k+1 is bounded by bn, and so E[Mn+1] = E E V (Xn+1) V * B γ n+1 − b n k=0 1 (X k ∈K) γ k+1 | Fn ≤ E E V (Xn+1) V * B γ n+1 | Xn = E P V (Xn) V * B γ n+1 ≤ E γV (Xn) + b V * B γ n+1 ≤ γEV (Xn) + b V * B γ n+1 < ∞. Therefore, if EV (X0) < ∞, then E | Mn |< ∞ for all n ≥ 0, and so Mn is a super-martingale with the filtration Fn. Now suppose that x is a fixed point in B. Then 1 ≤ V (x) V * B and 1 (x∈K) = 0. Therefore, using the bounded stopping time n ∧ τB for fixed n, the optional stopping theorem implies that 1 γ n P(τB > n) ≤ E V (Xn) V * B γ n 1 (τ B >n) ≤ EMn∧τ B ≤ EM0 = V (x) V * B . Therefore, P(τB > n) ≤ γ n V (x 0 ) V * B , and so summing over n, we have EτB ≤ 1 1 − γ · V (x) V * B . It follows that if x ∼ ν and νV < ∞, then EτB < ∞. Now suppose that νV < ∞, and define the measure µB(·) = Eν τ B −1 k=0 1 (X k ∈·) . We now have that µB(B) = Eν τB ≤ 1 1 − γ · νV V * B so the total mass of µB is bounded. Next, let ξB(·) = E1 (Xτ B ∈·) be the measure of the point where Xn hits B. Using M0 ≥ EMn ≥ E V (Xn∧τ B ) V * B γ n∧τ B 1τ B >n and letting n → ∞, we have M0 ≥ EV (Xτ B ) V * B and so EV (Xτ A ) ≤ V (X0). Therefore, ξBV ≤ νV ≤ ∞. Setting B = Aj ∩K c , and with compactness of Aj ∩K, the result follows. Our next lemma, which is the key to proving the first main theorem, relates an injection measure to its occupation and exit measure through the sub-stochastic kernels. Recall that, when a trajectory to exit is started, the threshold that determines when an exit occurs, κ, is chosen from a distribution η on where I is the identity kernel. Note that the true occupation and injection measures are obtained from those above by integrating over κ: µj(·) =´[ 0,1] µκ,j(·)η(dκ), ξj(·) =´[ 0,1] ξκ,j(·)η(dκ). Proof. For the first equality, the definition of the occupation measure gives µκ,j(·) = Eν j   τκ, j −1 k=0 1(X k ∈ ·)   Eν j ∞ k=0 1(X k ∈ ·) · 1(τκ,ν j > k) = ∞ k=0 Pν j (X k ∈ ·, τκ,ν j > k) = ∞ k=0 νjP k κ,j (·) Because νjP k κ,j (·) is the probability of a particle being in · at time k and not having left Aj at any time before k. The series on the third line above converges because Eτ ν 0 j < K by assumption (A0). The second equality is given by applying I −Pκ,j to the first. For the final equality, ξκ,j( · ) = Pν j (Xτ κ,j ∈ · ) = ∞ k=1 Pν j (X k ∈ · , τκ,j = k) = ∞ k=1 νjP k−1 κ,jP κ,j ( · ) = ∞ k=0 νjP k κ,j P κ,j = µκ,jPκ,j. We can now put the equalities in the last lemma together to derive, after integrating over κ, that a fixed point of Q must have a corresponding occupation measure which is a fixed point of P , as defined in 3.1. We then can establish existence and uniqueness of the fixed point of Q, ν * , since P is assumed to have a unique invariant measure. µκ,jPκ,jη(dκ). Furthermore, for any κ and j,Pκ,j +Pκ,j = Pκ and j ajξj = νQ, by construction. Therefore, we can subtract the second equation above from the first to get ν − νQ =ˆ[ 0,1] j ajµκ,j(I − Pκ)η(dκ) = µ(I − P ) Therefore, ν = νQ, if and only if µ(I − P ) = 0, in which case µ ∼ π, where π is the unique invariant distribution of P . Now, for a given occupation measure µj on Aj, define an injection measure νj on Aj by νj ∼ µj(I −Pj) recalling thatPj =´[ 0,1]P κ,j η(dκ). Then the occupation measure given by νj on Aj is, by Lemma 5.2, [0,1] νj(I +Pκ,j +P 2 κ,j + · · · )η(dκ) =ˆ[ 0,1] µj(I −Pj)(I +Pκ,j +P 2 κ,j + · · · )η(dκ) = µj(I −Pj)(I +Pj +P 2 j + · · · ) = µj Note that the infinite sum of operators converges, because for any B ⊂ Aj and any probability measure η, ∞ k=0 ηP k j (B) ≤ ∞ k=0 ηP k j (Aj) = ∞ k=0 Pη(τj ≥ k) ≤ K Where K is as in (A1). Therefore, given an occupation measure µj, the associated injection measure is µj(I −Pj), up to normalization. We can now show existence of the fixed point. Let π |A j be the un-normalized restriction of π to Aj. Define the injection measure on Aj by ν * j ∼ (π |A j )(I −Pj) and the weights by a * j = 1 W π(Aj) Eτ ν * j Where W = j π(A j ) Eτ ν * j . Then the above calculation show that if ν * = j a * j ν * j , then ν * −ν * Q ∼ π(I−P ) = 0. Uniqueness follows similarly. If νQ = ν, then we must have µP = µ, so µ ∼ π. But then νj = µj(I −Pj) ∼ (π |A j )(I −Pj), and since νj must be a probability measure by construction, this determines all the νj. Similarly, the weights must be given by the formula for a * j above, in order to have µ ∼ π, so they are determined as well. Therefore, the fixed point of Q is unique. Furthermore, it satisfies ν * V < ∞, because (π |A j )(I −Pj)V < ∞ for all j. This concludes our proof that the algorithm has the appropriate fixed distribution. Notice that, in a sense, the argument we used is really a Poisson equation argument. We use that µj satisfies the Poisson-like equation µj(I −Pj) = νj for each j to show that, µ(I − P ) = ν − νQ, which is like a Poisson equation on the whole space A. This equation is what lets us relate the fixed point of P to that of Q. In the rest of the proofs, we approach the problem of showing that the injection measures given by the algorithm actually approach the fixed point of Q. Convergence Proof for the Basic Algorithm We will first prove that the measures νQ n , which are the distributions found by the basic algorithm without finite approximations, do in fact converge to ν * as n → ∞, for any initial ν. Later we will improve our estimate of how fast the convergence is. The proof we give below has a simple intuition. Consider a particle following the exit kernel Q, whose position at the n-th exit from a stratum is Xn. Assumption (B1) implies that, after m exits, the particle has at least a certain probability of being in any stratum. This means that, for two such particles, a coupling can be constructed so they have a chance, bounded from below, of being in the same stratum. Assumption (A2) then implies that they have a probability of being at the same location on the next exit, for an appropriate coupling. Therefore, the operator that applies the algorithm m + 1 times, i.e. Q m+1 , is a contraction in total variation. With this intuition, we give the formal proof now. Proof of Theorem 3.2: First note that, because Q is a time-homogeneous Markov kernel, it suffices to consider is a contraction in the case where the initial measures are single-point delta distributions. Therefore, let ν 1 = x1, ν 2 = x2 where x1, x2 ∈ A and x1 = x2, and let n ≥ m. Let j1 = Idx(x1) and j2 = Idx(x2). Let X i 0 = xi, and X i n be the location of the n-th exit from a stratum starting from X i 0 , for i = 1, 2. X i 0 is in Aj i , so by (A1), with probability at least c, a particle started at X i 0 exits as if it started atνj i , That is, there exists measures η i j i on A c j i for i = 1, 2 such that ν i Q = δx i Q = cξj i + (1 − c)η i j i = cνj i Q + (1 − c)η i j i So applying Q n−1 to both sides, where n ≥ m, ν i Q n = cνj i Q n + (1 − c)η i j i Q n−1 . Now, by assumption (B1), the probability that a particle starting from the QSD in Aj i is in A k after n exits is minorized by ua * k , for any j, which means that νj i Q n (A k ) ≥ ua * k ∀i, k. Therefore, for i = 1, 2, ν i Q n (A k ) ≥ u · a * k . Therefore the distributions of Idx(X 1 n ), Idx(X 2 n ) are both minorized by cu times the same probability vector, a * k . Combining these two observations, there exists a coupling X 1 n , X 2 n of ν 1 Q n , ν 2 Q n such that P(Idx(X 1 n ) = Idx(X 2 n )) ≥ cu. Now, by (A2), conditioned on two particles starting in the same stratum A k , there exists a coupling such that the probability of their exit points being equal is ≥ c, because their exit distributions are both minorized by cξ k . Therefore, there exists a coupling X 1 n+1 , X 2 n+1 of ν 1 Q n+1 , ν 2 Q n+1 such that P X 1 n+1 = X 2 n+1 ≥ P Idx(X 1 n ) = Idx(X 2 n ) · P X 1 n = X 2 n | Idx(X 1 n ) = Idx(X 2 n ) ≥ cu · c = c 2 u. Therefore, ν 1 Q n+1 − ν 2 Q n+1 ≤ 1 − c 2 u = (1 − c 2 u) ν 1 − ν 2 . Basic Version -Long Term Convergence Rate Proving our more precise estimate of the rate of convergence in the long term will require more machinery. As we saw above, assumption (A2) allows us to show that coupling occurs if two particles start in the same stratum. If the difference between two injection measures, ν 1 and ν 2 , is mostly in their individual injection measures (i.e. the νj), but their weights are mostly the same, then we can conclude that they get closer when evolved by Q. This is because two particles distributed by ν 1 , ν 2 which are not already coupled have a good chance of being in the same stratum, and then (A2) says they have a chance of coupling on the next exit. But what if the difference is mostly in the weights? Then the particles have little chance to couple on the next exit, since they are most likely not in the same stratum. So we can't conclude that Q is a contraction. This can be thought of as the "problem case", as far as a coupling argument is concerned. We solve this issue with two tools. First, given the equilibrium ν * and a given measure ν, we want to decompose them into the the two above cases. We can start with the standard way of decomposing the measures into equal and mutually singular parts: ν = (1 − )ν + ν 1 ν * = (1 − )ν + ν 2 whereν, ν 1 , and ν 2 are probability measures with ν 1 ⊥ ν 2 and = ν − ν * . We can further decompose the mutually singular parts into a part that looks like the "easy case" above, and a part that looks like the "problem case". We get the following expression: ν 1 = (1 − γ) j b 1 j νj + γ j pjη 1 j ν 2 = (1 − γ) j b 2 j ν * j + γ j pjη 2 j(2) for some probability vectors p, b 1 , b 2 and measures η 1 , η 2 where b 1 ⊥ b 2 , η 1 ⊥ η 2 , and some γ ∈ [0, 1]. The first term represents the parts of ν 1 , ν 2 that have different weights. The second term represents the parts that have the same weights but orthogonal distributions on the strata. γ is the share of the second term relative to all of the TV distance between ν and ν * . We will handle the cases where γ is large or small separately, and get a bound for each. However, as explained above, Q may not actually bring the two measures closer together in the "problem case", where γ << 1. To fix this, we introduce the following new metric, which is equivalent to TV distance in the sense that they induce the same topology: dα(ν, ν * ) = ν − ν * T V (A) +αdw(a, a * ) where a, a * are the weight vectors of ν, ν * , and G * is the transition matrix between strata in equilibrium. The metric dw on weight vectors is given by dw(a, b) = m k=0 λ k a(G * ) k − b(G * ) k + ∞ k=m+1 λ m a(G * ) k − b(G * ) k where λ, m are as in (B2). The intuition behind the new metric is as follows. Even if we are in the problem case, the weights of our injection measure should get closer to the true weights, which should allow for coupling to occur at some later step. So we add a term to the standard TV-distance that should contract if the weights move approximately by G * under Q. We choose the metric dw on probability vectors because under it, G * is a contraction with constant λ. With the decomposition and metric defined, our strategy is now as follows: First, we consider the "easy case", where γ is close to 1. Then (A1) lets us conclude that coupling occurs on the next exit, so ν gets closer to ν * when acted on by Q. In the "problem case", where γ is close to 0, the weights of ν get closer to those of ν * because they move approximately by G * , so we still get contraction in dα. The complication we encounter is that the weights do not move exactly by G * , but by the transition matrix G given by ν. The result is that we only get local contraction of Q in dα. But we already have global convergence by 3.2. So we still get a rate of convergence that eventually applies. With the strategy now laid out, we can now proceed through the proof of 3.3. We begin by showing the decomposition (2). Lemma 5.3. Let ν = ajνj be an injection measure on A, and let ν * = j a * j ν * j be the fixed point of Q then there exist probability vectors p, b 1 ⊥ b 2 , measures η 1 ⊥ η 2 and γ ∈ [0, 1] such that (2) holds. Proof. Start by decomposing the weight vectors of ν, ν * into equal and orthogonal parts: a = (1 −ˆ )ā +ˆ b 1 a * = (1 −ˆ )ā +ˆ b 2 . with b 1 ⊥ b 2 . Similarly, we can decompose the injection measures on each stratum: νj = (1 − j )νj + j η 1 j ν * j = (1 − j )νj + j η 2 j . We can now substitute these expressions for the injection measures and weights into ν = ajνj, ν * = j a * j ν * j . We get ν 0 = (1 −ˆ )(1 − Z) jā j (1 − j ) 1 − Zν j +ˆ j b 1 j νj + (1 −ˆ )Z j pjη 1 j ν * = (1 −ˆ )(1 − Z) jā j (1 − j ) 1 − Zν j +ˆ j b * j ν * j + (1 −ˆ )Z j pjη 2 j where Z = jā j j and pj =ā j j Z . Setting γ = (1−ˆ )Ẑ +(1−ˆ )Z ,the result follows. As mentioned above, our proof strategy is complicated by the fact that the weight vectors of two injection measures move by different matrices under Q. However, if all a * j are non-zero, i.e. every stratum has some weight in equilibrium, then G, G * will be close if ν, ν * are close in total variation. So our next step is to bound the difference in weight vectors after applying Q, in the case where the transition matrices are close. Where S = 1 + λ + λ 2 + λ 3 + · · · = 1 1−λ . Proof. By decomposing the weight vectors a, a * into equal and orthogonal parts, as in the proof of Lemma 5.3, we have aG − a * G * = a(G − G * ) + (a − a * )G * = (1 −ˆ )ā(G − G * ) +ˆ b 1 (G − G * ) + (a − a * )G * . Therefore, by the triangle inequality, dw(aG, a * G * ) ≤ (1 −ˆ )dw(āG,āG * ) +ˆ dw(aG * , a * G * ) +ˆ dw(b 1 G, b 1 G * ). Now, G * is a contraction in dw, so dw(a 0 G * , a * G * ) ≤ λdw(a 0 , a * ). The construction of dw and assumption (B2) imply that dw(b 1 G, bG * ) ≤ S b 1 G − b 1 G * which is ≤ S G − G * ∞≤ sδ because b 1 is a probability vector. Using the definition of G, G * and assumption (A2), (ā(G − G * )) j = kā k (G jk − G * jk = kā k (ξ k (Aj) − ξ * k (Aj)) ≤ kā k ξj − ξ * j ≤ (1 − c) kā k νj − ν * j T V . Therefore, (1 −ˆ ) ā(G − G * )(G * ) k T V ≤ (1 − c)(1 −ˆ ) ā k k = (1 − c)γ ν − ν * T V Which implies that (1 −ˆ )dw(āG,āG * ) ≤ (1 − c)γS ν − ν * T V . Putting the inequalities just derived together gives the desired result. Now we are ready to show that Q acts as a local contraction in dα, for both small and large γ. We start with the case where γ is bounded away from 0. In this case, the idea is that, if two particles coupling ν and ν * are not at equal positions, they have a chance, bounded below, of being in the same stratum. (A2) then lets us assure that they can couple when moving by Q. The rest of the proof of the following lemma is simply keeping track of how all the terms in dα(νQ, ν * Q) behave. Lemma 5.5. Suppose that (A0)-(A2) and (B2) hold and fix β ∈ (0, 1). Then for initial ν and equilibrium ν injection measures, if γ > β and G − G * ∞< δ, dα(νQ, ν * Q) ≤ max (1 − βc + αS(1 − c) + αSδ, λ) · dα(ν, ν * ). Proof. Suppose ν = ν * , since otherwise the result is trivial. Let X, X * be a coupling of ν, ν * . By Lemma 5.3, the coupling can be chosen so that with probability γ ν − ν * , Idx(X) = Idx(X * ) and X = X * , so that P Idx(X) = Idx(X * ) | X = X * = P Idx(X) = Idx(X * ), X = X * P(X = X * ) = γ ν − ν * ν − ν * = γ ≥ β. Assumption (A2) now implies that if Y, Y * are a coupling of νQ, ν * Q = ν * , then they can be chosen so that P(Y = Y * | X = X * ) ≥ cγ ≥ cβ. Therefore, νQ − ν * ≤ (1 − cβ) ν − ν * . Combining this with Lemma 5.4 and the definition of dα implies that dα(νQ, νQ) ≤ (1 − βc + αS(1 − c) + αSδ) ν − ν * T V +αλdw(a, a * ) ≤ max 1 − βc + αS(1 − c) + αS G 0 − G * ∞, λ dα(ν 0 , ν * ). Now we can approach the case where γ is bounded away from 1 in a similar way. The idea this time is that, if we are close to the problem case, but G − G * is small, then the weights will get more accurate under Q and so the distance in dα should still contract, even if no coupling occurs. Lemma 5.6. Suppose that (B2) holds, and fix β ∈ (0, 1). Suppose that α < 1 S(1−c) . Then for any ν, ν * such that γ ≤ β, G − G * < δ, dα(νQ, ν * Q) ≤ (1 + αSδ) + (1 − β)αλ 1 + (1 − β)α dα(ν, ν * ). Proof. As before, we can assume that ν = ν * . First, as in the previous proof, we have νQ − ν * Q ≤ (1 − γc) ν − ν * and by the construction of γ, (1 + (αS(1 − c) − c)γ0 + αSδ) ν − ν * +αλdw(a, a * ) ν − ν * +dw(a, a * ) . a − a * = (1 − γ) ν − ν * ≥ (1 − β) ν − ν * =⇒ ν − ν * ≤ 1 1 − β a 0 − a * ≤ 1 1 − β d( Simplifying ρ gives the desired result. Notice that the contraction constants found in the previous two lemmas are < 1 if δ is sufficiently small. This is why we can conclude that Q is a local contraction in dα. With both cases now out of the way, we can put them together to get our next main convergence theorem. Proof of Theorem 3.3: Theorem 3.2 implies that, under the given assumptions, ν n − ν * → 0, where ν n = νQ n . Since, by assumption, a * j > 0 for all j, this implies that ν n j → ν * j for all j, in total variation, and therefore G n → G * , where G n is the transition matrix given by ν n . Therefore, there exists l ≥ 0, δ > 0 such that, for all n ≥ l and α < (1 + αSδ) + (1 − β)αλ 1 + (1 − β)α < 1. It follows that dα(ν n+1 , ν * ) ≤ q δ α,β · dα(ν n , ν * ) for n ≥ l. Furthermore, as l → ∞, the above holds for arbitrarily small δ. Also note that for λ < 1, (1+αSδ)+(1−β)αλ 1+(1−β)α ≥ 1+(1−β)αλ 1+(1−β)α > λ. Therefore, there exist qn ∈ (0, 1) such that dα(ν n+1 , ν * ) ≤ q n · dα(ν n , ν * ) for sufficiently large n, and q n → lim δ→0 q δ α,β = max 1 − βc + αS(1 − c), 1 + (1 − β)αλ 1 + (1 − β)α . Finally, optimizing the limit of q n over all choices of α, β proves the desired result. with all terms of the form | G i k − G * i k | replaced by 1, suffices. Now we can use the above lemma to show that γ is bounded in the eigenvector version. Lemma 5.8. Suppose that (A0)-(A2) and (B3) hold. Let νj be injection measures for 1, . . . , J, with transition matrix G and corresponding (normalized) eigenvector z. Let ν = j zjνj, and decompose ν, ν * as in (2). Then there exists r, independent of νj, such that γ ≥ 1 1 + r . Furthermore, as νj → ν * j in TV, r can be chosen to be arbitrarily close to r ∞ , as in Theorem 3.4. Proof. First, note that by (A2), G * , G ≥ cG, whereG is the transition matrix if the injection measure in Aj is the QSDνj. Therefore, (B3) applies to G and G * . Now writeˆ = z − a * and j = νj − ν * i ) i ≤ r i (zi ∧ a * i ) i where r = 2(1 − c) sup j (a * j ) sup i,k θ ik a * i E. We also now have that r → r ∞ as ν → ν * , because θ ik → e θ ik − 1. Now, observe that in the notation of Lemma 5.3, (1 −ˆ )Z (1 −ˆ )Z +ˆ = j (zj ∧ a * j ) j j (zj ∧ a * j ) j +ˆ and so the result follows. We can now show that the eigenvector version acts as a local contraction on injection measures. We choose to state the local contraction in another metric, for which the calculations are simpler. Given injection measures νj, with eigenvector weights zj and ν = j zjνj, define d(ν, ν * ) = j a * j νj − ν * j = j a * j j . Note that, since the measures νj determine the weights zj, this is a well-defined metric on the set of injection measures with eigenvector weights. The proof of Lemma 5.8 also implies thatˆ < rd(ν, ν * ) so d is equivalent to total variation distance for such choices of ν. Since we are in the case where the measures determine the weights, d can be thought of as the metric that only looks at the difference in measures, appropriately weighted. By moving between d(ν, ν * ) and ν − ν * , we can use a coupling argument and the boundedness of γ to show local contraction in d, as we do below. Lemma 5.9. Suppose that (A0)-(A2). Let ν = j z j ν j be the measure obtained from ν via the eigenvector injection measure method, with j = ν j − ν * j . Then j (zj ∧ a * j ) j ≤ q j (zj ∧ a * j ) j Where q := 1 − 1 1 + r c Proof. First, by Lemma 5.8 and the same coupling argument as in Lemma 5.5, νQ − ν * ≤ (1 − γc) ν − ν * ≤ q ν − ν * .(6) The individual injection measures of νQ are ν j , by construction of the algorithm. Furthermore, the weights of νQ are z, because zG = z. Therefore, νQ − ν * = j (zj ∧ a * j ) j +ˆ . We also have ν − ν * = j (zj ∧ a * j ) j +ˆ . Substituting the above expressions into (6), we get j (zj ∧ a * j ) j ≤ q j (zj ∧ a * j ) j − (1 − q)ˆ ≤ q j (zj ∧ a * j ) j . Algorithm 1 1The Injection Measure Method 1: N ← # iterations 2: M ← # Points per Strata 3: Initial weights a 0 j and strata measures ν 0 j Initial measure ν 0 = j a 0 j ν 0 j 4: for n ← 0 to N − 1 do 5: Theorem 3 . 2 . 32Under (A1) and (B1), for any n ≥ m, Q n+1 is a global contraction on probability measures on A, with contraction constant Theorem 3. 4 . 4Let ν 0 , ν 1 , ... be the total injection measures for a run of the eigenvector version. Suppose that (A0)-(A2) and (B3) hold. Let Figure 2 : 2Four choices of elliptical strata for the Maier-Stein system. We use setups with 3 and 5 vertically oriented strata, tilted strata, and 6 circular strata which cover the space more tightly Figure 3 : 31-dimensional histograms for the true u-marginal density (again from a high-resolution un-stratified run) and the approximations of it generated by the algorithm with the strata setups above. Note that, with some deviation, each setup produces quite an accurate approximation of the marginal density. Figure 4 : 42D Histogram of the density of the total injection measure, for the strata setups inFig. 2. The densities are calculated from the points generated on the last iteration of a 30-iteration run of the algorithm, with 3000 exits per iteration. Because we are using strata with hard boundaries, the injection measures lie on a portion of the boundaries of each ellipse. Figure 5 : 5Log-Log total variation error in u vs time, for each of the strata setups. The error values are averaged over 20 full runs of the algorithm for each setup. Suppose that (A0) and (A1) hold. Let νj be a distribution on Aj such that νjV < ∞. Fix κ ∈ [0, 1], and let τκ,j µκ,j, ξκ,j be the exit time, occupation, and exit measures for a trajectory in Aj started at νj, if the threshold is chosen to be κ. j(I −Pκ,j) = νj , and ξκ,j = µκ,jPj Lemma 5 . 4 . 54Suppose that (A0)-(A2) and (B2) hold. If G, G * are the transition matrices given by the exit measures for ν, ν * , and if G − G * ∞< δ, then dw(aG, a * G * ) ≤ S ((1 − c)γ + δ(1 − γ)) ν − ν * T V +λdw(a, a * ). a, a * ). Now, putting the above inequalities together with the result of Lemma 5.4 implies thatdα(ν, ν * ) ≤ 1 + (αS(1 − c) − c)γ + αSδ ν 0 − ν * T V +λdw(a 0 , a * ) ≤ ρ 0 dα(ν 0 , ν * ) where ρ = c) , G − G * ∞< δ and q δ α,β := max 1 − βc + αS(1 − c) + αSδ, λ, Lemma 5. 10 . 10Fix δ > 0. Under the assumptions of Lemma 5.9, if 2(1 − c)(sup i =k θ ik a * i )d(ν, ν * ) < δ, then d(ν , ν * ) ≤ q(1 + δ)d(ν, ν * ) In practice, the algorithm will build a finite approximation of ν n+1 given ν n , by sampling a finite number of starting points from ν n and calculating an exit point from each. If V (x) → ∞ as | x |→ ∞, then the condition is equivalent to requiring that P V (x) ≤ γ V (x) + b for some different γ ∈ (0, 1) and b > 0. We will use the slightly more general condition in Assumption A1. WhereG ij =ξ j (A i ) is the transition matrix between strata when the injection measures are the QSD's. Acknowledgments: We thank Jonathan Weare for introducing JCM to this general area. We also thank the NSF grant DMS-1613337 and both SAMSI (DMS-1638521) and Duke TRIPODS (CFF-1934964) for partial support of this work.Local Convergence of the Eigenvector VersionNow we turn to proving our final result, that the eigenvector version converges locally with a rate we can bound. The strategy again revolves around the decomposition(2). This time, however, we do not have to deal with the "problem case". The reason is that, as a result of the perturbation bound in (B3), we can show that the parameter γ is never too small when the weights are given by the eigenvector of the transition matrix G associated to {νj}. Therefore, we always get that coupling can occur, by assumption (A1).However, we also run into a new problem: this version of the algorithm cannot be represented as evolving the measure ν forward by a single Markov kernel, the way the basic version evolves by Q. The reason is that each of the eigenvector weights depends on all the measures νj. The eigenvector version does not act meaningfully on just one νj. Therefore, we cannot assure that no decoupling occurs in the step where the new weights are chosen. However, similarly to the previous proof, we can control the size of this decoupling in the case where the νj are already close to ν * j . Therefore, we will end up with a local convergence result.Our first goal is to show that, when using the eigenvector weights, the parameter γ is bounded below. This follows somewhat straightforwardly from assumption (B3), and the fact that the transition matrix entries are themselves determined by the injection measures. Note that the eigenvector weights for the equilibrium injection measure are simply given by a * , because by construction of the fixed point of Q, a * G * = a * .Lemma 5.7. Suppose that (B3) holds. Let G * be the transition matrix given by the fixed point measures νj, with principal eigenvector a * , i.e. a * G * = a * . Let G be another transition matrix on {1, . . . , J} with eigenvector z, such that zj > 0 for all j. Then for any δ > 0, there exist constants θ ik for i = k such thatProof. By (B3), there exist θ ik > 0 ∀i, k ∈ {1, ..., J} such that for all j ≤ J,where O(| G − G * |) represents terms which are order 1 or higher inSince there are only finitely many of these terms, setting θ ik to be the coefficient ofProof. Continuing on from Lemma 5.9, we havewhere the second line holds by (A2), and the last line holds because d(ν, ν * ) ≥Note, in particular, that if δ < 1 q − 1, then q(1 + δ) < 1, and so have have local contraction. It is now straightforward to use the equivalence of TV distance and d, when the weights are given by the eigenvector of G, to prove our final convergence result.Proof of Theorem 3.4: Suppose that the initial injection measure is ν 0 = j z 0 j ν 0 j , and that successive ν n are given by steps of the eigenvector version, with. Lemma 5.10 implies that, if 2(1−c)(sup i =k θ ik a * i )d(ν 0 , ν * ) < 1 q −1, then d(ν n , ν * ) → 0, and for all n, d(ν n+1 , ν * ) ≤ p n d(ν n , ν * ) where p n is the value q(1+δ), computed from ν = ν n . 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We show that existence and uniqueness of solutions to transported Monge-Ampère problem on complex compact toric manifold follows easily from the real theory of optimal transportation.

10.1007/s12220-020-00482-3

1906.07028

08f8b2338a4716de51587386ee7d7f048c5dbdf1

Optimal transport on completely integrable toric manifolds 17 Jun 2019 Szymon Myga Optimal transport on completely integrable toric manifolds 17 Jun 2019 We show that existence and uniqueness of solutions to transported Monge-Ampère problem on complex compact toric manifold follows easily from the real theory of optimal transportation. Introduction Let (X, ω) be a compact Kähler manifold of real dimension 2n, i.e. X is a complex manifold and one can find a hermitian metric on it whose fundemental form ω is closed, thus making (X, ω) into a symplectic manifold. We assume that X is toric -there is a real torus T k acting on it by automorphisms of ω. Such an action also generates a Lie algebra homomorphism from the Lie algebra of the torus t ≃ R k into the Lie algebra of the vector fields of X. This action can be extended to the holomorphic action of complexified torus T k c ≃ (C * ) k . We also assume that the action is completely integrable and effective. That is, we want the torus to be of greatest possible dimension (k = n) and we want the trivial automorphism to only come from the identity element. Lastly, we want the action to be Hamiltonian, so we assume that there is a moment map: an action invariant function m : X → (R n ) * , with (R n ) * being the Lie algebra dual to t, such that for every element t ∈ R n −d m(p), t = ω p (t # , ·) with t # being the vector field generated by t and m(p), t being the value of the linear form m(p) at t. In this setting one can prove that the image of X through m is a compact convex polytope in R n with non-empty interior. Moreover this image does not depend on the choice of particular ω in an invariant cohomology class. Suppose a probability measure with density 1/C < g(p) < C is given on a moment polytope P for some toric Kähler manifold (X, ω) with completely integrable torus action. Following the preprint [3] one can define a notion of complex transported Monge-Ampère measure M A g on X which corresponds to the Monge-Ampère measure that appears in the theory of optimal transporta-tion of measures. Then the natural question to ask is whether the equation M A g (φ) = µ has a unique (up to an additive constant) solution for any invariant measure µ that does not put any mass on polar sets, i.e. the sets that are −∞ loci of plurisubharmonic functions. This is a technical assumption that comes up when one tries to define a Monge-Ampère operator for singualr functions. A partial answer to the above question is provided in [3] where the authors prove the existence of solutions and uniqueness for a subclass of measures. In the setting sketched above we prove the following theorem: Theorem. For any invariant probability measure µ that does not put mass on polar sets there is an invariant φ ∈ P SH(X, ω) such that g(m φ )M A(φ) = µ, where m φ is a moment map for the torus action induced by φ. As already suggested in [3] the proof follows from the result of McCann [7], although not in a straightforward way. The only gap to fill from there is to ensure that appropriate notions of convergence for real and complex solution coincide. Towards this end we prove the following lemma: Lemma. If a unifiormly Lipschitz sequence of convex functions F n converges in a monotone way to a convex function F , then their Legendre transforms F * n converge to F * in W 1,∞ loc . As already mentioned, in the real setting this is a well studied equation that appears in the theory of optimal transportation. In the complex case, it comes up as an equation for Kähler-Ricci solitons on Fano varieties, although in that case the action might not be completely integrable. Acknowledgement The author would like to thank S lawomir Dinew for his guidance. The author was supported by Polish National Science Centre grant 2018/29/N/ST1/02817. 1 Background material Convex functions Here we want to recall a few facts about convex functions. For a convex function u : R n → R ∪ {+∞} we define its domain as the convex set {x : u(x) < ∞} and denote it as dom(u). For convenience we exclude the function u ≡ +∞ from the set of convex functions. For any convex function u on R n its Legendre transform is defined by u * (p) := sup x∈R n { x, p − u(x)}. It is a crucial notion in convex analysis. It is not hard to show that the Legendre transform u * is a convex lower semicontinuous function. The multivalued subgradient of u is a set-valued map defined on int(dom(u)) that attaches to a point the set of slopes of supporting planes at that point, namely y ∈ ∂u(x) ⇔ ∀z ∈ R n u(x) + y, z − x ≤ u(z). Since u is convex ∂u is always non-empty on int(dom(u)). It is single-vauled iff u is differentiable at x and at this point it is equal to ∇u(x). The notion of a subgradient is closely related to the notion of a Legendre trasform through the following equivalences x · p = u(x) + u * (p) ⇔ p ∈ ∂u(x) ⇔ x ∈ ∂u * (p). From this, one can see that in the case of a smooth strictly convex function the gradient of the function and of its Legendre transform are each other's bijective inverses. The most important fact concerning the differentiability of convex functions is the following one: The set of differentiability of u will be denoted by dom(∇u). Finally, we list the properties of convex subgradients that will be of use to us. This gives us the following corollary: Corollary 1.4. For any convex function u and for any compact subset K of int (dom(u)) the set ∂u(K) is bounded. Proof. Indeed, pick an ǫ and take appropriate δ x -balls at points x in K. This gives a covering of K. The only thing left to show is that at x / ∈ dom(∇u) the subgradient ∂u(x) is still bounded. But x is in int(dom(u)), if the subgradient would be unbounded then u would get arbitrarily big arbitrarily close to x, which cannot be since x lies a positive distance away from the boundary of dom(u). Lemma 1.5. Let u, v be two convex functions defined on some convex set C with non-empty interior. If {∇u = ∇v} is a subset of full measure of C then u ≡ v in int(C) modulo additive constant. Proof. If ρ ǫ is the standard mollifier then it is easy to verify that u * ρ ǫ and v * ρ ǫ are convex in C ǫ smooth and convegre locally uniformly to u and v respectively. Where the set C ǫ is the domain of definition of mollified function, i.e. {x ∈ C | dist(x, ∂C) > ǫ}. By smoothness, the almost everywhere equality of their gradients implies their equality everywhere and thus u * ρ ǫ and v * ρ ǫ must converge to the same function up to a constant. Convergence of convex functions The following facts about the convergence of convex functions will be of use. The most natural notion is the following. Lemma 1.6. If a sequence of convex functions {u k } converges locally uniformly to a function u, then u is convex. We would also like to say something about the convergence of subgradients. Definition. We say that the sequence of subgradients ∂u n converges graphically to subgradient ∂u if their graphs converge as sets, i.e. graph(∂u) = {(x, p) | ∃ (x n , p n ) ∈ R n × R n : p n ∈ ∂u n (x n ) and (x n , p n ) → (x, p)}. The following theorem of Attouch is a fundamental result concerning the graphical convergence of subgradients. 1. f n → f locally uniformly, 2. ∂f n → ∂f graphically and for some choice of p n ∈ ∂f n (x n ) and p ∈ ∂f (x) such that (x n , p n ) → (x, p) one has f n (x n ) → f (x). Remark. The first part of the orginal theorem is expressed in terms of "epigraphical" convergence, but is equivalent to locally uniform convergence (see [9,Theorem 7.17]). Finally, we would like to describe the relationship between the graphical convergence and pointwise convergence, for this we need the following definition: Definition. The sequence of set-valued maps S n is equicontinuous at point x with respect to subset X if for each positive ǫ there is a neighbourhood V of x such that for almost every n S n (y) ⊂ S n (x) + B(0, ǫ) for all y ∈ V ∩ X. The sequence is equicontinuous with respect to X if it is equicontinuous at each point of X. Remark. In [9] the above notion is called the asymptotic equi-outer-semicontinuity. Since we don't need other notions of equicontinuity we will just call that one the equicontinuity. Now the deisired relationship is the following: Theorem 1.8 ([9, Theorem 5.40]) . For the sequnece of set-valued maps S n , the map S and a set X any pair of the the following conditions implies the third: 1. S n is equicontinuous with respect to X, 2. S n converges graphically to S relative to X, 3. S n converges pointwise to S relative to X. The class of globally Lipschitz convex functions Definition. By support function of a bounded convex set P contatining zero we mean the function φ P (x) := sup p∈P x, p . If the set P is bounded then φ P is finite everywhere. Definition. We will denote by P the space of convex functions dominated by φ P , i.e. the set {u − convex | ∃ C : u ≤ φ P + C}. This is the set of convex functions whose Legendre transform is +∞ outisde P . The subset of P consisting of functions that also dominate φ P will be denoted by P + , in other words P + = {u ∈ P | ∃ C : u − C ≥ φ P }. Both sets can be equipped with the topology of pointwise convergence, which is equivalent to the topology of locally uniform convergence, by the virtue of uniform Lipschitz constant for all P. The set P + is dense in P. Moreover the approximating sequence can be chosen as nice as possible. Lemma 1.9 ([2, Lemma 2.2]) . Every φ ∈ P can be approximated by decreasing seqence of smooth strictly convex functions from P + . Optimal transport and Monge-Ampère equation We say that the function T : R n → R n transports probability measure µ to probability measure ν if for any Borel set A the following equality holds ν[A] = µ[T −1 (A)]. Alternatively we say that T pushes µ forward to ν and denote the push-froward measure by T # µ. In general there will be a lot of such maps, so it is natural to put some optimality constraints on them. The best understood contraint and in some cases the natural one is minimizing the quadratic cost, i.e. the transport map should minimize the following functional R d |x − T (x)| 2 dµ. In general there might not be a solution and if it exists it might not be unique, some regularity assumptions for the measures must be added. For example, one can assume that the measures have finite second moments and µ is absolutely continuous. In that case the solution exists and has a form of T = ∇φ for some convex function φ. For thorough discussion of this problem, the reader might consult [10]. Supposing that a solution exists, by the trasport condition we get χ A dν = A dν = (∇φ) −1 (A) dµ = χ A • ∇φ dµ That can easily be generalized to get that for any f ∈ C b (R n ) f dν = f • ∇φ dµ.(1) Here C b (R n ) denotes the set of continuous and bounded functions on R n . Suppose now that dν = g(x)dx for some density g(x) and φ is a C 2 function. By change of variables formula we get that f (∇φ(x)) dµ = f (∇φ(x))g(∇φ(x)) det D 2 φ dx and that provides one with a notion of solution to the transported Monge-Ampère equation M A R g (φ) := g(∇φ(x)) det D 2 φ = µ as long as the optimal transport map exists. As we mentioned, for any two probability measures the optimal transport solution might not exist. However, under a mild regularity assumption it is still possible to transport one to another through a subgradient of convex function, so that the condition (1) is still satisfied. This is the content of the following important theorem. Theorem 1.10 (McCann [7]). Let µ, ν be probability measures on R n and suppose that µ vanishes on Borel subsets of R n of Hausdorff diemnsion n − 1. Then there exists a convex function ψ on R n whose subgradient ∂ψ pushes µ forward to ν. ∂ψ is uniquely deterined µ-almost everywhere. Of course the assumption on the null sets of µ can not be abandoned. For example if µ = δ x and ν is not a point measure, then if A is such a set that 0 < ν[A] < 1 one gets that for any convex function φ, ν[A] = µ[(∂φ) −1 (A)] since the latter must always be either 0 or 1. Torus action As in the intrduction we are interested in completely integrable Kähler manifolds. In this setting the following results provide the correspondence between the Kähler geometry and convex functions. Proposition 1.11 ([5]). There is an open dense subset X 0 ⊂ X where the action of T n c is free, making X 0 diffeomorphic to (C * ) n . Every invariant Kähler form ω on X has a Kähler potential on X 0 , i.e. ω| X0 = 2i∂∂F for some F . The set X \ X 0 is given as a vanishing set of some holomorphic vector fields, so it must be analytic. If we introduce coordinates on X 0 coming from (C * ) n by L : e x+iy → x + iy, the invariance of potential means that the function F from the previous proposition depends only on x variable in R n and positive definiteness means that F must be convex. Moreover, nothing in the proof actually requires the form to be smooth, so the conclusion easily extends to forms with more singular coefficients, thus asserting that every closed positive and invarinat (1, 1)-current in the cohomolgy class [ω] will admits a convex potential. Proposition 1.12. For the symplectic form ω as above, the moment map is ∂F ∂x + c, with c being any constant vector in R n . Finally, we recall the theorem of Atiyah [1], Guillemin and Sternberg [6]: Theorem 1.13. The image of X through the moment map is a compact convex polytope in R n . In the case of completely integrable actions the polytopes that can arise as images of moment maps are called Delzant polytopes. Conversely for each Delzant polytope there exists a Kähler manifold with completely integrable torus action and a moment map that maps to this polytope. Toric pluripotential theory The class of plurisubharmonic functions that are torus invariant will be denoted by P SH tor (X, ω) = {φ ∈ P SH(X, ω) | ∀z ∈ X, t ∈ T n | φ(t · z) = φ(z)}. The results of the preovious section imply that to each such function corresponds a convex function on R n . More precisely, if set X 0 are coordinate map L are as in the previous subsection then for any v ∈ P SH tor (X, ω) the form ω v = ω + i∂∂v is still invariant and closed there, so restricting to X 0 there is a convex F v function given by F v • L = F 0 • L + v, with F 0 • L being the potential for ω. Of course if use the formula above to produce a plurisubharmonic function it will only be defined on X 0 , but since X \ X 0 is analytic, the function will extend to the whole X. Not every convex function can be a potential for an invariant Kähler form. If P is the Delzant polytope of the manifold (X, ω) then the following Propostion holds (see e.g. [4] for a proof). Proposition 1.14. The following are equivalent: For invariant plurisubharmonic funtion the two concepts are connected through the following proposition. Proposition 1.15. Let φ ∈ P SH tor (X), we identify X 0 with (C * ) n . Then for 1. v ∈ P SH tor (X), 2. F v ∈ P.any f ∈ C b (R n ) X0 (f • L) M A C (φ) = n! (2π) n R n f M A R (F φ ). The proof is just a straightforward computation (see e.g. [4]) in the smooth case and then the application of classical convergence theorems for convex and plurisubharmonic functions. g-Monge-Ampère measure Following the preprint [3] we define the complex g-Monge-Ampère measure or the complex transported Monge-Ampère measure as M A g (φ) = g(µ φ )M A(φ). From the Proposition 1.12 and the definition of the real transported Monge-Ampère measure it is not hard to see that 1.15 extends for smooth functions to transported measures. In the more general case, especially with the torus of smaller rank, the definition becomes more intricate. If we denote by E g the set of all P SH tor functions with full M A g mass, i.e. those functions for which X M A g (φ) = P g(p)dp = 1 then the following crucial continuity statement holds: Theorem 1.16 ([3], Theorem 2.7). If φ j is a sequence in E g decreasing to φ in E g then M A g (φ j ) → M A g (φ) in the weak topology of measures. Full rank existence and uniqueness Given a probability measure g(p)dp on P and any probability measure µ on R n , we would like to solve the equation M A R g (u) = µ in some appropriate sense. One can not apply McCann's theorem directly since for example µ = δ x would prevent the existence of the transport map, thus we must use the regularity of g. Suppose that we have a smooth strictly convex solution u, so that every term in M A R g (u) is well-defined and moreover so is ∇u * . By the fact that for any x and any p, ∇u(∇u * (p)) = p and ∇u * (∇u(x)) = x we define the solution through the change of variables formula. Thus R n f (x)g(∇u(x))M A R (u) = P f (∇u * (p))g(p)dp = R n f dµ(2) and a function u ∈ P such that the second equality holds for any contiunous bounded function f is defined to be a solution. The fact that there is such a solution follows easily from McCann's theorem. Suppose that φ is the convex function whose gradient transports g(p)dp (understood as a measure on R n ) to µ. By the regularity of g and McCann's theorem it must exist. Then ∇φ is defined g(p)dp-almost everywhere and since P is convex we can take φ to be +∞ outside of P . Thus after possibly fixing φ on ∂P so that it is lower semi-continuous, its Legendre transform φ * becomes unique and defined everywhere on R n and thus belongs to the class P since by lower semicontinuity φ * * = φ. The convex function u = φ * is the unique (up to additive constant) solution to the transported Monge-Ampère problem in the class P. Indeed, since ∇φ transports g(p)dp to µ it means that for any f ∈ C b (R n ) R n f dµ = P f (∇φ(p))g(p)dp = P f (∇u * (p))g(p)dp. If there was to be another solution v in the class P then its Legendre transform would have been +∞ on the complement of P and lower semicontinuous on its boundary and it would induce a transport of g(p)dp to µ, so by McCann's uniqueness theorem ∇u * = ∇v * g dp-almost everywhere, and since g > 0, by Lemma 1.5 we get u * = v * (mod R) everywhere on int(P ). Remark on uniqueness. Of course the uniqueness statement becomes false if we allow functions outside of class P. Suppose that µ = δ 0 , then the solution is obviously u = φ P , so that u * ≡ 0 on P . But now adding to u any convex function v such that min v = v(0) would also give a solution, since (u + v) * ≡ 0 on P . The complex case Corollary 2.1. The solution to the real problem in R n induces a unique solution to the g-Monge-Ampère problem on toric manifolds. Proof. Firstly, we notice that the fact that µ does not put any mass on pluripolar sets implies that X \ X 0 as an analytic set has no mass. Thus we can restrict the problem to X 0 . Moreover, since the measure is invariant, it can be interpreted as a measure on R n also denoted by µ. Suppose now we have a real solution F φ for the measure µ, then one suspects that φ = (F φ −F 0 )•L would be the solution for the corresponding invariant measure. Indeed, F φ is in P, so it must correspond to some invariant psh function. Moreover, for smooth strictly convex functions the formula (2) obiously translates by Proposition 1.15 to the complex setting. Finally, by Lemma 1.9 there exists a decreasing sequence F n of smooth strictly convex functions that decreases to F φ , so by smoothness and Theorem 1.16 M A R g (F n ) = M A C g (F n − F 0 ) converges weakly to M A C g (F φ ). Thus the only thing left to show is that M A R g (F n ) converges weakly to µ. Take f ∈ C b (R n ) and put f n := f • ∇F * n . We would like to show that f n converge almost everywhere to f . That would give us the desired assertion by the dominated convergence. First, let us prove that decreasing convergence of F n implies locally uniform convergence of F * n . Indeed, since F n 's are uniformly Lipschitz, their pointwise convergence implies locally uniform convergence. Now F ≤ F n implies F * n ≤ F * , take p ∈ intP and suppose that the supremum in F * (p) is realized by x * , thus F * (p) − F * n (p) = sup x∈R n { x, p − F (x)} + inf x∈R n {F n (x) − x, p } ≤ F n (x * ) − F (x * ). Thus F * n converges pointwise to F * . If K ⊆ int(P ) is compact then by Proposition 1.2 and Corollary 1.4 ∂F * (K) is compact and for every q ∈ K the supremum in F * (q) is realized by some y * in ∂F * (q), thus the convergence is locally uniform. Now, we will show that F * n converging locally uniformly to F * implies that ∇F * n converges to ∇F * almost everywhere and that would finish the proof. To do that we want to employ the Theorems 1.7 and 1.8 restricted to dom(∇F * ). Thus the only thing left to show is the equicontinuity of ∇F * n 's with respect to dom(∇F * ). In order to prove this we will first prove the following lemma: Now pick a positive η, starting from some n we get that 0 ≤ F * − F * n ≤ η over B(x, δ). We claim that ∇F * n (B(x, δ/2)) ⊂ B(∇F * (x), M + C) holds for some constant C, independent of F * n . Indeed, by convexity it is enough to estimate the gradients on the boundary of B(x, δ/2). Take a point y ∈ ∂B(x, δ/2) such that |∇F * n (y)| achieves maximum over ∂B(x, δ/2). The vector ∇F * n (y) must be pointed to the outside of B(x, δ/2) or at least be tangent to it. The "boundary" steepset case is F * n (y) = F * (y) − η, F * growing at best possible rate from y and tangent plane at F * n (y) touching F * at the boundary of B(x, δ), then ∇F * n (y) would become the steepest if that happened over shortest possible interval which would be of length δ/2. Thus finally |∇F * n (y)|≤ η + pδ/2, where p is the length of the longest vector in B(∇F * (x), M ). With the Lemma in hand the rest of the proof is straightforward. Suppose the sequence is not equicontinuous at some point x 0 ∈ intP ∩ dom(∇F * ). Thus there is a positive ǫ such that for any k there is y k ∈ B(x 0 , 1/k) ∩ dom(∇F * ) such that |∇F n(k) (x 0 ) − ∇F n(k) (y k )|> ǫ with n(k) being some subsequence of N. But by above lemma the set p k = {∇F * n(k) (y k )} is bounded, thus there must be a convergent subsequence, conviniently also named p k , such that p k k→∞ −−−→ p. But the set q k = {∇F * n(k) (x 0 )} is also bounded thus a subsequence must converge to some q such that |q −p|≥ ǫ. Thus we have two subsequences (y k , p k ) and (x 0 .q k ). By graphical convergence both of them must converge to some point in ∂F * (x 0 ), but this set is a singleton and that is a contradiction. Theorem 1 . 1 ( 11Rademacher's Theorem). Any convex function is differentiable on a subset of full measure of its domain. Proposition 1 . 2 ( 12Closedness of subgradients,[8, Theorem 24.4]). For any convex function u the graph of ∂u is a closed subset of R n × R n . Proposition 1 . 3 ( 13Boundedness of subgradients, [8, Corollary 24.5.1]). Pick any x ∈ int (dom(φ)), then for any positive ǫ there is a positive δ such that∂φ(B(x, δ)) ⊆ ∂φ(x) + B(0, ǫ),with B(x, δ) denoting the ball centered at x, with the radius δ. Theorem 1. 7 ([ 9 , 79Theorem 12.35]). For convex, lower semicontinuous functions f n and f the following are equivalent: Finally , the two Monge-Ampère measures coincide up to a constant. Specifically, the complex Monge-Ampère measure is defined as M A C (v)[A] := A (ω + i∂∂v) n and the real measure as M A R (F )[B] := |∂F (B)| for Borel sets A, B in C n and R n respectively. Here |·| is the Lebesgue measure. For C 2 convex functions it coincides with the measure M A R (F )[B] := B det(D 2 F ). Example. Finally, we would like to point out that if the assumptions of Mc-Cann's theorem are not satisfied for at least one of the measures there might not be a weak solution. For example, if µ = δ [−1,1]×{0} and ν = δ {0}×[−1,1] are measures on R 2 , then it is impossible to find a solution. Indeed, the only candidate is u(x, y) = |y| and it is easy to see that it can not be the solution since u * | {0}×[−1,1] ≡ 0. Lemma 2 . 2 . 22∇F * n are locally bounded independently of n. Proof. Take any point x ∈ intP ∩ dom(∇F * ) and pick a positive δ such that B(x, δ) is relatively compact in intP . By the boundedness of the subgradient (Corollary 1.4) there exists a positive M such that ∂F * (B(x, δ)) ⊆ B(∇F * (x), M ). . M F Atiyah, Convexity and commuting Hamiltonians. Bull. London Math. Soc. 141Atiyah, M. F. Convexity and commuting Hamiltonians. Bull. London Math. Soc. 14 (1982), no. 1, 1-15. Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties. R J Berman, B Berndtsson, Ann. Fac. Sci. Toulouse Math. 6Berman, R. J.; Berndtsson, B. Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties. Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 649-711. Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons. R J Berman, D Witt Nystrom, arXiv:1401.8264preprintBerman, R. J.; Witt Nystrom, D., Complex optimal transport and the pluripotential theory of Kähler-Ricci solitons, preprint arXiv:1401.8264. Toric pluripotnetial theory. D Coman, V Guedj, S Sahin, A Zeriahi, 10.4064/ap180409-3-7Ann. Polon. Math., on-line article. Coman, D.; Guedj, V.; Sahin, S.; Zeriahi, A. Toric pluripotnetial theory, Ann. Polon. Math., on-line article, DOI: 10.4064/ap180409-3-7 Kaehler structures on toric varieties. V Guillemin, J. Differential Geom. 402Guillemin, V. Kaehler structures on toric varieties. J. Differential Geom. 40 (1994), no. 2, 285-309. Convexity properties of the moment mapping. V Guillemin, S Sternberg, Invent. Math. 673Guillemin, V.; Sternberg, S. Convexity properties of the moment map- ping. Invent. Math. 67 (1982), no. 3, 491-513. Existence and uniqueness of monotone measure-preserving maps. R J Mccann, Duke Math. J. 802McCann, R. J. Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 (1995), no. 2, 309-323. R T Rockafellar, Convex Analysis. Princeton, N.J.Princeton University PressRockafellar, R. T. Convex Analysis, Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J. 1970 Variational Analysis. R T Rockafellar, R J Wets, Grundlehren der Mathematischen Wissenschaften. 317Springer-VerlagRockafellar, R. T.; Wets, R. J.-B. Variational Analysis, Grundlehren der Mathematischen Wissenschaften 317, Springer-Verlag, Berlin, 1998. Optimal transport. Old and new. C Villani, Grundlehren der Mathematischen Wissenschaften. BerlinSpringer-Verlag338Villani, C. Optimal transport. Old and new. Grundlehren der Mathema- tischen Wissenschaften 338, Springer-Verlag, Berlin, 2009.

In this paper, we show that, under appropriate conditions, there exists a quasinonexpansive extension of a mapping with an attractive point in the sense of Takahashi and Takeuchi[17]such that the fixed point set of the extension equals the attractive point set of the given mapping. Then using the quasinonexpansive extension, we establish some convergence theorems for approximating attractive points of a generalized hybrid mapping in the sense of Kocourek, Takahashi, and Yao [12].2010 Mathematics Subject Classification. 47J25, 47J20, 47H09.

2202.01419

08d12d95074b60816df57a746d29418c4d8c879d

A QUASINONEXPANSIVE EXTENSION OF A MAPPING WITH AN ATTRACTIVE POINT IN A HILBERT SPACE 3 Feb 2022 Koji Aoyama A QUASINONEXPANSIVE EXTENSION OF A MAPPING WITH AN ATTRACTIVE POINT IN A HILBERT SPACE 3 Feb 2022 In this paper, we show that, under appropriate conditions, there exists a quasinonexpansive extension of a mapping with an attractive point in the sense of Takahashi and Takeuchi[17]such that the fixed point set of the extension equals the attractive point set of the given mapping. Then using the quasinonexpansive extension, we establish some convergence theorems for approximating attractive points of a generalized hybrid mapping in the sense of Kocourek, Takahashi, and Yao [12].2010 Mathematics Subject Classification. 47J25, 47J20, 47H09. Introduction Let H be a Hilbert space, C a subset of H, and T : C → H a mapping. Takahashi and Takeuchi [17] introduced the notion of an attractive point of T ; see §2 for the definition of an attractive point. It is easy to verify that if T is quasinonexpansive, then every fixed point of T is an attractive point of T . Thus an attractive point is regarded as a generalization of a fixed point for a quasinonexpansive mapping. Takahashi and Takeuchi [17] also established a mean convergence theorem for an attractive point of a generalized hybrid mapping in the sense of Kocourek et al. [12]; see §2 for the definition of a generalized hybrid mapping. Such a mapping originates from a λ-hybrid mapping introduced in Aoyama et al. [2]; see also [4,5]. We know some existence and convergence results for attractive points of a generalized hybrid mapping and its variants; see, for example, [1,10,18,19]. In this paper, we prove that, under appropriate conditions, if a mapping T : C → H has an attractive point, then there exists a quasinonexpansive extensionT : H → H of T such that the set of fixed points (or asymptotic fixed points) ofT equals that of attractive points of T . Then using the quasinonexpansive extension, we derive convergence theorems for attractive points from those for fixed points of quasinonexpansive mappings. Moreover, we also obtain convergence results for attractive points of a generalized hybrid mapping. Preliminaries Throughout the present paper, H denotes a real Hilbert space, · , · the inner product of H, · the norm of H, C a nonempty subset of H, I the identity mapping on H, and N the set of positive integers. Strong convergence of a sequence {x n } in H to z ∈ H is denoted by x n → z and weak convergence by x n ⇀ z. Let T : C → H be a mapping. Then the set of fixed points of T is denoted by F(T ), that is, F(T ) = {z ∈ C : T z = z}. A point z ∈ H is said to be an asymptotic fixed point of T [15] if there exists a sequence {x n } in C such that x n − T x n → 0 and x n ⇀ z. The set of asymptotic fixed points of T is denoted byF(T ). It is clear that F(T ) ⊂F(T ). A point z ∈ H is said to be an attractive point of T [17] if T x − z ≤ x − z for all x ∈ C. The set of attractive points of T is denoted by A(T ), that is, A(T ) = x∈C {z ∈ H : T x − z ≤ x − z }. It is clear that C ∩ A(T ) ⊂ F(T ), and that A(T ) is closed and convex. Let T : C → H be a mapping and F a nonempty subset of H. Then T is said to be quasinonexpansive with respect to F [6] if T x − z ≤ x − z for all x ∈ C and z ∈ F ; T is said to be quasinonexpansive if F(T ) = ∅ and T x − z ≤ x − z for all x ∈ C and z ∈ F(T ); T is said to be nonexpansive if T x − T y ≤ x − y for all x, y ∈ C; T is said to be generalized hybrid [12] if there exist α, β ∈ R such that α T x − T y 2 + (1 − α) x − T y 2 ≤ β T x − y 2 + (1 − β) x − y 2 for all x, y ∈ C; T is said to be demiclosed at 0 if T z = 0 whenever {x n } is a sequence in C such that x n ⇀ z and T x n → 0; see, for example, [9]. It is clear that Moreover, under the assumption that C is closed and convex, we know the following: • If T is quasinonexpansive, then F(T ) is closed and convex; see [8, Theorem 1]; • if T is nonexpansive, then I − T is demiclosed at 0; see [9]. A generalized hybrid mapping has the following property: Lemma 2.1 ([19, Lemma 3.1]) . Let H be a Hilbert space, C a nonempty subset of H, T : C → H a generalized hybrid mapping, and {x n } a sequence in C such that x n − T x n → 0 and x n ⇀ z. Then z ∈ A(T ), that is, F(T ) ⊂ A(T ). Let D be a nonempty closed convex subset of H. It is known that, for each x ∈ H, there exists a unique point x 0 ∈ D such that x − x 0 = min{ x − y : y ∈ D}. Such a point x 0 is denoted by P D (x) and P D is called the metric projection of H onto D. It is known that the metric projection is nonexpansive; see [16] for more details. The following theorem is a direct consequence of [3, Theorem 5.5]; see also [14,Theorem 3.4]. Theorem 2.2. Let H be a Hilbert space, T : H → H a quasinonexpansive mapping, {α n } a sequence in (0, 1], {β n } a sequence in [0, 1], and {x n } a sequence defined by u, x 1 ∈ H and x n+1 = α n u + (1 − α n )[β n x n + (1 − β n )T x n ] for n ∈ N. Suppose thatF(T ) = F(T ), α n → 0, n α n = ∞, and lim inf n β n (1 − β n ) > 0. Then {x n } converges strongly to P F(T ) (u). Remark 2.3. In Theorem 2.2, the condition lim inf n β n (1 − β n ) > 0 is equivalent to the following: lim inf n β n > 0 and lim sup n β n < 1. The following theorem is a direct consequence of [13, Theorem 3.2]; see also [7]. Quasinonexpansive extensions In this section, we prove that, under appropriate assumptions, a mapping with an attractive point has a quasinonexpansive extension such that the set of fixed points (or asymptotic fixed points) is equal to that of attractive points (Lemma 3.4). We begin with the following: (3.1)T x = T x, x ∈ C; P A(T ) (x), otherwise. ThenT is an extension of T and quasinonexpansive with respect to A(T ). Moreover, A(T ) ⊂ F(T ). Proof. By the definition ofT , it is clear thatT is an extension of T . We show thatT is quasinonexpansive with respect to A(T ). Let x ∈ H and z ∈ A(T ). Suppose that z ∈ C. Since z is an attractive point of T , we have T x − z = T x − z ≤ x − z . On the other hand, suppose that z / ∈ C. Since P A(T ) is nonexpansive and z = P A(T ) (z), we have T x − z = P A(T ) (x) − P A(T ) (z) ≤ x − z . Therefore,T is quasinonexpansive with respect to A(T ). We next show that A(T ) ⊂ F(T ). Let z ∈ A(T ). Suppose that z ∈ C. SinceT is quasinonexpansive with respect to A(T ), we have T z − z ≤ z − z = 0, and hence z ∈ F(T ). On the other hand, suppose that z / ∈ C. Then we haveT z = P A(T ) (z) = z, and hence z ∈ F(T ). As a result, we conclude that A(T ) ⊂ F(T ). Proof. The equality F(T ) = {1} is obvious. We first show that A(T ) = {0}. Let x ∈ C. If x = 1, then |T x − 0| = |1| = |x − 0|; otherwise |T x − 0| = |−x| = |x − 0|. Thus 0 ∈ A(T ). On the other hand, suppose that z ∈ A(T ) and z = 0. Then z ∈ C. As a result, we have z ∈ C ∩ A(T ) ⊂ F(T ), and hence z = 1. However, since ( |T (1/2) − 1| = |−1/2 − 1| = 3/2 > 1/2 = |1/2 − 1| , we have z / ∈ A(T ), 1) If F(T ) ⊂ A(T ), then A(T ) = F(T ) andT is quasinonexpansive; (2) ifF(T ) ⊂ A(T ), thenF(T ) = F(T ), that is, I −T is demiclosed at 0. Proof. We first show (1). We know from Lemma 3.1 thatT is quasinonexpansive with respect to A(T ), and that A(T ) ⊂ F(T ). Thus it is enough to show that A(T ) ⊃ F(T ). Let z ∈ F(T ). If z ∈ C, then z =T z = T z, and hence z ∈ F(T ) ⊂ A(T ). If z / ∈ C, then z =T z = P A(T ) (z) ∈ A(T ). Consequently, it turns out that A(T ) ⊃ F(T ). We next show (2). Since F(T ) ⊂F(T ) ⊂ A(T ), it follows from (1) that A(T ) = F(T ). Thus it is enough to prove thatF(T ) ⊂ A(T ). Let z ∈F(T ). Then there exists a sequence {x n } in H such that x n −T x n → 0 and x n ⇀ z. We consider two cases, which might not be exclusive. (i) Suppose that there exists a subsequence {x n i } of {x n } such that x n i ∈ C for all i ∈ N. Then it follows that x n i − T x n i = x n i −T x n i → 0 and x n i ⇀ z. Thus, by assumption, we deduce that z ∈F( T ) ⊂ A(T ). (ii) Suppose that there exists a subsequence {x n i } of {x n } such that x n i / ∈ C for all i ∈ N. Then x n i − P A(T ) (x n i ) = x n i −T x n i → 0 and x n i ⇀ z. Since P A(T ) is a nonexpansive mapping on H, I − P A(T ) is demiclosed at 0. Hence z ∈ F(P A(T ) ) = A(T ). This completes the proof. Approximation of attractive points In this section, using lemmas in the previous section (Lemmas 3.1 and 3.4) and convergence theorems for quasinonexpansive mappings (Theorems 2.2 and 2.4), we obtain two convergence theorems for attractive points of a mapping satisfying the condition that every asymptotic fixed point is an attractive point, and as corollaries of them, we also obtain convergence results for attractive points of generalized hybrid mappings. (4.1) x n+1 = α n u + (1 − α n )[β n x n + (1 − β n )T x n ] for n ∈ N. Suppose that n α n = ∞, lim n α n = 0, and lim inf n β n (1−β n ) > 0. IfF(T ) ⊂ A(T ), then {x n } converges strongly to P A(T ) (u). Proof. LetT : H → H be an extension of T defined by (3.1). By the assumption thatF(T ) ⊂ A(T ), we see that F(T ) ⊂F(T ) ⊂ A(T ). Thus Lemma 3.4 implies thatF(T ) = F(T ) = A(T ) andT is quasinonexpansive. Moreover, since C is convex andT is an extension of T , it follows that x n+1 = α n u + (1 − α n )[β n x n + (1 − β n )T x n ] for all n ∈ N. Therefore we deduce from Theorem 2.2 that x n → P F(T ) (u) = P A(T ) (u). Using Theorem 4.1 and Lemma 2.1, we obtain the following corollary; see Takahashi, Wong, and Yao [19,Theorem 3.2]. . We can also check that x n+1 = α n x n + (1 − α n )T x n for all n ∈ N. Therefore Theorem 2.4 implies the conclusion. Finally, we obtain a weak convergence result for a widely more generalized hybrid mapping in the sense of [11] as a corollary of Theorem 4.4. Let C be a nonempty subset of a Hilbert space H and T : C → H a mapping. Recall that T is widely more generalized hybrid [11] if there exist α, β, γ, δ, ǫ, ζ, η ∈ R such that (4.3) α T x − T y 2 + β x − T y 2 + γ T x − y 2 + δ x − y 2 + ǫ x − T x 2 + ζ y − T y 2 + η x − T x − (y − T y) 2 ≤ 0 for all x, y ∈ C. Such a mapping T is called an (α, β, γ, δ, ǫ, ζ, η)-widely more generalized hybrid mapping. Using Theorem 4.4 and [10, Lemma 11], we obtain the following corollary; see [10,Theorem 14]. Corollary 4.5. Let H, C, and {α n } be the same as in Theorem 4.4. Let T : C → C be an (α, β, γ, δ, ǫ, ζ, η)-widely more generalized hybrid mapping with an attractive point and {x n } a sequence in C defined by x 1 ∈ C and (4.2) for n ∈ N. Suppose that (4.4) α + β + γ + δ ≥ 0, α + γ > 0, and ǫ + η ≥ 0 hold. Then {x n } converges weakly to some point in A(T ). Proof. [10,Lemma 11] shows thatF(T ) ⊂ A(T ). Thus Theorem 4.4 implies the conclusion. Remark 4.6. Corollary 4.5 is almost the same as [10,Theorem 14], except that α, β, γ, δ, ǫ, ζ, and η are assumed to satisfy (4.4) or (4.5) α + β + γ + δ ≥ 0, α + β > 0, and ζ + η ≥ 0 in [10, Theorem 14]. We can confirm that the conditions (4.4) and (4.5) are equivalent for an (α, β, γ, δ, ǫ, ζ, η)-widely more generalized hybrid mapping. • if A(T ) = ∅, then T is quasinonexpansive with respect to A(T ); • if T is a generalized hybrid mapping, then F(T ) ⊂ A(T ); • I − T is demiclosed at 0 if and only ifF(T ) = F(T ). Theorem 2 . 4 . 24Let H be a Hilbert space, T : H → H a quasinonexpansive mapping, {α n } a sequence in [0, 1], and {x n } a sequence defined by x 1 ∈ H and x n+1 = α n x n + (1 − α n )T x n for n ∈ N. Suppose thatF(T ) = F(T ) and lim inf n α n (1 − α n ) > 0. Then {x n } converges weakly to some point w ∈ F(T ). Lemma 3. 1 . 1Let H be a Hilbert space, C a nonempty subset of H, T : C → H a mapping with an attractive point, andT : H → H a mapping defined by Remark 3. 2 . 2In Lemma 3.1, one can verify that A(T ) = A(T ).The following example shows that A(T ) = F(T ) in Lemma 3.1. Example 3 . 3 . 33Let H = R and C = R \ {0}. Let T : C → C be a mapping defined by T x = 1, x = 1; −x, otherwise. Then F(T ) = {1} and A(T ) = {0}. Moreover, letT : H → H be a mapping defined by (3.1), that is,T x = 0, x = 0; T x, otherwise. Then F(T ) = {0, 1}. Therefore, A(T ) = F(T ). which is a contradiction. Therefore we conclude that A(T ) = {0}. We next show that F(T ) = {0, 1}. By definition,T 0 = 0 andT 1 = T 1 = 1. Thus {0, 1} ⊂ F(T ). If z / ∈ {0, 1}, then we haveT z = T z = −z = z. This means that {0, 1} ⊃ F(T ). Lemma 3 . 4 . 34Let H be a Hilbert space, C a nonempty subset of H, T : C → H a mapping with an attractive point, andT : H → H a mapping defined by(3.1). Then the following hold: Theorem 4 . 1 . 41Let H be a Hilbert space, C a nonempty convex subset of H, T : C → C a mapping with an attractive point, {α n } a sequence in (0, 1], {β n } a sequence in [0, 1], and {x n } a sequence in C defined by u, x 1 ∈ C and Corollary 4. 2 . 2Let H, C, {α n }, and {β n } be the same as in Theorem 4.1. Let T : C → C be a generalized hybrid mapping with an attractive point and {x n } a sequence in C defined by u, x 1 ∈ C and (4.1) for n ∈ N. Then {x n } converges strongly to P A(T ) (u). Proof. Lemma 2.1 shows thatF(T ) ⊂ A(T ). Thus Theorem 4.1 implies the conclusion. Remark 4 . 3 . 43Corollary 4.2 is almost the same as [19, Theorem 3.2], except that {α n } and {β n } are assumed to be sequences in (0, 1) in [19, Theorem 3.2]. Theorem 4 . 4 . 44Let H be a Hilbert space, C a nonempty convex subset of H, T : C → C a mapping with an attractive point, {α n } a sequence in [0, 1], and {x n } a sequence in C defined by x 1 ∈ C and n+1 = α n x n + (1 − α n )T x n for n ∈ N. Suppose that lim inf n α n (1 − α n ) > 0. IfF(T ) ⊂ A(T ), then {x n } converges weakly to some point in A(T ). Proof. LetT : H → H be an extension of T defined by (3.1). As in the proof of Theorem 4.1, Lemma 3.4 shows that a mappingT is a quasinonexpansive extension of T , and thatF(T ) = F(T ) = A(T ) AcknowledgmentThe author would like to acknowledge the financial support from Professor Kaoru Shimizu of Chiba University. Strong convergence theorem for nonexpansive mappings on star-shaped sets in Hilbert spaces. S Akashi, W Takahashi, Appl. Math. Comput. 219S. Akashi and W. Takahashi, Strong convergence theorem for nonexpansive mappings on star-shaped sets in Hilbert spaces, Appl. Math. Comput. 219 (2012), 2035-2040. Fixed point and ergodic theorems for λ-hybrid mappings in Hilbert spaces. K Aoyama, S Iemoto, F Kohsaka, W Takahashi, J. Nonlinear Convex Anal. 11K. Aoyama, S. Iemoto, F. Kohsaka, and W. Takahashi, Fixed point and ergodic theo- rems for λ-hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 11 (2010), 335-343. Strong convergence theorems for strongly relatively nonexpansive sequences and applications. 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Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces. T Kawasaki, W Takahashi, J. Nonlinear Convex Anal. 14T. Kawasaki and W. Takahashi, Existence and mean approximation of fixed points of generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 14 (2013), 71-87. Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. P Kocourek, W Takahashi, J.-C Yao, Taiwanese J. Math. 14P. Kocourek, W. Takahashi, and J.-C. Yao, Fixed point theorems and weak conver- gence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math. 14 (2010), 2497-2511. Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces. S.-Y Matsushita, W Takahashi, Fixed Point Theory Appl. S.-y. Matsushita and W. Takahashi, Weak and strong convergence theorems for rel- atively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2004), 37-47. 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W Takahashi, Y Takeuchi, J. Nonlinear Convex Anal. 12W. Takahashi and Y. Takeuchi, Nonlinear ergodic theorem without convexity for gen- eralized hybrid mappings in a Hilbert space, J. Nonlinear Convex Anal. 12 (2011), 399-406. Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces. W Takahashi, N.-C Wong, J.-C Yao, J. Nonlinear Convex Anal. 13W. Takahashi, N.-C. Wong, and J.-C. Yao, Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces, J. Nonlinear Convex Anal. 13 (2012), 745-757. Attractive points and Halpern-type strong convergence theorems in Hilbert spaces. J. Fixed Point Theory Appl. 17, Attractive points and Halpern-type strong convergence theorems in Hilbert spaces, J. Fixed Point Theory Appl. 17 (2015), 301-311. . 263-0043Aoyama) Aoyama Mathematical Laboratory, Konakadai, Inage-ku. Japan Email address: [email protected]) Aoyama Mathematical Laboratory, Konakadai, Inage-ku, Chiba, Chiba 263-0043, Japan Email address: [email protected]

This supplementary material contains some additional details regarding the velocity distribution in the molecular dynamics simulation, the power spectrum of the Fourier transform of the graphene samples, further details concerning the fractal dimension, the Left-passage probability test within the SLE theory, and finally some remarks on other atomistic membranes.

10.1038/srep22949

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Supplementary Material: Conformal Invariance of Graphene Sheets I Giordanelli Computational Physics for Engineering Materials Institute for Building Materials ETH Zürich Wolfgang-Pauli-Strasse 27 HIT CH-8093ZürichSwitzerland N Posé Computational Physics for Engineering Materials Institute for Building Materials ETH Zürich Wolfgang-Pauli-Strasse 27 HIT CH-8093ZürichSwitzerland M Mendoza Computational Physics for Engineering Materials Institute for Building Materials ETH Zürich Wolfgang-Pauli-Strasse 27 HIT CH-8093ZürichSwitzerland H J Herrmann Computational Physics for Engineering Materials Institute for Building Materials ETH Zürich Wolfgang-Pauli-Strasse 27 HIT CH-8093ZürichSwitzerland Departamento de Física Universidade Federal do Ceará Campus do Pici60455-760Fortaleza, CearáBrazil Supplementary Material: Conformal Invariance of Graphene Sheets This supplementary material contains some additional details regarding the velocity distribution in the molecular dynamics simulation, the power spectrum of the Fourier transform of the graphene samples, further details concerning the fractal dimension, the Left-passage probability test within the SLE theory, and finally some remarks on other atomistic membranes. Power spectrum We study some further properties of the graphene sheets, and especially its power spectrum. From the power spectrum of a graphene membrane of size 800Å at 300 K, we obtain an estimate of the Hurst exponent α (see Fig. S2) in agreement with the one obtained from the height-height correlation function, see Fig. 2 of the main text. Fractal dimension at higher temperatures We perform an additional study to see if the fractal dimension of the iso-height contour lines keeps being independent of the system size for higher temperatures. Indeed, Fig. S3 confirms the temperature independence of the system size for temperatures |q | Figure S2. Power spectrum of a given graphene sheet. One recovers the expected power law behavior. The solid line is a guide to the eye of a slope of −2(α + 1) with α = 0.68 ± 0.05. much higher than 600 K. Fractal dimension of the area The fractal dimension d a of the area that the contour lines enclose has also been measured (see Fig. S4), finding a value of d a = 1.82 ± 0.01, which is also independent of the temperature. Our results show that the iso-height contour lines and the area enclosed by them possess scale invariant properties. We have also studied the fractal dimension of the watershed (defined in Ref. 1 ) of each graphene sample. A temperature invariant fractal dimension of those lines is found to be d f = 1.07 ± 0.01. This result strengthens the analogy of graphene membranes with other landscapes at criticality. 2 Left-passage probability In order to have a further numerical evidence of the compatibility of the statistics of iso-height lines on graphene surfaces with SLE, we study a further property of the SLE paths. Their left-passage probabilities should follow the so-called Schramm's formula of Eq. (1) for chordal SLE curves. Such curves start at the origin and grow towards infinity. Therefore, they split the upper half-plane in two domains: the points that are at the left of the curve, and the ones that are at the right. Schramm provided an expression for the probability that the curve goes at the left of a given point in the upper half-plane, i.e. the point is on the right side of the curve: P κ (φ ) = 1 2 + Γ (4/κ) √ πΓ 8−κ 2κ cot(φ ) 2 F 1 1 2 , 4 κ , 3 2 , − cot(φ ) 2 ,(1) where Γ is the Gamma function and 2 F 1 is the hypergeometric function. Here we show that this formula is satisfied by the iso-height lines of graphene using the following method. We first define a set of sample points S, for which we estimate the probability of being at the right of our curves that have been mapped to infinity. 3 To estimate κ, we minimize the mean square deviation Q(κ) defined as: Q (κ) = 1 |S| ∑ z∈S [P(z) − P κ (φ )] 2 ,(2) where |S| is the cardinality of the set S, and P(z) the measured left-passage probability at the point z. The value of κ for which the minimum of Q is attained gives us an estimate of the diffusion coefficient of our iso-height lines. In Fig. S5, we can see that the results are in agreement with our previous estimates of κ = 2.27 ± 0.08, and also show that the left-passage probability satisfies Schramm's formula. Fig. S6 shows a simulation performed with silicon atoms in a honeycomb lattice. We can see that the suspended structure crumples and does not form a stable two-dimensional-like crystal, in contrast to graphene membranes (see inset of Fig. S6). The stability of the suspended graphene membrane is not found for the chemically similar suspended silicene membrane. Figure S6. Main panel: Simulation of suspended silicene, a two-dimensional honeycomb lattice with silicon atoms. The simulation has been performed with the Tersoff potential applied to silicon. 4 Inset panel: Graphene membrane simulated with the Tersoff potential. Other atomistic membranes 5/5 Figure S1 . S1Velocity distribution of carbon atoms of a quadratic graphene sheet of size 400Å after 200 ps simulation. The red line is the Maxwell-Boltzmann distribution at T = 300 K. Figure S3 . S3Fractal dimension of the iso-height contour lines computed with the yardstick method for different temperatures and fixed system size of 800Å. Figure S4 .Figure S5 . S4S5Fractal dimension of the area enclosed by the iso-height contour lines measured with the box-counting method. Main panel: The fractal dimension for different system sizes at T = 300 K. Inset: The fractal dimension for different temperatures for a system size of 800Å. (color online) Measured rescaled mean square deviation Q(κ)/Q min as a function of κ being Q min the minimum value of Q, for temperatures T = 100 K, T = 300 K, and T = 600 K. Inset: the measured left-passage probabilities are compared with Schramm's formula for κ = 2.24 (displayed as the solid line). New efficient methods for calculating watersheds. E Fehr, Journal of Statistical Mechanics: Theory and Experiment. 9007Fehr, E. et al. New efficient methods for calculating watersheds. Journal of Statistical Mechanics: Theory and Experiment 2009, P09007 (2009). Scaling relations for watersheds. E Fehr, D Kadau, N A M Araújo, J S AndradeJr, H J Herrmann, Phys. Rev. E. 8436116Fehr, E., Kadau, D., Araújo, N. A. M., Andrade Jr., J. S. & Herrmann, H. J. Scaling relations for watersheds. Phys. Rev. E 84, 036116 (2011). . N Posé, K J Schrenk, N A M Araújo, H Herrmann, J. Shortest path and Schramm-Loewner Evolution. Sci. Rep. 45495Posé, N., Schrenk, K. J., Araújo, N. A. M. & Herrmann, H. J. Shortest path and Schramm-Loewner Evolution. Sci. Rep. 4, 5495 (2014). New empirical approach for the structure and energy of covalent systems. J Tersoff, Phys. Rev. B. 375Tersoff, J. New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991-7000 (1988). 4/5

An rf superconducting quantum interference device (SQUID) array in an alternating magnetic field is investigated with respect to its effective magnetic permeability, within the effective medium approximation. This system acts as an inherently nonlinear magnetic metamaterial, leading to negative magnetic response, and thus negative permeability, above the resonance frequency of the individual SQUIDs. Moreover, the permeability exhibits oscillatory behavior at low field intensities, allowing its tuning by a slight change of the intensity of the applied field.

10.1063/1.2722682

cond-mat/0703400

48887d6d1cad4fd1b3619e3e3afd88a3f86b1af6

rf SQUID metamaterials 15 Mar 2007 N Lazarides G P Tsironis Department of Physics Institute of Electronic Structure and Laser Department of Electrical Engineering University of Crete Foundation for Research and Technology-Hellas P. O. Box 220871003HeraklionGreece Facultat de Fisica, Department d'Estructura i Constituents de la Materia Department of Physics Technological Educational Institute of Crete Universitat de Barcelona, Av. Diagonal 647, EP. O. Box 14071500, 08028Stavromenos, Heraklion, Crete, BarcelonaGreece, Spain and Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas University of Crete P. O. Box 220871003HeraklionGreece rf SQUID metamaterials 15 Mar 2007 An rf superconducting quantum interference device (SQUID) array in an alternating magnetic field is investigated with respect to its effective magnetic permeability, within the effective medium approximation. This system acts as an inherently nonlinear magnetic metamaterial, leading to negative magnetic response, and thus negative permeability, above the resonance frequency of the individual SQUIDs. Moreover, the permeability exhibits oscillatory behavior at low field intensities, allowing its tuning by a slight change of the intensity of the applied field. An rf superconducting quantum interference device (SQUID) consists of a superconducting ring interrupted by a Josephson junction (JJ). 1 When driven by an alternating magnetic field, the induced supercurrents around the ring are determined by the JJ through the celebrated Josephson relations. This system exhibits rich nonlinear behavior, including chaotic effects. 2 Recently, quantum rf SQUIDs have attracted great attention, since they constitute essential elements for quantum computing. 3 In this direction, rf SQUIDs with one or more zero and/or π ferromagnetic JJs have been constructed. 4 In this Letter we show that rf SQUIDs may serve as constitual elements for nonlinear magnetic metamaterials (MMs), i.e., artificial, composite, inherently non-magnetic media with (positive or negative) magnetic response at microwave frequencies. Classical MMs are routinely fabricated with regular arrays of split-ring resonators (SRRs), with operating frequencies up to the optical range. 5 Moreover, MMs with negative magnetic response can be combined with plasmonic wires that exhibit negative permittivity, producing thus left-handed (LH) metamaterials characterized by negative refraction index. Superconducting SRRs promise severe reduction of losses, which constrain the evanescent wave amplification in these materials. 6 Thus, metamaterials involving superconducting SRRs and/or wires have been recently demonstrated experimentally. 7 The effect of incorporating superconductors in LH transmission lines has been also studied. 8 Naturally, the theory of metamaterials has been extended to account for nonlinear effects. 9,10,11,12,13 Nonlinear MMs support several types of interesting excitations, e.g., magnetic domain walls, 11 discrete breathers, 12 and evnelope solitons. 13 Regular arrays of rf SQUIDs offer an alternative for the construction of nonlinear MMs due to the nonlinearity of the JJ. Very much like the SRR, the rf SQUID ( Fig. 1(b)) is a resonant nonlinear oscillator, and similarly it responds in a manner analogous to a magnetic "atom" in a time- varying magnetic field with appropriate polarization, exhibiting a resonant magnetic response at a particular frequency. The SRRs are equivalently RLC circuits in series, featuring a resistance R, a capacitance C and an inductance L, working as small dipoles. In turn, adopting the resistively and capacitively shunted junction (RCSJ) model for the JJ, 1 the rf SQUIDs are not dipoles but, instead, they feature an inductance L in series with an ideal Josephson element (i.e., for which I = I c sin φ, with φ the Josephson phase), shunted by a capacitor C and a resistor R ( Fig. 1(c)). However, the fields they produce are approximatelly those of small dipoles, although quantitatively they are affected by flux quantization in superconducting loops. Consider an rf SQUID with loop area S = πa 2 (radius a), in a magnetic field of amplitude H e0 , frequency ω, and intensity H ext = H e0 cos(ωt) perpendicular to its plane (t is the time variable). The field generates a flux Φ ext = Φ e0 cos(ωt) threading the SQUID loop, with Φ e0 = µ 0 SH e0 , and µ 0 the permeability of the vacuum. The flux Φ trapped in the SQUID ring is given (in normalized variables) by SQUID ring, and Φ 0 is the flux quantum. The dynamics of the normalized flux f is governed by the equation f = f ext + β i,(1)where f = Φ/Φ 0 , f ext = Φ ext /Φ 0 , i = I/I c , β = β L /2π ≡ LI c /Φ 0 , I isd 2 f dτ 2 + γ df dτ + β sin (2πf ) + f = f ext ,(2) where C and R is the capacitance and resistance, respectively, of the JJ, γ = Lω 0 /R, τ = ω 0 t, ω 2 0 = 1/LC, and f ext = f e0 cos(Ωτ ),(3) with f e0 = Φ e0 /Φ 0 , and Ω = ω/ω 0 . The small parameter γ actually represents all of the dissipation coupled to the rf SQUID. An approximate solution for Eq. (2) may be obtained for Ω close to the SQUID resonance frequency (Ω ∼ 1) in the non-hysteretic regime β L < 1. Following Ref. 14 we expand the nonlinear term in Eq. (2) in a Fourier -Bessel series of the form β sin (2πf ) = − ∞ n=1 (−1) n nπ J n (nβ L ) sin(2πnf ext ),(4) where J n is the Bessel function of the first kind, of order n. By substiting Eq. (3) in Eq. (4) and carrying out the Fourier -Bessel expansion of the sine term, one needs to retain only the fundamental Ω component in the expansion. 15 This leads to the simplified expression β sin (2πf ) ≃ D(f e0 ) cos(Ωτ ),(5) where D(f e0 ) = −2 ∞ n=1 (−1) n nπ J n (nβ L )J 1 (2πnf e0 ) . By substitution of Eq. (5) in Eq. (2), the latter can be solved for the flux f = f 0 cos(Ωτ + θ) in the loop, with f 0 = f e0 − D γ 2 Ω 2 + (1 − Ω 2 ) 2 , θ = tan −1 −γΩ 1 − Ω 2 ,(6) where θ is the phase difference between f and f ext . The dependence of D and f 0 on f e0 for low field intensity is illustrated in Figs. 1(a) and 1(b), respectively. For larger f e0 the coefficient D approaches zero still oscillating, while f 0 approaches a straight line with slope depending on Ω and γ. For γ ≪ 1 and not very close to the resonance, θ ≃ 0. It is instructive to express the γ = 0 solution as: f = ±|f 0 | cos(Ωτ ), |f 0 | = (f e0 − D)/|1 − Ω 2 |. (7) The plus (minus) sign, corresponding to a phase-shift of 0 (π) of f with respect to f ext , is obtained for Ω < 1 (Ω > 1). Thus, the flux f may be either in-phase (+ sign) or in anti-phase (-sign) with f ext , depending on Ω. This is confirmed by numerical integration of Eq. (2), as shown in Fig. 2, where we plot separately the three terms of Eq. (1) in time. The quantities f , f ext , and β i are shown for two periods T = 2π/Ω in each case, after they have reached a steady state. For Ω < 1 (Figs. 2(a) and 2(b)), the flux f (green-dashed curves) is inphase with f ext (blue-dotted curves), while for Ω > 1 (Figs. 2(c) and 2(d)) the flux f is in anti-phase with f ext . The other curves (red-solid curves) correspond to β i, the response of the SQUID to the applied flux. Away from the resonance, the response is (in absolute value) less than ( Fig. 2(a), for Ω = 0.63) or nearly equal ( Fig. 2(d), for Ω = 8.98) to the magnitude of f ext . However, close to resonance, the response β i is much larger than f ext , leading to a much higher flux f (Figs. 2(b) and 2(c) for Ω = 0.9 and Ω = 1.1, respectively). Moreover, in Fig. 2(c), f is in anti-phase with f ext , showing thus extreme diamagnetic (negative) response. The numerically obtained amplitudes f 0 (depicted as black circles for Ω = 0.9 and blue diamonds for Ω = 1.1 in Fig. 2(b)) are in fair agreement with the analytical expression, Eq. (6). The agreement becomes better for larger f e0 . We now consider a planar rf SQUID array consisting of identical units (Fig. 1(a)), and forming a lattice of unitcell-side d; the system is placed in a magnetic field H ext ≡ H perpendicular to SQUID plane. If the wavelength of H is much larger than d, the array can be treated as an effectivelly continuous and homogeneous medium. Then, the magnetic induction B in the array plane is where M = S I/d 3 is the magnetization induced by the current I circulating a SQUID loop, and µ r the relative permeability of the array. Introducing M into Eq. (8), and using Eqs. (1), (3), and (7), we get B = µ 0 (H + M ) ≡ µ 0 µ r H,(8)µ r = 1 +F (±|f 0 |/f e0 − 1) ,(9) whereF = π 2 (µ 0 a/L)(a/d) 3 . The coefficientF has to be very small (F ≪ 1), so that magnetic interactions between individual SQUIDs can be neglected in a first approximation. Recall that the plus sign in front of |f 0 |/f e0 should be taken for Ω < 1, while the minus sign should be taken for Ω > 1. In Fig. 3 we plot µ r both for Ω < 1 ( Fig. 3(a)) and Ω > 1 (Fig. 3(b)), for three different values of F . In real arrays, that coefficient could be engineered to attain the desired value. In both Figs. 3(a) and 3(b), the relative permeability µ r oscillates for low intensity fields (low f ext ), while it tends to a constant at larger f ext . In Fig. 3(a) (Ω < 1), the relative permeability µ r is always positive, while it increases with increasingF . In Fig. 3(b), however, µ r may assume both positive and negative values, depending on the value ofF . With appropriate choise ofF , it becomes oscillatory around zero (green-dashed curve in Fig. 3(b)) allowing tuning from positive to negative µ r with a slight change of f ext . In conclusion, we have shown that a planar rf SQUID array exhibits large magnetic response close to resonance, which may be negative above the resonance frequency, leading to effectivelly negative µ r . For low field intensities (low f ext ), µ r exhibits oscillatory behavior which gradually dissappears for higher f ext . This behavior may be exploited to construct a flux-controlled metamaterial (as opposed to voltage-controlled metamaterial demonstrated in Ref. 16 ). The physical parameters required for the rf SQUIDs giving the dimensionless parameters used above are not especially formidable. An rf SQUID with L ≃ 105 pH, C ≃ 80 f F , and I c ≃ 3 µA, would give β ≃ 0.15 (β L ≃ 0.94). For these parameters, a value of the resistance R ≃ 3.6 KΩ is required in order to have γ ≃ 10 −3 , used in the numerical integration of Eq. (2). However, our results are qualitatively valid for γ even an order of magnitude larger, in which case R ≃ 360 Ω. We note that ω 0 = ω p / √ β L , where ω p is the plasma frequency of the JJ. For the parameters considered above, where β L is slightly less than unity, the frequencies ω 0 and ω p are of the same order. However, ω p does not seem to have any special role in the microwave response of the rf SQUID. Du et al. have studied the quantum version of a SQUID array as a LH metamaterial, concluded that negative refractive index with low loss may be obtained in the quantum regime. 17 Consequently, µ r can be negative at some specific frequency range. However, their corresponding expression for µ r is linear, i.e., it does not depend on the amplitude of the applied flux, and thus it does not allow flux-tuning. Moreover, experiments with SQUID arrays in the quantum regime, where individual SQUIDs can be described as two-level systems, are much more difficult to realize. We acknowledge support from the grant "Pythagoras II" (KA. 2102/TDY 25) of the Greek Ministry of Education and the European Union, and grant 2006PIV10007 of the Generalitat de Catalunia, Spain. FIG. 1 : 1Schematic drawing of the SQUID array, along with the equivalent circuit for an rf SQUID in external flux Φext. FIG. 2 : 2the current circulating in the ring, I c is the critical current of the JJ, L is the inductance of the (Color online) (a) Coefficient D vs. the applied flux amplitude fe0, for β = 0.15 (red-solid curve); β = 0.10 (greendashed curve); β = 0.05 (blue-dotted curve). (b) The amplitude of the flux f0 vs. fe0, for Ω = 0.9 (red-solid curve), Ω = 1.1 (green-dashed curve), and γ = 0.001, β = 0.15. The black circles and blue diamonds correspond to the numerically obtained f0 for Ω = 0.9 and 1.1, respectively. FIG. 3 : 3(Color online) Time-dependence of the flux f (greendashed curves), the applied flux fext (blue-dotted curves), and the response β i (red-solid curves), for β = 0.15, FIG. 4 : 4(Color online) Relative permeability µr vs. fe0, for F = 0.01 (red-solid curves),F = 0.02 (green-dashed curves), F = 0.03 (blue-dotted curves), and Ω = 0.99 (a); 1.01 (b). K K Likharev, Dynamics of Josephson Junctions and Circuits. PhiladelphiaGordon and BreachK. K. Likharev, Dynamics of Josephson Junctions and Cir- cuits (Gordon and Breach, Philadelphia, 1986). . K Fesser, A R Bishop, P Kumar, Appl. Phys. Lett. 43123K. Fesser, A. R. Bishop, and P. Kumar, Appl. Phys. Lett. 43, 123 (1983). . M F Bocko, A M Herr, M J Feldman, IEEE Trans. Appl. Supercond. 73638M. F. Bocko, A. M. Herr, and M. J. Feldman, IEEE Trans. Appl. Supercond. 7, 3638 (1997). . T Yamashita, S Takahashi, S Maekawa, Appl. Phys. Lett. 88132501T. Yamashita, S. Takahashi, and S. Maekawa, Appl. Phys. Lett. 88, 132501 (2006). . T J Yen, W J Padilla, N Fang, D C Vier, D R Smith, J B Pendry, D N Basov, X Zhang, Science. 3031494T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, Science 303, 1494 (2004). . M C Ricci, H Xu, S M Anlage, R Prozorov, A P Zhuravel, A V Ustinov, cond-mat/0608737M. C. Ricci, H. Xu, S. M. Anlage, R. Prozorov, A. P. Zhuravel, and A. V. Ustinov, e-print cond-mat/0608737. . M C Ricci, N Orloff, S M Anlage, Appl. Phys. Lett. 8734102M. C. Ricci, N. Orloff, and S. M. Anlage, Appl. Phys. Lett. 87, 034102 (2005). . H Salehi, A H Majedi, R R Mansour, IEEE Trans. Appl. Supercond. 15996H. Salehi, A. H. Majedi, and R. R. Mansour, IEEE Trans. Appl. Supercond. 15, 996 (2005). . A A Zharov, I V Shadrivov, Y Kivshar, Phys. Rev. A. A. Zharov, I. V. Shadrivov, and Y. Kivshar, Phys. Rev. . Lett, 9137401Lett. 91, 037401 (2003); . M Lapine, M Gorkunov, K H Ringhofer, Phys. Rev. E. 6765601M. Lapine, M. Gorkunov, and K. H. Ringhofer, Phys. Rev. E 67, 065601 (2003). . N Lazarides, G P Tsironis, Phys. Rev. E. 7136614N. Lazarides and G. P. Tsironis, Phys. Rev. E 71, 036614 (2005); . I Kourakis, P K Shukla, Phys. Rev. E. 7216626I. Kourakis and P. K. Shukla, Phys. Rev. E 72, 016626 (2005). . I V Shadrivov, A A Zharov, N A Zharova, Y S Kivshar, Photonics Nanostruct. 469I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and Y. S. Kivshar, Photonics Nanostruct. 4, 69 (2006). . N Lazarides, M Eleftheriou, G P Tsironis, Phys. Rev. Lett. 97157406N. Lazarides, M. Eleftheriou, and G. P. Tsironis, Phys. Rev. Lett. 97, 157406 (2006). . I Kourakis, N Lazarides, G P Tsironis, cond-mat/0612615I. Kourakis, N. Lazarides, and G. P. Tsironis, e-print cond-mat/0612615. . S N Erné, H.-D Hahlbohm, H Lübbig, J. Appl. Phys. 475440S. N. Erné, H.-D. Hahlbohm, and H. Lübbig, J. Appl. Phys. 47, 5440 (1976). . A R Bulsara, J. Appl. Phys. 602462A. R. Bulsara, J. Appl. Phys. 60, 2462 (1986). . O Reynet, O Acher, Appl. Phys. Lett. 841198O. Reynet and O. Acher, Appl. Phys. Lett. 84, 1198 (2004). . C Du, H Chen, S Li, Phys. Rev. B. 74113105C. Du, H. Chen, and S. Li, Phys. Rev. B 74, 113105 (2006).

In order to obtain morphological information of unlabeled galaxies, we present an unsupervised machine-learning (UML) method for morphological classification of galaxies, which can be summarized as two aspects: (1) the methodology of convolutional autoencoder (CAE) is used to reduce the dimensions and extract features from the imaging data; (2) the bagging-based multiclustering model is proposed to obtain the classifications with high confidence at the cost of rejecting the disputed sources that are inconsistently voted. We apply this method on the sample of galaxies with H < 24.5 in CANDELS. Galaxies are clustered into 100 groups, each contains galaxies with analogous characteristics. To explore the robustness of the morphological classifications, we merge 100 groups into five categories by visual verification, including spheroid, early-type disk, late-type disk, irregular, and unclassifiable. After eliminating the unclassifiable category and the sources with inconsistent voting, the purity of the remaining four subclasses are significantly improved. Massive galaxies (M * > 10 10 M ) are selected to investigate the connection with other physical properties. The classification scheme separates galaxies well in the U-V and V-J color space and Gini-M 20 space. The gradual tendency of Sérsic indexes and effective radii is shown from the spheroid subclass to the irregular subclass. It suggests that the combination of CAE and multi-clustering strategy is an effective method to cluster galaxies with similar features and can yield high-quality morphological classifications. Our study demonstrates the feasibility of UML in morphological analysis that would develop and serve the future observations made with China Space Station telescope.

10.3847/1538-3881/ac4245

2112.13957

667f490a0fc3c21ad89f7ac7f5f00b2554f71238

AUTOMATIC MORPHOLOGICAL CLASSIFICATION OF GALAXIES: CONVOLUTIONAL AUTOENCODER AND BAGGING-BASED MULTICLUSTERING MODEL 30 Dec 2021 Chichun Zhou School of Engineering Dali University 671003DaliPeople's Republic of China Yizhou Gu School of Physics and Astronomy Shanghai Jiao Tong University 800 Dongchuan Road200240MinhangShanghaiPeople's Republic of China ZHOU ET AL Guanwen Fang Institute of Astronomy and Astrophysics Anqing Normal University 246133AnqingPeople's Republic of China Zesen Lin Department of Astronomy CAS Key Laboratory for Research in Galaxies and Cosmology University of Science and Technology of China 230026HefeiPeople's Republic of China School of Astronomy and Space Science University of Science and Technology of China 230026HefeiPeople's Republic of China AUTOMATIC MORPHOLOGICAL CLASSIFICATION OF GALAXIES: CONVOLUTIONAL AUTOENCODER AND BAGGING-BASED MULTICLUSTERING MODEL DRAFT VERSION JANUARY 3202230 Dec 2021Typeset using L A T E X twocolumn style in AASTeX61 Corresponding author: Guanwen Fang * Yizhou Gu and ChiChun Zhou contributed equally to this workGalaxy structure (622)Astrostatistics techniques (1886)Astronomy data analysis (1858) In order to obtain morphological information of unlabeled galaxies, we present an unsupervised machine-learning (UML) method for morphological classification of galaxies, which can be summarized as two aspects: (1) the methodology of convolutional autoencoder (CAE) is used to reduce the dimensions and extract features from the imaging data; (2) the bagging-based multiclustering model is proposed to obtain the classifications with high confidence at the cost of rejecting the disputed sources that are inconsistently voted. We apply this method on the sample of galaxies with H < 24.5 in CANDELS. Galaxies are clustered into 100 groups, each contains galaxies with analogous characteristics. To explore the robustness of the morphological classifications, we merge 100 groups into five categories by visual verification, including spheroid, early-type disk, late-type disk, irregular, and unclassifiable. After eliminating the unclassifiable category and the sources with inconsistent voting, the purity of the remaining four subclasses are significantly improved. Massive galaxies (M * > 10 10 M ) are selected to investigate the connection with other physical properties. The classification scheme separates galaxies well in the U-V and V-J color space and Gini-M 20 space. The gradual tendency of Sérsic indexes and effective radii is shown from the spheroid subclass to the irregular subclass. It suggests that the combination of CAE and multi-clustering strategy is an effective method to cluster galaxies with similar features and can yield high-quality morphological classifications. Our study demonstrates the feasibility of UML in morphological analysis that would develop and serve the future observations made with China Space Station telescope. INTRODUCTION Galaxy morphology is an important characteristic relevant to other key physical properties, such as the stellar mass, the color, the star-formation rate (SFR), the gas content, the galaxy age, and the environment (e.g., Dressler 1980;Kauffmann et al. 2003Kauffmann et al. , 2004Omand et al. 2014;Schawinski et al. 2014;Kawinwanichakij et al. 2017;Gu et al. 2018;Lianou et al. 2019). The morphological diversity of galaxies implies the difference in the evolutionary histories of galaxies. As the development of telescopes and instruments advances, the automatic morphological analysis and classification of galaxies are imperiously demanded to help understand galaxy formation rate and evolution. To exploit the potentials of galaxy morphologies from current and future surveys, we present an unsupervised machine-learning (UML) method for the morphological classification through the astronomical imaging data. The apparent visual morphology might be the earliest measurement of galaxies (Hubble 1926). Human classifiers can judge the morphological type of galaxies by a certain classification scheme (e.g, Hubble 1926;de Vaucouleurs 1959;van den Bergh 1960). Large projects can employ many human classifiers to give the label or probability of galaxy morphology (e.g., Lintott et al. 2011). Indeed, conventional visual classification may have some incompatible results due to subjective deviation. However, what's more important is that conventional visual classification is quite low inefficiency since galaxies should be inspected one by one. Except the visual classifications by eyes, many techniques are developed to quantify the galaxy structure and the morphology. To obtain the morphological classifications of galaxies, the key is to extract the morphological features from the raw images. Parametric measurements describe the light profiles of galaxies using the mathematical models with a set of parameters (e.g. Sérsic 1963;van der Wel et al. 2012;Lang et al. 2014). Nonparametric measurements design several model-independent parameters to describe some characteristics of galaxy morphologies (e.g. Abraham et al. 1994;Conselice 2003;Lotz et al. 2004;Freeman et al. 2013; Also see the review by Conselice 2014). Relying on the parameter spaces constituted of the morphological features, several approaches are proposed to distinguish the morphological types, such as the concentration, asymmetry, smooth-ness method (Conselice 2014), the principal component analysis (Scarlata et al. 2007), and the support-vector machine (Huertas-Company et al. 2008). The structural parameters reduce the complexity of morphological descriptions. However, it rejects the abundant information hidden in all pixels and may lead to failures of morphological distinction in some cases. The convolutional neural network is one of the most popular methods of supervised machine learning applied in astronomy to classify galaxy morphologies automatically (e.g., Huertas-Company et al. 2015;Dieleman et al. 2015;Walmsley et al. 2019). It can extract enormous amount of information contained in the pixels themselves hierarchically and have a good performance in mimicking the human perceptions with a high efficiency. However, it is a supervisedlearning method which means that it is highly dependent on the prelabeled training data set. By training the model on labeled data, the supervised-learning method is good at mimicking human perceptions. The methodology of UML might be another way to the morphological classification, which does not need a prelabeled training set labeled by human classifiers. Hence it has no subjective deviations of humans and may exploit a new angle of galaxy morphologies from the machine's view. The methodology of UML is ideally suited to the morphological analysis of Big Data surveys, which has been applied to the automatic classification of optical/NIR and radio images (e.g., Ralph et al. 2019;Galvin et al. 2020). Hocking et al. (2018) and Cheng et al. (2020) apply the growing neural gas algorithm (Fritzke 1995) to extract features from images and the hierarchical clustering technique is responsible for gathering the galaxies with similar features. The convolutional autoencoder (CAE) (Masci et al. 2011) is also another effective technique to extract features from images. The combination of autoencoder and clustering algorithm is able to obtain the reasonable results on strong lensing identification (Cheng et al. 2020) and morphological classification (Cheng et al. 2021). The upcoming China Space Station Telescope (CSST) will image the sky in 7 bands (N U V, u, g, r, i, z, and y) and vastly enrich the photometric data with a wide survey covered 17,500 deg 2 with 5σ depth of r = 26.0 mag and a deep survey covered 400 deg 2 with 5σ depth of r = 27.2. To pre-pare for and to exploit the imaging data of the CSST, we plan to develop an UML method for galaxy classifications. In this paper, we pioneer in the use of unsupervised classification in the five CANDELS fields. The CAE is trained for the extraction of unsupervised features from the raw imaging data. After that, to avoid the bias from one single clustering algorithm, a bagging-based multiclustering method is proposed which considers the results of three different clustering algorithms. Although at the cost of eliminating the disputed sources, galaxies are clustered into 100 groups under a comprehensive definition of similarity. As a result, the purity in each classification is significantly improved. To test the feasibility of our method, we merge 100 groups into five subclasses by visual verification, including spheroid (SPH), early-type disk (ETD), late-type disk (LTD), irregular (IRR), and unclassifiable (UNC). After discarding the disputed sources and the UNC category, we investigate the connection with colors and morphological parameters using the massive galaxies (M * > 10 10 M ). Moreover, by using the t-SNE visualize technique (van der Maaten & Hinton 2008), the comparisons between our result and that of CAN-DELS visual classifications (Kartaltepe et al. 2015) and the supervised deep-learning method (Huertas-Company et al. 2015) are given. It suggests that the proposed method, combination of CAE and multiclustering strategy, is an effective method to cluster galaxies with similar features and can yield high-quality morphological classifications, which are useful in other downstream tasks. This paper is organized as follows. The data set and our sample construction are described in Section 2. The method is described in Section 3, which includes the CAE and the bagging-based multiclustering method. We compare our clustering result with other physical properties of massive galaxies and the results of other works in Section 4. Main conclusions and outlooks are summarized in Section 5. When converting the effective radii of galaxy from observational scales (arcsec) to physical scales (kpc), we assume the cosmological parameters as following: H 0 = 70 km s −1 Mpc −1 , Ω m = 0.30, Ω Λ = 0.70. DATA AND SAMPLE SELECTION CANDELS have provided WFC3 and ACS photometry over ∼ 900 arcmin 2 in five fields: AEGIS, COSMOS, GOODS-N, GOODS-S, and UDS (Grogin et al. 2011;Koekemoer et al. 2011). The 3D-HST Treasury Program provides a large amount of the data products, refering to photometry (Skelton et al. 2014) and grism spectra (Momcheva et al. 2016). Momcheva et al. (2016) provide a "best" redshift catalog by merging their grism-based results with the photometric results in Skelton et al. (2014). Here, we give a brief introduction of the 3D-HST data set. The redshift of a galaxy is arranged in the order of spectroscopic redshift, grism redshift, and photometric redshift. If spectroscopic redshift is not available, then we adopt grism redshift. If grism redshift is not available, we use photometric redshift instead. Photometric redshifts are derived by the spectral Energy Distributions (SEDs) ranging from 0.3 to 8.0 µm with the EAZY code (Brammer et al. 2008). It applies the linear combination of seven galaxy spectral templates, wihch are the defaults in Brammer et al. (2008), including the five templates from the library of PÉGASE stellar population synthesis models (Fioc & Rocca-Volmerange 1997), one young, dusty template and one old, red galaxy template (see also Whitaker et al. 2011). The error of photometric redshift reaches ∆z/(1 + z) ≈ 0.02 on average. Grism redshifts are determined by the combined fitting of spectrum and photometry data, using a modified EAZY code. For most galaxies, their grism redshifts are of extremely high accuracy with ∆z/(1 + z) ≈ 0.003. Furthermore, spectroscopic redshifts are compiled from previous literature in the five fields (see Skelton et al. (2014) for detail field by field). The restframe colors are derived with the EAZY code at the same time. In addition, stellar masses are estimated with the FAST code (Kriek et al. 2009), by assuming exponentially declining star-formation histories, solar metallicity, Calzetti et al. (2000) dust attenuation law and Bruzual & Charlot (2003) stellar population synthesis models with a Chabrier (2003) initial mass function. The detailed morphological classification depends on the spatial resolution. In this paper, we use the HST/F160W (H-band) selected catalogs (v4.1.5) and H-band images from the 3D-HST project 1 (Skelton et al. 2014;Momcheva et al. 2016). We select all those galaxies with F160W < 24.5 mag, ensuring the galaxies being bright enough to obtain the reliable morphologies. An additional criterion is the flag A magnitude limit of F160W < 24.5 is imposed. use phot=1, which means that the object (1) is not too faint and not a star, (2) is not contaminated by a bright source, (3) is well exposed both in the F125W and F160W, (4) has a signal-to-noise ratio (S/N) > 3 in F160W images, and (5) has "noncatastrophic" photometric redshift and stellar population fits (Skelton et al. 2014). After removing the images containing abnormal values due to the bad pixels or locating in survey boundaries, our initial sample contains 47149 galaxies with a median redshift of ∼ 1.1. Figure 1 shows the distributions of H-band magnitudes in each of five fields. THE UNSUPERVISED METHOD FOR MORPHOLOGICAL CLASSIFICATION In this section, we introduce the main process for morphological classification. Our unsupervised method consists of two steps: feature extraction and multiclustering. Figure 2 gives an overview of the CAE architecture and multiclustering strategy. Firstly, we use the CAE to compress the dimension and extract morphological information of galaxies from the raw image. Then, we cluster galaxies into 100 groups by bagging-based multiclustering methods: three clustering methods vote on accepting or rejecting of a galaxy in a given group. The bagging-based multiclustering methods assures the classifications with a high degree of confidence, but at the cost of rejecting the disputed sources that are inconsistently voted. Data preprocessing Raw imaging data with large dimensions usually contain noises and useless information, although it contains all morphological information of galaxies. In this paper, the size of raw imaging data is 28×28 with a source in the center, corresponding to 1 .68 × 1 .68. Processing raw pixels of imaging data directly may have negative effects on the downstream tasks and is computing resources consuming. In order to avoid such effects and reduce the calculation amount, a maxmin normalization pretreatment is applied to each cutout. Namely, the flux of each pixel in float type is converted to nonnegative integers by the following procedure: new pixel = old pixel − m M − m × N ,(1) where M and m are the maximum and minimum fluxes of pixels, respectively, and x gives the largest integer that is smaller than x, for example 3.5 = 3. N takes 500 in the present paper, which is a given integer describing the degree of discretization. The choice of N would not significantly change our results. We show in Section 4.4 that N = 500 is a reasonable choice since the final classification is nearly unchanged when another value of N is adopted. Dimensionality reduction by CAE The CAE is a convolutional network involving the operations of convolution and pooling, which can extract informations from images effectively (Masci et al. 2011;Krizhevsky et al. 2012). It is dedicated to extract information from images and thus compress the dimension of features of photometric data hierarchically (Ng et al. 2011;Wang et al. 2016). In the CAE, operations of convolution and pooling encode the pixels and give an encoded feature with lower dimension. Operations of deconvolution and unpooling decode the encoded feature and reconstruct the pixels. Table 1 summarizes the outline of CAE architecture. In this work, the dimension of the input data is 28×28 (784). By using the CAE, we compress the dimension of features from 784 to a lower number and improve the efficiency of calculation. In this work, we choose to use 40 as the target dimension of the compressed features to balance the effectiveness and efficiency (i.e., the dimension of the hidden layer dim L6 = 40). After fixing the final feature number, the basic concept of the training is to reduce the differences between the input images and the re- constructed images, which is described by the loss function loss = 1 784n n i=1 28 j,k (ŷ j,k − y j,k ) 2 ,(2) where n is the number of subsamples in a batch, and y j,k andŷ j,k are input and reconstructed pixels in position j and k. By tuning the parameters in the networks to minimize the loss function, the CAE learns to encode the raw photometric data, which gives and decodes the compressed features. Figure 3 illustrates several reconstructed images by CAE with hidden-layer sizes of dim L6 = 40 and 100, together with the corresponding input images for comparison. Clearly, dim L6 = 40 is enough to recover the major morphological information of our galaxies. This method is useful in processing the raw imaging data of galaxies and can be used in other deep survey investigation. The bagging-based multiclustering models The clustering method is a class of UML methods that is efficient in handling large amount of unlabeled data and is good at clustering subsamples with similar properties. In this section, we describe the proposed bagging-based multiclustering strategy, which is considered to be an efficient method obtaining the high-quality subsamples with similar properties automatically. There are different clustering algorithms. However, we are cognizant of the problem that different clustering algorithms use different similarity definitions and techniques, possibly resulting in inconsistent clustering output even for the same data set. For one clustering method, to analyze similarities between subsamples from a single perspective may lead to misclustering. Therefore, we propose the bagging-based multiclustering method as shown in Figure 4: subsamples are finally clustered into one group if they are clustered into one group by all the clustering algorithms. That is, the strategy aims at providing morphological classifications with high confidence at the cost of rejecting a number of disputed subsamples. The key procedure to implement the bagging-based multiclustering strategy is to align labels generated by different clustering algorithms before voting, since labels assigned by different algorithms might be different and are meaningless in direct comparisons. To align labels, we set the labels given by the k-means clustering algorithm as the primary ones and assign them to every group clustered by the other two algorithms according to the k-means label with the highest fre-quency in that group. After aligning all labels given by all clustered algorithms, voting starts. Clustering algorithms can be divided into several categories (Elavarasi et al. 2011): (1) partition clustering algorithms that split subsamples into k partitions (Elavarasi et al. 2011), (2) hierarchical clustering algorithms that group subsamples into form a tree shaped structure (Murtagh 1983), (3) density-based clustering algorithms that grow the groups until the density in the neighborhood exceeds certain threshold (Han et al. 2011), (4) spectral clustering algorithms where groups are formed by partition subsamples using the similar- ity matrix (Meila & Verma 2001), and (5) grid based clustering algorithms that quantize the subsample space into cells with a grid structure (Han et al. 2011). In this work, our bagging based multi-clustering strategy consists of three typical clustering algorithms, i.e., the kmeans clustering algorithm (Hartigan & Wong 1979;Kanungo et al. 2002), the agglomerative clustering algorithm (AGG ;Murtagh 1983;Murtagh & Legendre 2014), and the balance iterative reducing and clustering using hierarchies (BIRCH; Zhang et al. 1996Zhang et al. , 1997Peng et al. 2018). In other words, each subsample is firstly voted by the k-means, the AGG, and the BIRCH algorithms, respectively, then the well clustered subsamples are accepted and grouped, while the disputed subsamples are rejected. These clustering algorithms are described briefly below: 1. The k-means clustering algorithm is a typical kind of partition clustering algorithm, which partitions N subsamples into k clusters based on the nearest mean distance. In this algorithm, subsamples are iteratively processed. Firstly, the algorithm selects k subsamples randomly as predefined clusters. Secondly, a new subsample is grouped into one of these predefined clusters by minimizing the distance between the subsample and the center of the cluster. Thirdly, the center of the cluster is reevaluated after the new subsample is added in. The process is repeated until no best clusters are found. 2. The AGG algorithm, a typical kind of hierarchical clustering algorithms, groups subsamples by constructing a tree-based representation of the subsamples. In the AGG algorithm, clustering happens in a bottom-up manner. Firstly, each subsample is considered as a singleton cluster or so called the leaf. Secondly, two similar clusters are combined into a new bigger cluster or so called the node. This procedure is iterated until all subsamples are member of one single cluster or so called the root. 3. The BIRCH algorithm, another typical kind of hierarchical clustering algorithms, summarizes a cluster of subsamples by using notions of clustering feature and represents a cluster hierarchy by using clustering feature tree. In the BIRCH algorithm, the clustering feature is essentially a summary of the statistics for the given cluster (Han et al. 2011) and a clustering feature tree is a height-balanced tree that stores the clustering features for a hierarchical clustering (Han et al. 2011). By scanning the entire database, BIRCH builds an initial clustering feature tree. Then, in clustering the leaf nodes of the clustering feature tree, sparse clusters are removed as outliers and dense clusters are grouped into larger one. The bagging of three typical kinds of clustering algorithms is executed in the multiclustering block of Fig 2. Although taking the intersection would reduce the sample size, it enables one to use a comprehensive similarity definition and provides a new classification scheme of galaxies with high confidence. This high-quality clustering subsamples can be used as a training sample in other downstream tasks such as the supervised machine-learning tasks. RESULTS AND DISCUSSION Results of morphology classification Based on the classification scheme described in Section 3, we classify galaxies with consistent voting by the three clustering algorithms into 100 groups. This strategy improves the purity of each group and provides a high-quality clustering subsamples at the cost of eliminating 22,249 (∼ 47%) disputed galaxies that are inconsistently voted in the voting model. In this approach, we mainly focus on the remaining 24,900 (∼ 53%) galaxies that are consistently voted. The excluded subsample will be considered in further studies. For example, by considering the well-clustered galaxies as prelabeled training data, we intend to develop supervised methods to group the remaining galaxies. For the sake of clar-ity, we eliminate these disputed galaxies from our following discussion. In this work, we use the t-Distributed Stochastic Neighbor Embedding (t-SNE), which is a technique that visualizes high-dimensional data by giving each datapoint a location in a two-or three-dimensional map (van der Maaten & Hinton 2008), to illustrate the clustering results. In Figure 5, we show the t-SNE diagrams before and after the voting for two groups (i.e., groups 35 and 69). Examples of sources that are well-clustered and are eliminated are marked out, and their cutouts in H band are also presented. To further investigate the connections between galaxy morphology and other galaxy properties, we merge the 100 groups into five subclasses, which are 6335 spheroids (SPHs), 3916 early-type disks (ETDs), 4333 late-type disks (LTDs), 9851 irregulars (IRRs), and 465 unclassifiable (UNC) sources, by visual inspection of images from each group. SPH galaxies are bulge-dominated, whereas LTD galaxies are disk-dominated. ETD galaxies are predominantly bulge-dominated, disk galaxies. IRR subclass includes galaxies with irregular structures or merger evidences. The remaining groups that cannot be attributed to any one of the above types are collected as UNC sources. The UNC subclass accounts for only < 1% of our sample. In order to illustrate the procedure of visual merger, we select 24 groups randomly and exhibit their distributions in the t-SNE graph based on raw images before and after the merger in Figure 6. Due to the small number of UNC sources, none of the groups belonging to this subclass are selected. Obviously, groups merged into the same subclass also tend to be clustered before the merger step expect for the IRR subclass, supporting the reliability of our visual inspection. The F160W cutouts of random examples at 0.5 < z < 2.5 for the five subclasses are shown in Figure 7. It is noteworthy that UNC sources seem to have relatively low S/N compared to the other four subclasses. This tendency is confirmed by further examination of the S/N distributions, which reveals that the majority of the UNC population has S/N 10 in H band, while the S/N in H band for most of the galaxies in other subclasses are > 10. Therefore, we conclude that the unclassifiable feature of UNC sources is mainly due to their low S/N. We will ignore this small population in the following analyses. Given these merged subclasses, we are able to study the distributions of key galaxy properties as a function of mor- phological type, as a test of the reliability of our classifications. Since there exists general correlations between morphologies and other physical properties of massive galaxies (M * > 10 10 M ; e.g., Ball et al. 2008;Gu et al. 2018), we utilize massive galaxies to explore the robustness of the morphological classifications. The distributions in the color-color diagram Observations reveal the morphology-color relation, especially at a fixed stellar mass, (e.g., Driver et al. 2006;Cameron et al. 2009;Mendez et al. 2011;Wake et al. 2012) and the morphology-SFR correlation (e.g., Wake et al. 2012;Bluck et al. 2014;Omand et al. 2014;Woo et al. 2015;Bren-nan et al. 2017). The relationship between the galaxy morphology and the color (or, alternatively, star formation) might be caused by the different properties of bulges and disks (Driver et al. 2006), while the bulge is considered as a reliable predictor of the state of quiescence (Lang et al. 2014;Teimoorinia et al. 2016). The UVJ diagram is a useful diagnosis to separate galaxies into star-forming and quiescent galaxies up to z = 2.5 (e.g., Williams et al. 2009;Straatman et al. 2016;Fang et al. 2018). Figure 8 exhibits the distribution of four morphological subclasses and and four corresponding groups of galaxies in the UVJ diagram. The wedge-shaped region is dominated by quiescent galaxies, whereas galaxies in the remaining region are recognized as star-forming galaxies. The ETDs, LTDs, and IRR subclass mainly stay in the star-forming region. However, the subclasses of ETDs and LTDs tend to invade into the quiescent region. Most of the SPH galaxies are regarded as quiescence. For the reason that quenching state is associated with bulge, our method can successfully strip the SPH galaxies out. As galaxy morphology transforms from IRR to SPH, it is found that galaxies tend to move into the wedged region. In general, it shows that there is a clear connection between UVJ colors and our morphological subclasses. Our classification is reliable from this perspective. The distribution of morphological parameters In this subsection, we directly link our classification to morphological parameters such as the Sérsic index, the effective radius, the Gini coefficient (G), and the second-order moment of the 20% brightest pixels (M 20 ). Parametric measurements Based on public HST/WFC3 F160W imaging, van der Wel et al. (2014) measured Sérsic index n and effective radius r e by modeling the galaxy as a single Sérsic profile (Sérsic 1963) using GALFIT (Peng et al. 2002). The histograms in the left panel of Figure 9 show the distribution of Sérsic index for four subclasses of galaxies. The median Sérsic indexes of IRR, LTD, ETD, and SPH subclasses are 0.8, 1.1, 1.5, and 3.4, respectively. It is clear to see that there is an increasing trend of Sérsic index from IRRs to SPHs. The increasing sequence is highly in agreement with our common understanding of galaxy morphologies. The histograms in the right panel of Figure 9 show the distribution of effective radii for four subclasses of galaxies. The median effective radii of IRR, LTD, ETD, and SPH subclasses are 4.0, 3.7, 2.8, and 1.5 kpc, respectively. The SPH galaxies feature smaller size than other three subclasses. The typical size is a decreasing function following the sequence from IRRs to SPHs. Nonparametric measurements The nonparametric measurements are also introduced. We take the measurements of Gini and M 20 using the program Morpheus developed by Abraham et al. (2007) on the Hband (F160W) images. The Gini coefficient (G) is defined as (Lotz et al. 2004) G = n i (2i − n − 1) |F i | F n(n − 1) ,(3) where F i is the pixel flux value sorted in ascending order, n is the total number of pixels uniquely assigned to a galaxy during object detection, andF is a mean flux for all the pixels. M 20 is defined as (Lotz et al. 2004) UNC UNC Figure 7. The stamps of galaxies in five subclasses (SPH, ETD, LTD, IRR, and UNC). Six random images are shown in two redshift bins, which illustrate that our method is capable of linking the clustered galaxies to the morphology types. The UNC category is ignored in the following discussion due to its low signal-to-noise ratio. M 20 = log i M i M tot , with i F i < 0.2F tot ,(4where M i = F i [(x i − x c ) 2 + (y i − y c ) 2 ] and M tot = N i=1 M i for the fluxes of the brightest 20% of light in a galaxy. In general, the Gini coefficient is a statistical coefficient to quantify the uniformity of light distribution, while M 20 traces the substructures in a galaxy, such as bars, spiral arms, and multiple cores. In Figure 10, we plot the distributions of four subclasses and four corresponding groups in the G-M 20 space. The sequence from IRRs to SPHs follows the orientation with increasing G and decreasing M 20 . The separation between SPH and IRR galaxies in the G-M 20 plane is clearly represented here, where IRR galaxies predominantly possess smaller G and larger M 20 and SPH galaxies have larger G and smaller M 20 . The results also prove that our classifications are closely aligned with nonparametric measurements. (cyan). The contour levels correspond to the 20%, 50%, and 80% of the specified galaxies from inner to outskirts. The distribution of some individual groups is overlapped, where the group id is marked in the square brackets. The SPH, ETD, LTD, and IRR categories have the marker differing, using red points, blue diamonds, green triangles, and cyan squares, respectively. The influences of different N and dim L6 We also examine the influences of two adopted parameters during the learning on the final classifications: (1) N in Equation (1) that describes the degree of discretization in the max-min normalization pretreatment, and (2) the number of features in the hidden-layer dim L6 (i.e., Layer 6 in Figure 2 and Table 1). In this work, we adopt N = 500 and dim L6 = 40 as fiducial values, the resulting t-SNE visualization graph for the four main subclasses is given in Figure 11, together with results based on different adopted values of these parameters (i.e., N = 700 and dim L6 = 100). The comparison between results derived from different combinations of the these two parameters indicate that the choice of a larger latent size of 100 or a larger degree of discretization of 700 gives no significant improvements. On the other hand, in Section 3.2 we have demonstrated that a hidden layer with feature number of 40 is enough to recover most of the morphological char-acteristics for our sample. Thus, our fiducial values of these two parameters are reasonable. Comparisons with visual classification and the supervised method In this section, we compare our results with other works in which galaxies are broadly split into the 5-tag case: SPH, ELD, LTD, IRR, and UNC. In the program of CANDELS visual classifications (Kartaltepe et al. 2015), ∼ 8000 galaxies in the GOODS-S field are classified by human visualization, where one image would be labeled by independent classifiers. Using these labeled galaxies as the training dataset, Huertas-Company et al. (2015) Unclassifiable (UNC): the remaining We cross match our sample with morphology catalogs released by these two works and perform direct comparisons via the t-SNE map in Figure Kartaltepe et al. (2015), since only catalog for GOODS-S field was released, we randomly select 1000 matched galaxies from the same field and exhibit the raw-image-based t-SNE map color coded by our classifications in Panel c, while the same map for the results from Kartaltepe et al. (2015) the t-SNE graph based on the final hidden-layer is also provided in the inset as a supplement. Since our results are not entirely consistent with those of previous studies, it is possible that some galaxies are classified as the UNC category in other works, as shown in Panels b and d of Figure 12. The computation of the t-SNE maps based on either the raw images or the encoded hiddenlayer naturally contains (nearly) all the morphological information of the galaxies used, thus one might expect that for a better classification, galaxies belonging to the same morphological type should be apparently more clustered, while the distributions of galaxies from different types should be more distinguishable and less mixed. Based on this simple and plausible deduction, the following conclusions can be reached by the t-SNE maps shown in Figure 12: (1) the similarity between Panels b and d suggests that the supervised deep-learning method developed by Huertas-Company et al. (2015) performs well using the training set from Kartaltepe et al. (2015), and (2) however, the classification results from our method generally performs better in identifying galaxies belonging to different morphological types than both of the above works, mainly due to the clearer clustering of galaxies from the same subclass. 4.6. Comparison with other UML method referenced by the supervised results Hocking et al. (2018) applied a UML technique that combines the growing neural gas algorithm (Fritzke 1995) and the hierarchical clustering technique to automatically segment and label galaxies. Because galaxies in Hocking et al. (2018) are clustered into 100-200 groups, we choose the 100-group case for an equivalent comparison with our 100-tag results (24,900 galaxies that are well-clustered). Given that the direct mapping between these results may not exist, we take the 5-tag classifications of Huertas-Company et al. (2015) as reference to see how consistent between the results from these two UML methods and the one from the supervised deep learning. However, we are not intending to claim a better classification scheme of Huertas-Company et al. (2015) than the other two, since in Section 4.5 we have demonstrated that our morphological results perform better than those of Huertas-Company et al. (2015) in term of clustering galaxies belonging to the same morphology in the t-SNE maps. We use our well-clustering sample (100 groups containing SPH, ETD, LTD, IRR, and UNC) to draw the following comparison. These galaxies can be matched by the object ID from the 3D-HST catalogs. We cross match our sample with those of Huertas-Company et al. (2015), and for each group of our unsupervised method, we can find a dominated class of the Huertas-Company et al. (2015) classifications (i.e., SPH, ELD, LTD, IRR or UNC). The proportions of the dominated class for each of the 100 groups are computed and illustrated in the left panel of Figure 13, color coded by the dominated classes of the groups. For our well-clustering sample, similar plot is computed for the 100-tag results of Hocking et al. (2018) and is shown in the right panel of Figure 13. The proportion weighted by the number in each group, which can be used to quantify the consistency between the classifications of Huertas-Company et al. (2015) and another ones, is ∼57% for our results, higher than ∼49% for the results of Hocking et al. (2018), suggesting that our morphological results are more consistent with those of the supervised method. We thus conclude that the voting strategy described in Section 3.3 improves the quality of clustering galaxies at the cost of the discard sources. CONCLUSIONS In this paper, we present a UML method that can automatically classify galaxies with similar morphology in deep filed surveys. The method consists of two steps: (1) the CAE is used to compress the dimensions from the raw data and extracts features, and (2) the bagging-based multiclustering method sorts out the assuring galaxies with analogous characteristics into one group. After discarding the galaxies with inconsistent results of voting, the remaining galaxies are well-clustered into 100 groups. To further investigate our galaxy morphologies, we merge the groups into five main categories (SPH, ELD, LTD, IRR, and UNC) by visual verification. After discarding the low-S/N UNC category, we utilize massive galaxies (M * > 10 10 M ) to explore the robustness of the morphological classifications by the connection with other physical properties. We show that this classification scheme is in a good agreement with other galaxy properties, including UVJ diag- respectively. The bottom panels show the randomly selected subsamples only from the GOODS-S field. Panels c and d are color coded by our results and those of CANDELS visual classifications (Kartaltepe et al. 2015), respectively. It shows that our results (i.e., Panels a and c), have more discrete distributions. noses and other morphological parameters. The comparison suggests that our method gives a robust result of morphological classification. Overall, the unsupervised method provides an independent feature extraction and galaxy classification only utilizing the monochromatic H-band images. It is able to obtain the reasonable clustering of galaxies with similar features first, and further increasing the efficiency of visual classification. Since a strict voting model is applied, the clustering subsamples with high quality can be used as a training sample in other downstream tasks such as the supervised machine learning. In the future, we intend to develop the techniques for multicolor images and apply them to the data processing from the Euclid and CSST. The techniques provide the results of galaxy classifications with few human intervention. We expect that the future work would help us to have a better understanding of the morphologies of galaxies as well as their formation and evolution. Figure 1 . 1H-band magnitude distribution in each of the five fields. Figure 2 . 2Schematic diagram of the convolutional autoencoder architecture and multi-clustering strategy. The CAE is firstly trained by minimizing the square mean error of pixels of raw images and decoded images (Eq. 2). Then, based on the encoded feature vectors given by the CAE, three clustering models vote on the labels of galaxies. Figure 3 . 3Top panels are the demonstrations of raw images of random selected galaxies. Middle and bottom panels are the reconstructed images using hidden-layer size 40 and 100, respectively. Three panels are separated by the horizontal lines. Figure 4 . 4An illustration of the bagging-based multiclustering models: the method based on voting. Red samples, such as the sample c, are not well clustered as the object c is disputed by methods 1 and 2 and thus are eliminated. Blue samples, such as object a and object b, and yellow samples, such as object d and object e, are well clustered as they are consistently voted by all the methods. Figure 5 . 5T-SNE diagrams for Groups 35 and 69 before (Panel b) and after (Panel c) the voting. Examples that are well-clustered and are eliminated due to inconsistent voting results by different clustering algorithms are marked out by green and red dashed circles, respectively.The H band cutouts of these examples are also given. Figure 6 . 6Illustration of the procedure of visual merger, showing the t-SNE visualization graph based on raw images of 24 randomly selected groups. Panel (a) is color-coded by the group IDs. Panel (b) is highlighted in different colors, representing the four merged groups by visualization (red: SPH; green: ETD; blue: LTD; cyan: IRR). Figure 8 . 8Rest-frame U − V versus V − J colors for galaxies that are merged into SPH (red), ETD (green), LTD (blue), and IRR proposed a supervised deep-learning method and then applied to other four fields. To define the morphological class, five parameters (i.e., f spheroid , f disk , f irr , f PS , f Unc ) for each galaxy are produced and retrieved through analysis of its H-band image. The definition of galaxy classifications is shown as follows (see Huertas-Company et al. 2015): 1. Spheroids (SPH): f spheroid > 2/3, f disk < 2/3, and f irr < 0.1 ; 2. Early-type Disks (ETD): f spheroid > 2/3, f disk > 2/3, and f irr < 0.1 ; 3. Late-type Disks (LTD): f spheroid < 2/3, f disk > 2/3, and f irr < 0.1; 4. Irregulars (IRR): f spheroid < 2/3 and f irr > 0.1. 12. For the supervised deeplearning results of Huertas-Company et al. (2015), 1000 matched galaxies from the CANDELS five fields are randomly selected and plotted in the t-SNE map based on the raw images, color coded by our classifications (Panel a) and those of Huertas-Company et al. (2015) (Panel b). For the visual classifications of Figure 9 .Figure 10 . 910are given in Panel d. In each panel, Distributions of Sérsic index (left) and effective radius (right) for galaxies that are merged into SPHs (red), ETDs (green), LTDs (blue), and IRRs (cyan). The median value of each subclass is denoted by the upper bar with the corresponding color. Distributions of galaxies in the G-M20 space for galaxies that are merged into SPHs (red), ETDs (green), LTD (blue), and IRR (cyan). The contour levels indicate the 20%, 50%, and 80% of the corresponding subclass from inner to outskirts. The symbols are the same as inFigure 8. Figure 11 . 11Comparisons between different preprocessing settings. The t-SNE visualization graphs are based on the hidden layers that are encoded from data preprocessed by difference strategies. Panel (a) shows the results of our fiducial strategy with a degree of discretization of N = 500 and a hidden-layer size of dimL6 = 40. Panel (b) shows the results of the strategy with N = 700 and dimL6 = 40. Panel (c) shows the results of the strategy with N = 700 and dimL6 = 100. It shows that our strategy is reasonable, since no significant improvements are given by other strategies. Figure 12 . 12The t-SNE visualization graphs based on raw images of the randomly selected subsamples. The embedded subdiagrams are the t-SNE visualization graphs based on the encoded hidden features of the CAE. The upper panels show the randomly selected galaxies from all the five CANDELS fields. Panels a and b are color coded by our results and those of the supervised method(Huertas-Company et al. 2015), Figure 13 . 13This work is based on observations taken by the 3D-HST Treasury Program (GO 12177 and 12328) with the NASA/ ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA Proportion of the dominated class based on the Huertas-Company et al. (2015) classifications for 100 groups of our sample from our results (left) and those of Hocking et al. (2018) (right). Each group is color coded by its dominated class, i.e., SPH (red), ETD (green), LTD (blue), IRR (cyan), or UNC (gray). The black dashed lines in each panel denote the weighted proportion of all the 100 groups. contract NAS526555. This work is supported by the National Natural Science Foundation of China (NSFC; Nos. 11673004, 62106033) and the National Basic Research Program of China (973 Program; 2015CB857004). 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We show that the nearring (Z[x], +, •) of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that are generated by subsets of {1, x, x 2 , x 3 }.

1709.01345

12a75b8f9ae4e57b84e1fd0dc38f1690cc9cdd04

July 28. 2018. 2010 Erhard Aichinger Institut für Algebra Institut für Algebra Johannes Kepler Universität Linz Altenbergerstraße 694040LinzAustria Erhard Aichinger Institut für Algebra Institut für Algebra Johannes Kepler Universität Linz Altenbergerstraße 694040LinzAustria Sebastian Kreinecker [email protected]. Johannes Kepler Universität Linz Altenbergerstraße 694040LinzAustria Sebastian Kreinecker Johannes Kepler Universität Linz Altenbergerstraße 694040LinzAustria Mathematics Subject Classification. 16Y30 (08A40) July 28. 2018. 2010SUBNEARRINGS OF (Z[x], +, •)and phrases nearringsinteger polynomialssubnearring membership problem We show that the nearring (Z[x], +, •) of integer polynomials, where the nearring multiplication is the composition of polynomials, has uncountably many subnearrings, and we give an explicit description of those nearrings that are generated by subsets of {1, x, x 2 , x 3 }. Motivation and results A basic algebraic operation on polynomials is composition: the composition of the polynomials f and g is defined by f • g (x) := f (g(x)). For example, (x 2 + 2) • (2x 3 − 1) = (2x 3 − 1) 2 + 2 = 4x 6 − 4x 3 + 3. This operation • is associative and satisfies the right distributive law (f + g) • h = f • h + g • h, and therefore (Z[x], +, •) is a nearring [8]. The ideal structure of this nearring was investigated in [2,4,5]. In the present paper, we provide some information on the set of subalgebras of this nearring. For this purpose, we describe the structure of the subnearrings generated by some simple sets of polynomials, such as the singleton {x 2 }. In this nearring, we find the polynomial 10 , and we will see that x 10 is not an element of this nearring. We will compute the nearring generated by each of the 16 subsets of {1, x, x 2 , x 3 } in Section 5. In other words, we describe those polynomials that can be obtained from, say, x 2 by using only addition, subtraction and composition. Taking a broader view, we will see that there are continuum many subnearrings of (Z[x], +, •) by detecting an infinite independent subset inside the nearring Z[x] (Theorem 4.1). x 2 • (x 2 + x 2 • x 2 • x 2 ) − x 2 • x 2 − x 2 • x 2 • x 2 • x 2 = (x 2 + x 8 ) 2 − x 4 − x 16 = 2x Preliminaries on nearring generation In this section, we state three lemmas that facilitate the description of the nearring generated by a given set of elements. For a nearring (N, +, •) and a subset F of N, we let F denote the subnearring generated by F . We are mainly concerned with (N, +, •) = (Z[x], +, •). If F = {f (x)} for some f (x) ∈ Z[x], then it is easy to see that F is contained in the subnearring Z[f (x)] := {q(f (x)) | q ∈ Z[x]} of (Z[x], +, •). However, we will see that Z[f (x)] is often strictly larger than f (x) . The following lemma helps to establish that a given set M is indeed the nearring generated by F . Lemma 2.2. Let N be a nearring, let a ∈ N be a right cancellable element of N, let e be a left identity of N, and let C be a subset of C(N). Let F be the subnearring of N generated by {a} ∪ C, and let G be the subnearring of N generated by {a, e} ∪ C. Then G = {n ∈ N | n • a ∈ F }. Proof. For this proof, we resort to some notions from universal algebra [3]. In the setting of universal algebra, we see N as an algebra (N, + N , − N , 0 N , • N ), where +, −, 0, • are the operation symbols and + N , − N , 0 N , • N denote their interpretations as finitary operations on N. The subnearring generated by a subset B of N can then be described as the set of all T N (b 1 , . . . , b k ), where k ∈ N 0 , T is a term over the variables X 1 , . . . , X k in the language {+, −, 0, •}, and b 1 , . . . , b k ∈ B [3, Theorem II.10.3(c)]. Here T N denotes the term function that the term T induces on N. For k ∈ N 0 , let F (X, Y 1 , . . . , Y k ) = F (X, Y ) denote the set of terms over the variables X, Y 1 , . . . , Y k , and F (X, Y 1 , . . . , Y k , Z) = F (X, Y , Z) denote the set of terms over the variables X, Y 1 , . . . , Y k , Z in the language of nearrings. Our first claim is: (2.1) ∀k ∈ N 0 ∀T ∈ F (X, Y , Z) ∀S ∈ F (X, Y ) ∃U ∈ F (X, Y ) ∀c ∈ C(N) k : T N (a, c, e) • N S N (a, c) = U N (a, c). We fix k ∈ N 0 and proceed by induction on the depth of the term T . If T = 0, then set U := 0. If T = X, then we let U(X, Y ) := X •S(X, Y ). We fix c ∈ C(N) k and compute T N (a, c, e) • N S N (a, c) = a • N S N (a, c) = U N (a, c). If T = Y j with j ∈ {1, . . . , k}, we set U(X, Y ) := Y j • S(X, Y ) (note that U(X, Y ) := Y j would also work), and if T = Z, we set U := S. For the induction step, suppose first that T = T 1 + T 2 . The induction hy- pothesis yields U 1 , U 2 ∈ F (X, Y ) with T N i (a, c, e) • N S N (a, c) = U N i (a, c) for i ∈ {1, 2} and for all c ∈ C(N) k , and thus T N (a, c, e) • N S N (a, c) = (T N 1 (a, c, e)+ N T N 2 (a, c, e)) • N S N (a, c) = T N 1 (a, c, e) • N S N (a, c) + N T N 2 (a, c, e) • S N (a, c) = U N 1 (a, c) + N U N 2 (a, c) for all c ∈ C(N) k . Now U := U 1 + U 2 is the required term. The case T = T 1 − T 2 is done similarly. Next, we suppose that T = T 1 • T 2 . Then for all c ∈ C(N) k , we have T N (a, c, e) • N S N (a, c) = (T N 1 (a, c, e) • N T N 2 (a, c, e)) • N S N (a, c) = T N 1 (a, c, e) • N (T N 2 (a, c, e) • N S N (a, c)). By the induction hypoth- esis, we have U 2 ∈ F (X, Y ) with T N 2 (a, c, e) • N S N (a, c) = U N 2 (a, c) for all c ∈ C(N) k . Thus T N 1 (a, c, e) • N (T N 2 (a, c, e) • N S N (a, c)) = T N 1 (a, c, e) • N U N 2 (a, c) for all c ∈ C(N) k . Using the induction hypothesis again (for T 1 ), we obtain c). This completes the proof of claim (2.1). Now we are ready to prove U ∈ F (X, Y ) with T N 1 (a, c, e) • N U N 2 (a, c) = U N (a,(2.2) G ⊆ {n ∈ N | n • N a ∈ F }. To this end, let g ∈ G. Then there is k ∈ N 0 , there are c 1 , . . . , c k ∈ C, and there is T ∈ F (X, Y , Z) with g = T N (a, c, e). Our goal is to show that g • N a ∈ F . To this end, we observe that g • N a = T N (a, c, e) • N S N (a, c) where S(X, Y ) = X. Now claim (2.1) yields U ∈ F (X, Y ) with T N (a, c, e) • N S N (a, c) = U N (a, c). Since U N (a, c) ∈ F , we obtain g • N a ∈ F , which completes the proof of (2.2). For establishing the converse inclusion, we define for each k ∈ N 0 a mapping R : F (X, Y ) → F (X, Y , Z) inductively by R(0) := 0, R(X) := Z, R(Y j ) := Y j for j ∈ {1, . . . , k}, R(T 1 • T 2 ) := T 1 • R(T 2 ), R(T 1 + T 2 ) := R(T 1 ) + R(T 2 ), R(T 1 − T 2 ) := R(T 1 ) − R(T 2 ) . Now we claim: (2.3) ∀T ∈ F (X, Y ) ∀c ∈ C(N) k : R(T ) N (a, c, e) • N a = T N (a, c). We proceed by induction on the depth of T . For T = 0, both sides of (2.3) evaluate to 0 N . For T = X, we compute R(X) N (a, c, e) • N a = e • N a = a = T N (a, c). For T = Y j , we notice that R(T ) = Y j and compute R(T ) N (a, c, e) • N a = c j • N a. Using that c j is constant, we have c j • N a = c j = T N (a, c). For the induction step, we see that the cases T = T 1 + T 2 and T = T 1 − T 2 can be done with routine calculations using the induction hypothesis and the right distributive law. Now let T = T 1 • T 2 . Then we compute (R( T 1 • T 2 )) N (a, c, e) • N a = (T 1 • R(T 2 )) N (a, c, e) • N a = (T N 1 (a, c) • N R(T 2 ) N (a, c, e)) • N a = T N 1 (a, c) • N (R(T 2 ) N (a, c, e) • N a). Using the induction hypothesis, the last expression is equal to T N 1 (a, c) • N T N 2 (a, c) = (T 1 • T 2 ) N (a, c) = T N (a, c) . This completes the proof of claim (2.3). Next, we prove (2.4) {n ∈ N | n • N a ∈ F } ⊆ G. Let n ∈ N be such that n • N a ∈ F . Then there is k ∈ N 0 , there are c 1 , . . . , c k ∈ C and there is a term T ∈ F (X, Y ) such that n • N a = T N (a, c). Hence, by claim (2.3), n • N a = R(T ) N (a, c, e) • N a. Since a is right cancellable, n = R(T ) N (a, c, e), and therefore n lies in the nearring generated by {a, c 1 , . . . , c k , e}, and thus in G. Setting C := ∅, we obtain the following consequence: Preliminaries from elementary number theory The proof of Theorem 4.1 will rely on number theoretic properties of the multinomial coefficients. For p ∈ Z[x] and i ∈ N 0 , we denote the coefficient of x i in the polynomial p by coeff(p, i), where coeff(p, i) = 0 if i > deg(p). For k = k 1 + · · · + k n , we abbreviate the multinomial coefficient k! k 1 !···kn! by k k 1 ,...,kn . Lemma 3.1 (Corollary 32.1 of [9]). Let p ∈ P, let n, s ∈ N, let j ∈ {1, . . . , n}, and let k, k 1 , . . . , k n ∈ N 0 be such that k = k 1 + . . . + k n . If p s divides k and gcd(k j , p) = 1, then p s divides k k 1 ,...,kn . Lemma 3.2. Let n, m ∈ N, and let l 1 , . . . , l n , k 1 , . . . , k n , k ∈ N 0 be such that k = k 1 + . . . + k n , k ≡ 2 0, and for all i ∈ {1, . . . , n}, l i ≡ 2 0. We assume that 2 m+1 − 2 = l 1 k 1 + . . . + l n k n . Then 2 divides k k 1 ,...,kn . Proof. Seeking a contradiction, we suppose that k k 1 ,...,kn is odd. Then we get from Lemma 3.1 that either k is odd, or for all j ≤ n, k j is even. Since k is even by assumption, all k j are even. Since l 1 , . . . , l n are also even, 4 divides l 1 k 1 +. . .+l n k n . But 4 does not divide 2 m+1 −2, which contradicts the assumption 2 m+1 − 2 = l 1 k 1 + . . . + l n k n . Uncountably many subnearrings We let P (N) denote the power set of N. (1) For each j ∈ N, p j is not an element of the nearring Φ(N \ {j}). (2) For all A, B ∈ P (N) we have Φ(A) ⊆ Φ(B) if and only if A ⊆ B. In particular, Φ is injective and hence |S| = 2 ℵ 0 . (3) (S, ⊆) contains a subset that is order isomorphic to (P (N), ⊆). Proof. (1) Let j ∈ N, let K = N \ {j} and let N * := {p i | i ∈ K} . We show that for each p ∈ N * , coeff(p, 2 j+1 − 2) is even. To this end, let p ∈ N * . Next, we define infinitely many unary operations on Z[x]; actually, for each i ∈ K, we define a unary operation p i on Z[x] by p i (q) = p i • q for q ∈ Z[x]. Let M be the subalgebra of (Z[x], +, −, 0, (p i ) i∈K ) that is generated by F = {p i | i ∈ K}. Then Lemma 2.1 yields N * = M. Hence there exists a term T p over the variables (X i ) i∈K in the language of M, which has +, −, (p i ) i∈K as operation symbols, such that p = T p M ((p i ) i∈K ). We will now use induction on the depth of T to show that for each term T , the coefficient of x 2 j+1 −2 in the polynomial T M ((p i ) i∈K ) is even. If T = X i with i ∈ K, then T M ((p i ) i∈K ) = p i , and the coefficient of x 2 j+1 −2 in p i is 0, hence even. If T = 0, the assertion is obvious. For the induction step, suppose first that T = T 1 + T 2 or T = T 1 − T 2 . Then the induction hypothesis yields coeff(T M 1 ((p i ) i∈K ), 2 j+1 −2) ≡ 2 0 and coeff(T M 2 ((p i ) i∈K ), 2 j+1 −2) ≡ 2 0. Therefore we have coeff(T M ((p i ) i∈K ), 2 j+1 −2) ≡ 2 0. Finally, we suppose that there exists i ∈ K such that T = p i (T 1 ). This means p = x 2 i+1 −2 • T M 1 ((p i ) i∈K ). Since M ⊆ Z[x 2 ] and T M 1 ((p i ) i∈K ) ∈ M, all monomials of T M 1 ((p i ) i∈K ) have even exponents, and therefore, there are n ∈ N and c 0 , . . . , c n ∈ Z such that T M 1 ((p i ) i∈K ) = n i=0 c i x 2i . By the multinomial theorem we get n j=0 c j x j 2 i+1 −2 = (k 0 , . . . , k n ) ∈ N n+1 0 k 0 + · · · + k n = 2 i+1 − 2 2 i+1 − 2 k 0 , . . . , k n n r=0 c kr r x 2rkr . A summand of the right hand side that contributes to the coefficient of 2 j+1 − 2 comes from a (k 0 , . . . , k n ) such that n r=0 2rk r = 2 j+1 − 2. By Lemma 3.2 (with k := 2 i+1 − 2), for such a (k 0 , . . . , k n ), the multinomial coefficient 2 i+1 −2 k 0 ,...,kn is even. Thus coeff((p i (T 1 )) M ((p i ) i∈K ), 2 j+1 − 2) ≡ 2 0. This concludes the induction step. Hence the coefficient of x 2 j+1 −2 in p = T p M ((p i ) i∈K ) is even. Therefore, p j = 1x 2 j+1 −2 ∈ N * . (2) The "if"-direction is obvious. For the "only if"-direction, we assume A ⊆ B, and we let j ∈ A such that j ∈ B. Then p j ∈ Φ(A) and by item (1), p j ∈ Φ(N \ {j}). Since Φ(B) ⊆ Φ(N \ {j}), we obtain p j ∈ Φ(B), and therefore Φ(A) ⊆ Φ(B). (3) By (2), the image of item (3) Φ is order isomorphic to (P (N), ⊆). As a consequence, the set of subnearrings of (Z[x], +, •) contains subsets of each of the following order types: infinite ascending chains, i.e., subsets order isomorphic to (N, ≤), infinite descending chains, i.e., subsets order isomorphic to ({z ∈ Z | z < 0}, ≤), uncountable linearly ordered subsets that are order isomorphic to (R, ≤), and uncountable antichains, i.e., subsets order isomorphic to (R, =). As another consequence, the nearring Φ(N), which is the nearring generated by {p i | i ∈ N}, is not finitely generated. However, the entire nearring Z[x] is finitely generated, with generators {1, x, x 2 , x 3 } [1, Corollary 4.3]. An example of a descending chain of finitely generated subnearrings of ( Z[x], +, •) is provided by (N i ) i∈N , where N i := {x 2 2 i } . Then for each i ∈ N, x 2 2 i+1 x 2 2 i +2 i = x 2 2 i ·2 2 i = x 2 2 i • x 2 2 i ∈ N i , and therefore N i+1 ⊆ N i . The inclusion is proper because N i+1 does not contain any nonzero polynomial p ∈ Z[x] with deg(p) < 2 2 i+1 , and thus x 2 2 i ∈ N i+1 . 5. Examples of subnearrings of (Z[x], +, •) In this section we describe all subnearrings of (Z[x], +, •) which are generated by an arbitrary subset of {1, x, x 2 , x 3 }. Let i, j ∈ N and y 1 , . . . , y j ∈ N 0 . We denote the set {x ∈ N 0 ∃k ∈ {1, . . . , j} : x ≡ i y k } by [≡ i ; y 1 , . . . ,, x, x 2 , x 3 }. a 0 a 1 a 2 a 3 p(x) = n i=0 c i x i ∈ Z[x] is an element of the nearring generated by Table 1. All possible subnearrings generated by 1, x, x 2 and x 3 5.1. The subnearrings generated by {x 2 , x 3 } and {x, x 2 , x 3 }. {a i x i | i ∈ {0, . . . , 3}} iff for all i ∈ N 0 : Content of 0 0 0 0 c i = 0 1 0 0 0 i > 0 ⇒ c i = 0 0 1 0 0 c 0 = 0 and (i > 1 ⇒ c i = 0) 1 1 0 0 i > 1 ⇒ c i = 0 0 0 1 0 c 0 = 0, c 2i+1 = 0, i > 0 ⇒ 2 s 2 (i)−1 | c 2i Theorem 5.4 1 0 1 0 c 2i+1 = 0, i > 0 ⇒ 2 s 2 (i)−1 | c 2i Theorem 5.5 0 1 1 0 c 0 = 0, i > 0 ⇒ 2 s 2 (i)−1 | c i Theorem 1.1 of [1] 1 1 1 0 i > 0 ⇒ 2 s 2 (i)−1 | c i Theorem 1.2 of [1] 0 0 0 1 i ∈ [≡ 6 ; 3] ⇒ c i = 0, i ∈ [≡ 6 ; 3] ⇒ 3 s 3 (i−1) 2 | c i , 2 | j∈A c j and 2 | j∈B c j Theorem 5.17 1 0 0 1 i ∈ [≡ 3 ; 0] ⇒ c i = 0, i ∈ [≡ 3 ; 0] ⇒ 3 ⌊ s 3 (i) 2 ⌋ | c i Theorem 5.19 0 1 0 1 i ∈ [≡ 2 ; 1] ⇒ c 2i+1 = 0, i ∈ [≡ 2 ; 1] ⇒ 3 s 3 (i−1) 2 | c i , 2 | j∈C c j and 2 | j∈D c j Theorem 5.18 1 1 0 1 3 ⌊ s 3 (i) 2 ⌋ | c iTheorem 5.1. A polynomial p = n i=0 c i x i ∈ Z[x] lies in the subnearring of (Z[x], +, •) that is generated by {x 2 , x 3 } if and only if c 0 = 0, c 1 = 0, and c 5 ≡ 2 0. Proof. We want to use Lemma 2.1. Let F := {x 2 , x 3 } and M := { n i=0 c i x i | c 0 = 0, c 1 = 0, 2 divides c 5 }. The conditions F ⊆ M and the closedness of M under +, − are obvious. Now we check: F •M ⊆ M. Let p = n i=0 c i x i ∈ M. We have to show that x 2 •p and x 3 •p lie in M. Let a 1 , . . . , a 2n ∈ Z such that x 2 •p = 2n i=0 a i x i and b 1 , . . . , b 3n ∈ Z such that x 3 • p = 3n i=0 b i x i . Now it is sufficient to show that a 5 ≡ 2 0 and b 5 ≡ 2 . Let j 1 , j 2 ∈ N \ {1} such that j 1 + j 2 = 5. 5 is odd and thus a 5 = 2c j 1 +j 2 . Therefore, a 5 ≡ 2 0. If j 1 , j 2 , j 3 ∈ N \ {1} then j 1 + j 2 + j 3 = 5. Therefore, b 5 = 0. What is left to show is that M ⊆ F . Since x 2 ∈ F , x 2 • x 2 = x 4 ∈ F . We know that x 3 ∈ F , and thus x 2 • (x 2 + x 3 ) − x 4 − x 2 • x 3 = 2x 5 ∈ F . Now we show by induction that for all i ≥ 6, x i ∈ F . For i = 6, x 2 • x 3 = x 6 ∈ F . For i = 8, x 2 • x 2 • x 2 = x 8 ∈ F . For i = 7, x 2 • (x 2 + 2x 5 ) − x 4 − x 2 • (2x 5 ) − (x 3 • (x 2 + x 3 ) − x 6 − x 3 • x 3 − 3x 8 ) = x 7 ∈ F . For i = 9, x 3 • x 3 = x 9 ∈ F . For i = 10, x 2 • (2x 5 ) − (x 3 • (x 2 + x 4 ) − x 4 − 3x 8 − x 3 • x 4 ) = x 10 ∈ F . For i = 11, x 3 • (x 3 + x 4 ) − x 9 − 3x 10 − x 3 • x 4 − (x 2 • (x 7 + x 4 ) − x 2 • x 7 − x 8 ) = x 11 ∈ F . For i = 12, x 3 • x 4 = x 12 ∈ F . For i = 13, we first show that 3x 17 ∈ F . This is true since x 3 • (x 4 + x 9 ) − x 12 − 3 · (x 2 • x 11 ) − x 3 • x 9 = 3x 17 ∈ F . Now we have that x 3 • (x 3 + x 7 ) − x 9 − 3x 17 − x 3 • x 7 − (x 2 • (x 2 + x 11 ) − x 4 − x 2 • x 11 ) = x 13 ∈ F . For the induction step we let i ≥ 14. If i is even, let j 2 = 2, otherwise j 2 = 3. Let j 1 := i−j 2 2 , then i = 2j 1 + j 2 . Since j 1 ≥ 6 the induction hypothesis yields that x j 1 , x 2j 1 , x j 2 and x 2j 2 +j 1 lie in F . Therefore we get (5.1) x 3 • (x j 1 + x j 2 ) − x 3 • x j 2 − x 3 • x j 1 − 3x 2j 2 +j 1 = 3x 2j 1 +j 2 ∈ F and also (5.2) x 2 • (x 2j 1 + x j 2 ) − x 2 • x 2j 1 − x 2 • x j 2 = 2x 2j 1 +j 2 ∈ F . Now we subtract (5.2) from (5.1) and get x 2j 1 +j 2 ∈ F , which concludes the proof. Proof. We have x 3 • (x + x 2 ) − x 3 − x 3 • x 2 − 3 · x 2 • x 2 − (x 2 • (x 2 + x 3 ) − x 4 − x 2 • x 3 ) = x 5 ∈ {x, x 2 , x 3 } . We are done, since {x 2 , x 3 } is a subset of {x, x 2 , x 3 } and by Theorem 5.1 we know that for all j ∈ N \ {1, 5}, we have x j ∈ {x 2 , x 3 } . { n i=1 c i x i | c 1 = 0 and 2 divides c 5 } ⊆ {x 2 , x 3 } . We are done, since 1 ∈ F and {x 2 , x 3 } ⊆ {1, x 2 , x 3 } . Now we show F • M ⊆ M. Let p = n i=0 c i x i ∈ M. We have to show that x 2 • p and x 3 • p lie in M. Let a 1 , . . . , a 2n ∈ Z such that x 2 • p = 2n i=0 a i x i and b 1 , . . . , b 3n ∈ Z such that x 3 • p = 3n i=0 b i x i . Now it is sufficient to show that a 1 = 0, b 1 = 0, a 5 ≡ 2 0 and b 5 ≡ 2 0. Since p ∈ M, c 1 = 0. Thus, a 1 = 2c 0 c 1 = 0. We have a 5 = 2c 1 c 4 + 2c 2 c 3 + 2c 0 c 5 . Therefore a 5 ≡ 2 0. Furthermore, b 1 = 3c 2 0 c 1 = 0 since c 1 = 0. Since c 1 = 0, b 5 = 6c 0 c 2 c 3 and this is divisible by 2. Proof. "⇒": By Lemma 2.2 we know that there exists q ∈ {1, x, x 2 } , such that p = q(x 2 ). By Theorem 1.2 of [1] it follows that c 2i is a multiple of 2 s 2 (i)−1 for all i ∈ N 0 . "⇐": Let q ∈ Z[x] such that q • x 2 = p. Then q lies in {1, x, x 2 } by Theorem 1.2 of [1] because s 2 (2i) = s 2 (i) for all i ∈ N 0 . Therefore Lemma 2.2 yields p ∈ {1, x 2 } . (1) x 3 a ∈ F . (2) If p(x) ∈ F then 3p(x)x 2·3 a + 3p(x) 2 x 3 a , 6p(x)x 2·3 a and 6p(x) 2 x 3 a lie in F . (3) The polynomials 3x 21 + 3x 15 , 3x 57 + 3x 33 , 3x 45 + 3x 63 , 6x 15 , 6x 33 , 6x 45 , 6x 63 lie in F . Proof. The first item can be proved by induction on a. For the second item we observe that x 3 • (p(x) + x 3 a ) − x 3 • (p(x)) − x 3 a+1 = p(x) 3 + 3p(x)x 2·3 a + 3p(x) 2 x 3 a + x 3 a+1 − p(x) 3 − x 3 a+1 = 3p(x)x 2·3 a + 3p(x) 2 x 3 a . If p(x) ∈ F we also have −p(x) ∈ F . Thus, (3p(x)x 2·3 a + 3p(x) 2 x 3 a ) − (−3p(x)x 2·3 a + 3p(x) 2 x 3 a ) = 6p(x)x 2·3 a ∈ F . Furthermore, 3p(x)x 2·3 a + 3p(x) 2 x 3 a + (−3p(x)x 2·3 a + 3p(x) 2 x 3 a ) = 6p(x) 2 x 3 a ∈ F . For item (3) we observe that setting p(x) := x 3 , a := 2 in item (2) yields that 3x 21 + 3x 15 and 6x 15 lie in F . Setting p(x) := x 3 , a := 3, item (2) yields that 3x 57 + 3x 33 and 6x 33 lie in F . Setting p(x) := x 9 , a := 3, item (2) yields that 3x 63 + 3x 45 , 6x 45 and 6x 63 lie in F . Lemma 5.9. Let F := {x 3 }, i, j ∈ [≡ 6 ; 3], l 1 , l 2 , l 3 ∈ N 0 , and o 1 , o 2 , o 3 , o 4 ∈ {0, 3} such that (5.3) 3 l 1 x j + 3 l 2 x i + o 1 x 15 + o 2 x 33 and (5.4) 3 l 3 x j + o 3 x 15 + o 4 x 33 lie in F . We define o 5 := 0 if o 1 = o 3 , 3 if o 1 = o 3 and o 6 := 0 if o 2 = o 4 , 3 if o 2 = o 4 . Then there exists t ∈ N such that 3 t x i + o 5 x 15 + o 6 x 33 ∈ F . Proof. A linear combination of (5.3) and (5.4) yields that there exist t ∈ N, c 1 , c 2 ∈ N such that 3 t x i + (3 c 1 o 1 − 3 c 2 o 3 )x 15 + (3 c 1 o 2 − 3 c 2 o 4 )x 33 ∈ F .3 l x i + a(i)x 15 + b(i)x 33 ∈ F . Proof. For the induction basis we used Mathematica [6]. The program is openly available on [7]. We get the following list of polynomials which lie in F as a part of the output. For the induction step, we let i ≥ 81. We consider the cases i ∈ [≡ 12 ; 9] and i ∈ [≡ 12 ; 3]. Case: i ∈ [≡ 12 ; 9]. Let r := i−3 2 . Then r ∈ [≡ 6 ; 3] and i = 2r + 3. Since i ≥ 81, r ≥ 39. By the induction hypothesis there exists a l 1 ∈ N 0 such that q(x) := 3 l 1 x r + a(r)x 15 + b(r)x 33 ∈ F . By item (2) of Lemma 5.8 we have 3q(x)x 2·3 + 3q(x) 2 x 3 ∈ F . Therefore, we have that (5.6) 3 l 1 +1 x r+6 + 3a(r)x 21 + 3b(r)x 39 + 3a(r) 2 x 33 + 2 · 3a(r)b(r)x 51 + 3b(r) 2 x 69 + 3 · 3 2l 1 x 2r+3 + 2 · 3 l 1 +1 a(r)x 18+r + 2 · 3 l 1 +1 b(r)x 36+r ∈ F . By (5.5) we have 3x 33 +9x 51 ∈ F and 6x 33 ∈ F . Hence 2·(3x 33 +9x 51 )−6x 33 = 18x 51 ∈ F . Since b(r) = 0 or b(r) = 3 this implies 6a(r)b(r)x 51 ∈ F . By (5.5), 6x 15 ∈ F , thus 2 · b(r)x 15 and 2 · b(r) 2 x 15 lie in F . By (5.5) we also have that 3x 15 + 3x 39 and 9x 69 + 3x 15 lie in F . Since 2 · (9x 69 + 3x 15 ) − 6x 15 = 18x 69 ∈ F we have 6b(r) 2 x 69 ∈ F . Hence, b(r) · (3x 15 + 3x 39 ) + b(r) 2 · (9x 69 + 3x 15 ) − 2 · b(r)x 15 − 2 · b(r) 2 x 15 − 6b(r) 2 x 69 = 3b(r)x 39 + 3b(r) 2 x 69 + (b(r) + b(r) 2 )x 15 ∈ F . Since b(r) = 0 or b(r) = 3 and 6x 15 ∈ F we get 3b(r)x 39 + 3b(r) 2 x 69 ∈ F . We also know by (5.5) that 3x 15 + 3x 21 ∈ F and hence a(r) · (3x 15 + 3x 21 ) ∈ F . Therefore, we get by subtracting 3b(r)x 39 + 3b(r) 2 x 69 , a(r) · (3x 15 + 3x 21 ) and 6a(r)b(r)x 51 from (5.6), 3 l 1 +1 x r+6 − 3a(r)x 15 + 3a(r) 2 x 33 + 3 2l 1 +1 x 2r+3 + 2 · 3 l 1 +1 a(r)x 18+r + 2 · 3 l 1 +1 b(r)x 36+r ∈ F . Since 6x 15 and 6x 33 lie in F and a(r) = 0 or a(r) = 3 we also get (5.7) 3 l 1 +1 x r+6 + a(r)x 15 + a(r)x 33 + 3 2l 1 +1 x 2r+3 + 2 · 3 l 1 +1 a(r)x 18+r + 2 · 3 l 1 +1 b(r)x 36+r ∈ F . Since 36 + r < 2r + 3 we know by the induction hypothesis that there exist l 2 , l 3 ∈ N 0 such that 3 l 2 x r+18 + a(r + 18)x 15 + b(r + 18)x 33 and 3 l 3 x r+36 + a(r + 36)x 15 + b(r + 36)x 33 lie in F . Therefore,y (5.5) 2 · (3 l 2 x r+18 + a(r + 18)x 15 + b(r + 18)x 33 ) − 2 · a(r + 18)x 15 − 2 · b(r + 18)x 33 = 2 · 3 l 2 x r+18 ∈ F and 2 · (3 l 3 x r+36 + a(r + 36)x 15 + b(r + 36)x 33 ) − 2 · a(r + 36)x 15 − 2 · b(r + 36)x 33 = 2 · 3 l 3 x r+36 ∈ F . Let L 1 := lcm(3 l 1 +1 , 3 l 2 , 3 l 3 ). Then there exists a k 1 ∈ N such that 3 k 1 ·3 l 1 +1 ≥ L 1 , 3 k 1 · 3 l 2 ≥ L 1 and 3 k 1 · 3 l 3 ≥ L 1 . By multiplying (5.7) with 3 k 1 and subtracting a(r) · 3 k 1 · 2 · 3 l 1 +1 x r+18 ∈ F and b(r) · 3 k 1 · 2 · 3 l 1 +1 x r+36 ∈ F we get (5.8) 3 l 1 +1+k 1 x r+6 + 3 k 1 a(r)x 15 + 3 k 1 a(r)x 33 + 3 2l 1 +1+k 1 x 2r+3 ∈ F . For all k ∈ N it holds that 3 k a(r) ≡ 6 a(r). Since 6x 15 and 6x 33 lie in F , we get (5.9) 3 l 1 +k 1 +1 x r+6 + 3 2l 1 +k 1 +1 x 2r+3 + a(r)x 15 + a(r)x 33 ∈ F . Now we have to distinguish between some cases for r according to their remainder modulo 72. Therefore, we consider the following table. We take as an example the first row of the table and read it in the following way. Let r ∈ [≡ 72 ; 9]. Then a(r) = 0 by the definition of a(r). We get that r + 6 ∈ [≡ 72 ; 15] and therefore the induction hypothesis yields a l 4 ∈ N 0 such that 3 l 4 x r+6 + 3x 15 ∈ F . Now we usẽ x for the variable x in the statement of Lemma 5.9. We setĩ := 2r + 3,j := r + 6, l 1 := l 1 + k 1 + 1,l 2 := 2l 1 + k 1 + 1,l 3 := l 4 ,õ 1 := a(r) = 0,õ 2 := a(r) = 0, o 3 := 3 andõ 4 := 0. By Lemma 5.9 we have that there exists at ∈ N 0 such that 3tx 2r+3 + 3x 15 ∈ F . Now we set t :=t and thus 3 t x 2r+3 + 3x 15 ∈ F . We have i = 2r + 3 ∈ [≡ 72 ; 21] and this satisfies our statement, since a(i) = 3 and b(i) = 0. The other 11 cases work in a similar way. Case: r ∈ a(r) Then r + 6 ∈ Then the induction hypothesis yields a l 4 ∈ N 0 such that . . . ∈ F Together with (5.9), Lemma 5.9 yields a t ∈ N 0 such that Hence in each case, there exists a t ∈ N 0 such that . . . ∈ F i ∈ a(i) b(i) [3 t x i + a(i)x 15 + b(i)x 33 ∈ F . Case: i ∈ [≡ 12 ; 3]. Let r := i−9 2 . Then r ∈ [≡ 6 ; 3] and i = 2r + 9. By the induction hypothesis, there exists a l 1 ∈ N 0 such that q(x) := 3 l 1 x r + a(r)x 15 + b(r)x 33 ∈ F . By item (2) of Lemma 5.8 we have 3q(x)x 2·9 + 3q(x) 2 x 9 ∈ F . Therefore, we have that (5.10) 3 l 1 +1 x r+18 + 3a(r)x 33 + 3b(r)x 51 + 3a(r) 2 x 39 + 2 · 3a(r)b(r)x 57 + 3b(r) 2 x 75 + 3 · 3 2l 1 x 2r+9 + 2 · 3 l 1 +1 a(r)x 24+r + 2 · 3 l 1 +1 b(r)x 42+r ∈ F . By (5.5) we have 3x 33 +3x 57 ∈ F and 6x 33 ∈ F . Hence 2·(3x 33 +3x 57 )−6x 33 = 6x 57 ∈ F . Thus 6a(r)b(r)x 57 ∈ F . By (5.5) we have that 3x 33 + 9x 75 and 3x 33 + 9x 51 lie in F . Since b(r) ∈ {0, 3} and 6x 33 ∈ F , b(r)(3x 33 + 3x 51 ) and b(r) 2 (3x 33 + 3x 75 ) lie in F . Therefore, we get that b(r)(3x 33 + 3x 51 ) + b(r) 2 (3x 33 + 3x 75 ) = 3b(r)x 51 + 3b(r) 2 x 75 + (3b(r) + 3b(r) 2 )x 33 ∈ F . Since 6x 33 ∈ F and 3b(r) + 3b(r) 2 ≡ 6 0, we get 3b(r)x 51 + 3b(r) 2 x 75 ∈ F . We also know by (5.5) that 3x 15 + 3x 39 ∈ F and hence a(r) 2 · (3x 15 + 3x 39 ) ∈ F . Therefore, we get by subtracting 3b(r)x 51 + 3b(r) 2 x 75 , a(r) 2 · (3x 15 + 3x 39 ) and 6a(r)b(r)x 57 from (5.10), 3 l 1 +1 x r+18 − 3a(r) 2 x 15 + 3a(r)x 33 + 3 2l 1 +1 x 2r+9 + 2 · 3 l 1 +1 a(r)x 24+r + 2 · 3 l 1 +1 b(r)x 42+r ∈ F . Since 6x 15 and 6x 33 lie in F and a(r) = 0 or a(r) = 3 we also get (5.11) 3 l 1 +1 x r+18 + a(r)x 15 + a(r)x 33 + 3 2l 1 +1 x 2r+9 + 2 · 3 l 1 +1 a(r)x 24+r + 2 · 3 l 1 +1 b(r)x 42+r ∈ F . Since 42 + r < 2r + 9 we know by the induction hypothesis that there exist l 2 , l 3 ∈ N 0 such that 3 l 2 x r+24 + a(r + 24)x 15 + b(r + 24)x 33 and 3 l 3 x r+42 + a(r + 42)x 15 + b(r + 42)x 33 lie in F . Therefore, 2 · (3 l 2 x r+24 + a(r + 24)x 15 + b(r + 24)x 33 ) − 2 · a(r + 24)x 15 − 2 · b(r + 24)x 33 = 2 · 3 l 2 x r+24 ∈ F and 2 · (3 l 3 x r+42 + a(r + 42)x 15 + b(r + 42)x 33 ) − 2 · a(r + 42)x 15 − 2 · b(r + 42)x 33 = 2 · 3 l 3 x r+42 ∈ F . Let L 2 := lcm(3 l 1 +1 , 3 l 2 , 3 l 3 ). Then there exists a k 2 ∈ N such that 3 k 2 ·3 l 1 +1 ≥ L 2 , 3 k 2 · 3 l 2 ≥ L 2 and 3 k 2 · 3 l 3 ≥ L 2 . By multiplying (5.11) with 3 k 2 and subtracting a(r) · 3 k 2 · 2 · 3 l 1 +1 x r+24 ∈ F and b(r) · 3 k 2 · 2 · 3 l 1 +1 x r+42 ∈ F we get (5.12) 3 l 1 +1+k 2 x r+18 + 3 k 2 a(r)x 15 + 3 k 2 a(r)x 33 + 3 2l 1 +1+k 2 x 2r+9 ∈ F . For all k ∈ N it holds that 3 k a(r) ≡ 6 a(r). Since 6x 15 and 6x 33 lie in F , we get (5.13) 3 l 1 +k 2 +1 x r+18 + 3 2l 1 +k 2 +1 x 2r+9 + a(r)x 15 + a(r)x 33 ∈ F . Now we have again to distinguish between some cases for r. Therefore, we consider the following Hence we have that in each case there exists a t ∈ N 0 such that 3 t x i + a(i)x 15 + b(i)x 33 ∈ F . This concludes the case i ∈ [≡ 12 ; 3]. and c(i) := 3 e(i) . Since i ∈ [≡ 6 ; 3] and i is therefore odd, the digit sum s 3 (i) is always odd. We proceed by induction on the digit sum of i. If s 3 (i) = 1, i is a power of 3. Hence by Lemma 5.8, x i ∈ F . If i = 3, a(i) = b(i) = 0. If i = 3 n with n ≥ 2, then i ∈ [≡ 72 ; 9, 27] and therefore a(i) = b(i) = 0. Now we assume s 3 (i) > 1. Then we choose h 1 , h 2 , k ∈ N such that i = 3 h 1 + 3 h 2 + k with k ∈ [≡ 6 ; 3] and s 3 (i) = s 3 (k) + 2. We have by the induction hypothesis, p(x) := c(k)x k + a(k)x 15 + b(k)x 33 lies in F . For all q(x) ∈ F , we have x 3 • (x 3 h 1 + x 3 h 2 + q(x)) = q(x) 3 + x 3 1+h 1 + x 3 1+h 2 + 3q(x)x 2·3 h 1 + 3q(x) 2 x 3 h 1 + 3q(x)x 2·3 h 2 + 3q(x) 2 x 3 h 2 + 3x 2·3 h 1 +3 h 2 + 3x 3 h 1 +2·3 h 2 + 6q(x)x 3 h 1 +3 h 2 ∈ F . We have q(x) ∈ F , hence also q(x) 3 ∈ F . By Lemma 5.8 we also have that x 3 h 1 +1 , x 3 h 2 +1 , 3x 2·3 h 1 +3 h 2 + 3x 3 h 1 +2·3 h 2 , 3q(x)x 2·3 h 1 + 3q(x) 2 x 3 h 1 and 3q(x)x 2·3 h 2 + 3q(x) 2 x 3 h 2 lie in F . Since x 3 • (x 3 h 1 + x 3 h 2 + q(x)) lies in F , we obtain 6q(x)x 3 h 1 +3 h 2 ∈ F . Therefore, we get by setting q(x) := p(x) that (5.14) 6c(k)x k+3 h 1 +3 h 2 + 6a(k)x 15+3 h 1 +3 h 2 + 6b(k)x 33+3 h 1 +3 h 2 ∈ F . Now we want to prove that 6a(k)x 15+3 h 1 +3 h 2 and 6b(k)x 33+3 h 1 +3 h 2 lie in F . By Lemma 5.8 we know that 3x 15 + 3x 21 ∈ F . Setting q(x) := 3x 15 + 3x 21 we get 18x 21+3 h 1 +3 h 2 + 18x 15+3 h 1 +3 h 2 ∈ F . Since s 3 (21 + 3 h 1 + 3 h 2 ) ≤ 5 and s 3 (21 + 3 h 1 + 3 h 2 ) is odd we get c(21 + 3 h 1 + 3 h 2 ) ∈ {1, 3, 9}. In all cases we get that 18x 21+3 h 1 +3 h 2 ∈ F and hence 18x 15+3 h 1 +3 h 2 ∈ F . By Lemma 5.8 we know 3x 33 + 3x 57 ∈ F . Setting q(x) := 3x 33 + 3x 57 yields 18x 57+3 h 1 +3 h 2 + 18x 33+3 h 1 +3 h 2 ∈ F . Since s 3 (57+3 h 1 +3 h 2 ) ≤ 5 and s 3 (57+3 h 1 +3 h 2 ) is odd we get c(57+3 h 1 +3 h 2 ) ∈ {1, 3, 9}. In all cases we get that 18x 57+3 h 1 +3 h 2 ∈ F and hence 18x 33+3 h 1 +3 h 2 ∈ F . Since a(k) ∈ {0, 3}, b(k) ∈ {0, 3}, 18x 15+3 h 1 +3 h 2 ∈ F and 18x 33+3 h 1 +3 h 2 ∈ F we get from (5.14) that 6c(k)x k+3 h 1 +3 h 2 ∈ F . This implies 2 · 3 1+e(k) x 3 h 1 +3 h 2 +k ∈ F . Since s 3 (3 h 1 + 3 h 2 + k) = 2 + s 3 (k), e(3 h 1 + 3 h 2 + k) = e(k) + 1. Therefore, (5.15) 2 · c(3 h 1 + 3 h 2 + k)x 3 h 1 +3 h 2 +k ∈ F . Hence, 2c(i)x i ∈ F . By Lemma 5.10 we have that there exists a j ∈ N 0 such that Proof. We have to show for all n ∈ N and for all n i=0 c i x i ∈ M, we have n i=0 c i x i ∈ F . We proceed by induction on n. For the induction basis let n = 3. Since x 3 ∈ F we also have c 3 x 3 ∈ F . Now let n > 3 and let n i=0 c i x i ∈ M. If n = 9, Lemma 5.8 yields that x 9 ∈ F and hence c 9 x 9 + c 3 x 3 ∈ F . If n = 15 we have to show that = 3 divides c 33 , 3 divides c 21 , 3 divides c 15 , 2 divides c 21 + c 15 and 2 divides c 33 . By Lemma 5.8 we know that 6x 33 ∈ F and therefore also c 33 x 33 ∈ F . The polynomial c 33 x 33 also lies in M by the definition of M. By the induction hypothesis we get that c 27 x 27 + c 21 x 21 + c 15 x 15 + c 9 x 9 + c 3 x 3 ∈ F . Hence we get (5.21). If n > 33 we define p(x) := n i=0 c i x i . If n ∈ [≡ 6 ; 3] we have c n = 0 and therefore the induction hypothesis yields that p(x) ∈ F . Now let n ∈ [≡ 6 ; 3]. Then we lies in {x 3 } because s 3 (3i) = s 3 (i) and c i =c 3i for all i ∈ N 0 . Therefore Lemma 2.3 yields p ∈ {x, x 3 } . (5.16) 3 j x i + a(i)x 15 + b(i)x 33 F . Lemma 2.1 ([1, Lemma 2.1]). Let (N, +, •) be a nearring, and let F and M be non-empty subsets of N. We assume that the following conditions hold:(1) F ⊆ M. (2) M ⊆ F .(3) M is closed under + and −. (4) F • M ⊆ M. Then M = F . Following [ 8 , 8Definitions 1.7 and 1.11], we call an element a in a nearring (N, +, •) right cancellable in N if for all x, y ∈ N with x • a = y • a, we have x = y; using the distributive law, one obtains that a is right cancellable if and only if x • a = 0 implies x = 0 for all x ∈ N. An element e ∈ N is a left identity if e • x = x for all x ∈ N, and an element c ∈ N is constant if c • 0 = c. If c is constant, then c • x = c for all x ∈ N. The set of constant elements of N will be denoted by C(N). In the nearring (Z[x], +, •), every p with deg(p) ≥ 1 is right cancellable, the polynomial x is the only left identity, and the constant elements of Z[x] are given by C(Z[x]) = {c 0 x 0 | c 0 ∈ Z}. If e is a left identity and F a subset of N, then sometimes the nearrings F and F ∪ {e} are closely related: Lemma 2. 3 . 3Let N be a nearring, let a ∈ N be a right cancellable element of N, let e be a left identity of N. Let F be the subnearring of N generated by {a}, and let G be the subnearring of N generated by {a, e}. Then G = {n ∈ N | n•a ∈ F }. Theorem 4 . 1 . 41For i ∈ N, let p i := x 2 i+1 −2 , let S be the set of all subnearrings of (Z[x], +, •) and let Φ : P (N) → S, where for each A ⊆ N, Φ(A) is the subnearring of Z[x] that is generated by {p i | i ∈ A}. Then we have: y j ] and the digit sum of the number a in base b by s b (a). For the remainder of this section let A, B, C, D defined by A := [≡ 24 ; 15, 21], B := [≡ 72 ; 3, 33, 45, 51, 57, 63] \ {3}, C := [≡ 8 ; 5, 7] and D := [≡ 24 ; 1, 11, 15, 17, 19, 21] \ {1}. The following table shows all possible combinations of subnearrings of (Z[x], +, •), which are generated by a subset of {1 Theorem 5. 2 . 2A polynomial p = n i=0 c i x i ∈ Z[x] lies in the subnearring of (Z[x], +, •) that is generated by {x, x 2 , x 3 } if and only if c 0 = 0. 5. 2 . 2The subnearring generated by {1, x 2 , x 3 }.Theorem 5.3. A polynomial p = n i=0 c i x i ∈ Z[x] lies in the subnearring of (Z[x], +, •) that is generated by {1, x 2 , x 3 } ifand only if c 1 = 0 and 2 divides c 5 . Proof. We want to use Lemma 2.1. Let F := {1, x 2 , x 3 } and M := { n i=0 c i x i | c 1 = 0 and 2 divides c 5 }. The conditions F ⊆ M and the closedness of M under +, − are obvious. Now we show M ⊆ F . By Theorem 5.1 we have 5. 3 . 3The subnearrings generated by {x 2 } and {1, x 2 }. Theorem 5.4. A polynomial p = n i=0 c i x i ∈ Z[x] lies in the subnearring of (Z[x], +, •) that is generated by {x 2 } if and only if c 0 = 0 and for all i ∈ N, 2 s 2 (i)−1 divides c 2i , and c 2i−1 = 0. Proof. "⇒": By Lemma 2.3 we know that there exists q = ⌊ n 2 ⌋i=0c i x i ∈ {x, x 2 } such that p = q(x 2 ). Hence p does not contain monomials of odd degree. By Theorem 1.1 of[1],c 0 = 0 and for all i ∈ N, 2 s 2 (i)−1 dividesc i = c 2i . "⇐": Let q ∈ Z[x] such that q • x 2 = p. Then q lies in {x, x 2 } by Theorem 1.1 of [1], because s 2 (2i) = s 2 (i) for all i ∈ N 0 . Therefore Lemma 2.3 yields p ∈ {x 2 } . i x i ∈ Z[x] lies in the subnearring of (Z[x], +, •) that is generated by {1, x 2 } if and only if for all i ∈ N, 2 s 2 (i)−1 divides c 2i , and c 2i−1 = 0. 5. 4 . 4The subnearrings generated by {x 3 }, {x, x 3 } and {1, x 3 }. The following definitions for M, a(i) and b(i) are just for this section. Let A, B be the subsets of N 0 defined at the beginning of section 5. Definition 5 . 6 .. 7 . 567Let M be the subset of Z[x] defined byM := n i=0 c i x i n ∈ N 0 , ∀i ∈ N 0 \ [≡ 6 ; 3] : c i Let a(i) and b(i) defined by a(i) := 0 if i ∈ A, 3 if i ∈ A and b(i) := 0 if i ∈ B, 3 if i ∈ B.Lemma 5.8. Let F := {x 3 }, a ∈ N and i ∈ N 0 . Then we have: By item 3 of 3Lemma 5.8, 6x 15 and 6x 33 lie in F . Since 3 c 1 o 1 − 3 c 2 o 3 ≡ 6 o 5 and 3 c 1 o 2 − 3 c 2 o 4 ≡ 6 o 6 we have 3 t x i + o 5 x 15 + o 6 x 33 ∈ F . Lemma 5.10. Let F := {x 3 }. For all i ∈ [≡ 6 ; 3] there exists l ∈ N 0 such that 3x 15 + 3x 33 + 3x 63 , 0, 0, 0, 0, 0, 3x 15 + 9x 69 , 0, 0, 0, 0, 0, 3x 33 + 9x 75 , . . .}. Lemma 5 . 11 . 511Let F := {x 3 }. For all i ∈ [≡ 6 ; 3] 15 x 15 + c 9 x 9 + c 3 x 3 ∈ F , knowing by the assumptions on c 15 that 3 c 15 and 2 divides c 15 . Hence, 6 divides c 15 . By Lemma 5.8 we have 6x 15 ∈ F and since c 3 x 3 and c 9 x 9 lie in F we get (5.18). If n = 21 we have to show that(5.19) c 21 x 21 + c 15 x 15 + c 9 x 9 + c 3 x 3 ∈ F ,knowing by the assumptions on c 21 and c 15 that 3 c 21 , 3 divides c 15 and 2 divides c 21 + c 15 . Case: 6 divides c 21 and 6 divides c 15 : Setting p(x) := x 9 and a := 1, Lemma 5.8 yields that 6x 21 ∈ F . Since 6x 15 , c 3 x 3 and c 9 x 9 lie in F we get (5.19). Case: 6 divides c 21 + c 15 and 6 does not divide c 21 : By Lemma 5.8 we have 3x 21 + 3x 15 ∈ F and since 6x 21 and 6x 15 lie in F and both c 21 and c 15 are elements of [≡ 6 ; 3], we obtain c 21 x 21 + c 15 x 15 ∈ F . Since c 3 x 3 and c 9 x 9 lie in F we get (5.19). If n = 27 we have to show that (5.20) c 27 x 27 + c 21 x 21 + c 15 x 15 + c 9 x 9 + c 3 x 3 ∈ F , knowing by the assumptions on c 21 and c 15 that 3 divides c 21 , 3 divides c 15 and 2 divides c 21 + c 15 . By Lemma 5.8 we know that x 27 ∈ F and therefore also c 27 x 27 ∈ F . Subtracting c 27 x 27 from 5.20 we get c 21 x 21 +c 15 x 15 +c 9 x 9 +c 3 x 3 ∈ M. By the induction hypothesis, c 21 x 21 + c 15 x 15 + c 9 x 9 + c 3 x 3 ∈ F . Hence, we get (5.20). If n = 33 we have to show that (5.21) c 33 x 33 + c 27 x 27 + c 21 x 21 + c 15 x 15 + c 9 x 9 + c 3 x 3 ∈ F , knowing by the assumptions on c 33 , c 21 and c 15 that 3 Theorem 5 . 519. A polynomial p = n i=0 c i x i ∈ Z[x] lies in the subnearring of (Z[x], +, •) that is generated by {1, x 3 } if and only if for all i ∈ N 0 the following conditions hold:(1) If i ∈ [≡ 3 ; 0] then c i = 0. (2) If i ∈ [≡ 3 ; 0] then 3 ⌊ s 3 (i) 2 ⌋ divides c i .Proof. "⇒": By Lemma 2.2 we know that there exists q ∈ {1, x, x 3 } such that p = q(x 3 ). By Theorem 1.3 of[1] it follows that p satisfies the conditions(1)and(2)because s 3 (3i) = s 3 (i) for all i ∈ N 0 . "⇐": Let q ∈ Z[x] such that q • x 3 = p. Then q lies in {1, x, x 3 } by Theorem 1.3 of [1] because s 3 (3i) = s 3 (i) for all i ∈ N 0 . Therefore Lemma 2.2 yields p ∈ {1, x 3 } . table. We read this table like the table before.Case: r ∈ a(r) Then r + 18 ∈ Then the induction hypothesis yields a l 4 ∈ N 0 such that . . . ∈ F Together with (5.13), Lemma 5.9 yields a t ∈ N 0 such that . . . ∈ F i ∈ a(i) b(i) [≡ 72 ; 9] 0 [≡ 72 ; 27] 3 l4 x r+18 3 t x 2r+9 [≡ 72 ; 27] 0 0 [≡ 72 ; 15] 3 [≡ 72 ; 33] 3 l4 x r+18 +3x 33 3 t x 2r+9 +3x 15 [≡ 72 ; 39] 3 0 [≡ 72 ; 21] 3 [≡ 72 ; 39] 3 l4 x r+18 +3x 15 3 t x 2r+9 +3x 33 [≡ 72 ; 51] 0 3 [≡ 72 ; 27] 0 [≡ 72 ; 45] 3 l4 x r+18 +3x 15 +3x 33 3 t x 2r+9 +3x 15 +3x 33 [≡ 72 ; 63] 3 3 [≡ 72 ; 33] 0 [≡ 72 ; 51] 3 l4 x r+18 +3x 33 3 t x 2r+9 +3x 33 [≡ 72 ; 3] 0 3 [≡ 72 ; 39] 3 [≡ 72 ; 57] 3 l4 x r+18 +3x 33 3 t x 2r+9 +3x 15 [≡ 72 ; 15] 3 0 [≡ 72 ; 45] 3 [≡ 72 ; 63] 3 l4 x r+18 +3x 15 +3x 33 3 t x 2r+9 [≡ 72 ; 27] 0 0 [≡ 72 ; 51] 0 [≡ 72 ; 69] 3 l4 x r+18 +3x 15 3 t x 2r+9 +3x 15 [≡ 72 ; 39] 3 0 [≡ 72 ; 57] 0 [≡ 72 ; 3] 3 l4 x r+18 +3x 33 3 t x 2r+9 +3x 33 [≡ 72 ; 51] 0 3 [≡ 72 ; 63] 3 [≡ 72 ; 9] 3 l4 x r+18 3 t x 2r+9 +3x 15 +3x 33 [≡ 72 ; 63] 3 3 [≡ 72 ; 69] 3 [≡ 72 ; 15] 3 l4 x r+18 +3x 15 3 t x 2r+9 +3x 33 [≡ 72 ; 3] 0 3 [≡ 72 ; 3] 0 [≡ 72 ; 21] 3 l4 x r+18 +3x 15 3 t x 2r+9 +3x 15 [≡ 72 ; 15] 3 0 Table 3 . 3Case: i ∈ [≡ 12 ; 3]. Distinction for r mod 72 in [≡ 6 ; 3] j∈A c j . Since p ∈ M we have 2 divides j∈A c j and therefore we are done. Lemma 5.16. Let F := {x 3 }. Then M ⊆ F .We can simplify j∈[≡ 72 ; 15,21,39,45,63,69] c j = AcknowledgementsThe authors thank Gleb Pogudin for discussions leading to Theorem 4.1.We have that c(i) = 3 e(i) and therefore c(i) is a multiple of gcd(2 · c(i), 3 j ). Then there are m, n ∈ Z such that m · 2 · c(i) + n · 3 j = c(i) and therefore we get with(5.15) and(5.16) that(5.17)c(i)x i + n · a(i)x 15 + n · b(i)x 33 ∈ F .Since c(i) is odd and m·2·c(i) is even we have that n is odd. Therefore there exists n 1 ∈ Z such that n = 2n 1 + 1. Since 6x 15 and 6x 33 lie in F and 2 · n 1 · a(i) and 2·n 1 ·b(i) are multiples of 6, we also get 2·n 1 ·a(i)x 15 ∈ F and 2·n 1 ·b(i)x 33 ∈ F . Now we computeand therefore we getLemma 5.12. If i ∈ [≡ 6 ; 3] then the following hold:Proof. For all i ∈ Z it holds thatand 3i ≡ 24 15 ⇔ (i ≡ 24 5 ∨ i ≡ 24 13 ∨ i ≡ 24 21). Since i ∈ [≡ 6 ; 3] we are done.Lemma 5.13. Let p = n i=0 c i x i ∈ M and let P = m j=0 C j x j with C 0 , . . . , C m ∈ Z such that P = p 3 . Then 2 divides j∈A C j .Proof. We have 6c i c j c k ≡ 2 0 for all i, j, k ∈ N 0 and 3c 2 i c j +3c i c 2 j ≡ 2 3c i c j +3c i c j ≡ 2 2 · (3c i c j ) ≡ 2 0 for all i, j ∈ N 0 . Furthermore, Proof. The following hold for all k ∈ Z:Since k ∈ [≡ 6 ; 3] the result follows.Proof. We have again 6c i c j c k ≡ 2 0 for all i, j, k ∈ N 0 and 3c 2 i c j + 3c i c 2 j ≡ 2 0 for all i, j ∈ N 0 . Furthermore, it holds that j∈B C j = x n + a(n)x 15 + b(n)x 33 lies in F . Since c n is a multiple of 3 s 3 (n)−1 2there exists m ∈ Z such that c n = m · 3 s 3 (n)−1 2. Since M is closed under +, −, and since q(x) also lies in M we get p(x) − m · q(x) ∈ M. Since deg(p(x) − m · q(x)) < n we know by the induction hypothesis that p(x) − m · q(x) ∈ F . Since m · q(x) ∈ F we also get p(x) ∈ F .lies in the subnearring of (Z[x], +, •) that is generated by {x 3 } if and only if for all i ∈ N 0 the following conditions hold: Generating polynomials using function composition. E Aichinger, Quaest. Math. 361E. Aichinger, Generating polynomials using function composition, Quaest. Math. 36 (2013), no. 1, 39-46. Composition algebras of polynomials. J L Brenner, Pacific J. Math. 1182J. L. Brenner, Composition algebras of polynomials, Pacific J. Math. 118 (1985), no. 2, 281-293. A course in universal algebra. S Burris, H P Sankappanavar, SpringerNew York Heidelberg BerlinS. Burris and H. P. Sankappanavar, A course in universal algebra, Springer New York Hei- delberg Berlin, 1981. Algunos aspectos de la teoría de casi-anillos de polynomios. J Gutierrez, 127Santander, SpainUniversidad de CantabriaPh.D. thesisJ. Gutierrez, Algunos aspectos de la teoría de casi-anillos de polynomios, Ph.D. thesis, Uni- versidad de Cantabria, Santander, Spain, 1988, p. 127. Ideals in the near-ring of polynomials. J Gutiérrez, C Ruiz De, Velasco, Near-rings and near-fields. OberwolfachJ. Gutiérrez and C. Ruiz de Velasco, Ideals in the near-ring of polynomials Z[X], Near-rings and near-fields (Oberwolfach, 1989), Math. Forschungsinst. Oberwolfach, Schwarzwald, 1995, pp. 91-94. Mathematica, Version 11. Wolfram Research, Inc1, Champaign, ILWolfram Research, Inc., Mathematica, Version 11.1, Champaign, IL, 2017. . S Kreinecker, Nearringgenerator , S. Kreinecker, Nearringgenerator, https://github.com/SebKrei/Nearringgenerator. G F Pilz, Near-rings. New York, OxfordNorth-Holland Publishing Company -Amsterdam2nd ed.G. F. Pilz, Near-rings, 2nd ed., North-Holland Publishing Company -Amsterdam, New York, Oxford, 1983. Divisibility of binomial and multinomial coefficients by primes and prime powers, A collection of manuscripts related to the Fibonacci sequence. D Singmaster, Fibonacci Assoc. D. Singmaster, Divisibility of binomial and multinomial coefficients by primes and prime powers, A collection of manuscripts related to the Fibonacci sequence, Fibonacci Assoc., Santa Clara, Calif., 1980, pp. 98-113.

In this paper we perform a systematic study of vacuum spatially flat anisotropic ((3 + D) + 1)-dimensional Einstein-Gauss-Bonnet cosmological models. We consider models which topologically are the product of two flat isotropic submanifolds with different scale factors. One of these submanifolds is three-dimensional and represents our 3D space and the other is D-dimensional and represents extra dimensions. We consider no ansatz of the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play and with the symmetry involved, the cases with D = 1, D = 2, D = 3 and D 4 have different dynamics due to different structure of the equations of motion. We analytically analyze equations of motion in all cases and describe all possibleregimes. It appears that the only regimes with nonsingular future asymptotes are the Kasner regime in General Relativity as well as exponential regimes. As of the past asymptotes, for a smooth transition only Kasner regime in Gauss-Bonnet is an option. With that at hand, we are down only to two viable regimes -"pure" Kasner regime (transition from highenergy (Gauss-Bonnet) to low-energy (General Relativity) Kasner regime) and a transition from high-energy Kasner regime to anisotropic exponential solution. It appears that these regimes take place for different signs of the Gauss-Bonnet coupling α: "pure" Kasner regime occur for α > 0 at low D and α < 0 for high D; anisotropic exponential regime is reached only for α > 0. So if we restrain ourselves with α > 0 solutions (that would be the case, say, if we identify α with inverse string tension in heterotic string theory), the only late-time regimes are Kasner for D = 1, 2 and anisotropic exponential for D 2. Also, low-energy Kasner regimes (a(t) ∝ t p ) have expansion rates for (3+1)-dimensional subspace ("our Universe") ranging from p = 0.5 (D = 1) to p = 1/ √ 3 ≈ 0.577 (D → ∞), which contradicts with dust-dominated Friedmann prediction (p = 2/3).

10.1103/physrevd.94.024046

1605.01456

6279a3532eac5ec064ffe1ffd7121746589d87ff

Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions. Vacuum case 4 May 2016 Sergey A Pavluchenko Programa de Pós-Graduação em Física Universidade Federal do Maranhão (UFMA) 65085-580São Luís, MaranhãoBrazil Cosmological dynamics of spatially flat Einstein-Gauss-Bonnet models in various dimensions. Vacuum case 4 May 2016PACS numbers: 04.20.Jb, 04.50.-h, 98.80.-k 2 In this paper we perform a systematic study of vacuum spatially flat anisotropic ((3 + D) + 1)-dimensional Einstein-Gauss-Bonnet cosmological models. We consider models which topologically are the product of two flat isotropic submanifolds with different scale factors. One of these submanifolds is three-dimensional and represents our 3D space and the other is D-dimensional and represents extra dimensions. We consider no ansatz of the scale factors, which makes our results quite general. With both Einstein-Hilbert and Gauss-Bonnet contributions in play and with the symmetry involved, the cases with D = 1, D = 2, D = 3 and D 4 have different dynamics due to different structure of the equations of motion. We analytically analyze equations of motion in all cases and describe all possibleregimes. It appears that the only regimes with nonsingular future asymptotes are the Kasner regime in General Relativity as well as exponential regimes. As of the past asymptotes, for a smooth transition only Kasner regime in Gauss-Bonnet is an option. With that at hand, we are down only to two viable regimes -"pure" Kasner regime (transition from highenergy (Gauss-Bonnet) to low-energy (General Relativity) Kasner regime) and a transition from high-energy Kasner regime to anisotropic exponential solution. It appears that these regimes take place for different signs of the Gauss-Bonnet coupling α: "pure" Kasner regime occur for α > 0 at low D and α < 0 for high D; anisotropic exponential regime is reached only for α > 0. So if we restrain ourselves with α > 0 solutions (that would be the case, say, if we identify α with inverse string tension in heterotic string theory), the only late-time regimes are Kasner for D = 1, 2 and anisotropic exponential for D 2. Also, low-energy Kasner regimes (a(t) ∝ t p ) have expansion rates for (3+1)-dimensional subspace ("our Universe") ranging from p = 0.5 (D = 1) to p = 1/ √ 3 ≈ 0.577 (D → ∞), which contradicts with dust-dominated Friedmann prediction (p = 2/3). I. INTRODUCTION It is already more then hundred years to Einstein's General Relativity, but the extra-dimensional models are even older. Indeed, the first attempt to construct extra-dimensional model was per- Presence in the Lagrangian of the corrections which are squared in curvature is one of the distinguishing features of the gravitational counterpart of string theories. Indeed, Scherk and Schwarz [7] were first to discover the potential presence of the R 2 and R µν R µν terms in the Lagrangian of the Virasoro-Shapiro model [8,9]. Curvature squared term of the R µνλρ R µνλρ type appears [10] in the low energy limit of the E 8 × E 8 heterotic superstring [11] to match the kinetic term for the Yang-Mills field. Later it was demonstrated [12] that the only combination of quadratic terms that leads to ghost-free nontrivial gravitation interaction is the Gauss-Bonnet (GB) term: L GB = L 2 = R µνλρ R µνλρ − 4R µν R µν + R 2 . This term, first found by Lanczos [13,14] (therefore it is sometimes referred to as the Lanczos term) is an Euler topological invariant in (3+1)-dimensional space-time, but not in (4+1) and higher dimensions. Zumino [15] extended Zwiebach's result on higher than squared curvature terms, supporting the idea that the low energy limit of the unified theory might have a Lagrangian density as a sum of contributions of different powers of curvature. In this regard Einstein-Gauss-Bonnet (EGB) gravity could be seen as a subcase of more general Lovelock gravity [16], but in current paper we restrain ourselves with only quadratic corrections and so to EGB case. Theories with extra dimensions have one thing in common -one needs to explain where these additional dimensions are "hiding" in, as we do not sense them, at least with current level of experiments. One of the ways to "hide" extra dimensions, as well as to recover 4D physics is to build so-called "spontaneous compactification" solution. Exact static solutions where the metric is a cross product of a (3+1)-dimensional manifold and a constant curvature "inner space", were discussed for the first time in [17], but with (3+1)-dimensional manifold being actually Minkowski (the generalization for a constant curvature Lorentzian manifold was done in [18]). In the context of cosmology, it is more interesting to consider a spontaneous compactification in the case where the four dimensional part is given by a Friedmann-Robertson-Walker metric. In this case it is completely natural to consider also the size of the extra dimensions as time dependent rather then static. Indeed in [20] it was explicitly shown that in order to have a more realistic model one needs to consider the dynamical evolution of the extra dimensional scale factor as well. In [18], the equations of motion for compactification with both time dependent scale factors were written for arbitrary Lovelock order in the special case of spatially flat metric (the results were further proven in [19]). The results of [18] were reanalyzed for the special case of 10 space-time dimensions in [21]. In [22], the existence of dynamical compactification solutions was studied with the use of Hamiltonian formalism. More recently, efforts on finding spontaneous compactifications have been done in [23] where the dynamical compactification of (5+1) Einstein-Gauss-Bonnet model was considered, in [24,25] with different metric ansatz for scale factors corresponding to (3+1)and extra dimensional parts, and in [26][27][28] where general (e.g. without any ansatz) scale factors and curved manifolds were considered. Also, apart from cosmology, the recent analysis focuses on properties of black holes in Gauss-Bonnet [29,30] and Lovelock [31,32] gravities, features of gravitational collapse in these theories [33][34][35], general features of spherical-symmetric solutions [36] and many others. In the context of finding exact solutions, the most common ansatz used for the functional form of the scale factor is exponential or power law. Exact solutions with exponential functions for both the (3+1)-and extra dimensional scale factors were studied for the first time in [37], and exponentially increasing (3+1)-dimensional scale factor and exponentially shrinking extra dimensional scale factor were described. Power-law solutions have been analyzed in [18,38] and more recently in [19,[39][40][41][42] so that there is an almost complete description (see also [43] for useful comments regarding physical branches of the solutions). Solutions with exponential scale factors [44] have been studied in detail, namely, models with both variable [45] and constant [46] volume, developing a general scheme for constructing solutions in EGB; recently [47] this scheme was generalized for general Lovelock gravity of any order and in any dimensions. Also, the stability of the solutions was addressed in [48], where it was demonstrated that only a handful of the solutions could be called "stable" while the remaining are either unstable or have neutral/marginal stability and so additional investigation is required. In order to find all possible regimes of Einstein-Gauss-Bonnet cosmology, it is necessary to go beyond an exponential or power law ansatz and keep the functional form of the scale factor generic. Of course in this case the equations of motion are much more complicated, but on the other hand, we are particularly interested in models which allow dynamical compactification, so it is natural to consider metric as a product of spatially three-dimensional part and extra-dimensional. In that case three-dimensional part represents "our Universe" and we expect for this part to expand while extra dimensional part should be suppressed in size with respect to three-dimensional one. In [26] it was found that there exists a phenomenologically sensible regime in the case when the curvature of the extra dimensions is negative and the Einstein-Gauss-Bonnet theory does not admit a maximally symmetric solution. In this case the three dimensional Hubble parameter and the extra dimensional scale factor asymptotically tend to the constant values. In [27] a detailed analysis of the cosmological dynamics in this model with generic couplings was performed. Recently this model was also studied in [28] where it was demonstrated that with an additional constraint on couplings Friedmann-type late-time behavior could be restored. In current paper, unlike [26][27][28], we consider both manifolds (three-dimensional and extradimensional) to be spatially flat and, similar to [26][27][28], put no ansatz on the behavior the of scale factors; also, to be as general as possible, perform all the analysis analytically. In this paper we consider only vacuum model, so neither matter nor even a boundary term (being just cosmological or Λ-term in the absence of curvature) are considered -we leave it to future consideration in a separate papers. Of particular relevance to our present analysis is [23] where authors performed numerical analysis of 5D EGB model with (3 + 2) splitting of the metric. Their approach was different from ours and so they have lost one of the branches while we provide full analysis of the system. 5 The structure of the manuscript is as follows: first we write down general equations of motion for Einstein-Gauss-Bonnet gravity, then we rewrite them for our symmetry ansatz. In the following sections we analyze them for D = 1, D = 2, D = 3, and general D 4 case, considering vacuum case in this paper only. Each case is followed by a small discussion of the results and properties of this particular case; after considering all cases we discuss their properties, generalities and differences, and draw conclusions. II. EQUATIONS OF MOTION As mentioned above, we consider the spatially flat anisotropic cosmological model in Einstein-Gauss-Bonnet gravity without any matter source. The equations of motion for such model include both first and second Lovelock contributions and could be easily derived from the general case (see e.g. [19]): 2    j =i (Ḣ j + H 2 j ) + {k>l} =i H k H l    + 8α    j =i (Ḣ j + H 2 j ) {k>l} ={i,j} H k H l + 3 {k>l> m>n} =i H k H l H m H n    = 0 (1) as ith dynamical equation. The first Lovelock term -Einstein-Hilbert contribution -is in first squared parenthesis and the second term -Gauss-Bonnet -is in second parenthesis; α is the coupling constant for Gauss-Bonnet contribution and we put the corresponding constant for Einstein-Hilbert contribution to unity. Also, since we consider spatially flat cosmological model, scale factors do not hold much physical sense and the equations are rewritten in terms of Hubble parameters H i =ȧ i (t)/a i (t). Apart from the dynamical equations we write down a constraint equation 2 i>j H i H j + 24α i>j>k>l H i H j H k H l = 0.(2) As mentioned in the Introduction, we want to investigate the particular case with the scale factors splitted in two parts -separately 3 dimensions 2 2Ḣ + 3H 2 + Dḣ + D(D + 1) 2 h 2 + 2DHh + 8α 2Ḣ DHh + D(D − 1) 2 h 2 + +Dḣ H 2 + 2(D − 1)Hh + (D − 1)(D − 2) 2 h 2 + 2DH 3 h + D(5D − 3) 2 H 2 h 2 + +D 2 (D − 1)Hh 3 + (D + 1)D(D − 1)(D − 2) 8 h 4 = 0;(3) the dynamical equation that corresponds to h: 2 3Ḣ + 6H 2 + (D − 1)ḣ + D(D − 1) 2 h 2 + 3(D − 1)Hh + 8α 3Ḣ H 2 + 2(D − 1)Hh+ + (D − 1)(D − 2) 2 h 2 + (D − 1)ḣ 3H 2 + 3(D − 2)Hh + (D − 2)(D − 3) 2 h 2 + 3H 4 + +9(D − 1)H 3 h + 3(D − 1)(2D − 3)H 2 h 2 + 3(D − 1) 2 (D − 2) 2 Hh 3 + + D(D − 1)(D − 2)(D − 3) 8 h 4 = 0;(4) and the constraint equation: 2 3H 2 + 3DHh + D(D − 1) 2 h 2 + 24α DH 3 h + 3D(D − 1) 2 H 2 h 2 + D(D − 1)(D − 2) 2 Hh 3 + + D(D − 1)(D − 2)(D − 3) 24 h 4 = 0.(5) Looking at (3) and (4) one can see that for D 4 the equations of motion contain the same terms, while for D = {1, 2, 3} the terms are different (say, for D = 3 terms with (D − 3) multiplier are absent and so on) and so should be the dynamics. We are going to study these four cases separately. As we mentioned in the Introduction, in this paper we are going to consider only vacuum case; the Λ-term case and possibly general case with perfect fluid with arbitrary equation of state we as well as effect of curvature are going to be considered in the following papers. 4Ḣ + 6H 2 + 2ḣ + 2h 2 + 4Hh + 8α 2(Ḣ + H 2 )Hh + (ḣ + h 2 )H 2 = 0, 6Ḣ + 12H 2 + 24α(Ḣ + H 2 )H 2 = 0,(6) 6H 2 + 6Hh + 24αH 3 h = 0. From (8) we can easily see that h = − H 1 + 4αH 2 ,(9) so that H and h always have opposite sign for α > 0, but they could have same sign in α < 0 case. We presented them in Fig. 1(a) -black for α > 0 and grey for α < 0. Also one can resolve (7) for vacuum case with respect toḢ to obtaiṅ H = − 2H 2 (1 + 2αH 2 ) 1 + 4αH 2 ;(10) after that with use of (10) one can solve (6) to geṫ h = − 2H 2 (8α 2 H 4 + 2αH 2 − 1) (1 + 4αH 2 )(16α 2 H 4 + 8αH 2 + 1) .(11) Now we can plotḢ andḣ versus H; we depicted them in Figs. 1(b, c). Panel (b) corresponds to α > 0 case and panel (c) -to α < 0; particular curves correspond to α = ±1. In these panels in black we putḢ(H) and in grey -ḣ(H). Now, let us handle non-singular asymptotic regimes for this case. From eqs. (9)-(11) we can find that lim H→0 h H = −1, lim H→0Ḣ H 2 = −2, lim H→∞ h H = 0, lim H→∞Ḣ H 2 = −1.(12) The solution of theḢ/H 2 = k equation is H(t) ∝ −1/(kt), remembering the definition of the power-law ansatz a(t) ∝ t p and comparing these two we find that p = −1/k, so that for H → 0 Kasner regime with p = 3 and we denote it in a similar way as above -K 3 -Kasner regime with we have p H = 0.5, p h = −0.5 so that p i = 3p H + p h = 1p = 3. There are no other regimes apart from these two for α > 0 (see Fig. 1(b)), but for α < 0 there are. One cannot miss that H → 0 asymptotes valid also for α < 0 (see Fig. 1(c)), as well as for H → ∞, so we denote them in the same way as in α > 0 case. The point H 2 0 = − 1 4α for α < 0 in vacuum case is physical singularity -one can check that the equations of motion are discontinuous at that point and with h,ḣ andḢ divergent the components of Riemann tensor are also divergent while H remains regular. The last fact makes it similar to non-standard singularities, so we denote this regime as nS (nonstandard singularity) and we will discuss them in the Discussion section. Final asymptotic regime for vacuum case is the stable point H → H 1 with H 2 1 = − 1 2α (see Fig. 1(c)). One can see from Eq. (9) that h(H 1 ) = H 1 so that it is isotropic solution. Also it givesḢ =ḣ ≡ 0 with H = h = 1 √ −2α which correspond to the exponential solution; as expected, expressions for Hubble exponents coincide with those obtained from exact solutions [45]. We denote this regime as E iso -as exponential isotropic solution. We summarize all regimes in Table I. Finally it could be useful to rewrite equations (10) and (11) in terms of Kasner exponentsfrom power-law ansatz a(t) = a 0 t p we can derive p = −H 2 /Ḣ and so retrieve expressions for p H and p h -Kasner exponents associated with three-and extra-dimensional parts respectively: In this notation both Kasner exponents depends only on one variable ξ whose sign indicate the sign of α. We presented (13) in Fig. 1 H → ∞ we have p H → 1 and p h → 0 and p → 3 as a result as well as p H = 0.5 with p h = −0.5 and so p = 1 at H = 0. As of irregularities, ξ = −0.5 correspond to the described above isotropic exponential solution (point E iso -indeed, for exponential solutions Kasner exponents diverge (see, e.g. [43] for the discussion of exponential and power-law solutions and their relations), p H = 1 2 × 4ξ + 1 2ξ + 1 , p h = 1 2 × 4ξ + 1 8ξ 2 + 2ξ − 1 with ξ = αH 2 .(13)α > 0 no K 3 → K 1 α < 0 H 2 < − 1 4α nS → K 1 − 1 2α > H 2 > − 1 4α nS → E iso H 2 > − 1 2α K 3 → E iso while ξ = 0.25 is a regular point -we can see that p H is regular there but p h is divergent -it is caused byḣ = 0 at that point (see Fig. 1(b)). Finally, physical nonstandard singularity depicted as p H = p h = 0 at ξ = −1/4. To conclude, in D = 1 vacuum case there are total four regimes but only two of them are nonsingular -K 3 → K 1 for α > 0 and K 3 → E iso for α < 0. Of these two only one could be called viable -K 3 → K 1 for α > 0 -since the other one suppose isotropisation of the entire space and this is not what we observe. IV. D = 2 CASE In this case the equations of motion take form (H-equation, h-equation and constraint correspondingly): 4Ḣ + 6H 2 + 4ḣ + 6h 2 + 8Hh + 8α 2(Ḣ + H 2 )(2Hh + h 2 ) + 2(ḣ + h 2 )(H 2 + 2Hh) + 3H 2 h 2 = 0,(14) 6Ḣ + 12H 2 + 2ḣ + 2h 2 + 6Hh + 8α 3(Ḣ + H 2 )(H 2 + 2Hh) + 3(ḣ + h 2 )H 2 + 3H 3 h = 0,(15)6H 2 + 12Hh + 2h 2 + 24α(2H 3 h + 3H 2 h 2 ) = 0.(16) If we solve (16) with respect to h we get h ± = − H 3 + 12αH 2 ± √ 6 − 36αH 2 + 144α 2 H 4 1 + 36αH 2 ;(17) one can see that the radicand is always positive and (one can easily verify it) both roots for h have different sign from H in α > 0 case (see Fig. 2(a)). One cannot also miss that the α < 0 case, presented in Fig. 2 H ∓ = − H 2 P ± 1 Q ± , h ∓ = 3H 2 (1 + 36ξ) 2 P ± 2 Q ± with ξ = αH 2 and D = 6 − 36ξ + 144ξ 2 , P ∓ 1 = 24192ξ 4 − 4896ξ 3 ± 575Dξ 3 + 648ξ 2 ± 624Dξ 2 + 66ξ ± 32Dξ − 3 ± 2D, P ∓ 2 = 2488320ξ 6 − 2446848ξ 5 ∓ 207360Dξ 5 + 145152ξ 4 ± 38016Dξ 4 + 39744ξ 3 ± ±6048Dξ 3 + 168ξ 2 ± 648Dξ 2 + 266ξ ∓ 80Dξ − 7 ± 3D, Q ∓ = 31104ξ 4 − 2880ξ 3 + 216ξ 2 ± 384Dξ 2 − 12ξ ± 32Dξ + 1.(18) Now we can plotḢ(H) andḣ(H) curves, we presented them in Fig. 2(c)-(f). In this figure we presentedḢ(H) curves in black andḣ(H) curves in grey and the cases are the following: α > 0, h = h + (c) panel, α > 0, h = h − (d) panel, α < 0, h = h + (e) panel, α < 0, h = h − (f) panel. Before having closer look on the panels, it is useful to find zeros ofḢ andḣ from expressions in (18): P + 1 = 0 ⇔ ξ = − 1 6 , ξ = ξ 0 = 3 √ 10 9 + 3 √ 100 36 + 7 36 ≈ 0.56276; P − 1 = 0 ⇔ ξ = − 1 36 ; P + 2 = 0 ⇔ ξ = ± 1 6 , ξ = 1 8 − √ 5 24 ≈ 0.03183, ξ = ξ 0 ; P − 2 = 0 ⇔ ξ = − 1 36 , ξ = 1 8 + √ 5 24 ≈ 0.21817.(19) Also it is useful to find vertical asymptotes forḢ andḣ: and α < 0 (b); both h ± branches are presented (see (17)). In panels (c)-(f) we presentedḢ(H) curves in black andḣ(H) curves in grey for the following cases: -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 h - h + h H -2 -1 0 1 2 -2 -1 0 1 h - h + h H H 0 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 H H H 1 2 3 H H, h . .α > 0, h = h + (c) panel, α > 0, h = h − (d) panel, α < 0, h = h + (e) panel, α < 0, h = h − (f) panel (see text for details). Q + = 31104ξ 4 − 2880ξ 3 + 216ξ 2 − 384Dξ 2 − 12ξ − 32Dξ + 1 = 0 11664Z 6 − 5616Z 5 + 612Z 4 − 72Z 3 + 12Z 2 − 48Z + 1 = 0 with Z = ξ/2, Z 1 ≈ 0.0209 ⇒ ξ 1 ≈ 0.0105, Z 2 ≈ 0.4690 ⇒ ξ 2 ≈ 0.2345; Q − = 0 ⇔ ξ = − 1 36 .(20)= ξ 0 /α into h + (17). The curves in (d) (α > 0, h = h − ) and (f) (α < 0, h = h − ) panels behave differently at early times but have the same asymptote at late times: H, h → 0, and they reach it as h − branch: lim H→0 h ± H = −3 ∓ √ 6 < 0.(21) Finally in (e) panel of Fig. 2 we presented the case α < 0, h = h + . One cannot miss singular behavior ofḣ at H = H 1 whileḢ is regular in this point. This is the same singularity we saw in Fig. 2 h H = ± √ α 2 − α 3α ; lim H→0Ḣ ± H 2 = 3 ∓ 2 √ 6; lim H→∞Ḣ ± H 2 = ±2 √ α 2 − 7α 9α ,(22) and the last limit is (21). From all four we can recover power-law behavior for both h ± branches in both H → 0 and H → ∞ limits; we present them in Table II. h± α lim H p H p h p i h + α > 0 0 − 1 3+2 √ 6 2 √ 6+5 7+3 √ 6 1 ∞ 9 5 − 6 5 3 α < 0 0 − 1 3+2 √ 6 2 √ 6+5 7+3 √ 6 1 ∞ 1 0 3 h − α > 0 0 1 2 √ 6−3 2 √ 6−5 3 √ 6−7 1 ∞ 1 0 3 α < 0 0 1 2 √ 6−3 2 √ 6−5 3 √ 6−7 1 ∞ 9 5 − 6 5 3 Finally, similar to the previous section it is useful to write down explicit expressions for Kasner exponents and plot them. As we noted in the previous section, Kasner exponent could be expressed in terms of Hubble and its derivative as p = −H 2 /Ḣ, then with use of (17) and (18) we can obtain p ± H = Q ± P ± 1 , p ± h = − (12ξ + 3 ± D) 2 Q ± 3P ± 2 .(23) We presented individual Kasner exponents as well as the expansion rate in Fig. 3. In (a) and (b) panels we presented the dynamics for h + branch -large-scale structure in (a) and fine structure in (b) panel. In (c) and (d) panels we have the same but for h − branch -large-scale structure in (c) and fine structure in (d). Black lines correspond to p H , solid grey -to p h and dashed grey depict the expansion rate 3p H + 2p h . From (a) and (c) panels of Fig. 3 we can immediately confirm Gauss-Bonnet Kasner regime as high-energy asymptote in all cases -and confirm corresponding Kasner exponents from Table II. Fig. 3 corresponds to h + branch -ξ < 0 part depicts α < 0 case while ξ > 0 corresponds to α > 0. Now we can compare ξ < 0 part of Fig. 3(b) with Fig. 2(e) -they both represent the same dynamical behavior but in different coordinates. One can see the isotropic exponential solution at ξ 1 = −1/6, but cannot see nonstandard singularity at ξ = −1/36. It is present inḢ,ḣ analysis but absent in p H , p h due to cancellations -this illustrates the fact that for full analysis we cannot rely on any metric ansatz and should perform it in maximally general case. Panel (b) of Continuing with comparisons, we compare ξ > 0 part of Fig. 3 P + 2 ) correspond toḣ = 0 between nonstandard singularities H 1 and H 2 from Fig. 2(c); zero-point ξ 5 ≈ 0.23449 from Q + = 0 (20) correspond to nonstandard singularity H 2 from Fig. 2(c) and finally asymptote ξ 6 ≈ 0.56276 (from roots of P + 2 ) correspond to the isotropic exponential solution at H 3 of Fig. 2(c). The h − branch has less abundant dynamics; one can simply map ξ < 0 part of Fig. 3 (d) into Fig. 2(f) while its ξ > 0 part also could be seen as Fig. 2(d) remembering thatḣ = 0 point in Fig. 2(d) occurs at ξ = 1/8 + √ 5/24 ≈ 0.21817 which comes from P − 2 roots. With this we mapped the dynamics in {Ḣ,ḣ} coordinates with {p H , p h } one. We saw that this mapping is not entirely equivalent as it could "lose" or "create" singularities. We summarize our findings for D = 2 regimes in Table III. In addition to columns we had in Table I, we added "Branch" column since starting from D = 2 case we have several branches for h solutions from constraint -e.g. two in D = 2 case (17). Also we denoted exponential regime with separate expansions rates from h + , α > 0 case as E 3+2 to distinguish it from isotropic exponential solution E iso . (19) and (20) nS α > 0 h + H < H 1 = ξ1 α from (20) K 1 → nS ξ2 α = H 2 > H > H 1 = ξ1 α from (20) nS → nS ξ0 α = H 3 > H > H 2 = ξ2 α from→ E 3+2 H > H 3 = ξ0 α from (19) K 3 → E 3+2 h − no K 3 → K 1 α < 0 h + H < H 1 = 1 6 √ −α K 1 → nS 1 √ −6α = H 2 > H > H 1 = 1 6 √ −α nS → E iso H > H 2 = 1 √ −6α K 3 → E iso To conclude, in D = 2 vacuum regime there are total 8 different regimes but only three of them are nonsingular -K 3 → K 1 , K 3 → E 3+2 and K 3 → E iso . The first of them is natural and the only regime for h − branch while the remaining occur for h + branch for either α > 0 (anisotropic) and α < 0 (isotropic) and with different bounds on H (see Table III). Also of these three only K 3 → K 1 and K 3 → E 3+2 could be viable -K 3 → E iso one expects isotropisation of all spatial dimensions and that is not what we observe nowadays. Let us also note that both exponential regimes appear within their stability ranges found in [48]. 4Ḣ + 6H 2 + 6ḣ + 12h 2 + 12Hh + 8α 6Ḣh(H + h) + 3ḣ(H 2 + h 2 + 4Hh) + 18H 2 h 2 + +18Hh 3 + 3h 4 + 6H 3 h = 0,(24) 6Ḣ + 12H 2 + 4ḣ + 6h 2 + 12Hh + 8α 3Ḣ(H 2 + 4Hh + h 2 ) + 6ḣH(H + h) + 6Hh 3 + +18H 2 h 2 + 18H 3 h + 3H 4 = 0,(25)6H 2 + 18Hh + 6h 2 + 24α(3H 3 h + 9H 2 h 2 + 3Hh 3 ) = 0.(26) Solving the constraint equation (26) for h, one gets h 1 = − 1 12αH ; h 2,3 = − 3 2 ± √ 5 2 H,(27) with "+" sign corresponds to h 2 and "-" to h 3 . One can see that h 2, 3 always have opposite sign from H while h 1 has opposite sign for α > 0 and the same for α < 0. H 1 = − 1 12α × 1728ξ 3 + 1 144ξ 2 + 12ξ + 1 ,ḣ 1 = − 1 144αξ × 1728ξ 3 + 1 144ξ 2 + 12ξ + 1 , H 2, 3 = 3ξ 2α P ∓ 1 Q ∓ ,ḣ 2, 3 = − 3ξ 2α P ∓ 2 Q ∓ with ξ = αH 2 and P ∓ 1 = 240 √ 5ξ 2 ∓ 528ξ 2 − 32 √ 5ξ ± 64ξ + √ 5 ∓ 1, P ∓ 2 = 312 √ 5ξ 2 ∓ 696ξ 2 − 40 √ 5ξ ± 88ξ + √ 5 ∓ 2, Q ± = 216 √ 5ξ 2 ∓ 504ξ 2 − 12 √ 5ξ ± 36ξ ∓ 1;(28) signs follow (27) notation: upper corresponds to "2" subscript while lower -to "3". Before analyzingḢ andḣ vs H 2 in a plot it is useful to find their zeros and asymptotes -roots of P ± 1, 2 and Q ± respectively. Here they are: P + 1, 2 : ξ = ξ 4 = √ 5 − 2 2(5 √ 5 − 11) ≈ 0.65451; ξ = ξ 3 = √ 5 − 2 6(5 √ 5 − 11) ≈ 0.21817; P − 1, 2 : ξ = ξ 2 = √ 5 + 2 2(5 √ 5 + 11) ≈ 0.09549; ξ = ξ 1 = √ 5 + 2 6(5 √ 5 + 11) ≈ 0.03183; Q + : ξ = ξ 3 ; Q − : ξ = ξ 1 ;(29) and numbering of ξ is arranged for growing values of ξ. Additionally,Ḣ 1 = 0 as well asḣ 1 = 0 give us ξ = −1/12 root. p H,(2,3) = p ± H = − H 2 H ± = − 2Q ± 3P ± 1 , p h,(2,3) = p ± h = − h 2 ± h ± = 3 ± √ 5 2 2 3 Q ± P ± 2 .(30) By taking appropriate limits from (30) and confirming them with the results of direct computations of limḢ/H 2 and limḣ/h 2 (similar to the previous sections) we can find power-law behavior in the limiting cases H → 0 and H → ∞; the results are presented in Table IV. Let us note that unlike previous cases, for D = 3 limits for α > 0 and α < 0 coincide. Fig. 4(a)) which corresponds to ξ > 0 we have singular K 3 → K 3 transition and that is what we see in Fig. 5(b) -one can see that at ξ = 0 we have p = 3, which makes it GB Kasner. For α < 0 ( Fig. 4(b)) we have isotropic exponential solution with GB Kasner from both sides -and that is exactly what we see from ξ < 0 part of Fig. 5 p i h 1 0 0 1 3 ∞ 1 0 3 h + 0 2 3 √ 5−3 3 √ 5−7 6 √ 5−12 1 ∞ 7−3 √ 5 5 √ 5−11 21 √ 5−47 13 √ 5−29 3 h − 0 − 2 3 √ 5+3 3 √ 5+7 6 √ 5+12 1 ∞ − 7+3 √ 5 5 √ 5+11 21 √ 5+47 13 √ 5+29h 1 α > 0 no K 3 → K 3 α < 0 K 3 → E iso (both branches) h 2 α > 0 H < ξ 3 α from (29) nS → K 1 ξ 4 α > H > ξ 3 α from (29) nS → E 3+3 H > ξ 4 α from (29) K 3 → E 3+3 α < 0 no K 3 → K 1 h 3 α > 0 H < ξ 1 α from (29) K 1 → nS ξ 2 α > H > ξ 1 α from (29) E 3+3 → nS H > ξ 2 α from (29) E 3+3 → K 3 α < 0 no K 1 → K 3 To conclude, in D = 3 vacuum model we have 10 different regimes, but some of them have more then one branch, like h 1 α < 0 case (see Fig. 4(b)), when both regimes are K 3 → E iso but in one of them H is increasing and in another it is decreasing. But on the other hand both subspaces are three-dimensional so we cannot discriminate them. Another interesting point is that of three branches of solutions one (h 1 ) is "independent" while two remaining are linked through some sort of "reversal". Exactly, all regimes in these two branches coincide up to "time reversal" -if in h + we have, say, nS → K 1 transition, in h − we have K 1 → nS. We link this feature with the structure of the solutions -the constraint equation (5) is cubic with respect to H and so the structure of its solutions affect solutions of the entire system. Of ten different regimes only three are nonsingular -K 3 → E iso with α < 0 on h 1 branch, K 3 → E 3+3 with α > 0 and K 3 → K 1 with α < 0 -both from h + branch. Of these three regimes isotropisation is not viable -we do observe discrimination between three and extra dimensions, so only two regimes (and both of them are from h + branch) remain. As we described, h − branch is reverse of h + , so there are no viable regimes on h − branch as well. VI. GENERAL D 4 CASE In the general case, we use general equations (3)-(5), but unlike previous cases we solve (5) with respect to H instead of h. The reason for it is that now (5) The procedure is similar to the previous cases -we solve (3) and (4) Panels (c)-(f) correspond to α > 0 while panels (g)-(i) -to α < 0. Black curves correspond tȯ H(h) and grey -toḣ(h). Now let us have a closer look on the panels. Remember, from Fig. 6(a, b) we found that for H > 0 one needs h < 0 in most cases, so we take closer look on h < 0 part. In (c) panel, which corresponds to H 1 , α > 0, we can clearly see K 1 → K 3 transition (we prove it with p H and p h graph below). In (d) panel, which correspond to H 2 with α > 0, the situation is more complicated -we have (with increase of absolute value for h) nS → K 1 , then nS → nS, E → nS and finally E → K 3 . One can clearly see that the exponential solution is unstable for H > 0, but is stable for H < 0 -see h > 0 part of Fig. 6(d); from [48] we know that, say, for (4 + 3) splitting (and it falls within D 4 case) there are three exponential solutions -isotropic which is stable for H > 0 and two anisotropic solutions with (4 + 3) splitting -one of them is stable for H > 0 and the other is unstable; we can assume that the general (D + 3) splitting has the same property -one stable (for H > 0) isotropic solution and two anisotropic solution with one of the stable for H > 0 and the other -for H < 0. This way, the exponential solution found is anisotropic; further on, our numerical investigation proves it. Next two panels -(e) and (f) -correspond to H 3 with α > 0. There are two regimes but they seem to be the same -K 3 → E. This exponential solution is also anisotropic and our numerical investigation proves it. One could mistake regime in h → 0 with K 1 but our investigation with p H and p h (see below) shows that it is K 3 . The remaining panels correspond to α < 0: in (g) panel (H 1 ) we can clearly see K 3 → K 1 regime; in (h) panel (H 2 ) there are two regimes with unstable isotropic exponential solution and both of them are E iso → K 3 . In the h → 0 limit one could assume that the regime is K 1 but analysis in (p H , p h ) coordinates reveals that it is K 3 . The mentioned unstable exponential solution turn into stable at h > 0 and the "correct" solution is namely h > 0 -indeed, from solid grey curve in Fig. 6(b) (which corresponds to the H 2 branch which we are dealing with right now) one can see that to have H > 0 in the vicinity of h = 0 we need h > 0. In this way, unstable isotropic solution is replaced with the stable one and the desciption follows the general scheme [48]. Finally, in (i) panel, which corresponds to H 3 , α < 0, we can see K 1 → K 3 -similar to (c) panel. Similar to the previous sections, we also want to make an analysis in Kasner exponents -(p H , us analyze them and compare with Fig. 6. Panel (a) of Fig. 7 clearly demonstrate K 1 → K 3 transition -all according to Fig. 6(c). The same but in opposite "direction" (h < 0 in Fig. 7(a) has p h > 0 while in (b) panel we have p h < 0that is why the "direction" is reversed) we can see K 3 → K 1 -again, all according to Fig. 6(g). Next two panels -(c) and (d) -represent H 2 , α > 0 case -we can see that the structure of regimes is as complicated as its counterpart in (Ḣ,ḣ) coordinates (see Fig. 6(d)). We can depict K 1 at h = 0 and after that nonstandard singularity (p H , p h → 0), which makes it nS → K 1 transition -minding p h < 0 at that region. After that we detect another nS, making nS → nS transition, all according to Fig. 6(d). Finally, after the second singularity we have vertical asymptote -exponential solution -and two surrounding regimes -E → nS and E → K 3 . Panels (e) and (f) correspond to the same branch H 2 but with α < 0 and the corresponding dynamics in (Ḣ,ḣ) is presented in Fig. 6(h). We can clearly see K 3 at h = 0 and h → ∞ as well as exponential solution inbetween. According to the sign of p h in each region we can easily restore the behavior -both regimes are E → K 3 , which agree with Fig. 6(h). Finally, H 3 branch is presented in the bottom row of Fig. 7 -(g) and (h) panels correspond to α > 0 while (i) -to α < 0. We can clearly see that regimes in (g) and (h) panels are "reversed" regimes in (e) and (f) panels -they are K 3 → E. Finally in (i) panel we have K 1 → K 3 . Now let us collect and list all regimes. Unlike previous sections we do not put conditions for them, just list them as they appear with growth of h. The results are presented in Table VI. Also they correspond only to regimes with H > 0 so one of them (H 2 , α < 0) coming from h > 0 part while others from h < 0. H 1 α > 0 K 1 → K 3 α < 0 K 3 → K 1 H 2 α > 0 nS → K 1 nS → nS E 3+D → nS E 3+D → K 3 α < 0 K 3 → E iso (both regimes) H 3 α > 0 K 3 → E 3+D (both regimes) α < 0 K 1 → K 3 To conclude, the choice of nonsingular late-time regimes in the general case is the same as in previous ones -GR Kasner and exponential solutions -either isotropic, or anisotropic. And with isotropic solutions violate observations, we are left with GR Kasner and anisotropic exponential solutions. The former of them presented in H 1 branch at α < 0 while the latter in H 3 branch at α > 0. VII. DISCUSSIONS After collecting all the results, it is time to summarize and discuss them. Before turning to the results in each particular D, let us describe the similarities of all cases. First, all asymptotic regimes are Kasner (GR or GB), exponential or singular. In the Introduction we mentioned that when looking for exact solutions, one usually consider either power-law or exponential ansatz for scale factor and our research proved that it is absolutely right decision -there are no nonsingular regimes in EGB cosmology apart from these two -at least with spatial splitting under consideration. Both of these two regimes are already well-described (see Introduction for appropriate citations) and exponential solutions for our spatial splitting fall into two categories -isotropic and anisotropic. For lower dimensions (D = 1, 2) exponential solutions are described in [45], for higher dimensions (D = 3, 4) -in [46]. From the results of [45] we know that for 5D EGB model (regardless of spatial splitting) there is only one stable vacuum exponential solution -isotropic, and that is exactly what we obtained. Next, in 6D there are only two stable vacuum exponential solutions -isotropic and the solution with (3 + 2) spatial splitting (i.e. spatial metric symmetry is given by {a 1 , a 1 , a 1 , a 2 , a 2 } -product of three-dimensional and two-dimensional isotropic subspaces). Again, this is exactly what we observe and the ratio of the Hubble parameters is in agreement with [46]. Next, 7D EGB model has [46] much more solutions but only two of them fit our (3 + D) spatial splitting -isotropic and (3 + 3) one. The latter has two branches and one of them is stable for H > 0 while the other -for H < 0. And again this is exactly what we see -as we work with H > 0 only, we detect one anisotropic solution to be stable and the other -unstable, plus isotropic solution. The same pattern -stable isotropic plus anisotropic with two branches -stable for H > 0 and stable for H < 0 -detected for D = 4 (see [46]) and judging from the results of current paper, all D 4 cases share the same pattern. Let us start with D = 1 case -we have found that in this case there is only one viable regime -K 3 → K 1 transition which happens for α > 0 regardless of the initial conditions. Regimes at α < 0 are either singular or exponential isotropic, which contradict observational data. The next case -D = 2 -has a bit more complicated structure. Indeed, unlike D = 1 case where we have only one branch of solutions, D = 2 case has two (see Eq. 17). So in this case we have two viable regimes -one of them is Kasner transition K 3 → K 1 and it happens on h − branch regardless of α and initial conditions. The second one is an anisotropic exponential solution E 3+2 with expanding three-dimensional space (H > 0) and contracting extra dimensions (h < 0). It takes place only in h + branch and there is a lower bound on the 4D expansion rate (see Table III). Actually, all cases started from this one have the same viable regimes but their distribution and the prerequisites are D = 1 D = 2 D = 3 D 4 H + K 3 → K 1 for α > 0 H 1 K 3 → K 1 for α < 0 H 1 K 3 → E iso for α < 0 H 1 K 3 → K 1 for α < 0 H 2 K 3 → K 1 for α > 0 H 2 K 3 → E iso for α < 0 K 3 → E iso for α < 0 K 3 → E iso for α < 0 H 2 K 3 → E 3+3 for α > 0 H − K 3 → K 1 for α > 0 H 3 K 3 → K 1 for α > 0 H 3 K 3 → E 3+D for α > 0 K 3 → E 3+2 for α > 0 K 3 → K 1 for α < 0 One can see that in D = 1 Kasner transition exists in both branches and in both with α > 0 while isotropic solution in one of them and with α < 0. In D = 2 Kasner transition exist in all three branches and with both signs; isotropic exponential solution exists in one of the branches with α < 0 while anisotropic exponential solution -in another branch with α > 0. We can see that in this approach different exponential solutions are not "mixing" inside the same branch, unlike approach we used in the main text -for D = 2 with "usual" approach both exponential solutions exist on the same h + branch (see Table III). Also for illustration purposes we summarized all nonsingular regimes in So let us look closer on Table VII. We can see that Kasner transitions K 3 → K 1 in low dimensions occur at α > 0 (D = 1), then for both signs (D = 2) for α and then for D 3 only for α < 0. This is the effect of more complicated dynamics in higher number of dimensions. Exponential solutions are always separated between branches -two different exponential solutions do not exist on the same branch. Also, exponential solutions, their abundance and stability are in exact agreement with the results of [45,47,48]. Of special interest are nonstandard singularities. This referred to a situation when some of the dynamical variables diverge while others are regular, at the same time curvature invariants diverge so it is a physical singularity (say, "suddenly", at some regular value of scale factor, curvature invariants diverge). This kind of singularity is "weak" by Tipler's classification [49], and "type II" in classification by Kitaura and Wheeler [50,51]. Recent studies of the singularities of this kind in the cosmological context in Lovelock and Einstein-Gauss-Bonnet gravity demonstrates [27,39,40,42,44] that their presence is not suppressed and they are abundant for a wide range of initial conditions and parameters. This is especially true for Bianchi-I-type (i.e. where all scale factors are different -diag(−1, a(t) 2 , b(t) 2 , c(t) 2 , d(t) 2 )) (4+1)-dimensional EGB model [40] where it was demonstrated that in vacuum case recollapse and nonstandard singularities are the only options for future behavior. Before concluding our results, two important notes regarding the viability of the regimes must be done. First of them regards GR Kasner regimes. We have found that D = 1 GR Kasner regime has p H = 0.5, D = 2 case -p H = 1 2 √ 6 − 3 ≈ 0.5266, and for D = 3 it is detected that p H = 2 3 √ 5 − 3 ≈ 0.5294; further, for general D 4 we derived p H = 1 3 − D + √ 3D 2 + 6D 3(D + 3) with lim D→∞ p H = 1 √ 3 ≈ 0.577.(31) One can see that the resulting Kasner exponent p H in K 1 gradually grows from 0.5 till approximately 0.577, and that is Kasner exponent which is detected by an observer living in (3 + 1)-dimensional space-time ("our Universe"). Transfering Kasner exponents to the expansion rate and remembering how scale factor depends on the equation of state in presence of the perfect fluid, we can write down p = 2/(3(1 + ω ef f )), so ω ef f = 2/(3p) − 1 -effective equation of state which corresponds to the expansion rate. One can clearly see that for D = 1 it is radiation (ω ef f = 1/3) and it becoming softer in the limit D → ∞: ω ef f = 2/ √ 3 − 1 ≈ 0.1547. We can see that all of them are different from 0 what one would assume from dust-dominated Friedmann stage, so that dust-dominated Friedmann behavior cannot be restored (unlike [28] where it could). The second note regards both Kasner and exponential solutions and it is linked to the Gauss-Bonnet coupling constant α. As we demonstrated, Kasner solutions exist for α > 0 at low D and for α < 0 at high D; isotropic exponential solutions exist only for α < 0 while anisotropic -only for α > 0. So that if there were a bounds on α, we could reject some of the regimes. And the situation with bounds on α are the following -from consideration of shear viscosity to entropy ratio as well as casuality violations and CFTs in dual gravity description there were obtained limits on α for 5D that α/2 9/100 [52,53] and α/2 −7/36 [54,55]; later they were updated for 7D [56][57][58] −5/16 α/2 3/16 and eventually for any D [57,58] (with the upper limit found earlier in [59]): − (3D + 11)(D + 1) 4(D + 5) 2 α 2 D(D + 1)(D 2 + 5D + 24) 4(D 2 + 3D + 26) 2 . From these constraints one can clearly see that we cannot abandon either α > 0 or α < 0; study of GB superconductors [60] also do not allow to discard either possibility. Considering black holes instabilities in 5D allow to lower upper limit to α < 1/24 [61] but it is still remains positive while in [62] α > 0 brought some instabilities. Overall, from all these studies we cannot disregard neither α > 0 nor α < 0. On the other hand, if we consider heterotic strings setup and so identify α with inverse string tension (see [63]), then we should use α > 0 only. In that case (if we restrain us with α > 0 only) the only regimes which are viable are: Kasner transitions K 3 → K 1 in D = 1, 2 and Kasner to anisotropic exponential transitions K 3 → E 3+D in D 2. VIII. CONCLUSIONS It is time to draw conclusions to the results of this paper. As we mention in the Introduction, when looking for the exact solutions in EGB and Lovelock gravity, usually power-law and exponential ansatz are considered. We have demonstrated that this approach is just -at least in EGB case and at least when looking for the solutions which allow dynamical compactification. Indeed, we have demonstrated analytically that there are no nonsingular regimes apart from power-law and exponential. And of power-law regimes, only Kasner regime is achieved (see [43] for (un)viability of another power-law regime). As of exponential regimes, their appearance, abundance and stability is in total agreement with theoretical predictions [45,47,48]. We have described distribution of all nonsingular regimes over branches and initial conditions. As we already mentioned, there are only two viable regimes -K 3 → K 1 Kasner regime and K 3 → E 3+D Kasner-to-exponential transition. The former of them occur for α > 0 at low D 2 and for α < 0 at high D 2 (at D = 2 it exist for both sings of α). On the contrary, anisotropic exponential solution stable for H > 0 and h < 0 (it corresponds to the expansion of our (3+1)dimensional Universe -that is what we observe and contraction of extra dimensions -we do not sense them) only for α > 0. To summarize the regimes, for α > 0 we have anisotropic exponential solutions at D 2 (there is no such solution for D = 1) as well as Kasner regime at D = 1, 2. On the contrary, for α < 0 we have only Kasner regime for D 2 and so there is no viable regime for D = 1. As we discussed earlier, current limits on α allow both signs so one needs to use additional reasoning to set either α > 0 (like appealing to inverse string tension in heterotic string theories) or α < 0. As we mentioned in the beginning, this is the first paper of the series, and we are about to finish similar paper but with boundary (cosmological, or Λ) term taken into account. Indeed, Λ-term case is broader and probably could offer more abundant dynamics, not to mention that one of the interpretations of current accelerated expansion of the Universe is the Λ-term. In future we are also going to consider matter in the form of the perfect fluid and probably the curvature of the manifolds. The former of them could change late-time asymptote for Kasner regime -we demonstrated that depending on D late-time power-law regime a(t) ∝ t p has 0.5 p 1/ √ 3 ≈ 0.577, which contradict observations. So we hope that the addition of matter in form of the perfect fluid could change this asymptote to favored by observations. As of the case with curved manifolds, we have considered that case numerically in [26][27][28] but analytical consideration is always more reliable. formed by Nordström [1] in 1914. It was a vector theory which unified Nordström's second gravity theory [2] with Maxwell's electromagnetism. Back then Einstein's General Relativity (GR) has not been fully formulated yet and so it was natural that this kind of theory arises. Later in 1915 Einstein introduced his theory [3], but still it took almost four years to prove that Nordström's theory was wrong. During Solar eclipse in 1919 there were performed measurements of light bending near the Sun and the deflection angle was in perfect agreement with GR while Nordström's theory, being scalar gravity, predicted zeroth deflection angle. Yet, the Nordström's idea on extra dimensions remained and in 1919 Kaluza proposed [4] similar model based on GR: in his model 5D Einstein equations could be decomposed into 4D Einstein equations plus Maxwell's electromagnetism. In order to perform such decomposition, extra dimension should be "curled" or compactified into a circle and "cylindrical conditions" should be imposed. Later in 1926, Klein introduced [5, 6] nice quantum mechanical interpretation of this extra dimension and so the theory called Kaluza-Klein was formally formulated. Back then their theory unified all known at that time interactions. With time, more interactions were known and it became clear that to unify them all more extra dimensions are needed. Nowadays, one of the promising theories to unify all interactions is M/string theory. ( 3 - 3dimensional isotropic subspace), which are suppose to represent our world and the remaining represent the extra dimensions (D-dimensional isotropic subspace). So we put H 1 = H 2 = H 3 = H and H 4 = . . . = H D+3 = h (D designs the number of additional dimensions) and the equations take the following form -dynamical equation that corresponds to H: III. D = 1 1CASE In this case the equations of motion take form (H-equation, h-equation and constraint correspondingly): FIG. 1 : 1Graphs illustrating the dynamics of D = 1 vacuum cosmological model. In (a) panel we present the behavior for h(H) from (9) -black for α > 0 and grey for α < 0. In panels (b) and (c) we presentedḢ(H) in black andḣ(H) in grey for α > 0 ((b) panel) and α < 0 ((c) panel). Finally in (d) panel we presented Kasner exponents p H in black, p h in grey and the expansion rate (3p H + p h ) in dashed grey; irregularities denoted as dashed black lines (see the text for more details). (b), has singularity at H 0 = ± − 1 36α . Now we can solve (14)-(15) with respect toḢ andḣ, substitute (17) to getḢ(H) andḣ(H) curves. The expressions forḢ(H) andḣ(H) are as follows: FIG. 2 : 2Dynamics of D = 2 vacuum model. In (a) and (b) panels we presented h(H) curve for α > 0 (a) FIG. 3 : 3(b) with Fig. 2(c) -ξ 2 ≈ 0.01046 from Q + = 0 (20) is zero-point for all Kasner exponents as it corresponds to nonstandard singularity H 1 from Fig. 2(c); two vertical asymptotes ξ 3 Dynamics of Kasner exponents in D = 2 vacuum model. In (a) and (b) panels we presented largescale structure (a) and fine structure (b) for h + branch while in (c) and (d) -the same but for h − branchlarge-scale in (c) and fine structure in (d). Black line corresponds to p H , solid grey -to p h and dashed grey -to the expansion rate 3p H + 2p h (see text for details). V. D = 3 3CASE In this case the equations of motion take form (H-equation, h-equation and constraint correspondingly): Now we can solve (24)-(25) with respect toḢ andḣ and substitute branches obtained(27) to get expressions forḢ(H 2 ) andḣ(H 2 ): FIG. 4 : 4The plot ofḢ (black curve) andḣ (grey curve) vs H 2 for D = 3 vacuum model. In (a) panel weplotted h 1 branch with α > 0, in (b) panel -h 1 branch with α < 0, (c) -h 2 , α > 0, (d) -h 2 , α < 0, (e) -h 3 , α > 0, (f) -h 3 , α < 0 (see text for details). Now we can plotḢ andḣ vs H 2 -see Fig. 4. There in (a) panel we presentedḢ(H 2 ) anḋ h(H 2 ) curves for h = h 1 branch and α > 0 while in (b) panel -the same branch but with α < 0 choice. Similar structure have two remaining rows -(c) panel corresponds to h = h 2 branch with α > 0 while (d) panel -h = h 2 with α < 0. Finally, (e) panel depicts graphs for h = h 3 branch with α > 0 and (f) -h = h 3 with α < 0. Let us have a closer look on the panels. In (a) panel (h = h 1 , α > 0) bothḢ andḣ are always negative;ḣ is also singular at H = 0. Additional studies of p h and p H (see below) reveal that H = 0 is Gauss-Bonnet Kasner singularity. In (b) panel (h = h 1 , α < 0) we have stable point at H 2 1 = − plot p H and p h versus ξ for all three branches of h; the corresponding plots are presented in Fig. 5. The first row ((a) and (b) panels) corresponds to first h 1 branch of Eq. (27), the second ((c) and (d) panels) -to h + and the last ((e) and (f) panels) to h − . The first column ((a), (c) and (e) panels) gives large-scale behavior while the second ((b), (d) and (f) panels)fine-scale in the vicinity of ξ = 0. From first column one can verify our limits for H → ∞ from FIG. 5 : 5Dynamics of Kasner exponents in D = 3 vacuum model. In (a) and (b) panels we presented largescale structure (a) and fine structure (b) for h 1 branch, in (c) and (d) -large-scale structure (c) and fine structure (d) for h + branch and in (e) and (f) panels we presented large-scale structure (e) and fine structure (f) for h − branch (see text for details). is cubic with respect to H but quartic with respect to h, and now it is simpler to solve it with respect to H. The general form of the solution is complicated, but we can plot resulting H(h) curves. Typical H(h) curves for D 4 case are given in Figs. 6 (a, b) -α > 0 on (a) and α < 0 on (b). In there we put three branches (H 1 , H 2 and H 3 -as three solutions of cubic equations) with different colors and linestyles -black, solid grey and dashed grey. One can see that the situation resemble D = 3 rather then D = 1 or D = 2. Exact curves inFig. 6(a, b)correspond to D = 6, but for any other D 4 typical behavior is the same, the difference lies only in the inclination of asymptotic h → ±∞ behavior. with respect toḣ andḢ, but unlike previous cases we substitute not h(H), but H(h) now. The difference is the following -in previous cases H was the "dynamical variable" and soḢ(H) leads the evolution whileḣ(H) followed. Now our "dynamical variable" is h and soḣ(h) leads whileḢ(h) follows. Similar to the H(h) functions, functional form ofḣ andḢ is too complicated to write them down, so we substitute H(h) into them and plot the resulting curvesḣ(h) andḢ(h) inFig. 6(c)-(i). FIG. 6 : 6Typical dynamics for D 4 case. In (a) and (b) panel we presented H(h) curves -α > 0 in (a) and α < 0 -in (b). Three different branches (H 1 , H 2 and H 3 ) are presented in three different linestylesblack, solid grey and dashed grey. In the remaining panels we presentedḢ (in black) andḣ (in grey) curves for α > 0 in (c)-(f) and for α < 0 in (g)-(i) panels. Panel (c) corresponds to H 1 , panel (d) -to H 2 , panels (e) and (f) -to H 3 -fine structure in the vicinity of h = 0 in (e) and large-scale structure in (f). For α < 0 (g) panel represents H 1 , (h) -H 2 and (i) -H 3 . (see text for details). FIG. 7 : 7p h ) coordinates. So we define p = −H 2 /Ḣ for both scale factors, use expressions forḢ andḣ and plot the resulting curves. They are presented in Fig. 7 with the same definitions as in previous figures -p H depicted by black line, p h -by solid grey and p by dashed grey. There (a) and (b) panels correspond to H 1 with α > 0 in (a) and α < 0 in (b); (c)-(f) panels depict H 2 : (c) and (d) are for α > 0 with large-scale structure in (c) and fine structure in (d); (e) and (f) are for α < 0fine structure in (e) and large-scale structure in (f); finally, panels (g)-(i) reflect H 3 branch -(g) and (h) for α > 0 -large-scale and fine structures respectively and (i) is panel for α < 0. Now let Typical dynamics of Kasner exponents for D 4 case. Panel (a) corresponds to H 1 , α > 0 case, panel (b) -to H 1 , α < 0; panels (c) and (d) show H 2 , α > 0 at large scale (c) and fine structure in the vicinity of h = 0 (d); (e) and (f) panels -H 2 branch for α < 0 with fine structure on (e) and large-scale structure on (f) panels; (g) and (h) panels depict H 3 for α > 0 -large-scale on (g) and fine structure on (h);finally, (i) panel corresponds to H 3 , α < 0 (see text for details). different. Indeed, D = 3 case has both Kasner transition on h 2 branch with α < 0 and anisotropic exponential solution -on the same branch but with α > 0. Starting from D 4 Kasner transition occurs on H 1 with α < 0 while anisotropic exponential expansion -on H 3 with α > 0; isotropic exponential solution is on H 2 and with α < 0 -so that for D 4 different exponential solutions resides on different branches plus stable Kasner is located on third branch. Just for comparison and without derivation (results obviously converge and the technics is the same as in Section V) we can provide the results for D = 1 and D = 2 if we solve constraint equation not for h, as we did in the corresponding sections, but for H, as we did for D 4 case. This could be useful for understanding of the structure of solution and its variation with varying number of extra dimensions. In the D = 1 case cubic equation reduced to quadratic so there are only two branches while in D = 2 there are three. We put these results into the first two columns of TABLE I : ISummary of D = 1 vacuum regimes.α Additional conditions Regimes Now let us have a closer look on the panels. The curves in (c) panel (α > 0, h = h + ) have two vertical asymptotes determined by Q + (20), so in Fig. 2(c) H 1 corresponds to ξ 1 and H 2to ξ 2 . From (c) panel one can clearly see that H 1 is singular attractor -indeed, for H 1 > H > 0 we have positiveḢ so that H → H 1 and for H 2 > H > H 1 we have negativeḢ so that H → H 1 again, and H = H 1 is singular. So that the only nonsingular regime exists for H > H 2 and this regime is exponential and anisotropic: h/H ≈ −0.722; its location is defined by positive root of P +1 (see(19)); to get explicit value for h/H ratio one can substitute H (b) at H = H 0 . So we have H = H 2 as a stable point and it corresponds to exponential isotropic solution, unlike situation with α > 0. For 0 < H < H 1 we face singularity at H = H 1 , so that isotropisation is reached only for H > H 1 . The value for H 2 is defined by roots of P + 1 (see(19)). Now let us address non-singular asymptotic regimes in this case. Similar to the previous section let us find the corresponding limits limH→∞ TABLE II : IIPower-law behavior in D = 2 TABLE III : IIISummary of D = 2 regimes.α Branch Additional conditions Regimes solution, but past asymptotes are different -for H 0 < H < H 1 it is nonstandard singularity while but the past asymptote is different -now it is GB Kasner regime K 3 .Finally, similarly to the previous sections, we rewrite equations of motion in terms of Kasner exponents p H and p h to see Kasner asymptotes. Following the definition p = −H 2 /Ḣ with use of1 12α ; this point corresponds to isotropic exponential expansion: H, h → − 1 12α . This regime is reached from both sides -H < H 1 and H > H 1 and in both cases the past regime is singular high-energy (Gauss-Bonnet) Kasner regime K 3 . In (c) panel (h = h 2 , α > 0), we have H, h → 0 solution with h H = − 3 2 + √ 5 2 (see (27)) for H 2 < H 2 0 = √ 5 + 3 24α ; this appears to be GR Kasner solution K 1 . For H 2 > H 2 0 we have stable point H 2 1 = √ 5 + 3 8α with exponential for H > H 2 it is GB Kasner regime K 3 . Exponential solution corresponds to H 2 = √ 5 + 3 8α , and from (27) we know that h/H = −3/2 + √ 5/2, so it is anisotropic exponential solution. In panel (d) ((h = h 2 , α < 0)), at H → 0, we have the same behavior as first regime in (c) panel: H, h → 0 with h H = − 3 2 + √ 5 2 , Last two cases correspond to h = h 3 : α > 0 in (e) panel and α < 0 in (f). One can see that they are in a sense "reverse" of h 2 regimes -first regime in (e) panel is singular: for H 2 < H 2 1 = 3 − √ 5 8α we have attractor H 2 = H 2 0 = 3 − √ 5 24α where bothḢ andḣ diverge, so it is K 1 → nS, which is directly opposite to what we saw in (c) panel. Two remaining regimes have exponential solution as past asymptotes (opposite to future asymptotes in (c) panel) and either non-standard singularity for H 0 < H < H 1 or GB Kasner for H > H 1 . One can clearly see the difference between (c) and (e) panels -future and past asymptotes interchange. The same is in (f) panel -there we have GR Kasner as past and GB Kasner as future asymptotes, exactly opposite to (d) panel. (27) and (28) we can write down exact expressions for p H and p h : p H,1 = 12ξ(144ξ 2 + 12ξ + 1) 1728ξ 3 + 1 , p h,1 = 144ξ 2 + 12ξ + 1 1728ξ 3 + 1 ; TABLE IV : IVPower-law behavior in D = 3h± lim H p H p h Table IV while IVfrom the second column -our limits for H → 0. Also one can see that the limits for H → ∞ coincide for α > 0 and α < 0.Similar to the previous sections let us make mappings between the dynamics in {Ḣ,ḣ} and {p H , p h } coordinates. The first row corresponds to h 1 branch and so to (a) and (b) panels ofFig. 4. For α > 0 ( (b). But there is an interesting feature -at exponential solution Kasner exponents diverge -and we see that both p H and p h are divergent -but their sum p is not. It is an artifact caused by the fact that the number of dimensions in both manifolds is the same -previously individual exponents did not cancel each other in this way at exponential solutions.The second and third rows corresponds to h ± branches. While describing them in {Ḣ,ḣ} coordinates we noted that they are "reverted" in a way. The same is true for the description in which is represented as interchanging between p H and p h in {p H , p h } coordinates in Figs. 5(d, f).We summarize our findings for D = 3 regimes inTable V. The denotations are similar to the previous case -Table III.{p H , p h } coordinates -one cannot miss similarity between Figs. 5(c, d) and Figs. 5(e, f). Both Figs. 5(d, f) in ξ < 0 have transition K 1 → K 3 -the same we see in Figs. 4(d, f) (h ± , α < 0) with the difference that in Fig. 4(f), which is (h − , α < 0), it is reverted with respect to time. In {p H , p h } coordinates, the difference is in interchanging between p H and p h . Similar effects we observe in ξ > 0 part -nonstandard singularities in Figs. 4(c, e) correspond to p H = p h ≡ 0 in Figs. 5(d, f) and the exponential solutions in Figs. 4(c, e) are mapped into vertical asymptotes in Figs. 5(d, f). Similar to the previously described case, the difference between Figs. 4(c, e) is "time reversal" TABLE V : VSummary of D = 3 regimes.Branch α Additional conditions Regimes TABLE VI : VISummary of D 4 regimes with H > 0.Branch α Regimes Table VII . VII TABLE VII : VIISummary of nonsingular regimes. Table VII . 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In[23], Sharir and Solomon showed that the number of incidences between m distinct points and n distinct lines in R 4 is O * m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n ,provided that no 2-flat contains more than s lines, and no hyperplane or quadric contains more than q lines, where the O * hides a multiplicative factor of 2 c √ log m for some absolute constant c. In this paper we prove that, for integers m, n satisfying n 9/8 < m < n 3/2 , there exist m points and n lines on the quadratic hypersurface in R 4such that (i) at most s = O(1) lines lie on any 2-flat, (ii) at most q = O(n/m 1/3 ) lines lie on any hyperplane, and (iii) the number of incidences between the points and the lines is Θ(m 2/3 n 1/2 ), which is asymptotically larger than the upper bound in (1), when n 9/8 < m < n 3/2 . This shows that the assumption that no quadric contains more than q lines (in the above mentioned theorem of [23]) is necessary in this regime of m and n. By a suitable projection from this quadratic hypersurface onto R 3 , we obtain m points and n lines in R 3 , with at most s = O(1) lines on a common plane, such that the number of incidences between the m points and the n lines is Θ(m 2/3 n 1/2 ). It remains an interesting question to determine if this bound is also tight in general.

1601.01817

a61d0ffb904b2fa6ac6f7567ba2e6f12d8685734

Highly incidental patterns on a quadratic hypersurface in R 4 * 8 Dec 2016 December 9, 2016 Noam Solomon Ruixiang Zhang Highly incidental patterns on a quadratic hypersurface in R 4 * 8 Dec 2016 December 9, 2016Combinatorial geometry, incidences In[23], Sharir and Solomon showed that the number of incidences between m distinct points and n distinct lines in R 4 is O * m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + m + n ,provided that no 2-flat contains more than s lines, and no hyperplane or quadric contains more than q lines, where the O * hides a multiplicative factor of 2 c √ log m for some absolute constant c. In this paper we prove that, for integers m, n satisfying n 9/8 < m < n 3/2 , there exist m points and n lines on the quadratic hypersurface in R 4such that (i) at most s = O(1) lines lie on any 2-flat, (ii) at most q = O(n/m 1/3 ) lines lie on any hyperplane, and (iii) the number of incidences between the points and the lines is Θ(m 2/3 n 1/2 ), which is asymptotically larger than the upper bound in (1), when n 9/8 < m < n 3/2 . This shows that the assumption that no quadric contains more than q lines (in the above mentioned theorem of [23]) is necessary in this regime of m and n. By a suitable projection from this quadratic hypersurface onto R 3 , we obtain m points and n lines in R 3 , with at most s = O(1) lines on a common plane, such that the number of incidences between the m points and the n lines is Θ(m 2/3 n 1/2 ). It remains an interesting question to determine if this bound is also tight in general. Introduction Let P be a set of m distinct points in R 2 and let L be a set of n distinct lines in R 2 . Let I(P, L) denote the number of incidences between the points of P and the lines of L; that is, the number of pairs (p, ℓ), such that p ∈ P , ℓ ∈ L and p ∈ ℓ. The classical Szemerédi-Trotter theorem [28] yields the worst-case tight bound I(P, L) = O m 2/3 n 2/3 + m + n . This bound clearly also holds in three, four, or any higher dimensions which can be easily proved by projecting the given lines and points onto some generic plane. Moreover, the bound will continue to be worst-case tight by placing all the points and lines in a common plane, in a configuration that yields the planar lower bound. In the groundbreaking paper of Guth and Katz [8], an improved bound has been derived for I(P, L), for a set P of m points and a set L of n lines in R 3 , provided that not too many lines of L lie in a common plane 1 . Specifically, they showed: Theorem 1.1 (Guth and Katz [8]). Let P be a set of m distinct points and L a set of n distinct lines in R 3 , and let s ≤ n be a parameter, such that no plane contains more than s lines of L. Then I(P, L) = O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n . Remark. When s = Θ( √ n), this bound is known to be tight, by a generalization to three dimensions of Elekes' planar construction of points and lines on an integer grid (see Guth and Katz [8] for the details). For smaller values of s, it is an open problem to give lower bounds or improve the upper bound, and the case s = O(1) is of particular interest. In Theorem 1.5 we give an improved upper bound, and it remains a question (see Question 4.1) whether it is tight. In a recent paper of Sharir and Solomon [23], the following analogous and sharper result in four dimensions was established. Theorem 1.2. Let P be a set of m distinct points and L a set of n distinct lines in R 4 , and let q, s ≤ n be parameters, such that (i) each hyperplane or quadric contains at most q lines of L, and (ii) each 2-flat contains at most s lines of L. Then I(P, L) ≤ 2 c √ log m m 2/5 n 4/5 + m + A m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n , where A and c are suitable absolute constants. When m ≤ n 6/7 or m ≥ n 5/3 , there is the sharper bound I(P, L) ≤ A m 2/5 n 4/5 + m + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 s 1/3 + n . In general, except for the factor 2 c √ log m , the bound is tight in the worst case, for any values of m, n, and for corresponding suitable ranges of q and s. The term m 2/3 n 1/3 s 1/3 comes from the planar Szemerédi-Trotter bound (2), and is unavoidable, as it can be attained if we densely pack points and lines into 2-flats, in patterns that realize the bound in (2). Likewise, the term m 1/2 n 1/2 q 1/4 comes from the bound of Guth and Katz [8] in three dimensions (as in Theorem 1.1), and is again unavoidable, as it can be attained if we densely "pack" points and lines into hyperplanes, in patterns that realize the bound in three dimensions. In this paper we show that the condition in assumption (i) of Theorem 1.2 that quadrics also do not contain too many lines, cannot be dropped, by proving the following theorem. Given integers m and n, there are k, α such that m = Θ(k 3+3α ) points and n = Θ(k 2+4α ). Substituting these values in Theorem 1.3, we obtain the following corollary. Corollary 1.4. For integers m, n, there is a configuration of m points and n lines in R 4 , such that all the points (resp., lines) are contained (resp., fully contained) in S, and (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in a common hyperplane is O(n/m 1/3 ), and (iii) the number of incidences between the points and lines is Ω(m 2/3 n 1/2 + m + n). S := {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 1 = x 2 2 + x 2 3 − x 2 4 } in R 4 , Remarks. (1) For integers m, n, satisfying n 9/8 < m < n 3/2 , the number incidences Ω(m 2/3 n 1/2 ) in Corollary 1.4 is asymptotically larger than the bound of Theorem 3 for the number of incidences O(m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n) = O(m 2/5 n 4/5 + m 5/12 n 3/4 + m + n) (as q = O(n/m 1/3 )). This implies that the condition in assumption (i) of Theorem 1.2 cannot be dropped, in this regime of m and n. (2) We note that the number of 2-rich points determined by n lines in R 4 is O(n 3/2 ), provided that at most O( √ n) of the lines lie on a common plane or regulus, 2 To see this, project the lines onto some (generic) hyperplane H, such that no two lines are projected onto the same line, and similarly, no two 2-rich points are projected onto the same 2-rich point, and such that at most O( √ n) lines lie on a common plane or regulus. Then, the number of 2-rich points in the configuration of n lines in R 4 is equal to the number of 2-rich points in the configuration of the projected lines onto H. By Guth and Katz [8], the number of 2-rich points determined by the projected lines is O(n 3/2 ), and therefore the same holds for the number of 2-rich points in the original configuration of lines in R 4 . We also notice that in a configuration of m points and n lines in R 4 , the 1-rich points (i.e., points that are incident to exactly one line) contribute at most m incidences. Therefore, in Corollary 1.4, as s = O(1), the assumption that m ≤ n 3/2 causes no loss of generality. Proof Techniques. It is a common practice to take geometric objects to be integer points on certain hypersurfaces (especially quadratic ones) and varieties passing through a lot of such points, in order to obtain lower bounds for their incidences. For some most recent applications of this method, see [24] [30] [33]. In this paper we obtain our incidence lower incidence bound by taking integer points and "low height" lines on the above hypersurface S. Projection to R 3 . As remarked above, Guth and Katz [8] proved that the number of incidences between m points and n lines in R 3 is I(P, L) = O m 1/2 n 3/4 + m 2/3 n 1/3 s 1/3 + m + n , provided that no plane contains more than s lines of L. By choosing a generic projection from R 4 to R 3 , we show that Corollary 1.4 directly implies the following Theorem. Theorem 1.5. For integers m, n, there is a configuration of m points and n lines in R 3 , such that (i) the number of lines in any common plane is s = O(1), and (ii) the number of incidences between the points and lines is Ω(m 2/3 n 1/2 + m + n). Remark. When n 3/4 ≪ m ≪ n 3/2 , the term m 2/3 n 1/2 dominates over m and n, showing that in this regime of m and n, the construction in Theorem 1.5, of m points and n lines with O(1) lines in a common plane, yields a super-linear number of incidences. As observed above, the bound of Guth and Katz [8] implies that the number of 2-rich points determined by the n lines is O(n 3/2 ), so the assumption that m ≤ n 3/2 causes no loss of generality. Background. Incidence problems have been a major topic in combinatorial and computational geometry for the past thirty years, starting with the Szemerédi-Trotter bound [28] back in 1983. Several techniques, interesting in their own right, have been developed, or adapted, for the analysis of incidences, including the crossing-lemma technique of Székely [27], and the use of cuttings as a divide-and-conquer mechanism (e.g., see [2]). Connections with range searching and related problems in computational geometry have also been noted, and studies of the Kakeya problem (see, e.g., [29]) indicate the connection between this problem and incidence problems. See Pach and Sharir [14] for a comprehensive survey of the topic. The simplest instances of incidence problems involve points and lines. Szemerédi and Trotter solved completely this special case in the plane [28]. Guth and Katz's second paper [8] provides a worst-case tight bound in three dimensions, under the assumption that no plane contains too many lines; see Theorem 1.1. Under this assumption, the bound in three dimensions is significantly smaller than the planar bound (unless one of m, n is significantly smaller than the other), and the intuition is that this phenomenon should also show up as we move to higher dimensions. The first attempt in higher dimensions was made by Sharir and Solomon in [20]. In a recent work, Sharir and Solomon [23] gave a tight bound in four-dimensions provided that the number of lines fully contained in a common hyperplane or quadric is bounded by a parameter q, and the number of lines fully contained in a common 2-flat is bounded by a parameter s. Whereas the condition that no common hyperplane contains more than a bounded number of lines was known to be necessary, it remained an open question whether the condition that the number of lines in a common quadric is bounded is necessary. In this paper, we show that when n 9/8 < m < n 3/2 , this condition is indeed necessary, by describing an explicit quadratic hypersurface in R 4 containing more incidences than the bound prescribed by the main theorem of [23]. This is the content of Theorem 1.3, and Corollary 1.4. We remark that in [30], another example of points on a quadratic hypersurface in F 4 with highly incidental pattern was noticed. There F is a finite field. Our current quadratic hypersurface and our counting techniques in R 4 are slightly different. The reader may find it interesting to compare the results here to the results in [30]. Another interesting remark is that in three dimensions, there are certain quadratic surfaces, called reguli, such that if one allows too many lines to lie on such a regulus, the number of 2-rich points determined by them can be larger than the Guth-Katz bound [8] of O(n 3/2 ). The quadratic hypersurface in R 4 presented in this paper can be thought of as a higher degree analogs of regulus. However, If one only cares about incidences between points and lines (instead of the number of 2-rich points determined by the lines), the existence of many lines on a regulus (or any quadratic surface in R 3 ) do not yield more than a linear number of incidences. Proof of Theorem 1.3 Proof. We start by recalling the quadric S = {(x 1 , x 2 , x 3 , x 4 ) ∈ R 4 | x 1 = x 2 2 + x 2 3 − x 2 4 },(5) on which the construction takes place, and define the set of points by P = {(x 1 , x 2 , x 3 , x 4 ) ∈ S | x i ∈ Z, i = 1, . . . , 4, |x 1 | ≤ 200k 2+2α , |x 2 |, |x 3 |, |x 4 | ≤ 100k 1+α },(6) and the set of lines L = {{x + tv | t ∈ R} ⊆ S | x = (x 1 , . . . , x 4 ), v = (v 1 , . . . , v 4 ), x i , v i ∈ Z, i = 1, . . . , 4, |x 1 | ≤ k 2+2α , |x 2 |, |x 3 |, |x 4 | ≤ k 1+α , k 1+2α 4 ≤ |v 1 | ≤ 8k 1+2α , |v 2 |, |v 3 | ≤ k α , v 2 4 = v 2 2 + v 2 3 , v 1 = 2x 2 v 2 + 2x 3 v 3 − 2x 4 v 4 , gcd(v 2 , v 3 , v 4 ) = 1, and |v 4 | ≥ k α 2 }, for any positive integer k and any α > 0. Since a point on S is uniquely determined by its last three coordinates, we have |P | = |{(x 2 , x 3 , x 4 ) ∈ Z 3 | |x 2 |, |x 3 |, |x 4 | ≤ 100k 1+α }| = Θ(k 3+3α ). The analysis of (an asymptotically tight bound on) the number of lines of L is a bit more involved. A line {x + tv | t ∈ R} in L (assuming x ∈ S, |x 1 | ≤ k 2+2α , |x 2 |, |x 3 |, |x 4 | ≤ k 1+α ) is fully contained in S if and only if v 1 = 2x 2 v 2 + 2x 3 v 3 − 2x 4 v 4 and v 2 4 = v 2 2 + v 2 3 . It follows by Benito and Varona [1, Theorem 1] that the number of primitive integer triples (v 2 , v 3 , v 4 ) (i.e., without a common divisor) satisfying v 2 4 = v 2 2 + v 2 3 , |v 2 |, |v 3 | ≤ k α , and |v 4 | ≥ k α 2 is Θ(k α ). For each such (v 2 , v 3 , v 4 ), we claim that there are Ω(k 3+3α ) (and trivially also O( |P |) = O(k 3+3α )) points x ∈ P , such that v 1 = 2x 2 v 2 + 2x 3 v 3 − 2x 4 v 4 satisfying k 1+2α 4 ≤ |v 1 | ≤ 8k 1+2α . Indeed, note that |v 2 |, |v 3 | ≤ |v 4 |. Choosing |x 2 |, |x 3 | ≤ |x 4 | 4 , k 1+α 2 ≤ |x 4 | ≤ k 1+α (there are at least k 3+3α 32 choices of such triples (x 2 , x 3 , x 4 )) implies that |2x 2 v 2 + 2x 3 v 3 | ≤ 2|x 2 ||v 2 | + 2|x 3 ||v 3 | ≤ 2 |x 4 | 4 (|v 2 | + |v 3 |) ≤ |x 4 ||v 4 |, Here |v 1 | ≥ |x 4 ||v 4 | ≥ k 1+2α 4 . The inequality |v 1 | ≤ 8k 1+2α is immediate. Moreover, each line ℓ satisfying the above conditions is incident to O(k) different points of P (and can thus be expressed in O(k) different ways as {x + tv | t ∈ R} ⊆ S, for |x 1 | ≤ k 2+2α , |x 2 |, |x 3 |, |x 4 | ≤ k 1+α ). Indeed, parameterize ℓ as {x + tv | t ∈ R} ⊂ S, where x, v satisfy |v 1 | ≥ k 1+2α 4 , |x 1 | ≤ k 2+2α , and v = (v 1 , v 2 , v 3 , v 4 ) is primitive (i.e., its coordinates do not have a common factor). Notice that if |t| > 8k, then the first coordinate of x + tv has absolute value greater than k 2+2α , and that if t ∈ Z, then x + tv ∈ Z 4 (since v is primitive and x ∈ Z 4 ). In either case, x + tv ∈ P . This implies that ℓ ∩ P ⊆ {x + tv | t ∈ Z, |t| ≤ 8k}, and thus |ℓ ∩ P | ≤ 16k = O(k) as claimed. Therefore, the total number of lines is Ω( k 3+3α k α k ) = Ω(k 2+4α ). It is easy to see that each line in L is incident to Ω(k) points in P . It follows that |L| = O(k 2+4α ). Hence |L| = Θ(k 2+4α ). Since each line has Θ(k) integer points in P on it, we have I(P, L) = Θ(k 3+4α ). We now bound the number of lines fully contained in any 2-flat, and then bound the number of lines on any hyperplane. The bounds will be uniform (i.e., independent of the specific 2-flat or hyperplane). Let π denote any 2-flat, and we analyze the number of lines that are fully contained in π ∩ S. We claim that S contains no planes, so π ⊂ S. Assume the contrary, then we parameterize π = {(u 1 s + r 1 t + w 1 , u 2 s + r 2 t + w 2 , u 3 s + r 3 t + w 3 , u 4 s + r 4 t + w 4 ) | s, t ∈ R}, for constants u i , r i , w i ∈ R, i = 1, 2, 3, 4 where (u 1 , u 2 , u 3 , u 4 ) and (r 1 , r 2 , r 3 , r 4 ) are both nonzero and not proportional to each other. Comparing the coefficients of quadratic terms in the identity u 1 s + r 1 t + w 1 ≡ (u 2 s + r 2 t + w 2 ) 2 + (u 3 s + r 3 t + w 3 ) 2 − (u 4 s + r 4 t + w 4 ) 2 , we deduce (u 2 , u 3 , u 4 ) and (r 2 , r 3 , r 4 ) are proportional to each other. Hence we may assume u 2 = u 3 = u 4 = 0. But this forces u 1 = 0, a contradiction. Therefore π is not contained in S. Thus the intersection π ∩ S is a curve of degree at most two, so there are at most two lines fully contained in π ∩ S. Next, we take any hyperplane H, and analyze the number of lines fully contained in S ∩ H. The surface S ∩ H is a quadratic 2-surface contained in H. We will use the classification of (real) quadratic surfaces in R 3 (see, e.g., Sylvester's original paper [26]), and distinguish between two cases. If the equation of H can be expressed as x 1 = ϕ(x 2 , x 3 , x 4 ), where ϕ is a linear form, then each point x ∈ H ∩ S satisfies the equations x 2 2 + x 2 3 − x 2 4 = ϕ(x 2 , x 3 , x 4 ), x ∈ H.(7) This is either a cone, i.e., is linearly equivalent to x 2 2 + x 2 3 − x 2 4 = 0, or a hyperboloid of one or two sheets, i.e., is linearly equivalent to x 2 2 + x 2 3 − x 2 4 = 1 or x 2 2 + x 2 3 − x 2 4 = −1, respectively. It is easy to verify (and well known) that there are no lines on the hyperboloid of two sheets. We therefore assume that S ∩ H is either a cone or a hyperboloid of one sheet. In these cases, there are at most two lines of L with any given direction that are fully contained in S ∩ H. Note that if a line {x + tv | t ∈ R} ∈ L is fully contained in S ∩ H, then v 1 = ϕ(v 2 , v 3 , v 4 ) (where we let ϕ denote the linear homogeneous part of ϕ), and v 2 4 = v 2 2 + v 2 3 (being the homogeneous part of degree two in t), |v 2 |, |v 3 | ≤ k α and |v 4 | ≥ k α . As observed above, there are O(k α ) such triples (v 2 , v 3 , v 4 ). Therefore, the number of lines in L that lie in S ∩ H is O(k α ). In the remaining case, the equation of H is of the form ϕ(x 2 , x 3 , x 4 ) = 0, where ϕ is a linear form. We can assume, without loss of generality, that the equation of H is x 2 = ψ(x 3 , x 4 ), where ψ is a linear form (the remaining case x 4 = 0 is simpler to handle). In this case, for every point x ∈ S ∩ H, we have x 1 = ψ(x 3 , x 4 ) 2 + x 2 3 − x 2 4 , x ∈ H. The classification of (real) quadratic surfaces implies that this can be an elliptic paraboloid, a parabolic cylinder or a hyperbolic paraboloid. An elliptic paraboloid contains no lines and the corresponding case is trivial. If S ∩ H is a parabolic cylinder, then all lines on it are parallel. It is straightforward that there are O(k 2+2α ) points in P that lie on it (by counting possible pairs (x 3 , x 4 )). Hence there are O(k 1+2α ) lines in L that are fully contained in S H. In the rest of the discussion we assume S ∩ H is a hyperbolic paraboloid. In this case, similarly to the case of the one-sheeted hyperboloid, there are at most two lines with the same direction. Moreover, the Finally, we show that for α < 1 2 , the number of incidences is (asymptotically) larger than Θ m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n , which is the bound of Theorem 3, with m = Θ(k 3+3α ), n = Θ(k 2+4α ), q = O(k 1+3α ), and s = O(1). We have There exists a projection from R 4 onto a hyperplane H ⊂ R 4 , such that at most s lines lie on any common plane in H. direction (v 1 , v 2 , v 3 , v 4 ) of any line on S ∩ H satisfies v 2 = ψ(v 3 , v 4 ) and v 2 4 = v 2 2 + v 2 3 (where Proof of Lemma 3.1. Let π 1 , . . . , π k denote the set of 2-flats containing at least two lines in L, then k ≤ n 2 . For a generic hyperplane H ⊂ R 4 , the projection p : R 4 → H maps π i onto a plane π ′ i contained in H. We pick, as we may, a hyperplane H, so that p is bijective on π 1 , . . . , π k . Denote by L ′ the set of projected lines in R 3 . It is easy to verify that the set of planes in H containing at least two lines in L ′ consists precisely of π ′ 1 , . . . , π ′ k . Moreover, the number of lines in L ′ that are contained in π ′ i is equal to the number of lines in L that are contained in π i , thus completing the proof. ✷ Discussion and open questions In Corollary 1.4, we show a concrete irreducible quadratic hypersurface S in R 4 , together with a set of m points and n lines that lie on S, for n 9/8 < m < n 3/2 , such that (i) the number of lines in any common 2-flat is O(1), (ii) the number of lines in any common hyperplane is O(n/m 1/3 ), and (iii) the number of incidences between the points and lines is Ω(m 2/3 n 1/2 ), which is asymptotically larger than Θ(m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n) in this regime of m and n. A natural question is to extend this result to other regimes by a similar construction. The condition (i) is natural and should not be hard to achieve, since if a plane is not contained in a quadratic hypersurface, then by the generalized version of Bézout's theorem [5] it can contain at most two lines. Here are a few natural questions that arise 1. Can we generalize our construction, such that in (ii) we are allowed to have a more general q, not necessarily ∼ n/m 1/3 , s.t. the number of lines in any common hyperplane is O(q), and we still get a lower bound of incidences asymptotically larger than Θ(m 2/5 n 4/5 + m 1/2 n 1/2 q 1/4 + m 2/3 n 1/3 + m + n)? 2. Can we find a similar construction when m < n 9/8 ? 3. How powerful is the natural generalization of this construction for R d , when d > 4? Notice that for d > 4, finding the precise bound for the number of incidences between a set P of m points and a set L of n lines in R d is already an interesting open question. It is probably too early for us to answer this question before we find the correct bound. 4. In three dimensions, it remains a question to determine if Theorem 1.5 is tight. We do not know the answer to this question yet. It seems to require new techniques. Theorem 1 . 3 . 13For each positive integer k and each α > 0, there exists m = Θ(k 3+3α ) points and n = Θ(k 2+4α ) lines on the quadratic hypersurface such that there are at most O(1) lines lying on any 2-flat and O(k 1+3α ) lines lying on any hyperplane, and I(P, L) = Θ(k 3+4α ). When s = Θ( √ n), this bound is tight, by a generalization to three dimensions of Elekes' construction of points and lines on an integer grid in the plane (see Guth and Katz [8] for the details). For smaller values of s, it is an open problem to give lower bounds or improve the upper bound, where the case s = O(1), is of particular interest. we let ψ denote the linear homogeneous part of ψ). Thus once we fix v 1 and "v 3 or v 4 " (depending on ψ), we have limited the possible direction (v 1 , v 2 , v 3 , v 4 ) in a set with ≤ 2 elements. Hence there are O(k 1+3α ) lines that are fully contained in S ∩ H. exponent is smaller than 3 + 4α for every α. Since both m and n are O(k 3+4α ), the claim is proved. ✷ 3 Proof of Theorem 1.5The proof of Theorem 1.5 follows easily by Corollary 1.4, together with the following lemma. Lemma 3. 1 . 1Let L be a set of n lines in R 4 such that at most s lines lie on a common 2-flat. Question 4. 1 . 1Let P be a set of m distinct points and L a set of n distinct lines in R 3 , and assume that no plane contains more than s = O(1) lines of L. Then what is a good or tight upper bound of I(P, L)? Would Om + n) suffice? The additional requirement in[8], that no regulus contains too many lines, is not needed for the bound given below. A regulus is a quadratic surface that is doubly ruled by lines. For more details about reguli, see e.g., Sharir and Solomon[22]. AcknowledgementsWe thank Micha Sharir for his invaluable advice, and the anonymous referees for their helpful comments. Pythagorian triples with legs less than n. M Benito, J L Varona, Journal of Computational and Applied Mathematics. 1431M. 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Sharir, On lines, joints, and incidences in three dimensions, J. Combinat. Theory, Ser. A 118 (2011), 962-977. Also in arXiv:0905.1583. Introduction to Intersection Theory in Algebraic Geometry. W Fulton, Expository Lectures from the CBMS Regional Conference Held at George Mason University. AMS Bookstore54W. Fulton, Introduction to Intersection Theory in Algebraic Geometry, Expository Lectures from the CBMS Regional Conference Held at George Mason University, June 27-July 1, 1983, Vol. 54. AMS Bookstore, 1984. Distinct distance estimates and low-degree polynomial partitioning. L Guth, arXiv:1404.2321Discrete Comput. Geom. 53L. Guth, Distinct distance estimates and low-degree polynomial partitioning, Discrete Comput. Geom. 53 (2015), 428-444. Also in arXiv:1404.2321. Algebraic methods in discrete analogs of the Kakeya problem, Advances Math. L Guth, N H Katz, arXiv:0812.1043v1225L. Guth and N. H. 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In this article, we study the problem of high-dimensional conditional independence testing, a key building block in statistics and machine learning. We propose an inferential procedure based on double generative adversarial networks (GANs). Specifically, we first introduce a double GANs framework to learn two generators of the conditional distributions. We then integrate the two generators to construct a test statistic, which takes the form of the maximum of generalized covariance measures of multiple transformation functions. We also employ data-splitting and cross-fitting to minimize the conditions on the generators to achieve the desired asymptotic properties, and employ multiplier bootstrap to obtain the corresponding p-value. We show that the constructed test statistic is doubly robust, and the resulting test both controls type-I error and has the power approaching one asymptotically. Also notably, we establish those theoretical guarantees under much weaker and practically more feasible conditions compared to the existing tests, and our proposal gives a concrete example of how to utilize some state-of-the-art deep learning tools, such as GANs, to help address a classical but challenging statistical problem. We demonstrate the efficacy of our test through both simulations and an application to an anti-cancer drug dataset. A Python implementation of the proposed procedure is available at https://github.com/ tianlinxu312/dgcit.

2006.02615

edad980b33f1c44759c2fedd6d8090e20abf7223

Double Generative Adversarial Networks for Conditional Independence Testing Chengchun Shi [email protected] Department of Statistics School of Economics and Political Science Department of Biostatistics and Epidemiology University of California at Berkeley London Tianlin Xu Department of Statistics School of Economics and Political Science Department of Biostatistics and Epidemiology University of California at Berkeley London Wicher Bergsma [email protected] Department of Statistics School of Economics and Political Science Department of Biostatistics and Epidemiology University of California at Berkeley London Lexin Li [email protected] Department of Statistics School of Economics and Political Science Department of Biostatistics and Epidemiology University of California at Berkeley London Double Generative Adversarial Networks for Conditional Independence Testing Editor:Conditional independenceDouble-robustnessGeneralized covariance mea- sureGenerative adversarial networksMultiplier bootstrap In this article, we study the problem of high-dimensional conditional independence testing, a key building block in statistics and machine learning. We propose an inferential procedure based on double generative adversarial networks (GANs). Specifically, we first introduce a double GANs framework to learn two generators of the conditional distributions. We then integrate the two generators to construct a test statistic, which takes the form of the maximum of generalized covariance measures of multiple transformation functions. We also employ data-splitting and cross-fitting to minimize the conditions on the generators to achieve the desired asymptotic properties, and employ multiplier bootstrap to obtain the corresponding p-value. We show that the constructed test statistic is doubly robust, and the resulting test both controls type-I error and has the power approaching one asymptotically. Also notably, we establish those theoretical guarantees under much weaker and practically more feasible conditions compared to the existing tests, and our proposal gives a concrete example of how to utilize some state-of-the-art deep learning tools, such as GANs, to help address a classical but challenging statistical problem. We demonstrate the efficacy of our test through both simulations and an application to an anti-cancer drug dataset. A Python implementation of the proposed procedure is available at https://github.com/ tianlinxu312/dgcit. Introduction Conditional independence (CI) is a fundamental concept in statistics and machine learning. Testing conditional independence is a key building block and plays a central role in a large variety of statistical learning problems, for instance, causal inference (Pearl, 2009), graphical models (Koller and Friedman, 2009), dimension reduction (Li, 2018), among many others. It is frequently used in a wide range of scientific and business applications, and we demonstrate its application with a cancer genetics example later. In this article, we aim at testing whether two random variables X and Y are conditionally independent given a set of confounding variables Z. That is, we test the hypotheses: H 0 : X ⊥ ⊥ Y | Z versus H 1 : X ⊥ ⊥ Y | Z,(1) given the observed data of n i.i.d. copies {(X i , Y i , Z i )} 1≤i≤n of (X, Y, Z). For our problem, X, Y and Z can all be multivariate. However, the main challenge arises when the confounding set of variables Z is multivariate and high-dimensional. As such, we primarily focus on the scenario where X and Y are univariate, and Z is multivariate and its dimension can potentially diverge to infinity. Meanwhile, our proposed method can be readily extended to the scenario of multivariate X and Y as well. Another challenge is the limited sample size compared to the dimensionality of Z. As a result, many existing tests may become ineffective, suffering from either an inflated type-I error, or not having enough power to detect the alternatives. See Section 2 for a detailed literature review. To deal with those challenges, we propose a testing procedure based on double generative adversarial networks (GANs, Goodfellow et al., 2014) for the CI testing problem in (1). GANs have recently stood out as a powerful approach for learning and generating random samples from a complex, high-dimensional data distribution. They have been successfully applied in numerous applications, ranging from image processing and computer vision, to sequential data modeling such as natural language, music, speech, and to medical fields such as DNA design and drug discovery; see Gui et al. (2020) for a review of the GANs applications. Moreover, there have recently emerged works studying the consistency and rate of convergence of the GANs estimators; see, e.g., Liang (2018); Chen et al. (2020). Our proposal involves two key components: a double GANs framework to learn two generators that approximate the conditional distribution of X given Z, and Y given Z, respectively, and a test statistic that is taken as the maximum of generalized covariance measures of multiple transformation functions of X and Y . We first show that our test statistic is doubly-robust, which offers an additional layer of protection against potential misspecification of the conditional distributions; see Theorems 2 and 3. We then show that the resulting test achieves a valid control of the type-I error asymptotically, and more importantly, under the set of conditions that are much weaker and practically more feasible compare to the existing tests; see Theorem 4. Besides, we prove that the power of our test approaches one asymptotically; see Theorem 5, and we demonstrate through simulations that it is more powerful than numerous competing tests empirically. In addition, we employ data splitting and cross-fitting that allow us to derive the asymptotic properties under minimal conditions on the generators, and employ multiplier bootstrap to obtain the corresponding p-value of the test. Our contributions are multi-fold. We develop a useful testing procedure for a fundamentally important statistical inference problem. We establish the statistical guarantees under much weaker conditions. We also give an example of how to utilize some state-of-the-art deep learning tools, such as GANs, to address a classical but challenging statistical problem. The rest of the article is organized as follows. Section 2 reviews some key existing CI testing methods. Section 3 develops the double GANs-based testing procedure. Section 4 derives the theoretical properties. Section 5 presents the simulations and a cancer genetics data example. Section 6 concludes the paper. The Appendix collects all technical proofs. Literature review on conditional independence testing There has been a large and growing literature on conditional independence testing; see Li and Fan (2019) for a review. Broadly speaking, the existing tests can be cast into four main categories, the metric-based tests (e.g., White, 2007, 2014;Wang et al., 2015;Pan et al., 2017;Wang et al., 2018), the conditional randomization-based tests (e.g., Candes et al., 2018;Bellot and van der Schaar, 2019), the kernel-based tests (e.g., Fukumizu et al., 2008;Zhang et al., 2011), and the regression-based tests (e.g., Hoyer et al., 2009;Shah and Peters, 2018). There are also some other types of tests (e.g., Bergsma, 2004;Berrett et al., 2019, to name a few). The metric-based tests typically employ some kernel smoothers to estimate the conditional characteristic function or the distribution function of Y given X and Z. Kernel smoothers, however, are known to suffer from the curse of dimensionality, and as such, these tests are usually not suitable when the dimension of Z is high. The conditional randomization-based tests require the knowledge of the conditional distribution of X|Z (Candes et al., 2018). If unknown, the type-I error rates of these tests rely critically on the quality of the approximation of this conditional distribution. Kernel-based tests are built upon the notion of maximum mean discrepancy (MMD, Gretton et al., 2012), and could have inflated type-I errors. Regression-based tests have valid type-I error control, but may suffer from inadequate power. Next, we discuss in detail the conditional randomization-based tests, in particular, the work of Bellot and van der Schaar (2019), the regression-based tests, and the MMD-based tests, as our proposal is related to and built on those methods. For each family of tests, we first lay out the main ideas, then discuss their potential limitations. Conditional randomization-based tests The family of conditional randomization-based tests is built upon the following basis. If the conditional distribution P X|Z of X given Z is known, then one can independently draw X (1) i ∼ P X|Z=Z i , for i = 1, . . . , n, where the superscript denotes the first round of draws. Besides, these samples are independent of the observed samples X i 's and Y i 's. Write X = (X 1 , . . . , X n ) , X (1) = (X (1) 1 , . . . , X (1) n ) , Y = (Y 1 , . . . , Y n ) , and Z = (Z 1 , . . . , Z n ) . Hereinafter we use boldface letters to denote data matrices that consist of n samples. Since the joint distributions of (X, Y , Z) and (X (1) , Y , Z) are the same under H 0 , any large difference between the two distributions can be interpreted as evidence against H 0 . Therefore, one can repeat the sample drawing process M times, i.e., X (m) i ∼ P X|Z=Z i , i = 1, . . . , n, m = 1, . . . , M . Write X (m) = (X (m) 1 , . . . , X (m) n ) . Then, for a given test statistic ρ = ρ(X, Y , Z), the associated p-value is p = 1 M M m=1 I ρ(X (m) , Y , Z) ≥ ρ(X, Y , Z) , where I(·) denotes the indicator function. Since the triplets (X, Y , Z), (X (1) , Y , Z), . . . , (X (M ) , Y , Z) are exchangeable under H 0 , the above p-value is valid, in the sense that it equals the significance level under the null, i.e., Pr (p ≤ α|H 0 ) = α, for any 0 < α < 1. In practice, however, P X|Z is rarely known. Bellot and van der Schaar (2019) proposed to approximate P X|Z using GANs. Specifically, they learned a generator G X (·, ·) from the observed data, then took Z i along with an independent noise variable as the input to obtain a sample X (m) i , which minimizes the divergence between the distributions of (X i , Z i ) and ( X (m) i , Z i ). They computed the p-value by replacing X (m) with X (m) = ( X (m) 1 , . . . , X (m) n ) . They called this test the generative conditional independence test (GCIT). By Theorem 1 of Bellot and van der Schaar (2019), the excess type-I error of this test is upper bounded as, Pr (p ≤ α|H 0 ) − α ≤ E d TV P X|Z , P X|Z = E sup A Pr(X ∈ A|Z) − Pr( X (m) ∈ A|Z) ≡ D,(2) where d TV is the total variation norm between two probability distributions P and Q such that d TV (P, Q) = sup A |P (A) − Q(A)|, the supremum is taken over all measurable sets A, and the expectations in (2) are taken with respect to Z. By definition, the error term D in (2) measures the quality of the conditional distribution approximation. Bellot and van der Schaar (2019) argued that this error term is negligible due to the capacity of deep neural networks in terms of estimating the conditional distribution. To the contrary, we find this approximation error is usually not negligible, and consequently, it may inflate the type-I error and invalidate the test. We consider a simple example to further elaborate this. Example 1 Suppose X is one-dimensional, and follows a simple linear regression model, X = Z β 0 + ε, where the error ε is independent of Z, and ε ∼ N (0, σ 2 0 ) for some σ 2 0 > 0. Suppose we know a priori that the linear regression model holds. We thus estimate β 0 by ordinary least squares, and denote the resulting estimator by β. For simplicity, suppose σ 2 0 is known too. For this simple example, we have the following result regarding the approximation error D. Proposition 1 Suppose the linear regression model holds, the dimension of Z is much smaller than the sample size n, and the derived distribution P X|Z is Normal(Z β, σ 2 0 I n ), where I n is the n × n identity matrix. Then D does not decay to zero. To facilitate the understanding of the convergence behavior of D, we sketch a few lines of the proof of Proposition 1. The complete proof is given in the Appendix. Let P X|Z=Z i denote the conditional distribution of X (m) i given Z i , which is Normal(Z i β, σ 2 0 ) in this example. If D = o(1), then, D ≡ n 1/2 E d 2 TV P X|Z=Z i , P X|Z=Z i = o(1).(3) In other words, in order to control the type-I error, GCIT requires the total variation distance measure in (3) to converge at a faster rate than n −1/2 . However, this rate cannot be achieved in general. In our Example 1, we have D ≥ c for some constant c > 0. Consequently, D in (2) is not o(1). Proposition 1 shows that, even if we know a priori that the linear model holds, D does not decay to zero as n tends to infinity. In practice, we do not have such prior model information. Then it would be even more difficult to estimate the conditional distribution P X|Z . Therefore, using GANs to approximate P X|Z does not guarantee a negligible approximation error. Regression-based tests The family of regression-based tests is built upon the generalized covariance measure, GCM(X, Y ) = 1 n n i=1 X i − E(X i |Z i ) Y i − E(Y i |Z i ) , where E(X|Z) and E(Y |Z) are the estimated condition means E(X|Z) and E(Y |Z), respectively, obtained by some supervised learner. When the prediction errors of E(X|Z) and E(Y |Z) satisfy certain convergence rates, Shah and Peters (2018) proved that GCM is asymptotically normal under H 0 , in which the asymptotic mean is zero, and the standard deviation can be consistently estimated by some standard error estimator, denoted by s(GCM). Therefore, at level α, we reject H 0 , if |GCM|/ s(GCM) exceeds the upper α/2th quantile of a standard normal distribution. Such a test can control the type-I error. Nevertheless, it may not have sufficient power to detect H 1 . Consider the asymptotic mean of GCM, which is GCM * (X, Y ) = E{X − E(X|Z)}{Y − E(Y |Z)}. The regression-based tests require |GCM * | to be nonzero under H 1 to have power. However, it may be difficult to satisfy such a requirement. We again consider a simple example. Example 2 Suppose X * , Y and Z are independent random variables. Besides, X * has mean zero, and X = X * g(Y ) for some function g. For this example, we have E(X|Z) = E(X), since both X * and Y are independent of Z, and so is X. Besides, E(X) = E(X * )E{g(Y )} = 0, since X * is independent of Y and E(X * ) = 0. Thus GCM * (X, Y ) = E{X − E(X)}{Y − E(Y |Z)} = 0 for any function g. On the other hand, X and Y are conditionally dependent given Z, as long as g is not a constant function. Therefore, for this example, the regression-based tests would fail to discriminate between H 0 and H 1 . MMD-based tests The family of MMD-based tests involves the maximum mean discrepancy as a measure of independence. For any two probability measures P , Q and a function space F, define MMD(P, Q|F) = sup f ∈F {Ef (W 1 ) − Ef (W 2 )} , where W 1 ∼ P, W 2 ∼ Q. Let H 1 , H 2 denote some function spaces of X and Y . Define φ XY = MMD(P XY , Q XY | H 1 ⊗ H 2 ), where ⊗ is the tensor product, P XY is the joint distribution of (X, Y ) whose definition does not rely on Z, and Q XY is the conditionally independent distribution with the same X and Y margins as P XY . Let X and Y be independent copies of X and Y , such that they are conditionally independent given Z. Then Q XY corresponds to the joint distribution of (X , Y ). Note that, to generate (X , Y ), we need to first sample Z according to P Z , then generate X and Y that follow P X|Z and P Y |Z , respectively. As such, Q XY depends on Z, and φ XY depends on Z through Q XY . Furthermore, since E{h 1 (X )h 2 (Y )} = E[E{h 1 (X )|Z}E{h 2 (Y )|Z}], we have, φ XY = sup h 1 ∈H 1 ,h 2 ∈H 2 E{h 1 (X)h 2 (Y )} − E{h 1 (X )h 2 (Y )} = sup h 1 ∈H 1 ,h 2 ∈H 2 E{h 1 (X)h 2 (Y )} − E[E{h 1 (X)|Z}E{h 2 (Y )|Z}] = sup h 1 ∈H 1 ,h 2 ∈H 2 E{h 1 (X)h 2 (Y )} − E[h 1 (X)E{h 2 (Y )|Z}] − E[{h 1 (X)|Z}h 2 (Y )] + E[E{h 1 (X)|Z}E{h 2 (Y )|Z}] = sup h 1 ∈H 1 ,h 2 ∈H 2 E h 1 (X) − E{h 1 (X)|Z} h 2 (Y ) − E{h 2 (Y )|Z} . As such, φ XY measures the average conditional association between X and Y given Z. Under H 0 , it equals zero, and hence an estimator of this measure can be used as a test statistic for H 0 . Moreover, if H 1 and H 2 are reproducing kernel Hilbert spaces (RKHSs), then φ XY has a closed form expression in terms of the reproducing kernels of the RKHS (Doran et al., 2014;Gretton et al., 2012), which makes the tests based on an estimator of φ XY easier to evaluate. A notable example of this family is the kernel MMD-based test (KCIT) of Zhang et al. (2011). We next further discuss this test. To control the type-I error asymptotically, KCIT requires the dimension d Z of Z to be fixed (Zhang et al., 2011, Proposition 5), since it uses the continuous mapping theorem to derive the limiting distribution of its test statistic. However, the continuous mapping theorem may not hold when d Z diverges with n. In addition, KCIT requires the 1 distance between the covariance operator and its empirical estimator to decay to zero. It remains unknown whether such an assertion holds as d Z diverges. By contrast, the test we develop allows d Z to diverge while maintaining the asymptotic control of the type-I error. This implies that our test is expected to have a better size control than KCIT when d Z is large. We later further verify this through numerical simulations. Moreover, the maximization of KCIT is done over unit balls in an RKHS, while our proposed test can deal with much more general function spaces such as those generated by neural networks. Consequently, the power of our test can be tailored to more general alternatives than KCIT. For instance, it is known that deep neural networks learn certain non-smooth functions at a faster rate than kernel methods (Imaizumi and Fukumizu, 2019). This implies that our test is expected to have a better power than KCIT under certain types of alternatives. A new double GANs-based testing procedure We propose a double GANs-based testing procedure for the conditional independence testing problem (1). Conceptually, our test integrates three families of tests that are based on conditional randomization, regression, and MMD. Meanwhile, our new test overcomes the limitations of the existing ones. Unlike the GCIT of Bellot and van der Schaar (2019) that only learned the conditional distribution of X given Z, we learn two generators G X and G Y to approximate the conditional distributions of both X given Z and Y given Z. We then integrate the two generators in an appropriate way to construct a doubly-robust test statistic. To ensure the theoretical properties of this test, we only require the root mean squared total variation norm to converge at a rate of n −κ for some κ > 1/4. Such a requirement is much weaker and practically more feasible than the condition in (3). Moreover, to improve the power of the test, we consider a collection of the generalized covariance measures, {GCM(h 1 (X), h 2 (Y )) : h 1 , h 2 }, for multiple combinations of transformation functions h 1 (X) and h 2 (Y ). We then take the maximum of all these GCMs as our test statistic. This essentially yields a type of maximum mean discrepancy measure φ XY . To see why this statistic can enhance the power, we quickly revisit Example 2. When g is not a constant function, there exists some nonlinear function h 1 such that h * 1 (Y ) = E{h 1 (X)|Y } is not a constant function of Y . Set h 2 = h * 1 . We then have GCM * = E[h 1 {X * g(Y )}{Y − E(Y )}] = Var{h * 1 (Y )} > 0, which enables us to discriminate H 1 from H 0 . We note that the maximum of GCMs yields MMD. Instead of using kernels, we have chosen GANs, because they have been shown to give good approximations of complex distributions (Imaizumi and Fukumizu, 2019). This allows the transformation functions h 1 and h 2 to be arbitrary function spaces. We set these function spaces to the class of neural networks in our implementation. In contrast, kernel based measures such as KCIT are limited to vector spaces of functions, which can be problematic for a high-dimensional conditioning variable (Doran et al., 2014). We also remark that, even though our proposal is built upon the existing CI tests, our test is far from a simple extension. The major challenge lies in how to properly utilize the GAN estimators for the purpose of high-dimensional conditional independence testing. Despite the fact that GANs are capable of approximating complex high-dimensional probability distributions, the GAN estimators have non-negligible bias that decays slower than the parametric root-n rate. Naively plugging the GAN estimators in the test statistic can invalidate the subsequent inference. We give a graphical overview of our proposed testing procedure in Figure 1. We first employ double GANs to compute the test statistic that is the maximum of the GCMs over multiple transform functions. We then employ multiplier bootstrap to compute the corresponding p-value. We next detail the main components of our testing procedure. Test statistic We begin with two function spaces, H 1 = h 1,θ 1 : θ 1 ∈ R d 1 and H 2 = h 2,θ 2 : θ 2 ∈ R d 2 , indexed by some parameters θ 1 and θ 2 , respectively. In our implementation, we set H 1 and H 2 to the classes of neural networks with a single-hidden layer, finitely many hidden nodes, and the sigmoid activation function. However, a broad range of other function spaces may be considered, as appropriate for the application at hand. We next randomly generate B func- tions, h 1,1 , . . . , h 1,B ∈ H 1 , h 2,1 , . . . , h 2,B ∈ H 2 , where we independently generate i.i.d. mul- tivariate normal variables θ 1,1 , . . . , θ 1,B ∼ N (0, 2I d 1 /d 1 ), and θ 2,1 , . . . , θ 2,B ∼ N (0, 2I d 2 /d 2 ). We then set h 1,b = h 1,θ 1,b , and h 2,b = h 2,θ 2,b , b ∈ [B] = {1, . . . , B}. Consider the following maximum-type test statistic, T = max b 1 ,b 2 ∈[B] σ −1 b 1 ,b 2 1 n n i=1 h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } , where σ 2 b 1 ,b 2 is the sampling variance estimator, σ 2 b 1 ,b 2 = 1 n − 1 n i=1 h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } − 1 n n i=1 h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } 2 . To compute T , we need to estimate the conditional means, E{h 1,b 1 (X)|Z} and E{h 2,b 2 (Y )|Z}, which can be done by applying some supervised learning methods. However, this needs to be performed for all b 1 , b 2 ∈ [B] . In theory, B should diverge to infinity to guarantee the power property of the test. As such, this approach is computationally very expensive. Instead, we propose to implement this step based on the generators G X and G Y estimated using GANs, which is much more efficient computationally. Specifically, we first randomly generate i.i.d. samples {v (m) i,X } M m=1 , {v (m) i,Y } M m=1 from mul- tivariate normal distribution, for i = 1, . . . , n. We then feed Z i and v (m) i,X into GANs to obtain the pseudo samples X (m) i = G X (Z i , v (m) i,X ), and feed Z i and v (m) i,Y to obtain Y (m) i = G Y (Z i , v (m) i,Y ), for i = 1, . . . , n, m = 1, . . . , M . These pseudo samples approximate the conditional distribution of X i and Y i given Z i , respectively. We then compute E{h 1,b 1 ( X i )|Z i } = 1 M M m=1 h 1,b 1 ( X (m) i ), E{h 2,b 2 (Y i )|Z i } = 1 M M m=1 h 2,b 2 ( Y (m) i ), Algorithm 1 Algorithm for computing the test statistic. Input: The number of transformation functions B, the number of pseudo samples M , and the number of data splits L. Step 1: Divide {1, . . . , n} into L folds I (1) , . . . , I (L) . Denote I (− ) = {1, . . . , n}\I ( ) . Step 2: For = 1, . . . , L, train two generators G ( ) (− ) , to approximate the conditional distributions of X|Z and Y |Z. X and G ( ) Y based on {(X i , Z i )} i∈I (− ) and {(Y i , Z i )} i∈I Step 3: For = 1, . . . , L and i ∈ I , generate i.i.d. random noises v (m) i,X M m=1 , v (m) i,Y M m=1 . Set X (m) i = G ( ) X Z i , v (m) i,X , and Y (m) i = G ( ) Y Z i , v (m) i,Y , m = 1, . . . , M . Step 4: Randomly generate h 1,1 , . . . , h 1,B ∈ H 1 and h 2,1 , . . . , h 2,B ∈ H 2 . Step 5: Compute the test statistic T . for b 1 , b 2 = 1, . . . , B. Plugging the estimated means into T produces the sample test statistic, T = max b 1 ,b 2 n −1/2 n i=1 ψ b 1 ,b 2 ,i , where (4) ψ b 1 ,b 2 ,i = σ −1 b 1 ,b 2 h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 X (m) i h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 Y (m) i . To help reduce the type-I error, we further employ a data splitting and cross-fitting strategy, which has been commonly used in statistical inferences in recent years (Romano and DiCiccio, 2019). That is, we use different subsets of data samples to learn GANs and to construct the test statistic. We begin by dividing the data into L folds of equal size. We use I ( ) to denote the set of indices of subsamples in the th fold, and I (− ) its complement. We next learn two generators G ( ) X and G ( ) Y , based on {(X i , Z i )} i∈I (− ) and {(Y i , Z i )} i∈I (− ) , to approximate the conditional distributions of X|Z and Y |Z, for = 1, · · · , L. Finally, for each and i ∈ I ( ) , we generate the pseudo samples X We summarize our procedure of computing the test statistic in Algorithm 1. Approximation of conditional distribution via GANs There are numerous GANs methods available for learning high-dimensional distributions. We adopt the proposal of Genevay et al. (2017) to learn the conditional distributions P X|Z and P Y |Z in our setting thanks to its competitive performance. Recall that P X|Z is the distribution of pseudo outcome generated by the generator G X given Z. We consider estimating P X|Z by optimizing min G X max c D c, (P X|Z , P X|Z ). Here D c, denotes the Sinkhorn loss function between two probability measures with respect to some cost function c and some regularization parameter > 0, D c, (µ, ν) = 2D c, (µ, ν) − D c, (µ, µ) − D c, (ν, ν), D c, (µ, ν) = inf π∈Π(µ,ν) x,y c(x, y) − H(π|µ ⊗ ν) π(dx, dy), where Π(µ, ν) is a set containing all probability measures π whose marginal distributions correspond to µ and ν, H is the Kullback-Leibler divergence, and µ ⊗ ν is the product measure of µ and ν. When = 0, D c,0 (µ, ν) measures the optimal transport of µ into ν with respect to the cost function c(·, ·) (Cuturi, 2013). When = 0, an entropic regularization is added to this optimal transport. As such, the objective function D c, is a regularized optimal transport metric, and the regularization is to facilitate the computation, so that D c, can be efficiently evaluated. Intuitively, the closer the two probability measures, the smaller the Sinkhorn loss. As such, maximizing the loss with respect to the cost function learns a discriminator that can better discriminate the samples generated between P X|Z and P X|Z . On the other hand, minimizing the maximum cost with respect to the generator G X makes it closer to the true distribution P X|Z . This yields the minimax formulation min G X max c D c, (P X|Z , P X|Z ) that we target. In practice, we approximate the cost and the generator based on neural networks. Integrations in the objective function D c, (P X|Z , P X|Z ) are approximated by sample averages. The conditional distribution of P Y |Z is estimated similarly. Bootstrap for the p-value Next, we propose a multiplier bootstrap method to approximate the distribution of T under H 0 and compute the corresponding p-value. Let ψ b 1 ,b 2 = n −1 n i=1 ψ b 1 ,b 2 ,i . The key obser- vation is that {ψ b 1 ,b 2 } B b 1 ,b 2 =1 are asymptotically multivariate normal with zero mean under H 0 ; see the proof of Theorem 4 for details. Consequently, T = max b 1 ,b 2 |n −1/2 n i=1 ψ b 1 ,b 2 ,i | is to converge to a maximum of normal variables in absolute values. To approximate this limiting distribution, we first estimate the covariance matrix of a B 2 -dimensional vector formed by {n −1/2 ψ b 1 ,b 2 } B b 1 ,b 2 =1 using the sample covariance matrix Σ, whose {b 1 + B(b 2 − 1), b 3 + B(b 4 − 1)}th entry is given by 1 n n i=1 (ψ b 1 ,b 2 ,i − ψ b 1 ,b 2 )(ψ b 3 ,b 4 ,i − ψ b 3 ,b 4 ), b 1 , b 2 , b 3 , b 4 = 1, . . . , B. We then generate i.i.d. random vectors with the covariance matrix equal to Σ. This can be achieved by generating i.i.d. standard normal variables {W i,j } i,j for 1 ≤ i · · · ≤ n and j = 1, · · · , J, then compute B 2 -dimensional normal random vectors W j whose {b 1 +B(b 2 −1)}th entry is given by n −1/2 n i=1 (ψ b 1 ,b 2 ,i − ψ b 1 ,b 2 )W i,j for j = 1, · · · , J. We next compute T j = W j ∞ , for j = 1, . . . , J, where · ∞ is the maximum element of a vector in absolute value, and J is the number of bootstrap samples. Finally, we use these maximum absolute values to approximate the distribution of T under the null hypothesis. This yields the p-value, p = J −1 J j=1 I( T ≥ T j ). We summarize this bootstrap procedure in Algorithm 2. Asymptotic theory To derive the theoretical properties of the test statistic T , we first introduce the concept of the "oracle" test statistic T * . If P X|Z and P Y |Z were known a priori, then one can draw {X (m) i } m and {Y (m) i } m from P X|Z=Z i and P Y |Z=Z i directly, and can compute the test statis- tic by replacing { X (m) i } m and { Y (m) i } m with {X (m) i } m and {Y (m) i } m . We call the resulting T * an "oracle" test statistic. We next establish the double-robustness property of T , which helps explain why our test can relax the requirement in (3). Roughly speaking, the doublerobustness means that T is asymptotically equivalent to T * when either the conditional distribution of X|Z, or that of Y |Z, is well approximated by GANs. It guarantees that T converges to T * at a faster rate than the estimated conditional distribution. In contrast, the convergence rate of the GCIT test statistic is the same as the rate of the estimated conditional distribution. For this reason, our procedure only requires a weaker condition. Theorem 2 (Double-robustness) Suppose M is proportional to n, and B = O(n c ) for some constant c > 0. Suppose min h 1 ∈H 1 ,h 2 ∈H 2 Var[{h 1 (X) − E{h 1 (X)|Z}}{h 2 (Y ) − E{h 2 (Y )|Z}}] ≥ c * for some constant c * > 0. Then, T − T * = o p (1), when E d 2 TV Q ( ) X (·|Z), Q X (·|Z) = o(log −1 n), or E d 2 TV Q ( ) Y (·|Z), Q Y (·|Z) = o(log −1 n). We note that the conditions on M and B are mild, as these are user-specified parameters. As we have mentioned, when both total variation distances converge to zero, the test statistic T converges at a faster rate than those total variation distances. Therefore, we can greatly relax the condition in (3) Since κ x , κ y > 0, the convergence rate of ( T − T * ) is faster than that in (5). To ensure √ n(T − T * ) = o p (1), it suffices to require κ x + κ y > 1/2. In contrast to (3), this rate is achievable. We consider two examples in Berrett et al. (2019) to illustrate this, while the condition holds in a much wider range of settings. Algorithm 2 Algorithm for computing the p-value. Input: The number of bootstrap samples J, and {ψ b 1 ,b 2 ,i } B,n b 1 ,b 2 =1,i=1 . Step 1: Generate i.i.d. standard normal variables W i,j for i = 1, · · · , n, j = 1, . . . , J. Step 2: Compute B 2 -dimensional normal random vectors W j whose {b 1 + B(b 2 − 1)}th entry is given by n −1/2 n i=1 (ψ b 1 ,b 2 ,i − ψ b 1 ,b 2 )W i,j and set T j = W j ∞ for j = 1, · · · , J. Step 3: Compute the p-value, p = J −1 J j=1 I( T ≥ T j ). Example 3 (Parametric setting) Suppose the parametric forms of Q X and Q Y are correctly specified. Then under certain regularity conditions, the requirement κ x + κ y > 1/2 holds if k x = O(n tx ) and k y = O(n ty ) for some t x + t y < 1/2, where k x and k y are the dimensions of the parameters defining the parametric models for Q X and Q y , respectively. Example 4 (Nonparametric setting with binary data) Suppose X, Y are binary variables. Then the requirement κ x + κ y > 1/2 holds if the mean squared prediction errors of the nonparametric estimators of the conditional means of X and Y given Z are O(n −tx ) and O(n −ty ) for some t x , t y , such that t x + t y > 1/2. We briefly remark that, there is no explicit specification on d Z in the statement of Theorem 3. It is implicitly imposed due to the requirement that κ x + κ y > 1/2, and d Z is allowed to diverge with the sample size. In addition, the condition κ x + κ y > 1/2 can be further relaxed to κ 1 , κ 2 > 0 using the theory of higher order influence functions (Robins et al., 2008. However, the resulting estimators would be considerably much more complicated, and thus we do not pursue those estimators. Next, we show that our proposed test can control the type-I error asymptotically. Theorem 4 Suppose the conditions in Theorem 2 hold. Suppose (5) holds for some κ x , κ y such that κ x + κ y > 1/2. Then, the p-value from Algorithm 2 satisfies that Pr(p ≤ α|H 0 ) = α + o(1). Next, to derive the asymptotic power of the test, we introduce the pair of hypotheses based on the notion of weak conditional independence (Daudin, 1980), H * 0 : E[cov{f (X), g(Y )|Z}] = 0, for any f ∈ L 2 X , g ∈ L 2 Y versus H * 1 : E[cov{f (X), g(Y )|Z}] = 0, for some f ∈ L 2 X , g ∈ L 2 Y , where L 2 X and L 2 Y denote the class of all squared integrable functions of X and Y , respectively. We note that conditional independence implies weak conditional independence, i.e., H 0 implies H * 0 , and H * 1 implies H 1 . We consider an example to further elaborate on the difference between weak CI and CI. Example 5 Let X, Y, Z be binary random variables with the distribution functions, Pr(X = 0, Y = 0|Z = 0) Pr(X = 0, Y = 1|Z = 0) Pr(X = 1, Y = 0|Z = 0) Pr(X = 1, Y = 1|Z = 0) = 1/6 1/3 1/3 1/6 , Pr(X = 0, Y = 0|Z = 1) Pr(X = 0, Y = 1|Z = 1) Pr(X = 1, Y = 0|Z = 1) Pr(X = 1, Y = 1|Z = 1) = 1/3 1/6 1/6 1/3 , and Z takes the value {0, 1} with equal probability. We can show that, for any x, y ∈ {0, 1}, E{Pr(X = x|Z)Pr(Y = y|Z)} = 1 2 × 1 2 = 1 4 , Pr(X = x, Y = y) = 1 2 Pr(X = x, Y = y|Z = 0) + Pr(X = x, Y = y|Z = 1) = 1 2 × 1 6 + 1 3 = 1 4 . By definition, this implies that X and Y are weakly conditionally independent given Z, since E[cov{f (X), g(Y )|Z}] = x,y f (x)g(y) Pr(X = x, Y = y) − E Pr(X = x|Z)Pr(Y = y|Z) = 0. However, Pr(X = 0, Y = 0|Z = 0) = Pr(X = 0|Z = 0)Pr(Y = 0|Z = 0), since the former equals 1/6, and the latter equals 1/4. As such, X and Y are not conditionally independent given Z. The next theorem shows that our proposed test is consistent against the alternatives in H * 1 , but not against all alternatives in H 1 . Theorem 5 Suppose the conditions in Theorem 4 hold, B = c 0 n c for some c 0 , c > 0, and X, Y are bounded random variables. Then the p-value from Algorithm 2 satisfies that Pr(p ≤ α|H * 1 ) → 1, as n → ∞. Finally, we remark that our test is constructed based on φ XY . Meanwhile, we may consider another test based on φ XY Z = MMD(P XY Z , Q XY Z |H 1 ⊗ H 2 ⊗ H 3 ), where P XY Z is the joint distribution of (X, Y, Z), Q XY Z = P X|Z P Y |Z P Z , and H 3 is a neural network class of functions of Z. This type of test is consistent against all alternatives in H 1 . However, in our numerical experiments, we find it less powerful compared to our test. This agrees with the observation by Li and Fan (2019) in that, even though the tests based on weak CI cannot fully characterize CI, they potentially enjoy an improved power. Numerical studies We begin with a discussion of some implementation details. We then carry out simulations to study the empirical size and power of the proposed test, and compare with several alternative methods. We further illustrate with an application to a cancer genetics example. Implementation details For the number of functions B in Algorithm 2, it represents a trade-off. By Theorem 5, B should be as large as possible to guarantee a good power. In practice, the computation complexity increases as B increases. Our numerical studies suggest that the value of B between 30 and 50 achieves a good balance between the power and the computational cost, and we fix B = 30. For the number of pseudo samples M , and the number of sample splittings L, we find the results are not overly sensitive to their choices, and thus we fix M = 100 and L = 3. Besides, we set the number of bootstrap samples J = 1000. For the GANs, we use a single-hidden layer neural network to approximate both the discriminator and the generator. The number of nodes in the hidden layer is set at 128. The dimension of the input noise v i,Y is set at 10. These tuning parameters are chosen following the common practice in the GANs literature, and also by investigating the goodness-of-fit of the resulting generator, which can be done by comparing the conditional histogram of the generated samples to that of the true samples. In our experiments, we (a) One random value of a g (b) Another random value of a g Figure 2: Conditional histograms. GANs are trained using data generated from the simulation study in Section 5.2. find such an approach yields GANs with satisfactory performances. More specifically, let d Z denote the dimension of Z, and µ Z the sample average n −1 i Z i . Let Y i = G Y (Z i , v i,Y ) denote a simulated sample to approximate the distribution of Y |Z = Z i obtained by the generator G Y . When G Y is accurate, we expect the conditional distribution of Y i and Y i given Z i are similar. As such, for any d Z -dimensional vector a, the histograms See Figures 2 (a) and (b) for the conditional histograms with two choices of a g . It is seen that the GANs fit the conditional density reasonably well. The fitted conditional distribution for P X|Z can be checked in a similar fashion. { Y i : a ( Z i − µ Z ) > 0} and {Y i : a (Z i − µ Z ) > 0} should be similar. We sample i.i.d. vectors {a g } g from Normal(0, I d Z ). For each g, we plot the histogram {Y i : a g (Z i − µ Z ) > 0} and { Y (m) i : a g (Z i − µ Z ) > 0}. Simulations We generate the data following the post nonlinear noise model similarly as in Zhang et al. (2011);Doran et al. (2014); Bellot and van der Schaar (2019), i.e., X = sin(a f Z + ε f ), and Y = cos(a g Z + bX + ε g ). The entries of a f , a g are randomly and uniformly sampled from [0, 1], then normalized to the unit 1 norm. The noise variables ε f , ε g are independently sampled from a normal distribution with mean zero and variance 0.25. In this model, the parameter b determines the degree of conditional dependence. When b = 0, H 0 holds, and otherwise H 1 holds. The sample size is set at n = 1000. We call our test DGCIT, short for double GANs-based conditional independence test. We compare it with the GCIT test of Bellot and van der Schaar (2019), the regression-based test (RCIT) of Shah and Peters (2018), the kernel MMD-based test (KCIT) of Zhang et al. (2011), and the classifier CI test (CCIT) of Sen et al. (2017). We first study the empirical size when b = 0. We vary the dimension of Z as d Z = 50, 100, 150, 200, 250, and consider two generation distributions. We first generate Z from a standard normal distribution, then from a Laplace distribution. We set the significance level at α = 0.05 and 0.1. Figure 3 reports the empirical size of the tests aggregated over 500 data replications. We make the following observations. First, the type-I error rates of our test and RCIT are close to or below the nominal level in nearly all cases. Second, KCIT fails in that its type-I error is considerably larger than the nominal level in all cases. We suspect it is due to the high-dimensional setting where d Z ≥ 50. We have experimented with d Z = 5, and found that KCIT is able to control the type-I error in that case. This is consistent with Proposition 5 of Zhang et al. (2011), which suggests that KCIT should work in a low-dimensional setting. Third, GCIT and CCIT both have inflated type-I errors in some cases. Take GCIT as an example. When Z is normal, d Z = 250 and α = 0.1, its empirical size is close to 0.15. This is consistent with our discussion in Section 2.1, since GCIT requires a rather strong condition to control the type-I error. We then study the empirical power when b > 0. We generate Z from a standard normal distribution, with d Z = 100, 200, and vary the value of b = 0.3, 0.45, 0.6, 0.75, 0.9 that controls the magnitude of the alternative. Figure 4 reports the empirical power of the tests over 500 data replications. We observe that our test is the most powerful, and the empirical power approaches 1 as b increases to 0.9, demonstrating the consistency of the test. Meanwhile, both GCIT and RCIT have no power in all cases. We do not report the power of KCIT, because as we have shown earlier, it cannot control the size, and thus its empirical power is not meaningful. Finally, we discuss the computation time. All experiments were run on a 16 N1 CPUs Google Cloud Computing platform. The wall clock time for running the entire GCIT test for one data replication was about 2.5 minutes. In contrast, the running time for CCIT was about 2 minutes, for KCIT about 30 seconds, and for GCIT and RCIT about 20 seconds. Anti-cancer drug data example We illustrate our proposed test with an anti-cancer drug dataset from the Cancer Cell Line Encyclopedia (Barretina et al., 2012). We concentrate on a subset, the CCLE data, that measures the treatment response of drug PLX4720. It is well known that the patient's cancer treatment response to drug can be strongly influenced by alterations in the genome (Garnett et al., 2012). This data measures 1638 genetic mutations of n = 472 cell lines, and the goal of our analysis is to determine which genetic mutation is significantly correlated with the drug response after conditioning on all other mutations. The same data was also analyzed in Tansey et al. (2018) and Bellot and van der Schaar (2019). We adopt the same screening procedure as theirs to screen out irrelevant mutations, which leaves a total of 466 potential mutations for our conditional independence testing. The ground truth is unknown for this data. Instead, we compare with the variable importance measures obtained from fitting an elastic net (EN) model and a random forest (RF) model as reported in Barretina et al. (2012). In addition, we compare with the GCIT test of Bellot and van der Schaar (2019). Table 1 reports the corresponding variable importance measures and the p-values, for 10 mutations that were also reported by Bellot and van der Schaar (2019). We see that, the p-values of the tests generally agree well with the variable important measures from the EN and RF models. Meanwhile, the two conditional independence tests agree relatively well, except for two genetic mutations, MAP3K5 and FTL3. GCIT concluded that MAP3K5 is significant (p < 0.001) but FTL3 is not (p = 0.521), whereas our test leads to the opposite conclusion that MAP3K5 is insignificant (p = 0.794) but FTL3 is (p = 0). Besides, both EN and RF place FTL3 as an important mutation. We then compare our findings with the cancer drug response literature. Actually, MAP3K5 has not been previously reported in the literature as being directly linked to the PLX4720 drug response. Meanwhile, there is strong evidence showing the connections of the FLT3 mutation with cancer response (Tsai et al., 2008;Larrosa-Garcia and Baer, 2017). Combining the existing literature with our theoretical and synthetic results, we have more confidence about the findings of our proposed test. Discussion In this article, we have developed a new inferential procedure for high-dimensional conditional independence testing, where the dimension of the conditional variables can diverge with the sample size. Our proposal utilizes a set of state-of-the-art deep learning tools to help address a classical statistics and machine learning problem. It integrates GANs, neural networks, cross-fitting and multiplier bootstrap. It achieves the asymptotic guarantees under much weaker conditions, and enjoys better empirical performances, when compared to the existing tests. As a tradeoff, our test is computationally more complicated. Nevertheless, the wall clock time for running the entire test for one data replication is in the order of a few minutes and is deemed reasonable. Finally, the computer code is publicly available on the GitHub repository: https://github.com/tianlinxu312/dgcit. Appendix A. Proofs We provide the proofs of Proposition 1, Theorems 3, 4, and 5. We omit the proof of Theorem 2, since it is similar to that of Theorem 3. We note that Theorems 2-5 are established under our choice of the function classes H 1 and H 2 , which are set to the classes of neural networks with a single-hidden layer, finitely many hidden nodes, and the sigmoid activation function, as used in our implementation. Meanwhile, our results can be extended to more general choices of the function classes. A.1 Proof of Proposition 1 Note that the total variation distance is bounded by 1. Suppose Ed TV ( P X|Z , P X|Z ) = o(1). Then we have d TV ( P X|Z , P X|Z ) = o p (1). By the dominated convergence theorem, we have Ed 2 TV ( P X|Z , P X|Z ) = o(1). By Theorem 1.2 of Devroye et al. (2018), we have d TV ( P X|Z , P X|Z ) is proportional to min   1, σ −1 0 n i=1 Z i ( β − β 0 ) 2   . It follows that 1 σ 0 E n i=1 {Z i ( β − β 0 )} 2 = o(1). Applying Theorem 1.2 of Devroye et al. (2018) again, we obtain that d TV ( P X|Z=Z i , P X|Z=Z i ) is proportional to min 1, σ −1 0 |Z i ( β − β 0 )| . Therefore, we obtain that, n i=1 Ed 2 TV P X|Z=Z i , P X|Z=Z i = o(1). Since the data is exchangeable, we have that, Ed 2 TV P X|Z=Z i , P X|Z=Z i = o(n −1 ).(6) This shows that when RHS of (2), i.e., E{d TV ( P X|Z , P X|Z )} is o(1), (6) holds. Next, we show (6) is violated in the linear regression example. By the data exchangeability, it suffices to show n i=1 Ed 2 TV { P X|Z=Z i , P X|Z=Z i } is not o(1). With some calculations, we obtain that, n i=1 E min 1, σ −2 0 |Z i ( β − β 0 )| 2 = n i=1 Eσ −2 0 |Z i ( β − β 0 )| 2 I σ −2 0 |Z i ( β − β 0 )| 2 ≤ 1 + n i=1 EI σ −2 0 |Z i ( β − β 0 )| 2 > 1 = n i=1 Eσ −2 0 |Z i ( β − β 0 )| 2 − n i=1 E σ −2 0 |Z i ( β − β 0 )| 2 − 1 I σ −2 0 |Z i ( β − β 0 )| 2 > 1 .(7) By the definition of β, we have n i=1 Eσ −2 0 |Z i ( β − β 0 )| 2 = 1 σ 2 0 E( β − β) Z Z( β − β) = 1 σ 2 0 Eε Z(Z Z) −1 Z ε, where ε = (ε 1 , · · · , ε n ) consist of i.i.d. copies of ε defined in Example 1. It follows that, n i=1 Eσ −2 0 |Z i ( β − β 0 )| 2 = 1 σ 2 0 Eε Z(Z Z) Z ε = 1 σ 2 0 trace Eεε Z(Z Z) −1 Z = trace EZ(Z Z) −1 Z = d Z ,(8) where d Z is the dimension of Z. Next, we show that, n i=1 Eσ −2 0 |Z i ( β − β 0 )| 2 I σ −2 0 |Z i ( β − β 0 )| 2 ≥ 1 = o(1),(9) or equivalently, Enσ −2 0 |Z i ( β − β 0 )| 2 I σ −2 0 |Z i ( β − β 0 )| 2 ≥ 1 = o(1). We have already shown that Enσ −2 0 |Z i ( β − β 0 )| 2 = d Z . By the dominated convergence theorem, it suffices to show that, nσ −2 0 |Z i ( β − β 0 )| 2 I σ −2 0 |Z i ( β − β 0 )| 2 ≥ 1 = o p (1). By definition, it in turn suffices to show that, Pr σ −2 0 |Z i ( β − β 0 )| 2 ≥ 1 → 0. This holds by Markov's inequality, as Eσ −2 0 |Z i ( β − β 0 )| 2 = d Z n → 0. Combining (9) together with (7) and (8) yields that, n i=1 E min 1, σ −2 0 |Z i ( β − β 0 )| 2 ≥ d Z − o(1) ≥ 1 − o(1), and hence n i=1 Ed 2 TV { P X|Z=Z i , Q (n) X (·|Z i )} ≥ 1 − o(1) . This completes the proof of Proposition 1. A.2 Proof of Theorem 3 We begin by providing an upper bound for the function classes H 1 and H 2 . Recall that both H 1 and H 2 are classes of neural networks with a single-hidden layer, finitely many hidden nodes, and the sigmoid activation function. Because of that, each function h 1,θ 1 ∈ H 1 and h 2,θ 2 ∈ H 2 can be represented as h 1,θ 1 (x) = M j=1 θ (1) 1,j sigmoid(x θ (2) 1,j ), h 2,θ 2 (x) = M j=1 θ (1) 2,j sigmoid(y θ (2) 2,j ), where θ 1 and θ 2 correspond to the sets of parameters (θ 2,j ) : 1 ≤ j ≤ M , respectively, and M is a finite integer. Note that the sigmoid function is bounded. As such, the functions h 1,θ 1 and h 2,θ 2 are uniformly bounded by M j=1 |θ (1) 1,j | and M j=1 |θ (2) 2,j |, respectively. Since we sample B many functions {h 1,θ b } B b=1 and {h 2,θ b } B b=1 , these functions are uniformly bounded by M max b,j |θ (1) b,j | + |θ (2) b,j | . Since these parameters θ 1 , θ 2 are sampled from standard normal distributions, and that Pr(W > t) = 1 √ 2π ∞ t exp − w 2 2 dw ≤ 1 √ 2π ∞ t w exp − w 2 2 dw = exp(−t 2 /2) √ 2π , for any t ≥ 1, we can show that max b,j |θ (1) b,j | + |θ (2) b,j | is upper bounded by √ log B , with probability approaching one. Note that B grows polynomially with respect to the sample size n. Therefore, we have that the functions in H 1 and H 2 are upper bounded by log n in absolute values. Define a test statistic T * * = max b 1 ,b 2 σ −1 b 1 ,b 2 1 n n i=1 h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 (X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) , where the σ b 1 ,b 2 is constructed based on { X (m) i } m and { Y (m) i } m , instead of {X (m) i } m and {Y (m) i } m . It suffices to show that | T − T * * | = O p (n −2κ log n), and |T * − T * * | = O p (n −2κ log n). Step 1. We first consider the difference | T − T * * |. For any sequences {a n } n , {b n } n , we have that, Consequently, we have | T − T * * | ≤ I 1 + I 2 + I 3 , where I 1 = max b 1 ,b 2 σ −1 b 1 ,b 2 1 n n i=1 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) , I 2 = max b 1 ,b 2 σ −1 b 1 ,b 2 1 n n i=1 h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 (X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) , I 3 = max b 1 ,b 2 σ −1 b 1 ,b 2 1 n n i=1 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) . If min σ b 1 ,b 2 ≥ c 0 for some constant c 0 > 0, then it suffices to show that I * j = O p (n −(κx+κy) log n), for j = 1, 2, 3, where I * 1 = max b 1 ,b 2 1 n n i=1 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) , I * 2 = max b 1 ,b 2 1 n n i=1 h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 (X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) , I * 3 = max b 1 ,b 2 1 n n i=1 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) . The number of folds L is finite, as such, it suffices to show that I ( ) j = O p (n −(κx+κy) log n), for j = 1, 2, 3 and = 1, . . . , L, where I ( ) 1 = max b 1 ,b 2 1 n i∈I ( ) 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) , I ( ) 2 = max b 1 ,b 2 1 n i∈I ( ) h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 (X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) , I ( ) 3 = max b 1 ,b 2 1 n i∈I ( ) 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) . We divide the rest of the proof into four sub-steps. We first show that I ( ) j = O p (n −(κx+κy) log n), for j = 1, 2, 3. Finally, we show Pr(min σ b 1 ,b 2 ≥ c 0 ) → 1 for some constant c 0 > 0. Step 1.1. Recall we have shown that the functions in H 1 and H 2 are bounded by log n in absolute values at the beginning of the proof of Theorem 3. By Bernstein's inequality, we have that, Pr M m=1 h 1,b (X (m) i ) − M E{h 1,b (X i )|Z i } ≥ t ≤ 2 exp − t 2 2(M log n + t √ log n/3) , for any b and i. Set t = 3(c + 2)M log n, where the constant c is as defined in the statement of Theorem 2. For a sufficiently large n, we have t √ log n/3 ≤ M log n/2. It follows that Pr M m=1 h 1,b (X (m) i ) − M E{h 1,b (X i )|Z i } ≥ 3(c + 2)M log n ≤ 2 n c+2 . By Bonferroni's inequality, we obtain that, Pr max b∈{1,··· ,B} max i∈{1,··· ,n} M m=1 h 1,b (X (m) i ) − M E{h 1,b (X i )|Z i } ≥ 3(c + 2)M log n ≤ Bn max b∈{1,··· ,B} max i∈{1,··· ,n} Pr M m=1 h 1,b (X (m) i ) − M E{h 1,b (X i )|Z i } ≥ 3(c + 2)M log n ≤ 2Bn n c+2 . Under the condition B = O(n c ), we obtain with probability 1 − O(n −1 ) that, max b∈{1,··· ,B} max i∈{1,··· ,n} M m=1 h 1,b (X (m) i ) − M E{h 1,b (X i )|Z i } ≤ O(1)n −1/2 log n,(11) as M is proportional to n, and O(1) denotes some positive constant. Similarly, we can show that, max b∈{1,··· ,B} max i∈I ( ) M m=1 h 1,b ( X (m) i ) − M x h 1,b (x) P ( ) X|Z=Z i (dx) ≤ O(1) √ n log n, with probability 1 − O(n −1 ). Combining this with (11), we obtain with probability 1 − O(n −1 ) that, max b∈{1,...,B} i∈I ( ) M m=1 h 1,b (X (m) i ) − h 1,b ( X (m) i ) −M x h 1,b (x) P X|Z=Z i (dx) − P ( ) X|Z=Z i (dx) ≤ O(1) √ n log n.(12) Conditional on Z i , the expectation of h 2,b 2 (Y i )−M −1 M m=1 h 2,b 2 (Y (m) i ) equals zero. Under the null hypothesis, the expectation of M −1 M m=1 {h 1,b 1 (X (m) i )−h 1,b 1 ( X (m) i )}{h 2,b 2 (Y i )− M −1 M m=1 h 2,b 2 (Y (m) i )} equals zero as well. Applying Bernstein's inequality again, we can show with probability tending to 1 that, I ( ) 1 ≤ O(1) σn −1/2 log 3/2 n + n −1 log 2 n ,(13) where σ 2 = max b 1 ,b 2 E 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) 2 ≤ max b 1 E 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) 2 log n. Let A denote the event in (12). The last term on the second line can be bounded from above by max b 1 ,i E 1 M M m=1 {h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i )} 2 I(A) log n (14) + max b 1 ,i E 1 M M m=1 {h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i )} 2 I(A c ) log n.(15) Since M is proportional to n, by (10), (14) is upper bounded by O(1)   n −1 log 2 n + max b∈{1,··· ,B} i∈I ( ) E x h 1,b (x) P ( ) X|Z=Z i (dx) − P X|Z=Z i (dx) 2    log n. By the boundedness of the function class H 1 , it can be further bounded from above by O(1) n −1 log 3 n + Ed 2 TV ( P ( ) X|Z , P X|Z ) log 2 n . The above quantity is of order O(n −2κx log 2 n). Consequently, (14) is of the order O(n −2κx log 2 n). Note that the event A occurs with probability at least 1 − O(n −1 ). By the boundedness of the function class H 1 , (15) is of the order O(n −1 log 2 n). Therefore, σ 2 is of the order O(n −2κx log 2 n). This implies that I ( ) 1 can be bounded from above by O(n −1/2−κx log 5/2 n), which in turn yields that I ( ) 1 = O p (n −κx−κy log n), since κ x , κ y < 1/2. Step 1.2. This step can be proven in a similar way as Step 1.1, and is omitted. Step 1.3. Under H 0 , the expectation of 1 |I ( ) | i∈I ( ) 1 M M m=1 h 1,b 1 (X (m) i ) − h 1,b 1 ( X (m) i ) 1 M M m=1 h 2,b 2 (Y (m) i ) − h 2,b 2 ( Y (m) i ) equals E x h 1,b 1 (x) P ( ) X|Z (dx) − P X|Z (dx) y h 2,b 2 (y) P ( ) Y |Z (dy) − P Y |Z (dy) . Similar to (16), its absolute value can be upper bounded by Ed TV P ( ) X|Z=Z i , P X|Z d TV P ( ) Y |Z=Z i , P Y |Z log n. Following Cauchy-Schwarz inequality, we have that, Ed TV P ( ) X|Z=Z i , P X|Z d TV P ( ) Y |Z=Z i , P Y |Z ≤ Ed 2 TV P ( ) X|Z=Z i , P X|Z Ed 2 TV P ( ) Y |Z=Z i , P Y |Z = O(n −(κx+κy) ). This yields that, max b 1 ,b 2 E x h 1,b 1 (x) P ( ) X|Z (dx) − P X|Z (dx) y h 2,b 2 (y) P ( ) Y |Z (dy) − P Y |Z (dy) = O(n −(κx+κy) log n). Following similar arguments as in Step 1.1, we obtain that, I ( ) 3 − max b 1 ,b 2 E x h 1,b 1 (x) P ( ) X|Z (dx) − P X|Z (dx) y h 2,b 2 (y) P ( ) Y |Z (dy) − P Y |Z (dy) = O p (n −(κx+κy) log n). Therefore, we obtain that I ( ) 3 = O p (n −(κx+κy) log n). Step 1.4. Recall that σ 2 b 1 ,b 2 is defined by 1 n − 1 n i=1 h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } − GCM{h 1,b 1 (X), h 2,b 2 (Y )} 2 . With some calculations, it is equal to 1 n − 1 n i=1 h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } 2 h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } 2 − n n − 1 GCM 2 {h 1,b 1 (X), h 2,b 2 (Y )},(17) where the estimated conditional expectation E is computed using GANs. Consider the second term GCM{h 1,b 1 (X), h 2,b 2 (Y )} in (17). Following similar arguments as in Steps 1.1 and 1.3, we have that, max b 1 ,b 2 GCM{h 1,b 1 (X), h 2,b 2 (Y )} − GCM {h 1,b 1 (X), h 2,b 2 (Y )} = O p (n −(κx+κy) log n), where GCM {h 1,b 1 (X), h 2,b 2 (Y )} equals 1 n n i=1 h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 (X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) . Similar to (12), we can show that, max b 1 ,b 2 GCM {h 1,b 1 (X), h 2,b 2 (Y )} − GCM * {h 1,b 1 (X), h 2,b 2 (Y )} = O p n −1/2 log n . Consequently, we have that, max b 1 ,b 2 |GCM{h 1,b 1 (X), h 2,b 2 (Y )} − GCM * {h 1,b 1 (X), h 2,b 2 (Y )}| = O p n −1/2 log n . Since the function classes H 1 and H 2 are bounded, both GCM and GCM * are bounded by log n in absolute values. Consequently, max b 1 ,b 2 GCM 2 {h 1,b 1 (X), h 2,b 2 (Y )} − GCM * 2 {h 1,b 1 (X), h 2,b 2 (Y )} = O p n −1/2 log 3/2 n .(18) Next, consider the first term in (17). Note that it can be represented by n n − 1 1 L L =1   1 |I | i∈I ( ) h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } 2 h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } 2   . Similar to (12), we can show that, max b 1 ,b 2 1 |I | i∈I ( ) h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } 2 h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } 2 − E h 1,b 1 (X 1 ) − E{h 1,b 1 (X 1 )|Z 1 } 2 h 2,b 2 (Y 1 ) − E{h 2,b 2 (Y 1 )|Z 1 } 2 = O p (n −1/2 log 3/2 n). Following similar arguments as in Steps 1.1 and 1.3, we can show that, max b 1 ,b 2 E h 1,b 1 (X 1 ) − E{h 1,b 1 (X 1 )|Z 1 } 2 h 2,b 2 (Y 1 ) − E{h 2,b 2 (Y 1 )|Z 1 } 2 − E [h 1,b 1 (X 1 ) − E{h 1,b 1 (X 1 )|Z 1 }] 2 [h 2,b 2 (Y 1 ) − E{h 2,b 2 (Y 1 )|Z 1 }] 2 = O p (n −c ), for some constant 0 <c < 1/2. It follows that, max b 1 ,b 2 1 |I | i∈I ( ) h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } 2 h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } 2 − E [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] 2 [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] 2 = O p (n −c ), and henceforth, max b 1 ,b 2 1 n n i=1 h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i } 2 h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i } 2 − E [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] 2 [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] 2 = O p (n −c ). Combining this together with (18) yields that, max b 1 ,b 2 σ 2 b 1 ,b 2 − n n − 1 Var [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] = O p (n −c ). Then, we have that, min b 1 ,b 2 Var [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] ≥ c * , for some constant c * > 0. Therefore, we have that min b 1 ,b 2 σ 2 b 1 ,b 2 ≥ 2 −1 c * , with probability tending to 1. Step 2. We next consider the difference |T * − T * * |, and show that it is of the order O p (n −2κ log n). Denote by σ * 2 b 1 ,b 2 the variance estimator with { X Following similar arguments as in Steps 1.1 and 1.3, we can show that, max b 1 ,b 2 σ 2 b 1 ,b 2 − n n − 1 Var [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] = O p (n −c ), max b 1 ,b 2 σ * 2 b 1 ,b 2 − n n − 1 Var [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] = O p (n −c ). This completes the proof of Theorem 3. A.3 Proof of Theorem 4 In the proof of Theorem 3, we have already shown that T − T * = O p (n −(κx+κy) log n). Following similar arguments as in Step 1.4, we can show that T * − T * * * = O p (n −(κx+κy) log n), where T * * * = max b 1 ,b 2 σ −1 b 1 ,b 2 | n −1 n i=1 h 1,b 1 (X i ) − 1 M M m=1 h 1,b 1 (X (m) i ) h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) , where σ 2 b 1 ,b 2 = n n − 1 Var [h 1,b 1 (X) − E{h 1,b 1 (X)|Z}] [h 2,b 2 (Y ) − E{h 2,b 2 (Y )|Z}] . By (11), following similar arguments as in the proof regarding the term I 1 in Theorem 3, we can show that T * * * − T * * * * = O p (n −(κx+κy) log n), where T * * * * = max b 1 ,b 2 σ −1 b 1 ,b 2 | n −1 n i=1 [h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i }] h 2,b 2 (Y i ) − 1 M M m=1 h 2,b 2 (Y (m) i ) . Similarly, we can show that T * * * * − T 0 = O p (n −(κx+κy) log n), where T 0 = max b 1 ,b 2 σ −1 b 1 ,b 2 | n −1 n i=1 [h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i }] [h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i }] . Therefore, we have shown that T − T 0 = O p (n −(κx+κy) log n). Since κ x + κ y > 1/2, we have that, √ n( T − T 0 ) = o p (log −1/2 n).(19) Define a B 2 × B 2 matrix Σ 0 whose {b 1 + B(b 2 − 1), b 3 + B(b 4 − 1)}th entry is given by cov σ −1 b 1 ,b 2 [h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i }] [h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i }] , σ −1 b 3 ,b 4 [h 1,b 3 (X i ) − E{h 1,b 3 (X i )|Z i }] [h 2,b 4 (Y i ) − E{h 2,b 4 (Y i )|Z i }] . In the following, we show that, sup t Pr √ n T 0 ≤ t|H 0 − Pr ( N (0, Σ 0 ) ∞ ≤ t) = o(1).(20) When B is finite, this is implied by the classical weak convergence results. When B diverges with n, we require B = O(n c ) for some constant c > 0. By the definition of σ b 1 ,b 2 , the variance of σ −1 b 1 ,b 2 [h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i }] [h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i }] is bounded from above by (n − 1)/n. Moreover, combining the boundedness of the function spaces H 1 and H 2 together with the definition of σ b 1 ,b 2 yields that, σ −1 b 1 ,b 2 [h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i }] [h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i }] : b 1 , b 2 ∈ {1, · · · , B} are uniformly bounded from infinity by O(log n), with probability tending to 1. We can show that (20) holds. This implies that, σ −1 b 1 ,b 2 n −1/2 n i=1 [h 1,b 1 (X i ) − E{h 1,b 1 (X i )|Z i }] [h 2,b 2 (Y i ) − E{h 2,b 2 (Y i )|Z i }] is asymptotically normal with zero mean. Combining (20) together with (19) yields that, Pr √ n T ≤ t|H 0 ≥ Pr N (0, Σ 0 ) ∞ ≤ t − 0 log −1/2 n − o(1), Pr √ n T ≤ t|H 0 ≤ Pr N (0, Σ 0 ) ∞ ≤ t + 0 log −1/2 n + o(1),(21) for any sufficiently small 0 > 0, where the little-o terms are uniform in t. Following similar arguments as in Step 1.4 and Step 2 of the proof of Theorem 3, we can show that Σ − Σ 0 ∞,∞ = O p (n −c ) for some constantc > 0. Following similar arguments for (21), we have that, Pr √ n T ≤ t|H 0 ≥ Pr N (0, Σ) ∞ ≤ t − 2 0 log −1/2 n| Σ − o(1), Pr √ n T ≤ t|H 0 ≤ Pr N (0, Σ) ∞ ≤ t + 2 0 log −1/2 n| Σ + o(1), for any sufficiently small 0 > 0. Since the little-o terms are uniform in t ∈ R, we obtain that, As such, the distribution of our test statistic can be well-approximated by that of the bootstrap samples. This completes the proof of Theorem 4. A.4 Proof of Theorem 5 We break the proof into two steps. In Step 1, we show that, under H * 1 , there exist two neural networks functions f (X) ∈ H 1 and g(Y ) ∈ H 2 , such that I(f, g) = E[f (X) − E{f (X)|Z}][g(Y ) − E{g(Y )|Z}] = 0, In Step 2, we prove the power of our test approaches one, as the sample size diverges to infinity. Step 1. We first observe that the measure I(f, g) = E[f (X) − E{f (X)|Z}][g(Y ) − E{g(Y )|Z}] is continuous in f and g. That is, for any f 1 , f 2 ∈ L 2 X and g 1 , g 2 ∈ L 2 Y , the difference I(f 1 , g 1 ) − I(f 2 , g 2 ) decays to zero as both E|f 1 (X) − f 2 (X)| 2 and E|g 1 (X) − g 2 (X)| 2 decay to zero. Under H * 1 , there exist functions f * ∈ L 2 X and g * ∈ L 2 Y , such that I(f * , g * ) = 0. Without loss of generality, assume f * and g * are bounded. Otherwise, we can find sequences of bounded functions {f * n } n and {g * n } n that converge to f * and g * under L 2 -norm, respectively. As a result, we would have I(f * n , g * n ) = 0 for some n. By Lusin's theorem, we can find a sequence of bounded and continuous functions {f * * n } n , such that lim n Pr(f * * n (X) = f * (X)) = 0. By dominated convergence theorem, it follows that f * * n converges to f * under L 2 -norm. Similarly, we can find a sequence of continuous functions {g * * n } n , such that g * * n converges to g * under L 2 -norm. This together with the fact that I(f, g) is continuous in (f, g) implies that there exist some continuous functions f * * and g * * , such that I(f * * , g * * ) = 0. A key observation here is that, the class of neural networks have universal approximation property. Since the support of X and Y are bounded, it follows from Theorem 1 of Cybenko (1989) that the class of single-layered neural networks with sigmoid activation function is dense in the class of bounded, continuous functions with a compact support. As such, we can find some neural network functions f * * * and g * * * such that I(f * * * , g * * * ) = 0. We then argue that there must exist f ∈ H 1 and g ∈ H 2 , such that I(f, g) = 0. Otherwise, f * * * and g * * * can be represented as linear combinations of neural network functions in H 1 , H 2 with finitely many number of parameters, and we would have I(f * * * , g * * * ) = 0 as a result. This completes Step 1. Step 2. We first show that I(h 1,θ 1 , h 2,θ 2 ) is a Lipschitz continuous function of (θ 1 , θ 2 ). Note that h 1,θ 1 (X) and h 2,θ 2 (Y ) are Lipschitz continuous functions of θ 1 and θ 2 , respectively. For any θ 1,1 , θ 1,2 ∈ R d 1 , θ 2,1 , θ 2,2 ∈ R d 2 , we have that, |I(h 1,θ 1 , h 2,θ 2 ) − I(h 1,θ 1 , h 2,θ 2 )| ≤ |E[h 1,1 (X) − E{h 1,1 (X)|Z} − h 1,2 (X) + E{h 2,1 (X)|Z}][h 2,1 (Y ) − E{h 2,1 (Y )|Z}]| (22) + |E[h 1,2 (X) − E{h 1,2 (X)|Z}][h 2,1 (Y ) − E{h 2,1 (Y )|Z} − h 2,2 (Y ) + E{h 2,2 (Y )|Z}]| . Since the class of functions in H 2 are upper bounded by O( √ log n) with probability tending to 1, the right-hand-side of (22) is bounded from above by O(1)E |h 1,1 (X) − E{h 1,1 (X)|Z} − h 1,2 (X) + E{h 2,1 (X)|Z}| log n, with probability tending to 1. By Jensen's inequality, the above quantity can be further bounded from above by O(1)E |h 1,1 (X) − h 1,2 (X)| 2 log n ≤ K θ 1,1 − θ 1,2 2 log n, for some constant K > 0. Following similar arguments, we can show that the right-handside of (23) is bounded from above by K θ 2,1 − θ 2,2 2 √ log n, for any θ 2,1 and θ 2,2 , with probability tending to 1. To summarize, conditional on the event that H 1 and H 2 are bounded function classes, we have shown that |I(h 1,θ 1 , h 2,θ 2 ) − I(h 1,θ 1 , h 2,θ 2 )| ≤ K ( θ 1,1 − θ 1,2 2 + θ 2,1 − θ 2,2 2 ) log n. Consequently, for any sufficiently small > 0, there exists a neighborhood N = {(θ 1 , θ 2 ) : θ j −θ * j 2 ≤ δ log −1/2 n} for some constant δ > 0 around (θ * 1 , θ * 2 ), such that I(h 1,θ 1 , h 2,θ 2 ) ≥ for any (θ 1 , θ 2 ) that belongs to this neighborhood. Since (θ 1,b , θ 2,b ) are generated from the multivariate normal distribution, and the dimensions d 1 and d 2 are finite, the probability that (θ 1,b , θ 2,b ) belongs to this neighborhood is strictly greater than O(log −c 1 n) for some constant c 1 > 0. Since B = c 0 n c , the probability that at least one pair of parameters (θ 1,b 1 , θ 2,b 2 ) belongs to this neighborhood approaches one. Consequently, we have that, max b 1 ,b 2 GCM * {h 1,b 1 (X), h 2,b 2 (Y )} ≥ , with probability tending to 1. Following similar arguments as in the proof of Theorems 3 and 4, we can show that |T − max b 1 ,b 2 GCM * {h 1,b 1 (X), h 2,b 2 (Y )}| = o p (1), and T j = o p (1). Consequently, both probabilities Pr(T < /2) and Pr( T j ≥ /2) converge to zero. Therefore, the probability that the p-value is greater than α is bounded by the probability that Pr(T < /2), which converges to zero. This completes the proof of Theorem 5. Figure 1 : 1Illustration of the conditional independence test with double GANs. independent of the observations in I ( ) given Z i . Such a cross-fitting strategy allows us to derive the asymptotic properties of the test under minimal conditions on the generators. =Y O(n −κy ), (5) for some constants 0 < κ x , κ y < 1/2 and any ∈ [L], where P |Z denote the conditional distributions approximated via GANs trained on the -th subset of data samples. The next theorem summarizes this discussion.Theorem 3 Suppose the conditions in Theorem 2. Furthermore, suppose (5) holds. Then, T − T * = O p n −(κx+κy) log n . Figure 3 : 3The empirical type-I error rate of various tests under H 0 . Left panels: α = 0.05, right panels: α = 0.1. Top panels: Z is normal, bottom panels: Z is Laplacian. Figure 4 : 4The empirical power of various tests under H 1 . Left panels: α = 0.05, right panels: α = 0.1. Top panels: d Z = 100, bottom panels: d Z = 200. n − b n |. ≤ t|H 0 ) − Pr( N (0, Σ) ∞ ≤ t| Σ)| ≤ o(1) + sup t |Pr( N (0, Σ) ∞ ≤ t + 2 log −1/2 n| Σ) − Pr( N (0, Σ) ∞ ≤ t − 2 0 log −1/2 n| Σ)|.By Theorem 1 ofChernozhukov et al. (2017), the term on the second line can be bounded by O(1) 0 log 1/2 B log −1/2 n, where O(1) denotes some positive constant. Since B = O(n c ), log 1/2 B log −1/2 n = O(1). As 0 grows to zero, this term becomes negligible. Consequently, we obtain that, sup t Pr √ n T ≤ t|H 0 − Pr N (0, Σ) ∞ ≤ t| Σ ≤ o(1). Table 1 : 1The variable importance measures of the elastic net and random forest models, versus the p-values of the GCIT and DGCIT tests for the anti-cancer drug example.BRAF.V600E BRAF.MC HIP1 FTL3 CDC42BPA THBS3 DNMT1 PRKD1 PIP5K1A MAP3K5 EN 1 3 4 5 7 8 9 10 19 78 RF 1 2 3 14 8 34 28 18 7 9 GCIT <0.001 <0.001 0.008 0.521 0.050 0.013 0.020 0.002 0.001 <0.001 DGCIT 0 0 0 0 0 0 0 0 0 0.794 } m . Using (10), the difference between T * and T * * is upper bounded byUnder H 0 , similar to(12), we can show that,To show |T * − T * * | = O p (n −2κ log n), it suffices to show thatfor some constantc > 0. Since both σ −1 b 1 ,b 2 and σ b 1 ,b 2 are bounded away from zero, it suffices to show that max b 1 ,b 2 | σ 2 b 1 ,b 2 − σ * 2 b 1 ,b 2 | = O p (n −c ). The cancer cell line encyclopedia enables predictive modelling of anticancer drug sensitivity. 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In this paper, we prove Kirchberg-type inequalities for any Kähler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kählerian twistor operators.

10.1016/j.geomphys.2005.01.009

0707.0202

00ad517a7eab1adf55a6640f2d4c3901d35cfbf8

Eigenvalues of the transversal Dirac Operator on Kähler Foliations 2 Jul 2007 Georges Habib [email protected] InstitutÉlie Cartan Université Henri Poincaré B.P. 23954506Nancy I, Vandoeuvre-Lès-Nancy CedexFrance Eigenvalues of the transversal Dirac Operator on Kähler Foliations 2 Jul 2007 In this paper, we prove Kirchberg-type inequalities for any Kähler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kählerian twistor operators. Introduction On a compact Riemannian spin manifold (M n , g M ), Th. Friedrich [Fri80] showed that any eigenvalue λ of the Dirac operator satisfies λ 2 ≥ n 4(n − 1) S 0 , (1.1) where S 0 denotes the infimum of the scalar curvature of M. The limiting case in (1.1) is characterized by the existence of a Killing spinor. As a consequence M is Einstein. K.D. Kirchberg [Kir86] established that, on such manifolds any eigenvalue λ satisfies the inequalities On a compact Riemannian spin foliation (M, g M , F ) of codimension q with a bundle-like metric g M such that the mean curvature κ is a basic coclosed 1-form, S.D. Jung [Jun01] showed that any eigenvalue λ of the transversal Dirac operator satisfies λ 2 ≥ q 4(q − 1) λ 2 ≥    m+1 4m S 0 if m is odd,K ∇ 0 , (1.2) where K ∇ 0 = inf M (σ ∇ + |κ| 2 ), here σ ∇ denotes the transversal scalar curvature with the transversal Levi-Civita connection ∇. The limiting case in (1.2) is characterized by the fact that F is minimal (κ = 0) and transversally Einstein (see Theorem 3.1). The main result of this paper is the following: Theorem 1.1 Let (M, g M , F ) be a compact Riemannian manifold with a Kähler spin foliation F of codimension q = 2m and a bundle-like metric g M . Assume that κ is a basic coclosed 1-form, then any eigenvalue λ of the transversal Dirac operator satisfies: λ 2 ≥ m + 1 4m K ∇ 0 if m is odd, (1.3) and λ 2 ≥ m 4(m − 1) K ∇ 0 if m is even. (1.4) The limiting case in (1.3) is characterised by the fact that the foliation is minimal and by existence of a transversal Kählerian Killing spinor (see Theorem 4.3). We refer to Theorem 4.4 for the equality case in (1.4). We point out that Inequality (1.3) was proved by S. D. Jung [JK03] with the additional assumption that κ is transversally holomorphic. The author would like to thank Oussama Hijazi for his support. Foliated manifolds In this section, we summarize some standard facts about foliations. For more details, we refer to [Ton88], [Jun01]. Let (M, g M ) be a (p + q)-dimensional Riemannian manifold and a foliation F of codimension q and let ∇ M be the Levi-civita connection associated with g M . We consider the exact sequence 0 −→ L ι −→ T M π −→ Q −→ 0, where L is the tangent bundle of T M and Q = T M/L ≃ L ⊥ the normal bundle. We assume g M to be a bundle-like metric on Q, that means the induced metric g Q verifies the holonomy invariance condition, L X g Q = 0, ∀X ∈ Γ(L), where L X is the Lie derivative with respect to X. Let ∇ be the connection on Q defined by: ∇ X s =    π [X, Y s ] , ∀X ∈ Γ(L) , π ∇ M X Y s , ∀X ∈ Γ(L ⊥ ) , where s ∈ Γ(Q) and Y s is the unique vector of Γ(L ⊥ ) such that π (Y s ) = s. The connection ∇ is metric and torsion-free. The curvature of ∇ acts on Γ(Q) by : R ∇ (X, Y ) s = ∇ X ∇ Y s − ∇ Y ∇ X s − ∇ [X,Y ] s, ∀X, Y ∈ χ (M) . The transversal Ricci curvature is defined by: ρ ∇ : Γ(Q) −→ Γ(Q) X −→ ρ ∇ (X) = q j=1 R ∇ (X, e j ) e j . Also, we define the transversal scalar curvature : σ ∇ = q i=1 g Q ρ ∇ (e i ) , e i = q i,j=1 R ∇ (e i , e j , e j , e i ) , where {e i } i=1,··· ,q is a local orthonormal frame of Q and R ∇ (X, Y, Z, W ) = g Q (R ∇ (X, Y )Z, W ), for all X, Y, Z, W ∈ Γ(Q). The foliation F is said to be transversally Einstein if and only if ρ ∇ = 1 q σ ∇ Id, with constant transversal scalar curvature. The mean curvature of Q is given by: κ (X) = g Q (τ, X) , ∀X ∈ Γ(Q), where τ = p l=1 II (e l , e l ) , with {e l } l=1,··· ,p is a local orthonormal frame of Γ(L) and II is the second fundamental form of F defined by: II : Γ(L) × Γ(L) −→ Γ(Q) (X, Y ) −→ II (X, Y ) = π ∇ M X Y . We define basic r-forms by : Ω r B (F ) = {Φ ∈ Λ r T * M| X Φ = 0 and X dΦ = 0, ∀X ∈ Γ(L)} , where d is the exterior derivative and X is the interior product. Any Φ ∈ Ω r B (F ) can be locally written as 1≤j 1 <···<jr≤q β j 1 ,··· ,jr dy j 1 ∧ · · · ∧ dy jr , where ∂ ∂x l β j 1 ,··· ,jr = 0, ∀l = 1, · · · , p. With the local expression of basic r- forms, one can verify that κ is closed if F is isoparametric (κ ∈ Ω 1 B (F )). For all r ≥ 0, d (Ω r B (F )) ⊂ Ω r+1 B (F ) . We denote by d B = d| Ω B (F ) where Ω B (F ) is the tensor algebra of Ω r B (F ) . We have the following formulas: d B = q i=1 e ⋆ i ∧ ∇ e i and δ B = − q i=1 e i ∇ e i + κ , where δ B is the adjoint operator of d B with respect to the induced scalar product and {e i } i=1,··· ,q is a local orthonormal frame of Q. The transversal Dirac operator on Kähler Foliations In this section, we start by recalling some facts on Riemannian foliations which could be found in [GK91a], [GK91b], [AG97], [Jun01]. For completeness, we also scketch a straightforward proof of Inequality ((1.2)) established in [Jun01] and end by recalling well-known facts (see [Kir86], [Kir96], [Hij94a], [Hij94b], [JK03]) on Kähler spin foliations. On a foliated Riemannian manifold (M, g M , F ) , a transversal spin structure is a pair (SpinQ, η) where SpinQ is a Spin q -principal fibre bundle over M and η a 2-fold cover such that the following diagram commutes: SpinQ × Spin q SpinQ M SOQ × SO q SOQ - ? η⊗Ad ? η - - 3 The maps SpinQ × Spin q −→ SpinQ, and SOQ × SO q −→ SOQ, are respectively the actions of Spin q and SO q on the principal fibre bundles SpinQ and SOQ. In this case, F is called a transversal spin foliation. We define the foliated spinor bundle by: S (F ) := SpinQ × ρ Σ q , where ρ : Spin q −→ Aut (Σ q ) , is the complex spin representation and Σ q is a C vector space of dimension N with N = 2 [ q 2 ] , where [ ] stands for the integer part. Recall that the Clifford multiplication M on S (F ) is given by: M : Γ(Q) × Γ(S(F )) −→ Γ(S(F )) (X, Ψ) −→ X · Ψ. There is a natural Hermitian product on S (F ) such that, for all X, Y ∈ Γ(Q), the following relations are true: X · Ψ, Φ = − Ψ, X · Φ , X ( Ψ, Φ ) = ∇ X Ψ, Φ + Ψ, ∇ X Φ , ∇ Y (X · Ψ) = (∇ Y X) · Ψ + X · (∇ Y Ψ) , where ∇ is the Levi-Civita connection on S (F ) and Ψ, Φ ∈ Γ(S(F )). The transversal Dirac operator [GK91a,GK91b] is locally given by: D tr Ψ = q i=1 e i · ∇ e i Ψ − 1 2 κ · Ψ, (3.1) for all Ψ ∈ Γ(S(F )). We can easily prove using Green's theorem [YT90] that this operator is formally self adjoint. Furthermore, in [GK91b] it is proved that if F is isoparametric and δ B κ = 0, then we have the Schrödinger-Lichnerowicz formula: D 2 tr Ψ = ∇ ⋆ tr ∇ tr Ψ + 1 4 K ∇ σ Ψ, where K ∇ σ = σ ∇ + |κ| 2 and ∇ ⋆ tr ∇ tr Ψ = − q i=1 ∇ 2 e i ,e i Ψ + ∇ κ Ψ, with ∇ 2 X,Y = ∇ X ∇ Y − ∇ ∇ X Y , for all X, Y ∈ Γ(T M) . Denote by P the transversal twistor operator defined by P : Γ(S(F )) ∇ tr −→ Γ(Q * ⊗ S(F )) π −→ Γ(ker M), where π is the orthogonal projection on the kernel of the Clifford multiplication M. With respect to a local orthonormal frame {e 1 , · · · , e q }, for all Ψ ∈ Γ(S(F )), one has PΨ = q i=1 e * i ⊗ (∇ e i Ψ + 1 q e i · D tr Ψ + 1 2q e i · κ · Ψ). (3.2) For any spinor field Ψ, one can easily show that q i=1 e i · P e i Ψ = 0. (3.3) Now we give a simple proof of the following theorem: Theorem 3.1 [Jun01] Let (M, g M , F ) be a compact Riemannian manifold with a spin foliation F of codimension q and a bundle-like metric g M with κ ∈ Ω 1 B (F ). Assume that δ B κ = 0 and let λ be an eigenvalue of the transversal Dirac operator, then λ 2 ≥ q 4(q − 1) K ∇ 0 . (3.4) Proof. For all Ψ ∈ Γ(S(F )), we have using Identities (3.2), (3.3) (3.1), |PΨ| 2 = |∇ tr Ψ| 2 − 1 q |D tr Ψ| 2 − 1 q ℜ(D tr Ψ, κ · Ψ) − 1 4q |κ| 2 |Ψ| 2 . For any spinor field Φ, we have that (Φ, κ · Φ) = −(κ · Φ, Φ) = −(Φ, κ · Φ), so the scalar product (Φ, κ · Φ) is a pure imaginary function. Hence for any eigenspinor Ψ of the transversal Dirac operator, we obtain M |PΨ| 2 + 1 4q M |κ| 2 |Ψ| 2 = M |∇ tr Ψ| 2 − 1 q M λ 2 |Ψ| 2 , from which we deduce (3.4) with the help of the Schrödinger-Lichnerowicz formula. Finally, we can easily prove in the limiting case that F is minimal i.e. κ = 0, and transversally Einstein. A foliation F is called Kähler if there exists a complex parallel orthogonal structure J : Γ(Q) −→ Γ(Q) (dimQ = q = 2m). Let Ω be the associated Kähler, i.e., for all X, Y ∈ Γ(Q), Ω(X, Y ) = g Q (J(X), Y ) = −g Q (X, J(Y )). The Kähler form can be locally expressed as Ω = 1 2 q i=1 e i · J(e i ) = − 1 2 q i=1 J(e i ) · e i , and for all X ∈ Γ(Q), we have [Ω, X] := Ω · X − X · Ω = 2J(X). Under the action of the Kähler form, the spinor bundle splits into an orthogonal sum S(F ) = m ⊕ r=o S r (F ), where S r (F ) is an eigenbundle associated with the eigenvalue iµ r = i(2r −m) of the Kähler form Ω. Moreover, the spinor bundle of a Kähler spin foliation carries a parallel anti-linear map j satisfying the relations: j 2 = (−1) m(m+1) 2 Id, [X, j] = 0, (jΨ, jΦ) = (Φ, Ψ), and we have jΨ r = (jΨ) m−r . For all X ∈ Γ(Q), we have p + (X) · S r (F ) ⊂ S r+1 (F ) and p − (X) · S r (F ) ⊂ S r−1 (F ), where p ± (X) = X∓iJ(X) 2 . We define the operator D tr by D tr Ψ = q i=1 J(e i ) · ∇ e i Ψ − 1 2 J(κ) · Ψ. The local expression of D tr is independant of the choice of the local frame and by Green's theorem [YT90], we prove that this operator is self-adjoint. On a Kähler spin foliation, the operators D tr and D tr satisfy: [Ω, D tr ] = 2 D tr , We should point out that Equations (3.7), (3.8) and (3.9) are true under the assumptions that F is isoparametric and δ B κ = 0. Now we define the two operators D + and D − by D + = 1 2 (D tr − i D tr ) and D − = 1 2 (D tr + i D tr ). (3.10) Furthermore, D tr splits into D + and D − , and we have the two exact sequences: Γ(S m (F )) D − −→ . . . Γ(S r (F )) D − −→ Γ(S r−1 (F )) D − −→ . . . Γ(S 0 (F )), (3.11) Γ(S 0 (F )) D + −→ . . . Γ(S r (F )) D + −→ Γ(S r+1 (F )) D + −→ . . . Γ(S m (F )P (r) : Γ(S r (F )) ∇ tr −→ Γ(Q * ⊗ S r (F )) πr −→ Γ(ker M r ), where M r is the transversal Clifford multiplication defined by M r : Γ(Q * ⊗ S r (F )) −→ Γ(S r−1 (F )) ⊕ Γ(S r+1 (F )) X ⊗ Ψ r −→ p − (X) · Ψ r ⊕ p + (X) · Ψ r . For all r ∈ {0, . . . , m} and Ψ r ∈ Γ(S r (F )), we have P (r) Ψ r = q i=1 e * i ⊗ (∇ e i Ψ r + a r p − (e i ) · D + Ψ r + b r p + (e i ) · D − Ψ r ), (4.1) where D ± = D ± + 1 2 p ± (κ) with a r = 1 2(r+1) and b r = 1 2(m−r+1) . For any spinor field Ψ r ∈ Γ(S r (F )), we can easily prove q i=1 e i · P (r) e i Ψ r = 0. (4.2) Remark 4.2 For any non zero eigenvalue λ of D tr , there exists a spinor field Ψ ∈ Γ(S(F )) called of type (r, r + 1), such that D tr Ψ = λΨ and Ψ = Ψ r + Ψ r+1 , with r ∈ {0, · · · , m − 1}. By using (3.10), (3.11) and (3.12) it follows that D − Ψ r = D + Ψ r+1 = 0, D − Ψ r+1 = λΨ r , D + Ψ r = λΨ r+1 and Ψ r L 2 = Ψ r+1 L 2 . Proof. Let ϕ be an eigenspinor of D tr . There exists an r such that ϕ r does not vanish. Let Ψ = 1 λ D − D + ϕ r + D + ϕ r , one can easily get that D tr Ψ = λΨ. λ 2 ≥ m + 1 4m K ∇ 0 . (4.3) If Ψ is an eigenspinor of type (r, r + 1) associated with an eigenvalue λ satisfying equality in (4.3), then r = m−1 2 , the foliation F is minimal and for all X ∈ Γ(Q), the spinor Ψ satisfies ∇ X Ψ + λ 2(m + 1) (X · Ψ − iεJ(X) ·Ψ) = 0, (4.4) where ε = (−1) m−1 2 , andΨ := (−1) r (Ψ r − Ψ r+1 ) . As a consequence m is odd and F is transversally Einstein with non negative constant transversal curvature σ ∇ . Proof. For all Ψ r ∈ Γ(S r (F )), using Identities (4.1) and (4.2), we have |P (r) Ψ r | 2 = q i=1 |P (r) e i Ψ r | 2 = q i=1 (P (r) e i Ψ r , ∇ e i Ψ r ) = q i=1 (∇ e i Ψ r + a r p − (e i ) · D + Ψ r +b r p + (e i ) · D − Ψ r , ∇ e i Ψ r ). Finally we obtain, |P (r) Ψ r | 2 = |∇ tr Ψ r | 2 − a r |D + Ψ r | 2 − b r |D − Ψ r | 2 . (4.5) Let λ be an eigenvalue of D tr and let Ψ an eigenspinor of type (r, r + 1). Applying Equality (4.5) to Ψ r , one gets |P (r) Ψ r | 2 = |∇ tr Ψ r | 2 − a r λ 2 |Ψ r+1 | 2 − a r λℜ(Ψ r+1 , p + (κ) · Ψ r ) − a r 4 |p + (κ) · Ψ r | 2 − b r 4 |p − (κ) · Ψ r | 2 . By the Schrödinger-Lichnerowicz formula and by the fact that Ψ r and Ψ r+1 have the same L 2 -norms, we get M |P (r) Ψ r | 2 + a r 4 M |p + (κ) · Ψ r | 2 + b r 4 M |p − (κ) · Ψ r | 2 = M ((1 − a r )λ 2 − 1 4 K ∇ σ )|Ψ r | 2 − a r λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ). (4.6) and 0 ≤ M ((1 − b r+1 )λ 2 − 1 4 K ∇ σ )|Ψ r+1 | 2 + b r+1 λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ). (4.11) Hence if λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ) ≤ 0, then by (4.11) λ 2 ≥ 1 4(1 − b r+1 ) K ∇ 0 , The antilinear isomorphism j sends S r (F ) to S m−r (F ). This allows the choice of µ r to be non negative (i.e. r ≥ m 2 ) where µ r is the eigenvalue associated with Ψ r . Then a careful study of the graph of the function 1 1−b r+1 , yields (4.8). On the other hand if λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ) > 0. Applying Equation (4.5) to the spinor jΨ, which is a spinor of type (m − (r + 1), m − r), we find the same inequalities as (4.10) and (4.11), then λ 2 > 1 1 − a r K ∇ 0 4 . As before we can choose µ m−(r+1) ≥ 0 (i.e. r ≤ m 2 − 1). A careful study of the graph of the function 1 1−ar gives Inequality (4.8). Now we discuss the limiting case of (4.8). As we have seen, it could not be achieved if λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ) > 0, so only the other case should be considered. By (4.7), one has M |P (r+1) Ψ r+1 | 2 + a r+1 4 M |p + (κ) · Ψ r+1 | 2 + b r+1 4 M |p − (κ) · Ψ r+1 | 2 − b r+1 λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ) = (1 − b r+1 ) M ( m 4(m−1) K ∇ 0 − 1 4(1−b r+1 ) K ∇ σ )|Ψ r+1 | 2 . Since m m−1 = inf r≥ m 2 1 1−b r+1 , and the l.h.s. of (4) is non negative, we deduce that κ = 0, P r+1 Ψ r+1 = 0 and m m−1 = 1 1−b r+1 so r = m 2 . It remains to show that Equation (4.9) holds. For this, take X = e j where {e j } j=1,··· ,q is a local orthonormal frame. For r = m 2 , and by definition of the Kählerian twistor operators, for all j ∈ {1, · · · , q}, we obtain ∇ e j Ψ r+1 + λ q (e j − iJe j ) · Ψ r = 0. ) S 0 if m is even. D tr D tr + D tr D tr = Theorem 4. 3 3Let (M, g M , F ) be a compact Riemannian manifold with a Kähler spin foliation F of codimension q = 2m and a bundle-like metric g M with κ ∈ Ω 1 B (F ) and δ B κ = 0. Then any eigenvalue λ of the transversal Dirac operator, satisfies Similarly applying (4.5) to Ψ r+1 , we obtainwhere K ∇ σ = σ ∇ + |κ| 2 . In order to get rid the term λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ), since the l.h.s. of (4.6) and (4.7) are non negative, dividing (4.6) by a r and (4.7) by b r+1 then summing up, we find by substituting the values of a r and b r+1 ,. Now, we discuss the limiting case of Inequality (4.3). Dividing (4.6) by a r and (4.7) by b r+1 then summing up as before, and substituting a r , b r+1 and λ 2 by their values, we easily deduce that κ = 0, P (r) Ψ r = 0 and P (r+1) Ψ r+1 = 0. Hence by (4.6), we find that λ 2 = 1 4(1−ar ) σ 0 = m+1 4m σ 0 where σ 0 = inf M σ ∇ , then r = m−1 2 and m is odd. It remains to prove that Ψ satisfies (4.4). For r = m−1 2 , by definition of the Kählerian twistor operators, for all j ∈ {1, · · · , q}, we obtainand ∇ e j Ψ r+1 + λ m + 1 p + (e j ) · Ψ r = 0.Summing up the two equations, we get (4.4) for X = e j . Using Ricci identity in (4.4), one easily proves that F is transversally Einstein.If Ψ is an eigenspinor of type (r, r+1) associated with an eigenvalue satisfying equality in (4.8), then r = m 2 , the foliation F is minimal and Ψ satisfies for all X ∈ Γ(Q),Proof. Let Ψ an eigenspinor of type (r, r + 1) associated with any eigenvalue λ of the transversal Dirac operator D tr . Recalling Equalities (4.6) and (4.7), we have 0 ≤ M ((1 − a r )λ 2 − 1 4 K ∇ σ )|Ψ r | 2 − a r λ M ℜ(Ψ r+1 , p + (κ) · Ψ r ), (4.10) Stabilité du Caractère Kählérien Transverse. A El Kacimi Alaoui, B Gmira, Israel J. Math. 101A. El Kacimi Alaoui and B. Gmira, Stabilité du Caractère Kählérien Transverse, Israel J. Math 101 (1997), 323-347. Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalar-Krümmung. Th, Friedrich, Math. Nach. 97Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalar- Krümmung, Math. Nach. 97 (1980), 117-146. On Spectral Flow of Transversal Dirac Operators and a theorem of Vafa-Witten. J F Glazebrook, F W Kamber, Ann. Glob. Anal. Geom. 9J.F. Glazebrook and F.W. Kamber, On Spectral Flow of Transver- sal Dirac Operators and a theorem of Vafa-Witten, Ann. Glob. Anal. Geom. 9 (1991), 27-35. Transversal Dirac families in Riemannian foliations. 140[GK91b] , Transversal Dirac families in Riemannian foliations, Com- mun. Math. Phy. 140 (1991), 217-240. Eigenvalues of the Dirac Operator On Compact Kähler Manifolds. O Hijazi, Commun. Math. Phys. 160O. Hijazi, Eigenvalues of the Dirac Operator On Compact Kähler Manifolds, Commun. Math. Phys. 160 (1994), 563-579. Twistor Operators and Eigenvalues of the Dirac Operator. Proceedings of the conference on Quaternionic-Kähler Geometry. the conference on Quaternionic-Kähler GeometryTrieste, Twistor Operators and Eigenvalues of the Dirac Operator, Proceedings of the conference on Quaternionic-Kähler Geometry (Trieste), 1994. Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation. S D Jung, Tae Ho Kang, J. Geom. Phys. 45S.D. Jung and Tae Ho Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003), 75-90. The first eigenvalue of the transversal Dirac operator. S D Jung, J. Geom. Phys. 39S.D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys. 39 (2001), 253-264. An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. K D Kirchberg, Ann. Glob. Anal. Geom. 3K.D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Glob. Anal. Geom. 3 (1986), 291-325. The first eigenvalue of the Dirac operator on Kähler manifolds. J. Geom. Phys. 4[Kir90] , The first eigenvalue of the Dirac operator on Kähler man- ifolds, J. Geom. Phys. 4 (1990), 449-468. Properties of Kählerian twistor-spinors and vanishing theorems. Math. Ann. 293[Kir92] , Properties of Kählerian twistor-spinors and vanishing the- orems, Math. Ann. 293 (1992), 349-369. Killing Spinors on Kähler Manifolds. 11[Kir96] , Killing Spinors on Kähler Manifolds, Ann. Glob. Anal. Geom. 11 (1996), 141-164. Ph, Tondeur, Foliations on Riemannian manifolds. New YorkSpringerPh. Tondeur, Foliations on Riemannian manifolds, Springer, New York, 1988. Green's theorem on a foliated Riemannian manifold and its applications. S Yorozu, T Tanemura, Acta. Math. Hung. 56S. Yorozu and T. Tanemura, Green's theorem on a foliated Rieman- nian manifold and its applications, Acta. Math. Hung. 56 (1990), 239-245.

The τ-lepton plays an important role in the physics program at the Large Hadron Collider (LHC). It offers a powerful probe in searches for New Physics. Spin of τ lepton represents an interesting phenomenological quantity which can be used for the sake of separation of signal from background or in measuring properties of New Particles decaying to τ leptons. A proper treatment of τ spin effects in the Monte Carlo simulations is important for understanding the detector acceptance as well as for the measurements of τ polarization and τ spin correlations.The TauSpinner package represents a tool which can be used to modify τ spin effects in any sample containing τ leptons. Generated samples of events featuring τ leptons produced from intermediate state W , Z, Higgs bosons can be used as an input. The information on the polarization and spin correlations is reconstructed from the kinematics of the τ lepton(s) (also ν τ in case of W -mediated processes) and τ decay products. No other information stored in the event record is needed. By calculating spin weights, attributed on the event-by-event basis, it enables numerical evaluation of the spin effects on experimentally measured distributions and/or modification of the spin effects. With TauSpinner, the experimental techniques developed over years since LEP 1 times may be used and extended for LHC applications.We review a selection of simple distributions which can be used to monitor the τ spin effects (polarization and spin correlations) in leptonic τ decays and hadronic τ decays with up to three pions. The main purpose is to provide basic benchmark distributions for validation of spin content of the user-prepared event sample and to visualize significance of the τ lepton spin polarization and correlation effects. The utility programs, demonstration examples for use of TauSpinner libraries, are prepared and documented. New methods, with respect to previous publications, for validation of such an approach are provided. Other topics like methods to evaluate TauSpinner systematic errors or sensitivity of experimental distributions to explore spin effects are also addressed, but are far from being exploited. Results of semi-analytical calculations, and some effects of QED bremsstrahlung, are shown as well.This approach is of particular interest for estimation of the theoretical systematic errors for implementation of spin effects in so-called embedded τ lepton samples, where Z → µµ events are selected from data and muons are replaced with simulated τ leptons. Such embedding techniques are used in several analyses at LHC for estimating dominant background from Z → ττ process to the Higgs boson H → ττ searches.

10.5506/aphyspolb.45.1921

1402.2068

0125e850d8e12ad723e47eec76ee54f759930e68

Application of TauSpinner for studies on τ-lepton polarization and spin correlations in Z, W and H decays at LHC 10 Feb 2014 February 2014 A Kaczmarska Institute of Nuclear Physics, PAN ul. Radzikowskiego 152KrakówPoland J Piatlicki The Faculty of Physics, Astronomy and Applied Computer Science Jagellonian University Reymonta 430-059CracowPoland T Przedziński The Faculty of Physics, Astronomy and Applied Computer Science Jagellonian University Reymonta 430-059CracowPoland E Richter-Was Institute of Nuclear Physics, PAN ul. Radzikowskiego 152KrakówPoland Institute of Physics Jagellonian University Reymonta 430-059CracowPoland CERN PH-TH CH-1211Geneva 23Switzerland Z Was Application of TauSpinner for studies on τ-lepton polarization and spin correlations in Z, W and H decays at LHC 10 Feb 2014 February 2014 The τ-lepton plays an important role in the physics program at the Large Hadron Collider (LHC). It offers a powerful probe in searches for New Physics. Spin of τ lepton represents an interesting phenomenological quantity which can be used for the sake of separation of signal from background or in measuring properties of New Particles decaying to τ leptons. A proper treatment of τ spin effects in the Monte Carlo simulations is important for understanding the detector acceptance as well as for the measurements of τ polarization and τ spin correlations.The TauSpinner package represents a tool which can be used to modify τ spin effects in any sample containing τ leptons. Generated samples of events featuring τ leptons produced from intermediate state W , Z, Higgs bosons can be used as an input. The information on the polarization and spin correlations is reconstructed from the kinematics of the τ lepton(s) (also ν τ in case of W -mediated processes) and τ decay products. No other information stored in the event record is needed. By calculating spin weights, attributed on the event-by-event basis, it enables numerical evaluation of the spin effects on experimentally measured distributions and/or modification of the spin effects. With TauSpinner, the experimental techniques developed over years since LEP 1 times may be used and extended for LHC applications.We review a selection of simple distributions which can be used to monitor the τ spin effects (polarization and spin correlations) in leptonic τ decays and hadronic τ decays with up to three pions. The main purpose is to provide basic benchmark distributions for validation of spin content of the user-prepared event sample and to visualize significance of the τ lepton spin polarization and correlation effects. The utility programs, demonstration examples for use of TauSpinner libraries, are prepared and documented. New methods, with respect to previous publications, for validation of such an approach are provided. Other topics like methods to evaluate TauSpinner systematic errors or sensitivity of experimental distributions to explore spin effects are also addressed, but are far from being exploited. Results of semi-analytical calculations, and some effects of QED bremsstrahlung, are shown as well.This approach is of particular interest for estimation of the theoretical systematic errors for implementation of spin effects in so-called embedded τ lepton samples, where Z → µµ events are selected from data and muons are replaced with simulated τ leptons. Such embedding techniques are used in several analyses at LHC for estimating dominant background from Z → ττ process to the Higgs boson H → ττ searches. Introduction The successful research programme of LHC experiments requires the careful analysis of a multitude of different final states. A broad spectrum of interesting observables have been developed over the years [1,2,3]. One of the physics quantities, which can be used for such purpose, is the spin state of the produced τ leptons. For the processes, like the charged or neutral Higgs boson production and their respective important backgrounds from single W or single Z production, the spin effects can be measured [4,5] and also used for optimising signal from background separation. The spin of the final state τ lepton carries information on the τ production processes and manifests itself in distributions of the τ decay products. There is a multitude of τ decay channels which are accessible experimentally. The dominant ones are τ ± → l ± ν l ν τ , τ ± → π ± ν τ , τ ± → ρ ± ν τ . The decay channels listed above represent more than 2/3 of the total τ lepton decay width. In all these channels spin effects manifest themselves in the energy spectrum of the visible τ decay products, but for each channel differently. In the past [6,7], the channel τ ± → a ± 1 ν τ was also often proposed for the τ spin measurements, but for this case, more sophisticated distributions were necessary. We skip discussion of this channel from the study presented here. We focus our paper on describing strategy for validating τ spin effects 1 in the analyses at LHC experiments, which can be performed with the help of the TauSpinner [8,9] program. For that purpose we recall simple distributions used for evaluation of τ spin effects at the LEP time [7,10,11], the fractions of τ lepton energy carried by its observable decay products, and provide several methods to verify if for the particular sample the spin effects can be observed. These distributions can be used as a validation check if spin effects were properly transmitted to the generated sample or provide important information for feasibility studies in planning of the experimental analysis. One should keep in mind that at LHC although fractions of the τ lepton's energy carried by its observable decay products is not directly measurable and thus of a limited use for the experimental analyses, it was partly adapted to LHC applications already in [12]. The τ + τ − pairs (or τν τ pairs) carry only small fraction of the colliding proton momenta and the τ leptons energies differ substantially from the beam energies, contrary to how it was in the case at LEP 1 where τ energies where strongly constrained by the beam energies. One should however not underestimate their usefulness for different Monte Carlo studies, thanks to their simplicity and direct sensitivity to the spin effects. The paper is organized as follows. We recall main properties of the TauSpinner algorithm in Section 2 and in Section 3 explain details on the event samples used for providing numerical results. In Section 4, we describe the properties of τ lepton spin effects which may be of interest at LHC and how they are transmitted to τ decay products. In subsections we discuss the semi-analytical formula for spectra of leptons and single π's from τ decays and the effects of QED bremsstrahlung on these spectra, which can be also described in semi-analytical form. In the following subsections we describe plots we propose for benchmarking spin effects and provide examples and short discussion on the numerical results. In Section 5, we describe technical details of installation of those example programs. The summary, Section 6, closes the paper. The complete set of automatically generated benchmarking plots is collected in Appendices of a preprint version of our paper. This documents the output from new, more advanced set of example programs for using TauSpinner libraries. TauSpinner brief description The TauSpinner is a program associated with Tauola++, enabling calculation of weights for the previously generated or constructed by other means events, for example like with embedding technique, where Z → µ + µ − events are selected from data and muons are replaced by the τ-leptons with simulated decays [13]. The events must feature kinematics of τ lepton production and decay products, but information on partons from which intermediate resonance decaying to the τ's was produced is assumed to be unknown, and therefore is not used. The algorithm calculates for each event, from this information alone, a spin weight corresponding to a presumed configuration, for example Higgs or Z/γ * production and decay. The part of the weight related to the production of τ lepton pair (or τ lepton and associated with its production ν τ in case of e.g. W mediated processes) is calculated only from the four-momenta of the τ − τ (τ − ν τ ) lepton pair. The information on flavours of the initial state partons, quarks or gluons, are assumed not to be available and are attributed stochastically on the basis of matrix elements for parton level hard processes and parton density functions (PDFs) of the user choice. As default, processes mediated by single W , Z/γ * production are assumed. Alternative processes, like Higgs-mediated processes, can be used as well, but then the intermediate state has to be explicit in the event record. For each τ lepton the decay part of the weight is calculated from the matrix element of the corresponding τ decay channel, as classified by the algorithm. For this purpose, the matrix elements of Tauola++ library [14] are used. To calculate this weight the four-momenta of τ decay products need to be boosted to τ rest-frame. This requires careful treatment of possible rounding errors. The weight constructed with the help of TauSpinner, W T , is separated into multiplicative components: production (wt σ prod ), decay (wt τ ± Γ decay ) and spin correlation/polarization (wt spin ): W T = wt σ prod wt τ + Γ decay wt τ − Γ decay wt spin wt spin = R i, j h i τ + h j τ − .(1) In the present note we use only the last component of the weight W T , the spin weight wt spin , leaving interesting discussion on the other ones (wt σ prod , wt τ + Γ decay and wt τ − Γ decay ) aside to other applications, namely ref. [9]. The definition of the spin correlation matrix R i, j and polarimetric vectors for the decay of τ leptons h i τ + , h j τ − is rather lengthy and also well known. Therefore we refer the reader to our previous publications [10,12,15] for detailed definitions. For the discussion presented here, it is important to recall only that h i τ + , h j τ − are defined completely from the kinematics of the corresponding τ decay products and R i, j from the τ production kinematics. For every event 0 < wt spin < 4 by construction. The average of wt spin taken over the unconstrained event sample, up to statistical error equals to 1. With the wt spin weight one can evaluate on event-by-event basis spin effects transmitted from the production to the decay of τ leptons. By definition wt spin = 1 if those effects are omitted. Consequently, reweighting each event with W T = 1/wt spin can remove spin effects from generated sample. Also, the cases when only part of spin effects is taken into account, more specifically the spin correlation but no effects due to vector and axial couplings to the intermediate Z/γ * state 2 , can be corrected with the help of the appropriate weights. On the other hand, the spin effects can be also removed completely or the missing parts installed. Analysed event samples In this paper, we use the samples of events from pp collision at 8 TeV center-of-mass energies, featuring final states of τ lepton pairs with a mass close to that of the Z or W , generated with Pythia8 Monte Carlo [16]. These samples, each of 10M events, are stored in HepMC format [17]. Essentially default 3 initialisation parameters of Pythia, are used and no selection criteria are applied on the kinematics of outgoing τ's. The decays of τ leptons are generated with Tauola++ initialized with standard options 4 . The samples are generated including spin effects (polarisation and correlations): we will refer to these samples as original (orig, pol, polarized) samples. Starting from the original samples, depending on the studied effects, the spin effects are removed, with the help of TauSpinner weights: unpolarized (unweighted, unpol) samples are obtained. For some of the presented results we have created events originating from the spin-0 resonance of the mass of Z boson and couplings of the Higgs boson, denoted as Φ resonance. This was performed for convenience using Z → ττ events generated with Pythia8 Monte Carlo and τ leptons decaying with Tauola++ configured for the scalar resonance decay. Such events, denoted in this paper as Φ → ττ events, serve to illustrate spin effects between τ pairs originating from the decay of vector or scalar boson of the same mass and width. Please note that TauSpinner weights could be as well used to reweight complete Z → ττ events to represent Φ → ττ events, however this procedure introduces large statistical fluctuations due to the large spread of the weights when reweighting for spin effects from vector to scalar resonance decays. As a very interesting example, we point to the case when spin effects are removed completely from the original sample and are reinstalled back with only spin correlations but not spin polarization effects. While reintroducing only the spin correlations, it is sufficient to use information on the four-momenta of τ leptons and their decay products. Reintroducing effects from polarization requires information on structure functions of partons forming decaying resonance. In the case of embedded Z → ττ samples [13], it means introducing theoretical uncertainty due to assumed PDF's parametrisation to the sample, which appriory, was free from such uncertainties. The theoretical systematic error for such approach can be assigned by comparing spin effects calculated from approximation which rely on four-momenta of τ leptons and their decay products with the one exploring full hard scattering parton level amplitudes. For some auxiliary tests, discussed in Section 4.5, we have also used 1M events from Drell-Yan pp → Z/γ * → ττ process generated within the virtuality interval of m ττ = 1.0-1.5 TeV. We would like to stress an important feature of this strategy, for removing or reintroducing spin effects. It represents a solution for the cases when the sample of events, which feature τ lepton decays, is generated and processed with CPU-intensive simulation of the detector response. There is no need to prepare another reference sample with spin effects excluded, as this effect can be introduced by weights calculated by TauSpinner. The solution may be very helpful to estimate the sensitivity of the sample to its spin content as one can profit from using correlated events to reduce the effects from statistical fluctuations. Finally, with this strategy, at very low CPU-cost the spin effects can be evaluated to validate correctness of the generation of the sample under scrutiny. Physics motivation of test observables and numerical results In the analysis of experimental data it is important to evaluate effects due to particular theoretical phenomena, incorporated in tools used in preparation of the experimental distributions where at the same time all experimental effects are taken into account. Only then, one can decide if the studied effects are sizeable and can be distinguished from effects such as eg. background contamination. Distortion of the energy spectra of decay products due to polarization of τ leptons and spin correlations are examples of such effect. Due to the short lifetime and their parity-violating decays, τ leptons are the only leptons whose spin information is transmitted to the observed decay products kinematics. In the τ lepton decay the neutrino(s) escape detection, so complete kinematics of all decay products cannot be reconstructed experimentally. We assume however that τ decay channel can be correctly determined and for the sake of definition of the test distribution, that the fraction of τ energy carried by all observable decay products combined can be used. This leads to relatively simple semi-observables even if still does not explore all correlations and energy fractions of the secondary decay products (ρ, a 1 ...). It would be optimal to measure energies of individual τ decay products and use all of them simultaneously achieving then substantial gain in the sensitivity to the spin. That is the case, for example, in τ ± → ρ ± ν τ → ν τ π ± π 0 decay channel. The difference between π ± and π 0 energies is determined by the spin of ρ which carries information on the spin of τ. This type of constraints is desirable to be included in any realistic studies, but substantially adds to the complexity of the τ decay response to its spin. Such effects are of course taken into account in TauSpinner algorithms but are not explored with the distributions we study in this paper. Let us remind that precision tests of the Standard Model were performed at LEP 1, with significant and well documented effort on experimental, theoretical and computational levels [19,20]. In particular, manifestation of τ lepton polarization in its all main decay channels was carefully explored [7], in the context of measuring the intermediate Z boson properties. In our discussion, we recall some of the phenomenological and technical considerations of that time [10,11], which may be useful for the LHC applications as well, as shown in ref. [12]. In the LEP 1 analyses, observables were at first limited to the fraction of τ energy carried by its observable decay products, x. Such fractions could have been used directly at LEP 1 experiments because τ energy was essentially equal to the beam energy. In general, in m τ ≪ M Z,W limit, the fraction x is independent from the boost and remain the same in the rest frame of intermediate Z (or W ) or in the lab frame. That is why it is of potential interest as a first step in preparation of the spin measurements at LHC experiments, or to validate the correctness of spin implementation in the generated samples. Even though x is not reconstructed experimentally, knowledge of spin effect to distributions in this variable can be used rather straightforward to estimate how the distributions, of the actual interest, will be modified. Because of the simplicity and direct relation to properties of the decay matrix elements, the x variable is also a good choice for Monte Carlo benchmarks 5 on polarization and spin correlation effects. Semi-analytical formulae The pattern of the τ response to spin can be studied by the Monte Carlo methods through its decay, taking into account the complexity of multi-dimensional signatures. We return to this solution later in the paper. In some cases, simple analytical formulae are nonetheless available. They can be quite helpful to visualize the effects in an intuitive way, even if some details of the distributions would be neglected. In case of τ ± → π ± ν τ and τ ± → l ± ν l ν τ decays formulae for energy spectra for visible decay products, neglecting mass and QED bremsstrahlung effects have been known for a long time [10]. For τ ± → π ± ν τ it is 1 + P × (2x − 1)(2) and for τ ± → ℓ ± ν l ν τ (ℓ = e, µ) it reads as 5 3 − 3x 2 + 4 3 x 3 − P × − 1 3 + 3x 2 − 8 3 x 3 ,(3) where P denotes τ polarization and x is a fraction of τ ± energy carried by π ± or ℓ ± . These analytic forms of the spectra can be extended to the case when effects of radiative corrections are taken into account. Such parametrization of the spectra for decay products of polarized τ leptons are given in ref. [11], formulae 6 A3 and A4. For tests presented here, we assume that bremsstrahlung in decays is not taken into account in the event generation or that its effect can be neglected. Also, that the mass terms (non neglible for muons) can be neglected. Otherwise, the semi-analytical formulae would become much more complicated. With time, as it was the case of LEP [7], these effects may become of interest as well. It is straightforward to introduce to formulae (2), (3) effects due to bremsstrahlung; not only in τ decay itself, but also in decay of intermediate Z or W bosons. With above assumptions, simple formulae as (2), (3) can be fitted to the histograms for the energy fractions x, to evaluate effective P polarization and conclude if the particular sample feature the spin effect and/or if this effect is big enough to be statistically significant. In case semi-analytical formula is not available for the distribution, or distribution is distorted by kinematical selection, one can use for fitting the linear combination of reference spectra corresponding to pure left-handed and right-handed τ leptons. This template fit technique was already used by LEP experiments [7]. Such a reference spectra can be obtained with the help of Monte Carlo methods. Spin correlations and polarization monitoring plots It is expected that in most cases of interest at the LHC, τ leptons are produced through the decay of intermediate states of W , Z/γ * or H bosons. Because of the detector properties, fraction of τ ± energy carried by visible decay products, respectively x 1 (τ + ) and x 2 (τ − ), is a natural choice for the monitoring variables. The example spectra of x 1,2 for the specific τ decay channels are shown in Fig. 1 for Z → τ + τ − decays, separately for leptonic, single π and 2π decay channels. For construction of these plots the original events sample, discussed in Section 3, was used. The unpolarized spectra were obtained from the original sample, using weights calculated by TauSpinner. From the comparison of the two spectra the effect of polarization can be evaluated. As shown in Fig. 1, the spectra and their sensitivity to the spin vary dramatically depending on the τ decay channel. Negative polarization leads to harder spectra in case of τ → ℓν ℓ ν τ decays, and softer in case of τ → πν τ . The effect is of 20% at the very end of the spectra. In Fig. 1 results from the fits to respectively formulae (2) and (3) are given: P orig = −0.142 ± 0.003 (P unweighted = 0.0006 ± 0.003 unpolarized) for leptonic mode and P orig = −0.145 ± 0.002 (P unweighted = 0.0003 ± 0.002 unpolarized) for single π decay mode. Note that the differences between the results for the leptonic and π modes, even though formula (3) is missing mass corrections for muons are small. This is because the first bins of the histograms were excluded from fitting. The nominal average τ polarization of sample used, as estimated by appropriate method of TauSpinner, reads P = −0.144 ± 0.001. The effect from the missing muon mass contributes less than -0.005 to the polarization obtained from the fit to distribution in leptonic channel, both for the original sample P pol and for the spin unweighted sample P unpol . Statistical errors on the fit results correspond to samples of 10M events, as discussed in Section 3. Although fit results are not precise (they are biased by approximations of analytic formulae (2), (3)), some distinguishing power between polarized and unpolarized sample is demonstrated. Statistical errors on the fitted values are calculated by the ROOT fitting package [21]. The spin effects show up differently depending on the particular τ decay channel, as shown in Fig. 1. As a consequence the spin correlation manifests itself differently depending on the particular cases of the τ + and τ − decay channels. In Fig. 2a and Fig. 2b we provide the two-dimensional plots (lego plots) for Z → ττ events with both τ ± → π ± ν τ and the distributions of invariant mass M vis for the all visible τ pairs decay products combined, Fig. 2c and Fig. 2d. Note that to a good approximation M vis = Qx 1 x 2 where Q denotes mass of the τ-pair. In case of spin-0 state Φ, the fast-fast (x 1,2 > 0.5) and slow-slow (x 1,2 < 0.5) pairs of π ± are disfavoured, whereas in Z/γ * case the fast-slow and slow-fast configurations are less populous. Each configuration of τ decay channels feature different spin response pattern. We refer reader to series of the plots in Appendices collecting automatically created numerical results respectively for W , Z/γ * and Φ cases and different configurations of the τ decay modes. The Appendices (attached to the preprint version of our paper) represent examples of output from programs described in Section 5. Fits to energy fractions, radiative corrections and experimental cuts If there is no kinematical selection with resulting correction, mass corrections are neglected and QED bremsstrahlung in decays of τ and Z → ττ is not present, analytic formulae for distributions of x 1 , x 2 are given by simple polynomial expressions; formulae (2) and (3). These formulae can be used to fit the distributions and extract the value of polarization P. The fit can be performed for both, polarized and unpolarized distributions (the second ones constructed from appropriately weighted events). The values of the parameter P obtained from the fit (average τ polarization) for the polarized and unpolarized distributions enables simple diagnostic if a given τ decay channel is sensitive to the spin effects. Fit errors on the parameter P provide an estimate on the statistical sensitivity to the spin effects. One can evaluate statistical significance of the spin determination for the leptonic and single π decay modes 7 as can be seen from Fig. 1. As it was explained in ref. [11], deformation of the spectra due to radiative corrections can be as big as the effect of τ polarization itself. Also our fit results given in caption of (c) τ → ρν τ Figure 1: The spectra of visible τ decay energy normalized to τ energy, x 1,2 . Spin effects included (red, solid line) and neglected (green, dashed line with triangles). The τ leptons are produced through Z decay close to the mass peak. The τ polarization P is obtained from the fit to the distributions constructed from Z → ττ sample for polarized and unpolarized (unweighted) cases. For the fit, the first bin in τ → πν case and first five bins in τ → lν l ν τ case where mass effects would be the largest, were omitted. may substantially differ in size 8 . The main deformations are for x 1,2 close to 0 or 1. In general case, when distributions are not sufficiently well described by formulae (2) and (3), Monte Carlo methods can be used to obtain the spectra and dependence on the polarization P similarly to how it was done for the measurements of the Z couplings performed at LEP [7]. (b) τ → πν τ ) τ )/E( ρ =E( From benchmarks toward realistic experimental distributions. Numerical results presented above were prepared in an idealized case, where no experimental selection was applied to the analyzed samples. We have relied on the unobservable fractions of visible τ ± energies, x 1 and x 2 . This is well suited for testing Monte Carlo programs and detector simulation samples. In Fig. 4, we show the impact of spin effects on experimentally observable and sensitive to spin quantity E π − /E vis for τ − → π − π 0 π 0 ν τ decay channel. As one can conclude effects of the spin are sizeable. The effect can be evaluated using the TauSpinner unweighting algorithm. The a 1 decay channel is less suitable for testing the programs or simulations because of more complex interpretation of different spectra. It indicates (d) Φ → τ + τ − ; τ ± → π ± ν τ Figure 2: The case of Z(Φ) → τ + τ − ; τ ± → π ± ν τ . On plots (a, b): lego plots of E π + /E τ + × E π − /E τ − and on (c, d): invariant mass distributions of visible τ + and τ − decay products are shown. In case when spin effects are included red (solid line), otherwise green (dashed line with triangles) is used. For plots (a, c): the τ leptons produced through Z decay are used. For plots (b, d): τ + τ − pairs from spin-0 state Φ are used. All distributions are normalized to unity. (b) Φ → τ + τ − ; τ ± → π ± ν τ further applications, more oriented to realistic studies than the ones we have collected in Appendices for technical purposes. For applications in experimental data analysis, one should address possible complications like background contamination or limited acceptance. The template fit technique [7], convenient at LHC, as demonstrated in [4], can be used for spin in this case as well. Figure 4: Example plots for effects of spin in τ − → π − π 0 π 0 ν τ ; the E π − /E vis distribution. Case when spin effects are included is denoted by red (solid line), for spin effects excluded, green (dashed line with triangles) is used. Left hand side plot is for W − → τ −ν τ production, right hand side plot for Z/γ * → τ − τ + . Note large statistical fluctuations for unpolarized distributions in W case obtained with unweighting procedure. It is because in case of 100 % polarization like for W → τν τ decays, the spin weight wt spin can approach zero. Its inverse used for unweighting polarization, can therefore become arbitrarily large, resulting in (integrable) singularity of the distribution. These large fluctuations indicate the limitation for use of weights method to remove spin effects from already generated events. (b) τ → µν µ ν τ (γ)(b) Z/γ * → τ − τ + ;τ − → π − π 0 π 0 ν τ Consistency checks For the TauSpinner algorithm the questions of theoretical systematic error are of a great importance. We do not plan to review this aspect of the program development now. Some results are already documented in [8,9], but more detailed studies will be needed when the precision requirements will become more strict than presently. The work with explicit multi-leg QCD matrix elements of appropriate form, like in Ref. [22], will be mandatory. It has been known for a long time [23,24], that predictions for the Drell-Yan processes must lead to the dependency on the polar and azimuthal angles of outgoing leptons in the center-of-mass frame of decaying resonance in the form of second order spherical harmonics. This feature leads to the broad spectrum of possible applications, from validating implementations of higher order QCD corrections in the Monte Carlo programs, to the indirect measurement of the mass of the W boson [25]. For shown here new tests, it is important to notice that in the process of preparing spin weights, TauSpinner calculates all ingredients of the effective Born parton level cross section, as described in [10,11,12,15]. Predictions for other observables or quantities of phenomenological interest, such as quark level forward-backward asymmetry or probability of a given quark flavour to originate a particular hard process event, can be obtained when executing the code. Because of mentioned above properties of QCD, formulae for polarization and other quantities, remain essentialy as at LEP. If the studied sample is generated by the Monte Carlo program and physics history entries (flavours and momenta of quarks entering hard process) are stored, one can directly use this information to retrieve properties of the electroweak matrix elements and hadronic interactions of the studied events sample to validate precision of the TauSpinner algorithms. The four-momenta and flavours of the incoming quarks can be used to calculate parton level forward-backward asymmetry or rate of production from distinct quark flavour. These results can then be compared with the similar quantities estimated from the weights calculated by the TauSpinner algorithms using kinematics of the τ decay products only, providing very interesting test on the precision of TauSpinner algorithms. Unfortunately information on four momenta and flavours of incoming quarks is usually available only for Monte Carlo with parton showers based on the leading logarithm approach. At the next to leading logarithm level [26] such information may be available as well, but it is not necessarily the case. One should mention here that because of the spin-1 nature of objects decaying to pair of leptons, the angular distributions of τ leptons in the rest frame of τ pair are described by spherical harmonics, of at most the order of two. This explains why higher order QCD corrections, contributing higher than second order spherical harmonics, must be small [23]. We have prepared following tests, supplementary to the ones of subsection 4.2, which exploit physics history entries. These tests may be particularly interesting if some kind of inconsistency is found in the analyzed sample and one is debugging its origin: A Test of kinematic reconstruction. In TauSpinner, to evaluate τ scattering angle θ * , an algorithm described in [27] is used. Resulting cosθ * is compared with cos θ of scattering angle calculated from physics history entry of the event record. The difference of the two results is monitored. B Test of electroweak Born cross section. For the sample featuring physics history entries, the scattering angle of the outgoing lepton in the hard process can be calculated and appropriate angular distribution plotted separately for each flavour of incoming quarks. This distribution, in the leading log approximation have functional form (1 + cos 2 θ + A cos θ). Coefficient in front of cos θ, defining size of forward-backward asymmetry A FB , can be obtained from the fit of this function to cos θ distribution obtained from the analysed sample. The same coefficient can be calculated with the help of TauSpinner algorithm. This calculation uses as an input information of parton density functions (PDFs) which is convoluted with the parton level matrix-element of the hard process. The average value of the coefficient A (defining size of the forward-backward asymmetry A FB ) can be therefore obtained independently from the algorithm responsible for calculating spin weights and in particular scattering angles. The comparison of the results obtained from the fit to cos θ distribution constructed from physics history entries of the events on one side, and of TauSpinner internal calculation of A (when only PDFs and virtuality of τ-pair is used) on the other side, provides tests for effective Born parameters consistency in the analysed sample and TauSpinner code. Results of this test depend also on the choice of PDFs and on the correctness of the TauSpinner algorithm for reconstruction of PDFs arguments (fractions of proton momenta carried by partons) from the kinematics of the τ's (used is virtuality and pseudorapidity of the ττ system). An example of such comparison is given in Fig. 5 for the case of Drell-Yan events with virtuality in the range of 1 -1.5 TeV. The fit gives A= 1.617 +/-0.002 for up quarks and A= 1.692 +/-0.003 for down quarks. From the TauSpinner calculation using Born amplitude, value of the A parameter averaged over the same sample (calculated from Born cross section) read respectively 1.613 and 1.691 with negligible statistical error 9 . For other choices of the virtualities range agreement was found to be of a similar quality. C Test of PDFs. To some level, previous test can be complemented with the direct test of PDFs. Fraction of sample events (production rates) with hard process of particular incoming quark flavour can be compared with that fraction attributed by the TauSpinner algorithm. In the second case, every event contribute, but with the weight proportional to quark level total cross sections multiplied by respective PDFs. For the example shown in Fig. 5 we obtained respectively: rate of down quarks 0.217 (TauSpinner 0.216) and rate of up quarks 0.766 (TauSpinner 0.762). The results for tests [B] and [C] depend on the PDF's used internally by TauSpinner and on effective Born-level crosssection used at parton quark level. In case of [B] dominant contribution comes from the odd power of axial couplings, whereas in case [C] even powers of the axial couplings dominate the result. That is why the two tests are to a large extent independent. D Partial polarization test. An option that the events sample features spin correlation of the two τ leptons, but not fully the polarization effects due to production of intermediate state Z/γ * couplings is of practical interest when constructing so called τ embedded sample from Z → µµ events selected from data. For example the sample features dominant spin effect, due to vector nature of the intermediate state, but is free of systematic error of the electroweak effective Born and of incoming quark PDFs. If only angular dependence of the polarization is neglected, the systematic error due to PDFs on the spin effects is reduced and much smaller systematic error due to the effective electroweak Born parameters remain. In both cases, relatively small neglected effects can be evaluated and introduced with the help of TauSpinner weights, see Section 5.4. In the discussion of numerical results presented above, samples without any kinematical cuts were used. However, one may be interested to test how our algorithm will perform if only a particular class of events, for example of high p T configurations only, is used. All the tests listed above can be then performed using sub-samples defined by the particular set of cuts. In such cases, the validity of the TauSpinner algorithms and of the parton shower algorithm used for the sample generation can be explored in more exclusive phase-space regions. Reference plots Let us now discuss briefly the large collection of automatically created plots prepared by our testing programs 10 . For the preprint version of our paper such plots are collected into rather lengthy appendices for the W , Z/γ * and spin-0 resonance Φ. The distributions shown, depend on the particular sample used. We have grouped the figures for each τ decay channel (case of W ) or for each pair of τ decay channels (case of Z) separately. For leptonic and single π decay channels results of the fits to spectra (2) or (3) are given. The input samples feature complete longitudinal spin effects. The TauSpinner weights were used to unpolarize the sample. For the case of Φ → ττ events the Z → ττ sample was used, but τ decays were regenerated instead of reweighted for better numerical stability. For each type of decaying resonance, we give specification of the sample used for the respective set of plots, reporting the number of the analyzed events with the decomposition into particular (pair of) τ decay channels and initialization of the generator used for sample preparation. We also plot the control distribution of invariant mass of ττ (τν τ ) system. This sanity plot verifies if the sample consists of events at the resonance peak or if substantial contribution from low energy or very high mass tails is included in the sample as well. Spin effects are different if events are taken at, above or below the Z peak. Afterwards come collection of plots to large extend following layouts of Fig. 1 and Fig. 2: • In case of W → τν τ decay only the one-dimensional distribution of the energy fraction carried by τ lepton is plotted, comparing the case of polarized and unpolarized samples. In the captions of the plots, similarly as in Fig. 1, the fitted value of the τ polarization P is given as a measure of spin sensitivity of analyzed samples. • In the case of the Z/γ * mediated processes and for each particular combination of τ + and τ − decay channel, the set of histograms are collected. The first is the two-dimensional lego plot constructed from the fractions of energies of τ + and τ − carried by their corresponding observable decay products. Analogous lego plot is also shown for the case when spin effects are removed with the help of weights calculated by TauSpinner. Ratio of the two distributions is given in the lego plot of the second row. It demonstrates the strength of the spin effect. On the right hand side of this lego plot the one-dimensional histogram for invariant mass, of all visible products of τ + and τ − combined, is given. It provides a convenient way of representing spin correlation effects in case of smaller samples, which may be insufficient to fill the two-dimensional distributions. The last two plots show single τ + and τ − decay product spectra respectively (each plot containing original sample, sample with modifications due to TauSpinner weights and their ratio). Spectra are normalized to unity. In the captions of the plots, see Fig. 1, the fitted value of τ polarization P is given as a measure of spin sensitivity of analyzed samples. Note that the groups of plots for the cases when decay channels for τ + and τ − are simply interchanged, coincide up to permutation of axes, unless some cuts are introduced by the user. • In case of Φ-mediated process, set of plots analogous to Z/γ * -mediated processes are given. The proposed set of benchmark plots can be extended further with the help of provided validation programs, in particular for the cases of partial implementation of polarization and spin correlations effects. Respective systematic errors can be evaluated. Technical details For the purpose of this paper, a directory TauSpinner/examples/applications 11 has been added to the previous distributions of Tauola++. It contains several tools used to produce the plots for this paper and to obtain necessary results. It was also extended with several tests that help validate TauSpinner. If Tauola++ is configured with all prerequisites needed to compile TauSpinner package, as well as TauSpinner examples 12 , compiling these additional programs should not require any further setup and can be done by executing make in applications directory. The applications directory In the following subsection we will briefly describe the sub-directories for this package and their use. Generating plots The main program, applications-plots.cxx, generates plots which are latter included in the pdf file (like of Appendix A). It uses the same algorithm as the one used in tau-reweight-test.cxx; part of the examples for TauSpinner included in Tauola++ tar-ball starting from version of November 2012. In this example code, input file events.dat is processed and for each event W T weight is calculated. The set of histograms is filled with weighted (to remove spin effects) and not weighted events, separately for each τ decay mode or τ pair decay mode combination. Histograming and plotting is done using the ROOT library [21] (also fits are performed with the help of RooFit library). This program can be used to recreate plots in the Appendices. For this, a datafile with W and Z which decay into τ's is needed. Note that since the template LaTeX file is prepared for both W and Z samples, this program can be executed on a single sample file containing both types of events or on two samples with separate W and Z events 13 . Only channels τ → µν µ ν, τ → eν e ν, τ → πν and τ → ρν are analysed. To run the program: • make sure that ROOT configuration is available through root-config, • execute make in TauSpinner/examples/applications directory, 11 In Tauola++ v1.1.4, released on 12 Dec 2013, this directory was called TauSpinner/examples/tauspinner-validation. All subsequent directories and programs have been renamed following the new convention. In particular, directories: applications-plots-and-paper, applications-rootfiles, applications-fits was respectively called tauspinner-validation-results, tauspinner-validation-plots, tauspinner-validation-fit and programs applications-plots.cxx, applications-comparison.cxx, applications-fits.cxx were called tauspinner-validation-plots.cxx, tauspinner-validation-comparison.cxx, tauspinner-validation-fit.cxx. While the naming of programs and subdirectories changed, the content of the programs remained the same. 12 Up-to-date instructions can be found on the Tauola++ website in the documentation to the most-recent version of the package [28]. 13 This program does not produce histograms stored in Appendix B. These plots require change of the PDGID of the Z boson so TauSpinner can calculate weight as if the intermediate boson is Higgs. This change is omitted from the example provided with the distribution tar-ball for simplicity. • verify that settings in file applications-plots.conf are correct, including the path to input file 14 , • execute ./applications-plots.exe applications-plots.conf. A set of plots will be generated in the directory indicated by the configuration file (the default one is applications-plots-and-paper) and a breakdown of the τ decay channels found in the sample will be written at the end of running the program. If the input file contains both W and Z decays, two sets of plots will be generated, each accompanied with summary of the W and Z events properties. The program also saves all histograms created during processing time to out.root file. This file can be used to archive the results for further analysis or to add fits to the plots. Adding fits The code for adding fits is provided in the subdirectory applications-fits. It is built along with other programs when executing make in applications directory. This tool adds fits to the histograms generated by applications-plots.exe using the formulae (2) and (3), results of the fits are stored in the rootfile. See the README file in this directory for details on how input files are processed. This program uses rootfiles from subdirectory applications-rootfiles. They are specified in the default configuration file applications-fits.conf as the input files of this program. The resulting plots, with added fit information on polarization, will be stored in applications-plots-and-paper directory. Previously generated plots will be overwritten. This can be changed in the configuration file with path to the output directory. As mentioned in Section 4.3, the fit can be applied not to the whole range but to the interval (x 1 , x 2 ), that is why an option to perform fits only in the limited range of [x min ,x max ] has been provided in the code and is controlled by the configuration file. Recreating figures 3 and 4 The subdirectory applications-rootfiles contains rootfiles of histograms necessary to reproduce all plots shown in our paper. These rootfiles are used by applications-fits.exe. Histograms for the plots that are not part of the Appendices are also stored in the rootfiles. Executing make will invoke code to generate the plots for Figures 3 and 4. Note that generation of these rootfiles requires different setup and different data samples than for any other plots. While necessary changes are straightforward, including such options would add to the already complex structure of the validation programs, thus they were skipped in the distribution. Additional tests and tools Two additional subdirectories: • test-bornAFB • test-ipol were added for further validation of the TauSpinner library. These tests are somewhat peripheral to the main topic of the paper, thus they were only documented in the README files of the corresponding sub-directories. The result of the first test is briefly discussed in Section 4.5, while the second one has not been presented here. It is, however, included in the package as a validation test of TauSpinner options (Ipol = 0, 1, 2, 3). The applications directory contains additional programs: • The hepmc-tauola-redecay.cxx, while not being an example for TauSpinner, can be used to process existing input file and remove τ decays substituting them with new ones generated by Tauola++. This tool can be used to generate unpolarized τ decays needed to verify different TauSpinner options (see Section 5.4). Note, that as with Tauola++, generation options are limited by the available information stored in the data files. • The applications-comparison.cxx, uses two input files. First one is considered as a reference. For the second one TauSpinner weights are used. The same set of histograms is produced for both input files and compared afterwards. This program can be used to validate TauSpinner options, as for example in case E described in Section 5.4. Details of how to use both programs are described in README of the directory. 14 Note that example file examples/events.dat can be used to verify if the program compiles and runs correctly. However, it contains only a sample of 100 Z → τ + τ − → π + π − ν τ ν τ events. Generating pdf file The subdirectory applications-plots-and-paper contains the LaTeX files, as well as all other files necessary to prepare Appendices of this paper. Executing make in this directory generates the pdf file as of our paper. Text of Appendices is stored in files: appendixA.tex and appendixB.tex. The user can thus easily re-attach results of the program run to the documentation of his own project starting from the template user-analysis.tex; the make user-analysis will include Appendices into short user-analysis.pdf. Final remarks It is possible to redo, for the sake of documenting results of one own tests, all figures and other numerical results of the Appendix A (that is also of user-analysis.pdf). In case the physics assumptions are substantially different than the one used for the present paper, shapes of the obtained distribution may differ as well. In every case the following step have to be followed: 1. generate a sample of W and Z decays to τ; τ decaying to µ, e, π and ρ; 2. run applications-plots.exe on this sample; 3. run applications-fits.exe on the resulting rootfile and store the output in applications-plots-and-paper subdirectory; 4. execute make in applications-plots-and-paper subdirectory. Further details on each of these steps, including more technical details on the output and input files, are given in the distribution tar-ball and in the README files located in TauSpinner/examples/applications directory and all of the subdirectories. The numerical results of whole paper can also be reconstructed. Scripts for most of the necessary operations are prepared and documented elsewhere in the paper or in README files. Input file formats Essentially any HepMC [17] file (saved in HepMC::IO_GenEvent format) can be processed 15 Files with events stored in different format can be either converted to HepMC or interfaced using methods described in TauSpinner documentation and used in the default example tau-reweight-test.cxx. Note that only the file applications-plots-and-paper/input-file-info.txt should be updated by the user with the information on the event sample processed. All other text files will be updated by the appropriate tools described in previous section. The content of these text files is included in the output file of pdf format, as shown in Appendix A.1. Rounding error recovering algorithm The τ leptons stored in data files can be ultra-relativistic. This may cause problems for the part of algorithm recalculating matrix elements for τ decays. For our example, there was no problem with errors from rounding numbers, but in general such problems are expected. The following correcting algorithm is prepared: 1. For each stable τ decay product its energy is recalculated from the mass and momentum. 2. The four-momentum of the τ is recalculated from the sum of four-momenta of its decay products. 3. The algorithm performs check if resulting operation doesn't introduce sizeable modifications, incompatible with rounding error recovery. If it does, a warning message is printed. This may indicate other than rounding error, difficulty with the production file. For example, some decay products not stored (eg. expected as non-observable soft photons). The algorithm is located in file applications/CorrectEvent.h. An example of its use is provided in applications/hepmc-tauola-redecay.cxx. By default, this algorithm is turned off. Package use cases This package can be used to validate several TauSpinner options representing different applications of TauSpinner. Such tests include, but are not limited to: (A) Applying longitudinal spin effects: adding spin effect to an unpolarized sample using weights WT calculated by TauSpinner. For this purpose, set Ipol=0 in the configuration file. (B) Removing spin effects: removing spin effects from the polarized sample using weights calculated by TauSpinner. This is the default option used for our figures. The weight 1/WT instead of WT should be used. Note, that regardless of whether Ipol=0 or 1, TauSpinner works in the same manner. The two options are distinguished at the level of the user program only (use 1/W T instead of W T to reweight events), as shown in our demo. (C) Working on the input file with spin correlations but without polarization: initialize TauSpinner with Ipol=2. In this case WT will represent correction necessary for implementation of the full longitudinal spin effects. Analogously, if the sample feature τ polarization, but polarization is missing dependence on the τ leptons directions, TauSpinner should be initialized with Ipol=3 and the missing dependence can be corrected with calculated weight WT. (D) Replacing spin effects of Z/γ * with the Higgs-like spin-0 state spin correlations: This could be realized with weights ( 1 wt spin to remove spin effects of Z/γ * times wt Φ spin to introduce spin effects of Φ) without modification of event kinematics. Due to large spread in the weights, this method introduces large statistical fluctuations. Alternatively, this can be realised by regenerating τ decays with Tauola++ configured for the τ pair originating from the scalar state resonance. (E) Validation: test is similar to test [A] , we apply spin effect to a sample without polarization. However, for this test we take the polarized sample and replace its τ decays by new, non-polarized ones using Tauola++. This allows to test different Ipol options as mentioned in test (C). It requires different setup and use of two input files. The TauSpinner should be executed on this new sample. The result should be then compared with the ones from original sample. Tools required to perform these steps are described in Section 5.1.4. The results of the tests B and D are presented in the Appendices of this paper. The details of test C are described in README of applications/test-ipol subdirectory. Tools, that can be adapted to perform tests A and E, have been provided as well (see Section 5.1.4). We have successfully performed tests A-E on samples generated with Pythia8 + Photos++ + Tauola++ (in some cases Pythia8 alone). Satisfactory results, of similar quality as discussed in our paper, sections 4.3 and 4.5 were always found. Further details for all of the cases listed above are given in the distribution tar-ball. Summary In this paper we presented the use of TauSpinner libraries for testing effects resulting from spin correlations and polarization of τ leptons in processes at LHC featuring W , Z and H decays. New example programs were developed and incorporated into program distribution tar-ball. The purpose of these programs is to analyze spin effects using information on the kinematics of τ decay products of events stored in a file. Moreover, they provide a convenient tool for validating the TauSpinner algorithms. As an important use case, this set of programs provides a method to evaluate systematic error on spin effect implementation in so called embedded τ samples, an experimental technique used for analyses at LHC experiments. For the purpose of presenting methodology, a set of kinematical distributions was selected and the physics properties of these distributions were explained on some example plots. Event samples featuring τ lepton decays of W and Z production at LHC energies were generated. The weights calculated by TauSpinner algorithms were used then to remove the effects due to polarization of decaying τ. The sample featuring no spin effects was also created on flight for comparisons. For studying the spin effects of the spin-0 intermediate state Φ, the Z → ττ sample was modified, namely the τ leptons decay products were removed and the decays were generated again, using spin density matrix of H → τ + τ − decay. The complete set of benchmark plots from analyses of these samples, graphical output from our program, is collected in the Appendices of the preprint version of our paper. What is shown in Appendix A are plots for W and Z decays, from executing our program on a single event file. With the additional run, one can prepare a set of plots shown in Appendix B, for the case when instead of spin effects from intermediate Z/γ * the Higgs couplings were used for the preparation of events file. As expected, effects of removed polarisation are present and spin correlations are of opposite sign to that of the Z/γ * case. The details of the program installation and use were given. Our example provides test that algorithms of TauSpinner used for calculating spin weights are equivalent to the ones in Tauola++, the τ decay library used for creation of initial samples. We have also demonstrated how physics history entries of event samples can be used to provide validation tests for algorithm of effective Born level kinematic reconstruction and cross-section calculations used in TauSpinner 16 . The easiest case to understand are the spin effects of τ ± → π ± ν τ decays. The spectra are affected by spin as discussed in ref. [10,12], that is why we have frequently used this decay channel for the example plots of the paper. Due to Z polarization there is clearly identifiable slope for the π ± 's energy spectrum. The spin correlations of the two τ's disfavour configurations when one of the π is hard and the other one is soft. For the Φ → ττ case the spin correlations effect is opposite. For other τ decay channels, effects of spin are more complex and we have presented results in the main body of the paper only for the τ's of polarization originating from Z → ττ decays. Pattern of spin correlations and single τ polarization effect depends on the τ decay channel. One can easily notice from the lego plots that it will be affected by the kinematical selection on the other τ decay products as well, biasing the observed τ polarization. One should keep in mind when performing above tests that features of presented distributions depend strongly on the analyzed event sample. If τ leptons predominantly originate from the decays of Z, W , or H, of virtualities close to the resonance peaks, and with the spin effects taken into account, the distributions should be similar to the ones presented in this paper. However, it might not always be the case. For results presented in this paper parameters of the electroweak interactions and the PDFs used were carefully tuned between events generation and TauSpinner analysis codes. If this is not the case particular patterns of discrepancies may appear. We will investigate this point in the future. To evaluate sensitivity to the spin the average τ lepton polarization fits to the histograms of the simple analytic distributions are provided. The effects of QED bremsstrahlung or mass corrections are not incorporated into the functions used for fits in case of leptonic τ decay. At LEP [11], it was shown that they may be of the similar size and shapes as polarization effects. We have also discussed that the alternative to analytical formule, Monte Carlo based template distributions are useful for fits and evaluation of spin effects. From the discussion presented in this paper we left aside discussion on functionality of matrix-element re-weighting like the one described in ref. [9] and recently being upgraded even further. This approach may be helpful to evaluate if spin effects present in a given sample can be helpful to distinguish different production mechanisms, ones combined with the effects of production distributions. In our present study, we however concentrated on discussing spin effects only. A Benchmark results The Appendix A with its subsections represents output 17 from a single execution of applications-plots.exe program. The numerical results of the Appendix are obtained from the events file generated by run of Pythia8 combined with Tauola++, details of initialization are given later in the Appendix. For the fits, all 100 bins except the first five in τ → lν l ν τ case and first one in τ → πν case, were used 18 . The figures in the main part of our paper are taken from the ones of Appendices, with somewhat improved graphic style for better readability. For the plots of Fig. 1 A.1 Input files The list of files and additional information on generation of events used for the plots: Input files from Pythia8165 + Tauola++ v1.1.4 + Photos++ v3.54 Sample size: 10Mevents Z, 10MEvents W+ W-pair, Equal BR for tau -> e, mu, pi, rho A.2 W decays The invariant mass distribution and break-down on the τ decay channels are shown for τν τ -pair originating from W decay. The spin effects should not depend on the virtuality of the W intermediate state, but this may be not the case if New Physics samples are studied. A.3 Z decays The invariant mass distribution and break-down on the τ decay channels are shown for ττ-pair originating from Z decay. The spin effects strongly depend on the virtuality of the Z/γ * intermediate state. Events were generated explicitly requiring virtuality of Zγ * within 88-94 GeV window. Minor contamination from some other process is nevertheless observed (we have not traced it back). In this section, we monitor effects of Higgs couplings for spin correlations and compare it with case of unpolarized τ's. As expected, one can observe strong opposite sign as in Z case, spin correlations, and there is no effect on single τ decay product spectra. The invariant mass distribution and break-down on the τ decay channels are shown for τν τ -pair originating from Φ decay. For this purpose Z → ττ events were used, with τ decay products removed and τ leptons decayed again using configuration of Tauola++ like for H → ττ decay. For the fits, all 100 bins except the first five in τ → lν l ν τ case and first one in τ → πν case, were used. Fig. 3support this observation. Depending on whether for calculation of x 1,2 bremsstrahlung photons are combined with the lepton or not, effect on τ polarization obtained from the fit of formulae(3) ) Z → τ + τ − ; τ ± → π ) Z → τ + τ − ; τ ± → π ± ν τ Figure 3 : 3Example plots for the effects of QED bremsstrahlung in leptonic decays of τ. Spectra of visible τ decay energy normalized to τ energy, x 1,2 , are shown. Spin effects and QED bremsstrahlung are excluded for green (dashed line), for spin effects included but QED bremsstrahlung excluded red (solid line) and finally both spin effects and QED bremsstrahlung included blue (dotted line). Distributions constructed for Z → ττ decays. Left-hand plot is for τ decays to electron, righthand plot for τ decays to muons. For the plots, histograms were rebinned. For the fits of histograms, all 100 bins except the first five (that is exactly as in fits forFig. 1) were used. The expression (3) was fitted to the spectra. For the e channel, P = 0.004 ± 0.002, −0.141 ± 0.002, −0.015 ± 0.002 respectively for unpolarized, polarized and polarized with bremsstrahlung effect included cases. For the µ channel, the analogous result reads P = −0.009 ± 0.002, −0.153 ± 0.002, −0.124 ± 0.002. ) W − → τ −ν τ ;τ − → π − π 0 π Figure 5 : 5Differential distribution in the hard scattering angle cos θ calculated from physics history entries in the event record, the superimposed fit of (1 + cos 2 θ + A cos θ) red line is shown. Histograms are normalized to8 3 (the integral of fit function in range [−1, 1]). The vituality of the ττ pair was restricted to the range 1-1.5 TeV. Photos configuration: --------------------------------Tauola::setRadiation(false); Figure A. 1 : 1Invariant mass distribution of τν τ -pair originating from W decay. Figure A. 2 :Figure A. 3 :Figure A. 4 : 234Fraction of τ energy carried by its visible decay products18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified.A.2.2The energy spectrum: τ ± → π ± Fraction of τ energy carried by its visible decay products18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified. Fraction of τ energy carried by its visible decay products. Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified. FigureFigureFigureFigureFigure A. 9 :FigureFigure A. 11 :FigureFigure A. 13 : 91113-, e-vs mu+, e+ 1251897 mu-, e-vs pi+ 1248820 pi-vs mu+, e+ 624890 pi-vs pi+ 1249843 mu-, e-vs rho+ 1249136 rho-vs mu+, e+ 624102 pi-vs rho+ 625707 rho-vs pi+ 624553 rho-vs rho+ A.3.1 The energy spectrum: τ − → µ − , e − vs τ + → µ + , e + pr od uc t ch ar ge -A.5: Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.2 The energy spectrum: τ − → µ − , e − vs τ + → π + pr od uc t ch ar ge -A.6: Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.3 The energy spectrum: τ − → π − vs τ + → µ + , e + pr od uc t ch ar ge -A.7: Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. A.8: Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.5 The energy spectrum: τ − → µ − , e − vs τ + → ρ + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.6 The energy spectrum: τ − → ρ − vs τ + → µ + , e + pr od uc t ch ar ge -A.10: Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.7 The energy spectrum: τ − → π − vs τ + → ρ + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.8 The energy spectrum: τ − → ρ − vs τ + → π + pr od uc t ch ar ge -A.12: Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions.A.3.9 The energy spectrum: τ − → ρ − vs τ + → ρ + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra. Red line is for original sample, green line is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified. B Φ decays: Z decay sample but with Higgs couplings used for spin Figure B. 1 :Figure B. 2 :Figure B. 3 :Figure B. 4 :Figure B. 5 :Figure B. 6 :Figure B. 7 :Figure B. 8 :Figure B. 9 : 123456789tau tau mass) in Phi (Z) decays log10(tau tau mass) in Phi (Z) decays Phi (Z) Events: 9959720 Total 2492521 mu-, e-vs mu+, e+ 1244305 mu-, e-vs pi+ 1245480 pi-vs mu+, e+ 621568 pi-vs pi+ 1243913 mu-, e-vs rho+ 1245332 rho-vs mu+, e+ 622021 pi-vs rho+ 622007 rho-vs pi+ 622573 rho-vs rho+ B.1 The energy spectrum: τ − → µ − , e − vs τ + → µ + , e + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.2 The energy spectrum: τ − → µ − , e − vs τ + → π + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.3 The energy spectrum: τ − → π − vs τ + → µ + , e + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.4 The energy spectrum: τ − → π − vs τ + → π + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.5 The energy spectrum: τ − → µ − , e − vs τ + → ρ + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.6 The energy spectrum: τ − → ρ − vs τ + → µ + , e + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.7 The energy spectrum: τ − → π − vs τ + → ρ + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.8 The energy spectrum: τ − → ρ − vs τ + → π + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra18 . Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified with whenever available superimposed result for the fitted functions. B.9 The energy spectrum: τ − → ρ − vs τ + → ρ + pr od uc t ch ar ge -Fractions of τ + and τ − energies carried by their visible decay products: two dimensional lego plots and one dimensional spectra. Red line (and left scattergram) is sample with spin effects like of Higgs, green line (and right scattergram) is for modified sample after removing polarisation using TauSpinner weights, black line is ratio original/modified. , the bottom-left plots Figs. A.5, A.8 and A.13, were selected. The lego plot (and visible mass plots) of Fig. 2 are shown as the top-left lego plots (and mid-right plots) of Figs. A.8 and B.4 respectively. Lines starting from '#' are comments. Empty lines are ignored # # Do not change the order of the parameters in this file! # Lines starting from '#' are comments. Empty lines are ignored # # Do not change the order of the parameters in this file! # ################################################################# # Input files (one per line) ../applications-rootfiles/out.W.root ../applications-rootfiles/out.Z.root ../applications-rootfiles/out.H.rootPhotos::setExponentiation(true); Photos::suppressAll(true); Photos::forceBremForDecay(2,23,15,-15); Photos::forceBremForDecay(2,-24,15,-16); Photos::forceBremForDecay(2,24,-15,16); Photos::createHistoryEntries(true,3); -- Fits performed using ROOT v5.34/14, RooFit v3.59 Configuration file used by the program: ################################################################# # Config file for tauspinner-validation-plots.exe # ################################################################# # ################################################################# # Input file (the example works on files in HepMC format) events.dat # Output directory applications-plots-and-paper # Number of events to be processed # 0 = process all events from the sample 0 # LHAPDF dataset MSTW2008nnlo90cl.LHgrid # TauSpinner -CMS energy (used in PDF calculation) check your units 8000 # TauSpinner -Ipol value (is the input sample polarized?) 1 # for further explanations on program options see our papers: # arXiv:1201.0117 arXiv:1212.2873 Configuration file used by the fitting program: ################################################################# # Config file for applications-fits.exe # ################################################################# # # Output directory ../applications-plots-and-paper # Fitting range for leptons begins at 0.05 # Fitting range for leptons ends at 1 # Fitting range for pions begins at 0.01 # Fitting range for pions ends at 1 The transverse τ spin effects are not yet installed in TauSpinner. Motivation for such natural extension is if directions of π 0 can be separated from the one of π ± . For introduction of such an extension, complete spin density matrix of the produced τ-pair has to be provided. This also requests some rather straightforward tests of kinematics. The other parts of the algorithm are already prepared. To τ leptons or to incoming quarks only.3 The configuration parameters are detailed in Appendix A.1, For the generation of multiphoton final state radiation in Z and W decays Photos Monte Carlo[18] was used.4 In general QED bremsstrahlung was not taken into account in τ decays. Only for preparation ofFig. 3, τ leptons were re-decayed with QED bremsstrahlung ON or OFF. Our definitions follow refs.[10,11,12]. At LEP1 time the x fraction was the actual observable; for the collisions of the center-of-mass energy close to the Z peak the τ lepton energies were close to the beam energies. The x variable received a lot of phenomenological attention.6 Unfortunately this review is known to have typing mistakes. It would be interesting to check how this observation is preserved in case when experimental cuts are applied. It is important to verify if such spectra with radiative corrections will be useful. Other effects, such as experimental cuts, may change shapes as well, making such theoretical improvements of a minimal interest only. In this case we concentrate on matching the electroweak parameters in initialization of Pythia and TauSpinner, hence the initial state hadronic effects were switched off. We have checked, that if more complete treatment is used, quality of the agreemet between A and A fit is degraded by ∼0.01, but shapes of the distributions become more complicated. These programs are included in the TauSpinner distribution tar-ball. See Section 5 for more details. Note however that it is user responsability to verify that HepMC file contains events with correctly structured information for TauSpinner to find outgoing τ leptons and their decay products. Text is adopted for the sake of paper preparation in a minor way only.18 If bremsstrahlung in τ decays would be present, the result of the fits would differ. For example, for leptonic channel (seeFig.3), the shift of ∼0.07 would be present. AcknowledgmentsWe thank Prof. Erez Etzion for inspiring comment that bremsstrahlung effects given in semi analytical form (as in the LEP time CALASY program) may be of interest for the LHC applications as well. We thank Dr. Ian Nugent for careful and critical reading of our manuscript and for discussions. We thank Dr. Will Davey for discussions as well.16In our paper this method was used for comparisons with Pythia8 results, which are of the leading logarithm precision level similarly as TauSpinner. Further extensions of this method is possible. Properties of factorization of exact QCD multiparton amplitudes need then to be used in the physics history entries of events stored by the reference generators. As one can see[22], such properties are present for QCD amplitudes, as it was the case of multiphoton amplitudes of QED which were used for τ lepton spin effects in KKMC Monte Carlo[29]of LEP applications. . JINST. 38004CMS Collaboration, JINST 3 (2008) S08004. . JINST. 38005LHCb Collaboration, JINST 3 (2008) S08005. . 1204.6720Eur.Phys.J. C72. 2062ATLAS Collaboration, Eur.Phys.J. C72 (2012) 2062, 1204.6720. . I Deigaard, CERN-THESIS-2012-091I. Deigaard, CERN-THESIS-2012-091. . J L Harton, Nucl. Phys. Proc. Suppl. 40J. L. Harton, Nucl. Phys. Proc. Suppl. 40 (1995) 463-473. . A Heister, ALEPH Collaborationhep-ex/0104038Eur. Phys. J. 20ALEPH Collaboration, A. Heister et al., Eur. Phys. J. C20 (2001) 401-430, hep-ex/0104038. . Z Czyczula, T Przedzinski, Z Was, 1201.0117Eur.Phys.J. C72. Z. Czyczula, T. Przedzinski, and Z. Was, Eur.Phys.J. C72 (2012) 1988, 1201.0117. . S Banerjee, J Kalinowski, W Kotlarski, T Przedzinski, Z Was, 1212.2873Eur.Phys.J. 732313S. Banerjee, J. Kalinowski, W. Kotlarski, T. Przedzinski, and Z. Was, Eur.Phys.J. C73 (2013) 2313, 1212.2873. . S Jadach, Z Was, Acta Phys. Polon. 15483S. Jadach and Z. Was, Acta Phys. Polon. B15 (1984) 1151, Erratum: B16 (1985) 483. . P Eberhard, B Van Eijk, J Fuster, S Jadach, A Lutz, C89-05-08.2P. Eberhard, B. van Eijk, J. Fuster, S. Jadach, A. Lutz, et al., CERN-EP-89-129, C89-02-20.1, C89-05-08.2. . T Pierzchala, E Richter-Was, Z Was, M Worek, hep-ph/0101311Acta Phys.Polon. 32T. Pierzchala, E. Richter-Was, Z. Was, and M. Worek, Acta Phys.Polon. B32 (2001) 1277-1296, hep-ph/0101311. . N Davidson, G Nanava, T Przedzinski, E Richter-Was, Z Was, 1002.0543Comput.Phys.Commun. 183N. Davidson, G. Nanava, T. Przedzinski, E. Richter-Was, and Z. Was, Comput.Phys.Commun. 183 (2012) 821-843, 1002.0543. . S Jadach, Z Was, R Decker, J H Kü, Comput. Phys. Commun. 76361S. Jadach, Z. Was, R. Decker, and J. H. Kü, Comput. Phys. Commun. 76 (1993) 361. . T Sjostrand, S Mrenna, P Skands, 0710.3820Comput. Phys. Commun. 178T. Sjostrand, S. Mrenna, and P. Skands, Comput. Phys. Commun. 178 (2008) 852-867, 0710.3820. . M Dobbs, J B Hansen, Comput. Phys. Commun. 134M. Dobbs and J. B. Hansen, Comput. Phys. Commun. 134 (2001) 41-46, https://savannah.cern.ch/projects/hepmc/. . N Davidson, T Przedzinski, Z Was, 1011.0937N. Davidson, T. Przedzinski, and Z. Was, 1011.0937. . G Altarelli, R Kleiss, C Verzegnassi, CERN-89-08-V-1G. Altarelli, R. Kleiss, and C. Verzegnassi, CERN-89-08, CERN-89-08-V-1. . G Altarelli, R Kleiss, C Verzegnassi, CERN-89-08-V-3G. Altarelli, R. Kleiss, and C. Verzegnassi, CERN-89-08, CERN-89-08-V-3. . A Van Hameren, Z Was, 0802.2182Eur.Phys.J. 61A. van Hameren and Z. Was, Eur.Phys.J. C61 (2009) 33-49, 0802.2182. . E Mirkes, Nucl.Phys. 387E. Mirkes, Nucl.Phys. B387 (1992) 3-85. . E Mirkes, J Ohnemus, hep-ph/9406381Phys.Rev. 50E. Mirkes and J. Ohnemus, Phys.Rev. D50 (1994) 5692-5703, hep-ph/9406381. . T Aaltonen, CDF Collaboration Collaboration1307.0770Phys.Rev. 8872002CDF Collaboration Collaboration, T. Aaltonen et al., Phys.Rev. D88 (2013) 072002, 1307.0770. . R Kleiss, Nucl. Phys. 347R. Kleiss, Nucl. Phys. B347 (1990) 67-85. . Z Was, S Jadach, Phys. Rev. 411425Z. Was and S. Jadach, Phys. Rev. D41 (1990) 1425. . N Davidson, G Nanava, T Przedzinski, E Richter-Was, Z Was, Tauola C++, N. Davidson, G. Nanava, T. Przedzinski, E. Richter-Was, and Z. Was, TAUOLA and TAUOLA C++ Interface source code and documentation aviable from http://wasm.web.cern.ch/wasm/C++.html or http://tauolapp.web.cern.ch. . S Jadach, B Ward, Z Was, hep-ph/9912214Comput.Phys.Commun. 130S. Jadach, B. Ward, and Z. Was, Comput.Phys.Commun. 130 (2000) 260-325, hep-ph/9912214.

Electron transport in realistic physical and chemical systems often involves the non-trivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a non-unitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here we present a general algorithm for implementing the action of any non-unitary operator on an arbitrary state on a quantum device. We show that any quantum operator can be exactly decomposed as a linear combination of at most four unitary operators. We demonstrate this method on a two-level system in both zero and finite temperature amplitude damping channels. The results are in agreement with classical calculations, showing promise in simulating non-unitary operations on intermediate-term and future quantum devices.

10.1103/physrevlett.127.270503

2106.12588

98239e83934ce9cb6e34fa934e6e3d3298740beb

Quantum Simulation of Open Quantum Systems Using a Unitary Decomposition of Operators Anthony W Schlimgen Department of Chemistry The James Franck Institute The University of Chicago 60637ChicagoILUSA Kade Head-Marsden John A. Paulson School of Engineering and Applied Sciences Harvard University 02138CambridgeMAUSA Leeann M Sager Department of Chemistry The James Franck Institute The University of Chicago 60637ChicagoILUSA Prineha Narang John A. Paulson School of Engineering and Applied Sciences Harvard University 02138CambridgeMAUSA David A Mazziotti Department of Chemistry The James Franck Institute The University of Chicago 60637ChicagoILUSA Quantum Simulation of Open Quantum Systems Using a Unitary Decomposition of Operators (Dated: Submitted June 23, 2021) Electron transport in realistic physical and chemical systems often involves the non-trivial exchange of energy with a large environment, requiring the definition and treatment of open quantum systems. Because the time evolution of an open quantum system employs a non-unitary operator, the simulation of open quantum systems presents a challenge for universal quantum computers constructed from only unitary operators or gates. Here we present a general algorithm for implementing the action of any non-unitary operator on an arbitrary state on a quantum device. We show that any quantum operator can be exactly decomposed as a linear combination of at most four unitary operators. We demonstrate this method on a two-level system in both zero and finite temperature amplitude damping channels. The results are in agreement with classical calculations, showing promise in simulating non-unitary operations on intermediate-term and future quantum devices. Introduction.-The time evolution of an electronic state is critical to predicting many important physical phenomena including exciton transport [1], molecular conductivity [2], chemical catalysis [3], quantum phase transitions [4,5], and magnetization [6]. Often these processes involve non-trivial interactions with a large environment, requiring the definition and treatment of open quantum systems [7][8][9][10][11][12][13][14][15]. Recent research has considered the simulation of open quantum systems on quantum devices [16][17][18][19][20]. While the complexity of these systems makes them ideal for quantum simulation, their dependence on non-unitary time propagation presents a challenge for quantum computers whose fundamental operations (gates) are unitary [16,17]. A variety of possible approaches to non-unitary time evolution are emerging [18,[21][22][23], including dilation methods [16,17,19,24,25], imaginary time evolution [20,26,27], time-dependent variational methods [28], and duality quantum algorithms [29]. Some of these methods, however, require the factorization of the non-unitary matrix [16,17,19,24,25] or the solution of a potentially large system of linear equations with assumptions of locality [20,26,27]. In this Letter we develop and implement a general quantum algorithm to simulate non-unitary time evolution on a quantum computer in which we decompose any quantum operator into the linear combination of at most four unitary operators. The decomposition is exact in the limit that a small parameter approaches zero. We show that this -limit can be efficiently evaluated by Richardson's extrapolation [30]. We demonstrate this algorithm on a two-level system in both zero and finite temperature amplitude damping channels. This formalism provides a completely general approach for simulating both unitary and non-unitary evolution of quantum systems. In contrast to other linear combination of unitaries approaches [23,29,31], our algorithm uses strictly unitary operators to represent the action of a non-unitary operator on a quantum state, and then performs quantum addition of known, prepared states. Furthermore, the formalism can be used in other contexts beyond evolution of quantum dynamics, such as in Hamiltonian simulation in quantum chemical applications, where the operators of interest are also often non-unitary. Theory.-The time evolution of an open quantum system can be performed using the Kraus formalism in the operator sum form, ρ(t) = i M i ρM † i ,(1) where the M i are Kraus maps corresponding to different environmental channels and ρ(t) is the system density matrix [6,32]. In general, the density matrix of a quantum system can be written as the outer product of the wavefunction with itself, ρ(t) = |ψ(t) ψ(t)|.(2) Any operator M can be decomposed into a Hermitian and anti-Hermitian component, S = 1 2 (M + M † )(3) and A = 1 2 (M − M † ),(4) such that M = S + A. The Hermitian and anti-Hermitian components can be written as the sum of two exponential unitary operators each, implemented on a quantum device as a sum of unitary operators. Importantly, we note that the decomposition results in strictly unitary operators, and that the only approximation arises in the choice of the expansion parameter which does not effect the unitarity of the implemented operators. The -limit can be accelerated by Richardson's deferred approach to the limit, also known as Richardson's extrapolation [30]. While a small is needed to obtain accurate S and A matrices, a sufficiently large is needed to resolve these matrices from the noise. Because the expansions of both S and A in in Eqs. (5) and (6) are even with their largest errors on the order of O( 2 ), the density matrix in Eq. (1), constructed from M = S + A, has an even expansion in with a similar error. Higher-order expansions of the density matrix can be generated by Richardson's extrapolation. The extrapolation employs a deferred approach to the limit where the density matrices at two different values of are used to obtain a better approximation to the → 0 limit. Formally, we have the following expression for the extrapolated density matrix S = lim →0 i 2 (e −i S − e i S ),(5)A = lim →0 1 2 (e A − e − A ),(6)ρ(0) ≈ ρ( 1 ) − ρ( 2 )r 2 1 − r 2(7) where 1 > 2 and the ratio r is equal to 1 / 2 . Unlike the input density matrices, the extrapolated ρ(0) is accurate to O( 4 ). Higher-order density matrices can be obtained by iterating Richardson's extrapolation with r 2 replaced by r n where the integer n denotes the order of the error of the input density matrices. Methods.-In order to implement the decomposition M = S + A on a quantum computer, we utilize the following quantum state preparation, | ψ = R · U |ψ ,(8) for, U =     S m 0 0 0 0 −S p 0 0 0 0 −A m 0 0 0 0 A p     ,(9) where S m = −S † p = ie −i S A m = A † p = e − A .(10) Each S and A operator is size 2 n , where n is the number of qubits, thus U is size 4x2 n . The operator U propogates the wavefunction, while the rotation matrix R adds the components of the prepared states in the appropriate manner, R = 1 2 r −r r r ,(11) for, r = I −I I I .(12) Finally, the populations are rescalled by classically after measurement on the quantum device. Since is generally less than 1, our approach requires a relatively large number of samples, as discussed below. Since U is diagonal, we can implement this operation using a quantum multiplexor, or uniformly controlled gate [33][34][35][36][37][38][39]. We utilize the uniformly controlled gate in Qiskit to propogate our initial wavefunctions [40,41]. We implement the quantum addition for the n = 3 case with an X gate on one qubit, and Hadamard gates on each of the other two qubits of the circuit. An example of this implementation is shown in Figure 1. We can lower the qubit complexity of the algorithm by noting that U can be implemented in parallel as 6 operators of size 2x2 n , which are block diagonal pairs of the S and A operators in Eq. (10). In this approach, each S and A operator of size 2 n is also implemented resulting in an additional 4 circuits. The measured matrix elements are subsequently summed classically to reconstruct the appropriate density matrix elements. If an operator is either purely Hermitian or anti-Hermitian, only one circuit is required. We use this reduced complexity implementation to generate the data shown here. An example circuit for a 2x2 n operator is shown in Fig. 2. While the three-qubit state preparation yields promising results for no-error and low-error simulations (see Supplemental Information), we observed significant drift using these circuits on the real, NISQ devices employed. As such, this paper utilizes the two-qubit preparation suited for these near-term, noisy devices. The use of a small in the expansions in Eqs. (5) and (6) requires a relatively large number of shots for accurate statistics on a quantum device. Our algorithm directly measures the diagonal elements of the propagated density matrix, scaled by . Since the expansion in Eqs. (5) and (6) generally requires < 1, and the populations are normalized to 1, these matrix elements are generally numerically small. In order to acheive good statistics in the population dynamics, many samples are therefore required on noisy devices. This drawback can be overcome by either repetition of the algorithm on shot-limited devices, or by using systems with larger available shot counts. An alternative solution to the issue of large shot counts is the use of a Richardson extrapolation based on simulations using larger as discussed above. We use a two-point Richardson extrapolation to generate populations dynamics on quantum device, as well as using a simulator. The extrapolation allows us to achieve accurate dynamics with significantly lower shot counts using 's on the order of unity ( = 1.15, 1.00), compared to simulations with smaller . Results.-To benchmark this method, we consider a twolevel system in a general amplitude damping channel. The Kraus operators are given by, M 0 = √ λ 1 0 0 √ e −γt M 1 = √ λ 0 √ 1 − e −γt 0 0 M 2 = √ 1 − λ √ e −γt 0 0 1 M 3 = √ 1 − λ 0 0 √ 1 − e −γt 0(13) where γ is the decay rate, and λ = 1 1+e − 1 K B T accounts for temperature dependence of the equilibrium state [42,43]. In the zero temperature limit, λ = 1 and the Kraus operator evolution reduces to requiring only M 0 and M 1 . For the finite temperature case, infinite temperature proves to be a good approximation for room temperature, where λ = 0.5 [42,43]. In this case, all four of the Kraus operators in Eq. 13 are required to capture the accurate time evolution. The initial density matrix is of the form, ρ(0) = 1 4 1 1 1 3 ,(14) and the decay rate γ = 1.52 × 10 9 s −1 . We chose this density matrix because of its decomposition into basis vectors which are easy to implement in a quantum circuit [17]; however, this need not be the initial density matrix. The dynamics in the zero temperature case are shown in Figure 3. The solid lines are the exact classical Kraus solution, and the dots are generated from the Qasm simulator using 2 19 shots or samples [41]. We generate the simulated dynamics using the expansion presented here with = 0.2. Using this value of , the mean absolute error between the decomposition computed classically and the exact populations is about 10 −3 for the cases studied here. Finally, the x's are results from ibmq_athens with a Richardson extrapolation [44]. We averaged the Richardson extrapolations for 10 repetitions of 2 13 shots with = 1.15, 1.00. Both the Qasm results using the small and the data generated from the device using the Richardson extrapolation are in good agreement with the exact solution. Initializing the system in the same initial state gives the finite temperature dynamics shown in Figure 4, where the classical Kraus evolution is again shown by the solid lines. The evolution using the Qasm simulator, 2 19 We present a decomposition that requires the implementation of a series of operators that are strictly unitary. Our decomposition is generally applicable for any operator, either unitary or non-unitary, and can be implemented as a block-diagonal operator. The numerical approximation in the expansion parameter is the only approximation in the decomposition, but this does not affect the unitarity of the resulting (anti-)Hermitian operators. Furthermore, we show that the convergence with can be accelerated with Richardson's extrapolation which aids the practical use of the algorithm on current devices. This method has general applicability to problems in quantum chemistry and physics in the realm of quantum computing beyond applications in open quantum system dynamics. For example, Hermitian operators such as Hamiltonians or dipole operators, which are common in quantum chemistry, and generally non-unitary, can be implemented with the decomposition presented here. In these cases, the approach can be simplified by implementing only the Hermitian operators in Eq. (5). Our approach would avoid Trotterization of these operators and presents an alternative that implements operators in an exponentiated unitary form which is asymptotically convergent. FIG. 1 . 1Circuit for preparing the wavefunction in Eq. 8, where the first gates represent the initial state preparation (in this case, the mixed state), the block diagonal propagation is highlighted in the dotted box, and the quantum adder is represented by the final three gates. X is the X-gate, H is the Hadamard gate, U3 is IBM's U3 gate, the 2-qubit gates are CNOT gates, and the final gates on each qubit are measurements.FIG. 2.Example circuit for preparing a wavefunction from a 2x2 n subblock of U , utilizing uniformly controlled gates, or multiplexing. The state preparation and the quantum addition are included in the U3 gates. This work is supported by the NSF RAISE-QAC-QSA, Grant No. DMR-2037783. D.A.M. also acknowledges the Department of Energy, Office of Basic Energy Sciences Grant DE-SC0019215. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team. shots, with a Richardson extrapolation ( = 1.15, 1.00) is shown by the dots. The simulated data show excellent agreement with the exact solution. FIG. 4. Excited (green) and ground (black) state populations of a two-level system in a general amplitude damping channel at room temperature where lines represent the exact classical Kraus evolution and dots represent the simulation results from Qiskit Qasm simulator using a Richardson extrapolation ( = 1.15, 1.00), with 2 19 shots Conclusions and Outlook.-Simulating open quantum systems with quantum hardware presents a challenge by requiring the implementation of non-unitary processes using only unitary gates.0 200 400 600 800 1000 Time (ps) 0.3 0.4 0.5 0.6 0.7 Population Ground state Exact Excited state Exact Ground state Qasm Excited state Qasm Quantum dynamics of exciton transport and dissociation in multichromophoric systems. W Popp, D Brey, R Binder, I Burghardt, 10.1146/annurev-physchem-090419-040306Annu. Rev. Phys. Chem. 72W. Popp, D. Brey, R. Binder, and I. Burghardt, "Quantum dynamics of exciton transport and dissociation in multichro- mophoric systems," Annu. Rev. Phys. 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The analysis of strings of n random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic (n → ∞) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.

2203.14773

9a7274c02fd738d42833b2961ecd85a01100c44b

The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis March 29, 2022 Guy Louchard Werner Schachinger Mark Daniel Ward The number of distinct adjacent pairs in geometrically distributed words: a probabilistic and combinatorial analysis March 29, 2022Geometrically distributed wordsNumber of distinct adjacent pairsEqual pairsDistinct pairsMomentsAsymptotic distribution 2010 Mathematics Subject Classification: 05A1660C0560F05 The analysis of strings of n random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic (n → ∞) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains. Introduction We follow the notation and setup of Archibald et al. [1]. Let us consider a string of n random variables Z 1 , Z 2 , . . . , Z n , with geometric distribution P(Z k = i) = P i := p q i−1 for i ≥ 1. In the sequel, all asymptotics will be related to n → ∞. We will use the following notations. Many of the results are obtained by studying the precise distribution of the number of times that (i, j) appears as a consecutive pair or couple in Z 1 , Z 2 , . . . , Z n , i.e., the number of k's such that X k = i and X k+1 = j. We define Asymptotic expressions for the moments are obtained by Mellin transforms applied to Harmonic sums (see for instance, Flajolet, Gourdon, Dumas [3] for a nice exposition). We can derive the dominant part of moments as well as the (tiny) periodic part in the form of a Fourier series. The paper is organized as follows: In Section 2 we present our main results, that is, asymptotic expressions for the variances of X (n) k , 1 ≤ k ≤ 3, and a result concerning asymptotic independence of X (n) i,i , i ∈ N. In Section 3 we conjecture some stronger forms of asymptotic independence, based on which we are able to derive the limiting distribution and asymptotics of higher moments of X (n) 1 . Section 4 is devoted to the proofs of these results, and to some considerations in support of a conjectured Gaussion limiting distribution of X (n) 3 . In Section 5 we use a combinatorial approach to derive exact expressions for first and second moments of X (n) k , 1 ≤ k ≤ 3. In an Appendix we collect our results pertaining to Mellin transforms. Main results In a private communication, B. Pittel observed that the asymptotic distribution of X and EX (n) 3 have also recently been obtained by [1], using generating functions of the sequences of expectations. One of our main results deals with asymptotics of Var X (n) i , 1 ≤ i ≤ 3, as n → ∞. Our approach simply consists in using Var X (n) 1 = i≥1 Var X (n) i,i + i =j Cov (X (n) i,i , X (n) j,j ), and similarly for Var X (n) 2 and Var X (n) 3 . This necessitates thorough investigation of the involved covariances. As it turns out, the main term of Var X + 1 2L =0 Γ χ 2 (np 2 ) − χ 2 1 − 2 − χ 2 ,(1) S (n) 2 := ln 2 L 2 ln(np 2 ) + ln 2 2L 2 (2γ + ln 2 + 2L) + ln(np 2 ) L 2 =0 Γ( χ)(np 2 ) − χ (1 − 2 − χ ) (2) − 1 L 2 =0 Γ( χ)(np 2 ) − χ (1 − 2 − χ ) Γ ( χ) Γ( χ) − L + 2 − χ ln 2 T (n) 2 := 2 L F 1 (0) + 2 L =0 Γ( χ)F 1 ( χ)(np 2 ) − χ ,(3) where F 1 (s) = i,k≥1 (q i + q k − pq i+k−1 ) −s − (q i + q k ) −s , and the constant term of T (n) 2 simplifies to 2 L F 1 (0) = − 2 L ln i,j≥1 1 − p q q i+j q i + q j . Then, as n → ∞, the variances of X (n) i , 1 ≤ i ≤ 3, satisfy Var X (n) 1 = S (n) 1 + O 1 √ n ,(4) Var X (n) 2 = S (n) 2 + T (n) 2 + O ln n √ n ,(5) Var X We leave it as an exercise to show that, for q → 0 (resp. q → 1), the limit is 2 ln 2 (resp. 4 ln 2). A question triggered by the observation that i =j Cov (X (n) i,i , X (n) j,j ) = O 1 n is: How "close to being independent" are X (n) i,i i∈N ? The following theorem provides a partial answer in that regard. Theorem 2.2 The random variables X (n) i,i , i ∈ N are asymptotically independent, in the sense that, for any k ∈ N, any subset I ⊆ N of size k, and any (x i ) i∈I ∈ {0, 1} k we have P(X (n) i,i = x i , i ∈ I) − i∈I P(X (n) i,i = x i ) = O 1 n ,(7) with implied constant depending on I only via k. as can easily be deduced from the theorem in [8]. Cases like the following for n = 5 and i = j, P(X (5) i,i = 1)P(X (5) j,j = 1) − P(X (5) i,i = X (5) j,j = 1) = P 2 i (4 − 2P i − 2P 2 i + P 3 i )P 2 j (4 − 2P j − 2P 2 j + P 3 j ) − P 2 i P 2 j (6 − 2P i − 2P j ) = P 2 i P 2 j (1−P i −P j )(10 + 4P i (1−P i ) + 4P j (1−P j )) + P i P j (12 + P i (2−P i )P j (2−p j )−2P 2 i −2P 2 j ) > 0 suggest that the inequality may be strict for |I| ≥ 2. This is different for the array X (n) i,j i,j≥1 , where both strictly positive and strictly negative correlations can be observed: For n = 3 and i = j, P(X (3) i,j = X (3) j,i = 1) − P(X (3) i,j = 1)P(X (3) j,i = 1) = P i P j (P i + P j ) − (2P i P j ) 2 = P i P j (P i + P j − 4P i P j ) > 0 holds for P i , P j small enough, and for different pairs (k i , m i ) i∈I , with |I| ≥ n, we clearly have P(X (n) k i ,m i = 1, i ∈ I) = 0 < i∈I P(X (n) k i ,m i = 1). 3 Further conjectures and results for pairs of identical letters 3 .1 Higher moments The proof of Theorem 2.1 (see Lemma 4.8) shows that lim n→∞ (Var X are given by (9), where V (n) j , j ≥ 1, are given by (8). We proceed as in Hitczenko and Louchard [5] and Louchard and Prodinger [6]. Let S n (θ) := ln(E(e θξ (n) )) = ∞ m=1 κ (n) m θ m m! be the cumulant generating function of ξ (n) . Furthermore let n 2 := np 2 /q 2 , and observe Ee θ [[ξ (n) i ≥1]] = 1 + (e θ − 1)(1 − e −n 2 q 2i ). By independence of ξ (n) i i≥1 , we get S n (θ) = ∞ i=1 ln 1 + e θ − 1 1 − e −n 2 q 2i = ∞ j=1 (−1) j+1 j (e θ − 1) j ∞ i=1 1 − e −n 2 q 2i j . Now let V (n) j := ∞ i=1 1 − e −n 2 q 2i j = ∞ i=1 j k=0 (−1) k j k e −kn 2 q 2i = ∞ i=1 j k=0 (−1) k j k e −kn 2 q 2i − j k=0 (−1) k j k = ∞ i=1 j k=1 (−1) k+1 j k 1 − e −kn 2 q 2i = j k=1 (−1) k+1 j k ∞ i=1 1 − e −kn 2 q 2i , where the asymptotics of the inner sum can be obtained using G(knp 2 ) from Appendix A.1. Setting again L := ln(1/q), this leads to V (n) j ∼ ln(np 2 ) 2L + γ 2L + 1 2 + 1 2L j k=2 (−1) k+1 j k ln k + 1 2L =0 j k=1 (−1) k j k k − χ 2 Γ χ 2 (np 2 ) − χ 2 ,(8) Finally the cumulants are given by finite linear combinations of the (V (n) j ) j≥1 , κ (n) m = [θ m ]m! ∞ j=1 (−1) j+1 j (e θ − 1) j V (n) j = m! m j=1 V (n) j (−1) j+1 j [θ m ](e θ − 1) j ,(9) explicit expressions for small m being κ (n) 1 = V (n) 1 , κ (n) 2 = V (n) 1 −V (n) 2 , κ (n) 3 = V (n) 1 −3V (n) 2 +2V (n) 3 , κ (n) 4 = V (n) 1 −7V (n) 2 +12V (n) 3 −6V (n) 4 . The sequence of (absolute values of) the coefficients, (1, 1, 1, 1, 3, 2, 1, 7, 12, 6, . . .), seems to coincide with sequence A028246 in the OEIS [14]. The cumulants now allow for computation of moments: The mean of X (n) 1 is given by EX (n) 1 ∼ V (n) 1 . This is identical to Archibald et al. [1,Thm. 2], see also (11). Our approach here is simple and general. Note that the mean does not rely on the state of Conjecture 3.1: the mean computation actually depends only on Lemma 4.3. Similarly, the variance of X (n) 1 is given by Var X (n) 1 ∼ V (n) 1 − V (n) 2 . After some algebra, we verify that this is identical to Thm 2.1 . Limiting distribution A conjecture weaker than Conjecture 3.1 is is given by (10). Conjecture 3.3 For any t ∈ R we have lim n→∞ [P(X (n) 1 ≤ t) − P(ξ (n) ≤ t)] = 0. Set again L = ln(1/q) and n 2 = np 2 /q 2 , set i * = ln(n 2 )/(2L) (implying q 2i * = 1/n 2 ), define η := i − i * , and use P(ξ (n) i = 0) = e −n 2 q 2i = exp(−e −2Lη ). This leads to P(ξ (n) k = 0, k > i) = exp −αe −2Lη , where α := q 2 1 − q 2 . As in Hitczenko and Louchard [5] and Louchard, Prodinger and Ward [7], we proceed by defining Ψ(η) := e −e −2Lη ∞ i=1 1 − e −e −2L(η−i) , and observing that, as n → ∞, we have P(X (n) 1 = i * + η) ∼ f (η) := ∞ v=0 Ψ(η − v + 1)e −αe −2L(η+1−v) r 1 <···<rv r j ≥2−v v i=1 1 − e −e −2L(η+r i ) e −e −2L(η+r i ) ,(10) P(X (n) 1 ≤ i * + η) ∼ F (η) := ∞ i=0 f (η − i). f (η) depends only on p. A simulation with p = 1/4 and 50000 simulated words for each n ∈ {10000, 11547, 13333, 15396} is given in Figure 2. The fit is excellent. A corresponding table of observed and theoretical nonperiodic mean and variance in the equal pairs case (as well as another table for the unequal pairs (circles), p = 1/4, number of simulated words = 50000 for each n ∈ {10000, 11547, 13333, 15396}. case) is given below, all results rounded to 3 decimal places. We defineX (n) j := 1 N N i=1 X (n),i j and s 2 j (n) := 1 N −1 N i=1 (X (n),i j −X (n) j ) 2 the sample mean and unbiased sample variance of a sample (X Remark 3.5 Here we briefly sketch, how we obtained the graph of f in Figure 2, where p = 1/4. As before, we use random variables ξ (n) i distributed Poisson(np 2 q 2(i−1) ), but now there is such a random variable for each i ∈ Z and each real n > 0. For fixed such n the random variables ξ (n) i i∈Z are assumed independent, and also the definition (n),i j ) N i=1 .ξ (n) := i≥1 ξ (n) i is used for real n > 0. We use i * = i * (n) = ln(np 2 /q 2 )/(2L) again. For any n satisfying i * + η ∈ Z, we have f (η) = lim k→∞ P ξ (nq −2k ) − k = i * + η = P i≥1 [[ξ (n) i ≥ 1]] + j≥0 ([[ξ (n) −j ≥ 1]] − 1) = i * + η = P i≥1 [[ξ (ν) i ≥ 1]] + j≥0 ([[ξ (ν) −j ≥ 1]] − 1) = 0 , where for n = ν = ν(η) := q 2(1−η) /p 2 we have i * + η = 0, and ξ (ν) i ∼ Poisson q 2(i−η) . We want a good approximation of f (η) only for η ∈ [− 3,5]. For such η we have P i>30 [[ξ (ν) i ≥ 1]] > 0 = 1 − i>30 e −q 2i−2η ≤ 1 − i>30 e −q 2i−10 = 1 − exp − q 52 1 − q 2 ≈ 7.28 · 10 −7 . and P j>7 ([[ξ (ν) −j ≥ 1]] − 1) < 0 = 1 − j>7 1 − e −q −2j−2η ≤ j>7 e −q 6−2j ≈ e −q −10 ≈ 1.94 · 10 −8 . So, up to an error smaller than 10 −6 , f (η) is given by P 30 i=1 [[ξ (ν) i ≥ 1]]+ 7 j=0 ([[ξ (ν) −j ≥ 1]]−1) = 0 = P 30 i=−7 [[ξ (ν) i ≥ 1]] = 8 = [z 8 ] 30 i=−7 1+(z−1) 1−e −q 2i−2η , where, for each fixed η, the latter coefficient can easily be computed using Maple. Theorem 3.6 (see [1,Thm. 2,Thm. 3]) Let L := ln(1/q) and χ := 2πi/L. Then, as n → ∞, the expectations of X (n) i , i ∈ {1, 3}, satisfy EX (n) 1 ∼ ln(np 2 ) 2L + 1 2 + γ 2L − 1 2L =0 Γ χ 2 (np 2 ) − χ/2 ,(11)EX (n) 3 ∼ ln 2 (np 2 ) 2L 2 + γ L 2 + 1 2L ln(np 2 ) + π 2 + 6γ 2 12L 2 + γ 2L − 1 12 − ln(np 2 ) L 2 =0 Γ( χ)(np 2 ) − χ(12)+ 1 L 2 =0 Γ ( χ)(np 2 ) − χ − 1 2L =0 (−1) Γ χ 2 (np 2 ) − χ/2 . 4 The probability of avoiding certain pairs via Markov chains. Two pairs (i, i) and (r, r) of identical letters The proofs of the theorems rest upon calculation of probabilities of avoiding certain pairs, which we will be doing by employing Markov chains. To illustrate that approach, we consider in greater detail the case of avoiding two fixed pairs (i, i) and (r, r), where i = r, in a sequence of length n. No distinction of letters different from i, r is necessary, so for our Markov chain we can use a finite state space S := {e, i, r, ∆}, where e := N \ {i, r} stands for "everything else", i.e., the set N \ {i, r} is lumped together, and ∆ denotes an additional cemetery state. The corresponding state diagram is e i r ∆ P i P e P r P e P r P i P i P r P e 1 1 From any realization (z k ) k≥1 of the i.i.d. sequence (Z k ) k≥1 we obtain a trajectory (y k ) k≥0 of this finite state Markov chain via (y 0 , y 1 , y 2 , . . .) = (e, φ(z 1 ), φ(z 2 ), φ(z 3 ), . . .), where φ(z k ) := ∆ if for some j < k we have (z j , z j+1 ) ∈ {(i, i), (r, r)}, and otherwise Those trajectories (y k ) n k=0 satisfying y n = ∆ are in correspondence to sequences (z k ) n k=1 that avoid the pairs (i, i) and (r, r). Using the transition matrix φ(z k ) = z k , z k ∈ {i,Π :=      P e P i P r 0 P e 0 P r P i P e P i 0 P r 0 0 0 1      , where P e := 1 − P i − P r , the sought probability is [1, 0, 0, 0] Π n [1, 1, 1, 0] t , respectively, using the restrictionΠ of Π to {e, i, r}, i.e.,Π :=    P e P i P r P e 0 P r P e P i 0    , and initial probability π(·) := [1, 0, 0] and column vector of all ones 1, that probability is P(X (n) i,i = X (n) r,r = 0) = πΠ n 1. A bound on such probability will now be derived in the following more general context. We fix a finite non-empty set of forbidden pairs I := {(k i , m i ) : i ∈ I} of size |I|, and let J := i∈I {k i , m i } = {j 1 , . . . , j |J| }, where j 1 < . . . < j |J| . Moreover we fix 0 < δ ≤ 1/2 and let D J δ := {x ∈ R |J| : x j ≥ 0 for j ∈ J, j∈J x j ≤ 1 − δ}. Lemma 4.1 Let ε := i∈I P k i P m i . Then P(X (n) k i ,m i = 0, i ∈ I) ≤ δ −1/2 e −εn/2(13) holds for (P j ) j∈J ∈ D J δ . Furthermore, there are functions λ 1 , C 1 and Φ n , n ≥ 1, depending on P j , j ∈ J, that are C ∞ and positive on an open set F satisfying D J δ ⊆ F, such that P(X (n) k i ,m i = 0, i ∈ I) = C 1 λ n 1 Φ n .(14) Remark 4.2 At several places we take the liberty to regard (P j ) j≥1 as variables (which is slight abuse of notation), to the effect, that several results in this section hold more generally also for strings of random variables with a distribution different from the geometric. The reader must be prepared to see expressions involving lim P j →0 , ∂ ∂P j , and functions of (P j ) j∈J being C ∞ in some domain, etc. all the time. Proof of Lemma 4.1. Assume P j > 0 for j ∈ J, as well as P e := 1 − j∈J P j ≥ δ. Note that ε ≤ j, ∈J P j P = (1 − P e ) 2 ≤ (1 − δ) 2 < 1 − δ. Define the matrixΠ with rows and columns indexed by the set J ∪ {e} (which we assume ordered, starting with e and followed by the elements of J in ascending order) viaΠ k,m := 0, (k, m) ∈ I, P m , else. We define a row vector w := [ √ P e , P j 1 , . . . , P j |J| ], satisfying w 2 = 1, and a diagonal matrix S := Diag(w), and the matrixΠ := SΠS −1 = S 11 t − i∈I e k i e t m i S, where the column vectors e j , j ∈ J ∪ {e}, denote the standard unit vectors in R |J|+1 , and observe, using the Frobenius norm Π F = k,m∈J∪{e}Π 2 k,m = 1 − i∈I P k i P m i , and π = (e e ) t , P(X (n) k i ,m i = 0, i ∈ I) = πΠ n 1 = wΠ n−1 w t ≤ w 2 2 Π n−1 2 ≤ Π n−1 F = (1 − ε) (n−1)/2 ≤ δ −1/2 (1 − ε) n/2 ≤ δ −1/2 e −εn/2 . Observe thatΠ is non-negative and primitive, therefore, by the Perron-Frobenius Theorem (see [13]), there is a unique positive eigenvalue λ 1 , that is strictly larger in modulus than any other eigenvalue, and corresponding strictly positive left and right eigenvectors u and v, such thatΠ n = λ n 1 uv vu+O(n |J| |λ 2 | n ) element-wise, where λ 2 is an eigenvalue of second largest modulus. This leads to P(X (n) k i ,m i = 0, i ∈ I) = (πv)(u1) uv λ n 1 + O(n |J| |λ 2 | n ). By setting one or more of (P j ) j∈J to zero, one or more of the non-dominant eigenvalues (λ k ) k≥2 become zero, but there is a non-negative primitive submatrix constructed from the non-zero columns (and corresponding rows) ofΠ, guaranteeing a unique positive eigenvalue larger in modulus than all other eigenvalues. As the row and column corresponding to state e will always be part of that submatrix, the first components u e and v e of u and of v will be positive. By continuity, these properties also hold in a neighbourhood of such (P j ) j∈J , which yields λ 1 being C ∞ in some superset F of D J δ , by the implicit function theorem, using the facts that the characteristic polynomial p(λ) ofΠ, considered as a function of (λ, (P j ) j∈J ), is C ∞ , and the derivative of p(λ) evaluated in a simple zero λ 1 is non-zero. On the set F, the components of 1 ue u and 1 ve v are C ∞ functions of (P j ) j∈J as well. We let C 1 := (πv)(u1) uv and Φ n : = 1 C 1 λ −n 1 P(X (n) k i ,m i = 0, i ∈ I). Those are positive C ∞ functions of (P j ) j∈J in a probably smaller superset of D J δ , that we still denote F. Note that primitivity ofΠ may cease to hold when P e = 0. Moreover note that |λ 2 | is continuous on D J δ , but need not be differentiable on that set. The bound (13) fits our needs when ε is large. Equation (14) is useful in the case of small ε, if asymptotics of λ 1 , C 1 and Φ n are known. In order to derive such asymptotics, we letΠ be the matrix obtained fromΠ by deleting row and column corresponding to state e. Left and right eigenvectors u = [1, β] and v = [1/P e , µ t ] t , with row vector β = (β j ) j∈J and column vector µ = (µ j ) j∈J , corresponding to the dominant eigenvalue λ 1 ofΠ, lead to equations λ 1 = P e (1 + j∈J β j ) = P e (1 + j∈J P j µ j ),(15)β = 1 λ 1 βΠ +p ,(16)µ = 1 λ 1 Π µ + 1 ,(17) with row vectorp = (P j ) j∈J , and with ascending order of indices in β, µ,p. We keep denoting the column vector of all ones of appropriate dimension by 1, and express C 1 in terms of β and µ as follows: C 1 = (πv)(u1) uv = 1 Pe (1 + j∈J β j ) 1 Pe + j∈J β j µ j = 1 + β1 1 + P e βµ .(18) Asymptotics up to any fixed order K of λ 1 , β, µ are conveniently computed via fixed point iteration as described by the following algorithm: Algorithm 1 Calculate asymptotics of λ 1 , β, µ up to fixed order. Require: K ≥ 0, k = 0,Π,p, λ = 1,β = [0, . . . , 0],μ = [1, . . . , 1] t while k < K dō β ← 1 λ βΠ +p μ ← 1 λ Πμ + 1 λ ← P e [1 +β1] k ← k + 1 end while return λ,β,μ The output λ,β,μ of the algorithm then satisfies λ 1 = λ + O * K+1 , β =β + O * K+1 , µ =μ + O * K+1 . Here and in the following the notation O * k always refers to the variables (P j ) j∈J , but not to P e . So, for instance, O * 4 is the same as O(γ 4 ), where γ = j∈J P j . The next lemma provides asymptotics of probabilities in the case of a single avoided pair. Lemma 4.3 The probabilities of avoiding the pair (i, i), resp. (i, r) for i = r, in a sequence of length n satisfy P(X (n) i,i = 0) = e −nP 2 i + O √ nP 2 i e − 2−δ 4 nP 2 i ,(19)P(X (n) i,r = 0) = e −nP i Pr + O P i P r e − 2−δ 4 nP i Pr ,(20) as n → ∞, uniformly for P i ∈ D {i} δ , resp. for (P i , P r ) ∈ D {i,r} δ . Proof. We first consider the forbidden pair (i, i). The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π = P e P i P e 0 , p(λ) = λ 2 − (1 − P i )λ − P i P e , λ 1 = 1 − P 2 i + P 3 i − 2P 4 i + O * 5 , C 1 = 1 + P 2 i − 2P 3 i + 6P 4 i + O * 5 , where we used Algorithm 1 (with K = 4) and (18). Following a suggestion by B.Salvy [12], we can easily derive λ 1 from p(λ), after replacing P e by 1 − P i . We add an extra variable v, carrying the weight of the P . :P . := vP . . We have the local expansion of the solution at 0 by using the Maple package gfun (see [11]): sol := gfun[algeqtoseries](p(λ), v, λ, pr), where pr denotes the precision of the expansion into v. We obtain the solutions as sol [1], sol [2] and we keep the solution close to 1. Denoting πΠ n 1 = C 1 λ n 1 + C 2 λ n 2 , with λ 2 the non-dominant eigenvalue ofΠ, we have C 1 λ 0 1 + C 2 λ 0 2 = 1, and therefore C 2 = −P 2 i + 2P 3 i − 6P 4 i + O * 5 , which leads to Φ n = 1 + C 2 C 1 λ 2 λ 1 n = 1 + O(P 2 i ) , uniformly in n. This is used in (14), together with C 1 = 1 + O(P 2 i ) and λ n 1 = e n ln λ 1 = e n(−P 2 i +P 3 i +O(P 4 i )) = e −nP 2 i (1 + O(nP 3 i )) leading to P(X (n) i,i = 0) = 1 + O(P 2 i ) + O(nP 3 i ) e −nP 2 i , for nP 3 i = O(1) , resp. for nP 2 i = O(n 1/3 ). Note that for fixed α, β > 0 the function x α e −βx is bounded for x > 0, implying nP 3 i e −nP 2 i = √ nP 2 i (nP 2 i ) 1/2 e − 2+δ 4 nP 2 i e − 2−δ 4 nP 2 i = O √ nP 2 i e − 2−δ 4 nP 2 i . Moreover also P 2 i e −nP 2 i = O √ nP 2 i e − 2−δ 4 nP 2 i holds, and (13) can be built in by observing that nP 2 i = Ω(n 1/3 ) implies δ −1/2 e − n 2 P 2 i = O √ nP 2 i e − 2−δ 4 nP 2 i . We have thus obtained (19). We now consider the forbidden pair (i, r) with i = r. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =    P e P i P r P e P i 0 P e P i P r    , p(λ) = λ 3 − λ 2 + P i P r λ, λ 1 = 1 − P i P r − P 2 i P 2 r − 2P 3 i P 3 r + O * 8 , C 1 = 1 + P i P r + 3P 2 i P 2 r + 10P 3 i P 3 r + O * 8 . Clearly, λ 1 , and therefore also C 1 and Φ n , are C ∞ functions of the coefficient P i P r of the characteristic polynomial p, meaning that the error term O * 8 is in fact O(P 4 i P 4 r ). Sufficiently accurate for our purposes are the asymptotics λ 1 = 1 − P i P r + O(P 2 i P 2 r ) and C 1 = 1 + O(P i P r ). One of the eigenvalues is 0, therefore a representation πΠ n 1 = C 1 λ n 1 + C 2 λ n 2 as before also holds in this case, with C 2 = O(P i P r ), and Φ n = 1 (14), taking care of error terms as above. + C 2 C 1 λ 2 λ 1 n = 1 + O(P i P r ), uniformly in n. All this, together with λ n 1 = e −nP i Pr (1 + O(nP 2 i P 2 r )), leads to (20) via The next corollary follows easily from equations (13), (19) and (20). Corollary 4.4 The variances of X (n) i,i and X (n) i,r for i = r satisfy Var X (n) i,i = e −nP 2 i − e −2nP 2 i + O √ nP 2 i e − 2−δ 4 nP 2 i ,(21)Var X (n) i,r = e −nP i Pr − e −2nP i Pr + O P i P r e − 2−δ 4 nP i Pr ,(22)as n → ∞, uniformly for P i ∈ D {i} δ , resp. for (P i , P r ) ∈ D {i,r} δ . Next we obtain asymptotics for the covariance Cov (X (n) i,i , X (n) r,r ) = P(X (n) i,i = X (n) r,r = 1) − P(X (n) i,i = 1)P(X (n) r,r = 1) = P(X (n) i,i = X (n) r,r = 0) − P(X (n) i,i = 0)P(X (n) r,r = 0). Lemma 4.5 For i = r and (P i , P r ) ∈ D {i,r} δ we have, for nP i P r (P i + P r ) 2 = O(1), Cov (X (n) i,i , X (n) r,r ) = O P i P r + nP i P r (P i + P r ) 2 P(X (n) i,i = 0)P(X (n) r,r = 0).(23)From (13) we derive Cov (X (n) i,i , X (n) r,r ) = O e − n 2 (P 2 i +P 2 r ) , that together with (23), where we use nP i P r (P i +P r ) 2 P(X (n) i,i = 0)P(X (n) r,r = 0) = O P i P r (nP 2 i +nP 2 r )e − n 2 (P 2 i +P 2 r ) = O P i P r e − 2−δ 4 n(P 2 i +P 2 r ) , implies the next corollary, since e − n 2 (P 2 i +P 2 r ) = O P i P r e − 2−δ 4 n(P 2 i +P 2 r ) , for nP i P r (P i + P r ) 2 = Ω(1). Corollary 4.6 For i = r, the covariance of X (n) i,i and X (n) r,r satisfies Cov (X (n) i,i , X (n) r,r ) = O P i P r e − 2−δ 4 n(P 2 i +P 2 r ) ,(24) as n → ∞, uniformly for (P i , P r ) ∈ D {i,r} δ . Proof of Lemma 4.5. We first find asymptotics of λ 1 and C 1 from P(X (n) i,i = X (n) r,r = 0) = C 1 λ n 1 Φ n , proceeding as in the previous lemma. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =    P e P i P r P e 0 P r P e P i 0    , p(λ) = λ 3 − P e λ 2 − [P e (P i + P r ) + P i P r ]λ − P i P r P e , λ 1 = 1 − P 2 i − P 2 r + P 3 i + P 3 r + O * 4 , C 1 = 1 − P 2 i − P 2 r + 2P 3 i + 2P 3 r + O * 4 . Again, we can also replace P e by 1 − P i − P r and use gfun. From Lemma 4.1 we know that λ 1 , C 1 and Φ n are C ∞ functions of P i , P r in some open superset F of D {i,r} δ , such that P(X (n) i,i = X (n) r,r = 0) = C 1 (P i , P r )[λ 1 (P i , P r )] n Φ n (P i , P r ) (25) holds for (P i , P r ) ∈ D {i,r} δ . In fact, we will only need that those functions are C 2 in the following. Note that P(X (n) i,i = 0) can be obtained from (25) as the limiting case P r → 0. Observe that we have lim P i →0 C 1 (P i ,Pr) C 1 (0,Pr)C 1 (P i ,0) = lim Pr→0 C 1 (P i ,Pr) C 1 (0,Pr)C 1 (P i ,0) = lim P i →0 Φn(P i ,Pr) Φn(0,Pr)Φn(P i ,0) = lim Pr→0 Φn(P i ,Pr) Φn(0,Pr)Φn(P i ,0) = 1, and therefore C 1 (P i ,Pr) C 1 (0,Pr)C 1 (P i ,0) = 1 + O(P i P r ) and Φn(P i ,Pr) Φn(0,Pr)Φn(P i ,0) = 1 + O(P i P r ). To see that the latter holds uniformly in n and (P i , P r ) ∈ D {i,r} δ , we start definingΠ :=Π − λ 1 uv vu, so thatΠ = λ 1 uv vu +Π, and πΠ n 1 = C 1 λ n 1 + πΠ n 1,(26) where we used that u and v are in the left resp. right kernel of the matrixΠ. Denoting the spectral radius of a square matrix A by ρ(A), we clearly have ρ(Π) = |λ 2 |, and since D {i,r} δ is compact, we have max (P i ,Pr)∈D {i,r} δ |λ 2 | λ 1 =: κ < 1. All components of 1 λ 1 κΠ are continuous, so there is a constant C such that 1 λ 1 κΠ F ≤ C on D {i,r} δ . By applying Lemma 4.7 below to the matrix 1 λ 1 κΠ , we obtain Φ n − 1 = πΠ n 1 C 1 λ n 1 = O(n 2 κ n ) = O(κ n ), for some κ <κ < 1, uniformly on D {i,r} δ . Similarly, we obtain ∂Φn ∂P i = O(κ n ), ∂Φn ∂Pr = O(κ n ), and ∂ 2 Φn ∂P i ∂Pr = O(κ n ), uniformly on D {i,r} δ , using, e. g., ∂πΠ n 1 ∂P i = 0≤j<n πΠ j ∂Π ∂P iΠ n−1−j 1, and again Lemma 4.7. Define Ψ n (P i , P r ) := Φn(P i ,Pr) Φn(0,Pr)Φn(P i ,0) − 1 and observe that lim n→∞ ∂ 2 ∂P i ∂Pr Ψ n = 0 holds uniformly on D {i,r} δ . Note that we have Ψ n (P i , 0) = Ψ n (0, P r ) = 0 for 0 ≤ P i , P r ≤ 1 − δ, yielding Ψ n (P i , P r ) = Ψ n (P i , P r ) − Ψ n (P i , 0) − Ψ n (0, P r ) + Ψ n (0, 0) = P i P r ∂ 2 Ψ n ∂P i ∂P r (p i , p r ) by the (bivariate) Mean Value Theorem, where 0 ≤ p i ≤ P i and 0 ≤ p r ≤ P r , see [10,Thm. 9.40]. DefiningC := max n≥1 max (p i ,pr)∈D {i,r} δ ∂ 2 Ψn ∂P i ∂Pr (p i , p r ) , we finally conclude |Ψ n (P i , P r )| ≤CP i P r for all n ≥ 1 and (P i , P r ) ∈ D {i,r} δ , establishing the uniformity claim. By our asymptotics for λ 1 , we similarly obtain ln λ 1 (P i , P r ) − ln λ 1 (P i , 0) − ln λ 1 (0, P r ) + ln λ 1 (0, 0) = P i P r ∂ 2 ln λ 1 ∂P i ∂P r (p i , p r ) = O(P i P r (P i + P r ) 2 ), leading to λ 1 (P i , P r ) λ 1 (P i , 0)λ 1 (0, P r ) = 1 + O P i P r (P i + P r ) 2 . We summarize P(X (n) i,i = X (n) r,r = 0)) P(X (n) i,i = 0)P(X (n) r,r = 0) = 1 + O P i P r + nP i P r (P i + P r ) 2 , finally arriving at (23). Lemma 4.7 Let A ∈ R k×k , with k ≥ 2, have spectral radius ρ(A) ≤ 1 and Frobenius norm A F = C . Then, with C := max(C , k) and C := 2C k−1 , we have A n F ≤ C n k−1 . Proof. There is an orthogonal matrix Q such thatĀ := QAQ −1 is upper triangular and satisfies ρ(Ā) = ρ(A) and Ā F = A F . Then A n F = Q −1Ān Q F = Ā n F . Since C ≥ k ensures that C/ √ 1≤ <k is increasing, we can show |(Ā n ) i,i | ≤ 1, |(Ā n ) i,i+1 | ≤ nC, |(Ā n ) i,i+2 | ≤ nC + n 2 C √ 2 2 ≤ n + 1 2 C √ 2 2 , |(Ā n ) i,i+3 | ≤ nC + 2 1 n 2 C √ 2 2 + n 3 C √ 3 3 ≤ n + 2 3 C √ 3 3 . Note that (Ā n ) i,i+3 is a sum of termsā i 0 ,i 1ā i 1 ,i 2 · · ·ā i n−1 ,in , where (i k ) n k=0 is an increasing sequence with i 0 = i and i n = i + 3. There is 1 = 2 0 way to have a single jump (of size 3) in this sequence, and it can happen at n 1 places. We have |ā i,i ·ā i,i+3 ·ā n−1− i+3,i+3 | ≤ C, because |ā k,k | ≤ 1 for any k, and |ā i,i+3 | ≤ Ā F ≤ C. There are 2 1 ways to have two jumps (two orderings of sizes 1 and 2) in this sequence, and there are n 2 ways to place each of them. We have |ā i,i ·ā i,i+1 ·ā m i+1,i+1 ·ā i+1,i+3 ·ā n−2− −m i+3,i+3 | ≤ C 2 2 , becausē a 2 i,i+1 +ā 2 i+1,i+3 ≤ Ā 2 F ≤ C 2 , and the product of those terms is maximized when they are both equal. There is 1 = 2 2 way to have three jumps (all of size 1) in this sequence, and there are n 3 ways to place them. We have |ā i,i+1 ·ā i+1,i+2 ·ā i+2,i+3 | ≤ C/ √ 3 3 , becauseā 2 i,i+1 +ā 2 i+1,i+2 +ā 2 i+2,i+3 ≤ Ā 2 F ≤ C 2 , and the product of those terms is maximized when they are all equal. More generally we have |(Ā n ) i,i+ | ≤ n+ −1 C/ √ ≤ n+k−2 k−1 C/ √ k − 1 k−1 for 1 ≤ i < i + ≤ k. We obtain Ā n F ≤ k n + k − 2 k − 1 C √ k − 1 k−1 ≤ 2C k−1 n k−1 , because of n+k−2 k−1 ≤ n k−1 for k ≥ 2 and n ≥ 1, and because of max k≥2 k (k−1) (k−1)/2 = 2, which completes the proof. The variance of X (n) 1 In this subsection we use the results on variances and covariances in the case of avoided pairs of identical letters, that we have derived so far, to furnish a proof of equation (4) is asymptotically given by Var X (n) 1 = i≥1 VarX (n) i,i + O 1 n = S (n) 1 + O 1 √ n , with S (n) 1 given in (1). In particular the contribution of covariances is negligible. Proof. Dealing with covariances first, note that (24) guarantees that the double sum of covariances i =r Cov (X (n) i,i , X (n) r,r ) makes a negligible contribution to the variance of X (n) 1 : We will use that k≥1 nP β k α e −nP β k = O(1) holds for α, β > 0.(27) With the help of (27), which will be proved in Appendix B, we find i≥1 r≥1 P i P r e − 2−δ 4 n(P 2 i +P 2 r ) = 1 n i≥1 (nP 2 i ) 1/2 e − 2−δ 4 nP 2 i r≥1 (nP 2 r ) 1/2 e − 2−δ 4 nP 2 r = O 1 n . This leads to i =r Cov (X (n) i,i , X (n) r,r ) = O( 1 n ). We now turn to i≥1 VarX (n) i,i . Observe that the sum of error terms from (21) satisfies i≥1 √ nP 2 i e − 2−δ 4 nP 2 i = O 1 √ n , by (27). Therefore, up to an error term O 1 √ n , the variance Var X (n) 1 equals i≥1 e −nP 2 i − e −2nP 2 i = i≥1 1 − e −2nP 2 i − i≥1 1 − e −nP 2 i = G(2np 2 ) − G(np 2 ), which can be evaluated using G from Appendix A.1, directly leading to S (n) 1 from (1). Contribution of covariances to the variance of X (n) 2 In this subsection we will prove the following lemma, which will also imply equations (5) and (6) of Theorem 2.1. Lemma 4.9 The variance of X (n) 2 is asymptotically given by Var X (n) 2 = i,j≥1 VarX (n) i,j + 2 i,j,k≥1 H(i, j, k) + O ln n √ n = S (n) 2 + T (n) 2 + O ln n √ n ,(28) where H(i, j, k) = (e nP i P j P k − 1)e −nP i P j −nP j P k , and S are given in (2) and (3). Only covariances Cov (X (n) i,j , X (n) j,k ), resp. Cov (X (n) j,i , X (n) k,j ), with i, j, k all different, and Cov (X (n) i,j , X (n) j,i ) with i, j different, contribute significantly to Var X (n) 2 . Proof. We start considering distinct forbidden pairs (i 1 , j 1 ), (i 2 , j 2 ), where we allow i 1 = j 1 or i 2 = j 2 or both, and are again interested in negligibility of covariance contributions. Let J := {i 1 , j 1 , i 2 , j 2 }, and assume P i > 0 for i ∈ J, as well as P e := 1 − i∈J P i ≥ δ. Define the matrixΠ with rows and columns indexed by the set J ∪ {e} (which we assume ordered, starting with e and followed by the elements of J in ascending order) viā Π i,j := 0, (i, j) ∈ {(i 1 , j 1 ), (i 2 , j 2 )}, P j , else, We will have to distinguish several cases, which however share some common features: The sought probability can be expressed as P(X (n) i 1 ,j 1 = X (n) i 2 ,j 2 = 0) = πΠ n 1 = C 1 λ n 1 Φ n , where, as previously observed, λ 1 , C 1 and Φ n for n ≥ 1 are C ∞ functions on an open superset of D J δ . Limits lim n→∞ Φ n = 1, lim n→∞ ∂Φn ∂P i 1 = 0, etc., will again be uniform for (P i ) i∈J ∈ D J δ . Denoting P(X (n) i 1 ,j 1 = 0) = C * λ n * Φ * n , P(X (n) i 2 ,j 2 = 0) = C • λ n • Φ • n , we observe lim P i →0 λ 1 λ * λ • = lim P i →0 C 1 C * C • = lim P i →0 Φ n Φ * n Φ • n = 1, for i ∈ J, leading to λ 1 λ * λ• = 1 + O i∈J P i , C 1 C * C• = 1 + O i∈J P i , and Φn Φ * n Φ • n = 1 + O i∈J P i , with implied constant independent of n. (This independence can be shown as in the proof of Lemma 4.5.) As we will see, more accurate representations for λ 1 , complementing those obtained by Algorithm 1, can always be found in the form λ 1 = 1 − P i 1 P j 1 − P i 2 P j 2 + Q + O * 4 , where Q = O * 3 and Q ≥ 0. We will observe, that in each of the cases Q = i,r,t:(i,r),(r,t)∈{(i 1 ,j 1 ),(i 2 ,j 2 )} P i P r P t(29) holds. Using λ * = 1 − P i 1 P j 1 + Q * + O(P 2 i 1 P 2 j 1 ) and λ • = 1 − P i 2 P j 2 + Q • + O(P 2 i 2 P 2 j 2 ) (depending on whether (i 1 , j 1 ) = (i, i) or (i, r), we have Q * = P 3 i or Q * = 0, and similarly for Q • , see the proof of Lemma 4.3), we will obtain in most of the cases ln λ 1 λ * λ • = Q − Q * − Q • + O(P i 1 P j 1 P i 2 P j 2 ),(30) where the error term needs justification in each of these cases. In some cases this is done by employing the MVT, as in the proof of Lemma 4.5. This results in the following expression for a quotient of probabilities, that directly leads to an expression for the covariance, where we denoteQ : = Q−Q * −Q • , P(X (n) i 1 ,j 1 = X (n) i 2 ,j 2 = 0) P(X (n) i 1 ,j 1 = 0)P(X (n) i 2 ,j 2 = 0) = λ 1 λ * λ • n C 1 C * C • Φ n Φ * n Φ • n = e n Q +O(P i 1 P j 1 P i 2 P j 2 ) 1 + O i∈J P i , Cov (X (n) i 1 ,j 1 , X (n) i 2 ,j 2 ) = (e nQ − 1) + O nP i 1 P j 1 P i 2 P j 2 + i∈J P i e nQ P(X (n) i 1 ,j 1 = 0)P(X (n) i 2 ,j 2 = 0), valid for nP i 1 P j 1 P i 2 P j 2 = O(1) . It will turn out that in some of the cases we haveQ = 0. In cases whereQ > 0 we always haveQ = O i∈J P i andQ ≤ 1−δ 2 ε, with ε := P i 1 P j 1 + P i 2 P j 2 . Using the latter, and (13), as well as e nQ − 1 ≤ nQe nQ , we obtain e nQ P(X (n) i 1 ,j 1 = 0)P(X (n) i 2 ,j 2 = 0) = O e − δ 2 nε , (e nQ − 1) P(X (n) i 1 ,j 1 = 0)P(X (n) i 2 ,j 2 = 0) − e −nε = O nQ √ nεe − δ 2 nε = O √ nQe − δ 4 nε . In case of nP i 1 P j 1 P i 2 P j 2 = Ω(1) we use (13) to obtain Cov (X (n) i 1 ,j 1 , X (n) i 2 ,j 2 ) = O e − n 2 ε = O nP i 1 P j 1 P i 2 P j 2 e − δ 4 nε , and all this results in Cov (X (n) i 1 ,j 1 , X (n) i 2 ,j 2 ) = (e nQ − 1)e −n(P i 1 P j 1 +P i 2 P j 2 ) + O (nP i 1 P j 1 P i 2 P j 2 + √ n i∈J P i )e − δ 4 n(P i 1 P j 1 +P i 2 P j 2 ) . We distinguish the following cases, only Cases 1, 5 and 6 involvingQ = 0, and Case 6 slightly deviating from the general pattern outlined above. Case 1: Pairs (i, r), (r, t) with i, r, t all different. The matrixΠ and its characteristic polynomial p are given bȳ Π =      P e P i P r P t P e P i 0 P t P e P i P r 0 P e P i P r P t      , p(λ) = λ 4 − λ 3 + P r (P i + P t )λ 2 − P i P r P t λ. Using Algorithm 1 and (18), we obtain λ 1 = 1 − P i P r − P r P t + P i P r P t + O * 4 , C 1 = 1 + P i P r + P r P t − 2P i P r P t + O * 4 . We can see that λ 1 = 1 − P i P r − P r P t + P r P i P t + P 2 r O * 2 holds, by noting that λ 1 is a C ∞ function of the coefficients P r (P i + P t ) and −P i P r P t of the polynomial p, and terms of order 2 or higher contribute P 2 r O * 2 . Thus, by the MVT, for some 0 < p i < P i , 0 < p t < P t , ln λ 1 λ * λ • = P i P t ∂ 2 ln λ 1 ∂P i ∂P t (p i , p t ) = P i P t P r (1 + O(P r )). So (30) is established withQ = P i P r P t , which indeed satisfiesQ ≤ 1 4 P r (P i +P t ) ≤ 1−δ 2 ε, since δ ≤ 1/2. Case 2a: Pairs (i, r), (i, t) with i, r, t all different. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =      P e P i P r P t P e P i 0 0 P e P i P r P t P e P i P r P t      , p(λ) = λ 4 − λ 3 + P i (P r + P t )λ 2 , λ 1 = 1 − P i P r − P i P t + O * 4 , C 1 = 1 + P i P r + P i P t + O * 4 . Again, λ 1 is a C ∞ function of the coefficient P i (P r + P t ), leading to λ 1 = 1 − P i P r − P i P t + P 2 i O * 2 , which we use to derive ln( λ 1 λ * λ• ) = P r P t ∂ 2 ln λ 1 ∂Pr∂Pt (p r , p t ) = O(P r P t P 2 i ), yielding (30) withQ = 0. Case 2b: Pairs (r, i), (t, i) with i, r, t all different. Here the matrix (call itΠ b ) can be seen to be a similarity transformation involving diagonal matrices of the transposed matrix (call itΠ a ) in Case 2a, more precisely, with p : = πΠ = [P e , (P i ) i∈I ], we havē Π b = Diag(p) −1Πt a Diag(p), leading to pΠ n−1 b 1 = pΠ n−1 a 1, and implying that p(λ), λ 1 , C 1 , and also the covariance, are the same as in Case 2a. Case 3: Pairs (i, i), (r, t) with i, r, t all different. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =      P e P i P r P t P e 0 P r P t P e P i P r 0 P e P i P r P t      , p(λ) = λ 4 − (1 − P i )λ 3 − (P i − P 2 i − P r P t )λ 2 + P i P r P t λ, λ 1 = 1 − P 2 i − P r P t + P 3 i + O * 4 , C 1 = 1 + P 2 i + P r P t − 2P 3 i + O * 4 . Denoting by λ • = lim P i →0 λ 1 the largest zero of λ 2 − λ + P r P t , and r(λ) = p(λ) λ , we compute r(λ • + P 2 i µ) = P 2 i λ • + P 2 i (P 2 i + 2P i λ • + 2λ • 2 − P i − λ • )µ + P 4 i (P i + 3λ • − 1)µ 2 + P 6 i µ 3 = 0, and conclude by the implicit function theorem, using λ • = 1 + O * 2 , that there is a unique C ∞ function µ of P i , P r , P t near the origin, satisfying µ(0, 0, 0) = −1, such that λ 1 = λ • + P 2 i µ. This leads to ∂ 2 λ 1 ∂P i ∂Pr = O(P i ), and similarly ∂ 2 λ 1 ∂P i ∂Pt = O(P i ), resulting in ln( λ 1 λ * λ• ) = O(P r P t P 2 i ), yielding (30). Case 4: Pairs (i, j), (r, t) with i, j, r, t all different. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =        P e P i P j P r P t P e P i 0 P r P t P e P i P j P r P t P e P i P j P r 0 P e P i P j P r P t        , p(λ) = λ 5 − λ 4 + (P i P j + P r P t )λ 3 , λ 1 = 1 − P i P j − P r P t + O * 4 , C 1 = 1 + P i P j + P r P t + O * 4 . Observe that ∂ 2 ln λ 1 ∂P i ∂Pr = O * 2 and ∂ 2 ln λ 1 ∂P j ∂Pt = O * 2 lead to ln( λ 1 λ * λ• ) = O(P i P j P r P t ), yielding (30). Case 5: Pairs (i, r), (r, i) with i, r different. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =    P e P i P r P e P i 0 P e 0 P r    , p(λ) = λ 3 − λ 2 + P i P r λ + P e P i P r , λ 1 = 1 − 2P i P r + P 2 i P r + P i P 2 r + O * 4 , C 1 = 1 + 2P i P r − 2P 2 i P r − 2P i P 2 r + O * 4 . Note that λ 1 is a C ∞ function of the coefficients P i P r and P i P r (1 − P i − P r ), leading to λ 1 = 1 − 2P i P r + P i P r (P i + P r ) + O(P 2 i P 2 r ), which, together with λ * = λ • = 1 − P i P r + O(P 2 i P 2 r ), we use to derive λ 1 λ * λ • = 1 + P 2 i P r + P i P 2 r + O(P 2 i P 2 r ). This is in accordance with (30), withQ = P 2 i P r + P i P 2 r = P i P r (P i + P r ) ≤ P i P r (1 − δ) = 1−δ 2 ε. Case 6a: Pairs (i, i), (i, r) with i, r different. The matrixΠ, its characteristic polynomial p, and asymptotics of λ 1 and C 1 are given bȳ Π =    P e P i P r P e 0 0 P e P i P r    , p(λ) = λ 3 − (1 − P i )λ 2 − P i P e λ, λ 1 = 1 − P 2 i − P i P r + P 2 i P r + P 3 i + O * 4 , C 1 = 1 + P 2 i + P i P r − 2P 2 i P r − 2P 3 i + O * 4 . We start deriving the more precise estimate λ 1 = 1 − P 2 i − P i P r + P 2 i P r + P 3 i + P 2 i O * 2 : Abbreviating σ = P i + P r , κ = P i − P 2 i , we use p(λ 1 ) = 0 to infer the existence of a function µ that satisfies λ 1 = 1 − κσ + P 2 i µ. Indeed, from 0 = λ 2 1 − (1 − P i )λ 1 − P i (1 − σ) = (1 − κσ + P 2 i µ) 2 − (1 − P i )(1 − κσ + P 2 i µ) − P i (1 − σ) = κ 2 σ 2 + σ(P i − κ − P i κ) + P 2 i µ(1 + P i − 2κσ) + P 4 i µ 2 = P 2 i (1 − P i ) 2 σ 2 + σP i + (1 + P i − 2κσ)µ + P 2 i µ 2 we conclude by the implicit function theorem that there is a unique C ∞ function µ of P i , P r near the origin, satisfying µ = O * 2 . Since lim Pr→0 λ 1 = λ * and lim Pr→0 λ • = 1, we have λ 1 λ * λ• = 1 + O(P r ). This estimate will now be refined. From λ * = 1 − P 2 i + P 3 i + O(P 4 i ) and λ • = 1 − P i P r + O(P 2 i P 2 r ) we deduce λ * λ • = 1 − P 2 i − P i P r + P 3 i + P 2 i O * 2 and λ 1 λ * λ • = λ * λ • + P 2 i P r + P 2 i O * 2 λ * λ • = 1 + P 2 i P r + P 2 i O * 2 = 1 + P 2 i P r + P 2 i P r O * 1 = 1 + P 2 i P r + O(P 2 i P r ). This is not quite (30), butQ = P 2 i P r = Pr 2 P 2 i + P i 2 P i P r ≤ 1−δ 2 ε is satisfied, and O(P 2 i P r ) turns out to be a sufficiently good substitute for O(P i 1 P j 1 P i 2 P j 2 ). Case 6b: Pairs (i, i), (r, i) with i, r different. Here the matrixΠ can be seen to be a similarity transformation of the transposed matrix in Case 6a, implying that p(λ), λ 1 , C 1 , and also the covariance, are the same as in Case 6a. We summarize the covariances Cov (X (n) i 1 ,j 1 , X (n) i 2 ,j 2 ), asymptotics valid for (P j ) j∈J ∈ D J δ , Cov (X (n) i,r , X (n) r,t ) = e nP i PrPt −1 e −n(P i Pr+PrPt) + O nP i P 2 r P t + √ nP i P r P t e − δ 4 n(P i Pr+PrPt) (Case 1) Cov (X (n) i,r , X (n) i,t ) = Cov (X (n) r,i , X (n) t,i ) = O nP 2 i P r P t + P i P r P t e − δ 4 n(P i Pr+P i Pt) (Cases 2) Cov (X (n) i,i , X (n) r,t ) = O nP 2 i P r P t + P i P r P t e − δ 4 n(P 2 i +PrPt) (Case 3) Cov (X (n) i,j , X (n) r,t ) = O nP i P j P r P t e − δ 4 n(P i P j +PrPt) (Case 4) Cov (X (n) i,r , X (n) r,i ) = e nP i Pr(P i +Pr) −1 e −2nP i Pr + O nP 2 i P 2 r + √ nP i P r e − δ 2 nP i Pr (Case 5) Cov (X (n) i,i , X (n) i,r ) = Cov (X (n) i,i , X (n) r,i ) = O nP 2 i P r + P i P r e − δ 4 n(P 2 i +P i Pr) (Cases 6) We continue showing that the multiple sums of error terms arising in (22) and Cases 1-6 are negligible. In addition to (27) we will also use that i,k≥1 (nP i P k ) α e −nP i P k = O(ln n) holds for α > 0.(31) Note that (31), proven in Appendix B, yields i,r≥1 P i P r e − 2−δ 4 nP i Pr = O ln n n , which settles (22), and also Case 5, where the double sum is O ln n √ n , and Case 4, with quadruple sum of order O ln 2 n n . Using P i P t ≤ √ P i P t , Case 1 can be reduced to bounding the sum 1 √ n i,r,t≥1 nP i P r nP r P t e − δ 4 n(P i Pr+PrPt) ≤ 1 √ n i,r≥1 nP i P r e − δ 4 nP i Pr = O ln n √ n , where for the inner sum (w.r.t. t) we used (27). Similarly Cases 2 give rise to triple sums of order O ln n n . The same is true for Case 3, which is seen by upper bounding the triple sums by 1 n i,r,t≥1 (nP 2 i ) α (nP r P t ) α e − δ 4 n(P 2 i +PrPt) = 1 n i≥1 (nP 2 i ) α e − δ 4 nP 2 i r,t≥1 (nP r P t ) α e − δ 4 nPrPt = O ln n n , where α ∈ {1/2, 1}. Finally, the following estimates i,r≥1 nP 2 i P r e − δ 4 n(P 2 i +P i Pr) = 1 √ n i≥1 (nP 2 i ) 1/2 e − δ 4 nP 2 i r≥1 nP i P r e − δ 4 nP i Pr = O ln n √ n ,(32) i,r≥1 P i P r e − δ 4 n(P 2 i +P i Pr) ≤ 1 n 3 4 i≥1 (nP 2 i ) 1/4 e − δ 4 nP 2 i r≥1 (nP i P r ) 1/2 e − δ 4 nP i Pr = O ln n n 3 4 , deal with Cases 6. The total contribution of error terms is therefore of order O ln n √ n . We are left with dealing with the sums of the main terms of Cases 1 and 5, and (22). Note that Case 1 has a twin case, Cov (X (n) i,r , X (n) r,t ) = Cov (X (n) r,i , X (n) t,r ). Denote H(i, j, k) = (e nP i P j P k − 1)e −nP i P j −nP j P k and H • (i, j) = (e nP i P j (P i +P j ) − 1)e −2nP i P j . Observe that i =j (e nP 2 i P j − 1)(e nP i P 2 j − 1)e −2nP i P j ≤ i,j nP 2 i P j e nP 2 i P j nP i P 2 j e nP i P 2 j e −2nP i P j ≤ i,j n 2 P 3 i P 3 j e −2δnP i P j = OH(i, j, k) + i,j≥1 |{i,j}|=2 H • (i, j) ∼ 2 i,j,k≥1 j ∈{i,k} H(i, j, k) = 2 i,j,k≥1 H(i, j, k) − 4 O ln n √ n i,j≥1 H(i, i, j) +2 O 1 √ n i≥1 H(i, i, i), where we have estimated two of the sums using (27) e −nP i Pr − e −2nP i Pr = i,r≥1 1 − e −2nP i Pr − i,r≥1 1 − e −nP i Pr =G(2np 2 ) −G(np 2 ), which, as we have seen, is an asymptotic equivalent of i,r≥1 Var X (n) i,r , is evaluated in Appendix A.3, confirming S (n) 2 as given in (2). This completes the proof of the lemma, and also proves (6), as we have seen, that multiple sums of covariances Cov (X (n) i 1 ,j 1 , X (n) i 2 ,j 2 ) with i 1 = j 1 , but i 2 = j 2 , are negligible. More than two pairs of identical letters We now turn to the case of k pairs (i 1 , i 1 ), . . . , (i k , i k ), allowing for k > 2. Lemma 4.11 Fix a set I := {i 1 , i 2 , . . . , i k } of size k, assuming i k < . . . < i 1 , and thus P i 1 < . . . < P i k . Let ε := i∈I P 2 i . Then we have P(X (n) i,i = 0, i ∈ I) = k+1 j=1 C j λ n j = O( 1 n ), for ε ≥ 3 ln n n , C 1 λ n 1 + O( 1 n ), for ε ≤ 1/4,(33) with all λ i different, and error terms holding uniformly in k. More precisely, we have λ 1 > |λ j | > 0 for 2 ≤ j ≤ k + 1, and −P i k < λ k+1 < −P i k−1 < λ k < . . . < −P i 1 < λ 2 < 0. Moreover, λ 1 = 1 − i∈I P 2 i + i∈I P 3 i + O(ε 2 ),(34)C 1 = 1 + O(ε),(35) again with error terms holding uniformly in k. Proof. As before, we let e := N \ I and P e := 1 − i∈I P i , and introduce the matrix Π =         P e P i 1 P i 2 · · · P i k P e 0 P i 2 · · · P i k P e P i 1 0 · · · P i k . . . . . . . . . . . . . . . P e P i 1 P i 2 · · · 0         . In order to find eigenvalues and corresponding left and right eigenvectors ofΠ, we have to solve the following systems, λ = P e 1 + j∈I β j λβ i = P i 1 + j∈I\{i} β j , i ∈ I λ = P e 1 + j∈I P j µ j λµ i = 1 + j∈I\{i} P j µ j , i ∈ I(36) Note that (µ i ) i∈I solves the right system if and only if (β i ) i∈I = (P i µ i ) i∈I solves the left system. From the left system we easily obtain β i = λP i P e (λ + P i ) , for i ∈ I,(37) and, upon inserting into the first equation of the left system, λ = P e + i∈I λP i λ + P i = P e + i∈I P i − i∈I P 2 i λ + P i = 1 − i∈I P 2 i λ + P i .(38) There are at most k + 1 different solutions to (38), those being exactly the eigenvalues ofΠ. Defining f (λ) := λ − 1 + i∈I P 2 i λ+P i , we observe the following k + 1 sign changes on the interval [−P i k , 1], i∈I P 2 i 1+P i = f (1) > 0 > f (0) = −P e , lim λ −P i f (λ) = −∞, lim λ −P i f (λ) = ∞, for i ∈ I, from which we obtain the result regarding the locations of the eigenvalues. We continue with the proof of (33). The first estimate, O( 1 n ), directly follows from (13). For the second, note that ε ≤ 1/4 implies P i k ≤ 1/2. We then use (26) and S and w as defined in the proof of Lemma 4.1. Then for some orthogonal matrix Q the matrixΠ := QSΠS −1 Q −1 is diagonal and satisfies ρ(Π) = |λ k+1 | < P i k ≤ 1/2, and |λ k+1−j | < P i k−j ≤ q j 2 for j ≥ 1, implying Π n F ≤ 1 1−q 2 −n . This leads to πΠ n 1 = w(SΠS −1 ) n−1 w t = wQ −1Πn−1 Qw t ≤ Π n−1 2 ≤ Π n−1 F ≤ 2 1−q 2 −n = O 1 n . Turning now to asymptotic expansions of λ 1 and C 1 , we first provide a convenient representation of the latter in the spirit of (38), starting from (18), C 1 = 1 + i∈I β i 1 + P e i∈I β 2 i P i = λ 1 P e + i∈I λ 2 1 P i (λ 1 +P i ) 2 = λ 1 λ 1 − i∈I P i λ 1 λ 1 +P i − λ 2 1 (λ 1 +P i ) 2 = 1 1 − i∈I P 2 i (λ 1 +P i ) 2 . (39) Note that asymptotic estimates of higher order than those given in (34) and (35) could easily be obtained by Algorithm 1, but as we need error terms uniformly in k, we choose another route. We assume ε ≤ 1/9 and observe f (1 − 3 2 ε) = − 3 2 ε + i∈I P 2 i 1− 3 2 ε+P i < − 3 2 ε + i∈I P 2 i 1− 3 2 ε ≤ − 3 2 ε + 6 5 ε ≤ 0, which implies λ 1 > 1 − 3 2 ε. Using λ 1 + P i ≤ 1 + 1/3 = 4/3 in equation (38), we obtain λ 1 = 1 − i∈I P 2 i λ 1 + P i ≤ 1 − 3 4 i∈I P 2 i = 1 − 3 4 ε. Next we employ 1 − x ≤ 1 1+x ≤ 1 − x + 2x 2 , holding for x ∈ [−1/2, 1], in λ 1 ≤ 1 − i∈I P 2 i 1 − 3 4 ε + P i ≤ 1 − i∈I P 2 i + i∈I P 3 i − 3 4 ε 2 , λ 1 ≥ 1 − i∈I P 2 i 1 − 3 2 ε + P i ≥ 1 − i∈I P 2 i + i∈I P 3 i − 3 2 ε 2 − 2 i∈I P 4 i + 6ε i∈I P 3 i − 9 2 ε 3 ≥ 1 − i∈I P 2 i + i∈I P 3 i − 4ε 2 , proving (34). Similarly, (35) follows from (39), using λ 1 + P i ≥ 1 − 3 2 ε ≥ 5/6: 1 ≤ C 1 = 1 − i∈I P 2 i (λ 1 +P i ) 2 −1 ≤ 1 − 36 25 i∈I P 2 i −1 ≤ 1 + 2ε. This completes the proof of the lemma. Proof of Theorem 2.2: We first consider x i = 0 for all i ∈ I. Letting ε := i∈I P 2 i again, by the previous lemma we have P(X (n) i,i = 0, i ∈ I) = C 1 λ n 1 + k+1 j=2 C j λ n j = i∈I e −n(P 2 i −P 3 i ) 1 + O(ε) + nO(ε 2 ) + O 1 n . By letting P j → 0 for j ∈ I \ {i}, we obtain P(X (n) i,i = 0) = e −n(P 2 i −P 3 i ) 1 + O(P 2 i ) + nO(P 4 i ) + O 1 n , and finally P(X (n) i,i = 0, i ∈ I) − i∈I P(X (n) i,i = 0) = i∈I e −n(P 2 i −P 3 i ) O(ε) + nO(ε 2 ) + O 1 n = O 1 n ,(40) using i∈I e −n(P 2 i −P 3 i ) ≤ e −P(X (n) i,i = x i , i ∈ I ) = P(X (n) i,i = 0, i ∈ I \ {j}) − P(X (n) i,i = 0, i ∈ I ), i∈I P(X (n) i,i = x i ) = i∈I \{j} P(X (n) i,i = 0) − i∈I P(X (n) i,i = 0), so, by taking the difference of these equations, we have P(X (n) i,i = x i , i ∈ I ) − i∈I P(X (n) i,i = x i ) = O 1 n . Similarly, by induction on κ := i∈I x i , we can prove that (40) holds for all I with |I | = k + 1 and all x ∈ {0, 1} k+1 . Clearly the error terms O 1 n may now suffer from dependence on |I|, but not on I, as the values {P i } i∈I did not enter the proof. Some insights regarding the dependence of λ 1 and C 1 on the matrixΠ Fix a finite non-empty set of forbidden pairs I := {(k i , m i ) : i ∈ I} and let J := i∈I {k i , m i }. We would like to better understand λ 1 and C 1 given in (14), as the examples in the proof of Lemma 4.9 constituted by Cases 1 to 6 suggest that there may be a simple relationship between λ 1 and C 1 . Usinḡ Π introduced shortly before Algorithm 1, we defineΠ := 1p −Π, i.e., Π k,m := P m , (k, m) ∈ I, 0, else, and use it to define ψ 1 :=p1 = j∈J P j = 1 − P e and ψ i+1 :=pΠ i 1 = k 0 ,...,k i :(k 0 ,k 1 ),...,(k i−1 ,k i )∈I P k 0 P k 1 · · · P k i , for i ≥ 1. Note that ψ 2 = ε, with ε introduced in Lemma 4.1, and ψ 3 is a generalization of Q introduced in (29). Moreover ψ i ≤ (1 − P e ) i holds for i ≥ 1. We can express λ 1 and C 1 in terms of (ψ i ) i≥2 as follows, λ 1 = 1 − ψ 2 + ψ 3 − (ψ 2 2 + ψ 4 ) + (3ψ 2 ψ 3 + ψ 5 ) − (2ψ 3 2 + 4ψ 2 ψ 4 + 2ψ 2 3 + ψ 6 ) + (10ψ 2 2 ψ 3 + 5ψ 2 ψ 5 + 5ψ 3 ψ 4 + ψ 7 ) − (5ψ 4 2 + 15ψ 2 2 ψ 4 + 15ψ 2 ψ 2 3 + 6ψ 2 ψ 6 + 6ψ 3 ψ 5 + 3ψ 2 3 + ψ 8 ) + (35ψ 3 2 ψ 3 + 21ψ 2 2 ψ 5 + 42ψ 2 ψ 3 ψ 4 + 7ψ 3 3 + 7ψ 2 ψ 7 + 7ψ 3 ψ 6 + 7ψ 4 ψ 5 + ψ 9 ) + O * 10 , C 1 = 1 + ψ 2 − 2ψ 3 + 3(ψ 2 2 + ψ 4 ) − 4(3ψ 2 ψ 3 + ψ 5 ) + 5(2ψ 3 2 + 4ψ 2 ψ 4 + 2ψ 2 3 + ψ 6 ) + O * 7 , and further terms of λ 1 can be extracted, using gfun, from the equation λ = 1 − vψ 1 1 − Ψ( v λ ) ,(41) where Ψ(z) = − i≥1 ψ i (−z) i , the solution being a power series in v, starting λ(v) = 1 − ψ 2 v 2 + ψ 3 v 3 + . . ., upon substituting v = 1. Furthermore an analogous power series for C 1 satisfies C(v) = λ(v) − vλ (v).(42) Also the generating function of the probabilities p (n) I := P(X (n) k i ,m i = 0, i ∈ I) can be expressed in terms of Ψ, and in fact again in terms of (ψ i ) i≥2 : n≥0 p (n) I z n = 1 1 − (1 − ψ 1 )z − Ψ(z) (43) = 1 + z + (1 − ψ 2 )z 2 + (1 − 2ψ 2 + ψ 3 )z 3 + (1 − 3ψ 2 + 2ψ 3 + ψ 2 2 − ψ 4 )z 4 + (1 − 4ψ 2 + 3ψ 3 − 2ψ 4 + 3ψ 2 2 + ψ 5 − 2ψ 2 ψ 3 )z 5 + O(z 6 ). For a proof of (41) we would start with (15) and (16), i.e., λ = P e (1 + β1), β = 1 λ βΠ +p , and replace P j with vP j for j ∈ J, leading to λ = (1 − vψ 1 )(1 + β1), β = v λ βΠ +p . Iterating the latter yields β = v λ k≥0 v λ kpΠ k , and plugging this into the former we obtain λ = (1 − vψ 1 ) 1 + k≥0 v λ k+1pΠ k 1 . Denoting F (z) := 1 + k≥0 z k+1pΠk 1 and usinḡ Π k = (1p −Π) k = (−Π) k + k−1 =0 (−Π) k−1− 1pΠ , we derive F (z) = 1 + k≥0 z k+1p (−Π) k 1 + k≥1 z k+1 k−1 =0 p(−Π) k−1− 1pΠ 1 = 1 + Ψ(z) + m≥0 z m+1pΠ 1 ≥0 z +1p (−Π) m 1 = 1 + Ψ(z) + (F (z) − 1)Ψ(z) = 1 + Ψ(z)F (z), from which we deduce F (z) = 1 1−Ψ(z) . For a proof of (42), which involves λ (v), we use (41), differentiating λ − λΨ( v λ ) − 1 + vψ 1 = 0 w.r.t. v, which leads to 0 = λ − λ Ψ( v λ ) − λ 1 λ − λ v λ 2 Ψ ( v λ ) + ψ 1 = λ 1−vψ 1 λ − 1 − λ v λ Ψ ( v λ ) + ψ 1 = λ λ − 1 − λ v λ (Ψ ( v λ ) − ψ 1 ), and finally to Ψ v λ − ψ 1 = λ λ − vλ .(44) Next we iterate µ = 1 λ vΠµ + 1 obtained from (17), which yields µ = 1 λ k≥0 v λ kΠ k 1. Note that we have βµ = k, ≥0 v λ k+1pΠ k 1 λ v λ Π 1 = 1 λ k, ≥0 v λ k+ +1pΠ k+ 1 = 1 λ m≥0 (m + 1) v λ m+1pΠ m 1 = 1 λ v λ F ( v λ ) = 1 λ v λ Ψ ( v λ ) (1 − Ψ( v λ )) 2 = vΨ ( v λ ) (1 − vψ 1 ) 2 . This we plug into (18), thus establishing (42), C(v) = 1 + β1 1 + (1−vψ 1 )βµ = λ 1−vψ 1 + (1−vψ 1 ) 2 βµ = λ 1 − vψ 1 + vΨ ( v λ ) = λ 1 + v λ λ−vλ = λ(v) − vλ (v). The proof of (43) starts with P n (X (n) k i ,m i = 0, i ∈ I) = pΠ n−1 1 for n ≥ 1, with p = [P e ,p], uses p(1p −Π) n−1 1 =pΠ n−1 1 = ψ n , and then uses ideas from the proof of (41). Let us finally remark that using the function Ψ(z) := − i≥2 ψ i (−z) i , equations (41) and (43) can be recast in the following, somewhat simpler forms, λ = 1 1 − Ψ( v λ ) , n≥0 p (n) I z n = 1 1 − z − Ψ(z) . We will meet the latter generating function again in Section 5, where, employing a combinatorial approach, we are able to show that in case of one or two forbidden pairs, the generating function is rational with a denominator of degree at most three, which allows for very explicit expressions for the coefficients. is Gaussian Limiting distribution of X Proof. Heuristic We assume asymptotic independence of X i,j , as the covariance total contribution is O(1) Let n 2 q u * = 1, u * = ln(n 2 )/L, andũ := u * . The total number of distinct X i,j cells up to diagonalũ is given bỹ u 1 (v − 1) − ũ 2 =ũ 2 2 −ũ 2 − ũ 2 ∼ ln(n) 2 2L 2 + O(ln(n)) , Now set η := u − u * , P[X i,j (0) = 1] ∼ e −e −Lη , u := i + j,ũ − u * = u * − u * =: η 1 , −1 ≤ η 1 ≤ 0, u + 1 − u * = u * + 1 − u * = u * − u * =: η 2 , 0 ≤ η 2 ≤ 1. The number of cells on diag- onal u is given by c(u) = u − 1 − [[even(u)]]. We must now add the contribution of full cells corresponding to diagonalũ + 1: Bin c(ũ + 1), 1 − e −e −Lη 2 , which is asymptotically Gaussian. Similarly for the contributions ofũ + i, i ≥ 2, we must now subtract the contribution of full cells corresponding to diagonalũ: − Bin c(ũ), e −e −Lη 1 , which is asymptotically Gaussian. Similarly for the contributions ofũ − i, i ≥ 1. So the asymptotic total random contribution is Gaussian. The result of a simulation with p = 1/4, n = 500000, and number of simulated words N = 200000 can be seen in Figure 3. The observed meanX is also shown in Figure 3. The fit is excellent. Another approach to the asymptotic distribution of X (n) 3 Let us turn to the couple (i, j). Let us first assume full independence. We first consider X j,k ). So we first proceed as in the (i, i) case. Next we will deal with X (n) 3 := i =j X (n) i,j i.e. subtract the contribution of the (i, i) couples. Finally, we will deal with the dependence property. We proceed as in Sec. 3.1. Set Q 2 := 1 q , L 2 := ln(Q 2 ) = − ln(q), n * := n 2 k, log := log Q 2 , L 1 = 2L 2 , P(X i,j (0) = 1) ∼P (i, j) = e −n 2 q i+j , V v = i j v k=0 (−1) k v k P (i, j) k = i j v k=1 (−1) k+1 v k 1 −P (i, j) k , i j 1 −P (i, j) k = u=2 (u − 1) 1 − e −n * q u = u=1 (u − 1) 1 − e −n * q u = u=1 u 1 − e −n * q u − u=1 1 − e −n * q u := V v,1 − V v,2 , The second summation leads, using the previous results on (i, i) to u=1 1 − e −n * q u = U 1 (Q 2 ) + log Q 2 k + β k (Q 2 ), where U 1 (Q 2 ) := ln(n 2 ) L 2 − 1 2 + γ L 2 , β k (Q 2 ) = − 1 L 2 ∈Z\{0} Γ 2i π L 2 e −2i π log Q 2 (kn 2 ) , This leads to V v,2 = − v k=1 (−1) k+1 v k [U 1 (Q 2 ) + log Q 2 k + β k (Q 2 )], omitting the periodic contribution, V v,2 = − U 1 (Q 2 ) + C 1 (v), C 1 (1) = 0, C 1 (2) = log Q 2 2 V 1,2 = − U 1 (Q 2 ) − β 1 (Q 2 ), V 2,2 = − U 1 (Q 2 ) + log Q 2 2 − 2 k=1 (−1) k+1 2 k β k (Q 2 ). From Appendix A.2, the first summation gives V v,1 = v k=1 (−1) k+1 v k ln(n 2 ) 2 + 2 ln(n 2 ) ln(k) + ln(k) 2 2L 2 2 + γ(ln(n 2 ) + ln(k)) L 2 2 − − π 2 /12 + γ 2 /2 L 2 2 + 1 12 +β v,1 , =U 2 (Q 2 ) + B v,1 +β v,1 , where U 2 (Q 2 ) = ln(n 2 ) 2 2L 2 2 + γ ln(n 2 ) L 2 2 − − π 2 /12 + γ 2 /2 L 2 2 + 1 12 = ln(n) 2 2L 2 2 + ln(n) γ + 2 ln (p) + 2 L 2 L 2 2 + 1/12 π 2 + 6 γ 2 + 23 L 2 2 + 24 γ ln (p) + 24 γ L 2 + 24 (ln (p)) 2 + 48 ln (p) L 2 L 2 2 , Now define B v,1 := v k=2 (−1) k+1 v k 2 ln(n 2 ) ln(k) + ln(k) 2 2L 2 2 + γ ln(k) L 2 2 = C 2 (v) ln(n) + C 3 (v), with, say, C 2 (1) = C 3 (1) = 0, and B 1,1 =0, B 2,1 = − 2 ln(n 2 ) ln(2) + ln(2) 2 2L 2 2 + γ ln(2) L 2 2 , β v,1 = v k=1 (−1) k+1 v k β k,1 ,β 1,1 = β 1,1 ,β 2,1 = 2β 1,1 − β 2,1 , β k,1 = 1 L 2 ∈Z\{0}   − ln(n 2 k) L 2 Γ 2i π L 2 + Γ 2i π L 2 L 2   e −2i π log(knp 2 ) , V 1,1 =U 2 (Q 2 ) + β 1,1 , V 2,1 =U 2 (Q 2 ) + B 2,1 +β 2,1 , Finally, omitting the periodic contribution, V 1 =V 1,1 + V 1,2 ∼ U 2 (Q 2 ) − U 1 (Q 2 ), V 2 =V 2,1 + V 2,2 ∼ U 2 (Q 2 ) − U 1 (Q 2 ) + B 2,1 + C 1 (2), V v ∼U 2 (Q 2 ) − U 1 (Q 2 ) + C 2 (v) ln(n) + C 4 (v), C 4 (v) = C 1 (v) + C 3 (v), C 4 (1) = 0, We can now compute the mean M 1 (X (n) 2 ) and V AR 1 (X (n) 2 ) (see (9)). We get M 1 (X (n) 2 ) = V 1 = U 2 (Q 2 ) − U 1 (Q 2 ) + β 1,1 − β 1 (Q 2 ), V AR 1 (X (n) 2 ) = κ 2 (X (n) 2 ) = V 1 − V 2 = −B 2,1 − log Q 2 2 + β 1,1 − β 1 (Q 2 ) −β 2,1 + 2 k=1 (−1) k+1 2 k β k (Q 2 ), = 2 ln(n 2 ) ln(2) + ln(2) 2 2L 2 2 + γ ln(2) L 2 2 − log Q 2 2 − β 1,1 + β 2,1 + β 1 (Q 2 ) − β 2 (Q 2 ) Subtraction of the contribution from all couples (i, i) We obtain for the non-periodic contribution M N P (X (n) 3 ) M N P (X (n) 3 ) := U 2 (Q 2 ) − U 1 (Q 2 ) − U 1 (Q 1 ) = ln(n) 2 2L 2 2 + ln(n) 1/2 L 2 + 2 γ + 4 ln (p) L 2 2 − 1/12 −π 2 − 6 γ 2 + L 2 2 − 24 γ ln (p) − 6 γ L 2 − 24 (ln (p)) 2 − 12 ln (p) L 2 L 2 2 , Next, for the periodic contribution M P (X (n) 3 ), we obtain M P (X (n) 3 ) = β 1,1 − β 1 (Q 2 ) − β 1 (Q 1 ) = 1 L 2 ∈Z\{0}   − ln(n 2 ) L 2 Γ 2i π L 2 + Γ 2i π L 2 L 2   e −2i π log Q 2 (np 2 ) + 1 L 2 ∈Z\{0} Γ 2i π L 2 e −2i π log Q 2 (np 2 ) + 1 L 1 ∈Z\{0} Γ 2i π L 1 e −2i π log Q 1 (np 2 ) . Similarly, the variance of X (without dependence property) is given by V AR 2 (X (n) 3 ) = 2 ln(n 2 ) ln(2) + ln(2) 2 2L 2 2 + γ ln(2) L 2 2 − log Q 2 2 − β 1,1 + β 2,1 + β 1 (Q 2 ) − β 2 (Q 2 ) − [log Q 1 2 + β 2 (Q 1 ) − β 1 (Q 1 )]. After some algebra, we can check that this is identical to Thm 2.1 (except for the term T (n) 2 , of course). Also the mean is identical with the expression given in Archibald and al. [1], Thm 3. Our approach is simple and general. Note again that the mean does not rely on the dependence property. The dominant terms, without the periodic components We have, successively (we keep only dominant terms), and setting X (n) i,j := X (n) i,j − E(X (n) i,j ), µ k (X) := E[(X − E(X)) k ], we have S n 2 (θ) = ∞ v=1 (−1) v+1 v e θ − 1 v V v , We first deal without dependence property, κ 2 (X (n) 2 ) = µ 2 (X (n) 2 ) = V AR 1 (X (n) 2 ) = −C 2 (2) ln(n) − C 4 (2), κ 3 (X (n) 2 ) = µ 3 (X (n) 2 ) = V 1 − 3V 2 + 2V 3 = [−3C 2 (2) + 2C 2 (3)] ln(n) + [−3C 4 (2) + 2C 4 (3)] = C 5 ln(n) + C 6 , say, but κ 2 (X (n) 2 ) = E     i =jX (n) i,j + iX (n) i,i   2   = κ 2 (X (n) 3 ) + κ 2 (X (n) 1 ) = κ 2 (X (n) 3 ) + C 7 , say , hence, V AR 2 (X (n) 3 ) = µ 2 (X (n) 3 ) = κ 2 (X (n) 3 ) = −C 2 (2) ln(n) − C 4 (2) − C 7 = −C 2 (2) ln(n) + C 8 , say , Now we calculate κ 3 (X (n) 2 ) = E     i =jX (n) i,j + iX (n) i,i   3   = κ 3 (X (n) 3 ) + κ 3 (X (n) 1 ) = κ 3 (X (n) 3 ) + C 9 , say , hence, κ 3 (X (n) 3 ) = µ 3 (X (n) 3 ) = C 5 ln(n) + C 6 − C 9 = C 5 ln(n) + C 10 , say . Now we deal with dependence property, σ 2 = V AR(X (n) 3 ) = κ 2 (X (n) 3 ) + T (n) 2 = −C 2 (2) ln(n) + C 11 , say ,Thenκ 3 (X (n) 3 ) is the new third cumulant, including the correlations, and T 3 is the correction, similar to T 2 , so thatκ 3 (X (n) 3 ) = κ 3 (X (n) 3 ) + T 3 = C 5 ln(n) + C 10 + T 3 . A bound such that |T 3 | = O(1) would be enough. We now have E exp θ X (n) 3 − M (X (n) 3 ) σ ∼ exp θ 2 2 + θ 3 3! C 5 ln(n) + C 10 + T 3 (−C 2 (2) ln(n) + C 11 ) 3/2 + . . . Let us, for instance, deal with the fourth cumulant. We have, without dependence κ 4 (X (n) 2 ) = V 1 − 7V 2 + 12V 3 − 6V 4 = C 12 ln(n) + C 13 , say , and µ 4 (X (n) 2 ) = κ 4 (X (n) 2 ) + 3κ 2 2 (X (n) 2 ), but µ 4 (X (n) 2 ) = µ 4 (X (n) 3 ) + µ 4 (X (n) 1 ) + 6µ 2 (X (n) 3 )µ 2 (X (n) 1 ), hence µ 4 (X (n) 3 ) = µ 4 (X (n) 2 ) − [µ 4 (X (n) 1 ) + 6µ 2 (X (n) 3 )µ 2 (X (n) 1 )] and µ 4 (X (n) 3 ) = κ 4 (X (n) 2 ) + 3κ 2 2 (X (n) 2 ) − µ 4 (X (n) 1 ) − 6µ 2 (X (n) 3 )µ 2 (X (n) 1 ); Now, with dependence, µ 4 (X (n) 3 ) = µ 4 (X (n) 3 ) + T 4 , κ 4 (X (n) 3 ) = κ 4 (X (n) 2 ) + 3µ 2 2 (X (n) 2 ) − µ 4 (X (n) 1 ) − 6µ 2 (X (n) 3 )µ 2 (X (n) 1 ) + T 4 − 3µ 2 2 (X (n) 3 ) − 6µ 2 (X (n) 3 )T 2 − 3T 2 2 = κ 4 (X (n) 2 ) + 3µ 2 2 (X (n) 1 ) − µ 4 (X (n) 1 ) − 6µ 2 (X (n) 3 )T 2 + T 4 − 3T 2 2 = C 14 ln(n) + C 15 + T 4 − 3T 2 2 , say. Again a bound such that |T 4 | = O(1) would be enough. We proceed similarly for all cumulants. This leads to the asymptotic Gaussian distribution. Combinatorial Pattern Matching Approach For a combinatorial approach, we utilize the methodology of the paper [2]. The full strength of this paper is not needed, because (in this case) we are only studying "reduced" sets of patterns, i.e., we are always analyzing patterns of length 2. So we only need to understand Sections 4.1 and 4.2 of [2]. We define X (n) as the total number of distinct (adjacent) pairs in a word Z 1 , . . . , Z n , and we have denote the roots of az 2 + bz + c. If these two roots are distinct, then X (n) = ∞ i=1 ∞ j=1 X (n) i,j Lemmas and notation about solving cubic formulas 1 az 2 + bz + c = 1 (1 − z/r)(1 − z/s) = s (s − r)(1 − z/r) + r (r − s)(1 − z/s) . The following lemma uses the notation from the "General cubic formula" section of https://en.wikipedia.org/wiki/Cubic_equation Lemma 5.2 Consider a, b, c and d = 1, and define ∆ 0 = b 2 − 3ac ∆ 1 = 2b 3 − 9abc + 27a 2 d C = ∆ 1 + ∆ 2 1 − 4∆ 3 0 2 1/3 ξ = −1 + √ −3 2 Let r = − 1 3a (b + C + ∆ 0 /C) and s = − 1 3a (b + ξC + ∆ 0 /(ξC)) and t = − 1 3a (b + ξ 2 C + ∆ 0 /(ξ 2 C)) denote the roots of az 3 + bz 2 + cz + d. If these three roots r, s, t are distinct, then 1 az 3 + bz 2 + cz + d = 1 (1 − z/r)(1 − z/s)(1 − z/t) = st (s − r)(t − r)(1 − z/r) + rt (r − s)(t − s)(1 − z/s) + rs (r − t)(s − t)(1 − z/t) . In all the instances that we use Lemma 5.2, the three roots r, s, t exist and are distinct, or they exist in the limit, i.e., there is a removable singularity that can be dealt with in a straightforward manner, by using continuity to remedy the removable singularity. Lemma 5.3 The roots of the polynomial in the denominators of the generating functions in Table 1 exist and are unique, or there is a removable singularity that can be defined by using continuity. Proof. The proof follows directly from Lemma 5.1. To understand the comment about removable singularity, consider (for instance) Table 1A. In the case when p = q = 1/2 and i = j = 1, we note that the "r" and "s" from Lemma 5.1 are identical, but nonetheless, the partial fraction decomposition holds in the limit. In this special case, the generating function 1/(1 − z + P i P j z 2 ) becomes 1 1 − z + z 2 /4 = 1 (1 − z/2) 2 = lim p→1/2 2p 2 (1 − 1 − 4p 2 ) 1 − 4p 2 1 − 2p 2 1− √ 1−4p 2 z − 2p 2 (1 + 1 − 4p 2 ) 1 − 4p 2 1 − 2p 2 1+ √ 1−4p 2 z In other words, the partial fraction decomposition and the coefficients of z n are valid in the limit in Table 1A, even in this special case. The other cases that have removable singularities are straightforward, and we do not comment on these further. Lemma 5.4 The roots of the polynomial in the denominators of the generating functions in Table 2 exist and are unique, or there is a removable singularity that can be defined by using continuity. Proof. The proof follows directly from Lemma 5.2. Given the values of a, b, c, d, we define the three roots r, s, t as in Lemma 5.2, and everything follows. To understand the comment about removable singularity, consider (for instance) Table 2E. In the case when p = q = 1/2 and i = k = 1, the "s" in Lemma 5.2 is undefined, but nonetheless, again Table 2E holds in the limit. In this special case, we have 1 1 − z + P 2 i z 2 1+P i z + P 2 k z 2 1+P k z = 2 + z 2 − z = lim p→1/2 (1 + pz)(1 + pz) × rs (r − t)(s − t)(1 − z/t) + rt (r − s)(t − s)(1 − z/s) + st (s − r)(t − r)(1 − z/r) In other words, as before, the partial fraction decomposition and the coefficients of z n are valid in the limit in this special case. Other cases that have removable singularities are similarly easy to handle. A gen. func. Lemma 5.5 For n ≥ 2, and for i = j, the probability that ij occurs (at least once) as an adjacent pattern in Z 1 , . . . , Z n is exactly 1 1−z+P i P j z 2 par. frac. 1− √ 1−4P i P j 2P i P j √ 1−4P i P j 1 − 2P i P j 1− √ 1−4P i P j z −1 − 1+ √ 1−4P i P j 2P i P j √ 1−4P i P j 1 − 2P i P j 1+ √ 1−4P i P j z −1 coeff. of z n (2P i P j ) n+1 √ 1−4P i P j 1 1− √ 1−4P i P j n+1 − 1 1+ √ 1−4P i P j n+1 B gen. func. 1 − z + P 2 i z 2 1+P i z −1 = 1+P i z 1−(1−P i )z−P i (1−P i )z 2 par. frac. 1 2 − 1+P i 2 √ (1−P i )(1+3P i ) 1 − −2P i (1−P i ) 1−P i + √ (1−P i )(1+3P i ) z −1 − 1 2 + 1+P i 2 √ (1−P i )(1+3P i ) 1 − −2P i (1−P i ) 1−P i − √ (1−P i )(1+3P i ) z −1 coeff. of z n 1 2 − 1+P i 2 √ (1−P i )(1+3P i ) −2P i (1−P i ) 1−P i + √ (1−P i )(1+3P i ) n − 1 2 + 1+P i 2 √ (1−P i )(1+3P i ) −2P i (1−P i ) 1−P i − √ (1−P i )(1+3P i ) n C gen. func. 1 1−z+P i P j z 2 +P k P z 2 par. frac. 1+ √ 1−4(PiPj+P k P ) 2 √ 1−4(PiPj+P k P ) 1− 2(PiPj+P k P ) 1− √ 1−4(PiPj+P k P ) z −1 − 1− √ 1−4(PiPj+P k P ) 2 √ 1−4(PiPj+P k P ) 1− 2(PiPj+P k P ) 1+ √ 1−4(PiPj+P k P ) z −1 coeff. of z n (2(P i P j +P k P )) n+1 √ 1−4(P i P j +P k P ) 1 1− √ 1−4(P i P j +P k P ) n+1 − 1 1+ √ 1−4(P i P j +P k P ) n+1 D gen. func. 1 − z + P i z(P i z+P z) 1+P i z −1 = 1+P i z 1−(1−P i )z+(P 2 i +P P i −P i )z 2 par. frac. 1 2 − 1+P i 2 √ (1−P i )(1+3P i )−4P i P 1 − −2P i (1−P i −P ) 1−P i + √ (1−P i )(1+3P i )−4P i P z −1 − 1 2 + 1+P i 2 √ (1−P i )(1+3P i )−4P i P 1 − −2P i (1−P i −P ) 1−P i − √ (1−P i )(1+3P i )−4P i P z −1 coeff. of z n 1 2 − 1+P i 2 √ (1−P i )(1+3P i )−4P i P −2P i (1−P i −P ) 1−P i + √ (1−P i )(1+3P i )−4P i P n − 1 2 + 1+P i 2 √ (1−P i )(1+3P i )−4P i P −2P i (1−P i −P ) 1−P i − √ (1−P i )(1+3P i )−4P i P nE[X (n) i,j ] = 1 − (2P i P j ) n+1 1 − 4P i P j 1 1 − 1 − 4P i P j n+1 − 1 1 + 1 − 4P i P j n+1 . Of course, for n < 2, we have E[X (n) i,j ] = 0. Proof. The proof of Lemma 5.5 is in subsection 5.3.1. E gen. func. 1 − z + P 2 i z 2 1+P i z + P 2 k z 2 1+P k z −1 = (1+P i z)(1+P k z) (1−z)(1+P i z)(1+P k z)+P 2 i z 2 (1+P k z)+P 2 k z 2 (1+P i z) a, b, c, d a = P i P k (P i + P k − 1), b = P 2 i + P i P k + P 2 k − P i − P k , c = P k + P i − 1, d = 1 par. frac. (1+P i z)(1+P k z) az 3 +bz 2 +cz+d = (1 + P i z)(1 + P k z) rs (r−t)(s−t)(1−z/t) + rt (r−s)(t−s)(1−z/s) + st (s−r)(t−r)(1−z/r) coeff. of z n (1+P i t)(1+P k t)rs (r−t)(s−t)t n + (1+P i s)(1+P k s)rt (r−s)(t−s)s n + (1+P i r)(1+P k r)st (s−r)(t−r)r n F gen. func. 1 − z + P 2 i z 2 1+P i z + P k P z 2 −1 = 1+P i z (1−z)(1+P i z)+P 2 i z 2 +P k P z 2 (1+P i z) a, b, c, d a = P i P k P , b = P i P i + P k P − P i , c = P i − 1, d = 1 par. frac. 1+P i z az 3 +bz 2 +cz+d = (1 + P i z) rs (r−t)(s−t)(1−z/t) + rt (r−s)(t−s)(1−z/s) + st (s−r)(t−r)(1−z/r) coeff. of z n (1+P i t)rs (r−t)(s−t)t n + (1+P i s)rt (r−s)(t−s)s n + (1+P i r)st (s−r)(t−r)r n G gen. func. 1 1−z+P i P j z 2 +P j P z 2 −P i P j P z 3 a, b, c, d a = −P i P j P , b = P i P j + P j P , c = −1, d = 1 par. frac. 1 az 3 +bz 2 +cz+d = rs (r−t)(s−t)(1−z/t) + rt (r−s)(t−s)(1−z/s) + st (s−r)(t−r)(1−z/r) coeff. of z n rs (r−t)(s−t)t n + rt (r−s)(t−s)s n + st (s−r)(t−r)r n H gen. func. 1 − z + 2P i P j z 2 1−P i P j z 2 − P 2 i P j z 3 1−P i P j z 2 − P i P 2 j z 3 1−P i P j z 2 −1 = 1−P i P j z 2 (1−z)(1−P i P j z 2 )+2P i P j z 2 −P 2 i P j z 3 −P i P 2 j z 3 a, b, c, d a = P i P j (1 − P i − P j ), b = P i P j , c = −1, d = 1 par. frac. Lemma 5.6 For n ≥ 2, the probability that ii occurs (at least once) as an adjacent pattern in Z 1 , . . . , Z n is exactly 1−P i P j z 2 az 3 +bz 2 +cz+d = (1 − P i P j z 2 ) rs (r−t)(s−t)(1−z/t) + rt (r−s)(t−s)(1−z/s) + st (s−r)(t−r)(1−z/r) coeff. of z n (1−P i P j t 2 )rs (r−t)(s−t)t n + (1−P i P j s 2 )rt (r−s)(t−s)s n + (1−P i P j r 2 )st (s−r)(t−r)r nE[X (n) i,i ] = 1 − 1 2 − 1 + P i 2 (1 − P i )(1 + 3P i ) −2P i (1 − P i ) 1 − P i + (1 − P i )(1 + 3P i ) n + 1 2 + 1 + P i 2 (1 − P i )(1 + 3P i ) −2P i (1 − P i ) 1 − P i − (1 − P i )(1 + 3P i ) n . Again, for n < 2, we have E[X Main results By adding the results from Lemmas 5.5 and 5.6, we establish the following theorem: Theorem 5.1 For n ≥ 2, the mean number of distinct (adjacent) pairs in a word Z 1 , . . . , Z n is exactly E[X (n) ] = ∞ i=1 j =i 1 − (2P i P j ) n+1 1 − 4P i P j 1 1 − 1 − 4P i P j n+1 − 1 1 + 1 − 4P i P j n+1 + ∞ i=1 1 − 1 2 − 1 + P i 2 (1 − P i )(1 + 3P i ) −2P i (1 − P i ) 1 − P i + (1 − P i )(1 + 3P i ) n + 1 2 + 1 + P i 2 (1 − P i )(1 + 3P i ) −2P i (1 − P i ) 1 − P i − (1 − P i )(1 + 3P i ) n For n < 2, we have E[X (n) ] = 0. ξ(z, t) = P i P j tz 2 . Analysis of the average number of distinct (adjacent) pairs The generating function of the decorated texts (with z marking the length of the words, and t marking the number of decorated occurrences of ij, and the coefficients are the associated probabilities) is T (z, t) = 1 1 − A(z) − ξ(z, t) = 1 1 − z − P i P j tz 2 , where A(z) = 1 is the probability generating function of the set A of all finite-length words. Now we use F (z, x) to denote the bivariate probability generating function of occurrences of ij (with z marking the length of the words, and x marking the number of occurrences of ij, and the coefficients are the associated probabilities), i.e., we define . . . , Z n has exactly k occurrences of ij as a subword)x k z n . F (z, x) := ∞ n=0 ∞ k=0 P (Z 1 , We know from inclusion-exclusion (see [4,Chapter 3] or [2]) that F (z, x) = T (z, x − 1), so we obtain F (z, x) = T (z, x − 1) = 1 1 − z − P i P j (x − 1)z 2 . The probability generating function of words with zero occurrences of pattern ij can be obtained by considering the case k = 0, corresponding to the coefficients of x 0 . To extract those coefficients, we can evaluate F (z, x) at x = 0, and we obtain [x 0 ]F (z, x) = F (z, 0) = 1 1 − z + P i P j z 2 , so, finally, the probability generating function of the words with at least one occurrence of ij is ∞ n=0 E[X (n) i,j ]z n = 1 1 − z − 1 1 − z + P i P j z 2 and it follows, using Table 1A, that Now we fix i and we analyze the occurrences of the pattern ii. The clusters have the form ii · · · i, i.e., they are all words that consist of 2 or more consecutive occurrences of i. So the generating function ξ(z, t) of the set of clusters C = {ii, iii, iiii, iiiii, . . .} becomes ∞ n=0 E[X (n) i,j ]z n = ∞ n=0 z n + 2P i P j (1 + 1 − 4P i P j ) 1 − 4P i P j ∞ n=0 2P i P j 1 + 1 − 4P i P j n z n − 2P i P j (1 − 1 − 4P i P j ) 1 − 4P i P j ∞ n=0 2P i P j 1 − 1 − 4P i P jξ(z, t) = P 2 i tz 2 1 − P i tz . The analysis is similar to the reasoning in subsection 5.3.1, and we get ∞ n=0 E[X (n) i,i ]z n = 1 1 − z − 1 1 − z + P 2 i z 2 1+P i z and then, using Table 1B, we have ∞ n=0 E[X (n) i,i ]z n = 1 1 − z − (1 + P i z)(−1 + P i + (1 − P i )(1 + 3P i ) ) 2 (1 − P i )(1 + 3P i ) 1 − −2P i (1−P i ) 1−P i + √ (1−P i )(1+3P i ) z + (1 + P i z)(−1 + P i − (1 − P i )(1 + 3P i ) ) 2 (1 − P i )(1 + 3P i ) 1 − −2P i (1−P i ) 1−P i − √ (1−P i )(1+3P i ) z and we conclude with the exact expression for E[X Analysis of the second moment of the number of distinct (adjacent) pairs Now we study the second moment of X (n) , namely, E[(X (n) ) 2 ]. We have (X (n) ) 2 = ∞ i=1 ∞ j=1 X (n) i,j ∞ k=1 ∞ =1 X (n) k, so the second moment is, by linearity of expectation, E[(X (n) ) 2 ] = ∞ i=1 ∞ j=1 ∞ k=1 ∞ =1 E[X (n) i,j X (n) k, ]. We break the analysis into 4 regimes, namely: • i = j and k = In the case i = j and k = , we have two possibilities, namely, either i = j = k = or i = j = k = . 5.4.1.1 i = j = k = In the case i = j = k = , we have X (n) i,j X (n) k, = X (n) i,i , so we get E[X (n) i,j X (n) k, ] = E[X (n) i,i ], which we already handled in Lemma 5.6. 5.4.1.2 i = j = k = In the case i = j = k = , we need to analyze the occurrences of the patterns ii and kk. The clusters each have the form ii · · · i or kk · · · k, i.e., they are all words that consist of 2 or more consecutive occurrences of i, or consist of 2 or more consecutive occurrences of k. So the generating function ξ(z, t, u) of the set of clusters C = {ii, iii, iiii, iiiii, . . . , kk, kkk, kkkk, kkkkk, . . .} becomes ξ(z, t, u) = P 2 i tz 2 1 − P i tz + P 2 k uz 2 1 − P k uz (with z marking the length of the words, and t marking the number of decorated occurrences of ii, and u marking the number of decorated occurrences of kk, and the coefficients are the associated probabilities). The methodology now proceeds in a very similar way to the method from Section 5.3.1, but ξ, T , and F all have an additional variable, as compared to that earlier (more simple) analysis. We have T (z, t, u) = 1 1 − A(z) − ξ(z, t, u) = 1 1 − z − P 2 i tz 2 1−P i tz − P 2 k uz 2 1−P k uz , and it follows that the probability generating function of occurrences of ii and kk (with z marking the length of the words, and x marking the number of occurrences of ii, and y marking the number of occurrences of kk, and the coefficients are the associated probabilities) is F (z, x, y) = T (z, x − 1, y − 1) = 1 1 − z − P 2 i (x−1)z 2 1−P i (x−1)z − P 2 k (y−1)z 2 1−P k (y−1)z . It follows that the probability generating function of the words with at least one occurrence of ii and at least one occurrence of kk is ∞ n=0 E[X (n) i,j X (n) k, ]z n = ∞ n=0 E[X (n) i,i X (n) k,k ]z n = 1 1 − z − F (z, 0, 1) − F (z, 1, 0) + F (z, 0, 0) = 1 1 − z − 1 1 − z + P 2 i z 2 1+P i z − 1 1 − z + P 2 k z 2 1+P k z + 1 1 − z + P 2 i z 2 1+P i z + P 2 k z 2 1+P k z The partial fraction decomposition for the second term is given in Table 1B. The third term is the same as the second term, using k instead of i. The partial fraction decomposition for the fourth term is given in Table 2E. 5.4.2 i = j and k = 5.4.2.1 i = j and k and are distinct The clusters each have the form ii · · · i or k , i.e., they are all words that consist of either 2 or more consecutive occurrences of i, or simply the word k . So ξ(z, t, u) of the set of clusters C = {ii, iii, iiii, iiiii, . . . , k } becomes ξ(z, t, u) = P 2 i tz 2 1 − P i tz + P k P uz 2 (with z marking the length of the words, and t marking the number of decorated occurrences of ij, and u marking the number of decorated occurrences of k , and the coefficients are the associated probabilities). It follows that T (z, t, u) = 1 1 − z − P 2 i tz 2 1−P i tz − P k P uz 2 , and F (z, x, y) = T (z, x − 1, y − 1) = 1 1 − z − P 2 i (x−1)z 2 1−P i (x−1)z − P k P (y − 1)z 2 . It follows that the probability generating function of the words with at least one occurrence of ij and at least one occurrence of k is ∞ n=0 E[X (n) i,j X (n) k, ]z n = 1 1 − z − F (z, 0, 1) − F (z, 1, 0) + F (z, 0, 0) = 1 1 − z − 1 1 − z + P 2 i z 2 1+P i z − 1 1 − z + P k P z 2 + 1 1 − z + P 2 i z 2 1+P i z + P k P z 2 The partial fraction decomposition for the second term is given in Table 1B. The partial fraction decomposition for the third term is given in Table 1A, using k and instead of i and j. The partial fraction decomposition for the fourth term is given in Table 2F. 5.4.2.2 i = j = k = The clusters each have the form ii · · · i or ii · · · i , i.e., they are all words that consist of 2 or more consecutive occurrences of i, or of 1 or more consecutive occurrences of i followed by . So ξ(z, t, u) of the set of clusters C = {ii, iii, iiii, iiiii, . . . , i , ii , iii , iiii , . . .} becomes ξ(z, t, u) = P 2 i tz 2 1 − P i tz + P i P uz 2 1 − P i tz = P i z(P i tz + P uz) 1 − P i tz , and F (z, x, y) = 1 1 − z − P i z(P i (x−1)z+P (y−1)z) 1−P i (x−1)z . It follows that the probability generating function of the words with at least one occurrence of ij and at least one occurrence of k is ∞ n=0 E[X (n) i,j X (n) k, ]z n = 1 1 − z − F (z, 0, 1) − F (z, 1, 0) + F (z, 0, 0) = 1 1 − z − 1 1 − z + P 2 i z 2 1+P i z − 1 1 − z + P i P z 2 + 1 1 − z + P i z(P i z+P z) 1+P i z The partial fraction decomposition for the second term is given in Table 1B. The partial fraction decomposition for the third term is given in Table 1A, using instead of j. The partial fraction decomposition for the fourth term is given in Table 1D. 5.4.2.3 i = j = = k The cluster have the form ki · · · i or ii · · · i, i.e., they are all words that consist of k followed by 1 or more consecutive occurrences of i, or of 2 or more consecutive occurrences of i. So ξ(z, t, u) of the set of clusters C = {ki, kii, kiii, kiiii, . . . , ii, iii, iiii, iiiii, . . .} becomes ξ(z, t, u) = P i P k uz 2 1 − P i tz + P 2 i tz 2 1 − P i tz = P i z(P k uz + P i tz) 1 − P i tz , and F (z, x, y) = 1 1 − z − P i z(P k (y−1)z+P i (x−1)z) 1−P i (x−1)z . It follows that the probability generating function of the words with at least one occurrence of ij and at least one occurrence of k is ∞ n=0 E[X (n) i,j X (n) k, ]z n = 1 1 − z − F (z, 0, 1) − F (z, 1, 0) + F (z, 0, 0) = 1 1 − z − 1 1 − z + P 2 i z 2 1+P i z − 1 1 − z + P i P k z 2 + 1 1 − z + P i z(P k z+P i z) 1+P i z The partial fraction decomposition for the second term is given in Table 1B. The partial fraction decomposition for the third term is given in Table 1A, using k instead of j. The partial fraction decomposition for the fourth term is given in Table 1D, using k instead of . ξ(z, t, u) = P i P j tz 2 + P k P uz 2 , and F (z, x, y) = 1 1 − z − P i P j (x − 1)z 2 − P k P (y − 1)z 2 . It follows that the probability generating function of the words with at least one occurrence of ij and at least one occurrence of k is = 1 1 − z − 1 1 − z + P i P j z 2 − 1 1 − z + P k P z 2 + 1 1 − z + P i P j z 2 + P k P z 2 The partial fraction decomposition for the second term is given in Table 1A. The partial fraction decomposition for the third term is given in Table 1A, using k and instead of i and j. The partial fraction decomposition for the fourth term is given in Table 1C. i, ]z n = 1 1 − z − F (z, 0, 1) − F (z, 1, 0) + F (z, 0, 0) = 1 1 − z − 1 1 − z + P i P j z 2 − 1 1 − z + P i P z 2 + 1 1 − z + P i P j z 2 + P i P z 2 Exactly as in section 5.4.4.1 above: The partial fraction decomposition for the second term is given in Table 1A. The partial fraction decomposition for the third term is given in Table 1A, using instead of j. The partial fraction decomposition for the fourth term is given in Table 1C, using i instead of k. 5.4.4.3 k = j and i and are distinct The clusters are ij, ij and j . So ξ(z, t, u) of the set of clusters C = {ij, ij , j } becomes ξ(z, t, u) = P i P j tz 2 + P j P uz 2 + P i P j P tuz 3 , and F (z, x, y) = 1 1 − z − P i P j (x − 1)z 2 − P j P (y − 1)z 2 − P i P j P (x − 1)(y − 1)z 3 . It follows that the probability generating function of the words with at least one occurrence of ij and at least one occurrence of j is 1−z+P j P z 2 + 1 1 − z + P i P j z 2 + P j P z 2 − P i P j P z 3 The partial fraction decompositions for the second and third terms are given in Table 1A, once using j and instead of i and j. The partial fraction decomposition for the fourth term is given in Table 2G. = 1 1 − z − 1 1 − z + P i P j z 2 − 1 1 − z + P k P j z 2 + 1 1 − z + P i P j z 2 + P k P j z 2 Exactly as in section 5.4.4.1 above: The partial fraction decomposition for the second term is given in Table 1A. The partial fraction decomposition for the third term is given in Table 1A, using k instead of i. The partial fraction decomposition for the fourth term is given in Table 1C, using j instead of . i,j ], which we already handled in Lemma 5.5. 5.4.4.7 i = and j = k are distinct The clusters each have the form ijiji . . . or jijij . . .. So ξ(z, t, u) of the set of clusters C = {ij, iji, ijij, ijiji . . . , ji, jij, jiji, jijij . . .} becomes ξ(z, t, u) = P i P j tz 2 1 − P i P j tuz 2 + P 2 i P j tuz 3 1 − P i P j tuz 2 + P j P i uz 2 1 − P i P j tuz 2 + P i P 2 j tuz 3 1 − P i P j tuz 2 , and F (z, x, y) = 1/(1 − z − ξ(z, x − 1, y − 1)). It follows that the probability generating function of the words with at least one occurrence of ij and at least one occurrence of k is = 1 1 − z − 1 1 − z + P i P j z 2 − 1 1 − z + P j P i z 2 + 1 1 − z + P i P j z 2 1−P i P j z 2 − P 2 i P j z 3 1−P i P j z 2 + P j P i z 2 1−P i P j z 2 − P i P 2 j z 3 1−P i P j z 2 The partial fraction decomposition for the second and for the third term is given in Table 1A. The partial fraction decomposition for the fourth term is given in Table 2H. Noting that there are no other singularities to the right of −1, 0 , the error term in the following expansion can be chosen O(n −β ) with any fixed β > 0. G(n) ∼ 1 2L ln(n) + γ 2L + 1 2 − 1 2L ∈Z\{0} Γ χ 2 n − χ/2 . A.2 Let G := i i 1 − e −nq i . This has the MT, with FS −1, 0 and χ := 2iπ L F (s) = − Q s (1 − Q s ) 2 Γ(s) − 1 L 2 s 3 + γ L 2 s 2 + − π 2 /12 + γ 2 /2 L 2 + 1 12 1 s − Γ( χ) L 2 1 (s − χ) 2 − Γ ( χ) L 2 (s − χ) , G ∼ ln(n) 2 2L 2 + γ L 2 ln(n)+ π 2 /12 + γ 2 /2 L 2 − 1 12 + 1 L 2 ∈Z\{0} −Γ 2i π L ln(n) + Γ 2i π L e −2i π ln(n)/L . A.4 Set G := i,j,k (e nP i P j P k − 1)e −nP i P j −nP j P k . This leads to the MT, with FS −1, 0 F (s) = i,j,k ∞ 0 (e xP i P j P k − 1)e −xP i P j −xP j P k x s−1 dx = j P −s j i,k ∞ 0 e −y(P i +P k −P i P k ) − e −y(P i +P k ) y s−1 dy = q s p s (q s − 1) Γ(s) i,k (P i + P k − P i P k ) −s − (P i + P k ) −s = q p 2s Γ(s) q s − 1 F 1 (s), where F 1 (s) = i,k (q i + q k − pq i+k−1 ) −s − (q i + q k ) −s = i≥1 q −is   (2 − P i ) −s − 2 −s + 2 j≥1 (1 + q j − pq i+j−1 ) −s − (1 + q j ) −s   . Note that F 1 (s), being a general Dirichlet series in the variable −s, is analytic at least for σ = s < 1, since, using the MVT, we have (1 + q j − pq i+j−1 ) −σ − (1 + q j ) −σ ≤ |σ| pq i+j−1 (1 + q j ) 1+σ , and therefore |F 1 (s)| ≤ 2|σ| i≥1 q i(1−σ) j≥0 P j (1 + q j ) 1+σ < ∞. Moreover, F 1 (0) = 0, so to the right of the fundamental strip we have the singularities F (s) − F 1 (0) Ls , and F (s) −p −2 χ Γ( χ) L F 1 ( χ) s − χ , for ∈ Z \ {0}. This leads to G = F 1 (0) L + 1 L ∈Z\{0} Γ( χ)F 1 ( χ)(np 2 ) − χ + O(n −β ), with any fixed β < 1, where the constant term simplifies to F 1 (0) L = 1 L ln i,k≥1 q i + q k q i + q k − pq i+k−1 . B Proofs of equations (27) and (31) The estimate (27) follows from the following general result: If for some c < 1 a set P = {x i : i ∈ N} satisfies x i > 0 and x i+1 x i ≤ c for i ∈ N, then x∈P x α e −x < ∞. For a proof observe that there is a constant C α > 0 such that x α e −x ≤ min(x α , C α x −α ) for x > 0. Letx := (C α ) 1/(2α) . Then x∈P x α e −x ≤ x∈P∩ ]0,x] x α + x∈P∩[x,∞[ C α x −α ≤x α i≥0 c i + C αx −α i≥0 c i = 2 √ C α 1 − c . The estimate (31) can be deduced from (27), using β = 1, observing i,k≥1 (nP i P k ) α e −nP i P k = ≥2 ( − 1)(n p q P ) α e j (m) :=[[ couple (i, j) appears m times in the string of size n]j (0) := [[ couple (i, j) appears at least once in the string of size n] j (m) = 1] ∼ e −λ λ m m! , where λ = nP i P j . Figure 1 . 1i , the double sum of covariances only contributing O( 1 n ). This is different for Var X stemming from the quadruple sum of covariances of different pairs, of order Θ(1). All of S in terms of Fourier series in ln(np 2 ). A plot of the constant term of T Theorem 2.1 Let L := ln(1/q) and χ := 2πi/L, where i denotes the imaginary unit. Figure 1 : 1Plot of 2(1 − q)F 1 (0), showing the dependence of the constant term 2 L F 1 (0) on q. Conjecture 2. 3 3The same kind of asymptotic independence as in Theorem 2.2 holds for X(n) k i ,m i i≥1 , when the sets {k i , m i } are pairwise disjoint. ]. This leads us to the following conjecture. Conjecture 3. 1 1For any k ∈ N we have lim n→∞ (E|X k − E|ξ (n) − Eξ (n) | k ) = 0. Theorem 3. 4 4If Conjecture 3.3 holds, the asymptotic distribution f (η) of X (n) 1 Figure 2 : 2Comparison between f (η) (line) and the simulation of X (n) 1 e a+b − 1) = (e a − 1) + (e b − 1) + (e a − 1)(e b − 1) imply i =j H • (i, j) = 2 i =j H(i, j, i) + O ln n n . Therefore we have 2 i,j,k≥1 |{i,j,k}|=3 Remark 4 .3 410 Along the lines of the two preceding proofs an independent proof of Theorem 3.6 could easily be furnished. We would use(13), (19), (20) to identify i≥1 (1−e −nP 2 i ) and i =j (1−e −nP i P j ) as asymptotic equivalents of EX ∼G(np 2 )−G(np 2 ), with G,G from Appendices A.1 and A.3. nε(1−P 1 ) , and the fact that e −x(1−P 1 ) (x + x 2 ) is bounded for x ≥ 0. Clearly, equation (40) holds for I = {i} and all x i ∈ {0, 1}. Assume that equation (40) has been shown for all I with |I| = k. Consider I with |I | = k + 1. Then, as we have just shown, equation (40) holds for I when i∈I x i = 0. It also holds when i∈I x i = 1: If x j = 1, x i = 0 for i ∈ I \ {j}, then 750.19 and observed variance s 2 3 (n) ≈ 130.05 are very close to EX Figure 3 : 3Comparison between Gaussian density f (x) (line) and the simulation of X(n) 3 (circles), with p = 1/4, n = 500000, and number of simulated words N = 200000. v ) which, as we have seen, is only due to i,j,k COV (X n,2 (θ) := ln (G n,2 (θ)) . |T . | = O(1). Lemma 5. 1 1Consider a, b and c = 1. Let r = Proof. The proof of Lemma 5.6 is in subsection 5.3.2. of distinct (adjacent) two letter patterns ij with i = j If we fix i = j and we analyze the occurrences of the pattern ij, then the only "cluster" (to use Bassino et al.'s terminology) is ij itself. So the generating function ξ(z, t) of the set of clusters C = {ij} becomes only (compare with (6) in Bassino et al.): of distinct (adjacent) two letter patterns ij with i = j • i = j and k =• i = j and k = • i = j and k = 5.4.1 i = j and k = 5.4.3 i = j and k = 5.4.3.1 k = and i and j are distinct Same as section 5.4.2.1 but with i and k exchanged, and with j and exchanged. 5.4.3.2 k = = i = j Same as section 5.4.2.2 but with i and k exchanged, and with j and exchanged. 5.4.3.3 k = = j = i Same as section 5.4.2.3 but with i and k exchanged, and with j and exchanged. 5.4.4 i = j and k = 5.4.4.1 i and j and k and are distinct The clusters are ij and k . So ξ(z, t, u) of the set of clusters C = {ij, k } becomes z+P i P j z 2 − 5.4.4.4 i = and k and j are distinct Same as section 5.4.4.3 but with i and k exchanged, and with j and exchanged.5.4.4.5= j and i and k are distinct The clusters are ij and kj. So, by the same analysis from section 5. , for ∈ Z \ {0}. 1FΓ − e −nq i+j = k≥0 (k + 1) 1 − e −nq k . This has the MT F (s) = − 1 (1 − q −s ) 2 Γ(s),with FS −1, 0 and singularities to the right of the fundamental strip (( χ) − (ln(n) + L)Γ ( χ) n − χ , again with error term O(n −β ) with any fixed β > 0. ln n − ln(nP ))(nP ) α e −nP = O(ln n), because of x α ln x = O(x α/2 ). See Theorem 3.6 for asymptotics of EX . Both simulations use p = 1/4. The sample size N for each row in the left table is 50000, and in the right table it is 200000, see also Figure 3.(n) 1 and EX (n) 3 n EX (n) 1X (n) 1 Var X (n) 1 s 2 1 (n) 10000 12.692 12.676 1.205 1.214 11547 12.942 12.927 1.205 1.206 13333 13.192 13.175 1.205 1.213 15396 13.442 13.427 1.205 1.211 n EX (n) 3X (n) 3 Var X (n) 3 s 2 3 (n) 500000 750.195 750.198 129.889 130.053 Table 1 : 1Table ofgenerating functions, partial fraction decompositions, and coefficients of z n , n ≥ 2, in each. Table 2 : 2Table ofgenerating functions, partial fraction decompositions, and coefficients of z n , n ≥ 2, in each. AcknowledgementsWe would like to thank B. Pittel for providing the Poisson distribution ofAppendix. Some Mellin transforms and some proofs A Some Mellin transformsWe will use the fundamental correspondence. Let a pole ξ = σ + it to the right of the fundamental strip :This induces an expansion atWe also need the rescaling rule:Recall the notations L := ln 1 q and χ := 2iπ L . We will use the rapid decrease property: Γ(s) decreases exponentially in the direction i∞:Also, this property is true for all other functions we encounter. So inverting the Mellin transforms is easily justified (see[6], Sec.4 for example). All our expressions are given with an error term n −β , with β > (χ), where χ is the rightmost pole used in our singular expansions. The number of distinct adjacent pairs in geometrically distributed words. M Archibald, A Blecher, C Brennan, A Knopfmacher, S Wagner, M D Ward, Discrete Mathematics and Theoretical Computer Science. 2242021M. Archibald, A. Blecher, C. Brennan, A. Knopfmacher, S. Wagner and M.D. Ward. The number of distinct adjacent pairs in geometrically distributed words. Discrete Mathematics and Theoret- ical Computer Science, 22(4), 2021. Counting Occurrences for a Finite Set of Words. F Bassino, J Clément, P Nicodème, Combinatorial Methods. ACM Transactions on Algorithms. 8331F. Bassino, J. Clément, and P. Nicodème Counting Occurrences for a Finite Set of Words: Combinatorial Methods. ACM Transactions on Algorithms, 8(3), article 31, 2012. . P Flajolet, X Gourdon, P , Dumas Mellin Transforms and Asymptotics: Harmonic sums Theoretical Computer Science. 144P. Flajolet, X. Gourdon and P. Dumas Mellin Transforms and Asymptotics: Harmonic sums Theoretical Computer Science, 144:3-58, 1995. Analytic Combinatorics. Cambridge. P Flajolet, R Sedgewick, P. Flajolet and R. Sedgewick. Analytic Combinatorics. Cambridge, 2009. Distinctness of compositions of an integer: a Probabilistic Analysis. P Hitczenko, G Louchard, Random Structures and Algorithms. 19P. Hitczenko and G. Louchard. Distinctness of compositions of an integer: a Probabilistic Anal- ysis. Random Structures and Algorithms, 19(3,4):407-437, 2001. Asymptotics of The Moments of Extreme-value related distribution functions. G Louchard, H Prodinger, Algorithmica. 46G. Louchard and H. Prodinger. Asymptotics of The Moments of Extreme-value related distribu- tion functions. Algorithmica, 46:431-467, 2006. The number of distinct values of some multiplicity in sequences of geometrically distributed random variables. G Louchard, H Prodinger, M D Ward, Discrete Mathematics and Theoretical Computer Science. G. Louchard, H. Prodinger and M.D. Ward. The number of distinct values of some multiplicity in sequences of geometrically distributed random variables. Discrete Mathematics and Theoretical Computer Science, AD:231-256, 2005. 2005 International Conference on Analysis of Algorithms On a Correlation Inequality of Farr. C Mcdiarmid, Combinatorics, Probability and Computing. 1C. McDiarmid. On a Correlation Inequality of Farr. Combinatorics, Probability and Computing, 1:157-160, 1992. . B Pittel, Technical reportPrivate communicationB. Pittel. Technical report. Private communication. W Rudin, Principles of Mathematical Analysis. McGraw-Hill3rd edW. Rudin. Principles of Mathematical Analysis, 3rd ed. McGraw-Hill, 1976. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. B Salvy, P Zimmermann, ACM Transactions on Mathematical Software. 202B. Salvy and P. Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163-177, 1994. . B Salvy, Private communicationB. Salvy. Private communication. E Seneta, Non-negative Matrices and Markov Chains. Springer2nd edE. Seneta. Non-negative Matrices and Markov Chains, 2nd ed. Springer, 1981. N J A Sloane, The On-Line Encyclopedia of Integer Sequences. N. J. A. Sloane. The On-Line Encyclopedia of Integer Sequences, http://oeis.org. Sequence A028246.

We prove that the stabilization of spaces functor-the classical construction of associating a spectrum to a pointed space by tensoring with the sphere spectrum-satisfies homotopical descent on objects and morphisms. This is the stabilization analog of the Quillen-Sullivan theory main result that the rational chains (resp. cochains) functor participates in a derived equivalence with certain coalgebra (resp. algebra) complexes, after restriction to 1connected spaces up to rational equivalence. In more detail, we prove that the stabilization of spaces functor participates in a derived equivalence with certain coalgebra spectra (where the stabilization construction naturally lands), after restriction to 1-connected spaces up to weak equivalence. This resolves in the affirmative the infinite case, involving stabilization and suspension spectra, of a question/conjecture posed by Tyler Lawson on (iterated) suspension spaces almost ten years ago. A key ingredient of our proof, of independent interest, is a higher stabilization theorem for spaces that provides strong estimates for the uniform cartesian-ness of certain cubical diagrams associated to n-fold iterations of the spaces-level stabilization map; this is the stabilization analog of Dundas' higher Hurewicz theorem.

1612.08623

e75f9db673607277b902fd569867f4f7249c6477

SUSPENSION SPECTRA AND HIGHER STABILIZATION 10 May 2017 Jacobson R Blomquist John E Harper SUSPENSION SPECTRA AND HIGHER STABILIZATION 10 May 2017arXiv:1612.08623v3 [math.AT] We prove that the stabilization of spaces functor-the classical construction of associating a spectrum to a pointed space by tensoring with the sphere spectrum-satisfies homotopical descent on objects and morphisms. This is the stabilization analog of the Quillen-Sullivan theory main result that the rational chains (resp. cochains) functor participates in a derived equivalence with certain coalgebra (resp. algebra) complexes, after restriction to 1connected spaces up to rational equivalence. In more detail, we prove that the stabilization of spaces functor participates in a derived equivalence with certain coalgebra spectra (where the stabilization construction naturally lands), after restriction to 1-connected spaces up to weak equivalence. This resolves in the affirmative the infinite case, involving stabilization and suspension spectra, of a question/conjecture posed by Tyler Lawson on (iterated) suspension spaces almost ten years ago. A key ingredient of our proof, of independent interest, is a higher stabilization theorem for spaces that provides strong estimates for the uniform cartesian-ness of certain cubical diagrams associated to n-fold iterations of the spaces-level stabilization map; this is the stabilization analog of Dundas' higher Hurewicz theorem. Introduction We have written this paper simplicially: in other words, "space" means "simplicial set" unless otherwise noted; see Dwyer-Henn [22] for a useful introduction to these ideas, together with Bousfield-Kan [10], Goerss-Jardine [29], and Hovey [36]. We work in the category of symmetric spectra (see Hovey-Shipley-Smith [37] and Schwede [45]), equipped with the injective stable model structure, so that "Smodules" means "symmetric spectra", which are the same as modules over the sphere spectrum S; in particular, fibrant S-modules enjoy the property of being Ω-spectra [37, 1.4] that are objectwise Kan complexes. For useful results and techniques involving spectra in closely related contexts, see Bousfield-Friedlander [9], Elmendorf-Kriz-Mandell-May [26], and Jardine [38]. 1.1. The spaces-level stabilization map. If X is a pointed space, the stabilization map has the form π * (X)→π s * (X) = colim r π * +r (Σ r X) (1.2) This comparison map between homotopy groups and stable homotopy groups comes from a spaces-level stabilization map of the form X →Ω ∞ Σ ∞ (X) (1.3) and applying π * to (1.3) recovers the map (1.2); here,Ω ∞ = Ω ∞ F (Definition 3.8) denotes the right-derived functor of the underlying 0-th space Ω ∞ = Ev 0 functor and Σ ∞ = S⊗− denotes the stabilization functor given by tensoring with the sphere spectrum S. 1.4. Iterating the stabilization map. With a spaces-level stabilization map in hand, it is natural to form a cosimplicial resolution of X with respect toΩ ∞ Σ ∞ of the form X / /Ω ∞ Σ ∞ (X) / / / / (Ω ∞ Σ ∞ ) 2 (X) / / / / / / (Ω ∞ Σ ∞ ) 3 (X) · · · (1.5) showing only the coface maps. The homotopical comonadK = Σ ∞Ω∞ , which is the derived functor of the comonad K = Σ ∞ Ω ∞ associated to the (Σ ∞ , Ω ∞ ) adjunction, can be thought of as encoding the spectra-level co-operations on the suspension spectra; compare with [6,7] for integral chains and iterated suspension. By analogy with the techniques in Bousfield-Kan [10], by iterating the spaceslevel stabilization map (1.3) Carlsson [12], and subsequently Arone-Kankaanrinta [2], study the cosimplicial resolution of X with respect toΩ ∞ Σ ∞ , and taking the homotopy limit of the resolution (1.5) produce theΩ ∞ Σ ∞ -completion map X→X ∧ Ω ∞ Σ ∞ (1.6) We will show that this completion map fits into a derived equivalence with comparison map the stabilization Σ ∞ functor (but landing in the naturally occurring category of coalgebraic objects). 1.7. The main result. In this paper we shall prove the following theorem, that the stabilization of spaces functor-the classical construction of associating a spectrum to a pointed space by tensoring with the sphere spectrum-satisfies homotopical descent on objects and morphisms. This is the stabilization analog of the Quillen-Sullivan theory main result [44,47] that the rational chains (resp. cochains) functor participates in a derived equivalence with certain coalgebra (resp. algebra) complexes, after restriction to 1-connected spaces up to rational equivalence. In more detail, we prove that the stabilization of spaces functor participates in a derived equivalence with certain coalgebra spectra (where the stabilization construction naturally lands), after restriction to 1-connected spaces up to weak equivalence. This resolves in the affirmative the infinite case, involving stabilization and suspension spectra, of a question/conjecture posed by Tyler Lawson on (iterated) suspension spaces almost ten years ago. Theorem 1.8. The fundamental adjunction (1.9) comparing pointed spaces to coalgebra spectra over the comonad K = Σ ∞ Ω ∞ via stabilization Σ ∞ S * Σ ∞ / / coAlg K lim∆ C o o (1.9) induces a derived adjunction of the form Map coAlgK (Σ ∞ X, Y ) ≃ Map S * (X, holim ∆ C(Y )) (1.10) that is a derived equivalence after restriction to the full subcategories of 1-connected spaces and 1-connectedK-coalgebra spectra; more precisely: (a) If Y is a 1-connectedK-coalgebra spectrum, then the derived counit map Σ ∞ holim ∆ C(Y ) ≃ −−→ Y associated to the derived adjunction (1.10) is a weak equivalence; i.e., stabilization Σ ∞ is homotopically essentially surjective on 1-connected coalgebra spectra overK, and hence homotopical descent is satisfied on such objects. (b) If X ′ is a 1-connected space, then the derived unit map X ′ ≃ −−→ holim ∆ C(Σ ∞ X ′ ) associated to the derived adjunction (1.10) is tautologically theΩ ∞ Σ ∞completion map X ′ →X ′ ∧ Ω ∞ Σ ∞ , and hence is a weak equivalence by [2,13]. In particular, the stabilization functor induces a weak equivalence Σ ∞ : Map h S * (X, X ′ ) ≃ Map coAlgK (Σ ∞ X, Σ ∞ X ′ ) (1.11) on mapping spaces; i.e., stabilization Σ ∞ is homotopically fully faithful on 1-connected spaces and hence homotopical descent is satisfied on such morphisms. Here, Map h S * (X, X ′ ) denotes the realization of the Dwyer-Kan [23] homotopy function complex. See Arone-Ching [1] and Hess [33] for a discussion of the homotopical descent ideas that have motivated and underlie much of our work, Carlsson-Milgram [14] and May [41] for related ideas, Edwards-Hastings [25] for a nice discussion of stabilization and abelianization in the spirit of this paper, and Behrens-Rezk [3] for an interesting survey of closely related ideas circulating in the spectral algebra context. 1.12. Classification and characterization theorems. The following are corollaries of the main result. Theorem 1.13 (Classification theorem). A pair of 1-connected pointed spaces X and X ′ are weakly equivalent if and only if the suspension spectra Σ ∞ X and Σ ∞ X ′ are weakly equivalent as derivedK-coalgebra spectra. Theorem 1.14 (Classification of maps theorem). Let X, X ′ be pointed spaces. Assume that X ′ is 1-connected and fibrant. (a) (Existence) Given any [15] for resolving the 0-connected case of the Francis-Gaitsgory conjecture [27], together with a powerful modification of that strategy developed in [6] for resolving the integral chains problem, together with Cohn's [17] work showing that this strategy of attack extends to homotopy coalgebras over the associated homotopical comonad (see ); essentially, we establish and exploit uniform cartesian-ness estimates to show that theΩ ∞ Σ ∞completion map studied in Carlsson [12], and subsequently in Arone-Kankaanrinta [2], participates in a derived equivalence between spaces and coalgebra spectra over the homotopical stabilization comonad, after restricting to 1-connected spaces. map φ in [Σ ∞ X, Σ ∞ X ′ ]K, there exists a map f in [X, X ′ ] such that φ = Σ ∞ (f ). (b) (Uniqueness) For each pair of maps f, g in [X, X ′ ], f = g if and only if Σ ∞ (f ) = Σ ∞ (g) in A key ingredient underlying our homotopical estimates is the higher stabilization theorem proved in this paper; it can be thought of as the stabilization analog of Dundas' [20] higher Hurewicz theorem estimates for integral chains that play a key role in [6] and underlies the main results in Dundas-Goodwillie-McCarthy [21]. The results of Hopkins [35], and the subsequent work of Goerss [28] and Klein-Schwanzl-Vogt [39] were motivating for us. This paper is a companion to our work in [7] that resolves the original question/conjecture of Tyler Lawson [40] on (iterated) suspension spaces; we regard this stabilization work as the stronger result: in the limit case we are throwing away more information, nevertheless, we find that homotopical descent on objects and morphisms still holds. 1.17. Commuting stabilization with holim of a cobar construction. Our main result reduces to proving our main technical theorem that the left derived stabilization functor Σ ∞ commutes, Σ ∞ holim ∆ C(Y ) ≃ holim ∆ Σ ∞ C(Y ) (1.18) up to weak equivalence, with the right derived limit functor holim ∆ , when composed with the cosimplicial cobar construction C and evaluated on 1-connected K-coalgebra spectra. 1.19. Organization of the paper. In Section 2 we outline the argument of our main result. In Section 3 we review the fundamental adjunction and then prove the main result. Sections 4 through 6 are important background sections; for the convenience of the reader we briefly recall some preliminaries on simplicial structures, the homotopy theory ofK-coalgebras, and the derived fundamental adjunction for stabilization. For the experts familiar with Arone-Ching [1], it will suffice to read Sections 2 and 3 for a complete proof of the main result. suggestions and useful remarks throughout this project. The second author would like to thank Bjorn Dundas, Bill Dwyer, Haynes Miller, and Crichton Ogle for useful suggestions and remarks and Lee Cohn, Mike Hopkins, Tyler Lawson, and Nath Rao for helpful comments. The second author is grateful to Haynes Miller for a stimulating and enjoyable visit to the Massachusetts Institute of Technology in early spring 2015, and to Bjorn Dundas for a stimulating and enjoyable visit to the University of Bergen in late spring 2015, and for their invitations which made this possible. The first author was supported in part by National Science Foundation grant DMS-1510640. Outline of the argument In this section we will outline the proof of our main result. Since the derived unit map is tautologically theΩ ∞ Σ ∞ -completion map X ′ →X ′ ∧ Ω ∞ Σ ∞ , which is proved to be a weak equivalence on 1-connected spaces in Carlsson [12] (see also Arone-Kankaanrinta [2]), proving the main result reduces to verifying that the derived counit map is a weak equivalence. The following are proved in Section 3.22. Theorem 2.1. If Y is a 1-connectedK-coalgebra spectrum and n ≥ 1, then the natural map holim ∆ ≤n C(Y ) −→ holim ∆ ≤n−1 C(Y ) (2.2) is an (n + 2)-connected map between 1-connected objects. Theorem 2.3. If Y is a 1-connectedK-coalgebra spectrum and n ≥ 0, then the natural maps holim ∆ C(Y ) −→ holim ∆ ≤n C(Y ) (2.4) Σ ∞ holim ∆ C(Y ) −→ Σ ∞ holim ∆ ≤n C(Y ) (2.5) are (n + 3)-connected maps between 1-connected objects. Proof. Consider the first part. By Theorem 2.1 each of the maps in the holim tower {holim ∆ ≤n C(Y )} n , above level n, is at least (n + 3)-connected. It follows that the map (2.4) is (n + 3)-connected. The second part follows from the first part. The following is a key ingredient for proving our main result. It provides estimates sufficient for verifying that stabilization commutes past the desired homotopy limits. Theorem 2.6. If Y is a 1-connectedK-coalgebra spectrum and n ≥ 1, then the natural map Σ ∞ holim ∆ ≤n C(Y ) −→ holim ∆ ≤n Σ ∞ C(Y ), (2.7) is (n + 5)-connected; the map is a weak equivalence for n = 0. The following resolves the original form of Tyler Lawson's question/conjecture. Theorem 2.8. If Y is a 1-connectedK-coalgebra spectrum, then the natural maps Σ ∞ holim ∆ C(Y ) ≃ −−→ holim ∆ Σ ∞ C(Y ) ≃ −−→ Y (2.9) are weak equivalences. Proof. For the case of the left-hand map, it is enough to verify that the connectivities of the natural maps (2.5) and (2.7) are strictly increasing with n, and Theorems 2.3 and 2.6 complete the proof. Consider the right-hand map. Since Σ ∞ C(Y ) ≃ F Σ ∞ C(Y ) and the latter is isomorphic to the cosimplicial cobar construction Cobar(F K, F K, F Y ), which has extra codegeneracy maps s −1 (Dwyer-Miller-Neisendorfer [24, 6.2]), it follows from the cofinality argument in Dror-Dwyer [18, 3.16] that the right-hand map in (2.9) is a weak equivalence. Proof of Theorem 1.8. We want to verify that the natural map Σ ∞ holim ∆ C(Y )→Y is a weak equivalence; since this is the composite (2.9), Theorem 2.8 completes the proof. Homotopical analysis In this section we recall the stabilization functor, together with related constructions, and prove Theorems 2.1 and 2.6. 3.1. Stabilization and the fundamental adjunction. The fundamental adjunction naturally arises by observing that Σ ∞ is equipped with a coaction over the comonad K associated to the (Σ ∞ , Ω ∞ ) adjunction; this observation, which remains true for any adjunction provided that the indicated limits below exist, forms the basis of the homotopical descent ideas appearing in Hess [33] and subsequently in Francis-Gaitsgory [27]. Consider any pointed space X and S-module Y , and recall that the suspension spectrum Σ ∞ (X) = S⊗X and 0-th space Ω ∞ (Y ) = Ev 0 (Y ) = Y 0 functors fit into an adjunction S * Σ ∞ / / Mod S Ω ∞ o o (3.2) with left adjoint on top. Associated to the adjunction in (3.2) is the monad Ω ∞ Σ ∞ on pointed spaces S * and the comonad K : = Σ ∞ Ω ∞ on S-modules Mod S of the form id η − → Ω ∞ Σ ∞ (unit), id ε ← − K (counit), (3.3) Ω ∞ Σ ∞ Ω ∞ Σ ∞ →Ω ∞ Σ ∞ (multiplication), KK m ← − K (comultiplication). and it follows formally that there is a factorization of adjunctions of the form S * Σ ∞ / / coAlg K / / lim∆ C o o Mod S K o o (3.4) with left adjoints on top and coAlg K →Mod S the forgetful functor. In particular, the suspension spectrum Σ ∞ X is naturally equipped with a K-coalgebra structure. To understand the comparison in (3.4) between S * and coAlg K it is enough to observe that lim ∆ C(Y ) is naturally isomorphic to an equalizer of the form lim ∆ C(Y ) ∼ = lim Ω ∞ Y d 0 / / d 1 / / Ω ∞ KY where d 0 = mid, d 1 = idm, m : Ω ∞ →Ω ∞ K = Ω ∞ Σ ∞ Ω ∞ denotes the K-coaction map on Ω ∞ (defined by m := ηid), and m : Y →KY denotes the K-coaction map on Y ; see, for instance, [6]. Definition 3.5. Let Y be a K-coalgebra. The cosimplicial cobar construction C(Y ) := Cobar(Ω ∞ , K, Y ) in (S * ) ∆ looks like C(Y ) : Ω ∞ Y d 0 / / d 1 / / Ω ∞ KY / / / / / / Ω ∞ KKY · · · (3.6) (showing only the coface maps) and is defined objectwise by C(Y ) n := Ω ∞ K n Y with the obvious coface and codegeneracy maps; see, for instance, the face and degeneracy maps in the simplicial bar constructions described in Gugenheim-May [32, A.1] or May [42,Section 7], and dualize. For instance, the indicated coface maps in (3.6) are defined by d 0 := mid and d 1 := idm. 3.7. Coalgebras over the homotopical comonadK. It will be useful to interpret the cosimplicialΩ ∞ Σ ∞ -resolution of X in terms of a cosimplicial cobar construction that naturally arises as a "fattened" version of (3.6); this leads to the notion of aK-coalgebra exploited in Cohn [17]. It is shown in 4.2,4.4], and subsequently exploited in Cohn [17], that the derived functorK := KF of the comonad K is very nearly a comonad itself with the structure maps m :K→KK and ε :K→F above. For instance, it is proved in [8] thatK defines a comonad on the homotopy category of Mod S , which is a reflection of the the fact thatK has the structure of a highly homotopy coherent comonad (see [8]); in particular,K has a strictly coassociative comultiplication m :K→KK and satisfies left and right counit identities up to factors of F ≃ id. In more detail, the homotopical comonadK makes the following diagrams K m / / m KK mid KK idm / /KKK FK idm / / FKK ( * ) FK FKK m / /KK ( * * ) KK commute; here, the map ( * ) is the composite FKK idεid − −− → F FK mid −−→ FK and the map ( * * ) is the composite KFK ididε − −− → KF F idm −−→ KF . Remark 3.11. Associated to the adjunction (Σ ∞ , Ω ∞ ) is a left K-coaction (or Kcoalgebra structure) m : Σ ∞ X→KΣ ∞ X on Σ ∞ X, defined by m = idηid), for any X ∈ S * . This map induces a corresponding leftK-coaction m : Σ ∞ X→KΣ ∞ X that is the composite Σ ∞ X m − → KΣ ∞ X = KidΣ ∞ X − → KF Σ ∞ X The following notion of a homotopyK-coalgebra, exploited in Cohn [17], captures exactly the leftK-coaction structure that stabilization Σ ∞ X of a pointed space X satisfies; this is precisely the structure being encoded by the cosimplicialΩ ∞ Σ ∞ resolution (1.5). Definition 3.12. A homotopyK-coalgebra (orK-coalgebra, for short) is a Y ∈ Mod S together with a map m : Y →KY in Mod S such that the following diagrams Y m / / m K Y mid KY idm / /KK Y F Y idm / / FKY ( * ) F Y F Y commute; here, the map ( * ) is the composite FKY idεid − −− → F F Y mid −−→ F Y .C(Y ) :Ω ∞ Y d 0 / / d 1 / /Ω ∞K Y / / / / / /Ω ∞KK Y · · ·( 3.14) (showing only the coface maps) and is defined objectwise by C(Y ) n :=Ω ∞Kn Y = Ω ∞ F (KF ) n Y with the obvious coface and codegeneracy maps; for instance, in (3.14) the indicated coface maps are defined by d 0 := mid and d 1 := idm (compare with (3.6)). It is important to note that the cosimplicial resolution (1.5) of a pointed space X with respect to stabilizationΩ ∞ Σ ∞ , built by iterating the spaces-level stabilization map (1.3), is naturally isomorphic to the map X → C(Σ ∞ X); in other words, the homotopical comonadK can be thought of as encoding the spectra-level cooperations on the suspension spectra. Remark 3.15. It may be helpful to note, in particular, that the cosimplicial cobar construction is encoding the fact that the derived functorΩ ∞ has a naturally occurring rightK-coaction map m :Ω ∞ →Ω ∞K that makes the following diagrams 3.17. Higher stabilization. The purpose of this section is to prove Theorem 3.20; see Ching-Harper [16] and Goodwillie [30] for notations and definitions associated to cubical diagrams. Definition 3.18. Let f : N→N be a function and W a finite set. A W -cube X is f -cartesian (resp. f -cocartesian) if each d-subcube of X is f (d)-cartesian (resp. f (d)-cocartesian); here, N denotes the non-negative integers. Ω ∞ m / / m Ω ∞K mid Ω ∞K idm / /Ω ∞KKΩ ∞ m / /Ω ∞K ( * * ) Ω ∞Ω∞ commute; here, the map ( * * ) is the composite Ω ∞ FK ididε − −− → Ω ∞ F F idm −−→ Ω ∞ F . The following proposition, proved in [7], will be helpful in organizing the proof of the higher stabilization theorem below; compare with Dundas-Goodwillie-McCarthy [21, A.8.3]. Proposition 3.19 (Uniformity correspondence). Let k ≥ 1 and W a finite set. A W -cube of pointed spaces is (k(id + 1) + 1)-cartesian if and only if it is ((k + 1)(id + 1) − 1)-cocartesian. The following theorem plays a key role in our homotopical analysis of the derived counit map below; it also provides an alternate proof, with stronger estimates, of the result in [2,12] that theΩ ∞ Σ ∞ -completion map X→X ∧ Ω ∞ Σ ∞ is a weak equivalence for any 1-connected space X. It is motivated by Dundas [20, 2.6] and is the infinite or limit case of the closely related higher Freudenthal suspension theorem [7]. Theorem 3.20 (Higher stabilization theorem). Let k ≥ 1, W a finite set, and X a W -cube of pointed spaces. If X is (k(id + 1) + 1)-cartesian, then so is X→Ω ∞ Σ ∞ X. Proof. Consider the case |W | = 0. Suppose X is a W -cube and X ∅ is k-connected. We know by Freudenthal suspension, which can be understood as a consequence of the Blakers-Massey theorem (see, for instance, [21, A.8.2]), that the map X ∅ →ΩΣX ∅ is (2k +1)-connected. More generally, it follows by repeated application of Freudenthal suspension that the map X ∅ →Ω ∞ Σ ∞ X ∅ is a (2k + 1)-connected map between k-connected spaces. Consider the case |W | ≥ 1. Suppose X is a W -cube and X is (k(id + 1) + 1)cartesian. Let's verify that X→Ω ∞ Σ ∞ X is (k(id + 1) + 1)-cartesian (|W | + 1)-cube. It suffices to assume that X is a cofibration W -cube; see [30, 1.13] and [16, 3.4]. Let C be the iterated cofiber of X and C the W -cube defined objectwise by C V = * for V = W and C W = C. Then X→C is ∞-cocartesian. Consider the commutative diagram X / / ( * ) C Ω ∞ Σ ∞ X / /Ω ∞ Σ ∞ C (3.21) of |W |-cubes. Let's verify that ( * ) is (k(|W | + 2) + 1)-cartesian as a (|W | + 1)-cube of pointed spaces. We know that X is ((k + 1)(id + 1) − 1)-cocartesian by Proposition 3.19, and in particular, C is ((k + 1)(|W | + 1) − 1)-connected. For d < |W |, any (d + 1) dimensional subcube of X is ((k+1)(d+2)−1) = ((k+1)(d+1)+k)-cocartesian and any d dimensional subcube of X is ((k + 1)(d + 1) − 1)-cocartesian. So if X|T is some d-subcube of X with T not containing the terminal set W , then X|T →C|T = * is (k + 1)(d + 1)-cocartesian by [30, 1.7]. Furthermore, even if T contains the terminal set W , we know that X|T →C|T is still (k + 1)(d + 1)-cocartesian by [30, 1.7]; this is because (k + 1)(d + 1) < (k + 1)(|W | + 1) − 1 since k ≥ 1 and d < |W |. Hence X|T →C|T is (k + 1)(d + 1)-cocartesian for any d-subcube X|T of X. It follows easily from higher Blakers-Massey [30, 2.5] that X→C is (k(|W | + 2) + 1)-cartesian. We know that Σ ∞ X→Σ ∞ C is ∞-cocartesian and hence ∞-cartesian; thereforẽ Ω ∞ Σ ∞ X→Ω ∞ Σ ∞ C is ∞-cartesian. Also, C→Ω ∞ Σ ∞ C is at least (k(|W | + 2) + 1)cartesian since C→Ω ∞ Σ ∞ C is (2[(k+1)(|W |+1)−1]+1)-connected by Freudenthal suspension; this is because the cartesian-ness of C→Ω ∞ Σ ∞ C is the same as the connectivity of the mapΩ |W | C→Ω |W |Ω∞ Σ ∞ C (by considering iterated homotopy fibers). Putting it all together, it follows from diagram (3.21) and [30, 1.8] that the map ( * ) is (k(|W |+2)+1)-cartesian. Doing this also on all subcubes gives the result. Z 0 Z 1 s 0 o o Z 2 · · · Z n s 0 o o s 1 o o the n-truncation of Z; in particular, Y 0 is the object (or 0-cube) Z 0 . We often refer to Y n as the codegeneracy n-cube associated to Z. The following is proved in Carlsson [13, Section 6], Dugger [19], and Sinha [46, 6.7], and plays a key role in this paper; see also Dundas-Goodwillie-McCarthy [21] and Munson-Volic [43]; it was exploited early on by Hopkins [35]. Remark 3.26. We follow the conventions and definitions in Bousfield-Kan [10], together with [6] and Ching-Harper [15] for the various models of homotopy limits. The associated ∞-cartesian (n + 1)-cube built from Z, denoted Z : P([n])→S * , is defined objectwise by Z V := holim BK T =∅ Z T , for V = ∅, Z V , for V = ∅. Proposition 3.28 (Uniformity of faces). Let Z ∈ (S * ) ∆ and n ≥ 0. Assume that Z is objectwise fibrant. Let ∅ = T ⊂ [n] and t ∈ T . Then there is a weak equivalence (iterated hofib)∂ T {t} Z ≃ Ω |T |−1 (iterated hofib)Y |T |−1 in S * , where Y |T |−1 denotes the codegeneracy (|T | − 1)-cube associated to Z. Proof. This is proved in [15]; compare Goodwillie [31, 3.4] and Sinha [46, 7.2]. Theorem 3.29. Let Y be aK-coalgebra and n ≥ 1. Consider the ∞-cartesian (n + 1)-cube C(Y ) in S * built from C(Y ). If Y is 1-connected, then (a) the cube C(Y ) is (2n + 5)-cocartesian in S * , (b) the cube Σ ∞ C(Y ) is (2n + 5)-cocartesian in Mod S , (c) the cube Σ ∞ C(Y ) is (n + 5)-cartesian in Mod S . Proof. Consider part (a) and let W = [n]. Our strategy is to use the higher dual Blakers-Massey theorem in Goodwillie [30, 2.6] to estimate how close the W -cube C(Y ) in S * is to being cocartesian. We know from the higher stabilization theorem, Theorem 3.20, on iterations of the stabilization map applied toΩ ∞ Y , together with the uniformity enforced by Proposition 3.28, that for each nonempty subset V ⊂ W , the V -cube ∂ W W −V C(Y ) is (|V |+2)-cartesian; since it is ∞-cartesian by construction when V = W , it follows immediately from Goodwillie [30, 2.6] that C(Y ) is (2n+5)cocartesian in S * , which finishes the proof of part (a). Part (b) follows from the fact that Σ ∞ : S * →Mod S preserves cocartesian-ness. Part (c) follows easily from [16, 3.10]. Proof of Theorem 2.6. We want to estimate how connected the comparison map Σ ∞ holim ∆ ≤n C(Y ) −→ holim ∆ ≤n Σ ∞ C(Y ), is, which is equivalent to estimating how cartesian Σ ∞ C(Y ) is; Theorem 3.29(c) completes the proof. Proposition 3.30. Let Y be aK-coalgebra and n ≥ 1. Denote by Y n the codegeneracy n-cube associated to the cosimplicial cobar construction C(Y ) of Y . If Y is 1-connected, then the total homotopy fiber of Y n is (2n + 1)-connected. Proof. This follows immediately from the proof of Theorem 3.29, together with Proposition 3.28. Proof of Theorem 2.1. The homotopy fiber of the map (2.2) is weakly equivalent tõ Ω n of the total homotopy fiber of the codegeneracy n-cube Y n associated to C(Y ) by Proposition 3.23, hence by Proposition 3.30 the map (2.2) is (n + 2)-connected. As an immediate corollary of Theorem 2.1, we get the following. Theorem 3.31. If Y is a 1-connectedK-coalgebra spectrum, then the homotopy spectral sequence E 2 −s,t = π s π t C(Y ) =⇒ π t−s holim ∆ C(Y ) converges strongly; compare with Ching-Harper [15]. Proof. This follows from the connectivity estimates in Theorem 2.1. These types of homotopy spectral sequences have been studied in Bousfield-Kan [11], Bendersky-Curtis-Miller [4] and Bendersky-Thompson [5]. Background on simplicial structures It will be useful to recall, in this background section, the simplicial structures on pointed spaces and S-modules. where the pointed mapping space hom * (X, X ′ ) in S * is Hom S * (X, X ′ ) pointed by the constant map; see [29,II.3]. Definition 4.2. Let Y, Y ′ be S-modules and K a simplicial set. The tensor product Y⊗K in Mod S , mapping object hom Mod S (K, Y ) in Mod S , and mapping space Hom Mod S (Y, Y ′ ) in sSet are defined by Y⊗K := Y ∧ K + hom Mod S (K, Y ′ ) := Map(K + , Y ′ ) Hom Mod S (Y, Y ′ ) n := hom Mod S (Y⊗∆[n], Y ′ ) where Map(K + , Y ′ ) denotes the function S-module; see [37, 2.2.9]. We sometimes drop the S * and Mod S decorations from the notation and simply write Hom and hom. Proof. This is proved, for instance, in [29,II.3] and [37]. Recall that the stabilization adjunction (Σ ∞ , Ω ∞ ) in (3.2) is a Quillen adjunction with left adjoint on top; in particular, for X, Y ∈ S * there is an isomorphism hom(Σ ∞ X, Y ) ∼ = hom(X, Ω ∞ Y ) (4.4) in Set, natural in X, Y . The following proposition, which follows from Goerss-Jardine [29,II.2.9], verifies that the stabilization adjunction (3.2) meshes nicely with the simplicial structure. Hom(Σ ∞ X, Y ) ∼ = Hom(X, Ω ∞ Y ) in sSet, natural in X, Y , that extends the adjunction isomorphism in (4.4); (c) there is an isomorphism Ω ∞ hom(K, Y ) ∼ = hom(K, Ω ∞ Y ) in S * , natural in K, Y . (d) there is a natural map σ : Ω ∞ (Y )⊗K→Ω ∞ (Y⊗K) induced by Ω ∞ . (e) the functors Σ ∞ and Ω ∞ are simplicial functors (Remark 4.6) with the structure maps σ of (a) and (d), respectively. Remark 4.6. For a useful reference on simplicial functors in the context of homotopy theory, see Hirschhorn [34, 9.8.5]. The following proposition is fundamental to this paper. d 1 / / Hom Y, FKY ′ / / / / / / Hom Y, FKKY ′ · · · and is defined objectwise by Hom Y, FK • Y ′ n := Hom Y, FK n Y ′ = Hom Y, F (KF ) n Y ′ with the obvious coface and codegeneracy maps. Remark 5.3. This is simply the resolution in Arone-Ching [1], but "fattened-up" by F . For instance, on the level of hom-sets (simplicial degree 0), let's verify that s 0 d 1 = id on Hom(Y, F Y ′ ) . Start with f : Y →F Y ′ and consider the commutative diagram Y f / / F Y ′ idm / / F KF Y ′ idεid 2 ( * ) F F Y ′ mid Y f / / F Y ′ F Y ′TheY m / / ηid KF Y id 2 f / / ηid 3 KF F Y ′ idmid / / ηid 4 KF Y ′ ηid 3 / / ηid 3 F KF Y ′ F Y idm / / F KF Y id 3 f / / idεid 2 ( * ) F KF F Y ′ id 2 mid / / idεid 3 F KF Y ′ idεid 2 F KF Y ′ idεid 2 F F Y id 2 f / / mid F F F Y ′ idmid / / mid 2 F F Y ′ mid F F Y ′ mid F Y F Y idf / / F F Y ′ mid / / F Y ′ F Y ′ Y ηid O O f / / F Y ′ ηid 2 O O F Y ′ The composite along the upper horizontal and right-hand vertical maps is s 0 d 0 f and the composite along the bottom horizontal maps is f ; the diagram commutes verifies that s 0 d 0 = id. Remark 5.7. Note that there are natural zigzags of weak equivalences Hom coAlgK (Y, Y ′ ) ≃ holim ∆ Hom Y, FK • Y ′ Definition 5.8. Let Y, Y ′ beK-coalgebra spectra. A derivedK-coalgebra map f of the form Y →Y ′ is any map in (sSet) ∆res of the form f : ∆[−] −→ Hom Y, FK • Y ′ . A topological derivedK-coalgebra map g of the form Y →Y ′ is any map in (CGHaus) ∆res of the form g : ∆ • −→ Map Y, FK • Y ′ . Background on the derived fundamental adjunction Here we review the stabilization version of the derived fundamental adjunction; see also Arone-Ching [1] (which is based on ideas described in Hess [33]), Ching-Harper [15], and Cohn [17]. The derived unit is the map of pointed spaces of the form X→ holim ∆ C(Σ ∞ X) corresponding to the identity map id : Σ ∞ X→Σ ∞ X; it is tautologically theΩ ∞ Σ ∞completion map X→X ∧ Ω ∞ Σ ∞ studied in Carlsson in [12] (see also Arone-Kankaanrinta [2]). which on objects is the map X → Σ ∞ X and on morphisms is the map [X, X ′ ]→[Σ ∞ X, Σ ∞ X ′ ]K which sends [f ] to [Σ ∞ f ]. Proposition 6.7. Let X ∈ S * and Y ∈ coAlgK. The adjunction isomorphisms associated to the (Σ ∞ , Ω ∞ ) adjunction induce well-defined isomorphisms Hom Σ ∞ X, FK • Y ∼ = −−→ Hom X,Ω ∞K• Y of cosimplicial objects in sSet, natural in X, Y . Proposition 6.8. If X is a pointed space, then there is a zigzag of weak equivalences X ∧ Ω ∞ Σ ∞ ≃ holim ∆ C(Σ ∞ X) ≃ Tot res C(Σ ∞ X) in S * , natural with respect to all such X. Definition 6.9. A pointed space X isΩ ∞ Σ ∞ -complete if the natural coaugmentation X ≃ X ∧ Ω ∞ Σ ∞ is a weak equivalence. in sSet; applying realization, together with [15, 6.14] finishes the proof. Proposition 6.11. Let X, X ′ be pointed spaces. If X ′ isΩ ∞ Σ ∞ -complete and fibrant, then there is a natural zigzag Σ ∞ : Map(X, X ′ ) ≃ −−→ Map coAlgK (Σ ∞ X, Σ ∞ X ′ ) of weak equivalences; applying π 0 gives the map [f ] → [Σ ∞ f ]. Definition 3 . 8 . 38Denote by η : id→F and m : F F →F the unit and multiplication maps of the simplicial fibrant replacement monad F on Mod S (Blumberg-Riehl [8, 6.1]). It follows thatΩ ∞ := Ω ∞ F andK := KF are the derived functors of Ω ∞ and K, respectively. The comultiplication m :K→KK and counit ε :K→F maps Definition 3. 13 . 13Let Y be aK-coalgebra. The cosimplicial cobar construction C(Y ) := Cobar(Ω ∞ ,K, Y ) in (S * ) ∆ looks like Remark 3.16. It may be helpful to note that the counit map(3.10) 3. 22 . 22Homotopical estimates and codegeneracy cubes. Here we prove Theorems 2.1 and 2.6. The following calculates the layers of the Tot tower; see, for instance, Bousfield-Kan [10, X.6.3]. Proposition 3 . 23 . 323Let Z be a cosimplicial pointed space and n ≥ 0. There are natural zigzags of weak equivalences hofib(holim ∆ ≤n Z→ holim ∆ ≤n−1 Z) ≃ Ω n (iterated hofib)Y n where Y n denotes the canonical n-cube built from the codegeneracy maps of Proposition 3 . 24 . 324Let n ≥ 0. The composite P 0 ([n]) ∼ = P ∆[n] −→ ∆ ≤n res ⊂ ∆ ≤n is left cofinal (i.e., homotopy initial). Here, P 0 ([n]) denotes the poset of all nonempty subsets of [n] and P ∆[n] denotes the poset of non-degenerate simplices of the standard n-simplex ∆[n]; see [29, III.4]. Proposition 3 . 25 . 325If X ∈ M ∆ is objectwise fibrant, then the natural maps holim BK ∆ ≤n X ≃ −−→ holim BK P ∆[n] X ∼ = holim BK P0([n]) X in M are weak equivalences; here, M is any simplicial model category. Definition 3 . 27 . 327Let Z be a cosimplicial pointed space and n ≥ 0. Assume that Z is objectwise fibrant and denote by Z : P 0 ([n])→S * the compositeP 0 ([n]) → ∆ ≤n → ∆ → S * Definition 4 . 1 . 41Let X, X ′ be pointed spaces and K a simplicial set. The tensor product X⊗K in S * , mapping object hom S * (K, X) in S * , and mapping space Hom S * (X, X ′ ) in sSet are defined by X⊗K := X ∧ K + hom S * (K, X ′ ) := hom * (K + , X ′ ) Hom S * (X, X ′ ) n := hom S * (X⊗∆[n], X ′ ) Proposition 4 . 3 . 43With the above definitions of mapping object, tensor product, and mapping space the categories of pointed spaces S * and S-modules Mod S are simplicial model categories. Proposition 4 . 5 . 45Let X be a pointed space, Y an S-module, and K, L simplicial sets. Then(a) there is a natural isomorphism σ : Σ ∞ (X)⊗K ∼ = −−→ Σ ∞ (X⊗K); (b) there is an isomorphism Definition 5. 4 . 4The realization functor | − | : sSet→CGHaus for simplicial sets is defined objectwise by the coend X → X × ∆ ∆ (−) ; here, ∆ n in CGHaus denotes the topological standard n-simplex for each n ≥ 0 (see [29, I.1.1]).Definition 5.5. Let X, Y be pointed spaces. The mapping space functor Map is defined objectwise by realization Map(X, Y ) := | Hom(X, Y )| of the indicated simplicial set. Definition 5.6. Let Y, Y ′ beK-coalgebra spectra. The mapping spaces of derived K-coalgebra maps Hom coAlgK (Y, Y ′ ) in sSet and Map coAlgK (Y, Y ′ ) in CGHaus are defined by the restricted totalizations Hom coAlgK (Y, Y ′ ) := Tot res Hom Y, FK • Y ′ Map coAlgK (Y, Y ′ ) := Tot res Map Y, FK • Y ′ of the indicated objects. Definition 6 . 1 . 61The derived counit map associated to the fundamental adjunction (1.9) is the derivedK-coalgebra map of the formΣ ∞ holim ∆ C(Y ) → Y with underlying map Σ ∞ Tot res C(Y ) −→ F Y (6.2)corresponding to the identity map id : Tot res C(Y )→Tot res C(Y )(6.3) in S * , via the adjunctions[6, 5.4] and (Σ ∞ , Ω ∞ ). In more detail, the derived counit map is the derivedK-coalgebra map defined by the HomTot res C(Y ), C(Y ) (6.4) ∼ = Hom Σ ∞ Tot res C(Y ), FK • Yin (sSet) ∆res , where ( * ) corresponds to the map (6.3) in S * .Proposition 6.5. Let X, X ′ be pointed spaces. There are natural morphisms of mapping spaces of the formΣ ∞ : Hom(X, X ′ )→ Hom coAlgK (Σ ∞ X, Σ ∞ X ′ ), Σ ∞ : Map(X, X ′ )→ Map coAlgK (Σ ∞ X, Σ ∞ X ′ ),in sSet and CGHaus, respectively. Proposition 6.6. There is an induced functor Σ ∞ : Ho(S * )→Ho(coAlgK) Proposition 6.10. There are natural zigzags of weak equivalencesMap coAlgK (Σ ∞ X, Y ) ≃ Map(X, holim ∆ C(Y )) in CGHaus; applying π 0 gives the natural isomorphism [Σ ∞ X, Y ]K ∼ = [X, holim ∆ C(Y )].Proof. There are natural zigzags of weak equivalences of the formHom(X, holim ∆ C(Y )) ≃ Hom X, Tot res C(Y ) ∼ = Tot res Hom X,Ω ∞K• Y ∼ = Tot res Hom Σ ∞ X, FK • Y ∼ = Hom coAlgK (Σ ∞ X, Y ) the homotopy category ofK-coalgebra spectra.Theorem 1.15 (Characterization theorem). AK-coalgebra spectrum Y is weakly equivalent to the suspension spectrum Σ ∞ X of some 1-connected space X, via de- rivedK-coalgebra maps, if and only if Y is 1-connected. 1.16. Strategy of attack and related work. We are leveraging a line of attack developed in Ching-Harper composite along the upper horizontal and right-hand vertical maps is s 0 d 1 f and the composite along the bottom horizontal maps is f ; the diagram commutes verifies that s 0 d 1 = id. Similarly, on the level of hom-sets (simplicial degree 0), let's verify that s 0 d 0 = id on Hom(Y, F Y ′ ). Start with f : Y →F Y ′ and consider the commutative diagram The underlying map of a derivedK-coalgebra map f is the map f 0 : Y →F Y ′ that corresponds to the map f 0 : ∆[0]→Hom(Y, F Y ′ ). Every derivedK-coalgebra map f determines a topological derivedK-coalgebra map |f | by realization.Definition 5.9. The homotopy category ofK-coalgebra spectra (compare, [1, 1.15]), denoted Ho(coAlgK), is the category with objects theK-coalgebras and morphism sets [X, Y ]K from X to Y the path components [X, Y ]K := π 0 Map coAlgK (X, Y ) of the indicated mapping spaces. Definition 5.10. A derivedK-coalgebra map f of the form Y →Y ′ is a weak equivalence if the underlying map f 0 : Y →F Y ′ is a weak equivalence. Proposition 5.11. Let Y, Y ′ beK-coalgebra spectra. A derivedK-coalgebra map f of the form Y →Y ′ is a weak equivalence if and only if it represents an isomorphism in the homotopy category ofK-coalgebras. Acknowledgments. The authors would like to thank Michael Ching for helpfulProposition 4.7. Consider the monad Ω ∞ Σ ∞ on pointed spaces S * and the comonad Σ ∞ Ω ∞ on S-modules Mod S associated to the adjunction (Σ ∞ , Ω ∞ ) in(3.2). The associated natural transformationsProof. This is an exercise left to the reader; compare [15, Proof of 3.16].Background on the homotopy theory ofK-coalgebrasIn this section we recall the homotopy theory ofK-coalgebras developed in Arone-Ching[1]; in fact, we use a tiny modification exploited in Ching-Harper[16]and Cohn[17]. The expert already familiar with[1]may wish to skip this background section.A morphism ofK-coalgebra spectra from Y to Y ′ is a map f :in Mod S commute. This motivates the following homotopically meaningful cosimplicial resolution ofK-coalgebra maps. 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In gauge theories, the physical, experimentally observable spectrum consists only of gaugeinvariant states. This spectrum can be different from the elementary spectrum even at weak coupling and in the presence of the Brout-Englert-Higgs effect.We demonstrate this for an SU(3) gauge theory with a single fundamental Higgs, a toy theory for grand-unified theories. The manifestly gauge-invariant approach of lattice gauge theory is used to determine the spectrum in four different channels. It is found to be qualitatively different from the elementary one, and especially from the one predicted by standard perturbation theory.The result can be understood in terms of the Fröhlich-Morchio-Strocchi mechanism. In fact, we find that analytic methods based on this mechanism, a gauge-invariant extension of perturbation theory, correctly determines the spectrum, and gives already at leading order a reasonably good quantitative description. Together with previous results this supports that this approach is the analytic method of choice for theories with a Brout-Englert-Higgs effect.

10.1016/j.aop.2018.08.018

1804.04453

3d970621aee0f94a60c43612790ea06cb8912c28

The spectrum of an SU(3) gauge theory with a fundamental Higgs field Axel Maas Institute of Physics NAWI Graz University of Graz Universitätsplatz 58010GrazAustria Pascal Törek Institute of Physics NAWI Graz University of Graz Universitätsplatz 58010GrazAustria The spectrum of an SU(3) gauge theory with a fundamental Higgs field In gauge theories, the physical, experimentally observable spectrum consists only of gaugeinvariant states. This spectrum can be different from the elementary spectrum even at weak coupling and in the presence of the Brout-Englert-Higgs effect.We demonstrate this for an SU(3) gauge theory with a single fundamental Higgs, a toy theory for grand-unified theories. The manifestly gauge-invariant approach of lattice gauge theory is used to determine the spectrum in four different channels. It is found to be qualitatively different from the elementary one, and especially from the one predicted by standard perturbation theory.The result can be understood in terms of the Fröhlich-Morchio-Strocchi mechanism. In fact, we find that analytic methods based on this mechanism, a gauge-invariant extension of perturbation theory, correctly determines the spectrum, and gives already at leading order a reasonably good quantitative description. Together with previous results this supports that this approach is the analytic method of choice for theories with a Brout-Englert-Higgs effect. I. INTRODUCTION Non-Abelian gauge theories in combination with scalars are compelling theories to study. Of special interest is the case of an SU(2) gauge group with a single scalar field in the fundamental representation, since this is the gauge-Higgs sector of the standard model. The physical spectrum of these kind of theories needs to be gauge invariant. This, almost tautological, insight has a realization which is far from obvious in the standard model. In QCD confinement takes care of this issue [1], while for QED dressings by Dirac phases create observable states [2]. In the weak sector the same necessity applies [3][4][5][6][7]. At first sight, this seems surprising, as a perturbative description using the BRST-invariant, but still gauge-dependent, elementary states of the Lagrangian, the W , the Z, the Higgs, and the fermion fields, describes experimental results remarkably well [8]. The, subtle, reason for this is the mechanism described by Fröhlich, Morchio, and Strocchi (FMS) [6,7]: Under certain conditions, realized in the standard model, the properties of the physical states can be mapped to the gauge-dependent states which appear in the Lagrangian. This FMS mechanism has been confirmed in lattice calculations for the scalar sector [9,10]. An extensive review on this (and more concerning field theories with scalars) can be found in [11]. However, the conditions mentioned are quite specific, and the standard model is special to fulfill them. Especially, the weak gauge group is the same as the global custodial symmetry group. In general beyond-standardmodel theories this is not the case, and they therefore potentially not meet these requirements [12]. Then, discrepancies between the actual physical spectrum and the elementary one, and thus the one described by perturbation theory, may arise. Investigations of explicit examples have found both types of behaviors [11,13,14]. * [email protected] † [email protected] In particular, this can imply that the low-lying observable spectrum is different from the standard model, even if a model features perturbatively the W and Z bosons and a light Higgs. Such theories would therefore not be suitable extensions of the standard model. The FMS mechanism can be used to create the analytic tool of gauge-invariant perturbation theory (GIPT) [11]. This tool has been applied to SU(N ) gauge theories with scalars in different representations in [14]. Such theories are of particular interest as their structures is typical for so-called grand-unified theories (GUTs), which are one of the candidates for beyond-standard-model theories. This lead to analytical predictions of the spectrum, which generically disagree with the elementary one. The primary aim of this work is to check these analytical predictions. This requires a manifestly gaugeinvariant approach which is capable of treating nonperturbative physics. We choose this method to be the lattice. The required resources forced us to concentrate on a particular case, an SU(3) gauge theory with a fundamental scalar. The technicalities and details of the lattice simulations can be found in Section II. This includes how the spectroscopy of gauge-invariant operators is performed, and how gauge-variant quantities, like the propagators of elementary fields and the running gauge coupling, are obtained. This section can be skipped entirely if only the results are of interest. In Section III we concentrate on the properties of the theory. We present the phase diagram of the theory, and show in which regions Brout-Englert-Higgs (BEH) physics or QCD-like physics takes place. We then compute the spectrum of gauge-invariant states as well as the spectrum of elementary fields in the Higgs-like region of the phase diagram. Finally, in Section V the primary aim of this work will be achieved, the test of FMS mechanism and GIPT. To this end, we first show that standard perturbation theory is not able to describe the physics of the theory even qualitatively. Then, to be self-contained, we first rehearse the predictions of the spectrum [14]. Finally, we compare the results of the lattice simulations to the predictions of arXiv:1804.04453v2 [hep-lat] 16 Apr 2018 GIPT. We find that, already at leading order, all channels are qualitatively correctly described, and even the quantitative agreement is good in all channels where the lattice results are reasonably reliable. This strongly supports GIPT as the analytic tool for this type of theories. This agrees with all other available results, especially in the standard model [11]. Preliminary and related results of this work can be found in [14][15][16][17]. II. TECHNICALITIES We consider an SU(3) gauge theory with a single scalar in the fundamental representation of the gauge group. The theory has therefore a global U(1) custodial symmetry [14]. A. Preliminaries The action of the theory on a 4-dimensional, Euclidean, isotropic, hypercubic lattice with lattice constant a, and volume V = L 4 , is given by [18] S[U, φ] = x φ(x) † φ(x) + λ φ(x) † φ(x) − 1 2 − κ ±4 µ=±1 φ(x) † U µ (x) φ(x +μ) + β 3 µ<ν Re tr 1 − U µν (x) ,(1) The first sum runs over all lattice sites x = (x 1 , x 2 , x 3 , x 4 ), x i = 0, 1, . . . , L − 1 andμ denotes the unit vector in the µ-direction. The first term of the action is the Wilson gauge action with the plaquette variable U µν (x), which is a product of four link variables U µ (x) forming a closed loop, i.e., (2) and is essentially the field-strength tensor squared plus O(a 2 )-corrections in the naive continuum limit a → 0. The links are related to the gauge fields by U µ (x) = exp(iaA c µ (x)T c ), with 2T c being the Gell-Mann matrices. Thus, the links are elements of the gauge group SU (3). Note, that U −µ (x) ≡ U µ (x −μ) † . U µν (x) = U µ (x) U ν (x +μ) U µ (x +ν) † U ν (x) † , Both, the scalar field as well as the links, obey periodic boundary conditions, i.e., φ(x +νL) = φ(x) , U µ (x) = U µ (x +νL) . ( Under gauge transformations the scalar field and the gauge links transform as φ(x) → g(x) φ(x) , U µ (x) → g(x) U µ (x) g(x +μ) † ,(4) with g(x) ∈ SU (3). The scalar field also transforms under global custodial U(1) transformations exp(iα) ∈ U(1). The Equation (1) is invariant under these transformations, making the action gauge and custodial invariant. In total three parameters appear in the action (1): β is the inverse gauge coupling, λ is the coupling for the self-interaction of the scalar fields, and κ is related to the square of the inverse bare mass. Those lattice parameters are related to the continuum ones by β = 6 g 2 , a 2 m 2 0 = 1 − 2λ κ − 8 , λ c = λ κ 2 ,(5) where m 0 is the bare mass and λ c the bare self-interaction of the corresponding continuum theory. In order to generate configurations we use one multihit Metropolis sweep for the links, where 5 attempts are made to update one link by standard techniques [19] before moving to the next link, and one subsequent Metropolis sweep for the scalar field using a Gaußian proposal. We tuned the widths of the proposals adaptively to achieve a 50% acceptance rate for both updates. After every 5 sweeps through the lattice, a projection step of the gauge links to SU(3) matrices is performed by a standard Gram-Schmidt procedure [19] in order to keep rounding errors under control. A list of all lattice parameter sets with the corresponding lattice volumes and number of configurations is given in Appendix A in Table III. B. Techniques for gauge-invariant quantities For the spectroscopy we use the zero-momentum projected interpolators listed in Table I with distinct J P C U (1) quantum numbers, where the lower index is the quantum number of the global custodial group U(1), which only acts on the scalar field. The parity P , the charge parity C, and the total angular momentum J, are assigned to the interpolators by their transformation properties under the octahedral symmetry group O h , which is the discrete symmetry group of an isotropic lattice. A method on how these quantum numbers are assigned to the interpolators according to the irreducible representations of the octahedral group can be found in, e.g., [20] and [21]. Note, that we only use the spatial-directions µ = 1, 2, 3 for the operators, since we are interested in the propagation of the state in Euclidean time-direction µ = 4. In Table I several of the interpolators can be viewed as bound states of the scalar and the gauge bosons in the language of a naive constituent interpretation (we only discuss the 'atomic' interpolators here): • O 0 ++ 0 1 describes a two-scalar bound state. (1) µ , L (2) µ , and L (3) µ can be found in the main text. We perform a zero-momentum projection for all interpolators. We use the notation x x x = (x1, x2, x3), and t = x4. Name Interpolator J PC U(1) O 0 ++ 0 1 (t) 1 L 3 x x x φ(x x x, t) † φ(x x x, t) 0 ++ 0 O 0 ++ 0 2 (t) O 0 ++ 0 1 (t) O 0 ++ 0 1 (t) 0 ++ 0 O 0 ++ 0 3 (t) 1 L 3 x x x 3 µ,ν=1,µ<ν Re tr Uµν (x x x, t) 0 ++ 0 O 1 −− 0 1,µ (t) i L 3 x x x φ(x x x, t) † Dµ φ(x x x, t) 1 −− 0 O 1 −− 0 2,µ (t) O 0 ++ 0 1 (t) O 1 −− 0 1,µ (t) 1 −− 0 O 1 −− 0 3,µ (t) 3 ν=1 O 1 −− 0 1,ν (t) O 1 −− 0 1,ν (t) O 1 −− 0 1,µ (t) 1 −− 0 O 1 −− 0 4,µ (t) 1 L 3 x x x Im L (1) µ (x x x, t) 1 −− 0 O 1 −− 0 5,µ (t) 1 L 3 x x x Im L (2) µ (x x x, t) 1 −− 0 O 1 −− 0 6,µ (t) 1 L 3 x x x Im L (3) µ (x x x, t) 1 −− 0 O 0 ++ 0 4 (t) 3 µ,=1 O 1 −− 0 1,µ (t) O 1 −− 0 1,µ (t) 0 ++ 0 O 2 ++ 0 (t) 1 L 3 x x x Re tr U12(x x x, t) + U23(x x x, t) − 2 U13(x x x, t) 2 ++ 0 O 0 −+ 0 (t) 1 L 3 x x x 3 µ =ν =γ =ρ=1 tr Uµν (x x x, t) Uγρ(x x x, t) 0 −+ 0 O 0 ++ 1 (t) 1 L 3 x x x 3 µ,ν=1 ijk φi (Dµφ)j (DµDν Dν φ) k (x x x, t) 0 ++ 1 O 1 −− 1 µ (t) 1 L 3 x x x 3 ν=1 ijk φi (Dµφ)j (Dν Dν φ) k (x x x, t) 1 −− 1 covariant derivative defined as D µ φ(x) = U µ (x)φ(x +μ) − U µ (x −μ) † φ(x −μ) 2 . (6) • O 0 ++ 0 3 is a scalar gaugeball. • O 1 −− 0 4,5,6,µ are vector gaugeball interpolators as defined in [20]. Several definitions are needed to define the spatially summed quantities L W µνρ (x) = tr U µ (x)U µ (x +μ)U ν (x + 2μ)U µ (x +μ +ν) † ×U ρ (x +μ +ν)U µ (x +ν +ρ) † U ρ (x +ν) † U ν (x) † ,(7) and linear combinations thereof, L (1) µνρ = W +µ+ν+ρ + W +µ+ν−ρ + W +µ−ν+ρ + W +µ−ν−ρ −W −µ+ν+ρ − W −µ+ν−ρ − W −µ−ν+ρ − W −µ−ν−ρ , L (2) µνρ = W +µ+ν+ρ + W +µ+ν−ρ + W +µ−ν+ρ − W +µ−ν−ρ +W −µ+ν+ρ + W −µ+ν−ρ − W −µ−ν+ρ − W −µ−ν−ρ , L (3) µνρ = W +µ+ν+ρ − W +µ+ν−ρ + W +µ−ν+ρ − W +µ−ν−ρ +W −µ+ν+ρ − W −µ+ν−ρ + W −µ−ν+ρ − W −µ−ν−ρ ,(8) where we skipped the spacetime argument x for brevity. The last step is to build the following linear combinations of Equation (8) to build the interpolators that give the vector representation, J = 1, and negative parity P : L (1) = L (1) 123 + L (1) 132 , L(1)231 + L (1) 213 , L(1)312 + L(1) 321 , L (2) = L(2) 123 + L (2) 321 , L 231 + L (2) 132 , L 312 + L (2) 213 , L (3) = L(3)123 + L(3) 213 , L 231 + L(3)321 , L(3)312 + L(3) 132 . Taking the imaginary parts of these quantities yields interpolators with negative charge parity C. Vector gaugeball interpolator with other P and C quantum numbers could be constructed from the definitions given above [20]. However, we are particularly interested in the 1 −− 0 gaugeball for reasons illustrated in the next subsection where this quantum number channel is discussed. • O 0 −+ 0 , and O 2 ++ 0 are a pseudo-scalar gaugeball, and a tensor gaugeball, respectively, see [22]. • O 0 ++ 1 and O 1 −− 1 µ are the only interpolators with an open U(1)-quantum number. We assigned a U(1) charge of 1/3 to the scalar field φ. The continuum versions are discussed in [14] and the corresponding lattice versions are O 0 ++ 1 (t) = 1 L 3 x x x 3 µ,ν=1 ijk φ i (D µ φ) j (D µ D ν D ν φ) k (x x x, t) = 1 16 L 3 x x x 3 µ,ν=1 ijk φ i (x x x, t) × U µ (x x x, t) φ(x x x +μ, t) − U µ (x x x −μ, t) † φ(x x x −μ, t) j × U µ (x x x, t) U ν (x x x +μ, t) U ν (x x x +μ +ν, t) φ(x x x +μ + 2ν, t) − U µ (x x x −μ, t) † U ν (x x x −μ, t) U ν (x x x −μ +ν, t) φ(x x x −μ + 2ν, t) + U µ (x x x, t) U ν (x x x +μ −ν, t) † U ν (x x x +μ − 2ν, t) † φ(x x x +μ − 2ν, t) − U µ (x x x −μ, t) † U ν (x x x −μ −ν, t) † U ν (x x x −μ − 2ν, t) † φ(x x x −μ − 2ν, t) k ,(10)O 1 −− 1 µ (t) = 1 L 3 x x x 3 ν=1 ijk φ i (D µ φ) j (D ν D ν φ) k (x x x, t) = 1 8 L 3 x x x 3 ν=1 ijk φ i (x x x, t) × U µ (x x x, t) φ(x x x +μ, t) − U µ (x x x −μ, t) † φ(x x x −μ, t) j × U ν (x x x, t) U ν (x x x +ν, t) φ(x x x + 2ν, t) + U ν (x x x −ν, t) † U ν (x x x − 2ν, t) † φ(x x x − 2ν, t) k .(11) We employ a variational analysis [23][24][25] in order to get access to the energy levels of the different quantum states in the respective J P C U(1) -channels. Therefore, we compute a time-sliced matrix of cross correlators for a set of basis interpolators O i , i = 1, 2, . . . , N , defined as C ij (t) = 1 L L−1 t =0 O i (t )O † j (t + t ) c = 1 L L−1 t =0 O i (t ) − O i (t ) × O † j (t + t ) − O † j (t + t ) ,(12) where we subtracted the vacuum contribution O i (t) from the correlator, i.e., we only consider the connected contributions · · · c of the correlator. This is necessary since states with the quantum numbers J P C = 0 ++ mix with the vacuum, which has exactly these quantum numbers. One can show that the eigenvalues of the matrix (12) behave as λ k (t) ∼ e −aE k t , k = 0, 1, . . . , N − 1 [26]. Thus, the energy levels can be extracted as aE k (t + 1 2 ) = ln λ k (t) λ k (t + 1) .(13) Since all fields appearing in the action (1) obey periodic boundary conditions, the propagation in t and L − t of all the interpolators O i is identical and thus we fit the eigenvalues to λ k (t) = A (1) k cosh aE (1) k (t − L/2) + A (2) k cosh aE (2) k (t − L/2) ,(14) to extract the numerical values of the energy levels. We take into account a possible excitation of the level E k , since heavier states still can contribute for small values of t to this level after the variational analysis. The bound states of Table I are expected to have a finite extent. Approximating them with point-like operators can therefore create an overlap problem. Therefore, we smear all our fields. For the links we apply stout smearing according to the procedure in [27]. We choose this approach due to fact that with this method a projection back to the gauge group is not necessary. The new link after one stout smearing step is U µ (x) = e iQµ(x) U µ (x) ,(15) where Q µ (x) is a hermitian and traceless matrix given by Q µ (x) = i 2 Ω µ (x) † − Ω µ (x) − 1 3 tr Ω µ (x) † − Ω µ (x) , Ω µ (x) = ν =µ ρ µν C µν (x) † U µ (x) † ,(16) where the so-called staples C µν (x) enter: C µν (x) = U ν (x +μ) U µ (x +ν) † U ν (x) † + U ν (x +μ −ν) † U µ (x −ν) † U ν (x −ν) .(17) We set ρ µ4 = ρ 4µ = 0, ρ ij = ρ in Equation (16) since we want to measure correlations in the Euclidean time direction and thus only spatial links are allowed to be smeared. In all our simulations we set ρ = 0.1, see [27]. Of course, this procedure can be iterated. Therefore, the new link after (n + 1) stout smearing steps is given by U (n+1) µ (x) = e iQ (n) µ (x) U (n) µ (x) .(18) The scalar field is APE smeared in our case. After (n + 1) APE smearing steps the field is then [28] φ (n+1) (x) = 1 7 φ (n) (x) + ±4 µ=±1 U (n) µ (x) φ (n) (x +μ) ,(19) where the n-times stout smeared links U (n) µ (x) enter in the smearing procedure of the scalar. We usually perform 300 + 10L updates to drive the system into equilibrium. Between the measurements of the observables we drop 3L configurations for decorrelation. We also performed several independent runs with different random number seeds for each parameter set to further reduce correlations. The integrated autocorrelation time for the plaquette is τ int ≈ 1/2, i.e., close to the minimal value, for the parameter sets we analyzed. Therefore, no significant correlations between subsequent measurements of observables are detected. We usually studied V = 8 4 , 12 4 , 16 4 , and 20 4 lattices to perform a finite-size analysis of the resulting masses in several quantum number channels. We typically have O 10 5 configurations at hand to compute the correlation functions. E.g., in Section III B we used V = 8 4 with 320000 configurations, V = 12 4 with 240000 configurations, V = 16 4 with 120000 configurations, and V = 20 4 with 190000 configurations. The errors of the correlators are computed throughout by a standard Jackknife procedure, and for secondary observables we use the method of error propagation unless stated otherwise. C. Techniques for gauge-variant quantities In order to compute propagators of elementary fields we need to fix a gauge. Without this procedure the propagators would be zero [29]. Determining these propagators is relevant, as they will be an important building block of GIPT. They will also provide additional support that we probe the theory at weak coupling and our observed results are not genuine strong-coupling effects. • Local and global gauge fixing We locally fix to minimal Landau gauge as described in [30,31] by the so-called stochastic overrelaxation method [32]. Additionally we use the Cabibbo-Marinari trick [33] and the method of maximal trace [34] for reunitarization of the links. To accomplish the so-called 't Hooft-Landau gauge condition [35], which gives rise to a vacuum expectation value of the scalar field, we have to fix also the global direction of the scalar field. We want to perform a global gauge transformation such that the space-time averageφ of the scalar field point into some direction n: gφ φ = n withφ = 1 V x φ(x) and g ∈ SU(3) ,(20) where we set n i = δ i,3 without loss of generality. We use two consecutive SU(3) rotations, i.e., gφ = g 2 g 1φ = n , g 1 , g 2 ∈ SU(3) .(21) Without loss of generality we assume a normalized vector |φ| = 1 in the following. The first transformation g 1 has the task to rotate the first component ofφ to zero: g 1φ =    g 11 1 g 12 1 0 − g 12 1 g 11 1 0 0 0 1      φ 1 φ 2 φ 3    =    0 φ 2 φ 3    =φ , with g 11 1 2 + g 12 1 2 = 1 .(22) The second transformation g 2 then rotates the second component ofφ to zero: g 2φ =    1 0 0 0 g 11 2 g 12 2 0 − g 12 2 g 11 2       0 φ 2 φ 3    =    0 0 φ 3    =φ , with g 11 2 2 + g 12 2 2 = 1 .(23) Solving these equations for the matrix elements g nm i with the normalization constraint gives the desired transformation matrix g. To summarize, the following steps have to be performed: 1. Normalizeφ, φ = 1. Compute g 11 1 = 1 + φ 1 2 φ 2 2 − 1 2 , g 12 1 = −φ 1φ 2 φ 2 2 .(24) 3. Compute Re g 11 2 = 1 1 + φ 2 2 φ 3 2 Re φ 3 1 + Im φ 3 2 Re φ 3 2 −1 , Im g 11 2 = Imφ 3 Reφ 3 Re g 11 2 , g 12 2 = −φ 3 φ 3 2 g 11 1 g 11 1 2φ 2 g 11 2 .(25) 4. Construct g = g 2 g 1 from the previous steps. 5. Apply the global gauge transformations to the scalar and gauge fields: φ(x) → g φ(x) , U µ (x) → g U µ (x) g † , ∀ x, µ . (26) With this procedure the gauge is now completely fixed to the minimal 't Hooft Landau gauge. Note that we use less gauge-fixed configurations than for the spectrum calculations, as gauge-fixing is expensive in terms of computing time while at the same time the quantities we study are much less noisy. E.g., for the situation in Section IV we use for the 8 4 lattice 16000, for the 12 4 lattice 12000, for the 16 4 lattice 4700, and for the 20 4 lattice 5500 gauge-fixed configurations. • Propagators We are interested in the propagator of the gauge bosons, the scalars and the ghosts. The latter will be needed to determine the running gauge coupling. Due to the isotropic lattice, we can take the trace over the Euclidean Lorentz-indices of the gauge-field propagator D bc µν p 2 = A b µ (p) A c ν (−p) , with the momentum p µ = 2 a sin π L k µ , k µ = 0, 1, . . . , L/2. Further, the propagator is proportional to δ bc in the minimal 't Hooft Landau gauge, and thus D bc µν = δ bc D c µν , with D c p 2 = 4 µ=1 A c µ (p) A c µ (−p) , c = 1, 2, . . . , 8 . (27) For the scalar propagator we split the field φ into its real and imaginary parts and use the notation φ = 6 . Then, we define the propagator as 1 √ 2 φ 1 + i φ 2 , φ 3 + i φ 4 , φ 5 + i φD ij p 2 = φ i (p) φ j (−p) , i, j = 1, 2, . . . , 6 . (28) Again, in the minimial 't Hooft Landau gauge this propagator is diagonal, i.e., D ij p 2 = D i p 2 δ ij . As it is discussed in [14] we expect, for the vev-choice n i = δ i, 3 , that only the propagator D 5 (p 2 behaves like a massive propagator and the remaining ones correspond to the propagation of massless particles in the Landau gauge. The ghost field propagator G ab (x, y) = c a (x)c b (y) can be computed by inverting the Faddeev-Popov operator M ab (x, y). On the lattice this operator is a linear combination of links mixed with the generators of the gauge group. This will be done using the methods described in [36]. • Running gauge coupling Having computed the gauge field propagator (27) and the ghost propagator on the lattice, the running coupling α b can be extracted for every value of b = 1, 2, . . . , N 2 −1 in the miniMOM scheme [37,38] as α b p 2 = p 6 α µ 2 G b p 2 , µ 2 2 D b p 2 , µ 2 ,(29) where µ is the renormalization scale. Note that, this is a renormalization-scale invariant combination. • Renormalization of the scalar propagator We need to define a renormalization scheme for the scalar propagator D ij p 2 . To this end we follow [39,40], which assumes that the renormalization of the propagator works qualitatively as in the perturbative case [35]. Thus, there are two renormalization constants: The multiplicative wave function renormalization Z i and an additive mass renormalization δm 2 i . This yields the renormalized scalar propagator in minimal 't Hooft Landau gauge, D r i p 2 = 1 Z i p 2 + m r i 2 + Π i p 2 + δm 2 i ,(30) for i = 1, 2, . . . , 6 and where m r i is the renormalized mass of the i th particle and Π i p 2 is the corresponding self energy which is obtained from the unrenormalized propagator (28) as Π i p 2 = 1 − p 2 D i p 2 D i p 2 .(31) Thus, the self energy measures essentially the deviation from the tree-level propagator, i.e., D i p 2 = 1 p 2 + Π i p 2 .(32) Note, that the tree-level mass m i is implicitly included in the self-energy. The scheme we use to fix the renormalization constants is: D r i µ 2 = 1 µ 2 + m r i 2 , dD r i p 2 dp p 2 =µ 2 = − 2µ µ 2 + m r i 2 2 ,(33) where µ is again the renormalization scale. Therefore, the renormalized propagator and its derivative are given by their tree-level values at p 2 = µ 2 . From these equations the renormalization constants Z and δm can be derived. The renormalization constants are determined numerically by linear interpolation between two physical momenta along the x-axis, with the value of µ inside the interval (p 1 , p 2 ). The derivative of the self-energy is obtained by analytically deriving the linear interpolation between the momenta points. We only choose values for µ such that 0 < p 1 < µ < p 2 < 2/a. Note that both the gauge boson propagator and the ghost propagator require only a single multiplicative renormalization. • Position-space propagators One can also compute form the momentum-space propagators the position-space correlators, also called Schwinger functions. The position-space correlator ∆(t) is computed by [41] ∆(t) = 1 aπL L−1 p4=0 cos 2πp 4 L t D p 2 4 ,(34) for a field with propagator D p 2 . Note that, the propagator D p 2 4 is evaluated at zero spatial momentum, as is indicated by the argument p 2 4 , and the sum extends over the whole momentum range including the parts of the propagator reproduced by periodicity. III. THE PHYSICS OF AN SU(3) GAUGE THEORY WITH A FUNDAMENTAL SCALAR A. Phase diagram of the theory Since the perturbative breaking pattern in our case is SU(3) → SU(2) and thus the gauge group is not fully broken, the Osterwalder-Seiler-Fradkin-Shenker argument [4,42] does not apply. Therefore, this theory may or may not have separated phases and a possibly rich phase structure. We expect (at least) two regions of the phase diagram: Due to the non-Abelian nature of our theory defined in Equation (1), a QCD-like region (QLR) where QCD-like physics takes place and due to the Higgs sector we also expect a region with BEHlike physics (HLR). Since we are especially interested in a situation with a perturbatively accessible BEH effect [6,9,10,43] we scanned the phase diagram using the quantity [44] φ 2 = 1 V x φ(x) 2 = 1 V 2 x,y φ(x) † φ(y) ,(35) withφ being the space-time average of the scalar field. This quantity is gauge-dependent, and thus determined after fixing to minimal 't Hooft Landau gauge gauge. If the BEH effect is active [17,44]. Examples of how this quantity behaves can be found in [17]. φ2 V →∞ − −−− → const. > 0, while without φ2 ∼ 1/V V →∞ − −−− → 0 To scan the phase diagram quickly, we performed simulations for V = 4 4 , 6 4 , 8 4 , and 12 4 lattices for randomly distributed parameters β, κ, and λ. We measured the quantity defined in Equation (35) on 1000 gauge-fixed configurations for each random parameter set and lattice size. Then, the volume dependence of this observable was used to decide to which region the parameter point belongs to. This lead to the results shown in Figure 1. The corresponding data can be found in Table VIIa and Table VIIb. (35). The red dots show a BEH effect in minimal 't Hooft Landau gauge, putting them in the HLR, while the blue triangles do not, meaning that they are in the QLR of the phase diagram. This is an update to the phase diagram in [17]. B. Physical spectrum In what follows, we focus on a set of parameters in the Higgs-like region, since our main target is to test the analytical predictions of the FMS mechanism in the end. We choose a point close to the boundary of the two regions of the phase diagram given by β = 6.85535, κ = 0.456074, λ = 2.3416. This choice is motivated by the simulation results of the SU(2) theory, where the smallest lattice spacings, i.e., the largest cutoffs, have been found [22]. We have also studied the spectrum for different sets of lattice parameters, which are listed in Table III in Appendix A. As far as a statistically reliable signal could be obtained we did not observe any qualitative differences. Hence, this set of parameters yields a suitable representative for the spectrum. In the following, we investigate individually all the quantum number channels which are listed in Table I. , which contain only scalar fields, are smeared ten times as they are statistically very noisy due to the fact that the vacuum carries the same quantum numbers. For the same reason, we smear the gaugeball operator O 0 ++ 0 3 ten times as well. However, the interpolator O 0 ++ 0 4 , which is a scattering state built form two 1 −− 0 operators, seems to be less noisy, and it was therefore only needed to smear it four and five times. Table VI in Appendix A). The extrapolated mass is am 0 ++ 0 = 0.68 (2). Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) - ( ) ( + ) ̀( )( The resulting effective mass as a function of time is plotted in Figure 2. We plot the energy of the lowest state (ground state), for each volume, and the second energy level (first excited state) for the largest volume. The effective masses and their errors, listed in the legend of Figure 2, are obtained by fitting the the mean, upper, and lower value of the eigenvalues by a double-cosh, see Equation (14), for each volume. The resulting fit parameters are listed in Table IV in Appendix A. Note that because of the large statistical noise we do not show data points for t > 6. The volume dependence of the ground state mass is also plotted in Figure 2. We see that this state has a moderate dependence on the lattice size. Nevertheless, a fit of the lattice masses as a function of the volume, am 0 ++ 0 (V ) = am 0 ++ 0 + δ e −γ V , can be performed and gives the gray error band (see Table VI in Appendix A for the numerical values). We conclude that the dimensionless ground state mass in this channel is am 0 ++ 0 = 0.68 (2) which is below the 2 am 1 −− 0 threshold, i.e., the elastic threshold, as it can be seen in the discussion of the 1 −− 0 channel below. Since our analysis below suggests that this is the only open decay channel, this implies that the 0 ++ ground state is a stable particle. The next-level state has an approximated mass of am 0 ++ 0 ≈ 0.9(1) which is compatible with the 2am 1 −− 0 = 0.78(2) threshold scattering state expected from the pro- cess 0 ++ 0 → 1 −− 0 + 1 −− 0 . However, much more statistics for all volumes would be needed to make a definite statement. The next expected states are the ones with mass 2am 0 ++ 0 and with 2am 0 ++ 0 + p rel , where p rel is a non-zero relative momentum. However, these states are relatively heavy and only noisy signals around this region have been found and thus no definite results are available. • 1 −− 0 channel For this channel a suitable basis of operators was found to be O 1 −− 0 1,µ,(3) , O 1 −− 0 1,µ,(4) , O 1 −− 0 2,µ,(3) , O 1 −− 0 2,µ,(4) , O 1 −− 0 3,µ,(3) , O 1 −− 0 3,µ,(4) .(37) The vector gaugeball interpolators O 1 −− 0 4,µ , O 1 −− 0 5,µ , and O 1 −− 0 6,µ were too noisy even for the largest used smearing level as can be seen from the effective masses in Figure 5 below. However, those states seem to be very high up in the spectrum and thus it is a justified assumption that they do not alter the infrared spectrum of the theory. In the top of Figure 3, we show the energy levels obtained from the variational analysis with the crosscorrelation matrix built from the basis interpolators (37). The lowest energy level is shown for each lattice volume. As in the previous discussion the second energy level is only shown for the largest volume and for t < 6. Again we list the effective masses in the legend of this figure, which are obtained by the same fit strategy as in the Table IV in Appendix A. Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) - ( ) ( + ) ̀( ) ( + ) Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = ( ) Ԃ = ( ) ( ) ( ) / ++ + -- ++ + --+ The ground state has almost no volume dependence, hence the infinite volume extrapolated ground state mass is am 1 −− 0 = 0.39 (1), see the bottom of Figure 3 and Table IV. Hence, the singlet vector state is lighter than the sin- . This is possible since the operators can have overlap with such states, even if carrying zero total momentum. The energy levels E can be extracted from [20] sinh 2 E(L, k) 2 = sinh 2 m 2p 2 + 3 i=1 sin 2 π L k i ,(38) where m 2p is the mass of the two-particle state and the relative lattice momentum is p rel Here we show the spectroscopy results of several gaugeball states. All the results shown below share that the signals are very noisy. This makes determinations of their masses comparatively unreliable. Still, all the masses seem to be well above the lowest lattice mass in the spectrum, i.e., above am 1 −− 0 . Figure 4 shows the effective masses of the 0 −+ 0 pseudoscalar gaugeball in the top panel and the 2 ++ 0 gaugeball in the bottom panel as a function of Euclidean time for several lattice volumes. We do not show data points for t > 3 and t > 2 respectively, since these regions are dominated by noise even though we used 10-times smeared operators. The effective masses in both channels are around am 0 −+ 0 ≈ am 2 ++ 0 ≈ 2.0, i.e., above the lattice cutoff. These approximate masses are of course just crude estimates. We also performed a variational analysis with sets of different smeared operators in these channels. However, this procedure did not improve the signal substantially and therefore we do not show the results here. Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = Ԃ = Ԃ = Ԃ = ( ) ( + ) ( ) ( + ) U (1) = -+ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = Ԃ = Ԃ = Ԃ = ( ) ( + ) ( ) ( + ) U (1) = ++ Some, but not all, possible decay channels for the two states with the available channels are: . Nonetheless, this is very speculative since more statistics and more operators including better overlap with the decay channels would be needed to make precise statements. Of course, another option is that those signals are just too noise dominated. In Figure 5 we show the effective masses of the three 1 −− 0 gaugeballs, L (1) , L (2) , and L (3) , for V = 8 4 , 12 4 , 16 4 and 20 4 lattices. As before, we do not plot the whole time region in all the plots due to the large fluctuations of the correlators and thus the effective masses. The results are shown for 10-times smeared operators as before. -0 −+ 0 channel: two 1 −− 0 in a p-wave -2 ++Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = Ԃ = Ԃ = Ԃ = ) + ) ) + ) U (1) = --( ( ) ) Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = Ԃ = Ԃ = Ԃ = ) + ) ) + ) U (1) = --( ( ) ) Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = Ԃ = Ԃ = Ԃ = ) + ) ) + ) U (1) = --( ( ) ) Even though the signals are again noisy we deduce that the effective masses of the three 1 −− 0 gaugeballs are approximately 7-times larger than the extracted ground state mass in this channel, and thus well above the lattice cutoff. As already argued in the discussion of the 1 −− 0 channel, they do not alter the ground state and thus the infrared spectrum, since they are too high up in the spectrum to generate any significant contribution. We are well aware that the effective mass plateau of three points which are still inclined, are probably still contaminated by excited state contributions, and higher statistics would be needed. In Figure 6 we present results for the effective masses of the 0 ++ ±1 (left plot) and 1 −− ±1 (right plot) channels for different lattice volumes. In both channels we performed a variational analysis with different smearing levels of the corresponding operators: In the scalar sector the basis consists of 6-to 10-times smeared interpolators, whereas in the vector sector we included 8-to 10-times smeared interpolators in the basis. The effective masses 1 of both states are listed in the legends of the figure and the corresponding values from the fit are given in Table VI in Appendix A. 1 Note that, m 0 ++ +1 = m 0 ++ −1 , i.e., particle and anti-particle states. Thus the effective mass given in the left-hand side of Figure 6 is the mass of the particle and anti-particle. Certainly, the same is true for the 1 −− ±1 channel. The scalar sector is dominated by noise and only the points for t < 3 are reliable. The mass in this channel is roughly am 0 ++ ±1 ≈ 2. Of course this is only a coarse estimation and larger statistics as well as larger lattices can alter the result. The vector channel is not so much dominated by noise and thus more time slices can be used for the fit (t < 6). However, from V = 8 4 to V = 16 4 the effective mass seems to drop but for the largest lattice slightly rises again. Again, more statistics could still change this behavior. Nonetheless, we estimate a mass of am 1 −− ±1 ≈ 0.8 (2). Higher levels were unaccessible due to the amount of statistical noise. Besides increasing the statistics also including more operators could improve the result in both cases. • Summary of the spectrum We summarize the computed spectrum of states in Figure 7. The filled boxes correspond to the ground states, the empty boxes are the elastic thresholds for the scalar and vector singlet channels as discussed above, and the dashed lines are the estimated ground state masses of the 0 ++ ±1 , 0 −+ 0 , 2 ++ 0 channels. Where available, results for other lattice parameters can be found in the appendix. Of course, it would be important to track the development of the spectrum along lines of constant physics, even though we find qualitatively the same situation everywhere in the HLR. Due to the fine-tuning character of the theory, the required numerical resources for this purpose, also given that at least three states have to be determined reliably for this, unfortunately exceed our resources by far. IV. GAUGE-VARIANT OBSERVABLES AND RUNNING GAUGE COUPLING A. Spectrum from tree-level perturbation theory For future reference, we briefly rehearse the spectrum of the theory at tree-level perturbation theory, see [14] for details. For this we use a continuum setup, and employ 't Hooft-Landau gauge. We split the scalar field into its vev and a fluctuation part ϕ around the vev φ(x) = v √ 2 n + ϕ(x) .(39) The spectrum then contains one real-valued massive scalar degree of freedom and 8 would-be Goldstone modes. The non-Goldstone Higgs boson and the wouldbe Goldstones can be described in a gauge-covariant manner without specifying n by h ≡ √ 2 Re(n † φ) and ϕ ≡ φ − Re(n † φ)n = ϕ − Re(n † ϕ)n, respectively. However, without loss of generality, in the following we will usually make the explicit choice n i = δ i, 3 . Rewriting the scalar kinetic term of the Lagrangian by splitting the Higgs field into the vev and the fluctuation part, we obtain (D µ φ) † (D µ φ) = ∂ µ ϕ † ∂ µ ϕ + g 2 v 2 2 n † T a T b n A µ a A b µ + √ 2gv Im(n † T a ∂ µ ϕ)A a µ + . . . ,(40) where we have the usual [35] mass matrix for the gauge bosons in the first line and the mixing between the longitudinal parts of the gauge bosons and the Goldstone bosons in the second line. Note that only those gauge bosons mix with the Goldstone bosons which acquire a mass, i.e., which correspond to the broken generators of the gauge group. These mixing terms are removed by the 't Hooft Landau gauge fixing condition [35]. The mass matrix (M 2 A ) ab of the gauge bosons is already diagonal for our choice of n, and is given by, (M 2 A ) ab = g 2 v 2 2 n † {T a , T b }n = g 2 v 2 4 diag 0, 0, 0, 1, 1, 1, 4 3 ab .(41) Thus, we obtain 3 massless gauge bosons, 4 degenerated massive gauge bosons with mass m A = 1 2 gv and one with mass M A = 4/3 m A . Moreover, the elementary Higgs field has a mass m 2 h = λ 2 c v 2 , where λ c is the four-Higgs coupling, i.e., the term λc 2 (φ † φ) 2 in the continuum setup. The situation is now that which is, in an abuse of language, usually called 'spontaneously broken' in case the system experiences the BEH effect. The breaking pattern reads SU(3) → SU (2). With respect to the subgroup SU(2) the gauge bosons are in the adjoint representation (massless), a fundamental and an anti-fundamental representation (mass m A ) and a singlet representation (mass M A ), explaining their degeneracy pattern. B. Spectrum from the lattice Here we again study the same set of lattice parameters as before, as again the behavior is representative for all other cases. • Gauge-field propagators Let us now focus on the propagator of the gauge bosons D c p 2 , c = 1, 2, . . . , 8, defined in Equation (27). The lattice momenta p µ = 2πk µ /L are along the links, and along all possible diagonals of the lattice, i.e., (k, 0, 0, 0), (k, k, 0, 0), (k, k, k, 0), and (k, k, k, k), k = 0, 1, . . . , L/2. In the top panel of Figure 8 we show the propagators, evaluated on a 20 4 lattice, of the perturbatively 3 massless modes (c = 1, 2, 3, black circles), of the perturbatively 4 degenerate massive modes (c = 4, 5, 6, 7, red squares), and the perturbatively heaviest mode (c = 8, blue diamonds). Those are plotted as a function of the absolute value of the physical momentum |p| ≡ √ p µ p µ . Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ( = -) Ԃ ( = -) = ( ) Ԃ ( = ) = ( ) - | | ( ) ( ) | | Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ԂԂ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ԂԂ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ԂԂ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ԂԂ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ԂԂ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ ԂԂ Ԃ The degenerate modes are averaged over to improve the statistics. The dashed lines are fits according to one-loop-inspired fit formulas D c p 2 = Z p 2 A V p 4 + p 2 am c 2 + b 2 p 2 1 + d 2 ln p 2 +Λ 2 Λ 2 γ , c = 1, 2, 3 , D c p 2 = Z p 2 + ln p 2 + b 2 d + am c 2 , c = 4, 5, 6, 7, 8 ,(42) where Z are wave function renormalization constants and am c is the effective mass in lattice units. The first term in the first line is a pure finite-volume effect. The logarithmic corrections of leading loop-corrections are taken into account for both cases. A list of the fit parameters can be found in Table V in Appendix A. Also fits with the tree-level propagators have been performed but those fit functions did not resolve the UV-behavior well. Only for coarser lattices, i.e., larger am 1 −− 0 masses, the tree-level form is a good fit ansatz at least for the massive modes. Those larger masses dominate, such that the logarithmic corrections only play a minor role, see [17]. The effective masses extracted for the different sectors are listed in the legend of the figure for the 20 4 lattice. The fitted effective masses for the perturbatively massless modes are indeed very small and comparable to zero. This suggests a Coulomb-like behavior, although corrections deep in the infrared may still alter this. In the bottom panel of Figure 8 the ratio of data to the fit is shown. For the massive modes the fit according to (42) shows only small deviations from the data for the whole momentum range, whereas larger deviations for the massless modes are visible towards the infrared. The latter is accounted for in the fit as a finite-volume effect, which is to be expected for a massless mode. The extracted masses from the fits for c = 4, 5, 6, 7 (red diamonds) and c = 8 (green triangles) are shown in the top panel of Figure 9 Table VI. The ratio of the lighter and heavier mass is m A /M A = 0.89(5) which is in good agreement with the tree-level ratio of 3/4 ≈ 0.87, see Equation (41). Together with the (almost) masslessness of the propagator in the unbroken sector this implies that the spectrum of the elementary fields coincides with the one expected from perturbation theory, especially of three massless and five massive states. In the bottom panel of Figure 9 we show the ratio of the masses of the 4 degenerate gauge bosons to the mass of the heaviest gauge boson, i.e., m A /M A , as a function of the lattice mass of the singlet vector state am 1 −− 0 obtained for different lattice parameter sets, see Table III. The dashed line is the prediction of tree-level perturbation theory, i.e., m A /M A = 3/4, see Equation (41). Almost all the data points we studied are in good agreement with this prediction signaling that next-to-leading-order effects should only play a minor role. • Space-time correlators / Schwinger functions We also computed the space-time correlators or Schwinger functions ∆ c (t), c = 1, 2, . . . , 8, as described in Section II C, along the lines of [22]. Again, Schwinger functions where degeneracies are expected, i.e., c = 1, 2, 3, and c = 4, 5, 6, 7, are averaged over. 2 The resulting effective masses obtained from ln ∆ c (t)/∆ c (t + 1) for different lattice volumes are given in Figure 10. The errors are computed from the propagators by the method of error propagation. The top left panel shows the effective mass as a function of Euclidean time for the heaviest mode, the top right panel the effective mass of the 4 degenerate massive modes, and the remaining panel shows a plot of the effective mass for the 3 degenerate massless modes. Due to the relatively large error bars for the massive modes for t > 6, we do not show those points here. The dashed lines in the top panel correspond to the space-time correlator obtained by inserting the corresponding fit functions (42) into the definition of the lattice space-time correlator (34). From the maximum values of the effective mass curves one can deduce the masses for each volume. The results are given in the legend of each plot. Of course, the errors are still too large and more statistics is needed to make a final statement. But the trend is clear and the obtained masses are in agreement within the large error bars with the ones obtained from the fits of the propagators with the functions defined in Equation (42). Furthermore, the effective masses of the particles in the unbroken subsector (c = 1, 2, 3) tend to zero for V → ∞. Table VI. Bottom: Ratio of the masses of the 4 degenerate lightest gauge bosons m A to the mass of the heaviest gauge boson M A as a function of all the lattice parameter sets we studied, see Table III. The dashed line is the prediction from tree-level perturbation theory, i.e., m A /M A = 3/4, see Equation (41). • Scalar-field propagator In the scalar sector we computed the renormalized propagators of the real components of the scalar field D r i p 2 , i = 1, 2, 3, 4, 6, as described in Section II C. We choose the arbitrary dimensionless renormalization scale to be aµ = 0.85 for each propagator. Under the assumption that the pole scheme works [11,40] we set the renormalized masses to am r = am 0 ++ 0 for the perturbatively massive propagator (i = 5) and to am r = 0 for the perturbatively massless propagators (i = 1, 2, 3, 4, 6). The degenerate massless renormalized propagators are averaged over to increase the statistics. Having determined both renormalization constants, the renormalized propagator D r i can be computed. The result is shown in Figure 11, where again both, the perturbatively massless (black circles) and massive (red squares) modes are shown. Both propagators show the expected behaviors, namely the ones of a massless and a massive propagator. In order to extract the effective masses, the spacetime correlators need to be computed. Unfortunately, the statistics is too low at this point and thus the error bars too large to extract the effective mass from the Schwinger functions. Therefore, no results on this are presented here. However, in [10] 3 it was found that under similar circumstances the effective mass agreed reasonably with the renormalized mass, supporting that the employed scheme acts like a pole scheme. Still, this will require further scrutiny, as this is not necessarily always the case [40]. • Running gauge coupling and the ghosts The ghost propagators are all very close to the one of a massless particle, and thus close to perturbation theory. There is only a little deviation towards the infrared, which is larger the smaller the associated gauge boson mass is. As a consequence, the running coupling is mainly dominated by the gauge-boson propagator. Thus, we show here only the latter, the renormalized running gauge coupling α c p 2 for the different sectors in Figure 12: The perturbatively massless sector (black circles), the sector with the 4 degenerate massive modes (red squares), and the sector with the heaviest mode (blue diamonds). The coupling to the massive modes show the typical behavior already seen for the SU(2) case [10]. The coupling of the massless modes is infrared (mildly) enhanced, and does not (yet) saturate. Still, because all propagators are rather close to the perturbative ones, so are the gauge couplings. In particular, all unify, implementing the simple picture of a grand-unified theory, in the ultraviolet. Only at small momenta the BEH effect induces the differences. The suppression of the massive couplings can be interpreted as a decoupling of the massive modes from the massless dynamics. However, this statement is only true for the gauge sector, as the gaugeinvariant physics of Section III B does not show any sign of this separation. The couplings stay small throughout the whole momentum range, signaling that leading order GIPT should already be quite reliable. This is also supported by the fact that the propagators can be fitted well with one-loop expressions. Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) ( ) ( + ) ̀ ) + ) = Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) Ԃ = = ( ) ( ) ( + ) ̀ ) + ) = Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ V. TEST OF THE FRÖHLICH-MORCHIO-STROCCHI MECHANISM Thus, this section is dedicated to test GIPT and the underlying FMS mechanism [6,7]. To this end, we first recapitulate the predictions of GIPT for this theory. The generalization of these predictions to general SU(N ) gauge groups can be found in [14]. A. Predictions from gauge-invariant perturbation theory The gauge-invariant, and thus experimentally observable, spectrum consists of states which are either singlets or non-singlets with respect to the custodial U(1) group, see Section II B. • U(1)-singlet states Let us start the discussion with the U(1)-singlet states: A gauge-invariant composite operator describing a scalar, positive (charge-) parity boson, i.e., J P C U(1) = 0 ++ 0 , is O 0 ++ 0 (x) = φ † φ (x) . We apply the FMS mechanism and expand the correlator first in Higgs fluctuations, using Equation (39), and then the resulting propagators to leading order in standard perturbation theory, yielding [14] O 0 ++ 0 (x)O 0 ++ 0 (y) † = v 4 2 + v 2 h(x)h(y) tl + h(x)h(y) 2 tl + · · · ,(43) where 'tl' means 'tree level'. Here, the Higgs field h is identified with φ 5 , see Equation (28). The second term on the right-hand side of Equation (43) describes the propagation of a single elementary Higgs boson and the third term describes two non-interacting Higgs bosons propagating both from x to y. Comparing poles on both sides of Equation (43) predicts the mass of the left-hand side, and thus of the observable particle. This scalar boson should therefore have a mass equal to the mass of the elementary Higgs m h . Also, a next state should exist in this quantum number channel which is a scattering state of twice this mass. Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Ԃ Next, consider a singlet vector operator O 1 −− 0 µ (x) = i φ † D µ φ (x) . The same expansion yields [14] O 1 −− 0 µ (x)O 1 −− 0 ν (y) † = v 4 g 2 12 A µ (x) 8 A ν (y) 8 tl + · · · .(44) The poles of the right-hand side are at the mass M A of the heaviest gauge boson A 8 µ . All the remaining states which can be constructed from the fields expand to trivial scattering states, e.g., gaugeball states, and thus do not give rise to stable particles, see also Table I. Table I. The lattice versions are given in Equations (10) and (11). Applying the FMS mechanism and employing a tree-level analysis to the bound state correlators of both operators yields, after a cumbersome calculation [14], a ground state mass of 2m A for both quantum number channels. This arises as in leading order a product of propagators of one of the massless elementary gauge boson as well as two gauge bosons with mass m A contribute to this state. We expect also a higher order excitation with mass 2m A + M A in both channels. There exists, of course, an anti-particle of the same mass but opposite U(1) charge for both channels as well. • Summary The prediction for the physical (gauge-invariant) spectrum obtained from leading-order gauge-invariant perturbation as well as the gauge-variant spectrum from standard perturbation theory are summarized in Table II. B. Comparison between the spectra We focus again on the lattice parameter set β = 6.85535, κ = 0.456074, and λ = 2.3416 for our investigations. From the findings of the previous subsection and the predictions of the gauge-invariant physical spectrum, we are able to check the predictions of gauge-invariant perturbation theory utilizing the FMS mechanism explicitly. In the 0 ++ 0 channel we found one stable state with a mass of am 0 ++ 0 = 0.68 (2) which is the one that we expect to find from the Schwinger function of the renormalized scalar propagator. While consistent, the results are still too strongly affected by statistical errors to provide an unambiguous conclusion. The remaining states in this 0 + h m h 1 0 O 0 ++ 0 m h - 1 ±1 O 0 ++ ±1 2m A 2m A + M A 1/1 1 − A 1,2,3 µ 0 3 0 O 1 −− 0 µ M A - 1 A 4,...,7 µ m A 4 ±1 O 1 −− ±1 µ 2m A 2m A + M A 1/1 A 8 µ M A 1 channel are high up in the spectrum and consistent with trivial scattering states. In the 1 −− 0 channel we extracted a ground state lattice mass of am 1 −− 0 = 0.39 (1). The mass extracted from the heaviest gauge boson is aM A = 0.36 (1). The volume dependence of these masses is shown on the left-hand side of Figure 13. They are already in pretty good agreement. However, at leading order, those masses should be equal, but there remains a bit more than a 1σ discrepancy between them. There are several explanations for that: First, the prediction relies on the smallness of the Higgs fluctuations and on the applicability of standard perturbation theory. Higher-order effects or genuine nonperturbative effects could explain this deviation. Another possibility is that this discrepancy of the masses could stem from finite volume and discretization effects. We observe that the larger the mass of the lightest state is 4 , i.e., the larger the lattice spacing a is and thus the larger the physical volume is, the better is the agreement with the vector boson mass, see right-hand side of Figure 13 and [17]. Lastly, also the infinite volume extrapolation we used, see Table VI, does not take into account that the broken sector of the theory still interacts weakly with the unbroken sector, i.e., the sector of massless particles. The extrapolation we used does not take this into account and more sophisticated fitting procedures could change the results slightly. In view of these additional systematic uncertainties, the results is already quite well in agreement with the prediction. At the same time, the right-hand side of Figure 13 shows that the result is not a coincidence, and that the agreement is generic. It contains all our results in the HLR, see Table III The open U(1) quantum number channels, i.e., the 0 ++ ±1 and 1 −− ±1 channels, still suffer from low statistics. The extracted ground state of the vector state would be consistent with both, the ground state, 2am A = 0.64 (2), and the predicted next-level state, am A + aM A = 1.00(3) since am 1 −− ±1 ≈ 0.8 (2). The scalar state has a mass of am 0 ++ ±1 ≈ 2.0(1) and is relatively high up in the spectrum. Thus, we neither can confirm nor disprove the FMS prediction in these channels firmly, even though the results are consistent. Nevertheless, the mass for the vector is already significantly smaller than one would naively expect from a simple constituent model or from ordinary perturbation theory. These would place the mass at at least 3m h ≈ 2.0(1). This is especially important, as the lightest such particle is stable and in a realistic theory could serve as a dark matter candidate. Summarizing, in Figure 14 we show the physical spectrum of the theory for different quantum number channels and compare it to the predictions from GIPT. The filled, blue boxes correspond to the ground states, the empty, blue boxes are the elastic thresholds for the scalar and vector singlet channels as discussed above, and the dashed, blue lines are the approximate ground state masses of the 0 ++ ±1 , 0 −+ 0 , and 2 ++ 0 . The red boxes are the predictions of leading-order gauge-invariant perturbation theory for the ground states. The overall agreement shows that the spectrum is, even at leading order, well predicted. VI. SUMMARY AND CONCLUSIONS Summarizing, we have presented a detailed study of the spectroscopy of an SU(3) gauge theory with a fundamental scalar, a toy model for grand-unified theories. To this end, we determined the spectrum using leadingorder standard perturbation theory, leading-order gaugeinvariant perturbation theory utilizing the FMS mechanism [11], and in a full non-perturbative lattice investigation. As in the general case [14] the predictions form standard perturbation theory and gauge-invariant perturbation theory disagree qualitatively. As was already seen in the exploratory study [17], it is found that the predictions of gauge-invariant perturbation theory describe the spectrum obtained from the lattice not only qualitatively, but within some 10% quantitatively, wherever the lattice results were statistically sufficiently significant. Given the remaining systematic uncertainties from the lattice and the fact that the analytical computations were done at leading order, this is quite an impressive agreement. In particular, this worked even in cases where the lowest order in the analytical calculation vanished surprisingly good. At the same time the results of standard perturbation theory are even qualitatively off, especially the theory shows strong evidence for a mass gap of the order the GUT scale. This included also the pure gauge sector, which could be argued to have light gaugeballs in standard approaches [35,45]. This strongly suggest that only gauge-invariant perturbation theory is adequate in describing the theory, and its dynamics, analytically. This is in agreement with all other comparative studies, see [11] for a review of those. However, this work is the first systematic study over a range of parameters and in several channels simultaneously for a theory which was expected to show qualitative disagreement to standard methods. This strongly suggest that the physics picture behind the FMS mechanism is the correct one to describe theories with a BEH effect. Moreover, this implies that gaugeinvariant perturbation theory is the analytical toolkit to work with for these theories, and that BSM predictions should be performed using it. Of course, as always, this is evidence, and further investigations will be necessary for a firm establishing of these conclusions. But given the additional effort need for gauge-invariant perturbation theory in comparison to standard perturbation theory, there is little reason not to use it to make BSM predictions. ACKNOWLEDGMENTS PT has been supported by the FWF doctoral school W1203-N16. The computational results presented have been obtained using the Vienna Scientific Cluster (VSC), the HPC center at the University of Graz and the Graz University of Technology. In this appendix, we collect all the parameter sets of the phase diagram points were we performed simulations for different lattice sizes in order to obtain data for spectroscopy and for propagators of gauge-variant fields. Additionally, we list the fit parameters which we obtained and which are used in the figures shown in Section III B and V . Lattice parameters sets and some observables Here, we provide the numerical values for the ground state energy levels in the vector singlet channel in lattice units am 1 −− 0 , the masses of the 4 degenerate gauge bosons m A and the heaviest gauge boson mass aM A also in lattice units, the average plaquette defined for gauge group SU(3) as U P = 1 18 V x µ<ν Re tr U µν (x) ,(A1) where U µν (x) is defined in Equation (2), for several values of the lattice couplings β, κ and λ. We also provide the expectation values ofφ 2 defined in Equation (35) as well as for the length of the scalar field |φ| given by |φ| = 1 V x φ(x) † φ(x) .(A2) Only such values are shown for which a BEH effect was found. We do not list higher energy levels in this channel as well as the lattice masses in the scalar singlet channel, since only for the main simulation point defined in Section III B enough statistics was gained. There, the mass am 0 ++ 0 was below the elastic threshold and also the higher levels were not to noisy to draw conclusions. Also, points have not been included where no BEH effect was found and/or where the singlet vector mass was above 1 in lattice units. The errors listed in Table III are obtained by fitting the lower and upper bounds of the eigenvalues for the gaugeinvariant case and of the propagators in the gauge-variant case. Subsequently, the method of error propagation is used. Systematic errors are not included. Tables of fit parameters In this section we show the fit parameters used in the figures shown in Section III B and V for the parameter values β = 6.855350, κ = 0.456074, and λ = 2.341600. All the errors are obtained as described previously. We use fit routines provided by Mathematica [46] throughout. Table III. Numerical values of the ground state energy level in the 1 −− 0 channel, the masses of the gauge-variant vector states aM A and am A , the plaquette expectation value U P , φ 2 , and |φ| , for various values of β, κ, and λ in the Higgs-like region of the phase diagram. We also list all the lattice volumes we studied for these values and how many (gauge-fixed) configurations we used. Table IV. Fit parameters from a double-cosh fit of the eigenvalues, λ(t) = A cosh am (1) eff (t − L/2) + B cosh am (2) eff (t − L/2) , obtained from a variational analysis for several lattice volumes V = L 4 in the scalar and vector channels. The dash indicates that only a single-cosh fit has been used. Figure 1. Parameters on the left are in the QLR and on the right in the HLR. J P C U1 V Level µ is a gauge boson dressed with two scalar fields. The gauge bosons appear in the lattice version of the Figure 1 . 1The phase diagram of the theory according to the value of analysis of Section II B yielded a statistically reliable and stable result for the operator set (5) ,(36) where the number in the brackets of the lower index denotes the smearing levels of the operators. Including other or more operators did not improve the result. Figure 2 . 2Top: Results of the variational analysis in the 0 ++ 0 channel. The first energy levels are shown for the V = 8 4 , 12 4 , 16 4 , and 20 4 lattices, whereas the second energy level (green triangles) is only shown for the largest volume for a clear display. The dashed lines are obtained by double-cosh fits of the eigenvalues. The lowest fitted energy values are listed in the legend as effective masses. Bottom: First energy level of the 0 ++ 0 channel as a function of the inverse lattice size. The gray bands are the error bands obtained by fits of the lower and upper bounds of the masses (see Figure 3 . 3Top: Result of the variational analysis in the 1 −− 0 channel. The first energy levels are shown for the V = 8 4 , 12 4 , 16 4 , and 20 4 lattices, whereas the second energy level (green triangles) is only shown for the largest volume for a clear display and for t < 6. The dashed lines are obtained by double-cosh fits of the eigenvalues except for the smallest volume where we used a single-cosh fit. The lowest extracted fitted energy values are listed in the legend as effective mass. Bottom: First and second energy level of the 1 −− 0 channel as a function of the inverse lattice size. The gray bands are the error bands obtained by fits of the lower and upper bounds of the masses (seeTable VIin Appendix A). The dashed blue lines are the expected masses of the next-level states am 0 The fit parameters can be found in i = 2πk i /L, k i = −L/2 + 1, . . . , L/2. In the continuum limit this equation turns into the familiar energy-momentum relation E(p p p) = m 2 + p p p 2 .The ordering of the states depends on the value of the masses of the . Besides the ground state, we also show on the right-hand side ofFigure 3the volume dependence of the second level (blue triangles) with its error band as well as the expected next-level states am 0 ++ 0 + am 1 −− 0 and p rel + am 0 ++ 0 + am 1 −− 0 (dashed blue lines, upper and lower bounds) with p p p rel = (2π/L, 0, 0), i.e., the smallest possible relative momentum. It seems that the mass of the second state is consistent with the is not in agreement with the state including relative momentum. All other energy levels are too noisy to comment on them. Figure 4 . 4In the top panel the effective mass of the pseudoscalar gaugeball is shown as a function of Euclidean time. In the bottom panel the effective mass of the tensor gaugeball is plotted. For both, the results are shown for V = 8 4 , 12 4 , 16 4 , and 20 4 for 10-times smeared fields. a s-wave The masses in both channels are compatible with the last option at both points, i.e., a decay in 1 −− 1 and 1 −− −1 in an s-wave (see below) Figure 5 . 5The three panels show the effective masses of the 1 −− 0 gaugeballs L (1) , L (2) and L (3) as a function of Euclidean time t < 3. The results are shown for V = 8 4 , 12 4 , 16 4 , and 20 4 for 10-times smeared fields. • 0 0++ ±1 and 1 −− ±1 open U(1) channels Finally, we study quantum number channels with an open U(1) quantum number, i.e., the 0 ++ ±1 and 1 −− ±1 states. At least the lightest state with non-vanishing U(1) charge is necessarily stable, as this custodial charge is conserved in the theory. Figure 6 . 6On the left-hand side the effective mass of the 0 ++ ±1 state is shown as a function of Euclidean time t < 3. On the right-hand side the effective mass of the 1 −− ±1 state is plotted for t < 6. For both, the results are shown for V = 8 4 , 12 4 , 16 4 , and 20 4 lattices. The dashed lines in the left and right panels are results of single-and double-cosh fits, respectively. Figure 7 . 7Spectrum of the theory for the lattice parameter set β = 6.85535, κ = 0.456074, λ = 2.3416. The description is given in the main text. Dashed levels are only estimates. Figure 8 . 8Top: Plot of the gauge-boson propagators on a 20 4 lattice for the 3 perturbatively massless modes (black circles), the 4 degenerate perturbatively massive modes (red squares), and the heaviest mode (blue diamonds) as a function of the absolute value of the physical momentum |p|. The dashed lines are results of the fits described in the main text. Bottom: Here, the data points are divided by the corresponding fitted values as a function of |p| and thus shows the qualities of the fits. as a function of the inverse lattice size L. The extrapolated infinite volume values are m A = 0.32(1) for the 4 degenerate massive and M A = 0.36(1) for the heaviest gauge boson, see the legend in the figure and Figure 9 . 9Top: Masses of the 4 degenerate massive (red diamonds) and the heaviest (green triangles) gauge bosons as a function of the inverse lattice size L. The gray areas are the corresponding error bands obtained from a fit to am+α e −γ V , see Figure 10 . 10Effective masses from the space-time correlators for the heaviest mode (top left panel), the 4 degenerate modes (top right panel), and the 3 massless modes (bottom left panel), for 8 4 , 12 4 , 16 4 , and 20 4 lattices. The masses in the legends in each panel are obtained by taking the maximum value of the functions for each volume. The dashed lines in the top panels correspond to the space-time-transformed fit functions (42). Figure 11 .Figure 12 . 1112Perturbatively massless and massive scalar propagators for a 20 4 lattice and for aµ = 0.85. The renormalized masses are am r = 0 for the (averaged) massless mode and am r = am 0 Renormalized running coupling for the three different sectors. The renormalization has been performed such that the couplings agree with the perturbative one for large momenta, see[10] for details. The lattice couplings are in this case β = 6.85535, κ = 0.456074, and λ = 2.3416. • U(1)-non-singlet statesLet us now focus on states with open U(1) quantum numbers. Since the corresponding charge is conserved, the lightest such state is absolutely stable. The scalar as well as the vector non-singlet states, Figure 13 . 13Left: Effective masses of the singlet vector channel obtained from the gauge-variant propagator and from a variational analysis of gauge-invariant operators as a function of the inverse lattice size. The discrepancy of the infinite volume extrapolated values is discussed in the main text. Right: Ratio of the heaviest vector boson mass M A and the ground state mass in the 1 −− 0 channel, m 1 −− 0 , as a function of the lattice mass am 1 −− 0 . The points are obtained from simulations at points in the phase diagram in the HLR, see Table III. The dashed line is the GIPT prediction. Table II. Left: Gauge-variant spectrum of an SU(3) gauge theory with a single scalar field in the fundamental representation. We set the direction n of the vev to ni = δi,3. Right: Prediction of the gauge-invariant (physical) spectrum of the theory using leading-order gauge-invariant perturbation theory. Here m h denotes the mass of the elementary Higgs field, M A is the mass of the heaviest elementary gauge boson and m A the mass of the degenerated lighter massive gauge bosons. We assign a custodial U(1) charge of 1/3 to the scalar field φ. The column 'N.l. state' (next-level state) lists the masses of possible additional bound states or resonances. Whether these states are indeed bound states or resonances or only nontrivial scattering states can not be decided here. Trivial scattering states are ignored. elementary spectrum gauge-invariant spectrum J P Field Mass Deg. U(1) Op. Mass N.-l. state Deg. . In this plot the dimensionless ratio of the vector boson mass M A to the singlet vector mass m 1 −− 0 as a function of the lattice singlet vector mass am 1 −− 0 is shown. The dashed line at M A /m 1 −− 0= 1 reflects agreement with the prediction in the vector channel. All in all, good agreement to the GIPT prediction is found, in particular for larger physical volumes, corre-sponding due to the fixed lattice volumes to larger lattice masses of the singlet vector state and thus larger a. Figure 14 . 14Physical (gauge-invariant) spectrum of the theory (blue boxes) compared to the predictions from the FMS mechanism to leading order (red boxes) for the lattice parameter set β = 6.85535, κ = 0.456074, λ = 2.3416. 4 , 11200 , 112),(12 4 , 19500 , 1170)4 , 120000, 3000),(20 4 , 190000, 5473) 7.912200 0.493113 3.423300 ( 8 4 , 11200 , 112), (12 4 , 24950 , 1479)4 , 25000 , 1500) 8.172900 0.490558 6.483650 ( 8 4 , 25000 , 250), (12 4 , 10000 , 250)4 , 10000 , 250), (20 4 , 10000 , 400) 8.433600 0.488003 9.544000 ( 8 4 , 10000 , 100), (12 4 , 10000 , 1000)4 , 5775 , 579), (20 4 , 1981 , 261) 9.590550 0.444462 0.411800 ( 8 4 , 100000, 1000), (12 4 , 102000, 3400)4 , 102000, 3400), (20 4 , 130848, 2802) 9.607400 0.174193 0.030100 ( 8 4 , 11200 , 112), (12 4 , 25000 , 1500)4 , 25000 , 1500), (20 4 , 1543 , 172) 10.05222 0.420352 0.717362 ( 8 4 , 20000 , 200), (12 4 , 100000, 1000)4 , 42494 , 1000), (20 4 , 9195 , 749) masses, as well as the extrapolation of the gauge-variant lattice masses of the gauge-boson propagator D c p 2 . In all those cases the fit function is am eff (V ) = am + α e −γ V . Table I . IList of interpolators used for our spectroscopic analysis. Definitions of the objects Dµ, L Table I . IThese quantities are built from the Wilson loop operator Fit-range [tmin, tmax] am(1) eff am (2) eff A B 0 ++ 0 8 4 1 st [1, 4] 0.55(1) 1.44(1) 0.1156(3) 0.0031(1) 12 4 1 st [1, 6] 0.65(1) 1.55(1) 0.0180(4) 0.0009(1) 16 4 1 st [1, 6] 0.70(1) 1.45(3) 0.0044(1) 0.000007(2) 20 4 1 st [2, 7] 0.67(3) 1.08(5) 0.0010(1) 0.00003(2) 8 4 2 nd [1, 4] 0.95(4) 1.4(2) 0.0282(2) 0.002(1) 12 4 2 nd [1, 6] 0.80(1) 1.4(2) 0.0045(5) 0.00030(15) 20 4 2 nd [2, 6] 0.90(10) 1.3(1) 0.00013(6) 0.000003(2) 1 −− 0 8 4 1 st [2, 4] 0.42(1) − 0.337(1) − 12 4 1 st [2, 6] 0.39(1) 1.50(4) 0.159(1) 0.0003(1) 16 4 1 st [2, 8] 0.39(1) 1.5(2) 0.072(1) 0.000003(2) 20 4 1 st [2, 9] 0.39(1) 1.4(1) 0.033(1) 0.0000005(5) 8 4 2 nd [2, 4] 0.99(1) − 0.026(1) − 12 4 2 nd [2, 4] 1.02(2) 1.7(2) 0.018(1) 0.00026(1) 16 4 2 nd [2, 6] 1.01(1) 1.9(2) 0.0010(1) 0.0000003(2) 20 4 2 nd [2, 6] 1.03(2) 1.4(1) 0.0003(1) 0.000002(1) 0 ++ ±1 20 4 1 st [1, 3] 2.0(1) − 2.3(1) · 10 −9 − 1 −− ±1 8 4 1 st [2, 6] 0.87(1) 1.65(1) 0.0236(1) 0.0017(1) 12 4 1 st [2, 6] 0.75(5) 1.35(5) 0.0025(1) 0.0005(2) 16 4 1 st [2, 6] 0.67(1) 1.27(2) 0.0007(1) 0.00006(1) 20 4 1 st [2, 6] 0.95(5) 1.60(5) 0.00006(6) 0.00000002(1) Table Table VIIa . 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We investigate a systematic error coming from higher excited state contributions in the energy shift of light nucleus in the two-nucleon channel by comparing two different source calculations with the exponential and wall sources. Since it is hard to obtain a clear signal of the wall source correlation function in a plateau region, we employ a large quark mass as the pion mass is 0.8 GeV in quenched QCD. We discuss the systematic error in the spin-triplet channel of the two-nucleon system, and the volume dependence of the energy shift.IntroductionWe carried out an exploratory study of the direct calculation of the binding energy of the light nuclei with the atomic mass number less than or equal to four in quenched lattice QCD [1, 2]. These studies were followed by several calculations[3][4][5][6][7][8][9]. All the recent calculations at m π > 0.3 GeV, which were obtained from the calculations with the exponential or gaussian source, suggest the existence of a bound state in the two-nucleon channels.HALQCD[10]suggested that there is a sizable systematic error in the energy shift in the twonucleon channels obtained from the ratio of the correlation functions. They compared the two results with the exponential and wall sources, and found discrepancies in the effective energy shifts. However, it is well known that the wall source needs the longest temporal extent to obtain a plateau even in the single nucleon mass. In this comparison a high precision calculation is necessary.The purpose of this work is to investigate the systematic error coming from excited states by comparing the exponential and wall source calculations in the spin-triplet two-nucleon channel in a high precision calculation using a large quark mass of m π = 0.8 GeV in the quenched approximation.To determine a plateau of the ratio of the correlation functions, we focus on an important condition in the direct calculation, which will be explained below, though it is trivial in lattice QCD calculation. The results in this report are the updated ones from the last conference[11]. All the results in this report are preliminary.

10.1051/epjconf/201817505019

1710.08066

5c75d2e7ab8dfab42da66c69e1886d865794c92a

Comparison of different source calculations in two-nucleon channel at large quark mass Takeshi Yamazaki Faculty of Pure and Applied Sciences University of Tsukuba 305-8571TsukubaIbarakiJapan Center for Computational Sciences University of Tsukuba 305-8577TsukubaIbarakiJapan RIKEN Advanced Institute for Computational Science 650-0047KobeHyogoJapan Ken-Ichi Ishikawa Department of Physics Hiroshima University 739-8526Higashi-Hiroshima, HiroshimaJapan Yoshinobu Kuramashi Center for Computational Sciences University of Tsukuba 305-8577TsukubaIbarakiJapan RIKEN Advanced Institute for Computational Science 650-0047KobeHyogoJapan Pacs Collaboration Comparison of different source calculations in two-nucleon channel at large quark mass 10.1051/epjconf/201817505019 We investigate a systematic error coming from higher excited state contributions in the energy shift of light nucleus in the two-nucleon channel by comparing two different source calculations with the exponential and wall sources. Since it is hard to obtain a clear signal of the wall source correlation function in a plateau region, we employ a large quark mass as the pion mass is 0.8 GeV in quenched QCD. We discuss the systematic error in the spin-triplet channel of the two-nucleon system, and the volume dependence of the energy shift.IntroductionWe carried out an exploratory study of the direct calculation of the binding energy of the light nuclei with the atomic mass number less than or equal to four in quenched lattice QCD [1, 2]. These studies were followed by several calculations[3][4][5][6][7][8][9]. All the recent calculations at m π > 0.3 GeV, which were obtained from the calculations with the exponential or gaussian source, suggest the existence of a bound state in the two-nucleon channels.HALQCD[10]suggested that there is a sizable systematic error in the energy shift in the twonucleon channels obtained from the ratio of the correlation functions. They compared the two results with the exponential and wall sources, and found discrepancies in the effective energy shifts. However, it is well known that the wall source needs the longest temporal extent to obtain a plateau even in the single nucleon mass. In this comparison a high precision calculation is necessary.The purpose of this work is to investigate the systematic error coming from excited states by comparing the exponential and wall source calculations in the spin-triplet two-nucleon channel in a high precision calculation using a large quark mass of m π = 0.8 GeV in the quenched approximation.To determine a plateau of the ratio of the correlation functions, we focus on an important condition in the direct calculation, which will be explained below, though it is trivial in lattice QCD calculation. The results in this report are the updated ones from the last conference[11]. All the results in this report are preliminary. Important condition of direct calculation In the direct calculation [1][2][3][4][5][6][7][8][9] of the two-nucleon channel, the energy shift ∆E NN = 2m N − E NN with the nucleon mass m N and two-nucleon ground state energy E NN is determined from a plateau region of the ratio of the correlation functions R(t) = C NN (t)/C 2 N (t) with the two-nucleon correlation function C NN (t) in the spin-triplet channel and the single nucleon correlation function C N (t). An important condition of this determination is that ∆E NN should be determined in a region where both C NN (t) and C 2 N (t) have each plateau. It means that it is not enough to determine a plateau region from only R(t), but we need to investigate plateaus for C NN (t) and C 2 N (t). If one chooses a plateau from only R(t), it might cause an incorrect determination of ∆E NN , as discussed in the later sections. For example, when statistics is not enough, R(t) using the wall source has a plateau like behavior in early t region, where C NN (t) and C 2 N (t) do not have plateaus. In the following sections, we shall call the minimum t of the plateau region for C NN (t), C N (t), and R(t) as t S N , t S NN , and t S R , respectively, using the source S = E (exponential) or W (wall). Simulation parameters We calculate the two-nucleon correlation function C NN (t) in the spin-triplet channel as well as the single nucleon correlation function C N (t) in the quenched approximation. In this calculation, we employ Iwasaki gauge action at β = 2.416, corresponding to a = 0.128 fm [12]. The quark propagators are calculated with a tad-pole improved Wilson action with c SW = 1.378 at κ ud = 0.13482 corresponding to m π = 0.8 GeV and m N = 1.62 GeV. The actions and parameters are the same as in our previous works [1,2]. The temporal lattice size is fixed to 64, while the spatial size L is chosen to be 16, 20, and 32. In order to compare results with different source operators, we employ the exponential and wall sources. The exponential source at the time slice t is defined by q ′ (x, t) = q(x, t) + A ∑ y x exp(−B|y − x|)q(y, t),(1) where q(x, t) is the local quark field. The parameters A and B are chosen to obtain an early plateau of the effective nucleon mass on each volume. At the sink time slice, each nucleon operator is projected to zero momentum using the local quark field as in our previous calculations [2,5,7]. The number of the measurement of the correlation functions is tabulated in Table 1. L Results In this section we present the results for twice the effective nucleon mass 2m eff N , the effective twonucleon energy E eff NN , and the effective energy shift ∆E eff NN evaluated from C N (t), C NN (t), and R(t), respectively, on each volume. The volume dependence of ∆E NN is also presented. Figure 1. Effective twice nucleon mass 2m eff N (circle) and effective two-nucleon energy E eff NN (square) in the spin-triplet channel using the exponential (left panel) and wall (right panel) sources on L = 20. The vertical dot-dashed lines denote t S N and t S NN for S = E, W explained in the text. The horizontal dashed lines express the values of each plateau in the exponential source in both the panels. L = 20 First we present the results of 2m eff N , E eff NN , and ∆E eff NN in both the exponential and wall sources on L = 20 as a typical result. Figure 1 shows the results for 2m eff N and E eff NN using the exponential (left panel) and wall (right panel) sources. The results of the exponential source have plateaus, which start from t = 12 as denoted by vertical dot-dashed line in the left panel. It means that t E N = t E NN = 12 in this case. The horizontal dashed lines in black and red represent the values of the plateaus for 2m eff N and E eff NN , respectively. The wall source results need longer t than the exponential source to have plateaus as shown in the right panel of Fig. 1. We determine t W N = 17 and t W NN = 16 from each plateau region, which are expressed by vertical dot-dashed lines. The same horizontal lines as in the left panel are shown in the right panel. Those lines are in good agreement with each plateau. As discussed in Sec. 2, t S R , the minimum t of the plateau region of ∆E eff NN , should be larger or equal to t S N and t S NN . Thus, t E R = t E N = t E NN , and t W R = t W N , in this case. Figure 2 presents that ∆E eff NN of the exponential source has a reasonable plateau after t E R . On the other hand, the result of the wall source has a non-monotonic t dependence in t < 15. One might choose t W R ∼ 14, if it is determined from only the wall source data in Fig. 2. Moreover, if the statistics is much smaller than the current calculation, the data around t = 5 would be also regarded as a plateau. However, the data in the small t region contain excited state contributions as shown in the right panel of Fig. 1. It suggests that it is easy to mistake the plateau region, when it is determined from only ∆E eff NN , especially in the case where ∆E eff NN has a non-monotonic t dependence, like the wall source data in the current study. In the wall source, while the data in t ≥ t W R has the large error, it agrees with the plateau value of the exponential source within the error. L = 16 The results for 2m eff N and E eff NN using the exponential and wall sources are plotted in the left and right panels of Fig. 3, respectively. The results are similar to the ones on L = 20 in the previous subsection. Figure 3. The same figures as Fig. 1, but in the L = 16 case. The data of the exponential source have plateaus, which start from t = 12, and the ones of the wall source need longer t to have plateaus. It is noted that comparing with the results on L = 20 and 16 we observe 0.02% finite volume effect in m N on this volume of the spatial extent 2.0 fm. The results of ∆E eff NN are shown in Fig. 4. The exponential source has a reasonable plateau after t E R = t E N = t E NN as in the L = 20 case. The t dependence of the wall source in the smaller t region becomes larger as the volume decreases comparing with the result in Fig. 2. While the error of the wall source is large after t W R = t W N , the data is consistent with the plateau value of the exponential source. It is noted that on L = 16 the consistent results with the exponential and wall sources are not obtained even in 2m eff N , when the number of the measurement of the wall source is half of the current calculation. This suggests that a huge statistics is necessary to obtain statistically stable result from the wall source even in the single nucleon mass. L = 32 The left panel of Fig. 5 presents that the results for 2m eff N and E eff NN with the exponential source are similar to the ones in the L = 16 and 20 cases. On the other hand, the wall source results look different from the ones on the other volumes. The result of E eff NN with the wall source in t ≤ 20 is larger than the one with the exponential source represented by the red dashed line. One of the reasons is that the contribution of the two-nucleon scattering state with almost zero relative momentum, which corresponds to the first excited state in this system, becomes relatively larger than the one of the ground state in C NN (t) with the wall source as the volume increases. Another reason is that the energy of the first excited state is larger than 2m N in this system, where one bound state exists [2]. Thus, it is harder to obtain the same plateau as the one of the exponential source, expressed by the red dashed line in the right panel of Fig. 5, from the wall source as the volume increases. From the data, we cannot determine t NN of the wall source. In the following analysis, it is assumed that t W N = t W NN in the wall source result. From the above reasons, it is expected that on much larger volumes than the current calculation E eff NN of the wall source would become larger than 2m eff N in a large t region. Then, it would go down to agree with the plateau value of the exponential source in much larger t region. Figure 6 shows the results of ∆E eff NN with both the sources. It is surprising that the wall source result has a mild t dependence in the small t region, although the data for 2m eff N and E eff NN largely depend on t in the same region as shown in the right panel of Fig. 5. If a plateau of ∆E eff NN is determined from only the wall source data in smaller statistics than the present calculation, one might choose much smaller t region as a plateau than t W R . While it is not as good as the smaller volumes, we observe a plateau after t E R in the exponential source result on this volume. Although the wall source result has large error after t W R , it is not inconsistent with the plateau of the exponential source. We expect that the plateau of E eff NN with the wall source is obtained in a region of the much larger t than the smaller volumes. In order to confirm this expectation, it is an important future work to observe clear signal of the wall source after t W R . From the comparisons including the ones in the smaller volumes, we conclude that the results using the exponential and wall source are consistent with each other in each plateau region. Thus, contaminations of excited states in ∆E eff NN obtained from the plateau region are negligible in our calculation. Figure 5. The same figures as Fig. 1, but in the L = 32 case. Volume dependence The result of ∆E NN on the three volumes with the exponential source are plotted in Fig. 7 together with our previous result [2]. We neglect the wall source data in the following due to the much larger error. The result of the current calculation denoted by the filled circle has much smaller statistical error, and is reasonably consistent with the fit curve using the previous data, so that the result indicates that the existence of a bound state in this system. Recently HALQCD Collaboration suggested that the volume dependence of ∆E NN obtained from the direct calculation is too small comparing to the one expected from the effective range expansion [13]. However, this argument is assumed that the effective range expansion is valid in p 2 < 0 region in the continuum theory, and there is no finite volume effect in the two-nucleon interaction. In the comparison between the expectation in the ideal case and the lattice data, there could be several sources of systematic errors, such as finite lattice spacing and finite volume effects, which may deform the two-nucleon interaction. In order to understand the current situation, it is an important future work to investigate such systematic errors in the ∆E NN calculation. It is noted that even if there is a finite volume effect in ∆E NN , which cannot be treated by the finite volume method [14,15], we consider that the signal of the existence of the bound state is meaningful in our calculation, because we discuss the existence in the infinite volume limit, so that our result does not contain the finite volume effect. Summary We have carried out the high precision calculation of the spin-triplet two-nucleon channel at the large quark mass, corresponding to m π = 0.8 GeV in the quenched approximation to investigate a systematic error of ∆E NN coming from excited states by comparing the results with the two different source calculations using the exponential and wall sources on the three volumes. Though it might be a trivial, we discuss the important condition to calculate ∆E NN . When the condition is satisfied, the two sources give the consistent results of ∆E eff NN in each plateau region, while the wall source data has the large error due to the late plateau. From this comparison, we have concluded that the systematic error from higher excited states is negligible in our calculation. There are several important future works, such as comparing the current result with the one obtained from the generalized eigenvalue problem [16], and investigations of systematic errors in ∆E NN . It is also an important future work to clarify the qualitative difference between the direct calculation and HALQCD method in the point of view of the definitions of the scattering amplitude in quantum field theory and quantum mechanics [17]. Figure 4 . 4The same figures asFig. 2, but in the L = 16 case. Figure 6 . 6The same figures asFig. 2, but in the L = 32 case. Figure 7 . 7Volume dependence of the energy shift ∆E NN using the exponential source. The horizontal axis is one over volume. The filled circles denote ∆E NN on L = 16, 20, 32 in the current calculation. The open symbols and dashed curve are the results in our previous work[2]. The star symbol expresses the experimental value of the deuteron binding energy. Table 1. Numbers of the measurement on each L with the exponential (Exp) and wall sources.16 20 32 Exp 6,272,000 5,504,000 4,736,000 Wall 8,307,200 8,960,000 4,473,600 EPJ Web of Conferences 175, 05019 (2018) https://doi.org/10.1051/epjconf/201817505019 Lattice 2017 AcknowledgementsNumerical calculations for the present work have been carried out on the FX10 supercomputer system at Information Technology Center of the University of Tokyo, on the COMA cluster system under the "Interdisciplinary Computational Science Program" of Center for Computational Science at University of Tsukuba, on the Oakforest-PACS system of Joint Center for Advanced High Performance Computing, on the computer facilities of the Research Institute for Information Technology of Kyushu University, and on the FX100 and CX400 supercomputer systems at the Information Technology Center of Nagoya University. This research used computational resources of the HPCI system provided by Information Technology Center of the University of Tokyo through the HPCI System Research Project (Project ID: hp160125). We thank the colleagues in the PACS Collaboration for providing us the code used in this work. This work is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (No. 16H06002). . T Yamazaki, PACS-CS CollaborationY Kuramashi, PACS-CS CollaborationA Ukawa, PACS-CS CollaborationPhys. Rev. 81111504T. Yamazaki, Y. Kuramashi, A. Ukawa (PACS-CS Collaboration), Phys. Rev. D81, 111504 (2010) . T Yamazaki, PACS-CS CollaborationY Kuramashi, PACS-CS CollaborationA Ukawa, PACS-CS CollaborationPhys. Rev. 8454506T. Yamazaki, Y. Kuramashi, A. Ukawa (PACS-CS Collaboration), Phys. Rev. D84, 054506 (2011) . S Beane, NPLQCD CollaborationPhys. Rev. 8554511S. Beane et al. (NPLQCD Collaboration), Phys. Rev. D85, 054511 (2012) . S Beane, NPLQCD CollaborationE Chang, NPLQCD CollaborationS Cohen, NPLQCD CollaborationW Detmold, NPLQCD CollaborationH Lin, NPLQCD CollaborationPhys.Rev. 8734506S. Beane, E. Chang, S. Cohen, W. Detmold, H. Lin et al. (NPLQCD Collaboration), Phys.Rev. D87, 034506 (2013) . T Yamazaki, K I Ishikawa, Y Kuramashi, A Ukawa, Phys. Rev. 8674514T. Yamazaki, K.i. Ishikawa, Y. Kuramashi, A. Ukawa, Phys. Rev. D86, 074514 (2012) . E Berkowitz, CalLat CollaborationT Kurth, CalLat CollaborationA Nicholson, CalLat CollaborationB Joo, CalLat CollaborationE Rinaldi, CalLat CollaborationM Strother, CalLat CollaborationP M Vranas, CalLat CollaborationA Walker-Loud, CalLat CollaborationPhys. Lett. 765285E. Berkowitz, T. Kurth, A. Nicholson, B. Joo, E. Rinaldi, M. Strother, P.M. Vranas, A. Walker- Loud (CalLat Collaboration), Phys. Lett. B765, 285 (2017) . T Yamazaki, K I Ishikawa, Y Kuramashi, A Ukawa, Phys. Rev. 9214501T. Yamazaki, K.i. Ishikawa, Y. Kuramashi, A. Ukawa, Phys. Rev. D92, 014501 (2015) tion). K Orginos, NPLQCD CollaboraA Parreno, NPLQCD CollaboraM J Savage, NPLQCD CollaboraS R Beane, NPLQCD CollaboraE Chang, NPLQCD CollaboraW Detmold, NPLQCD CollaboraPhys. Rev. 92114512K. Orginos, A. Parreno, M.J. Savage, S.R. Beane, E. Chang, W. Detmold (NPLQCD Collabora- tion), Phys. Rev. D92, 114512 (2015) . M L Wagman, F Winter, E Chang, Z Davoudi, W Detmold, K Orginos, M J Savage, P E Shanahan, 1706.06550M.L. Wagman, F. Winter, E. Chang, Z. Davoudi, W. Detmold, K. Orginos, M.J. Savage, P.E. Shanahan (2017), 1706.06550 . T Iritani, HALQCD CollaborationJHEP. 10101T. Iritani et al. (HALQCD Collaboration), JHEP 10, 101 (2016) . T Yamazaki, PACSK I Ishikawa, PACSY Kuramashi, PACSA Ukawa, PACSPoS. 2016108T. Yamazaki, K.I. Ishikawa, Y. Kuramashi, A. Ukawa (PACS), PoS LATTICE2016, 108 (2017) . A , CP-PACS CollaborationAli Khan, CP-PACS CollaborationPhys. Rev. 6554505A. Ali Khan et al. (CP-PACS Collaboration), Phys. Rev. D65, 054505 (2002) . T Iritani, S Aoki, T Doi, T Hatsuda, Y Ikeda, T Inoue, N Ishii, H Nemura, K Sasaki, Phys. Rev. 9634521T. Iritani, S. Aoki, T. Doi, T. Hatsuda, Y. Ikeda, T. Inoue, N. Ishii, H. Nemura, K. Sasaki, Phys. Rev. D96, 034521 (2017) . M Lüscher, Commun. Math. Phys. 105153M. Lüscher, Commun. Math. Phys. 105, 153 (1986) . M Lüscher, Nucl. Phys. 354531M. Lüscher, Nucl. Phys. B354, 531 (1991) . M Lüscher, U Wolff, Nucl. Phys. 339222M. Lüscher, U. Wolff, Nucl. Phys. B339, 222 (1990) . T Yamazaki, Y Kuramashi, 1709.09779T. Yamazaki, Y. Kuramashi (2017), 1709.09779

The phase diagram of a system constituted of neutrons, protons, Λ-hyperons and electrons is evaluated in the mean-field approximation in the complete three-dimensional space given by the baryon, lepton and strange charge. It is shown that the phase diagram at sub-saturation densities is strongly affected by the electromagnetic interaction, while it is almost independent of the electric charge at supra-saturation density. As a consequence, stellar matter under the condition of strangeness equilibrium is expected to experience a first as well as a second-order strangeness-driven phase transition at high density, while the liquid-gas phase transition is expected to be quenched. An RPA calculation indicates that the presence of this critical point might have sizable implications for the neutrino propagation in core-collapse supernovae. PACS numbers: 26.50.+x, 26.60.-c 21.65.Mn, 64.10.+h, 64.60.Bd,

10.1103/physrevc.87.055809

1301.0390

4f52a4dd0dbe3cb5320e0e5b73ec2b1725dd2fe3

Strangeness-driven phase transition in (proto)-neutron star matter 13 May 2013 December 12, 2013 F Gulminelli UMR6534 CNRS ENSICAEN 14050Caen cédexLPCFrance Ad R Raduta IFIN-HH POB-MG6Bucharest-MagureleRomania M Oertel LUTH CNRS Observatoire de Paris Université Paris Diderot 5 place Jules Janssen92195MeudonFrance J Margueron Institut de Physique Nucléaire IN2P3-CNRS Université Paris-Sud F-91406Orsay cedexFrance Strangeness-driven phase transition in (proto)-neutron star matter 13 May 2013 December 12, 2013 The phase diagram of a system constituted of neutrons, protons, Λ-hyperons and electrons is evaluated in the mean-field approximation in the complete three-dimensional space given by the baryon, lepton and strange charge. It is shown that the phase diagram at sub-saturation densities is strongly affected by the electromagnetic interaction, while it is almost independent of the electric charge at supra-saturation density. As a consequence, stellar matter under the condition of strangeness equilibrium is expected to experience a first as well as a second-order strangeness-driven phase transition at high density, while the liquid-gas phase transition is expected to be quenched. An RPA calculation indicates that the presence of this critical point might have sizable implications for the neutrino propagation in core-collapse supernovae. PACS numbers: 26.50.+x, 26.60.-c 21.65.Mn, 64.10.+h, 64.60.Bd, I. INTRODUCTION Supernova explosions following the gravitational collapse of a massive star (M > ∼ 8M ⊙ ) are among the most fascinating events in the universe as they radiate as much energy as the sun is expected to emit over its whole life span [1]. Nuclear physics is an essential ingredient in the numerical simulations which aim to describe these events, since realistic astrophysical descriptions of the collapse and post-bounce evolution rely on the accuracy of the implementation of weak processes and equation(s) of state (EOS) [2,3]. Determining the EOS constitutes a particularly difficult task since phenomenology ranges from a quasi-ideal un-homogeneous gas to strongly interacting uniform matter and, potentially, deconfined quarkmatter. The situation is even more difficult if phase transitions are experienced, since mean-field models fail in such situations [4]. The Coulomb-quenched liquid-gas (LG) phase transition taking place at densities smaller than the nuclear saturation density (n 0 = 0.16 fm −3 ) is, probably, the most notorious and best understood case [5][6][7][8][9]. At highest densities, a quark-gluon plasma is expected, but predictions on the exact location of the transition are strongly model dependent [10]. In the intermediate density domain simple energetic considerations show that additional degrees of freedom may be available, such as hyperons, nuclear resonances, mesons or muons [11]. The possibility that the onset of hyperons could pass via a first order phase transition in neutron stars has been evoked in Ref. [12], using a relativistic mean field model (RMF), and in Ref. [13], a phase transition between phases with different hyperonic species has been observed for cold matter. The possibility of a first order phase transition to hyperonic matter in effective RMF models has been discussed in Refs. [14][15][16], too. Within the latter models, the phase transition region is located at sub-saturation densities, and is thus not relevant for star matter. Using a simple two-component (n, Λ) model, we have recently studied the complete phase diagram of strange baryonic matter showing that it exhibits a complex structure with first and second order phase transitions [17]. However, the exploratory calculation of Ref. [17] neglects the fact that in addition to baryon number B and strangeness S, the charge Q and lepton L quantum numbers are also populated. The thermodynamics of the complete system should thus be studied in the four-dimensional space of the associated charges n B , n S , n L , n Q . The strict electroneutrality constraint n Q = 0, necessary to obtain a thermodynamic limit [18], makes the physical space three-dimensional. As it is known from the EOS studies at sub-saturation density [19], the introduction of the charge degree of freedom can have a very strong influence on the phase diagram and cannot be neglected. In this work we therefore introduce a four-component model constituted of neutrons, protons, electrons and Λhyperons. Electrons are treated as an ideal gas. We present, in sec. II of this paper, the thermodynamics and phase transition of the n, p, e and Λ system, and discuss the influence of the Coulomb interaction. The consequence of the phase transition on the cooling of proto-neutron stars, through the neutrino mean free path, is qualitatively discussed in sec. III. Finally, we present our conclusions in sec. IV. II. THERMODYNAMICS OF A N, P, Λ SYSTEM WITH ELECTRONS In the widely used mean-field approximation [11,[20][21][22][23][24][25][26] the total baryonic energy density is given by the sum of the mass, kinetic and potential energy density functionals which represents a surface in the threedimensional space defined by the baryon, strange and charge density given, in our case, by n B = n n + n p + n Λ , n S = −n Λ and n Q = n p . In the non-relativistic formalism valid in the considered domains of density and temperature it reads e B = i=n,p,Λ n i m i c 2 +h 2 2m i τ i + e pot (n n , n p , n Λ ) (1) The single-particle densities are given by the Fermi integrals n i = 4π h 3 2m i β 3 2 F 1 2 (βμ i ); τ i = 8π 3 h 5 2m i β 5 2 F 3 2 (βμ i ), (2) where F ν (η) = ∞ 0 dx x ν 1+exp(x−η) is the Fermi-Dirac integral, β = T −1 is the inverse temperature, m i is the effective i-particle mass andμ i is the effective chemical potential of the i-species self-defined by the single-particle density. A. The model A full thermodynamics characterization of the system is provided by the pressure P B = T s B − e B + i µ i n i together with the entropy density s B in mean-field, s B = i,p,Λ 10h 2 6 m i βτ i − n i βμ i .(3) The thermodynamical definition n i . = ∂P ∂µi | β allows to infer the relation among the chemical potentials µ i and the effective parametersμ i as µ i =μ i + m i c 2 + U i , with U i = ∂e pot /∂n i . Within the numerical applications we shall use the potential energy density proposed by Balberg and Gal [27], e pot (n n , n p , n Λ ) = i,j={n,p,Λ} (a ij n i n j + b ij t i t j n i n j (4) +c ij 1 n i + n j (n γij +1 i n j + n γij+1 j n i ) , accounting for nucleon-nucleon, nucleon-Λ and Λ-Λ interactions. t i denotes the third isospin component of particle i. In the non-strange sector the form of the interaction is the same as in the widely used Lattimer-Swesty [28] EOS. Let us mention that the observation of a neutron star (PSR J 1614-2230) with a mass of almost two solar masses [29] imposes stringent constraints on the hyperonic interaction in dense neutron star matter. The maximum mass for a n, p, Λ + e system as studied in the present manuscript is 2.04M ⊙ with the parameter set BG I for the coupling constants (see Table I) in agreement with the mass of PSR J 1614-2230. Including all hyperonic degrees of freedom, the maximum neutron star mass obtained with parametrisation BG I decreases and becomes slightly too low. However, the qualitative results discussed here about the thermodynamics of the system and the consequences on the neutrino mean free path are independent of the parametrisation used. In particular, the same qualitative results are obtained with the parametrisations from Ref. [21], in agreement with the mass of PSR J 1614-2230 even upon including all the different hyperons. Quantitative differences are very small, such that we have chosen here to use for numerical applications one parametrisation from the original paper by Balberg and Gal [27], BG I. B. Instabilities and phase-transition Matter stability with respect to phase separation can be checked in any point of the extensive variable space by analyzing the eigen-values of the curvature matrix [5,30,31] , C ij = ∂ 2 f ({n l } l={i,j,k} )/∂n i ∂n j , where i, j, k = B, S, Q and f = e tot − T s tot is the total freeenergy. The occurrence of, at least, one negative eigenvalue in a certain domain of (n B , n S , n Q ) means that the system is unstable versus phase separation. The associated 3-dimensional Gibbs construction can be reduced to a simpler 1-dimensional Maxwell construction [5] by performing a Legendre transformation with respect to two out of the three chemical potentials µ B = µ n , µ S = −µ Λ + µ n and µ Q = µ p − µ n . We have chosen to work in the hybrid ensemble (n B , µ S , µ Q ) defined by: f baryon (n B , µ S , µ Q ) = f baryon − µ S n S − µ Q n Q ,(5) If the associated equation of state µ B = ∂f baryon (n B , µ S , µ Q )/∂n B as a function of n B presents a slope inversion, the relation µ B (n B ) is three-valued within a given interval of µ B . Then a Maxwell equal-area construction on this function allows defining two values n (1) B , n(2) B , which are characterized by the complete Gibbs equilibrium conditions (P, µ B , µ S , µ Q ) (1) = (P, µ B , µ S , µ Q ) (2) for a multicomponent system . The hyper-surface connecting the two points (n B , n S , n Q ) (1) ,(n B , n S , n Q ) (2) is the usual Gibbs construction. This procedure is independent of the choice of the densities (here: n S , n Q ) to be Legendre-transformed, provided the order parameter has a non-vanishing component along the remaining density (here: n B ). If this was not the case, that is if there was no jump in n B at the phase transition, n (1) B = n (2) B , the information on the phase transition could not be extracted from the hybrid ensemble eq.(5). We have verified that this is never the case, and the phase transition we will identify always separates a more diluted (lower n B ) from a denser (higher n B ) phase. For a generic physical multi-component system, this is not always the case and in the general case the convexity properties of the free energy have to be examined with care in order to identify phase transitions in such systems. In particular for our specific physics application of strangeness phase transition, we will explicitly show that the charge density is almost unaffected by the phase transition. This means that the concavity of the free energyf baryon (µ B , µ S , n Q ) as a function of n Q is extremely small. This means that working in that statistical ensemble would have rendered the observation of the phase transition very difficult. We stress that the one-dimensional Maxwell construction in the hybrid ensemble eq. (5), as long as a n B jump occurs through the phase transition as it does here, is strictly equivalent to the complete Gibbs construction. In particular, the pressure, as function of one density with the other densities kept constant, has not a constant value in the mixed phase [5]. For instance, P (n B ) at fixed n S , n Q is not constant in the mixed phase region. On the contrary, a Maxwell construction on P (n B ) or µ B (n B ) at constant values of n S , n Q is never theoretically justified. The upper part of Fig. 1 illustrates the projection of the T = 0 phase diagram in the n B − n S plane for µ S = 0. The arrows mark the direction of phase separation which, in case of phase coexistence, coincides with the order parameter. Two phase-coexistence domains may be identified. The one lying along n S = 0 at sub-saturation density corresponds to the well known LG like phase transition taking place in dilute nuclear matter [5]. The second domain lies at supra-saturation densities (n B > ∼ 2n 0 ), and the direction of phase separation is dominated by the strange density. In the density range shown by the figure, this domain is not upper limited in ρ B . We observe a bending at very high density meaning that the domain is finite as for the sub-saturation LG transition, but since we do not consider these densities as being realistically described within the present model, we refrain from showing the whole domain here. This transition is consistent with our previous findings within a simpler 2D model [17], though it obviously depends on the assumed strengths of the N N, N Y and Y Y -interactions. It is at first sight surprising to observe that the coexistence border of the latter transition is given by simple straight lines. In principle the coexistence borders of a first order phase transition with three conserved charges are given by two surfaces in the three dimensional space. For their projection on the n B − n S plane to be given by a one-dimensional curve, these surfaces have to be perpendicular to that plane, that is independent of n Q . The observed independence on the electric charge shows that the strange charge is the dominant order parameter for this transition. However, we can expect that some dependence on the electric charge would arise if charged strange particles were included, because of the correlation which would then exist between n Q and n S . The middle part of Fig. 1 illustrates the projection of the phase coexistence domains in the n B − n Q plane for µ S = 0, corresponding to the strangeness-equilibrium condition which is relevant for star matter. As discussed above, the coexistence domain being three-dimensional this representation depends on the value of the third variable given by µ S (or n S ). The well-known isospin dependence of the LG phase transition occurring at n S = 0 [5] is apparent. We have just noticed that the order parameter of the strangeness phase transition is given by a combination of the strange and baryonic density. Not surprisingly, the direction of phase separation of this transition in the n B −n Q plane is thus dominated by the baryonic density. The order parameter component in the direction of the electric charge can be understood as due to the correlation between the different densities. We are facing a transition between a relatively diluted, non-strange phase to a relatively dense, more strange one. Since Λ's are neutral, the positively charged component of the baryonic density is relatively less important in the dense phase, which explains the slope of the separation direction. Fig. 2 offers the complementary image on how the two phase coexistence domains look like when plotted with respect to µ S − µ B . In this case the condition µ S = 0 is released and alternative arbitrary constraints on µ Q = −50, 0, 50 MeV are imposed. The absence of Λ-hyperons at sub-saturation densities makes the conjugated chemical potential undefined. Mathematically, this means that any µ Λ ≤ (U Λ + m Λ c 2 ) is possible. This makes µ S span a semi-infinite domain lower limited by (µ n − U Λ − m Λ c 2 ). Reminding that -in µ n − µ p coordinates -the nuclear matter LG phase coexistence is figured by a curve whose extremities are the critical points, it is easy to understand that, by fixing µ Q , one fixes both µ n and µ p . Now, one can straightforwardly identify the semi-infinite horizontal coexisting lines as the ones corresponding to LG. The strangeness-driven phase coexistence at fixed µ Q appears in µ S − µ B as two merged semi-infinite linear segments. The merging point corresponds to the state where the equilibrium counter-part of the dense phase jumps from vacuum (low µ S ) to a dilute mixture (high µ S ). C. Influence of the Coulomb interaction We now turn to investigate the influence of Coulomb effects on the phase diagram. For simplicity, we will con- Electrons are coupled to charged baryons through the electromagnetic interaction, which can modify the baryonic phase diagram. However, the charge neutrality condition n Q = 0 makes the associated chemical potential µ Q ill-defined and keeps the problem threedimensional [18]. Since in homogeneous matter with the condition n Q = 0 the Coulomb interaction exactly vanishes [18], the total mean field pressure can be written as a sum over independent terms P B + P L + P γ + P ν + .... We shall still concentrate on the P B contribution, as the other terms do not affect the convexity properties of the thermodynamical potential on which phase transitions rely. We have constructed the full phase diagram of the (npΛe) system in the (n B , n S , n L ) space using the hybrid ensemblē f baryon (n B , µ S , µ L ) = f baryon − µ S n S − µ L n L .(6) In practice, the charge neutrality condition gives n p = n e , which allows to infer the electron chemical potential, µ e , via n e = 1 3π 2 µ ē hc 3 1 + µ −2 e π 2 T 2 − 1 2 m 2 e c 4 ,(7) and to obtain µ L = µ p − µ n + µ e . Note that n e = n e − − n e + = n L (= n p ) stands here for the net electron density. If neutrinos were in equilibrium, then µ L would correspond to the electron neutrino chemical potential. In particular, in a cold neutron star in β-equilibrium we would have µ L = 0 fixing n e . In core collapse events, on the other hand, neutrinos cannot be considered in equilibrium in the major part of the system. Here we want to study the entire phase diagram, not restricting to βequilibrium, and we therefore leave µ L free. Neutrinos, even in equilibrium, would not change the phase properties, and are thus neglected for the sake of simplicity in the present discussion. The lowest panel of Fig. 1 depicts the phase coexistence regions of the (npΛe) system at T = 0 and µ S = 0. In agreement with the results of Ref. [19,32], a strong Coulomb quenching of the LG-phase transition is obtained. However, the coexistence domain of the strangeness-driven phase transition is practically unmodified. This can be easily understood from the fact that the effect of the neutrality condition n Q = 0 on the two phase transitions is very different. The phase transition at sub-saturation density has the total baryonic density as order parameter. At such densities, n B is strongly correlated to n p because of the nuclear sym-metry energy which favorizes symmetric n n = n p matter. The phase transition thus implies a discontinuity in n p = n e , which is strongly disfavored by the huge electron incompressibility [19]. At supersaturation densities the order parameter is given by n S which is very loosely correlated to n Q . The phase transition thus does not imply any strong change in the electron distribution and the presence of electrons thus does not influence much the phase diagram. The temperature dependence of the phase diagram along µ S = 0 is presented in Fig. 3. We can observe that the direction of phase separation is almost independent of T . More interesting, starting from a finite value of T , a critical point appears and survives up to very high temperature. On Fig. 4 the critical temperature and the electron fraction Y e = n e /n B are shown as a function of baryon density. These values are typically reached within the cooling proto-neutron star, meaning that effects of criticality should be experienced. III. EFFECT OF THE PHASE-TRANSITION ON THE NEUTRINO MEAN FREE PATH The cooling of proto-neutron stars is mostly driven by neutrino diffusion during the first seconds. To explore the consequence of criticality for the cooling of proto-neutron star, we therefore turn to calculate the mean free path for the neutrino scattering off n, p, and Λ particles including the long-range correlations, essential for the study of criticality, in the linear response approximation. In the non-relativistic limit for the baryonic components, the mean free path at temperature T of a neutrino with initial energy E ν is given by 1/λ = 1/λ V + 1/λ A [33,34] where the contribution of the vector channel is defined as, 1 λ V (E ν , T ) = G 2 F 16π 2 (1 + cos θ)S V (q, T )(1 − f ν (k 3 ))dk 3 ,(8) and that of the axial channel, 1 λ A (E ν , T ) = G 2 F 16π 2 (3 − cos θ)S A (q, T )(1 − f ν (k 3 ))dk 3 .(9) In Eqs. (8)-(9), G F is the Fermi constant, θ is the angle between the initial and final neutrino momentum (=k 3 ), q is the transferred energy-momentum, q = (ω, q), and f ν is the Fermi-Dirac distribution of the outgoing neutrino. S V (S A ) are the dynamical response function in the vector (axial) channel. Since this study is focused on the impact of the density fluctuations close to the critical point, only the vector channel is considered. For densities close to the critical point, spin-density fluctuations are however expected to be small [35,36]. The dynamical response function in the vector channel is defined as S V (q, T ) = − 2 π 1 1 − exp(−ω/T ) × c n V c p V c Λ V Π V (q, T )    c n V c p V c Λ V    ,(10) where Π V (q, T ) is the vector-polarization matrix for the three species n, p, and Λ, given by the Lindhard functions in the case of the mean-field approximation and by the solution of the Bethe-Salpeter equations in the case of the mean-field+RPA approximation [33,34,36]. The vector coupling constants are set to be: -1 (n), 0.08 (p), -1 (Λ) [37]. The residual p-h interaction is derived from the potential energy (5) and is closely related to the curvature matrix without electrons [31]. The neutrino mean free path along an arbitrary Y e = 0.2981 trajectory in the phase diagram which passes by the critical point (see dot-dashed line in Fig. 3(a)) is shown in Fig. 5. As expected, the RPA correlations strongly reduce the neutrino mean free path close to the critical point, similar to the critical opalescence effect observed for the photon scattering off matter in critical water. The ratio of the neutrino mean free path in mean-field+RPA approximation over that at the mean-field level is shown in panel (b) exploring different neutrino energies around the neutrino chemical potential defined at beta equilibrium. The effect of the RPA correlations around the critical point is almost independent of the neutrino energy in agreement with the interpretation as critical opalescence. IV. CONCLUSIONS To conclude, we have studied the phase diagram of a mixture constituted of interacting neutrons, protons and Λ-hyperons under the condition of strangenessequilibrium, relevant for supernovae and neutron star physics. At supra-saturation densities, a strangenessdriven phase transition can take place, depending on the assumed strengths of nucleon-Λ and Λ-Λ interactions [17]. This second transition survives the screening effect of electrons and persists over a large domain of temperatures such that it may have an impact on star phenomenology. For a first study of this equation of state (EoS) within core-collapse supernovae, see [38]. In addition to the EoS, linear response theory shows that the neutrino mean-free path dramatically decreases close to the critical point of this phase transition, which occurs in a thermodynamic domain accessible to newly-born protoneutron stars. These results present a first step, and quantitative results might be somewhat modified in the presence of other strange-and non-strange baryonic, leptonic or mesonic degrees of freedom. This work is in progress and it will make the subject of a forthcoming publication. The result of the mean-field approximation is compared to the mean-field+RPA. (b) The ratio of the mean free path within mean-field+RPA over mean-field approximation is shown. FIG. 1 : 1(Color online) Borders of the phase-coexistence domains at T=0 and µS = 0. Upper (middle): (n, p, Λ)-mixture in nB − nS (nB − nC ) coordinates. Lower: (n, p, Λ, e)-mixture in nB − nL coordinates. Red: liquid-gas phase transition of non-strange dilute nuclear matter; blue: non-strange to strange phase transition. The arrows mark the directions of phase separation.sider only electrons and neglect other charged leptons or mesons. online) Borders of the phase-coexistence domains corresponding to the (n, p, Λ)-mixture at T=0 and µ Q = −50, 0, 50 MeV in µ B − µ S coordinates. Red: liquid-gas phase transition of non-strange dilute nuclear matter; blue: phase transition from non-strange to strange compressed baryonic matter. online) Borders of the strangeness driven phase transition domain corresponding to the neutral net-charge (n, p, Λ, e)-mixture at T=10, 20, 30 MeV and µ S = 0 in n B − n L coordinates. The dot-dashed line marks, for T =20 MeV, a path of constant Y e = 0.298. online) Electron fraction, Y e , and n B at the corresponding critical temperature for µ S = 0. online) (a) Neutrino mean free path for the scattering off n, p, and Λ at T = 20 MeV along a constant-Y e = 0.2981 trajectory in the phase diagram for E ν = µ ν , µ ν ± T as a function of the baryonic density ρ. 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Inclusive and differential cross sections of single top quark production in association with a Z boson are measured in proton-proton collisions at a center-of-mass energy of 13 TeV with a data sample corresponding to an integrated luminosity of 138 fb −1 recorded by the CMS experiment. Events are selected based on the presence of three leptons, electrons or muons, associated with leptonic Z boson and top quark decays. The measurement yields an inclusive cross section of 87.9 +7.5 −7.3 (stat) +7.3 −6.0 (syst) fb for a dilepton invariant mass greater than 30 GeV, in agreement with standard model (SM) calculations and represents the most precise determination to date. The ratio between the cross sections for the top quark and the top antiquark production in association with a Z boson is measured as 2.37 +0.56 −0.42 (stat) +0.27 −0.13 (syst). Differential measurements at parton and particle levels are performed for the first time. Several kinematic observables are considered to study the modeling of the process. Results are compared to theoretical predictions with different assumptions on the source of the initial-state b quark and found to be in agreement, within the uncertainties. Additionally, the spin asymmetry, which is sensitive to the top quark polarization, is determined from the differential distribution of the polarization angle at parton level to be 0.54 ± 0.16 (stat) ± 0.06 (syst), in agreement with SM predictions.

10.1007/jhep02(2022)107

2111.02860

35c93a91b1112293733e5ea995dd2a0e3dcc2725

Inclusive and differential cross section measurements of single top quark production in association with a Z boson in proton-proton collisions at √ s = 13 TeV The CMS Collaboration *-BY-4.0 license 2022/02/18 17 Feb 2022 W Adam J W Andrejkovic T Bergauer S Chatterjee K Damanakis M Dragicevic A Escalante Del Valle R Frühwirth M Jeitler N Krammer L Lechner D Liko I Mikulec P Paulitsch F M Pitters J Schieck R Schöfbeck D Schwarz S Templ W Waltenberger C.-E Wulz V Chekhovsky A Litomin V Makarenko M R Darwish E A De Wolf T Janssen T Kello A Lelek H Rejeb Sfar P Van Mechelen S Van Putte N Van Remortel F Blekman E S Bols J D&apos;hondt M Delcourt H El Faham S Lowette S Moortgat A Morton D Müller A R Sahasransu S Tavernier W Van Doninck D Beghin B Bilin B Clerbaux G De Lentdecker L Favart A Grebenyuk A K Kalsi K Lee M Mahdavikhorrami I Makarenko L Moureaux L Pétré A Popov N Postiau E Starling L Thomas M Vanden Bemden C Vander Velde P Vanlaer T Cornelis D Dobur J Knolle L Lambrecht G Mestdach M Niedziela C Roskas A Samalan K Skovpen M Tytgat B Vermassen L Wezenbeek A Benecke A Bethani G Bruno F Bury C Caputo P David C Delaere I S Donertas A Giammanco K Jaffel Sa Jain V Lemaitre K Mondal J Prisciandaro A Taliercio M Teklishyn T T Tran P Vischia S Wertz G A Alves C Hensel A Moraes P Rebello Teles W L Aldá Júnior M Alves Gallo Pereira M Barroso Ferreira Filho H Bran-Dao Malbouisson W Carvalho J Chinellato E M Da Costa G G Da Silveira D De Jesus Damiao S Fonseca De Souza C Mora Herrera K Mota Amarilo L Mundim H Nogima A Santoro S M Silva Do Amaral A Sznajder M Thiel F Torres Da Silva De Araujo A Vilela Pereira C A Bernardes L Calligaris T R Fernandez Perez Tomei E M Gregores D S Lemos P G Mercadante S F Novaes Sandra S Padula A Aleksandrov G Antchev R Hadjiiska P Iaydjiev M Misheva M Rodozov M Shopova G Sultanov A Dimitrov T Ivanov L Litov B Pavlov P Petkov A Petrov T Cheng T Javaid M Mittal L Yuan M Ahmad G Bauer C Dozen Z Hu J Martins Y Wang K Yi E Chapon G M Chen H S Chen M Chen F Iemmi A Kapoor D Leggat H Liao Z.-A Liu V Milosevic F Monti R Sharma J Tao J Thomas-Wilsker J Wang H Zhang J Zhao A Agapitos Y An Y Ban C Chen A Levin Q Li X Lyu Y Mao S J Qian D Wang J Xiao M Lu Z You X Gao H Okawa Y Zhang Z Lin M Xiao C Avila A Cabrera C Florez J Fraga J Mejia Guisao F Ramirez J D Ruiz Alvarez C A Salazar González Z Antunovic M Kovac T Sculac V Brigljevic D Ferencek D Majumder M Roguljic A Starodumov T Susa A Attikis K Christoforou E Erodotou A Ioannou G Kole M Kolosova S Konstantinou J Mousa C Nicolaou F Ptochos P A Razis H Rykaczewski H Saka M Finger M Finger Jr A 13 Kveton Universidad San Francisco De Quito Ecuador E Quito Carrera Jarrin A Lotfy M A Mahmoud S Bhowmik R K Dewanjee K Ehataht M Kadastik S Nandan C Nielsen J Pata M Raidal L Tani C Veelken S Bharthuar E Brücken F Garcia M.SJ Havukainen R Kim T Kinnunen K Lampén S Lassila-Perini T Lehti M Lindén L Lotti M Martikainen J Myllymäki H Ott E Siikonen J Tuominen Tuominiemi P Luukka H Petrow T Tuuva C Amendola M Besancon F Couderc M Dejardin D Denegri J L Faure F Ferri S Ganjour P Gras G Hamel De Monchenault P Jarry B Lenzi E Locci J Malcles J Rander A Rosowsky M Ö Sahin A Savoy-Navarro M Titov G B Yu S Ahuja F Beaudette M Bonanomi A Buchot Perraguin P Busson A Cappati C Charlot O Davignon B Diab G Falmagne S Ghosh R Granier De Cassagnac A Hakimi I Kucher J Motta M Nguyen C Ochando P Paganini J Rembser R Salerno U Sarkar J B Sauvan Y Sirois A Tarabini A Zabi A Zghiche J.-L Agram J Andrea D Apparu D Bloch G Bourgatte J.-M Brom E C Chabert C Collard D Darej J.-C Fontaine U Goerlach C Grimault A.-C Le Bihan E Nibigira P Van Hove E Asilar S Beauceron C Bernet G Boudoul C Camen A Carle N Chanon D Contardo P Depasse H El Mamouni J Fay S Gascon B. IlleM Gouzevitch I B Laktineh H Lattaud A Lesauvage M Lethuillier L Mirabito S Perries K Shchablo V Sordini L Torterotot G Touquet M Vander Donckt S Viret I Lomidze T Toriashvili Z Tsamalaidze V Botta L Feld K Klein M Lipinski D Meuser A Pauls N Röwert J Schulz M Teroerde A Dodonova D Eliseev M Erdmann P Fackeldey B Fischer S Ghosh T Hebbeker K Hoepfner F Ivone L Mastrolorenzo M Merschmeyer A Meyer G Mocellin S Mondal S Mukherjee D Noll A Novak T Pook A Pozdnyakov Y Rath H Reithler J Roemer A Schmidt S C Schuler A Sharma L Vigilante S Wiedenbeck S Zaleski C Dziwok G Flügge W Haj Ahmad O Hlushchenko T Kress A Nowack C Pistone O Pooth D Roy A Stahl T Ziemons A Zotz H Aarup Petersen M Aldaya Martin P Asmuss S Baxter M Bayatmakou O Behnke A Bermúdez Martínez S Bhattacharya A A Bin Anuar K Borras D Brunner A Camp-Bell A Cardini C Cheng F Colombina S Consuegra Rodríguez G Correia Silva V Danilov M De Silva L Didukh G Eckerlin D Eckstein L I Estevez Banos O Filatov E Gallo A Geiser A Giraldi A Grohsjean M Guthoff A Jafari N Z Jomhari H Jung A Kasem M Kasemann H Kaveh C Kleinwort R Kogler D Krücker W Lange J Lidrych K Lipka W Lohmann R Mankel I.-A Melzer-Pellmann M Mendizabal Morentin J Metwally A B Meyer M Meyer J Mnich A Mussgiller Y Otarid AdánD Pérez D Pitzl A Raspereza B Ribeiro Lopes J Rübenach A Saggio A Saibel M Savitskyi M Scham V Scheurer S Schnake P Schütze C Schwanenberger M Shchedrolosiev R E Sosa Ricardo D Stafford N Tonon M Van De Klundert R Walsh D Walter Q Wang Y Wen K Wichmann L Wiens C Wissing S Wuchterl R Aggleton S Albrecht S Bein L Benato P Connor K De Leo M Eich F Feindt A Fröhlich C Garbers E Garutti P Gunnellini M Hajheidari J Haller A Hinzmann G Kasieczka R Klanner T Kramer V Kutzner J Lange T Lange A Lobanov A Malara A Nigamova K J Pena Rodriguez M Rieger O Rieger P Schleper M Schröder J Schwandt J Sonneveld H Stadie G Steinbrück A Tews I Zoi J Bechtel S Brommer M Burkart E Butz R Caspart T Chwalek W De Boer A Dierlamm A Droll K El Morabit N Faltermann M Giffels J O Gosewisch A Gottmann F Hartmann C Heidecker U Husemann P Keicher R Koppenhöfer S Maier M Metzler Th. MüllerS Mitra M Neukum A Nürnberg G Quast K Rabbertz J Rauser D Savoiu M Schnepf D Seith I Shvetsov H J Simonis R Ulrich J Van Der Linden R F Cube M Wassmer M Weber S Wieland R Wolf S Wozniewski S Wunsch G Anagnostou G Daskalakis T Geralis A Kyriakis D Loukas A Stakia M Diamantopoulou D Karasavvas G Karathanasis P Kontaxakis C K Koraka A Manousakis-Katsikakis A Panagiotou I Papavergou N Saoulidou K Theofilatos E Tziaferi K Vellidis E Vourliotis G Bakas K Kousouris I Papakrivopoulos G Tsipolitis A Zacharopoulou M Csanad K Farkas M M A Gadallah S Lökös P Major K Mandal A Mehta G Pasztor A J Rádl O Surányi G I Veres M Bartók G Bencze C Hajdu D Horvath F Sikler V Veszpremi S Czellar D Fasanella J Karancsi J Molnar Z Szillasi D Teyssier P Raics Z L Trocsanyi B Ujvari T Csorgo F Nemes T Novak S Choudhury J R Komaragiri D Kumar L Panwar P C Tiwari S Bahinipati C Kar P Mal T Mishra V K Muraleedharan Nair Bindhu A Nayak P Saha N Sur S K Swain D Vats S Bansal S B Beri V Bhatnagar G Chaudhary S Chauhan N Dhingra R Gupta A Kaur M Kaur S Kaur P Kumari M Meena K Sandeep J B Singh A K Virdi A Ahmed A Bhardwaj B C Choudhary M Gola S Keshri A Kumar M Naimuddin P Priyanka K Ranjan A Shah M Bharti R Bhattacharya S Bhattacharya D Bhowmik S Dutta S Dutta B Gomber M Maity P Palit P K Rout G Saha B Sahu S Sarkar M Sharan B Singh S Thakur P K Behera S C Behera P Kalbhor A Muhammad R Pradhan P R Pujahari A Sharma A K Sikdar D Dutta V Jha V Kumar D K Mishra K Naskar P K Netrakanti L M Pant P Shukla T Aziz S Dugad M Kumar S Banerjee R Chudasama M Guchait S Karmakar S Kumar G Majumder K Mazumdar S Mukherjee K Alpana S Dube B Kansal A Laha S Pandey A Rane A Rastogi S Sharma H Bakhshiansohi E Khazaie M Zeinali S Chenarani S M Etesami M Khakzad M Mohammadi Najafabadi M Grunewald Infn Sezione Di Bari ItalyUniversità Bari Di Bari ItalyPolitecnico Bari Di Bari Italy M Bari Abbrescia R Aly C Aruta A Colaleo D Creanza N De Filippis M De Palma A Di Florio A Di Pilato W Elmetenawee L Fiore A Gelmi M Gul G Iaselli M Ince S Lezki G Maggi M Maggi I Margjeka V Mastrapasqua S My S Nuzzo A Pellecchia A Pompili G Pugliese D Ramos A Ranieri G Selvaggi L Silvestris F M Simone Ü Sözbilir R Venditti P Verwilligen Infn Sezione Di Bologna Bologna, ItalyUniversità Di Bologna Italy G Bologna Abbiendi C Battilana D Bonacorsi L Borgonovi L Brigliadori R Campanini P Capiluppi A Castro F R Cavallo M Cuffiani G M Dallavalle T Diotalevi F Fabbri A Fanfani P Giacomelli L Giommi C Grandi L Guiducci S Lo Meo L Lunerti S Marcellini G Masetti F L Navarria A Perrotta F Primavera A M Rossi T Rovelli G P Siroli S Albergo S Costa A Di Mattia R Potenza A Tricomi C Tuve G Barbagli A Cassese R Ceccarelli V Ciulli C Civinini R D&apos;alessandro E Focardi G Latino P Lenzi M Lizzo M Meschini S Paoletti R Seidita G Sguazzoni L Viliani L Benussi S Bianco D Piccolo Infn Sezione Di Genova ItalyUniversità Genova Di Genova Italy M Genova Bozzo F Ferro R Mulargia E Robutti S Tosi A Benaglia G Boldrini F Brivio F Cetorelli F De Guio M E Dinardo P Dini S Gennai A Ghezzi P Govoni L Guzzi M T Lucchini M Malberti S Malvezzi A Massironi D Menasce L Moroni M Paganoni D Pedrini B S Pinolini S Ragazzi N Redaelli T Tabarelli De Fatis D Valsecchi D Zuolo Infn Sezione Di Napoli Napoli, Italy, UniversitàDi Napoli &apos; Napoli, ItalyFederico Ii&apos; B Università G Marconi Roma Italy S Buontempo F Carnevali N Cavallo A De Iorio F Fabozzi A O M Iorio L Lista S Meola P Paolucci B Rossi C Sciacca Infn Sezione Di Padova Padova, ItalyUniversità Di Padova Padova, ItalyUniversità Di Trento Italy P Trento Azzi N Bacchetta D Bisello P Bortignon A Bragagnolo R Carlin P Checchia T Dorigo U Dosselli F Gasparini U Gasparini G Grosso S Y Hoh L Layer E Lusiani M Margoni A T Meneguzzo J Pazzini P Ronchese R Rossin F Simonetto G Strong M Tosi H Yarar M Zanetti P Zotto A Zucchetta G Zumerle Infn Sezione Di Pavia Pavia, ItalyUniversità Di Pavia Pavia Italy C Aime&apos; A Braghieri S Calzaferri D Fiorina P Montagna S P Ratti V Re C Riccardi P Salvini I Vai P Vitulo Infn Sezione Di Perugia Perugia, ItalyUniversità Di Perugia PerugiaItaly P Asenov G M Bilei D Ciangottini L Fanò M Magherini G Mantovani V Mariani M Menichelli F Moscatelli A Piccinelli M Presilla A Rossi A Santocchia D Spiga T Tedeschi P Azzurri G Bagliesi V Bertacchi L Bianchini T Boccali E Bossini R Castaldi M A Ciocci V D&apos;amante R Dell&apos;orso M R Di Domenico S Donato A Giassi F Ligabue E Manca G Mandorli FigueiredoD Matos A Messineo F Palla S Parolia G Ramirez-Sanchez A Rizzi G Rolandi S Roy Chowdhury A Scribano N Shafiei P Spagnolo R Tenchini G Tonelli N Turini A Venturi P G Verdini Infn Sezione Di Roma Rome, ItalySapienza Università Di Roma Italy P Rome Barria M Campana F Cavallari D Del Re E Di Marco M Diemoz E Longo P Meridiani G Organtini F Pandolfi R Paramatti C Quaranta S Rahatlou C Rovelli F Santanastasio L Soffi R Tramontano Infn Sezione Di Torino Torino, ItalyUniversità Di Torino N Amapane R Arcidiacono S Argiro M Arneodo N Bartosik R Bellan A Bellora J Berenguer Antequera C Biino N Cartiglia S Cometti M Costa R Covarelli N Demaria B Kiani F Legger C Mariotti S Maselli E Migliore E Monteil M Monteno M M Obertino G Ortona L Pacher N Pastrone M Pelliccioni G L Pinna Angioni M Ruspa K Shchelina F Siviero V Sola A Solano D Soldi A Staiano M Tornago D Trocino A Vagnerini S Belforte V Candelise M Casarsa F Cossutti A Da Rold G Della Ricca G Sorrentino F Vazzoler S Dogra C Huh B Kim D H Kim G N Kim J Kim J Lee S W Lee C S Moon Y D Oh S I Pak B C Radburn-Smith S Sekmen Y C Yang H Kim D H Moon B Francois T J Kim J Park S Cho S Choi Y Go B Hong K Lee K S Lee J Lim J Park S K Park J Yoo J Goh A Gurtu H S Kim Y Kim J Almond J H Bhyun J Choi S Jeon J Kim J S Kim S Ko H Kwon H Lee S Lee B H Oh M Oh S B Oh H Seo U K Yang I Yoon W Jang D Y Kang Y Kang S Kim B Ko J S H Lee Y Lee J A Merlin I C Park M.SY Roh D Ryu I J Song S Watson Yang S Ha H D Yoo M Choi H Lee Y Lee I Yu T Beyrouthy Y Maghrbi K Dreimanis V Veckalns M Ambrozas A Carvalho Antunes De Oliveira A Juodagalvis A Rinkevicius G Tamulaitis M A T Wan N Abdullah Z Yusli Zolkapli J F Benitez A Castaneda Hernandez M León Coello J A Murillo Quijada A Sehrawat L Valencia Palomo G Ayala H Castilla-Valdez E De La Cruz-Burelo I Heredia-De La Cruz R Lopez-Fernandez C A Mondragon Herrera D A Perez Navarro A Sánchez Hernández S Carrillo Moreno C Oropeza Barrera F Vazquez Valencia I Pedraza H A Salazar Ibarguen C Uribe Estrada J Mijuskovic N Raicevic D Krofcheck P H Butler A Ahmad M I Asghar A Awais M I M Awan H R Hoorani W A Khan M A Shah M Shoaib M Waqas V Avati L Grzanka M Malawski H Bialkowska M Bluj B Boimska M Górski M Kazana M Szleper P Zalewski K Bunkowski K Doroba A Kalinowski M Konecki J Krolikowski M Araujo P Bargassa D Bastos A Boletti P Faccioli M Gallinaro J Hollar N Leonardo T Niknejad M Pisano J Seixas O Toldaiev J Varela S Afanasiev D Budkouski I Golutvin I Gorbunov V Karjavine V Korenkov A Lanev A Malakhov V Matveev V Palichik V Perelygin M Savina D Seitova V Shalaev S Shmatov S Shulha V Smirnov O Teryaev N Voytishin B S Yuldashev A Zarubin I Zhizhin G Gavrilov V Golovtcov Y Ivanov V Kim E Kuznetsova V Murzin V Oreshkin I Smirnov D Sosnov V Sulimov L Uvarov S Volkov A Vorobyev Yu Andreev A Dermenev S Gninenko N Golubev A Karneyeu D Kirpichnikov M Kirsanov N Krasnikov A Pashenkov G Pivovarov A Toropin V Epshteyn V Gavrilov N Lychkovskaya A Nikitenko V Popov A Stepennov M Toms E Vlasov A Zhokin O Bychkova M Chadeeva M Danilov A Oskin P Parygin E Popova V Andreev M Azarkin I Dremin M Kirakosyan A Terkulov A Belyaev E Boos V Bunichev M Dubinin L Dudko A Ershov V Klyukhin M K Jayananda B Kailasapathy D U J Sonnadara D D C Wickramarathna T K Aarrestad D Abbaneo J Alimena E Auffray G Auzinger J Baechler P Baillon D Barney J Bendavid M Bianco A Bocci T Camporesi M Capeans Garrido E A Yetkin A Cakir K Cankocak Y Komurcu S Sen S Cerci I Hos B Kaynak S Ozkorucuklu H Sert D Sunar Cerci C Zorbilmez Ukraine B Grynyov L Levchuk D Anthony E Bhal S Bologna J J Brooke A Bundock E Clement D Cussans H Flacher J Goldstein G P Heath H F Heath L Kreczko B Krikler S Paramesvaran S Seif El Nasr-Storey V J Smith N Stylianou K Walkingshaw Pass R White K W Bell A Belyaev C Brew R M Brown D J A Cockerill C Cooke K V Ellis K Harder S Harper M.-L Holmberg J Linacre K Manolopoulos D M Newbold E Olaiya D Petyt T Reis T Schuh C H Shepherd-Themistocleous I R Tomalin T Williams R Bainbridge P Bloch S Bonomally J Borg S Breeze O Buchmuller V Cepaitis G S Chahal D Colling P Dauncey G Davies M Della Negra S Fayer G Fedi G Hall M H Hassanshahi G Iles J Langford L Lyons A.-M Magnan S Malik A Martelli D G Monk J Nash M Pesaresi D M Raymond A Richards A Rose E Scott C Seez A Shtipliyski A Tapper K Uchida T Virdee M Vojinovic N Wardle S N Webb D Winterbottom K Coldham J E Cole A Khan P Kyberd I D Reid L Teodorescu S Zahid S Abdullin A Brinkerhoff B Caraway J Dittmann K Hatakeyama A R Kanuganti B Mcmaster N Pastika M Saunders S Sawant C Sutantawibul J Wilson R Bartek A Dominguez R Uniyal A M Vargas Hernandez A Buccilli S I Cooper D Di Croce S V Gleyzer C Henderson C U Perez P Rumerio C West A Akpinar A Albert D Arcaro C Cosby Z Demiragli E Fontanesi D Gastler S May J Rohlf K Salyer D Sperka D Spitzbart I Suarez A Tsatsos S Yuan D Zou D Acosta P Avery D Bourilkov L Cadamuro V Cherepanov F Errico R D Field D Guerrero B M Joshi M Kim E Koenig J Konigsberg A Korytov K H Lo K Matchev N Menendez G Mitselmakher A Muthirakalayil Madhu N Rawal D Rosenzweig S Rosenzweig J Rotter K Shi J Sturdy J Wang E Yigitbasi X Zuo T Adams A Askew R Habibullah V Hagopian K F Johnson R Khurana T Kolberg G Martinez H Prosper C Schiber O Viazlo R Yohay J Zhang M M Baarmand S Butalla T Elkafrawy M Hohlmann R Kumar Verma D Noonan M Rahmani F Yumiceva M R Adams H Becerril Gonzalez R Cavanaugh S Dittmer O Evdokimov C E Gerber D A Hangal D J Hofman A H Merrit C Mills G Oh T Roy S Rudrabhatla M B Tonjes N Varelas J Viinikainen X Wang Z Wu Z Ye M Alhusseini K Dilsiz L Emediato R P Gandrajula O K Köseyan J.-P Merlo A Mestvirishvili J Nachtman H Ogul Y Onel A Penzo C Snyder E Tiras O Amram B Blumenfeld L Corcodilos J Davis M Eminizer A V Gritsan S Kyriacou P Maksimovic J Roskes M Swartz T Á Vámi A Abreu J Anguiano C Baldenegro Barrera P Baringer A Bean A Bylinkin Z Flowers T Isidori S Khalil J King G Krintiras A Kropivnitskaya M Lazarovits C Le Mahieu C Lindsey J Marquez N Minafra M Murray M Nickel C Rogan C Royon R Salvatico S Sanders E Schmitz C Smith J D Tapia Takaki Q Wang Z Warner J Williams G Wilson S Duric A Ivanov K Kaadze D Kim Y Maravin T Mitchell A Modak K Nam F Rebassoo D Wright E Adams A Baden O Baron A Belloni S C Eno N J Hadley S Jabeen R G Kellogg T Koeth S Lascio A C Mignerey S Nabili C Palmer M Seidel A Skuja L Wang K Wong D Abercrombie G Andreassi R Bi W Busza I A Cali M. D&apos;Y Chen Alfonso J Eysermans C Freer G Gomez Ceballos M Goncharov P Harris M Hu M Klute D Kovalskyi J Krupa Y.-J Lee C Mironov C Paus D Rankin C Roland G Roland Z Shi G S F Stephans J Wang Z Wang B Wyslouch R M Chatterjee A Evans J Hiltbrand Sh Jain M Krohn Y Kubota J Mans M Revering R Rusack R Saradhy N Schroeder N Strobbe M A Wadud K Bloom M Bryson S Chauhan D R Claes C Fangmeier L Finco F Golf C Joo I Kravchenko M Musich I Reed J E Siado G R Snow W Tabb F Yan A G Zecchinelli G Agarwal H Bandyopadhyay L Hay I Iashvili A Kharchilava C Mclean D Nguyen J Pekkanen S Rappoccio A Williams G Alverson E Barberis Y Haddad A Hortiangtham J Li G Madigan B Marzocchi D M Morse V Nguyen T Orimoto A Parker L Skinnari A Tishelman-Charny T Wamorkar B Wang A Wisecarver D Wood S Bhattacharya J Bueghly Z Chen A Gilbert T Gunter K A Hahn Y Liu N Odell M H Schmitt M Velasco R Band R Bucci M Cremonesi A Das N Dev R Goldouzian M Hildreth K Hurtado Anampa C Jessop K Lannon J Lawrence N Loukas D Lutton J Mariano N Marinelli I Mcalister T Mccauley C Mcgrady K Mohrman C Moore Y Musienko R Ruchti A Townsend M Wayne A Wightman M Zarucki L Zygala B Bylsma B Cardwell L S Durkin B Francis C Hill M Nunez Ornelas K Wei B L Winer B R Yates F M Addesa B Bonham P Das G Dezoort P Elmer A Frankenthal B Greenberg N Haubrich S Higginbotham A Kalogeropoulos G Kopp S Kwan D Lange D Marlow K Mei I Ojalvo J Olsen D Stickland C Tully S Malik S Norberg A S Bakshi V E Barnes R Chawla S Das L Gutay M Jones A W Jung S Karmarkar D Kondratyev M Liu G Negro N Neumeister G Paspalaki S Piperov A Purohit J F Schulte M Stojanovic J Thieman F Wang R Xiao W Xie J Dolen N Parashar A Baty T Carnahan M Decaro S Dildick K M Ecklund S Freed P Gardner F J M Geurts A Kumar W Li B P Padley R Redjimi W Shi A G Stahl Leiton S Yang L Zhang Y Zhang A Bodek P De Barbaro R Demina J L Dulemba C Fallon T Ferbel M Galanti A Garcia-Bellido O Hindrichs A Khukhunaishvili E Ranken R Taus B Chiarito J P Chou A Gandrakota Y Gershtein E Halkiadakis A Hart M Heindl O Karacheban I Laflotte A Lath R Montalvo K Nash M Osherson S Salur S Schnetzer S Somalwar R Stone S A Thayil S Thomas H Wang H Acharya A G Delannoy S Fiorendi S Spanier O Bouhali M Dalchenko A Delgado R Eusebi J Gilmore T Huang T Kamon H Kim S Luo S Malhotra R Mueller D Overton D Rathjens A Safonov N Akchurin J Damgov V Hegde S Kunori K Lamichhane S W Lee T Mengke S Muthumuni T Peltola I Volobouev Z Wang A Whitbeck E Appelt S Greene A Gurrola W Johns A Melo H Ni K Padeken F Romeo P Sheldon S Tuo J Velkovska M W Arenton B Cox G Cummings J Hakala R Hirosky M Joyce A Ledovskoy A Li C Neu C E Perez Lara B Tannenwald S White Yerevan Physics Institute Institute for Nuclear Problems EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) Tumasyan Institut für Hochenergiephysik Yerevan, Vienna, MinskArmenia A., Austria, Belarus Universiteit Antwerpen AntwerpenBelgium Vrije Universiteit Brussel BrusselBelgium Université Libre de Bruxelles BruxellesBelgium Ghent University GhentBelgium Centro Brasileiro de Pesquisas Fisicas Institute for Nuclear Research and Nuclear Energy Universidade Estadual Paulista (a), Universidade Federal do ABC (b) Université Catholique de Louvain Louvain-la-NeuveBelgium, Brazil, Brazil, Brazil Bulgarian Academy of Sciences SofiaBulgaria University of Sofia SofiaBulgaria Department of Physics Beihang University BeijingChina Institute of High Energy Physics State Key Laboratory of Nuclear Physics and Technology Tsinghua University Beijing, BeijingChina, China Peking University BeijingChina Institute of Modern Physics and Key Laboratory of Nuclear Physics and Ion-beam Applica-tion (MOE) -Fudan University Sen University Guangzhou, ShanghaiChina, China Zhejiang University HangzhouZhejiangChina, China Universidad de Los Andes BogotaColombia Universidad de Antioquia MedellinColombia Faculty of Electrical Engineering Faculty of Science, Split Mechanical Engineering and Naval Ar-chitecture, Split, Croatia D. Giljanovic, N. Godinovic , D. Lelas , I. Puljak University of Split University of Split Croatia Institute Rudjer Boskovic ZagrebCroatia University of Cyprus NicosiaCyprus Charles University PragueCzech Republic 16 Center for High Energy Physics (CHEP-FU) Academy of Scientific Research and Technology of the Arab Republic of Egypt Escuela Politecnica Nacional Egyp- H. Abdalla 14 , Y. Assran 15QuitoEcuador National Institute of Chemical Physics and Biophysics Department of Physics Fayoum University El-FayoumEgypt, Estonia P. Eerola , L. Forthomme , H. Kirschenmann , K. Osterberg , M. Voutilainen Helsinki Institute of Physics University of Helsinki Helsinki, HelsinkiFinland, Finland Lappeenranta University of Technology LappeenrantaFinland Laboratoire Leprince-Ringuet, CNRS/IN2P3, Ecole Polytechnique, Institut Polytech-nique de Paris IRFU Université Paris-Saclay Gif-sur-Yvette, PalaiseauFrance, France UMR 7178 Université de Strasbourg CNRS IPHC StrasbourgFrance Institut de Physique des 2 Infinis de Lyon (IP2I ) Georgian Technical University Villeurbanne, TbilisiFrance, Georgia I. Physikalisches Institut RWTH Aachen University AachenGermany III. Physikalisches Institut A RWTH Aachen University AachenGermany III. Physikalisches Institut B Deutsches Elektronen-Synchrotron RWTH Aachen University Aachen, HamburgGermany, Germany University of Hamburg HamburgGermany Institute of Nuclear and Particle Physics (INPP) Karlsruher Institut fuer Technologie KarlsruheGermany NCSR Demokritos Aghia ParaskeviGreece Kapodistrian University of Athens AthensGreece National Technical University of Athens AthensGreece Ioánnina, Greece K. Adamidis, I. Bestintzanos, I. Evangelou , C. Foudas, P. Gianneios, P. Katsoulis, P. Kokkas, N. Manthos, I. Papadopoulos , J. Strologas MTA-ELTE Lendület CMS Particle and Nuclear Physics Group University of Ioánnina Eötvös Loránd University BudapestHungary Institute of Nuclear Research ATOMKI Institute of Physics Wigner Research Centre for Physics, Budapest University of Debrecen Debrecen, DebrecenHungary, Hungary, Hungary Indian Institute of Science (IISc) Karoly Robert Campus MATE Institute of Technology Gyongyos, BangaloreHungary, India National Institute of Science Education and Research HBNI BhubaneswarIndia Panjab University ChandigarhIndia Saha Institute of Nuclear Physics University of Delhi DelhiIndia HBNI KolkataIndia Indian Institute of Technology Madras MadrasIndia Bhabha Atomic Research Centre MumbaiIndia Tata Institute of Fundamental Research-A MumbaiIndia Indian Institute of Science Education and Research (IISER) Institute for Research in Fundamental Sciences (IPM) Tata Institute of Fundamental Research-B Pune, TehranMumbaiIndia, India, Iran INFN Sezione di Catania a University College Dublin DublinCatania, ItalyIreland INFN Sezione di Firenze a , Firenze, Italy Università di Catania b CataniaItaly Università di Firenze b FirenzeItaly INFN Sezione di Milano-Bicocca a , Milano, Italy, Università di Milano-Bicocca b INFN Laboratori Nazionali di Frascati Frascati, MilanoItaly, Italy INFN Sezione di Pisa a , Pisa, Italy, Università di Pisa b , Pisa, Italy, Scuola Normale Superiore di Pisa c , Pisa, Italy, Università di Siena d , Siena Università della Basilicata c PotenzaItaly, Italy INFN Sezione di Trieste a , Trieste, Italy Università del Piemonte Orientale c NovaraItaly Università di Trieste b TriesteItaly Kyungpook National University DaeguKorea Institute for Universe and Elementary Particles Chonnam National University KwangjuKorea Hanyang University SeoulKorea Korea University SeoulKorea Department of Physics, Seoul Kyung Hee University SeoulRepublic of Korea, Korea Sejong University SeoulKorea Seoul National University SeoulKorea University of Seoul SeoulKorea Department of Physics Yonsei University SeoulKorea College of Engineering and Technology Sungkyunkwan University SuwonKorea Mid-dle East (AUM) American University EgailaDasmanKuwait, Kuwait Riga Technical University RigaLatvia National Centre for Particle Physics, Universiti Malaya, Kuala Lumpur, Malaysia N. Bin Norjoharuddeen , W Universidad de Sonora (UNISON) Vilnius University VilniusLithuania, Mexico Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City Universidad Iberoamericana Mexico CityMexico, Mexico Benemerita Universidad Autonoma de Puebla PueblaMexico University of Montenegro PodgoricaMontenegro University of Auckland AucklandNew Zealand National Centre for Physics, Quaid-I-Azam University University of Canterbury Christchurch, IslamabadNew Zealand, Pakistan Faculty of Computer Science Electron-ics and Telecommunications Institute of Experimental Physics Faculty of Physics National Centre for Nuclear Research, Swierk AGH University of Science and Technology KrakowPoland, Poland Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa University of Warsaw WarsawPoland, Portugal Institute for Nuclear Research Petersburg Nuclear Physics Institute, Gatchina (St. Petersburg) Joint Institute for Nuclear Research Dubna, MoscowRussia, Russia, Russia Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC 'Kur-chatov Institute', Moscow Russia P.N. Lebedev Physical Institute Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow Institute of Physics and Technology, Moscow, Russia T. Aushev National Research Nuclear University 'Moscow Engineering Physics Institute' (MEPhI) Moscow State University Moscow, Moscow, MoscowRussia, Russia, Russia University of Colombo Sri Lanka Department of Physics, Matara Sri Lanka W.G.D. Dharmaratna , K. Liyanage, N. Perera, N. Wickramage CERN, European Organization for Nuclear Research, Geneva University of Ruhuna Switzerland Istanbul Technical University IstanbulTurkey Institute for Scintillation Materials of National Academy of Science of Ukraine, Kharkov Istanbul University IstanbulTurkey National Scientific Center Kharkov Institute of Physics and Technology KharkovUkraine Rutherford Appleton Laboratory University of Bristol Bristol, DidcotUnited Kingdom, United Kingdom Imperial College LondonUnited Kingdom Brunel University UxbridgeUnited Kingdom Baylor University WacoTexasUSA Catholic University of America WashingtonDCUSA The University of Alabama TuscaloosaAlabamaUSA Boston University BostonMassachusettsUSA Brown University ProvidenceRhode IslandUSA University of Florida GainesvilleFloridaUSA Florida State University TallahasseeFloridaUSA University of Illinois at Chicago (UIC) Florida Institute of Technology Melbourne, ChicagoFlorida, IllinoisUSA, USA The University of Iowa Iowa City IowaUSA Johns Hopkins University BaltimoreMarylandUSA The University of Kansas LawrenceKansasUSA Lawrence Livermore National Laboratory, Livermore Kansas State University ManhattanKansas, CaliforniaUSA, USA University of Maryland College ParkMarylandUSA Massachusetts Institute of Technology CambridgeMassachusettsUSA University of Minnesota MinneapolisMinnesotaUSA State University of New York at Buffalo, Buffalo University of Nebraska-Lincoln LincolnNebraska, New YorkUSA, USA Northeastern University BostonMassachusettsUSA Northwestern University EvanstonIllinoisUSA University of Notre Dame Notre DameIndianaUSA The Ohio State University ColumbusOhioUSA Princeton University PrincetonNew JerseyUSA University of Puerto Rico MayaguezPuerto RicoUSA Purdue University West LafayetteIndianaUSA Purdue University Northwest HammondIndianaUSA Rice University HoustonTexasUSA University of Rochester RochesterNew YorkUSA The State University of New Jersey PiscatawayNew JerseyUSA University of Tennessee KnoxvilleTennesseeUSA Texas A&M University College StationTexasUSA Texas Tech University LubbockTexasUSA Vanderbilt University NashvilleTennesseeUSA University of Virginia CharlottesvilleVirginiaUSA Wayne State University DetroitMichiganUSA Inclusive and differential cross section measurements of single top quark production in association with a Z boson in proton-proton collisions at √ s = 13 TeV The CMS Collaboration *-BY-4.0 license Journal of High Energy Physics 2022/02/18 17 Feb 202210.1007/JHEP02(2022)107* See Appendix D for the list of collaboration members Inclusive and differential cross sections of single top quark production in association with a Z boson are measured in proton-proton collisions at a center-of-mass energy of 13 TeV with a data sample corresponding to an integrated luminosity of 138 fb −1 recorded by the CMS experiment. Events are selected based on the presence of three leptons, electrons or muons, associated with leptonic Z boson and top quark decays. The measurement yields an inclusive cross section of 87.9 +7.5 −7.3 (stat) +7.3 −6.0 (syst) fb for a dilepton invariant mass greater than 30 GeV, in agreement with standard model (SM) calculations and represents the most precise determination to date. The ratio between the cross sections for the top quark and the top antiquark production in association with a Z boson is measured as 2.37 +0.56 −0.42 (stat) +0.27 −0.13 (syst). Differential measurements at parton and particle levels are performed for the first time. Several kinematic observables are considered to study the modeling of the process. Results are compared to theoretical predictions with different assumptions on the source of the initial-state b quark and found to be in agreement, within the uncertainties. Additionally, the spin asymmetry, which is sensitive to the top quark polarization, is determined from the differential distribution of the polarization angle at parton level to be 0.54 ± 0.16 (stat) ± 0.06 (syst), in agreement with SM predictions. Introduction The electroweak production of a top quark or antiquark in association with a Z boson, the tZq process, was recently observed in proton-proton (pp) collisions at a center-of-mass energy of 13 TeV at the CERN LHC by both the CMS and ATLAS experiments [1,2]. The process has unique features that make it a suitable probe for several interactions in the standard model (SM) of particle physics. Figure 1 shows representative leading-order (LO) Feynman diagrams of tZq, where stands for an electron or muon, including also off-shell photons (γ * ) and the possibility of nonresonant dilepton emission to correctly account for interference effects. Throughout the text, unless stated otherwise, tZq stands collectively for the top quark and antiquark production, including nonresonant dilepton emission. Because of the pure electroweak nature of tZq production, corrections to the cross section arising from quantum chromodynamics (QCD) are typically small. As a result, the study of tZ, tWb, and WWZ couplings in tZq production is not primarily affected by QCD uncertainties [3]. This makes an analysis of tZq production advantageous in comparison to the associated production of a top quark-antiquark pair (tt) and a Z boson (ttZ), where the tt is produced via a QCD interaction. The top quark is strongly polarized in this process because of its electroweak production mechanism. Measurement of the top quark polarization in the tZq process provides complementary information to the existing studies of the top quark electroweak interactions [4][5][6][7][8]. Furthermore, the tZq process offers the possibility of measuring the top quark and antiquark production cross sections separately, as well as their ratio. These measurements yield potential sensitivity to different parameterizations of the parton distribution functions (PDFs) of the proton. Figure 1: Representative LO Feynman diagrams for the tZq production process. The production mechanism of nonresonant lepton pairs (lower right) is included in the signal definition to correctly account for interference effects. q b q ′ t Z/γ ⋆ t W q b q ′ t Z/γ ⋆ b W q b Z/γ ⋆ q ′ t q ′ W q b Z/γ ⋆ q ′ t q W q b q ′ Z/γ ⋆ t W W q b q ′ ℓ + ℓ − t W W Previous measurements of the inclusive tZq cross section σ tZq in leptonic final states have reached a precision of about 15% [1, 2, 9, 10] and are in agreement with the SM predictions at next-to-LO (NLO), σ SM tZq = 94.2 +1.9 −1.8 (scale) ± 2.5 (PDF) fb, for a dilepton invariant mass greater than 30 GeV [10]. The systematic uncertainty associated with the energy scale used in the calculations arises from variations of the factorization and renormalization scales. The calculation is performed in the five-flavor scheme (5FS), where the b quark content of the proton is described by the appropriate PDF. No differential measurement of the tZq process has been reported so far. This paper presents the most precise measurement of σ tZq to date, as well as the first measurement of differential cross sections for the tZq process. Data from pp collisions at √ s = 13 TeV collected by the CMS experiment, corresponding to an integrated luminosity of 138 fb −1 [11], are used in this analysis. The improved precision on σ tZq in this measurement compared to the previous results [1, 2, 9, 10] is due to the larger data sample, an optimized lepton identification, and the use of various control regions in data [1]. The inclusive measurement is extended to report the first separate determination of the Z boson associated production cross sections of the top quark and antiquark, as well as their ratio. The studies are performed in final states with three leptons (electrons or muons), including also a small contribution from sequential τ lepton decays. Two selected leptons of same flavor and opposite charge are assumed to come from the Z boson decay, while the third lepton is associated with the leptonic decay of the W boson produced in the top quark decay. Both the inclusive and differential cross section measurements heavily rely on multivariate classifiers to separate the tZq signal from various background processes including ttZ. The results are obtained by performing maximum likelihood fits on distributions that are obtained from the responses of the classifiers. Tabulated results are provided in HEPData [12]. The differential distributions are extracted at both parton and particle levels using a likelihoodbased unfolding (as detailed in, e.g., Ref. [13]). Measured observables at parton level are the transverse momenta, p T , of the top quark, the Z boson, and the lepton from the top quark decay, together with the invariant masses of the three leptons and the t + Z system. The azimuthal angular distance between the two leptons from the Z boson decay, as well as the cosine of the top quark polarization angle, are also measured. The differential measurement of the top quark polarization angle is used to determine the top quark spin asymmetry. Additionally, the p T and absolute pseudorapidity, |η|, of the jet corresponding to the light-flavor quark that recoils against the virtual W boson (q in Fig. 1), denoted as the recoiling jet or j', are measured at the particle level. Results are compared with predictions using the four-flavor scheme (4FS), where the incoming b quark is produced in the gluon-splitting process, and the 5FS. The paper is organized as follows: the CMS detector is briefly introduced in Section 2. Section 3 is devoted to the data and simulated samples, and the identification and selection requirements applied to the reconstructed objects. A description of the event selection and reconstruction is presented in Section 4, while the estimation of the backgrounds is discussed in Section 5. Discussion of systematic uncertainties affecting the presented measurements follows in Section 6. Sections 7 and 8 are dedicated to the description of the methodology and the obtained results relevant to the inclusive and differential measurements, respectively. Finally, the paper is summarized in Section 9. The CMS detector, data, and simulated samples The central feature of the CMS apparatus [14] is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Silicon pixel and strip trackers, a lead tungstate crystal electromagnetic calorimeter (ECAL), and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections, reside within the solenoid. Forward calorimeters extend the η coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionization detectors embedded in the steel flux-return yoke outside the solenoid. The data events used in the analysis correspond to the pp collisions recorded by the CMS experiment in 2016-2018. Events are required to pass several selection criteria defined at trigger level including the presence of either one, two, or three leptons (electrons or muons) [15]. The combination of these triggers yields a trigger selection efficiency close to 100% in the full phase space relevant to the presented study. In order to compare the recorded and selected data with SM predictions, dedicated sets of simulated samples are employed, with consistent modeling of the running conditions for each data-taking year. The tZq process requires the presence of a bottom quark in the initial state. This can be described using the 5FS as shown in Fig. 1, where the b quark production depends on the proton PDF. The Monte Carlo (MC) event simulation produced in the 5FS is preferable in the calculation of the total production cross sections [16]. The modeling of the kinematic properties of the particles in the final state is, however, expected to be more precise in the 4FS, where the b quark is explicitly required to be associated with the gluon-splitting process at the matrix element (ME) level [16]. This directly leads to the presence of a second b quark that is produced with relatively small p T . On the other hand, the additional vertex in the 4FS ME leads to an increased uncertainty related to the renormalization and factorization scales used in the calculation. Examples of LO and NLO Feynman diagrams corresponding to the tZq production in the 4FS are shown in Fig. 2. In the extraction procedure applied to the tZq events, the prediction of the signal process is based on the 4FS calculations and is normalized to the production cross section obtained in the 5FS. An alternative tZq signal sample is generated in the 5FS with the same generator as used in the simulation of the default tZq sample, and is compared to the unfolded results at the parton and particle levels. Figure 2: An example of the LO (left) and NLO (right) Feynman diagrams for the tZq production process in the 4FS. The Z/γ * interference term is included in the MC event simulation. q g q ′ Z/γ ⋆ t b b W g gq q ′ Z/γ ⋆ t b b W q q ′ In the case of the NLO generation, the gg-and qq-initiated processes are possible, with an additional quark or gluon present in the final state. Using MADGRAPH5 aMC@NLO (v2.4.2) [17,18], the tZq signal events are generated at NLO precision in perturbative QCD, such that processes initiated by gluon (gg) and quark (qq) pairs are included and the radiation of an additional gluon is allowed. The nonresonant dilepton production and Z/γ * interference is also included in the simulation. The same ME generator is used to simulate the dominant background processes: associated production of a W and Z boson (WZ), tt production in association with a Z (ttZ) or W (ttW) boson, production of a photon in association with a Z (Zγ) or W (Wγ) boson or with a top quark (tγ), production of three electroweak gauge bosons (VVV), and production of four top quarks (tttt). Other background processes, which are simulated at LO with MADGRAPH [17], include single top quark production in association with a Higgs boson (tHq), with a Higgs boson and an additional W boson (tHW), and with a W and Z boson (tWZ). Additional background processes considered include tt production in association with a photon (tt γ), with two electroweak gauge bosons (ttVV), with one electroweak gauge boson and a Higgs boson (ttVH), and with two Higgs bosons (ttHH). For processes with the associated production of two Z bosons (ZZ), as well as tt production in association with a Higgs boson (ttH), the POWHEG v2 [19][20][21][22] generator is used at NLO in QCD. The MCFM [23] generator (v7.0.1) is used for the MC event simulation at LO for the gluon-initiated ZZ production (gg → ZZ). In the measurement of the lepton misidentification rate (see Section 5), simulated samples with the Drell-Yan (DY) and tt production processes are used. The processes are simulated at NLO with MADGRAPH5 aMC@NLO and POWHEG v2, respectively. Simulated events are processed with PYTHIA (v8.2) [24] to model the fragmentation and the parton shower. The FxFx [18] merging scheme is used to avoid double counting associated with the MC event simulation in the same phase space due to the ME generation of extra partons at NLO with MADGRAPH5 aMC@NLO. A set of CP5 tuning parameters [25] is used for the parton shower, hadronization, and modeling of the underlying event in the 2017 and 2018 MC samples, as well as of the tZq, ttZ, ttW, ttH, tt γ, Zγ, and tttt MC samples in all three years. The CUETP8M1 [26,27], CUETP8M2, and CUETP8M2T4 [28] [31], are simulated with PYTHIA. The simulated events are reweighted according to the distribution of the number of interactions in each bunch crossing corresponding to a total inelastic pp cross section of 69.2 mb [32]. The simulation of the CMS detector is performed with GEANT4 [33]. Reconstruction and identification of physics objects To reconstruct the physics objects described below, the same algorithms are applied to simulated events and data. The particle-flow (PF) algorithm [34] is used to reconstruct and identify photons, electrons, muons, and charged and neutral hadrons in an event, with an optimized combination of information from the various elements of the CMS detector. The missing transverse momentum vector p miss T is computed as the negative vector p T sum of all the PF candidates in an event [35]. The candidate vertex with the largest value of summed p 2 T of all physics objects assigned to this vertex is taken to be the primary vertex (PV) of the pp interaction. Jets are reconstructed by clustering the PF candidates using the anti-k T algorithm [36,37] with a distance parameter of R = 0.4. Charged particles identified as originating from pileup interactions are discarded and an offset correction is applied to correct for remaining contributions. Jet energy corrections are derived from simulation to bring the measured response of jets to that of particle-level jets on average. In situ measurements of the momentum balance in dijet, photon+jet, Z+jet, and multijet events are used to account for any residual differences in the jet energy scale between data and simulation [38]. The jet energy resolution in simulation is corrected to match the one observed in data. Additional selection criteria are applied to each jet to remove jets potentially dominated by anomalous contributions from various subdetector components, or misreconstruction. Jets are required to have p T > 25 GeV, |η| < 5, and be separated from any identified lepton by ∆R = √ (∆η) 2 + (∆φ) 2 > 0.4, where ∆η and ∆φ are the pseudorapidity and azimuthal angular separation between the jet and the lepton, respectively. The relatively loose selection criterion applied to the jet |η| is necessary for reconstructing the light quark jet in the tZq process, which is predominantly produced in the forward region of the detector (see Fig. 1). Jets that are reconstructed within the acceptance of the CMS pixel detector (|η| < 2.4 for 2016, |η| < 2.5 for 2017 and 2018) are denoted as central jets. Using the DEEPJET algorithm [39][40][41], central jets containing b hadrons are identified as btagged jets. The b tagging requirement used in the analysis corresponds to a b quark jet selection efficiency of about 85% for jets with p T > 30 GeV as estimated in simulated tt events. An associated misidentification rate of 1% for jets arising from u, d, or s quarks and gluons, and 15% for jets arising from c quarks is obtained for those events. The electron momentum is estimated by combining the energy measurement in the ECAL with the momentum measurement in the tracker. The momentum resolution for electrons with p T ≈ 45 GeV from Z → ee − decays is within 1.7-4.5%. The resolution is generally better in the barrel than in the endcap region, and depends on the bremsstrahlung energy emitted by the electron as it traverses the material in front of the ECAL [42]. Electrons are selected within |η| < 2.5. Muons are reconstructed within |η| < 2.4 using drift tubes, cathode strip chambers, and resistive plate chambers. Association of muon objects to reconstructed tracks that are measured in the silicon tracker yields the relative p T resolution of 1% in the barrel and 3% in the endcaps for muons with p T up to 100 GeV, and of better than 7% in the barrel for muons with p T up to 1 TeV [43]. Leptons originating from decays of electroweak bosons are referred to as "prompt", while those originating from hadron decays, as well as misidentified leptons from jets or hadrons, are collectively referred to as "nonprompt". A strong separation between prompt and nonprompt leptons is obtained by using a set of discriminating variables based on the reconstructed properties of leptons and jets. A relative isolation variable is defined as the scalar p T sum of all PF objects inside a cone of ∆R = 0.3 around the direction of the lepton, excluding the lepton itself, divided by the p T of the lepton [44,45]. A relative isolation parameter computed with a cone size that decreases for higher lepton p T values is also used. The isolation variables are corrected for pileup effects. The reconstructed transverse and longitudinal impact parameters, as well as the signed impact parameter significance, of the tracks associated with the leptons, computed with respect to the PV position, are used to determine the consistency of the leptons originating from the PV. A number of variables discriminating between prompt and nonprompt leptons use information about the reconstructed jet with the smallest ∆R with respect to the identified lepton, requiring ∆R < 0.4. This jet is used to compute the number of charged particles matched to the jet, the ratio of the jet p T to the lepton p T , the lepton momentum projected on the transverse plane to the reconstructed jet direction, as well as the output discriminator value of the DEEPJET b tagging algorithm. In addition, the muon segment compatibility criteria [43] are used for selected muons, while the output discriminator value of the electron identification algorithm is used for electrons [42]. The aforementioned variables are combined into a multivariate analysis (MVA) based discriminant (lepton MVA), which is trained and evaluated with the TVMA package [46]. A boosted decision tree (BDT) algorithm is trained on a large sample of simulated prompt leptons originating from the tZq, ttZ, and ttW processes, as well as nonprompt leptons taken from simulated tt events. The requirement on the lepton MVA value corresponds to a prompt lepton selection efficiency of about 95%, while rejecting 98% of the nonprompt leptons, as evaluated from MC simulation for leptons with p T > 25 GeV. The lepton MVA and its training discussed here are an extension and reoptimization of a similar MVA used in the first observation of the tZq process by CMS [1]. Leptons that pass the requirement on the lepton MVA are labeled as "tight" leptons and are selected for further analysis. Leptons that fail this requirement are subjected to additional criteria, including requirements on the relative lepton isolation and the DEEPJET discriminator value of the jet that is closest to the lepton. Leptons that are either tight or satisfy those additional criteria are labeled as "loose" leptons and are used in the estimation of the nonprompt-lepton background from control samples in data (as discussed in Section 5). Event reconstruction and signal selection Selected events are required to contain exactly three tight leptons and at least two jets, of which at least one is b tagged. The three leptons, ordered according to their p T , must have a p T of at least 25, 15, and 10 GeV, respectively. Two of the three leptons are required to form a pair with electric charge of opposite sign and same lepton flavor (OSSF). Furthermore, the invariant mass of the OSSF pair must be compatible with the Z boson mass within 15 GeV. In case of an ambiguity, the OSSF pair with the mass closest to the Z boson mass is chosen. Events that satisfy the aforementioned conditions define the signal region. For tZq events with three prompt leptons the selection efficiency is about 20%. For the inclusive and differential measurements the signal region is furthermore divided into subregions based on the number of jets and b-tagged jets. In order to study the background prediction, we define dedicated control regions that are complementary to the signal region. They are discussed in detail in Section 5. A good discrimination between the tZq process and various backgrounds contributing to the signal region is achieved by using MVA techniques. In the final step of the measurement, the output score of an MVA is used in maximum likelihood fits to extract the inclusive and differential tZq cross sections. A full event reconstruction, described below, is performed to obtain a set of additional variables used as inputs for the MVA to improve the separation between signal and background events. Identified physics objects are used to compute several observables for the differential cross section measurement. The four-momentum of the neutrino in the decay of the W boson originating from the top quark is reconstructed similarly to Ref. [47]. First, the lepton that is not associated with the OSSF lepton pair, denoted as t , is assigned to the W boson. Then, a W boson mass constraint on the system of the p miss T and p T ( t ) is imposed. This leads to two solutions, or in some cases one solution, for the neutrino four-momentum. The top quark candidate is reconstructed by combining the four-momenta of the neutrino solution(s), the t , and a b-tagged jet. In the case of an ambiguity, the combination that gives the top quark candidate mass closest to the value of 172.5 GeV is chosen. A particular feature of the tZq process is the recoiling jet that is often radiated in the forward detector region. This jet is identified with an efficiency of about 86% by selecting the jet with the highest p T , excluding b-tagged jets. The most powerful discriminating variables used in the event classification are obtained from the event reconstruction and correspond to the scalar p T sum of all jets, the maximum invariant mass and maximum p T of any two-jet system, the maximum DEEPJET score of any jet, the number of jets in the event, and the |η| of the recoiling jet. Predicted distributions of these variables compared to data are shown in Fig. 3. Other discriminating variables are the reconstructed top quark mass, the invariant mass of the OSSF lepton pair, the angle between j' and the b-tagged jet associated with the top quark decay, the number of b-tagged jets, the scalar sum of the p T of all selected leptons and p miss T , and p miss T itself. The transverse mass of the reconstructed W boson (m W T ) is also included in the event classification and is defined as: m W T = 2p miss T p T ( t ) [1 − cos ∆φ],(1) where ∆φ is the difference in azimuthal angle between p T ( t ) and p miss The last bins include the overflows. The lower panels show the ratio of the data to the sum of the predictions. The vertical lines on the data points represent the statistical uncertainty in the data; the shaded area corresponds to the total uncertainty in the prediction; the gray area in the ratio indicates the uncertainty related to the limited statistical precision in the prediction. polarization angle cos(θ pol ) is defined similarly to Ref. [8] as: cos(θ pol ) = p(q ) · p( t )) | p(q )|| p( t )| ,(2) where p(q ) and p( t )) are the three-momenta of the recoiling quark and the lepton from the top quark decay, respectively. The asterisks indicate that the three-momenta are measured in the top quark candidate rest frame. The polarization P of the top quark is related to the spin asymmetry as A = 1 2 Pa , where a refers to the spin-analyzing power of the lepton associated with the top quark decay and is equal to unity in LO calculations [48,49]. The spin asymmetry A is related to the differential cross section as a function of cos(θ pol ) by: dσ d cos(θ pol ) = σ tZq 1 2 + A cos(θ pol ) .(3) Background determination Several background contributions to the signal region are studied, divided into two main categories. The first contains processes that include three genuine prompt leptons. Events in the second category contain at least one nonprompt lepton, and therefore enter the signal region by virtue of imperfect nonprompt-lepton rejection. Background contributions from the first category are modeled using the MC simulations, whereas the backgrounds from the second are estimated using a technique based on control samples in data. The production of WZ bosons is an important source of background events, especially for events with a small number of reconstructed jets or b-tagged jets. The inclusive production cross section of this process is both predicted and measured with high precision [50]. In order to validate the predictions obtained for WZ production with additional jets, a dedicated data control region is defined with similar lepton identification requirements as used in the signal region, but vetoing events containing a b-tagged jet. Additionally, p miss T > 50 GeV is required, accounting for the reconstructed missing momentum originating due to the neutrino coming from the W boson decay. Figure 4 shows the predicted jet multiplicity and m W T distributions compared to data in this control region. Good agreement in the overall normalization and shape of the presented distributions is observed. In the signal region, about 30% of the simulated WZ events have a jet containing a bottom quark. The other 70% enter the signal region because of misidentification in the heavy-flavor jet tagging algorithm, with jets originating from light quarks and c quarks each contributing about half. The modeling of the WZ process with b quarks is subject to an uncertainty that is not constrained in the control region because there is a negligible fraction of events with an additional b quark. A dedicated study of this uncertainty was performed in DY events [51], resulting in an additional uncertainty of 20% assigned to the normalization of WZ selected events containing an additional b quark in the signal region. The dominant background in the subregions with a large number of jets or b-tagged jets comes from the ttZ process. The modeling and normalization of ttZ is validated in two distinct ways. In the signal region, good separation of this process from tZq is achieved with the MVA technique. Hence, the signal region contains an implicit control region for ttZ at low MVA discriminant values. Additionally, a dedicated control region is defined that requires an event to contain four leptons, with an OSSF pair compatible with the Z boson mass. If a second OSSF pair is present, it is required not to be compatible with the Z boson mass (within 15 GeV) in order to suppress the contribution from the ZZ process. The distributions of the number of b-tagged jets in data and the prediction for this ttZ-enriched control region are shown in the lower-left plot of Fig. 4. The data exceed the prediction especially for the bin with two b-tagged jets where the ttZ contribution is large. This observed underprediction is consistent with previous measurements [51]. The background from ZZ events in the final states involving three leptons consists of events where both Z bosons decay leptonically, but one of the leptons is not reconstructed or does not satisfy the lepton selection requirements. The ZZ control region requires the presence of four leptons that are used to form two OSSF pairs, both of them compatible with the Z boson mass within 15 GeV. The distributions of the number of jets in data and the prediction for this ZZ-enriched control region are shown in the lower-right plot of Fig. 4. The Zγ process can represent a background to the signal region via conversion of the photon to an electron-positron pair in the detector material. In such a process, the converted photon may transfer a large part of its momentum to one of the two leptons. This leads to the production of one lepton of sufficient p T to pass the selection criteria, with the other lepton failing those requirements. The leptonic decay of the Z boson yields the additional two leptons needed to satisfy the three-lepton selection. The selected events for the Zγ control region must contain three tight leptons whose combined invariant mass must be compatible with the Z boson mass within 15 GeV, whereas any pair of leptons is required to fail this invariant mass constraint. With this selection, a pure Zγ region is obtained, allowing the validation of the modeling of photon conversions in the detector material. Figure 5 displays the distributions for the number of selected muons in an event (left) and the three-lepton invariant mass (right) from the data and prediction for the Zγ control region. The plots show the contributions from "external" photon conversions, where a real photon converts into a pair of (mostly) electrons from its interaction in the detector material, and so-called "internal" conversions, where a virtual photon decays into a pair of leptons. Also note that in all figures, Zγ is the major contribution to the background category labeled 'Xγ', with only minor contributions from other processes involving photons. Other sources associated with the prompt-lepton backgrounds lead to much smaller contributions and are also estimated from simulation. They mainly include tWZ, ttH, and ttW events, as well as the production of three massive electroweak bosons (VVV). The second major category of background includes those containing at least one nonprompt lepton. These arise from either tt dileptonic events or DY production with an additional selected lepton that is either misidentified from a jet or a genuine lepton from hadron decay. This contribution is estimated from data using the so-called "tight-to-loose ratio" method [45]. The key feature of this method is the measurement of the probability for a nonprompt lepton satisfying the loose-quality definition to pass the tight-selection criteria. This probability, the "misidentification rate", is measured in a kinematic region enriched in QCD multijet events. In order to estimate the nonprompt-lepton background contribution to the signal region, the measured misidentification rate is used to compute a transfer factor, which is applied to events in a region with similar selection criteria as the signal region, except that at least one of the three loose leptons is not identified as a tight lepton. In the first step of this procedure, the method is validated using simulated samples. The misidentification rate is measured in simulated QCD multijet events and applied to simulated tt and DY events. A good description of nonprompt leptons in simulation is obtained in terms of all kinematic variables used in the multivariate classifier for signal extraction, as well as the classifier output score itself (shown in Appendix A). This indicates that the method can be used Number of events / 15 GeV Number of jets to predict the nonprompt-lepton kinematic properties in tt and DY events with misidentification rates measured in QCD multijet events. Data tZq WZ Nonprompt ZZ/H γ X Multiboson Z t t )X t t( Next, the misidentification rate is measured in a multijet-enriched data sample. The event selection criteria in this measurement aim at selecting events containing nonprompt or misidentified leptons. Events are required to have exactly one loose lepton, at least one jet with p T > 30 GeV that does not overlap with the lepton within ∆R = 0.7, and p miss T < 20 GeV, in order to suppress contamination from processes containing prompt leptons. The misidentification rate is defined as the ratio of the number of events with a nonprompt lepton passing the tight selection to the number of events with a nonprompt lepton passing Number of events / 1.50 GeV and the invariant mass of the three-lepton system (right) for the data (points) and predictions (colored histograms). The contributions in the simulation from "external" photon conversions, where a real photon converts into a pair of (mostly) electrons from its interaction in the detector material, and so-called "internal" conversions, where a virtual photon decays into a pair of leptons, are shown separately. The lower panels show the ratio of the data to the sum of the predictions. The vertical lines on the data points represent the statistical uncertainty in the data; the shaded area corresponds to the total uncertainty in the prediction; the gray area in the ratio indicates the uncertainty related to the limited statistical precision in the prediction. Data tZq (ext.) γ X (int.) γ X ZZ/H Nonprompt WZ )X t t( Z t tData tZq (ext.) γ X (int.) γ X ZZ/H Nonprompt WZ )X t t( Z t t the loose selection. Prompt-lepton contamination from electroweak processes is subtracted using simulation. The misidentification rate is calculated separately for electrons and muons, and binned in p T and |η| to take into account the kinematic properties of nonprompt leptons passing the tight-lepton definition. Finally, the method is validated in data using a dedicated nonprompt-lepton control region. This region is defined similarly to the signal region, with the exception that either an OSSF pair is vetoed or the invariant mass of the OSSF pair is required to be incompatible with the Z boson mass. The nonprompt-lepton control region is defined for events with exactly one b-tagged jet and two or three jets, as these represent the events in the signal region for which the nonprompt-lepton background is sizable. The result of this validation is shown in Fig. 6, where the contribution labeled "nonprompt" is obtained using the approach described above. There is good agreement between the data and the nonprompt-lepton background prediction obtained from the control region in data. Systematic uncertainties The inclusive and differential measurements are affected by similar sources of experimental and theoretical uncertainties. The measurements follow the same approach to assessing the systematic uncertainties, modulo some small differences that are motivated by different fitting procedures. The trigger efficiency is the probability of events that pass the analysis selection criteria to satisfy any of the trigger selection requirements. This efficiency is measured in a data sample where events are required to pass a set of reference triggers uncorrelated with any of the trigger requirements used in this analysis. An efficiency consistent with 100% is measured in both simulation and data, and therefore no correction is applied. Driven by the limited statistical precision of the trigger efficiency measurement, a systematic uncertainty of 2% is assigned to cover potential differences between data and simulation. This systematic uncertainty affects all processes equally, but because of its statistical origin, it is considered to be uncorrelated across the data-taking years. 1 − 0.8 − 0.6 − 0.4 − 0.2 − 0 0.2 The integrated luminosity measured in each data-taking period is used to normalize the predictions obtained from simulation with an associated systematic uncertainty of 1.2-2.5%, which is partially correlated across the data-taking years [11,52,53]. The pileup uncertainty alters the distribution of the number of pp collisions per bunch crossing. It is estimated by varying the total pp inelastic cross section in all simulated samples by ±4.6% [32], affecting both the shape of the distributions that are fit in the signal extraction and the normalization of the predictions. This source of uncertainty is fully correlated across the data-taking years and processes. In the 2016 and 2017 data sets, a too-early response of triggers related to the ECAL led to the mistaken selection of data events from the previous bunch crossing. In order to account for this effect, the simulated events are reweighted as a function of the p T and η of selected jets and photons with the corresponding uncertainties. This source of uncertainty is fully correlated across all 2016 and 2017 simulated samples, but absent for the 2018 simulation. Several uncertainties arise from the reconstruction and identification of various physics objects. Data-to-simulation scale factors are derived to correct the efficiencies of prompt-lepton reconstruction and identification, as well as the b tagging efficiency of reconstructed jets. The scale factors are varied within the associated uncertainties, affecting both the shape and normalization of the derived predictions in simulation. The uncertainties in the scale factors are split into a statistical part originating from the finite statistical precision of the methods used to obtain them, and a systematic part originating from the methodology itself. The first part is considered to be uncorrelated between the data-taking years, while the latter part is treated as fully correlated. The four-momentum of each selected jet is varied to account for the uncertainty in the jet energy resolution and the jet energy scale [38]. These variations are consistently propagated to p miss T and the b tagging efficiency scale factors, and are considered to be partially correlated across all data-taking years and processes. An additional uncertainty, related to the unclustered energy, is taken into account by varying all unclustered contributions to p miss T within their respective resolutions and propagating these changes to p miss T . This uncertainty is considered to be uncorrelated among the data-taking years. The tight-to-loose ratio method that is used for the estimation of the nonprompt-lepton background from control samples in data was verified and no bias in the shapes of various variable distributions was observed (as shown in Appendix A and Fig. 6). Therefore, no shape uncertainty is assigned to the nonprompt-lepton background estimate from data. Based on the level of agreement in the nonprompt-lepton control regions in data, a conservative normalization uncertainty of 30% is applied to this process, which covers the discrepancies (as shown in Fig. 6). It is considered correlated across the data-taking years. Theoretical uncertainties include systematic effects associated with the renormalization and factorization scales at ME level, as well as with the PDFs used in the simulation. The former uncertainties are propagated to the final measurement by varying both scales independently up and down by a factor of two, avoiding the case where one scale is varied up while the other is varied down. The latter is propagated by reweighting the simulation using the corresponding variations in the NNPDF sets [29,30]. Both types of systematic uncertainty are treated as fully correlated across the data-taking years, but while the PDF uncertainty is also correlated across all processes, the QCD scale uncertainties are considered uncorrelated between QCDand electroweak-induced processes. These sources affect both the production cross section and the acceptance of all simulated processes. The systematic effect in the cross section is not taken into account for the ttZ, WZ, ZZ, and Zγ processes, and only the acceptance effects are considered. Instead, a global normalization uncertainty is assigned to each of these processes. An uncertainty of 10% is applied to the WZ, ZZ, and Zγ processes, which is larger than the typical uncertainty from dedicated measurements [50,54]. This covers any difference in the considered phase space and is based on the study of control regions. The ttZ cross section is measured to a precision of 8% [51]. However, there is some tension between the theoretical prediction and the measurement. In this analysis, a normalization uncertainty of 15% is assigned to the ttZ process to cover this effect. The theoretical uncertainties associated with the renormalization scales for the initial-and final-state radiations (ISR and FSR) are estimated in a similar way, independent of the other scale uncertainties, by varying the corresponding scales up and down by a factor of two. Both sources of uncertainty are fully correlated across the data-taking years, but while the FSR uncertainty is also correlated across all processes, the ISR uncertainty is treated as uncorrelated between QCD-induced (ttZ) and electroweak-induced (tZq) processes. The uncertainties related to the choice of the color-reconnection model used in the parton shower and to the underlying event tune are estimated for tZq and ttZ with additional MC samples, produced with different color-reconnection models and varied underlying event tunes, respectively [25,55]. Both are considered correlated across the data-taking years and processes. In the differential measurement, choices used in estimating the background uncertainties are slightly different with respect to the inclusive measurement to give more freedom to the differential fit. The theoretical uncertainties in PDFs, ISR, FSR, and the renormalization and factorization scales, are considered for the tZq and ttZ processes. No a priori normalization uncertainty is applied to WZ, ZZ, and Zγ events since the normalizations are kept freely floating in the fit. In addition, a normalization uncertainty of 25% is applied to the triboson processes [56]. For other rare processes involving top quarks, such as tWZ and tt in association with two additional bosons, a normalization uncertainty of 50% is assigned. The impact of the dominant sources of systematic uncertainties in the inclusive tZq cross section measurement is discussed in more detail in Section 7.2 7 Inclusive cross section measurement Signal extraction The measurement of the inclusive tZq cross section is performed by fitting the distribution of the BDT discriminant used to separate the tZq signal from the various backgrounds. A binary BDT classifier is trained using the TVMA [46] package. The discriminating input variables are similar to those used in an earlier CMS measurement of the tZq process [1]. The most powerful ones were discussed in Section 4 and shown in Fig. 3. The measurement uses a maximum likelihood fit of the predicted signal and background contributions to the data, binned in the distribution of the BDT discriminant. The likelihood L to be maximized consists of the product of bin-by-bin Poisson probabilities P for observing a given number of data events in each bin [57]: L = ∏ i P (y obs i |y exp i ) ∏ k p(θ k |θ k ),(4) where y obs σ tZq , w, θ = s i σ tZq , θ + ∑ j b i,j w j , θ ,(5) where s i is the expected number of tZq events in the ith bin, which depends on the targeted cross section σ tZq and nuisance parameters θ to describe the uncertainties in the prediction. The variable b i,j denotes the number of expected events from the jth background process in the ith bin, which depends on its normalization w j and nuisance parameters θ. The factors p(θ k |θ k ) in Eq. (4) represent the probability of obtaining a best fit valueθ k for the kth nuisance parameter, given its a priori value θ k . This probability is log-normally distributed for normalization uncertainties and has a Gaussian density for shape uncertainties. For the inclusive measurement, the signal region is subdivided into three subregions based on the number of jets and b-tagged jets: exactly 1 b-tagged jet and 2 or 3 jets (dominated by the WZ and nonprompt backgrounds), 1 b-tagged jet and ≥4 jets (dominated by the ttZ and WZ backgrounds), and ≥2 b-tagged jets (dominated by the ttZ background). The fit is performed simultaneously for all considered data-taking years and event categories. The corresponding BDT discriminant distributions are shown in Fig. 7 for both the prefit (left) and postfit (right) normalizations. The control regions discussed in Section 5 are included in the fit, which allows a better constraint on the relevant systematic uncertainties in the background processes, especially their normalizations. The distributions used in the fit are the transverse W boson mass for the WZ control region, the number of jets for the ZZ and Zγ control region, the number of b-tagged jets for the ttZ control region, and finally the total event yield for the nonprompt control region. All sources of systematic uncertainties discussed in Section 6 are treated as nuisance parameters in the fit, with a consistent treatment of all correlations between various uncertainties. Results The predicted cross section for the tZq process, where the Z boson decays to a pair of electrons, muons, or τ leptons, is σ SM tZq = 94.2 +1.9 −1.8 (scale) ± 2.5 (PDF) fb [10]. The calculation is performed at NLO in the 5FS and also includes nonresonant lepton-pair production with m > 30 GeV. The signal strength is defined as the ratio of the observed to the predicted tZq cross sections and is measured in the signal region defined in Sections 4 and 7.1. The result is: µ = σ tZq σ SM tZq = 0.933 +0.080 −0.077 (stat) +0.078 −0.064 (syst), Using the predicted cross section mentioned above, this signal strength corresponds to the measured cross section of σ tZq = 87.9 +7.5 −7.3 (stat) +7.3 −6.0 (syst) fb. Combining the statistical and systematic uncertainties in quadrature, the measured tZq cross section has a precision of 11%, which is an improvement over the previous measurements of this process [1, 2]. This is partly due to the smaller integrated luminosity of 77 fb −1 in earlier measurements. Another improvement comes from a broader definition of the signal by including events with two or more b-tagged jets, and events with at least four selected jets. The latter subregion, whose distributions are shown in the middle plots of Fig. 7, provides an important contribution to the improved sensitivity of this measurement. Additional gain comes from the improved performance of the lepton MVA and the looser selection criteria applied to this discriminant. The working point used in this analysis has a signal efficiency and background rejection of about 95% and 98%, respectively (as discussed in Section 3), whereas for the typical tighter working point, these numbers are 90% and 99%, respectively. The loosening of the lepton MVA selection criteria allowed better constraints on the relevant systematic uncertainties in the nonprompt-lepton background prediction using the dedicated control regions. The dominant systematic uncertainties affecting the inclusive cross section measurement are shown in Fig. 8. These uncertainties include the effects on acceptance from varying the renormalization and factorization scales associated with the tZq and ttZ processes, the normalization uncertainties in all the considered processes discussed in Section 6, and several other sources related to the b tagging efficiency correction, color reconnection, and parton showering. The impact of the color-reconnection model choice on the tZq signal strength is one-sided since it results from using an alternative model as opposed to a model parameter varied up and down. All sources of systematic uncertainty not shown in Fig. 8 have smaller impacts than the dominant ones discussed here. The best fit values of the nuisance parameters are all within one standard deviation of their expected values, and the measured impacts on the signal strength are generally in agreement with the corresponding expected values. The measured normalization of ttZ events is shifted by about one standard deviation with respect to the theoretical prediction, and indicates an underprediction of this background. This is expected and is consistent with the previously published results [51]. The increased contribution from ttZ events after the fit results in a good agreement of prediction to data in both the ttZ-dominated signal subregion (lower plots in Fig. 7) and the ttZ control region with four leptons. Figure 8: Summary of the dominant systematic uncertainties affecting the inclusive tZq cross section measurement. The left column lists the sources of systematic uncertainty, treated as nuisance parameters in the fit, in order of importance. In the middle column, the black points with the horizontal bars show for each source the difference between the observed best fit value (θ) and the nominal value (θ 0 ), divided by the expected standard deviation (∆θ). The right column plots the change in the tZq signal strength µ if a nuisance parameter is varied one standard deviation up (red), or down (blue). The gray, red, and blue bands display the same quantity as their corresponding markers, but using a simulated data set where all nuisance parameters are set to their expected values. Figure 9 displays some noteworthy kinematic prefit distributions in the tZq-enriched region, where in addition to the signal selection detailed in Section 4, the BDT score is required to be greater than 0.5. The number of tZq signal and background events passing this selection are estimated to be 252 and 264, respectively, implying that the contribution of the signal in this tZq-enriched region is about 49%. Furthermore, these plots show a good sensitivity to the kinematic properties of the tZq signal and a good agreement between the data and simulation, with the exception of the m(3 ) variable, as discussed further in Section 8. The imperfect simulation of reconstruction inefficiencies and detector acceptance effects (combined and labeled as "detector level" effects) could result in discrepancies between the shape and normalization of the measured distributions and the simulated ones. The good agreement between the measured and simulated distributions in the figure shows that these possible effects are relatively small in this phase space. Furthermore, the number of events in the data associated with the tZq process means that a differential cross section measurement is possible, once the detector-level effects are corrected for, as described in Section 8. In addition to the inclusive tZq cross section, the production cross sections of a top quark (tZq( + strengths for the separate top quark and antiquark cross sections are: 2 − 1 − 0 1 2 θ ∆ ) / 0 θ - θ ( Z µ tZq( + t ) = 1.02 +0.10 −0.09 (stat) +0.07 −0.06 (syst), µ t Zq( − t ) = 0.79 +0.15 −0.14 (stat) +0.09 −0.08 (syst). Using the MADGRAPH5 aMC@NLO 5FS predictions for the cross sections, these signal strengths translate into: The relative systematic uncertainty is reduced in the ratio measurement due to partial correlations. Although currently dominated by the statistical uncertainties, these results show promise for a future precise determination (at the high-luminosity LHC) of the top quark to antiquark production cross section ratio in the rare process tZq, similarly to what has already been obtained for t-channel single top quark production [58]. σ tZq( + t ) = 62.2 +5.9 −5.7 (stat) +4.4 −3.7 (syst) fb, σ t Zq( − t ) = 26.1 +4.8 Measurements of the differential cross sections and the spin asymmetry Differential tZq cross section measurements are performed as functions of several observables at the parton and particle levels, as defined in Section 8.1. The selected observables are potentially sensitive to beyond-SM effects and can provide information on the modeling of the tZq process. Observables based only on lepton kinematic properties are the transverse momenta of the Z boson, p T (Z), and the lepton from the top quark, p T ( t ), the invariant mass of the three leptons, m(3 ), and the difference in azimuthal angle between the two leptons from the Z boson decay, ∆φ( , ). Other selected variables rely on the top quark reconstruction and include the cosine of the top quark polarization angle, cos(θ pol ), and the invariant mass of the t + Z system, m(t, Z). The last two observables are the p T (j ) and |η(j )| at the particle level of the recoiling jet. A likelihood-based unfolding procedure is performed to separately measure the cross section σ tZq, k , in each kinematic region defined by each generator-level bin k ∈ {1, 2, ...}, where the generator level here corresponds to truth information at either the parton or particle levels, defined in the next section. The unfolding procedure accounts for the finite resolution and limited acceptance of the detector and, at the parton level, for hadronization effects. Compared to the inclusive measurement, not one but multiple signal parameters associated with the different generator-level bins are extracted in a multidimensional maximum likelihood fit. The first term in Eq. (5) is therefore replaced by a sum over signal contributions from the generator-level bins, s i σ tZq , θ → ∑ k s i, k σ tZq, k , θ .(6) To extract the signal in multiple kinematic regions requires not only the separation between the tZq and background processes, but also the mutual separation of the tZq contributions coming from different generator-level bins corresponding to the different kinematic regions. For this reason, an alternative categorization of events in the signal region with respect to the inclusive measurement and a more elaborate classifier have been developed. Parton-and particle-level definitions The binning of the various observables at the generator level is optimized based on a trade-off between the expected number of tZq events in each bin at the detector level, the bin width, and the stability and purity of the response matrix that relates the generator-level distributions to the detector-level distributions in the simulated tZq signal. The stability is based on all reconstructed events and defined as the fraction of events from a generator-level bin that are observed in the corresponding detector-level bin. The purity is based on all reconstructed events and defined as the fraction of events from a detector-level bin that belong to the corresponding generator-level bin. Observables associated only with leptons generally have good measurement resolution, and a total of four bins is chosen. This is also motivated by the number of events in the data set and the purity and stability values above 95%. For observables involving jets, a total of three bins is chosen to account for the poorer resolution compared to the lepton observables, and to lessen the effects from statistical fluctuations. The corresponding purity and stability values are above 55%. With this choice of binning, the application of a regularization procedure was found to be unnecessary. Examples of two response matrices, for the p T (Z) at the parton level and p T (t) at the particle level, are shown in Fig. 10. Generator-level definitions at the parton and particle levels are used. At the parton level, the measurement is performed in the full phase space of events with three prompt leptons. Partonlevel objects are defined based on event generator particles after ISR and FSR and before hadronization. The generated on-shell top quark is selected and the lepton from its decay is identi-fied. The leptons that are not associated with the decay products of the top quark are assigned to the OSSF lepton pair. The quark that recoils against the virtual W boson is identified as the quark with flavor u, d, s, or c. In the case of an ambiguity, the one with the highest p T is chosen. The particle-level definition aims at minimizing the dependency on the choice of the generator and reducing the uncertainty associated with the extrapolation to the detector level. A collection of so-called "dressed" leptons is defined through a clustering process involving prompt electrons or muons and photons that do not arise from hadronic decays using the anti-k T algorithm with R = 0.1 [36,37]. Jets are clustered from all stable (lifetime >30 ps) particles, excluding prompt leptons but including neutrinos from hadronic decays, with the anti-k T algorithm and R = 0.4. Using the ghost-clustering method [59], b hadrons are scaled to have an infinitely small momentum and included in the clustering process. A jet is labeled as a btagged jet if such a ghost b hadron is clustered inside it. The p miss T is defined as the vector p T sum of all neutrinos from W, Z, or prompt τ decays. Further selection criteria on p T , η, and ∆R(j, ) are applied to these objects, and the events are selected and reconstructed using the same requirements and algorithms as in the signal region at the detector level. The measurement at the particle level is hence performed in a fiducial phase space, leading to the fact that the absolute differential cross sections are reduced by a factor of about two with respect to the parton-level measurements. Reconstructed tZq events outside the fiducial phase space make up about 7% of all reconstructed tZq events. They are included in the signal extraction and scaled in the fit with the integrated differential cross section. Signal extraction and fit strategy In the measurement of the differential tZq cross sections, a multiclass neural network, implemented via the TENSORFLOW [60] package, is used. The neural network contains 22 input and five output nodes to distinguish tZq events from ttZ, WZ, t(t )X, and all other backgrounds. The most important input variables are shown in Fig. 3. The separation into multiple output nodes is based on the distinct nature of the different backgrounds and provides an improved isolation of tZq events compared to a binary classifier. For the signal extraction, events in the signal region (defined in Section 4) are additionally required to have fewer than four central jets. Events with four or more central jets are dominated by ttZ events and used as a control region (ttZ → 3 ). The remaining events in the signal region are split into three subregions (four for observables only involving leptons) based on the value of the observable at the detector level, using the bin ranges from the definition of the parton-and particle-level bins. As a result, the tZq contribution from each parton-and particle-level bin of the considered observable is enriched in the corresponding subregion at the detector level. To isolate the tZq events from various background events, each subregion is binned in the neural network score of the tZq output node. As an example, the prefit and postfit p T (j ) and m(3 ) distributions in the signal region are shown in Figs. 11 and 12, respectively. To further constrain the normalization of various backgrounds, additional complementary control regions with three or four leptons are defined, as described in Section 5, and included in the signal extraction. Events in these control regions are additionally required to have at least two jets to minimize the uncertainty associated with the extrapolation of the various backgrounds to the signal region. Events in the ttZ → 3 control region are fit as a function of the neural network score of the ttZ output node to separate the ttZ process, while events in the WZ control regions are fit as a function of m W T . The Zγ control region consists of two bins that are included in the fit, which are determined from the number of selected electrons. Events in the ttZ → 4 control region are separated into three bins in the fit, as a function of the number of b-tagged jets in the event. Events in the ZZ and nonprompt-lepton control regions are fit in one bin. Figure 11: Prefit (upper) and postfit (lower) distributions of the neural network score from the tZq output node for events in the signal region with fewer than four jets, used for the p T (j ) differential cross section measurement at the particle level. The data are shown by the points and the predictions by the colored histograms. The vertical lines on the points represent the statistical uncertainty in the data, and the hatched region the total uncertainty in the prediction. The events are split into three subregions based on the value of p T (j ) measured at the detector level. Three different tZq templates, defined by the same intervals of p T (j ) at the particle level and shown in different shades of orange and red, are used to model the contribution of each particle-level bin. Reconstructed tZq events that are outside the fiducial phase space are labeled as "tZq (others)" and represent a minor contribution. The lower panels show the ratio of the data to the sum of the predictions, with the gray band indicating the uncertainty from the finite number of MC events. Results The measured absolute differential cross sections are shown in Figs. 13-15. The last bin of each distribution also includes the overflow contributions. Measurement Prediction Figure 13: Absolute differential cross sections as a function of p T (Z) measured at the parton (upper left) and particle levels (upper right), as well as a function of p T (j ) (lower left) and |η(j )| (lower right) at the particle level. The observed values are shown as black points, with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for the tZq process are based on events simulated in the 5FS (green) and 4FS (blue). The p-values of the χ 2 tests are given to quantify their compatibility with the measurement. The lower panels show the ratio of the simulation to the measurement. The full covariance matrix is obtained from the fit and the normalized differential cross sections, shown in Figs. [16][17][18], are calculated by dividing each absolute differential cross section value by the sum of the values from all the bins. In general, observables that include quarks in their definition are measured to a precision of around 20-30% in each bin. For observables that are associated with leptons only the uncertainty goes down to 15% in some bins. As a cross-check, the differential cross section of each distribution is integrated, extrapolated, and compared to the result of the inclusive measurement. In all cases, the results are in good agreement within the associated uncertainties. The measured distributions are compared with theoretical predictions of tZq at NLO in QCD for the 4FS and 5FS. Uncertainties in these predictions include the effects from the ME factorization and renormalization scales, PDF, ISR, and FSR, as discussed before. For the normalization Measurement Prediction Figure 14: Absolute differential cross sections at the parton (left) and particle level (right) measured as a function of ∆φ( , ) (upper), p T ( t ) (middle) and m(3 ) (lower). The observed values are shown as black points, with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for the tZq process are based on events simulated in the 5FS (green) and 4FS (blue). The p-values of the χ 2 tests are given to quantify their compatibility with the measurement. The lower panels show the ratio of the simulation to the measurement. Measurement Prediction Figure 15: Absolute differential cross sections at the parton (left) and particle level (right) measured as a function of p T (t) (upper), m(t, Z) (middle) and cos(θ pol ) (lower). The observed values are shown as black points, with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for the tZq process are based on events simulated in the 5FS (green) and 4FS (blue). The p-values of the χ 2 tests are given to quantify their compatibility with the measurement. The lower panels show the ratio of the simulation to the measurement. of the 4FS a cross section of 73.6 ± 6.2 (scale) ± 0.4 (PDF) fb is used, as obtained from the MAD-GRAPH5 aMC@NLO generator. Although the 5FS predicts larger cross sections [61,62] as compared with the 4FS, the absolute differential cross sections measurements are compatible with both calculations within the uncertainties. The two methods yield similar results for the normalized differential cross sections, both of which are compatible with the measurement. The only exception is the normalized m(3 ) differential cross section, shown in the lower plots of Fig. 17, where neither scheme is able to describe the measurement around 175 GeV. The level of agreement between the unfolded measurements and the theoretical predictions is quantified using p-values from a χ 2 test, summarized in Table B.1, where the full covariance matrix for the measurement and each prediction is considered. Measurement Prediction Figure 16: Normalized differential cross sections measured as a function of p T (Z) at the parton (upper left) and particle level (upper right), as well as a function of p T (j ) (lower left) and |η(j )| (lower right) at the particle level. The observed values are shown as black points, with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for the tZq process are based on events simulated in the 5FS (green) and 4FS (blue). The p-values of the χ 2 tests are given to quantify their compatibility with the measurement. The lower panels show the ratio of the simulation to the measurement. Measurement Prediction Figure 18: Normalized differential cross sections measured at the parton (left) and particle level (right) as a function of p T (t) (upper), m(t, Z) (middle) and cos(θ pol ) (lower). The observed values are shown as black points, with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for the tZq process are based on events simulated in the 5FS (green) and 4FS (blue). The p-values of the χ 2 tests are given to quantify their compatibility with the measurement. The lower panels show the ratio of the simulation to the measurement. In the measurement of the spin asymmetry, the fit of cos(θ pol ) at the parton level is reparameterized according to Eq. (3) such that the spin asymmetry is directly used as a free parameter in the fit. Apart from the spin asymmetry, the total cross section is left freely floating in the fit as well, to account for the overall normalization. The spin asymmetry is measured as A = 0.54 ± 0.16 (stat) ± 0.06 (syst), compatible with SM predictions of A 4FS = 0.44 (in the 4FS) and A 5FS = 0.45 (in the 5FS) from MADGRAPH5 aMC@NLO simulations at NLO. Uncertainties in the SM predictions from ISR, FSR, PDFs, and renormalization and factorization scales at ME level were found to be negligible with respect to those of the measured value. Additional prefit and postfit results, the extracted distribution, and the likelihood as a function of the spin asymmetry are given in Appendix C. The uncertainties in both the differential cross section and spin asymmetry measurements are dominated by the statistical component. The leading systematic uncertainties come from the experimental uncertainties, including the background modeling, b tagging efficiency, and lepton identification. For measurements using observables that include quarks in their definition, the jet energy scale also represents an important source of systematic uncertainty. The leading theoretical uncertainties are associated with the renormalization and factorization scales at ME level, and FSR effects. Summary Inclusive and differential cross section measurements of single top quark production in association with a Z boson (tZq) are presented using events with three leptons (electrons or muons). The data sample for this measurement was collected by the CMS experiment at the LHC in proton-proton collisions at a center-of-mass energy of 13 TeV and corresponds to an integrated luminosity of 138 fb −1 . Including nonresonant lepton pairs, an inclusive cross section of σ tZq = 87.9 +7.5 −7.3 (stat) +7.3 −6.0 (syst) fb is obtained for dilepton invariant masses greater than 30 GeV. This result is the most precise inclusive tZq cross section measurement to date, with a relative precision about 25% better than previously published results. For the first time, the inclusive tZq cross sections are also measured separately for top quark and antiquark production, obtaining σ tZq( + t ) = 62.2 +5.9 −5.7 (stat) +4.4 −3.7 (syst) fb and σ t Zq( − t ) = 26.1 +4.8 −4.6 (stat) +3.0 −2.8 (syst) fb, respectively, with the ratio of 2.37 +0. 56 −0.42 (stat) +0.27 −0.13 (syst). The measured values compared to the theoretical predictions are summarized in Fig. 19. The differential tZq cross sections are measured for the first time at the parton and particle levels using a binned maximum-likelihood-based unfolding. The studied observables are the transverse momenta of the top quark, the Z boson, and the lepton associated with the top quark decay, as well as the invariant masses of the three leptons and the t + Z system. Also used as observables are the difference in azimuthal angle between the two leptons from the Z boson decay, the cosine of the top quark polarization angle, and, at the particle level, the transverse momentum and absolute pseudorapidity of the recoiling jet. The results are mostly compatible with the standard model predictions using both the four-and five-flavor schemes, while the sensitivity is not sufficient to show a preference for one or the other. From the differential distribution of the top quark polarization angle, the top quark spin asymmetry is measured to be A = 0.54 ± 0.16 (stat) ± 0.06 (syst), in agreement with the standard model prediction. Observed / Predicted A ℓ R (̅ tZq (ℓ − t ) tZq (ℓ + t ) ) ̅ tZq (ℓ − t ) tZq (ℓ + t ) tZq (inclusive) Measurement Theory Stat. unc. Total unc. (13 TeV) -1 138 fb Figure 19: Measured values of the inclusive tZq cross section (first row), top quark and antiquark cross sections (second and third rows) and their ratio (fourth row), together with the top quark spin asymmetry in the tZq process (last row). Each row shows the ratio between the observed and the SM predicted values. The black points show the central values, while the red and blue bars refer to the statistical and total uncertainties in the measurements, respectively. Uncertainties in the predictions are indicated by the gray bands. CMS Acknowledgments We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the success of the CMS effort. In addition, we gratefully acknowledge the computing centers and personnel of the Worldwide LHC Computing Grid and other centers for delivering so effectively the computing infrastructure essential to our analyses. Finally, we acknowledge the enduring support for the construction and operation of the LHC, the CMS detector, and the supporting computing infrastructure provided by the follow- References [1] CMS Collaboration, "Observation of single top quark production in association with a Z boson in proton-proton collisions at √ s = 13 TeV", Phys. Rev. Lett. 122 (2019) 132003, doi:10.1103/PhysRevLett.122.132003, arXiv:1812.05900. [2] ATLAS Collaboration, "Observation of the associated production of a top quark and a Z boson in pp collisions at √ s = 13 TeV with the ATLAS detector", JHEP 07 (2020) 124, doi:10.1007/JHEP07(2020)124, arXiv:2002.07546. [4] ATLAS Collaboration, "Search for anomalous couplings in the Wtb vertex from the measurement of double differential angular decay rates of single top quarks produced in the t channel with the ATLAS detector", JHEP 04 (2016) 023, doi:10.1007/jhep04(2016)023, arXiv:1510.03764. [5] CMS Collaboration, "Measurement of top quark polarisation in t-channel single top quark production", JHEP 04 (2016) 073, doi:10.1007/jhep04(2016)073, arXiv:1511.02138. [6] ATLAS Collaboration, "Probing the Wtb vertex structure in t-channel single top quark production and decay in pp collisions at √ s = 8 TeV with the ATLAS detector", JHEP 04 (2017) 124, doi:10.1007/jhep04(2017)124, arXiv:1702.08309. [7] ATLAS Collaboration, "Analysis of the Wtb vertex from the measurement of triple-differential angular decay rates of single top quarks produced in the t channel at √ s = 8 TeV with the ATLAS detector", JHEP 12 (2017) 017, doi:10.1007/jhep12(2017)017, arXiv:1707.05393. [8] CMS Collaboration, "Measurement of differential cross sections and charge ratios for t-channel single top quark production in proton-proton collisions at √ s = 13 TeV", Eur. Phys. J. C 80 (2020) 370, doi:10.1140/epjc/s10052-020-7858-1, arXiv:1907.08330. [12] HEPData record for this analysis, 2021. doi:10.17182/hepdata.105865. [13] CMS Collaboration, "Measurement of inclusive and differential Higgs boson production cross sections in the diphoton decay channel in proton-proton collisions at √ s = 13 TeV", JHEP 01 (2019) 183, doi:10.1007/jhep01(2019)183, arXiv:1807.03825. Figure A.1 shows the results of the misidentification-rate optimization and validation using the tight-to-loose ratio method, as described in Section 5. This cross-check is fully simulationbased: the misidentification rate is measured in simulated QCD multijet samples and evaluated separately in simulated DY and tt samples. The event BDT discriminant distributions are shown for (left) simulated trilepton events from the DY process, and (right) simulated tt events. The black points are the nonprompt-lepton predictions from the simulation, and the colored histograms the predictions using the tight-to-loose ratio method applied to the same simulated events. The lower panels give the ratio of the two predictions. The good agreement between the two predictions for the two different types of simulated events validates the method for determining the nonprompt-lepton background. Pred. A Validation of the misidentification-rate method in simulation MC Obs. Stat. uncertainty B Hypothesis tests of differential cross sections The level of agreement of the measured differential cross sections with the theory predictions is quantified with p-values of χ 2 tests as summarized in Table B.1. The full covariance matrix for the measurement and each prediction is considered. C Extraction of the top quark spin asymmetry Additional material on the likelihood fit used for the extraction of the spin asymmetry is given. The prefit and postfit distributions are shown in Fig. C.1 where a good agreement with the data is visible. The extracted parton-level bins and the likelihood as a function of the spin asymmetry are shown in Fig. C Figure C.1: Prefit (upper) and postfit (lower) distributions of the neural network score from the tZq output node for events in the signal region with fewer than four jets, used for the measurement of the spin asymmetry from the cos(θ pol ) distribution at the parton level. The data are shown by the points and the predictions by the colored histograms. The vertical lines on the points represent the statistical uncertainty in the data, and the hatched region the total uncertainty in the prediction. The events are split into three subregions based on the value of cos(θ pol ) measured at the detector level. Three different tZq templates, defined by the same intervals of cos(θ pol ) at parton level and shown in different shades of orange and red, are used to model the contribution of each parton-level bin. The lower panels show the ratio of the data to the sum of the predictions, with the gray band indicating the uncertainty from the finite number of MC events. Table B.1: Summary of the p-values from the χ 2 test between the unfolded measurements and theoretical predictions from the 4FS and 5FS. The test is performed on the measurements of the absolute and normalized differential cross sections at the parton and particle levels for the observables given in the first column. All numbers are given in percent. Observable D The CMS Collaboration quark polarization is linked to the polarization of the lepton from its decay and can be measured with respect to the direction of the recoiling quark. The cosine of the top quark Figure 3 : 3Distributions of the most powerful discriminating variables in the signal region for the data (points) and predictions (colored histograms), including the scalar p T sum of all jets (upper left), the maximum invariant mass of any two-jet system (upper right), the maximum DEEPJET score of any jet (middle left), the maximum p T value of any two-jet system (middle right), the number of jets in the event (lower left), and the |η| of the recoiling jet (lower right). Figure 4 : 4Distributions of the transverse W boson mass (upper left) and the number of selected jets (upper right) for the WZ-enriched control region, the number of b-tagged jets (lower left) in the ttZ-enriched control region, and the number of jets (lower right) in the ZZ-enriched control region for the data (points) and predictions (colored histograms). The lower panels show the ratio of the data to the sum of the predictions. The vertical lines on the data points represent the statistical uncertainty in the data; the shaded area corresponds to the total uncertainty in the prediction; the gray area in the ratio indicates the uncertainty related to the limited statistical precision in the prediction. Figure 5 : 5Distributions in the Zγ-enriched control region of the number of selected muons (left) Figure 6 : 6Distributions of the number of selected muons per event (left) and the output score of the multivariate classifier used for the signal extraction in the inclusive cross section measurement (right) in a control region enriched with nonprompt leptons for data (points) and predictions (colored histograms). The lower panels show the ratio of the data to the sum of the predictions. The vertical lines on the data points represent the statistical uncertainty in the data; the shaded area corresponds to the total uncertainty in the prediction; the gray area in the ratio indicates the uncertainty related to the limited statistical precision in the prediction. observed and expected numbers of events in the ith bin, respectively. The number of expected events in bin i depends on the signal and background predictions as: y exp i Figure 7 : 7Distributions of the event BDT discriminant in the signal region for data (points) and predictions (colored histograms). The results are shown for prefit (left) and postfit (right) distributions in mutually exclusive signal subregions: exactly one b-tagged jet, 2-3 jets (upper); exactly one b-tagged jet, ≥4 jets (middle); and ≥2 b-tagged jets (lower). The lower panels show the ratio of the data to the sum of the predictions. The vertical lines on the data points represent the statistical uncertainty in the data; the shaded area corresponds to the total uncertainty in the prediction; the gray area in the ratio indicates the uncertainty related to the limited statistical precision in the prediction. Figure 9 : 9Prefit distributions at the detector level of some of the important variables used in the tZq analysis from a tZq-enriched region for the data (points) and predictions (colored histograms). The selection criteria discussed in Section 4 have been used, along with the requirement that the event BDT discriminant be greater than 0.5. The variables shown are as follows: transverse momentum of the lepton associated with the decay of the top quark (upper left), number of muons in the event (upper right), reconstructed transverse momentum of the Z boson (lower left) and transverse mass of the W boson (lower right). The lower panels show the ratio of the data to the sum of the predictions. The vertical lines on the data points represent the statistical uncertainty in the data; the shaded area corresponds to the total uncertainty in the prediction; the gray area in the ratio indicates the uncertainty related to the limited statistical precision in the prediction. R = 2.37 +0.56 −0.42 (stat) +0.27−0.13 (syst). Figure 10 : 10Response matrices for p T (Z) at the parton level (left) and p T (t) at the particle level (right) for tZq events in the full and visible phase space, respectively. The expected number of reconstructed events is given for each bin. The color indicates the transition probability for an event in a generator-level bin to have a reconstructed value corresponding to a given detectorlevel bin. The efficiency times acceptance values of reconstructing events are plotted in the middle panels. The lower panels show the stability and purity values as defined in the text. The vertical bars on the points give the statistical uncertainty and the horizontal bars show the bin width. Figure 12 : 12Prefit (upper) and postfit (lower) distributions of the neural network score from the tZq output node for events in the signal region with fewer than four jets, used for the m(3 ) differential cross section measurement at the parton level. The data are shown by the points and the predictions by the colored histograms. The vertical lines on the points represent the statistical uncertainty in the data, and the hatched region the total uncertainty in the prediction. The events are split into four subregions based on the value of m(3 ) measured at the detector level. Four different tZq templates, defined by the same intervals of m(3 ) at the parton level and shown in different shades of orange and red, are used to model the contribution of each parton-level bin. The lower panels show the ratio of the data to the sum of the predictions, with the gray band indicating the uncertainty from the finite number of MC events. Parton-level ∆ϕ (ℓ , ℓ ′) [rad] Particle-level ∆ϕ (ℓ , ℓ ′) [rad] Figure 17 : 17Parton-level ∆ϕ (ℓ , ℓ ′) [rad]Particle-level ∆ϕ (ℓ , ℓ ′) [rad] Normalized differential cross sections measured at the parton (left) and particle level (right) as a function of ∆φ( , ) (upper), p T ( t ) (middle) and m(3 ) (lower). The observed values are shown as black points, with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for the tZq process are based on events simulated in the 5FS (green) and 4FS (blue). The p-values of the χ 2 tests are given to quantify their compatibility with the measurement. The lower panels show the ratio of the simulation to the measurement. ing funding agencies: BMBWF and FWF (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, FAPERGS, and FAPESP (Brazil); MES and BNSF (Bulgaria); CERN; CAS, MoST, and NSFC (China); MINCIENCIAS (Colombia); MSES and CSF (Croatia); RIF (Cyprus); SENESCYT (Ecuador); MoER, ERC PUT and ERDF (Estonia); Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRI (Greece); NK-FIA (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); MSIP and NRF (Republic of Korea); MES (Latvia); LAS (Lithuania); MOE and UM (Malaysia); BUAP, CIN-VESTAV, CONACYT, LNS, SEP, and UASLP-FAI (Mexico); MOS (Montenegro); MBIE (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal); JINR (Dubna); MON, RosAtom, RAS, RFBR, and NRC KI (Russia); MESTD (Serbia); SEIDI, CPAN, PCTI, and FEDER (Spain); MOSTR (Sri Lanka); Swiss Funding Agencies (Switzerland); MST (Taipei); ThEPCenter, IPST, STAR, and NSTDA (Thailand); TUBITAK and TAEK (Turkey); NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA). [ 3 ] 3C. Degrande et al., "Single-top associated production with a Z or H boson at the LHC: the SMEFT interpretation", JHEP 10 (2018) 005, doi:10.1007/JHEP10(2018)005, arXiv:1804.07773. [ 9 ] 9ATLAS Collaboration, "Measurement of the production cross section of a single top quark in association with a Z boson in proton-proton collisions at 13 TeV with the ATLAS detector", Phys. Lett. B 780 (2018) 557, doi:10.1016/j.physletb.2018.03.023, arXiv:1710.03659. [10] CMS Collaboration, "Measurement of the associated production of a single top quark and a Z boson in pp collisions at √ s = 13 TeV", Phys. Lett. B 779 (2018) 358, doi:10.1016/j.physletb.2018.02.025, arXiv:1712.02825. [11] CMS Collaboration, "Precision luminosity measurement in proton-proton collisions at √ s = 13 TeV in 2015 and 2016 at CMS", Eur. Phys. J. C 81 (2021) 800, doi:10.1140/epjc/s10052-021-09538-2, arXiv:2104.01927. Figure A. 1 : 1The event BDT discriminant distributions for simulated (left) trilepton DY events and (right) tt events. The black points give the nonprompt-lepton predictions from the simulation, while the colored histograms represent a similar prediction estimated with the misidentification rate method applied to the same events. The lower panels plot the ratio of the observed MC prediction to the prediction from the misidentification rate method. The vertical bars on the points show the statistical uncertainty in the observed MC distribution. The hatched bands represent the total uncertainty in the misidentification rate predictions and the shaded band its statistical component in the ratio. Figure C. 2 : 2The left plot shows the measured absolute cos(θ pol ) differential cross section at the parton level used in the extraction of the top quark spin asymmetry. The generator-level bins are parameterized according to Eq. (3), shown as a dashed black line in the plot, such that the spin asymmetry is directly used as a free parameter in the fit. The observed values of the generator-level bins are shown as black points with the inner and outer vertical bars giving the systematic and total uncertainties, respectively. The SM predictions for events simulated in the 5FS (dashed green line) and 4FS (solid blue line) are plotted as well. The hatched regions indicate the corresponding uncertainties, respectively. The lower panel displays the ratio of the MC prediction to the measurement. On the right, the negative log likelihood in the fit for the spin asymmetry A is shown when considering only the statistical uncertainties (dashed red line) or the combined statistical and systematic uncertainties (solid black line). The dotted black lines indicate the one (inner) and two (outer) standard deviation confidence intervals, respectively. Individuals have received support from the Marie-Curie program and the European Research Council and Horizon 2020 Grant, contract Nos. 675440, 724704, 752730, 758316, 765710, 824093, 884104, and COST Action CA16108 (European Union); the Leventis Foundation; the Alfred P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formationà la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the F.R.S.-FNRS and FWO (Belgium) under the "Excellence of Science -EOS" -be.h project n. 30820817; the Beijing Municipal Science & Technology Commission, No. Z191100007219010; the Ministry of Education, Youth and Sports (MEYS) of the Czech Republic; the Deutsche Forschungsgemeinschaft (DFG), under Germany's Excellence Strategy -EXC 2121 "Quantum Universe" -390833306, and under project number 400140256 -GRK2497; the Lendület ("Momentum") Program and the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the New National Excellence ProgramÚNKP, the NKFIA research grants 123842, 123959, 124845, 124850, 125105, 128713, 128786, and 129058 (Hungary); the Council of Science and Industrial Research, India; the Latvian Council of Science; the Ministry of Science and Higher Education and the National Science Center, contracts Opus 2014/15/B/ST2/03998 and 2015/19/B/ST2/02861 (Poland); the Fundação para a Ciência e a Tecnologia, grant CEECIND/01334/2018 (Portugal); the National Priorities Research Program by Qatar National Research Fund; the Ministry of Science and Higher Education, projects no. 14.W03.31.0026 and no. FSWW-2020-0008, and the Russian Foundation for Basic Research, project No.19-42-703014 (Russia); the Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia María de Maeztu, grant MDM-2015-0509 and the Programa Severo Ochoa del Principado de Asturias; the Stavros Niarchos Foundation (Greece); the Rachadapisek Sompot Fund for Postdoctoral Fellowship, Chulalongkorn University and the Chulalongkorn Academic into Its 2nd Century Project Advancement Project (Thailand); the Kavli Foundation; the Nvidia Corporation; the SuperMicro Corporation; the Welch Foundation, contract C-1845; and the Weston Havens Foundation (USA). .2. Data / Pred.1 10 2 10 3 10 4 10 Number of events / bin Data Nonprompt )X t t( γ X Multiboson ZZ/H WZ Z t t tZq[0.65, 1] tZq[0, 0.65] tZq[−1, 0] Uncertainty cos(Θ * pol. ) < 0 0 < cos(Θ * pol. ) < 0.65 cos(Θ * pol. ) > 0.65 CMS (13 TeV) -1 138 fb Prefit 0 0.1 0.4 1|0 0.1 0.4 1|0 0.1 0.4 1 Neural network score of tZq output node 0.6 0.8 1 1.2 1.4 Data / Pred. Stat. uncertainty 1 10 2 10 3 10 4 10 Number of events / bin Data Nonprompt )X t t( γ X Multiboson ZZ/H WZ Z t t tZq[0.65, 1] tZq[0, 0.65] tZq[−1, 0] Uncertainty cos(Θ * pol. ) < 0 0 < cos(Θ * pol. ) < 0.65 cos(Θ * pol. ) > 0.65 CMS (13 TeV) -1 138 fb Postfit 0 0.1 0.4 1|0 0.1 0.4 1|0 0.1 0.4 1 Neural network score of tZq output node 0.6 0.8 1 1.2 1.4 Stat. uncertainty t )), top antiquark (tZq( − t )), and their ratio R are determined. 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Gary , M. Gordon, G. Hanson , G. Karapostoli , O.R. Long , N. Manganelli, M. Olmedo Negrete, W. Si , S. Wimpenny, Y. Zhang University of California, San Diego, La Jolla, California, USA J.G. Branson, P. Chang , S. Cittolin, S. Cooperstein , N. Deelen , D. Diaz , J. Duarte , R. Gerosa , L. Giannini , J. Guiang, R. Kansal , V. Krutelyov , R. Lee, J. Letts , M. Masciovecchio , F. Mokhtar, M. Pieri , B.V. Sathia Narayanan , V. Sharma , M. Tadel, A. Vartak , F. Würthwein , Y. Xiang , A. Yagil University of California, Santa Barbara -Department of Physics, Santa Barbara, California, USA N. Amin, C. Campagnari , M. Citron , A. Dorsett, V. Dutta , J. Incandela , M. Kilpatrick , J. Kim , B. Marsh, H. Mei, M. Oshiro, M. Quinnan , J. Richman, U. Sarica , F. Setti, J. Sheplock, P. Siddireddy, D. Stuart, S. Wang California Institute of Technology, Pasadena, California, USA A. Bornheim , O. Cerri, I. Dutta , J.M. Lawhorn , N. Lu , J. Mao, H.B. Newman , T.Q. Nguyen , M. Spiropulu , J.R. Vlimant , C. Wang , S. Xie , Z. Zhang , R.Y. Zhu . Pittsburgh, Usa J Pennsylvania, S Alison, M B An, P Andrews, T Bryant, A Ferguson, C Harilal, T Liu, M Mudholkar, A Paulini, W ; J P Sanchez, W T Cumalat, A Ford, E Hassani, R Macdonald, A Patel, C Perloff, K Savard, K A Stenson, S R Ulmer, Wagner, Boulder, Colorado, USA; Ithaca, New York, USACarnegie Mellon University ; Terrill University of Colorado Boulder ; Cornell UniversityCarnegie Mellon University, Pittsburgh, Pennsylvania, USA J. Alison , S. An , M.B. Andrews, P. Bryant , T. Ferguson , A. Harilal, C. Liu, T. Mudholkar , M. Paulini , A. Sanchez, W. Terrill University of Colorado Boulder, Boulder, Colorado, USA J.P. Cumalat , W.T. Ford , A. Hassani, E. MacDonald, R. Patel, A. Perloff , C. Savard, K. Stenson , K.A. Ulmer , S.R. Wagner Cornell University, Ithaca, New York, USA . S Alexander, X Bright-Thonney, Y Chen, D J Cheng, S Cranshaw, J Hogan, J R Monroy, D Patterson, J Quach, M Reichert, A Reid, W Ryd, J Sun, P Thom, R Wittich, Zou, Fermi, Batavia, Illinois, USANational Accelerator LaboratoryAlexander , S. Bright-Thonney , X. Chen , Y. Cheng , D.J. Cranshaw , S. Hogan, J. Monroy , J.R. Patterson , D. Quach , J. Reichert , M. Reid , A. Ryd, W. Sun , J. Thom , P. Wittich , R. Zou Fermi National Accelerator Laboratory, Batavia, Illinois, USA . M Albrow, M Alyari, G Apollinari, A Apresyan, A Apyan, S Banerjee, L A T Bauerdick, D Berry, J Berryhill, P C Bhat, K Burkett, J N Butler, A Canepa, G B Cerati, H W K Cheung, F Chlebana, K F Di Petrillo, V D Elvira, Y Feng, J Freeman, Z Gecse, L Gray, D Green, S Grünendahl, O Gutsche, R M Harris, R Heller, T C Herwig, J Hirschauer, B Jayatilaka, S Jindariani, M Johnson, U Joshi, T Klijnsma, B Klima, K H M Kwok, S Lammel, D Lincoln, R Lipton, T Liu, C Madrid, K Maeshima, C Mantilla, D Mason, P Mcbride, P Merkel, S Mrenna, S Nahn, J Ngadiuba, V O&apos;dell, V. Papadimitriou, K. Pedro , C. Pena 60 , O. Prokofyev, F. Ravera , A. Reinsvold Hall , L. Ristori , E. Sexton-Kennedy , N. Smith , A. Soha , W.J. Spalding , L. Spiegel, S. Stoynev , J. Strait , L. Taylor , S. Tkaczyk, N.V. Tran , L. Uplegger , E.W. Vaandering , H.A.Madison, WI, Wisconsin, USAWeber University of Wisconsin -MadisonM. Albrow , M. Alyari , G. Apollinari, A. Apresyan , A. Apyan , S. Banerjee, L.A.T. Bauerdick , D. Berry , J. Berryhill , P.C. Bhat, K. Burkett , J.N. Butler, A. Canepa, G.B. Cerati , H.W.K. Cheung , F. Chlebana, K.F. Di Petrillo , V.D. Elvira , Y. Feng, J. Freeman, Z. Gecse, L. Gray, D. Green, S. Grünendahl , O. Gutsche , R.M. Harris , R. Heller, T.C. Herwig , J. Hirschauer , B. Jayatilaka , S. Jindariani, M. Johnson, U. Joshi, T. Klijnsma , B. Klima , K.H.M. Kwok, S. Lammel , D. Lincoln , R. Lipton, T. Liu, C. Madrid, K. Maeshima, C. Mantilla , D. Mason, P. McBride , P. Merkel, S. Mrenna , S. Nahn , J. Ngadiuba , V. O'Dell, V. Papadimitriou, K. Pedro , C. Pena 60 , O. Prokofyev, F. Ravera , A. Reinsvold Hall , L. Ristori , E. Sexton-Kennedy , N. Smith , A. Soha , W.J. Spalding , L. Spiegel, S. Stoynev , J. Strait , L. Taylor , S. Tkaczyk, N.V. Tran , L. Uplegger , E.W. Vaandering , H.A. Weber University of Wisconsin -Madison, Madison, WI, Wisconsin, USA Also at Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC 'Kurchatov Institute. K Black, T Bose, C Caillol, S Dasu, I De Bruyn, P Everaerts, F Fienga, C Galloni, H He, M Herndon, A Hervé, U Hussain, A Lanaro, A Loeliger, R Loveless, J Sreekala, A Mallampalli, A Mohammadi, D Pinna, A Savin, V Shang, V Sharma, W H Smith, D Teague, S Trembath-Reichert, W Vetens, † , Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd University. Also at INFN Sezione di PaviaWien, Austria; Alexandria, Egypt; Bruxelles, Belgium; Campinas, Brazil; Beijing, China; Beijing, China; Nova Andradina, Brazil; Nanjing, China; Iowa City, Iowa, USA; Moscow, Russia; Dubna, Russia; Cairo, Egypt; Suez, Egypt; Cairo, Egypt; West Lafayette, Indiana, USA; Mulhouse, France; Tbilisi, Georgia; Erzincan, Turkey; Geneva, Switzerland; Aachen, Germany; Hamburg, Germany; Isfahan, Iran; Cottbus, Germany; Juelich, Germany; Assiut, Egypt; Gyongyos, Hungary; Debrecen, Hungary; Debrecen, Hungary; Budapest, Hungary; Budapest, Hungary; Bhubaneswar, India; Bhubaneswar, India; Ludhiana, India, LUDHIANA, India; Solan, India; Hyderabad, India; Santiniketan, India; Mumbai, India; Hamburg, Germany; Tehran, Iran; Behshahr, Iran; Bari, Italy; Bologna, Italy; Catania, Italy; Napoli, Italy; Napoli, Italy; PERUGIA, Italy; Riga, Latvia; Mexico City, Mexico; Paris-Saclay, Gif-sur-Yvette, France; Moscow, Russia; Moscow, Russia; Tashkent, Uzbekistan; St. Petersburg, Russia; Gainesville, Florida, USA; London, United Kingdom; Moscow, Russia; Moscow, Russia; Pasadena, California, USA; Novosibirsk, Russia; Belgrade, Serbia; Sri Lanka, Nilaveli, Sri Lanka; Pavia, Italy; Athens, Greece; Lausanne, Switzerland; Zurich, Switzerland; Vienna, Austria; Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS, Annecyle-Vieux, France; Sirnak, Turkey; Nicosia, Turkey; Konya, Turkey; Istanbul, Turkey; Adiyaman, Turkey; Istanbul, Turkey; Konya, Turkey; Yozgat, Turkey; Istanbul, Turkey; Istanbul, Turkey; Kars, Turkey; Istanbul, Turkey; Ankara, Turkey; Istanbul, Turkey; Brussel, Belgium; Southampton, United Kingdom; Didcot, United Kingdom; Durham, United Kingdom; Clayton, Australia; TORINO, Italy; St. Paul, Minneapolis, USA; Karaman, Turkey; Cairo, Egypt; Bingol, Turkey; Tbilisi, Georgia; Sinop, Turkey; KAYSERI, Turkey; Shanghai, China; Doha, Qatar; Daegu, KoreaAlso at Imperial College2TU Wien ; Also at Institute of Basic and Applied Sciences, Faculty of Engineering, Arab Academy for Science, Technology and Maritime Transport ; Also at Université Libre de Bruxelles ; Also at Universidade Estadual de Campinas ; Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil 6: Also at The University of the State of Amazonas, Manaus, Brazil 7: Also at University of Chinese Academy of Sciences ; Also at Department of Physics, Tsinghua University ; Also at Nanjing Normal University Department of Physics ; Now at The University of Iowa ; Also at Joint Institute for Nuclear Research ; Also at Cairo University ; Also at Suez University ; Now at British University in Egypt ; Also at Purdue University ; Also at Université de Haute Alsace ; Also at Tbilisi State University ; Also at Erzincan Binali Yildirim University ; Also at RWTH Aachen University, III. Physikalisches Institut A ; Also at University of Hamburg ; Also at Isfahan University of Technology ; Also at Brandenburg University of Technology ; Also at Physics Department, Faculty of Science, Assiut University ; University of Debrecen ; Also at Punjab Agricultural University ; Also at Shoolini University ; Also at University of Hyderabad ; Also at University of Visva-Bharati ; Also at Sharif University of Technology ; Also at Department of Physics, University of Science and Technology of Mazandaran ; Università di Napoli Federico II ; Also at Riga Technical University ; Also at Consejo Nacional de Ciencia y Tecnología ; Also at Institute for Nuclear Research ; Now at National Research Nuclear University 'Moscow Engineering Physics Institute' (MEPhI ; Also at St. Petersburg State Polytechnical University ; Also at University of Florida ; California Institute of Technology ; University of Belgrade ; Also at Trincomalee Campus, Eastern University ; Also at National and Kapodistrian University of Athens ; Also at Universität Zürich ; Also at Şırnak University ; Also at Near East University, Research Center of Experimental Health Science ; Also at Konya Technical University ; Also at Piri Reis University ; Also at Adiyaman University ; Also at Ozyegin University ; Also at Necmettin Erbakan University ; Also at Bozok Universitetesi Rektörlügü ; Also at Marmara University ; Also at Milli Savunma University ; Also at Kafkas University ; Also at Istanbul Bilgi University ; Also at Hacettepe University ; Also at Istanbul University -Cerrahpasa, Faculty of Engineering ; Also at Vrije Universiteit Brussel ; Also at School of Physics and Astronomy, University of Southampton ; Also at Rutherford Appleton Laboratory ; Also at IPPP Durham University ; Also at Monash University, Faculty of Science ; Also at Università di Torino ; Also at Bethel University ; Also at Karamanoglu Mehmetbey University ; Also at Ain Shams University ; Also at Bingol University ; Also at Georgian Technical University ; Also at Sinop University ; Also at Erciyes University ; Also at Institute of Modern Physics and Key Laboratory of Nuclear Physics and Ionbeam Application (MOE) -Fudan University ; Also at Texas A&M University at Qatar ; Also at Kyungpook National UniversityAlso at P.N. Lebedev Physical InstituteAlso at Laboratoire dK. Black , T. Bose , C. Caillol, S. Dasu , I. De Bruyn , P. Everaerts , F. Fienga , C. Galloni, H. He, M. Herndon , A. Hervé, U. Hussain, A. Lanaro, A. Loeliger, R. Loveless, J. Madhusudanan Sreekala , A. Mallampalli, A. Mohammadi, D. Pinna, A. Savin, V. Shang, V. Sharma , W.H. Smith , D. Teague, S. Trembath-Reichert, W. Vetens †: Deceased 1: Also at TU Wien, Wien, Austria 2: Also at Institute of Basic and Applied Sciences, Faculty of Engineer- ing, Arab Academy for Science, Technology and Maritime Transport, Alexandria, Egypt 3: Also at Université Libre de Bruxelles, Bruxelles, Belgium 4: Also at Universidade Estadual de Campinas, Campinas, Brazil 5: Also at Federal University of Rio Grande do Sul, Porto Alegre, Brazil 6: Also at The University of the State of Amazonas, Manaus, Brazil 7: Also at University of Chinese Academy of Sciences, Beijing, China 8: Also at Department of Physics, Tsinghua University, Beijing, China 9: Also at UFMS, Nova Andradina, Brazil 10: Also at Nanjing Normal University Department of Physics, Nanjing, China 11: Now at The University of Iowa, Iowa City, Iowa, USA 12: Also at Institute for Theoretical and Experimental Physics named by A.I. Alikhanov of NRC 'Kur- chatov Institute', Moscow, Russia 13: Also at Joint Institute for Nuclear Research, Dubna, Russia 14: Also at Cairo University, Cairo, Egypt 15: Also at Suez University, Suez, Egypt 16: Now at British University in Egypt, Cairo, Egypt 17: Also at Purdue University, West Lafayette, Indiana, USA 18: Also at Université de Haute Alsace, Mulhouse, France 19: Also at Tbilisi State University, Tbilisi, Georgia 20: Also at Erzincan Binali Yildirim University, Erzincan, Turkey 21: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 22: Also at RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany 23: Also at University of Hamburg, Hamburg, Germany 24: Also at Isfahan University of Technology, Isfahan, Iran 25: Also at Brandenburg University of Technology, Cottbus, Germany 26: Also at Forschungszentrum Jülich, Juelich, Germany 27: Also at Physics Department, Faculty of Science, Assiut University, Assiut, Egypt 28: Also at Karoly Robert Campus, MATE Institute of Technology, Gyongyos, Hungary 29: Also at Institute of Physics, University of Debrecen, Debrecen, Hungary 30: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 31: Also at MTA-ELTE Lendület CMS Particle and Nuclear Physics Group, Eötvös Loránd Uni- versity, Budapest, Hungary 32: Also at Wigner Research Centre for Physics, Budapest, Hungary 33: Also at IIT Bhubaneswar, Bhubaneswar, India 34: Also at Institute of Physics, Bhubaneswar, India 35: Also at Punjab Agricultural University, Ludhiana, India, LUDHIANA, India 36: Also at Shoolini University, Solan, India 37: Also at University of Hyderabad, Hyderabad, India 38: Also at University of Visva-Bharati, Santiniketan, India 39: Also at Indian Institute of Technology (IIT), Mumbai, India 40: Also at Deutsches Elektronen-Synchrotron, Hamburg, Germany 41: Also at Sharif University of Technology, Tehran, Iran 42: Also at Department of Physics, University of Science and Technology of Mazandaran, Behshahr, Iran 43: Now at INFN Sezione di Bari (a), Università di Bari (b), Politecnico di Bari (c), Bari, Italy 44: Also at Italian National Agency for New Technologies, Energy and Sustainable Eco- nomic Development, Bologna, Italy 45: Also at Centro Siciliano di Fisica Nucleare e di Struttura Della Materia, Catania, Italy 46: Also at Scuola Superiore Meridionale, Università di Napoli Federico II, Napoli, Italy 47: Also at Università di Napoli 'Federico II', Napoli, Italy 48: Also at Consiglio Nazionale delle Ricerche -Istituto Officina dei Materiali, PERUGIA, Italy 49: Also at Riga Technical University, Riga, Latvia 50: Also at Consejo Nacional de Ciencia y Tecnología, Mexico City, Mexico 51: Also at IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette, France 52: Also at Institute for Nuclear Research, Moscow, Russia 53: Now at National Research Nuclear University 'Moscow Engineering Physics Insti- tute' (MEPhI), Moscow, Russia 54: Also at Institute of Nuclear Physics of the Uzbekistan Academy of Sciences, Tashkent, Uzbekistan 55: Also at St. Petersburg State Polytechnical University, St. Petersburg, Russia 56: Also at University of Florida, Gainesville, Florida, USA 57: Also at Imperial College, London, United Kingdom 58: Also at P.N. Lebedev Physical Institute, Moscow, Russia 59: Also at Moscow Institute of Physics and Technology, Moscow, Russia 60: Also at California Institute of Technology, Pasadena, California, USA 61: Also at Budker Institute of Nuclear Physics, Novosibirsk, Russia 62: Also at Faculty of Physics, University of Belgrade, Belgrade, Serbia 63: Also at Trincomalee Campus, Eastern University, Sri Lanka, Nilaveli, Sri Lanka 64: Also at INFN Sezione di Pavia (a), Università di Pavia (b), Pavia, Italy 65: Also at National and Kapodistrian University of Athens, Athens, Greece 66: Also at Ecole Polytechnique Fédérale Lausanne, Lausanne, Switzerland 67: Also at Universität Zürich, Zurich, Switzerland 68: Also at Stefan Meyer Institute for Subatomic Physics, Vienna, Austria 69: Also at Laboratoire d'Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS, Annecy- le-Vieux, France 70: Also at Şırnak University, Sirnak, Turkey 71: Also at Near East University, Research Center of Experimental Health Science, Nicosia, Turkey 72: Also at Konya Technical University, Konya, Turkey 73: Also at Piri Reis University, Istanbul, Turkey 74: Also at Adiyaman University, Adiyaman, Turkey 75: Also at Ozyegin University, Istanbul, Turkey 76: Also at Necmettin Erbakan University, Konya, Turkey 77: Also at Bozok Universitetesi Rektörlügü, Yozgat, Turkey 78: Also at Marmara University, Istanbul, Turkey 79: Also at Milli Savunma University, Istanbul, Turkey 80: Also at Kafkas University, Kars, Turkey 81: Also at Istanbul Bilgi University, Istanbul, Turkey 82: Also at Hacettepe University, Ankara, Turkey 83: Also at Istanbul University -Cerrahpasa, Faculty of Engineering, Istanbul, Turkey 84: Also at Vrije Universiteit Brussel, Brussel, Belgium 85: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 86: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 87: Also at IPPP Durham University, Durham, United Kingdom 88: Also at Monash University, Faculty of Science, Clayton, Australia 89: Also at Università di Torino, TORINO, Italy 90: Also at Bethel University, St. Paul, Minneapolis, USA 91: Also at Karamanoglu Mehmetbey University, Karaman, Turkey 92: Also at Ain Shams University, Cairo, Egypt 93: Also at Bingol University, Bingol, Turkey 94: Also at Georgian Technical University, Tbilisi, Georgia 95: Also at Sinop University, Sinop, Turkey 96: Also at Erciyes University, KAYSERI, Turkey 97: Also at Institute of Modern Physics and Key Laboratory of Nuclear Physics and Ion- beam Application (MOE) -Fudan University, Shanghai, China 98: Also at Texas A&M University at Qatar, Doha, Qatar 99: Also at Kyungpook National University, Daegu, Korea

Solid insight into the physics of the inner Milky Way is key to understanding our Galaxy's evolution, but extreme dust obscuration has historically hindered efforts to map the area along the Galactic mid-plane. New comprehensive near-infrared time-series photometry from the VVV Survey has revealed 35 classical Cepheids, tracing a previously unobserved component of the inner Galaxy, namely a ubiquitous inner thin disk of young stars along the Galactic mid-plane, traversing across the bulge. The discovered period (age) spread of these classical Cepheids implies a continuous supply of newly formed stars in the central region of the Galaxy over the last 100 million years.

10.1088/2041-8205/812/2/l29

1509.08402

c6f8a7ab5b1e64c2c409ea7108c75e1598cd5fab

The VVV Survey reveals classical Cepheids tracing a young and thin stellar disk across the Galaxy's bulge 28 Sep 2015 I Dékány Instituto Milenio de Astrofísica SantiagoChile Instituto de Astrofísica Facultad de Física Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 4860SantiagoChile D Minniti Instituto Milenio de Astrofísica SantiagoChile Departamento de Física Facultad de Ciencias Exactas Universidad Andres Bello República 220SantiagoChile Vatican Observatory V00120Vatican City StateItaly D Majaess Saint Mary's University HalifaxNova ScotiaCanada Mount Saint Vincent University HalifaxNova ScotiaCanada M Zoccali Instituto Milenio de Astrofísica SantiagoChile Instituto de Astrofísica Facultad de Física Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 4860SantiagoChile G Hajdu Instituto Milenio de Astrofísica SantiagoChile Instituto de Astrofísica Facultad de Física Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 4860SantiagoChile J Alonso-García Instituto Milenio de Astrofísica SantiagoChile Unidad de Astronomía Universidad de Antofagasta Av. U. de Antofagasta 02800AntofagastaChile M Catelan Instituto Milenio de Astrofísica SantiagoChile Instituto de Astrofísica Facultad de Física Pontificia Universidad Católica de Chile Av. Vicuña Mackenna 4860SantiagoChile W Gieren Instituto Milenio de Astrofísica SantiagoChile Departamento de Astronomía Universidad de Concepción 160-CCasilla, ConcepciónChile J Borissova Instituto Milenio de Astrofísica SantiagoChile Instituto de Física y Astronomía Universidad de Valparaíso Av. Gran Bretaña 1111ValparasoChile The VVV Survey reveals classical Cepheids tracing a young and thin stellar disk across the Galaxy's bulge 28 Sep 2015Subject headings: Galaxy:generalGalaxy:bulgeGalaxy:diskGalaxy:stellar con- tentstars: variables: Cepheids Solid insight into the physics of the inner Milky Way is key to understanding our Galaxy's evolution, but extreme dust obscuration has historically hindered efforts to map the area along the Galactic mid-plane. New comprehensive near-infrared time-series photometry from the VVV Survey has revealed 35 classical Cepheids, tracing a previously unobserved component of the inner Galaxy, namely a ubiquitous inner thin disk of young stars along the Galactic mid-plane, traversing across the bulge. The discovered period (age) spread of these classical Cepheids implies a continuous supply of newly formed stars in the central region of the Galaxy over the last 100 million years. Introduction The inner Milky Way is dominated by a peanut-shaped bulge (McWilliam & Zoccali 2010;Wegg & Gerhard 2013) that flares up from a prominent Galactic bar (Nakada et al. 1991), and its structural and kinematical properties are consistent with a formation scenario driven by the instabilities of a multi-component stellar disk (Ness et al. 2013;Zoccali et al. 2014;Gardner et al. 2014;Ness et al. 2014;Di Matteo et al. 2015). The majority of its constituent stars are very ( 8 billion years) old (Ortolani et al. 1995;Zoccali et al. 2003;Brown et al. 2010), implying that its formation occurred at early epochs of the Milky Way's evolution. Although intermediate-age stars are also present in the bulge (van Loon et al. 2003;Bensby et al. 2013), their origin, nature and ubiquity are poorly understood, owing partly to sizable age uncertainties and possible biases from small sample sizes and contamination by foreground disk stars. Yet it is well known that the innermost core (R 300 pc) of the Galaxy hosts stars with ages ranging from a few million to several billion years (e.g., Serabyn & Morris 1996;van Loon et al. 2003;Matsunaga et al. 2011). This "nuclear bulge" has rather distinct physical properties from the rest of the bulge and is related to the Central Molecular Zone, but neither its transition to the surrounding bulge regions nor the triggering process for star formation and its history are well understood (see, e.g., Launhardt et al. 2002). Our global picture of the inner Galaxy is linked primarily to observations of the outer bulge, i.e., at Galactic latitudes higher than ∼ 2 • . That deficit arises from extreme obscuration by interstellar dust, high source density, and confusion with foreground disk populations. The properties of the bulge region at low latitudes have thus remained largely unexplored, leaving ambiguities concerning the interplay of the nuclear bulge and the various components of the boxy bulge and surrounding outer stellar disk. The VISTA Variables in the Vía Láctea (VVV) ESO Public Survey (Minniti et al. 2010) presents a means to ameliorate the situation by opening the time-domain in the near-infrared. That is particularly important as classes of variable stars, such as classical Cepheids, are indicators of young stellar populations (e.g., Catelan & Smith 2015), and can yield critical insight as demonstrated here. Discovery and Characterisation of the Cepheids We performed a comprehensive near-infrared variability search using VVV Survey data collected between 2010 and 2014, in a ∼ 66 square degree area in the central bulge (−10.5 • l +10 • , −1.7 • b +2 • , aligned with VVV image borders). We excluded two fields lying toward the nuclear bulge owing to extreme source crowding. Time-series photometry for ∼ 10 8 point sources were analyzed, with up to 70 K s -band measurements per object, together with color information from 1-5 independent measurements in the Z, Y , J, and H bands. The data processing, calibration, and light-curve analysis were conducted in the same way as in our previous study on the Twin Cepheids beyond the bulge (Dékány et al. 2015), and are based on standard VVV Survey data products (Minniti et al. 2010;Irwin et al. 2004;Catelan et al. 2013). We detected a sample of approximately 3 · 10 5 objects which displayed putative light variations, and scanned the results for Cepheids possessing pulsation periods in the range of 4-50 days. The lower limit mitigates confusion between the pulsation modes (e.g., Macri et al. 2015), while the upper limit was constrained by the sampling of the photometric time-series. We found 655 fundamental-mode Cepheid candidates based on their periods, amplitudes, and asymmetric light-curve shapes. A large fraction of these objects were only detected in the H and K s bands given the extreme reddening. For computing the distances and extinctions of the Cepheids, we employed the periodluminosity (PL) relations (a.k.a. Leavitt Law, Monson et al. 2012;Matsunaga et al. 2009) adopted in our previous study (Dékány et al. 2015), adjusted to match the latest and most accurate distance modulus for the Large Magellanic Cloud (LMC, Pietrzyński et al. 2013). The color excess was converted to an extinction using a selective-to-absolute extinction ratio found towards the Galactic Center (GC, Nishiyama et al. 2009), once converted into the VISTA photometric system (see Dékány et al. 2015). For each object, a distance and extinction were computed under both the assumptions that the target is a classical and a type II Cepheid. We subsequently identified the correct solution and class upon further analysis. Following the approach adopted in our previous study (Dékány et al. 2015), uncertainties tied to the distance and extinction were estimated by Monte Carlo simulations. The distances exhibit 1-3% precision and 8-10% accuracy. The latter is dominated by the uncertainty in the adopted extinction curve. The systematic uncertainty might be larger, possibly up to ∼ 20%, if variations in the extinction's wavelength dependence, either as a function of Galactic longitude or along the sight-line, significantly exceed those found previously toward the inner bulge (Nishiyama et al. 2009). Type II Cepheids, i.e. old, low-mass, He-shell burning pulsating stars in the classical instability strip (Catelan & Smith 2015), populate the same range of periods as classical Cepheids. The two Cepheid classes share similar near-infrared light-curves (see, e.g. Catelan & Smith 2015;Matsunaga et al. 2011;Dékány et al. 2015). Since the bulge contains a sizable population of type II Cepheids with a centrally concentrated distribution (Soszyński et al. 2011), our sample is expected to be dominated by these objects. Therefore, additional observational information is required to identify classical Cepheids in our sample. The critical information is conveyed by an extinction map of the bulge, based on the analysis of its red clump stars (Gonzalez et al. 2012). Classical and type II Cepheids have similar intrinsic colors but rather different luminosities (Monson et al. 2012;Matsunaga et al. 2009), and consequently their PL relations yield similar extinctions but very different distances when applied to the same object. A Cepheid's type can be determined if its distance and extinction are inconsistent with the cumulative extinction up to the bulge in its direction, computed under one of the two possible assumptions for its type (Matsunaga et al. 2011;Dékány et al. 2015). Figure. 1a-b show the difference between the extinction values predicted by the PL relations and the bulge extinction map, normalized by its uncertainty as a function of distance, when extinction and distance are computed under the assumption that all detected Cepheids are either classical or type II. There are several Cepheids in our sample for which the computed extinction was within 3σ agreement with, or higher than the predicted cumulative extinction up to the bulge towards their sight-lines, but for which the type II assumption yields short distances (3 − 4 kpc), making them bona fide classical Cepheid candidates. While most of them reside in the disk at the far side of the bulge as expected, we found 35 stars (green points in Fig. 1) that are likewise located within the bulge volume (see, e.g., Wegg & Gerhard 2013). These 35 stars could be type II Cepheids only if they were located by chance beyond thick dust clouds of small angular sizes (< 1 ), otherwise the 2 ×2 resolution of the VVV reddening map would be sensitive to those. To investigate this unlikely possibility, we examined the color images and analyzed the color-magnitude diagrams (CMD's) of the stellar fields around these Cepheids, in search of anomalously high gradients in the foreground extinction. In all cases, the CMDs are inconsistent with the presence of such nearby foreground clouds, down to angular sizes of ∼ 10 , where the surface density of the disk population becomes too low for drawing such conclusion. That limiting angular scale would correspond to physical sizes of 0.2 pc, smaller than a typical Bok globule (Das et al. 2015). The probability for such chance alignment is negligible, and that it affects all 35 objects is virtually zero. Consequently, we rule out the possibility that they are nearby type II Cepheids in the Galactic disk, and classify them as classical Cepheids located inside the bulge volume. Their light-curves are presented in Fig. 2, while Fig. 3 conveys their spatial distribution. Their properties are summarized in Table 1. By reversing the arguments above, 425 new type II Cepheids are identified in the central bulge ( Fig. 1a-b, red points), with extinctions consistent with predictions from the VVV reddening map (Gonzalez et al. 2012). If these stars were classical Cepheids, they would be located far beyond the bulge, and their corresponding extinctions would be inconsistent with such large distances. The detailed study of these objects and the rest of the new classical Cepheids beyond the bulge will be presented in forthcoming papers. To bolster the classification of the classical and type II Cepheids in our sample, the distributions of their amplitudes and positions were compared. The middle panel of Fig. 1 displays the period-amplitude diagram of these objects with previously known classical and type II Cepheids. Although these parameters do not allow us to unambiguously separate the two types, particularly at shorter periods, the locus of the newly discovered classical Cepheid population agrees with that of other classical Cepheids in the Local Group. Furthermore, the right panel of Fig. 1 compares the spatial distributions of classical and type II Cepheids identified in this study. While the type II objects are expectedly concentrated towards the GC (Soszyński et al. 2011), the classical Cepheids are rather evenly distributed. The lack of classical Cepheids toward small longitudes, and the slightly offset peak from zero longitude in the density of type II Cepheids, arise from the central gap in the coverage (cf. Fig. 3). The sample of 35 new classical Cepheids is incomplete owing to the bright magnitude limit of the VVV Survey. Brighter longer-period Cepheids (Monson et al. 2012) can be saturated in the VVV images in the absence of significant foreground extinction. The asymmetry in the classical Cepheids' observed number density distribution, with more objects at the far side of the bulge (Fig. 3), is an observational bias owing to that saturation limit. The typical K s limit where saturation cannot be calibrated out is ∼ 11 mag. The faint detection limit towards the central bulge is K s ∼ 15.5 mag. These estimates are time-varying, and depend on the seeing, sky transparency, and crowding. Figure 4 conveys the detection threshold in period-extinction parameter space for classical Cepheids, at various distances. Short-period objects near the 4 d lower limit are detected throughout the entire range of observed extinction. However, nearly half of our sample would be overlooked in the absence of significant reddening if they were located at the near side of the bulge. Consequently, the asymmetry in the objects' observed spatial distribution, with more Cepheids further away (Fig. 3), is an observational bias owing to the saturation limit of the VVV Survey. On the other hand, the observed small vertical spread of the classical Cepheids cannot be attributed to a similar observational bias. While long-period classical Cepheids further from the Galactic plane and occupying the bulge would remain undetected by the VVV Survey, more than half our sample would still be detected if they were located at considerably higher Galactic latitudes. That is apparent when comparing the distributions of Fig. 4 and the mean bulge extinction as a function of Galactic latitude in the VVV extinction map (Gonzalez et al. 2012, see also Fig. 3, lower panel). Bulge Cepheids exhibiting periods less than 15 days would be detected up to latitudes of 1 • at most longitudes, while those with periods below 8 days would be found at latitudes typically up to 2 • . The inner thin disk The 35 new classical Cepheids inside the bulge volume span across the entire longitudinal range of our study and lie in close proximity to the Galactic mid-plane. The standard deviation of their vertical distances from it is only 22 pc, with the farthest Cepheid being only 82 pc above it. This vertical distribution is less than estimates of relatively nearby classical Cepheids (Majaess et al. 2009), whereby the latter exhibit a scale height of ∼ 75 pc. All classical Cepheids discovered are younger than 100 Myr, since a Cepheid's pulsation period is closely linked to its age (Bono et al. 2005). The youngest Cepheid observed may be ∼ 25 million years old (Bono et al. 2005). We cannot exclude the possible presence of even younger and brighter Cepheids, which would be saturated in the VVV Survey. Our objects thus trace an underlying young and thin inner stellar disk along the Galactic plane. Although their census may be incomplete, it is noteworthy that both short-and long-period Cepheids were detected, while in the nuclear bulge only Cepheids with periods close to 20 days are known. The period spread implies that they originate from continuous star formation along the mid-plane in the central Galaxy over the last ∼ 100 million years. A limiting factor that hinders a detailed analysis of the spatial distribution of Cepheids in the inner Galaxy is the uncertainty in the wavelength dependence of interstellar extinction, i.e., the properties of interstellar dust particles along the sight-line. However, the bulk of the classical Cepheids in our sample remain within 3 kpc of the GC if we consider different extinction curves (e.g., Cardelli et al. 1989;Fitzpatrick 1999), even if they have not been observed to be valid towards the inner bulge. For instance, adopting the "standard" Galactic extinction curve (Cardelli et al. 1989) would shift the mean Cepheid distance ∼ 1.5 kpc closer, and only 2 objects would become further than 3 kpc from the GC (in the near disk). The discovery of a thin star-forming inner disk across the bulge is immune to the effects of potentially anomalous extinction. Cepheid extinction estimates derived from PL relations were compared to the VVV extinction map (Gonzalez et al. 2012). The comparison is nearly split between values that generally agree, and Cepheids that exhibit significant (> 3σ) positive deviations from this map's predictions. These anomalies may originate from localized absorbing material rather than scale height differences between the two populations. Since bulge red clump stars are detected to small angular radii (∼ 10 ) around all the Cepheids, most of the material should be distributed inside the bulge's volume and along the sight-lines of objects with extreme extinction. Several of these Cepheids are in the angular vicinity ( 30 ) of ionized hydrogen bubbles, embedded young stellar objects, or extended mid-infrared objects visible in images of the GLIMPSE (Churchwell et al. 2009) surveys (see Table 1). Conclusion The presence of numerous classical Cepheids demonstrate that the central Galaxy contains a very young (< 100 Myr old) stellar population spanning well outside the nuclear bulge. Our results transcend earlier evidence for intermediate-age stars in the bulge volume (e.g., van Loon et al. 2003;Bensby et al. 2013) by exploring low-latitude regions where studies of stellar ages were unobtainable. A hitherto unobserved component of the inner Galaxy was identified, namely a young inner thin disk along the Galactic mid-plane. The confined vertical extent, together with the wide longitudinal breadth of the Cepheids' spatial distribution, suggest that this stellar disk has a smooth transition from both the nuclear bulge and the Galactic thin disk that encompasses the bulge region. The findings discussed here are in general agreement with recent numerical results concerning the ages of stellar constituents residing in the inner Galaxy. In simulated disk galaxies forming stars from cooling gas inside a dark matter halo, a sizable population of young and metal-rich stars emerged in close proximity to the plane, within a radius of ∼ 2 kpc from the centers. Having a median age of ∼ 1 Gyr and a large age dispersion, these simulated stars show evidence that the presence of young stars in an old bulge can follow naturally from a model galaxy's evolution (Ness et al. 2014). Yet continued investigations are needed to assess whether the Cepheids in the inner disk were born in situ, possibly from star formation stemming from gas inflow towards the GC (e.g., Kormendy & Kennicutt 2004), or if they originate from further out, either being formed at the bar ends (Phillips 1996) or captured from the outer disk by a growing bar (Martinez-Valpuesta et al. 2006). Constraining the Cepheids' orbits by kinematical measurements would supply critical information for resolving this point. Another important question is whether the young stars in the inner disk evolve dynamically as old bulge stars once did, and thus will eventually flare into the boxy outer bulge. Such a secular process could supply younger metal-rich stars to the predominantly old outer bulge (Bensby et al. 2013). Given that disk galaxies with small bulges such as ours are common, understanding their stellar components' fundamental properties, interactions, and coevolution is key in our quest to understand galaxy evolution as a whole. We gratefully acknowledge the use of data from the ESO Public Survey program 179.B-2002, taken with the VISTA telescope, and data products from the Cambridge Astronomical Survey Unit. The authors acknowledge the following funding sources: BASAL CATA PFB-06, the Chilean Ministry of Economy's ICM grant IC120009, FIC-R Fund (project 30321072), CONICYT-PCHA (Doctorado Nacional 2014-63140099), CONICYT Anillo ACT 1101, and FONDECYT projects 1141141, 1130196, 3130552, 1120601, and 1150345. Facilities: ESO:VISTA. (Gonzalez et al. 2012) divided by the uncertainty, as a function of the distance. Error bars show 1σ uncertainty ranges. Green points: the 35 classical Cepheids inside the bulge volume; red points: new type II bulge Cepheids with solid classifications; black points: rest of the sample. Black points at short distances in this figure represent candidate classical Cepheids beyond the bulge, while those at larger distances are type II Cepheid candidates outside the bulge or objects with uncertain classification. Panel b: As panel a, but showing values computed under the assumption that all Cepheids are of the classical type. Panel c: K s total amplitude vs log P diagram of classical Cepheids from the LMC (Persson et al. 2004) and the Galactic field (Monson & Pierce 2011) (blue crosses), previously known type II Cepheids in the bulge (Soszyński et al. 2011, K s amplitudes derived from VVV data) and in Galactic globular clusters (Matsunaga et al. 2006) (red triangles), and new type II Cepheids from our study (orange triangles). Green points show the new classical Cepheids in the inner Galaxy. Panel d: Histograms of the discovered classical and type II Cepheid sample as a function of Galactic longitude. Orange symbols mark the Twin Cepheids (Dékány et al. 2015) beyond the bulge, yellow symbols represent the classical Cepheids in the nuclear bulge (Matsunaga et al. 2011(Matsunaga et al. , 2015. The Galactic bar (Gonzalez et al. 2012) is outlined by a red curve, whereas the red "x" marks the GC (Dékány et al. 2013). The solid and dashed cyan lines denote the longitudinal limits of our study and the direction of the gap in coverage towards the GC, respectively. The bottom panel shows the classical Cepheids' positions in Galactic coordinates, overlaid on the VVV bulge K s extinction map (Gonzalez et al. 2012). Black lines denote the borders of the surveyed area. Dashed lines delineate the faint detection limits, whereas solid lines convey saturation limits for Cepheids at three different distances, illustrating the distance to the GC (Dékány et al. 2013) and the corresponding near and far edges of the bulge (Wegg & Gerhard 2013). Blue points mark the classical Cepheids discovered here, blue circles show the values of classical Cepheids in the nuclear bulge (Matsunaga et al. 2011(Matsunaga et al. , 2015. Fig . 1.-The selection of classical Cepheids. Panel a: Under the assumption that all Cepheids in our sample are of type II, this figure shows the deviation of their computed extinction relative to the VVV reddening map Fig. 2 . 2-K s -band phase diagrams of the classical Cepheids in the inner Galaxy (on arbitrary magnitude scale). Curves denote Fourier series fits to the data, identifiers are shown on the left. Fig. 3 . 3-Top panel : positions of the classical Cepheids projected onto the Galactic plane and their errors (green symbols), overlaid on Robert Hurt's illustration of the Milky Way. Fig . 4.-Approximate VVV Survey detection limits, showing the range of pulsation periods and foreground K s extinction where classical Cepheids in the inner Galaxy can be detected. Table 1 . 1Catalog of the classical Cepheids. Note. -Errors are given in parentheses. a tot. : Ks total amplitude; d: Heliocentric distance; A Ks : Ks extinction; A Ks,B : extinction from Gonzalez et al. (2012); T1/T2: quantities under the assumptions that the object is classical or type II Cepheid, respectively. Objects in the immediate vicinity of the Cepheids: [1]: DOBASHI 7236 (dark nebula); [2]: [CWP2007] CN 35 (HII bubble); [3]: Rahman & Murray (2010) SFC02 star forming complex; [4]: [CWP2007] CS 96 (HII bubble); [5]: SSTGLMC G003.9104+00.0010 (young stellar object) * A Ks,T 1 has > 3σ positive deviation from A Ks ,B .Known ionozed hydrogen bubble close to the Cepheid.Elevated diffuse mid-infrared emission around the Cepheid at 8-and 24-µm GLIMPSE images.Extended bright object close to the Cepheid in 8-and 24-µm GLIMPSE images.Young stellar object(s) close to the Cepheid.Obj. RA Dec. Period a tot. d T 1 A Ks ,T 1 d T 2 A Ks ,T 2 A Ks,B σA Ks,RC Ks H − Ks J − Ks Note [J2000.0] [J2000.0] [days] [mag.] [kpc] [mag.] [kpc] [mag.] [mag.] [mag.] [mag.] [mag.] [mag.] 1 17:22:23.24 -36:21:41.5 7.5685 0.22 10.99 (0.96) 2.07 4.40 (0.40) 1.94 1.44 0.21 12.01 (.02) 1.31 (.07) 3.68 (.03) d 2 17:21:16.05 -36:43:25.2 6.3387 0.21 9.97 (1.02) 2.59 4.11 (0.43) 2.47 1.59 0.20 12.57 (.02) 1.62 (.10) 4.65 (.07) 3 17:20:14.62 -37:11:16.0 5.0999 0.20 9.55 (0.91) 1.84 4.07 (0.40) 1.72 1.21 0.22 12.04 (.02) 1.16 (.08) 3.21 (.04) 4 17:26:34.72 -35:16:24.1 13.4046 0.23 9.57 (0.77) 2.88 3.49 (0.28) 2.74 2.02 0.21 11.70 (.02) 1.81 (.05) 5.24 (.09) 5 17:25:29.70 -34:45:45.9 12.3267 0.29 10.23 (0.80) 2.78 * 3.78 (0.30) 2.64 1.79 0.12 11.86 (.02) 1.75 (.04) 5.01 (.07) d 6 17:26:43.41 -34:58:25.6 9.8383 0.22 9.99 (1.15) 4.50 * 3.83 (0.44) 4.37 1.77 0.40 13.86 (.02) 2.80 (.11) . . . 7 17:26:54.24 -35:01:08.2 4.2904 0.20 10.79 (0.95) 3.23 * 4.74 (0.44) 3.11 1.64 0.32 13.93 (.02) 2.01 (.05) . . . 8 17:30:46.64 -34:09:04.4 4.7272 0.13 9.46 (0.84) 3.13 * 4.09 (0.38) 3.01 1.15 0.21 13.41 (.02) 1.95 (.06) . . . 9 17:28:15.86 -34:32:27.2 7.0235 0.22 8.50 (0.75) 3.78 * 3.44 (0.31) 3.65 1.67 0.15 13.27 (.02) 2.35 (.06) . . . e,[1] 10 17:38:42.96 -31:44:55.7 11.9663 0.28 10.56 (0.83) 2.81 3.92 (0.31) 2.67 1.61 0.22 12.01 (.03) 1.77 (.04) 4.38 (.09) 11 17:36:44.46 -32:04:38.6 8.3220 0.17 6.58 (0.65) 4.40 * 2.59 (0.26) 4.27 1.31 0.33 13.09 (.03) 2.74 (.07) . . . 12 17:40:41.72 -30:48:46.9 10.6634 0.22 9.89 (0.90) 2.91 3.74 (0.34) 2.78 1.73 0.27 12.13 (.04) 1.83 (.07) 5.26 (.08) 13 17:40:25.15 -31:04:50.5 4.7004 0.16 7.92 (0.69) 2.19 3.42 (0.31) 2.07 1.79 0.22 12.10 (.04) 1.38 (.06) 3.94 (.05) 14 17:42:20.00 -30:14:50.7 23.9729 0.42 9.09 (0.91) 4.16 * 3.02 (0.29) 4.01 1.51 0.25 12.05 (.04) 2.61 (.07) . . . d 15 17:41:15.13 -30:07:17.7 13.3262 0.33 9.31 (1.00) 2.84 * 3.40 (0.36) 2.70 1.22 0.19 11.62 (.04) 1.79 (.10) 5.26 (.14) d 16 17:51:05.72 -26:38:18.3 6.3496 0.17 9.62 (0.81) 2.95 3.96 (0.34) 2.82 1.77 0.34 12.85 (.03) 1.84 (.05) 5.26 (.07) dey 17 17:51:13.77 -26:48:55.9 12.9488 0.25 10.25 (0.90) 3.11 * 3.76 (0.33) 2.96 1.41 0.27 12.13 (.03) 1.95 (.06) . . . e 18 17:49:41.42 -27:27:14.6 10.4762 0.20 9.81 (0.81) 2.25 3.72 (0.31) 2.12 1.46 0.35 11.48 (.03) 1.43 (.06) 4.03 (.04) 19 17:50:30.49 -27:13:46.7 12.6433 0.29 8.28 (0.72) 3.70 * 3.05 (0.26) 3.56 1.59 0.55 12.30 (.03) 2.32 (.05) . . . 20 17:53:16.07 -26:28:26.9 5.9995 0.21 7.57 (0.65) 3.62 * 3.15 (0.28) 3.50 1.35 0.36 13.08 (.03) 2.25 (.05) . . . 21 17:52:21.66 -26:31:19.3 11.9921 0.27 9.73 (0.85) 3.23 * 3.61 (0.31) 3.09 1.69 0.29 12.25 (.03) 2.02 (.06) 5.29 (.08) d 22 17:51:43.80 -26:31:11.3 5.3407 0.15 7.78 (0.68) 3.57 * 3.30 (0.30) 3.45 1.47 0.27 13.25 (.03) 2.22 (.05) . . . b,[2] 23 17:55:44.68 -25:00:30.2 6.7911 0.20 9.95 (0.88) 2.77 * 4.05 (0.37) 2.64 1.38 0.32 12.64 (.03) 1.73 (.07) 5.01 (.10) 24 17:58:26.68 -23:52:08.3 4.5517 0.16 9.10 (0.87) 3.12 * 3.96 (0.39) 3.00 1.22 0.23 13.37 (.02) 1.94 (.07) 5.44 (.22) 25 18:03:31.12 -22:21:14.0 10.9622 0.16 8.70 (0.71) 3.55 * 3.28 (0.27) 3.41 1.92 0.13 12.45 (.02) 2.22 (.04) . . . dey 26 18:09:14.03 -20:03:21.4 24.8683 0.42 7.28 (0.72) 5.64 * 2.40 (0.23) 5.49 0.83 0.24 13.00 (.02) 3.52 (.04) . . . de,[3] 27 17:22:10.10 -36:44:18.8 14.8719 0.33 9.55 (0.95) 2.77 * 3.43 (0.34) 2.63 1.47 0.28 11.44 (.02) 1.75 (.09) 5.19 (.05) b,[4] 28 17:26:00.10 -35:15:15.0 18.2396 0.21 8.80 (0.71) 2.88 3.05 (0.24) 2.73 2.39 0.07 11.09 (.02) 1.82 (.04) 5.21 (.10) 29 17:32:14.07 -33:23:59.5 9.9061 0.21 9.86 (0.82) 1.69 3.78 (0.32) 1.56 1.41 0.19 11.01 (.02) 1.08 (.07) 2.98 (.06) 30 17:40:51.51 -30:24:53.2 7.9311 0.17 9.75 (0.94) 2.11 3.87 (0.38) 1.97 1.60 0.23 11.72 (.04) 1.33 (.09) 3.70 (.06) 31 17:40:24.58 -31:01:32.9 17.3711 0.32 10.57 (0.93) 2.50 3.70 (0.32) 2.36 1.94 0.29 11.18 (.04) 1.59 (.06 ) 4.62 (.09) 32 17:50:17.54 -27:08:13.3 7.0503 0.14 10.22 (0.88) 2.47 4.14 (0.37) 2.34 1.62 0.52 12.35 (.03) 1.55 (.06) 4.44 (.06) 33 17:55:24.20 -25:30:22.3 15.4472 0.37 10.24 (0.88) 2.69 3.65 (0.31) 2.54 1.44 0.33 11.46 (.03) 1.70 (.06) 5.14 (.05) 34 17:54:40.25 -25:34:39.5 17.1620 0.41 8.82 (0.78) 3.17 * 3.09 (0.27) 3.02 1.42 0.30 11.47 (.03) 2.00 (.06) 5.86 (.09) y,[5] 35 17:56:01.96 -25:15:44.9 13.4540 0.25 9.78 (0.85) 2.55 * 3.57 (0.31) 2.41 1.14 0.20 11.42 (.03) 1.61 (.07) 4.59 (.04) b d e y . 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It is of crucial importance to be able to identify the location of atmospheric pollution sources in our planet. Global models of atmospheric transport in combination with diverse Earth observing systems are a natural choice to achieve this goal. It is shown that the ability to successfully reconstruct the location and magnitude of an instantaneous source in global chemical transport models (CTMs) decreases rapidly as a function of the time interval between the pollution release and the observation time. A simple way to quantitatively characterize this phenomenon is proposed based on the effective -undesired-numerical diffusion present in current Eulerian CTMs and verified using idealized numerical experiments. The approach presented consists of using the adjoint-based optimization method in a state-of-the-art CTM, GEOS-Chem, to reconstruct the location and magnitude of a realistic pollution plume for multiple time scales. The findings obtained from these numerical experiments suggest a time scale of 2 days after which the accuracy of the adjoint-based optimization methodology is compromised considerably in current global CTMs. In conjunction with the mean atmospheric velocity, the aforementioned time scale leads to an estimate of a length scale of about 1700km, downwind from the source, beyond which measurements, in conjunction with current global CTMs, may not be successfully utilized to reconstruct continuous-in-time sources. The approach presented here can be utilized to characterize the capabilities and limitations of adjoint-based optimization inversions in other regional and global Eulerian CTMs. arXiv:1311.6315v1 [math.NA]

1311.6315

46b2b4023dbebabc01c82c8e92f13d97c50c8bc5

Quantifying the loss of information in source attribution problems using the adjoint method in global models of atmospheric chemical transport Mauricio Santillana School of Engineering and Applied Sciences Harvard University CambridgeMassachusetts Quantifying the loss of information in source attribution problems using the adjoint method in global models of atmospheric chemical transport Generated using V3.2 of the official AMS L A T E X template-journal page layout FOR AUTHOR USE ONLY, NOT FOR SUBMISSION! It is of crucial importance to be able to identify the location of atmospheric pollution sources in our planet. Global models of atmospheric transport in combination with diverse Earth observing systems are a natural choice to achieve this goal. It is shown that the ability to successfully reconstruct the location and magnitude of an instantaneous source in global chemical transport models (CTMs) decreases rapidly as a function of the time interval between the pollution release and the observation time. A simple way to quantitatively characterize this phenomenon is proposed based on the effective -undesired-numerical diffusion present in current Eulerian CTMs and verified using idealized numerical experiments. The approach presented consists of using the adjoint-based optimization method in a state-of-the-art CTM, GEOS-Chem, to reconstruct the location and magnitude of a realistic pollution plume for multiple time scales. The findings obtained from these numerical experiments suggest a time scale of 2 days after which the accuracy of the adjoint-based optimization methodology is compromised considerably in current global CTMs. In conjunction with the mean atmospheric velocity, the aforementioned time scale leads to an estimate of a length scale of about 1700km, downwind from the source, beyond which measurements, in conjunction with current global CTMs, may not be successfully utilized to reconstruct continuous-in-time sources. The approach presented here can be utilized to characterize the capabilities and limitations of adjoint-based optimization inversions in other regional and global Eulerian CTMs. arXiv:1311.6315v1 [math.NA] Introduction Understanding and quantifying the fate of anthropogenic and natural emissions of chemically active gases in the atmosphere is an important endeavour in our changing climate. In particular, many efforts have been directed to the quantification of large-scale spatial and temporal variations of sources and sinks of gases such as carbon dioxide, methane and nitrous oxide, due to their direct role in changing the radiative properties of the atmosphere, see Kasibhatla et al. (2000); Enting (2002); Ciais et al. (2010) and the multiple references therein. Having reliable estimates of these sources and sinks is of crucial relevance in the evaluation of global policies such as the Kyoto protocol, designed to curb emissions of green-house gases. Global chemical transport models (CTMs) have been designed to simulate the dynamics of the concentration fields of chemicals under the influence of atmospheric transport and chemical reactions. In conjunction with Earth observing systems such as satellite retrievals and monitoring stations, CTMs provide a natural modeling framework to estimate the strength and location of such chemical sources in a top-down fashion. In this context, CTMs are expected to have the capability of identifying pollu-tion sources within the time scales of relevance to global atmospheric chemical transport events: weeks, for intercontinental transport of pollutants, and months to a year, for inter-hemispheric transport. It is shown in this study that modeling frameworks that use the adjoint-based optimization method to identify the strength and location of pollution sources, as implemented in current CTMs, may not achieve their goal in relevant time scales. Our ability to reconstruct a pollution source using global CTMs depends mainly on three factors: (i) the quality of our measurement systems, (ii) the appropriateness of our mathematical model and our inversion approach, and (iii) our ability to numerically approximate the mathematical model and the inversion approach using computers (Enting 2002). In this study, it is assumed that a perfect observation system is in place (no noise) at every location in the atmosphere, thus ignoring errors coming from (i), and it is also assumed that the mathematical model (equations (1)) approximates well the dynamics of the atmosphere, thus ignoring the issue of the appropriateness of the mathematical model in (ii). Specifically, the efficacy of the adjoint-based optimization methodology is studied in a practical and realistic computational framework (iii). Eulerian global CTMs simulate the chemical composition of the atmosphere by numerically solving a set of coupled partial differential equations of the form, ∂C i ∂t + u · ∇(C i ) = R i + s i (x, t)(1) where C i is the concentration of chemical species i, u is the wind velocity field (obtained from a global atmospheric circulation model), R i is the effective chemical production rate (typically a function of the mass fractions of other chemicals), and s i describes local emissions and nonchemical sinks. A source attribution problem in a global CTM can be stated as follows, given a set of observations of the state of the atmosphere (typically concentration fields) and atmospheric wind fields for a time period T = t f − t 0 > 0, find the best emission and deposition rates, s i (x, t), for the time period T , such that when they are utilized as parameters in the CTM, the simulated chemical concentrations C i (t) are consistent with observations. Finding the solution (the best emission and depostion rates) to this inverse problem is achieved by minimizing the misfit between observations and model results, frequently using gradientbased iterative optimization methods. The adjoint-based optimization method studied here, is a widely accepted methodology to calculate the gradients needed to identify the direction of steepest descent in the minimization of a cost function that typically quantifies the misfit between model simulations and measurements. This methodology has been widely utilized in the reconstruction of emission sources using global CTMs, see for example (Henze et al. 2007;Zhang et al. 2009;Henze et al. 2009;Kopacz et al. 2009Kopacz et al. , 2010Zhang et al. 2011;Wecht et al. 2011). In practical terms, the calculation of the gradient at each iteration using the adjoint is achieved by integrating the CTM forward-in-time and back-in-time with reversed winds (Talagrand and Courtier 1987). Thus, the efficacy of the adjoint-based optimization methodology depends directly on the properties of the numerical schemes utilized in the CTM, along with the properties of atmospheric flow. In particular, the presence of numerical errors in Eulerian advection schemes utilized to integrate equations (1) cause the true solution of equations (1) and the (computational) numerical approximation to the solution, to diverge as the integration time t increases (LeVeque 2002). This numerical divergence also takes place between the true adjoint and the numerical calculation of the adjoint, and thus, in the approximation of the gradient to be utilized in the minimization process. As a consequence, it is expected that the efficacy of the numerical solution of source attribution problems, using the adjoint-based optimization method, will lead to unreliable source reconstructions for large enough simulation times. It is shown in this work that unreliable source reconstructions may happen in global Eulerian CTM simulation times of about 2 days, a considerably short time scale in the context of global atmospheric dynamics (compare it to a week, a typical time scale for inter-continental transport of pollution plumes). An idealized accident-type source reconstruction problem is utilized to investigate this and in general the capabilities and limitations of the adjoint-based optimization methodology. The aforementioned time scale in combination with the mean atmospheric velocity leads to a length scale of about 1700km (or about 3 grid boxes downwind), downwind from the source, beyond which measurements may not be successfully utilized to solve source attribution problems in current global CTMs. A simple way to estimate this crucial time scale is proposed here for any regional or global Eulerian CTM, based on the "effective" numerical diffusion present in the model, which happens to be much greater than the numerical diffusion determined by the order of the advection scheme, due to the chaotic properties of atmospheric flow. This paper is organized as follows, the idealized source attribution problem is introduced in section 2, the adjointbased optimization method, as well as the technical details of its numerical implementation are presented in section 3. The numerical results of our computational experiments along with their implications in real-world applications, including data assimilation methodologies, are discussed in section 4. A source attribution problem An idealized accident-type release source attribution problem for an inert gas is studied here in order to evaluate the efficacy of the adjoint-based methodology to reconstruct local-in-time and space perturbations. Thus, from here on R i = 0 in equations (1). An observation system capable of providing exact (synthetic) observations of the atmosphere everywhere at time t f (obtained directly from a forward simulations using the CTM) is assumed. The problem consists of finding the best field of instantaneous emissions at time t = t 0 , s i (x, t 0 ), such that the misfit between observations and the simulation results, at time t f , is minimized. The problem is simplified by placing the instantaneous release at a vertical height of approximately 4 km above the ground at time t = 0. This choice minimizes the effects of planetary boundary layer mixing processes, as represented (by sub-grid parametrizations) in global CTM (Lin et al. 2008). By having the same number of observations, p (all grid boxes in the domain at t = t f ), as the number of discrete instantaneous sources to be de-termined, N (all grid boxes in the domain at t = 0), the usual ill-posed nature of the inversion problem caused by the under-determination of parameters when p N in real-life Earth observations systems, is removed (Bocquet 2005b,a). This fact, in combination with the numerical stability provided by the use of the numerical approximation of the continuous adjoint (see section 3) as a means to calculate the gradients in the minimization process, makes it possible for the idealized experiments under examination to be solved without the need of prior information about the plume or a regularization technique (Bocquet 2005b,a). The aforementioned problem is solved numerically, using a three dimensional state-of-the-art CTM, GEOS-Chem (Bey et al. 2001). GEOS-Chem is a state-of-the-art CTM, driven by GEOS-5 analyzed meteorological data from the NASA Global Modeling and Assimilation Office (GMAO). The GEOS-Chem adjoint-based optimization was developed by (Henze et al. 2007). In order to produce synthetic observations, an inert tracer plume is propagated for different simulation times in GEOS-Chem, with 4 • x 5 • and 2 • x 2.5 • horizontal resolutions, using the high-order advection scheme native of GEOS-Chem (Lin and Rood 1996). The true initial plume concentration was set to be twice that of the constant global background in all other grid cells, with a horizontal extent of approximately 1330 km × 1680 km (3 x 3 grid cells), and a vertical extent of one vertical pressure level at 4km (20th pressure level) above the ground. The horizontal location of the initial plume is shown in Figures 1 and 2 (Top left plot). Through successive iterations, the adjoint-based optimization methodology generates an "optimized" initial condition estimate. The maximum number of iterations was set to 99 to keep our experiments within the practical limit set by our computational resources. We performed six numerical experiments for each of the two spatial resolutions, assimilating data after 3, 12, 24, 48, 96, and 168 hours. Since any continuous emission field in time can be thought of as a sum of instantaneous releases, the structural results of our numerical investigation are valid in constant emission flux reconstructions and more generally, in data assimilation frameworks that use Eulerian CTMs and the adjoint-based optimization method to estimate source locations and/or magnitudes. The idealized numerical problem studied here is called an "identical twin" experiment in the inverse modeling community (Kaipio and Somersalo 2004), since synthetic observations are produced using the same forward model utilized in the inversion approach. It is well known that assessing the performance of inverse modeling strategies using an identical twin experiment leads to optimistic results and constitutes what the inverse modeling community calls an "inverse crime" (Kaipio and Somersalo 2004). Indeed, in identical twin experiments, the exact solution of the minimization problem lives in the parameter space (the space of initial conditions in our specific problem) of the modeling framework, and thus, with an appropriate methodology one should be able to recover the optimal solution. From this perspective, the idealized numerical experiments studied here assess only the algorithmical capabilities and limitations of the adjoint-based inversion methodology, as implemented in CTMs, and can be interpreted as optimistic since the further complications of real-life source reconstructions are ignored. The adjoint-based optimization method Inverse modeling and data assimilation approaches involve finding an optimal set of model parameters that best match observations (Enting (2002); Bennet (2002); Wunsch (2006)). In order to find this optimal set of parameters (in our source attribution problem we find the best initial condition), one minimizes the distance or misfit between the model results and observations. This minimization can be performed using a gradient based approach. As shown in (Talagrand and Courtier 1987) the "adjoint equations" of the model can be used to compute explicitly the functional derivative (or the gradient) of the distance function, between observations and model results, with respect to the initial conditions. The computation of one gradient requires one forward-in-time integration of the full model equations over the time interval on which the observations are available, followed by one backward-in-time integration of the adjoint equations. Subsequent gradients required for a steepest descent algorithm in a minimization routine are computed in a similar fashion. When R i = 0 in equations (1), the forward model becomes a linear transport problem with sources and sinks. The adjoint operator of the transport operator is again a linear transport operator with reversed winds and reversed time integration (Bocquet 2005a;Hourdin et al. 2006). A subtle point arises when implementing the adjoint-based optimization method numerically, since one faces the dilemma of whether using the adjoint of the discrete approximation to the transport operator or the discretization of the continuous adjoint (Vukicevic et al. 2001;Bocquet 2005b,a;Hourdin et al. 2006;Liu and Sandu 2008). This issue has been previously studied not only in the context of atmospheric chemistry modeling (Gou and Sandu 2011), but in the context of oceanography and weather modeling (Sirkes and Tziperman 1997). While the adjoint of the discrete approximation of the transport may produce better pointwise results in sensitivity analyses, it can be very unstable numerically and potentially lead to non-physical sensitivity results, specially when using stabilized high-order advec- tion schemes that are not time symmetric (Hourdin et al. 2006). The superior numerical stability and global convergence properties of the numerical approximation to the continuous adjoint make it a preferred choice in practical implementations of atmospheric transport. As a consequence, the adjoint operator is calculated using the same transport routine as in the forward simulations (Vukicevic et al. 2001;Hourdin et al. 2006;Henze et al. 2007;Gou and Sandu 2011) and thus, the efficacy of this inversion approach inherits the capabilities and limitations of the transport routine. a. Numerical diffusion versus atmospheric eddy diffusion Numerical errors in transport routines of Eulerian global CTMs have been shown to be significant, compromising the quality of simulation results, at spatial resolutions larger than 1 • × 1 • even with high-order advection schemes (Wild and Prather, 2006). Moreover, the decay and spatial broadening observed in Eulerian global CTM simulations of realistic pollution plumes signal the presence of excessive numerical diffusion. This undesired numerical diffusion is shown to be much higher than expected by the order of the advection scheme, and shown to be nearly governed by the theoretical upper limit, set by the value of the local finite-time Lyapunov exponent, almost independently of the grid-box size, in Rastigeyev et al. (2010) for cur- rently practical spatial resolutions. As a consequence, high values of numerical diffusion are observed in regions with high values of finite-time Lyapunov exponents, i.e. in regions where wind shear is high and where chaotic mixing (and thus where loss of information) is pronounced. This excessive numerical diffusion will eventually compromise our ability to recover the plume information (location and magnitude) in the multiple iterations of the adjoint-based optimization process. Some amount of diffusion is to be expected in any atmospheric transport simulation. Indeed, turbulent mixing, consequence of the multi-scale chaotic velocity field in the atmosphere, can be thought of and is often parametrized as an eddy diffusion term in meso-scale Eulerian atmospheric simulations and represented by stochastic perturbations on the velocity field in Lagrangian global CTMs (Gifford 1982;Sillman et al. 1990;Mauzerall et al. 1998;Pisso et al. 2009). Studies combining Lagrangian trajectories and aircraft measurements of actual passive tracer plumes have placed the values of this eddy diffusivity around 10 4 m 2 s −1 in the horizontal, and 0.3 − 1 m 2 s −1 in the vertical (Pisso et al. 2009). In Eulerian global CTMs, however, the values of the "effective" horizontal numerical diffusion are estimated to be approximately 10 5 m 2 s −1 (Pisso et al., 2009), an order of magnitude larger than those estimated by aircraft studies. Specifically, this horizontal numerical diffusion in GEOS-Chem can be estimated 1 to be about 3.6 × 10 6 m 2 s −1 , using equation (13) in (Rastigeyev et al., 2010) for realistic plumes in the mid-latitudes of sizes of ∼ 1000 km. This value is in fact two orders of magnitude larger than those reported in aircraft studies which is why horizontal eddy diffusion terms are not considered, as subgrid parametrizations, in Eulerian global CTMs. Studying the effect of this excessive numerical diffusion on source attribution problems is the main motivation of this study. The effective (numerical) equations governing the dynamics of a pollution plume in global Eulerian CTMs can be incorporated in the mathematical model (1) as: ∂C i ∂t + u · ∇(C i ) = ∇ · (D h ∇(C i )) + R i + s i (x, t) (2) where the "effective" numerical diffusivity matrix D h is, in fact, a function of the grid size, ∆x, the time step, ∆t, the characteristics of the flow field given by u(x, t), and the spatial extent of the pollution plume with respect to the grid size (LeVeque 2002; Majda and P. 1999;Rastigeyev et al. 2010). Numerical Results A gradual process to investigate the impact of the effective numerical diffusion in the solution of the aforementioned idealized source attribution problems was designed. First, one iteration of the adjoint-based optimization was performed in order to identify the geographic area of influence, i.e. where information came from, as calculated by the numerical advection scheme in our numerical experiments for multiple time-scales. Since no prior knowledge of the pollution plume was assumed, the first meaningful iteration of the adjoint-based optimization consists of utilising the resulting concentration field of the forwardly advected idealized plume as an initial condition, and integrate it back-in-time using the advection scheme, with reversed winds, for exactly the same time period t f − t 0 = t f . In the absence of numerical errors, and due to the linearity of the transport operator in equations (1), this procedure should reconstruct the location and magnitude of the original plume at time t = 0. a. Geographic region of influence and loss of information The obtained results from this single iteration of the adjoint-based optimization, displayed in Figures 1 and 2 for spatial resolutions 4 • × 5 • and 2 • × 2.5 • respectively, show the numerical reconstruction of the geographic region 1 For this estimate we assumed that the decay rate for a spatial resolution of 4 • × 5 • is governed by the Lyapunov exponent λ, thus α ≈ 9 −5 sec −1 , the plume width W ∼ 10 6 m, and the characteristic length scale where the concentration of the plume decays to zero r b ≈ ∆x = 4 × 10 5 m. of influence, A h (t 0 ), also called the numerical sensitivity region, at ∼ 4 km altitude (at time t = 0). Note that for a simulation time t f = 3h, the numerical region of influence coincides with the original region of the plume in both spatial resolutions. As the simulation time t f (time between release and observations) increases, the numerical sensitivity region broadens more and more (the boundary of this geographic region was drawn where the reconstructed source was 100 times smaller than the original excess concentration). The broadening -a direct consequence of the numerical diffusion-is slightly anisotropic reflecting the underlying structure of the atmospheric winds characterized by the jet stream in mid-latitudes. Using estimates of the local finite-time Lyapunov exponent for mid-latitudes in GEOS-Chem from Rastigeyev et al. (2010), for 4 • × 5 • and 2 • × 2.5 • spatial resolutions, we can infer approximate values of D h ∼ 3.5 × 10 6 m 2 s −1 and D h ∼ 1.4 × 10 6 m 2 s −1 for the numerical horizontal diffusion for each resolution respectively, in the chosen geographic location of the idealized plume. Assuming horizontal isotropic diffusion only, an estimate of the "diffusive deformation" of the numerical region of influence can be calculated by adding the length scale √ D h t to each the boundary lengths of the plume at time t. We can thus expect that the area of the numerical region of influence will have increased its size nearly four-fold in 4 • ×5 • simulations and three-fold in 2 • ×2.5 • , after a day of forward simulation (and a day of back integration), since the tracer concentration will have spread ∼ 1000 km on each of the four boundaries of the plume. As shown in the Sensitivity Area (1 day) panel of Figures 1 and 2, this estimate captures well the behaviour of the numerical results. For two-days and four-days forward simulation times, this estimate suggests a 4.5-fold and 6-fold (for 4 • × 5 • simulations) and 4-fold and 5.4-fold (for 2 • × 2.5 • simulations) broadening of the numerically reconstructed plume, respectively. In fact, the introduced deformation estimates of the area of the numerical region of influence appear to be a lower limit of the actual one shown in our numerical experiments in the panels of Figure 1. The disagreement, however, is not significant. Moreover, it is shown in the next section that this estimates of the -undesired-numerical spatial deformation of the original plume can be used to estimate the loss of information of the full adjoint-based optimization. While this estimate of the deformation of the numerical sensitivity region, using only (spatially and time averaged) horizontal isotropic broadening, may appear incomplete (local finite-time Lyapunov exponent values, from which we estimated the spatially and time averaged diffusivity constant in the diagonal of D h , change as a function of time and region of space), it seems appropriate when cal- culating the effective horizontal broadening of the horizontal component of the boundaries of the plume (and thus the deformed area of the plume) and it is consistent with the findings shown in Figure 3 of (Rastigeyev et al. 2010), where 2D plume decay estimates (and thus plume broadening estimates) approximate well the plume decay even in 3D simulations in GEOS-Chem. Note that when the reconstructed numerical area of influence differs significantly from the original area of influence, as is the case for t > 2 days (Figures 1 and 2), the gradient information provided by the adjoint operator is of poor quality and as a consequence, the adjoint-based optimization fails to produce substantial improvements in subsequent iterations (Figures 11 and 10). This statement is investigated quantitatively in the following section. b. Full adjoint-based optimization results In order to quantify the effects of the aforementioned numerical diffusion on the full adjoint-based optimization, the adjoint framework in GEOS-Chem was used to produce an optimal reconstruction of the plume's location and magnitude through successive iterations. The best reconstructed plumes produced by the optimization algorithm (Zhu et al. 1994) after 99 iterations are shown, for both spatial resolutions, in Figures 3 and 4 for the 20th pressure level, and in Figures 5 and 6 for a normalized zonallyintegrated side-view. The total number of iterations was considered appropriate based on the limits of computational resources in practical situations. For a time simulation of t f = 3 hr, the adjoint-based optimization produced excellent reconstructions for both spatial resolutions. Note however, that as the total simulation time increased in our simulations, the quality of the reconstructions decreased rapidly. This is shown in the multiple panels in Figures 3, 4, 5, and 6. Quantitatively speaking, since the adjoint-based optimization method is formulated in the inner-product space of square integrable functions, it is natural to use the induced L 2 -norm to evaluate the performance of the method. In Figure 7 we show the relative L 2 -error between the true initial plume and the reconstructed plume as a function of simulation time for both spatial resolutions. For short simulation times, the reconstructions are excellent. As the simulation time increases, the relative error increases very rapidly. In fact, according to this metric, relative errors above 70% are incurred in both resolutions in 2-day simulations. c. Estimation of the loss of information A simple calculation to estimate of the relative error observed in the adjoint-based source reconstructions is proposed here based on the horizontal deformation of the numerical region of influence, due to undesired numerical diffusive processes, as a function of time. Estimates of the numerical broadening of the area of influence, for GEOS-Chem, were discussed in section a. In the approach proposed here, it is assumed that the loss of information (specifically our inability to reconstruct the initial condition accurately) increases as the numerically reconstructed area of influence, A h (t 0 ), spreads when compared to the original area of the plume A(t 0 ). Thus, if the numerical area of influence has broadened two-fold (A h (t 0 ) = 2A(t 0 )), it is assumed that one can only recover 50% of the original information (of the location and magnitude of the initial condition) through successive iterations, leading to a 50% relative error in the adjoint-based reconstruction, calculated as relative error(%) = 100 × (1 − A(t 0 )/A h (t 0 )). If the numerical area of influence has tripled (A h (t 0 ) = 3A(t 0 )), then one can only recover 33% of the information, leading to a 66% relative error, and so on. This estimate is plotted in Figure 7. The loss of information estimates for both 4 • ×5 • and 2 • ×2.5 • spatial resolutions are very similar and thus only one representative curve was plotted. Figure 7 illustrates that this simple calculation captures well the salient features of the experimental results obtained in the numerical adjoint-based reconstructions. Note that for simulations of less than 2-days, the adjoint-based iterative process produces better results than the proposed estimate. For longer simulation times the estimate seems appropriate if not slightly optimistic. The quality of the gradient information provided by the adjoint operator is more accurate, as discussed before, for shorter simulations. As a consequence, the BFGS gradientbased minimization routine utilized in GEOS-Chem (Zhu et al. 1994), is capable of producing excellent results by decreasing the normalized value of the cost function by 15 orders of magnitude for the 4 • × 5 • , 3 hr simulations, and nearly by 14 orders of magnitude for the 2 • × 2.5 • , 3 hr simulations. This is shown in Figures 10 and 11. As the simulation time increases though, the gradient information loses quality rapidly and leads to significantly poorer performances in the minimization processes. Indeed, for 7-day simulations, the minimization only achieves an 8-order of magnitude reduction of the normalized cost function, leading to very poor reconstructed plumes. The center of mass and total mass relative errors were calculated, between the reconstructed plume and the true plume, for each spatial resolution and simulation period, in order to provide a more complete characterization of the numerical errors. The results are shown in Figures 8 and 9. For these calculations, an error in the center of mass equal to the maximal radius of the plume (1680 km) was considered a 100% relative error. Errors above 100% are observed in simulations longer that 2 days. The total mass relative error does not reach very high values, for 7-day simulations these are of the order of 20 − 30%, as shown in Relative L 2 -error (%) between the reconstructed initial condition and the true solution as a function of the simulation time for both 4 • × 5 • and 2 • × 2.5 • spatial resolutions. The estimate on the loss of information is obtained assuming that information is lost proportionally to the excess area of the numerical area of influence when compared to the true area of influence. Figure 9, due to the conservative properties of the advection scheme. In simulations longer than 2 days, the total plume mass was consistently over-estimated. In all of the numerical experiments unphysical negative concentrations in the reconstructed plume were not observed. Figures 3, 4, 5, and 6 show that the reconstructed plume is deformed both in horizontal and the vertical components. This is due to vertical atmospheric convective processes represented as sub-grid parametrization in GEOS-Chem. d. Relevance in continuous-in-time source reconstructions In the context of reconstruction of continuous-in-time sources or top-down flux source attribution problems, one way to interpret the implications of the local-in-time findings presented here, is to calculate the product of the time scale, associated with a given relative error upper limit, and the mean flow velocity, to find the distance between the continuous sources and observation sites, beyond which measurements may not successfully be used for the reconstruction of the magnitude and location of the source in Eulerian CTMs. This reconstruction will be compromised since the signal to noise ratio will be smaller and smaller in subsequently further observation sites. In the experiments presented here, assuming a mean atmospheric flow of 10 m s −1 , and a 2 day time scale, the resulting length scale for which measurements may not be successfully used to recon- struct a source is about 1700 km, equivalent to 3 grid boxes in a 4 • × 5 • simulation. This means that if a continuous source is to be successfully reconstructed using the adjointbased optimization method in current global CTMs, then a continuous observation system, closer than 3 grid cells (in a 4 • × 5 • simulation), should be in place down wind from the source. Moreover, the aforementioned length scale naturally suggests a spatial density over the globe below which sources may not be accurately reconstructed in global Eulerian CTMs in practical spatial resolutions. Depending on the tracer, this spatial density of observation sites may or may not be realistic with current Earth observing systems. Conclusions It is shown that the ability to reconstruct the location and magnitude of an accident-type, instantaneous inert pollution plume, using the adjoint-based optimization method in a global Eulerian CTM, decreases rapidly as the the time between release and observations increases. The obtained numerical results suggest a time scale of 2 days after which significant numerical errors (> 70%) compromise the ability of the adjoint-based optimization method to track information accurately back to the pollution source. This time scale is shorter than the time scales of inter-continental transport for example, for which it would be desirable to have a reliable methodology to reconstruct sources. It is shown that the quality of the gradient utilized to minimize the misfit between observations and simulation results, used by the adjoint-based optimization method, decays fast as a function of simulation time. As a consequence of this fact, a simple way to quantitatively characterize the loss of information in the adjoint-based optimization is proposed based on the -undesired-deformation of the numerical area of influence caused by the effective diffusion in the CTM (which is much larger than expected by the order of the advection scheme due to the chaotic nature of atmospheric flow). This characterization of the loss of information is shown to successfully describe the behavior of the numerical relative error observed between the true solution and the numerical reconstructions, as a function of simulation time. The approach presented in this study exhibits a structural (algorithmical) limitation also present in data assimilation systems utilizing the adjoint to obtain the gradient for successive optimization iterations in source attributions problems. Since identical twin experiments were utilized to evaluate the performance of the adjoint-based optimization methodology, and measurement errors and model errors were ignored, one should expect the results obtained here to be optimistic in the context of real-life source attribution problems. The use of regularization techniques as well as a dense observation network near the source(s) to be reconstructed may help counter balance this structural numerical limitation of the adjoint method, as it is currently implemented in global CTMs. Inversion approaches not relying on gradient information should be more intensely studied in this context (Kaipio and Somersalo 2004). Fig. 1 . 1Area of influence at ∼ 4 km of altitude (20th pressure level) as a function of simulation time, as reconstructed by the first iteration of the adjoint-based optimization method for 4 • × 5 • numerical simulations. Latitude vs Longitude. Fig. 2 . 2Area of influence at ∼ 4 km of altitude (20th pressure level) as a function of simulation time, as reconstructed by the first iteration of the adjoint-based optimization method for 2 • × 2.5 • numerical simulations. Latitude vs Longitude. Fig. 3 . 3Reconstructed plumes at ∼ 4 km of altitude (20th pressure level) as a function of simulation time, as reconstructed by the adjoint-based optimization method for 4 • × 5 • numerical simulations. Latitude vs Longitude. Fig. 4 . 4Reconstructed plumes at ∼ 4 km of altitude (20th pressure level) as a function of simulation time, as reconstructed by the adjoint-based optimization method for 2 • × 2.5 • numerical simulations Latitude vs Longitude. Fig. 5 . 5Normalized, zonally-integrated reconstructed plumes as a function of simulation time, as reconstructed by the adjoint-based optimization method for 4 • × 5 • numerical simulations. Latitude vs Pressure level (atm) Fig. 6 . 6Normalized, zonally-integrated reconstructed plumes as a function of simulation time, as reconstructed by the adjoint-based optimization method for 2 • × 2.5 • numerical simulations. Latitude vs Pressure level (atm) Fig. 7 . 7Fig. 7. Relative L 2 -error (%) between the reconstructed initial condition and the true solution as a function of the simulation time for both 4 • × 5 • and 2 • × 2.5 • spatial resolutions. The estimate on the loss of information is obtained assuming that information is lost proportionally to the excess area of the numerical area of influence when compared to the true area of influence. Fig. 8 . 8Center of mass relative error as a function of simulation time. 100% error corresponds to a reconstructed center of mass at a distance equal to the maximal radius of the plume (1680 km) away from the true center of mass. Fig. 9 . 9Total mass relative error as a function of simulation time. Fig. 10 . 10Behavior of the cost function as a function of the number of iterations for multiple simulations times for 4 • × 5 • spatial resolution. Logarithm of the normalized cost function vs number of iterations. Fig. 11 . 11Behavior of the cost function as a function of the number of iterations for multiple simulations times for 2 • × 2.5 • spatial resolution. Logarithm of the normalized cost function vs number of iterations. Acknowledgments. A Bennet, Inverse Modeling of the Ocean and Atmosphere. Cambridge University PressBennet, A., 2002: Inverse Modeling of the Ocean and At- mosphere. Cambridge University Press. Global modeling of tropospheric chemistry with assimilated meteorology: Model description and evaluation. I Bey, Journal of Geophysical Research. 10696Bey, I., et al., 2001: Global modeling of tropospheric chem- istry with assimilated meteorology: Model description and evaluation. Journal of Geophysical Research, 106, 23,073-23,096. 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I Enting, Cambridge University PressEnting, I., 2002: Inverse Problems in Atmospheric Con- stituent Transport. Cambridge University Press. 1982: Horizontal diffusion in the atmosphere: A lagrangian dynamical theory. F Gifford, Atmos. Environ. 16Gifford, F., 1982: Horizontal diffusion in the atmosphere: A lagrangian dynamical theory. Atmos. Environ., 16, 502-512. Continuous versus discrete advection adjoints in chemical data assimilation with cmaq. T Gou, A Sandu, Atmos. Environ. Gou, T. and A. Sandu, 2011: Continuous versus discrete advection adjoints in chemical data assimilation with cmaq. Atmos. Environ. Development of the adjoint of geos-chem. D Henze, A Hakami, J H Seinfeld, Atmos. Chem. Phys. 7Henze, D., A. Hakami, and J. H. Seinfeld, 2007: Develop- ment of the adjoint of geos-chem. Atmos. Chem. Phys., 7, 2413-2433. Inverse modeling and mapping u.s. air quality influences of inorganic pm2.5 precursor emissions with the adjoint of geos-chem. D Henze, J H Seinfeld, D T Shindell, Atmos. Chem. Phys. 9Henze, D., J. H. Seinfeld, and D. T. Shindell, 2009: In- verse modeling and mapping u.s. air quality influences of inorganic pm2.5 precursor emissions with the adjoint of geos-chem. Atmos. Chem. Phys., 9, 5877-5903. Eulerian backtracking of atmospheric tracers. ii: Numerical aspects. F Hourdin, O Talagrand, A Idelkadi, Quarterly Journal of the Royal Meteorological Society. 132Hourdin, F., O. Talagrand, and A. Idelkadi, 2006: Eule- rian backtracking of atmospheric tracers. ii: Numerical aspects. Quarterly Journal of the Royal Meteorological Society, 132, 585-603. J Kaipio, E Somersalo, Statistical and Computational Inverse Problems. SpringerKaipio, J. and E. Somersalo, 2004: Statistical and Compu- tational Inverse Problems. Springer. 2000: Inverse methods in global biogeochemical cycles. P Kasibhatla, M Heimann, P Rayner, N Mahowald, R G Prinn, Geophys. Monogr. Series. D. E. H.114324Kasibhatla, P., M. Heimann, P. Rayner, N. Mahowald, R. G. Prinn, and D. E. H. (Eds), 2000: Inverse meth- ods in global biogeochemical cycles. Geophys. Monogr. Series, 114, 324. A comparison of analytical and adjoint bayesian inversion methods for constraining asian sources of co using satellite (mopitt) measurements of co columns. M Kopacz, D J Jacob, D K Henze, C L Heald, D G Streets, Q Zhang, J. Geophys. Res. 114Kopacz, M., D. J. Jacob, D. K. Henze, C. L. Heald, D. G. Streets, and Q. Zhang, 2009: A comparison of analytical and adjoint bayesian inversion methods for constraining asian sources of co using satellite (mopitt) measurements of co columns. J. Geophys. Res., 114. Global estimates of co sources with high resolution by adjoint inversion of multiple satellite datasets (mopitt, airs, sciamachy, tes). M Kopacz, Atmos. Chem. Phys. 10Kopacz, M., et al., 2010: Global estimates of co sources with high resolution by adjoint inversion of multiple satellite datasets (mopitt, airs, sciamachy, tes). Atmos. Chem. 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Estimation of mixing in the troposphere from lagrangian trace gas reconstructions during long-range pollution plume transport. I Pisso, E Real, K S Law, B Legras, N Bousserez, J L Attié, H Schlager, J. Geophys. Res. 114Pisso, I., E. Real, K. S. Law, B. Legras, N. Bousserez, J. L. Attié, and H. Schlager, 2009: Estimation of mix- ing in the troposphere from lagrangian trace gas recon- structions during long-range pollution plume transport. J. Geophys. Res., 114. Resolving intercontinental pollution plumes in global models of atmospheric transport. Y Rastigeyev, R Park, M Brenner, D Jacob, J. Geophys. Res. 115Rastigeyev, Y., R. Park, M. Brenner, and D. Jacob, 2010: Resolving intercontinental pollution plumes in global models of atmospheric transport. J. Geophys. Res., 115. A regional scale model for ozone in the united states with subgrid representation of urban and power plant plumes. S Sillman, L Logan, S Wofsy, J. Geophys. Res. 9557315748Sillman, S., L. Logan, and S. 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Hecht, 2001: Properties of advection algorithms in the context of variational data assimilation. Mon. Weather Rev. Validation of tes methane with hippo aircraft observations: implications for inverse modeling of methane sources. K J Wecht, Atmos. Chem. Phys. Discuss. 11Wecht, K. J., et al., 2011: Validation of tes methane with hippo aircraft observations: implications for inverse modeling of methane sources. Atmos. Chem. Phys. Dis- cuss., 11, 27 887-27 911. C Wunsch, Discrete Inverse and State Estimation Problems. Cambridge University PressWunsch, C., 2006: Discrete Inverse and State Estimation Problems. Cambridge University Press. Intercontinental source attribution of ozone pollution at western u.s. sites using an adjoint method. L Zhang, D J Jacob, M Kopacz, D K Henze, K Singh, D A Jaffe, Geophys. Res. Lett. 36Zhang, L., D. J. Jacob, M. Kopacz, D. K. Henze, K. Singh, and D. A. Jaffe, 2009: Intercontinental source attribu- tion of ozone pollution at western u.s. sites using an ad- joint method. Geophys. Res. Lett., 36. Improved estimate of the policyrelevant background ozone in the united states using the geos-chem global model with 1/2 degrees x 2/3 degrees horizontal resolution over north america. L Zhang, Atmospheric Environment. 45Zhang, L., et al., 2011: Improved estimate of the policy- relevant background ozone in the united states using the geos-chem global model with 1/2 degrees x 2/3 degrees horizontal resolution over north america. Atmospheric Environment, 45, 6769-6776. Lbfgs-b: a limited memory fortran code for Tech. rep. C Zhu, R Byrd, P Lu, J Nocedal, Northwestern UniversityZhu, C., R. Byrd, P. Lu, and J. Nocedal, 1994: Lbfgs-b: a limited memory fortran code for Tech. rep. Northwestern University.

In the framework of the theory of partially fluidized granular flows we study the formation of longitudinal structures observed experimentally by Forterre and Pouliquen in a flow down a rough inclined plane. We show that the formation of longitudinal structures is related to the positive feedback between the fluidization rate and the lateral stress (side pressure), which leads to a convective instability. Our theory explains main experimental features, such as appearance and amplification of the structure at some distance from the outlet and non-stationary behavior of the structures. 45.70.Ht, 45.70.Qj, 83.70.Fn The dynamics of granular media has been an active area of research for physicists [1] and engineers[2]. One of the most interesting phenomena pertinent to many granular systems is the transition from a static equilibrium to a granular flow. There has been a number of experimental studies of flows in large sandpiles[3,4]as well as in thin layers of grains on inclined surfaces[5][6][7][8]. Recent experiments with granular flows on a rough inclined plane revealed a new striking phenomenon: formation of nonstationary longitudinal structures at some distance downstream from an outlet[8]. Authors of[8]proposed an explanation of the instability mechanism based on analogy with thermal convection in fluids: a rapid shear granular flow leads to the increase of the granular temperature near the rough bottom. Because of the intrinsic dissipative nature of collisions between particles, the granular temperature should decay away from the bottom, creating necessary conditions for the "thermal granular convection" (cf.[10,11]). Using hydrodynamic equations obtained from the kinetic theory for dilute granular gases, this instability was studied analytically and numerically[9]. Although shear flow activated thermal granular convection could be a useful concept for the interpretation of experimental results, the theory [9] did not address two important observations: (1) experiments as well as simulations[12]show that fluctuations of velocity are more significant near the free surface of the granular flow, and so the granular temperature may in fact be higher at the top rather than at the bottom, (2) longitudinal structures appear at some distance downstream from the outlet and exhibit complex spatio-temporal dynamics.In this Letter we demonstrate that the observed transverse instability can be described within the framework of the continuum theory of partially fluidized flows proposed by us in Refs.[13,14]without invoking the concept of granular temperature more appropriate for dilute gas-like granular flows. In this theory, we introduce an equation for the order parameter which characterizes the phase state of the granular matter. In a certain range of parameters, the shear flow described by these equations, is unstable with respect to transverse perturbations. We show that this instability is of convective nature, i.e. small perturbations grow downstream while remaining small in the laboratory frame. Thus, the "rolls" appear at some finite distance from the outlet. This distance depends on the noise level and flow conditions. Since the pattern structure is determined by random fluctuations near the outlet, the resulting pattern is always non-stationary in the laboratory frame, very similar to that observed experimentally.Model. The starting point of our theory is the momentum conservation equationwhere v i are the components of velocity, ρ 0 = const is the density of material (in the following we set ρ 0 = 1), g is acceleration of gravity, and D/Dt = ∂ t + v i ∂ xi denotes the material derivative. Since the relative density fluctuations are small, the velocity obeys the incompressibility condition ∇ · v = 0. The stress tensor is represented as a sum of the hydrodynamic part proportional to the flow strain rate and the strain-independent part,where η has the meaning of the viscosity coefficient. At this point we introduce the order parameter (OP) ρ which describes the "phase state" of the granular matter: it varies from 0 in the "liquid" phase to 1 in the "solid" phase. We interpret the OP as the relative local density of static contacts among the grains. In the solid state (ρ = 1) the strain-independent part should coincide with the "true" static stress tensor σ 0 ij for the immobile grain configuration in the same geometry, and in the completely fluidized state (ρ = 0) we should have σ s ij = −Πδ ij (Π is the hydrodynamic pressure). According to our assumption, the off-diagonal elements of the strain-independent part of the stress tensor obey the conditions σ s ij = ρσ 0 ij for i = j. In the solid state the normal 1

cond-mat/0203189

137eb88c11cd1ca5b6a2756e136ef71194f87d80

Formation of longitudinal structures in granular flows 8 Mar 2002 Igor S Aranson Argonne National Laboratory 9700 South Cass Avenue60439ArgonneIL Lev S Tsimring Institute for Nonlinear Science University of California San Diego, La Jolla92093-0402CA Formation of longitudinal structures in granular flows 8 Mar 2002(February 7, 2020) In the framework of the theory of partially fluidized granular flows we study the formation of longitudinal structures observed experimentally by Forterre and Pouliquen in a flow down a rough inclined plane. We show that the formation of longitudinal structures is related to the positive feedback between the fluidization rate and the lateral stress (side pressure), which leads to a convective instability. Our theory explains main experimental features, such as appearance and amplification of the structure at some distance from the outlet and non-stationary behavior of the structures. 45.70.Ht, 45.70.Qj, 83.70.Fn The dynamics of granular media has been an active area of research for physicists [1] and engineers[2]. One of the most interesting phenomena pertinent to many granular systems is the transition from a static equilibrium to a granular flow. There has been a number of experimental studies of flows in large sandpiles[3,4]as well as in thin layers of grains on inclined surfaces[5][6][7][8]. Recent experiments with granular flows on a rough inclined plane revealed a new striking phenomenon: formation of nonstationary longitudinal structures at some distance downstream from an outlet[8]. Authors of[8]proposed an explanation of the instability mechanism based on analogy with thermal convection in fluids: a rapid shear granular flow leads to the increase of the granular temperature near the rough bottom. Because of the intrinsic dissipative nature of collisions between particles, the granular temperature should decay away from the bottom, creating necessary conditions for the "thermal granular convection" (cf.[10,11]). Using hydrodynamic equations obtained from the kinetic theory for dilute granular gases, this instability was studied analytically and numerically[9]. Although shear flow activated thermal granular convection could be a useful concept for the interpretation of experimental results, the theory [9] did not address two important observations: (1) experiments as well as simulations[12]show that fluctuations of velocity are more significant near the free surface of the granular flow, and so the granular temperature may in fact be higher at the top rather than at the bottom, (2) longitudinal structures appear at some distance downstream from the outlet and exhibit complex spatio-temporal dynamics.In this Letter we demonstrate that the observed transverse instability can be described within the framework of the continuum theory of partially fluidized flows proposed by us in Refs.[13,14]without invoking the concept of granular temperature more appropriate for dilute gas-like granular flows. In this theory, we introduce an equation for the order parameter which characterizes the phase state of the granular matter. In a certain range of parameters, the shear flow described by these equations, is unstable with respect to transverse perturbations. We show that this instability is of convective nature, i.e. small perturbations grow downstream while remaining small in the laboratory frame. Thus, the "rolls" appear at some finite distance from the outlet. This distance depends on the noise level and flow conditions. Since the pattern structure is determined by random fluctuations near the outlet, the resulting pattern is always non-stationary in the laboratory frame, very similar to that observed experimentally.Model. The starting point of our theory is the momentum conservation equationwhere v i are the components of velocity, ρ 0 = const is the density of material (in the following we set ρ 0 = 1), g is acceleration of gravity, and D/Dt = ∂ t + v i ∂ xi denotes the material derivative. Since the relative density fluctuations are small, the velocity obeys the incompressibility condition ∇ · v = 0. The stress tensor is represented as a sum of the hydrodynamic part proportional to the flow strain rate and the strain-independent part,where η has the meaning of the viscosity coefficient. At this point we introduce the order parameter (OP) ρ which describes the "phase state" of the granular matter: it varies from 0 in the "liquid" phase to 1 in the "solid" phase. We interpret the OP as the relative local density of static contacts among the grains. In the solid state (ρ = 1) the strain-independent part should coincide with the "true" static stress tensor σ 0 ij for the immobile grain configuration in the same geometry, and in the completely fluidized state (ρ = 0) we should have σ s ij = −Πδ ij (Π is the hydrodynamic pressure). According to our assumption, the off-diagonal elements of the strain-independent part of the stress tensor obey the conditions σ s ij = ρσ 0 ij for i = j. In the solid state the normal 1 In the framework of the theory of partially fluidized granular flows we study the formation of longitudinal structures observed experimentally by Forterre and Pouliquen in a flow down a rough inclined plane. We show that the formation of longitudinal structures is related to the positive feedback between the fluidization rate and the lateral stress (side pressure), which leads to a convective instability. Our theory explains main experimental features, such as appearance and amplification of the structure at some distance from the outlet and non-stationary behavior of the structures. The dynamics of granular media has been an active area of research for physicists [1] and engineers [2]. One of the most interesting phenomena pertinent to many granular systems is the transition from a static equilibrium to a granular flow. There has been a number of experimental studies of flows in large sandpiles [3,4] as well as in thin layers of grains on inclined surfaces [5][6][7][8]. Recent experiments with granular flows on a rough inclined plane revealed a new striking phenomenon: formation of nonstationary longitudinal structures at some distance downstream from an outlet [8]. Authors of [8] proposed an explanation of the instability mechanism based on analogy with thermal convection in fluids: a rapid shear granular flow leads to the increase of the granular temperature near the rough bottom. Because of the intrinsic dissipative nature of collisions between particles, the granular temperature should decay away from the bottom, creating necessary conditions for the "thermal granular convection" (cf. [10,11]). Using hydrodynamic equations obtained from the kinetic theory for dilute granular gases, this instability was studied analytically and numerically [9]. Although shear flow activated thermal granular convection could be a useful concept for the interpretation of experimental results, the theory [9] did not address two important observations: (1) experiments as well as simulations [12] show that fluctuations of velocity are more significant near the free surface of the granular flow, and so the granular temperature may in fact be higher at the top rather than at the bottom, (2) longitudinal structures appear at some distance downstream from the outlet and exhibit complex spatio-temporal dynamics. In this Letter we demonstrate that the observed transverse instability can be described within the framework of the continuum theory of partially fluidized flows proposed by us in Refs. [13,14] without invoking the concept of granular temperature more appropriate for dilute gas-like granular flows. In this theory, we introduce an equation for the order parameter which characterizes the phase state of the granular matter. In a certain range of parameters, the shear flow described by these equations, is unstable with respect to transverse perturbations. We show that this instability is of convective nature, i.e. small perturbations grow downstream while remaining small in the laboratory frame. Thus, the "rolls" appear at some finite distance from the outlet. This distance depends on the noise level and flow conditions. Since the pattern structure is determined by random fluctuations near the outlet, the resulting pattern is always non-stationary in the laboratory frame, very similar to that observed experimentally. Model. The starting point of our theory is the momentum conservation equation ρ 0 Dv i /Dt = ∂σ ij ∂x j + ρ 0 g i , j = 1, 2, 3.(1) where v i are the components of velocity, ρ 0 = const is the density of material (in the following we set ρ 0 = 1), g is acceleration of gravity, and D/Dt = ∂ t + v i ∂ xi denotes the material derivative. Since the relative density fluctuations are small, the velocity obeys the incompressibility condition ∇ · v = 0. The stress tensor is represented as a sum of the hydrodynamic part proportional to the flow strain rate and the strain-independent part, σ ij = η ∂v i ∂x j + ∂v j ∂x i + σ s ij ,(2) where η has the meaning of the viscosity coefficient. At this point we introduce the order parameter (OP) ρ which describes the "phase state" of the granular matter: it varies from 0 in the "liquid" phase to 1 in the "solid" phase. We interpret the OP as the relative local density of static contacts among the grains. In the solid state (ρ = 1) the strain-independent part should coincide with the "true" static stress tensor σ 0 ij for the immobile grain configuration in the same geometry, and in the completely fluidized state (ρ = 0) we should have σ s ij = −Πδ ij (Π is the hydrodynamic pressure). According to our assumption, the off-diagonal elements of the strain-independent part of the stress tensor obey the conditions σ s ij = ρσ 0 ij for i = j. In the solid state the normal stresses σ 0 ii do not in general coincide. For the weaklyfluidized state (ρ → 1) the normal stresses are close to the static values, however some dependence on the order parameter ρ (i.e. degree of fluidization) may appear. In the first order in 1−ρ one writes for the diagonal elements of the stress tensor σ s ii = σ 0 ii + α i (1 − ρ) + O((1 − ρ) 2 )(3) where α i characterizes the response of the normal stresses on fluidization. Since fluidization is accompanied by the decrease in the number of static contacts among granules (i.e. dilution), the normal stresses should decrease with fluidization, i.e. α i > 0. It also agrees with the observation in Ref. [8] that the crests of the surface deformations correspond to a more dilute granular state. According to Ref. [13,14], the equation for the order parameter ρ is taken in the forṁ ρ + v∇ρ = ∇ 2 ρ + ρ(1 − ρ)(ρ − δ),(4)where δ = (φ − φ 0 )/(φ 1 − φ 0 ) is the control parameter, φ = max |σ 0 mn /σ 0 nn | is the maximum ratio of shear to normal stresses in the bulk, and tan −1 φ 1,2 are static and dynamic repose angles characterizing the granular material. Parameter φ which enters the Mohr-Coloumb yield condition [2], in the context of our theory is equivalent to a melting temperature in the theory of phase transition. Following the analysis of Ref. [13], we consider a layer of dry cohesionless grains on an inclined rough surface (see Fig.1). However, now we assume that the layer thickness h can vary in both x and y directions. The momentum conservation equation (1) in the stationary regime yields the force balance conditions σ xz,x + σ yz,y + σ zz,z = g cos ϕ, (5) σ xx,x + σ xy,y + σ xz,z = −g sin ϕ, σ xy,x + σ yy,y + σ yz,z = 0, where the subscripts after commas mean partial derivatives, and z = 0 corresponds to the bottom of the layer. We assume that variations of the layer thickness along the y direction are small. Accordingly, there will be a small component of velocity along y direction and corresponding stress components σ yz and σ yy . In the avalanche problem considered in Ref. [13] these terms were irrelevant. We show later that the transverse flux is crucial for the explanation of longitudinal structures. In the first order in h x , h y Eqs. (5)-(7) yield σ zz ≈ g cos ϕ(z − h) , σ xz ≈ −g sin ϕ(z − h), σ yz = h z dzσ yy,y .(8) Here we used the conditions σ xx = const, σ xy = const and σ zz = σ xz = σ yz = 0 on free surface z = h. For the chute geometry Fig. 1 parameter δ in Eq. (4) can be specified. For h = const there is a simple relation between shear and normal stresses, σ xz = − tan ϕσ zz . The most "unstable" yield direction is parallel to the inclined plane, i.e. φ = |σ xz /σ zz | and δ = δ 0 = (tan ϕ − φ 0 )/(φ 1 − φ 0 ). If h is a slowly varying function of x and y, one obtains for the parameter δ: δ = δ 0 − βh x + O(h 2 x , h 2 y )(9) where β = 1/(φ 1 − φ 0 ) (see [13,14] for more details). Thin layer solutions of Eq. (4) for the chute geometry Fig. 1 are subject to the following boundary conditions (BC): no-flux condition ρ z = 0 at the free surface z = h, and no-slip condition ρ = 1 at the bottom of the chute z = 0 (a granular medium is assumed to be in a solid phase near the rough surface). The velocity profiles corresponding to a stationary profile of ρ(z), can be easily found from Eq. (2),(3), η ∂v x ∂z = −g sin ϕ(z − h) − ρσ 0 xz = −g sin ϕ(1 − ρ)(z − h), η ∂v y ∂z = − h z α y ∂ y ρdz(10) In the derivation of Eq. (10) we used σ s yz = 0 for a flat layer. The components of the flux J = h 0 v(z)dz are J x = − g sin ϕ η h 0 dz z 0 (1 − ρ(z ′ ))(z ′ − h)dz ′ J y = − α y η h 0 dz z 0 dz ′′ h z ′′ ∂ y ρdz ′(11) For thin weakly-fluidized layers (see [13,14]), we can look for solution of Eq.(4) in the form ρ = 1 − A sin π 2h z + h.o.t.,(12) where A ≪ 1 is a slowly varying function of t, x, and y. Substituting ansatz (12) into Eq. (4) and applying orthogonality conditions [13,14]), we obtain A t = ΛA + ∇ 2 ⊥ A + 8(2 − δ) 3π A 2 − 3 4 A 3 −μh 2 A∂ x A (13) where ∇ 2 ⊥ = ∂ 2 x + ∂ 2 y , Λ = δ − 1 − π 2 4h 2 ,μ = (3π 2 − 16)g sin ϕ/3π 3 η = 0.146g sin ϕ/η. The mass conservation yields ∂h ∂t = −∇ · J = −∂ x J x − ∂ y J y(14) where the flux components J x,y calculated from Eq. (11) in the thin layer approximation, are Our model has two control parameters, the downhill mass flux J 0 at the outlet x = 0, and δ. For large enough J 0 and δ, Eqs. (13),(14) possess a steady-state solution A = A 0 , h = h 0 (see Fig.2). The stability of this solution can be examined by substituting the ansatz {A, h} = {A 0 , h 0 } + {ζ, ξ} exp(λt + ik x x + ik y y), where {ζ, ξ} is a small perturbation. For long-wave perturbations (k x , k y ≪ 1) in the leading order one obtains for the growth rate λ J x = µh 3 A , J y = −α 1 h 2 Ah y + α 2 h 3 A y (15) µ = 2(π 2 − 8)g sin ϕ/ηπ 3 ≈ 0.12g sin ϕ/η, α 1 = α y (π 3 + 8π − 48)/2π 3 η ≈ 0.13α y /η, α 2 = 8α y /π 3 η ≈ 0.26α y /η.λ = ik x µh 2 0 A 0 + Bk 2 y + ...(16) where B = π 2 α 2 [16(2−δ)/3π−3A 0 ] −1 −α 1 h 2 0 A 0 . As seen from Eq. (16), pure longitudinal perturbations k y = 0 are neutrally stable, but for B > 0 the perturbations with transverse component of the wave number can be unstable. For pure transverse perturbations (k x = 0) the instability is aperiodic (λ is real), as in thermo-convection in ordinary fluids. The physical meaning of the instability is the following: decrease of ρ (increase of fluidization) corresponds to the decrease of the lateral pressure (−σ yy ) which in turns lead to the increase of the height h and further fluidization. The phase diagram of the instability on δ − J 0 plane is shown in Fig. 2. The transverse instability exists in the band restricted from above by the condition B = 0 and from below by the existence of steady-state solution [19]. Complexity of λ and asymmetry of the problem in xdirection signal the possibility of the convective nature of the instability: the perturbations may grow in the moving frame and decay in the laboratory frame (see, e.g. [15,16]). As a test for convective instability one has to examine the value of λ for the saddle point of the function λ(k x , k y ) in the complex k x plane, i.e. ∂λ/∂k x = 0. In fact, it is easy to see that if the instability is weak (for α y ≪ 1) and the downhill flux J 0 is finite, the instability has to be convective. In the presence of persistent fluctuations, e.g. at the flow outlet, the perturbation will be spatially amplified down the flow. Let us estimate the spatial growth rate. Consider the unstable perturbations caused by a small stationary yperiodic force with wavenumber k 0 localized at x = 0. Calculations show that the linearized solution ζ will be described by the integral ζ ∼ ∞ −∞ exp[ikr] dk x λ(k) (1 − exp[λ(k)t])(17) The non-stationary part ∼ exp[iλ(k)t] decays in time due to the convective character of the instability, and the integral yields ζ ∼ exp[ik 0 y + ik * x](18) where k * is found from λ(k x = k * , k 0 ) = 0. It is easy to see from Eq.(16) that, for small k 0 , k * ≈ −iBk 2 0 /µh 2 0 A 0 . Since k * is imaginary, perturbations from a stationary source at x = 0 grow exponentially downhill. For random perturbations introduced at the outlet the typical scale of the pattern will be determined by the most unstable wavenumber in the transverse direction. However, the pattern will remain non-stationary due to intrinsically random nature of the noise. We studied Eqs. (13), (14) numerically. The simulations were performed in a large system, more than 2000 dimensionless units in x-direction (downhill), and 200 units in y-direction. Fixed flux boundary conditions were imposed at the outlet at x = 0. Selected results are presented in Figs. 3,4. Figure 3 illustrates spatial amplification of perturbations downstream and formation of longitudinal structures. As it follows from our simulations, far away from the outlet the structures remain nonstationary and exhibit spatio-temporal dynamics which are very similar to observed in Ref. [8]. The profiles of h and A vs y are shown in Fig. 4. In agreement with experiment, crests in h corresponds to crests in A, i.e. to more fluidized regions of flow. We demonstrated that formation of longitudinal structures in granular flow down a rough inclined plane is the result of noise amplification and saturation due to the convective instability. The mechanism of the instability is related to the dependence of the local pressure on the fluidization rate controlled by the order parameter. We conjectured a simple linear relation between these quantities. It would be of interest to verify this relation in physical experiment and molecular dynamic simulations. The convective character of the instability suggests that the flow can be controlled by small systematic perturbations introduced at the outlet. In recent paper [20], a novel fingering instability in a thin granular layer inside a horizontal rotating cylinder was reported. We believe that the nature of this insta-bility fundamentally is the same as described here. The only important difference is that due to backflow the perturbation can propagate upstream, and thereby the convective instability becomes "global". This may explain why this fingering instability occurs in a "short" system as compared with inclined layer experiment [8]. This research was supported by the Office of the Basic Energy Sciences at the US DOE, grants W-31-109-ENG-38 and DE-FG03-95ER14516. Simulations were performed at the National Energy Research Scientific Computing Center. PACS: 45.70.-n, 45.70.Ht, 45.70.Qj, 83.70.Fn FIG. 1 . 1Schematic representation of a chute geometry. z-axis is normal to the chute bottom, dashed line is parallel to the direction of gravity. FIG. 2 . 2Phase diagram in δ − J0 plane. FIG. 3 .FIG. 4 . 34Grey-scale snapshots of h(x, y) (white corresponds to larger h), for δ = 1.5, µ = 0.025, β = 3.14, J0 = 1.75, αy = 0.04. Equilibrium value of the layer thickness h0 ≈ 4.45. Profiles of h(y) and A(y) at x = 2000. . H M Jaeger, S R Nagel, R P Behringer, Rev. Mod. Phys. 681259H.M. Jaeger, S.R. Nagel, and R.P. Behringer, Rev. Mod. Phys. 68, 1259 (1996); . L Kadanoff, Rev. Mod. Phys. 71435L. Kadanoff, Rev. Mod. Phys. 71, 435 (1999); . P G De, Gennes Rev. Mod. Phys. 71374P. G. de Gennes Rev. Mod. Phys. 71, S374 (1999). R M Nedderman, Statics and Kinematics of Granular Materials. Cambridge, EnglandCambridge University PressR.M. Nedderman, Statics and Kinematics of Granu- lar Materials, (Cambridge University Press, Cambridge, England, 1992) . R A Bagnold, Proc. Roy. Soc. London A. 225219ibid.R.A. Bagnold, Proc. Roy. Soc. London A 225, 49 (1954); ibid., 295, 219 (1966) J Rajchenbach, Physics of Dry Granular Media. H. Hermann, J.-P. Hovi, and S. LudingDordrechtKluwer421J. Rajchenbach, in Physics of Dry Granular Media, eds. H. Hermann, J.-P. Hovi, and S. Luding, p. 421, (Kluwer, Dordrecht, 1998); . A Daerr, S Douady, Nature. 399241A. Daerr and S. Douady, Nature (London) 399, 241 (1999) . A Daerr, Phys. Fluids. 132115A. Daerr, Phys. Fluids 13, 2115 (2001) . O Pouliquen, Phys. Fluids. 11542O. Pouliquen, Phys. Fluids, 11, 542 (1999) . Y Forterre, O Pouliquen, Phys. Rev. Lett. 865886Y. Forterre and O. Pouliquen, Phys. Rev. Lett. 86, 5886 (2001) . Y Forterre, O Pouliquen, cond-mat/0108517Y. Forterre and O. Pouliquen, cond-mat/0108517. . R D Wildman, J M Huntley, D J Parker, Phys. Rev. Lett. 863304R.D. Wildman, J.M. Huntley, and D.J. Parker, Phys. Rev. Lett. 86, 3304 (2001) . X He, B Meerson, G Doolen, Phys. Rev. E. 65R30301X. He, B. Meerson and G. Doolen, Phys. Rev. E 65 030301(R) (2002) . L E Silbert, Phys. Rev. E. 6451302L.E. Silbert et al, Phys. Rev. E 64, 051302 (2002). . I S Aranson, L S Tsimring, Phys. Rev. E. 6420301I.S. Aranson and L.S. Tsimring, Phys. Rev. E. 64, 020301 (R) (2001) . I S Aranson, L S Tsimring, cond-mat/0109358I.S. Aranson and L.S. Tsimring, cond-mat/0109358 . L D Landau, E M Lifshitz, Physical Kinetics. Pergamon PressL.D.Landau and E.M.Lifshitz, Physical Kinetics, Perga- mon Press, New York, 1981 . I S Aranson, L Aranson, L Kramer, A Weber, Phys. Rev. A. 462992I.S. Aranson, L. Aranson, L. Kramer, and A. Weber, Phys. Rev. A 46, 2992 (1992) . J P Wittmer, M E Cates, P J Claudine, J. Phys. II France. 7J.P. Wittmer, M.E. Cates, and P.J. Claudine, J. Phys. II France 7, 39, (1997); . L Vanel, Phys. Rev. Lett. 841439L. Vanel et al, Phys. Rev. Lett. 84, 1439 (2000) . M E Cates, Phys. Rev. Lett. 811841M.E. Cates et al, Phys. Rev. Lett. 81, 1841 (1998) Our phase diagram is somewhat different from that in Ref. [8] shown in opening width hg -angle variables. However, the direct comparison is difficult because the opening hg in Ref. 08] is not explicitly related to the flux J0. According to Ref. [8] significant variation in hg practically does not change the stationary thickness of the flow h, and, thereforeOur phase diagram is somewhat different from that in Ref. [8] shown in opening width hg -angle variables. However, the direct comparison is difficult because the opening hg in Ref. [8] is not explicitly related to the flux J0. According to Ref. [8] significant variation in hg prac- tically does not change the stationary thickness of the flow h, and, therefore, J0. . A Q Shen, Phys. Fluids. 14462A. Q. Shen, Phys. Fluids, 14, 462 (2002).

We investigate the minimal singularities of metrics on a big line bundle L over a projective manifold when the stable base locus Y of L is a submanifold of codimension r ≥ 1. Under some assumptions on the normal bundle and a neighborhood of Y , we give a explicit description of the minimal singularity of metrics on L. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle L is not semi-ample, however it is nef and big.

10.5802/afst.1628

1612.08212

00eeb2ca1a59630776e733b1ff9318662214f74f

ON METRICS WITH MINIMAL SINGULARITIES OF LINE BUNDLES WHOSE STABLE BASE LOCI ADMIT HOLOMORPHIC TUBULAR NEIGHBORHOODS 17 Nov 2018 Genki Hosono Takayuki Koike ON METRICS WITH MINIMAL SINGULARITIES OF LINE BUNDLES WHOSE STABLE BASE LOCI ADMIT HOLOMORPHIC TUBULAR NEIGHBORHOODS 17 Nov 2018 We investigate the minimal singularities of metrics on a big line bundle L over a projective manifold when the stable base locus Y of L is a submanifold of codimension r ≥ 1. Under some assumptions on the normal bundle and a neighborhood of Y , we give a explicit description of the minimal singularity of metrics on L. We apply this result to study a higher (co-)dimensional analogue of Zariski's example, in which the line bundle L is not semi-ample, however it is nef and big. Introduction The purpose of this paper is to investigate metrics with minimal singularities on a big line bundle L on a projective manifold X. Metrics with minimal singularities have been introduced in [DPS,Definition 1.4] as a weak analytic analogue of the so-called Zariski decomposition. There exists a metric with minimal singularities uniquely up to certain equivalence of singularities when L is pseudo-effective [DPS,Theorem 1.5]. Indeed, the equilibrium metric h e of any C ∞ Hermitian metric h on L has minimal singularities (see Example 2.2.2). On a higher-dimensional variety, a line bundle does not necessarily admit the Zariski decomposition. Nakayama constructed an example of a line bundle which admits no Zariski decomposition even after any modification [N,IV,§2.6]. Nakayama's example is constructed as the relative tautological bundle on certain projective space bundle over an abelian variety. Boucksom [B] posed a decomposition called divisorial Zariski decomposition, in which the negative part of a big line bundle L is identified with the divisorial (i.e. one-codimensional) part of the singularities of a metric with minimal singularities on L. From this point of view, it is important for a study of the Zariski decomposition to investigate the higher-codimensional part of the singularities of metrics with minimal singularities in detail. In [K1], the second author explicitly described the metrics with minimal singularities for Nakayama's example we mentioned above. We also investigate the case where the line bundle L is nef (and big) and thus L has no negative part in the sense of Zariski decompositions. In this case, our main interest is in the semi-positivity of the line bundle, i.e. whether L admits a C ∞ metric with semipositive curvature or not. In [K2], the second author studied the metrics with minimal singularities on a line bundle called Zariski's example, which is known to be nef and big, but not semi-ample. As a result, it was shown that Zariski's example admits a C ∞ Hermitian metric with semi-positive curvature. In this paper, we investigate the metrics of L with minimal singularities for more general cases than both [K1] and [K2]. Our main result has the following application: Theorem 1.0.1. Take two general quadric surfaces Q 1 and Q 2 in P 3 and fix general N points p 1 , p 2 , . . . , p N in Q 1 ∩ Q 2 (N ≥ 12). Denote by π : X := Bl {p 1 ,p 2 ,...,p N } P 3 → P 3 the blow-up of P 3 at these N points, and by D 1 and D 2 the the strict transform of Q 1 and Q 2 , respectively. Define a line bundle L by L := π * O P 3 (1) ⊗ O X (D 1 ). Then, the local weight function ϕ min,L of a metric with minimal singularities h min,L of L (i.e. ϕ min,L is a locally defined function such that h min,L = e −ϕ min,L ) is written as ϕ min,L (z, y) = N − 12 N − 8 · log(|z 1 | 2 + |z 2 | 2 ) + O(1) on a neighborhood of every point of Y := D 1 ∩ D 2 , where y is a coordinate of Y and z = (z 1 , z 2 ) is a system of local defining functions of Y . We have that ϕ min,L is locally bounded on X \ Y . When N = 12, the line bundle L in this theorem is nef and big, but not semi-ample. Hence it can be regarded as a higher-dimensional analogue of Zariski's example. In this case, we will show that L admits a C ∞ Hermitian metric with semi-positive curvature (see §6.2 for detail), which can be regarded as a two-codimensional analogue of [K2,Theorem 1.1]. In what follows, L denotes a big line bundle on a projective manifold X. We study the metric with minimal singularities when (X, L) satisfies the following condition. Condition 1.0.2. (i) The stable base locus Y = B(L) of L is a smooth (i.e. nonsingular) compact subvariety of codimension r ≥ 1, (ii) the normal bundle N Y /X of Y admits a direct sum decomposition N Y /X = N 1 ⊕ N 2 ⊕ · · · ⊕ N r into r negative line bundles. Nakayama's example satisfies Condition 1.0.2. For the pair (X, L) with Condition 1.0.2, we define the following convex set: L = α = (α 1 , α 2 , . . . , α r ) ∈ R r ≥0 |α| ≤ 1, and c 1 (L| Y ) + r λ=1 α λ c 1 (N −1 λ ) is pseudo-effective , where |α| := r λ=1 α λ . In [K1], a metric with minimal singularities on Nakayama's example was described explicitly by using L on a neighborhood of Y . It is easily observed that a metric with minimal singularities is locally bounded on the complement X \ Y of Y (see Example 2.2.3). Hence our interest is in the behavior of metrics with minimal singularities near Y . We always take a system of local defining functions z = (z 1 , z 2 , . . . , z r ) of Y so that, for each λ, the subbundle of N −1 Y /X generated by dz λ corresponds to the direct component N −1 λ . Our main result is stated as follows: Theorem 1.0.3. Let X be a projective manifold, L be a big line bundle on X, and Y = B(L) be the stable base locus of L. Assume that Y admits a holomorphic tubular neighborhood (see below) and (X, L, Y ) satisfies Condition 1.0.2. Assume also that L| Y ⊗ N −1 λ and K −1 Y ⊗ N −1 λ are positive for every λ = 1, 2, . . . , r. Take C ∞ Hermitian metrics 2 h L| Y on L| Y and h N λ on N λ satisfying Θ h L| Y ⊗h −1 N λ > 0 for every λ. Then the local weight function ϕ min,L of a metric with minimal singularities h min,L is written as ϕ min,L (z, y) = log max α∈ L r λ=1 |z λ | 2α λ · e (ϕα)e(y) + O(1) on a neighborhood of any given point of Y , where we are formally regarding 0 0 as 1, y is a coordinate of Y , z = (z 1 , z 2 , . . . , z r ) is a system of local defining functions of Y as above, and (ϕ α ) e is the local weight function of the equilibrium metric of h L| Y ⊗ h −α 1 N 1 ⊗ h −α 2 N 2 · · · ⊗ h −αr Nr (see §2 for the notion of the "metric" h L| Y ⊗ h −α 1 N 1 ⊗ h −α 2 N 2 · · · ⊗ h −αr Nr for real α λ 's). A complex submanifold Y ⊂ X is said to have a holomorphic tubular neighborhood if there exist a neighborhood V of Y in X, a neighborhood V of the zero section in N Y /X and a biholomorphism i : V → V such that i| Y coincides with the natural isomorphism. Note that the description of the singularity of ϕ min,L as in Theorem 1.0.3 does not depend on the choice of the coordinates (up to O (1)). This theorem is a generalization of the main result of [K1]. Moreover it is also a generalization of [K2] in higher codimensional cases. In this theorem, Condition 1.0.2 and the condition that Y admits a holomorphic tubular neighborhood are essential and can not be dropped (see §6.3). When Y is an abelian variety (as in Nakayama's example), we have a sufficient condition for the existence of a holomorphic tubular neighborhood of Y by Grauert's theory on a neighborhood of an exceptional subvariety [G] (see §5). As a result, we have the following theorem: Theorem 1.0.4. Let X be a projective manifold, let L be a big line bundle on X, and let Y = B(L) be the stable base locus of L. Assume Condition 1.0.2. Assume also that Y is an abelian variery, L| Y ⊗N −1 λ is positive for every λ = 1, 2, . . . , r, and that N λ ∼ = N µ for every λ and µ. Then the local weight function ϕ min,L of a metric with minimal singularities h min,L is written as ϕ min,L (z, y) = log max α∈ L r λ=1 |z λ | 2α λ + O(1) on a neighborhood of any given point of Y , where y is a coordinate of Y and z = (z 1 , z 2 , . . . , z r ) is a system of local defining functions of Y as above. When L| Y is pseudo-effective, it is natural to ask whether (h min,L )| Y is a metric with minimal singularities on L| Y . For (X, L, Y ) in Theorem 1.0.3, it follows by definition that the convex set L includes the origin 0 if L| Y is pseudo-effective. In this case, it is directly deduced from Theorem (1) , which means that h min,L | Y has minimal singularities. Therefore we have the following: Corollary 1.0.5. Let X, L, and Y be those in Theorem 1.0.3. Assume that L| Y is pseudo-effective. Then h min,L | Y is a singular Hermitian metric of L| Y with minimal singularities. 1.0.3 that h min,L | Y ≤ (h L| Y ) e · e O The proof of Theorem 1.0.3 is based on the arguments in [K2]. We first study a special case where X is a projective space bundle over Y and L is the relative tautological 3 bundle. After that, we apply the exact description of metrics with minimal singularities for this spacial case to the study of general (X, L, Y ) by using, what we call, the maximum construction technique (here we use the assumption of a holomorphic tubular neighborhood). The organization of the paper is as follows. In §2, we introduce fundamental notation and recall some facts on projective space bundles and singular Hermitian metrics. In §3, we show the main result in the special case where X is the total space of a projective space bundle. In §4, we prove Theorem 1.0.3 in general. In §5, we give a sufficient condition for the existence of a holomorphic tubular neighborhood by using Grauert's theory. Here we also show Theorem 1.0.4. In §6, we give several examples. Preliminaries 2.1. Notations on projective space bundles. Let Y be a compact complex manifold. Let M 1 , M 2 , . . . , M r and M r+1 be holomorphic line bundles on Y . Let {U j } j be an open cover of Y . Assume that every U j is sufficiently small so that M λ | U j is trivial for every λ and j. Then there exist local holomorphic trivializations given by sections s j,λ ∈ H 0 (U j , M λ ). Denote by E the vector bundle M 1 ⊕ M 2 ⊕ · · · ⊕ M r+1 . Then (s j,λ ) λ forms a holomorphic frame of E on U j . Let (ξ j,λ ) λ be the dual frame of (s j,λ ) λ . We fix the notation on P(E) as follows. Let us denote by P(E) the projective space bundle of hyperplanes of E over Y , i.e. P(E) := y (E * y \ 0)/C * . We will denote the bundle of lines by P(E) in this paper. Let π denote the natural projection P(E) → Y . We will use the notation P(E)| U j to denote π −1 (U j ). By using homogeneous coordinates, ([x j,1 : x j,2 : · · · : x j,r+1 ], y) denotes the point [x j,1 ξ j,1 + x j,2 ξ j,2 + · · · + x j,r+1 ξ j,r+1 ] ∈ P(E) y on P(E)| U j . Here P(E) y denotes the fiber π −1 (y). Let U (λ) j be an open set {([x j,1 ξ j,1 + x j,2 ξ j,2 + · · · + x j,r+1 ξ j,r+1 ], y) | y ∈ U j , x j,λ = 0} of P(E). Note that {U (λ) j } j,λ forms an open cover of P(E). The tautological line bundle O P(E) (1) on P(E) is defined by setting its fiber on ([ξ], y) as E y /Ker ξ, where ξ denotes an element of E * y \ 0. Let Γ λ be the divisor of P(E) defined as P(M 1 ⊕ M 2 ⊕ · · · M λ · · · ⊕ M r+1 ). The following fact is obtained by a simple computation. Lemma 2.1.1. (i) [Γ λ ] ⊗ π * M λ = O P(E) (1), where [Γ λ ] denotes the line bundle defined by the divisor Γ λ . (ii) N Y /X = r λ=1 O P(E) (1)| Y ⊗ M −1 λ . 2.2. Singular Hermitian metrics. In this subsection, we review some properties of singular Hermitian metrics on line bundles. Definition 2.2.1. Let X be a (possibly non-compact) complex manifold and let L be a line bundle on X. A singular Hermitian metric h on L is defined as a metric of L with the form s 2 h = |s| 2 e −φ on U for each trivialization L| U ∼ = U × C, where φ ∈ L 1 loc (U). In this situation, we will write as h = e −φ and call φ as a local weight. Note that φ is a collection of a function defined on small open sets. The curvature of a singular Hermitian metric h = e −φ is defined as a (1,1)-current Θ h = √ −1∂∂φ. A singular Hermitian metric h = e −φ is semi-positively curved (or h admits semi-positive curvature) if its local weight φ is plurisubharmonic on the set where φ is defined. In this case, its curvature is non-negative as a (1,1)-current. Let h 1 and h 2 be singular Hermitian metrics on L. We say that h 1 is more singular than h 2 when, for every relatively compact set U, there is a constant C > 0 such that the inequality h 1 ≥ Ch 2 holds on U. In this case we write h 1 sing h 2 . We say that h 1 and h 2 have equivalent singularities (written h 1 ∼ sing h 2 ) when both h 1 sing h 2 and h 1 sing h 2 hold. A semi-positively curved singular Hermitian metric h on L has minimal singularities if h sing h ′ for any semi-positively curved singular Hermitian metric h ′ . When X is compact, h sing h ′ holds if and only if there exists a constant C such that h 1 ≥ Ch 2 on X. To investigate singular Hermitian metrics, it will be convenient to consider globally defined functions corresponding to their local weights. For this reason, we introduce the notion of θ-plurisubharmonic functions here. Let θ be a smooth real (1,1)-form. We say that a function u ∈ L 1 loc (X) is a θ-plurisubharmonic function when the inequality θ + i∂∂u ≥ 0 holds as currents. We denote the set of θ-plurisubharmonic functions on X by PSH(X, θ). Let L be a holomorphic line bundle on X. Fix a smooth Hermitian metric h 0 on L with curvature θ. Then there is a one-to-one correspondence between θ-plurisubharmonic functions u and semi-positively curved singular Hermitian metrics h 0 ·e −u on L. We define θ-plurisubharmonic functions with minimal singularities similarly to the case of metrics. Namely, a θ-plurisubharmonic function u has minimal singularities (in PSH(X, θ)) if, for every θ-plurisubharmonic function u ′ , there exists a (local) constant C such that u ≥ u ′ +C on each compact set. For an R-line bundle L (i.e. a formal "line bundle" corresponding to an R-divisor), a notion of singular Hermitian metric on L is well-defined formally in this sense. Example 2.2.2. Assume X is compact and L is pseudo-effective, i.e. L admits a semipositively curved singular Hermitian metric. Fix a smooth metric h with curvature θ. Then, the function defined by V θ := sup{v ∈ PSH(X, θ) | v ≤ 0} is θ-plurisubharmonic. It is easily observed that V θ has minimal singularities in P SH(X, θ). The corresponding singular Hermitian metric h · e −V θ is denoted by h e , which is called the equilibrium metric. Example 2.2.3. Fix a smooth Hermitian metric h 0 on L. Let f 1 , f 2 , . . . , f N ∈ H 0 (X, L) be global holomorphic sections of L. Then we define a singular Hermitian metric h by the formula f 2 h := f 2 h 0 N j=1 f j 2 h 0 . In this manner, we obtain a semi-positively curved singular Hermitian metric h which is smooth on the Zariski open set N j=1 {f j = 0}. If L is big, there exist a finite number of sections f 1 , . . . , f N ∈ H 0 (X, L m ) for sufficiently large m such that {f 1 = · · · = f N = 0} = B(L) ( [Laz,2.1.21]). Here we denote the tensor product L ⊗m by L m . Then, we have a singular Hermitian metric on L m which is smooth on X \ B(L). By taking m-th root, we can define a singular Hermitian metric on L (we call it a Bergman-type metric on L obtained by f 1 , . . . , f N ). Example 2.2.4. Let X be a compact complex manifold and let L be a line bundle. Fix a smooth volume form dV on X and a smooth metric h = e −φ on L. Let θ be the curvature of h. We define a θ-plurisubharmonic function V φ,B by V φ,B := V h,B := sup 1 m log |f | 2 h m m ∈ Z, f ∈ H 0 (X, L m ), X |f | 2 h m dV ≤ 1 . The corresponding singular Hermitian metric on L and its local weight are denoted by h B = e −φ B . By Proposition 2.2.5 below, h B has minimal singularities when L is big. We use this construction when L is a Q-line bundle with a smooth metric h = e −φ , that is, for some integer m > 0, L m is an ordinary line bundle and h m = e −mφ is a smooth metric on L m . In this case, we take the smallest integer m > 0 such that L m is a Z-line bundle and define φ B by (1/m)(mφ) B . To compare the metrics h e and h B , we need the following proposition. Proposition 2.2.5 ([K2, Lemma 2.10]). Let X be a projective manifold and let L be a big line bundle. Let h = e −φ be a smooth Hermitian metric on L. Fix a smooth volume form dV on X. Then, there is a constant C such that the inequality V φ,B − C ≤ V θ ≤ V φ,B holds. Before starting the proof, we shall explain how we use Proposition 2.2.5 in §3. We shall apply it to a family of Q-line bundles of the form L α := L α 1 1 ⊗ L α 2 2 ⊗ · · · ⊗ L α r+1 r+1 , where L λ are Z-line bundles, α λ ≥ 0 and α 1 + · · · + α r+1 = 1. Let e −φ λ be a fixed smooth metric on L λ and let m be the smallest positive integer such that (L α ) m is a Z-line bundle. Then, the local weight mφ α := m λ α λ φ λ defines a smooth metric on (L α ) m . The constant C in Proposition 2.2.5 depends only on C 1 , C 2 and C(φ), which will be defined in the proof below. We note that constants C 1 and C 2 are independent of the choice of line bundles, and C(φ) only depends on the differences ( sup B ′′ j − inf B ′′ j )φ, where {B ′′ j } j is a open cover of X consisting of open balls. Thus there exists a constant C 3 depending on the metrics e −φ λ and independent of α, such that V mφ α ,B − log(C 1 C 2 ) − mC 3 ≤ V mθα ≤ V mφ α ,B . Dividing by m, we have that V φ α ,B − 1 m log(C 1 C 2 ) − C 3 ≤ V θα ≤ V φ α ,B . In conclusion, there exists a constant C such that we have (1) V φα,B − C ≤ V θα ≤ V φα,B for every α such that α λ ≥ 0 and α 1 + α 2 · · · + α r+1 = 1. Proof of Proposition 2.2.5. First we prove the inequality V θ ≤ V φ,B . Since L is big, there exits a singular Hermitian metric ψ + on L such that its curvature is a Kähler current, i.e. Θ ψ + ≥ ǫω for some ǫ > 0 and some Kähler form ω. Define a θ-plurisubharmonic function V + by ψ + = φ + V + . We may assume that V + ≤ 0. Let V ℓ : = (1 − 1/ℓ) V θ + (1/ℓ)V + and φ ℓ := φ + V ℓ . Then the curvature of the metric e −φ ℓ is a Kähler current. Now consider the following approximations: V φ,B,m := sup * 1 m log |f | 2 mφ f ∈ H 0 (X, L m ), X |f | 2 e −mφ dV ≤ 1 , and V φ ℓ ,B,m := sup * 1 m log |f | 2 mφ ℓ f ∈ H 0 (X, L m ), X |f | 2 e −mφ ℓ dV ≤ 1 . Then we have that V ℓ + V φ ℓ ,B,m ≤ V φ,B,m . Applying Demailly's approximation theorem ([D2, Theorem 13.21]) to φ ℓ , we have that V φ ℓ ,B,m ≥ V ℓ − C/m, where C is independent of ℓ and m. Hence we have V ℓ − C m ≤ V φ,B,m ≤ V φ,B . By letting m → ∞, we obtain V ℓ ≤ V φ,B . After that, letting ℓ → ∞, we have that V θ ≤ V φ,B . Next we prove the inequality V φ,B − C ≤ V θ . Fix a collection of open coordinate balls B ′ j ⊂ B ′′ j ⊂ B j such that {B ′ j } j is an open cover of X and the radii of B ′ j , B ′′ j and B j are 1/2, 1 and 2 respectively. Fix a local trivialization of L. Take f ∈ H 0 (X, L m ) with X |f | 2 e −mφ dV ≤ 1. Then, for every p ∈ B ′ j , we have that |f (p)| 2 ≤ 1 π n (1/2) 2n /n! |z−p|<1/2 |f | 2 dλ ≤C 1 C 2 · e m sup B ′′ j φ |z−p|<1/2 |f | 2 e −mφ dV ≤C 1 C 2 · e m sup B ′′ j φ . Here we write the constants as C 1 := 1 π n (1/2) 2n /n! and C 2 : = sup B ′′ j dλ/dV . It follows that |f (p)| 2 e −mφ(p) ≤ C 1 C 2 · exp(m(sup B ′′ j φ − φ(p))) ≤ C 1 C 2 · exp(m(sup B ′′ j − inf B ′′ j )φ). Thus we have that 1 m log |f (p)| 2 mφ ≤ (log C 1 C 2 )/m + sup B ′′ j − inf B ′′ j φ. It follows that V φ,B,m (p) ≤ (log C 1 C 2 )/m + sup B ′′ j − inf B ′′ j φ. The right-hand side is estimated by using the constants C 1 , C 2 and a constant C(φ) depends only on φ. Taking the supremum over m, we have that V φ,B (p) ≤ log(C 1 C 2 ) + C(φ). We denote this constant by C. Considering all B j , we have V φ,B − C ≤ V θ for some constant C. Projective bundles 3.1. Settings in the case of P r -bundle. Let Y be a projective manifold. Let M 1 , M 2 , · · · , M r and M r+1 be line bundles. We assume that the first r line bundles M 1 , . . . , M r are ample (we do not assume the ampleness of M r+1 ). Define a manifold X by X := P(M 1 ⊕ M 2 ⊕ · · · ⊕ M r+1 ) and a line bundle L on X by L := O P(M 1 ⊕M 2 ⊕···⊕M r+1 ) (1). Let us recall that P(E) denotes the projective space bundle of hyperplanes of E. Let π denote the natural projection X → Y . We regard Y as a submanifold of X via the inclusion P(M r+1 ) ⊂ P(M 1 ⊕ M 2 ⊕ · · · ⊕ M r+1 ) induced by the projection M 1 ⊕ M 2 ⊕ · · · ⊕ M r+1 → M r+1 . Let h λ (1 ≤ λ ≤ r+1 ) be smooth Hermitian metrics on M λ and θ λ be the curvature forms of h λ . Here we assume that every h λ (1 ≤ λ ≤ r) has a positive curvature, i.e. the curvature form θ λ is a positive (1,1)-form for every λ = 1, . . . , r. Let us denote by h L = e −ϕ L the naturally induced metric on L from h 1 , . . . , h r+1 by considering the Euler sequence. We denote by θ L the curvature of h L . Let L be a convex set defined in §1 as follows: L = α = (α 1 , . . . , α r ) ∈ R r ≥0 |α| ≤ 1, and c 1 (L| Y ) + r λ=1 α λ c 1 (M λ ⊗ L| −1 Y ) is pseudo-effective , where |α| denotes α 1 + α 2 + · · · + α r . Here we use the direct decomposition N Y /X = r λ=1 (L ⊗ π * M −1 λ )| Y (see Lemma 2.1.1). The following theorem is the main result of this section. Theorem 3.1.1. Let Y , M λ , X, L and h λ be as above. For an r-tuple of nonnegative real numbers α = (α 1 , . . . , α r ) with α 1 + α 2 + · · · + α r ≤ 1 and a real number α r+1 := 1 − α 1 −α 2 − · · · − α r , define a function u α (x) on X as follows: u α (x) := α 1 log |s 1 | 2 h 1 + α 2 log |s 2 | 2 h 2 + · · · + α r+1 log |s r+1 | 2 h r+1 + π * V θα . Here, s λ denotes the canonical section of a divisor Γ λ , h λ denotes the metric on the line bundle [Γ λ ] defined by h L /π * h λ , and θ α denotes the (1,1)-form r+1 λ=1 α λ θ λ . Then, (i) u α is a θ L -plurisubharmonic function. (ii) For every fixed x ∈ X, there exists the maximum value of the function u α (x) of α on α ∈ L . (iii) Define a function V (x) on X by V (x) := max (α 1 ,...,αr)∈ L u α (x). Then V is upper semicontinuous and θ L -plurisubharmonic. (iv) V is a θ L -plurisubharmonic function with minimal singularities. 3.2. The relation between Theorem 3.1 and Theorem 1.2. Before proving Theorem 3.1.1, we shall explain the relation between Theorem 3.1.1 and Theorem 1.0.3. We assume that X, Y and L are those in Theorem 1.0.3. We construct a "projective space bundle model" ( X, Y , L), to which we apply Theorem 3.1.1. We define X by P( I Y ⊕ N * Y /X ), Y by P(I Y ), and L by O P(I Y ⊕N * Y /X ) (1) . In §4, we use a trick called the maximum construction to get a metric with minimal singularities on L from that on L. To check that ( X, Y , L) satisfies the assumption of Theorem 3.1.1, we have to choose appropriate line bundles M λ on Y as in the following lemma. M λ = L| Y ⊗ N −1 λ for λ = 1, . . . , r and M r+1 = L| Y , one have that X ∼ = P(M 1 ⊕ · · · ⊕ M r+1 ) and L = O P(M 1 ⊕M 2 ⊕···⊕M r+1 ) (1). Proof. We have that P(M 1 ⊕ · · · ⊕ M r+1 ) = P(L| Y ⊗ (N −1 1 ⊕ · · · ⊕ N −1 r ⊕ I Y )) ∼ = P(N −1 1 ⊕ · · · ⊕ N −1 r ⊕ I Y ) = P(N * Y /X ⊕ I Y ) = X. We also have that O P(M 1 ⊕M 2 ⊕···⊕M r+1 ) (1) = O P(N −1 1 ⊕N −1 2 ⊕···⊕N −1 r ⊕I Y ) (1) ⊗ π * M r+1 = L. Note that, under this choice, we have N Y /X = r λ=1 (L ⊗ π * M −1 λ )| Y . To use the maximum construction argument in §4, we have to use the following lemma. Lemma 3.2.2. In this situation, one have SB( L) ⊂ Y . Proof. Recall that s λ is the canonical section of the line bundle [Γ λ ] = π * M −1 λ ⊗ L (Lemma 2.1.1). For every global section f ∈ H 0 (Y, π * M m λ ) (m ≥ 0), we have that s m λ ⊗ π * f ∈ H 0 ( X, L m ). By the assumption of Theorem 1.0.3, the line bundle L| Y ⊗ N −1 λ is ample for every λ = 1, . . . , r. Therefore, for sufficiently large m, there exist global sections of L whose common zero is Γ λ . Using this argument for every λ, we have that B( L) ⊂ Y . 3.3. Proof of Theorem 3.1.1. 3.3.1. The outline of the proof. We obtain (i) easily from the construction of u α . We will prove (ii) and (iii) in §3.3.2. Now we explain the outline proof of (iv). Fix a Kähler form ω on Y . Define functions V Q and V Q B on X by V Q := sup α=(α 1 ,...,αr) α∈ L ∩Q r * u α and V Q B := sup α=(α 1 ,...,αr) α∈ L ∩Q r * r+1 λ=1 α λ log |s λ | 2 h λ + π * V h α 1 1 h α 2 2 ···h α r+1 r+1 ,B , where V h α 1 1 h α 2 2 ···h α r+1 r+1 ,B is a function on Y defined as in Example 2.2.4 with respect to the volume form ω n on Y . In §3.3.4, we prove that V θ L ,B ≤ V Q B + C holds for some constant C, where V θ L ,B is also defined as in Example 2.2.4 (we specify the volume form on X later in §3.3.3). Then we have that, for some C ′ ≥ 0, V θ L ,B ≤ V Q B + C ≤ V Q + C ′ ≤ V + C ′ . Here, the second inequality follows from the equation (1) before the proof of Proposition 2.2.5. 3.3.2. Proof of Theorem 3.1.1 (ii) and (iii). In this subsection, we will show the upper semicontinuity of V . For simplicity of notation, we write V α instead of V θα = sup{ψ ∈ P SH(Y, θ α ) | ψ ≥ 0}. We will show the following proposition. Proposition 3.3.1. The function F : L × Y → R ∪ {∞} : F (α, y) := V α (y) is upper semi-continuous. From this proposition and compactness of L , a standard argument shows (ii) and (iii) of Theorem 3.1.1. In this subsection, we write α ≤ β when α λ ≤ β λ for every λ. To prove Proposition 3.3.1, we need the following lemma. Lemma 3.3.2. Let α and β be points in L . (i) If α ≤ β, Vα 1−|α| ≤ V β 1−|β| . (ii) lim β↓α V β 1−|β| = Vα 1−|α| , where lim β↓α means the limit as β approaches to α under the condition α ≤ β. Proof. (i) Consider the local weights ϕ α := r λ=1 α λ ϕ λ + (1 − |α|)ϕ r+1 . First, we use the equation ϕ β + 1 − |β| 1 − |α| · V α = 1 − |β| 1 − |α| · (ϕ α + V α ) + r λ=1 β λ − 1 − |β λ | 1 − |α λ | · α λ · ϕ λ . As the right-hand side is plurisubharmonic, we have that 1−|β| 1−|α| · V α is θ β -plurisubharmonic. Since this function is non-positive, we obtain 1 − |β| 1 − |α| · V α ≤ V β by the definition of V β . (ii) Take a sequence {β (ν) } ∞ ν=1 , β (ν) = (β (ν) 1 , β(ν) 2 , . . . , β (ν) r ), with β (ν) λ ↓ α λ for every λ = 1, 2, . . . , r. We shall prove lim ν→∞ V β (ν) 1 − |β (ν) | = V α 1 − |α| . By (i), the inequality lim ν→∞ V β (ν) 1−|β (ν) | = Vα 1−|α| holds. Hence it is sufficient to prove the converse inequality. Let us consider the local weight 1 1 − |β (ν) | · (ϕ β (ν) + V β (ν) ) = ϕ r+1 + r λ=1 β (ν) λ 1 − |β (ν) | · ϕ λ + V β (ν) 1 − |β (ν) | . By the right-hand side, this weight is clearly decreasing in ν. Moreover, by focusing on the left-hand side, we have that this weight is plurisubharmonic. Therefore the limit ϕ α /(1 − |α|) + lim ν→∞ V β (ν) /(1 − |β (ν) |) is also plurisubharmonic. As the function (1 − |α|) · lim ν→∞ V β (ν) 1−|β (ν) | is non-positive and θ α -plurisubharmonic, we have that (1 − |α|) · lim ν→∞ V β (ν) 1 − |β (ν) | ≤ V α . Proof of Proposition 3.3.1. Fix a point (α 0 , y 0 ) ∈ L × Y . We shall prove the upper semicontinuity of F at (α 0 , y 0 ). First, we treat the case where |α 0 | = 1. It is sufficient to prove lim sup L ×Y ∋(α,y)→(α 0 ,y 0 ) V α (y) ≤ V α 0 (y 0 ). Since the forms θ 1 , θ 2 . . . , θ r are positive, θ α 0 is also positive. Thus V α 0 (y 0 ) = 0 and upper semicontinuity is trivial in this case. Next, we treat the case when |α 0 | < 1. In this case, there exists ε > 0 such that α 0 + ε := (α 0 1 + ε, α 0 2 + ε, . . . , α r + ε) lies in the interior of L . Take a sufficiently small neighborhood U α 0 ,ε in L of α 0 such that every point α in U α 0 ,ε satisfies α ≤ α 0 + ε. It follows from Lemma 3.3.2 (i) that lim sup L ×Y ∋(α,y)→(α 0 ,y 0 ) V α (y) 1 − |α| = lim sup Uα 0 ,ε×Y ∋(α,y)→(α 0 ,y 0 ) V α (y) 1 − |α| ≤ lim sup Y ∋y→y 0 V α 0 +ε (y) 1 − |α 0 + ε| holds. By the upper semicontinuity of V α 0 +ε 1−|α 0 +ε| , we have lim sup L ×Y ∋(α,y)→(α 0 ,y 0 ) V α (y) 1 − |α| ≤ V α 0 +ε (y 0 ) 1 − |α 0 + ε| . Letting ǫ → 0, we have lim sup L ×Y ∋(α,y)→(α 0 ,y 0 ) V α (y) 1 − |α| ≤ V α 0 (y 0 ) 1 − |α 0 | , which shows the upper semicontinuity of the function F (α, y)/(1 − |α|) near (α 0 , y 0 ). By multiplying by the continuous function (α, y) → 1 − |α|, we have that F itself is also upper semicontinuous. 3.3.3. Integral formula. In the following, we consider the local coordinates on U (r+1) j defined as in §2. Recall that we defined homogeneous fiber coordinates [x j,1 : x j,2 : · · · : x j,r+1 ] on U (r+1) j . We define the fiber coordinate z 1 , . . . , z r on U (r+1) j as z λ := x j,λ /x j,r+1 . Then, we have s λ = z λ (1 ≤ λ ≤ r) and s r+1 = 1. We rewrite V Q B as follows: V Q B = sup ℓ λ ,m∈Z ℓ/m∈ L 2ℓ 1 m log |z 1 | + · · · + 2ℓ r m log |z r | + (ϕ ℓ/m ) B (y) , where ϕ ℓ/m is defined by (ℓ 1 /m)ϕ 1 + (ℓ 2 /m)ϕ 2 + · · · + (ℓ r /m)ϕ r + (1 − (ℓ 1 + ℓ 2 + · · · + ℓ r )/m)ϕ r+1 and φ B denotes a local weight corresponding to a function V φ,B . Here we regard z λ as a holomorphic function on U (r+1) j . Define a volume form dV on X by setting dV := π * ω n n! ∧ e ϕ 1 +···+ϕ r+1 (|z 1 | 2 e ϕ 1 + · · · + |z r | 2 e ϕr + e ϕ r+1 ) r+1 ( √ −1) r dz 1 ∧ dz 1 ∧ . . . ∧ dz r ∧ dz r on U (r+1) j . Here, ω is a fixed Kähler form on Y as in §3.3.1. A simple calculation shows that this form extends to a smooth volume form on X. In the proof of Theorem 3.1.1, we need the following integral formula. Here we integrate a function F = |z 1 | 2t 1 · · · |z r | 2tr e −ϕ L = |z 1 | 2t 1 · · · |z r | 2tr |z 1 | 2 e ϕ 1 + · · · + |z r | 2 e ϕr + e ϕ r+1 by using the volume form dV . Lemma 3.3.3. (z 1 ,...,zr)∈C r |z 1 | 2t 1 · · · |z r | 2tr · e ϕ 1 +···+ϕ r+1 (|z 1 | 2 e ϕ 1 + · · · + |z r | 2 e ϕr + e ϕ r+1 ) r+2 ( √ −1) r dz 1 ∧ dz 1 ∧ . . . ∧ dz r ∧ dz r = (2π) r Γ(1 + t 1 ) · · · Γ(1 + t r )Γ(2 − (t 1 + · · · + t r )) Γ(r + 2)e ϕt . Proof. To make ideas clear, we first prove the case that r = 2. In this case, the equation we want to prove is as follows: (z 1 ,z 2 )∈C 2 |z 1 | 2t 1 |z 2 | 2t 2 · e φ 1 +φ 2 +φ 3 (|z 1 | 2 e φ 1 + |z 2 | 2 e φ 2 + e φ 3 ) 4 ( √ −1) 2 dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 = (2π) 2 Γ(1 + t 1 )Γ(1 + t 2 )Γ(2 − (t 1 + t 2 )) Γ(4)e φt , where φ t := t 1 φ 1 + t 2 φ 2 + (1 − (t 1 + t 2 ))φ 3 . Write z 1 = a 1 e iθ 1 and z 2 = a 2 e iθ 2 and define A λ by A λ := e φ λ (λ = 1, 2, 3). Then we have C 2 |z 1 | 2t 1 |z 2 | 2t 2 · A 1 A 2 A 3 (|z 1 | 2 A 1 + |z 2 | 2 A 2 + A 3 ) 4 ( √ −1) 2 dz 1 ∧ dz 1 ∧ dz 2 ∧ dz 2 =(2π) 2 2 2 R 2 >0 a 1+2t 1 1 a 1+2t 2 2 A 1 A 2 A 3 (a 2 1 A 1 + a 2 2 A 2 + A 3 ) 4 da 1 da 2 . Next, we use the following polar coordinates (s, θ): s := A 1 A 3 a 2 1 + A 2 A 3 a 2 2 1/2 ∈ R ≥0 , θ := Arctan √ A 2 a 2 √ A 1 a 1 ∈ [0, π/2]. Note that these coordinates are written as a 1 = A 3 /A 1 ·s cos θ and a 2 = A 3 /A 2 ·s sin θ. Then we have R 2 >0 a 1+2t 1 1 a 1+2t 2 2 A 1 A 2 A 3 (a 2 1 A 1 + a 2 2 A 2 + A 3 ) 4 da 1 da 2 =A −t 1 1 A −t 2 2 A t 1 +t 2 −1 3 ∞ s=0 s 3+2t 1 +2t 2 (s 2 + 1) 4 ds π/2 θ=0 (cos θ) 1+2t 1 (sin θ) 1+2t 2 dθ. We denote this value by I. We will calculate the integration in s and θ respectively. First we consider the integration in s. To compute, we use the substitution σ = s 2 . Then, ∞ s=0 s 3+2t 1 +2t 2 (s 2 + 1) 4 ds = 1 2 ∞ σ=0 σ 1+t 1 +t 2 (σ + 1) 4 dσ = 1 2 B(2 + t 1 + t 2 , 2 − t 1 − t 2 ). At the last equality, we use the formula [OLBC,5.12.3] for the beta function. Next, we consider the integration in θ. By the formula [OLBC,5.12.2], we have π/2 θ=0 (cos θ) 1+2t 1 (sin θ) 1+2t 2 dθ = 1 2 B(1 + t 1 , 1 + t 2 ). By [OLBC,5.12.1], we have (2π) 2 2 2 I = (2π) 2 A −t 1 1 A −t 2 2 A t 1 +t 2 −1 3 Γ(2 − t 1 − t 2 )Γ(1 + t 1 )Γ(1 + t 2 ) Γ(4) . The proof in the case that r = 2 is complete. Now we will prove the theorem in the general case. Since the proof is almost the same, we only explain the essential points. We use the coordinate change z λ = a λ e iθ λ and get the expression of a 1 , . . . , a r . Then we use the r-dimensional polar coordinate a 1 = A r+1 /A 1 · s cos θ 1 a 2 = A r+1 /A 2 · s sin θ 1 cos θ 2 a 3 = A r+1 /A 3 · s sin θ 1 sin θ 2 cos θ 3 . . . a r−1 = A r+1 /A r−1 · s sin θ 1 sin θ 2 sin θ 3 · · · sin θ r−2 cos θ r−1 a r = A r+1 /A r · s sin θ 1 sin θ 2 sin θ 3 · · · sin θ r−2 sin θ r−1 , where A λ = e ϕ λ . The determinant of the Jacobian matrix is written as A r+1 A 1 · · · A r+1 A r s r−1 (sin θ 1 ) r−2 (sin θ 2 ) r−3 · · · (sin θ r−2 ) 1 . Finally we use formulae of the beta function to deduce the conclusion. 3.3.4. Proof of Theorem 3.1.1 (iv). As we described in §3.3.1, we shall prove V θ L ,B ≤ V Q B + C. Fix a global section F ∈ H 0 (X, L m ) with X |F | 2 h m L dV = 1. We will decompose F into orthogonal components using the following claim. (X, L m ) = H 0 (X, O X (m)) = H 0 (Y, S m (M 1 ⊗· · ·⊗M r+1 )) ∼ = ℓ 1 +···+ℓ r+1 =m H 0 (Y, M ℓ 1 1 ⊗ · · · ⊗ M ℓ r+1 r+1 ) is the orthogonal decomposition with respect to the L 2 -norm defined by Hermitian metric h m L of L m and the volume form dV . Here, S m E denotes the m-th symmetric tensor of E. Proof. By the decomposition above, we have injective morphisms H 0 (Y, M ℓ 1 1 ⊗ · · · ⊗ M ℓ r+1 r+1 ) → H 0 (X, L m ) for every (r + 1)-tuple of non-negative integers ℓ = (ℓ 1 , . . . , ℓ r+1 ) with ℓ 1 + · · · + ℓ r+1 = m. In the following, M ℓ 1 1 ⊗ · · · ⊗ M ℓ r+1 r+1 will be denoted by M ℓ . This morphism maps f ℓ ∈ H 0 (Y, M ℓ ) to s ℓ 1 1 · s ℓ 2 2 · · · s ℓ r+1 r+1 π * f ℓ . We will prove that, for any ℓ = (ℓ 1 , . . . , ℓ r+1 ) and ℓ ′ = (ℓ ′ 1 , . . . , ℓ ′ r+1 ) with ℓ = ℓ ′ , two sections β := s ℓ 1 1 · s ℓ 2 2 · · · s ℓ r+1 r+1 π * f ℓ and β ′ := s ℓ ′ 1 1 · s ℓ ′ 2 2 · · · s ℓ ′ r+1 r+1 π * f ℓ ′ are orthogonal. We regard β and β ′ as holomorphic functions via the local trivialization. Then, by the equations s λ = z λ (λ = 1, 2, . . . , r) and s r+1 = 1, it follows that X β, β ′ h m L dV = X (z ℓ 1 1 · z ℓ 2 2 · · · z ℓr r π * f ℓ ) · (z ℓ ′ 1 1 · z ℓ ′ 2 2 · · · z ℓ ′ r r π * f ℓ ′ )e −mϕ L dV = y∈Y z∈π −1 (y) F (z)( √ −1) r dz 1 ∧ dz 1 ∧ . . . ∧ dz r ∧ dz r ω n n! , where F (z) = (z ℓ 1 1 · · · z ℓr r π * f ℓ )(z ℓ ′ 1 1 · · · z ℓ ′ r r π * f ℓ ′ )e −mϕ L e ϕ 1 +···+ϕ r+1 (|z 1 | 2 e ϕ 1 + · · · + |z r | 2 e ϕr + e ϕ r+1 ) r+1 . Write z λ = s λ r iθ λ . Considering integration in θ λ 's, we have that it becomes 0 if ℓ = ℓ ′ . Therefore, two sections are orthogonal for different ℓ and ℓ ′ . Let us decompose F ∈ H 0 (X, L m ) into the sum of the components β ℓ = s ℓ 1 1 · s ℓ 2 2 · · · s ℓ r+1 r+1 ·π * f ℓ , where f ℓ ∈ H 0 (Y, M ℓ 1 1 ⊗ · · · ⊗ M ℓ r+1 r+1 ), according to the orthogonal decomposition obtained in the claim. By the orthogonality, we have X |β ℓ | 2 h m L dV ≤ X |F | 2 h m L dV (= 1). Next we estimate the norm of f ℓ . Claim 3.3.5. There exist constants C 1 and C 2 independent of m such that y∈Y |f ℓ (y)| 2 e −m(ϕ ℓ/m ) ω n n! ≤ C 1 C m 2 X |β ℓ | 2 e −mϕ L dV, where ϕ ℓ/m stands for ℓ 1 m ϕ 1 + ℓ 2 m ϕ 2 + · · · + ℓr m ϕ r + (1 − ℓ 1 +···+ℓr m )ϕ r+1 as in §3.3.3. Proof. Let z ℓ := z ℓ 1 1 · z ℓ 2 2 · · · z ℓr r . Then we have z ℓ π * f ℓ = β ℓ under the trivialization. We estimate the right-hand side from below. We have that X |β ℓ | 2 e −mϕ L dV = X |z ℓ f ℓ (x)| 2 e −mϕ L dV = y∈Y |f ℓ (x)| 2 z∈π −1 (y) |z ℓ | 2 e −mϕ L dP ω n n! , where dP is the measure on the fiber π −1 (y) defined as dP := e ϕ 1 +···+ϕ r+1 (|z 1 | 2 e ϕ 1 + · · · + |z r | 2 e ϕr + e ϕ r+1 ) r+1 ( √ −1) r dz 1 ∧ dz 1 ∧ . . . ∧ dz r ∧ dz r . By Hölder's inequality, we have that z∈π −1 (y) |z ℓ | 2/m · e −ϕ L dP ≤ π −1 (y) |z ℓ | 2 · e −mϕ L dP 1 m · π −1 (y) 1 · dP m−1 m. 14 A straightforward computation similar to the proof of Lemma 3.3.3 shows that the value of the integral π −1 (y) dP in the right-hand side is a constant independent of y, which we will denote by I. By Lemma 3.3.3, the integral in the left-hand side is equal to (2π) r Γ 1 + ℓ 1 m · · · Γ 1 + ℓr m Γ 2 − ℓ 1 +···+ℓr m Γ(r + 2)e ϕ ℓ . As Γ(t) is bounded for 1 ≤ t ≤ 2 from below, there exists a positive constant C > 0 such that Γ(t) ≥ C for 1 ≤ t ≤ 2. Combining these estimates, we have that (2π) r C r+1 e −ϕ ℓ Γ(r + 2) ≤ z∈π −1 (y) |z ℓ | 2 e −mϕ L dP 1 m · I m−1 m . Thus we obtain X |β ℓ | 2 e −mϕ L dV = y∈Y |f ℓ (y)| 2 z∈π −1 (y) |z ℓ | 2 e −mϕ L dP ω n n! ≥ y∈Y |f ℓ (y)| 2 I −(m−1) (2π) r C r+1 e −ϕ ℓ Γ(r + 2) m ω n n! = (2π) rm C rm I −(m−1) Γ(r + 2) m y∈Y |f ℓ (y)| 2 e −mϕ ℓ ω n n! . By the previous estimates and the definition of the Bergman-type metric, we have 1 m log |f ℓ | 2 ≤ (ϕ ℓ ) B + log C 1 + m log C 2 m . To prove the desired inequality V θ L ,B ≤ V Q B + C, we will estimate the norm of the section F = ℓ z ℓ π * f ℓ from above. Assume ℓ = (ℓ 1 , . . . , ℓ r ) satisfies ℓ/m ∈ L . By the argument as in the proof of [K2,Proposition 2.5(2)], we have 2 m log |F | ≤ max ℓ 2ℓ 1 m log |z 1 | + · · · + 2ℓ r m log |z r | + (φ ℓ /m) B + 1 m log C 1 + log C 2 . Let C := log C 1 + log C 2 . Then the right-hand side is estimated by V Q B + C by definition. Since m is arbitrary, the supremum of the left-hand side over m and F is V θ L ,B , and the proof is complete. Proof of the main results In this section, we prove Theorem 1.0.4 and Theorem 1.0.3. Let X, Y , and L be those as in Theorem 1.0.3 (in particular, we assume that Y admits a holomorphic tubular neighborhood). The idea of the proof is based on [K2]: we first construct a new "projective space bundle model" ( X, Y , L) from (X, Y, L), and construct a metric of L with minimal singularities by using the metric on L as in §3. See also §3.2 for the relation between the models (X, Y, L) and ( X, Y , L). 4.1. The projective space bundle model ( X, Y , L). Let X, Y , and L be as in Theorem 1.0.3. Denote by X the total space of the projective space bundle P(I Y ⊕ N * Y /X ), by Y the subvariety P(I Y ), and by L the line bundle O P(I Y ⊕N * Y /X ) (1) ⊗ π * L| Y , where π : X → Y is the natural projection. Note that X = P(I Y ⊕ N Y /X ) and Y = P(I Y ). It is easily observed that X is a compactification of the normal bundle N Y /X , and from this point of view, Y is regarded as a zero section of N Y /X . Therefore, by the assumption on the existence of a holomorphic tubular neighborhood, there exists a neighborhood V of Y in X, V of Y in X and a biholomorphic map i : The proof is based on [K2, §3]. First we prove the following proposition as a higher codimensional analogue of [K2, Proposition 3.1]: V ∼ = V with i| Y = π| Y . Proposition 4.1.2. By shrinking V suitably, one have the following: (1) (a version of Rossi's theorem) The natural map H 1 (V, O V ) → H 1 (V, O V /I n Y ) is injec- tive for some n ≥ 1, where I Y the defining ideal sheaf of Y ⊂ V . (2) If H 1 (Y, S ℓ N * Y /X ) vanishes for every ℓ ≥ 1, then the groups Pic 0 (V ) and Pic 0 (Y ) are isomorphic. Proof. See the proof of [K2, Proposition 3.1] (We intrinsically use Rossi's theorem [R,Theorem 3]. Here we remark that, by Lemma 4.1.3 below, we may assume that V is a strongly pseudoconvex domain which has Y as a maximal compact set). Lemma 4.1.3. There exists a strongly pseudoconvex holomorphic tubular neighborhood V of Y which has Y as a maximal compact analytic set. Proof. As Y admits a holomorphic tubular neighborhood, it is sufficient to show the lemma by assuming X = X. Take a C ∞ Hermitian metric h N −1 λ on N −1 λ with positive curvature. Taking a local coordinate y of Y and pulling it back by π, we regard (z, y) = (z 1 , z 2 , . . . , z r , y) as a local coordinates system of X, where z λ is a fiber coordinate of N λ . By considering the sublevel set of the function Φ : N Y /X → R defined by Φ(z, y) := r λ=1 |z λ | 2 e ϕ λ (y) , where ϕ λ is the local weight of h N −1 λ , the lemma follows. Proof of Proposition 4.1.1. First note that K −1 Y ⊗ S ℓ N * Y /X = α∈Z r ≥0 ,|α|=ℓ K −1 Y ⊗ r λ=1 N −α λ λ holds. As K −1 Y ⊗ N −1 λ and N µ are positive for each λ, µ = 1, 2, . . . , r, it follows from Kodaira's vanishing theorem that H 1 (Y, S ℓ N * Y /X ) vanishes for each ℓ ≥ 1. Thus, by Proposition 4.1.2, it is sufficient to show that L| Y ∼ = π| * Y L holds. The line bundle O P(I Y ⊕N * Y /X ) (1) corresponds to the divisor P(N * Y /X ) ⊂ P(I Y ⊕N * Y /X ), which does not intersect Y . Therefore we have L| Y = O P(I Y ⊕N * Y /X ) (1)| Y ⊗ π| * Y L = π| * Y L. 4.2. Minimal singular metrics on L and L. Let X, Y , and L be those in Theorem 1.0.3, and let X, Y , L, V , and V be those in the previous subsection. Here we prove the following: Proposition 4.2.1. Let h be a metric of L with minimal singularities and h be a metric of L with minimal singularities. Then h| V ∼ sing (i −1 ) * h| V holds at each point in V . The proof of Proposition 4.2.1 is based on the "maximum construction technique" which is also used in the proof of [K2,Theorem 1.2]. Fix a C ∞ metric h ∞ of L and denote by θ the curvature tensor Θ h∞ . Set ϕ V := sup{ϕ ∈ P SH(V, θ| V ) | ϕ ≤ 0 on V } and ϕ X := sup{ϕ ∈ P SH(X, θ) | ϕ ≤ 0 on X}. We first show the following: Lemma 4.2.2. It holds that ϕ V ∼ sing ϕ X . In particular, the restriction of a metric of L with minimal singularities to V has singularities equivalent to the metric h ∞ | V · e −ϕ V . Proof. As the inequality ϕ X | V ≤ ϕ V is easily obtained, all we have to do is to show the existence of a constant C with ϕ V ≤ ϕ X | V +C. As L is big and B(L) = Y , we can take an integer m ≥ 1 and sections f 1 , f 2 , . . . , f ℓ ∈ H 0 (X, L m ) such that the common zero of these sections is Y [Laz,2.1.21]. Denote by h a the Bergman type metric on L constructed from f 1 , f 2 , . . . , f ℓ (see Example 2.2.3 ). Define the function ϕ a by h a = h ∞ · e −ϕa . We may assume that ϕ a ≤ 0 holds on V . Fix a relatively compact open neighborhood V 0 ⋐ V of Y and set C 1 := − inf V \V 0 ϕ a . Consider a θ-plurisubharmonic function ϕ := max{ϕ V − C 1 , ϕ a }. As ϕ V − C 1 ≤ −C 1 ≤ ϕ a holds on V \ V 0 , we have ϕ = ϕ a on each point in V \ V 0 . Thus we can extend ϕ to whole X by defining ϕ(x) := ϕ a (x) for each x ∈ X \V 0 . It is clear from the construction that ϕ ∈ P SH(X, θ). Set C 2 := max X ϕ. Then, as ϕ − C 2 ≤ 0, we obtain that ϕ − C 2 ≤ ϕ X . Therefore it holds that ϕ V − C 1 − C 2 ≤ ϕ − C 2 ≤ ϕ X . Proof of Proposition 4.2.1. Take a C ∞ Hermitian metric h ∞ on L and h ∞ on L with h ∞ | V = i * h ∞ (here we used Proposition 4.1.1). By Lemma 4.2.2, it follows that ϕ V ∼ sing ϕ X , where ϕ V and ϕ X are as above. Set ϕ V := sup{ϕ ∈ P SH( V , Θ h∞ | V ) | ϕ ≤ 0 on V } and ϕ X := sup{ϕ ∈ P SH( X, Θ h∞ ) | ϕ ≤ 0 on X}. By the arguments in §3.2, we can apply Lemma 4.2.2 also to the projective space bundle model ( X, L) to obtain that ϕ V ∼ sing ϕ X . As h ∞ | V = i * h ∞ , we have that ϕ V = i * ϕ V . Therefore it follows that ϕ X ∼ sing i * ϕ X . Let X be a complex manifold and let Y ⊂ X be a compact complex submanifold of codimension r. In this section, we investigate when Y admits a holomorphic tubular neighborhood V in X. In particular, we here study a higher codimensional analogue of Grauert's theorem ([G], the case of r = 1, see also Theorem 5.1.4 below). 5.1. A higher codimensional analogue of Grauert's theorem. In this subsection, we show the following: Proposition 5.1.1. Assume that N Y /X admits a direct decomposition N Y /X = N 1 ⊕ N 2 ⊕ · · · ⊕ N r into r negative line bundles. Assume also that H 1 (E, O P(N Y /X ) (ν)) = 0 and H 1 (E, T E ⊗ O P(N Y /X ) (ν)) = 0 hold for every ν ≥ 1, where E is the total space of the projective space bundle P(N Y /X ). Then Y admits a holomorphic tubular neighborhood. Note that Proposition 5.1.1 is the Grauert's theorem when r = 1. We will consider a blow-up p : W → X of X along Y and apply the Grauert's theorem to E ⊂ W to show this proposition, where we are regarding E as the exceptional divisor (see [D1,Proposition 12.4]). For this purpose, we first show the following: Lemma 5.1.2. Assume that E admits a holomorphic tubular neighborhood in W . Then Y also admits a holomorphic tubular neighborhood in X. Proof. Denote by Y the zero section of π : N Y /X → X. Take a neighborhood V of Y . We denote by W the blow-up p : W → V of V along Y and by E ⊂ W the exceptional set. By the assumption (and by shrinking X if necessary), we may assume that there exists a biholomorphic map F : W → W with F | E = q| E , where q : O P(N Y /X ) (−1) → E(= P(N Y /X )) is the natural projection (here we are regarding W as a neighborhood of the zero section E of the line bundle O P(N Y /X ) (−1), see also [D1,Proposition 12.4]). First, let us construct a holomorphic map g : V → Y with g| Y = id Y such that the diagram W F / / W p / / V g E q| E / / E / / Y is commutative. It is clear that such function g is uniquely determined in the set-theoretic sense. It also follows from a standard argument that this g is a continuous map. It is also clear that g| V \Y is biholomorphic and that g| Y = id Y . Thus the existence of the holomorohic map g follows from Riemann's extension theorem (see Lemma 5.1.3 below). Next we show that the biholomorphic map f : V \ Y ∼ = V \ Y induced by F | W \ E extends to the biholomorphism f : V ∼ = V with f | Y = π| Y . Note that, by construction, the fibration structures π| V : V → Y and g : V → Y are preserved by f . Therefore, by a simple topological argument, it follows that there uniquely exists a continuous map f : V → V with f | V \ Y = f , and that this f satisfies f | Y = π| Y . The regularity of f and f −1 is shown again by Lemma 5.1.3 below. Lemma 5.1.3. Let M and N be complex manifolds, let Z ⊂ M be a submanifold with codimension greater than or equal to 1, and let h : M → N be a continuous map. Assume that h| M \Z is holomorphic. Then h is holomorphic on M. Proof. Take a point z ∈ Z and an open ball U ′ ⊂ N with coordinate system η = (η 1 , η 2 , . . . , η n ) around h(z) (n := dim N). We may assume that U ′ = {|η| < ε} for some ε > 0. Take a sufficiently small neighborhood U ⊂ M of z so that U ⊂ h −1 (U ′ ) (and thus h(U) ⊂ U ′ ). In what follows, we show the lemma by replacing M with U, N with U ′ , and h with h| U (in particular, we are regarding N as an open ball of C n ). It is sufficient to show each function h λ is holomorphic on z, where h = (h 1 , h 2 , · · · , h n ) is the decomposition by the coordinate system η = (η 1 , η 2 , . . . , η n ). As h λ is continuous (and thus it is locally bounded), we may assume that the L 2 -norm of h λ | U \Z is bounded by shrinking U if necessary. Therefore it follows from Riemann's extension theorem that we can extend h λ | U \Z to a holomorphic function h λ : U → C. Since U \ Z ⊂ U is a dense subset, we conclude that h λ = h λ , which proves the lemma. By Lemma 5.1.2, all we have to do is to investigate when E admits a holomorphic tubular neighborhood in W . We apply the following Grauert's theorem to this problem: Theorem 5. 1.4 ([G,Sats 7,p. 363], see also [CM,Theorem 4.4]). Let M be a complex manifold and let Z ⊂ M be a strongly exceptional subvariety of pure codimension 1. Proof of Proposition 5.1.1. By the assumption, Lemma 5.1.2 and Theorem 5.1.4, all we have to do is to show that E ⊂ W is an exceptional subset (in the sense of Grauert). By [Lau,Theorem 4.9,6.12], [G,Satz 8,p. 353], and [HR,Lemma 11] (see also [CM,Theorem 3.6]), it is sufficient to see the following two conditions: (i) N E/W is negative, and (ii) O W /I 2 E ∼ = O W /I E 2 , where I E ⊂ O W and I E ⊂ O W are the defining ideal sheaves of E and E, respectively. (i) follows from N −1 E/W = O P(N Y /X ) (1) = O P(N −1 1 ⊕N −1 2 ⊕···⊕N −1 r ) (1) and the assumption that N λ is negative. (ii) follows from the condition H 1 (E, T E ⊗O P(N Y /X ) (1)) = 0 (see [CM,Proposition 1.10,1.11]). 5.2. A sufficient condition for the existence of a holomorphic tubular neighborhood. In this subsection, we show the following lemma as an application of Proposition 5.1.1: Lemma 5.2.1. Let X be a complex manifold and let Y be a compact complex submanifold. Assume that N Y /X admits a direct decomposition N Y /X = N 1 ⊕ N 2 ⊕ · · · ⊕ N r into r negative line bundles. Assume also the following three conditions: (i) N λ ∼ = N µ for each λ and µ, (ii) N −1 λ ⊗ K −1 Y ⊗ T Y is Nakano positive, and (iii) N −1 λ ⊗ K −1 Y is ample for each λ. Then Y admits a holomorphic tubular neighborhood in X. Note that, when T Y is holomorphically trivial, conditions (ii) and (iii) are automatically satisfied. Proof. By Proposition 5.1.1, it is sufficient to show H 1 (E, O P(N Y /X ) (ν)) = 0 and H 1 (E, T E ⊗ O P(N Y /X ) (ν)) = 0 for every ν ≥ 1, where E := P(N Y /X ). Note that it follows from condition (i) that E ∼ = Y × P r . By the relative Euler sequence 0 → I E → p| * E N Y /X ⊗ O P(N Y /X ) (1) → T E/Y → 0, it turns out that it is sufficient to show the following four vanishing assertions: H 1 (E, O P(N Y /X ) (ν)) = 0, H 2 (E, O P(N Y /X ) (ν)) = 0, H 1 (E, O P(N Y /X ) (ν + 1) ⊗ p| * E N λ ) = 0, and H 1 (E, O P(N Y /X ) (ν) ⊗ p| * E T Y ) = 0 for each ν ≥ 1. By Nakano's vanishing theorem, the problem is reduced to show Nakano positivity for the following three vector bundles: K −1 E ⊗ O P(N Y /X ) (1), K −1 E ⊗ O P(N Y /X ) (2) ⊗ p| * E N λ , and K −1 E ⊗ O P(N Y /X ) (1) ⊗ p| * E T Y . As K −1 E ∼ = p| * E (N r ⊗ K −1 Y ) ⊗ O P(N Y /X ) (r) holds (N := N 1 ∼ = N 2 ∼ = . . . ∼ = N r ), these three bundles are written as O P(N Y /X ) (r + 1) ⊗ p| * E (N r ⊗ K −1 Y ), O P(N Y /X ) (r + 2) ⊗ p| * E (N r+1 ⊗ K −1 Y ), and O P(N Y /X ) (r + 1) ⊗ p| * E (T Y ⊗ N r ⊗ K −1 Y ), respectively. In what follows, we show the Nakano positivity for these three bundles. First let us note that (1), where Pr 1 is the first projection E ∼ = P r × Y → P r . Let us denote by h the metric on this line bundle which is the pullback of the Fubini-Study metric by Pr 1 . By tensoring h and a metric on N −1 ⊗ K −1 Y with positive curvature, we have the positivity for the bundles O P(N Y /X ) (1)⊗p| * E N = Pr * 1 O P rO P(N Y /X ) (r + 1) ⊗ p| * E (N r ⊗ K −1 Y ) = Pr * 1 O P r (r + 1) ⊗ p| * E (N −1 ⊗ K −1 Y ) and O P(N Y /X ) (r + 2) ⊗ p| * E (N r+1 ⊗ K −1 Y ) = Pr * 1 O P r (r + 2) ⊗ p| * E (N −1 ⊗ K −1 Y ). Finally we show the Nakano positivity for F := O P(N Y /X ) (r + 1) ⊗ p| * E (T Y ⊗ N r ⊗ K −1 Y ) = O P(I Y ) (r + 1) ⊗ p| * E (T Y ⊗ N −1 ⊗ K −1 Y ). Take a metric h ′ on T Y ⊗ N −1 ⊗ K −1 Y with Nakano positive curvature. Then h r+1 ⊗ p| * E h ′ is a metric on F , with curvature (r + 1)Θ h ⊗ Id F + p| * E Θ h ′ , which is easily seen to be Nakano positive. 5.3. Proof of Theorem 1.0.4. By Lemma 5.2.1, Y admits a holomorphic tubular neighborhood in X. By Theorem 1.0.3, there exists a metric h min,L with minimal singularities whose local weight ϕ min,L is written in the form ϕ min,L (z, y) = log max α∈ L |z α | 2 e (ϕα)e(y) + O(1). By choosing metrics e ϕα(y) as in [K1, §2.2], we obtain that (ϕ α ) e (y) = ϕ α (y) holds and ϕ α (y) depends continuously on (y, α), which proves the assertion. 6. Examples 6.1. Nakayama's example. Nakayama's example (X, L, Y ) is the example which admits no Zariski decomposition even after modifications [N,IV,§2.6] (see also [K1, §1]). In this example, the manifold X is a total space of the projective space bundle π : X := P(M 1 ⊕ M 2 ⊕ M 3 ) → Y over an abelian surface Y , where M 1 and M 2 are ample line bundles on Y and M 3 is a line bundle on Y . The line bundle L is the inverse of the tautological line bundle: i.e. L := O P(M 1 ⊕M 2 ⊕M 3 ) (1). Then the stable base locus of L is the subset P(M 3 ) ⊂ X, which we are regarding as Y here. As it is clearly observed, this example (X, L, Y ) is a special case of those in §3. Therefore we can apply Theorem 3.1.1 to this example. In particular, by using the metrics as in the proof of Theorem 1.0.4, we can reprove the main result in [K1]. 6.2. Zariski's example and its higher (co-)dimensional analogues and proof of Theorem 1.0.1. In [K2, §4.2], the second-named author applied its main result (=Theorem 1.0.4, 1.0.3 of this paper for r = 1) to Zariski's and Mumford's example (X, L, Y ), in which L is nef and big however not semi-ample, and showed the semi-positivity of L (i.e. the existence of a C ∞ Hermitian metric on L with semi-positive curvature). Here we construct an example which can be regarded as a higher-codimensional analogue of Zariski's example and apply Theorem 1.0.4 to it. In what follows, we only consider the case of r = 2 for simplicity. Take two general quadric surfaces Q 1 and Q 2 in P 3 . Then we may assume that the intersection C := Q 1 ∩Q 2 is a smooth elliptic curve and Q 1 and Q 2 intersects transversally along C. Fix N points p 1 , p 2 , . . . , p N in C. Denote by π : X := Bl {p 1 ,p 2 ,...,p N } P 3 → P 3 the blow-up of P 3 at these N points, by Y the strict transform of C, by D 1 and D 2 the the strict transform of Q 1 and Q 2 , respectively, by E λ the exceptional divisor π −1 (p λ ) for each λ, by E the divisor N λ=1 E λ , and by H the pull-back π * O P 3 (1). Note that D λ ∈ |2H −E|. Let us consider the line bundle L := O X (H + D 1 ) = O X (3H − E) on X. As H is big and D 1 is effective, L is also big. It is also observed that Bs |L| ⊂ Y holds, since H is base point free and Bs |H| ⊂ Y by construction. As a simple computation shows that the intersection number (L.Y ) is equal to 12 − N, we conclude that L is nef if and only if 12 ≥ N. First let us consider the case of N = 12. In this case, we may assume that L| Y is a general (and thus non-torsion) element of Pic 0 (Y ) by choosing p 1 , p 2 , . . . , p 12 generically. Then it is easily observed that B(L) = Y holds, and therefore that L is not semi-ample, hoverer L is nef and big. In this sense, we can regard this example (X, Y, L) as an analogue of Zariski's example with r = 2. As D 1 and D 2 intersects transversally along Y , we obtain the decomposition N Y /X = N D 1 /X | Y ⊕ N D 2 /X | Y = O X (D 1 )| D 1 | Y ⊕ O X (D 2 )| D 2 | Y = O X (D 1 )| Y ⊕ O X (D 2 )| Y By denoting N λ := O X (D λ )| Y for each λ = 1, 2, it holds that N 1 ∼ = N 2 , (D λ .Y ) = 2(H.Y ) − (E.Y ) = 8 − 12 < 0, and deg Y L| Y ⊗ N −1 λ = 0 − (8 − 12) > 0. Therefore we can apply Theorem 1.0.4, Corollary 1.0.5, and also Lemma 5.2.1 to our (X, L, Y ). By Corollary 1.0.5, we have that h min,L | Y is bounded, where h min,L is a metric of L minimal singularities (here we use that fact that L| Y admits a C ∞ Hermitian metric with zero curvature, since L| Y is a flat line bundle). Therefore we can conclude that h min,L is bounded. Note that we can moreover show that the semi-positivity of L (i.e. that we can choose h min,L as a C ∞ Hermitian metric) by applying Lemma 5.2.1 and use the "regularized maximum construction" technique as in [K2,Corollary 3.4]. Next let us consider the case of N > 12. In this case, L is not nef . By the argument as above, we also have B(L) = Y , deg N λ = 8 − N < 0, and deg (L| Y ⊗ N −1 λ ) = (12 − N) − (8 − N) = 4 > 0. Thus we can apply Theorem 1.0.4 to (X, Y, L) also in this case. As the computation shows that L = α = (α 1 , α 2 ) ∈ R 2 ≥0 N − 12 N − 8 ≤ |α| ≤ 1 , it follows from Theorem 1.0.4 that the local weight function ϕ min,L of a metric h min,L with minimal singularities can be written as ϕ min,L (z, y) = log max α∈ L r λ=1 |z λ | 2α λ + O(1) = N − 12 N − 8 · log(|z 1 | 2 + |z 2 | 2 ) + O (1) on a neighborhood of any point of Y , where y is a coordinate of Y and z = (z 1 , z 2 , . . . , z r ) is a system of local defining functions of Y . In particular, in this case, ϕ min,L has analytic singularities along Y . Note that similar example can be constructed in general dimension by considering some points blow-up of a del Pezzo manifold of degree 1 (see [F] for example. For the choice of the counterpart of the divisors D ν 's above, see [K3, §6.3]). 6.3. An example in [BEGZ]. The above two examples satisfies Condition 1.0.2 (ii) and the condition that Y admits a holomorphic tubular neighborhood. On the other hand, the example (X, Y, L) in [BEGZ,Example 5.4] does not satisfy these conditions. In [BEGZ]'s example, a metric of L with minimal singularities is unbounded and actually has singularities along Y (i.e. local weight function equals to −∞ on Y ), however the Lelong number of the local weight is 0 for every point in X (see also [K2,Example 4.2]). In particular, the conclusion of Theorem 1.0.3 does not hold in this example. Lemma 3.2.1. Let N λ be line bundles as in Theorem 1.0.3. If one take The direct decomposition of H 0 (X, L m ) induced by the isomorphisms H 0 If one choose V sufficiently small, one have that i * L ∼ = L. 4. 3 . 3Proof of Theorem 1.0.3. Theorem 1.0.3 follows from Theorem 3.1.1 and Proposition 4.2.1. 5. A sufficient condition for the existence of a holomorphic tubular neighborhood and Proof of Theorem 1.0.4 Assume that H 1 (Z, N −ν Z/M ) = 0 and H 1 (Z, T Z ⊗ N −ν Z/M ) = 0 hold for every ν ≥ 1.Then Z has a holomorphic tubular neighborhood in M. Acknowledgment.The authors would like to thank Dr. Tatsuya Miura for his helpful suggestions on the change of variables in the proof of Lemma 3.3.3. The first author is supported by Program for Leading Graduate Schools, MEXT, Japan. He is also supported by the Grant-in-Aid for Scientific Research (KAKENHI No.15J08115). The second author is supported by the Grant-in-Aid for Scientific Research . Divisorial Zariski decompositions on compact complex manifolds. S Boucksom, Ann. Sci.École Norm. Sup. 4S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci.École Norm. Sup. (4) 37(1) (2004), 45-76. Monge-Ampère equations in big cohomology classes. S Boucksom, P Eyssidieux, V Guedj, A Zeriahi, Acta Math. 205S. Boucksom, P. Eyssidieux, V. Guedj, A. Zeriahi, Monge-Ampère equations in big co- homology classes. Acta Math. 205 (2010), 199-262. Neighborhoods of Analytic Varieties, Monografías del Instituto de Matemática y Ciencias Afines, 35. Instituto de Matemática y Ciencías Afines, IMCA, Lima; Pontificia Universidad Católica del Perú. C Camacho, H , Movasati , LimaC. Camacho, H, Movasati, Neighborhoods of Analytic Varieties, Monografías del Instituto de Matemática y Ciencias Afines, 35. Instituto de Matemática y Ciencías Afines, IMCA, Lima; Pontificia Universidad Católica del Perú, Lima, 2003. Complex Analytic and Differential Geometry, monograph. J.-P Demailly, J.-P. Demailly, Complex Analytic and Differential Geometry, monograph, 2012, available at http://www-fourier.ujf-grenoble.fr/~demailly. Analytic methods in algebraic geometry. J.-P Demailly, Surveys of Modern Mathematics. 1International PressJ.-P. Demailly, Analytic methods in algebraic geometry, Surveys of Modern Mathematics, 1. International Press, Somerville, MA, 2012. Pseudo-effective line bundles on compact Kähler manifolds. J.-P Demailly, T Peternell, M Schneider, Internat. J. Math. 12J.-P. Demailly, T. Peternell, M. Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math. 12 (2001), 689-741. Classification theories of polarized varieties. T Fujita, London Mathematical Society Lecture Note Series. 155Cambridge University PressT. Fujita, Classification theories of polarized varieties. London Mathematical Society Lecture Note Series, 155, Cambridge University Press, Cambridge (1990). Über Modifikationen und exzeptionelle analytische Mengen. H Grauert, Math. Ann. 146H. Grauert,Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146 (1962), 331-368. 22 On the equivalence of imbeddings of exceptional complex spaces. H Hironaka, H Rossi, H , Math. Ann. 156H. Hironaka, H. Rossi, H, On the equivalence of imbeddings of exceptional complex spaces, Math. Ann. 156 (1964) 313-333. Minimal singular metrics of a line bundle admitting no Zariski-decomposition. T Koike, Tohoku Math. J. 672T. Koike, Minimal singular metrics of a line bundle admitting no Zariski-decomposition, Tohoku Math. J. (2) Volume 67, Number 2 (2015), 297-321. On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated. T Koike, Ann. Inst. Fourier (Grenoble). 655T. Koike, On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated, Ann. Inst. Fourier (Grenoble) Volume 65, Number 5 (2015), 1953-1967. T Koike, arXiv:1606.01837Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle. T. Koike, Higher codimensional Ueda theory for a compact submanifold with unitary flat normal bundle, arXiv:1606.01837. Normal two-dimensional singularities. H B Laufer, Ann. Math. Stud. 71University of Tokyo PressH. B. Laufer, Normal two-dimensional singularities, Ann. Math. Stud., 71. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. R Lazarsfeld, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Positivity in algebraic geometry. Results in Mathematics and Related AreasR. Lazarsfeld, Positivity in algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzge- biete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. Zariski decomposition and abundance. N Nakayama, MSJ Mem. 14Springer-Verlag3rd Series. A Series of Modern Surveys in MathematicsN. Nakayama, Zariski decomposition and abundance, MSJ Mem. 14, Mathematical Society of Japan, Tokyo, 2004. 3rd Series. A Series of Modern Surveys in Mathematics], 48. Springer-Verlag, Berlin, 2004. F W Olver, D M Lozier, R F Boisvert, C W Clark, Digital Library of Mathematical Functions: Online Companion to NIST Handbook of Mathematical Functions (CUP). National Insitute of Standards and Technology. F. W. Olver, D. M. Lozier, R. F. Boisvert, and C. W. Clark, eds., Digital Library of Mathematical Functions: Online Companion to NIST Handbook of Mathematical Functions (CUP). National Insitute of Standards and Technology, 2010. http://dlmf.nist.gov. H Rossi, Strongly pseudoconvex manifolds, Lectures in Modern Analysis and Applications I. 103H. Rossi, Strongly pseudoconvex manifolds, Lectures in Modern Analysis and Applications I, Lec- ture Notes in Mathematics 103, 10-29 (1969). The University of Tokyo, 3-8-1 Komaba. Graduate School of Mathematical Sciences. Meguro-kuGraduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for all N ∈ N, there is a tile set that uniquely self-assembles into an N × N square shape at temperature 1 with optimal program-size complexity of O(log N/ log log N ) (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it works even when the placement of tiles is restricted to the z = 0 and z = 1 planes. This result affirmatively answers an open question from Cook, Fu, Schweller (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SICOMP 2007), and perhaps of independent interest.

10.1007/978-3-319-21999-8_5

1411.1122

9e06b1708c85fd95a01005f9300587f9d454d283

Optimal program-size complexity for self-assembly at temperature 1 in 3D 5 Nov 2014 David Furcy [email protected]. Computer Science Department University of Wisconsin-Oshkosh 54901OshkoshWIUSA Samuel Micka [email protected]. Computer Science Department Computer Science Department University of Wisconsin-Oshkosh 54901OshkoshWIUSA University of Wisconsin-Oshkosh 54901OshkoshWIUSA Scott M Summers [email protected]. Optimal program-size complexity for self-assembly at temperature 1 in 3D 5 Nov 20141 Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for all N ∈ N, there is a tile set that uniquely self-assembles into an N × N square shape at temperature 1 with optimal program-size complexity of O(log N/ log log N ) (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it works even when the placement of tiles is restricted to the z = 0 and z = 1 planes. This result affirmatively answers an open question from Cook, Fu, Schweller (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SICOMP 2007), and perhaps of independent interest. Introduction The simplest mathematical model of nanoscale tile self-assembly is Erik Winfree's abstract Tile Assembly Model (aTAM) [14]. The aTAM extends classical Wang tiling [13] in that the former bestows upon the latter a mechanism for sequential "growth" of a tile assembly. Very briefly, in the aTAM, the fundamental components are un-rotatable, translatable square "tile types" whose sides are labeled with (alpha-numeric) glue "colors" and (integer) "strengths". Two tiles that are placed next to each other bind if both the glue colors and the strengths on their abutting sides match and the sum of their matching strengths sum to at least a certain (integer) "temperature". Self-assembly starts from a "seed" tile type, typically assumed to be placed at the origin, and proceeds nondeterministically and asynchronously as tiles bind to the seedcontaining assembly one at a time. In this paper, we work in a three-dimensional variant of the aTAM in which tile types are unit cubes and growth proceeds in a noncooperative manner. Tile self-assembly in which tiles may be placed in a noncooperative fashion is often referred to as "temperature 1 self-assembly". Despite the arcane name, this is a fundamental and ubiquitous form of growth: it refers to growth from growing and branching tips in Euclidean space, where each new tile is added if it can bind on at least one side. Note that a more general form of cooperative growth, where some of the tiles may be required to bind on two or more sides, leads to highly non-trivial behavior in the aTAM, e.g., Turing universality [14] and the efficient self-assembly of N × N squares [1,11] and other algorithmically specified shapes [12]. Doty, Patitz and Summers conjecture [6] that the shape or pattern produced by any 2D temperature 1 tile set that uniquely produces a final structure is "simple" in the sense of Presburger arithmetic [10]. However, their conjecture is currently unproven and it remains to be seen if noncooperative self-assembly in the aTAM can achieve the same computational and geometric expressiveness as that of cooperative self-assembly. In this paper, we specifically focus on a problem that is very closely related to that of finding the minimum number of distinct tile types required to self-assemble an N × N square, i.e., its tile complexity (or program-size complexity), at temperature 1. The tile complexity of an N × N square at temperature 1 has been studied extensively. In 2000, Rothemund and Winfree [11] proved that the tile complexity of an N × N square at temperature 1 is N 2 , assuming the final structure is fully connected, and at most 2N − 1, otherwise (they also conjectured that the lower bound, in general, is 2N − 1). A decade later, Manuch, Stacho and Stoll [9] established that, assuming no mismatches are present in the final assembly, the tile complexity of an N × N square at temperature 1 is 2N −1. Shortly thereafter, and quite surprisingly, Cook, Fu and Schweller [5] showed that the tile complexity of an N × N square at temperature 1 is O(log N ) if tiles are allowed to be placed in the z = 0 and z = 1 planes (here, an N × N square is actually a full 2D square in the z = 0 plane with additional tiles above it in the z = 1 plane). Technically speaking, the aforementioned, just-barely-3D construction of Cook, Fu and Schweller is actually a general transformation that takes as input a 2D temperature 2 "zig-zag" tile set, say T , and outputs a corresponding 3D temperature 1 tile set, say T ′ , that simulates T . In this transformation from T to T ′ , the tile complexity increases by O(log g), where g is the number of unique north/south glues in the input tile set T . Since the number of north/south glues in the standard 2D aTAM base-2 binary counter is O(1), Cook, Fu and Schweller use their transformation to produce several tile sets, which, when wired together appropriately and combined with "filler" tiles, self-assemble into an N × N square at temperature 1 in 3D with O(log N ) tile complexity. Of course, it is well-known that the tile complexity of an N ×N square at temperature 2 is O log N log log N [1], which, as Cook, Fu and Schweller point out in [5], is achievable using a zig-zag counter with an optimallychosen base, say b, which satisfies log N log log N < b < 2 log N log log N , rather than in base b = 2. However, using currently-known techniques, counting in base b at temperature 2 requires having a tile set with Θ(b) unique north/south glues, whence the zig-zag transformation of Cook, Fu and Schweller cannot be used to get O log N log log N tile complexity for an N × N square at temperature 1 in 3D. Moreover, the optimal encoding scheme of Soloveichik and Winfree [12] and the base conversion technique of Adleman et. al. [1] do not work correctly at temperature 1 and they also cannot be simulated by the Cook, Fu and Schweller construction without an Ω log N log log N blowup in tile complexity. Thus, Cook, Fu and Schweller, at the end of section 4.4 in [5], pose the following question: is it possible to achieve the tile complexity bound of O log N log log N for an N × N square at temperature 1 in 3D? In Theorem 4.1, the main theorem of this paper, we answer the previous question in the affirmative, i.e., we prove that the tile complexity of an N ×N square at temperature 1 in 3D is O log N log log N (in our construction, tiles are placed only in the z = 0 and z = 1 planes of Z 3 ). Our tile complexity matches a corresponding lower bound dictated by Kolmogorov complexity (see [7] for details on Kolmogorov complexity), which was established by Rothemund and Winfree in 2000, and holds for all "algorithmically random" values of N [11] 1 . Thus, our construction yields optimal tile complexity for the self-assembly of N × N squares at temperature 1 in 3D, for all algorithmically random values of N . To achieve optimal tile complexity, we adapt the optimal encoding technique of Soloveichik and Winfree [12] (which, itself, is based on the base-conversion scheme of [1]) to work at temperature 1 in 3D. Our 3D temperature 1 optimal encoding technique, described in Section 3, is perhaps of independent interest. Definitions In this section, we give a brief sketch of a 3-dimensional version of Winfree's abstract Tile Assembly Model. 3D abstract Tile Assembly Model Let Σ be an alphabet. A 3-dimensional tile type is a tuple t ∈ (Σ * × N) 6 , e.g., a unit cube with six sides listed in some standardized order, each side having a glue g ∈ Σ * × N consisting of a finite string label and a non-negative integer strength. In this paper, all glues have strength 1. There is a finite set T of 3-dimensional tile types but an infinite number of copies of each tile type, with each copy being referred to as a tile. A 3-dimensional assembly is a positioning of tiles on the integer lattice Z 3 and is described formally as a partial function α : Z 3 T . Two adjacent tiles in an assembly bind if the glue labels on their abutting sides are equal and have positive strength. Each assembly induces a binding graph, i.e., a grid graph whose vertices are (positions of) tiles and whose edges connect any two vertices whose corresponding tiles bind. If τ is an integer, we say that an assembly is τ -stable if every cut of its binding graph has strength at least τ , where the strength of a cut is the sum of all of the individual glue strengths in the cut. A 3-dimensional tile assembly system (TAS) is a triple T = (T, σ, τ ), where T is a finite set of tile types, σ : Z 3 T is a finite, τ -stable seed assembly, and τ is the temperature. In this paper, we assume that |dom σ| = 1 and τ = 1. An assembly α is producible if either α = σ or if β is a producible assembly and α can be obtained from β by the stable binding of a single tile. In this case we write β → T 1 α (to mean α is producible from β by the binding of one tile), and we write β → T α if β → T * 1 α (to mean α is producible from β by the binding of zero or more tiles). When T is clear from context, we may write → 1 and → instead. We let A [T ] denote the set of producible assemblies of T . An assembly is terminal if no tile can be τ -stably bound to it. We let A [T ] ⊆ A [T ] denote the set of producible, terminal assemblies of T . A TAS T is directed if |A [T ]| = 1. Hence, although a directed system may be nondeterministic in terms of the order of tile placements, it is deterministic in the sense that exactly one terminal assembly is producible. For a set X ⊆ Z 3 , we say that X is uniquely produced if there is a directed TAS T , with A [T ] = {α}, and dom α = X. For N ∈ N, we say that S 3 N ⊆ Z 3 is a 3D N × N square if {0, . . . , N − 1} × {0, . . . , N − 1} × {0} ⊆ S 3 N ⊆ {0, . . . , N − 1} × {0, . . . , N − 1} × {0, 1}. In other words, a 3D N × N square is at most two 2D N × N squares, one stacked on top of the other. In the spirit of [11], we define the tile complexity of a 3D N × N square at temperature τ , denoted by K τ 3DSA (N ), as the minimum number of distinct 3D tile types required to uniquely produce it, i.e., K τ 3DSA (N ) = min n T = (T, σ, τ ) , |T | = n and T uniquely produces S 3 N † . Notation for figures In the figures in this paper, we use big squares to represent tiles placed in the z = 0 plane and small squares to represent tiles placed in the z = 1 plane. A glue between a z = 0 tile and z = 1 tile is denoted as a small black disk. Glues between z = 0 tiles are denoted as thick lines. Glues between z = 1 tiles are denoted as thin lines. 3 Optimal encoding at temperature 1 A key problem in algorithmic self-assembly is that of providing input to a tile assembly system (e.g., the size of a square, the input to a Turing machine, etc.). In real-world laboratory implementations, as well as theoretical constructions, input to a tile system is typically provided via a (possibly large) collection of "hard-coded" seed tile types that uniquely assemble into a convenient "seed structure," such as a line of tiles that encodes some input value. Unfortunately, in practice, it is more expensive to manufacture different types of tiles than it is to create copies of each tile type. Thus, it is critical to be able to provide input to a tile system using the smallest possible number of hard-coded seed tile types. Consider the problem of constructing a tile set that uniquely self-assembles from a single seed tile into a "seed row" that encodes an n-bit binary string, say x. The most straightforward way to do this is to construct a set of n unique tile types that deterministically assemble into a line of tiles of length n, where each tile in the line represents a different bit of x. This simple construction encodes one bit of x per tile, whence its tile complexity is O(n). Note that, in this example, each tile type is an element of a set of size n, yet each tile type encodes only 1 bit of information, instead of the optimal O(log n) bits. Is there a more efficient encoding construction? The optimal encoding constructions of Adleman et al. [1], and Soloveichik and Winfree [12] are more efficient methods of encoding input to a tile set. These constructions are based on the idea that each seed row tile type should encode k = O(log n) bits -instead of a single bit -of x, which means that O(n/ log n) unique tile types suffice to uniquely self-assemble into a seed row that encodes the bits of x. Unfortunately, now the bits of x are no longer conveniently represented in distinct tiles. Fortunately, if k is chosen carefully, then it is possible to use a tile set of size O(n/ log n) to "extract" the bits of x into a more convenient one-bit-per-tile representation, which can be used to seed a binary counter or a Turing machine simulation. Up until now, all known optimal encoding constructions (e.g., [1,12]) required cooperative binding (that is, temperature τ ≥ 2). In what follows, we propose an optimal encoding construction (based on the construction of Soloveichik and Winfree [12]) that works at temperature τ = 1 and is "just barely" 3D, i.e., tiles are only placed in the z = 0 and z = 1 planes. Setup Let x = x n−1 x n−2 . ..x 1 x 0 be the input string, where x i ∈ {0, 1}. Let m = ⌈n/k⌉, where k is the smallest integer satisfying 2 k ≥ n/ log n. We write x = w 0 w 1 ...w m−2 w m−1 , where each w i is a k-bit block. Note that w 0 is padded to the left with leading 0's, if necessary. In the figures in this section, a green tile represents a starting point for some portion of an assembly sequence, and a red tile represents an ending point. † One subtle difference between our 3D definition of K and the original 2D definition of the tile complexity of an N × N square, given by Rothemund and Winfree in [11], is that they assume a fully-connected final structure, whereas we do not. Figure 1: The perimeter of the first extraction region is hard-coded to self-assemble like this. In this example, the four bumps along the top (from left to right) represent the bits 1, 0, 0 and 1, respectively. The green tile (bottom tile in the penultimate column) is the single seed tile for our entire optimal encoding construction. Overview of the construction We extract each of the m k-bit blocks within a roughly rectangular region of space of width O(k) and height O(m). We refer to this region of space as a "block extraction region" (or simply "extraction region"). For each 0 ≤ i < m, we extract block w i in extraction region i. Each extraction region, other than the first and last ones, assembles via a series of gadgets (small groups of tiles that carry out a specific task). We encode the k bits of a k-bit block as a series of geometric bumps along a path of tiles that makes up the top border of an extraction region. A bump in the z = 0 plane represents the bit 0 and a bump in the z = 1 plane represents the bit 1. The end result of our construction is an assembly in which each bit of x is encoded in its own bit-bump (see Figure 10 for an example). We extract the k-bit blocks in order, starting with the first block w 0 , which represents the most significant bits of x. Normally, to carry out this sort of activity at temperature 1 (i.e., to enforce the ordering of tile placements), one has to encode the order of placement directly into the glues of the tiles. However, for our construction, this would essentially mean encoding the number of the block that is being extracted into the glues of the tiles that fill in its extraction region. Unfortunately, doing so, at least in the most straightforward way, results in an increase in tile complexity from the optimal O(n/ log n) to Ω(n 2 / log 2 n). Therefore, in our construction, we encode the number of the block that is being extracted as a geometric pattern along a vertical path of tiles that runs along the right side of each extraction region. We call this special geometric pattern the "block number." Then we use a special gadget called the "block-number gadget" to search for this pattern. Within an extraction region, the block number determines which block gets extracted next. Basically, the path along which the block number is encoded blocks the placement of m − 1 special tiles, each of which tries to initiate the extraction of a particular k-bit block. We call these special tiles "extraction tiles." Since the first extraction region is hard-coded (see below), the first block does not have an extraction tile associated with it. Within any given extraction region, exactly one extraction tile will not be blocked. The one extraction tile that is not blocked by the block number gadget will initiate the extraction of the k bits of the block to which it corresponds. In our construction, we hard-code the assembly of the first and last extraction regions. What this means is that, in each of these extraction regions, a single-tile-wide path assembles the perimeter and then we use a filler tile to fill in the interior. For this step, it is crucial to first assemble the perimeter of the extraction region and then use a filler tile to tile the interior. Note that, if one were to uniquely tile every location in the first (or last) extraction region, then the tile complexity of the construction would be Ω(mk), which is not optimal. Tiling the perimeter of either the first or last extraction region can be done with O(m + k) unique tile types (see Figure 1 for the example of the first extraction region). All extraction regions other than the first and last ones are constructed using a general set of gadgets. In the second extraction region, which is the first generally-constructed extraction region, the block-number gadget determines that w 1 is the next block to be extracted by "searching" for the block number position. When the block number is found, a path of tiles, initiated by the extraction tile for w 1 , is allowed to assemble (see Figure 2 for an example of this process). In general, for extraction region i, for all 1 ≤ i < m − 1, the path along which the block number is encoded geometrically hinders the placement of all extraction tiles (a) The block-number gadget determines the next block to extract by "searching" for the position of the block number (i.e., the position of the notch in which the red tile is ultimately placed). (b) The path initiated by the extraction tile for w 1 "jumps" over the block-number gadget and grows a hook to block a subsequent gadget. (c) The path initiated by the extraction tile for w 1 continues growing upward and eventually finds the top of the block-number gadget. The upward growth of this path is blocked by a portion of the previous extraction region. (d) Once at the top of the block-number gadget, the path initiated by the extraction tile for w 1 "jumps" over a portion of the previous extraction region and starts extracting the bits of w 1 along the top of the second extraction region. that correspond to blocks w 1 , ..., w i−1 , w i+1 , ..., w m−2 . Each extraction tile initiates the extraction of the k-bit block to which it corresponds (see Figure 3). We use a set of "bit-extraction" gadgets to extract a k-bit block into a one-bit-per-bump representation (the bit extraction gadgets are collectively referred to as the "extraction gadget"). Our bit extraction gadgets are basically 3D, temperature 1 versions of the "extract bit" tile types in Figure 5.7a of [12]. After a block, say w i , for i > 0, is extracted, the block number is geometrically "incremented", i.e., its position is translated up by a small constant amount (notice the position of the white "hook" at the bottom of Figure 3). We do this in two phases. First, the current position of the block number is found and then it is incremented and translated. Figure 4 shows how the current position of the block number is detected using a zig-zag path of tiles. Figure 5 shows how the current position of the block number is geometrically incremented. After the block number has been updated, a series of gadgets geometrically propagate the position of the block number to the right through the remainder of extraction region i so that it is advertised to extraction region i + 1. This is shown in Figures 6 and 7. Technically, we geometrically propagate the block number position through the rest of the extraction region using a series of gadgets. Logically, however, we do this in two phases, which are iterated: "up" propagation and "down" propagation. The "up" propagation phase grows from the position of the block number up to (and is blocked by) a previous portion of the assembly. This is shown in Figure 6. The "down" propagation phase grows from the top of the previous (up) propagation phase back down to the position of the block number. The upward Figure 4: A path of tiles searches for the block number, represented by a notch in a previous portion of the assembly. The red tile "knows" that it found the position of the block number because it was allowed to be placed. Figure 5: The block number is geometrically incremented. The green tile "jumps" over the previous gadget that found the position of the block number and grows a hook of tiles to represent the updated block number. Notice that the new hook of tiles is two tiles higher than the previous hook (shown in black), which corresponds to the two rows of tiles that each block takes up in the blocknumber gadget. Figure 9) that will fill in the bottom row of the current extraction region before the next extraction region begins. Figure 9: The bottom row of the extraction region is tiled by a special gadget with O(m) tile complexity. After the bottom row of the extraction region is tiled, the next extraction region is initiated. Notice that the red tile in this figure belongs to the same row of tiles as the red tile in Figure 1 but the position of the block number has moved up, which means the extraction tile for the next block (in this case, w 2 ) will be allowed to assemble and all other extraction tiles will be blocked. growth of each up propagation phase is blocked in the z = 0 plane but not in the z = 1 plane. However, this is switched for the last up propagation phase. In other words, the last up propagation phase may continue its upward growth, which signals the end of the extraction region, but its z = 1 growth is blocked. In Figure 8, the last up propagation phase is allowed to continue its upward growth in the z = 0 plane. The last up propagation phase initiates the assembly of a special gadget that fills in the bottom row of the current extraction region before the next extraction region begins. The reason we do this is to ensure that, when the entire extraction process is done (i.e., when all n bits have been extracted into a one-bit-per-bump representation), the bottom row of the assembly is completely filled in. Figure 9 shows an example of how this gadget tiles the remaining perimeter of an extraction region. Note that the tile complexity of this gadget is the size of the perimeter of an extraction region, i.e., O(m). The final extraction region, like the initial extraction region, is hard-coded to assemble its perimeter via a single-tile-wide path. The tiles that comprise the final extraction region "know" to stop the extraction process and possibly initiate the growth of some other logical component of a larger assembly, e.g., a binary counter or a Turing machine simulation in which the extracted bits of x, along the top of each of the m extraction regions, are used as input. The end result of our optimal encoding construction is a roughly rectangular assembly of tiles with height O(m) and width O(n), where each bit of x is encoded as a bump (either in the z = 0 or z = 1 plane) along the top of the rectangle, with four "spacer" tiles to the left and right of each bit-bump. Figure 10 shows the result of our optimal encoding construction with four extraction regions. Complete details for this construction are in the appendix (see Section 6.1). Tile complexity To establish the tile complexity bound of O(n/ log n) for our construction, we use the following technical lemma. Figure 10: This is an example of our optimal encoding construction using n = 16 and k = 4. Note that this does not correspond to an actual instance of our optimal encoding construction because if n = 16, then the smallest value of k satisfying 2 k ≥ n/ log n is k = 2. The bit string encoded along the top is 1001001100111000. All of the empty spaces in the z = 0 plane are filled in with the same filler tile. Lemma 3.1. Let 1 < n ∈ Z + and m = ⌈n/k⌉, where k is the smallest integer satisfying 2 k ≥ n/ log n. Then m = O(n/ log n). Proof. Assume the hypothesis. First, note that, by our choice of k, 2 k < 2n/ log n. Then we have m = n k ≤ ⌈n/ log(n/ log n)⌉ = ⌈n/(log n − log log n)⌉ < ⌈n/(log n − (log n)/2)⌉ ≤ 2n/ log n + 1 ≤ 2n/ log n + n/ log n = 3n/ log n = O(n/ log n) The tile complexity of our construction is the sum of the tile complexities of all of the gadgets that assemble all extraction regions. We will first analyze the tile complexity of the extract bit gadgets. Recall that these gadgets convert a k-bit binary string, encoded as a strength-1 glue, into a one-bit-per-bump representation. The first bitextraction gadget accepts a k-bit binary string, converts the most significant bit of the block into the appropriate bump and then outputs a (k − 1)-bit binary string. The latter is the input for the second bitextraction gadget. This process is iterated k times (once for each bit). For a given n and our choice of k (as described above), the number of distinct extract bit gadgets needed in our construction can be computed as: 2 {0, 1} 0 + {0, 1} 1 + · · · + {0, 1} k−1 = 2 1 + 2 + · · · + 2 k−1 = 2 2 k − 1 < 2 · 2 k < 4n/ log n = O (n/ log n) Since each bit-extraction gadget is comprised of O(1) unique tile types, the total tile complexity for the extraction gadgets is O(n/ log n). It is easy to see that all other gadgets in our construction can be implemented using O(m) unique tile types (see Section 6.1 for details). Thus, by Lemma 3.1, the tile complexity of our construction is O(n/ log n), which is optimal for all algorithmically random values of n. Optimal self-assembly of squares at temperature 1 in 3D In this section, we describe how to use our 3D temperature 1 optimal encoding construction to prove the following theorem. Proof. Our proof is constructive. Figure 11a shows how we build an N ×N square using two counters C1 and C2 and two filler regions F1 and F2. Counter C1 is a zig-zag counter whose construction is described in the appendix (see Section 6.2). Counter C2 is identical to C1 after a 90-degree clockwise rotation. Each counter is seeded with a value produced by an optimal encoding region (OER for short). The full construction for F1 is depicted in Figure 12. F2 is a smaller, mirror-image of F1 with minor modifications to properly connect all of the pieces of the square. Both F1 and F2 are essentially squares, except for two hooks needed to stop the horizontal and vertical growths of each filler region, namely, one eight-tile hook encroaching on and another one-tile hook protruding from each filler region (see Figure 12). These hooks require simple modifications of the OER regions (see Figure 11b) that are all located in the hard-coded (i.e., first and last) block extracting regions of OER1 and OER2. Note that F1 is also missing a two-tile wide rectangle region on its left that is used up by the vertical connector that initiates the assembly of OER2 immediately after the assembly of C1 terminates. Figure 11c shows the assembly sequence for the whole square, while Figure 11b zooms in on the region of the square where OER1, F1, OER2 and F2 all interact. (a) Overall square construction. (b) Detail of the region of the square where OER1, F1, OER2 and F2 meet. (c) The assembly sequence for the whole square is shown with red arrows starting from the seed tile (the red square located in OER1). The central region in this sub-figure is shown in more detail in Figure 11b to the left. First, we compute the tile complexity of our construction as the sum of the tile complexities of all of the components that make up the N × N square. Let n = ⌈log N ⌉. If k denotes the smallest integer satisfying 2 k ≥ n/ log n and m is defined as ⌈n/k⌉, then the tile complexity of each OER is O(n/ log n), as proved in Section 3.3. Furthermore, the tile complexity of each binary counter is O(1) (see Section 6.2). Finally, the tile complexity of each filler region is O(1), since each colored gadget in Figure 12 has tile complexity O(1). Therefore, the tile complexity of our square construction is dominated by that of the OERs and is therefore The gray tiles in this figure do not belong to F1. They are all added to the N ×N square assembly before F1 starts assembling and they determine the height and width of F1, both of which are adjustable in the following way: • The height of F1 is always a multiple of four (i.e., the total height of each pink plus red gadget), plus the number of purple rows at the top, which can be hard-coded to any value in {1, 2, 3, 4}, together with a corresponding increase in the height of the top-left (gray) hook. • The width of F1 is always a multiple of two (i.e., the width of each pink gadget) plus the width of each orange gadget, which is either three (by deleting the column occupied by the white tiles) or four (as shown). Therefore, this construction gives us two knobs, namely the number of purple rows and the width of the orange gadget, to assemble filler regions of any height and width, respectively. log log N . Second, we need to prove that our tile system is directed and does produce an N × N square. The assembly sequence depicted in Figure 11c demonstrates that our tile system uniquely produces a square. To make sure that this square has width N , we need to pick the initial value i of the counters and adjust the size of the filler regions as follows. The width of OER1, C1 and F2 in our construction, and thus also the height of OER2, C2 and F2 is 6n + 4. The height of OER1, and thus also the width of OER2, is 2m + 7 (see, for example, Figure 10). Therefore, the height of C1 (and thus also the width of C2) must be equal to N − (2m + 7 + 6n + 4) = N − 2 ⌈log N ⌉ k − 6⌈log N ⌉ − 11. Let us denote this value by h(N ). Our construction in Section 6.2 gives us two knobs to control the height of any n-bit counter: the initial value i of the counter and the number r of rooftop rows, where r ∈ {1, 2, 3, 4}. Since each value from i to the final value of the counter 2 n − 1 (inclusive) takes up four rows of tiles, we must have ⌊ h(N ) 4 ⌋ = 2 n − i and r = 1 + h(N ) mod 4. Therefore, for both C1 and C2, the initial value of the counter is 2 n − ⌊ h(N ) 4 ⌋. Finally, the correct height and width of F1 are obtained by setting the two knobs described in Figure 12 to 1 + (2m + 7 + h(N ) − 1) mod 4 and 4 − (2m + 7 + h(N ) − 2) mod 2, respectively. Similarly, the correct height and width of F2 are obtained by setting the second knob to 4 − (6n + 4) mod 2 and the first knob to 1 + (6n + 4 − 1) mod 4. Conclusion In this paper, we developed a 3D temperature 1 optimal encoding construction, based on the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree [12]. We then used our construction to answer an open question of Cook, Fu and Schweller [5], namely, we proved that K 1 3DSA (N ) = O log N log log N , which is the optimal tile complexity for all algorithmically random values of N . We propose a future research direction as follows. Consider a generalization of the aTAM, called the two-handed [2] (a.k.a., hierarchical [3], q-tile, multiple tile [4], polyomino [8]) abstract Tile Assembly Model (2HAM). A central feature of the 2HAM is that, unlike the aTAM, it allows two "supertile" assemblies, each consisting of one or more tiles, to bind. In the 2HAM, an assembly of tiles is producible if it is either a single tile, or if it results from the stable combination of two other producible assemblies. Now define the quantity K τ 3D2SA (N ), as the minimum number of distinct 3D tile types required to uniquely produce it in the 2HAM at temperature τ . An interesting problem would be to determine if K 1 3D2SA (N ) = O log N log log N . Appendix In this section, we provide details on two of our constructions, namely the optimal encoding and the zig-zag counter, that did not fit in the main body of the paper. Throughout this section, we assume that the glues on each tile are implicitly defined to ensure deterministic assembly. Optimal encoding construction The assembly of an extraction region is initiated by the "block-number" gadget (see Figure 13b). The blocknumber gadget is initiated in one of two ways. On the one hand, the final tile placed in the hard-coded path of tiles that assembles the first extraction region may initiate the assembly of the block-number gadget. On the other hand, the block-number gadget may be initiated by the final tile placed by the final gadget to assemble in the previous extraction region, known as the "floor" gadget, which we describe later (see Figure 27a). The block-number gadget uses a zig-zag assembly pattern, i.e., it grows in an alternating left-to-right and right-to-left pattern. This zig-zag growth pattern is five tiles wide because of the spacing requirements of our construction. Each zig-zag represents an attempt to place an "extraction tile" for a k-bit block at a special location, which is denoted by the dotted tiles above the red tile in the block-number gadget in Figure 13b. When an extraction tile is placed, the corresponding bits for that specific k-bit string are extracted by a subsequent series of gadgets. The extraction tile for each k-bit block is obstructed (i.e., geometrically hindered from being placed) by a previous portion of the construction, except for the extraction tile for the current k-bit block (highlighted in red in Figure 13b). Finally, the block-number gadget is prevented from growing down any further by the tiles placed below it. These tiles either are part of the perimeter of the first (hard-coded) extraction region or they were placed by a gadget called the "repeating hook gadget" (described below) during the assembly of the previous extraction region. The placement of these blocking tiles ensures that only the correct k-bit block is extracted within a given extraction region. Note that we use the same block-number gadget to initiate the extraction of each of the m distinct k-bit blocks, whence its tile complexity is O(m). (a) The shaded gadget in the extraction region represents the block-number gadget, pictured on the right in Figure 13b. The zig-zag pattern of assembly described above for the block-number gadget is a common property of many of the gadgets in our constructions. The basic idea is that, as the zig-zag path assembles, it will first "zig" in one direction, for a small constant number of tiles (depending on the spacing requirements of the gadget) and then it will "zag" back in the opposite direction. The final tile in the zag direction tries to grow the zag portion of the path by one more tile (in the zag direction), and also initiates the next zig-zag iteration. In all but one case, the extra zag tile is prevented from being placed (i.e., a tile from a previous portion of the construction is already placed at this location) and the zig-zag pattern simply continues. However, in one case, the zig-zag pattern is blocked and there is an empty location at which the extra zag tile is placed. This non-cooperative assembly algorithm allows various gadgets in our construction to "know" when they have reached a certain "stopping point". Moreover, most of the time, the gadgets that implement this zig-zag assembly algorithm can be implemented in O(1) tile complexity. Therefore, throughout the following discussion, unless noted otherwise, all gadgets are implemented using O(1) unique tile types. (a) The shaded gadget in the extraction region represents the block-hook gadget, pictured on the right in Figure 14b. The block-number gadget places the correct extraction tile based on the block number position, which is encoded in the geometry of a previous portion of the construction. The placement of an extraction tile either initiates the assembly of the "block-hook" gadget (see Figure 14b) or initiates a different gadget that assembles the perimeter for the final extraction region via a hard-coded path of tiles. First, the block-hook gadget assembles an L-shape path above a portion of the previous extraction region, in the z = 1 plane, and then builds, in the z = 0 plane, a geometric pattern of tiles in the shape of a one-tile-wide hook that will later stop the downward growth of the "hook-seeking gadget" (see Figure 20b). Then the block-hook gadget initiates the assembly of the "up-extraction" gadget (see Figure 15b). Finally, the block-hook gadget is responsible for determining the starting point (in the north-south direction) for a subsequent gadget called the "hook-initiating" gadget (see Figure 21b). Note that, as the construction continues, from k-bit block to k-bit block (except for the first and last blocks, which are hard-coded), the hook of tiles initially assembled by the block-hook gadget is translated up by two tiles (one translation per extraction region). The location of this hook essentially represents which k-bit block to extract next. As mentioned above, the final tile of the block-hook gadget initiates the up-extraction gadget. The up-extraction gadget assembles upward, in a zig-zag pattern (as described above), parallel to the blocknumber gadget. The top-left tile of each zig-zag pattern tries to grow left but is blocked by a portion of the block-number gadget (this is the zag portion of the path). When the up-extraction gadget grows to the row immediately above the top row of the block-number gadget, the red tile shown in Figure 15b is placed. This tile initiates the "extraction-jump" gadget (see Figure 16b). Subsequently, the upward growth of the up-extraction gadget is blocked by tiles from the previous extraction region. The extraction-jump gadget grows a path of tiles in the z = 1 plane, above the up-extraction gadget and to the right of a portion of the previous extraction region. In our construction, we have O(m) distinct block-hook gadgets, up-extraction gadgets and extraction-jump gadgets (one for each k-bit block). The final tile of the extraction-jump gadget initiates the "extraction" gadget (see Figure 17a). (a) The shaded gadget in the extraction region represents the extraction-jump extraction gadget, pictured on the right in Figure 16b. The extraction gadget extracts the current k-bit block into a one-bit-per-bump representation along the top of the current extraction region. A bump in the z = 0 plane is representative of a 0 and a bump in the z = 1 plane is representative of a 1. These bumps can be seen clearly in Figure 18. Note that the extraction gadget is the result of concatenating k "bit-extraction" gadgets together (see Figure 18). The bit-extraction gadgets are "temperature 1" versions of the "extract bit" (temperature 2) tile types from Figure 5.7a of [12]. The tile complexity of the extraction gadget is O(m) (see our discussion at the end of Section 3.3) and we use the same extraction gadget to extract all k-bit blocks. After the extraction gadget finishes extracting the k bits of the current block, it initiates the assembly of the "ceiling" gadget (see Figure 19a). The ceiling gadget assembles a path of tiles from right to left, placing its final tile under the starting point of the extraction gadget (see the red tile in Figure 19a). Note that, as it assembles toward its ending point, the ceiling gadget places a tile at a carefully chosen location in the z = 1 plane, the purpose of which is to block a portion of a subsequent gadget, known as the "repeating-up" gadget (see Figure 22b), which we describe later. Due to spacing constraints, this special z = 1 tile is placed in the column of tiles that is one tile to the left of the penultimate bit-bump of the current extraction region. The placement of this special z = 1 tile signals a subsequent gadget to assemble the remaining perimeter of the current extraction region and then initiate the assembly of the next extraction region (the gadgets that carry out these tasks will be discussed below). We use a single ceiling gadget in our construction (i.e., the same one is used in all of the m − 2 generally-constructed extraction regions) and its tile complexity is proportional to the width of an extraction region, which is O(k) and thus O(m). The final tile in the ceiling gadget initiates the assembly of the "hook-seeking" gadget (see Figure 20b). The hook-seeking gadget starts growing from the final tile that was placed by the ceiling gadget and grows down in a zig-zag pattern, parallel to the previous up-extraction gadget. The hook-seeking gadget grows down until it finds the location of the block hook (see Figure 14b). The assembly of the hook-seeking gadget is very similar to the block-number gadget, i.e., the placement of certain tiles is blocked until the block hook is reached, at which point the downward growth of the hook-seeking gadget is blocked and a special tile is allowed to be placed to the left of the hook-seeking gadget (in the space directly above the block-hook gadget). Once placed, this special tile initiates the assembly of the "hook-initiating" gadget (see Figure 21b). The same hook-seeking gadget is used in all generally-constructed extraction regions. First, the hook-initiating gadget assembles a path of tiles in the z = 1 plane directly above a portion of the previous hook-seeking gadget. Then, it assembles a group of tiles in the shape of a two-tile-wide hook in the z = 0 plane. This hook of tiles will block the downward assembly of the subsequent "repeating-down" gadget (see Figure 24b). When the first repeating-down gadget in the current extraction region is blocked by the hook-initiating gadget, the former will "know" the location of the latter and thus will initiate the assembly of a translated version of the hook (translated up by two tiles). The same hook-initiating gadget is used in all generally-constructed extraction regions. The final tile of the hook-initiating gadget initiates the assembly of the "repeating-up" gadget ( Figure 22b). The repeating-up gadget is three tiles wide and uses a zig-zag pattern of assembly to search for the top of the previous repeating-down gadget (or the top of the hook-seeking gadget in the case of the first occurrence of the repeating-up gadget). This top is found when the top-left tile of the zig-zag pattern can place a special tile in one of the locations denoted with a dotted outline (under the red tile) in Figure 22b. Further upward zig-zag growth of the repeating-up gadget is blocked by the ceiling gadget. The tile two tiles south of the big red tile in Figure 22b grows north one location in the z = 0 plane and is forced to make a decision: either (1) assemble into the z = 1 plane and initiate the "initiate-repeating-down" gadget (see Figure 23b), or (2) if such growth is blocked in the z = 1 plane, grow north one more location and initiate the assembly of the "initiate-next-extraction-region" gadget (see Figure 26b). The same repeating-up gadget is used in all generally-constructed extraction regions. (a) The shaded gadget in the extraction region represents the initiate-repeating-down gadget, pictured on the right in Figure 23b. The assembly of the initiate-repeating-down gadget is initiated by the repeating-up gadget. It is basically a line of tiles that assembles in the z = 1 plane and "jumps" over the top row of the previous repeating-up gadget. An important property of the initiate-repeating-down gadget is that it can only form when a certain tile in the previous repeating-up gadget is not blocked (by a tile in the ceiling gadget) in the z = 1 plane. The final tile of the initiate-repeating-down gadget initiates the assembly of another repeating-down gadget. The same initiate-repeating-down gadget is used in all generally-constructed extraction regions. (a) The shaded gadget in the extraction region represents the repeating-down gadget, pictured on the right in Figure 24b. The assembly of a repeating-down gadget is initiated by the final tile of the initiate-repeating-down gadget. The purpose of the repeating-down gadget is to find either the repeating-hook gadget or the hookinitiating gadget (note that the latter scenario only occurs with the first repeating-down gadget within each extraction region). When the hook is found, the repeating-down gadget places a tile at a special location, namely one of the locations denoted by a dotted outline above the red tile in Figure 24b. The red tile placed at this special location initiates the assembly of the "repeating-hook" gadget (see Figure 25b). The same repeating-down gadget is used in all generally-constructed extraction regions. (a) The shaded gadget in the extraction region represents the repeating-hook gadget, pictured on the right in Figure 25b. (b) The repeating-hook gadget. Figure 25: Overview of the repeating-hook gadget and its location in the construction. The repeating-hook gadget assembles a path of tiles in the z = 1 plane in order to avoid a portion of the previous repeating-down gadget. This z = 1 path assembles right and then down and ultimately assembles a group of tiles in the shape of a hook in the z = 0 plane, similar to the shape of the hook-initiating gadget. The repeating-hook gadget is initiated by each repeating-down gadget. The main purpose of the repeating-hook gadget is to geometrically propagate the block number position, via the hook of tiles, through the current extraction region. The hook shape of the repeating-hook gadget will also serve to block the downward assembly of the next repeating-down gadget. Note that the final, rightmost hook within an extraction region will serve to block the downward assembly of the block-number gadget of the next extraction region. The same repeating-hook gadget is used in all generally-constructed extraction regions. (a) The shaded gadget in the extraction region represents the initiate-next-extraction-region gadget, pictured on the right in Figure 26b. (b) The initiate-next-extraction-region gadget. Figure 26: Overview of the initiate-next-extraction-region gadget and its location in the construction. In the case where the repeating-up gadget is blocked in the z = 1 plane (by a particular tile in the ceiling gadget, as described above), it cannot initiate the assembly of another repeating-down gadget. However, in this case, because of the geometry of the ceiling gadget, the repeating-up gadget may grow a path of tiles in the z = 0 plane up and underneath part of the ceiling gadget, much like how a highway runs directly underneath an overpass. This is essentially the "signal" from the ceiling gadget to the repeating-up gadget that the current extraction region is almost completed. Note that this signal is hard-coded into the geometry of the ceiling gadget for every extraction region (an obvious consequence of the fact that we use a single ceiling gadget in all generally-constructed extraction regions). The red tile in Figure 22b, in this case, is blocked from growing into the z = 1 plane, but is unblocked on its north side in the z = 0 plane and therefore initiates the assembly of the "initiate-next-extraction-region" gadget (see Figure 26b). The initiate-next-extraction-region gadget is a short horizontal path of tiles in the z = 0 plane, the last of which initiates the assembly of the "floor" gadget (see Figure 27a). The same initiate-next-extraction-region gadget is used in all generally-constructed extraction regions. The assembly of the floor gadget is initiated by the last tile placed by the initiate-next-extraction-region gadget. The floor gadget serves two purposes: it first places tiles along the bottom row of the current extraction region and then it initiates the assembly of the next extraction region by initiating the block-number gadget for the next extraction region. See Figure 27a for an example of how the floor gadget assembles. Since this gadget must assemble a path of tiles of length O(perimeter of an extraction region), its tile complexity is O(m). We use the same floor gadget in all generally-constructed extraction regions. Construction of a zig-zag counter In this section, we describe the construction for the binary, n-bit, zig-zag counter that we use in our square construction. 2 An example assembly with n = 3 is depicted in Figure 28. The initial value of the counter is encoded as a geometric pattern of bit-bumps. This seed row, which is part of another construction (in our case, an optimal encoding region or OER), appears as bit-bumps sticking out on the north side of the row of gray tiles at the bottom of Figure 28. In our example, the initial value of the counter is 000. The assembly of the counter starts at the single north glue drawn in orange and sticking out of the OER in the bottom-right corner of the figure. The assembly proceeds by alternating increment rows, assembling from right to left (in blue in the figure) and copy rows, assembling from left to right (in green in the figure). The counter stops when the maximum n-bit value is reached, at which point it assembles one additional increment row (in blue) and one flat roof (i.e., with no bumps on the north side), shown as white tiles in the figure. The tile complexity of this construction, which is described in detail in the rest of this section, is O(1). The counter construction begins with a right wall, that is, the gadget depicted in Figure 29b, that will serve to block the growth of the next copy row. But first, the right-wall gadget initiates an increment row. The three gadgets needed for the increment rows are shown at the bottom of Figure 29. The main gadget in this group, depicted in Figure 29(e), increments each bit (from 0 to 1 or from 1 to 0). Note that the bit advertised by the previous row is not only incremented but also shifted by two tiles to the left. The second gadget in this group is the copy gadget depicted in Figure 29(d). This gadget is used to leave the bits unchanged in the increment row once the rightmost 0-bit has been incremented and no carry needs to be propagated. Again, the copied bits are shifted by two tiles to the left. This shift also happens with the third gadget in this group, depicted in Figure 29(f), which is specific to the least significant bit of the counter: the notch (i.e., the two missing tiles in the top-right corner of the gadget) will serve as the starting point for a later gadget. The left-wall gadget, depicted in Figure 29a, is initiated only when the bottom-left tile in any one of the increment gadgets is allowed to grow south. This wall is used to mark the end of the current increment row and initiate the next copy row. The two gadgets needed for the copy rows are shown at the top of Figure 30. Both gadgets in this group copy each bit (unchanged) and shift the corresponding bit-bump two tiles to the right to compensate for the leftward shift performed in the previous increment row. Each copy row starts with the gadget in Figure 30a that copies the most significant bit of the counter. The other gadget in this group, depicted in Figure 30b, copies all other bits. This gadget also detects the end of the copy row when its bottom-right tile is allowed to grow south (and is simultaneously blocked in its rightward movement in the z = 1 plane by the right wall that was assembled at the beginning of the previous increment row). At that point, the right-wall foundation gadget (see Figure 29(c)) takes over and initiates another iteration of the increment row/copy row construction by building a right wall. The next group of gadgets in our construction are used to detect that the maximum value has been reached, that is, when all n bits are equal to 1. These gadgets are modified copies of all of the gadgets that we have described so far. The only difference between each copy and the original gadget is that the new gadget remembers that the most significant bit of the counter has already been incremented to the value 1. These gadgets are not depicted individually since they are identical to the red, blue and green gadgets except for, say, a prime being added to their glue labels. These gadgets, shown with bold outlines in Figure 28, are used exclusively in the "top-half" of the counter construction, or as soon as the "msb right copy" gadget has incremented the most significant bit from 0 to 1 (see the row labelled "1 0 0 (inc)" in Figure 28). To complete the construction of the counter as a perfect rectangle, we need to build a flat roof on top. This roof construction starts at the south glue of the bottom-left tile in the "copied and modified" (bold) increment gadget. This glue initiates the assembly of the "msb eave" gadget (see Figure 30(c)), which makes up the topmost left wall and allows the roof to assemble. First, the "middle bottom roof" gadget (see Figure 30(d)) is repeated from left to right to form a single row of (white) tiles with no bit-bumps on its north side. Second, the main roof gadget (see Figure 30(e)) is hard-coded to assemble between 1 and 4 rows of tiles (again, with a flat top). The height of this last gadget depends on the target height h of the counter in the following way: the bottommost row in the main roof gadget rounds up the total height of the counter to a multiple of 4 (note that the "middle bottom roof" row and the bottom row of the main roof gadget together play the role of the last green, copy row). Then the number of additional rows in the main roof gadget must be equal to h modulo 4. Therefore, the number of full rows of white tiles in the main roof gadget must be equal to 1 + h mod 4. Of course, the height of the "msb eave" gadget must also be adjusted to match the height of the main roof gadget. In conclusion, this construction uses an n-bit counter to build a rectangle with width 6n + 4 and height 4(2 n − i) + r, where i is the initial value of the counter and r is equal to the height of the counter modulo 4. Figure 2 : 2This sequence of figures shows how the position of the block number is found. The black tiles correspond to tiles of the previous extraction region. Figure 3 : 3The bits of the current block are represented as bumps along the top of the extraction region that is currently being assembled. Figure 6 : 6A series of gadgets geometrically propagate the position of the block number through the rest of the extraction region. This figure shows two of the gadgets. The first one assembles upward until it is blocked by a previous portion of the assembly. The second one assembles horizontally and to the right as it jumps over the top row of the previous gadget. Figure 7 : 7The position of the block number is propagated through the rest of the extraction region. Figure 8 : 8The last up propagation phase detects when it has reached the end of the extraction region and initiates a perimeter gadget (see Figure 11 : 11Construction of an N × N square, where m is O log N log log N .The counters C1 and C2 (in medium gray) are identical up to rotation. So are their seed rows, each of which is the output of an optimal extraction region (OER1 and OER2, respectively, in light gray). F1 and F2 (in dark gray) are filler regions. Figure 12 : 12Detailed construction for the F1 filler region inFigure 11 ( b ) bThe block-number gadget. Figure 13 : 13Overview of the block-number gadget and its location in the construction. ( b ) bThe block-hook gadget. Figure 14 : 14Overview of the block-hook gadget and its location in the overall construction. (a ) )The shaded gadget in the extraction region represents the up-extraction gadget, pictured on the right inFigure 15b.(b) The up-extraction gadget. Figure 15 : 15Overview of the up-extraction gadget and its location in the overall construction. ( b ) bThe extraction-jump gadget. Figure 16 : 16Overview of the extraction-jump gadget and its location in the overall construction. extraction gadget (seeFigure 18for a detailed view of the large squares in thisfigure).(b) The shaded gadget in the extraction region represents the extraction gadget, pictured above inFigure 17a. Figure 17 : 17Overview of the extraction gadget and its location in the overall construction. Figure 18 : 18Magnified view of the bit-extraction gadget to clarify the differences between bumps representing 0s and 1s. shaded gadget in the extraction region represents the ceiling gadget, pictured above inFigure 19a. Figure 19 : 19Overview of the ceiling gadget and its location in the overall construction. shaded gadget in the extraction region represents the hook-seeking gadget, pictured on the right inFigure 20b.(b) The hook-seeking gadget. Figure 20 : 20Overview of the hook-seeking gadget and its location in the construction. The shaded gadget in the extraction region represents the hook-initiating gadget, pictured on the right inFigure 21b.(b) The hook-initiating gadget. Figure 21 : 21Overview of the hook-initiating gadget and its location in the construction. (a ) )The shaded gadget in the extraction region represents the repeating-up gadget, pictured on the right inFigure 22b.(b) The repeating-up gadget. Figure 22 : 22Overview of the repeating-up gadget and its location in the construction. (b) The initiate-repeating-down gadget. Figure 23 : 23Overview of the initiate-repeating-down gadget and its location in the construction. (b) The repeating-down gadget. Figure 24 : 24Overview of the repeating-down gadget and its location in the construction. The shaded gadget in the extraction region represents the floor gadget, pictured above inFigure 27a. Figure 27 : 27Overview of the floor gadget and its location in the overall construction. Figure 28 : 28Counter construction. The gray tiles at the bottom of the counter are part of the optimal extraction region that produces the seed value. All other tiles are part of the gadgets shown inFigures 29 and 30. The assembly of the counter starts at the orange glue in the bottom-right corner of the figure. Figure 29 :Figure 30 : 2930First set of gadgets used in the construction of the zig-zag counter. In each gadget, the black arrows indicate the entry and exit points of the gadget. eave gadget: the two-tile-wide leftmost column can vary in height from 2 to 5 tiles (5 tiles are shown above) (d) Middle bottom roof gadget (e) Main roof gadget: The roof itself can vary in height from 1 to 4 tiles (3 are shown here) Second set of gadgets used in the construction of the zig-zag counter. In each gadget, the black arrows indicate the entry and exit points of the gadget. Technically, Rothemund and Winfree established the 2D self-assembly case, but their proof easily generalizes to 3D selfassembly. Our construction is different from the one described in[5]. That paper describes a general procedure for converting any 2D temperature 2 zig-zag tile system into a 3D temperature 1 tile system. For example, one difference is in the scaling factor in the vertical dimension, that is, how many rows of tiles are needed in the temperature 1 tile system to represent a single increment row or copy row in the temperature 2 tile system. In our construction, this scaling factor is equal to 2, while it is equal to 4 in the conversion procedure described in[5]. Of course, our construction only produces binary counters and does not apply to any other zig-zag tile system. Running time and program size for self-assembled squares. Leonard M Adleman, Qi Cheng, Ashish Goel, Ming-Deh A Huang, STOC. Leonard M. Adleman, Qi Cheng, Ashish Goel, and Ming-Deh A. Huang, Running time and program size for self-assembled squares, STOC, 2001, pp. 740-748. Sarah Cannon, Erik D Demaine, Martin L Demaine, Sarah Eisenstat, Matthew J Patitz, Robert Schweller, Scott M Summers, Andrew Winslow, Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science. the Thirtieth International Symposium on Theoretical Aspects of Computer ScienceTwo hands are better than one (up to constant factorsSarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert Schweller, Scott M. Summers, and Andrew Winslow, Two hands are better than one (up to constant factors), Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science, 2013, pp. 172-184. Parallelism and time in hierarchical self-assembly. Lin Ho, David Chen, Doty, SODA 2012: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms. SIAMHo-Lin Chen and David Doty, Parallelism and time in hierarchical self-assembly, SODA 2012: Proceed- ings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2012, pp. 1163-1182. Complexities for generalized models of self-assembly. Qi Cheng, Gagan Aggarwal, Michael H Goldwasser, Ming-Yang Kao, Robert T Schweller, Pablo Moisset De Espanés, SIAM Journal on Computing. 34Qi Cheng, Gagan Aggarwal, Michael H. Goldwasser, Ming-Yang Kao, Robert T. Schweller, and Pablo Moisset de Espanés, Complexities for generalized models of self-assembly, SIAM Journal on Com- puting 34 (2005), 1493-1515. Temperature 1 self-assembly: Deterministic assembly in 3D and probabilistic assembly in 2D. Matthew Cook, Yunhui Fu, Robert T Schweller, SODA 2011: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM. Matthew Cook, Yunhui Fu, and Robert T. Schweller, Temperature 1 self-assembly: Deterministic assem- bly in 3D and probabilistic assembly in 2D, SODA 2011: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2011. Limitations of self-assembly at temperature 1. David Doty, Matthew J Patitz, Scott M Summers, Theoretical Computer Science. 412David Doty, Matthew J. Patitz, and Scott M. Summers, Limitations of self-assembly at temperature 1, Theoretical Computer Science 412 (2011), 145-158. M Li, P Vitányi, An introduction to Kolmogorov complexity and its applications. New YorkSpringer VerlagThird EditionM. Li and P. Vitányi, An introduction to Kolmogorov complexity and its applications (Third Edition), Springer Verlag, New York, 2008. Polyomino-safe DNA self-assembly via block replacement. Chris Luhrs, Lecture Notes in Computer Science. Ashish Goel, Friedrich C. Simmel, and Petr Sosík14SpringerChris Luhrs, Polyomino-safe DNA self-assembly via block replacement, DNA14 (Ashish Goel, Friedrich C. Simmel, and Petr Sosík, eds.), Lecture Notes in Computer Science, vol. 5347, Springer, 2008, pp. 112-126. Two lower bounds for self-assemblies at temperature 1. Ján Manuch, Ladislav Stacho, Christine Stoll, Journal of Computational Biology. 176Ján Manuch, Ladislav Stacho, and Christine Stoll, Two lower bounds for self-assemblies at temperature 1, Journal of Computational Biology 17 (2010), no. 6, 841-852. Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, welchem die addition als einzige operation hervortritt. Mojżesz Presburger, Compte-rendus du premier Congrès des Mathématiciens des pays Slaves. WarsawMojżesz Presburger,Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, welchem die addition als einzige operation hervortritt, Compte-rendus du premier Congrès des Mathématiciens des pays Slaves, Warsaw, 1930, pp. 92-101. The program-size complexity of self-assembled squares (extended abstract. W K Paul, Erik Rothemund, Winfree, STOC '00: Proceedings of the thirty-second annual ACM Symposium on Theory of Computing. Paul W. K. Rothemund and Erik Winfree, The program-size complexity of self-assembled squares (ex- tended abstract), STOC '00: Proceedings of the thirty-second annual ACM Symposium on Theory of Computing, 2000, pp. 459-468. Complexity of self-assembled shapes. David Soloveichik, Erik Winfree, SIAM J. Comput. 366David Soloveichik and Erik Winfree, Complexity of self-assembled shapes, SIAM J. Comput. 36 (2007), no. 6, 1544-1569. Proving theorems by pattern recognition -II. Hao Wang, The Bell System Technical Journal XL. 1Hao Wang, Proving theorems by pattern recognition -II, The Bell System Technical Journal XL (1961), no. 1, 1-41. Algorithmic self-assembly of DNA. Erik Winfree, California Institute of TechnologyPh.D. thesisErik Winfree, Algorithmic self-assembly of DNA, Ph.D. thesis, California Institute of Technology, June 1998.

We show that there exist scalar field theories with well-defined one-particle states in general D dimensional nonstationary curved spacetimes whose propagating modes are localized on d ≤ D dimensional hypersurfaces, and the corresponding stress tensor resembles the bare cosmological constant λ B in the D dimensional bulk. We show that nontrivial d = 1 dimensional solutions correspond to λ B < 0. Considering free scalar theories we find that for d = 2 the symmetry of the parameter space of classical solutions corresponding to λ B = 0 is O(1, 1) which enhances to Z 2 ×Diff(R 1 ) at λ B = 0. For d > 2 we obtain O(d−1, 1), O(d−1)×Diff (R 1 ) and O(d−1, 1)×O(d−2)×Diff (R 1 ) corresponding to, respectively, λ B < 0, λ B = 0 and λ B > 0.

10.1093/ptep/ptaa039

1909.05593

ed1ba30ea60163ef5123b94f7e524c90a6578e8a

Quantum of the bare cosmological constant Sep 2019 Farhang Loran Department of Physics Isfahan University of Technology 84156-83111IsfahanIran Quantum of the bare cosmological constant 12Sep 2019 We show that there exist scalar field theories with well-defined one-particle states in general D dimensional nonstationary curved spacetimes whose propagating modes are localized on d ≤ D dimensional hypersurfaces, and the corresponding stress tensor resembles the bare cosmological constant λ B in the D dimensional bulk. We show that nontrivial d = 1 dimensional solutions correspond to λ B < 0. Considering free scalar theories we find that for d = 2 the symmetry of the parameter space of classical solutions corresponding to λ B = 0 is O(1, 1) which enhances to Z 2 ×Diff(R 1 ) at λ B = 0. For d > 2 we obtain O(d−1, 1), O(d−1)×Diff (R 1 ) and O(d−1, 1)×O(d−2)×Diff (R 1 ) corresponding to, respectively, λ B < 0, λ B = 0 and λ B > 0. Introduction In this paper we propose novel scalar field theories, the scalar creepers, in general nonstationary curved spacetime whose stress tensor resemble a bare cosmological constant and the corresponding one-particle states are localized on lower dimensional hypersurfaces. The particle interpretation of states of ordinary quantum field theory (QFT) in Minkowski spacetime can be extended to QFT in stationary curved spacetime. However, there is a lore that a physically meaningful notion of particles do not exist for QFT in a general nonstationary curved spacetime except in an approximate or asymptotic sense. So, some authors argue that the particle interpretation of states should not be considered as an essential feature of QFT [1]. The scalar creepers do not describe ordinary matter field but they have well-defined one particle states in general. Another question in QFT is the existence of matter fields whose propagating modes are localized on d < D dimensional hypersurfaces of the D-dimensional spacetime [2,3]. We show that the propagator of the one-particle states of the scalar creepers are delta-function localized on d dimensional hypersurfaces. In general d ≤ d * , where d * is the number of linearly independent nowhere-zero vector fields in the spacetime which can be computed in the homotopy theory [4]. The scalar creepers are not ordinary matter fields. Their stress tensor resembles a bare cosmological constant, i.e., they all act like perfect fluid with equation of state w = −1. An unsettled issue in QFT in curved spacetime is the cosmological constant problem [5]. Considering a general four dimensional nonstationary curved spacetime (M 4 , g) 1 local Lorentz symmetry implies that the expectation value of the quantum vacuum stress-energy tensor of ordinary fields T µν = − ρ g µν where ρ ∼ Λ 4 is the vacuum energy density and Λ is the high energy cutoff of the ordinary QFT. So T µν adds 8πG ρ ∼ M −2 Pl Λ 4 to the effective cosmological constant whose observed value is λ eff ∼ 10 −122 M 2 Pl . Since λ eff = λ B + 8πG ρ this requires an incredible fine-tuning of λ B . Recently Wang and Unruh have shown that the cosmological constant problem can be resolved if fluctuations of ρ are taken into account and λ B has taken a large negative value −λ B ≫ Λ 2 [6,7]. We show the stress tensor of d = 1 dimensional creepers resembles λ B < 0, and in the simplest d ≥ 2 dimensional models we find that the symmetry of the parameter space of classical solutions corresponding to λ B < 0 is O(d −1, 1) which enhances to O(d −1) ×Diff(R 1 ) at λ B = 0, indicating a phase transition. In summary, the scalar creepers have the following properties: 1. Similarly to ordinary scalars, they are natural extensions of scalars in Minkowski spacetime to curved spacetime. Their actions are diffeomorphism invariant. 2. They have a well-defined notion of one-particle states in nonstationary curved spacetimes, localized to d ≤ D dimensional hypersurfaces without using warp factors or potential wells, hence the moniker. 3. Their stress tensor resembles a bare cosmological constant, i.e., they all act like perfect fluid with equation of state w = −1. So they do not describe ordinary matter field. In section 2, we define scalar creepers as scalar fields whose quanta are confined to d ≤ D dimensional flat hypersurfaces of (M D , η) and examine their existence in (M D , g) in section 3. We discuss their application to cosmology in section 4 and recapitulate the main results in section 5. Scalar creepers in Minkowski spacetime In this section we define the scalar creepers in (M D , η) (defined in footnote 1) by giving their classical action. For quantization, we compute the propagator of non-interacting creepers by path integral and show that the propagating modes are localized on d ≤ D dimensional subspaces. The action of d dimensional scalar creepers in (M D , η) is obtained by removing (D − d) terms corresponding to directions perpendicular to a d dimensional hypersurface from the kinetic term of the action of ordinary scalar field theory. It is given by S = − d D x d−1 a,b=0 1 2 η ab ∂φ ∂x a ∂φ ∂x b + V (φ) ,(1) where V (φ) = 1 2 m 2 φ 2 + V int (φ),(2) m 2 is a constant, and V int (φ) gives self-interaction. Although we have dropped kinetic terms corresponding to directions perpendicular to the hypersurface, we are still considering the scalar field as a function of all spacetime coordinates, i.e., φ = φ(x , x ⊥ ),(3) where x a := x a for a = 0, · · · , d − 1, and x a ⊥ := x a for a = d, · · · , D − 1. Henceforth we study the non-interacting fields and set V int (φ) = 0. The classical field equation is given by δS/δφ = 0, in which, δS δφ = (d) − m 2 φ,(4) and (d) := d−1 a,b=0 η ab ∂ 2 ∂x a ∂x b .(5) For d = D, Eq.(1) is the ordinary scalar field theory in Minkowski spacetime. For d < D, the classical field equation (d) − m 2 φ = 0 is not deterministic if not meaningless altogether because it is silent about the behaviour of the classical field in directions x ⊥ perpendicular to the hypersurface. But the classical fields do not participate in particle physics. The particle interpretation of physical states comes from quantum fields whose correlation function is given by the path integral [8], D F (x − x ′ ) := Z −1 Dφ e iS φ(x)φ(x ′ ),(6) where Z := Dφ e iS is the partition function. Eq.(4) implies that (d) − m 2 D F (x − x ′ ) = −iZ −1 Dφ δe iS δφ(x) φ(x ′ ).(7) Since δφ(x ′ )/δφ(x) = δ D (x−x ′ ), where δ D denotes Dirac delta-function in D-dimensions, integration by parts gives (d) − m 2 D F (x − x ′ ) = iδ D (x − x ′ ),(8) whose solution is D F (x − x ′ ) = D (d) F (x − x ′ )δ D−d (x ⊥ − x ′ ⊥ ),(9) where D (d) F (x − x ′ ) denotes the celebrated Feynman propagator in d-dimensional Minkowski space- time [8]. Eq. (9) shows that the path-integral and the correlation function (6) are well-defined although the action (1) is not classically viable since it does not include kinetic terms corresponding to directions normal to the d dimensional hypersurface. So the action (1) together with the path-integral (6) describe scalar fields whose one-particle states are delta-function localized on a d dimensional subspace. Scalar creepers in general spacetimes In this section we define scalar creepers in general spacetimes and show that, in addition to their ability to stick to lower dimensional hypersurfaces, they hold a well-defined notion of vacuum state and one-particle states in general nonstationary curved spacetimes, in contrast to ordinary scalar field theories [1]. Thus, before discussing the scalar creepers we review the ordinary scalar field theory briefly in order to identify the main obstacles in computing their propagator via path integral in general spacetimes. To see the problem with ordinary scalar field theory, consider the minimally coupled massless scalar field theory in (M D , g) (defined in footnote 1) whose action is given by S := − 1 2 d D y e g µν ∂ µ φ∂ ν φ,(10) where e := |det g| and ∂ µ := ∂ ∂y µ . By introducing the tetrad e µ a satisfying η ab e µ a e ν b = g µν and the vector fields e a := e µ a ∂ µ , the action (10) can be written as S := − 1 2 d D y e η ab (e a φ) (e b φ) ,(11) where e a φ := e µ a ∂ µ φ is diffeomorphism invariant. Although the action (11) is similar to Eq.(1) we cannot, in general, compute the corresponding path integral and obtain the correlation function explicitly. The difficulty can be seen from the expression δS δφ = eη ab e a e b + (∇ µ e a µ )e b φ,(12) where ∇ µ denotes the Levi-Civita connection, and we have used the identity ∇ µ v µ = e −1 ∂ µ (e v µ ).(13) Comparing Eq.(12) with equations (4) and (5) reveals the roots of the difficulty: the vector fields e a are not necessarily divergence free and they do not commute with each other in general. In order to define the creepers in general nonstationary curved spacetimes we need to replace the tetrad e a in Eq.(11) with a set of nowhere-zero vector fields v a := v a µ ∂ µ such that [v a , v b ]φ := (v a µ ∇ µ v b ν − v b µ ∇ µ v a ν ) ∂ ν φ = 0,(14) and ∇ µ v a µ = 0.(15) In subsection 3.1 we show that the vector fields v a exist locally though they do not exist globally. We introduce the scalar creepers in subsection 3.2 and define their one-particle states in subsection 3.3. Geometry A straightforward approach to obtain the vector fields v a satisfying equations (14) and (15) is to work with coordinate systems x µ used in unimodular gravity [9,10] in which e = 1. In these coordinates v a µ = δ a µ where δ denotes the Kronecker delta, i.e., v a := v a µ ∂ µ = ∂ ∂x a .(16) To confirm this proposition we only need to use the identities v a µ ∇ µ v b ν − v b µ ∇ µ v a ν = v a µ ∂ µ v b ν − v b µ ∂ µ v a ν ,(17)∇ µ v µ = e −1 ∂ µ (e v µ ).(18) Henceforth by x-coordinates we mean a coordinate system in which e = 1. In order to find such coordinates explicitly we rearrange the coordinates y µ as (y 0 , y) and suppose that x := y and x 0 (y 0 , y) := y 0 y 0 ref dz e(z, y),(19)where y 0 ref = y 0 ref ( y) is a reference point. One verifies that d D x = e d D y which implies that the determinant of the metric in the x-coordinates equals -1. Furthermore ∂ ∂x 0 is timelike as long as ∂ 0 is timelike because ∂ ∂x 0 = e −1 ∂ 0 ,(20)∂ ∂x i = −e −1 ∂ i x 0 ∂ 0 + ∂ i .(21) As an example consider the four dimensional Schwarzschild spacetime in Kruskal-Szekeres coordinates ds 2 = 4r −1 e −r dudv − r 2 dΩ 2 ,(22) where dΩ 2 := dc 2 1−c 2 + (1 − c 2 )dϕ 2 , c := cos ϑ, 2 and r = 1 + W , in which W := W (−uve −1 ) denotes the Lambert W function. Thus, uv = e r (1 − r). Eq.(19) (with u playing the role of y 0 ) reads x 0 = − 2W 3v (W 2 + 3W + 3),(23) where we have supposed that y 0 ref ( y) = 0, i.e., the reference point is located on the event horizon r = 1. W (−uve −1 ) is real-valued for uv ≤ 1 or equivalently for r ≥ 0. Closed FRW universe So far we have shown that in a D dimensional spacetime there always exist D vector fields satisfying equations (14) and (15) locally. For defining creepers we need to know how many of them exist globally. The existence of nowhere-zero vector fields is a question in homotopy theory. Take n dimensional spheres S n for instance. We know that for n = 1, 3 there exist exactly n nowhere-zero vector fields while there is no such vector fields on S 2 [4]. In this subsection we show that in a D = 4 dimensional FRW universe with closed spatial sections only two nowhere-zero vector fields satisfying equations (14) and (15) coexist globally. Suppose that ds 2 = −dt 2 + ω(t) 2 ds 2 Σ ,(24) where Σ, whose line element ds 2 Σ is t-independent, is diffeomorphic to a three sphere S 3 [11]. The x 0 coordinate is given by Eq.(19) x 0 = t dt ′ ω(t ′ ) D−1 ,(25) and consequently v 0 := ω(t) 1−D ∂ t . The vector field v 0 exists and its divergence is zero except for the beginning and the end when ω(t) = 0. Now we focus on the space section Σ, which we model by S 3 , S 3 := X 1 , X 2 , X 3 , X 4 4 i=1 X i 2 = 1 ,(26) embedded in (R 4 , δ). S 3 is parallelizable, i.e., there exist three independent vector fields on S 3 , v 1 := (−X 2 , X 1 , −X 4 , X 3 ),(27)v 2 := (−X 3 , X 4 , X 1 , −X 2 ),(28)v 3 := (−X 4 , −X 3 , X 2 , X 1 ).(29) The divergence of v i is zero but [v i , v j ] = −2 3 k=1 ǫ ijk v k , i, j = 1, 2, 3.(30) Thus, in an FRW universe with closed spatial sections there exist at most two vector fields, e.g., v 0 and v 1 satisfying equations (14) and (15) simultaneously. In the coordinate system (θ, φ 1 , φ 2 ) given by X 2I−1 = r I cos φ I ,(31)X 2I = r I sin φ I ,(32) where I = 1, 2, and r 1 := cos θ and r 2 := sin θ, we have ds 2 Σ = dθ 2 + cos 2 θdφ 2 1 + sin 2 θdφ 2 2 ,(33) and v 1 = ∂ ∂φ 1 + ∂ ∂φ 2 .(34) Action In a D dimensional spacetime (M D , g) (defined in footnote 1) with d * ≤ D linearly independent nowhere-zero vector fields v a satisfying equations (14) and (15), we give the d ≤ d * dimensional creepers' action by S := d D y eL(φ; v a ),(35) in which the creeper φ = φ(y 0 , · · · , y D−1 ) is a scalar field, e := |det g|, L(φ; v a ) := − 1 2 d−1 a,b=0 η ab (v a φ) (v b φ) − V (φ),(36) and the potential V (φ) is given by Eq. (2). Following the definition of vector fields, v a φ := v a µ ∂ µ φ is a scalar, therefore the Lagrangian density L(φ; v a ) is also a scalar though it is independent of the spacetime metric g and S is diffeomorphism invariant off-shell. In the x-coordinates the action (35) is given by S = − d D x 1 2 d a,b=0 η ab ∂φ ∂x a ∂φ ∂x b + V (φ) ,(37) resembling the action (1). Using equations (15) and (18) one verifies that the classical equation of motion is given by δS δφ = 0, where δS δφ = e (d) φ − ∂V (φ) ∂φ ,(38)and (d) φ := η ab v a (v b φ) .(39) Later in subsection 3.3 we use this result to compute the correlation function by path integral. Stress tensor Now we couple the scalar creepers to Einstein's gravity. For this purpose we first recall the definition of the stress tensor and the consequences of the off-shell diffeomorphism invariance in ordinary matter field theory. In a theory whose action is S = d D y eL the stress tensor is defined by T µν := − 2 e δS δg µν .(40) A diffeomorphism is given by the map y → y ξ such that δ ξ y := y ξ − y = ξ(y). We suppose that the vector field ξ(y) is zero on the boundary and drop all the boundary terms subsequently. The corresponding variation of a scalar φ, a vector v, the one-form dφ, the tensor g and the volume element e are given by the Lie derivatives δ ξ φ(y) = −ξ µ ∂ µ φ(y),(41)δ ξ (∂ µ φ(y)) = −∂ µ (ξ ν ∂ ν φ(y)) = ∂ µ (δ ξ φ) ,(42)δ ξ v µ = −ξ ν ∇ ν v µ + v ν ∇ ν ξ µ ,(43)δ ξ g µν = −∇ ν ξ µ − ∇ µ ξ ν ,(44)δ ξ e = −∇ µ ξ µ .(45) Since L is a scalar, similarly to Eq.(41) we have δ ξ L = −ξ µ ∂ µ L which together with Eq.(45) results in the off-shell diffeomorphism invariance δ ξ S = 0. For ordinary matter fields we have δ ξ S = d D y δS δg µν δ ξ g µν + d D y e ∂L ∂ (∂ µ φ) δ ξ (∂ µ φ(y)) + ∂L ∂φ δ ξ φ .(46) Using equations (41) and (42) together with the classical field equation ∂L ∂φ = ∇ µ ∂L ∂ (∂ µ φ) ,(47) one verifies that the second term on the right hand side of Eq.(46) vanishes. Furthermore, by using Eq.(44) and integration by parts we obtain d D y δS δg µν δ ξ g µν = d D y e ξ ν ∇ µ T µν .(48) Thus in ordinary scalar theory, the off-shell diffeomorphism invariance together with the classical field equation imply that the stress tensor is conserved on-shell, i.e., ∇ µ T µν | on−shell = 0.(49) Similarly, the Einstein-Hilbert action S EH (in the metric formulation) depends on the spacetime metric only and δ ξ S EH = d D y δS EH δg µν δ ξ g µν .(50) Defining the Einstein's tensor by G µν := 2 e δS EH δg µν ,(51) and using Eq.(44) and δ ξ S EH = 0, we conclude that ∇ µ G µν = 0. On the other hand, Einstein's field equation given by δ δg µν (S EH + S) = 0,(52) reads G µν = T µν . Thus, Eq.(49) shows that ordinary scalar theories whose actions are diffeomorphism invariant off-shell can be coupled to Einstein's gravity on-shell. In the case of scalar creepers, L(φ; v a ) is independent of the metric g µν and Eq.(35) together with the identity δe = 1 2 eg µν δg µν imply that the creepers' stress tensor is given by T µν := − 2 e δS δg µν = L(φ; v a ) g µν ,(53) which resembles a perfect fluid with equation of state w = −1. Coupling the scalar creepers to Einstein's gravity results in the field equation δ δg µν (S EH + S) = 0,(54) which gives G µν = T µν . Consistency of this equation with the identity ∇ µ G µν = 0 requires that ∇ µ T µν | on−shell = 0,(55) Therefore a consistent coupling to gravity requires L(φ; v a )| on−shell = constant.(56) In other words, the scalar creepers couple to gravity similarly to a cosmological constant term. We study this phenomenon in section 4. We should not have expected to obtain Eq.(55) as a direct consequence of the off-shell diffeomorphism invariance of S and the classical field equation (38) similarly to Eq.(49), because there exist solutions to the classical field equation for which the on-shell value of L(φ; v a ) is not constant. In fact we can not obtain Eq.(55) by following those steps that led us to Eq.(49) although L(φ; v a ) is a scalar and consequently δ ξ S = 0 off-shell. This can be verified by noting that δ ξ S = d D y δS δg µν δ ξ g µν + d D y δS δφ δ ξ φ + d D y eη ab (v a φ) [ξ, v b ]φ,(57) where we have used equations (43) to obtain the last term which is new compared to Eq.(46). By integration by parts and using equations (15), (38), (39) and (41) we obtain d D y eη ab (v a φ) [ξ, v b ]φ = d D y eδ ξ L(φ; v a ) − d D y e δS δφ δ ξ φ.(58) So the third term in Eq.(57) cancels the δS/δφ term therein and by inserting Eq.(58) in Eq.(57) we obtain δ ξ S = d D y δS δg µν δ ξ g µν + d D y eδ ξ L(φ; v a ).(59) Thus there is no contribution from the classical field equation. After using equations (53), (44) and (41) in Eq.(59) we simply end up with δ ξ S = 0. In summary, a scalar creeper on a d ≤ D dimensional hypersurface is defined by equations (35) and (36). The creepers are perfect fluids with equation of state w = −1 and consequently they do not correspond to ordinary matter field. In order to couple the scalar creepers to gravity their Lagrangian should be constant on-shell. Quantization The propagator of the d dimensional scalar creepers in a (M D , g) can be computed by using the classical action in the path integral. For V int (φ) = 0 the path integral D F (y, y ′ ) := Z −1 Dφ e iS φ(y)φ(y ′ ),(60) together with Eq.(38) give (d) − m 2 D F (y, y ′ ) = − i Z Dφ e(y) −1 δe iS δφ(y) φ(y ′ ) = i e(y) δ D (y − y ′ ),(61) in spite of the fact that we have not defined the path integral, especially the temporal (Feynman) boundary conditions yet. We have only assumed that the path integral exists and the integration by parts is applicable. To compute D F (y, y ′ ) we note that the creepers' action in the x-coordinate system given by Eq.(37) is identical to Eq.(1), the operator (d) defined in Eq.(39) is given by (d) = d−1 a,b=0 η ab ∂ 2 ∂x a ∂x b ,(62) similarly to Eq.(5), and 1 e δ D (y−y ′ ) = δ D (x−x ′ ). Therefore Eq.(61) in the x-coordinates is equivalent to Eq. (8). Consequently the correlation function is given by Eq.(9), and the corresponding particle description is also applicable here, though the d ≤ D dimensional subspace is embedded in a nonstationary spacetime. Precisely we can consider an auxiliary action S = − 1 2 d d x d−1 a,b=0 η ab ∂Φ(x ) ∂x a ∂Φ(x ) ∂x a + m 2 Φ(x ) 2 ,(63) whose second quantization indicates that there exist a vacuum state |0 with respect to the "time coordinate" x 0 (cf. Eq.(20)) and a set of creation and annihilation operators a † (p) and a(p) with commutation relations [a(p), a(q)] = 0,(64)[a(p), a † (p ′ )] = (2π) d−1 δ d−1 (p − p ′ ),(65)[a † (p), a † (p ′ )] = 0,(66) such that a(p) |0 = 0 [8]. The field operator is given by Φ (x ) := ∞ −∞ d d−1 p (2π) d−1 2E(p) a(p)e −ip·x + a † (p)e ip·x ,(67) where x := (x 0 , x), p := (E(p), p), E(p) := p · p + m 2 , p · x := E(p)x 0 − p · x, and the inner product u · w := d−1 a=1 u a w a .(68) We postulate, and define the path integral accordingly, that D (d) F in Eq.(9) equals the corresponding Feynman propagator given by 3 θ(x 0 − x ′ 0 ) 0 Φ (x )Φ (x ′ ) 0 + x ↔ x ′ ,(69) which gives the amplitude for particles with "positive frequency" to propagate from x ′ to x and from x to x ′ for x 0 > x ′ 0 and x 0 < x ′ 0 respectively [8]. Consequently, we have a notion of vacuum state, creation and annihilation operators and "time-ordering" of n-point functions with respect to the "time-coordinate" x 0 , although Eq.(20) demonstrates that the vector field v 0 is not necessarily timelike everywhere. In particular, for d = 2 and m = 0, Eq.(37) reads S = 2 d D x ∂ + φ∂ − φ,(70) in which ∂ ± := 1 2 ∂ ∂x 0 ± ∂ ∂x 1 . This theory can be interpreted as a c = 1 conformal field theory [12] embedded in (M D , g). Application to cosmology In section 3 we defined the scalar creepers in (M D , g) and observed that they have well-defined one-particle states localized to d dimensional hypersurfaces of the bulk. In this section we couple the scalar creepers to gravity and investigate their contribution to the dark energy. In subsection 3.2.1 we showed that the scalar creepers can be coupled to gravity consistently if ∇ µ T µν | on−shell = 0.(71) This condition together with Eq.(53) implies that the creepers can be coupled to Einstein's gravity only if L(φ; v a ) is constant on-shell. In this way, T µν resembles the bare cosmological constant term λ B in the Einstein field equation suggesting that λ B = −8πG L(φ; v a )| on−shell .(72) For the classical solution φ =φ = constant, such that dV (φ) dφ φ=φ = 0,(73) we have L| on−shell = −V (φ).(74) Similarly to ordinary scalar field theories, such solutions are not interesting because they need a fine-tuned potential according to the requirements (72), (73) and (74) [13]. Thus, we dismiss such solutions and suppose that the creepers are not constant on-shell, though we seek classical solutions such that L is constant on-shell, corresponding to a bare cosmological constant. So we demand φ on−shell = constant, L on−shell = constant.(75) In the following we study d = 1 and d ≥ 1 creepers separately. In the one dimensional case we verify that the requirements (75) imply that V (φ) is necessarily zero and λ B < 0. For d ≥ 1 we confine our study to free creepers with V (φ) = 0 and identify the parameter space of classical solutions satisfying the requirements (75). We discover that for d = 2 the parameter space enjoys an O(1, 1) symmetry which enhances to Z 2 × Diff(R 1 ) at λ B = 0, while for d > 2 the symmetry is O(d − 1, 1), O(d − 1) × Diff(R 1 ) and O(d − 1, 1) × O(d − 2) × Diff(R 1 ) respectively for λ B < 0, λ B = 0 and, λ B > 0. d = 1 dimensional scalar creepers For d = 1 the field equation (38) together with Eq.(72) imply thaṫ φ 2 = λ 0 − λ B 8πG ,(76)V (φ) = λ B + λ 0 8πG ,(77) where λ 0 is an integration constant, andφ := δS δ (v 0 φ) = v 0 φ,(78) is the momentum conjugate to φ. Recalling Eq.(2) we note that there exists no function V (φ) except for V (φ) = 0 satisfying (75), (76) and (77). Therefore λ 0 = −λ B , and Eq.(72) reads, λ B = −4πGφ 2 < 0.(79) To estimate the value of λ B in the D = 4 model, we note that since V (φ) = 0 the only energy scale at hand is M Pl . So we can rewrite the action (35) and the corresponding classical field equation in terms of the dimensionless fieldφ := M −1 Pl φ and the dimensionless coordinatesȳ := M Pl y as S = 1 2 d 4ȳ e v 0φ 2 ,(80)v 2 0φ = 0,(81) wherev 0 := M −1 Pl v 0 . The classical solutions isφ = κ(x 0 − x 0 ), wherex 0 := M Pl x 0 and x 0 ∈ R is an integration constant. So it is natural 4 to assume that |κ| ∼ 1 and consequently |λ B | ∼ 1M 2 Pl .(82) Thus equation (72) satisfies the conditions required in the references [6,7], i.e., λ B < 0 and − λ B ≫ Λ 2 ,(83) where Λ ∼ 10 −14 M Pl is the high energy cutoff for the ordinary quantum field theory [14]. It is important to note that this result is independent of how we choose v 0 and x 0 . This result has two consequences. First of all we can identifyφ with a continuous and monotonic time which according to Eq.(79) does not commute with λ B [15]. Secondly, since |κ| ∼ 1 quite naturally, we conclude that the Wang-Unruh approach to the cosmological constant problem provides an anthropic explanation of the hierarchy problem: since |λ B | ∼ 1M 2 PL and λ eff ∼ 10 −122 M 2 Pl we should have Λ ≪ 1 in Planck units [6,7]. d ≥ 2 dimensional scalar creepers Suppose V (φ) = 0. In this case the action (36) is invariant under adding an arbitrary constant φ 0 to φ. So in the following we study the equivalence classes of classical solutions defined accordingly, i.e., throughout this subsection, φ stands for the set of all fieldsφ such thatφ(y) − φ(y) is constant. For d ≥ 2 the classical solutions to the field equation (d) φ = 0 satisfying the requirements (75) are given by φ cl = d−1 a=0 κ a x a + f (z),(84) in which f is a smooth function and z := d−1 a=0 K a x a ,(85) such that K · K = 0, K · κ = 0, κ · κ = (4πG) −1 λ B ,(86) where, for example, K · κ := d−1 a,b=0 η ab K a κ b .(87) In the following we investigate the parameter space of the classical solutions (84) indicated by equations (86) for λ B < 0, λ B = 0 and λ B > 0 separately. Classical solutions corresponding to λ B < 0 Since κ · κ = (4πG) −1 λ B we can choose κ = ( |κ · κ|, 0, · · · , 0), which together with K · κ = 0 imply that K 0 = 0. Using this result in K · K = 0 we obtain K = 0. Thus the classical solution is φ cl = d−1 a=0 κ a x a ,(88) the parameter space includes only κ a 's, and it is symmetric under the action of the group O(d−1, 1). Classical solutions corresponding to λ B = 0 In this case κ · κ = 0. Therefore in Eq.(84) we can absorb the term d−1 a=0 κ a x a to f (z), and without loss of generality claim that the classical solution is φ cl = f (z),(89) where f is a smooth function and z := d−1 a=0 K a x a ,(90) such that K · K = 0, i.e., K 2 0 = d−1 a=1 K 2 a .(91) Thus the symmetry of the parameter space has two factors; O(d − 1) for the K-subspace and the diffeomorphism z → f (z). Classical solutions corresponding to λ B > 0 Now we can satisfy κ · κ = (4πG) −1 λ B by choosing κ = (0, |κ · κ|, 0, · · · , 0). In this way, K · κ = 0 implies that K 1 = 0. Using this result in K · K = 0 we obtain K 0 = 0 for d = 2 and K 2 0 = d−1 a=2 K 2 a , for d > 2.(92) Thus for d = 2 the classical solution is φ cl = κ 0 x 0 + κ 1 x 1 such that κ 2 1 − κ 2 0 = (4πG) −1 λ B and similarly to λ B < 0, the parameter space is symmetric under the action of the group O(1, 1). For d > 2 the classical solution is given by Eq.(84) and the symmetry of the parameter space has three factors; O(d − 1, 1) for the κ-subspace, O(d − 2) for the K-subspace and diffeomorphisms z → f (z). Concluding Remarks Second quantization of quantum field theory in four dimensional Minkowski spacetime has been successful in modelling particle physics. Thinking about our universe as a four dimensional spacetime conceivably embedded in a higher dimensional geometry, it is reasonable to seek a quantum field theory describing particles localized to a nonstationary curved hypersurface embedded in a nonstationary curved spacetime. In [16] we have shown that a fermionic field theory exists in four dimensions with a consistent particle interpretation in general nonstationary curved spacetime whose one-particle states are localized on two dimensional subspaces. In this work, we have shown that there exist scalar field theories whose one-particle states are well-defined in general D dimensional nonstationary curved spacetimes and their quanta are localized on d ≤ D dimensional subspaces. Therefore, in addition to providing a notion of scalar particles in nonstationary curved spacetimes, this construction might be of some interest in the brane-world models of a four dimensional universe embedded in a higher dimensional bulk, see, e.g., [2,3] and references therein. In particular, the "massless" quantum field theories localized on two dimensional subspaces can be interpreted as c = 1 conformal field theories embedded in the bulk. The main obstacle to the particle interpretation of states of a quantum field theory in general is the absence of a preferred notion of time translations in nonstationary spacetimes [1]. We have noted that this problem can be circumvented by distinguishing the time-ordering of field operators arising in the path integral formalism from the timelike directions of the spacetime. The so-called time-ordering occurring in the path integral can be attributed to the relative signs in the kinetic term. So, it can be separated from the spacetime geometry by using the action (35) whose field equation does not engage the spacetime metric. The path integral implies that the correlation function satisfies Eq.(61). We have been able to solve this equation, and explicate the path integral accordingly, by using a coordinate system denoted by x, in which the volume element e d D y equals d D x. Such coordinate systems are familiar in unimodular gravity [9,10]. We observed that the Feynman propagator mimics the propagator of an auxiliary scalar field confined to a d dimensional flat hypersurface of a D dimensional Minkowski spacetime. So, we interpreted it accordingly as the propagating amplitude of one-particle states localized on the d dimensional hypersurface. These scalars do not describe ordinary matter as can be seen from their stress tensor T µν = Lg µν in which L denotes the Lagrangian density. They all act like perfect fluid with equation of state w = −1. By coupling to gravity, conservation of the stress tensor implies that L is constant onshell. So these scalars add to the cosmological constant. We approached this problem classically and considered the on-shell value of L as the bare cosmological constant λ B . The d = 1 case is almost unique. Requiring the existence of nontrivial solutions to the classical field equation implies that the potential term is zero. So the only freedom in writing the action is to choose a nowhere-vanishing vector field which is asymptotically timelike. In an FRW universe the vector field is timelike everywhere so the d = 1 creeper can be considered as a continuous and monotonic time which does not commute with λ B [15]. Furthermore, since in this model λ B < 0 it might be of some interest in the recent approaches to the cosmological constant problem [6,7]. The existence of higher dimensional creepers depends on the number of linearly independent nowhere-vanishing vector fields in the spacetime which is a question in homotopy theory [4]. Such creepers are not unique because we need to choose the potential term and also the (asymptotically) spacelike vector fields which together with the (asymptotically) timelike vector field give the action. For example in an FRW universe whose spatial sections are three dimensional spheres, d > 2 dimensional creepers do not exist but we can define d = 2 dimensional creepers and for that purpose we need to choose one spacelike vector field from the three dimensional tangent space. Of course, it is reasonable to use the SO(3) symmetry of the tangent space in order to consider all of these choices equivalent to each other. We studied d ≥ 2 massless and non-interacting creepers and investigated the parameter space of the classical solutions corresponding to a bare cosmological constant. Contrary to the d = 1 creepers, these models allow both positive and negative values for the bare cosmological constant with a signature of phase transition at λ B = 0. For d > 2 the corresponding parameter spaces have different symmetries: O(d − 1, 1) for λ B < 0, O(d − 1) × Diff(R 1 ) for λ B = 0 and O(d − 1, 1) × O(d − 2) × Diff(R 1 ) for λ B > 0. The symmetries of the parameter space for d = 2 are O(1, 1) for λ B = 0 and O(1) × Diff(R 1 ) at λ B = 0. (M D , g) stands for a D dimensional spacetime whose metric in y-coordinates is g. Similarly we denote a D dimensional Minkowski space whose metric in x-coordinates is η by (M D , η). η = diag(−1, 1, · · · , 1) stands for the Minkowski metric. We are replacing ϑ with c in order to make e independent of ϑ. θ(x) denotes the Heaviside step function. θ(x) equals 1 and 0, for x > 0 and x < 0 respectively. Since the transition rates in particle physics are extremely slow compared to 1M Pl , there should exist some other physical processes, inevitably in the gravity side, whose rates are of order 1M Pl . The dimensionless gravitational degrees of freedom are the metric components g µν and the creeperφ, hence we are suggesting v 0φ as the natural candidate. AcknowledgmentsThe author is indebted to B. Azad, S. Lakzian, and M. R. Koushesh for fruitful discussions. R M Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics. University of Chicago PressR. M. Wald, "Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics", University of Chicago Press, 1994. TASI lectures on extra dimensions and branes. C Csaki, hep-ph/0404096From fields to strings. *Shifman, M.2C. Csaki, "TASI lectures on extra dimensions and branes," In *Shifman, M. (ed.) et al.: From fields to strings, vol. 2* 967-1060 [hep-ph/0404096]. Localization of matter and cosmological constant on a brane in anti-de Sitter space. B Bajc, G Gabadadze, hep-th/9912232Phys. Lett. B. 474282B. Bajc and G. Gabadadze, "Localization of matter and cosmological constant on a brane in anti-de Sitter space," Phys. Lett. B 474, 282 (2000) [hep-th/9912232]. Vector fields on spheres. J F Adams, Ann. of Math. 752603J. F. Adams, "Vector fields on spheres," Ann. of Math. (2) 75, No. 3, 603 (1962). The Cosmological Constant Problem. S Weinberg, Rev. Mod. Phys. 611S. Weinberg, "The Cosmological Constant Problem," Rev. Mod. Phys. 61, 1 (1989). Fine-tuning of the cosmological constant is not needed. Q Wang, arXiv:1904.09566gr-qcQ. Wang, "Fine-tuning of the cosmological constant is not needed," arXiv:1904.09566 [gr-qc]. Vacuum fluctuation, micro-cyclic "universes" and the cosmological constant problem. Q Wang, W G Unruh, arXiv:1904.08599gr-qcQ. Wang and W. G. Unruh, "Vacuum fluctuation, micro-cyclic "universes" and the cosmological constant problem," arXiv:1904.08599 [gr-qc]. An Introduction to quantum field theory. M E Peskin, D V Schroeder, Addison-WesleyM. E. Peskin and D. V. Schroeder, "An Introduction to quantum field theory," Addison-Wesley, 1995. Cosmological constant and fundamental length. J L Anderson, D Finkelstein, Am. J. Phys. 39901J. L. Anderson and D. Finkelstein, "Cosmological constant and fundamental length," Am. J. Phys. 39, 901 (1971). A note on classical and quantum unimodular gravity. A Padilla, I D Saltas, arXiv:1409.3573Eur. Phys. J. C. 7511gr-qcA. Padilla and I. D. Saltas, "A note on classical and quantum unimodular gravity," Eur. Phys. J. C 75, no. 11, 561 (2015), [arXiv:1409.3573 [gr-qc]]. S W Hawking, G F R Ellis, The Large Scale Structure of Space-Time. Cambridge University PressS. W. Hawking and G. F. R. Ellis, "The Large Scale Structure of Space-Time," Cambridge University Press, 1973. Applied Conformal Field Theory. P H Ginsparg, hep-th/9108028P. H. Ginsparg, "Applied Conformal Field Theory", hep-th/9108028. Beyond the Cosmological Standard Model. A Joyce, B Jain, J Khoury, M Trodden, arXiv:1407.0059Phys. Rept. 568astro-ph.COA. Joyce, B. Jain, J. Khoury and M. Trodden, "Beyond the Cosmological Standard Model," Phys. Rept. 568, 1 (2015) [arXiv:1407.0059 [astro-ph.CO]]. Infrared and ultraviolet cutoffs of quantum field theory. J M Carmona, J L Cortés, hep-th/0012028Phys. Rev. D. 6525006J. M. Carmona and J. L. Cortés, "Infrared and ultraviolet cutoffs of quantum field theory," Phys. Rev. D 65, 025006 (2002). [hep-th/0012028]. 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The ability of a classifier to recognize unknown inputs is important for many classification-based systems. We discuss the problem of simultaneous classification and novelty detection, i.e. determining whether an input is from the known set of classes and from which specific class, or from an unknown domain and does not belong to any of the known classes. We propose a method based on the Generative Adversarial Networks (GAN) framework. We show that a multi-class discriminator trained with a generator that generates samples from a mixture of nominal and novel data distributions is the optimal novelty detector. We approximate that generator with a mixture generator trained with the Feature Matching loss and empirically show that the proposed method outperforms conventional methods for novelty detection. Our findings demonstrate a simple, yet powerful new application of the GAN framework for the task of novelty detection. * This work was done while both authors were with Intel Corporation.

1802.10560

40511630113ce02fce40d13f03f3ca85d78361d8

Novelty Detection with GAN Mark Kliger [email protected] Shachar Fleishman [email protected] Amazon Amazon Novelty Detection with GAN The ability of a classifier to recognize unknown inputs is important for many classification-based systems. We discuss the problem of simultaneous classification and novelty detection, i.e. determining whether an input is from the known set of classes and from which specific class, or from an unknown domain and does not belong to any of the known classes. We propose a method based on the Generative Adversarial Networks (GAN) framework. We show that a multi-class discriminator trained with a generator that generates samples from a mixture of nominal and novel data distributions is the optimal novelty detector. We approximate that generator with a mixture generator trained with the Feature Matching loss and empirically show that the proposed method outperforms conventional methods for novelty detection. Our findings demonstrate a simple, yet powerful new application of the GAN framework for the task of novelty detection. * This work was done while both authors were with Intel Corporation. Introduction In recent years we have witnessed incredible progress in AI, largely due to the success of Deep Learning, and more specifically Supervised Deep Learning. One of the basic requirements from a good supervised learning algorithm is generalization -the ability to classify input that is reasonably similar to the training data. However, usually there are no requirements whatsoever on how the classifier should behave for new types of input that differ substantially from the data that are available during training. In fact for such novel input the algorithm will produce erroneous output and classify it as one of the classes that were available to it during training. Ideally, we would like that the classifier, in addition to its generalization ability, be able to detect novel inputs, or in other words, we would like the classifier to say, "I don't know." Novelty detection can be defined as the task of recognizing that test data differs in some manner from the data that was available during training. The problem of novelty detection arises in many fields and is closely related, but not identical, to the problems of anomaly detection (also outlier detection), where the goal is to recognize anomalous examples in a dataset Chandola et al. (2009). Novelty detection is a hard problem that has received relatively little attention in the ML literature. Nevertheless, in our opinion novelty detection should be a central part of every recognition system. For a comprehensive in-depth review of the topic of novelty detection, we refer the reader to Pimentel et al. (2014). Popular conventional novelty detection methods such as Probabilistic methods Pimentel et al. (2014), which perform density estimation of the examples from a nominal class, or Domainbased methods Schölkopf et al. (2001); Tax & Duin (2004), which re-formulate novelty detection as a "one-class classification" problem and define a boundary around the nominal class, do not scale well to large high-dimensional datasets. Another class of novelty detection methods is distance-based methods, which assume that the nominal data are tightly clustered, while novel data occur far from their nearest neighbors. These methods require computationally expensive clustering or nearest neighbors search. One of the major drawbacks of conventional novelty detection methods is that at test time, they are separated from the classification algorithm and therefore increase the overall computational and design complexity of the system. Furthermore, combining novelty detection with a multi-class classifier into a single algorithm may leverage important intra and inter-class information in the training data which can benefit both tasks. In this paper we are interested in methods for simultaneous classification and novelty detection. A popular heuristic approach to simultaneous classification and novelty detection is thresholding the maximum of estimated class probability Hendrycks & Gimpel (2017), or alternatively, to threshold the entropy of the estimated probability distribution. Another practical approach is to collect "backgroundclass" samples, that is, samples that are not from the nominal set and hopefully represent novel data. In this case, novelty detection can be reduced to a supervised learning problem. Unfortunately, this solution requires collecting a large set of background inputs -a time-consuming task and moreover, it is very difficult to sample a large enough background class that will represent all possible novel examples. We embrace the "background-class" sampling idea and ask is it possible to generate novel data? To the best of our knowledge, no method for novelty detection is based on generating novel examples. In this paper we examine whether Generative Adversarial Networks (GAN), a popular generative framework that can be used to generate novel examples. The GAN framework was proposed by Goodfellow et al. (2014), as a generative modeling method, mostly used for generating realistic samples of natural images. More specifically, GAN is an approach to generative modeling where two models are trained simultaneously: a generator and a discriminator. The task of the discriminator is to classify an input as either the output of the generator ("fake" data), or actual samples from the underlying data distribution ("real" data). The goal of the generator is to produce outputs that are classified by the discriminator as "real", or as coming from the underlying data distribution. In some formulations of GANs Odena (2016); ; Springenberg (2015), the discriminator is trained to classify data not only into two classes "real" and "fake", but rather into multiple classes. If the "real" data consists of K classes, then the output of the discriminator is K + 1 class probabilities where K probabilities corresponds to K known classes, and the K + 1 probability correspond to the "fake" class. In this paper, we propose to use a multi-class GAN framework for simultaneous classification and novelty detection. If during training the generator generates a mixture of nominal data and novel data, the multi-class discriminator learns to discriminate novel data from nominal data and essentially became a novelty detector. At test time, when the discriminator classifies a real example to the K + 1 th class, i.e., class which represented "fake examples" during training, this example is most likely a novel example and not from one of the K nominal classes. In fact, we prove that in this case the discriminator become an optimal novelty detector (for a given false positive rate). We approximate such a mixture generator with a generator trained with specifically designed loss functions. In Section 2 we provide background and a theoretical justification to our proposed method and its connection to existing novelty detection methods. We validate the proposed method by a set of experiments in Section 3. Novelty detection with GANs Novelty detection In the novelty detection task, we assume that during inference, the classifier may be tested on data samples, which do not belong to the nominal data distribution p data (x), or in other words, novel data from p novel (x) and not from one of the K classes that the classifier is supposed to classify. If we had access to both p data (x) and p novel (x), the densities of the nominal data and novel data, then according to Neyman-Pearson's Lemma (Lehmann & Romano, 2005), the optimal novelty detection test for a given false positive rate α can be achieved by thresholding the likelihood ratio p data (x) p novel (x) at an appropriate value. In practice we don't know both distributions. A "second best" solution would have to have access to both nominal and novel examples, then novelty detection can be reduced to a supervised binary classification problem. However, in practice we do not have access to novel examples. One of the standard approaches to novelty detection is to estimate a level set of the nominal density p data (x) > α, and to declare test points outside of the estimated level set as novel. Density level set estimation is equivalent to assuming that novelties are uniformly distributed on the support of p data (x). Unfortunately, these methods are difficult to implement since they require estimating the high-dimensional density of the nominal data. Level set methods are closely related to one-class classification methods Schölkopf et al. (2001); Tax & Duin (2004) and in general can be reduced to binary classification problem between nominal data and artificially generated sample from uniform distribution on the data domain, as was discussed by Steinwart et al. (2005). The authors showed that in this case it is possible to obtain an optimal (for a given false positive rate) novelty detector. Indeed, if we have access to nominal data distribution p data (x) and a mixture of the form πp novel (x) + (1 − π)p data (x) then πp novel (x) + (1 − π)p data (x) p data (x) = πp novel (x) p data (x) + (1 − π)(1) which leads to an appropriately scaled optimal novelty detector. Therefore, in the SSND case, novelty detection can still be reduced to a supervised classification problem. This is in contrast to the density estimation problem in which no novel data is available. Of course the problem remains to estimate the likelihood ratio from the training data. While semi-supervised novelty detection provides interesting insights, how can these ideas be applied to the fully unsupervised novelty detection problem where only nominal data are available during training? Level set methods can be solved by generating artificial examples from uniform distribution on data domain and formulating binary classification problem. But, is it possible to generate artificial examples from some non-uniform distribution which better represent novel data than uniform distribution? and Is it possible to generate examples from the mixture of nominal and non-uniform distributions to apply the SSND ideas? And finally, is it possible to combine these ideas for novelty detection with multi-class classification? In the next sections we demonstrate how we can leverage the GAN framework to generate examples from the mixture of nominal and non-uniform distributions and use the GAN's discriminator as a simultaneous multi-class classifier and novelty detector. Generative Adversarial Networks (GANs) Generative Adversarial Networks Goodfellow et al. (2014) is a recently proposed approach for generative modeling. The main idea behind GANs is to have two competing differentiable functions, usually implemented as neural network models. One model, which is called the generator G(z; θ G ), maps a noise sample z sampled from some prior distribution p(z) to the "fake" sample x = G(z; θ G ); x should be similar to a "real" sample sampled from the nominal data distribution p data (x). The objective of the other model called the discriminator D(x; θ D ) is to correctly distinguish generated samples from the training data which samples p data (x). This is a minimax game between the two models with a solution at the Nash equilibrium. For an excellent overview on GANs, see Goodfellow (2016). Unfortunately, there is no closed-form solution for such problems. Therefore, the solution is approximated using an iterative gradient-based optimization of the generator and the discriminator functions. The discriminator is optimized by maximizing: max θ D E x∼p data (x) [log D(x; θ D )] + E z∼p(z) [log(1 − D(G(z; θ G ); θ D ))](2) and the generator is optimized by minimizing: min θ G E z∼p(z) [log(1 − D(G(z; θ G ); θ D ))](3) For a fixed generator G(z) we can analytically derive that the optimal discriminator will take the form: D * G (x) = p data (x) p data (x) + p g (x)(4) Since the introduction of Generative Adversarial Networks by Goodfellow et al. (2014), multiple variants of GANs were published in the literature. GANs were applied to various interesting tasks such as realistic image generation Radford et al. (2015), text-to-image generation Reed et al. (2016), video generation Vondrick et al. (2016), image-to-image generation Isola et al. (2016), image inpainting Pathak et al. (2016), super-resolution Ledig et al. (2017), and many more. In almost all of these applications, only the generator G is used at a test time, while the discriminator D is trained for the sake of training the generator and is discarded at test time. In contrast, introduced a GAN based semi-supervised classifier (SSL-GAN). They showed that in case where only a small fraction of the real examples have labels, and the bulk of the real data is unlabeled, their GAN framework is able to train a powerful multi-class classifier, which is the discriminator D. In order to improve the GAN convergence, proposed to optimize the generator by minimizing a Feature-Matching loss: L F M (x) = min θ G ||E x∼p data (x) [f (x)] − E z∼p(z) [f (G(z; θ G ))]||(5) where f (x) is an intermediate layer of the fixed discriminator used as a feature representation of x. Optimizing a generator using the Feature Matching loss results in samples which are not of the best visual quality, but the resulting multi-class discriminator performs well for supervised classification and for semi-supervised classification as well. Mixture Generator for Novelty Detection In the semi-supervised settings of multi-class classification, showed that SSL-GAN trained with the Feature Matching loss (eq.5) improves the classification accuracy of the discriminator. This improvement occurs even though the generated samples are not of the best visual quality. Dai et al. (2017) addressing the question "why does the Feature Matching loss improves the semi-supervised discriminator" made two important observations: In Proposition 1 Dai et al. (2017) show that when the generator is perfect, i.e. p g (x) = p data (x), it does not improve the generalization performance of the SSL-GAN discriminator over the supervised learning training without GAN. In Proposition 2 Dai et al. (2017) defined the complement generator, a generator which generates samples with feature representation f (x) distributed in a complementary region to the distribution support of the features of the real data. They show that under mild assumptions when training a multi-class discriminator with the complement generator the discriminator places the real-classes boundaries in low-density areas of the features distributions of the real data. The conclusion is that only a bad generator, where p g (x) = p data (x) and which generates samples outside of the high-density areas of the real data distribution, is able to improve semi-supervised learning as was demonstrated by . The results of and Dai et al. (2017) led us to propose the following definition of a mixture generator: Definition 1. Mixture generator is a generator with a mixture distribution p g (x) = πp other (x) + (1 − π)p data (x) of the true data distribution p data (x) and some other distribution p other (x), such that there exists a non-empty region Ω where {∀x ∈ Ω, p other (x) > p data (x), p data (x) ≤ }, for some > 0. In other words the mixture generator is a generator that generates a mixture of true data distribution p data (x) and some other data p other (x), where at least part of the p other (x) probability mass is concentrated in lower-density regions of p data (x). In general, any generator distribution p g (x) that is different from p data (x), can be represented as a unique mixture of p data (x) and some other distribution, which by itself cannot be represented as a (nontrivial) mixture of p data (x) with another distribution (see Proposition 5 by Blanchard et al. (2010)). However not any generator with p g (x) = p data (x) is a mixture generator. The requirement is that at least part of the p other (x) probability mass is concentrated in lower-density regions of p data (x) excludes cases where p g (x) = p data (x) but generates samples in high-density regions of p data (x) as happens in the case of mode collapse. The mixture generator can be viewed as a relaxed version of the theoretical complement generator defined by Dai et al. (2017): every complement generator is a mixture generator with π = 1 and where p other (x) and p data (x) have disjoint support, but the opposite is not true. We can also consider the degenerate mixture generator: in the degenerate case p other (x) is a uniform distribution over the real data domain and π = 1 (assuming p data (x) is not uniform by itself). In this case no learning for such generator is required, and the generator only need to be able to generate uniformly distributed samples in data domain. For novelty detection we would like the generator to generate samples from the (unknown) distribution of the novel data p novel (x). This allows us to solve the novelty detection problem using the following observation: Proposition 1. For a fixed mixture generator G, if p other (x) defines a distribution of the novel data, i.e. p other (x) = p novel (x), then the optimal discriminator D * G (x) is also an optimal (for a given false positive rate) novelty detector. The proof of Proposition 1 follows trivially from the definition of the mixture generator, eq. 4 of the optimal discriminator for a fixed generator, and eq. 1 of the optimal novelty detector as in the SSND case. Indeed, if p g (x) = πp novel (x) + (1 − π)p data (x) then we can invert eq. 4 and we have 1 − D * G (x) D * G (x) = p g (x) p data (x) = πp novel (x) + (1 − π)p data (x) p data (x) = π p novel (x) p data (x) + (1 − π)(6) which as in eq. 1 is an optimal novelty detector for a given false positive rate. The question is which mixture generator is capable of generating data from the unknown distribution p novel (x) of the real novel data. Unfortunately, without any knowledge of the specific distribution and without labeled real examples from this distribution, or at least unlabeled examples from a mixture of real novel data and nominal data, it is impossible to design a generator which will generate samples from it 2 . Nevertheless, we can train a mixture generator with specific properties of p other (x) which will help novelty detection. In the case of the degenerate mixture generator, i.e. when the generator generates samples uniformly distributed over data domain, the discriminator in eq. 6 is a level set novelty detector. As we mentioned in section 2.1 such reduction of the level-set novelty detection to a classification problem is well known in the literature (Steinwart et al. (2005) and references therein). However, it is clear that sampling novel examples from a uniform distribution in a very high dimensional space is not an efficient strategy for training novelty detector. Therefore, we want a mixture generator that will generate novel data distributed in nearby surrounding or at low-density regions of the true data manifold. To train such a mixture generator we need a loss function that encourage the generator to do that. For example Dai et al. (2017) proposed a loss function based on the Kullback-Leibler (KL) divergence between the distributions of the generator distribution and distribution of the data. More specifically they propose to minimize: min θ G −H p g (x; θ G ) + E x∼pg(x;θ G ) log p data (x)I[p data (x) > ] + L F M (x; θ G )(7) where H(·) is the entropy function and I[·] is the indicator function. This loss function is designed to produce a generator with a distribution which on one hand has support which does not intersect with high density regions of the real data (second term), but still close to the data manifold (Feature Matching loss as a third term). Unfortunately, to train a generator with the loss function in eq. 7, we need to estimate p data (x), the same problem which we wanted to avoid in conventional novelty detection methods. In their paper Dai et al. (2017) experimentally demonstrated on two synthetic datasets that a generator that is trained with a Feature Matching loss eq. 5 is able to generate both samples that fall onto the data manifold, and samples which are scattered in the nearby surrounding of the data manifold, i.e. a mixture of real data and a data outside of data manifold. The empirical results of Dai et al. (2017) suggest that a generator that is trained with the Feature Matching loss only, is by itself a mixture generator. Moreover, we know from the results of Dai et al. (2017) that when the generator generates samples that are scattered around and in the low-density areas of the data manifold, the discriminator classification boundaries becomes tighter, thus improving the discriminator's ability to detect real novelties. In Section 3 we empirically demonstrate that when a multi-class discriminator trained with a mixture generator that was trained with the Feature Matching loss, has an impressive ability to detect real novel examples. To summarize, we propose to train a GAN in the multi-class setting with the Feature Matching loss or some other similar loss which facilitates the generator to be a mixture generator as in Definition 1 to solve the problem of simultaneous classification and novelty detection. When the multi-class discriminator of the network classifies an example as "fake", it means that most likely this example is a novel example, otherwise the class with the highest probability is the classification result. The novelty detection scores are the "fake" class probability or a related quantity such as D G (x) 1−D G (x) , the ratio between "real" and "fake" class-probabilities. The novelty detection scores are thresholded to achieve a desired false positive rate. The GAN based training produce a unified multi-class classifier and novelty detector with minimal additional computation overhead at inference time. Moreover, using unlabeled samples as in improve both the multi-class classifier and the novelty detector. Experiments We perform an experimental evaluation of the proposed GAN-based novelty detection method and compare it to several methods for simultaneous classification and novelty detection on the MNIST and CIFAR10 datasets. As a comparison metric we use the Area Under the Receiver Operating Characteristic curve (AUROC) of novelty detection scores. The first method that we compare to is based on the estimated class probabilities of a multi-class classifier. Hendrycks & Gimpel (2017) suggested to use the maximum of the estimated (softmax) probability as a baseline for out-of-distribution data (i.e. novelty) detection score. Another popular novelty score is the entropy of the estimated probabilities, which takes into account all the predicted class probabilities. These scores are highly correlated. In addition, we compare to methods that rely on nearest-neighbors distance analysis in feature space derived from the trained multi-class classifier (e.g. last convolutional layer of CNN). We use normalized k-NN distance as a novelty score as described by Ding et al. (2014). The kNN novelty score of a test data point x is a distance to the k th nearest neighbor in the nominal training dataset N N k (x), where N N k () denotes the k nearest neighbors, normalized by k-NN distance between the N N k (x) to its k nearest neighbors N N k (N N k (x)): d(x, N N k (x)) d(N N k (x), N N k (N N k (x))) . For k = 1, d is an l 2 distance in a feature space. For k > 1, d denotes the average distance to the k nearest neighbors. We evaluate this method using k = {1, 5}. Additionally we experimented with the OCSVM and the SVDD methods Schölkopf et al. (2001); Tax & Duin (2004) implemented in LIBSVM library Chang & Lin (2011). We tried various kernels and parameters values, but failed to achieve competitive results, even relative to the kNN methods. Our findings are supported by Ding et al. (2014) where kNN methods outperformed domain-based methods on multiple datasets. The fact that domain-based method struggling in applications involving high-dimensional spaces was also noted by Pimentel et al. (2014). For the GAN based novelty detection score which we call ND-GAN, we trained the SSL-GAN model as described by using modified publicly available implementation of SSL-GAN 3 and using Theano Theano Development Team (2016). The mixture generator was trained with the Feature-Matching loss eq.5, and the novelty score of the ND-GAN was defined as the ratio: D G (x) 1−D G (x) , the ratio between"real" class and "fake" class probabilities. To make a fair comparison, when comparing to the competing methods, we trained the multi-class classifiers with the same architectures as the discriminator. MNIST We performed two experiments with MNIST dataset, novelty detection from other datasets and a holdout novelty detection test. First, we identify novelties with respect to the MNIST datasets by running MNIST trained classifiers on the Omniglot dataset of images of handwritten characters from Lake et al. (2015), and notMNIST dataset images typeface characters from Bulatov (2011). Figure 1. We see that the ND-GAN novelty score outperforms the other novelty detection methods in all experiments. In the holdout experiment, we train an SSL-GAN model and multi-class classifier with the same architecture as discriminator for each of the ten digits. Every model is trained on nine out of the ten classes. During testing, we compute the novelty scores for all of the classes, including the holdout class which is considered to be novel (negative). For the holdout class testing is performed on both the train and test examples in order to balance the number of nominal and novel examples. In Table 1 we present the AUROC of ND-GAN and the other competing methods. We see that in seven out of ten holdout experiments, ND-GAN outperforms the other methods, as well as the mean result. Krizhevsky (2009). The CIFAR100 dataset contains 100 fine and 20 coarse categories, which differ from the CIFAR10 categories (small overlaps in the data such as lion-cat, wolf-dog, truck-bus, etc. contributes equally to the error of all of the methods). For the ND-GAN novelty detector, we employ the architecture as in the SSL-GAN framework with 3000 labeled examples from each class. For the discriminator a 9 layer deep convolutional network with dropout and weight normalization was used. The generator has a 4 layer deep CNN with batch normalization. For the other methods (kNN, entropy and max-probability) the same architecture as The CIFAR10-train-set was used to train the model. The CIFAR10 test-set was used as a nominal data and the CIFAR100 test-set as a novel data. in the discriminator was used. The k-NN distances are computed from the 192 dimensional feature vectors of last convolution layer. Figure 2 depicts a ROC curves in which the CIFAR10 test set (nominal examples) is compared against the whole CIFAR100 test set (novel examples). We see that the ND-GAN novelty score performs comparably to other methods. We also compare each of the 20 coarse categories in CIFAR100 separately. The results of this experiment are presented in Table 2. We see that in 13 out of 20 coarse CIFAR100 categories ND-GAN outperforms the other methods. Blanchard et al. (2010) discussed a Semi-Supervised Novelty Detection (SSND) problem, where in addition to the training dataset of nominal examples we have access to an unlabeled dataset that contains a mixture of nominal and novel examples. Figure 1 : 110000 MNIST test images are used as positive examples, while images from Omniglot and notMNIST datasets are used as a negative examples. Finally, as a controlled experiment we use the ICDAR 2003 Lucas et al. (2003) dataset of handwritten Latin letters and digits. The ICDAR 2003 digits which From left to right, ROC curves and AUROC numbers, of training a model on MNIST digits and testing for novelty against the Omniglot, notMNIST. In the ICDAR-2003 dataset, digits were used as nominal examples while letters were used as novel examples. are different from MNIST, serve as an independent validation dataset. For the GAN-based novelty detector we follow the SSL-GAN framework and use only 100 labeled examples from each class. Both the generator and the discriminator have 5 fully connected hidden layers each. Weight normalization Salimans & Kingma (2016) was used and Gaussian noise was added to the output of each layer of the discriminator. To evaluate the other methods (kNN, entropy and max-probability), we trained a standard supervised network with the same architecture. The k-NN distances were computed from the 250 dimensional feature vectors of the last fully connected layer. For the ICDAR 2003 datasets the following ambiguous letters {o, O, i, I, z, Z, S, s,l} were excluded from the experiment. The ROC curves of the experiments are depicted in Figure 2 : 2CIFAR10 vs CIFAR100 experiment: Table 1 : 1MNIST holdout experiment: comparison of different novelty detection methods on the MNIST dataset. In each of the ten tests, the model was trained with the standard MNIST training set, excluding one hold-out digit. The AUROC is computed from a balanced test-set in which the nine digits are considered nominal data and the hold-out digit is novel.ND-GAN 0.992 0.982 0.966 0.976 0.936 0.989 0.947 0.978 0.976 0.967 0.971 3.2 CIFAR10 vs CIFAR100 In the CIFAR experiment we train classifiers on the CIFAR10 train datatset and test for novelties on images from the CIFAR10 test datasets (nominal examples) and the CIFAR100 dataset (novel examples)HoldOut 0 1 2 3 4 5 6 7 8 9 mean Entropy 0.966 0.984 0.965 0.961 0.947 0.957 0.955 0.970 0.974 0.958 0.964 max prob. 0.965 0.984 0.964 0.961 0.946 0.957 0.954 0.970 0.973 0.958 0.963 1-NN 0.916 0.921 0.862 0.844 0.680 0.863 0.865 0.818 0.857 0.840 0.846 5-NN 0.975 0.952 0.943 0.930 0.811 0.918 0.919 0.941 0.928 0.918 0.924 Table 2 : 2CIFAR10 vs CIFAR100 experiment: comparison of different novelty detection methods on the 20 CIFAR100 coarse classes. The AUROC is computed from a balanced CIFAR10 test-set as a nominal data and train and test data of the CIFAR100 coarse classes as a novel data. Cherti et al. (2016) empirically demonstrated that a GAN that was trained with a regular loss function is able to generate out-of-class realistic images, e.g., a GAN trained on MNIST digits generated Latin letters https://github.com/openai/improved-gan Conclusion and future workThe ability to identify novelties or say "I don't know" is an important tool for many classificationbased systems. In this work, we propose to solve a problem of simultaneous classification and novelty detection within the GAN framework. We propose to use a GAN with a mixture generator to turn this problem into a supervised learning problem without collecting "background-class" data.In the case where a mixture generator generates samples from a mixture of nominal data distribution and novel data distribution, we showed that the GAN's discriminator is an optimal novelty detector. We approximate that generator with a mixture generator trained with the Feature Matching loss. This mixture generator generates samples scattered around and in the low-density areas of the data manifold, and this makes a multi-class discriminator a powerful novelty detector. We empirically validate that the performance of the proposed solution is comparable to several popular novelty detection methods, and sometimes outperforms them.Clearly, evaluation of the proposed framework on more challenging datasets is required. 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In this contribution, we discuss three situations in which complete integrability of a three dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three dimensional sphere and a particular case associated with the Kepler problem in S 3 .

10.1088/1751-8113/41/30/304030

0711.4915

8abd8daef561ff5e421abe71310483671a4d8b13

Classical and quantum integrability in 3D systems 30 Nov 2007 M Gadella [email protected] Departamento de Física Teórica. Facultad de Ciencias 47011ValladolidSpain J Negro [email protected] Departamento de Física Teórica. Facultad de Ciencias 47011ValladolidSpain G P Pronko [email protected] Institute for High Energy Physics Moscow regProtvinoRussia Institute of Nuclear Physics National Research Center "Demokritos" AthensGreece M Santander Departamento de Física Teórica. Facultad de Ciencias 47011ValladolidSpain Classical and quantum integrability in 3D systems 30 Nov 2007arXiv:0711.4915v1 [math-ph] Version 27Nov07:12:45, ultima revision MS In this contribution, we discuss three situations in which complete integrability of a three dimensional classical system and its quantum version can be achieved under some conditions. The former is a system with axial symmetry. In the second, we discuss a three dimensional system without spatial symmetry which admits separation of variables if we use ellipsoidal coordinates. In both cases, and as a condition for integrability, certain conditions arise in the integrals of motion. Finally, we study integrability in the three dimensional sphere and a particular case associated with the Kepler problem in S 3 . Introduction. For a system whose configuration space has dimension n, integrability in the sense of Liouville-Arnold requires the existence of a number n of constants of motion which should be in involution (the Hamiltonian itself is one of them). Integrable systems are rare, yet many systems which are important from a physical standpoint turn out to be integrable, with the outstanding examples of the harmonic oscillator and the Kepler problem, which in fact are superintegrable (they have more than n functionally independent constants of motion, albeit not all of them are in involution). A type of integrability appears to be particularly relevant: the quadratic integrability. A standard natural Hamiltonian has a kinetic part which is quadratic in the momenta. If all n − 1 additional constants of motion have a similar kind of structure, we will speak of quadratic integrability (understanding quadratic as 'at most quadratic', i.e., allowing possibly for constants of motion which are linear in the momenta). For a 2D system in any constant curvature configuration space, quadratic integrability means the existence of a single constant of motion quadratic in the momenta; this case is simpler as there is no any extra condition ensuring the additional constants are in involution [1,26]. For any space of constant curvature, integrable systems have a Hamiltonian which is separable in confocal (or general elliptic) coordinates in that space [2,5]. These coordinates include generic ones (as the elliptic coordinates in the sphere S 2 or the Euclidean plane E 2 ) as well as all its possible degenerations or limiting cases. This also holds in the 3D case [14,8]. It is generally believed that a similar connection among quadratic integrability and the nD version of confocal coordinates does somehow hold, but to the best of our knowledge a proof is not available in the literature. In this work we explore some aspects of quadratic integrability for a system in 3D configuration spaces E 3 or S 3 . The results should be considered as a stage towards studying how precisely integrable systems in Euclidean or in any constant curvature 3D space are related to separable systems in the generic version of ellipsoidal coordinates, associated to confocal quadrics [16,17]. We discuss mainly the Euclidean case, but extension of these results to spaces with any constant curvature seems to be possible, and as a hint in that direction, we discuss the integrability with axial symmetry and the Kepler problem in a space with positive constant curvature S 3 (see also [13]). 3D systems with axial symmetry. In this first part, we consider a three dimensional system in Euclidean space with axial symmetry around an axis given by the unit vector n = (n 1 , n 2 , n 3 ). The classical analysis of this system makes use of canonical coordinates (p, x) = (p i , x i ), i = 1, 2, 3 and Poisson brackets {A, B} = ∂A ∂p k ∂B ∂x k − ∂A ∂x k ∂B ∂p k . We want to characterize those systems with axial symmetry which are completely integrable. For these systems dynamics is described by a Hamiltonian of the form H = p 2 2m + U (x) ,(1) where U (x) is a time independent potential. Axial symmetry requires that the angular momentum L around the symmetry axis L = n · (x × p)(2) be a constant of motion, i.e., {H, L} = 0. Complete quadratic integrability requires the existence of a third independent constant of motion H 1 , which is in involution with H and L, i.e., {H, H 1 } = {L, H 1 } = 0 and has a 'Hamiltonian form' in general position dependent yet quadratic in the momenta, plus a time independent 'potential', with form H 1 = 1 2m p i g ik (x) p k + Φ(x) .(3) The tensor g ik (x), which determines the 'kinetic' part of the constant of motion H 1 has to be a Killing tensor for the Euclidean metric. Its determination can be done by assuming that the 'kinetic' terms in (1) and (3) commute among themselves and also with the angular momentum L, with respect to the Poisson bracket. If we choose the angular momentum direction as the z axis, then the most general expression for [20]: p i g ik (x) p k isp i g ik (x) p k = L 2 1 + L 2 2 + αp 2 3 + β(p 1 L 2 − p 2 L 1 ) + γp 3 L 3(4) to which of course any linear combination of the quadratic Casimirs of the Euclidean algebra, p 2 1 + p 2 2 + p 2 3 and p 1 L 1 + p 2 L 2 + p 3 L 3 can be added. Here we will discuss mainly the family with α < 0, β = 0, γ = 0, for which p i g ik (x) p k = L 2 1 + L 2 2 − a 2 p 2 3 (5) with a a positive constant. If now the angular momentum direction is along a general unit vector n = (n 1 , n 2 , n 3 ), by direct computation we get for the Killing tensor g ik (x) the expression g ik (x) = δ ik (x · n) 2 − x · n(x i n k + x k n i ) + (x 2 − a 2 )n i n k ,(6) For more details, see [9]. The commutation of L with H and H 1 restricts the form of the functions U and Φ, collectively called 'potential' terms to: U (x) = U (x 2 , (x · n) 2 ), Φ(x) = Φ(x 2 , (x · n) 2 )(7) (note that the condition of axial symmetry around the n axis implies that U and Φ cannot depend on the azimuthal angle ϕ) and the commutation of H and H 1 leads to the equations: ∂ i Φ(x) = g ik (x)∂ k U (x) , i, k = 1, 2, 3 ,(8) where we always assume summation over repeated indices. In order to obtain the solutions of (8), we diagonalize the matrix with components g ik (x). Its eigenvalues λ(x) and eigenvectors A(x) are obtained from the matrix equation g ik (x) − λ(x)δ ik A k (x) = 0 .(9) The eigenvalue equation (9) has the following solutions: λ ± (x) = x 2 − a 2 2 ± x 2 − a 2 2 2 + a 2 xn 2 , λ 0 (x) = xn 2 ,(10) with corresponding eigenvectors given by A ± i (x) = ∂ i λ ∓ (x) , A 0 i (x) = n × x x 2 − xn 2 = ∂ i ϕ(x) ,(11) where the index i = 1, 2, 3 labels the component of the corresponding vector field and ϕ is the azimuthal angle around the n-axis. These vectors form a new coordinate basis, {∇λ − , ∇λ + , ∇ϕ} referred to which the matrix g ik (x) is diagonal. Taking into account that, due to the geometric symmetry, U and Φ cannot depend on ϕ, (8) becomes ∂ i λ + ∂ + Φ + ∂ i λ − ∂ − Φ = g ik (∂ k λ + ∂ + U + ∂ k λ − ∂ − U ) ,(12) where ∂ ± stand for ∂ ∂λ± . In the basis {∇λ − , ∇λ + , ∇ϕ} that diagonalizes the matrix g ik (x), equation (12) decouples in ∂ + Φ = λ − ∂ + U , ∂ − Φ = λ − ∂ + U .(13) This means that Φ − λ + U = −f (λ + ), Φ − λ − U = −g(λ − ) ,(14) where f (λ + ) and g(λ − ) are arbitrary functions on their respective variables. These equations give an expression for the 'potentials' as follows: U = f (λ + ) − g(λ − ) λ + − λ − , Φ = λ − λ + − λ − f (λ + )− λ + λ + − λ − g(λ − ) .(15) These are the most general expressions for the potentials U (x) and Φ(x) compatible with our choice for H 1 and with the requirement of H, H 1 , L being in involution. A comment is here in order: the presence of a constant of motion which is firstorder in the momenta, i.e. is a Noether constant, means that the system is invariant under rotations around the axis n. Then this situation reduces to one that is essentially two-dimensional, with a single azimutal coordinate ϕ added to the ordinary 2D elliptic coordinates in a fixed plane containing the axis n. In the 3D Euclidean space these coordinates are the spheroidal (oblate) cordinates [22], which play the role of 3D separation coordinates for the system, as we will show explicitly in the next section. Integration. The integration of the classical system with axial symmetry with respect to the n axis is based in the following idea: If e i = h(p, x), i = 1, 2, 3 are three independent functions with {e i , e j } = 0, then, there exists a function F (x, e) with e = (e 1 , e 2 , e 3 ) and p k = ∂F (x, e) ∂x k .(16) The function F (x, e) is the characteristic function in the Hamilton-Jacobi approach. In our case, the chain rule and (16) gives the following expression for the momentum p = (p 1 , p 2 , p 3 ): p = ∇λ + ∂ + F (λ + , λ − , ϕ) + ∇λ − ∂ − F (λ + , λ − , ϕ) + n × x x 2 − (x · n) 2 ∂ ϕ F (λ + , λ − , ϕ) .(17) Then, the kinetic part of H can be written in the following form: p 2 = (∇λ + ) 2 (∂ + F ) 2 + (∇λ − ) 2 (∂ − F ) 2 + ℓ 2 x 2 − (x · n) 2 = 2m (E − U ) ,(18) where ℓ is the value of the integral of motion L corresponding to the angular momentum around n, i.e., (n × x) · p = n · L = ℓ = ∂ ϕ F (λ + , λ − , ϕ)(19) and E is a given value of the constant of motion H. For the kinetic term of H 1 , we obtain a similar expression: p i g ik p k = λ − (∇λ + ) 2 (∂ + F ) 2 + λ + (∇λ − ) 2 (∂ − F ) 2 + (x · n) 2 ℓ 2 x 2 − (x · n) 2 = 2m(E 1 − Φ) , (20) where E 1 is a constant value of H 1 . With some calculations [9], we obtain the partial derivatives of the function F (λ + , λ − , ϕ) in terms of its arguments: (∂ + F ) 2 = 1 4λ + (λ + + a 2 ) 2m (λ + (E − U ) − (E 1 − Φ)) − λ + l 2 λ + + a 2 ,(21)(∂ − F ) 2 = 1 4λ − (λ − + a 2 ) 2m (λ − (E − U ) − (E 1 − Φ)) − λ − l 2 λ − + a 2 .(22) Due to (14), (21) does not depend on λ − and (22) does not depend on λ + . From this simple idea, we conclude that the function F (λ + , λ − , ϕ) is of the form: F (λ + , λ − , ϕ) = A(λ + ) + B(λ − ) + ℓϕ ,(23) where A(·) and B(·) are functions of one variable only. Thus, we have separated variables in the Hamilton-Jacobi generating function. This permits us to obtain the equations of motion in terms of the variables (λ + , λ − , ϕ). The final result is [9] λ + λ + ∂ + A(λ + ) +λ − λ − ∂ − B(λ − ) = 4 ṁ λ + λ + (λ + + a 2 )∂ + A(λ + ) +λ − λ − ∂ − (λ − + a 2 )B(λ − ) = 0 − la 2 4 λ + λ + (λ + + a 2 ) 2 ∂ + A(λ + ) +λ − λ − ∂ − (λ − + a 2 ) 2 B(λ − ) =φ ,(24) where the upper dot represents the derivative with respect time. These equations are at least formally integrable. Quantum case. Canonical quantization of the functions H, L, H 1 give respective operators that we represent with the same symbols. Then, complete integrability means that [H, L] = [H 1 , L] = [H, H 1 ] = 0 ,(25) where [A, B] = AB −BA denotes the commutator of the operators A and B. Equation (25) implies the existence of simultaneous eigenfunctions ψ(x) of these three operators: (H − E)ψ = (H 1 − E 1 )ψ = (L − ℓ)ψ = 0 ,(26) or equivalently, − ∆ψ(x) = 2m(E − U (x))ψ(x) (27) −∆ 1 ψ(x) = 2m(E 1 − Φ(x))ψ(x) (28) −i∂ ϕ ψ(x) = ℓ ψ(x) ,(29) where, ∆ = ∂ k ∂ k , ∆ 1 = ∂ j g jk (x)∂ k .(30) We can express (27) and (28) in terms of the variables λ ± . Using (29) we obtain respectively, − 4λ + (λ + +a 2 )ψ ++ + 2(a 2 +3λ + )ψ + − l 2 λ + λ + +a 2 ψ = 2m [(E−U )λ + −(E 1 −Φ)] ψ ,(31) − 4λ − (λ − +a 2 )ψ −− + 2(a 2 +3λ − )ψ − − l 2 λ − λ − +a 2 ψ = 2m [(E−U )λ − −(E 1 −Φ)] ψ .(32) Note that (31) and (32) depend only on λ + and λ − respectively. Then, the wave function ψ(x) can be factorized as ψ(x) = ψ + (λ + ) ψ − (λ − ) e iℓϕ(33) so that (30), (31) and (29) can be written as equations depending solely on the variables (λ + , λ − , ϕ) respectively. Once we have solved equations (30) and (31), we have solved the problem of finding solutions of (26) or, equivalently, (27)(28)(29). In order to finding solutions, we have to give explicit expressions for the potentials U and Φ. These expressions must be obtained from (15) and the form of the functions f (λ + ) and g(λ − ). In order to illustrate the search for solutions of (31) and (32), let us propose a simple although nontrivial choice of f (λ + ) and g(λ − ) so that (30) and (31) are exactly solvable. Here, we propose f (λ + ) = Φ − λ + U = 0 , g(λ − ) = Φ − λ − U = −Q(λ − + a 2 ) ,(34) where Q is a constant. This choice give the following form for the potentials: U (x) = −Q λ − + a 2 λ + − λ − , Φ(x) = −Q λ + (λ − + a 2 ) λ + − λ − .(35) The steps to solve (30) and (31) with these potentials are the following: i) From (10), we conclude that λ + is always positive meanwhile the values of λ − lie on the interval [−a 2 , 0]. This suggest the following change of variables: λ + = a 2 sinh 2 α and λ − = −a 2 sin 2 β. The new coordinates (α, β, ϕ) are known as the oblate spherical coordinates. This change of variables transforms (30) and (31) into two new wave functions [9]. ii) The new change of variables t := sinh α and u := sin β and the introduction of the new parameters, E := 2ma 2 E, q := 2ma 2 Q, E 1 := ℓ 2 + 2mE 1 , G := E 1 + E and q ′ = q − E, with t = iα transform equations (30) and (31) respectively into (1 − α 2 ) d 2 ψ + (α) dα 2 − 2α dψ + (α) dα + G − ℓ 2 1 − α 2 − E(1 − α 2 ) ψ + (α) ,(36)(1 − u 2 ) d 2 ψ − (u) du 2 − 2u dψ − (u) du + G − ℓ 2 1 − u 2 + q ′ (1 − u 2 ) ψ − (u) = 0 .(37) Equations (36-37) are spheroidal equations, a type of second order differential equation which has been studied [21]. A brief discussion of the solutions for (36-37) and therefore for (31-32) can be found in [9]. General 3D integrable systems. In this section, we intend to generalize the previous discussion to the case in which no (Noether) symmetry is present. As in the previous case, we shall discuss the classical point of view first and then its quantum counterpart. The key of the solution to this general case will be the choice of a proper system of coordinates, as we shall see. This justifies the next subsection. The ellipsoidal coordinates. The generic ellipsoidal coordinate system in the Euclidean three dimensional space is determined as follows. Fix three positive numbers a, b and c with a > b > c, and consider the one-parameter family of quadrics with equation x 2 a 2 + ξ + y 2 b 2 + ξ + z 2 c 2 + ξ = 1 ,(38) Through any given point (x, y, z) in Euclidean 3D space, there passes precisely three such quadrics, corresponding to the values λ, µ, ν of the parameter ξ which lie in the intervals − a 2 < ν < −b 2 < µ < −c 2 < λ .(39) One quadric is an ellipsoid (because a 2 + λ > b 2 + λ > c 2 + λ > 0) x 2 a 2 + λ + y 2 b 2 + λ + z 2 c 2 + λ = 1 ,(40) other is an one sheeted hyperboloid (because a 2 + µ > b 2 + µ > 0 > c 2 + µ) x 2 a 2 + µ + y 2 b 2 + µ + z 2 c 2 + µ = 1 ,(41) and the last is a two sheeted hyperboloid (because a 2 + ν > 0 > b 2 + ν > c 2 + ν) x 2 a 2 + ν + y 2 b 2 + ν + z 2 c 2 + ν = 1 .(42) Solving (40-42) for x 2 , y 2 and z 2 , we obtain the parametrization of 3D Euclidean space in terms of ellipsoidal coordinates, λ, µ and ν: x 2 = (λ + a 2 )(µ + a 2 )(ν + a 2 ) (b 2 − a 2 )(c 2 − a 2 ) ,(43)y 2 = (λ + b 2 )(µ + b 2 )(ν + b 2 ) (c 2 − b 2 )(a 2 − b 2 ) ,(44)z 2 = (λ + c 2 )(µ + c 2 )(ν + c 2 ) (a 2 − c 2 )(b 2 − c 2 ) .(45) Inversion of (43-45) gives ellipsoidal coordinates in terms of Cartesian coordinates. Note that ellipsoidal coordinates are only one-to-one in the interior of each octant of the Euclidean space determined by the three coordinate planes x = 0, y = 0, z = 0 through the origin. Classical 3D integrable systems. Let us study the situation from classical point of view first. Consider a three dimensional system with Hamiltonian given by H = p 2 2m + U (x) and look for two additional independent integrals of motion of the form: H 1 = 1 2m p i g ij 1 (x)p j + U 1 (x) , H 2 = 1 2m p i g ij 2 (x)p j + U 2 (x) ,(46)with {H, H 1 } = {H, H 2 } = {H 1 , H 2 } = 0, where again the brackets are the Poisson brackets. The tensors g ij 1 (x) and g ij 2 (x) can be obtained using the hypothesis of commutativity of the kinetic parts of H, H 1 and H 2 (due to their structure, we could extend the name 'Hamiltonian' to H i , i = 1, 2 too). The condition {p i g ik 1 (x)p k , p 2 } = 0(47) implies that p i g ik 1 (x)p k should be a quadratic function of the momenta. A particular choice follows from a further restriction of H 1 , H 2 being invariant under any reflection, so that both can be expressed as a linear combination of the squares L 2 i of the components L i of the angular momentum and p 2 i of the linear the momentum p i . There is some arbitrariness in the choice of H 1 , for which we set p i g ik 1 (x)p k := L 2 − p 2 1 (b 2 + c 2 ) − p 2 2 (a 2 + c 2 ) − p 2 3 (a 2 + b 2 ) ,(48) where a, b and c are the constants which specify the system of ellipsoidal coordinates, see (39). In matrix form, the tensor g ik 1 (x) defined by (48) can be written as g ik 1 (x) =   z 2 + y 2 − (a 2 + c 2 ) −xy −xz −xy x 2 + z 2 − (a 2 + c 2 ) −yz −xz −yz x 2 + y 2 − (a 2 + b 2 )   . (49) Analogously, the commutativity of the kinetic part of H 2 with those of H and H 1 , {p i g ik 2 (x)p k , p i g ik 1 (x)p k } = 0(50) fixes the linear combination of the components of L 2 and of p 2 in H 2 as p i g ik 2 (x)p k = −(L 2 1 a 2 + L 2 2 b 2 + L 2 3 c 2 ) + p 2 1 b 2 c 2 + p 2 2 a 2 c 2 + p 2 3 a 2 b 2 ,(51) with matrix form given by g ik 2 (x) =   −c 2 y 2 − b 2 z 2 + b 2 c 2 c 2 xy b 2 xz c 2 xy −c 2 x 2 − a 2 z 2 + a 2 c 2 a 2 yz b 2 xz a 2 yz −b 2 x 2 − a 2 y 2 + a 2 b 2   .(52) Matrices (49) and (52) are symmetric and therefore diagonalizable. In addition, these matrices commute and therefore admit a diagonal form in the same basis. This basis is given by E λ := (∂ x λ, ∂ y λ, ∂ z λ) , E µ := (∂ x µ, ∂ y µ, ∂ z µ) , E ν := (∂ x ν, ∂ y ν, ∂ z ν) .(53) These vectors are mutually orthogonal [10]. Note that λ, µ and ν are functions of the Cartesian coordinates, so that the partial derivatives in (53) make sense. The eigenvalues of (49) and (52) are easy to obtain and are given in the following table: g ij (x) g ij 1 (x) g ij 2 (x) E λ 1 µ + ν µν E µ 1 λ + ν λν E ν 1 λ + µ λµ(54) Thus, we have determined the kinetic parts of H 1 , H 2 in terms of ellipsoidal coordinates. Our next goal is to obtain the most general form of the 'potential' terms U (x), U 1 (x) and U 2 (x). From the commutation relations {H, H 1 } = {H, H 2 } = {H 1 , H 2 } = 0, we obtain the relations, ∂ i U 1 = g ik 1 ∂ k U , ∂ i U 2 = g ik 2 ∂ k U , g ik 1 ∂ k U 2 = g ik 2 ∂ k U 1 .(55) Then, after a calculation that makes use of the chain rule involving ellipsoidal and Cartesian coordinates, the form (53) of the eigenvectors of matrices g ik 1 (x) and g ik 2 (x) and the orthogonality of these vectors, we can obtain the most general form of the potentials U , U 1 and U 2 satisfying {H, H 1 } = {H, H 2 } = {H 1 , H 2 } = 0 [10]. This is: U = l(λ) (λ − ν)(λ − µ) + m(µ) (µ − ν)(µ − λ) + n(ν) (ν − µ)(ν − λ) ,(56)U 1 = (µ + ν) l(λ) (λ − ν)(λ − µ) + (ν + λ) m(µ) (µ − ν)(µ − λ) + (λ + µ) n(ν) (ν − µ)(ν − λ) ,(57)U 2 = (µν) l(λ) (λ − ν)(λ − µ) + (νλ) m(µ) (µ − ν)(µ − λ) + (λµ) n(ν) (ν − µ)(ν − λ) ,(58) where l(λ), m(µ) and n(ν) are arbitrary functions of their arguments. This is the most general form of the potentials compatible with complete integrability. The quantum case. We shall briefly comment the procedure here; details can be found in [10]. First of all, we obtain the quantum operators H, H 1 and H 2 by canonical quantization of their classical counterparts (46). Then, complete integrability means that these three Hamiltonians commute with each other. Therefore, we can find wave functions ψ(x) such that (H − E)ψ = (H 1 − E 1 )ψ = (H 2 − E 2 )ψ = 0 .(59) We can prove that the use of ellipsoidal coordinates λ, µ and ν, the form for the potentials given in (58) and the factorization ψ(x) = ψ(λ)φ(µ)ϕ(ν) show that equations (59) are equivalent to three wave functions solely in the variables λ, µ and ν respectively, so that complete separation of variables is also achieved in the quantum case [10]. The equation for λ gives (ψ λ denotes the derivative of ψ with respect to λ): 4(λ + a 2 )(λ + b 2 )(λ + c 2 )ψ λλ + 2[a 2 b 2 + a 2 c 2 + b 2 c 2 + 2λ(a 2 + b 2 + c 2 ) + 3λ 2 ]ψ λ +2m(λ 2 E − λE 1 + E 2 − l(λ))ψ = 0 .(60) The other two equations are just obtained by replacing λ by µ and ν respectively. Equation (60) can be written in a more compact form by means of an of a change of variable to the auxiliary variable t. This is given by means of the condition: λ ′ (t) = 2 (λ + a 2 )(λ + b 2 )(λ + c 2 ) .(61) In terms of t, equation (60) has the follwing form d 2 ψ dt 2 + 2m[λ 2 (t)E − λ(t)E 1 + E 2 − l(λ(t))]ψ = 0 ,(62) where l(λ) has already appeared in (56-58). In addition to (62), there are two other equations, one for µ and the other for ν. These three equations are similar and, in particular, all depend on the function λ(t) even in the free particle case, l(λ) = m(µ) = n(ν) = 0. Integrability in spaces of constant curvature. There are three types of homogeneous three dimensional manifolds with constant curvature and Riemannian positive definite metric, which are the sphere S 3 , the Euclidean plane E 3 and the hyperboloid H 3 (for general references on integrability in spaces of constant curvature, see [30]). In this section, we shall deal with the standard S 3 with curvature equal to 1. As is well known, S 3 can be realized as the submanifold X 2 1 + X 2 2 + X 2 3 + X 2 4 = 1 in a 4D Euclidean ambient space, with the induced metric. A convenient coordinate system for S 3 is provided through stereographic coordinates, the point on S 3 parametrized by x ∈ R 3 ∪ ∞ being given by: X = (X, X 4 ) = 2χx x 2 + χ 2 , x 2 − χ 2 x 2 + χ 2 .(63) where χ is a parameter which should be different from zero. With this choice x = 0 corresponds to the sphere's 'South Pole' (0, 0, 0, −1), and the projection is made from the North Pole (0, 0, 0, 1) to the plane X 4 = 1−χ. No actual generality would be lost if we fix a particular value for χ; the preferred choice χ = 2 corresponds to projecting over the plane tangent to the sphere's south pole and expressions are generally more clear for this choice, because in this case near the South Pole the coordinates x approach the ordinary Cartesian coordinates in the tangent E 3 with the same scaling as lengths on the sphere (a rescaling is required for other values of χ) and neglecting terms which are higher order in x brings the Euclidean expression with the correct factors, as obvious for instance in (65)-(66). The Lagrangian for the free evolution on S 3 can be taken in terms of the ambient space coordinates as: L 0 = m 2 Ẋ 2 +Ẋ 2 4 ,(64) which can be expressed in stereographic coordinates as L 0 = m 2 4χ 2ẋ 2 (x 2 + χ 2 ) 2 ,(65) Alternatively, we may start from the angular momentum tensor in ambient space, with components M αβ := X αẊβ −Ẋ α X β and define the free lagrangian in S 3 as (proportional to) the square of this angular momentum tensor; this leads to the same free lagrangian. The canonical conjugate momenta associated to the stereographic coordinates x i is p i = m 4χ 2 (x 2 + χ 2 ) 2ẋ i .(66) and the Legendre transformation provides the following free Hamiltonian: H 0 = 1 2m 1 4χ 2 p 2 (x 2 + χ 2 ) 2 .(67) The symmetry group for S 3 is SO (4) and (67) is invariant with respect to the sphere isometries in SO(4). Then, H 0 has 6 'kinematic' integrals of motion, which are generators of SO(4); in terms of the stereographic parametrization these generators are: L i = ǫ ijk x j p k , K i = 1 2χ 2x i p · x − p i (x 2 − χ 2 ) ,(68) with Poisson brackets {L i , L j } = −ǫ ijk L k , {K i , K j } = −ǫ ijk L k , {L i , K j } = −ǫ ijk K k .(69) (these close also an so(4) algebra as expected). There are two Casimirs given by K 2 + L 2 and K · L. The free Hamiltonian is proportional to the first Casimir: H 0 = 1 2m (K 2 + L 2 ) .(70) while the second Casimir vanishes. Then, we are going to pose in S 3 the same question studied for the flat case in section 1. Assume the existence of a symmetry axis, that, for the moment, we can identify with the z-axis. Then, consider a Hamiltonian with kinetic term given by H 0 plus a time independent potential U (x), H = 1 2m (K 2 + L 2 ) + U (x) .(71) Then, if this system has to be completely integrable, we need finding a new 'Hamiltonian' H 1 with 'kinetic' part T 1 = (2m) −1 p i g ij (x)p j and 'potential' Φ(x), H 1 = T + Φ(x), independent of H and such that {H, L 3 } = {H 1 , L 3 } = {H, H 1 } = 0. As in Section 1, the kinetic term T 1 is obtained under the condition that it commutes with H 0 and L 3 . There is an S 3 analog of the most general 'kinetic' term (4), but we simply write down the version for S 3 of the Euclidean Killing tensor (5) leads to the constant of motion: H 1 = 1 2m (L 2 1 + L 2 2 − a 2 K 2 3 ) + Φ(x) ,(72) where a is a constant playing a role fully similar to those in (5). If the symmetry axis were arbitrary in the direction given by the unitary axis n = (n 1 , n 2 , n 3 ), the matrix g ij has the following form in terms of the stereographic parametrization: g ij (x) = δ ij (x · n) 2 − x · n(x i n j + x j n i ) + n i n j x 2 − a 2 4χ 2 2x i x · n − n i (x 2 − χ 2 ) 2x j x · n − n j (x 2 − χ 2 ) . (73) The requirement {H, H 1 } = 0 leads to the version for S 3 of equation (8), which in this case reads ∂ i Φ(x) = G ij (x)∂ j U (x) , with G ij (x) = g ij (x) (x 2 + χ 2 ) 2 .(74) We shall denote the eigenvalues of G ij (x) by {µ + , µ − , µ 3 }. The two former are the roots of the quadratic equation z 2 − P z − Q = 0, where P := x 2 − α 2 [(x 2 − χ 2 ) 2 + 4(x · n) 2 ] (x 2 + χ 2 ) 2 , Q := (x · n) 2 α 2 (x 2 + 1) 2(75) and µ 3 = Q/α 2 . As in the flat case described in section 1, the eigenvalues µ ± together with the azimuthal angle ϕ define a coordinate system, which is the analogous for the sphere S 3 of the spheroidal oblate coordinates [24]. As in the flat case, their corresponding eigenvectors have components ∂ i µ − and ∂ i µ + , i, j = 1, 2, 3 respectively, i.e., G ij ∂ j µ − = µ + ∂ i µ − , G ij ∂ j µ + = µ − ∂ i µ + .(76) Finally, following a procedure similar to that studied in section 1, we can show that equations (76) provide the most general form for the 'potentials' U (x) and Φ(x) on the variables µ ± . The final result is: U = f (µ + ) − g(µ − ) µ + − µ − , Φ = µ − f (µ + ) − µ + g(µ − ) µ + − µ − ,(77) where f (µ + ) and g(µ − ) are arbitrary functions. Note that symmetry prevents that U and Φ to depend on the azimuthal angle ϕ, as constancy of L 3 means invariance under rotations with z-axis. Application to the Kepler problem. In this subsection, we shall discuss a particular case of very special importance: the Kepler problem in the standard sphere S 3 with curvature equal to 1, which provides an example of the type of situation just described (recall a factor which dimensionally is a lenght square has been taken as equal to 1, and hence is invisible in the expressions). For more details on the Kepler problem in spaces with constant curvature, see [3,4,7,11,18,23,25]. In the sphere S 3 (and also in any dimension n > 3), this system is an example of a maximally superintegrable system. Restricting attention to the 3D case, the Kepler Hamiltonian is H = 1 2m (K 2 + L 2 ) − k tan(r)(78) where the kinetic part is (71) and the potential term depends only on the intrinsic distance r in S 3 to the potential center and k > 0 for the atractive case. The potential, with center at sphere's south pole, can be expressed in terms of the ambient space coordinates as − k tan(r) = k X 4 X 2 1 + X 2 2 + X 2 3(79) and in terms of stereographic coordinates the Kepler Hamiltonian is: H = 1 2m (K 2 + L 2 ) + k 1 2χ x 2 − χ 2 |x| .(80) where the generators in the kinetic part are given in (68). This Hamiltonian possesses 3 integrals of motion linear in the momenta (the three components of the angular momentum L) and a further 3 integrals of motion which are quadratic in the momenta (the components of the Laplace-Runge-Lenz vector A): A = K × L + mk x |x| ,(81) Of course there are 2 independent relations among the seven constants of motion H, L, A, reducing to five functionally independent constants, which entitles the Kepler problem in S 3 to be maximally superintegrable. These relations are: A 2 = m 2 k 2 + 2mH − L 2 L 2 , A · L = 0 (82) The algebra of Poisson brackets for the components of L and A has the following form: {A i , A j } = ǫ ijk L k (2mH − 2L 2 ) , {L i , A j } = −ǫ ijk A k .(83) Should these commutation relations be computed in a sphere with curvature κ, the term −2L 2 in the first Poisson bracket would appear as −2κL 2 , making expressions dimensionally correct. This term displays clearly the effects due to the curvature of the configuration space, when it is compared (83) with the commutation relations for the corresponding algebra in the flat case, which are {A i , A j } = ǫ ijk L k 2mH , {L i , A j } = −ǫ ijk A k .(84) From (83) we see that the components of L and A do not have the commutation relations of a Lie algebra, because of the presence of higher order terms in the Poisson bracket; the quadratic term L k H is already present in the flat case, but on the curved sphere S 3 cubic terms L k L 2 appear as well. In the flat Euclidean case, this algebra has been studied under the name of Higgs algebra [12]. Then we consider this system as an example of the situation discussed in the previous section: the Kepler Hamiltonian admits two additional constant of motion which are also in involution: the components of the angular momentum anf of the Runge-Lenz vector along any fixed direction. If we take the z axis, then we get H, L 3 , A 3 as three constants of motion in involution. Now we note that the Kepler problem has axial symmetry around any axis. In particular, the third constant A 3 in the Kepler problem in Euclidean belong indeed to the family (6) with the values α = 0, β = 1, γ = 0 for the parameters. While the coordinate system behind the family α = 0, β = 0, γ = 0 were oblate spheroidal coordinates in E 3 and their analogous in S 3 , the family α = 0, β = 0, γ = 0 turns out to be separable in parabolic coordinates in E 3 and in its analogous for S 3 . To end, we mention the possibility, which exists in S 3 , to rescale the Runge-Lenz vector by a factor which depends on the integrals of motion (so the property of being a constant is not disturbed) in such a way that the Poisson brackets of new components, together with those of angular momentum closes a Lie algebra so (4). A rescaling ensuring this [25] is: R = A L 2 + m H 2 + k 2 − H −1/2 ,(85) (the rescaling factor is always positive, no matter neither the value nor the sign of H) and the components of L and R turn out to have the so(4) commutation relations with respect to the Poisson bracket (compare to (69)): {L i , L j } = −ǫ ijk L k , {R i , R j } = −ǫ ijk L k , {L i , R j } = −ǫ ijk R k .(86) For these commutation relations (86) the first Casimir reads R 2 + L 2 . Direct computations reveals that this Casimir is: R 2 + L 2 = m H + H 2 + k 2(87) Either by solving this equation for H or by a direct tedious but straightfoward computation, this allows us to write Hamiltonian (80) in terms of this Casimir: 2mH = R 2 + L 2 − m 2 k 2 R 2 + L 2 .(88) This discussion has been purely classical. We simply mention that this last property can be also discussed in the quantum case, where now the new global so(4) Lie algebra symmetry allows us to derive the energy spectrum first found by Schrödinger for the Kepler problem in S 3 [27,28,29,15]. Full details on this new derivation can be found in [25]; we must recall this approach is conceptually very different from the standard Pauli discussion for the Euclidean Kepler problem, where there is a different Lie algebra in each energy eigenspace, with its isomorphism class depending on the energy sign. Concluding remarks. We have discussed a special class of three dimensional completely integrable either in the flat Euclidean 3D space as well as in the standard sphere S 3 ; these systems are required to have a rotational symmetry axis, so they have axial symmetry. Further to the Hamiltonian H, there are two additional constants for such a system: L, the angular momentum with respect the symmetry axis, and H 1 = 1/(2m)p i g ij 2 (x)p j + Ψ(x). The general form for the tensor g ij 2 (x) which guarantees that the 'kinetic' parts of H and H 2 commute among themselves and with L is given, and two special cases are mentioned; we discuss one of them, corresponding geometrically to separability in oblate spheroidal coordinates in full detail; the other corresponds to separation of variables in parabolic coordinates. Integrability determines the 'potentials' V (x), Ψ(x) to have a particular form in terms of undetermined functions of the coordinates, and the equations of motion are written as three equations each one involving one coordinate (separation of variables). In the quantum case, one of the three separated wave equations is trivial and each of the other two can be easily transformed into a spheroidal wave equation, for which the solutions have been studied. If no symmetry conditions are imposed a similar study can be performed. Separation of variables can be achieved in this case, as one could expect, in terms of the (general) ellipsoidal coordinates. In the quantum case, we obtain three similar wave equations, one for each of the variables, each one in terms of a different function. These functions appear in the form of the potentials as a consequence of the integrability condition. Finally, we have carried the analysis of the systems with axial symmetry to the three dimensional sphere S 3 in the classical case. The results obtained are quite similar to the flat ones, in agreement to the idea that results in this area for the constant curvature spaces are essentially 'the same' as in the flat case. This is illustrated with the particular case of the three dimensional Kepler problem. Extension to higher dimensions should be also possible [6,19] Acknowledgements.Partial financial support is acknowledged to the Junta de Castilla y León Project Maximal superintegrability on N -dimensional curved spaces. A Ballesteros, F J Herranz, M Santander, T Sanz-Gil, J. Phys. A. 36A. Ballesteros, F.J. Herranz, M. Santander, T. Sanz-Gil, Maximal superintegrability on N -dimensional curved spaces, J. Phys. A 36, L93-99 (2003). Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation. S Benenti, J. Math. Phys. 38S. Benenti, Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation, J. Math. Phys. 38, 6578-6602 (1997). Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S 2 and the hyperbolic plane H 2. J F Cariñena, M F Rañada, M Santander, J. Math. Phys. 46J.F. Cariñena, M.F. Rañada, M. Santander, Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere S 2 and the hyperbolic plane H 2 , J. Math. Phys. 46, 052702, 1-25 (2005). Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton. J F Cariñena, M F Rañada, M Santander, J. Phys.A To appear. J.F. Cariñena, M.F. Rañada, M. Santander, Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton, J. Phys.A To appear (2007). Conformal Killing tensors with vanishing torsion and the separation of variables in the Hamilton-Jacobi equation. M Crampin, Diff. Geom. Appl. 18M. Crampin, Conformal Killing tensors with vanishing torsion and the separation of variables in the Hamilton-Jacobi equation, Diff. Geom. Appl. 18, 87-102 (2003). Two families of superintegrable and isospectral potentials in two dimensions. B Demircioglu, Ş Kuru, M Onder, A Vercin, J. Math. Phys. 432133B. Demircioglu, Ş. Kuru, M. Onder and A. Vercin, Two families of superintegrable and isospectral potentials in two dimensions, J. Math. Phys. 43, 2133 (2002). On the planetary motion in the 3-Dim standard spaces M 3 κ of constant curvature κ. P Dombrowski, J Zitterbarth, Demonstratio Mathematica. 24P. Dombrowski, J. Zitterbarth, On the planetary motion in the 3-Dim standard spaces M 3 κ of constant curvature κ, Demonstratio Mathematica 24, 375-458 (1991). Superintegrability in classical mechanics. N W Evans, Phys. Rev. A. 41N.W. Evans, Superintegrability in classical mechanics, Phys. Rev. A 41, 5666-76 (1990). . M Gadella, J Negro, G P Pronko, J. Phys. A: Math. Theor. 4010791M. Gadella, J. Negro and G.P. Pronko, J. Phys. A: Math. Theor., 40 (2007), 10791. Integrable systems in ellipsoidal coordinates. M Gadella, M Ioffe, J Negro, G P Pronko, to appearM. Gadella, M. Ioffe, J. Negro, G.P. Pronko, Integrable systems in ellipsoidal coordinates, to appear. L García-Gutiérrez, M Santander, arXiv:0707.3810Levi-Civita regularization and geodesic flows for the 'curved' Kepler problem. math-phL. García-Gutiérrez and M. Santander, Levi-Civita regularization and geodesic flows for the 'curved' Kepler problem, arXiv:0707.3810 [math-ph] The Higgs algebra and the Kepler problem in R3. V V Gritsev, Yu A Kurochkin, J. Phys. A. 33V.V. Gritsev and Yu.A. Kurochkin, The Higgs algebra and the Kepler problem in R3 J. Phys. A 33, 4073-4079, (2000). F J Herranz, A Ballesteros, Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications 2, 010. 22F.J. Herranz, A. Ballesteros, Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature, SIGMA (Symmetry, Integrability and Geometry: Methods and Applications 2, 010, 22p (2006). Available online at http://www.emis.de/journals/SIGMA/2006/ Paper010/ Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space. J T Horwood, R G Mclenaghan, R G Smirnov, Comm. Math. Phys. 259J.T. Horwood, R.G. McLenaghan, R.G. Smirnov, Invariant classification of orthogonally separable Hamiltonian systems in Euclidean space, Comm. Math. Phys. 259, (2005) 679- 709. A note on the Kepler problem in a space of constant negative curvature. L Infeld, A Schild, Phys. Rev. 67L. Infeld and A. Schild, A note on the Kepler problem in a space of constant negative curvature, Phys. Rev. 67, 121-122 (1945). Superintegrability in threedimensional Euclidean space. E G Kalnins, G C Williams, W Miller, G S Pogosyan, J. Math. Phys. 40E.G. Kalnins, G.C. Williams, W. Miller, G.S. Pogosyan, Superintegrability in three- dimensional Euclidean space, J. Math. Phys. 40, (1999) 708-725. Complete sets of invariants for dynamical systems that admit a separation of variables. E G Kalnins, J M Kress, W Miller, G S Pogosyan, J. Math. Phys. 43E.G. Kalnins, J.M. Kress, W. Miller, G.S. Pogosyan, Complete sets of invariants for dynamical systems that admit a separation of variables, J. Math. Phys. 43, (2002) 708-725. Kepler's problem in constant curvature spaces. V V Kozlov, A O Harin, Celest. Mechanics. 54V.V. Kozlov and A.O. Harin, Kepler's problem in constant curvature spaces, Celest. Mechanics 54, (1992) 393-399. Intertwined isospectral potentials in an arbitrary dimension. 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Hydrogen atom as an eigenvalue problem in 3D spaces of constant curvature and minimal length. L M Nieto, H C Rosu, M Santander, Mod. Phys. Lett. A. 14L.M. Nieto, H.C. Rosu and M. Santander, Hydrogen atom as an eigenvalue problem in 3D spaces of constant curvature and minimal length, Mod. Phys. Lett. A 14, 2463-2469, (1999). Triorthogonal systems in spaces of constant curvature in which the equation ∆ 3 u + λu = 0 allows a complete separation of variables. M N Olevski, Mat. Sb. 2769In RussianM.N. Olevski, Triorthogonal systems in spaces of constant curvature in which the equation ∆ 3 u + λu = 0 allows a complete separation of variables, Mat. Sb. 27 (69), 379-426 (1950) (In Russian). Kepler problem in the constant curvature space. G P Pronko, arXiv:0705.3111v2math-phG.P. Pronko, Kepler problem in the constant curvature space, arXiv:0705.3111v2 [math-ph] From oscillator(s) and Kepler(s) potentials to general superintegrable systems in spaces of constant curvature. 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Integrable problems of celestial mechanics in spaces of constant curvature. T G Vozmischeva, Astrophysics and Space Science Library. 295Kluwer Academic PubT.G. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, Astrophysics and Space Science Library, 295. (Kluwer Academic Pub., Dordrecht, 2003).

The spiral is one of Nature's more ubiquitous shape: it can be seen in various media, from galactic geometry to cardiac tissue. In the literature, very specific models are used to explain some of the observed incarnations of these dynamic entities. Barkley [1,2]first noticed that the range of possible spiral behaviour is caused by the Euclidean symmetry that these models possess.In experiments however, the physical domain is never perfectly Euclidean. The heart, for instance, is finite, anisotropic and littered with inhomogeneities. To capture this loss of symmetry (and as a result model the physical situation with a higher degree of accuracy), LeBlanc and Wulff introduced forced Euclidean symmetry-breaking (FESB) in the analysis, via two basic types of perturbations: translational symmetrybreaking (TSB) and rotational symmetry-breaking terms. In[3,4], they show that phenomena such as anchoring and quasi-periodic meandering can be explained by combining Barkley's insight with FESB.In this article, we provide a fuller characterization of spiral anchoring by studying the effects of n simultaneous TSB perturbations, where n > 1.

10.1007/s00332-007-9000-z

math/0512511

5a77ea7dd4aec56c2fdd78b28f766e4b445b000c

Spiral anchoring in media with multiple inhomogeneities: a dynamical system approach Spiral anchoring in media with multiple inhomogeneities 2 19 Jan 2006 P Boily [email protected] V G Leblanc E Matsui Department of Mathematics and Statistics University of Ottawa K1N 6N5OttawaCanada University of Ottawa K1N 6N5OttawaCanada Spiral anchoring in media with multiple inhomogeneities: a dynamical system approach Spiral anchoring in media with multiple inhomogeneities 2 19 Jan 2006Submitted to: Journal of Nonlinear Science ‡ Present address: Institute of the Environment,AMS classification scheme numbers: 34C2037G4037L1037N2592E20 The spiral is one of Nature's more ubiquitous shape: it can be seen in various media, from galactic geometry to cardiac tissue. In the literature, very specific models are used to explain some of the observed incarnations of these dynamic entities. Barkley [1,2]first noticed that the range of possible spiral behaviour is caused by the Euclidean symmetry that these models possess.In experiments however, the physical domain is never perfectly Euclidean. The heart, for instance, is finite, anisotropic and littered with inhomogeneities. To capture this loss of symmetry (and as a result model the physical situation with a higher degree of accuracy), LeBlanc and Wulff introduced forced Euclidean symmetry-breaking (FESB) in the analysis, via two basic types of perturbations: translational symmetrybreaking (TSB) and rotational symmetry-breaking terms. In[3,4], they show that phenomena such as anchoring and quasi-periodic meandering can be explained by combining Barkley's insight with FESB.In this article, we provide a fuller characterization of spiral anchoring by studying the effects of n simultaneous TSB perturbations, where n > 1. Introduction Spiral waves have been observed in a variety of experimental contexts, ranging from the well-known Belousov-Zhabotinsky chemical reaction to the electrical potential in cardiac tissue [1,2,[5][6][7][8][9][10][11][12][13][14][15][16]. In this last case, spiral waves are believed to be a precursor to several fatal cardiac arrythmias (e.g. ventricular tachycardia and ventricular fibrillation) [13,17,18]. A thorough understanding of the various dynamical properties of spiral waves is therefore warranted. One of the most interesting and fruitful approaches in recent years to the study of spiral waves has been to use the theory of equivariant dynamical systems to derive finite-dimensional models for many of the observed dynamical states and bifurcations of spirals. The pioneer of this approach was Barkley, who realized that the experimentallyobserved transition from rigid rotation to quasi-periodic meandering and drifting could be explained using only the underlying symmetries (the group SE(2) of all planar translations and rotations) of the governing reaction-diffusion partial differential equations: he derived an ad hoc system of 5 ordinary differential equations with SE(2) symmetry which model a Hopf bifurcation from a rotating wave, and then showed that this finite-dimensional system replicated the experimentally-observed transition to meandering and drifting [1,2,19]. Sandstede, Scheel and Wulff later proved a general center manifold reduction theorem for relative equilibria and relative periodic solutions in spatially extended infinite-dimensional SE(2)-equivariant dynamical systems, thereby providing mathematical justification for Barkley's approach [20][21][22][23][24]. One of the advantages of this equivariant dynamical systems approach is that one can often give universal, model-independent explanations of many of the observed dynamics and bifurcations of spiral waves. For example, the above-mentioned Hopf bifurcation from rigid rotation to quasi-periodic meandering and drifting has been observed in both numerical simulations [16] and in actual chemical reactions [5]. Another example is the anchoring/repelling of spiral waves on/from a site of inhomogeneity, which has been observed in numerical integrations of an Oregonator system [8], in photo-sensitive chemical reactions [7] and in cardiac tissue [11]. Using a modelindependent approach based on forced symmetry-breaking, LeBlanc and Wulff showed that anchoring/repelling of rotating waves is a generic property of systems in which the translation symmetry of SE(2) is broken by a small perturbation [3]. Similarly, some dynamics of spiral waves observed in anisotropic media (e.g. phase-locking and/or linear drifting of meandering spiral waves) have been shown to be generic consequences of rotational symmetry-breaking [4,12,[25][26][27][28]. Consider as a paradigm a system of reaction-diffusion partial differential equations ∂u ∂t = D · ∇ 2 u + f (u) (1.1) where u is a k-vector valued function of time and two-dimensional space, D is a matrix of diffusion coefficients and f : R k −→ R k is a smooth reaction term. Many of the phenomena in which spiral waves are observed experimentally are modeled by systems of the form (1.1). Moreover, Scheel has proved that systems of this form can admit time-periodic, rigidly rotating spiral wave solutions [29]. Implicit in the form of equations (1.1) is the fact that the medium of propagation is completely homogeneous and isotropic. Mathematically, this is represented by the invariance of (1.1) under the transformations u(t, x) −→ u(t, x 1 cos θ − x 2 sin θ + p 1 , x 1 sin θ + x 2 cos θ + p 2 ), (1.2) where (θ, p 1 , p 2 ) ∈ S 1 × R 2 and x ∈ R 2 [30,31]. The group of all transformations of the form (1.2) is isomorphic to the special Euclidean group SE(2) of all planar translations and rotations. When studying the effects of inhomogeneities on the propagation of spiral waves, one must consider a larger class of models than (1.1), since inhomogeneous media do not possess Euclidean invariance. For example, one might consider systems of the form ∂u ∂t = D · ∇ 2 u + f (u) + λ g(u, x 2 , λ) (1.3) which are perturbations of (1.1). Such systems could model a spatially extended reaction-diffusion medium in which there is one site of inhomogeneity (with circular symmetry) centered at the origin of R 2 . For instance, the Oregonator model which is used to study spiral anchoring in [8] is of the form (1.3). When λ = 0, (1.3) has rotational symmetry about the origin, but does not possess any translation symmetry. This phenomenon is called forced translational symmetry-breaking; it is studied in detail in [3]. In this paper, we use a similar equivariant dynamical systems approach to study the problem of spiral wave dynamics (specifically, with regards to anchoring/repelling) in media in which there are several sites of inhomogeneities (as opposed to just one site), of which cardiac tissue is an important example. We will make several simplifying assumptions which are meant to make the analysis more tractable. First, we assume that the inhomogeneities consist of a finite number of "sources" which are localized near distinct sites ζ 1 , . . . , ζ n in the plane. Second, we will assume that these n sources of inhomogeneity are independent in the following sense: we introduce n independent real parameters λ 1 , . . . , λ n which give some measure of the relative "amplitudes" of the sources. In particular, when all the λ i are zero except, say λ i * = 0, then there is only one source of inhomogeneity localized near the point ζ i * . In that case, we will make a third simplifying assumption: the single inhomogeneity is circularly symmetric around the point ζ i * . The following is an example of a class of reaction-diffusion partial differential equations which are perturbations of (1.1) and which might model such a situation: u t =D∆u + f (u) + n j=1 λ j D j ( x − ζ j 2 , λ)∆u + f j (u, x − ζ j 2 , λ) ,(1.4) where the functionsD j , f j are bounded and smooth enough. The goal of this paper is to provide a detailed analysis of a larger class of abstract dynamical systems which share the symmetry properties of (1.4): (S1) when λ 1 = · · · = λ n = 0, the systems are invariant under the action (1.2) of the group SE(2), (S2) when all the λ i are zero except λ i * , the systems have rotational symmetry about the point ζ i * , but they do not generically possess translation symmetries, (S3) when two or more of the λ i are non-zero, the systems do not generically possess any of the symmetries (1.2) except for the identity. Our results will apply to the subclass LC 0 of systems whose members also generate a smooth local semi-flow on a suitable function space [20][21][22][23][24], as well as some technical conditions which will be specified as we proceed. The paper is organized as follows. In the second section, we derive the center bundle equations of the semi-flow of a system in LC 0 , near a hyperbolic rotating wave. We state and prove our main results in the third section: to wit, spiral anchoring is generic in a parameter wedge. Then, we provide a visual criterion characterizing the anchoring wedges in the case n = 2. Finally, we perform numerical experiments demonstrating the validity of our results. Reduction to the Center Bundle Equations Let X be a Banach space, U ⊂ R n a neighborhood of the origin and Φ t,λ be a smoothly parameterized family (parameterized by λ ∈ U) of smooth local semi-flows on X. Let SE(2) = C+SO(2) denote the group of all planar translations and rotations, and let a : SE(2) −→ GL(X) (2.1) be a faithful and isometric representation of SE(2) in the space of bounded, invertible linear operators on X. For example, if X is a space of functions with planar domain, a typical SE(2) action (such as (1.2) in the preceding section) is given by (a(γ)u)(x) = u(γ −1 (x)), γ ∈ SE(2). We will parameterize SE(2) as follows: SE(2) ∼ = C × S 1 , with multiplication given by (p 1 , ϕ 1 ) · (p 2 , ϕ 2 ) = (e iϕ 1 p 2 + p 1 , ϕ 1 + ϕ 2 ), ∀ (p 1 , ϕ 1 ), (p 2 , ϕ 2 ) ∈ C × S 1 . For fixed ξ ∈ C, we define the following subgroup of SE (2): SO(2) ξ = { (ξ, 0) · (0, θ) · (−ξ, 0) | θ ∈ S 1 } which is isomorphic to SO(2), and represents rotations about the point ξ. We will assume the following symmetry conditions on the family Φ t,λ of semi-flows. Hypothesis 1 There exists n distinct points ξ 1 , . . . , ξ n in C such that if e j denotes the j th vector of the canonical basis in R n , then ∀ u ∈ X, α = 0, t > 0, Φ t,αe j (a(γ)u) = a(γ)Φ t,αe j (u) ⇐⇒ γ ∈ SO(2) ξ j , and Φ t,0 (a(γ)u) = a(γ)Φ t,0 (u), ∀ γ ∈ SE(2). Hypothesis 1 basically states that (a) when λ = 0, the semi-flow Φ t,0 is SE(2)equivariant; (b) when λ = 0 is near the origin and along the j th coordinate axis of R n , the semi-flow is only SO(2) ξ j -equivariant (i.e. it only commutes with rotations about the point ξ j ), and (c) when λ is not as in (a) or (b), the semi-flow has (generically) trivial equivariance. We are interested in the effects of the forced symmetry-breaking on normally hyperbolic rotating waves. Therefore, we will assume the following hypothesis. Hypothesis 2 There exists u * ∈ X and Ω * in the Lie algebra of SE (2) such that e Ω * t is a rotation and Φ t,0 (u * ) = a(e Ω * t )u * for all t. We also assume that the set { λ ∈ C | |λ| ≥ 1 } is a spectral set for the linearization a(e −Ω * )DΦ 1,0 (u * ) with projection P * such that the generalized eigenspace range(P * ) is three dimensional. For sake of simplicity, we will only be interested in one-armed spiral waves; therefore, we assume that u * in hypothesis 2 has trivial isotropy subgroup. While hypotheses 1 and 2 hold for a large variety of spirals (such as decaying spirals), there is also a large family of spirals for which they don't (including Archimedean spirals) [29]. § Let LC 0 be the collection of all abstract dynamical systems that do satisfy them, as well as all other hypotheses required in order for the center manifold theorems of [20][21][22][23] to hold, and let Φ t,λ be produced by some member of LC 0 . It follows that for λ near the origin in R n , the essential dynamics of the semi-flow Φ t,λ near the rotating wave reduces § However, even in the case of Archimedean spirals (for which hypothesis 1 fails), finite-dimensional center-bundle equations which share the symmetries of the underlying abstract dynamical systems have been shown to possess a definite predictive value in terms of possible dynamics and bifurcations of these spiral waves [1][2][3][4][5]. to the following ordinary differential equations on the bundle C × S 1 (see [32] for more details):ṗ = e iϕ V + G p (p, p, ϕ, λ) ϕ = ω + G ϕ (p, p, ϕ, λ) (2.2) where V is a complex constant, ω = 0 is a real constant, G p and G ϕ are smooth, uniformly bounded in p, and such that G p (p, p, ϕ, 0) = 0 and G ϕ (p, p, ϕ, 0) = 0. If λ is near the origin, we can re-scale time along orbits of (2.2) to geṫ p = e iϕ v + G(p, p, ϕ, λ) ϕ = 1 (2.3) where G is smooth, uniformly bounded in p, and such that G(p, p, ϕ, 0) ≡ 0. Of course, G is not completely arbitrary because of the symmetry conditions in hypothesis 1. A simple computation and Taylor's theorem lead to the following. p = e iϕ(t) v + n j=1 λ j H j ((p − ξ j )e −iϕ(t) , (p − ξ j )e iϕ(t) , λ) (2.4) where, without loss of generality, ϕ(t) = t, v ∈ C, λ = (λ 1 , . . . , λ n ), and the functions H j are smooth and uniformly bounded in p. A 2π−periodic solution p λ of (2.4) is called a perturbed rotating wave of (2.4). Define the average value [p λ ] A = 1 2π 2π 0 p λ (t) dt. (2.5) If the Floquet multipliers of p λ all lie within (resp. outside) the unit circle, we shall say that [p λ ] A is the anchoring (resp. repelling, or unstable anchoring) center of p λ . In the following section, we will perform an analysis of anchoring of perturbed rotating waves of (2.4) for parameter values near λ = 0. Analysis of the Center Bundle Equations Equations (2.4) represent the dynamics near a normally hyperbolic rotating wave for a parameterized family Φ t,λ of semi-flows satisfying the forced-symmetry breaking conditions in hypothesis 1. We start with a brief review of the case n = 1 which was studied in detail in [3], and then present new results on the general n case. The Case n = 1 In this case, we may assume without loss of generality that ξ 1 = 0, so that (2.4) has the formṗ = e it v + λH(pe −it , pe it , λ) (3.1) where λ ∈ R is small. By writing w = pe −it + iv, this system becomeṡ w = −iw + λ H(w, w, λ) (3.2) where H(w, w, λ) = H(w − iv, w + iv, λ). The following theorem is proved in [3]. p(t) = (−iv + O(λ)) e it , ϕ(t) = t. (3. 3) The origin [p] A = 0 is an anchoring center if aλ < 0; it is a repelling center if aλ > 0. Remark 3.2 In the case where the semi-flow Φ t,λ is generated by a system of planar reaction-diffusion partial differential equations, the solution (3.3) represents a wave which is rigidly and uniformly rotating around the origin in the plane. In the case where aλ < 0, the rotating wave is locally asymptotically stable. When aλ > 0, the rotating wave is unstable (see [8] for an experimental characterization of this phenomenon in an Oregonator model). The Case n > 1 One might think that the combination of many perturbations would just combine the effects of each perturbation, so that spirals would be observed anchoring at each of the centers, but we shall see that this is not usually the case. By re-labeling the indices in (2.4) if necessary, we can temporarily shift our point of view so that ξ 1 plays the central role in the following analysis. Then, under the co-rotating frame of reference z = p − ξ 1 + ie it v, (2.4) becomeṡ z =ṗ − e it v = e it n j=1 λ j H j (z − ζ j )e −it − iv, (z − ζ j )e it + iv, λ ,(3.4) where ζ j = ξ j − ξ 1 for j = 1, . . . , n. When λ 1 = 0 and λ 2 = · · · = λ n = 0, we find ourselves in the situation described in the previous subsection. Now, set ε = λ 1 , µ 1 = 1 and λ j = µ j ε for j = 2, . . . , n and µ = (µ 2 , . . . , µ n ) ∈ R n−1 . Then (3.4) can be viewed as a perturbation of the corresponding equation in the case n = 1. Note that ζ 1 = 0 and λ = (1, µ)ε. Equation (3.4) rewrites aṡ z = εe it n j=1 µ j H j (z − ζ j )e −it − iv, (z − ζ j )e it + iv, (1, µ)ε . (3.5) LetĤ j (w, w, ε, µ) = H j w − iv, w + iv, (1, µ)ε for j = 1, . . . n. Then (3.5) becomeṡ z = εe it K(ze −it , ze it , t, ε, µ) (3.6) where K(w, w, t, ε, µ) = n j=1 µ jĤj (w − ζ j e −it , w − ζ j e it , ε, µ) is 2π−periodic in t. Set α 1 = D 1 H 1 (−iv, iv, 0) . The time−2π map P of (3.6) is given by P (z, z, ε, µ) = z + 2πε α 1 z + O |z| 2 + O ε, µ 2 , . . . , µ n (3.7) near z = 0 and (ε, µ) = (0, 0). Hyperbolic fixed points of (3.7) correspond to hyperbolic 2π−periodic solutions of (3.6), and so to perturbed rotating waves of (2.4), that is, the path traced by the solution wave need not be circular. As z = 0 is not generally a fixed point of (3.7), these perturbed rotating waves may not be centered at ξ 1 . Indeed, let B(z, z, ε, µ) = α 1 z + O |z| 2 + O ε, µ 2 , . . . , µ n (3.8) be the function inside the square brackets in (3.7). Note that B(0, 0, 0, 0) = 0 and that, generically, D 1 B(0, 0, 0, 0) = α 1 = 0. By the implicit function theorem, there is a unique smooth function z(ε, µ) defined near (ε, µ) = (0, 0) with z(0, 0) = 0 and B z(ε, µ), z(ε, µ), ε, µ ≡ 0 (3.9) near z = 0. This leads to the following theorem. Theorem 3.3 Let α 1 be as in the preceding discussion, with Re(α 1 ) = 0. If the parameters are small enough to satisfy the conditions outlined in the proof below, the time−2π map (3.7) has a unique family of hyperbolic fixed points, whose stability is exactly determined by the sign of ε Re(α 1 ). Proof: Let B be as in (3.8) and z(ε, µ) be the unique continuous function solving the equation B = 0 for small parameter values, as asserted above. When ε = 0, any point of R 2 is a non-hyperbolic fixed point of P and so, from now on, we will assume that ε = 0. If that is the case, and if ε and µ are small enough, the eigenvalues ω 1,2 (ε, µ) of DP (z(ε, µ), ε, µ) satisfy |ω 1,2 (ε, µ)| 2 = 1 + 4πε Re(α 1 ) + εO(ε, µ) = 1, since Re(α 1 ) = 0. In other words, the fixed point z(ε, µ) is hyperbolic. When ε Re(α 1 ) < 0, the eigenvalues lie inside the unit circle and the fixed point is asymptotically stable; otherwise, it is unstable. ✷ We are now able to formulate and prove the following result. W 1 = {(λ 1 , . . . , λ n ) ∈ R n : |λ j | < W 1,j |λ 1 |, W 1,j > 0, for j = 1 and λ 1 near 0 } such that for all 0 = λ ∈ W 1 , (2.4) has a unique perturbed rotating wave S 1 λ , with center [S 1 λ ] A generically away from ξ 1 . Furthermore, [S 1 λ ] A is a center of anchoring when λ 1 Re(α 1 ) < 0. Proof: For j = 1, let W 1,j > 0 be such that the conclusion of theorem 3.3 holds for any µ j with |µ j | < W 1,j . Let W 1 be as stated in the hypothesis. If (ε, µ) is such that the time−2π map (3.7) has a hyperbolic fixed point z(ε, µ) near 0, then (3.5) has a hyperbolic 2π−periodic orbitz ε,µ (t) centered at a point near z = ζ 1 = 0. For j = 1, let λ 1 = ε = 0 be small enough and set λ j = µ j ε. Then λ ∈ W 1 , as |λ j | = |µ j ||ε| < W 1,j |λ 1 | for j = 1, andz ε,µ (t) is a 2π−periodic orbit for the parameter λ, which we denote by z λ (t). Since p = z − ie it v + ξ 1 , (2.4) has a unique perturbed rotating wave S 1 λ , with [S 1 λ ] A = 1 2π 2π 0 z λ (t) − ie it v + ξ 1 dt = ξ 1 + [z λ ] A . If 0 = λ ∈ W is such that µ j = λ j /ε = 0 is fixed for j = 2, . . . , n, then [z λ ] A = O(1) as λ 1 → 0 and so [S 1 λ ] A = ξ 1 , generically. The conclusion about the stability of S 1 λ follows directly from theorem 3.3. ✷ Remark 3.5 When λ approaches the λ 1 −axis away from the origin, [S 1 λ ] A → ξ 1 . On the other hand, when the parameter values stray outside of W 1 , all that can generically be said with certainty is that solutions of (2.4) locally drift away from ξ 1 , which cannot then be a center of anchoring. After drifting, the spiral may very well get anchored at some point far from ξ 1 , depending on the global nature of the perturbation functions H j in (2.4). The preceding results have been achieved by considering (2.4) under a co-rotating frame of reference around ξ 1 . Of course, since the choice for ξ 1 was arbitrary, corresponding results must also be achieved, in exactly the same manner, when the viewpoint shifts to another ξ k . For j = 1, . . . , n, let α j = D 1 H j (−iv, iv, 0) be the anchoring coefficients of (2.4). Theorem 3.6 Let k ∈ {1, . . . , n}. If Re(α k ) = 0, then there exists a wedge-shaped region near λ = 0 of the form W k = {(λ 1 , . . . , λ n ) ∈ R n : |λ j | < W k,j |λ k |, W k,j > 0, for j = k and λ k near 0 } such that for all 0 = λ ∈ W k , (2.4) has a unique perturbed rotating wave S k λ , with center [S k λ ] A generically away from ξ k . Furthermore, [S k λ ] A is a center of anchoring when λ k Re(α k ) < 0. Clearly, the remark that appears after the proof of theorem 3.4 still holds. Characterization of Spiral Anchoring (n = 2) In the previous section, we described the (local) behaviour of spiral anchoring in small wedges around the parameter coordinate axes. In this section, we present a fuller characterization of spiral wave anchoring for the case n = 2. Let 0 = ξ ∈ R 2 , Λ 0 = (λ 1 , 0), Λ ξ = (0, λ 2 ) ∈ R 2 and let P : R 2 × R 2 → R 2 be a real analytic map with P (x, 0) = x, DP (x, 0) = I 2 for all x ∈ R 2 , satisfying the following conditions: for η ∈ {0, ξ}, (P1) ∃ ω * > 0 such that P (η, Λ η ) ≡ 0, for all ||Λ η || < ω * ; (P2) the eigenvalues of DP (η, Λ η ) lie both outside or both inside the unit circle for all 0 = ||Λ η || < ω * ; (P3) there is a wedge region w η surrounding the coordinate axis generated by Λ η in parameter space (see figure 1) in which P has a (locally) unique manifold x η (λ) such that, for all λ ∈ w η , (a) P (x η (λ), λ) ≡ x η (λ); (b) x η (λ) → η as λ approaches the coordinate axis away from the origin; (c) x η (λ) shares its stability with η in (P2). When the hypotheses of theorem 3.6 hold, the associated time−2π map (3.7) (viewed in real coordinates) satisfies (P1)−(P3). Numerous questions cannot be answered by local analysis alone. For instance: (i) Can the wedges overlap? What does that imply for anchoring in (2.4)? (ii) Can a wedge contain its "opposite" coordinate axis? (iii) If the wedges do not overlap, what is the nature of their complement? (iv) If there is a complement with non-trivial measure, what kinds of dynamics can be expected as the parameter vector λ traces a circle around the origin in parameter space? We will provide answers to these questions by first studying a specific map, then extending our results to the general mapping. A Specific Mapping Consider the mapping P : R 2 × R 2 → R 2 given by P (x, λ) = x + 2π λ 1 F 0 (x) + λ 2 G ξ (x) ,(4.1) where 0 = ξ ∈ R 2 , and F 0 , G ξ are real analytic functions of x, λ ∈ R 2 . Such a map is obtained by truncating the λ−terms of order ≥ 2 from the time−2π map (3.7), for instance. According to theorem 3.6, the jacobians DF 0 (0) and DG ξ (ξ) have a particular structure. Proposition 4.1 If F 0 (0) = 0, G ξ (ξ) = 0, and if DF 0 (0) = a −b b a and DG ξ (ξ) = c −d d c where a, c = 0, then there exists ω * > 0 such that the map defined by (4.1) satisfies the conditions (P1)−(P3). The search for fixed points. Define A : R 2 → M 2 (R) by A(x) = F 0 (x) G ξ (x) . (4.2) Then,x is a fixed point of (4.1) forλ ∈ R 2 if and only if A(x) ·λ = 0, that is if and only ifλ ∈ Lx = ker A(x). Let (x,λ) be such a pair. According to the implicit function theorem, as long as det D x P (x,λ) − I = 4π 2 det λ 1 DF 0 (x) +λ 2 DG ξ (x) = 0, (4.3) there is a neighbourhood W ofλ and a unique analytic function X : W → R 2 such that X(λ) =x and A(X(λ)) · λ ≡ 0 for all λ ∈ W. By construction, X(λ) is a fixed point of (4.1) for all λ ∈ W . If dim Lx = 0 as a manifold, then Lx = {0}. Consequently, the preceding implicit function theorem construction fails, which contradicts property (P3). We need thus only investigate fixed pointsx for which dim Lx = 0. As the quantities under consideration are analytic, it can further be assumed that rank A(x) = 1 and dim Lx = 1. We now show how to optimally extend the wedge regions w η using property (P3). Let (x * , λ * ), (x * , λ * ) ∈ R 2 × (R 2 − {0}) be such that x * , x * are fixed points of (4.1), λ * ∈ L x * , λ * ∈ L x * , and (4.3) is satisfied for both pairs. According to the implicit function theorem, there are open neighbourhoods W * , W * of λ * , λ * ∈ R 2 respectively, and a pair of unique real analytic functions X * : W * → R 2 , X * : W * → R 2 for which X * (λ * ) = x * , X * (λ * ) = x * and A(X * (Λ)) · Λ ≡ 0, for all Λ ∈ W * , A(X * (Λ)) · Λ ≡ 0, for all Λ ∈ W * . Lemma 4.2 If Λ * * ∈ W * * = W * ∩ W * is such that X * (Λ * * ) = X * (Λ * * ) , then X * = X * on W * * Proof: The assertion follows from the uniqueness of the real analytic functions X * , X * in the implicit function theorem. ✷ Denote the punctured open disc of radius ω * centered at the origin by B(0, ω * ). Let η ∈ {0, ξ}, ω * > 0 be as in proposition 4.1 and 0 = Λ η ∈ w η be a point on the appropriate coordinate axis, as in properties (P1) and (P2). According to these same properties, η is a fixed point of (4.1) for Λ η and det (D x P (η, Λ η ) − I) = 0. built in the Fundamental Theorem of ODE [33]), containing w η ∩ B(0, ω * ) and for which there is a unique real analytic function X η : W η → R 2 satisfying x η = X η | Wη , where x η is as in property (P3). Sincex is a fixed point of (4.1) for 0 =λ wheneverλ ∈ Lx, W η is described (in polar coordinates) by either one of W η = {(r, θ) : 0 < r < ω * and s η − ϕ − η < θ < s η + ϕ + η } W η = {(r, θ) : 0 < r < ω * and θ ∈ [0, 2π]} where ϕ − η , ϕ + η ∈ (0, π/2] and s η = 0 if η = 0, π/2 if η = ξ. ,(4.4) In the latter case, we will say that W η is catastrophe-free. In the former case, the quantities ϕ − η , ϕ + η ∈ (0, π/2] are called the fore-angle and post-angle of W η , respectively (see figure 2). The implicit function theorem fails to extend A(X η (λ)) · λ ≡ 0 (that is, it fails to extend W η ) at (x * , λ * ) if either (C1) det (λ * 1 DF 0 (x * ) + λ * 2 DG ξ (x * )) = 0 and X η (λ) → x * as λ → λ * , or (C2) X η (λ) → ∞ as λ → λ * . Such events will be referred to as fold and ∞−catastrophes, respectively, or catastrophes, collectively. Let 0 < ρ < ω * and set γ ρ (s) = ρ (cos(s), sin(s)) ⊤ . (4.5) Assume W η is not catastrophe-free. Starting at (ρ, s η ) ∈ W η , denote the angles in (0, π/2] measuring the first clockwise and the first counter-clockwise occurrence of a catastrophe along γ ρ by θ − η and θ + η respectively. Then, ϕ ± η = s η ± θ ± η . 4.1.2. Fold bifurcation points. Modulo a simple regularity condition (see below), (C1) is equivalent to the existence of a fold bifurcation curve in parameter space for (4.1). Indeed, in that case, (x * , λ * ) is a solution of P (x, λ) − x = 0, det (D x P (x, λ) − I) = 0. (4.6) If the (full) Jacobian of the left-hand side of (4.6) has rank 3 at that point, (4.6) has a fold bifurcation curve through λ * [34]. Such solutions are in one-to-one correspondence with regular solutions of A quick computation shows that (4.7) can be written as A(x) · λ = 0, det (D x [A(x) · λ]) = 0.A(x) · λ = 0, λ ⊤ Q(x)λ = 0,(4.8) where Let x * be a fixed point of (4.1) and denote K x = {λ : λ ⊤ Q(x)λ = 0}. Generically, K x * consists of a single line or a pair of intersecting lines through the origin in parameter space. Writing L x = L x ∩ B(0, ω * ) and K x = K x ∩ B(0, ω * ), we can summarize the situation with the following proposition. Q(x) = B(x) 1 2 C(x) 1 2 C(x) E(x) and B(x) = det DF 0 (x) C(x) = det DH 1 (x) + det DH 2 (x) E(x) = det DG ξ (x). Proposition 4.3 If (x * , λ * ) is a regular solution of (4.7) with {0} = L x * ⊆ K x * , then (x * , λ) is a fold bifurcation point of (4.1) for all λ ∈ L x * . The Visual Criterion Set Z = {(x, λ) : P (x, λ) = x and λ ∈ L x = {0}} and κ(Z) = {x : ∃ λ = 0 such that (x, λ) ∈ Z}. By construction, κ(Z) is the zero-set of det A(x) in R 2 and 0, ξ ∈ κ(Z). Generically, κ(Z) is a collection C of isolated planar curves, whose constituents come in two varieties: bounded or unbounded. ¶ Denote this partition by C = C B ⊔ C ∞ and let C 0 , C ξ be the curves in C for which 0 ∈ C 0 and ξ ∈ C ξ . Let γ ρ : [0, 2π] → R 2 be the circle of radius ρ around the origin, parameterized as in (4.5). For each (x, λ) ∈ Z, define L ρ x and K ρ x as the intersection of that circle with L x and K x , respectively, and let P ρ : R 2 × [0, 2π] → R 2 be given by P ρ (x, s) = x + 2πρ cos(s)F 0 (x) + sin(s)G ξ (x) . (4.10) Then L ρ x consists of two antipodal points {±α x,ρ }, and the fixed points (x, s) of P ρ are in one-to-one correspondence with the 'lines' of fixed points (x, L x ) of P for which L x = {0} (see proposition 4.3). Set Z ρ = {(x, s) ∈ R 2 × [0, 2π] : P ρ (x, s) = x and γ ρ (s) ∈ L ρ x } and κ ρ (Z ρ ) = {x : (x, s) ∈ Z ρ }. By construction, κ ρ (Z ρ ) = κ(Z). Thus for each C ∈ C, κ −1 ρ (C) is a branch of fixed points in the bifurcation diagram of P ρ . According to section 4.1.1, the converse also holds: each branch of fixed points in the bifurcation diagram of P ρ projects down via κ ρ to a curve in C. The existence and location of fold catastrophes cannot be read directly from C, but the next proposition remedies that situation. Let (x * , α) ∈ R 2 × (R 2 − {0}) be such that α = ρ, α ∈ L ρ x * ⊆ K ρ x * . Recall that A(x * ) = 0. Then, (A j,1 (x * ) A j,2 (x * )) = 0 for some j ∈ {1, 2} and L x * = {λ : A j,1 (x * )λ 1 + A j,2 (x * )λ 2 = 0}. The function Γ j : R 2 → R 2 defined by Γ j (x) = A 2 j,2 (x)B(x) − A j,1 (x)A j,2 (x)C(x) + A 2 j,1 (x)E(x) ,(4. 11) ¶ Indeed, were any such curves to intersect at x * , P would undergo a transcritical bifurcation along κ −1 (x * ). Such bifurcations are not generically permitted by (C1) and (C2). where B, C and E are as in (4.9), is called the j−fold bifurcation function of (4.1). Let R j be the zero-set of Γ j (x) in R 2 . We shall say that x * is a transverse intersection of κ(Z) and R j if det A(x * ) = Γ j (x * ) = 0 and rank D det A(x * ) Γ j (x * ) = 2. Proposition 4.4 Let j ∈ {1, 2}. If x * is a transverse intersection of κ(Z) and R j such that (A j,1 (x * ) A j,2 (x * )) = 0 and either (1) B(x * ) = 0 and A j,1 (x * )C(x * ) − A j,2 (x * )E(x * ) = 0 or (2) B(x * ) = 0 and C(x * ) 2 − 4B(x * )E(x * ) ≥ 0, then P ρ undergoes a fold catastrophe at (x * , s * ) for all s * such that γ ρ (s * ) = ±α x * ,ρ . Proof: By re-labeling the terms if necessary, we may assume A j,1 = 0. There are then two possibilities. (i) If B = 0 and A j,1 C − A j,2 E = 0, then (ii) If B = 0 and C 2 − 4BE ≥ 0, then K x * = {λ : λ 1 λ 2 C + λ 2 2 E = 0} = {λ : λ 2 = 0 or λ 1 C + λ 2 E = 0}.K x * = λ : λ 1 = −C ± √ C 2 − 4BE 2B λ 2 . In this case, 4BΓ j = (−2A j,2 B + A j,1 C) 2 − A 2 j,1 (C 2 − 4BE) = −2A j,2 B + A j,1 C + √ C 2 − 4BE · −2A j,2 B + A j,1 C − √ C 2 − 4BE = 0 and so − A j,2 A j,1 = −C + √ C 2 − 4BE 2B or − A j,2 A j,1 = −C − √ C 2 − 4BE 2B . In either cases, L x * is contained in K x * ; thus {0} = L x * ⊆ K x * and {0} = L ρ x * ⊆ K ρ x * . As x * is a transverse intersection of κ(Z) and R j , it is also a regular solution of (4.8); (x * , L x * ) then consists of fold bifurcation points of (4.1), according to proposition 4.3. The desired conclusion follows from {±α x * ,ρ } = L ρ x * = L x * ∩ γ ρ and from the correspondence between fixed points of P ρ and 'lines' of fixed points of P . ✷ By construction, the bifurcation diagram of P ρ is 2π−periodic in s. Consequently, elements of C B must be (bounded) loops and elements of C ∞ must give rise to two ∞−catastrophes. Moreover, the number of fold catastrophes on any given C ∈ C B cannot be odd as C could not be a loop were that the case. Finally, note that catastrophes cannot occur at 0 or ξ as this would contradict (P2) and (P3). The Bifurcation Diagrams Let C 0 , C ξ , s 0 and s ξ be as defined previously. Set η ∈ {0, ξ}. By definition, C η goes through η at s = s η . By (P3), the wedges' angles ϕ ± η lie in (0, π) or (0, π] (when they exist), according to whether they record fold or ∞−catastrophes, respectively. Set ν 1 = ϕ + 0 + ϕ − ξ and ν 2 = ϕ − 0 + ϕ + ξ . Then W 0 and W ξ overlap (i) in all four quadrants if and only if ν 1 , ν 2 > π/2; (ii) in the first and third quadrants if and only if ν 1 > π/2 and ν 2 ≤ π/2, and in the second and fourth quadrants if and only if ν 1 ≤ π/2 and ν 2 > π/2. If ν j = π/2, the wedges do not overlap but their complement has zero measure in a neighbourhood of the origin. When the wedge angles ϕ ± η do not exist, W η is a deleted neighbourhood of the origin in parameter space. 4.3.1. The case C 0 = C ξ In this instance, it is sufficient to understand the bifurcation diagrams along a single curve: the full picture can then be obtained by combining the diagrams corresponding to C 0 and C ξ . When C η ∈ C B , there are two (essentially) distinct generic possibilities. (i) If there is no fold catastrophe along C η , then the angles ϕ ± η do not exist and W η is catastrophe-free deleted neighbourhood of the origin in parameter space. (ii) If there are 2k fold catastrophes along C η , k > 0, then the angles ϕ ± η are well-defined: s η ∓ ϕ ± η are the s−values of the first fold catastrophes occurring respectively before and after η along C η . When C η ∈ C ∞ , there are two (essentially) distinct generic possibilities. (i) If there is no fold catastrophe along C η , then the angles ϕ ± η are well defined and s η ∓ ϕ ± η are the s−values of the ∞−catastrophes occurring respectively before and after η via C η . (ii) If there are k fold catastrophes along C η , k > 0, then the angles ϕ ± η are welldefined: if all the fold catastrophes lie on one side of η (say s > s η ) along C η then s η ∓ ϕ ± η are the s−values of the ∞−catastrophes occurring before η and the first fold catastrophe after η along C η , respectively (or vice-versa). Otherwise, s η ∓ ϕ ± η are the s−values of the first fold catastrophes occurring respectively before and after η along C η . Some corresponding qualitative bifurcation diagrams are shown in figure 3. 4.3.2. The case C 0 = C ξ In this instance, the bifurcation diagram must pass through 0 at s = 0 and ξ at s = π/2. When C 0 = C ξ ∈ C B , the number of fold catastrophes along the curve is even; there are then three (essentially) distinct generic possibilities. (i) If there is no fold catastrophe along C 0 = C ξ , then the angles ϕ ± 0 and ϕ ± ξ do not exist and W 0 = W ξ are catastrophe-free deleted neighbourhoods of the origin in parameter space. (ii) If there is an odd number of fold catastrophes between the origin and ξ along C 0 = C ξ , then the angles ∓ϕ ± 0 and π/2∓ϕ ± ξ are well-defined: they are the s−values of the first fold catastrophes occurring respectively before and after 0 and ξ via C 0 = C ξ . (iii) If there is an even number of fold catastrophes between 0 and ξ along C 0 = C ξ , the situation is much as described in (2), save for the fact that C 0 = C ξ is not a loop in the bifurcation diagram of P ρ . When C 0 = C ξ ∈ C ∞ , there are two (essentially) distinct generic possibilities. (i) If there is no fold catastrophe along C 0 = C ξ , then the angles ϕ ± 0 and ϕ ± ξ are well defined and s η ∓ ϕ ± η are the s−values of the ∞−catastrophes occurring respectively before and after 0 and ξ via C 0 = C ξ . (ii) If there are k fold catastrophes along C 0 = C ξ , k > 0, then the angles ϕ ± 0 and ϕ ± ξ are well-defined: if no fold catastrophe lies between 0 and ξ along C 0 = C ξ then ∓ϕ ± 0 and π/2 ∓ ϕ ± η are determined as in the case C η ∈ C ∞ , item (2) (see p. 18). If there are fold catastrophes between 0 and ξ along C 0 = C ξ , then ϕ + 0 and π/2 − ϕ − ξ are the s−values of the first fold catastrophes occurring respectively after 0 and before ξ along C ξ . Some corresponding qualitative bifurcation diagrams are shown in figure 4. The General Mapping The mapping (4.1) is not the most general mapping satisfying (P1)−(P3); one should instead study maps of the form P(x, λ) = x + 2π λ 1 F 0 (x, λ 1 ) + λ 1 λ 2 J (x, λ) + λ 2 G ξ (x, λ 2 ) ,(4.12) where ξ = 0 ∈ R 2 , F 0 , J and G ξ are real analytic in their variables and the jacobians D x F 0 (0, λ 1 ) and D x G ξ (ξ, λ 2 ) have the particular form prescribed by proposition 4.5, which is analogous to proposition 4.1. Proposition 4.5 If F (0, λ 1 ) ≡ 0, G ξ (ξ, λ 2 ) ≡ 0, and if D x F 0 (0, λ 1 ) = a(λ 1 ) −b(λ 1 ) b(λ 1 ) a(λ 1 ) and D x G ξ (ξ, λ 2 ) = c(λ 2 ) −d(λ 2 ) d(λ 2 ) c(λ 2 ) , where a, b, c, d : R → R are continuous in their variables and a(0), c(0) = 0, then there exists ω * > 0 such that the map defined by (4.12) satisfies conditions (P1)−(P3). Define A : R 2 × R 2 → M 2 (R) by A(x, λ) = F 0 (x, λ 1 ) + λ 2 2 J (x, λ) λ 1 2 J (x, λ) + G ξ (x, λ 2 ) ; (4.13) Figure 4. Partial bifurcation diagrams of P ρ when C 0 = C ξ . The squares represent 0 and ξ, and the circles and arrows indicate fold and ∞−catastrophes, respectively. The apparent self-intersection is an artifact of the projection on the x − s plane; it does not, in fact, occur. fixed points of (4.12) are then in one-to-one correspondence with solutions of A(x, λ) · λ = 0. (4.14) Set F(x, λ) = det A(x, λ). According to Taylor's theorem, there are appropriate functions K 10 , K 01 such that F(x, λ) = F(x, 0) + λ 1 K 10 (x, λ) + λ 2 K 01 (x, λ). Letx be such that F(x, 0) = 0, det D x F(x, 0) = 0 and A(x, 0) = 0. Then, by the implicit function theorem, there is a neighbourhood V ⊆ R 2 of the origin and a unique analytic function X : V → R 2 such that X(0) =x, F(X(λ), λ) ≡ 0 and rank A(X(λ), λ) = 1 ∀ λ ∈ V. Define Lx = {λ ∈ V : A(X(λ), λ) · λ = 0}. A simple rank argument shows that Lx is defined via a single equation in two real variables, with a regular solution at the origin; consequently, as a manifold, Lx is one-dimensional. Let Lx = ker A(x, 0). Then, there is a small neighbourhood U ⊆ B(0, ω * ) of the origin in parameter space for which {(X(λ), λ) : λ ∈ U ∩ Lx} is a deformation of {(x, λ) : λ ∈ U ∩ Lx} : both 'curves' can be parameterized by the same λ j , j = 1, 2. The preceding discussion shows that the fixed points of (4.12) are in one-to-one correspondence with the fixed points of the (already studied) truncated map P T (x, λ) = x + 2π λ 1 F 0 (x, 0) + λ 2 G ξ (x, 0) . (4.15) Fold bifurcations persist under small perturbations [35,36]. Similarly, a generic unbounded curve remains unbounded under small perturbations. Indeed, in the real projective plane, an element of C ∞ meets the line at infinity in two points. Generically, these two points are distinct and a small perturbation will not change that fact, i.e the perturbed curve is still an element of C ∞ . In the non-generic case where the two points at infinity are equal, a small perturbation will either cause the points to separate or to vanish entirely (reminescent of a fold bifurcation of points at infinity), i.e the perturbed curve either stays in C ∞ or becomes finite. Thus, catastrophes generically persist: as a result, the bifurcation diagrams of (4.12) and (4.15) are (locally) topologically equivalent for small parameter values λ. Consequently, (4.12) has wedge-like regions W η corresponding to the wedge regions W η of (4.1). Finally, note that since P is the 'linearization' of P at the origin with respect to λ, the wedge regions W η of (4.1) provide tangential 'cones' for the corresponding wedge-like regions W η of (4.12). Numerical Simulations and Examples In this section, we illustrate and interpret the results of the preceding sections through various examples. As such, the emphasis lies with qualitative observations rather than with precise numerical analysis. First, we study systems of PDE from a (naive) numerical perspective: we observe spiral anchoring, as well as hysteresis and homotopy of the spiral tip. Finally, we provide a few examples of mappings of the form (4.12) together with their zero-level sets and partial bifurcation diagrams. PDE, FESB and Semi-Flows In this section, we examine systems of partial differential equations giving rise to semiflows satisfying the FESB equivariance described in section 2. Each g j (x), alone, breaks translational symmetry but preserves rotational symmetry about (9, 0) (for j = 1) or (−10, 5 √ 3) (for j = 2). Note that both perturbations are uniformly bounded on R 2 and that they go to 0 as x → ∞. Under these conditions, the flow of (5.1) near a normally hyperbolic rotating wave is equivalent to the flow of some center bundle equation (2.4). Thus, if spiral waves anchor at all, they will generically do so away from either perturbation center. This is confirmed in figure 5, in which the transients anchor at what would be an otherwise unremarkable location. u t = 1 ς u − 1 3 u 3 − v + φ 1 + ∆u, v t = ς(u + β − γv − φ 2 ), (5.1) where φ j (x) = √ 2 cos(0.05π)0.12f (x 1 − c 1,j , x 2 − c 2,j ), j = 1, 2,(5. We now present the results of simulations on a reaction-diffusion system with 4 TSB perturbations. Set u t = 1 ς u − 1 3 u 3 − v + φ 1 + ∆u, v t = ς(u + β − γv + φ 2 ),(5.3) where ς = 0.3, β = 0.6, γ = 0.5, and where φ 1 ,φ 2 are inhomogeneous terms which depend on x ∈ R 2 and are defined by φ 1 (x) = g 1 (x) + g 2 (x) = 0.12f 1 (x 1 − 9, x 2 ) − 0.10f 2 (x 1 + 1, x 2 − 10), φ 2 (x) = g 3 (x) + g 4 (x) = −0.12f 1 (x 1 + 10, x 2 − 5 √ 3) + 0.08f 3 (x 1 − 10, x 2 − 10), where A 1 = 0.12, A 2 = −0.10, B 1 = −0.12, B 2 = 0.08, f j (x) = exp a j (x 2 1 + x 2 2 ) , j = 1, 2, 3, a 1 = −0.00086, a 2 = −0.0008 and a 3 = −0.0009. Each g j (x), alone, breaks translational symmetry but preserves rotational symmetry about c 1 = (9, 0) (for j = 1), c 2 = (−1, 10) (for j = 2), c 3 = (−10, 5 √ 3) (for j = 3) and c 4 = (10, 10) (for j = 4). Note that the four perturbations are uniformly bounded on R 2 and that they go to 0 as x → ∞. As predicted, anchoring takes place away from the c j , j = 1, . . . , 4. The transients in figure 6 appear to first (hyperbolically) approach some manifold along which they travel to the anchored perturbed rotating wave; this will be the topic of an upcoming paper. (5.4) where f = 1.4, q = 0.002, ς = 0.05 and φ is an inhomogeneous term which depends on x ∈ R 2 . When φ ≡ 0, (5.4) has full Euclidean symmetry. In the following simulations, φ is the sum of two Gaussian bells: Homotopy and hysteresis of rotating waves Following [8], define the modified Oregonator u t = 1 ς u − u 2 − (f v + φ) u−q u+q + ∆u, v t = (u − v) + 0.6∆v,φ(x) = α 1 exp − (x 1 , x 2 ) − (15, 15) 2 β 1 2 + α 2 exp − (x 1 , x 2 ) − (18.75, 15) 2 β 2 2 , with α 1 , α 2 , β 1 , β 2 ∈ R, and β 1 , β 2 = 0. Each g j (x), alone, breaks translational symmetry but preserves rotational symmetry about c 1 = (15, 15) (for j = 1) or c 2 = (18.75, 15) (for j = 2). Note that both perturbations are uniformly bounded on R 2 and that they go to 0 as x → ∞. Set β 1 = β 2 = 1 and ρ * = 0.01. Along the path α(τ ) = γ 1 (τ ) = ρ * (cos(τ ), sin(τ )) ⊤ in parameter space, (5.4) undergoes a homotopy of perturbed rotating waves, whose tip paths deform continuously from a circle centered at c 1 , when τ = 0, to a circle centered at c 2 , when τ = π/2 (see figure 7). Along the path α(τ ) = γ 2 (τ ) = 1 10 γ 1 (τ ), however, the homotopy is replaced by hysteresis. As the parameters vary along the path, (5.4) has an anchored perturbed rotating wave whose tip path deforms continuously from a circle centered at c 1 . At ℵ 1 ∈ γ 2 , the rotating wave jumps (discontinuously) to another anchored perturbed rotating wave, whose tip path deforms continuously from a circle centered at c 2 . Following γ 2 in the opposite direction leads to similar behaviour, this time with the discontinuous jump taking place at ℵ 2 ∈ γ 2 , as can be seen in figure 8. Consequently, there must be a third unstable rotating wave (which escapes detection by direct means) appearing and then disappearing at ℵ 1 and ℵ 2 , respectively, in saddle-node bifurcations. In a sense, both of these occurrences have been predicted by the analysis provided in section 4; consult, for instance, the first and third bifurcation diagrams on p. 20. Wedges and Catastrophes In this final section, we provide a partial catalogue of (partial) bifurcation diagrams for mappings of the form P (x, λ) = x + 2π λ 1 F 0 (x) + λ 2 G ξ (x) ,(5.5) where λ ∈ R 2 , ξ = (2, 2) ⊤ , F 0 (x) = 2x 1 − x 2 + a i,j x i 1 x j 2 x 1 + 2x 2 + b i,j x i 1 x j 2 · f 0 (x), G ξ (x) = 7 − 3x 1 − x 2 2 + c i,j (x 1 − 2) i (x 2 − 2) j 5 − 3x 1 + x 2 2 + d i,j (x 1 − 2) i (x 2 − 2) j · g ξ (x) , a i,j , b i,j , c i,j , d i,j ∈ R, i + j > 1, and f 0 and g ξ are continuous functions such that (5.5) satisfies proposition 4.1; we can then use the visual criterion of section 4.2 to under- stand the nature of the bifurcation diagram of the associated map P 0.01 . Furthermore, the wedge angles can be read directly from the bifurcation diagram. In the figures of this section, C 0 and C ξ are shown in black or gray. Fold catastrophes are indicated by circles, ∞−catastrophes by arrows and the squares mark both the origin and ξ. The Elowyn-Bonhomme map The Elowyn-Bonhomme (EB) map is obtained from (5.5) by setting f 0 ≡ g ξ ≡ 1, a 1,1 = b 2,0 = c 0,2 = d 1,1 = d 2,0 = 1, a 0,2 = −1 and all other coefficients to 0; the corresponding κ(Z) is shown in figure 8. In this instance, C 0 ∈ C B and C ξ ∈ C ∞ . For the EB map, ω * = 12 37π . Let ρ = 0.01 < ω * . Using a pseudoarc length continuation algorithm (see [37] for details), a partial bifurcation diagram of P ρ (ignoring all fixed point branches but those through η at s = s η , for η ∈ {0, ξ}) is built: the results can be seen in figure 9. There are 6 fold catastrophes: two along C 0 and four along C ξ . Their location can be recovered directly from κ(Z) and R j , j = 1, 2, with the help of proposition 4.4: in figure 9, the six intersections that satisfy the appropriate hypotheses are marked with circles. Each corresponds to one of the six fold catastrophes observed in figure 7. Furthermore, two ∞−catastrophes occur via C ξ . The interesting values for the EB map are compiled in table 1 above. We continue by providing examples that highlight the various possibilities. . On the left: zero-level set κ(Z) for the EB map. On the right: fixed point branches of P 0.01 (the apparent self-intersection on C 0 is due to a projection onto the x − s plane). Both C 0 (the curve through 0) and C ξ (the curve through ξ) are shown in black. The first example a 2,0 = 0 a 1,1 = 1 a 0,2 = −1 b 2,0 = 1 b 1,1 = 0 b 0,2 = 0 c 2,0 = 0 c 1,1 = 0 c 0,2 = 1 d 2,0 = 1 d 1,1 = 1 d 0,2 = 0 f 0 (x) = exp (−(x 2 1 + x 2 2 )/10) g ξ (x) = exp (− ((x 1 − 2) 2 + (x 2 − 2) 2 ) /14). See figure 11 for a portion of κ(Z) and a partial bifurcation diagram of P 0.01 . By construction, the zero-level set for this first example is exactly the zero-level set of the EB map; however, their bifurcation diagrams are not topologically equivalent (compare with figure 9). In this instance, the wedge angles record fold catastrophes on C 0 (black) ∈ C B and ∞−catastrophes on C ξ (gray) ∈ C ∞ . Note further that this map provides an instance when the anchoring wedges overlap. See figure 12 for a portion of κ(Z) and a partial bifurcation diagram of P 0.01 . In this instance, C 0 (black), C ξ (gray) ∈ C ∞ , and ϕ ± 0 , ϕ − ξ record fold catastrophes while ϕ + ξ records an ∞−catastrophe. See figure 13 for a portion of κ(Z) and a partial bifurcation diagram of P 0.01 . In this instance, C 0 (black), C ξ (gray) ∈ C B , and the wedge angles all record fold catastrophes. See figure 14 for a portion of κ(Z) and a partial bifurcation diagram of P 0.01 . In this instance, C 0 = C ξ ∈ C B , the wedge angles all record fold catastrophes and the anchoring wedges overlap. See figure 15 for a portion of κ(Z) and a partial bifurcation diagram of P 0.01 . In this instance, C 0 = C ξ ∈ C ∞ and the wedge angles all record fold catastrophes. Then, according to proposition 4.5, P satisfies (P1)−(P3). By the preceding discussion, the bifurcation diagrams of the EB map P is topologically equivalent to that of P when λ is close enough to the origin. As an example, let 0.01 < ω * = 12 37π and define P 0.01 as the restriction of P to the circle γ 0.01 (s) = 0.01 (cos(s), sin(s)) ⊤ in parameter space. Compare the diagram shown (on the left) in figure 16 with the corresponding diagram of section 5.2.1. Finally, the wedge regions W 0 of the EB map and W 0 of the revisited EB map are shown (on the right) in figure 16, illustrating the last remark of section 4. Theorem 3. 1 1Let a = Re(D 1 H(0, 0, 0)), whereH is as in(3.2). If a = 0, then for all λ = 0 small enough, (3.1) has a hyperbolic rotating wave Theorem 3. 4 4Suppose the hypotheses of theorem 3.3 are satisfied. Then there exists a wedge-shaped region near λ = 0 of the form Figure 1 . 1Wedges in parameter space corresponding to property (P3). Lemma 4.2 then implies the existence of a maximal open region W η , defined as a union of open sets W ⊆ B(0, ω * ) (in much the same way as the maximal interval is Figure 2 . 2Wedge angles, with optimal wedge-like regions in parameter space. define H 1 , H 2 : R 2 → R 2 by H 1 (x) = I 10 A(x) + I 01 A(x)Î e 1 , H 2 (x) = I 10 A(x) + I 01 A(x)Î e 2 . ( a ) 0 a0If λ 2 = 0, then L x * = {(λ 1 , 0) : λ 1 A j,1 = 0} = {0} since A j,1 = 0. But this contradicts the assumption dim L x * = 1. (b) If λ 1 C + λ 2 E = Figure 3 . 3Partial bifurcation diagrams of P ρ when C 0 = C ξ . Only one branch is shown. The square represents the origin or ξ and the circles and arrows indicate fold and ∞−catastrophes, respectively. Figure 5 . 5Anchoring in (5.1) with perturbations as in (5.2). The spiral tip paths are plotted in black, the anchored perturbed rotating wave is shown in gray, and the squares indicate the location of the perturbation centers. The computations are carried out on a two-dimensional square domain [−30, 30] 2 with 200 grid points to a side and time-step ∆t = 0.005 and Neumann boundary condition, using a 5-point Laplacian and i) an explicit Runge-Kutta 2−stage method of order two in section 5.1.1, and ii) Matsui's fourth-order Runge-Kutta code based on Barkley's EZ-Spiral in section 5.1.2. Throughout, centers of anchoring are found via fast Fourier transforms of the tip data. 5.1.1. Spiral anchoring Consider the following small perturbation of the FitzHugh-Nagumo equations: f (x) = exp −0.00086 x 2 1 + x 2 2 . Figure 6 . 6Anchoring in the FitzHugh-Nagumo equations (5.3). The spiral tip paths are plotted in black, the anchored perturbed rotating wave is shown in gray and the black squares indicate the location of the perturbation centers. When α = (α 1 , 0) = 0, (5.4) is SO(2) c 1 −equivariant. Similarly, (5.4) is SO(2) c 2 − equivariant when α = (0, α 2 ) = 0 and trivially equivariant when α 1 , α 2 = 0. Figure 7 . 7Homotopy of the spiral tip path in (5.4). The first spatial coordinates of the anchoring centers are plotted against τ ; compare this image with the first bifurcation diagram on p. 20. Figure 8 . 8Hysteresis of the spiral tip path in(5.4). The first spatial coordinates of the anchoring centers are plotted against τ . The question marks interpolate (roughly) the unstable rotating waves. Compare this image with the third bifurcation diagram on p. 20. Figure 9 9Figure 9. On the left: zero-level set κ(Z) for the EB map. On the right: fixed point branches of P 0.01 (the apparent self-intersection on C 0 is due to a projection onto the x − s plane). Both C 0 (the curve through 0) and C ξ (the curve through ξ) are shown in black. Figure 10 . 10Intersections of the zero-level sets κ(Z) (thick black lines) and R 1 and R 2 (thin gray lines) of the Elowyn-Bonhomme problem. The squares represent 0 and ξ; the points that satisfy the hypotheses of proposition 4.4 are marked with circles. The light gray region is part of the planar set for which C(x) 2 − 4B(x)E(x) < 0. Figure 11 . 11The first example: κ(Z) (left), bifurcation diagram of P 0.01 (right). Figure 12 . 12The second example: κ(Z) (left), bifurcation diagram of P 0.01 (right). Figure 13 . 13The third example: κ(Z) (left), bifurcation diagram of P 0.01 (right). Figure 14 . 14The fourth example: κ(Z) (left), bifurcation diagram of P 0.01 (right). Figure 15 . 15The fifth example: κ(Z) (left), bifurcation diagram of P 0.01 (right). Figure 16 . 16Fixed point branches of P 0.01 (left); anchoring wedges W 0 (gray) and W 0 (black) for the Elowyn-Bonhomme maps (right).5.2.7. The Elowyn-Bonhomme map revisitedLet ξ, F 0 and G ξ be as in the original EB map (see p. 26) and consider the map P defined via (4.12), with F 0 (x, λ 1 ) = F 0 (x) Table 1 . 1EB map catastrophes along C 0 and C ξ .Curve Type x * s * Wedge Angle C 0 Fold (1.2483, −0.1286) ⊤ 5.9809 ϕ − 0 ≈ 0.3023 C 0 Fold (0.2269, −3.4760) ⊤ 0.2308 ϕ + 0 ≈ 0.2308 C ξ Fold (0.3371, 3.1473) ⊤ 1.1020 ϕ − ξ ≈ 0.4688 C ξ Fold (2.2769, 0.2982) ⊤ 2.1125 ϕ + ξ ≈ 0.5417 C ξ Fold (−3.2933, 6.1024) ⊤ 1.1581 n.a. C ξ Fold (5.6733, −1.2807) ⊤ 1.9267 n.a. C ξ Infinity n.a. 1.0172 n.a. C ξ Infinity n.a. 2.3562 n.a. Most of the analysis can be extended and adapted to the general case n ≥ 2, but at the cost of substantial algebraic complications. Linear stability analysis of rotating spiral waves in excitable media. D Barkley, Phys. Rev. Lett. 68Barkley D 1992 Linear stability analysis of rotating spiral waves in excitable media Phys. Rev. Lett. 68 2090-3 Euclidean symmetry and the dynamics of rotating spiral waves. D Barkley, Phys. Rev. Lett. 76Barkley D 1994 Euclidean symmetry and the dynamics of rotating spiral waves Phys. Rev. Lett. 76 164-7 Translational symmetry-breaking for spiral waves. V G Leblanc, C Wulff, J. Nonlin. Sc. 10LeBlanc V G and Wulff C 2000 Translational symmetry-breaking for spiral waves J. Nonlin. Sc. 10 569-601 Rotational symmetry-breaking for spiral waves. V Leblanc, Nonlinearity. 15LeBlanc V G 2002 Rotational symmetry-breaking for spiral waves Nonlinearity 15 1179-203 Transition from simple rotating chemical spirals to meandering and traveling spirals. G Li, Q Ouyang, V Petrov, H L Swinney, Phys. Rev. Lett. 77Li G, Ouyang Q, Petrov V and Swinney H L 1996 Transition from simple rotating chemical spirals to meandering and traveling spirals Phys. Rev. Lett. 77 2105-9 Spiral wave dynamics under pulsatory modulation of excitability. S Grill, V Zykov, S C Müller, J. Phys. Chem. 100Grill S, Zykov V S and Müller S C 1996 Spiral wave dynamics under pulsatory modulation of excitability J. Phys. Chem. 100 19 082-8 Spiral waves on circular and spherical domains of excitable medium. V S Zykov, S C Müller, Physica D. 97Zykov V S and Müller S C 1996 Spiral waves on circular and spherical domains of excitable medium Physica D 97 322-32 Attraction and repulsion of spiral waves by localized inhomogeneities in excitable media. A P Muñuzuri, V Pérez-Muñuzuri, Pérez-Villar , V , Phys. Rev. E. 58Muñuzuri A P, Pérez-Muñuzuri V and Pérez-Villar V 1998 Attraction and repulsion of spiral waves by localized inhomogeneities in excitable media Phys. Rev. E 58 R2689-92 The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle Arch. N Wiener, A Rosenblueth, Inst. Card. De Mexico. 16Wiener N and Rosenblueth A 1946 The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle Arch. Inst. Card. De Mexico 16 205-65 Interaction of rotating spiral waves with a boundary Biophys. Y A Yermakova, A M Pertsov, 31Yermakova Y A and Pertsov A M 1987 Interaction of rotating spiral waves with a boundary Biophys. 31 932-40 Stationary and drifting spiral waves of excitation in isolated cardiac muscle. J M Davidenko, A V Persov, R Salomonsz, W Baxter, J Jalife, Nature. 355Davidenko J M, Persov A V, Salomonsz R, Baxter W and Jalife J 1992 Stationary and drifting spiral waves of excitation in isolated cardiac muscle Nature 355 349-51 Frequency locking of meandering spiral waves in cardiac tissue. B Roth, Phys. Rev. E. 57Roth B J 1998 Frequency locking of meandering spiral waves in cardiac tissue Phys. Rev. E 57 R3735-8 F X Witkowski, L J Leon, P A Penkoske, W R Giles, M L Spanol, W Ditto, A T Winfree, Spatiotemporal evolution of ventricular fibrillation Nature. 392Witkowski F X, Leon L J, Penkoske P A, Giles W R, Spanol M L, Ditto W L and Winfree A T 1998 Spatiotemporal evolution of ventricular fibrillation Nature 392 78-82 Ventricular fibrillation: Mechanisms of initiation and maintenance. J Jalife, Annu. Rev. Physiol. 62Jalife J 2000 Ventricular fibrillation: Mechanisms of initiation and maintenance Annu. Rev. Physiol. 62 25-50 Making sense of a heart gone wild. D Mackenzie, Science. 303Mackenzie D 2004 Making sense of a heart gone wild Science 303 786-7 Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation. D Barkley, M Kness, L S Tuckerman, Phys. Rev. Lett. 42Barkley D, Kness M and Tuckerman L S 1990 Spiral-wave dynamics in a simple model of excitable media: The transition from simple to compound rotation Phys. Rev. Lett. 42 2489-92 A Winfree, Cardiac Electrophysiology, From Cell to Bedside. in Zipes and JalifePhiladelphia: Saunderssecond editionWinfree A T 1995 in Zipes and Jalife (eds.) Cardiac Electrophysiology, From Cell to Bedside second edition (Philadelphia: Saunders) pp. 379-89 . J Keener, J Sneyd, Mathematical Physiology IAM. SpringerKeener J and Sneyd J 1998 Mathematical Physiology IAM (New York: Springer) D Barkley, I G Kevrekedis, A dynamical system approach to spiral wave dynamics Chaos. 4Barkley D and Kevrekedis I G 1994 A dynamical system approach to spiral wave dynamics Chaos 4 453-60 Center manifold reduction for spiral waves. B Sandstede, A Scheel, C Wulff, C. R. Acad. Sci. 324Sandstede B, Scheel A and Wulff C 1997 Center manifold reduction for spiral waves C. R. Acad. Sci. 324 153-8 Dynamics of spiral waves on unbounded domains using center-manifold reductions. B Sandstede, A Scheel, C Wulff, J. Diff. Eq. 141Sandstede B, Scheel A and Wulff C 1997 Dynamics of spiral waves on unbounded domains using center-manifold reductions J. Diff. Eq. 141 122-49 Bifurcations and dynamics of spiral waves. B Sandstede, A Scheel, C Wulff, J. Nonlin. Sc. 9Sandstede B, Scheel A and Wulff C 1999 Bifurcations and dynamics of spiral waves J. Nonlin. Sc. 9 439-78 Dynamical behavior of patterns with euclidean symmetry in. B Sandstede, A Scheel, C Wulff, Pattern Formation in Continuous and Coupled Systems. Golubitsky, Luss and StrogatzBerlinSpringer-VerlagSandstede B, Scheel A and Wulff C 1999 Dynamical behavior of patterns with euclidean symmetry in Golubitsky, Luss and Strogatz (eds.) Pattern Formation in Continuous and Coupled Systems (Berlin: Springer-Verlag) Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders and drifts Doc. B Fiedler, B Sandstede, A Scheel, C Wulff, Math. 1Fiedler B, Sandstede B, Scheel A and Wulff C 1996 Bifurcation from relative equilibria of noncompact group actions: Skew products, meanders and drifts Doc. Math. 1 479-555 Approximate analytical solutions to the bidomain equations with unequal anisotropy ratios. B Roth, Phys. Rev. E. 55Roth B J 1997 Approximate analytical solutions to the bidomain equations with unequal anisotropy ratios Phys. Rev. E 55 1819-26 P Boily, Spiral waves and the dynamical system approach. Boily P [arXived] Spiral waves and the dynamical system approach Boily P [forthcoming] Epicyclic drifting in media with multiple inhomogeneities: a dynamical system approach. Boily P [forthcoming] Epicyclic drifting in media with multiple inhomogeneities: a dynamical system approach Boily, Spiral wave dynamics under combined translational and rotational symmetry-breaking: a dynamical system approach. Boily P [forthcoming] Spiral wave dynamics under combined translational and rotational symmetry-breaking: a dynamical system approach Bifurcation to spiral waves in reaction-diffusion systems SIAM. A Scheel, J. Math. Anal. 29Scheel A 1998 Bifurcation to spiral waves in reaction-diffusion systems SIAM J. Math. Anal. 29 1399-418 Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems Ph. C Wulff, Universität BerlinD. thesis FreieWulff C 1996 Theory of Meandering and Drifting Spiral Waves in Reaction-Diffusion Systems Ph.D. thesis Freie Universität Berlin . H Dym, H P Mckean, Fourier Series and Integrals. Academic PressDym H and McKean H P 1972 Fourier Series and Integrals (New York: Academic Press) P Boily, Spiral Wave Dynamics Under Full Euclidean Symmetry-Breaking: A Dynamical System Approach Ph. D. thesis University of OttawaBoily P 2006 Spiral Wave Dynamics Under Full Euclidean Symmetry-Breaking: A Dynamical System Approach Ph.D. thesis University of Ottawa M W Hircsh, S Smale, Differential Equations, Dynamical Systems, and Linear Algebra. San DiegoAcademic PressHircsh M W and Smale S 1974 Differential Equations, Dynamical Systems, and Linear Algebra (San Diego: Academic Press) Kuznetsov Yu, Elements of Applied Bifurcation Theory. BerlinSpringer-VerlagKuznetsov Yu 1995 Elements of Applied Bifurcation Theory (Berlin: Springer-Verlag) J Guckenheimer, P Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New YorkSpringerGuckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (New York: Springer) Introduction to Applied Non-Linear Dynamical Systems and Chaos. S Wiggins, SpringerNew YorkWiggins S 1990 Introduction to Applied Non-Linear Dynamical Systems and Chaos (New York: Springer) Numerical solutions of bifurcation and nonlinear eigenvalue problems in Applications of Bifurcation Theory pp. H Keller, Keller H B 1977 Numerical solutions of bifurcation and nonlinear eigenvalue problems in Applications of Bifurcation Theory pp. 359-84

The grand unified theories are theoretically well motivated, but they typically have less direct indications on the low energy physics and it is not easy to test them. Here, we discuss a scenario of them which naturally solves the so-called doublet-triplet splitting problem and, at the same time, generally predicts characteristic collider phenomenology. Then, we may get a hint on the breaking of the grand unified symmetry at the on-going and next-generation collider experiments.I. INTRODUCTIONSince a resonance consistent with the standard model (SM) Higgs field was reported [1], most people consider that the SM is now being confirmed, at least as an effective theory valid below the TeV scale. Then, the next question that we ask is what will come as the physics beyond the SM. The reason we ask it is that the SM still has some problems and puzzles, such as the hierarchy problem and the charge quantization problem. Here, we emphasize that the latter requires a tuning at least as fine as 10 −10 to explain why the hydrogen atom is (almost) neutral. Thus, if we mind the former problem (as often happens), the latter also should be taken care. A simple solution to the latter is to extend the gauge group to a semi-simple one. Looking back the history of the physics, which is that of the unification, it is reasonable to take the idea of the grand unification [2] seriously.Supposing the unification of the three forces of the SM, the matter fields are also to be unified. This unification works perfectly for the SM fermions: the five multiplets in each generation are unified into two in SU (5) grand unified theories (GUTs). This is not trivial at all actually, as it becomes clear when people try to unify the electroweak SU (2) × U (1) symmetry into SU (3). Thus, this success strongly supports the idea of the grand unification. It is also to be commented that the idea can easily accommodate other ideas to solve other SM problems: the supersymmetry (SUSY) for the hierarchy problem, the conventional seesaw mechanism [3] for the tiny neutrino masses and the Leptogenesis [4] for the baryon asymmetry of the universe. And if the R-parity is assumed, as usual in SUSY models, the candidate dark matter is supplied. In addition, in the minimal model, the three running gauge couplings become almost the same value at a superheavy scale, called the GUT scale: 2 × 10 16 GeV. This success of the gauge coupling unification (GCU) is so impressive that many people tend to believe the SUSY-GUTs. In this way, GUTs, especially SUSY-GUTs, potentially solve many of the problems/puzzles in the SM and give an amazing by-product.On the other hand, the grand unification fails to unify the Yukawa interactions and the Higgs field. The former is insisted as a consequence of the fermion unification and is not necessarily bad for the third generations but not good for the lighter fermions. This issue is often called the wrong GUT relation and is to be taken care in model building, while it is relatively easy to solve (see for example Ref.[5]). The latter requires a SU (5) partner of the SM doublet Higgs fields. The minimal choice is to introduce a color-triplet partner to embed them into the fundamental representation of the SU (5) group. In SUSY models[26], the triplet partner generates effective dimension five operators that contribute to the nucleon decay[6]. In order to make the proton lifetime long enough without tuning, the triplet partner should be much heavier than the GUT scale. It is not an easy task to realize naturally the mass splitting between such a superheavy triplet and the weak scale doublet which originate from a common (SU (5)) multiplet. This rather severe issue is called the doublet-triplet (DT) splitting problem and is one of the biggest problem in the SUSY-GUTs. When we consider the grand unification seriously, these issues have to be dealt with.Since the idea of the grand unification is so attractive, this problem has been attacked by many researchers for long time, and several solutions have been proposed[7][8][9][10][11]. They, however, all require some extension of the matter content, the grand unified gauge group and/or spacetime geometry. These extensions bring rather large ambiguity on the gauge couplings around the GUT scale due to the threshold corrections and so on. Then, what usually done are just to forget the ambiguity, to assume the corrections are aligned not to affect the GCU or, at most, to make models so that the GCU is kept. At this stage, the GCU is no longer a success but just a constraint in model building. Here, however, we would like to stress that the success of the GCU is just a by-product, and even without it the idea of the grand unification is attractive enough, as mentioned above.Next, let us discuss indications of the grand unification on the low energy physics that we can detect. The most famous one is the nucleon decay. It is actually impressive prediction, but the information that we would get will be rather little and thus it would be nice if there are some characteristic predictions on the collider

1304.7166

8a72ef567a8c08fa670f889c0ee9333e41b12ce7

Grand Unified Theories and Higgs Physics 2013. February, 2013 T Yamashita School of Medicine Aichi Medical University 480-1195NagakuteJAPAN Grand Unified Theories and Higgs Physics Toyama International Workshop on Higgs as a Probe of New Physics 2013. February, 2013arXiv:1304.7166v1 [hep-ph] 1 The grand unified theories are theoretically well motivated, but they typically have less direct indications on the low energy physics and it is not easy to test them. Here, we discuss a scenario of them which naturally solves the so-called doublet-triplet splitting problem and, at the same time, generally predicts characteristic collider phenomenology. Then, we may get a hint on the breaking of the grand unified symmetry at the on-going and next-generation collider experiments.I. INTRODUCTIONSince a resonance consistent with the standard model (SM) Higgs field was reported [1], most people consider that the SM is now being confirmed, at least as an effective theory valid below the TeV scale. Then, the next question that we ask is what will come as the physics beyond the SM. The reason we ask it is that the SM still has some problems and puzzles, such as the hierarchy problem and the charge quantization problem. Here, we emphasize that the latter requires a tuning at least as fine as 10 −10 to explain why the hydrogen atom is (almost) neutral. Thus, if we mind the former problem (as often happens), the latter also should be taken care. A simple solution to the latter is to extend the gauge group to a semi-simple one. Looking back the history of the physics, which is that of the unification, it is reasonable to take the idea of the grand unification [2] seriously.Supposing the unification of the three forces of the SM, the matter fields are also to be unified. This unification works perfectly for the SM fermions: the five multiplets in each generation are unified into two in SU (5) grand unified theories (GUTs). This is not trivial at all actually, as it becomes clear when people try to unify the electroweak SU (2) × U (1) symmetry into SU (3). Thus, this success strongly supports the idea of the grand unification. It is also to be commented that the idea can easily accommodate other ideas to solve other SM problems: the supersymmetry (SUSY) for the hierarchy problem, the conventional seesaw mechanism [3] for the tiny neutrino masses and the Leptogenesis [4] for the baryon asymmetry of the universe. And if the R-parity is assumed, as usual in SUSY models, the candidate dark matter is supplied. In addition, in the minimal model, the three running gauge couplings become almost the same value at a superheavy scale, called the GUT scale: 2 × 10 16 GeV. This success of the gauge coupling unification (GCU) is so impressive that many people tend to believe the SUSY-GUTs. In this way, GUTs, especially SUSY-GUTs, potentially solve many of the problems/puzzles in the SM and give an amazing by-product.On the other hand, the grand unification fails to unify the Yukawa interactions and the Higgs field. The former is insisted as a consequence of the fermion unification and is not necessarily bad for the third generations but not good for the lighter fermions. This issue is often called the wrong GUT relation and is to be taken care in model building, while it is relatively easy to solve (see for example Ref.[5]). The latter requires a SU (5) partner of the SM doublet Higgs fields. The minimal choice is to introduce a color-triplet partner to embed them into the fundamental representation of the SU (5) group. In SUSY models[26], the triplet partner generates effective dimension five operators that contribute to the nucleon decay[6]. In order to make the proton lifetime long enough without tuning, the triplet partner should be much heavier than the GUT scale. It is not an easy task to realize naturally the mass splitting between such a superheavy triplet and the weak scale doublet which originate from a common (SU (5)) multiplet. This rather severe issue is called the doublet-triplet (DT) splitting problem and is one of the biggest problem in the SUSY-GUTs. When we consider the grand unification seriously, these issues have to be dealt with.Since the idea of the grand unification is so attractive, this problem has been attacked by many researchers for long time, and several solutions have been proposed[7][8][9][10][11]. They, however, all require some extension of the matter content, the grand unified gauge group and/or spacetime geometry. These extensions bring rather large ambiguity on the gauge couplings around the GUT scale due to the threshold corrections and so on. Then, what usually done are just to forget the ambiguity, to assume the corrections are aligned not to affect the GCU or, at most, to make models so that the GCU is kept. At this stage, the GCU is no longer a success but just a constraint in model building. Here, however, we would like to stress that the success of the GCU is just a by-product, and even without it the idea of the grand unification is attractive enough, as mentioned above.Next, let us discuss indications of the grand unification on the low energy physics that we can detect. The most famous one is the nucleon decay. It is actually impressive prediction, but the information that we would get will be rather little and thus it would be nice if there are some characteristic predictions on the collider physics in addition. Unfortunately, since the GUT scale is so high, the decoupling theorem [12] makes it hopeless to detect the effects in most of the SUSY-GUTs. Here, we would like to introduce a scenario of the SUSY-GUTs where the DT splitting problem is naturally solved and an extraordinary collider phenomenology is generally predicted [13]. Interestingly, this scenario, in a sense, can be regarded as an effective field theoretical description of the SUSY-GUTs embedded into the heterotic string theory [14] which could treat the quantum gravity and explain the numbers of our spacetime [15] and of the generations [16]. It is quite exiting if we can get some informations on the GUT breaking which might indicate the string theory at the on-going and next-generation collider experiments. II. SUSY GRAND GAUGE-HIGGS UNIFICATION In this section we review the scenario proposed in Ref. [13]. In the scenario, the Hosotani mechanism [17], which works in higher dimensional gauge theories, is applied to break the GUT symmetry [18]. In the Hosotani mechanism, the symmetry breaking occurs by the extradimensional component of the gauge field which is a higher dimensional vector field. Thus, in this model, the Higgs field is unified with the gauge field, and it is often called the gauge-Higgs unification especially when it is applied to the electroweak symmetry breaking. In the present scenario, it is applied to the grand unified symmetry breaking [27] and named as grand gauge-Higgs unification [18]. An important point is that, in this case, the order parameter is not the extradimensional component itself which is valued on the algebra, but the Wilson loop which is valued on the group and thus free from the traceless condition. Because of it, interestingly, a kind of the so-called missing VEV [9] can be realized and the DT splitting problem is naturally solved even in SU (5) models [13]. In this way, the application of the Hosotani mechanism to the GUT breaking in SUSY-GUTs looks attractive. Naively thinking, the application to the GUT breaking seems reasonable since the Higgs field that is unified with the gauge field behaves as an adjoint field. Actually, at the first stage of the study of this mechanism, it was applied to the GUT breaking (or simpler toy model) [17,19]. Unfortunately, however, chiral fermions can not be accommodated in these models and thus these are phenomenologically less interesting. After the orbifold symmetry breaking [11] becomes famous among researchers who works on phenomenological model building, this mechanism have been applied mainly to the electroweak symmetry breaking [20]. This is because the orbifold symmetry breaking can extract fundamental-representational components (with respect to the remaining subgroup of the original gauge group) from the adjoint representation (with respect to the original group), besides chiral fermions from the higher-dimensional fermions. Furthermore, in such models, the Higgs field are free from the quadratically divergent radiative corrections to the mass term, thanks to the higher dimensional gauge symmetry [21]. In any case, now we know the orbifold symmetry breaking to realize chiral fermions, and thus it is interesting to examine the application to the GUT breaking. It might seem straightforward, but we immediately meet a difficulty. Namely, the adjoint scalar fields (with respect to the remaining gauge symmetry) originated from the extra-dimensional components tend to be projected out by the orbifold action when chiral fermions are realized. This difficulty is shared with the heterotic string theory [14] and, fortunately, a method, called diagonal embedding method [22], to evade the difficulty is known. In Ref. [18], it is pointed out that the same method can be applied in a field theoretical setup and thus we have an advantage that it is much easier to calculate the quantum corrections that tell us the positions of vacua. By this, the symmetry breaking pattern is controlled by the dynamics described by the field theory irrelevantly to the ultraviolet theory, in contrast to the orbifold breaking where it is chosen by hand. In addition, as mentioned above, the DT splitting problem, when the SUSY version is considered, is naturally solved. Thus, this scenario is theoretically well motivated. Interestingly, this scenario generically gives particular predictions also on the collider phenomenology. It is existence of light adjoint chiralmultiplets with masses of the SUSY-breaking scale. The reason is as follows. The adjoint Higgs field is a part of the gauge field and thus massless at the tree level. Since the symmetry that ensures the masslessness is broken by the compactification, the adjoint field gets mass corrections via the quantum corrections. As the radiative corrections would be vanishing when the SUSY was not broken, the mass corrections are proportional to the SUSY breaking scale. In SUSY models, there are the SUSY partners of the adjoint scalar which would be degenerate with the scalar if the SUSY is exact and thus again has masses of at most the SUSY-breaking scale. Then, the whole the adjoint chiralmultiplet is predicted to be light (if the SUSY-breaking scale is around the electroweak scale, as often expected) while the components of SU (5)/SU (3) × SU (2) × U (1) are eaten when the unified SU (5) gauge group is broken. Thus, color octet, weak triplet and singlet chiralmultiplets [28] will appear in the effective theory below the compactification scale (which is assumed to be around the GUT scale), and they may be observed in the on-going and next-generation collider experiments. An immediate consequence of the adjoint chiralmultiplets is that the GCU is disturbed. As mentioned above, however, the GCU should be treated just as a constraint instead of a success. It is easy to recover the GCU by adding further chiralmultiplets. An example which is easily realized in this scenario and we consider here is two vectorlike pairs of (1, 2) −1/2 , one of (¯3, 1) −2/3 and one of (1, 1) 1 , with which the GCU is realized at the GUT scale and the unified gauge coupling is remains in the perturbative region: α G ∼ 0.3. Since the strong interaction is no longer asymptotically free (irrelevantly to the choice of the additional fields to recover the GCU), the QCD corrections are enhanced and thus the colored particles tend to be rather heavy in this scenario. Although it is also interesting study to examine the extraordinary pattern of the mass spectrum of the colored particles for the hadron colliders, here we concentrate on the colorless fields: the singlet and the triplet. These additional fields couples to the two Higgs doublets of the minimal SUSY SM (MSSM). These couplings push the SM-like Higgs mass by the tree level F -term contribution and thus the rather heavy Higgs mass around 125GeV can be easily realized. In addition, they cause mixing between the MSSM doublet Higgs fields and the adjoint fields which result in modification of the coupling of the SM-like Higgs fields [24]. Such corrections may be measured at the linear collider. In the next section, we will discuss these issues in more detail. III. PHENOMENOLOGY In order to examine the colorless sector, it is convenient to consider an effective theory where the Higgs sector of the MSSM is extended with the singlet S and the triplet ∆. The superpotential, in this case, is given as W = µH u H d + µ ∆ tr(∆ 2 ) + µ S 2 S 2 + λ ∆ H u ∆H d + λ S SH u H d ,(1) where H u and H d are the MSSM doublet Higgs supermultiplets. Note that there are no self-couplings among S and ∆ although such couplings are allowed by the symmetry at the level of the effective theory. This is because S and ∆ originate from the gauge field. Furthermore, this fact also insists the two additional couplings λ ∆ and λ S to be related to the gauge couplings so that they are unified at the GUT scale (with appropriate group theoretical factors). Thus, this model is quite predictive (up to the dimensionful parameters). For instance, taking the above example of the additional chiralmultiplets to recover the GCU, the running of the gauge couplings are determined. The unified gauge coupling is used to fix the boundary values of λ ∆ and λ S at the GUT scale, and we get λ ∆ = 1.1, λ S = 0.26,(2) at the weak scale (within the 1-loop approximation). Using these predicted parameters, we can calculate the SM-like Higgs mass, the charged Higgs mass in therms of the CP-odd Higgs mass, deviations of the SM-like Higgs couplings from the corresponding SM values, and so on. Here, we just mention that the deviations are typically of order a few percents and thus can be tested at the linear collider. The details of the results will be shown in Refs. [24,25]. IV. SUMMARY In this article, we introduce the supersymmetric version [13] of the grand gauge-Higgs unification scenario [18] where the grand unified gauge symmetry is broken by the Hosotani mechanism [17]. Interestingly, in this scenario, the doublet-triplet splitting problem can be solved naturally even in SU (5) models [13], thanks to the phase nature of the Hosotani mechanism. In addition, it generally predicts the existence of light adjoint chiralmultiplets: the color octet, the weak triplet and the neutral singlet. Their mass is around the supersymmetrybreaking scale, which is often assumed to be the TeV scale, and thus there is a chance to detect them at the on-going and next-generation collider experiments. Due to the color octet chiralmultiplet, the QCD is no longer asymptotic free, and the QCD corrections are typically enhanced. This suggests that the additional colored particles become rather heavy. Thus, we concentrate on the colorless fields [24], though it is also an interesting work to examine the mass spectrum of the colored particles. Then, the effective theory of this scenario becomes the one with an extended Higgs sector: the neutral triplet and singlet are added. Since these are unified to the gauge field, they do not have self couplings and their couplings are related to the unified gauge coupling. This fact makes the model very predictive. For instance, we can calculate the SM-like Higgs mass, the charged Higgs mass in therms of the CP-odd Higgs mass, deviations of the SM-like Higgs couplings from the corresponding SM values, and so on, up to the ambiguity due to the dimensionful parameters. Although the details are referred to Refs. [24,25], we emphasize that the linear collider is expected since the deviations are typically of order a few percents which are in its reach. FIG. 1 : 1An example of running couplings. 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Note that the Higgs field that is unified with the gauge field is an adjoint Higgs field, in this case, and the SM doublet Higgs field is introduced as a matter field. In this scenario, models generally have a Z2 symmetry under which these adjoint multiplets change the sign. 13In this scenario, models generally have a Z2 symmetry under which these adjoint multiplets change the sign [13]. Thus, this scenario can give a theoretical background of (SUSY). inert models [23], though we consider a model where this Z2 symmetry is brokenThus, this scenario can give a theoretical background of (SUSY) inert models [23], though we consider a model where this Z2 symmetry is broken.

This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along nonorthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.1. generate U ∼ U ([0, 1]).2. Set V = k−U K and X = Φ −1 (V), with Φ the inverse of the cumulative normal distribution. 3. Generate Z ′ ∼ N (0, I d ) independent on U.4. Set vX + I − vv T Z ′ . We suggest to implement the last term in the last step as Z ′ − v(v · Z ′ ) which requires O(d) operation rather than O(d 2 ).

10.1142/s0219024911006772

1004.5037

d926cb8926dc652362761224e596cc2eb837eaa5

Convenient Multiple Directions of Stratification * 28 Apr 2010 Benjamin Jourdain Bernard Lapeyre Piergiacomo Sabino Convenient Multiple Directions of Stratification * 28 Apr 2010arXiv:1004.5037v1 [q-fin.CP]Monte Carlo methodsvariance reductionstratification methods This paper investigates the use of multiple directions of stratification as a variance reduction technique for Monte Carlo simulations of path-dependent options driven by Gaussian vectors. The precision of the method depends on the choice of the directions of stratification and the allocation rule within each strata. Several choices have been proposed but, even if they provide variance reduction, their implementation is computationally intensive and not applicable to realistic payoffs, in particular not to Asian options with barrier. Moreover, all these previously published methods employ orthogonal directions for multiple stratification. In this work we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a comparable variance reduction. In addition, we study the accuracy of optimal allocation in terms of variance reduction compared to the Latin Hypercube Sampling. We consider the directions obtained by the Linear Transformation and the Principal Component Analysis. We introduce a new procedure based on the Linear Approximation of the explained variance of the payoff using the law of total variance. In addition, we exhibit a novel algorithm that permits to correctly generate normal vectors stratified along nonorthogonal directions. Finally, we illustrate the efficiency of these algorithms in the computation of the price of different path-dependent options with and without barriers in the Black-Scholes and in the Cox-Ingersoll-Ross markets.1. generate U ∼ U ([0, 1]).2. Set V = k−U K and X = Φ −1 (V), with Φ the inverse of the cumulative normal distribution. 3. Generate Z ′ ∼ N (0, I d ) independent on U.4. Set vX + I − vv T Z ′ . We suggest to implement the last term in the last step as Z ′ − v(v · Z ′ ) which requires O(d) operation rather than O(d 2 ). Introduction The main purpose of Monte Carlo (MC) simulations is to compute integrals numerically. It is frequently the only alternative for solving problems in applied sciences and notably for financial applications. The pricing of derivative contracts and value-at-risk calculations for risk-management purposes typically require numerical simulations. However, the MC method for high-dimensional problems is a demanding computational task and a considerable number of studies have been devoted to increase its efficiency via variance reduction techniques. This paper investigates the use of multiple directions of stratification as a variance reduction technique for MC simulations of path-dependent options driven by high-dimensional Gaussian vectors. The precision of the method depends on the choice of the partitions of the space and the allocation of the number of samples within each strata. Usually, the strata are polyhedrons delimited by hyperplanes orthogonal to a few direction vectors. Several choices have been proposed: Glasserman et al. [8] select the directions for the stratification of linear projections based on the quadratic approximation of the integrand or payoff function. In contrast, Etoré et al. [4] find the directions by adaptive techniques. These two approaches provide a high variance reduction but their implementation can be computationally intensive and the former one cannot be applied to more realistic payoff functions such as Asian options with barrier at each time step. Moreover, these two methods suppose orthogonal directions for multiple stratification. In this work, we investigate the use of algorithms producing convenient directions, generally non-orthogonal, combining a lower computational cost with a variance reduction that is comparable to the above mentioned methods. In addition, we study the accuracy of optimal allocation, combined with the above stratification techniques, in terms of variance reduction, compared to "fixed" allocation procedures such as Latin Hypercube Sampling (LHS). We consider the directions produced by the Linear Transformation (LT) decomposition introduced by Imai and Tan [9] and the Principal Component Analysis (PCA). Moreover, we propose a new procedure based on the Linear Approximation (LA) of the "explained" variance of the payoff function by the use of the law of total variance. Notably, we design a novel algorithm that permits to correctly generate multivariate normal random vectors stratified along non-orthogonal directions. We illustrate the efficiency of the proposed algorithms and their combination for the computation of the price of different path-dependent options with and without barriers in the Black-Scholes (BS) and in the Cox-Ingersoll-Ross (CIR) models. In the former dynamics, it turns out that the LA and the LT approaches return the same first order direction while this vector is almost parallel to the one obtained by the GHS technique even in the case of Asian options with a barrier at expiry. This justifies the application and the good performance of the LA (and LT) if the barrier is at each monitoring time. Consequently, the approaches return the same variance reduction and the LA (LT) is easier to implement and has a lower computational cost. We repeat our numerical investigation in the CIR framework where we find explicit solutions for the LT and LA directions. In order to find a further direction, we compute the first principal component of the sampled covariance matrix of the price process obtained by a MC estimation via a pilot run. In both BS and CIR dynamics, LT and LA return remarkable variance reduction with a low computational cost. We also show that in some setting the stratification along multiple directions can be more efficient than stratifying along a single one. In particular, the combination of the LA (LT) direction and a non-orthogonal direction, notably the first principal component, can even outperform the variance reduction of two orthogonal directions in the case of barrier options. Finally, as far as the allocation of the samples is concerned, in any case the LHS displays a considerable higher computational time and has always a lower variance reduction as compared to the use of a convenient direction of stratification with optimal allocation. The paper is organized as follows. Section 2 reviews the main ideas of stratification and the motivations of this study. Section 3 presents the new algorithm that permits the stratification along non-orthogonal directions. Section 4 discusses the use of convenient stratification directions and in particular, contains the presentation of the LT decomposition and the introduction of the LA procedure. In Section 5 we explain the financial applications and find the explicit solutions for the LA and the LT methods both for the BS and the CIR dynamics. In Section 6 the variance reductions and the computational costs of the proposed technique are illustrated by numerical experiments. Finally, Section 7 concludes the paper by summarizing the most important findings. Stratified Sampling and Linear Projections Stratified Sampling is a general variance reduction technique that consists of drawing the observations from specific partitions of the sample space. More specifically, suppose we want to compute by MC simulations an expectation of the form E[g(Y)] where g : R d → R is a Borel function and Y is a R d -valued random vector with the assumption that E[g(Y) 2 ] < ∞. Consider a stratification variable X and let A 1 , . . . , A K be disjoint subsets of the real line for which P K k=1 {X ∈ A k } = 1. Then E[g(Y)] = K ∑ k=1 E[g(Y)|X ∈ A k ]P(X ∈ A k ) = K ∑ k=1 E[g(Y)|X ∈ A k ]p k(1) where p k = P(X ∈ A k ), k = 1, . . . , K. The stratified estimator with N S draws is defined as: K ∑ k=1 p k 1 n k n k ∑ j=1 g(Y kj ) = 1 N S K ∑ k=1 p k q k n k ∑ j=1 g(Y kj ),(2) where n k are the number allocations in the k-th stratum and q k = n k /N S is their fraction in the k-th stratum and Y kj are independent draws from the conditional distribution of Y given X ∈ A k . Its variance is given by ∑ K k=1 p 2 k σ 2 k n k where σ k is the conditional variance of g(Y) given X ∈ A k . This estimator may be more efficient than the usual MC sample mean estimator of a random sample of size N S . The potential higher efficiency of the former estimator critically depends on the allocation rule and the choice of the partition of the sample space. The optimal allocation rule is the one that minimizes the variance of the stratified sampling estimator given the partition of the state space and the constraint ∑ K k=1 q k = 1. It is given by: q k = p k σ k ∑ K k=1 p k σ k .(3) The probabilities p k are known whereas generally the conditional variances are not known. They can be estimated in a pilot run and then used in a second stage to determine the stratified estimator. This is not the optimal procedure and more sophisticated techniques can be employed, see for example Etoré and Jourdain [5]. We focus our attention on MC simulation driven by high-dimensional Gaussian vectors that are of particular interest in financial applications. As such, we consider in the following only normal random variables. Stratifying Linear Projections: 1-dimensional Setting We begin with a general description of stratifying a linear projection of a Gaussian random vector. Suppose Z is a d dimensional centered Gaussian random vector, Z ∼ N (0, Σ Z ) and then consider the stratification variable X as the linear projection of Z over a fixed direction v ∈ R d , X = v · Z. X is also Gaussian with variance v · Σ Z v. This choice permits to partition the sample space R d into strata defined by S k,v = x ∈ R d , x · v ∈ A k .(4) Due to the Gaussian structure of the random variables we can generate Z stratified along the direction v in the following way. Consider a general Gaussian random vector Y = (Y 1 , Y 2 ): Y = (Y 1 , Y 2 ) ∼ N µ 1 µ 2 , Σ 11 Σ 12 Σ 21 Σ 22(5) and denote L (Y 1 |Y 2 = x) the law of Y 1 given Y 2 = x, it is possible to prove (see for instance Glasserman [7]) that L(Y 1 | Y 2 = x) = N µ 1 + Σ 12 Σ −1 22 (x − µ 2 ) , Σ 11 − Σ 12 Σ −1 22 Σ 21 .(6) where we assume that Σ 22 is invertible. Adapting the above result for Z given X = v · Z and Var[X] = v · Σ Z v = 1 we have L (Z |X = x ) = N Σ Z v v · Σ Z v x, Σ Z − Σ Z vv T Σ Z v · Σ Z v = N Σ Z vx, Σ Z − Σ Z vv T Σ Z .(7) If we consider Σ Z = I d the above equation becomes: L (Z |X = x ) = N vx, I d − v T v .(8) The conditional covariance matrix D = I d − v T v does not depend on x and since D is an orthogonal projection matrix, we have DD T = D. Due to this result, we do not need to compute the Cholesky (or a general square-root) matrix of D to sample from the conditional distribution of Z given X. These observations give an easy and simple algorithm to generate K samples of Z stratified along the direction v. Suppose now that A k is the interval between the quantiles of order k−1 K and of order k K of the standard normal distribution. We can sample from Z given Z · v ∈ A k in the following steps: Stratifying Linear Projections: Multidimensional Setting We start with the case of orthogonal directions and consider a matrix V ∈ R d×d ′ , d ′ ≤ d, whose columns are the direction vectors, such that V T V = I d ′ . Following the notation introduced above we have: X = V T Z (9) where now X is d ′ dimensional. Moreover, Z X ∼ N 0 0 , Σ Z Σ Z V V T Σ Z V T Σ Z V(10) Consequently L (Z |X = x ) = N Σ Z V V T Σ Z V −1 x, Σ Z − Σ Z V V T Σ Z V −1 V T Σ Z(11) where we assume that V T Σ Z V is invertible. In the case Σ Z = I d we have L (Z |X = x ) = N V V T V −1 x, I d − V V T V −1 V T .(12) Hence, if we adopt orthogonal directions V T V = I d ′ the algorithm to stratify Z given X = V T Z is a simple multidimensional version of the algorithm illustrated before where now we should stratify the d ′ dimensional hypercube [0, 1] d ′ . Suppose, for example, that we stratify the j-th coordinate of the hypercube, j = 1, . . . , d ′ , into K j intervals of equal length so that we have a total number of K 1 × · · · × K d ′ equiprobable strata. In this multidimensional setting we can sample from Z given X = V T Z ∈ A k , where A k = ∏ d ′ j=1 Φ k j −1 K j , k j K j , in the following steps: 1. generate U = (U 1 . . . , U d ′ ) with independent components each of law U ([0, 1]). Set V j = k j −U j K j with j ∈ {1, . . . , d ′ } and k j ∈ {1, . . . , K j }. 3. Set X = (X 1 . . . , X d ′ ), X j = Φ −1 (V j ). 4. Generate Z ′ ∼ N (0, I d ) independent of U. 5. Set VX + I d − VV T Z ′ . We now investigate the possibility to stratify over different directions that can be non-orthogonal either. When the directions are not orthogonal the components of X are not independent since Var[X] = VV T = I d ′ and the previous multidimensional algorithm cannot be adopted anymore. A first way yo approach this problem may be to assume X L = C X ǫ with ǫ ∼ N (0, I d ′ ) independent on Z and C X ∈ R d ′ ×d ′ such that Var[X] = C X C T X , and use the following slight modification of the above algorithm. 1. generate U = (U 1 . . . , U d ′ ) with independent components each of law U ([0, 1]). 2. Set V j = k j −U j K j with j = {1, . . . , d ′ } and k j = {1, . . . , K j }. 3. Set ǫ = (ǫ 1 . . . , ǫ d ′ ), ǫ j = Φ −1 (V j ). 4. Generate Z ′ ∼ N (0, I d ) independent of U. 5. Set V(C X X ) −1 ǫ + (I d − V V T V −1 V T )Z ′ . However, although mathematically correct, this algorithm stratifies the marginals of the random vector ǫ that has independent components. This construction does not consider the fact that the marginals of X are not independent and the introduction of the dependence can affect this partial stratification in complicated ways (see Glasserman [7]). Stratification along non-orthogonal directions In this section we show how to generate multivariate normal random vectors, Z ∼ N (0, I d ), stratified along non-orthogonal directions. We prove the following proposition: 1 a − 2 −(e 1 ·e 2 )e 1 ·Z f ′ 2 ≤ Z · f 2 ≤ a + 2 −(e 1 ·e 2 )e 1 ·Z f ′ 2 . Based on the results of the Section 2 and the properties of the conditional expectation, the previous expression equals: CE g Z + f 1 Φ −1 Φ(a − 1 ) + U 1 Φ(a + 1 ) − Φ(a − 1 ) − f 1 · Z 1 1ã− 2 (U 1 )≤f 2 ·Z≤ã + 2 (U 1 )(16) where C = Φ(a + 1 ) − Φ(a − 1 ) P a − 1 ≤ Z · e 1 ≤ a + 1 a − 2 ≤ Z · e 2 ≤ a + 2 . The expected value is then: 1 0 E g Z + f 1 Φ −1 Φ(a − 1 ) + u 1 Φ(a + 1 ) − Φ(a − 1 ) − f 1 · Z 1 1ã− 2 (u 1 )≤f 2 ·Z≤ã + 2 (u 1 ) du 1 = 1 0 E g Z + f 1 Φ −1 Φ(a − 1 ) + u 1 Φ(a + 1 ) − Φ(a − 1 ) − f 1 · Z + f 2 Φ −1 Φ(ã − 2 (u 1 )) + U 2 Φ(ã + 2 (u 1 )) − Φ(ã − 2 (u 1 )) − f 2 · Z × Φ(ã + 2 (u 1 )) − Φ(ã − 2 (u 1 )) du 1 = E g Z + f 1 Φ −1 Φ(a − 1 ) + U 1 Φ(a + 1 ) − Φ(a − 1 ) − f 1 · Z + f 2 Φ −1 Φ(ã − 2 (U 1 )) + U 2 Φ(ã + 2 (U 1 )) − Φ(ã − 2 (U 1 )) − f 2 · Z × Φ(ã + 2 (U 1 )) − Φ(ã − 2 (u 1 ) ) . Rearranging the terms in Z we get equation (13) for d ′ = 2. The result for d ′ direction is obtained iterating the steps above. Convenient Directions Given an allocation rule, the crucial point in the stratification of linear projections is the choice of the directions of stratification. Indeed, stratified sampling eliminates the sampling variability across strata without affecting the sampling variability within strata. Good directions are characterized by their higher capacity to dissect the state space into strata where the integrand function is nearly constant. In the following we describe the approaches that we adopt in order to find the directions of stratification. Principal Component Directions Suppose we want to find the singled-factor approximation of a d-dimensional Gaussian random vector X ∼ N (0, Σ) that maximizes the variance of v · X. This is equivalent to the following optimization problem: arg max v =1 v · Σv(17) Suppose λ 1 ≥ · · · ≥ λ d represent the eigenvalues of Σ in increasing order, and e 1 , . . . , e d their associated eigenvectors, then the optimization above is solved by v * = e 1 an eigenvector associated to the largest eigenvalue λ 1 . As e 1 produces the linear combination e 1 · X that best captures the variability of the components of X. We may choose this vector as the first direction of stratification. In the case we would consider multiple stratification, we can iterate the optimization above. This means that we would consider e j , j = 1, . . . , d, associated to the j-th eigenvalue, as the j-th direction of stratification. Indeed, in the statistical literature, the linear combinations e j · X, j = 1, . . . , d, are called the principal components of X. The variance explained by the first k ≤ d principal components is the ratio: ∑ k i=1 λ i ∑ d i=1 λ i Finally, we note that this procedure based on the PCA only produces orthogonal directions. Law of Total Variance and GHS Directions In this section we illustrate the law of total variance and we briefly describe the strategy to select optimal directions illustrated in Glasserman et al. [8]. Given two random vectors X 1 and X 2 of dimension d 1 and d 2 , respectively, and a function g : 2 ] < ∞, the law of total variance reads as: R d 1 → R, if E[g(X)Var [g(X 1 )] = E [Var[g(X 1 )|X 2 ]] + Var[E[g(X 1 )|X 2 ]].(18) Usually, in the context of linear model, the two terms are known as the "unexplained" and the "explained" components of the variance, respectively. In our case, X 1 is a standard normal random vector Z and X 2 = v · Z where v ∈ R d . It is well known that stratification eliminates the "explained" component of the variance up to terms with order o(1/N S ), where N S is the total number of draws (see for instance Lemma 4.1 in Glasserman et al. [8]). Hence, a good direction candidate is the one that maximizes the "explained" component of the variance or minimizes the "unexplained" part. Such an optimal direction is then the solution of the following optimization problem: v * = arg min v∈R d , v =1 R d Var g(Z) v · Z = x p X (x)dx,(19) where p X is the density of X = v · Z. The approach proposed in Glasserman et al. [8] is to adopt directions that are optimal for the quadratic approximation of the logarithm of the integrand function. Glasserman et al. [8] considered g(z) = exp 1 2 z · Bz with B non-singular symmetric matrix whose eigenvalues λ 1 , . . . , λ d are all less than 1/2. Now number the eigenvalues and eigenvectors of the matrix B so that λ 1 1 − λ 1 2 ≥ λ 2 1 − λ 2 2 ≥ λ d 1 − λ d 2 . (20) Glasserman et al. [8] proved that the optimal direction v * is the eigenvector e 1 of the matrix B associated with the eigenvalue λ 1 . When one considers multiple stratification, the j-th optimal direction is the eigenvector e j associated with the eigenvalue λ j . Since the directions are the eigenvectors of the matrix B, the GHS approach only produces orthogonal directions. When the logarithm of the integrand function is not quadratic, one could evaluate its Hessian at the certain point. Glasserman et al. [8] proposed to calculate the Hessian at a point used for an importance sampling procedure. This last operation might be really computationally expansive, in particular if d is large. It depends on a non-convex optimization procedure and cannot always be easily applied to realistic situations arising in finance. In addition, in financial applications, payoff functions (integrand functions) are far to be quadratic. In contrast, Etoré et al. [4] found the directions by adaptive techniques that in some cases outperform the above approach. However, the numerical procedure still remains computationally intensive. These drawbacks motivate our study where our main purpose is to investigate convenient multiple stratification directions that provide comparable variance reductions with a notable advantage from the computational point of view. Linear Approximations In this section we describe a different approach, that we name Linear Approximation (LA), in order to find convenient directions for the stratification of linear projections. Suppose g ∈ C 1 , this approach is based on a linear approximation of the function g that leads to an approximation of the "unexplained" component of the variance. Then, we can approximate the optimization problem (19) as: R n ∇g(0) · Var Z Z · v = x ∇g(0)p X (x)dx,(21) where we also use the approximation ∇g( E[Z Z · v = x]) ≈ ∇g(E[Z]) , that is we evaluate the gradient at the expected value of Z (zero for each component) instead of its conditional one. The solution of the optimization problem (21) is given by the following proposition: Proposition 2. The optimal direction v * of the optimization problem (21) is: v * = ± ∇g(0) ∇g(0) (22) Proof. Developing equation (21) we get: R d ∇g(0) · Var Z X = x ∇g(0)p X (x)dx = R d ∇g(0) · (I − v T v)∇g(0)p X (x)dx = ∇g(0) 2 − ∇g(0) · v T v∇g(0).(23) The minimization problem is equivalent to maximize the second term that can be written as (∇g(0) · v) 2 . The maximum of this dot product is attained when the two vectors are parallel. The optimal direction is then obtained by normalization. Multiple directions in the LA procedure can be produced calculating the gradient at different points. For example, we might iteratively consider Z 2 = ∇g (∇g(0)) , . . . , Z d ′ = ∇g (∇g(Z d ′ −1 ) ) in order to capture higher order components. We remark that the LA approach does provide non-orthogonal directions. Linear Transformations The LT procedure, proposed by Imai and Tan [9], is originally conceived to enhance the accuracy of simulation techniques that employ low-discrepancy sequences also known as Quasi-Monte Carlo (QMC) methods. Indeed, given Z ∼ N (0, I d ), the variance of the MC estimation of the expected value E[g(Z)] does not change if we replace Z by Aǫ where ǫ ∼ N (0, I d ) and A is a d × d orthogonal matrix, AA T = I d , while the choice of A can deeply affect the accuracy of QMC simulations (see for instance Papageorgiou [14]). The Imai and Tan's choice is such that A minimizes the effective dimension in the truncation sense defined in Caflisch et al. [3] of the integrand function. In our context, the columns of A will be chosen as the orthogonal directions of stratification. We briefly describe the LT algorithm. Consider a d dimensional normal random vector T ∼ N (µ; Σ), a vector w = (w 1 , . . . , w d ) ∈ R d and let f (T) = ∑ d i=1 w i T i be a linear combination of T. Let C be such that Σ = CC T and assume ǫ ∼ N (0, I d ) with T L = Cǫ. The LT approach considers C as C = C LT = C CH A, with C CH the Cholesky decomposition of Σ. Then, in the linear case, we can define: g A (ǫ) := f (C CH Aǫ) = d ∑ k=1 α k ǫ k + µ · w,(24) where α k = C LT ·k · w = A ·k · B, k = 1 . . . , d and B = (C CH ) T w while C ·k and A ·k are the k-th columns of the matrix C and A, respectively. In the linear case, setting A * ·1 = ± B B ,(25) with arbitrary remaining columns with the only constrain that AA T = I d , leads to the following expression: g A (ǫ) = µ · w ± B ǫ 1 .(26) This is equivalent to reduce the effective dimension in the truncation sense to 1 and this means to maximize the variance of the first component ǫ 1 . In a non-linear framework, we can use the LT construction, which relies on the first order Taylor expansion of g A : g A (ǫ) ≈ g A (ǫ) + d ∑ l=1 ∂g A (ǫ) ∂ǫ l ∆ǫ l .(27) The approximated function is linear in the standard normal random vector ∆ǫ ∼ N (0, I d ) and we can rely on the considerations above. The first column of the matrix A * is then: A ·1 * = arg max A ·1 ∈R d ∂g A (ǫ) ∂ǫ 1 2 (28) Since we have already maximized the variance contribution for ∂g A (ǫ) ∂ǫ 1 2 , in order to improve the method using adequate columns we might consider the expansion of g about d − 1 different points. More precisely Imai and Tan [9] propose to maximize: (25) provides an easy solution at each step, the correct procedure requires that the column vector A ·k * is orthogonal to all the previous (and future) columns. Imai and Tan [9] propose to chooseǫ =ǫ 1 = E[ǫ] = 0,ǫ 2 = (1, 0, . . . , 0), . . .ǫ k = (1, 1, 1, . . . , 0, . . . , 0), where the k-th point has k − 1 leading ones. Sabino [16] illustrated an economic and convenient implementation of the LT algorithm by an iterative QR decomposition that we will use to find the directions of stratification. This method is computationally more expensive than the LA and it is not clear if it admits a solution when the sequence of expansion points is different from the one described above. A ·k * = arg max A ·k ∈R d ∂g A (ǫ k ) ∂ǫ k 2 (29) subject to A ·k * = 1 and A ·j * · A ·k * = 0, j = 1, . . . , k − 1, k ≤ d. Although equation Financial Applications In this section we illustrate how to calculate the convenient directions introduced above in the context of option pricing. We consider two price-dynamics: • BS dynamics for M risky assets with constant volatilities: dS i (t) = rS i (t) dt + σ i S i (t) dW i (t) , S i (0) = S i0 , i = 1, . . . , M,(30) S i (t) denotes the i-th asset price at time t, σ i represents the volatility of the i-th asset return, r is the risk-free rate, and W (t) = (W 1 (t) , . . . , W M (t)) is a M-dimensional Brownian motion such that dW i (t)dW k (t) = ρ ik dt, i, k = 1, . . . , M. When M = 1 we simply denote S(t) = S 1 (t). • CIR dynamics: dS(t) = α (µ − S(t)) dt + σ S(t)dW(t), S(0) = S 0 ,(31) with S 0 , α, µ, σ positive constants. We impose the condition 2αµ > σ 2 in order to ensure that S(t) remains positive. Applying the risk-neutral pricing formula (see Lamberton and Lapeyre [12]), the calculation of the price at time t of any European derivative contract with maturity date T boils down to the evaluation of an (discounted) expectation: a(t) = exp (−r(T − t)) E [ψ| F t ],(32) the expectation is under the risk-neutral probability measure and ψ is a generic F T -measurable variable that determines the payoff of the contract. We show how to derive the convenient directions of stratification for the following derivative contracts: 1. discretely monitored Asian basket options: a (t) = exp (−r(T − t)) E   M ∑ i=1 N ∑ j=1 w ij S i t j − K S + F t   Option on a Basket(33) where x + = max(x, 0), t 1 < t 2 · · · < t N = T is a time grid, the coefficients w ij satisfy ∑ i,j w ij = 1 and K S is the strike price. When N = 1 and M > 0 the option is known as basket option while if M = 1 and N > 0 it is simply known as Asian option. 2. Asian option with knock-out barrier at expiry T: a (t) = exp (−r(T − t)) E   1 N N ∑ j=1 S t j − K S + 1 1 S(T)<B F t  (34) where B represents the value of the barrier. 3. Asian option with knock-out barrier at each monitoring time: a (t) = exp (−r(T − t)) E   1 N N ∑ j=1 S t j − K S + 1 1 S(t j )<B ,∀j=1,...,N F t  (35) where B represents the value of the barrier. Linear Transformation in the Black-Scholes Market Suppose the BS dynamics with constant volatilities and a time grid t 1 < t 2 · · · < t N = T, the elements of the autocorrelation matrix Σ B of the Brownian motion are (Σ B ) jn = min(t j , t n ), j, n = 1, . . . , N. Moreover, denote Σ A the a covariance matrix whose elements are ( Σ A ) im = σ i ρ im σ m , i, m = 1, . . . , M, and consider Σ MN = Σ B ⊗ Σ A where ⊗ denotes the Kronecker product. Given ǫ ∼ N(0, I MN ) and C LT = C CH A such that C CH (C CH ) T = Σ MN and AA T = I MN , the payoff of an Asian basket option can written as: ψ = (g(ǫ) − K S ) + where g(ǫ) = MN ∑ k=1 exp µ k + MN ∑ l=1 C LT kl ǫ l(36) and µ k = ln(w k 1 k 2 S k 1 (0)) + r − σ 2 k 1 2 t k 2(37) where the indexes k 1 and k 2 are k 1 = (k − 1)moduloM + 1, k 2 = ⌊(k − 1)/M⌋ + 1, respectively and ⌊x⌋ denotes the greatest integer less than or equal to x. Since the Asian payoff function is not everywhere differentiable, the LT procedure is applied to its differentiable part g (or g − K S ). This is done also for the other barrier-style Asian options, hence we obtain the same directions of stratification for the three types of derivative contracts. Hereafter we detail the adopted procedure: 1. Expand g up to the first order: g(ǫ) ∼ = g(ǫ) + N M ∑ l=1 N M ∑ i=1 exp µ i + N M ∑ k=1 C LT ikǫk C LT il ∆ǫ l(38) 2. Forǫ = 0 find the first column of the optimal matrix A: g(ǫ) ∼ = g(0) + N M ∑ l=1 N M ∑ i=1 exp (µ i ) C LT il ∆ǫ l(39) Set α l = ∑ N M i=1 exp (µ i ) C LT il = ∑ N M m=1 ∑ N M i=1 exp (µ i ) C CH im A ml and set u (1) = (e µ 1 , . . . , e µ MN ) T and B (1) = (C CH ) T u (1) then the first column is A * ·1 = ± B (1) B (1) .(40) 3. The p-th optimal column is found considering the p-th expansion point of the strategy. This results in: g(ǫ) ∼ = g(ǫ p ) + N M ∑ l=1 N M ∑ i=1 exp µ i + p−1 ∑ k=1 C * ik C LT il ∆ǫ l(41) where C * ik = (C CH A * k ) i , k < p have been already found at the p − 1 previous steps and A ·p * must be orthogonal to all the other columns. Also define u (p) = exp µ 1 + ∑ p−1 k=1 C * 1k , . . . , exp µ MN + ∑ p−1 k=1 C * MNk T and B (p) = (C CH ) T u (p) , then the solution is A * ·p = ± B (p) B (p) .(42) We remark that at each time step all the columns must be orthogonalized (see Sabino [15,16]) Linear Transformation in the CIR Market We extend the procedure described in the previous section with the assumption of a CIR dynamics. Consider an equally spaced time-grid whose time step is denoted by ∆t, the Euler scheme of the CIR dynamic is: S j = S j−1 + α µ − S j−1 ∆t + σ S j−1 ∆t Z j , j = 1, . . . N,(43) where Z is a Gaussian vector of N independent standard random variables. The Asian payoff is: ψ = (h(Z) − K S ) + with h(Z) = 1 N N ∑ j=1 S j (Z).(44) As done in the BS setting, we find the LT-based convenient directions of stratification applying the LT technique to the differentiable part of the payoff function of an Asian option (in this dynamics we only consider options on a single asset). This is done also for the other barrier-style Asian options, so that we have the same directions of stratification for the three types derivative contracts. Applying the LT decomposition the Euler scheme becomes S j = S j−1 + α µ − S j−1 ∆t + σ S j−1 ∆t N ∑ m=1 A jm ǫ m , j = 1, . . . N,(45) the computation of the first direction of LT decomposition consists in the following steps: 1. Compute the partial derivatives ∂S j ∂ǫ 1 , j = 1, . . . , N: ∂S j (0) ∂ǫ 1 = 1 − α∆t + σ 2 ∆t S j−1 N ∑ m=1 A jm ǫ m ∂S j−1 ∂ǫ 1 + σ ∆tS j−1 A j1 ǫ=0 .(46) Now denote p (1) j = ∂S j (0) ∂ǫ 1 , α(1)j−1 = 1 − α∆t + σ 2 ∆t S j−1 ∑ N m=1 A jm ǫ m ǫ=0 and β (1) j−1 = σ ∆tS j−1 (0), we have p (1) j = p (1) j−1 α (1) j−1 + β (1) j−1 A j1 .(47)p (1) j = j ∑ m=1 w (1) m (j)A m1 , j = 1, . . . , N,(48) where the components of vector w (1) (j), that depends on j, are: w (1) m (j) = β (1) m−1 j−1 ∏ i=m α (1) i .(49) The superscripts indicate the number of the direction under consideration and the proof can be obtained by iteration. Remark 3. Note that w (1) j (j) = β (1) j−1 with the assumption that ∏ i∈∅ α (1) i = 1 and w (1) m (j + 1) = α (1) j w (1) m (j), ∀j, m. 2. Denoteh(ǫ) = h(Z) = h(Aǫ) then ∂h(0) ∂ǫ 1 = 1 N N ∑ j=1 p (1) j .(50)Corollary 1. ∂h ∂ǫ l ǫ 1 =0 in equation (50) is a linear combination of the rows of A: N ∑ j=1 p (1) j = N ∑ j=1 t (1) j A j1 , ∀N ∈ N,(51) where t (1) j = β (1) j−1 1 + N−1 ∑ l=j l ∏ i=j α (1) i .(52) As for Proposition 3, the proof can be obtained by iteration. Remark 4. t (1) N = β (1) N−1 = w (1) N (N). 3. The first optimal direction is established by the following theorem. Theorem 1. The first column of the matrix A, solution of the LT optimization problem, in the case of Asian options assuming the Euler discretization of the CIR model is: A ·l * = t (1) t (1) ,(53) with t being the vector defined in Corollary 1. Proof. Knowing that the scalar product t (1) · A ·1 attains the maximum when the two vectors are parallel, we can conclude that the optimal A * ·1 is proportional to t (1) . After normalization the optimum solution is given by equation (53). Remark 5. We observe that, if Z = 0, after some algebra, the Euler discretization is simply S j − µ = (1 − α∆t) S j−1 − µ (54) then S j = (1 − α∆t) j (S 0 − µ) + µ(55) We use the results of this remark to simplify the computational cost to find the first direction of stratification. 4. In order to compute the remaining optimal columns we need to repeat the procedure illustrated in steps 1 to step 3. As far as the calculation of the l-th column is concerned, one needs to evaluate ∂S j (ǫ l ) ∂ǫ l and accordingly the quantities p (l) j , α (l) j , β(l) j , ∀j, and the components of the vectors w (l) and t (l) . All the results in Proposition 3, Corollary 1 and Theorem 1 remain valid while now considering the quantities with superscripts l. The orthogonal directions LT are then obtained by orthogonalization. Linear Approximation in the Black-Scholes Market Hereafter we describe how to find the directions of the LA technique in the case of a BS dynamics. Since the payoff function is not differentiable, as for the LT method we consider only the differentiable part g − K S . The gradient has components: ∂g(ǫ) ∂ǫ m = MN ∑ k=1 C km exp µ k + MN ∑ l=1 C kl ǫ l , then, ∇g(0) =    ∑ MN k=1 C k1 e µ 1 . . . ∑ MN k=1 C kMN e µ MN    and in general ∇g(f fl) =    ∑ MN k=1 C k1 e µ 1 +ǫ 1 . . . ∑ MN k=1 C kMN e µ MN +ǫ MN    . (56) In the above derivation we assume that C = C CH since we do not need to introduce any orthogonal matrix and the Cholesky decomposition of the autocorrelation matrix of a Brownian motion is explicitly known. It turns out that the LT and the LA methods return the same first order direction. Nevertheless, the latter approach can produce different directions changing the value at which the gradient is calculated. In contrast, the LT procedure admits solution only assuming the starting points strategy described above. Hence, the LA is more flexible and in particular the new algorithm does not require an incremental QR decomposition to find the new directions. Indeed, if we would look for orthogonal directions a unique orthogonalization would be required; consequently, the LA computational cost is much lower. Moreover, the mathematical derivation is simpler. Linear Approximation in the CIR Market We now illustrate how to apply the new LT approach for the derivative contracts above in CIR dynamics. Consider the Euler discretization scheme in equation (43) and compute the following partial derivatives for j, l = 1, . . . , N: ∂S j ∂Z l = 1 − α∆t + σ 2 ∆t S j−1 Z j ∂S j−1 ∂Z l + σ ∆tS j−1 δ jl , then ∂S j (0) ∂Z l = (1 − α∆t) ∂S j−1 (0) ∂Z l + σ ∆tS j−1 (0)δ jl ,(57) and the gradient is ∇S j (0) =                  (1 − α∆t) j−1 σ √ ∆tS 0 (1 − α∆t) j−2 σ ∆tS 1 (0) . . . (1 − α∆t) σ ∆tS j−2 (0) σ ∆tS j−1 (0) 0 . . . 0                  .(58) Due to Proposition 2, the LA first optimal direction is given by the normalized sum of ∇S j (0), j = 1, . . . , N. Further directions are obtained by iterating this procedure with a starting points rule. Alternatively, we can choose the evaluation points as in the LT strategy or the components of the l-th direction for the starting point of the gradient for the l + 1-direction. Numerical Illustrations We now illustrate the results developed in the previous sections through examples and numerical experiments. As mentioned before, we consider the BS and the CIR dynamics and different exotic path-dependent options. All the numerical procedures have been implemented in MATLAB on a computer with Intel Pentium M, 1.60 GHz, 1 GB RAM. In the numerical illustrations we consider K = 1000 strata and N S = 2 × 10 6 total number of scenarios so that for orthogonal directions we have a constant allocation rule (which, in this case, coincides the proportional rule as the strata are equiprobable) with 2000 random draws in each stratum (const in the tables). When we consider non-orthogonal directions the constant allocation rule is not proportional anymore since the strata are not equiprobable. For the optimal allocation rule (opt), the standard deviations have been computed by a first pilot run and then they have been used in a second stage to determine the stratified estimator. We report the estimated variances and the total computational times with constant and optimal allocation. We compare the variances employing the directions of stratification returned by GHS (see Glasserman et al. [8]), LT, LA, the PCA and their combination. Note that the GHS procedure requires the calculation of an importance sampling direction that is a computationally demanding task. In our experiments we report the variances due to the stratification only in order to compare the relative efficiency of the pure stratification methods. As far as the PCA directions are concerned, they consist of the eigenvectors associated to the highest eigenvalues of the autocorrelation matrix of the multi-dimensional Brownian motion that drives the BS dynamics. In contrast, since the CIR dynamics is not Gaussian, in a first pilot run with a 2000-sample we compute the MC estimation of the autocorrelation matrix of the price dynamics and then calculate its eigenvectors and values. We employ a Euler scheme that always takes the positive value of the square-root term because it was shown that this exhibits the smallest discretization bias among Euler CIR-discretizations (see Andersen [2]). Even if this dynamics is not normal, the i-th step price, given the i − 1-th one, is normal and this can justify the use of the PCA in the CIR dynamics. We consider the multiple combination of two directions of stratification. Our algorithm and considerations are also applicable to additional directions but, due to the so called curse of dimensionality, this would require a higher number of strata and hence a higher number of total samples that would considerably increase the computational burden. Finally, we compare these stratified estimators to LHS-based estimators (see Owen [13] or Stein [17] for more on this topic). Stein [17] proved that LHS eliminates the variance of the additive part of the integrand (payoff) function and hence produces an important variance reduction when coupled with LA or LT. Unfortunately, it is difficult to numerically compute the asymptotic variance in the central limit theorem for the LHS estimator. LHS is characterized by a fixed multiple allocation rule that has a high computational cost. Our purpose is to compare this very high-dimensional allocation rule to one with a lower dimension where we can adopt optimal allocation. In addition, the ex- pectation of interest E[ψ(Z)] is equal to E[ψ(OZ)] where O is a general orthogonal matrix. In a standard MC simulation the variance of the two estimators does not depend on O but in contrast, the accuracy of LHS-based estimators critically depend on the choice of O. Our simulations adopt the orthogonal matrix produced by the LT decomposition that has been shown to be an efficient choice (see Sabino [16]). Asian Options in the Black-Scholes Market Our first example is the pricing of arithmetic Asian options on a single risky security defined by equation (33) with M = 1. For simplicity we assume that the time grid is regular with time steps t i = i∆t, i = 1, . . . , N. This permits a simple derivation of the PCA and the Cholesky decomposition of the autocorrelation matrix of the Brownian motion (see Åkesson and Lehoczky [1]). Table 1(a) reports the input parameters used in the simulation with different moneyness of the options. We remind that in this setting LT and LA provide the same first order direction. Tables 5-7 report the numerical results obtained and the total computational times: all the procedures return unbiased estimates of the option prices while giving remarkably different variances. All the stratified techniques give a variance reduction that is particularly remarkable with the GHS and the LA (LT) methods. The PCA orthogonal directions (one dimensional and two dimensional) give a modest effect also taking into account the computational times. The main observation is that GHS and LA (LT) show the same computational cost and the same variance reduction. Both LA and GHS give a remarkable variance reduction, of a factor of more than 100 in the case of constant allocation and of several hundreds in the case of optimal allocation. However, given the parameters in Table 1(a), we stress the fact that the computational time required for the calculation of the direction is really a small part of the total time requested for all the proposed procedures. In contrast, with a really high problem dimension (i.e. a dimension 1000 typical in financial applications), the solution of the GHS optimization problem becomes a hard task depending on the starting guess and its computational burden has a relevant influence. In contrast, the LA (LT) algorithm consists in a simple vector O(N) calculation that is feasible even in high-dimensional problems. Table 2(a) reports the angles (in degrees) between the discussed directions. The GHS and LA directions are almost parallel meaning that the GHS algorithm is not so sensitive to the moneyness and this justifies the equal performance in terms of variance reduction of the LA method. As mentioned before, the PCA direction does not furnish a relevant variance direction and hence the non-orthogonal 2-dimensional stratification that employs such a direction always returns a lower accuracy than the GHS or LA methods. Moreover, the orthogonal GHS or LA bi-dimensional stratifications give variance reductions that are about 4 times lower than the corresponding one-dimensional ones. We remind that the two settings have the same number of strata so that we can conclude that the second order direction has a lower impact on the variance reduction and, with these directions of stratification, it is more efficient to employ a stratified MC estimator with a single direction. We conclude the study for the simple Asian options with the comparison between the accuracies of the LHS and the stratified sampling with a single direction with optimal allocation. The results shown in Table 5 illustrate that the LHS never outperforms the optimal allocation. Indeed, the LHS-based variance is at least two times the variance obtained with the stratified estimator with optimal allocation. Moreover, the computational cost is a lot higher, almost twice as high as the times needed for the optimal allocation. All these arguments strongly favor the use of convenient directions with optimal allocation. We modify the Asian option example by adding a knock-out barrier at expiration T or at each sampling date so that the option pays nothing if the asset price is above the barrier. Due to the discontinuous payoff of barrier options, the GHS optimization problem is a demanding task especially when the barrier is at each time step (indeed Glasserman et al. [8] did not elaborate this possibility). In contrast, the LA (LT) focuses only the continuous part of the payoff function. Table 1(b) reports the input parameters used in the simulation with different moneyness and barriers. The values of the barriers should be larger than the strike prices but not too high otherwise the pricing problem would almost boil down into the case without barrier. Also for barrier options (barrier at expiry), we notice that GHS and LA give directions of stratification that are almost parallel as illustrated in Table 2(b). This justifies the approximation of the LA method and its use for stratified MC to price the two types of barrier options. In addition, the GHS algorithm is not applicable to Asian options with a complete barrier. Different approaches should be employed in order to improve the stratification efficiency for barrier-style options, as suggested in Etoré et al. [4], but these are nevertheless computationally expensive and use orthogonal directions. The stratified MC does not return variances as low as for plain Asian options, especially when the barrier is close to the strike price. For example, the case of Asian options with barrier B = 80 (both at expiry and at each sampling date) and with strike K S = 55 displays a variance reduction of several hundreds with a computational time that ranges between 22% and 55% higher than the standard MC. However, when the barrier and the strike price are K S = 50 and B = 60, respectively, the variance reduction is lower with an extra effort ranging from 22% and 50% with respect to the standard MC. The numerical simulation of the prices of Asian basket options with a barrier close to the strike price, both at expiry and at all the monitoring times, shows that stratifying along multiple directions can be worthwhile. Indeed, if K S = 50 and B = 60, the multiple stratification enhances the accuracy of the estimation compared to the use of a single direction. In particular, the highest variance reduction is achieved with the choice of non-orthogonal directions (LA-PCA) with optimal allocation. In this setting the variance reduction is of an order 100, with barrier at expiry, or 40, with barrier at each monitoring time, and is several times higher compared to the other setting of stratification. Finally, even for Asian barrier options the LHS never outperforms the technique that displays the smallest variance with optimal allocation. These considerations suggest that the use of multiple non-orthogonal directions can be worthwhile. However, finding many different multiple directions is not a simple task. Basket Options in the Black-Scholes Market In this example the stratification estimator once more improves the accuracy of the standard MC method. Indeed, in the BS market, the financial features of basket options are almost the same as those of arithmetic Asian options. The main difference between the two is that for Asian options the Gaussian variables are correlated by the autocovariance matrix of a single Brownian motion while for basket options the dependence is measured by the covariance matrix among the asset returns. In addition, both payoffs contain a (weighted) average of the exponential of a Gaussian random vector. Table 8 shows that for all the considered exercise prices, the stratification using the LA (LT) with and without optimal allocation has a remarkable variance reduction comparable to the one given by the GHS algorithm with the same computational considerations as in the Asian option example. Indeed, these two directions are almost parallel (see Table 3(b)). The PCA-based direction has again a modest effect in terms of variance reduction and the stratification over a single linear projection produces a better accuracy than the one that exploits two directions. Finally, the LHS estimator neither achieves a higher variance reduction than the stratified estimator with a single LA direction with optimal allocation nor does it require a lower computational effort. Asian Options in the CIR Market As a last example we consider arithmetic Asian options on a single asset in a CIR dynamics whose depicted parameters (in Table 4(a)) are chosen in order to ensure positive prices (2αµ > σ 2 ). In this setting the LA method and the LT decomposition do not provide the same stratification direction and the GHS algorithm is really difficult to apply. However, as illustrated in Table 4 the directions returned by the LT and LA are almost parallel. In any case the derivation of the LA solution and its implementation are much easier. Since the CIR model is neither a Gaussian nor a lognormal process, the PCA decomposition is not applicable. However, in order to obtain a further direction we estimate a PCA-like direction as explained at the beginning of this section. Tables 9-11 show that both the LA and LT algorithms give remarkable variance reductions. The best accuracies are obtained with the stratification along a single direction which attains a reduction of an order of several hundreds, both with a constant and optimal allocation rule. The extra cost for the computational time is only 20%. As in the BS setting, the PCA approach is less efficient and requires a higher computational cost due to calculation of the sampled autocovariance matrix of the price process. Also in this situation the solution employing two orthogonal or non-orthogonal directions provides a variance reduction. Unfortunately, this choice never provides an accuracy as precise as the one obtained by a single direction. Moreover, the use of the fixed LHS-allocation rule never enhances the accuracy of the simulation more than the best low-dimensional stratification method with optimal allocation. As in the BS example, we add a knock-out barrier at expiry or at each monitoring time. For this latter option we must chose a barrier level that is much higher than the strike price. Indeed, due to the high variability of the CIR dynamics, with a low barrier value the option would easily knock-out producing a zero-valued price. As already mentioned, in the example of barrier options we adopt the same convenient directions of stratification that we would consider without the barrier since the LA and LT approaches do not take into account the non-differentiable part of the payoff. Tables 6 and 7 illustrate the results of this numerical investigation. The variance reduction is not as efficient as the case without barrier but in contrast, the use of multiple directions improves the efficiency of the simulation without highly influencing the computational cost. In addition, the combination of non-orthogonal directions can achieve a better variance reduction. Indeed, the combination of LA-PCA directions (LT and LA are almost parallel) returns a variance that ranges from 10 to 30 times lower than that with standard Monte Carlo. Moreover, this estimated variance is always at least equal, for K S = 100, B = 170 with barrier at each monitoring time, or lower than the variance obtained with different combinations of stratifying directions and barrier levels. Finally, as in all examples, the LHS sampling coupled with LT does not provide a convenient alternative to stratification over few directions with optimal allocation. Concluding Remarks and Future Perspectives In this paper we have investigated the use of convenient multidimensional directions of stratification in order to enhance the accuracy of Monte Carlo methods. We have discussed directions of stratification that are easy to derive and display variance reductions that are comparable to those introduced by Glasserman et al. [8]. These solutions do not require a complex calculation and can be applied in really high-dimensional problems without an extra cost. In contrast, the use of the Glasserman et al. [8] or Etoré et al. [4] methods risk to be computationally unfeasible and are based only on orthogonal directions. Indeed, the LT and the LA directions are computed under convenient approximations that lead to simple matrix operations and vector norms. Moreover, we have proved an algorithm that allows to correctly generate Gaussian vectors stratified along non-orthogonal directions. Our numerical experiments demonstrate that the proposed convenient directions return remarkable variance reductions both in BS, where the proposed techniques display the same variance reduction as those given by GHS, and in the CIR dynamics. In particular, the use of multiple non-orthogonal directions can be worthwhile for barrier style options. Moreover, in this work we show that the use of a few convenient directions of stratification with optimal allocation always outperform LHS (even in its LT-enhanced form) especially in terms of computational burden. A natural extension would be the combination with importance sampling procedures like the Robust Adaptive Technique recently proposed by Jourdain and Lelong [10] for Gaussian random vectors. In addition, due to its simple derivation and its affinity with the Fox's greedy rule (see Fox [6]), it would be interesting to investigate how to apply the LA procedure to derive a Quasi-Monte Carlo version of discretization schemes for stochastic volatility models like those proposed by Andersen [2] and Jourdain and Sbai [11]. Remark 2 . 2The third term in α(1) is zero, nevertheless we show its expression because the results below still hold when we compute the vector α (l) of parameters in the l-th step, where we consider ǫ l = (1, 1, . . ., 1 l−1 times , 0, . . . , 0), l = 1, . . . , N. Proposition 3. The solution of the recurrence equation (47) is a linear combination of the rows of A: Table 1 : 1Input Parameters in the BS dynamics(a) Arithmetic Asian Options S 0 K S N r σ T 50 45, 50, 55 64 0.05 0.3 1 (b) Arithmetic Asian Barrier Options S 0 K S B N r σ T 50 50, 55 60, 70, 80 16 0.05 0.1 1 Table 2 : 2Angles between the Stratifying Directions in degrees(a) Arithmetic Asian Options K S = 45 K S = 50 K S = 55 LA-GHS 1.35 1.04 1.74 LA-PCA 54.62 52.73 51.60 GHS-PCA 56.60 53.83 53.30 (b) Arithmetic Asian Barrier Options K S = 50 K S = 55 B = 60 B = 70 B = 70 B = 80 LA-GHS 0.37 0.37 0.75 0.75 LA-PCA 51.95 51.95 51.89 51.89 GHS-PCA 51.67 51.67 51.10 51.10 Table 3 : 3Input Parameters and Angles between Directions of Stratification for Basket Options.(a) Input Parameters. M S 0 ρ r σ T 40 Linear 20-60 0.5 0.05 Linear 0.1 − 0.4 1 (b) Angles in degrees K S = 30 K S = 40 K S = 50 LA-GHS 2.76 3.11 2.52 LA-PCA 64.74 65.04 65.19 GHS-PCA 62.29 62.02 62.47 Table 4 : 4Input Parameters and Angles between Directions of Stratification in the CIR dynamics.(a) Input Parameters S 0 N r α µ σ T 100 64 0.05 1.5 1 0.8 1 (b) Angles in degrees for Asian Options K S = 90 K S = 100 K S = 110 LA-LT 1.00 1.00 1.00 LA-PCA 43.72 43.72 43.72 LT-PCA 44.24 41.52 41.53 Table 5 : 5Results for Arithmetic Asian Options in the BS dynamics.Price 1 Dir Table 6 : 6Results for Arithmetic Asian Options with a Barrier at Expiry in the BS dynamics.Price 1 Dir Table 7 : 7Results for Arithmetic Asian Options with a Complete Barrier in the BS dynamics.Price 1 Dir 2 dirs MC LA PCA LA PCA LA-PCA LHS const opt const opt const opt const opt const opt K S = 50 B = 60 1.22 var 2.42 0.85 0.23 2.42 2.39 0.54 0.12 1.23 0.92 0.53 0.07 0.77 time 1 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.56 ×1.22 ×3.80 K S = 50 B = 70 1.89 var 4.76 0.14 0.0047 4.75 4.75 0.16 0.02 1.29 1 1.52 0.02 0.15 time 11.17 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.56 ×1.22 ×3.81 K S = 55 B = 70 0.19 var 0.47 0.041 0.00087 0.47 0.46 0.06 0.0038 0.22 0.06 0.14 0.0036 0.04 time 1 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.56 ×1.22 ×3.85 K S = 55 B = 80 0.2 var 0.55 0.0015 0.000059 0.55 0.53 0.05 0.0038 0.22 0.06 0.056 0.0048 0.002 time 1 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.54 ×1.14 ×1.56 ×1.22 ×3.83 Table 8 : 8Results for Basket Options in the BS dynamics.Price 1 Dir Table 9 : 9Results for Asian Options in the CIR dynamics.Price 1 Dir 2 dirs MC LT LA PCA LT PCA LA-PCA LHS const opt const opt const opt const opt const opt const opt K S = 90 15.63 var 427.73 1.85 1.09 1.54 0.9 115.73 106.85 9.3 2.28 51.21 40.61 9.13 4.62 1.08 time 1 ×1.2 ×1.22 ×1.2 ×1.22 ×1.5 ×1.6 ×1.2 ×1.22 ×1.5 ×1.6 ×1.55 ×1.55 ×2.76 K S = 100 10.6 var 310.11 1.49 0.67 1.22 0.54 97.22 69.7 8.75 1.73 53.03 25.73 8.92 1.66 1.02 time 1 ×1.2 ×1.22 ×1.2 ×1.22 ×1.5 ×1.6 ×1.2 ×1.22 ×1.5 ×1.6 ×1.55 ×1.554 ×2.75 K S = 110 6.95 var 212.19 1.18 0.37 0.29 82.25 54.28 8.72 1.26 40.34 20.69 8.29 2.22 0.9 time 1 ×1.2 ×1.22 ×1.2 ×1.22 ×1.5 ×1.6 ×1.2 ×1.22 ×1.5 ×1.6 ×1.55 ×1.55 ×2.76 Table 10 : 10Results for Arithmetic Asian Options with a Barrier at Expiry in the CIR dynamics.Price 1 Dir 2 dirs MC LT LA PCA LT PCA LA-PCA LHS const opt const opt const opt const opt const opt const opt K S = 100 B = 110 2.63 var 60.43 45.76 17.77 45.78 17.22 55.81 38.69 26.19 9.17 40.61 12.49 20.23 3.08 39.46 time 1 ×1.2 ×1.22 ×1.2 ×1.22 ×1.5 ×1.6 ×1.2 ×1.22 ×1.5 ×1.6 ×1.55 ×1.55 ×2.91 K S = 110 B = 120 1.82 var 41.64 32.64 8.1 32.55 7.85 38.69 26.43 20.76 5.77 26.4 6.27 11.95 1.26 28.52 time 1 ×1.2 ×1.22 ×1.2 ×1.22 ×1.5 ×1.6 ×1.2 ×1.22 ×1.5 ×1.6 ×1.55 ×1.55 ×2.87 K S = 100 B = 120 3.46 var 81.21 34.54 20.77 31.61 20.3 53.62 33.82 37.05 15.01 50.05 15.57 21.19 4.5 48.97 time 1 ×1.2 ×1.22 ×1.2 ×1.22 ×1.5 ×1.6 ×1.2 ×1.22 ×1.5 ×1.6 ×1.55 ×1.55 ×2.87 Table 11 : 11Results for Arithmetic Asian Options with a Complete Barrier in the CIR dynamics. ×1.21 ×1.1 ×1.21 ×1.1 ×1.33 ×1.23 ×1.21 ×1.1 ×1.33 ×1.23 ×1.33 ×1.23 ×2.82 K S = 110 ×1.21 ×1.1 ×1.21 ×1.1 ×1.33 ×1.23 ×1.21 ×1.1 ×1.33 ×1.23 ×1.33 ×1.23 ×2.85 K S = 100 ×1.21 ×1.1 ×1.21 ×1.1 ×1.33 ×1.23 ×1.21 ×1.1 ×1.33 ×1.23 ×1.33 ×1.23 ×2.79Price 1 Dir 2 dirs MC LT LA PCA LT PCA LA-PCA LHS const opt const opt const opt const opt const opt const opt K S = 100 B = 180 2.84 var 42.98 25.39 7.44 25.38 7.32 37.52 27.98 22.4 6.25 30.14 16.5 21.37 5.57 22.58 time 1 B = 180 1.1 var 14.03 9.51 2.05 9.59 2.05 12.68 8.37 8.63 1.73 11.33 4.76 8.35 1.58 8.79 time 1 B = 170 1.79 var 23.7 15.86 4.65 15.96 4.58 21.59 15.21 10.75 3.44 15.97 8.75 13.73 3.47 15.08 time 1 Discrete Eigenfuction Expansion of Multi-Dimensional Brownian Motion and the Ornstein-Uhlenbeck Process. 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The measurement of B s -meson branching fractions is a fundamental tool to probe physics beyond the Standard Model. Every measurement of untagged time-integrated B s -meson branching fractions is model-dependent due to the time dependence of the experimental efficiency and the large lifetime difference between the two B s mass eigenstates. In recent measurements, this effect is bundled in the systematics. We reappraise the potential numerical impact of this effect -we find it to be close to 10% in real-life examples where new physics is a correction to dominantly Standard-Model dynamics. We therefore suggest that this model dependence be made explicit, i.e. that B s branching-fraction measurements be presented in a two-dimensional plane with the parameter that encodes the model dependence. We show that ignoring this effect can lead to over-constraining the couplings of new-physics models. In particular, we note that the effect also applies when setting upper limits on non-observed B s decay modes, such as those forbidden within the Standard Model.

10.1016/j.physletb.2018.07.041

1804.03591

0dc0a56a406af5f05b16de3cf9b68999d693d627

On the model dependence of measured B s -meson branching fractions Francesco Dettori Oliver Lodge Laboratory University of Liverpool LiverpoolUK Diego Guadagnoli Laboratoire d'Annecy-le-Vieux de Physique Théorique UMR5108 Université de Savoie Mont-Blanc et CNRS B.P. 110F-74941Annecy-le-Vieux CedexFrance On the model dependence of measured B s -meson branching fractions LAPTH-013/18 The measurement of B s -meson branching fractions is a fundamental tool to probe physics beyond the Standard Model. Every measurement of untagged time-integrated B s -meson branching fractions is model-dependent due to the time dependence of the experimental efficiency and the large lifetime difference between the two B s mass eigenstates. In recent measurements, this effect is bundled in the systematics. We reappraise the potential numerical impact of this effect -we find it to be close to 10% in real-life examples where new physics is a correction to dominantly Standard-Model dynamics. We therefore suggest that this model dependence be made explicit, i.e. that B s branching-fraction measurements be presented in a two-dimensional plane with the parameter that encodes the model dependence. We show that ignoring this effect can lead to over-constraining the couplings of new-physics models. In particular, we note that the effect also applies when setting upper limits on non-observed B s decay modes, such as those forbidden within the Standard Model. Introduction -The branching fractions of B s mesons belong to the most sensitive probes of physics beyond the Standard Model (SM) in low-energy, high-intensity experiments. Their precise measurement is of prime importance to establish possible new physics or else to constrain models beyond the SM. However, the comparison between measurements and theory predictions of B s -meson branching fractions presents some subtleties due to the sizeable lifetime difference ∆Γ s between the two mass eigenstates of the B 0 s −B 0 s system [1]. First of all, in the absence of flavour tagging the measured branching fraction will be the average of the B 0 s andB 0 s branching fractions, due to their fast mixing. Secondly, since the theoretically calculated branching fraction is usually defined as the CP average between the flavour eigenstates before any oscillation, a ∆Γ s -dependent correction is required for it to be compared to the experimental values [1,2]. Both effects are proportional to a model-and channel-dependent factor known as A f ∆Γ (f denotes the final state). So, in general, the comparison between measurements and theoretical predictions involves an assumption about this factor. A third model-dependent bias is introduced by the non-perfect time acceptance of real experiments, again because of the sizeable lifetime difference ∆Γ s . This effect is discussed in [3], where it is quantified as a 1-3% correction. 1 In experimental measurements this effect was first appreciated in Ref. [6] (see also Ref. [4]), and in recent results this model-dependent correction is accounted for in the systematic error. Aim of the present paper is twofold: (i) we reappraise the relevance of this effect with respect to existing literature, as we find an O(7%) correction in a realistic example. We accordingly advocate that experiments report explicitly the correlation of the result with the value of the model-dependent parameter (A f ∆Γ , or any other parameter correlated with it), even when the effect is smaller than the statistical uncertainty; (ii) we emphasise that this effect has implications when setting bounds on new-physics couplings, especially in decay modes where new physics is not a correction, but the bulk of the dynamics. In such cases, not properly tracking this effect may even lead to constraints that qualitatively depart from the dynamics actually at play, as we discuss in a specific example related to present-day anomalies in flavour data. We begin by shortly reviewing the basic observation in Ref. [1]. One starts from the time-dependent untagged decay rate for a B s into a final state f , defined as [7] Γ(B s (t) → f ) ≡ Γ(B 0 s (t) → f ) + Γ(B 0 s (t) → f ) = R f H e −Γ H t + R f L e −Γ L t = = (R f H + R f L )e −Γst coshy s t τ Bs + A f ∆Γ sinh y s t τ Bs ,(1) where, in standard notation [8], Γ s = 1/τ Bs is the average between the widths, Γ H and Γ L , of the two mass eigenstates in the B s system. The parameter y s = Γ L −Γ H 2Γs = ∆Γs 2Γs quantifies the generic size of effects due to the B s -system width difference, y s = 0.061(4) [9]. Finally A f ∆Γ = R f H −R f L R f H +R f L depends on the final state and is related to the underlying dynamics, hence being model-dependent. The time-integrated branching ratio is then obtained by integrating eq. (1): B ave (B s → f ) = 1 2 ∞ 0 Γ(B s (t) → f ) dt = (R f H + R f L ) τ Bs 2 1 + A f ∆Γ y s 1 − y 2 s . ( As noted in Ref. [1], this is different from the theoretical branching fraction, which is usually calculated as CP -averaged at time zero: B th (B s → f ) ≡ τ Bs 2 Γ(B s (t) → f ) | t=0 ,(3) so that even with a perfect experiment, a model-dependent correction is needed to compare with the time-integrated branching fraction, B ave : B th (B s → f ) = 1 − y 2 s 1 + A f ∆Γ y s B ave (B s → f ) .(4) Time-dependent efficiencies -However, experiments are not perfect. In particular, the integral of the rate over the meson proper time is sampled according to a time-dependent efficiency. Hence, the experimentally measured branching fraction is actually B exp (B s → f ) = N obs N ε exp = 1 2ε exp ∞ 0 ε(t) Γ(B s (t) → f ) dt(5) where ε(t) is the time-dependent efficiency of the apparatus, ε exp is the time-averaged efficiency with which the observed yield, N obs , is corrected, and N is the total number of mesons produced to which the experiment normalises. Unless ε(t) is perfectly constant, the apparatus efficiency introduces an extra dependence on A f ∆Γ , and the latter makes the measurement of eq. (5) model dependent. This dependence cannot be factorised and accounted for as in eq. (4) as it rests on the explicit functional form of the efficiency. Intuitively, the rates of the two physical eigenstates will not be sampled uniformly, and this will distort the more the physical decay distribution, the more the two lifetimes differ. As a consequence, the measured admixture is not as given by the r.h.s. of eq. (2), and the dependence on A f ∆Γ in the relation between the calculated and the measured branching fraction is not as simple as given in eq. (4). This bias could be simply corrected for if A f ∆Γ could be univocally fixed for each given decay channel f . However A f ∆Γ depends on the short-distance structure of the decay, hence it is in general different in models of new physics with respect to the SM. For example, within the SM for the B s → µ + µ − decay one has A µµ ∆Γ = +1, i.e. that the decay occurs mostly through the heavier B s eigenstate (R L = 0) [10]. This assumes negligible CP violation in mixing and in the interference between decays with and without mixingan assumption that turns out to be robust. However, the B 0 s → µ + µ − decay could receive contributions beyond the SM from semileptonic scalar and pseudoscalar couplings, whose current bounds do not actually exclude any A µµ ∆Γ value in the whole range [−1, +1] [10,11]. One clear way to expose the measurements' dependence on the value of A f ∆Γ , and the ensuing model dependence would be to present measurements as a function of the assumed value for A f ∆Γ . Of course, such practice is not always necessary. Notably, if the mixture of the heavy and light eigenstates is known for a given final state, the effect can be properly accounted for in the experimental efficiency. For example, A f ∆Γ = 0 for flavour-specific decays. Furthermore, this effect is diluted or absent in decay rates where the SM contribution is precisely known and dominant. This effect can instead be prominent in rare decays, whose branching fractions can receive large contributions from new physics. We now illustrate such effect with a concrete example (see also [3]). While the functional form of the time-dependent efficiency can be non-trivial, to estimate the size of the bias one may assume a simple step function ε(t) = θ(t − t 0 ), i.e. ε = 0 for t < t 0 and ε = 1 elsewhere. With this function one gets 1 2 ∞ 0 ε(t) Γ(B s (t) → f ) dt = (R f H + R f L ) τ Bs 2 e −Γst 0 1 − y 2 s cosh (Γ s y s t 0 ) (1 + A f ∆Γ y s ) + sinh (Γ s y s t 0 ) (y s + A f ∆Γ ) , (6) which clearly reduces to eq. (2) for t 0 = 0. One can accordingly define the bias δ with respect to the branching ratio obtained with constant efficiency as the function δ(A f ∆Γ , y s , ε exp ) ≡ B exp (B s → f ) B ave (B s → f ) = e −Γst 0 ε exp cosh (Γ s y s t 0 ) + sinh (Γ s y s t 0 ) y s + A f ∆Γ 1 + A f ∆Γ y s , where the efficiency correction appears explicitly as in eq. (5). This efficiency is estimated ε exp (A a ) = ∞ 0 ε(t) Γ a (B s (t) → f dt ∞ 0 Γ a (B s (t) → f dt (7) where Γ a is the time-dependent width under the assumption A f ∆Γ = A a . Here we posit that the experimenter can estimate ε(t) with good accuracy from auxiliary measurements, typically from control channels, or else from Monte Carlo simulations. The bias will be therefore a function of A a : δ(A f ∆Γ , y s , A a ) = cosh (Γ s y s t 0 ) + sinh (Γ s y s t 0 ) ys+A f ∆Γ 1+A f ∆Γ ys cosh (Γ s y s t 0 ) + sinh (Γ s y s t 0 ) ys+Aa 1+Aa ys(8) which is by construction equal to 1 when the assumed value A a for A f ∆Γ coincides with the physical one. Hence in practice ε exp has to be calculated for each value of A a , so that for the same experimental event yield the branching fraction can be properly estimated for an assumed model. We illustrate the numerical impact of the bias δ in Fig. 1. Here δ is shown as a function of A a , under the hypothesis that the physical A f ∆Γ = 1, and for two realistic values of t 0 . In this example the bias amounts to overestimating the measured branching fraction with respect to the real one: as soon as the assumed value of A f ∆Γ , A a , departs from the physical value, the bias δ is larger than 1. This is as expected. In fact, with the considered efficiency function, estimating ε exp with A a < +1 means that one is undersampling the heavy eigenstate, the only one actually contributing if the physical A f ∆Γ = +1. As a consequence, ε exp in eq. (5) is smaller than the correct value that one would obtain for the physical A f ∆Γ = +1. As the figure shows, for values as low as t 0 = 0.5τ Bs the bias can be as large as ∼ 7%. Conversely, if one assumes that the inefficiency is for high proper-time values, ε(t) = θ(t 0 − t), then the bias will be in the opposite direction. In general, in real experiments one can expect inefficiencies both at low and at high proper-time values, so that the convolution with the expected time distribution will be performed by means of Monte Carlo simulations. Current status -In the majority of recent B s branching fraction measurements, the effect of the possible model dependence generated by a time-dependent efficiency has been treated as a systematic uncertainty, e.g. see Refs. [12][13][14][15]. On the other hand, only in very few examples is the effect treated as full-fledged dependence -which is what we advocate. An example of such treatment is the latest LHCb measurement of B(B 0 s → µ + µ − ) [16], where the branching fraction is quoted for the SM assumption (A f ∆Γ = 1), and corrections for A f ∆Γ = {0, −1} are reported. The size of the variation is respectively +4.6% (A f ∆Γ = 0) and +10.9% (A f ∆Γ = −1). This is displayed in Fig. 2 where the three values are shown in the two-dimensional plane of branching fraction and A f ∆Γ , together with the SM prediction [17]. We also note that Ref. [16] reports a measurement of the B 0 s → µ + µ − effective lifetime (τ µµ ) [10,18,19], which is in turn directly sensitive to A µµ ∆Γ itself. Therefore the two observables could already be represented in a two-dimensional plane, although the current τ µµ measurement would translate into A µµ ∆Γ = 8 ± 11, whose central value lies in the non-physical region but with large uncertainty. An illustrative example of such a correlated measurement is again in Fig. 2. In particular, the lines labelled "future contours" represent 1-and 2-σ contours assuming the current central value of the branching fraction with A µµ ∆Γ = 1, and a tenfold smaller uncertainties with respect to the LHCb measurement [16]. Biases on the Wilson coefficients -Neglecting the discussed variation can lead to an over-constraining of the theory parameter space, notably in models with sizeable scalar or pseudo-scalar contributions (with arbitrary phases), as illustrated by the following example. Let us consider a shift to the Wilson coefficients C S,P of the operators O S = e 2 16π 2 (sP R b)(¯ ) , O P = e 2 16π 2 (sP R b)(¯ γ 5 ) ,(9) that can give sizeable contributions to the B 0 s → µ + µ − rate. Let us assume they fulfil the constraint C S = −C P , as generally expected for new physics above the electroweak symmetry-breaking scale [20]. The B(B 0 s → µ + µ − ) prediction as a function of C S , and corrected by the factor (1 + A f ∆Γ y s )/(1 − y 2 s ) (see eq. (4)), is displayed in Fig. 3 for two choices of A µµ ∆Γ . The first choice is A µµ ∆Γ = +1, shown as a red dashed curve. The latest LHCb measurement corresponding to this value of A µµ ∆Γ is shown as a yellow dashed horizontal band. The upper line of this band and the red dashed curve intersect at C S −0.25 which may be taken as a 1σ bound on C S . However, A µµ ∆Γ = A µµ ∆Γ (C S ) [10]: the theory prediction corrected for this dependence, again through the (1+A f ∆Γ (C S ) y s )/(1−y 2 s ) factor, is displayed as a solid red curve. Concurrently, also the experimental measurement is a function of A µµ ∆Γ as we have discussed. In the figure we show as a solid green band the measurement for A µµ ∆Γ = −0.56, which corresponds to C S −0.28, the value at which the theory prediction and the experimental central value +1σ intersect. It is this C S value that should be taken as the correct 1σ bound on C S . We see that the difference between the two bounds, obtained respectively for A µµ ∆Γ = +1 and the correct A µµ ∆Γ , is of O(10%). Of course, the size of the effect just described will depend on the relative importance of scalar operators in the process being constrained. While intuitively the size O(10%) of the experimental bias -concretely, the variation of the branching-ratio measurement with A f ∆Γ -is expected to provide an upper bound on the size of the corresponding bias on Wilson coefficients, we would like to put forward an example where the latter bias turns out to be larger. This example is relevant in view of the existing discrepancies in flavour physics, and underlines the necessity of precisely tracking the theory that is being constrained (hence assumed), as soon as the measured A f ∆Γ in a given decay mode B s → f should differ from the assumed one. This in turn highlights the importance of effective-lifetime measurements, pointed out in [10,18,19], that are a probe of A f ∆Γ . Let us consider the effective-theory description emerging from present-day discrepancies in b → sµµ data, in particular by the lepton universality violation (LUV) tests R K and R K * measurements [21,22]. Among the preferred explanations in terms of shifts to the Wilson coefficients of the b → s effective Hamiltonian, an important one is the scenario with opposite contributions to the operators O 9 ∝ (sγ α L b) (μγ α µ) and O 10 ∝ (sγ α L b) (μγ α γ 5 µ). In particular a shift δC µ 9 = −δC µ 10 −13%|C SM 10 | ≈ −0.5 to the C µ 9(10),SM Wilson coefficients is preferred [23,24]. The structure resulting from such shifts, (sγ α L b) (μγ αL µ), has a (V − A) × (V − A) form and as such is very suggestive from the point of view of the ultraviolet dynamics, e.g. it can be straightforwardly rewritten in terms of SU (2) Linvariant fields [20,25]. Since the effective scale of such structure lies typically above the electroweak scale, the fermion fields involved will in general not be aligned with the mass basis. Hence, below the electroweak symmetry-breaking scale, such structure, introduced to account for LUV, will also generate lepton flavour violating dynamics, whose size is related to the measured amount of LUV [26]. From this argument, the analogous (V − A) × (V − A) operator (sγ α L b) (¯ γ αL ) would contribute to processes such as B s → − + , if a similar structure with the appropriate flavour indices is also favoured to explain LUV. Such argument does not forbid contributions from scalar operators of comparable size. Actually, constraints on scalar contributions (for recent analyses see [27,28]) are substantially weakened to the extent that a shift to C 10 is at play, as we discuss next. 2 In any of the B s → − + decays, contributions from the Wilson coefficients of the operators O 9 ≡ e 2 16π 2 (sγ α L b) (¯ γ α ) , O 10 ≡ e 2 16π 2 (sγ α L b) (¯ γ α γ 5 ) , O S ≡ m b e 2 16π 2 (sP R b) (¯ ) , O P ≡ m b e 2 16π 2 (sP R b) (¯ γ 5 ) ,(10) are of the form (see e.g. [29]) B(B s → + 1 − 2 ) ∝ (1 −m 2 )|F P +M C 10 | 2 + (1 −M 2 )|F S −mC 9 | 2 ,(11) wherem ≡m 2 −m 1 ,M ≡m 1 +m 2 , with hats denoting that the given mass is normalized by M Bs , and where F S,P ≈ M Bs C S,P . A sizeable departure in A f ∆Γ from unity would signal accordingly sizeable contributions from C S,P . In particular, C P could partly cancel (depending on its phase, which is unconstrained) the contribution from C 10 so that the measured signal would actually be due to C S dominantly, and this is the Wilson coefficient that the measurement would constrain in reality. In these circumstances, if one insisted with the assumption A f ∆Γ = +1, one would, instead, interpret the branching-ratio measurement as a constraint to C 10 , under the hypothesis that scalar contributions are negligible. So, the combination of Wilson coefficients that is actually constrained by a B s → f decay measurement needs be carefully tracked as soon as A f ∆Γ is measured and departs from unity. 3 In short, it will be important to present future experimental measurements in a twodimensional plane of the branching fraction and either A f ∆Γ or another observables correlated with it, such as the effective lifetime. A quite useful example is Ref. [30], where the limit is quoted for A f ∆Γ = {−1, 1}, thus allowing a handy extrapolation to any scenario with shifts to the operators in the second line of eq. (10). Other considerations -It is clear that if time information is available and the statistics are sufficient to perform a time-dependent analysis, the effect described in this paper is no longer present as the time-dependent efficiency can be convoluted with the correct time distribution. Secondly, this effect is even more relevant when combining different experimental measurements, as different apparatuses can have a different time-dependent efficiency and thus a different dependence on A f ∆Γ . In third place, since this effect depends experimentally on the apparatus efficiency and not on the yield, it is also present when setting limits on branching fractions; for example, it does apply to limits on channels forbidden in the SM and, as we argued, it may be a large effect there. Finally, we note that this effect was presented here for the case of B s mesons but in fact it is more general. The measurement of a branching fraction of a meson that oscillates is model dependent if 1. the experiment is realistic, i.e. ε(t) is not constant over the whole proper-time range; 2. the final state f is available to both mass eigenstates; 3. the difference in lifetime between the mass eigenstates is not negligible with respect to the meson average lifetime. In practice the last condition is realized only for B s mesons so far. In fact, while for B s mesons ∆Γ s is sizeable compared to Γ s , this is not true for B d or D 0 mesons. In the other relevant case of K 0 mesons, the difference in lifetimes between K S and K L is so large that branching fractions are directly reported for the two mass eigenstates rather than for the flavour ones. If one had to report branching fractions for the K 0 andK 0 the effect here described would be maximal. Summary -Every measurement of a B s untagged time-integrated branching fraction is model dependent due to the time dependence of the experimental efficiency [1,3]. We show with two real-life examples that this dependence can be as large as O(10%), and argue that it needs be properly tracked. We accordingly suggest that B s branching-fraction measurements be presented in a two-dimensional plane with the parameter A f ∆Γ or another observable correlated with it, even in the case the latter would not be yet measurable. We also argue that theoretical predictions within a given model should be compared with the measured value of the branching fraction corresponding to the A f ∆Γ value calculated assuming the same model. These practices should also be carried out for upper limits on the branching fraction of non-observed channels, notably those forbidden in the SM, where new physics is dominant, rather than just a correction. Ignoring this effect may lead to over-constraining new-physics couplings, or even to constraints that qualitatively depart from the dynamics actually at play. Figure 1 : 1The bias δ as a function of the assumed value for A f ∆Γ , A a , for a decay with A f ∆Γ = 1. The efficiency function is modelled as a step function θ(t − t 0 ), with two realistic t 0 values. by making a definite assumption about A f ∆Γ , namely as Figure 2 : 2LHCb measurement of the B 0 s → µ + µ − branching fraction vs. A µµ ∆Γ (blue squares)[16]. The respective SM predictions are also reported (red circle). Black ellipses show 1-and 2-σ contours of a possible future measurement of the two observables simultaneously (see text). Figure 3 : 3Red lines: theory predictions as a function of a scalar Wilson-coefficient shift C S = −C P , for A µµ ∆Γ = +1 (dashed) and respectively A ∆Γ (C S ) (solid). Horizontal bands: experimental ranges for A µµ ∆Γ = +1 (yellow dashed), and respectively A µµ ∆Γ (C S ), whereC S corresponds to the filled dot in the figure. See text for more details. The effect is also mentioned in[1] (see sec. V). In the specific context of the B 0 s → µ + µ − measurement[4], this effect was subsequently developed in Ref.[5] and by one of the authors. Sensitivity of rare decays to scalar operators is warranted by the fact that the fermion mass necessary to perform the chiral flip may actually be a large mass, at variance with the SM case. Sizeable scalar contributions are accordingly ubiquitous as soon as the bosonic sector is enlarged with respect to the sheer SM content.3 We emphasise that our argument holds for LU and lepton-flavour conserving decays alike. AcknowledgementsThe authors are indebted to Tim Gershon and Patrick Koppenburg for crucial remarks on the first preprint version of the manuscript. We also acknowledge useful comments from Peter Stangl. 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In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular realization by a complete matching on the Hasse diagram of a triangulation of the manifold.

0803.2616

db73b67c21dca941fa4ea9724f5902c0fba46bd7

COMBINATORIAL REALIZATION OF THE THOM-SMALE COMPLEX VIA DISCRETE MORSE THEORÝ 18 Mar 2008 Etienne Gallais COMBINATORIAL REALIZATION OF THE THOM-SMALE COMPLEX VIA DISCRETE MORSE THEORÝ 18 Mar 2008arXiv:0803.2616v1 [math.GT] In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular realization by a complete matching on the Hasse diagram of a triangulation of the manifold. Introduction R. Forman defines a combinatorial analog of smooth Morse theory in [4], [5], [6] for simplicial complexes and more generally for CW-complexes. Discrete Morse theory has many applications (computer graphics [10], graph theory [3]). An important problem is the research of optimal discrete Morse functions in the sense that they have the minimal number of critical cells ( [8], [9], [18] for the minimality of hyperplane arrangements). Thanks to a combinatorial Morse vector field V , Forman constructs a combinatorial Thom-Smale complex (C V , ∂ V ) whose homology is the simplicial homology of the simplicial complex. The differential is defined by counting algebraically Vpaths between critical cells. Nevertheless, the proof of ∂ V • ∂ V = 0 is an indirect proof (see [5]). We give two proofs of ∂ V • ∂ V = 0, one which focusses on the geometry and another one which focusses on the algebraic point of view (compare with [2] and [19]). In fact, the algebraic proof gives also the property that the combinatorial Thom-Smale complex is a chain complex homotopy equivalent to the simplicial chain complex. After that, we investigate one step forward the relation between the smooth Morse theory and the discrete Morse theory. We prove that any Thom-Smale complex has a combinatorial realization. We use this to prove that any Spin c -structures on a closed oriented 3-manifold can be realized by a complete matching on a triangulation of this manifold. This article is organised as follows. In section 2, we recall the discrete Morse theory from the viewpoint of combinatorial Morse vector field. Section 2.2.3 is devoted to the proofs of ∂ V • ∂ V = 0 and that the Thom-Smale complex is a chain complex homotopy equivalent to the simplicial chain complex. In section 3, we prove that any combinatorial Thom-Smale complex is realizable as a combinatorial Thom-Smale complex. In section 4, we obtain as a corollary the existence of triangulations with complete matchings on their Hasse diagram and prove that any Spin c -structures on a closed oriented 3-manifold can be realized by such complete matchings. Discrete Morse theory 2.1. Combinatorial Morse vector field. First of all, instead of considering discrete Morse functions on a simplicial complex, we will only consider combinatorial Morse vector fields. In fact, working with discrete Morse functions or combinatorial vector fields is exactly the same [5,Theorem 9.3]. In the following, X is a finite simplicial complex and K is the set of cells of X. A cell σ ∈ K of dimension k is denoted σ (k) . Let < be the partial order on K given by σ < τ iff σ ⊂ τ . Given a simplicial complex, one associates its Hasse diagram: the set of vertices is the set of cells K, an edge joins two cells σ and τ if σ < τ and dim(σ) + 1 = dim(τ ). Definition 2.1. A combinatorial vector field V on X is an oriented matching on the associated Hasse diagram of X that is a set of edges M such that (1) any two distincts edges of M do not share any common vertex, (2) every edge belonging to M is oriented toward the top dimensional cell. A cell which does not belong to any edge of the matching is said to be critical. Remark. The original definition of a combinatorial vector field is the following one: given a matching on the Hasse diagram define V : K → K ∪ {0} σ → V (σ) = τ iff (σ, τ ) is an edge of the matching and σ < τ, 0 otherwise. We will use both these points of view in the following. A V -path of dimension k is a sequence of cells γ : σ 0 , σ 1 , . . . , σ r of dimension k such that (1) σ i = σ i+1 for all i ∈ {0, . . . , r − 1}, (2) for every i ∈ {0, . . . , r − 1}, σ i+1 < V (σ i ). A V -path γ is said to be closed if σ 0 = σ r , and non-stationary if r > 0. Definition 2.2. A combinatorial vector field V which has no non-stationary closed path is called a combinatorial Morse vector. In this case, the corresponding matching is called a Morse matching. The terminology Morse matching first appeared in [2]. Remark. Let V be a combinatorial (resp. combinatorial Morse) vector field. It we remove an edge from the underlying matching, it remains a combinatorial (resp. combinatorial Morse) vector field (there are two extra critical cells). 2.2. The combinatorial Thom-Smale complex. 2.2.1. Definition of the combinatorial Thom-Smale complex. The following data are necessary to define the combinatorial Thom-Smale complex (see [5]). First, let X be a finite simplicial complex, K its set of cells and V a combinatorial Morse vector field. Suppose that every cell σ ∈ K is oriented. Let γ : σ 0 , σ 1 , . . . , σ r be a V -path. Then the multiplicity of γ is given by the formula m(γ) = r−1 i=0 − < ∂V (σ i ), σ i >< ∂V (σ i ), σ i+1 > ∈ {±1} where for every cell σ, τ , < σ, τ >∈ {−1, 0, 1} is the incidence number between the cells σ and τ (see [11]) and ∂ is the boundary map when we consider X as a CWcomplex. In fact, one can think of the multiplicity as checking if the orientation of the first cell σ 0 moved along γ coincides or not with the orientation of the last cell σ r . Let Γ(σ, σ ′ ) be the set of V -paths starting at σ and ending at σ ′ and Crit k (V ) be the set of critical cells of dimension k. Definition 2.3. The combinatorial Thom-Smale complex associated with (X, V ) is (C V * , ∂ V ) where: (1) C V k = σ∈Crit k (V ) Z.σ, (2) if τ ∈ Crit k+1 (V ) then ∂ V τ = σ∈Crit k (V ) n(τ, σ).σ where n(τ, σ) = e σ<τ < ∂τ, σ > γ∈Γ(e σ,σ) m(γ) Thus, this complex is exactly in the same spirit as the Thom-Smale complex for smooth Morse functions (see section 3): it is generated by critical cells and the differential is given by counting algebraically V -paths. Theorem 2.4 (Forman [5]). ∂ V • ∂ V = 0. Theorem 2.5 (Forman [5]). (C V * , ∂ V ) is homotopy equivalent to the simplicial chain complex. In particular, its homology is equal to the simplicial homology. We will give a direct proof of both of these theorems. The proof of Theorem 2.4 is done by looking at V -paths and understanding their contribution to ∂ V • ∂ V . Then, we prove Theorem 2.5 (which gives another proof of Theorem 2.4) using Gaussian elimination (this idea first appears in [2], see also [19]). Theorem 2.4. Let X be a simplicial complex, K be the set of its cells and K n the set of cells of dimension n. The proof is by induction on the number on edges belonging to the Morse matching. Initialization: matching with no edge. In this case, every cell is critical and the combinatorial Thom-Smale complex coincides with the well-known simplicial chain complex. Therefore, ∂ V • ∂ V = 0. Heredity: suppose the property is true for every matching with at most k edges defining a combinatorial Morse vector field. Let V be a combinatorial Morse vector field with corresponding matching consisting of k + 1 edges. In particular, there is no non-stationary closed V -path. Let (σ, τ ) be an edge of this matching with σ < τ and let V be the combinatorial Morse vector field corresponding to the original matching with the edge (σ, τ ) removed. By induction hypothesis ∂ V • ∂ V = 0. In particular, for every n ∈ N, every τ ∈ K n+1 and every ν ∈ K n−1 when there is a cell σ 1 ∈ K n such that there is a V -path from an hyperface of τ to σ 1 and another V -path from an hyperface of σ 1 to ν there exists another cell σ 2 ∈ K n with the same property so that their contribution to ∂ V • ∂ V are opposite. Proof of First, we will prove that ∂ V • ∂ V = 0 when the chain complex is with coefficients in Z/2Z and after we will take care of signs. Suppose that the distinguished edge of the matching (σ, τ ) is such that dim(σ) + 1 = dim(τ ) = n + 1. Therefore, C V i = C V i for i = n, n + 1 and ∂ V |C V i = ∂ V |C V i for i / ∈ {n, n+1, n+2}. So we have ∂ V •∂ V (µ) = 0 for all µ ∈ K −(K n ∪K n+1 ∪K n+2 ). Remark that it is also true for every σ ′ ∈ Crit n (V ) that ∂ V • ∂ V (σ ′ ) = 0 (since with respect to V it is true and σ ′ = σ). There are two cases left. Case 1. Let τ ′ ∈ Crit n+1 (V ) . To see that ∂ V • ∂ V (τ ′ ) = 0 we must consider two cases. First case is when the two V -paths which annihilates don't go through σ. Then, nothing is changed and contributions to ∂ V • ∂ V (τ ′ ) cancel by pair. The second case is when at least one the V -path which cancel by pair for ∂ V go through σ. The V -paths which go from τ ′ to ν are of two types: those who go via σ and the others. Let τ ′ → σ 2 → ν be a juxtaposition of two V -paths which cancel with the juxtaposition of V -path τ ′ → σ → ν. Since ∂ V • ∂ V (τ ) = 0, there must be a critical cell σ 1 such that the juxtaposition of V -paths τ → σ → ν and τ → σ 1 → ν cancels. Therefore, when considering ∂ V , three juxtapositions of V -paths disappear and one is created: τ ′ → (σ → τ ) → σ 1 → ν. It cancels with τ ′ → σ 2 → ν. It may happens that two juxtapositions of V -paths go through σ but this case works exactly in the same way. Case 2. This case is similar to the previous case. Let ς be a cell in K n+2 . There are two cases to see that ∂ V • ∂ V (ς) = 0. The first case is when the two Vpaths whose contributions are opposite don't go through τ . Then, nothing is changed and contributions to ∂ V • ∂ V (τ ′ ) cancel by pair. The second case is when the V -path which disappears is replaced by exactly a new one which goes through the edge (σ, τ ). The result follows similarly. Note that to deal with this two cases we used the fact that there is no nonstationary closed V -path (and so V -path). Now, let's deal with the signs. We will only consider the case 1. above, other cases work similarly. Denote n(α → β) (resp. n(α → β)) the sign of the contribution in the differential ∂ V (resp. ∂ V ) of a path going from α to β where both cells are critical of consecutive dimension. While considering V , we have by induction hypothesis (2.1) n(τ ′ → σ 2 ).n(σ 2 → ν) = −n(τ ′ → σ).n(σ → ν) and (2.2) n(τ → σ 1 ).n(σ 1 → ν) = −n(τ → σ).n(σ → ν) Since the juxtaposition of the V -paths τ ′ → σ 2 → ν don't go through σ we have that (2.3) n(τ ′ → σ 2 ).n(σ 2 → ν) = n(τ ′ → σ 2 ).n(σ 2 → ν) By definition of the multiplicity of paths we have (2.4) n(τ ′ → σ 1 ) = n(τ ′ → σ).(− < ∂τ, σ >).n(τ → σ 1 ) Combining equations (2.1)-(2.4) we obtain the following equalities n(τ ′ → σ 1 ).n(σ 1 → ν) = n(τ ′ → σ).(− < ∂τ, σ >).n(τ → σ 1 ).n(σ 1 → ν)(2.4) = n(τ ′ → σ).(− < ∂τ, σ >).n(τ → σ 1 ).n(σ 1 → ν) = n(τ ′ → σ). < ∂τ, σ > .n(τ → σ).n(σ → ν) (2.2) = (< ∂τ, σ > .n(τ → σ)).n(τ ′ → σ).n(σ → ν) = n(τ ′ → σ).n(σ → ν) by definition = −n(τ ′ → σ 2 ).n(σ 2 → ν) (2.1) = −n(τ ′ → σ 2 ).n(σ 2 → ν) (2.3) which concludes the proof of the theorem. [19]. Given a matching between two cells σ < τ , we would like to remove the following short complex (which is acyclic) 0 → Z.τ <∂τ,σ> − −−−− → Z.σ → 0 where ∂ is the boundary operator of the simplicial chain complex. To do this, we use Gaussian elimination (see e.g. [1]): Lemma 2.6 (Gaussian elimination). Let C = (C * , ∂) be a chain complex over Z freely generated. Let b 1 ∈ C i (resp. b 2 ∈ C i−1 ) be such that C i = Z.b 1 ⊕ D (resp. C i−1 = Z.b 2 ⊕ E). If φ : Z.b 1 → Z.b 2 is an isomorphism of Z-modules, then the four term complex segment of C (2.5) . . . → C i+1 0 @ α β 1 A − −−− → b 1 D 0 @ φ δ γ ε 1 A − −−−−− → b 2 E " µ ν " −−−−−→ C i−2 → . . . is isomorphic to the following chain complex segment (2.6) . . . → C i+1 0 @ 0 β 1 A −−−→ b 1 D 0 @ φ 0 0 ε − γφ −1 δ 1 A − −−−−−−−−−−−− → b 2 E " 0 ν " −−−−−→ C i−2 → . . . Both these complexes are homotopy equivalent to the complex segment (2.7) . . . → C i+1 " β " − −− → D " ε − γφ −1 δ " − −−−−−−−−− → E " ν " − −− → C i−2 → . . . Here we used matrix notation for the differential ∂. Proof. Since ∂ 2 = 0 in C, we obtain φα + δβ = 0 and µφ + νγ = 0. By doing the following change of basis A = 1 φ −1 δ 0 1 on b 1 D and B = 1 0 −γφ −1 1 on b 2 E we see that the complex segments 2.5 and 2.6 are isomorphic. Then, we remove the short complex 0 → b 1 φ → b 2 → 0 which is acyclic. Now, we are ready to prove Theorem 2.5. Proof of Theorem 2.5. Like for the proof of Theorem 2.4, we make an induction on the number of edges belonging to the matching defining the combinatorial Morse vector field. Let X be a simplicial complex, K be the set of its cells. Initialization: matching with no edge. In this case, there is nothing to prove since the combinatorial Thom-Smale complex is exactly the simplicial chain complex. Heredity: suppose the property is true for every matching with at most k edges defining a combinatorial Morse vector field. Let V be a combinatorial Morse vector field whose underlying matching consists of k + 1 edges. Let σ (n) < τ (n+1) be an element of this matching and V be the combinatorial Morse vector field equal to V with the matching σ < τ removed (it is actually a combinatorial Morse vector field). So, C V i = C V i for all i = n, n + 1 and ∂ V = ∂ V when restricted to C V i for all i / ∈ {n, n + 1, n + 2}. Moreover, we have the following equalities: (∂ V ) |C V n+1 = (∂ V ) |C V n+1 and ∂ V |C V n = ∂ V |C V n . By induction hypothesis, the combinatorial Thom- Smale complex (C V * , ∂ V ) is a chain complex homotopy equivalent to the simplicial chain complex of X. Thus, the combinatorial Thom-Smale complex associated with V is equal to the one of V except on the following chain segment (where ε = (∂ V ) |C V n |C V n+1 ): (2.8) . . . → C V n+2 0 @ α ∂ V 1 A − −−−− → τ C V n+1 0 @ < ∂τ, σ > δ γ ε 1 A − −−−−−−−−−−−− → σ C V n " µ ∂ V " − −−−−−− → C V n−1 → . . . Since X is a simplicial complex we have < ∂τ, σ >∈ {±1}. Applying lemma 2.6, we obtain the following new combinatorial chain complex which is homotopic to the combinatorial Thom-Smale complex of V (2.9) . . . → C V n+2 " ∂ V " − −−− → C V n+1 " α " − −− → C V n " ∂ V " − −−− → C V n−1 → . . . where α = ε − γ < ∂τ, σ > δ = ∂ V . Thus, the only thing to prove is that α = ∂ V over C V n+1 . To do this, we investigate ∂ V . There are two types of contributions. First type correspond to V -paths which do not go through σ, and they are counted in ε. Second type are V -paths which go through σ. They begin at an hyperface of a critical cell τ ′ and go through σ: this is the contribution of δ. Then, they jump to τ : this is the contribution of < ∂τ, σ >. Finally they begin at an hyperface of τ and go to a critical cell in C V n : this is the contribution of γ. It remains to check that the sign is correct, but this is exactly the same as in the first proof of Theorem 2.4. Corollary 2.7. Let X be a finite simplicial complex, C = (C * , ∂) be the corresponding simplicial chain complex and M be a matching (σ i < τ i ) i∈I on its Hasse diagram defining a combinatorial vector field V . Then the following properties are equivalent: (1) M is a Morse matching, (2) for any sequence (σ i1 < τ i1 ), (σ i2 < τ i2 ), . . . , (σ i |I| < τ i |I| ) such that i j = i k if j = k, Gaussian eliminations can be performed in this order. In particular, any sequence of Gaussian eliminations corresponding to M lead to the same chain complex which is the combinatorial Thom-Smale complex of V . Proof. 1 ⇒ 2 This is an immediate consequence of the proof of Theorem 2.5 and the fact that it leads to the Thom-Smale complex associated to (X, V ). 2 ⇒ 1 It is enough to show that there is no non-stationary closed path under the hypothesis. Suppose there is a closed V -path γ : σ 1 , . . . , σ r , σ 1 and consider any sequence of Gaussian elimination which coincides with (σ j < V (σ j )) until step r. In particular, r ≥ 3 since X is a simplicial complex. Let V be the corresponding combinatorial vector field. Let γ ′ : σ 1 , . . . , σ r be the V -path with length decrease by one. Then < ∂ r−1 V (σ r ), σ r >=< ∂V (σ r ), σ r > +m(γ ′ ) Since m(γ ′ ) = ±1, < ∂ r−1 V (σ r ) , σ r > is not invertible over Z and the Gaussian elimination cannot be performed (see lemma 2.6). This is a contradiction. Relation between smooth and discrete Morse theories In this section, we investigate the link between smooth and discrete Morse theories. We first recall briefly the main ingredients of smooth Morse theory. In particular, we describe the Thom-Smale complex and prove the following: Theorem 3.1 (Combinatorial realization). Let M be a smooth closed oriented Riemannian manifold and f : M → R be a generic Morse function. Suppose that every stable manifold has been given an orientation so that the smooth Thom-Smale complex is defined. Then, there exists a C 1 -triangulation T of M and a combinatorial Morse vector field V on it which realize the smooth Thom-Smale complex (after a choice of orientation of each cells of T ) in the following sense: (1) there is a bijection between the set of critical cells and the set of critical points, (2) for each pair of critical cells σ p and σ q such that dim(σ p ) = dim(σ q ) + 1, V -paths from hyperfaces of σ p to σ q are in bijection with integral curves of v up to renormalization connecting q to p, (3) this bijection induce an isomorphism between the smooth and the combinatorial Thom-Smale complexes. Throughout this section, we follow conventions of Milnor ([12], [13]). Definition 3.2. A smooth map f : M → R is called a Morse function if at each critical point p of f , D 2 f (p) is non-degenerate. More generally, a Morse function on a (smooth) cobordism (M ; M 0 , M 1 ) is a smooth map f : M → [a, b] such that (1) f −1 (a) = M 0 , f −1 (b) = M 1 ,(2)f (x) = f (p) − k i=1 x 2 i + n i=k+1 x 2 i and v has coordinates v(x) = (−x 1 , . . . , −x k , x k+1 , . . . , x n ). Given any Morse function, there always exists a gradient-like vector field (see [13]). In the following, we shall abreviate "gradient like vector field" by "gradient ". Thus, when needed, we will assume that we have chosen one. Given any x 0 ∈ M , we consider the following Cauchy problem γ ′ (t) = v(γ(t)) γ(0) = x 0 and call integral curve (denoted γ x0 ) the solution of this Cauchy problem. The stable manifold of a critical point p is by definition the set W s (p, v) := {x ∈ M | lim t→+∞ γ x (t) = p}. The unstable manifold of a critical point p is by definition the set {x ∈ M | lim t→−∞ γ x (t) = p}. When stable and unstable manifolds are transverse (this is called Morse-Smale condition), we called v a Morse-Smale gradient: such gradient always exists in a neighbourhood of a gradient (see e.g. [16]). We shall call a Morse function f generic if we have chosen for f a Morse-Smale gradient. To define the smooth Thom-Smale complex we need the following data: • a generic Morse function f , • an orientation of each stable manifold. Under these conditions, the number of integral curves of v up to renormalization (that is γ x ∼ γ y iff there exists t ∈ R such that γ x (t) = y) connecting two critical points of consecutive index is finite. Moreover, when we consider an integral curve from q to p where ind(p) = ind(q) + 1, it carries a coorientation induced by the orientation of the stable manifold and the orientation of the integral curve. One can move this coorientation from p to q along the integral curve and compare it with the orientation of the stable manifold of q. This gives the sign which is carried by the integral curve connecting q to p. The Thom-Smale complex (C f * , ∂ f ) is defined as: (1) C f k = p∈Crit k (f ) Z.p, (2) if p ∈ Crit k (f ) then ∂p = q∈Crit k−1 (f ) n(p, q).q where n(p, q) is the algebraic number of integral curves up to renormalization connecting q to p. The proof of this theorem can be extracted from [13]. Elementary cobordisms. In this subsection, we will prove that we can realize combinatorially the smooth Thom-Smale complex of any elementary cobordisms. Thus, by cutting the manifold M into elementary cobordism we will obtain the first part of Theorem 3.1: there exists a bijection between the set of critical cells and the set of critical points. We will only consider C 1 -triangulation of manifolds for technical reasons (see [21]). So, whenever we use the word triangulation it means C 1 -triangulation. A triangulation of a n + 1-cobordism (M n+1 ; M 0 , M 1 ) is a triplet (T ; T 0 , T 1 ) such that T is a C 1 -triangulation of M , T 0 (resp. T 1 ) is a subcomplex of T which is a C 1 -triangulation of M 0 (resp. M 1 ). A combinatorial Morse vector field V on a triangulated n+1-cobordism (T ; T 0 , T 1 ) is a combinatorial Morse vector field on T such that no cells of T 1 is critical and every cell of T 0 is critical. Definition 3.5. Let V be a combinatorial Morse vector field on a triangulated n + 1-cobordism (T ; T 0 , T 1 ). V satisfies the ancestor's property if given any n-cell σ 0 ∈ T 0 , there exists an n-cell σ 1 ∈ T 1 and a V -path starting at σ 1 and ending at σ 0 . Remark. There is a key difference between integral curves up to renormalization of a gradient v and V -paths. Given a point x ∈ M , there is only one solution to the Cauchy problem. Moreover, the past and the future of a point pushed along the flow is uniquely determined. A contrario given a cell σ, there are (in general) many V -paths starting at σ. The ancestor's property caracterises n + 1cobordism equipped with a combinatorial Morse vector field which knows its history in maximal dimension (n, n + 1). To prove that elementary cobordism can be realized , we need a combinatorial description of being a deformation retract. Let X be a simplicial complex and σ be an hyperface of τ which is free (that is σ is a face of no other cell). In this case, we say that X collapses to X − (σ ∪ τ ) by an elementary collapsing and write X ց X − (σ ∪ τ ). A collapsing is a finite sequence of such elementary collapsings. In particular, a collapsing defines a matching on the Hasse diagram of the simplicial complex. Moreover, one can prove that X − (σ ∪ τ ) is a deformation retract of X. Proposition 3.6. Let X be a simplicial complex and X 0 be a subcomplex. Suppose X ց X 0 . Then the matching given by this collapsing defines a combinatorial Morse vector field whose set of critical cells is the set of cells of X 0 . Proof. The only thing to check is that there is no non-stationary closed path. Since elementary collapsings are performed by choosing a free hyperface of a cell, there is no non-stationary closed path. Let ∆ m = (a 0 , . . . , a m ) be the standard simplex of dimension m. The cartesian product X = ∆ m × ∆ n is the cellular complex whose set of cells is {µ × ν} where µ (resp. ν) is a cell of ∆ m (resp. ∆ n ) (see [22]). Lemma 3.8. Let X 1 = ∆ k be the standard simplicial complex of dimension k and X 0 be a simplicial subdivision of X 1 . Consider the CW-complex which is equal to the cartesian product ∆ k × ∆ 1 and where we subdivide ∆ k × {0} so that it is equal to X 0 . Then, there exists a simplicial subdivision X of this CW-complex such that X |∆×{i} = X i for i ∈ {0, 1}. Moreover for i ∈ {0, 1} there exists a collapsing X ց X i and the combinatorial Morse vector field associated V i satifies the ancestor's property on (X; X i , X i+1 ) (j is the class in Z/2Z). Proof. The simplicial subdivision and the collapsing is constructed by induction on k. If k = 0, choose a new vertex in the interior of the simplex ∆ 0 × ∆ 1 and the elementary collapsing ∆ 1 ց {i} gives the two collapsing ∆ 0 × ∆ 1 ց ∆ 0 × {i} for i ∈ {0, 1}. In particular, the corresponding combinatorial Morse vector field satifies the ancestor's property. Suppose the lemma is true until rank k − 1. At rank k, let Y be the corresponding CW-complex and x be a point in the interior of the cell of dimension k + 1. By induction hypothesis Y |∂∆ k ×∆ 1 admits a simplicial subdivision. Therefore, Y |∂(∆ k ×∆ 1 ) admits a simplicial subdivision denoted Z (just add the simplexes ∆ k × {i} which are equal to X i for i ∈ {0, 1}). The simplicial subdivision X is given by making the join of the simplicial subdivision of the boundary over {x}: X = Z * {x}. Now, the collapsing X ց X 0 is performed in three steps. Step 1. The cell σ ∈ X 1 of dimension k is the free hyperface of the cell σ * {x}. We do the following elementary collapsing: (3.1) X ց X − (σ ∪ σ * {x}) Step 2. By induction hypothesis, X |∂∆ k ×∆ 1 ց X |∂∆ k ×{0} . Performing the join over x induces the following collapsing: (3.2) X |∂∆ k ×∆ 1 * {x} ց X |∂∆ k ×{0} * {x} Step 3. It remains to collapse X 0 * {x} on X 0 . Let y be a vertex in X 0 which is a vertex of the original simplex X 1 . Since X 0 is a simplicial subdivision of ∆ k , there exists a collapsing X 0 ց {y}. This collapsing gives the following collapsing: (3.3) X 0 * {x} ց X 0 ∪ ({y} × {0}) * {x} ց X 0 Combining collapsings (3.1), (3.2) and (3.3) gives X ց X 0 . The corresponding combinatorial Morse vector field satisfies the ancestor's property by construction. The collapsing X ց X 1 is constructed in the same way and conclusions of lemma follows. Remark. The proof of lemma 3.8 is by induction. Let δ (j) be the j-th skeleton of ∆ k . Denote by X (j) (resp. X (j) i ) the simplicial complex X |δ (j) ×∆ 1 (resp. (X i ) |δ (j) ×∆ 1 ). For i ∈ {0, 1}, the collapsing X ց X i can be restricted to X (j) ց X (j) i for any 0 ≤ j ≤ k and the induced combinatorial Morse vector field satisfies the ancestor's property. The next two lemmas are technical lemmas. The first one is the basic tool to glue together triangulated cobordisms. The second one will be useful to construct a combinatorial realization of a cobordism with exactly one critical point and is a generalization of lemma 3.8. . . , a n ) be the standard simplex of dimension n and δ n−1 = ( a 0 , . . . , a n ) be the hyperface which does not contain a 0 . In particular ∆ n = {a 0 } * δ n−1 . Then there exists a simplicial subdivision X of the cartesian product ∆ m × ∆ n such that • X |∆ m ×δ n−1 is a simplicial subdivision without any new vertex given by lemma 3.7, • X |∆ m ×{a0} = ∆ m , • X ց X |(∂∆ m ×∆ n )∪(∆ m ×{a0}) . Moreover, for each simplex ∆ 1 i = (a 0 , a i ) (i = 0), • X |∆ m ×∆ 1 i coincides with the simplicial complex given by lemma 3.8, • the collapsing X ց X |(∂∆ m ×∆ n )∪(∆ m ×{a0}) restricted to X |∆ m ×∆ 1 i coincides with the collapsing of lemma 3.8, • the induced combinatorial Morse vector field satisfies the ancestor's property on (X |∆ m ×∆ 1 i ; X |∆ m ×{a0} , X |∆ m ×{ai} ). Proof. The proof is by induction on k = m + n > 0. At rank k = 1 there are two cases. The case m = 0 and n = 1 is trivial: there is nothing to prove. The case m = 1 and n = 0 is given by lemma 3.8. Suppose the lemma is true until rank k − 1. Let (m, n) ∈ N 2 be such that m + n = k. We will first subdivide the boundary of ∆ m × ∆ n . Since ∂(∆ m × ∆ n ) = (∂∆ m × ∆ n ) ∪ (∆ m × ∂∆ n ) = (∂∆ m × ∆ n ) ∪ (∆ m × ({a 0 } * ∂δ n−1 )) ∪ (∆ m × δ n−1 ) we define for each cellular complex above a simplicial subdivision. 12ÉTIENNE GALLAIS • The simplicial subdivision of ∆ m × δ n−1 is given by Proposition 3.7: in particular, we do not create any new vertex. • The induction hypothesis gives a simplicial subdivision of (∂∆ m × ({a 0 } * δ n−1 )) ∪ (∆ m × ({a 0 } * ∂δ n−1 )). Let x be a point which is in the interior of the (m + n)-cell of ∆ m × ∆ n . The simplicial subdivision X of ∆ m × ∆ n is given by making the cone over {x} of the simplicial subdivision of the boundary of ∆ m × ∆ n . By construction we have the following collapsing (3.6) X |{x} * (∆ m ×δ n−1 ) ց X |{x} * ∂(∆ m ×δ n−1 ) which is realized by a downward induction on the dimension of cells of ∆ m ×(δ n−1 − ∂δ n−1 ): every cell σ ∈ ∆ m × (δ n−1 − ∂δ n−1 ) is a free hyperface of {x} * σ. The induction hypothesis says that there exits a simplicial subdivision Y of ∆ u × ∆ v such that Y ց Y |(∂∆ u ×∆ v )∪(∆ u ×{a0}) whenever u + v < k (a 0 is the first vertex of ∆ v ). In fact, we have also the following collapsing since the construction is made by induction: Y ց Y |∆ u ×{a0} Therefore, we have the following collapsings (3.7) X |∂∆ m ×({a0} * δ n−1 ) ց X |∂∆ m ×{a0} (3.8) X |∆ m ×{a0} * ∂δ n−1 ց X |∆ m ×{a0} Collapsing (3.6) followed by the cone over x of the collapsing (3.7) and the cone over x of the collapsing (3.8) give the following collapsing: In case n = 1, this construction is the same as the one of lemma 3.8. (1) the stable manifold of p is a subcomplex of T denoted T s p and T ց T s p ∪ T 0 , (2) there is a cell σ p of dimension ind(p) such that p ∈ σ p ⊂ T s p and T s p − σ p ց (T s p ∩ T 0 ) In particular, the combinatorial Morse vector field given by these two collapsings has exactly one critical cell σ p outside cells of T 0 . Proof. Suppose a Morse-Smale gradient v for f is fixed. Let W s (p, v) be the corresponding stable manifold of p. We follow the proof of Milnor which proves that M 0 ∪ W s (p, v) is a deformation retract of M (see the proof of Theorem 3.14 [12]). Let C be a (small enough) tubular neighbourhood of W s (p, v). The original proof consists of two steps. First, M 0 ∪ C is a deformation retract of M : this is done by pushing along the gradient lines of v. Then, M 0 ∪ W s (p, v) is a deformation retract of M 0 ∪ C. We prove the theorem in two steps. First step: construction of a good triangulation of C. The tubular neighbourhood C is diffeomorphic to D k × D n−k (for i ∈ N * , D i is the unit disk in R i ). Thanks to this diffeomorphism, the stable manifold is identified with D k × {0} and the adherence of the unstable manifold is identified with {0} × D n−k . Triangulate the stable manifold by the standard simplex ∆ k and denote σ p its interior (so T s p = σ p ). We triangulate D n−k by choosing an arbitrary triangulation of ∂D n−k = S n−k−1 and considering D n−k as the cone over its center {0}: this gives a triangulation of D n−k . The triangulation of σ p × D n−k is the following one: choose a simplicial subdivision of σ p × ∂D n−k without any new vertex given by proposition 3.7. Then, triangulate the cartesian product σ p × D n−k with the triangulation of σ p × ∂D n−k already fixed thanks to lemma 3.10: • for each simplex ν ∈ ∂D n−k , the lemma constructs a triangulation of σ p × ({0} * ν), • for each pair of simplexes (ν 0 , ν 1 ) ∈ (∂D n−k ) 2 , the simplicial subidivisions of σ p × ({0} * ν i ) coincides over σ p × ({0} * (ν 0 ∩ ν 1 )). Let T C be the triangulation of σ p × D n−k constructed above. By construction, we have the following collapsing The manifold (∂C + , V ) is a manifold with boundary which is triangulated. The gradient lines of v starting at any point of this manifold are transverse to it: we push along the gradient lines of v the triangulation until it meets M 1 . It gives a triangulation of (M (3.9) T C ց T s p ∪ T C |∂σp×D n−k∂C+ 1 , M V 1 ) which is a submanifold of M 1 with boundary. This triangulation is C 1 since pushing along the flow in this case is a diffeomorphism. Then, we get a product cobordism (with boundary) with triangulation of the top and the bottom already fixed: lemma 3.9 gives a triangulation of this cobordism with the desired collapsing. The same construction holds for (M C 0 , V ) (we suppose that the triangulation of V × [0, 1] is the same as the one given above). Let T be the corresponding triangulation of M . Then, we have the following collapsing (3.10) T ց T 0 ∪ T C Conclusion. The composition of collapsings (3.10) and (3.9) give T ց T 0 ∪ T s p Since T s p = σ p we get the following collapsing: T s p −σ p ց ∂T s p . Thus a combinatorial Morse vector field which satifies the conclusion of the theorem has been constructed. Nevertheless, note that the triangulation above in not C 1 : the triangulation of the stable manifold done by ∆ k gives only a topological triangulation. To correct this, push the level M 0 (denote this level M ′ 0 ) along the gradient line a little inside the cobordism so that the stable manifold can be C 1 -triangulated by the standard simplex. Then, we endow the cobordism whose boundary is M 0 ∪ M ′ 0 with a C 1triangulation given by Lemma 3.8. Corollary 3.12. Let f be a generic Morse function on a Riemannian closed manifold M . Then, there exists T a C 1 -triangulation of M and a combinatorial Morse vector field V defined on T such that for every k ∈ N the set of critical poins of index k is in bijection with the set of critical cells of dimension k. Proof. Since the Morse function f is generic, we have that for any critical points p = q, f (p) = f (q). Let a 1 < a 2 < . . . < a l be the ordered set of critical values of f . For each k ∈ {1, . . . , l}, let ε k > 0 be small enough so that the cobordism (M a k ; M a k − , M a k + ) = f −1 ([a k − ε k , a k + ε k ]); f −1 (a k − ε k ), f −1 (a k + ε k ) is a cobordism with exactly one critical point. Define for k ∈ {1, . . . , l − 1} the product cobordisms • the set of critical points of index k of g coincides with the one of f for every k ∈ N, • for each pair of critical points p and q of successive index, the set of integral curves up to renormalization connecting q to p for g is in bijection with the corresponding set for f (we suppose here that Morse-Smale gradients have been chosen for f and for g), • this bijection induces an isomorphism between the Thom-Smale complexes of f and g (we suppose that orientations of stable manifolds have been chosen). Thus, we suppose that f : M n → R is a generic self-indexed Morse function i.e. for every k ∈ N, for every p ∈ Crit k (f ), f (p) = k. In particular f (M ) = [0, n]. We suppose whenever we need it that a Morse-Smale gradient v for f is given. (M b k ; M a k−1 + , M a k − ) = f −1 ([a k−1 + ε k−1 , a k − ε k ]); f −1 (a k−1 + ε k−1 ), f −1 (a k − ε k ) The manifold M is equal to: M a1 ∪ M b1 ∪ . . . ∪ M b l−1 ∪ M a l One more time, we will cut M in cobordisms (almost) elementary and control combinatorially the behavior of V -paths. For i ∈ {0, . . . , n} choose 0 < ε i < 1/2. Remark. Theorem 3.11 is proved in the case where there is exactly one critical point. This proof extends directly to the case of k critical points of the same index under the condition that tubular neighbourhoods of stable manifolds are chosen to be disjoints one from each other. For i ∈ {0, . . . , n}, let (M i ; M i −, , M i + ) be the cobordism (f −1 ([i − ε i , i + ε i ]); f −1 (i − ε i ), f −1 (i + ε i )) Similarly, define (M i,i+1 ; M i + , M i+1 − ) the product cobordism equal to (f −1 ([i + ε i , i + 1 − ε i+1 ]); M i + , M i+1 − ) Then M = M 0 ∪ M 0,1 ∪ M 1 ∪ . . . ∪ M n Let p be a critical point of index k and C(p) be a tubular neighbourhood (small enough) of the stable manifold of p in the corresponding cobordism. Denote ∂C − (p) (resp. ∂C + (p)) the submanifold diffeomorphic to ∂D k × D n−k (resp. D k × ∂D n−k ). Denote σ p the critical cell of dimension k corresponding to p (see Theorem 3.11). Hypothesis on the triangulation of ∂C + (p). (1) stable manifolds of critical points of index k + 1 intersect ∂C + (p) along a subcomplex of dimension k and intersect σ p × ∂D n−k along cells of dimension k of the type σ × {a i } where a i is a vertex of ∂D n−k , (2) each integral curve up to renormalization γ from p to q ∈ Crit k+1 (f ) intersects ∂C + (p) in the interior of a k-cell σ γ ∈ σ p × ∂D n−k , (3) given two distinct integral curves up to renormalization γ and γ ′ from p to critical points of index k + 1 then σ γ = σ γ ′ . Remark. The first hypothesis is satisfied by choosing small enough ε k and since stable and unstable manifolds are transverse. For such a small enough ε k the last hypothesis will be satisfied. The second hypothesis is automatically satisfied if the first hypothesis is satisfied. In each triangulated cobordism (M k ; M k − , M k + ), stable manifolds of critical points of index k are subcomplexes. Following notations of Theorem 3.11, we have the following collapsing T s p − σ p ց ∂T s p Using lemma 3.9, we obtain the following collapsing M k−1,k ց M k−1 + which can be restricted to the stable manifold of p since it is a submanifold of M k − . With respect to the stable manifold, the combinatorial Morse vector field satisfies the ancestor's property. Let γ be an integral curve of v up to renormalization from q ∈ Crit k−1 (f ) to p. It intersects ∂C + (q) in a point which by hypothesis belongs to a cell σ q × {a γ }. There is a 1 − 1 correspondance between the set of integral curves up to renormalization from q to p (with ind(p) = ind(q) + 1) and V -paths from hyperfaces of σ p to σ q given by γ ←→ σ γ . From σ γ , there is a unique V -path ending at σ q . Since V satisfies the ancestor's property in the stable manifold and σ γ is a cell of dimension k−1 there is an ancestor of σ γ which is an hyperface of σ p . This gives a V -path between an hyperface of σ p to σ q which corresponds to γ. We endow each critical cell with the orientation of the corresponding stable manifold and every other cell is endowed with an arbitrary orientation. By construction, the multiplicity of V -path coincides with the sign of the corresponding gradient path and the theorem follows. Complete matchings and Euler structures In this section, we use Theorem 3.1 to prove the following: given a closed oriented 3-manifold and an Euler structure on it, there is a triangulation such that a complete matching on the Hasse diagram of a triangulation realizes this Euler structure. Complete matchings. Definition 4.1. A complete matching on a graph is a matching such that every vertex belongs to an edge of the matching. As a corollary of theorem 3.1 we obtain: Proof. Since M is a closed smooth manifold of dimension 3 we have χ(M ) = 0 where χ denotes the Euler characteristic. Take a pointed Heegaard splitting of M (Σ g ; α = (α 1 , . . . , α g ), β = (β 1 , . . . , β g ); z) of genus g so that there is an n-uplets of intersection points x between the α's and the β's which defines a bijection between the sets α and β. It is always possible to find such a pointed Heegaard splitting after a finite number of isotopies of the α's and β's curves (see [7]). The Morse function f corresponding to this Heegaard splitting has one critical point of index 0 and 3 and g critical points of index 1 and 2. Denote the set of index 1 (resp. 2) critical points by {q i } g i=1 (resp. {p i } g i=1 ) where q i (resp. p i ) corresponds to α i (resp. β i ) for all i ∈ {1, . . . , g}. The n-uplet of intersection points x = (x 1,i1 , . . . , x n,in ) gives for each j ∈ {1, . . . , g} an integral curve connecting q j to p ij . The point z gives an integral curve connecting the index 0 critical point to the index 3 critical point. Take a combinatorial realization (T, V ) as given by Theorem 3.1 of (M, f ). Then to each point x i,ji correspond now a V -path γ from an hyperface of the critical cell σ pi j to σ qi : we change the matching along this path so that both τ pi j and σ qi are no more critical cells. If γ : σ 0 , . . . , σ r = σ qi , then do the following: • match σ 0 with τ , • for every i ∈ {1, . . . , r} match σ i with V (σ i−1 ). Now suuppose that z belongs to the interior of a 2-cell τ z (if not, subdivide T ). Denote by ς the critical cell of dimension 3 and by υ the critical cell of dimension 0. There is by construction a V -path γ : τ 0 , . . . , τ r = τ z from an hyperface of ς to τ z since z is in the stable manifold of the index 3 critical point. We modify the matching along γ this way: • match τ 0 with ς, • for every i ∈ {1, . . . , r} match τ i with V (τ i−1 ). In fact, it is no more a matching since τ z belongs to two edges of the matching. Nevertheless, τ z belongs to the unstable manifold of υ the critical cell of dimension 0. By the construction done in Theorem 3.1, the tubular neighbourhood of the critical point of index 0 is equal to D 3 = ∂D 3 * {0}. The triangulation of this tubular neighbourhood is given by making the cone over {0} = υ of a triangulation of ∂D 3 . We modify the matching as follows. Let ν denotes the critical 0-cell and suppose τ z * ν is the tetrahedron ABCD where A corresponds to ν. [20]. Let (M, ∂M ) be a smooth manifold of dimension n and T be a C 1 -triangulation of M . Suppose ∂M = ∂ 0 M ∂ 1 M be such that χ(M, ∂ 0 M ) = 0 and let T i be equal to T |∂iM for i ∈ {0, 1}. Denote K the set of cells of T and K i the set of cells of T i for i ∈ {0, 1}. For each cell σ ∈ K, let sgn(σ) be equal to (−1) dim(σ) and pick a σ a point in the interior of σ. An Euler chain in (T, T 0 ) is a one-dimensional singular chain ξ in T with the boundary of the form σ∈K−K0 sgn(σ)a σ . Since χ(M, ∂ 0 M ) = 0, the set of Euler chains is non-empty. Given two Euler chains ξ and η, the difference ξ − η is a cycle. If ξ − η = 0 ∈ H 1 (M ) then we say that ξ and η are homologous. A class of homologous Euler chains in (T, T 0 ) is called a combinatorial Euler structure on (T, T 0 ). Let Eul(T, T 0 ) be the set of Euler structures on (T, T 0 ). If ξ is an Euler chain, denote by [ξ] its class as a combinatorial Euler structure. Euler chains behave well with respect to the subdivision of a triangulation: this allows us to consider the set Eul(M, ∂ 0 M ) of Euler structures on (M, ∂ 0 M ). Taking ξ an element of this set means choosing a triangulation (T, T 0 ) of (M, ∂ 0 M ) and considering an Euler chain on (T, T 0 ). Remark. Let C be a complete matching on a C 1 -triangulation (T, T 0 ) of (M, ∂ 0 M ). Then it defines an Euler chain [ξ c ] ∈ Eul(T, T 0 ): orient every edge of the matching from odd dimensional cells to even dimensional cells. Complete matchings are special Euler chains that do not pass through a cell more than one time. Remark. In fact, this bijection is an H 1 (T )-isomorphism, but we'll make no use of it. Let us describe the construction of Turaev in the case ∂M = ∅. Let T be a C 1 -triangulation of M and T ′ be the first barycentric subdivision of T . We recall the definition of the vector field F 1 with singularities on M . For a simplex a of the triangulation T , let a denotes its barycenter. If A =< a 0 , a 1 , . . . , a p > is a simplex of the triangulation T ′ , where a 0 < a 1 < . . . < a p are simplexes of T , then, at a point x ∈ Int(A), F 1 (x) = 0≤i<j≤p λ i (x)λ j (x)(a j − x) Here λ 0 , λ 1 , . . . , λ p are barycentric coordinates in A and a j − x is the image of the tangent vector a j − x ∈ T x A via the homeomorphism between T and M . Every barycenter of each cell of T is a singular point of F 1 . Let M be a matching on the Hasse diagram of T : in particular, every edge of the matching connects two singular points. Turaev proved that the index of F 1 in a neighbourhood of every edge of the Hasse diagram (thought as embedded in M in the obvious way) is equal to zero. Thus, if we think about combinatorial (not necessarily Morse) vector field on T as a Matching on its Hasse diagram, it encodes a desingularization of the vector field F 1 (where critical points of F 1 remain if the corresponding cell is critical). This can be done in the case of Euler chain. More precisely, if ξ is an Euler chain, let F ξ denote an extension of F 1 to a non-singular vector field. Turaev proved that the homotopy [F ξ ] ∈ V ect(M ) depends only on [ξ] ∈ Eul(T ). The map ρ is defined by ρ([ξ]) = [F ξ ]. In the general case where ξ corresponds to a matching (instead of a complete matching), we still denote by ρ the map which assigns to ξ the vector field F ξ given by the construction above. We refer to [20] in the case ∂M = ∅. This theorem is a step toward Heegaard-Floer homology [14,15]. Recall the first steps of the construction of Heegaard-Floer homology for closed, oriented 3manifolds. Choose a pointed Heegaard splitting M = U 0 ∪ Σ U 1 (that is choose a Morse function) and fix a Spin c -structure on M given by an n-uplets of intersection points between the α's and the β's. Turaev proved that Spin c -structures are in bijection with Euler structures (and so the set vect(M )). Thus, Theorem 4.5 together with Theorem 3.1 give for a closed oriented 3-manifold a combinatorial realization of both the Spin c -structure and the pointed Heegaard splitting. The hard part remaining is to understand how holomorphic disks can be combinatorially realized. The following two lemmas will be useful to prove Theorem 4.5. 3. 1 . 1Smooth Morse theory. Let M be a smooth closed oriented Riemannian manifold of dimension n. Given a smooth function f : M → R, a point p ∈ M is said critical if Df (p) = 0. Let Crit(f ) be the set of critical points. At a critical point p, we consider the bilinear form D 2 f (p). The number of negative eighenvalues of D 2 f (p) is called the index of p (denoted ind(p)). We denote Crit k (f ) the set of critical points of index k. all critical points of f are interior (lie in M − (M 0 ∪ M 1 )) and are nondegenerate. For technical reasons, we must consider the following object:Definition 3.3. Let f be a Morse function on a cobordism (M n ; M 0 , M 1 ). A vector field v on M n is a gradient-like vector field for f if (1) v(f ) > 0 throughout the complement of the set of critical points of f , (2) given any critical point p of f there is a Morse chart in a neighbourhood U of p so that Theorem 3. 4 . 4The homology of the Thom-Smale complex is equal to the singular homology of M . Proposition 3.7 ([17, Prop. 2.9]). The cartesian product ∆ m ×∆ n has a simplicial subdivision without any new vertex. More generally, the cartesian product of two simplicial complexes has a simplicial subdivision without any new vertex. Lemma 3 . 9 . 39Let (T M i , T N i ) be two C 1 -triangulations of the pair (M, N ) where N k is a submanifold (possibly with boundary) of M n (k ≤ n). Then, there exists a C 1 -triangulation T of (M × [0, 1], N × [0, 1]) such that (T |M×{i} , T |N ×{i} ) = (T M i , T N i ) for i ∈ {0, 1} and 2 collapsings Moreover, the induced combinatorial Morse vector fields V satifies the ancestor's property on the cobordisms (T |N ×[0,1]; T N 0 , T N 1 ) and (T ; T M 0 , T M 1 ). Proof. First, suppose N = ∅.Both triangulations T 0 and T 1 are C 1 -triangulation of the same manifold therefore they have a common simplicial subdivision T 1/2[21] (this is where we use the fact that triangulations are C 1 -triangulations). Subdivide∆ 1 = [0, 1] in two standard simplexes [0, 1/2] and [1/2, 1]. Lemma 3.8 gives a C 1triangulation of M × [0, 1/2] (resp. M × [1/2, 1]) denoted T [0,1/2] (resp. T [1/2,1] ). The union T [0,1/2] ∪ T [1/2,1] is a triangulation of M × [0, 1] denoted T . By construction, T |M×{i} = T M i for i ∈ {0, 1} and we have the two following collapsings: Composing these two collapsings give the desired collapsing and lemma 3.8 give the ancestor's property. In the case where the submanifold N is non-empty, the construction above gives a triangulation T of the pair (M ×[0, 1], N ×[0, 1]) and we have (T |M×{i} , T |N ×{i} ) = (T M i , T N i ) for i ∈ {0, 1}. The collapsing T ց T 0 can be restricted to T |N ×[0,1]. We remove from the matching edges corresponding to T N ×[0,1] ց T N ×{0} to obtain the desired collapsing. Again lemma 3.8 give the ancestor's property. Lemma 3 . 10 . 310Let (m, n) be a pair of positive integers. Let ∆ n = (a 0 , . X ց X |(∂∆ m ×∆ n )∪({x} * (∆ m ×{a0})) Finally there exists a collapsing {x} * (∆ m × {a 0 }) ց ∆ m × {a 0 } (by choosing a vertex y ∈ ∆ m and considering the collapsing ∆ m ց {y}) which gives the result. Theorem 3 . 11 . 311Let f be a generic Morse function on a cobordism (M ; M 0 , M 1 ) with exactly one critical point p of index k. Then, there exists a C 1 -triangulation of the cobordism (T ; T 0 , T 1 ) such that Second step: combinatorial realization of the first retraction. Let T 0 be a triangulation of M 0 which coincides over M 0 ∩ C with the triangulation above. Consider the following submanifolds with boundary: ∂C − = M 0 ∩ C, M C 0 = M 0 − Int(∂C − ) and ∂C + = ∂C − Int(∂C − ). Let V be ∂C − ∩ ∂C + : it is diffeomorphic to ∂D k × ∂D n−k = S k−1 × S n−k−1 a manifold of dimension n − 2. Theorem 3 . 311 gives for k = 1, . . . , l a combinatorial realization of the cobordism (M a k ; M a k − , M a k + ). Lemma 3.9 gives a combinatorial realization of each cobordism(M b k ; M a k−1 + , M a k − ) for k = 1, . . . , l − 1 (with the convention that M a0 = ∅).Then, we construct a C 1 -triangulation of M and define on it a combinatorial vector field. It is in fact a combinatorial Morse vector field since along V -paths we only can go down and the conclusion of the corollary follows.3.3. Proof of Theorem 3.1. Since f is generic, we use the Rearrangement Theorem [13, Theorem 4.8] to consider g a generic self-indexed Morse function such that −1,n ∪ M n For all i ∈ {0, . . . , n}, (M i ; M i − , M i + ) is a cobordism with |Crit i (f )| critical points of index i (maybe there is no critical point). The triangulation of M is constructed in the following way: (1) triangulation of cobordisms (M i ; M i − , M i + ) for all i ∈ {0, . . . , n} given by Theorem 3.11, (2) triangulation of cobordisms (M i,i+1 ; M i + , M i+1 − ) for all i ∈ {0, . . . , n} given by lemma 3.9. Corollary 4 . 2 . 42Let M be a closed smooth manifold of dimension 3. Then there exits a C 1 -triangulation of M such that a complete matching on its Hasse diagram exists. The collapsing ∂D 3 * {0} ց {0} gives in particular the following matching on ABCD: (BCD, ABCD), (BC, ABC) and (B, AB) (A is critical). Modify the matching by (A, AB), (B, BC) and (ABC, ABCD). Then, BCD (which is τ z ) is critical. This gives a complete matching over T . 4.2. 2 . 2Homologous vector fields. By a vector field on (M, ∂ 0 M ) we mean (except in clearly mentioned case) a non-singular continuous vector field of tangent vectors on M directed into M on ∂ 0 M and directed outwards on ∂ 1 M . Since χ(M, ∂ 0 M ) = 0, there exists such vector fields on (M, ∂ 0 M ). Vector fields u and v on (M, ∂ 0 M ) are called homologous if for some closed ball B ⊂ Int(M ) the restriction of the fields u and v are homotopic in the class of non-singular vector fields on M − Int(B) directed into M on ∂ 0 M , outwards on ∂ 1 M , and arbitrarily on ∂B. Denote by vect(M, ∂ 0 M ) the set of homologous vector fields on (M, ∂ 0 M ) and the class of a vector field u is denoted by [u]. 4.2.3. The canonical bijection. Turaev proved the following: 18ÉTIENNE GALLAIS Theorem 4.3 (Turaev [20]). Let (M, ∂ 0 M ) be a smooth pair such that dim(M ) ≥ 2. For each C 1 -triangulation (T, T 0 ) of the pair (M, ∂ 0 M ) there exists a bijection ρ : Eul(T, T 0 ) → vect(M, ∂ 0 M ) 4. 3 . 3M-realization of Euler structures. Definition 4.4. Let [ξ] ∈ Eul(M, ∂ 0 M ). We say that [ξ] has a M-realization if there exists a C 1 -triangulation T of (M, ∂M ) and a matching η on the corresponding Hasse diagram such that [η] = [ξ]. Theorem 4. 5 . 5Any Euler structure on a smooth oriented closed riemannian 3manifold has an M-realization. 4.2. Euler structures and homologous vector fields. Throughout this subsection we use conventions of Turaev [20]. 4.2.1. Combinatorial Euler structures. Complete matchings have an interpretation as Euler chains. First, we recall Euler structures as defined by Turaev Proof. We follow notations of Theorem 3.11. The two collapsings T ց T s p ∪ T 0 and T s p − σ p ց (T s p ∩ T 0 ) define a combinatorial Morse vector field with only one critical cell. Let ξ be the corresponding one singular chain. Thus, the barycenter of this cell (which is p) must be a critical point of ρ(ξ). It remains to check that outside a small neighbourhood of p, ρ(ξ) is homologous to the Morse-Smale gradient v. Since outside the tubular neighbourhood C of the stable manifold the triangulation is constructed by pushing it along gradient lines of v, we can use lemma 4.6 to see that ρ(ξ) is homologous to v outside the tubular neighbourhood C of the stable manifold. In a small ball neighbourhood (which is a Morse chart at p) of the critical point p, the vector field F 1 coincides with the Morse-Smale gradient v. Since T C ց σ p ∪ (T 0 ∩ T C ), ρ(ξ) is homologous to v outside the Morse chart of p . The fact that the index of ρ(ξ) at p is equal to k is a consequence of the definition of F 1 .Proof of Theorem 4.5. We apply Theorem 3.1 to obtain a C 1 -triangulation of the 3-manifold and a combinatorial Morse vector field which realizes combinatorially the Thom-Smale complex. Then, the construction done in corollary 4.2 defines a matching which in turns defines an Euler chain ξ. The map ρ sends ξ to a non-singular vector field which is by construction homologous to the Morse-Smale gradient of f . Thus, to prove the theorem for any [v] ∈ vect(M ), it remains to find a pointed Heegaard splitting (Σ g ; α = (α 1 , . . . , α g ), β = (β 1 , . . . , β g ); z) of the 3-manifold M such that an n-uplet of intersection points x corresponds to a given [v] ∈ vect(M ). Finally,[15,Lemma 5.2] tells that any [v] ∈ vect(M ) can be realized in such a way. This concludes the proof. D Bar-Natan, arXiv.org:math/0606318Fast Khovanov homology computations. D. Bar-Natan, Fast Khovanov homology computations, arXiv.org:math/0606318, 2006. On discrete Morse functions and combinatorial decompositions. K Manoj, Chari, Formal power series and algebraic combinatorics. Vienna21752016MR MR1766262Manoj K. Chari, On discrete Morse functions and combinatorial decompositions, Discrete Math. 217 (2000), no. 1-3, 101-113, Formal power series and algebraic combinatorics (Vienna, 1997). MR MR1766262 (2001g:52016) Discrete Morse theory and graph braid groups. 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MR MR2113019Ann. of Math. 257016, Holomorphic disks and topological invariants for closed three-manifolds, Ann. of Math. (2) 159 (2004), no. 3, 1027-1158. MR MR2113019 (2006b:57016) On the Novikov complex for rational Morse forms. A V Pajitnov, Ann. Fac. Sci. Toulouse Math. 6MR MR1344724 (97f:57033A. V. Pajitnov, On the Novikov complex for rational Morse forms, Ann. Fac. Sci. Toulouse Math. (6) 4 (1995), no. 2, 297-338. MR MR1344724 (97f:57033) Introduction to piecewise-linear topology. C P Rourke, B J Sanderson, MR MR0350744Springer-Verlag503236New YorkC. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, New York, 1972. MR MR0350744 (50 #3236) M Salvetti, S Settepanella, arXiv.org:0705.2874Combinatorial Morse theory and minimality of hyperplane arrangements. M. Salvetti and S. Settepanella, Combinatorial Morse theory and minimality of hyperplane arrangements, arXiv.org:0705.2874, 2007. Morse theory from an algebraic viewpoint. 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Campus de Tohannic -BP 573, 56017 Vannes, FRANCE Current address: Laboratoire de Mathématiques Jean Leray (LMJL) UFR Sciences et Techniques, 2 rue de la Houssinières -BP 92208. 44322Université de Bretagne SudLaboratoire de Mathématiques et Applications des Mathmatiques (LMAM)Laboratoire de Mathématiques et Applications des Mathmatiques (LMAM), Univer- sité de Bretagne Sud, Campus de Tohannic -BP 573, 56017 Vannes, FRANCE Current address: Laboratoire de Mathématiques Jean Leray (LMJL) UFR Sciences et Tech- niques, 2 rue de la Houssinières -BP 92208, 44 322 Nantes Cedex 3, FRANCE E-mail address: [email protected]

We apply a general theory describing the dynamics of supervised learning in layered neural networks in the regime where the size p of the training set is proportional to the number of inputs N , as developed in a previous paper, to several choices of learning rules. In the case of (on-line and batch) Hebbian learning, where a direct exact solution is possible, we show that our theory provides exact results at any time in many different verifiable cases. For non-Hebbian learning rules, such as Perceptron and AdaTron, we find very good agreement between the predictions of our theory and numerical simulations. Finally, we derive three approximation schemes aimed at eliminating the need to solve a functional saddle-point equation at each time step, and assess their performance. The simplest of these schemes leads to a fully explicit and relatively simple non-linear diffusion equation for the joint field distribution, which already describes the learning dynamics surprisingly well over a wide range of parameters.

cond-mat/9909428

0875e047f1f169c9c420ddbfc09128051ea6b349

Dynamics of Learning with Restricted Training Sets II. Tests and Applications 29 Sep 1999 September 28th 1999 A C C Coolen Department of Mathematics King's College London Strand The Neural Computing Research Group Aston University WC2R 2LS, B4 7ETLondon, BirminghamUK, UK D Saad Department of Mathematics King's College London Strand The Neural Computing Research Group Aston University WC2R 2LS, B4 7ETLondon, BirminghamUK, UK Dynamics of Learning with Restricted Training Sets II. Tests and Applications 29 Sep 1999 September 28th 1999 We apply a general theory describing the dynamics of supervised learning in layered neural networks in the regime where the size p of the training set is proportional to the number of inputs N , as developed in a previous paper, to several choices of learning rules. In the case of (on-line and batch) Hebbian learning, where a direct exact solution is possible, we show that our theory provides exact results at any time in many different verifiable cases. For non-Hebbian learning rules, such as Perceptron and AdaTron, we find very good agreement between the predictions of our theory and numerical simulations. Finally, we derive three approximation schemes aimed at eliminating the need to solve a functional saddle-point equation at each time step, and assess their performance. The simplest of these schemes leads to a fully explicit and relatively simple non-linear diffusion equation for the joint field distribution, which already describes the learning dynamics surprisingly well over a wide range of parameters. Introduction In a previous paper [1] we have applied the formalism of dynamical replica theory [2] to analyse the dynamics of supervised learning in perceptrons with restricted training sets. For an introduction into the area of the dynamics of learning in layered neural networks, a guide to the relevant references, as well as a proper discussion of the peculiarities of the dynamics of learning with restricted as opposed to infinite raining sets, we refer to [1]. The microscopic variables in the learning process are the components of the weight vector J of the 'student' network. The 'teacher' network, which defines the task to be learned by the student, is characterised by a weight vector B. Learning proceeds on the basis of answers to be given to questions ξ ∈D ⊆ {−1, 1} N , according to a dynamical rule for J which is defined in terms of a function G[x, y], where x = J · ξ and y = B · ξ are the student and teacher fields, respectively. The randomly composed setD of questions, of size p = αN , is called the training set. If α < ∞ as N → ∞ the learning dynamics will be nontrivial. Firstly, the data iñ D will be recycled in due course, which generates complicated correlations and non-Gaussian local field distributions, and allows the system to improve its performance partly by memorizing answers rather than by learning the underlying rule (hence the difference between training-and generalization errors). Secondly, the actual composition of the randomly drawn training set introduces an element of frozen disorder into the problem, which will have to be averaged out. The analysis described in [1] resulted in a general macroscopic theory, describing the behaviour of such learning processes with α < ∞ in the limit N → ∞ (infinite systems) and on finite time-scales, in terms of deterministic laws for macroscopic observables. The theory applies to both on-line learning (where weight updates are made after each presentation of an input vector from the training set), and to batch learning (where weight updates are themselves averages over the full training set), as well as to arbitrary learning rules (i.e. arbitrary functions G[x, y]). In this paper we apply this general theory to various different specific choices of learning rules. One of these, (on-line and batch) Hebbian learning, provides an excellent benchmark test for our theory, since for this simple rule exact solutions are known, even for the regime of restricted training sets [4]. We find that our theory is fully exact for batch execution, and that it succeeds in predicting exactly the evolution of several macroscopic observables, including the generalisation error and moments of the joint field distribution for student and teacher fields, in the on-line case (although here full exactness is difficult to assess, and not a priori guaranteed). A preliminary presentation of some of the results in this paper (those involving Hebbian learning) was given in [3]. For non-Hebbian error-correcting learning rules, such as on-line and batch versions of Perceptron learning and AdaTron learning, no exact solutions are known at present with which to confront our theory; instead we here compare the predictions (with regard to the evolution of training-and generalization errors and the joint field distribution) of the full theory, as well as of a number of simple approximations of our equations, with the results of carrying out extensive numerical simulations in large (size N = 10, 000) neural networks. We find, surprisingly, that even the simplest of these approximations, which does not require solving any saddle point equations and takes the form of a fully explicit non-linear diffusion equation for the joint field distributions P [x, y], describes the simulation experiments remarkably well. Employing the more sophisticated (and thereby more CPU intensive) approximations, or, at the other end of the spectrum, a numerical solution of the full macroscopic theory, leads to increasingly accurate quantitative predictions for the evolution of the relevant macroscopic observables of the learning process, with deviations between theory and numerical experiment which are of the order of magnitude of the finite size effects in the simulations. We close our paper with a discussion of the strengths and weaknesses of the approach used, and an outlook on future work on the dynamics of learning with restricted training sets, involving the present and possibly other formalisms. Summary of the Theory and its Properties In this section we will be brief in order to avoid inappropriate duplication of the material in [1], to which we refer for full details. The Macroscopic Laws Our macroscopic observables are Q = J 2 , R = B·J , and the joint distribution of student and teacher fields (or 'activations') P [x, y] = δ[x−J ·ξ]δ[y −B ·ξ] D . For N → ∞ all these quantities are found to obey deterministic and self-averaging equations. We define f [x, y] = dxDy P [x|y]f [x, y], where Dy = (2π) − 1 2 e − 1 2 y 2 dy, and the following averages (the function Φ[x, y] will be specified below): U = Φ[x, y]G[x, y] V = xG[x, y] W = yG[x, y] Z = G 2 [x, y](1) For on-line learning (which is a stochastic process) we have found, using the prescriptions of dynamical replica theory (in replica symmetric ansatz) [1]: d dt Q = 2ηV + η 2 Z d dt R = ηW (2) d dt P [x|y] = 1 α dx ′ P [x ′ |y] δ[x−x ′ −ηG[x ′ , y]]−δ[x−x ′ ] − η ∂ ∂x P [x|y] [U (x−Ry)+W y] + 1 2 η 2 Z ∂ 2 ∂x 2 P [x|y] − η V −RW−(Q−R 2 )U ∂ ∂x P [x|y]Φ[x, y](3) For batch learning (which is a deterministic process) we found: d dt Q = 2ηV d dt R = ηW (4) d dt P [x|y] = − η α ∂ ∂x [P [x|y]G[x, y]] − η ∂ ∂x P [x|y] [U (x−Ry)+W y] − η V −RW−(Q−R 2 )U ∂ ∂x P [x|y]Φ[x, y](5) From the solution of the above closed sets of equations for the trio {Q, R, P } (one of which is a function) follow the familiar training-and generalization errors E t = θ[−xy] and E g = π −1 arccos[R/ √ Q]. The auxiliary order parameters generated in the replica calculation, the spin-glass order parameter q and the function M [x|y], are calculated at each time-step by solving the following saddle-point equations: (x−Ry) 2 + (qQ−R 2 )(1− 1 α ) = 2(qQ−R 2 ) 1 2 + 1 B DyDz z x ⋆ (6) P [X|y] = Dz δ[X −x] ⋆(7) with B = qQ−R 2 Q(1−q) f [x,Φ[X, y] = Q(1−q)P [X|y] −1 Dz X −x ⋆ δ[X −x] ⋆(9) We refer to [1] for full derivations of the above equations. Properties of the Macroscopic Laws Some useful properties of our theory are independent of which learning rule G[x, y] is used. The first of these is that in the limit α → ∞, which corresponds to the case of infinite training sets, our theory reduces to the simpler formalism initiated in [5] and elaborated in papers like [6,7], which is built on assuming P [x, y] to be of a Gaussian form (this only happens as α → ∞) [1]. Secondly, for any given q the solution M [x|y] of the functional saddle-point problem (7) is unique, and can even be obtained as the fixed-point of a converging nonlinear functional map [1]. Thirdly, we note that the first conditional moment x(y) = dx xP [x|y] of P [x|y] of the joint field distribution obeys a simple equation, which is obtained from (3) and (5) upon multiplication by x, followed by integration over x: d dt [x(y) − Ry] = η α dx P [x|y]G[x, y] + ηU [x(y)−Ry](10)dx ′ 2π e ix ′ (k ′ −k)−iηkG[x ′ ,y] − 1 − iηk(W−U R)y + ηkU ∂ ∂k logP [k|y] − 1 2 η 2 k 2 Z − iηk V −RW−(Q−R 2 )U qQ−R 2P [k|y] Dz zM [k + iBz|y] M [iBz](12) This representation will be particularly convenient when applying our theory to Hebbian learning rules. We can derive more explicit results for the special class of locally Gaussian solutions, defined as P [x|y] = e − 1 2 [x−x(y)] 2 /∆ 2 (y) ∆(y) √ 2π For such distributions the functional saddle-point equation (7) can be solved, giving M [x, y] = e − 1 2 [x−x(y)] 2 /σ 2 (y) σ(y) √ 2π(13) with ∆ 2 (y) = σ 2 (y)+B 2 σ 4 (y). For such solutions to exist, the conditional moments x(y) and ∆(y) must obey the following equation: −ik d dt x(y)− 1 2 k 2 d dt ∆ 2 (y) = 1 α du √ 2π e − 1 2 [u−ik∆(y)] 2 −ikηG[x(y)+u∆(y),y] − 1 −iηk {W y + U [x(y)−Ry]} − 1 2 k 2 η 2 Z + 2ηU ∆ 2 (y) + 2ησ 2 (y) V −RW−(Q−R 2 )U Q(1−q)(14) The simple form of (13) allows us to calculate many objects explicitly. In particular: x ⋆ = x(y) + zBσ 2 (y) Φ[x, y] = x − x(y) Q(1−q)[1+B 2 σ 2 (y)] Finally, for many types of learning rules there are symmetry properties of our macroscopic equations to be exploited. Note that in the most common types of learning rules no distinction is being made between system errors of the type {x > 0, y < 0} and those where {x < 0, y > 0}. This translates into the following property of the function G[x, y] (note that by the nature of the learning processes under study G[x, y] can depend on y only via sgn(y)): G[x, y] = sgn(y) θ[xy]G + [|x|] + θ[−xy]G − [|x|](15) In particular we find for the most common learning recipes: Hebbian : G + [u] = 1, G − [u] = 1 Perceptron : G + [u] = 0, G − [u] = 1 AdaTron : G + [u] = 0, G − [u] = u The form (15) implies the symmetry G[−x, y] = G[x, −y], which turns out to allow for self-consistent solutions of the macroscopic equations with the following property at any time: P [−x|y] = P [x| − y](16) Combination of (16) with (7) and (9) shows that the measure M [x|y] and the function Φ[x, y] must consequently have the following symmetry properties: M [−x|y] = M [x| − y], Φ[−x, y] = −Φ[x, −y] This, in turn, guarantees that the right-hand sides of equations (3) and (5) indeed preserve the symmetry (16) of the field distribution, as claimed. Benchmark Tests: Hebbian Learning In the special case of the Hebb rule, G[x, y] = sgn[y], where weight changes ∆J never depend on J, one can write down an explicit expression for the weight vector J at any time, and thus for the expectation values of our observables. We choose as our initial field distribution a simple Gaussian one, resulting from an initialization process which did not involve the training set: P 0 [x|y] = e − 1 2 (x−R 0 y) 2 /(Q 0 −R 2 0 ) 2π(Q 0 −R 2 0 )(17) Careful averaging of the exact expressions for our observables over all 'paths' {ξ(0), ξ(1), . . .} taken by the question/example vector through the training setD (for on-line learning), followed by averaging over all realizations of the training setD of size p = αN , and taking the N → ∞ limit, then leads to the following exact result [4]. For on-line Hebbian learning one ends up with: Q = Q 0 + 2ηtR 0 2 π + η 2 t + η 2 t 2 1 α + 2 π R = R 0 + ηt 2 π (18) P [x|y] = dx 2π e − 1 2x 2 [Q−R 2 ]+ix[x−Ry]+ t α [e −iηx sgn[y] −1](19) For batch learning a similar calculation 1 gives: Q = Q 0 + 2ηtR 0 2 π + η 2 t 2 1 α + 2 π R = R 0 + ηt 2 π (20) P [x|y] = e − 1 2 [x−Ry−(ηt/α) sgn[y]] 2 /(Q−R 2 ) 2π(Q−R 2 )(21) Neither of the two field distributions is of a fully Gaussian form (although the batch distribution is at least conditionally Gaussian). Note that for both on-line and batch Hebbian learning we have dx xP [x|y] = Ry + ηt α sgn[y](22) The generalization-and training errors are, as before, given in terms of the above observables as E g = π −1 arccos[R/ √ Q] and E t = DydxP [x|y]θ[−xy]. We thus have exact expressions for both the generalization error and the training error at any time and for any α. The asymptotic values, for both batch and on-line Hebbian learning, are given by lim t→∞ E g = 1 π arccos 1 1 + π/2α (23) lim t→∞ E t = 1 2 − 1 2 Dy erf |y| α π + 1 √ 2α(24) As far as E g and E t are concerned, the differences between batch and on-line Hebbian learning are confined to transients. Clearly, the above exact results (which can only be obtained for Hebbian-type learning rules) provide excellent and welcome benchmarks with which to test general theories such as the one investigated in the present paper. Batch Hebbian Learning We compare the exact solutions for Hebbian learning to the predictions of our general theory, turning first to batch Hebbian learning. We insert into the equations of our general formalism the Hebbian recipe G[x, y] = sgn[y]. This simplifies our dynamic equations enormously. In particular we obtain: U = 0, V = x sgn(y) , W = 2/π For batch learning we consequently find: d dt Q = 2ηV d dt R = η 2/π d dt P [x|y] = − η α sgn(y) ∂ ∂x P [x|y] − ηy 2 π ∂ ∂x P [x|y] − η(V −R 2 π ) ∂ ∂x P [x|y]Φ[x, y] Given the initial field distribution (17), we immediate obtain V 0 = R 0 2/π. From the general property dx P [x|y]Φ[x, y] = 0 and the above diffusion equation for P [x|y] we derive an equation for the quantity V = x sgn(y) , resulting in d dt V = η/α + 2η/π, which subsequently allows us to solve Q = Q 0 + 2ηtR 0 2 π + η 2 t 2 1 α + 2 π R = R 0 + ηt 2 π(25) Furthermore, it turns out that the above diffusion equation for P [x|y] meets the requirements for having conditionally-Gaussian solutions, i.e. P [x|y] = e − 1 2 [x−x(y)] 2 /∆ 2 (y) ∆(y) √ 2π , M [x|y] = e − 1 2 [x−x(y)] 2 /σ 2 (y) σ(y) √ 2π provided the y-dependent average x(y) and the y-dependent variances ∆(y) and σ(y) obey the following three coupled equations: x(y) = Ry+ ηt α sgn(y) d dt ∆ 2 (y) = 2η 2 tσ 2 (y) αQ(1−q) ∆ 2 (y) = σ 2 (y)+B 2 σ 4 (y) The spin-glass order parameter q is to be solved from the remaining scalar saddle-point equation (6). With help of identities like x ⋆ = x(y) + zBσ 2 (y), which only hold for conditionally-Gaussian solutions, one can simplify the latter to η 2 t 2 α + α Dy ∆ 2 (y) + (qQ−R 2 )(α−1) = α 2 qQ−R 2 Q(1−q) +1 Dy σ 2 (y) We now immediately find the solution ∆ 2 (y) = Q−R 2 , σ 2 (y) = Q(1−q), q = [αR 2 +η 2 t 2 ]/αQ P [x|y] = e − 1 (this solution is unique). If we calculate the generalization error and the training error from (25) and (26), respectively, we recover the exact expressions E g = 1 π arccos     R 0 +ηt 2 π Q 0 +2ηtR 0 2 π +η 2 t 2 1 α + 2 π     (27) E t = 1 2 − 1 2 Dy erf   |y|[R 0 +ηt 2 π ]+ ηt α 2[Q 0 −R 2 0 + η 2 t 2 α ]  (28) Comparison of (25,26) with (20,21) shows that for batch Hebbian learning our theory is fully exact. This is not a big feat as far as Q and R (and thus E g ) are concerned, whose determination did not require knowing the function Φ[x, y]. The fact that our theory also gives the exact values for P [x|y] and E t , however, is less trivial, since here the disordered nature of the learning dynamics, leading to non-Gaussian distributions, is truly relevant. On-Line Hebbian Learning We next insert the Hebbian recipe G[x, y] = sgn[y] into the on-line equations (2,3). Direct analytical solution of these equations, or a demonstration that they are solved by the exact result (18,19), although not ruled out, has not yet been achieved by us. The reason is that here one has conditionally Gaussian field distributions only in special limits. Numerical solution is in principle straightforward, but will be quite CPU intensive (see also a subsequent section). For small learning rates the on-line equations reduce to the batch ones, so we know that in first order in η our on-line equations are exact (for any α, t). We now show that the predictions of our theory are fully exact (i) for Q, R and E g , (ii) for the first moment (22) of the conditional field distribution, and (iii) for all order parameters in the stationary state. At intermediate times we construct an approximate solution of our equations in order to obtain predictions for P [x|y] and E t . As before we choose a Gaussian initial field distribution. Many (but not all) of our previous simplifications still hold, e.g. U = 0, V = x sgn(y) , W = 2/π, Z = 1 (Z did not occur in the batch equations). Thus for on-line learning we find: d dt Q = 2ηV + η 2 d dt R = η 2/π The previous derivation of the identities d dt V = η/α+2η/π and V 0 = R 0 2/π still applies (just replace the batch diffusion equation by the on-line one), but the resultant expression for Q is different. Here we obtain: Q = Q 0 + 2ηtR 0 2 π + η 2 t + η 2 t 2 1 α + 2 π R = R 0 + ηt 2 π(29) Comparing (29) with (18) reveals that also for on-line Hebbian learning our theory is exact with regard to Q and R, and thus also with regard to E g . Upon using V = ηt/α+R 2/π, the on-line diffusion equation simplifies to d dt P [x|y] = 1 α P [x−η sgn(y)|y]−P [x|y] −ηy 2 π ∂ ∂x P [x|y]+ 1 2 η 2 ∂ 2 ∂x 2 P [x|y]− η 2 t α ∂ ∂x P [x|y]Φ[x, y]x(y) = dx xP [x|y] = Ry + ηt α sgn[y] Comparison with (22) shows also this prediction to be correct. We now turn to observables which involve more detailed knowledge of the function Φ[x, y]. Our result for x(y) and the identity x ⋆ = B −1 ∂d dt logP [k|y] = 1 α e −iηk sgn(y) −1 − iηky 2 π − 1 2 η 2 k 2 − ikη 2 t αP [k|y] qQ−R 2 Dz zM [k+iBz|y] M [iBz|y](30)with logP 0 [k|y] = −ikR 0 y− 1 2 k 2 (Q 0 −R 2 0 ) , and with the two saddle-point equationŝ P [k|y] = DzM [k+iBz|y] M [iBz|y] (31) η 2 t 2 α 2 + Dy dx P [x|y][x−x(y)] 2 +(1− 1 α )(qQ−R 2 ) = 2Q(1−q)+ 1 B 2 DyDz ∂ 2 ∂z 2 logM [iBz|y] (32) Since the fields x grow linearly in time (see our expression for x(y)) the equations (30,32,31) cannot have proper t → ∞ limits. To extract asymptotic properties we have to turn to the rescaled distribution Q[k|y] =P [k/t|y]. We define v(y) = (η/α) sgn(y)+ηy 2/π. Careful integration of (30), followed by inserting k → k/t and by taking the limit t → ∞, produces: logQ ∞ [k|y] = −ikv(y) − iη 2 k α 1 0 du lim t→∞ t qQ−R 2 Dz zM [uk/t+iBz|y] Q ∞ [uk|y]M [iBz|y](33) with the functional saddle-point equation Q[k|y] = DzM [k/t+iBz|y] M [iBz|y](34) The rescaled asymptotic system (33,34) admits the solution Q[k|y] = e −ikv(y)− 1 2 k 2∆2 ,M [k|y] = e −ikx(y)− 1 2 k 2σ2 t with the asymptotic values of B,∆,σ and q determined by solving the following equations: ∆ = Bσ 2∆ = η 2 α lim t→∞ t qQ−R 2 B = lim t→∞ qQ−R 2 Q(1−q) η 2 /α 2 +∆ 2 + (1−α −1 ) lim t→∞ (qQ−R 2 )/t 2 = 2B 2σ2 lim t→∞ Q(1−q)/t Inspection shows that these four asymptotic equations are solved by so that lim lim t→∞∆ = η/ √ α,t→∞P t [k/t|y] = e −ikη α −1 sgn(y)+y √ 2/π − 1 2 η 2 k 2 /α(35) Comparison with (18,19) shows that this prediction (35) is again exact. Thus the same is true for the asymptotic training error. Finally, in order to arrive at predictions with respect to P [x|y] and E t for intermediate times (without rigorous analytical solution of the functional saddle-point equation), and in view of the conditionally-Gaussian form of the field distribution both at t = 0 and at t = ∞, it would appear to make sense for us to approximate P [x|y] and M [x|y] by simple conditionally Gaussian distributions at any time: P [x|y] = e − 1 2 [x−x(y)] 2 /∆ 2 ∆ √ 2π , M [x|y] = e − 1 2 [x−x(y)] 2 /σ 2 σ √ 2π with the (exact) first moments x(y) = Ry +ηtα −1 sgn(y), and with the variance ∆ 2 self-consistently given by the solution of: ∆ 2 = σ 2 +B 2 σ 4 B = qQ−R 2 Q(1−q) d dt ∆ 2 = η 2 α +η 2 + 2η 2 tσ 2 αQ(1−q) α∆ 2 + η 2 t 2 α + (qQ−R 2 )(α−1) = ασ 2 2 qQ−R 2 Q(1−q) +1 The solution of the above coupled equations behaves as ∆ 2 = Q − R 2 + η 2 t/α + O(t 3 ) (t → 0) ∆ 2 = (Q−R 2 )[1+O(t −1 )] (t → ∞) for short and long times, respectively (note Q−R 2 ∼ t 2 as t → ∞). Thus we obtain a simple approximate solution of our equations, which extrapolates between exact results at the temporal boundaries t = 0 and t = ∞, by putting ∆ 2 = Q−R 2 +η 2 t/α with Q and R given by our previous exact result (29), one obtains E g = 1 π arccos R √ Q E t = 1 2 − 1 2 Dy erf |y|R+ηt/α ∆ √ 2 (36) We can also calculate the student field distribution P (x) = Dy P [x|y], giving P (x) = e − 1 2 [x+ ηt α ] 2 /(∆ 2 +R 2 ) 2 2π(∆ 2 +R 2 ) 1−erf R[x+ ηt α ] ∆ 2(∆ 2 +R 2 ) + e − 1 2 [x− ηt α ] 2 /(∆ 2 +R 2 ) 2 2π(∆ 2 +R 2 ) 1+erf R[x− ηt α ] ∆ 2(∆ 2 +R 2 )(37) In figure 1 we compare the predictions for the generalization and training errors (36) of the approximate solution of our equations with the results obtained from numerical simulations of on-line Hebbian learning for N = 10, 000 (initial state: Q 0 = 1, R 0 = 0; learning rate: η = 1). All curves show excellent agreement between theory and experiment. For E g this is guaranteed by the exactness of our theory for Q and R; the agreement found for E t is more surprising, in that these predictions are obtained from a simple approximation of the solution of our equations. We also compare the theoretical predictions made for the distribution P [x|y] with the results of numerical simulations. This is done in figure 2, where we show the fields as observed at time t = 50 in simulations (N = 10, 000, η = 1, R 0 = 0, Q 0 = 1) of on-line Hebbian learning, for three different values of α. In the same figure we draw (as dashed lines) the theoretical prediction (22) for the y-dependent average of the conditional x-distribution P [x|y]. Finally we compare the student field distribution P (x), as observed in simulations of on-line Hebbian learning (N = 10, 000, η = 1, R 0 = 0, Q 0 = 1) with our prediction (37). The result is shown in figure 3, for α ∈ {4, 1, 0.25}. In all cases the agreement between theory and experiment, even for the approximate solution of our equations, is quite satisfactory. General Approximation Schemes All three approximation schemes presented in this section aim at providing alternatives to calculating the effective measure M [x|y] at each time step from the functional saddle-point equation. Since this calculation cannot (yet) be done analytically, it constitutes a significant numerical obstacle in working out the predictions of our theory. Each scheme preserves both normalisation and symmetries of the probability density P [x, y] and its marginals, as well as the relation dx P [x|y]Φ[x, y] = 0 for all y. In the first two approximation schemes, a large α expansion and a conditionally-Gaussian saddlepoint approximation, all Gaussian integrals representing the disorder in the problem can be done analytically; this leads to a significant reduction in CPU time when solving our equations numerically (especially the large α approximation is extremely simple and fast, as it does not even involve a saddlepoint equation for q). We only work out the equations for on-line learning; the batch laws follows as usual upon expanding the equations in powers of η and retaining only the linear terms. Large α Approximation Our first approximation scheme is obtained upon taking into account the finite nature of the training set (i.e. the disordered nature of the dynamics) in first non-trivial order. The amount of disorder is effectively measured by the parameter B, or, equivalently, by the deviation of the value of the spinglass order parameter q from its naive value R 2 /Q. Putting B = 0 in the saddle-point equation (7) immediately Upon inserting (38) as an ansatz into the saddle-point equation (7) , one easily shows that M [x|y] = P [x|y] e − 1 2 B 2 [x−x(y)] 2 + 1 2 B 2 x 2 (y)−x(y) 2 +O(B 3 )(39) with the abbreviations x(y) = dx P [x|y]x x 2 (y) = dx P [x|y]x 2 (the second O(B 2 ) term in the exponent of (39), being independent of x, just reflects the normalisation requirements). This result enables us, in turn, to expand the function Φ[x, y] which controls the nontrivial term in our diffusion equation for P [x|y]. Note that from the definition of B it follows that Q(1−q) = 1 2 B −2 [ 1+4B 2 (Q−R 2 )−1], which gives Φ[x, y] = x − x(y) Q−R 2 + O(B 2 ) With this expression we can write our approximate equations in explicitly closed form (i.e. without any remaining saddle-point equations). The relevant scalar functions become U = G[x, y][x−x(y)] Q−R 2 V = xG[x, y] W = yG[x, y] Z = G 2 [x, y](40) For on-line learning we find: d dt Q = 2ηV + η 2 Z d dt R = ηW (41) d dt P [x|y] = 1 α dx ′ P [x ′ |y] δ[x−x ′ −ηG[x ′ , y]]−δ[x−x ′ ] − η ∂ ∂x P [x|y] [U (x−Ry)+W y] + 1 2 η 2 Z ∂ 2 ∂x 2 P [x|y] − η V −RW Q−R 2 − U ∂ ∂x P [x|y][x−x(y)](42) From the solution of the above equations follow, as always, the training-and generalization errors E t = Dydx P [x|y]θ[−xy] and E g = π −1 arccos[R/ √ Q]. The resulting theory is obviously exact in the limit α → ∞ (see [1]), by construction. Conditionally-Gaussian Approximation Our basic idea here is a variational approach to solving the functional saddle-point problem (valid for any α), i.e. to carry out the functional extremisation only within the restricted family of conditionally Gaussian measures M [x|y] (which, together with q, characterises the saddle-point): M [x|y] = e − 1 2 [x−x(y)] 2 /σ 2 (y) σ(y) √ 2π Note that this does not imply the stronger statement that P [x|y] itself is taken to be of a conditionally-Gaussian form (as in the case of the approximation used for on-line Hebbian learning). Extremisation of the original replica-symmetric functional Ψ[q, {M }] (see [1]) within the conditionally-Gaussian family of functions results in the requirement that the two y-dependent moments x(y) and σ 2 (y) be given by x(y) = dx xP [x|y], ∆ 2 (y) = dx x 2 P [x|y] − x 2 (y) = σ 2 (y)+B 2 σ 4 (y) Now we can again calculate all relevant averages which involve the effective measure M [x|y] exactly. In particular: x ⋆ = x(y) + zBσ 2 (y) B = qQ−R 2 Q(1−q) Φ[x, y] = e − 1 2 [x−x(y)] 2 /∆ 2 (y) ∆(y) √ 2πP [x|y] (x − x(y))σ 2 (y) Q(1−q)∆ 2 (y) For on-line learning this results in the following approximated theory: U = DyDu uσ 2 (y)G[x(y)+u∆(y), y] Q(1−q)∆(y) V = xG[x, y] W = yG[x, y] Z = G 2 [x, y](43)d dt Q = 2ηV + η 2 Z d dt R = ηW (44) d dt P [x|y] = 1 α dx ′ P [x ′ |y] δ[x−x ′ −ηG[x ′ , y]]−δ[x−x ′ ] − η ∂ ∂x P [x|y] [U (x−Ry)+W y] + 1 2 η 2 Z ∂ 2 ∂x 2 P [x|y] − ησ 2 (y) V −RW−(Q−R 2 )U √ 2πQ(1−q)∆ 5 (y) ∆ 2 (y)−(x−x(y)) 2 e − 1 2 [x−x(y)] 2 /∆ 2 (y)(45) The remaining order parameter q is calculated at each time-step by solving (x−Ry) 2 + (qQ−R 2 )(1− 1 α ) = 2 qQ−R 2 Q(1−q)+1 Partially Annealed Approximation In order to construct our third and final approximation we return to an earlier stage of the derivation of the present formalism (see [1]), and rewrite the functional saddle-point equation in a form where the replica limit n → 0 has not yet been taken, i.e. for all x, y : . We can now define in a natural way an annealed approximation of our theory upon replacing the complicated n = 0 functional saddle-point equation (7) by the much simpler n = 1 version: P [x|y] = Dz M n [x|y]e Bz[x−x(y)] dx ′ M n [x ′ |y]e Bz[x ′ −x(y)] n−1 Dz dx ′ M n [x ′ |y]e Bz[x ′ −x(P [x|y] = Dz M [x|y]e Bz[x−x(y)] Dz dx ′ M [x ′ |y]e Bz[x ′ −x(y)] The z-integrations can immediately be carried out, and the resulting equation solved for M [x|y], giving: M [x|y] = P [x|y] e − 1 2 B 2 [x−x(y)] 2 dx ′ P [x ′ |y] e − 1 2 B 2 [x ′ −x(y)] 2 ,(46) Averages involving the effective measure M [x|y] are thus written explicitly in terms of P [x|y], and we are left with the following approximate theory: U = Φ[x, y]G[x, y] V = xG[x, y] W = yG[x, y] Z = G 2 [x, y](47)d dt Q = 2ηV + η 2 Z d dt R = ηW (48) d dt P [x|y] = 1 α dx ′ P [x ′ |y] δ[x−x ′ −ηG[x ′ , y]]−δ[x−x ′ ] − η ∂ ∂x P [x|y] [U (x−Ry)+W y] + 1 2 η 2 Z ∂ 2 ∂x 2 P [x|y] − η V −RW−(Q−R 2 )U ∂ ∂x P [x|y]Φ[x, y](49)with Φ[X, y] = 1 Q(1−q) Dz      dx P [x|y] e − 1 2 [B(x−x(y))−z] 2 − 1 2 [B(X−x(y))−z] 2 (X − x) dx P [x|y] e − 1 2 [B(x−x(y))−z] 2 2      As always, B = qQ−R 2 /Q(1−q). The remaining spin-glass order parameter q is calculated at each time-step by solving (x−Ry) 2 + (qQ−R 2 )(1− 1 α ) = 2(qQ−R 2 ) 1 2 + 1 B DyDz z dx P [x|y] e − 1 2 [B(x−x(y))−z] 2 x dx P [x|y] e − 1 2 [B(x−x(y))−z] 2 From the solution of the above equations follow the training-and generalization errors E t = θ[−xy] and E g = π −1 arccos[R/ √ Q]. It should be emphasised that the present approximation is not equivalent to (and should be more accurate than) a full annealed treatment of the disorder in the problem; the latter would have affected not only the equation for M [x|y] but also the saddle-point equation for q (hence the name partially annealed approximation). Non-Hebbian Rules: Theory versus Simulations Henceforth we will always assume initial states with specified values for R 0 and Q 0 but without correlations with the training set, i.e. P 0 [x|y] = e − 1 2 [x−R 0 y] 2 /(Q 0 −R 2 0 ) 2π(Q 0 − R 2 0 ) This implies that the student could initially have some knowledge of the rule to be learned, if we wish, but will never know beforehand about the composition of the training set. We will inspect the learning dynamics generated upon using two of the most common non-Hebbian (error-correcting) learning rules: Perceptron : G Note that in the case of AdaTron learning the cases η ≤ 1 and η > 1 give rise to qualitatively different behaviour of the first term in the diffusion equation (3). For η < 1 the learning process, aiming at the situation where xy > 0 never occurs, remedies inappropriate student fields by slowly moving them towards (but not immediately across) the decision boundary. For η > 1 the adjustments made to the student fields could move them well into the region at the other side of the decision boundary. The case η = 1 is special, in that changes to the student fields tend to move them precisely onto the decision boundary. The student field distribution consequently develops a δ-peak at the origin, in perfect agreement with what can be observed in numerical simulations (see e.g. the figures referring to on-line AdaTron learning with η = 1 in [1]): η = 1 : d dt P [x|y] = 1 α δ(x) dx ′ θ[−x ′ y]P [x ′ |y] − P [x|y]θ[−xy] + . . . In fact the same occurs for all η ≤ 1: about half of the probability weight of P [x|y] will in due course become concentrated in an increasingly thin ridge along the decision boundary x = 0. This is illustrated in figure 4, for η = 1 2 . Since such a singular behaviour (although in principle accurately described by our equations) will be difficult to reproduce when solving the equations numerically, using finite spatial resolution, we will in this paper only deal with the case of η > 1 for AdaTron learning. Large α and Conditionally-Gaussian Approximations Our first approximated theory (the large α approximation) is very simple, with neither saddle-point equations to be solved nor nested integrations. As a result, numerical solution of the macroscopic equations is straightforward and fast. In figures 5 (on-line perceptron learning) and 6 (on-line Adatron learning) we compare the results of solving the coupled equations (40,41,42) numerically for finite values of α, plotting the generalisation-and training errors as functions of time, with results obtained from performing numerical simulations. As could have been expected, the large α approximation under-estimates the amount of disorder in the learning process, which immediately translates into under-estimation of the gap between E t and E g (which is its fingerprint). It is also clear from these figures that, although at any given time the quality of the predictions of this approximation does improve when α increases (as indeed it should), and although there is surely qualitative agreement, reliably accurate quantitative statements on the values of the training-and generalisation errors are confined to the regime ηt ≤ α. Yet, surprisingly, the agreement obtained is very good, even for ηt > α. Apparently the present approximation does still capture the main characteristics of the (non-Gaussian) joint field distribution. This is illustrated quite clearly and explicitly in figures 7 and 8, where we compare for a fixed time t = 10 the student and teacher fields as measured during numerical simulations (for N = 10, 000, drawn as dots in the (x, y) plane) for the p = αN questions ξ µ in the training setD, to the theoretical predictions for the joint field distribution P [x, y] (drawn as contour plots). We will not at this stage attempt to explain the surprising effectiveness of the large α approximation for small values of α (note that figures 5 and 6 even suggest an increase in accurateness as α is lowered below α = 1). This would require a systematic mathematical analysis of the non-linear diffusion equation (42), which we consider to be beyond the scope of the present paper. The conditionally-Gaussian approximation again involves no nested integrals, and its equations can therefore still be solved numerically in a reasonably fast way, but it does already require the solution (at each infinitesimal time step) of a scalar saddle-point equation to determine the spin-glass order parameter q. Approximations of this type work extremely well for the simple Hebbian learning rules, as we have seen earlier. However, numerical solution of the coupled equations (43,44,45) shows quite clearly that for the more sophisticated non-Hebbian rules such as Perceptron and AdaTron, which are of an error correcting nature (i.e. where where changes are made only when student and teacher disagree), the conditionally-Gaussian approximation is less accurate than the previously investigated large α approximation, in spite of the fact that the latter involved much simpler equations. Apparently the generally non-Gaussian nature of the conditional distribution P [x|y], and thereby of the measure M [x|y], is of crucial importance. It is not good enough to try getting away with allowing the y-dependent averages x(y) and variances ∆(y) to be non-trivial functions. With conditionally-Gaussian measures M [x|y] it turns out that generating the right width of the conditional distributions P [x|y] inevitably introduces tails for P [x|y] which spill into the xy < 0 region, which are found to be absent in error-correcting learning rules such as Perceptron and Adatron. This picture is consistent with figures 7 and 8, where we can observe that for any fixed value of the teacher field y the remaining marginal distribution for x is generally not symmetric around its (y-dependent) average. We conclude that the conditionally-Gaussian approximation is generally inferior to the large α approximation. We will not waste paper by producing large numbers of graphs to illustrate this explicitly and comprehensively, but we will rather draw the conditionally-Gaussian predictions together with those of the other approximations and of the full theory, by way of illustration. Partially Annealed Approximation and Full Equations The partially annealed approximation and the full theory are both expected to improve upon the large α approximation (note that the partially annealed approximation can be seen as an improved version of the large α approximation, similar in structure but valid also for small α, i.e. large B). Although the partially annealed approximation does not involve a functional saddle-point equation to be solved (which improves numerical speed), it shares with the full theory the appearance of nested (Gaussian) integrals, namely those appearing in the function Φ[x, y] and in the saddle-point equation for q. Thus, solution of both the full theory and of the partially annealed approximation involves a significant amount of CPU time (avoiding standard instabilities of discretised diffusion equations sets further limits on the maximum size of the time discretisation, dependent on the field resolution [8]), which implies that we have to reduce our ambition and restrict the number of experiments to a few typical ones. We will thus investigate two examples, both with α = 1: on-line Perceptron learning with η = 1 2 , and on-line AdaTron learning with η = 3 2 . We solve numerically the full equations of our theory, i.e. the macroscopic dynamical laws (2,3) with the order parameters calculated at each time step by solving (6,7), and show in figure 9 the training and generalisation errors as functions of time together with the corresponding values as measured during numerical simulations, with systems of size N = 10, 000. In addition, we plot in the same picture, for comparison, the training-and generalisation errors obtained by numerical solution of the three approximated theories as derived in the previous section. In comparing curves we have to take into account that those describing the large α approximation were generated upon solving the diffusion equation with a significantly higher numerical field resolution (∆x = 0.015) than the others (where we used ∆x = 0.05), because of CPU limitations. A restricted field resolution is likely to be more critical at large times, where the probability weight in the xy < 0 region, responsible for the residual error and for the non-stationarity of the dynamics, is highly concentrated close to the decision boundary x = 0. Especially for large times, we should therefore expect the full theory, the conditionally-Gaussian approximation, and the partially annealed approximation to all three perform better in reality than what is suggested by the numerical solutions of their equations as shown in figure 9. This is particularly true for AdaTron learning, where even for η > 1 (where we do not expect to observe a δ-singularity) the field distributions still tend to develop a jump discontinuity at x = 0. It turns out that the curves of the full theory and those of the partially annealed approximation are very close (virtually on top of one another for the case of Perceptron learning) in figure 9; apparently for the learning times considered here there is no real need to evaluate the full theory. Finally, we show in figure 10 for both the full theory and for the simulation experiments the two distributions P ± (x) = dy P [x, y]θ[±y] for the student fields, given a specified sign of the teacher field y (and thus a given teacher output), corresponding to the same experiments. Note that P (x) = P + (x) + P − (x). The pictures in figure 10 again illustrate quite clearly the difference between learning with restricted training sets and learning with infinite training sets: in the former case the desired agreement xy > 0 between student and teacher is achieved by a qualitative deformation of P [x|y], away from the initial Gaussian shape, rather than by adaptation of the first and second order moments. Our restricted resolution numerics obviously have difficulty in reproducing the discontinuous behaviour of P ± (x) near x = 0 for on-line Adatron learning (as expected), which explains why in this regime the simplest large α approximation (which can be numerically evaluated with almost arbitrarily high field resolution) appears to outperform the more sophisticated versions of the theory (which CPU limitations force us to evaluate with rather limited field resolution), according to figure 9. We conclude from the results in this section that our full theory indeed gives an adequate description of the macroscopic process, and that the partially annealed approximation is almost equivalent in performance to the full theory. As mentioned before, the conditionally-Gaussian approximation performs generally poorly (except, as we have seen earlier, for the simple Hebian rule). Which of the remaining three versions of our theory to use in practice will clearly depend on the accuracy constraints and available CPU time of the user, with the full theory at the higher end of the market (in principle very accurate, but almost too CPU expensive to work out and exploit properly), with the large α approximation on the lower end (reasonably acurate, but very cheap), and with the annealed approximation as a sensible compromise in between these two. Discussion Our aim in this sequel paper was to work out the general theory developed in [1] for several supervised (on-line and batch, linear and non-linear) learning scenarios in single-layer perceptrons, to develop a number of systematic approximations from the full set of equations, and to test the theory and its approximations against both exactly solvable benchmarks and extensive numerical simulations. The theory, built on the dynamical replica formalism [2], was designed to predict the evolution of trainingand generalisation errors, via a non-linear diffusion equation for the joint distribution of student and teacher fields, in the regime where the size p of the (randomly composed) training set scales as p = αN , with 0 < α < ∞ and where N denotes the number of inputs. In this regime the input data will in due course be recycled, as a result of which complicated correlations develop between the student weights and the realisations of the data vectors (with their corresponding teacher answers) in the training set; the student fields are no longer described by Gaussian distributions, training-and generalisation errors will no longer be identical, and the more traditional and familiar statistical mechanical formalism as developed for infinite training sets consequently breaks down. We have first worked out our equations explicitly for the special case of Hebbian learning, where the availability of exact results, derived directly from the microscopic equations, allows us to perform a critical test of the theory. For batch Hebbian learning we can demonstrate explicitly that our theory is fully exact. For on-line Hebbian learning, on the other hand, proving or disproving full exactness requires solving a non-trivial functional saddle-point equation analytically, which we have not yet been able to do. Nevertheless, we can prove that our theory is exact (i) with respect to its predictions for Q, R and E g , (ii) with respect to the first moments of the conditional field distributions P [x|y] (for any y ∈ ℜ), and (iii) in the stationary state. In order to also generate predictions for intermediate times we have constructed an approximate solution of our equations, which is found to describe the results of performing numerical simulations of on-line Hebbian learning essentially perfectly. No exact benchmark solution is available for non-Hebbian (i.e. non-trivial) learning rules, leaving numerical simulations as the only yardstick against which to test our theory. Motivated by the need to solve a functional saddle-point equation at each time step in the full theory, and by the presence of nested integrations, we have constructed a number of systematic approximations to the original equations. We have compared the predictions of the full theory and of the three approximation schemes with one another and with the results obtained upon performing numerical simulations of non-linear learning rules, such as Perceptron and AdaTron, in large perceptrons (of size N = 10, 000), with various values of learning rates η and relative training set sizes α. One of the approximations, a conditionally-Gaussian saddle-point approximation in the spirit of the particular approximation that was found to work perfectly for Hebbian learning, turned out to perform badly for general non-Hebbian rules. The other two approximations, the large α approximation and the partially annealed approximation, each have their specific usefulness; the former is extremely simple and fast, whereas the latter is overall more accurate, but more expensive in its CPU requirements (so that in practice its true accurateness cannot always be realised). Yet, the large α approximation still works remarkably well, even for small α, in spite of it being so simple that it can be written as a fully explicit set of equations for Q, R and the joint field distribution P [ . The observed accurateness of these simple equations in the small α regime suggests that for α → 0 the leading term in the diffusion equation for P [x|y] is the first term in the right-hand side, which reflects the direct effect of pattern recycling, and which indeed has not been approximated. For a discussion of further theoretical developments and refinements of the present dynamical replica formalism we refer to [1]. We believe that our theory offers an efficient tool with which to analyse and predict the outcome of learning processes in single-layer networks. In particular, for those who are primarily interested in the progress and the outcome of learning processes there is no real need to understand the full details of the derivation in [1]; as in this paper, one can simply adopt the macroscopic laws of [1] (or one of the two appropriate approximations, to save CPU time) as a starting point, and just apply them to the learning rules as hand. Generalization to multi-layer networks (with a finite number of hidden nodes) is straightforward, although numerically intensive [9]. The case of noisy teachers can also be studied with an appropriate extension of the present formalism [10], involving a joint distribution of three rather than two fields (namely those of student, 'clean' teacher, and 'noisy' teacher). In the example applications worked out so far in this paper (Hebbian learning, Perceptron learning and AdaTron learning) our formalism has been found to be either exact or an excellent approximation. It is not realistic to expect that simpler theories can be found with a similar level of accuracy. While putting the finishing touch to this manuscript a preprint was communicated [11] in which the authors apply the cavity method to the present problem. They manage to keep their theory relatively simple by restrict themselves in several serious ways (to batch learning only, and to gradient descent learning rules in order to use FDT relations) and by applying their theory only to a linear learning rule. Here also the present theory would have been both simpler and exact. An exact theory for both on-line and batch learning and for arbitrary learning rules can be constructed [12] using a suitable adaptation of the path integral methods as in [13], with the obvious appeal of full mathematical rigour at any time, but in describing transients it is going to be more complicated than the present one as it will be built around macroscopic observables with two time arguments rather than one (correlation-and response functions) and will take the form of an effective single weight process with a dynamics with coloured stochastic noise and retarded self-interactions. Multiplication of this equation by x followed by integration over x, together with usage of the general properties dx {P [x|y]Φ[x, y]} = 0 and dx xP 0 [x|y] = R 0 y, gives us the average of the conditional distribution P [x|y] at any time: Figure 1 : 1On-line Hebbian learning, simulations versus theoretical predictions, for η = 1 and α ∈ {0.25, 0.5, 1.0, 2.0, 4.0} (N = 10, 000). Upper curves: generalization errors as functions of time. Lower curves: training errors as functions of time. Circles: simulation results for E g ; diamonds: simulation results for E t . Solid lines: corresponding predictions of dynamical replica theory. Figure 2 :Figure 3 : 23Comparison between simulation results for on-line Hebbian learning (system size N = 10, 000) and dynamical replica theory, for η = 1 and α ∈ {0.5, 1.0, 2.0}. Dots: local fields (x, y) = (J ·ξ, B ·ξ) (calculated for questions in the training set), at time t = 50. Dashed lines: conditional average of student field x as a function of y, as predicted by the theory, x(y) = Ry + (ηt/α) sgn(y). Simulations of on-line Hebbian learning with η = 1 and N = 10, 000. Histograms: student field distributions measured at t = 10 and t = 20. Lines: theoretical predictions for student field distributions. α = 4 (upper), α = 1 (middle), α = 0.25 (lower). y)] n with x(y) = dx xP [x|y]. In our full (quenched disorder) calculation we find ourselves with the effective measure M [x|y] = lim n→0 M n [x|y]. In contrast, an alternative calculation, whereby the quenched average over all training sets would have been replaced by an annealed average over all training sets, would have led us to the value n = 1 rather than n = 0: M [x|y] = M 1 [x|y] Figure 4 : 4Numerical simulations of on-line Adatron learning, with N = 10,000, α = 1 and η = 1 2 . The scatter plots show the observed student and teacher fields (x, y) = (J ·ξ, B ·ξ) at times t = 5 (upper left), t = 10 (upper right), t = 15 (lower left) and t = 20 (lower right), as measured during simulations for the data in the training setD, drawn as points in the (x, y) plane. Note the development over time of an increasingly narrow 'ridge' along the line x = 0. [x, y] = sgn(y)θ[−xy] AdaTron : G[x, y] = |x| sgn(y)θ[−xy] Figure 5 : 5Comparison between the large α approximation of the theory and numerical simulations of on-line perceptron learning with N = 10, 000 and η = 1. Markers: training errors E t (circles) and generalisation errors E g (squares); finite size effects in the simulation data are of the order of the marker size. Lines: theoretical predictions for training errors (solid) and generalisation errors (dashed) as functions of time, according to the approximated theory. Training set sizes: α = 4 (upper left), α = 2 (upper right), α = 1 (lower left), and α = 0.5 (lower right). Figure 6 : 6Comparison between the large α approximation of the theory and numerical simulations of on-line Adatron learning with N = 10, 000 and η = 2. Markers: training errors E t (circles) and generalisation errors E g (squares); finite size effects in the simulation data are of the order of the marker size. Lines: theoretical predictions for training errors (solid) and generalisation errors (dashed) as functions of time, according to the approximated theory. Training set sizes: α = 4 (upper left), α = 2 (upper right), α = 1 (lower left), and α = 0.5 (lower right). Figure 7 : 7Comparison between the large α approximation of the theory and numerical simulations of on-line Perceptron learning, with N = 10, 000 and η = 1. Scatter plots (left): observed student and teacher fields (x, y) = (J·ξ, B·ξ) as measured at time t = 10 during simulations, for the data inD, drawn in the (x, y) plane. Contour plots (right): corresponding predictions for the joint field distribution P [x, y], according to the approximated theory. Training set sizes: α = 0.5, 1, 2, 4 (from top to bottom). Figure 8 : 8Comparison between the large α approximation of the theory and numerical simulations of on-line Adatron learning with N = 10, 000 and η = 2. Scatter plots (left): observed student and teacher fields (x, y) = (J ·ξ, B ·ξ) as measured at time t = 10 during simulations, for the data inD, drawn in the (x, y) plane. Contour plots (right): corresponding predictions for the joint field distribution P [x, y], according to the approximated theory. Training set sizes: α = 0.5, 1, 2, 4 (from top to bottom). Figure 9 : 9Comparison between the full numerical solution of our equations, as well as the three approximations of the theory, and the results of doing numerical simulations of on-line learning with N = 10, 000 and α = 1. Markers: training errors E t (circles) and generalisation errors E g (squares); finite size effects are of the order of the size of te markers. Lines: theoretical predictions for training errors (lower) and generalisation errors (upper) as functions of time, according to the theory. The different line types refer to: full equations (solid), annealed approximation (dashed), conditionally-Gaussian approximation (dashed-dotted) and large α approximation (dotted) (note: the dashed and solid curves fall virtually on top of one another). Left picture: Perceptron learning, with η = 1 2 . Right picture: AdaTron learning, with η = 3 2 . Figure 10 : 10Comparison between the full numerical solution of our equations and the results of doing numerical simulations of on-line learning with N = 10, 000 and α = 1. Histograms: conditional student field distributions P ± (x) = dyP [x, y]θ[±y] as measured at time t = 5. Smooth curves: corresponding theoretical predictions. Upper pictures: Perceptron learning, with η = 1 (left: P − (x), right: P + (x)). Lower pictures: AdaTron learning, with η = 3 2 (left: P − (x), right: P + (x)). Without loss of generality we can always normalize M according to dx M [x|y] = 1 for all y ∈ ℜ. From the physical meaning of q follows R 2 /Q ≤ q ≤ 1. After q and M [x, y] have been determined,y, z] ⋆ = dx M [x|y]e Bxz f [x, y, z] dx M [x|y]e Bxz (8) the key function Φ[x, y] in (1,3,5) is calculated as where we have also used the built-in property dx P[x|y]Φ[x, y] = 0 for all y. we could rewrite our macroscopic equations into Fourier language, i.e. in terms of P [k|y] = dx e −ikx P [x|y] andM [k|y] = dx e −ikx M [x|y]. The functional saddle-point equation, givingM [k], then becomesPAlternatively [k|y] = DzM [k+iBz|y] M [iBz|y] (11) and the diffusion equation takes the form d dt logP [k|y] = 1 α dk ′P [k ′ |y] P [k|y] ∂z logM [iBz|y] allow us to rewrite all remaining equations in Fourier representation, i.e. in terms ofP [k|y] = dx e −ikx P [x|y] andM [k|y] = dx e −ikx M [x|y]: gives lim B→0 M [x|y] = P [x|y], so we writeM [x|y] = P [x|y] [1 + ℓ>0 B ℓ m ℓ [x|y]], dx P [x|y]m ℓ [x|y] = 0 (38) x, y] only. For on-line learning these equationscan be simplified to P [x ′ |y] δ[x−x ′ −ηG[x ′ , y]]−δ[x−x ′ ] + 1 2 η 2 ∂ 2 ∂x 2 P [x|y] G 2 [x ′ , y ′ ] [x|y] G[x ′ , y ′ ] yy ′ + (x−x(y))(x(y ′ )−Ry ′ ) + (x−Ry)(x ′ −x(y ′ )) Q − R 2 with f [x, y] =Dydx P [x|y]f [x, y] and x(y) = dx xP [x|y]d dt Q = 2η xG[x, y] + η 2 G 2 [x, y] d dt R = η yG[x, y] d dt P [x|y] = 1 α dx ′ −η ∂ ∂x P Note that in[4] only the on-line calculation was carried out; the batch calculation can be done along the same lines. [x−Ry−(ηt/α) sgn(y)] 2 /(Q−R 2 ) 2π(Q−R 2 )(26) King's College London. Acc Coolen, D Saad, preprint KCL-MTH-99-32Coolen ACC and Saad D 1999 King's College London preprint KCL-MTH-99-32 . Acc Coolen, S N Laughton, D Sherrington, Phys. Rev. B. 538184Coolen ACC, Laughton SN and Sherrington D 1996 Phys. Rev. B 53 8184 Acc Coolen, D Saad, On-Line Learning in Neural Networks D. Saad (Ed). Cambridge: U.P.Coolen ACC and Saad D 1998 in On-Line Learning in Neural Networks D. Saad (Ed) (Cambridge: U.P.) . H C Rae, P Sollich, Acc Coolen, J. Phys. A: Math. Gen. 323321Rae HC, Sollich P and Coolen ACC 1999 J. Phys. A: Math. Gen. 32 3321 . W Kinzel, P Rujan, Europhys. Lett. 13473Kinzel W and Rujan P 1990 Europhys. Lett. 13 473 . M Biehl, H Schwarze, Europhys. Lett. 20733Biehl M and Schwarze H 1992 Europhys. Lett. 20 733 . O Kinouchi, N Caticha, J. Phys. A: Math. Gen. 256243Kinouchi O and Caticha N 1992 J. Phys. A: Math. Gen. 25 6243 . W H Press, B P Flannery, S A Teukolsky, W T Vetterling, Numerical Recipes in C. Press WH, Flannery BP, Teukolsky SA and Vetterling WT 1988 Numerical Recipes in C (Cam- bridge: U.P.) . Acc Coolen, D Saad, Y S Xiong, in preparationCoolen ACC, Saad D and Xiong YS 1999 in preparation Cwh Mace, Acc Coolen, Proc. NIPS*99. NIPS*99Mace CWH and Coolen ACC 1999 to be published in Proc. NIPS*99 . K Y Wong, Li S Tong, Y W , cond-mat/9909004Wong KY, Li S and Tong YW 1999 preprint cond-mat/9909004 . J A Heimel, Acc Coolen, in preparationHeimel JA and Coolen ACC 1999 in preparation . H Horner, Z. Phys. B86. 291371Z. Phys.Horner H 1992 Z. Phys. B86 291; Z. Phys. B87 371

The superconducting properties of the recently discovered double Fe2As2 layered high-Tc superconductor RbCa2Fe4As4F2 with Tc ≈ 30 K have been investigated using magnetization, heat capacity, transverse-field (TF) and zero-field (ZF) muon-spin rotation/relaxation (µSR) measurements. Our low field magnetization measurements and heat capacity (Cp) reveal an onset of bulk superconductivity with Tc ∼ 30.0(4) K. Furthermore, the heat capacity exhibits a jump at Tc of ∆Cp/Tc=94.6 (mJ/mole-K 2 ) and no clear effect of applied magnetic fields was observed on Cp(T) up to 9 T between 2 K and 5 K. Our analysis of the TF-µSR results shows that the temperature dependence of the magnetic penetration depth is better described by a two-gap model, either isotropic s+s-wave or s+d-wave than a single gap isotropic s-wave or d-wave model for the superconducting gap. The presence of two superconducting gaps in RbCa2Fe4As4F2 suggests a multiband nature of the superconductivity, which is consistent with the multigap superconductivity observed in other Fe-based superconductors, including ACa2Fe4As4F2 (A=K and Cs). Furthermore, from our TF-µSR study we have estimated an in-plane penetration depth λ ab (0) =231.5(3) nm, superconducting carrier density ns = 7.45 × 10 26 m −3 , and carrier's effective-mass m * = 2.45me. Our ZF µSR measurements do not reveal a clear sign of time reversal symmetry breaking at Tc, but the temperature dependent relaxation between 150 K and 1.2 K might indicate the presence of spin-fluctuations. The results of our present study have been compared with those reported for other Fe pnictide superconductors.

10.7566/jpsj.87.124705

1802.07334

9589610f5e94d7e2d0ea099f8b9ef704e8cd50c3

Multigap superconductivity in RbCa 2 Fe 4 As 4 F 2 investigated using µSR measurements (Dated: February 22, 2018) D T Adroja ISIS Facility OX11 0QXRutherford Appleton Laboratory, Chilton, Didcot OxonUnited Kingdom Physics Department Highly Correlated Matter Research Group University of Johannesburg PO Box 5242006Auckland ParkSouth Africa F K K Kirschner Department of Physics Clarendon Laboratory University of Oxford Parks RoadOX1 3PUOxfordUnited Kingdom F Lang Department of Physics Clarendon Laboratory University of Oxford Parks RoadOX1 3PUOxfordUnited Kingdom M Smidman Center for Correlated Matter Department of Physics Zhejiang University 310058HangzhouChina A D Hillier ISIS Facility OX11 0QXRutherford Appleton Laboratory, Chilton, Didcot OxonUnited Kingdom Zhi-Cheng Wang Department of Physics State Key Lab of Silicon Materials Zhejiang University 310027HangzhouChina Guang-Han Cao Department of Physics State Key Lab of Silicon Materials Zhejiang University 310027HangzhouChina G B G Stenning ISIS Facility OX11 0QXRutherford Appleton Laboratory, Chilton, Didcot OxonUnited Kingdom S J Blundell Department of Physics Clarendon Laboratory University of Oxford Parks RoadOX1 3PUOxfordUnited Kingdom Multigap superconductivity in RbCa 2 Fe 4 As 4 F 2 investigated using µSR measurements (Dated: February 22, 2018)numbers: 7470Xa7425Op7540Cx The superconducting properties of the recently discovered double Fe2As2 layered high-Tc superconductor RbCa2Fe4As4F2 with Tc ≈ 30 K have been investigated using magnetization, heat capacity, transverse-field (TF) and zero-field (ZF) muon-spin rotation/relaxation (µSR) measurements. Our low field magnetization measurements and heat capacity (Cp) reveal an onset of bulk superconductivity with Tc ∼ 30.0(4) K. Furthermore, the heat capacity exhibits a jump at Tc of ∆Cp/Tc=94.6 (mJ/mole-K 2 ) and no clear effect of applied magnetic fields was observed on Cp(T) up to 9 T between 2 K and 5 K. Our analysis of the TF-µSR results shows that the temperature dependence of the magnetic penetration depth is better described by a two-gap model, either isotropic s+s-wave or s+d-wave than a single gap isotropic s-wave or d-wave model for the superconducting gap. The presence of two superconducting gaps in RbCa2Fe4As4F2 suggests a multiband nature of the superconductivity, which is consistent with the multigap superconductivity observed in other Fe-based superconductors, including ACa2Fe4As4F2 (A=K and Cs). Furthermore, from our TF-µSR study we have estimated an in-plane penetration depth λ ab (0) =231.5(3) nm, superconducting carrier density ns = 7.45 × 10 26 m −3 , and carrier's effective-mass m * = 2.45me. Our ZF µSR measurements do not reveal a clear sign of time reversal symmetry breaking at Tc, but the temperature dependent relaxation between 150 K and 1.2 K might indicate the presence of spin-fluctuations. The results of our present study have been compared with those reported for other Fe pnictide superconductors. I. INTRODUCTION The discovery of high temperature superconductivity in fluorine-doped LaFeAsO (1111-family) with a transition temperature of T c ∼26 K by Kamihara et al. has generated a considerable research interest world-wide to understand the nature of the superconductivity in this new class of compounds 1 . It was realized soon after the discovery that the T c of Fe-based superconductors can be increased up to 56 K as observed in Gd 0.8 Th 0.2 FeAsO 2 , Sr 0.5 Sm 0.5 FeAsF 3 and Ca 0.4 Nd 0.6 FeAsF 4 . Until this discovery, high temperature superconductivity in cuprates, created the impression that only Cu-O planes are pivotal for understanding the mechanism of high temperature superconductivity 5,6 . Of course, the Fe-based superconductors do not contain Cu-O planes and some of the materials are even O free, for example, FeSe (11-family, T c =8 K at ambient pressure and 46 K in applied pressure), LiFeAs (111-family, T c = 18 K), CaFeAs 2 (112-family, T c = 20 K) and ThFeAsN (T c =30 K) [7][8][9][10] . Another highly investigated family of Fe-based superconductors is hole (i.e. K) and electron (i.e. Co, Ni, Rh and Pd) doped BaFe 2 As 2 (122-family), which have a body centered tetragonal ThCr 2 Si 2 -type structure (I4/mmm), where the ubiquitous Fe 2 As 2 layers of the Fe arsenide superconductors lie between the alkaline/alkaline earth atom layers shown in Fig. 1 [11][12][13][14][15] . Recently superconductivity with T c ∼35 K has been reported in CaAFe 4 As 4 (A = K, Rb, Cs, 1144-family) 16 and these materials consist of different arrangements of the layers along the c-axis also displayed in Fig. 1. In this structure, the alternating arrangement of the A and Ca layers leads to two inequivalent As sites either side of the Fe sheets. The crystallographically inequivalent position of the Ca and A atoms changes the space group from I4/mmm (as for the 122-family) to P4/mmm. Further, the different valence attraction from Ca 2+ and A 1+ layers to Fe 2 As 2 1.5− and the different ionic radii leads to different lengths of the As-Fe bonds, which was proposed to be an important parameter for controlling the T c of Fe-based superconductors 17 . Stoichiometric CaKFe 4 As 4 is intrinsically near optimal hole doping 16,19 and does not exhibit a high temperature structural phase transition 20 . Similar to the optimally doped 122 compounds, probes of the gap structure and inelastic neutron scattering results strongly suggest the presence of a fully gapped s ± state 21,22,[24][25][26] . Further angle-resolved photoemission spectroscopy (ARPES) measurements of CaKFe 4 As 4 re- port the presence of four superconducting gaps on different sheets of the Fermi surface 21,24 , which has been explained using a four-band s ± -wave Eliashberg theory emphasizing the important role of antiferromagnetic spin fluctuations 27 . Very recently Zhi-Cheng Wang et al. 28,29 have discovered high temperature superconductivity at 29-33 K in ACa 2 Fe 4 As 4 F 2 (A=K, Rb, Cs, 12442-family). The crystal structure is displayed in Fig. 1, where the Fe 2 As 2 layers are now sandwiched between A atoms on one side and Ca 2 F 2 on the other, again leading to two distinct As sites above (As 1 ) and below (As 2 ) the Fe-plane as in 1144. These materials are also situated near to optimal doping. The electronic structure and magnetic properties of KCa 2 Fe 4 As 4 F 2 have been calculated based on first-principle calculations and discussed in relation to the Fe-pnictide superconductors 30 . There are ten bands crossing the Fermi level in the nonmagnetic (NM) state, resulting in six hole-like Fermi surface (FS) sheets along the Γ-Z line and four electron-like FS sheets along the X − P line. The shape of the FS is more complicated than other FeAs-based superconductors, showing multiband character. Furthermore the fixed spin moment calculations and the comparisons between total energies of different magnetic phases indicate that KCa 2 Fe 4 As 4 F 2 has a strong tendency towards magnetism, i.e. the stripe antiferromagnetic state. It has been found that the self-hole-doping suppresses the spin-density wave (SDW) state, inducing superconductivity in the parent compound KCa 2 Fe 4 As 4 F 2 30 . The pairing symmetry of the Cooper pairs in a superconductor is manifested in an energy gap in the single-particle excitation spectrum. The superconduct-ing gap structure is an important characteristic for a superconductor. There is experimental evidence that cuprate-based unconventional superconductors have distinct d-wave nodal gap symmetry compared with conventional phonon-mediated superconductors which have nodeless s-wave gap 5,6 . On the other hand the superconducting gap symmetry in iron-based superconductors is rather more diverse and the subject of ongoing debate 12,13,31,32 . Whereas nodeless gap structures have been observed in some of the doped 122family [11][12][13][14][15] , 1144-family 21,22,[24][25][26] , A x Fe 2 Se 2 (A=K, Cs) 33 . More interestingly the single crystal µSR study on FeSe reveals a nodeless gap (anisotropic-swave) along the c-axis, but one nodal and one isotropic (s+d-wave ) gap in the ab-plane 44 . To understand the mechanism of unconventional superconductivity and develop realistic theoretical models of Fe-based superconductors it is very important to study the pairing symmetry and the nature of the superconducting gap. There is no general consensus on the nature of pairing in iron-based superconductors leading to a variety of possibilities ranging from s ++ wave to s ± , to d wave. Furthermore it is also important to investigate whether time-reversal symmetry (TRS) in the superconducting state is preserved or not as well as the role of spin-fluctuations. Broken symmetry can modify the physics of a system and nature of the pairing, thereby resulting in novel and uncommon behavior. Muon-spin rotation and relaxation (µSR) is an ideal and sensitive microscopic technique to investigate the properties of the superconducting state. Transverse field (TF) µSR provides information on the field distribution in the superconducting state and hence information on the penetration depth and gap symmetry. On the other hand zero-field (ZF) µSR allows the detection of very small internal fields and hence can provide direct information about whether TRS is preserved. Recently we have investigated the nature of the superconducting gap and TRS in ACa 2 Fe 4 As 4 F 2 (A=K and Cs) compounds using µSR measurements 45,46 . We found two superconducting gaps with at least one nodal gap in these compounds, but no clear sign of TRS breaking. It is therefore important to investigate the gap symmetry and TRS in A=Rb compound. Here we report TF-and ZF-µSR measurements of the A=Rb compound. Our study shows that the superfluid density derived from the depolarization rate of the TF-µSR fits better to two isotropic gaps following a s+s-wave model and ZF-µSR does not reveal any clear sign of TRS breaking below T c . II. EXPERIMENTAL DETAILS The sample was characterized using powder x-ray diffraction (XRD), magnetic susceptibility and heat capacity measurements. The heat capacity was measured using a Quantum Design Physical Property Measurement System (PPMS) between 1.8 and 80 K. A standard thermal relaxation method was used with a sample mass of 8 mg. The DC magnetization measurements were carried out using a Quantum Design Magnetic Property Measurement System (MPMS). Muon spin relaxation/rotation (µSR) experiments were carried out on the MuSR spectrometer at the ISIS pulsed muon source of the Rutherford Appleton Laboratory, UK 47 . The µSR measurements were performed in transverse−field (TF), zero-field (ZF) and longitudinal field modes. A powder sample of RbCa 2 Fe 4 As 4 F 2 was mounted on a silver (99.999%) sample holder. The sample was cooled under He-exchange gas in a He-4 cryostat operating in the temperature range of 1.5 K−300 K. TF−µSR experiments were performed in the superconducting mixed state in an applied field of 40 mT, well above the lower critical field of µ 0 H c1 ∼ 20 mT (see Fig.2c) of this material. Data were collected in the field−cooled (FC) mode, where the magnetic field was applied above the superconducting transition temperature and the sample was then cooled down to base temperature. Muon spin rotation and relaxation is a dynamic method that allows one to study the nature of the pairing symmetry in superconductors 48,49 . The vortex state in the case of type-II superconductors gives rise to a spatial distribution of local magnetic fields; which demonstrates itself in the µSR signal through a relaxation of the muon polarization. Zero-field (ZF) µSR measurements were performed from 1.2 K to 150 K in the longitudinal geometry. We also performed longitudinal field µSR measurements at 1.2 K and 35 K. The µSR data were analyzed using WiMDA 50 . III. RESULTS AND DISCUSSIONS The analysis of the powder x-ray diffraction at 300 K reveals that the sample is single phase and crystallizes in the tetragonal crystal structure with space group I4/mmm (No. 139, Z = 2) as shown in Fig. 1 28,29 . The refined values of the lattice parameters are a = 3.8716(1)Å and c = 31.667(1)Å. The low-field magnetic susceptibility measured in an applied field of 1 mT shows an onset of diamagnetism below 30 K indicating that superconductivity occurs at 30 K and the superconducting volume fraction is close to 100% at 10 K [ Fig. 2(a)]. This result confirms the bulk nature of superconductivity with T c = 30 K in RbCa 2 Fe 4 As 4 F 2 , which is comparable to T c = 33.3 K and 29 K observed in ACa 2 Fe 4 As 4 F 2 (A=K and Cs), respectively 28,29 . The magnetization isotherm M (H) curve at 3 K [ Fig. 2(b)] shows typical behaviour for type-II superconduc- Fig. 2(c)]. The upper critical field (µ 0 H c2 ) measurements using the field dependent resistivity reveals the slope dµ 0 H c2 /dT∼-13.9 T/K at T c 29 and the Pauli limit is µ 0 H P = 1.84T c = 55.2 T 51 . Further using the orbital limiting upper critical field, 30 kT. This value of H c2 gives the coherence length ξ=(Φ 0 /(2πH c2 )) 1/2 =1.04 nm, where Φ 0 = 2.07x10 −15 Tm 2 is the magnetic flux quantum. The specific heat (C p ) is displayed in Fig. 2(d) for zero field and an applied fields up to 9 T (Fig.2(f)). A clear anomaly is observed in the zero field C p corresponding to the superconducting transition at around 30.4(4) K. The jump in C p was estimated by linearly extrapolating the data above and below T c , yielding a jump of ∆C p /T c = 94.6 (mJ/mol K 2 ), which is smaller than 150 (mJ/mol K 2 ) observed in KCa 2 Fe 4 As 4 F 2 28 . To shed light on the nature of the gap symmetry we also performed field dependent heat capacity measurements up to a field of 9 T. We found that the heat capacity is almost independent of applied field between 2 K and 5 K. show the corresponding maximum entropy plots, respectively. It is clear that above T c the µSR spectra show a very small relaxation mainly from the quasi-static nuclear moments, and the internal field distribution is very sharp and centered near the applied field. However at 1.2 K µSR spectra show strong damping and the internal field distribution has two components, one very sharp near the applied field and one very broad which is shifted lower than applied field. The observed decay of the µSR signal with time below T c is due to the inhomogeneous field distribution of the flux-line lattice. We attribute the narrow component at the applied field to muons stopping in the silver sample holder, which indicates that the field distribution within the vortex lattice is described well by one Gaussian centered at a field below 40 mT. We have used an oscillatory decaying Gaussian function to fit the TF−µSR time dependent asymmetry spectra: µ 0 H c2 (0)=0.73(dH c2 /dT) T c T c , we have esti- mated µ 0 H c2 (0)=0.A(t) = A 1 e −σ 2 t 2 /2 cos(γ µ B 1 t + φ) + A 2 cos(γ µ B 2 t + φ), (1) where γ µ /2π = 135.5 MHz/T is the muon gyromagnetic ratio, σ is the Gaussian relaxation rate, φ is the phase, which is related to the detector geometry, A 1 and A 2 are the magnitudes of the terms from the sample and silver holder respectively, while B 1 and B 2 are respective internal fields. We grouped all detectors in 8 groups and all the groups were fitted simultaneously using the WIMDA software. The total amplitudes for each group of detectors were fixed. Furthermore, we first estimated the value of A 1 ≈0.7 and A 2 ≈0.3 by fitting the 1.2 K data and kept them fixed during the analysis allowing us to extract the temperature dependence of the relaxaton rate σ(T ). Equation 1 contains the total relaxation rate σ from the superconducting fraction of the sample; there are contributions from the vortex lattice (σ sc ) and nuclear dipole moments (σ nm ) (see Fig.4b inset), where the latter is assumed to be constant over the entire temperature range [where σ = (σ 2 sc + σ 2 nm )]. The contribution from the vortex lattice, σ sc , was determined by quadratically subtracting the background nuclear dipolar relaxation rate (σ nm =0.138(5)µs −1 ) obtained from the spectra measured above T c . As the applied field (40 mT) is much less than the upper critical field (µ 0 H c2 > 7 T), σ sc can be directly related to the effective penetration depth λ eff using the following equation 52 : σ sc /γ µ = 0.0609Φ 0 /λ 2 eff ,(2) where Φ 0 is the magnetic flux quantum. This relation between σ sc and λ eff is valid for 0.13/κ 2 <<(H/H c2 )<<1, where κ=λ/ξ 70 52 . Since RbCa 2 Fe 4 As 4 F 2 has a twodimensional layered crystal structure with large separation between Fe 2 As 2 -layers, the out of plane penetration depth (λ c ) is much larger than that in the plane (λ ab ), so that the effective penetration depth can be estimated as λ eff = 3 1 4 λ ab 53 . Furthermore the penetration depth is directly related to the normalized superfluid density, n ns . In our analysis we modelled the temperature dependent normalized superfluid density using the following equation 54 n ns (T ) = λ −2 ab (T, ∆) λ −2 ab (0) = 1+ 1 π 2π 0 ∞ ∆(T,ϕ) ∂f ∂E EdEdφ E 2 − ∆ 2 (T, ϕ) ,(3)where f = [1 + exp (−E/k B T )] −1 is the Fermi function. The temperature and angular dependence of the gap is given by ∆(T, ϕ)=∆ 0 δ(T /T c )g(ϕ), whereas g(ϕ) refers to the angular dependence of the superconducting gap function and ϕ is the azimuthal angle along the Fermi in the internal field below T c as shown in Fig. 4(b). From the analysis of the observed temperature dependence of λ −2 ab , using different models for the gap, the nature of the superconducting gap can be probed. We have analyzed the temperature dependence of λ −2 ab based on five different models, the single gap isotropic s-wave and line nodal d-wave models, as well as isotropic s+s-wave, s+d-wave and d+d-wave two-gap models. It was clear from the analysis that single-gap models did not fit the data (fits are not shown). The fits to the λ −2 ab data with various two-gap models using Eq. (3) are shown by lines in Fig. 4(a) and the estimated fit parameters are given in Table. I. It is clear from the goodness of fitted χ 2 values given in Table. I that the d+d-wave model does not fit the data very well. On the other hand the isotropic s+s-wave, s+d-wave models show good fits to the λ −2 ab data. The value of χ 2 = 3.0 for s+s-wave model is slightly less than 3.1 for s+d-wave model. The estimated parameters for the s+s (s+d)-wave model show one larger gap ∆ 1 (0) = 8.15 (8.08) (meV) and another much smaller gap ∆ 2 (0) = 0.88 (0.92) (meV). The smaller gap is a nodal gap in the s+d-wave model. Our µSR analysis suggests that an s+s-wave model explains better the temperature dependence of the superfluid density than an s+d-wave model. The value of λ ab (0) = 231.5±3 nm and T c = 29.19±0.04 K were estimated from the s+s-wave fit. The estimated value of 2∆ 1 (0)/k B T c = 6.48 from the s+s-wave fit is larger than the value 3.53 expected for BCS superconductors 59 , indicating the presence of strong coupling and unconventional superconductivity in RbCa 2 Fe 4 As 4 F 2 . On the other hand for the smaller gap the value 2∆ 1 (0)/k B T c = 0.7 is much smaller than the BCS value. The two-gap nature, one larger and another smaller than the BCS value, are commonly observed in Fe-based superconductors 60,61 as well as in Bi 4 O 4 S 3 62 . The observation of two isotropic gaps and nodeless superconductivity in RbCa 2 Fe 4 As 4 F 2 is very similar to that observed in CaKFe 4 As 4 , where clear evidence is found for multigap nodeless superconductivity with an s ± pairing state 21,22,[24][25][26] . Recently we have observed two gaps in ACa 2 Fe 4 As 4 F 2 (A=K and Cs) 45,46 and ThFeAsN 63 , but at least one gap appears to be nodal in these compounds. Two superconducting gaps (one larger and another smaller) were also observed in SrFe 1.85 Co 0.15 As 2 , with T c = 19.2 K in an STM study 64 . Moreover combined ARPES and µSR studies on Ba 1−x K x Fe 2 As 2 with T c = 32.0 K also revealed the presence of two gaps (∆ 1 = 9.1 meV and ∆ 2 = 1.5 meV) 65 . The recent µSR study on FeSe single crystals revealed that the superconducting gap is most probably anisotropic s-wave (nodeless) along the crystallographic c-axis, but it fits better to a two-gap s+d-wave model with one nodal gap in the ab-plane 44 . Furthermore, nodal superconductivity has been observed in cuprate superconductors 66,67 and the recently discovered quasi-1D Cr-based superconductors, A 2 Cr 3 As 3 (A = K and Cs) 68,69 . As with other phenomenological parameters characterizing a superconducting state, the penetration depth can also be related to microscopic quantities. Within London theory 48 , λ 2 L = λ 2 eff = m * c 2 /4πn s e 2 , where m * = (1 + λ e−ph )m e is the effective mass and n s is the density of superconducting carriers. Within this simple picture λ L is independent of magnetic field. λ e−ph is the electron-phonon coupling constant, which can be estimated from Θ D and T c using McMillan's relation 70 λ e−ph = 1.04 + µ * ln(Θ D /1.45T c ) (1 − 0.62µ * ) ln(Θ D /1.45T c ) + 1.04 ,(5) where µ * is the repulsive screened Coulomb parameter and usually assigned as µ * = 0.13. As we do not have heat capacity above 80 K for the present Rb-sample, we first estimated the value of Θ rmD for KCa 2 Fe 4 As 4 F 2 by fitting the heat capacity data between 50 K and 300 K to the Debye model 28 , which gave Θ K D =366 K. Then using a scaling factor 71 , which incorporates the differing molecular weight and unit-cell volume, we estimated Θ Rb D =351.6 K (similar for the Cs-sample Θ Cs D =344.3 K). For RbCa 2 Fe 4 As 4 F 2 we have used T c = 29.19 K together with µ * = 0.13 and have estimated λ e−ph = 1. 45. This value of λ e−ph is very similar to 1.38 for LiFeAs 72 , 1.53 for PrFeAsO 0.60 F 0.12 73 and 1.2 for LaO 0.9 F 0.1 FeAs 74 . On the other hand for many Fe-based superconductors (11family and 122-family) and HTSC cuprates (YBCO-123) smaller values of λ e−ph =0.02 to 0.2 and 0.02, respectively have been reported 75 . Further assuming that roughly all the normal state carriers (n e ) contribute to the superconductivity (i.e., n s ≈ n e ) and using the value of λ ab (0) = 231.5±3 nm, we have estimated the superconducting carrier density n s and effective-mass enhancement m * to be n s = 7.45×10 26 carriers/m 3 , and m * = 2.45m e , respectively. We also estimated these parameters for ACa 2 Fe 4 As 4 F 2 (A=K and Cs) samples (see Table-II) for comparison. Zero-field µSR measurements were performed from 1.2 K to 150 K and the results are displayed in Fig.5(a) for four selected temperatures. The data were fitted with the sum of a Lorentzian and Gaussian relaxation function A 0 (t) = A(aexp(−Λt) + (1 − a)exp(−σ 2 ZF t 2 /2)) + A bg ,(6) where A bg is the temperature independent background arising from muons stopping on the sample holder. The value of A bg =5.898(8)% and a=0.367 were estimated by fitting the 150 K data and were kept fixed during the analysis. At high temperature the relaxation is dominated by Gaussian decay, while at low temperature the relaxation changes to a Lorentzian decay. Moreover, there is a gradual decrease of initial asymmetry (A) with decreasing temperature, which suggests the development of fast component, which relaxes faster than the resolution of the experiment. The asymmetry exhibits a small drop below 70 K, which could be due to a competing magnetic/structural phase or related to some unknown phase transition and needs further investigation. The temperature dependence of Λ and σ ZF increases with decreasing temperature between 150 K and 75 K, followed by a weak temperature dependence between 75 K and 25 K. Below 25 K both Λ and σ ZF show a moderate temperature dependence. These results suggest the presence of weak magnetic fluctuations, but neither quantity shows a detectable anomaly upon passing through T c , indicating an absence of time reversal symmetry breaking. However, since Λ(T ) and σ ZF (T ) show some temperature dependence, a weak increase of the relaxation due to time reversal symmetry breaking cannot be entirely ruled out 69,76,77 . Furthermore we also performed longitudinal fields (LF) measurements at 1.2 K and 35 K in applied LF of 25, 40 and 50 mT and the data of 40 mT field are shown in the inset of Fig. 5a. At all applied longitudinal fields, the data showed negligible relaxation (i.e the asymmetry is almost constant with time), indicating very weak spin-fluctuations which require a very small LF field to decouple the µSR signal. The correlation between T c and σ sc (0) (or λ −2 ab (0)) observed in µSR studies has suggested a new empirical framework for classifying superconducting materials 78 . Here we explore the role of muon spin relaxation rate/penetration depth in the superconducting state for the characterisation and classification of superconducting materials as first proposed by Uemura et al. 78 . In particular we focus upon the Uemura classification scheme which considers the correlation between the superconducting transition temperature, T c , and the effective Fermi temperature, T F , determined from µSR measurements of the penetration depth 79 . Within this scheme strongly correlated "exotic" superconductors, i.e. high T c cuprates, heavy fermions, Chevrel phases and the organic superconductors, form a common but distinct group characterised by a universal scaling of T c with T F such that 1/10> (T C /T F ) >1/100 (Fig. 6). For conventional BCS superconductors 1/1000> (T c /T F ). Considering the value of T c /T F = 0.04 for RbCa 2 Fe 4 As 4 F 2 (see Fig. 6), this material can be classified as an exotic superconductor, according to Uemura's classification 78 . Furthermore we have also plotted the data of ACa 4 Fe 2 As 4 F 2 (A=K and Cs) in Fig. 6, which also belong to the same class. It has been found that the jump in the heat capacity ∆C p /T c at T c is also related to T c for electron and hole doped BaFe 2 As 2 superconductors 13,80 . We have plotted the heat capacity jump of ACa 2 Fe 4 As 4 F 2 (A=K and Rb) on the scaling plot shown in Fig.7. It is clear that for A=K and Rb compounds the heat capacity jump also follows this trend suggesting a common relation between ∆C p /T c ∼T c 2 , the so-called BNC scaling 80 . IV. CONCLUSIONS In conclusion, we have presented magnetization, heat capacity and transverse field (TF) and zero-field (ZF) muon spin rotation (µSR) measurements in the normal and the superconducting state of RbCa 2 Fe 4 As 4 F 2 , which has a double Fe 2 As 2 layered tetragonal crystal structure. Our magnetization and heat capacity measurements confirmed the bulk superconductivity with T c = 30.0 (4) K. From the TF µSR we have determined the muon depolarization rate in the FC mode associated with the vortex-lattice. The temperature dependence of the superfluid density fits better to a two-gap model, with either an isotropic s+s-wave or an s+d-wave gap, than to single gap isotropic s−wave or d-wave models. The s+s-and s+d-wave model fits give a goodness of fit (χ 2 ) value of 3.0 and 3.1, respectively, suggesting than an s+s-wave model is an appropriate for the gap structure of RbCa 2 Fe 4 As 4 F 2 . Furthermore, the value (for the larger gap) of 2∆ 1 (0)/k B T c = 6.48±0.08 obtained from the s+s-wave model fit is larger than 3.53, expected for BCS superconductors, indicating the presence of strong coupling superconductivity, that is supported through a larger value of λ e−ph , in RbCa 2 Fe 4 As 4 F 2 . Moreover, two superconducting gaps have also been observed in the Fe-based families of superconductors, including in other ACa 2 Fe 4 As 4 F 2 (A=K and Cs) compounds and hence our observation of two gaps is in agreement with the general trend observed in Fe-based superconductors. It is an open question why the A=Rb material is more consistent with two isotropic gaps, while A=K and Cs have at least one nodal gap despite the ionic size (lattice pa-TABLE I. Fitted parameters obtained from the fit to the σsc(T ) data of RbCa2Fe4As4F2 (as shown in Fig. 4(a)) using different gap models. Model Tc Gap value Gap ratio w λ −2 ab (0) χ 2 K ∆1(0), ∆2(0) (meV) 2∆(0)/kBTc µm −2 s+s wave 29.19 (4) 8. 15(1); 0.88(1) 6.48; 0.70 0.79(1) 17.37 (12) 3.0 s+d wave 29.19 (5) 8.08 (2) rameters and unit cell volume) increasing, while T c decreases linearly, going down the alkali atom group from K to Cs. Further confirmation of the presence of two gaps and their symmetry in RbCa 2 Fe 4 As 4 F 2 could be found from angle-resolved photoemission spectroscopy (ARPES) study and TF-µSR study on single crystals, for H c-axis and H ab-plane, of RbCa 2 Fe 4 As 4 F 2 . ACKNOWLEDGEMENT This work is supported by EPSRC grant EP/N023803. F.K.K.K. thanks Lincoln College, Oxford, for a doctoral studentship and National Key R and D Program of China (Grant No. 2017YFA0303100). DTA would like to thank the Royal Society of London for the UK-China Newton mobility funding. DTA and ADH would like to thank CMPC-STFC, grant number CMPC-09108, for financial support. . 1. (Color online) The tetragonal crystal structure of RbCa2Fe4As4F2. For comparison we have also given the crystal structure of RbFe2As2, CaRbFe4As4 (right side) and CaFeAsF (left side bottom)28,29 . FIG. 2 . 2(Color online) (a) Low-field dc-magnetic susceptibility measured in zero-field cooled (ZFC) and field cooled (FC) modes in an applied field of 1 mT. (b) The isothermal field dependence of magnetization at 3 K. (c) The isothermal field dependence of the magnetization at low fields at 3 K. The solid line shows a linear fit to the low field data. (d) Temperature dependence of heat capacity (Cp) versus temperature in zero field. (e) Cp vs T in an expanded scale near Tc. The solid lines show the linear fit above and below Tc and the vertical line shows the jump in the heat capacity at Tc. (f) (Cp) versus temperature in various applied magnetic fields up to 9 T. tivity. The lower critical field H c1 obtained from the M vs H plot at 3 K by linear fitting the data between 0 and 15 mT is about 20 mT [ FIG. 3 . 3(Color online)Muon spin rotation (µSR) measurements of RbCa2Fe4As4F2 in a transverse field of 40 mT at (a) 33.7 K (above Tc) and (b) 1.2 K (below Tc). The solid line shows a fit using Eq.(1). (c) and (d) display the corresponding maximum entropy spectra (above and below Tc) and the red lines show fits using one (c) and two (d) Gaussian functions. Figures 3 (a) and (b) show the TF−µSR precession signals above and below T c obtained in FC mode with an applied field of 40 mT (well above H c1 ∼ 20 mT but below H c2 >> 7 T, at 3 K, see Fig.2b) and Figs. 3 (c) and (d) FIG. 4 .Figure 4 44(Color online) (a) Temperature dependence of λ −2 ab of RbCa2Fe4As4F2. λ −2 ab of FC mode (symbols), where the lines are the fits to the data using Eq. 3 for various two-gap models. The solid black line shows the fit using an isotropic s + s-wave model with ∆1(0) = 8.15±0.01 meV and ∆2(0) = 0.88±0.01 meV, the dotted blue line shows the fit to an s+d-wave model with ∆1(0) = 8.08 ± 0.02 meV and ∆2(0) = 0.92 ± 0.01 meV and the dashed-dotted green line shows the fit to a d+d-wave model with ∆1(0) = 14.05±0.26 meV and ∆2(0) = 1.26±0.02 meV. The inset shows low temperature data in an expanded scale. (b) The normalized internal field shift as a function of temperature. The inset shows temperature dependence of total relaxation rate σ and the dotted line shows the temperature independent contribution of nuclear depolarization rate σnm. surface. We have used the BCS formula for the temperature dependence of the gap, which is given by δ(T /T c ) =tanh[(1.82)(1.018(T c /T − 1 )) 0.51 ] 55 . g(ϕ) 56,57 is given by (a) 1 for s−wave gap [also for s + s wave gap], (b) |cos(2ϕ)| for an d−wave gap with line nodes 54,55,58 . For the two-gap analysis, we have used a weighted sum of the two components of the resulting normalized superfluid density: n ns = wn ns (∆ 1 , T ) + (1 − w)n ns (∆ 2 (a) shows the temperature dependence of λ −2 ab , measured in an applied field of 40 mT. λ −2 ab increases with decreasing temperature confirming the presence of a flux-line lattice and indicates a decrease of the magnetic penetration depth with decreasing temperature. Further below 10 K λ −2 ab shows an upturn indicating multigap behavior. The onset of diamagnetism below the superconducting transition can be seen through the decrease FIG. 5. (Color online) (a) Zero-field µSR spectra at four selected temperatures. The solid red lines show the fit described in the text. (b)-(c) The fit parameters versus temperature of zero-field µSR spectra of RbCa2Fe4As4F2. The dotted vertical line shows the transition temperature. The inset in (a) shows the spectra measured in an applied longitudinal field of 40 mT. FIG. 6 . 6(Color online) A schematic representation of the Uemura plot of superconducting transition temperature Tc against effective Fermi temperature TF . The dotted squares shows the region where ACa4Fe2As4F2 (A=K, Rb and Cs) compounds are located. The solid cyan square, solid blue circle and dark green horizontal triangle show the points for A=K, Rb and Cs, respectively. The "exotic" superconductors fall within a common band for which 1/100< Tc/TF <1/10, indicated by the region between two red color dashed lines in the figure. The solid black line correspond to the Bose-Einstein condensation temperature (TB).79 . The positions of A=K, Rb and Cs on the plot indicate that these materials belong to exotic superconductors family. online) ∆Cp/Tc vs Tc for the 122-family of FeAs-based superconductors from Ref.13,80 . The half filled circles red and blue colors (normalized by a factor 2 for comparison between 1244-family (4 Fe-atoms per formula unit) and 122-family (2 Fe-atoms per formula unit) are for ACa2Fe4As4F2 (A=K and Rb) compounds, respectively. The solid line is the fit to a quadratic power law 13 . and FeTe 1−x Se x 34 , the signatures of nodal superconducting gaps have been reported in LaOFeP 35 , LiFeP 36 ,KFe 2 As 2 37,38 , BaFe 2 (As 1−x P x ) 2 39 , BaFe 2−x Ru x As 2 40 and FeSe 41 . Furthermore applied pressure and dop- ing or chemical pressure change the gap symmetry from nodeless to nodal in Ba 0.65 Rb 0.35 Fe 2 As 2 42 and in BaFe 2−x Ni x As 2 43 TABLE II. 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With the fundamental stress mechanism of accretion disks identified-correlated MHD turbulence driven by the magneto-rotational instability-it has become possible to make numerical simulations of accretion disk dynamics based on well-understood physics. A sampling of results from both Newtonian 3-d shearing box and general relativistic global disk MHD simulations is reported. Among other things, these simulations have shown that: contrary to long-held assumptions, stress is continuous through the marginally stable and plunging regions around black holes, so that rotating black holes can give substantial amounts of angular momentum electromagnetically to surrounding matter; the upper layers of accretion disks are primarily supported by magnetic pressure, potentially leading to interesting departures from local black-body emitted spectra; and initially local magnetic fields in accretion flows can, in some cases, spontaneously generate large-scale fields that connect rotating black holes to infinity and mediate strong relativistic jets.

10.1017/cbo9780511794254.022

0709.1489

00b0e11fe8250e62885f3904ae976b34c0e6c440

Making black holes visible: accretion, radiation, and jets 10 Sep 2007 J U L I A N Department of Physics and Astronomy Johns Hopkins University 21218BaltimoreMDUSA H K R O L I K Department of Physics and Astronomy Johns Hopkins University 21218BaltimoreMDUSA Making black holes visible: accretion, radiation, and jets 10 Sep 2007arXiv:0709.1489v1 [astro-ph] With the fundamental stress mechanism of accretion disks identified-correlated MHD turbulence driven by the magneto-rotational instability-it has become possible to make numerical simulations of accretion disk dynamics based on well-understood physics. A sampling of results from both Newtonian 3-d shearing box and general relativistic global disk MHD simulations is reported. Among other things, these simulations have shown that: contrary to long-held assumptions, stress is continuous through the marginally stable and plunging regions around black holes, so that rotating black holes can give substantial amounts of angular momentum electromagnetically to surrounding matter; the upper layers of accretion disks are primarily supported by magnetic pressure, potentially leading to interesting departures from local black-body emitted spectra; and initially local magnetic fields in accretion flows can, in some cases, spontaneously generate large-scale fields that connect rotating black holes to infinity and mediate strong relativistic jets. Prolog: the classical view of accretion disks It has been understood for decades that accretion through disks can be an extremely powerful source of energy for the generation of both photons and material outflows. When the central object is a black hole, the gravitational potential at the center of the disk is relativistically deep, so that the amount of energy that might be released per unit rest-mass accreted can be a substantial fraction of unity. If the central black hole spins, an additional store of tappable energy resides in its rotation. At the same time, however, the physical processes by which matter inflow is transmuted into observable outputs has long remained extremely murky. For matter to move inward, it must somehow lose its orbital angular momentum. That there might be some sort of inter-ring friction seems plausible, given the orbital shear, but the way this frication is generally envisaged is in terms of an imaginary "viscosity coefficient" famously parameterized by Shakura & Sunyaev (1973) as αc s h, for local sound speed c s and vertical scale-height h. This ansatz is based entirely on dimensional analysis: the (unknown) local stress is set equal to a dimensionless number α times the local pressure solely because the local pressure has the same units as stress. Although there is no particular reason why the stress, measured in local pressure units, should always have the same value, α is very frequently assumed to be a constant at all places and at all times. Moreover, despite the fact that ordinary molecular viscosity fails miserably to explain the friction, it is often assumed that the stress is some sort of intrinsically dissipative kinetic process whose operation is at least analogous to that of conventional viscosity. Inter-ring torques do work, moving energy outward. Simultaneously, matter moves inward, carrying its orbital energy. In a steady-state disk, where the mass inflow rate is the same at all radii, these two energy fluxes do not cancel. Instead, there is a net amount of energy that must be deposited in each ring. If one thinks of the torques as due to some sort of kinetic process like viscosity, it is natural to suppose that this energy imbalance is deposited as heat. The radial profile of this heating in a steady-state disk can be easily written down when the disk is in steady state, provided one is able to guess a boundary condition at the inner edge of the disk (we will return to this issue in greater detail later). Integrating over the radial heating profile then immediately yields the total amount of energy per accreted rest-mass that could, in principle, be used for radiation, i.e., the radiative efficiency. Unfortunately, its value depends strongly on the guessed boundary condition. When the matter density is large enough (as it often should be), atomic collisional processes can efficiently transform the heat into photons, which can then escape after diffusing vertically through the disk. In conditions of high density and large optical depth, the emergent spectrum should be nearly thermal, an argument that has led many to assume that the spectrum is locally Planckian. Note the use of "it is natural to suppose" and "can" and "should be" and generic terminology such as "collisional processes" here; the specific mechanisms for all these steps are as little understood as the actual torque is when analyzed through the α-model. How exactly to make use of the black hole's rotational energy has been in a similarly unsatisfactory state. Although Blandford & Znajek (1977) pointed out thirty years ago that magnetic fields can in principle efficiently convey black hole rotational energy from deep within the black hole's ergosphere all the way to infinity, knowledge about this mechanism's details has been almost as scanty as for accretion. The particular solution found by Blandford and Znajek and extended by Phinney (1984) assumes that the magnetic field is essentially force-free, with (almost) negligible matter inertia everywhere (including in the plane of the accretion disk) and is valid only for small spin parameter a/M . In addition, nothing in that theory specifies the strength of the magnetic field, or explains how it came to extend to infinity. Genuine disk physics I would not have painted such a gloomy picture if I did not intend quickly to adopt a very different attitude and report on some very significant recent progress. As a result of this progress, many of the mysteries bemoaned in the preceding section can now be substantially solved by the application of well-understood physical processes. In some cases, the main stumbling block is not lack of physics knowledge but lack of computing power. With this recently-gained understanding, it has become possible to outline a program by which the entire process, from mechanics of mass inflow to photon generation to jet launching, might reasonably be followed as a sequence of connected events. The story behind these advances begins fifteen years ago when Steven Balbus and John Hawley pointed out that weak magnetic fields destabilize an orbiting disk and lead to the rapid growth of MHD turbulence . Orbital shear is what drives this "magneto-rotational instability"; orbital shear also enforces a correlation in the resulting MHD turbulence such that the magnetic stress −B r B φ /4π is always on average positive. That is, the shear itself ensures that the magnetic forces transmit angular momentum outward. It is important to note that magnetic stress, unlike viscosity, is not intrinsically dissipative. The local rate of heat generation is not always proportional to the local stress. On the other hand, it is intimately connected to dissipation. Nonlinear mode-mode interactions can convey turbulent energy from the relatively large lengthscales on which the turbulence is stirred to much shorter lengthscales on which a variety of genuinely dissipative processes can act efficiently. When integrated over a large enough volume (vertically-integrated through the disk and wide enough to make averaging well-defined), the heating rate must agree with the one predicted by the time-steady disk picture described above, but there is no requirement for it to match up with the stress rate locally, either in time or space. The radiative output can then be described by taking a position-and time-dependent heating rate from turbulence calculations and solving the radiation transfer equation using physical opacities. If the vertical structure of the disk is known well-enough, non-LTE effects in the disk atmosphere can be incorporated, and departures from locally Planckian spectra can be predicted. To find the global radial profile of heating and radiation, it is still necessary to understand that inner boundary condition. Thirty years ago, heuristic arguments based on hydrodynamic intuition pointed toward a boundary condition requiring the stress to be zero at and inside the marginally stable orbit, but even then it was recognized that if magnetic fields were important, a different choice might be necessary (Page & Thorne 1974). Now that we know magnetic fields are essential to the entire accretion process, what Page and Thorne saw as a back-of-the-mind worry is now front-and-center. Fortunately, as we shall shortly see, it is now possible to calculate, rather than guess, what happens to the magnetic stress in the marginally stable region. If we can compute the magnetic field in the marginally stable region, then we can also compute the magnetic field even closer to the black hole's event horizon. Part of its nature is determined by dynamics within the accretion flow itself; part may be constrained by boundary conditions at infinity, where it is possible that some fieldlines are anchored. Numerical simulations The only fly in the ointment is that analytic methods for calculating the properties of any turbulent system are extremely limited in their power. Numerical simulation is really the only tool we have for examining fully-developed turbulence, and it has its own limitations. Nonetheless, some fifteen years after the first attempt to study numerically the properties of MHD turbulence in accretion disks, a great deal has been learned. The simulations that have been done to date can all be divided into two classes: shearing boxes and global disks. To make a shearing box, imagine cutting out from a complete disk a narrow radial annulus of limited azimuthal extent. When its azimuthal length is small compared to a radian, it can be well approximated as straight along the tangential direction. Rather than describing the orbital motion by a rotational frequency Ω(r) = Ω 0 (r/r 0 ) −3/2 for radius r and annular central radius r 0 , we can instead approximate it by writing the orbital speed as v y = r 0 Ω 0 [1 − (1/2)(r/r 0 − 1)], for r/r 0 − 1 ≪ 1. The equations of motion can then be written in the rotating frame with appropriate centrifugal and Coriolis terms. At a similar level of approximation, the vertical gravity is g z = zΩ 2 0 . Shearing boxes are best for wide dynamic range studies of the turbulent cascade, well-resolved exploration of internal vertical structure within the disk, and tracing disk thermodynamics. The advantage for the latter subject is that the relatively large dynamic range in lengthscale for turbulent dynamics allows one to localize comparatively well where dissipation occurs and then follow the diffusion of radiation away from its source regions. In a global disk simulation, one places a large amount of mass on the grid in an initial state of hydrodynamic equilibrium. To avoid noise propagation across the boundary of the problem area, the outer boundary is placed well outside the outer edge of the initial mass distribution. When angular momentum begins to flow outward through the disk as a result of the MHD turbulent torques, matter from the inner part moves inward while a small amount of mass on the outside moves outward, soaking up the angular momentum that has been carried outward to it. Thus, these global disks are truly "accretion disks" only in their inner portions. For this reason, it is necessary to locate their initial centers far enough outside the innermost stable circular orbit (the ISCO, i.e., the radius of marginal stability) that there can be a reasonable radial dynamic range for the accretion flow proper. In the current state-of-the-art, shearing box simulations employ 3-d Newtonian dynamics in the MHD approximation including radiation forces. By means of integrating both an internal energy and a total energy equation, they can track the numerical dissipation rate as a function of time in each cell of the simulation; this numerical dissipation rate is assumed to mimic the physical dissipation and is used to increase the heat content of the cells where and when it occurs. Radiation transfer is computed in the approximation of flux-limited diffusion, using thermally-averaged opacities . Global disk simulations are best for following the inflow dynamics, the radial profile of magnetic stress, the surface density profile that the stress produces, and non-local magnetic field effects. They also permit identification of typical global structures (the main disk body, disk "coronae", etc.) and the study of jets. The most physically complete global disk simulations now available use 3-d fully general relativistic dynamics in the MHD approximation, but they take no account of radiation. In one version (Gammie et al. 2003), the total energy equation is integrated, so energy is rigorously conserved; in another (De Villiers & Hawley 2003), an internal energy equation is solved, permitting numerical energy losses. The advantage of the former method is that numerical dissipation does not lead to energy loss; its disadvantage is that physical radiation losses don't occur either. The advantages and disadvantages of the latter method are more or less reversed, if one is willing to accept numerical energy losses as approximating genuine radiative losses. In all cases, both shearing box and global simulations, the magnetic field is nearly always assumed to have zero net flux. This choice is made largely because it's the simplest and involves the smallest number of arbitrary choices: the initial field on the boundary of the simulation is always zero. On the other hand, it is possible that there can be largescale fields running through real accretion flows, and they may have substantial effects on the character of those flows; this question remains to be investigated. Selected Results The body of this talk will be devoted to a brief summary of some of the principal achievements of these simulations. Consistent with my title, I will focus on three topics: what we have learned from the global simulations about the radial profile of stress (and possibly of dissipation); what shearing box simulations have shown us about the vertical structure of disks, and how that can influence the character of the emitted spectrum; and how magnetically-driven accretion can (or maybe not) launch relativistic jets. Radial stress profiles The De Villiers-Hawley simulation code is designed to do an excellent job of conserving angular momentum and propagating magnetic fields reliably, but is less good at conserving energy. Consequently, we believe its description of the electromagnetic stress and its relation to mass inflow should be reliable, but it is much more difficult to use the data from these simulations to predict dissipation and the radiation that may follow from it. For the time being, then, we can discuss the stress with some confidence, while using it as an indirect and approximate indicator of dissipation. Figure 1 shows the instantaneous shell-integrated electromagnetic stress (i.e., −b r b φ + ||b|| 2 u r u φ , for magnetic four-vector b µ and four-velocity u µ ) evaluated in the local fluidframe at late-times in two simulations, one with a non-rotating black hole, the other with a black hole having spin parameter a/M = 0.9. For comparison, the figure also shows the stress predicted by the Novikov-Thorne model (i.e., a time-steady disk with a zero-stress inner boundary condition) when the accretion rate is the same as the timeaveraged value in the corresponding simulation. As can readily be seen, the zero-stress boundary condition fails drastically to describe what actually happens. In both cases, the electromagnetic stress continues quite smoothly inward through the marginally stable region. In the Schwarzschild case, the stress rises slowly inward until just outside the event horizon, and then plummets as the event horizon is approached. In the Kerr case, the stress rises sharply upward and does not diminish even very near the horizon. It is easy to understand qualitatively both stress profiles. Because there is no reason for the magnetic field to disappear suddenly at the ISCO, while orbital shear continues to stretch any radial components in the azimuthal direction, there is no physical mechanism to eliminate magnetic stress there (Krolik 1999, Gammie 1999; rather, the stress continues and, if anything, strengthens. The contrast in behavior between the spinning and non-spinning cases can just as easily be understood when one thinks of stress as momentum flux. The electromagnetic stress is nothing else than an outward flux of angular momentum, carried in the electromagnetic field. A non-rotating black hole has no angular momentum to give up, so it cannot act as a source for outgoing electromagnetic angular momentum flux; on the other hand, the angular momentum of a rotating black hole can be tapped, and we see this process in action here. Those concerned by an outflow of anything from an event horizon should have their qualms removed by the recognition that there is nothing to prevent a rotating black hole from swallowing negative angular momentum, i.e., angular momentum corresponding to rotation in the opposite sense. This is, of course, completely equivalent to releasing positive angular momentum. Indeed, deep in the ergosphere, the electromagnetic energy-at-infinity is frequently negative (Krolik et al. 2005). As previously discussed, although the work done by magnetic stress does not necessarily correspond to any particular local rate of heating, there is a relationship on a more globally-averaged level. Thus, the curves of integrated fluid-frame stress shown in Figure 1 hint that dissipation may also continue smoothly across the marginally stable orbit, also contrary to the guessed boundary condition of the Novikov-Thorne model. A further suggestion that this is so comes from a different argument. Many of the specific physical mechanisms of dissipation in this context are associated with regions of high current density. If, for example, there is a (small: we make the MHD approximation, after all) uniform resistivity η in the fluid, the local heating rate is η||J|| 2 , where J µ is the electric current four-vector and ||J|| is its scalar magnitude. In fact, the dissipation may be even more closely associated with ||J|| 2 than this simple guess would suggest because there are a number of plasma instabilities that create anomalous resistivity precisely where the current is strong. Maps of the current density show that it is strongly concentrated toward the center of the accretion flow, rising rapidly into the plunging region inside the ISCO (Hirose et al. (2004)). To the degree that current density indicates candidate regions for rapid magnetic energy losses, this signal, too, suggests that there may be a great deal of dissipation in and within the marginally stable region. The continuation of stress through the plunging region can also be looked at from a different point of view: in the language of the Shakura-Sunyeav α model. Their argument from dimensional analysis was that the time-averaged vertically-integrated stress should be comparable to the time-averaged vertically-integrated pressure. Our data confirm that this is so, provided one interprets "comparable" loosely. In the disk body, that is, at radii well outside the ISCO and well inside the initial pressure maximum (beyond which there is no accretion), the time-average ratio at a single radius of vertically-integrated stress to vertically-integrated pressure is generally in the range ∼ 0.01-0.1. However, the instantaneous value of this ratio can easily change by factors of several over an orbital period. Moreover, if one tracks this ratio from somewhat outside the ISCO to well inside, there is a consistent trend for the ratio between stress and pressure to increase. As shown in Figure 1, the stress generally increases inward in this region; because the radial speed of the accretion flow also increases inward, the vertically-integrated density and pressure of the matter tend to decrease. The result is that the ratio of stress to pressure increases sharply, often rising by factors of 10-100 from the disk body to deep inside the plunging region. Thus, a description of inflow dynamics near the ISCO in terms of a constant α parameter is strongly in conflict with the results of these simulations. Claims (as are often made) based on assuming a constant value of α within this region are therefore on very shaky ground. Internal vertical structure Having thus emphasized the consequences of magnetically-driven accretion in the inner part of the accretion flow, it is now time to turn to its implications for the internal structure of accretion disks at larger radii. The best tool for studying this problem is shearing box simulations that both accurately conserve energy and follow radiation transfer (as described in Hirose et al. 2006). In this review, we will briefly discuss two of the pricipal results of these simulations: their implications for thermal stability in disks and the fact that disk upper layers are generically supported primarily by magnetic fields. In their classic paper on the α-model, Shakura and Sunyaev (1973) also predicted that the inner regions of all disks surrounding black holes in which the accretion rate is more than a small fraction of the Eddington rate should be dominated by radiation pressure. Three years later (Shakura & Sunyaev 1976), the same two authors demonstrated that, within the approximation scheme of the vertically-integrated α-model, radiation pressure dominance leads directly to thermal instability. If so, the standard equilibrium solution for the region from which most of the light is generated in the brightest accreting black hole systems is unstable. To this day, no satisfactory resolution to the question of, "What actually happens in these circumstances?" has emerged. One of the principal motivations for our program of simulating shearing boxes with radiation generation and transport is to answer this question. At this stage, there is progress to report, but not yet any firm answer to the big question. When the gas pressure is dominant, it is clear that a truly stable steady-state can be found. Figure 2a illustrates this point by showing the "light-curve" of a shearing box disk segment in which the radiation pressure p r is only about 20% of the gas pressure p g . The fluctuations in radiative output are only at the tens of percent level. On the other hand, increasing radiation pressure does tend to drive fluctuations. When Figure 3. Vertical profiles of several kinds of pressure in a simulation whose volume-and time-averaged ratio of radiation pressure to gas pressure was ≃ 0.2 ). The thick curves are time-averaged over fifty orbital periods starting from the end of initial transients; the thin curves are the initial conditions. Magnetic pressure is shown by the solid curves, radiation by the dotted curves, and gas pressure by the dash-dot curves. the radiation and gas pressures are comparable (Fig. 2b), the output flux varies over a range of a factor of 3-4. Intriguingly, although p r /p g at its greatest is above the threshold for instability suggested by Shakura and Sunyaev, and stays there for as long as 5 cooling times, the disk exhibits large limit-cycle oscillations, but no unstable runaway. We are actively investigating whether the instability remains under control at still higher values of p r /p g (Hirose et al., in preparation). A consistent result of all vertically-stratified shearing box studies is that their upper layers are magnetically-dominated (Miller & Stone 2000; and as shown in Fig. 3, taken from Hirose et al. 2006). Although the data shown here are from a particular gasdominated simulation, more recent work studying shearing boxes with radiation pressure comparable to gas pressure ) and radiation pressure considerably greater than gas pressure (Hirose et al., in preparation) shows very much the same pattern: independent of whether gas or radiation pressure dominates near the midplane, by a few scale-heights from the center, magnetic pressure is larger than either one. A somewhat surprising corollary of magnetic dominance in the upper layers is that "coronal" heating is rather limited. Strong hard X-ray emission is so commonly seen from accreting black holes, no matter whether the central black hole has a mass ∼ 1M ⊙ or ∼ 10 9 M ⊙ , that somewhere in the system there must be a region of intense heating with only small matter density and optical depth. Otherwise, there would be no way to heat electrons to the ∼ 100 keV temperatures required to produce the X-rays. This region is generally called the accretion disk "corona", and it is often thought of in conceptual terms derived from experience with the Solar corona: it is imagined that somehow magnetic field loops emerge buoyantly from the nearby disk, twist and cross, and release energy at reconnection points. Unfortunately for this popular scenario, these simulations, the first to treat the dynamics of the upper layers of disks in a consistent fashion, show no sign of anything In this simulation, the mean gas and radiation pressures were comparable, but there were order unity fluctuations. The solid curve pertains to those times when the radiation presssure was particularly high, the dashed curve to those times in which it was particularly low. resembling it. The very fact that magnetic fields dominate the energy density of the upper strata of disks means that these regions are comparatively quiet, and the strong magnetic fields themselves prevent any twisting or field line crossing that might lead to reconnection. Rather, orbital shear stretches radial field components into azimuthal field with great regularity and smoothness (Hirose et al. 2004). At the same time, the nonlinearly-saturated Parker instability (i.e., magnetic buoyancy counter-balanced by magnetic tension) supports smooth "plateaus" of field interrupted by occasional narrow cusps ). The smoothness and steadiness of the magnetic field in this kind of corona tends to suppress dissipation. Indeed, the dissipation in shearing box segments of accretion disks is typically confined to the central regions of the disk. Figure 4 illustrates this fact in a shearing box whose radiation and gas pressures were, on a time-averaged basis comparable, but in which the ratio of radiation to gas pressure fluctuated over the range 0.5-2. Whether the energy content of the disk was high (and the radiation pressure dominated the gas pressure) or low (and the ratio went the other way), the dissipation was still confined to the inner few scale-heights of the disk. Although magnetic dominance in the upper layers of disks likely does not lead to strong coronal heating, it does have other potentially important observational consequences. Chief among them is the fact that when magnetic pressure gradients replace gas pressure gradients as the matter's principal support against gravity, the density of the gas must fall. Because the photosphere of the disk is located where magnetic support is so important, we immediately infer that previously-estimated photospheric densities were too high, and that LTE may not be enforced as thoroughly as previously thought (Blaes et al. 2006). The locally-emitted spectrum may therefore have larger deviations from Planckian, and if these features are sufficiently strong, may be visible in the disk-integrated spectrum. An example is shown in Figure 5; when magnetic support is properly included in the atmosphere structure, a prominent CVI edge appears. Still further departures from conventional spectral predictions may arise from the fact that at any given moment, the atmosphere can depart substantially from the usual picture of plane-parallel symmetry. Figure 5. Output spectrum from a gas-dominated shearing box at 55 • from the local vertical direction. The solid curve shows the spectrum predicted when the atmosphere is magneticallysupported, the dashed curve shows the predicted spectrum when magnetic support is neglected (Blaes et al. 2006). The surprising quietness of the magnetically-dominated regions of shearing box atmospheres motivates a search for other places to supply the intense heating required to explain the observed hard X-ray emission. Better places to look might include the plunging region, which is both strongly magnetized and highly dynamical, and the region just above it from which jets are launched. Jet launching For many years, two models have dominated thinking about the launching of jets: the Blandford-Znajek mechanism (Blandford & Znajek 1977) and the Blandford-Payne scheme (Blandford & Payne 1983). The two models share two central elements: largescale poloidal magnetic fields that extend from infinity and pierce the midplane of the accretion flow; and dynamically-enforced rotation. They are distinguished by whether the rotation is enforced by space-time frame-dragging (Blandford-Znajek) or the orbiting matter of the accretion disk (Blandford-Payne). Because the latter depends in an essential way on accretion, whereas the other (at least in principle) does not, they are also distinguished by whether the inertia of matter is significant. Lastly, they differ in the source of power for the jets: orbital energy of accreting matter for Blandford-Payne, the rotational kinetic energy of the black hole itself for Blandford-Znajek. Both models' dependence on externally-imposed large-scale fields is problematic because the most natural way to bring magnetic fields into either the inner parts of accretion disks or all the way to the black hole event horizon is by advection along with accreted fluid. However, we have no way of knowing whether such large-scale connections might survive the many orders of magnitude in compression suffered by the matter; reconnection might destroy large-scale connections far from the black hole. On the other hand, if even a small fraction of the flux survives, over time it might still build up to be significant. This remains an open question. The simulations we have done so far have all assumed zero large-scale field, primarily as a result of our effort to minimize the number of arbitrarily chosen free parameters. Table 1. Jet Poynting flux efficiency in rest-mass units ηEM , as a function of black hole spin parameter a/M (Hawley & Krolik 2006). These numbers can be compared with the radiative efficiency predicted by the Novikov-Thorne model, i.e., the specific binding energy of a particle in the innermost stable circular orbit. Interestingly, we have found that even when the magnetic field in the accreting matter has no net flux at all, and therefore no externally-imposed connections to infinity, under the right circumstances it can spontaneously create such connections within a limited volume. To be specific, when the accretion flow contains closed dipolar field loops large enough that the outer ends are accreted long (≫ 10 3 GM/c 3 ) after the inner ones, a jet is automatically created from the flux provided by the inner half of the field loop (McKinney & Gammie 2004, Hawley & Krolik 2006. When flux is brought toward the black hole along with the accretion flow, as soon as field lines begin to thread the event horizon, matter drains off them, and the field lines, freed of the matter's inertia, float upward. Because a centrifugal barrier prevents any matter with non-zero angular momentum from penetrating into a cone surrounding the rotation axis, there is little inertia above these field lines. The field lines then rapidly expand upward. This process of filling the region around the rotation axis with magnetic field ceases only when there is enough field intensity that the magnetic pressure distribution reaches equilibrium. If the central black hole spins, the portions of the field lines within the ergosphere are forced to rotate along with the black hole, imposing a twist on the field lines. The result is Poynting flux travelling outward through the evacuated cone around the rotation axis. When the black hole rotates rapidly, the electromagnetic luminosity can be quite sizable. Table 1 presents that luminosity, normalized by the rest-mass accretion rate, as a function of black hole spin. The radiative efficiency predicted by the Novikov-Thorne model is also given in that table in order to provide a standard of comparison. As can be seen, when the black hole rotates rapidly, the jet efficiency becomes comparable to the putative radiative efficiency. Thus, consistent with what many have long speculated, black hole rotation does seem to enhance jet luminosity. On the other hand, black hole spin may not be the only relevant parameter. For example, large dipolar loops are not the only imaginable field structure for an accretion flow. One could just as easily imagine narrower dipolar loops, or quadrupolar loops (loops that don't cross the equatorial plane) or toroidal loops. These other geometries are in general less favorable to jet support than the initial form explored, the large dipolar loops (Beckwith, Hawley & Krolik, in preparation). Real systems are likely to exhibit some mixture of these sorts of field topologies, and that mixture could easily vary from one object to another, or from one time to another in a single object. Some of the observed variability in jets may conceivably reflect varying field structures in the matter fed to the central black hole. Conclusions Thanks to the fundamental discovery that stresses in accretion disks come from correlated MHD turbulence, driven by the magneto-rotational instability, we can now begin to speak with confidence about a number of aspects of their operation. With the aid of ever-more-detailed and realistic numerical simulations, we have taken the first steps toward connecting their internal dynamics with observable properties. In this talk, advances in this direction in three areas have been reported: • We now see that angular momentum transport is accomplished quasi-coherently, by magnetic stress. Because this mechanism is not a kinetic process like viscosity, it is not intrinsically dissipative, although the associated MHD turbulence does eventually dissipate. Moreover, far from ceasing at the innermost stable circular orbit, as has been generally assumed for more than thirty years, magnetic stress continues through the marginally stable region and deep into the plunging region. When the black hole spins, the stress can be continuous all the way to the event horizon. At the very least, the ability of electromagnetic stresses to carry angular momentum away from the black hole and into the accretion flow means that the spin-up rate of black holes can be rather less than would have been estimated on the basis of accreting matter with the specific angular momentum of the last stable orbit. It is also possible, although quantitative determination of this effect remains a job for the future, that these extended stresses lead to extended dissipation as well, augmenting the radiative efficiency of black hole accretion beyond the traditional values. • Detailed study of the vertical structure of disks subject to the MRI has shown that their upper layers are supported primarily by magnetic pressure gradients. In addition, these upper layers can be far from the smooth time-steady plane-parallel condition in which they are commonly imagined. As a result, the density at the photosphere is likely to be rather smaller than previously estimated, and the locally-emitted spectrum may have significant departures from black-body form. Ongoing work promises to clear up the long-standing mystery of whether radiationdominated disk regions are thermally unstable, and if they are, what happens in the nonlinear stage of this instability. • As had been initially pointed out in the mid-1970s, when large-scale magnetic fields pass close by rotating black holes, it is possible for very energetic relativistic jets to be driven, deriving their energy from the rotational kinetic energy of the black hole itself. We can now begin to compute the detailed structure of these jets, as functions of both space and time. In addition, we now see that it is possible to create large-scale magnetic field threading the ergosphere of the black hole and stretching out to infinity from much smaller-scale field embedded in the accretion flow-but not all small-scale field structures are capable of doing this. I am indebted to my many collaborators on the work reported here. In alphabetical order, they are: Kris Beckwith, Omer Blaes, Shane Davis, Jean-Pierre De Villiers, John Hawley, Shigenobu Hirose, Ivan Hubeny, Yawei Hui, Jeremy Schnittman, and Jim Stone. This work was partially supported by NSF Grants AST-0313031 and AST-0507455 and NASA ATP Grant NAG5-13228. Figure 1 . 1Fluid-frame stress, integrated over spherical shells. Upper panel shows a simulation with a non-rotating black hole, lower panel one with a black hole having a spin parameter a/M = 0.9. Solid curves are the electromagnetic stress as found in the simulations at a particular time; dotted curves are the prediction of the Novikov-Thorne model for an accretion rate matching the time-average of that simulation. Figure 2 . 2Heating and radiative output from two shearing box simulations. Left panel shows a case in which pr/pg ≃ 0.2 (taken fromHirose et al. 2006); right panel a case in which 0.5 pr/pg 2, depending on the time within the simulationKrolik et al. 2007). In both figures, the solid curve shows the volume-integrated dissipation rate, while the dashed curve shows the radiative output. After initial transients, heating and radiative flux are nearly identical. Figure 4 . 4Time-averaged vertical profiles of the dissipation rate in a shearing box simulation. A powerful local shear instability in weakly magnetized disks. I -Linear analysis. S A Balbus, J F Hawley, Ap. J. 376Balbus, S.A. & Hawley, J.F. 1991 A powerful local shear instability in weakly magnetized disks. I -Linear analysis. Ap. J. 376, 214-223 Magnetic Pressure Support and Accretion Disk Spectra Ap. O M Blaes, S W Davis, S Hirose, J H Krolik, J M Stone, J. 645Blaes, O.M., Davis, S.W., Hirose, S., Krolik, J.H. & Stone, J.M. 2006 Magnetic Pressure Support and Accretion Disk Spectra Ap. J. 645, 1402-1407 Surface Structure in an Accretion Disk Annulus with Comparable Radiation and Gas Pressure Ap. O M Blaes, S Hirose, J H Krolik, J. 664Blaes, O.M., Hirose, S. & Krolik, J.H. 2007 Surface Structure in an Accretion Disk Annulus with Comparable Radiation and Gas Pressure Ap. J. 664, 1057-1071 Hydromagnetic flows from accretion discs and the production of radio jets M. R D Blandford, D G Payne, N.R.A.S. 199Blandford, R.D. & Payne, D.G. 1983 Hydromagnetic flows from accretion discs and the production of radio jets M.N.R.A.S. 199, 883-903 Electromagnetic extraction of energy from Kerr black holes M. R D Blandford, R L Znajek, N.R.A.S. 179Blandford, R.D. & Znajek, R.L. 1977 Electromagnetic extraction of energy from Kerr black holes M.N.R.A.S. 179, 433-456 A Numerical Method for General Relativistic Magnetohydrodynamics Ap. , J.-P De Villiers, J F Hawley, J. 589De Villiers, J.-P. & Hawley, J.F. 2003 A Numerical Method for General Relativistic Mag- netohydrodynamics Ap. J. 589, 458-480 Efficiency of Magnetized Thin Accretion Disks in the Kerr Metric Ap. C F Gammie, J. Letts. 522Gammie, C.F. 1999 Efficiency of Magnetized Thin Accretion Disks in the Kerr Metric Ap. J. Letts. 522, L57-60 HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics Ap. C F Gammie, J C Mckinney, G Tóth, J. 589Gammie, C.F., McKinney, J.C. & Tóth, G. 2003 HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics Ap. J. 589, 444-457 A powerful local shear instability in weakly magnetized disks. II -Nonlinear evolution. J F Hawley, S A Balbus, Ap. J. 376Hawley, J.F. & Balbus, S.A. 1991 A powerful local shear instability in weakly magnetized disks. II -Nonlinear evolution. Ap. J. 376, 223-233 Magnetically Driven Jets in the Kerr Metric Ap. J F Hawley, J H Krolik, J. 641Hawley, J.F. & Krolik, J.H. 2006 Magnetically Driven Jets in the Kerr Metric Ap. J. 641, 103-116 Magnetically Driven Accretion Flows in the Kerr Metric. II. Structure of the Magnetic Field Ap. S Hirose, J H Krolik, J.-P De Villiers, J F Hawley, J. 606Hirose, S., Krolik, J.H., De Villiers, J.-P. & Hawley, J.F. 2004 Magnetically Driven Accretion Flows in the Kerr Metric. II. Structure of the Magnetic Field Ap. J. 606, 1083- 1097 Vertical Structure of Gas Pressure-dominated Accretion Disks with Local Dissipation of Turbulence and Radiative Transport Ap. S Hirose, J H Krolik, J M Stone, J. 640Hirose, S., Krolik, J.H. & Stone, J.M. 2006 Vertical Structure of Gas Pressure-dominated Accretion Disks with Local Dissipation of Turbulence and Radiative Transport Ap. J. 640, 901-917 Magnetized Accretion inside the Marginally Stable Orbit around a Black Hole Ap. J H Krolik, J. Letts. 515Krolik, J.H. 1999 Magnetized Accretion inside the Marginally Stable Orbit around a Black Hole Ap. J. Letts. 515, L73-76 Magnetically Driven Accretion Flows in the Kerr Metric. IV. Dynamical Properties of the Inner Disk Ap. J H Krolik, J F Hawley, S Hirose, J. 622Krolik, J.H., Hawley, J.F. & Hirose, S. 2005 Magnetically Driven Accretion Flows in the Kerr Metric. IV. Dynamical Properties of the Inner Disk Ap. J. 622, 1008-1023 Thermodynamics of an Accretion Disk Annulus with Comparable Radiation and Gas Pressure Ap. J H Krolik, S Hirose, O M Blaes, J. 664Krolik, J.H., Hirose, S. & Blaes, O.M. 2007 Thermodynamics of an Accretion Disk Annulus with Comparable Radiation and Gas Pressure Ap. J. 664, 1045-1056 A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole Ap. J C Mckinney, C F Gammie, J. 611McKinney, J.C. & Gammie, C.F. 2004 A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole Ap. J. 611, 977-995 The Formation and Structure of a Strongly Magnetized Corona above a Weakly Magnetized Accretion Disk Ap. K A Miller, J M Stone, J. 534Miller, K.A. & Stone, J.M. 2000 The Formation and Structure of a Strongly Magnetized Corona above a Weakly Magnetized Accretion Disk Ap. J. 534, 398-419 Disk-Accretion onto a Black Hole. D N Page, K S Thorne, Time-Averaged Structure of Accretion Disk Ap. J. 191Page, D.N. & Thorne, K.S. 1974 Disk-Accretion onto a Black Hole. Time-Averaged Structure of Accretion Disk Ap. J. 191, 499-506 . E S Phinney, unpublished Cambridge University Ph.D. thesisPhinney, E.S. 1983 unpublished Cambridge University Ph.D. thesis Black holes in binary systems. Observational appearance. N I Shakura, R A Sunyaev, Astron. Astrophys. 24Shakura, N.I. & Sunyaev, R.A. 1973 Black holes in binary systems. Observational appear- ance. Astron. Astrophys. 24, 337-355. A theory of the instability of disk accretion on to black holes and the variability of binary X-ray sources, galactic nuclei and quasars M. N I Shakura, R A Sunyaev, N.R.A.S. 175Shakura, N.I. & Sunyaev, R.A. 1976 A theory of the instability of disk accretion on to black holes and the variability of binary X-ray sources, galactic nuclei and quasars M.N.R.A.S. 175, 613-632

Some established views of the solar magnetic cycle are discussed critically, with focus on two aspects at the core of most models: the role of convective turbulence, and the role of the 'tachocline' at the base of the convection zone. The standard view which treats the solar cycle as a manifestation of the interaction between convection and magnetic fields is shown to be misplaced. The main ingredient of the solar cycle, apart from differential rotation, is instead buoyant instability of the magnetic field itself. This view of the physics of the solar cycle was already established in the 1950s, but has been eclipsed mathematically by mean field turbulence formalisms which make poor contact with observations and have serious theoretical problems. The history of this development in the literature is discussed critically. The source of the magnetic field of the solar cycle is currently assumed to be located in the 'tachocline': the shear zone at the base of the convection zone. While the azimuthal field of the cycle is indeed most likely located at the base of the convection zone, it cannot be powered by the radial shear of the tachocline as assumed in these models, since the radiative interior does not support significant shear stresses. Instead, it must be the powered by the latitudinal gradient in rotation rate in the convection zone, as in early models of the solar cycle. Possible future directions for research are briefly discussed.

10.1007/978-90-481-9787-3_5

1004.4545

d91919319c380acec715fb14069698ca6bcfa5af

Theories of the solar cycle : a critical view H C Spruit Max Planck Institute for Astrophysics Box 131785741GarchingGermany Theories of the solar cycle : a critical view Some established views of the solar magnetic cycle are discussed critically, with focus on two aspects at the core of most models: the role of convective turbulence, and the role of the 'tachocline' at the base of the convection zone. The standard view which treats the solar cycle as a manifestation of the interaction between convection and magnetic fields is shown to be misplaced. The main ingredient of the solar cycle, apart from differential rotation, is instead buoyant instability of the magnetic field itself. This view of the physics of the solar cycle was already established in the 1950s, but has been eclipsed mathematically by mean field turbulence formalisms which make poor contact with observations and have serious theoretical problems. The history of this development in the literature is discussed critically. The source of the magnetic field of the solar cycle is currently assumed to be located in the 'tachocline': the shear zone at the base of the convection zone. While the azimuthal field of the cycle is indeed most likely located at the base of the convection zone, it cannot be powered by the radial shear of the tachocline as assumed in these models, since the radiative interior does not support significant shear stresses. Instead, it must be the powered by the latitudinal gradient in rotation rate in the convection zone, as in early models of the solar cycle. Possible future directions for research are briefly discussed. The role of convective turbulence For a star to generate a self-sustained magnetic field, it is sufficient that it rotate differentially. This differs from the traditional view of dynamos in stars, which holds that in addition to the shear flow due to differential rotation, a small scale velocity field has to be imposed in order to 'close the dynamo cycle', thus enabling a selfsustained field independent of initial conditions. Convection can provide such a velocity field, and in fact convection has become such an integral part of thinking about dynamos in stars that the subject of 'stellar magnetic fields' has been almost synonymous with 'convective dynamos' for decades (for reviews see e.g. Weiss 1981-1997, Rüdiger and Hollerbach 2004, Tobias 2005, for recent texts Brandenburg 2009, Jones et. al. 2009, Charbonneau 2005. Whether or not such a dynamo Figure 1: Rate of increase of the azimuthal field strength as a function of heliographic latitude, due to the observed differential rotation acting on an assumed uniform poloidal field. process can take place in principle is a separate matter. From the observations it is evident, however, that it is not the way the solar cycle works. Instead, as I will argue below, the cycle operates on dynamic instability of the magnetic field itself. Convection plays only an indirect role, namely by maintaining the differential rotation of the envelope from which the cycle derives its energy. Mechanism of the solar cycle as inferred from observations The common ingredient in all dynamo models such as those for the Earth's magnetic field or the solar cycle is the generation of a toroidal (azimuthally directed) field by stretching ('winding-up') of the lines of a poloidal field (e.g. Elsasser 1956). This is 'the easy part'. It produces a field that increases in strength linearly with time and is proportional to the imposed initial field. To produce a cyclic, self-sustained field as observed there must be a second step that turns some of the toroidal field into a new poloidal component, which is again wound up, completing a field-amplification cycle that becomes independent of initial conditions. The particular process by which the new poloidal field is generated distinguishes the models from each other. In early models of the solar cycle that were popular in their time (Babcock 1961, 1963, Leighton 1969 observations of the emergence of active regions were used to infer the nature of the process responsible for this key step in the dynamo cycle. These models proposed that the increasing toroidal field eventually becomes unstable, erupting to the surface to form the observed active regions (Cowling 1953, Elsasser 1956, Babcock 1961. The equatorward drift of the main zone of activity reflects the latitude dependence of the time taken for the toroidal field to reach the point of buoyant instability (Babcock 1961(Babcock , 1963. This is illustrated with a simple model in Fig. 1. In this sketch, a uniform poloidal field is assumed to be stretched passively by the latitudinal differential rotation as observed on the surface of the Sun. Helioseismic observations (see review Figure 2: Closing of the dynamo cycle by active region emergence. Left: sub-surface field produced by stretching of a poloidal field B p by differential rotation (equator rotates faster). Coriolis forces during emergence of a stretch of the field (broken) to the surface causes displacements of the footpoints, observed at the surface as 'tilt' of the active regions (circles). At depth, this produces a new poloidal field component of opposite sign. by Howe, 2009) show that this pattern of rotation also holds to a fair approximation inside the convection zone. The azimuthal field becomes unstable to buoyant rise when a critical strength of ∼ 10 5 G is reached (Schüssler et al. 1994). This happens first at the latitude where the rate of increase of the field is largest, around a latitude of 60 • in the simple model of Fig. 1. This agrees with observations (Altrock 2010), though initially only small-scale magnetic activity without sunspots is produced. As time progresses, the field also becomes unstable at lower latitudes, producing an equatorward drift of the zone of activity. For reasons unknown, sunspots form only below a latitude of around 40 • . As Fig. 1 implies, Babcock's model also predicts a poleward propagating branch. Such a branch (but without sunspots) is actually present on the Sun (the 'poleward rush', Leroy & Trellis, 1974, Altrock 2010. Its observational status and interpretation are not entirely clear, however. The process of emergence of an active region has been studied in great detail for more than a century. A small patch of fragmented magnetic fields with mixed polarities appears and expands as more flux emerges (Fig. 3). The surroundings of this patch remain unaffected by this process. The mix of polarities then separates into two clumps, the polarities traveling in opposite directions to their destination, ignoring the convective flows in the region. This striking behavior is the opposite of diffusion. To force it into a diffusion picture, one would have to reverse the arrow of time. Instead of opposite polarities decaying by diffusing into each other, they segregate out from a mix. The MHD equations are completely symmetric with respect to the sign of the magnetic field, however. There are no flows (no matter how complex) that can separate fields of different signs out of a mixture. This rules out a priori all models attempting to explain the formation of sunspots and active regions by turbulent diffusion. For (From Zwaan, 1978) recent such attempts, which actually ignore the observations they are trying to explain, see Kitiashvili et al. (2010), Brandenburg et al. (2010). The observations, instead, demonstrate that the orientation and location of the polarities seen in an active region must already be have been present in the initial conditions: in the layers below the surface from which the magnetic field traveled to the surface. The fragmented state near the surface in the early stages of the eruption process is only temporary. The intuitive 'rising tree' picture (Zwaan, 1978) illustrates this (Fig. 4). The observed fragmentation and subsequent formation of spots from a horizontal strand of magnetic field below the surface has recently been reproduced in striking realism in full 3-D radiative MHD simulations (Cheung et al. 2008, Rempel, this volume). The axes of active regions are observed to be tilted with respect to the east-west direction. This was attributed to the action of Coriolis forces during the emergnece process, and identified with the generation of the new poloidal field component that closes the dynamo cycle by Leighton (1969, Later developments Models like Leighton's thus made a direct connection between observations of the active regions that make up the solar cycle and the functioning of the cycle as a whole. One might have expected that this natural state of affairs would have led to a further development of the theoretical ideas in continued contact with the observations. But this has not been the case. Instead, the development of these ideas has been eclipsed for several decades by the parallel development of turbulent mean field formalisms for the solar cycle. These ideas postulated mathematically tractable equations which were claimed to represent the physics of the interaction between magnetic fields and convection in some statistical sense. They relied on theoretical assumptions like cascades in wavenumber space, correlation functions to represent the interaction between magnetic fields and flows, and an assumed separation of length scales between mean fields and fluctuations. Just looking at the data as described above, it is difficult see how a separation would be accomplished. What is more, the data themselves already contain more detailed and more critical information on the functioning of the cycle than is present in mean field models. The dominance of these formalisms in the astrophysical literature (thousands of papers) has led to a particularly sterile theoretical view of the solar cycle, supported neither by a sound theoretical foundation of the equations used nor making much contact with the observations. In addition, it has had the effect of obscuring an important fact, namely that no turbulence needs to be imposed at all for dynamo action to take place. A system that is completely laminar in the absence of magnetic fields can produce dynamo action from shear and magnetic instability alone (cf. Spruit 2002). A well studied and very successful example of such a dynamo process is the MRI turbulence observed in numerical simulations of accretion disks (e.g. Hawley et al. 1996). The models by Babcock and Leighton are just another example where magnetic instability is the key element in closing the field amplification cycle. These kinds of magnetic cycle are intrinsically non-linear (i.e. not 'kinematic' in dynamo parlance): their functioning depends on the finite amplitude of the field generated. This is because the time scale of the magnetic instabilities that close the dynamo cycle depends on field strength. The conditions for self-sustained field generation to occur by differential rotation and instability alone, the properties of the magnetic field produced in this way, and its observable consequences all reflect the nature of the magnetic instability involved. In the case of the solar cycle: the properties of magnetic buoyancy. It is sometimes argued that such a process just brings about an 'alpha effect', so that one just has to use a set of equations that incorporate such an effect. Neither the fact that a poloidal field component can appear by a process changing the direction of an initially toroidal field, however, nor the fact that turbulent mean field equations contain a term describing such an effect, are justifications for using these equations. An understanding of the solar cycle, or any other dynamo process, requires physics to be found out first, rather than assumed in some parametrized form. The idea that insight about the solar cycle can be obtained from the solutions of such models has been an impediment to real progress, however tempting the equations may have looked. Justification for this critical view is found in the history of ideas about the solar cycle; this is done in the following section. I briefly discuss there how mean field thinking has led to a systematic disconnect between theory and observations. In all likelihood this would not have been necessary if the observations and their interpretation in models such as Leighton's (1969), had been taken more serious. Failure of convective dynamos models of the solar cycle The turbulent view of magnetic field generation in convective stellar envelopes holds that the generation of a new poloidal field from the toroidal field produced by differential rotation should be seen as due to the effect of convection acting on a magnetic field in a rotating fluid (Parker 1955, 1979, Steenbeck et al. 1966, Weiss 1981. A consequence of this model is therefore that dynamo action takes place throughout the convection zone. In this model the equatorward drift of the main belt of activity during the cycle reflected the (at the time unknown) radial gradient in the rotation rate, not the observed latitudinal differential rotation that were key to the Babcock and Leighton models. This is because mean field dynamo equations naturally lead to dynamo waves traveling in a direction perpendicular to surfaces of constant rotation rate. The observed equatorward drift of activity during the cycle therefore required the rotation rate to have a predominantly radial gradient, with rotation increasing inward. Finally, it was noted that the Lorentz force limits the action of convection on magnetic fields when the field strength reaches equipartition with the kinetic energy of the convective 'eddies' (e.g. Proctor and Weiss 1982). This predicts that the field in the solar envelope should be an intermittent turbulent field, with intrinsic field strengths of a few thousand Gauss. Predictions Mean field dynamo models thus made three testable predictions. (They are found in many of the texts of the 70's and 80's, where they appear mostly as accepted consensus rather than as testable predictions): • The dynamo action takes place through interaction with turbulent convection, • The rotation rate in the convection zone depends mainly on radius, it increases with depth. • The field strength does not exceed equipartition with convective energy densities (few thousand Gauss). These predictions have never agreed with the phenomenology of the solar cycle very well. One of the important observations is Hale's polarity law: the fact that magnetic fields are not present on the surface in a random 'turbulent' form, but appear in a strikingly systematic way, as bipolar active regions oriented eastwest, with one of the polarities systematically leading (in the direction of rotation). On top of the east-west orientation, the leading polarity is systematically shifted towards the equator compared with the following polarity (Joy's law, see sketch in Fig. 2). To circumvent these observations, dynamo theories ignore heliographic longitude (of sunspot locations, for example); parameters of the model are then adjusted to fit the remaining data (the 'butterfly diagram'). Given the reduced nature of these data and the degrees of freedom of the models, this process is usually successful. The price paid is that most observations of active region phenomenology have to be declared irrelevant, when in fact they provide the most telling evidence about the operation of the cycle. This attitude has remained an integral part of mainstream dynamo thinking. Attempts have been made to reconcile the magnetic eruption view of active regions with the role of convective turbulence in the mean field dynamo view. Weiss (1964) proposed that magnetic fields rise from the interior, but that turbulence takes over in bringing about observations like the formation of sunspots. In this view, sunspots would form by random walk of magnetic fields in a turbulent flow. This proposal thus kept Cowling's view of active regions as emerging from below, but effectively discarded the observational evidence that led to this idea in the first place, namely the formation of sunspots. Meyer et al (1974) explicitly repeated the view that sunspots form by random walk of magnetic field lines in convective turbulence. This has been challenged by observers, who noted that sunspots do not form randomly but in a strikingly deterministic way, as described above (e.g. Zwaan 1978). The motion of active region magnetic fields independent of and opposing surface flows is documented by virtually all observations of active region formation (e.g. Tarbell et al. 1990, Strous et al. 1996. The consequence, namely that the magnetic field itself, rather than convective turbulence, forms active regions has always contradicted the role of convection assumed in mean field models. Since then, helioseismic measurements of the internal rotation have shown the second prediction to be wrong as well: the differential rotation is in fact mainly in latitude. The radial gradient is weak, and where it is present it is mostly of the wrong sign. The third prediction can be tested somewhat more indirectly by making use of the many clues given by observations of active regions. A major step forward in the interpretation of this phenomenology are the simulations of flux bundles rising from the base of the convection by Moreno-Insertis (1986), D'Silva and Choudhuri (1993), Fan et al. (1994), Schüssler et al. (1994). In these simulations, a horizontal (azimuthal, zonal) bundle of magnetic field lines at the base of the convection zone is allowed to become unstable and rise to the surface. The degrees of freedom in these calculations are the magnetic flux of the bundle (set by the value observed for a typical active region), and its initial field strength. The results show that several key characteristics of active regions can be reproduced simultaneously by such magnetic flux loops emerging from the interior of the convection zone: the time scale for emergence of an active region, the heliographic latitude range of emergence, and the degree of tilt of active region axes. For all these phenomena, agreement between simulations and observation points to the same value of the field strength: about 10 5 G. This is also the field strength at which instability is predicted to set in (Schüssler et al. 1994). It can thus be identified with Babcock's (then still unquantified) critical field strength. Within this picture, a further piece of evidence that would otherwise be a disconnected observation finds its natural place. After formation of a spot, its position drifts a bit in latitude and longitude ('proper motion'), with a random component superimposed on a systematic drift. The random component varies quasiperiodically on a time scale of a few days, decaying with time (e.g. Herdiwijaya et al.1997). For a field strength of 10 5 G at the base, the Alfvén travel time along the flux strand from the base to the surface is around 3 days. The random proper motions of spots are thus neatly interpreted as reflecting the 'settling' of a sunspot to its equilibrium position after the eruption process is completed. [In addition to this random component there is a systematic drift in longitude, corresponding to the increasing separation between the two polarities. This was explained as due to the tension in the sub-surface magnetic field by van Ballegooijen (1982), and reproduced in simulations of rising flux tubes (Caligari et al. 1995)]. Assessment of the turbulent convective dynamo view The success of the rising flux tube simulations was not immediately seen as a threat to mean field models. In line with the status of active region phenomenology in mean field models, the eruption process would simply be of marginal significance: it would just be some secondary manifestation of the mean field dynamo operating in deeper layers. The success of the simulations, if it is not accidental, has ominous consequences, however, since the agreement with each of the observations only holds if the field strength in the deep interior of the convection zone is around 10 5 G. The energy density in such a field is at least two orders of magnitude larger than the kinetic energy of the convective turbulence invoked in mean field models. If taken serious as a diagnostic, the observed mode of emergence of active regions thus implies that the third prediction also fails. The situation is more serious, however, since at this field strength the rising flux tubes are so strong that convective turbulence can have little effect on them. The positions of active regions on the surface must consequently correspond reasonably with those of their anchors at the base, also explaining the regularity with which active regions follow Hale's and Joy's laws, and the proper motions of sunspots mentioned above. Active regions as seen at the surface are therefore not a manifestation of a convective mean field dynamo, even if it were to exist somewhere in the convection zone. This is a major setback for this theory, since observations of active regions are the dominant source of information we have about the solar cycle, and virtually the only source ever used for mean field parameter fitting. Considering these spectacular failures of the turbulent mean field dynamo paradigm for the solar cycle it is useful to reflect for a moment how it could have survived for several decades, and is still going strong (cf. Rüdiger and Hollerbach 2004, Brandenburg 2005, Tobias 2005, Jones et al. 2009, Charbonneau 2005. Major problems like the lack of a stable theoretical foundation for the equations used, and the lack of connection with most of the relevant observations should have been reasons to pause and reflect on the basis of the enterprise. Instead such problems, when faced at all, were usually countered with these arguments: i) in a complicated problem one has to start somewhere, ii) the key to a dynamo cycle is the recreation of a poloidal component, and this is included in the mean field dynamo equation. The first argument was perfectly reasonable at the time of the formulation of mean field electrodynamics, (c.f. Parker 1955, Steenbeck et al. 1966), but half a century hence it is beginning to wear a bit thin. The second argument is essentially a semantic kludge. At a sufficiently abstract level, the nature of the solar cycle as a combination of winding up and an alpha-effect need not be contested. What should have been considered more critically, however, is the question how much more the use of mean field dynamo formalisms provides, above just the alpha-effect that was already put in as an assumption from the beginning. The applied mathematical attractions of solving this equation in countless variations of geometry and parameters appears to have led to a misplaced sense of reality. The failure of the theory to show demonstrable progress by providing increasing contact with observations should have been a warning here. The mean field dynamo model for the solar cycle is best regarded as a mirage. Remarkably, it still keeps a substantial community busy searching for an oasis. The theoretical basis of mean field electrodynamics has always been problematic. The series expansions used to derive mean field equations, for example are known to diverge unconditionally at large Reynolds numbers. In the mean time, it is now also being called into question by the results from high resolution numerical simulations of magnetic fields driven by imposed small scale forces. From these simulations it is becoming clear that large scale fields do not appear from small scales as expected (e.g. Cattaneo and Hughes, 2010), at least not under the generic conditions where the mean field dynamo equation was applied. The occurrence of small scale dynamos (i.e. the exponential growth of magnetic energy on small scales) in the interaction of a magnetic field with turbulence is now also somewhat in doubt. Standard wisdom (the 'proper' view, Brandenburg 2010), that an appropriately complex velocity field is sufficient to produce a selfsustained smallscale magnetic field turns out to be incorrect. Selfsustained fields have been found in such flows when the viscosity is larger than the magnetic diffusivity (magnetic Prandtl number P m > 1, e.g. Schekochihin et al. 2005). For the case P m < 1 however, conflicting results are reported on the presence or absence of selfsustaining fields, depending also on numerical method. In the case P m > 1, the actual operation mechanism of the field amplification does not agree with conventional ideas based on cascades in wavenumber space (Schekochihin et al. 2005). Kinematic models of small-scale dynamo action, i.e. models using an imposed velocity field, which have guided much of previous thinking, do not provide guidance in this context since they are equivalent to assuming an infinite magnetic Prandtl number. Tachocline dynamos An somewhat older idea to reconcile mean field models with flux emergence observations is that the dynamo works as a turbulent mean field dynamo near the base of the convection zone (Galloway andWeiss 1981, Parker 1993). After its discovery, the narrow shear zone below the convection zone, the tachocline, was quickly identified as a region of choice to operate a mean field dynamo. It contained the strong radial gradient needed in mean field models to produce the drift of active latitudes during the cycle (e.g. Dikpati 2006 and references therein), and required no conceptual adjustments to the models developed before. This idea, however does not make physical sense. It assumes that the shear zone can be exploited in the same way as the shear between two moving plates in the laboratory. Turbulence generated by the shear exerts stress on both plates, the energy put into the system by the work done against this boundary stress can be tapped to maintain turbulence and a magnetic field. In the Sun, stresses can be maintained in the convection zone by the rapid momentum exchange due to convective flows. On the other side, however, in the stable stratification on the interior side of the tachocline, the stress that can be supported by fluid motions is many orders of magnitude weaker, since the stratification is very stable on this side (in terms of the buoyancy N 2 , the interior is 10 6 times more stable than the convection zone is unstable). This means that the analogy with shear maintained between two moving plates is incorrect (resembling the Zen exercise to clap with one hand). If the analogy is incorrect, what is then the cause of the tachocline? The tachocline is more appropriately treated as a shear flow driven only by the latititudinal dependence of the rotation rate on the convective side, with a free-slip surface on the interior side. The velocities in the tachocline can just be an 'imprint' into the interior of the differential rotation with latitude in the convection zone. This has been proposed early on after the discovery of the tachocline by Spiegel and Zahn (1992), who studied the long-term evolution of the rotation in the interior under the effect of a (weak) viscous stress. A more complete analysis of this problem, which stresses the importance of baroclinicity and thermal diffusion, is the 'gyrotropic pumping' model of McIntyre (2007). In a model by Forgács-Dajka & Petrovay (2001) the tachocline is also seen as an imprint of the convection zone on the interior, but this model involves turbulence of unspecified origin inside the tachocline. The radial gradient in the tachocline is thus useless for driving a dynamo, since it does not support any significant stress with which a field could be amplified. The latitudinal gradient in the tachocline can of course be tapped, but this is no different from what the bulk of the convection zone can do. Its shallow imprint into the interior does not add much, and defeats the original idea of using the (radial) tachocline shear to drive a dynamo. New directions 4.1 Compromises The mean field paradigm holds that, after ignoring or averaging out most of the surface observations, the bit that is left is still a useful validation of the theory. This point of view is still popular in the literature on the solar cycle. Another view appears to follow more of an 'adiabatic adjustment' approach: attempts are made to incorporate elements like the disregarded observations mentioned in section 1.1, and physics like buoyant instability of the magnetic field into mean field turbulence, as gradual adjustments of the formalism. This view thus attempts to accommodate intrinsically incompatible elements into a mean field approach without questioning its status as the underlying fundamental theory. The kind of compromises this leads to looks ugly. In one such attempt to conciliate mean field theory with observations, the emergence of strong magnetic flux tubes is in fact acknowledged to account for phenomena observed at the surface. However, the surface fields are then seen as a separate phenomenon, not representative of the real solar cycle. The real cycle takes place, unseen, somewhere in the convection zone in the manner demanded by the mean field dynamo equations (e.g. Tobias 2009). Another proposal (Brandenburg 2005) postulates the existence of a shallow surface layer (a few Mm depth). This layer contains the puzzling observations, again in some form of mean field dynamo, while at the same time shielding the turbulent mean field dynamo happening below it from our view. 'Turbulent pumping' is advanced as achieving this. Observations of active region emergence (Fig. 3) that refute such ideas (however vague) are ignored. Weak fields A more progressive view within the convective turbulence category is the idea (Durney et al. 1993, Cattaneo, 1999 that turbulent interaction between field and convection is observed at the surface of the Sun in the form of the so-called weak or inner-network fields. These appear as fields of mixed polarity, short life times and low intrinsic strength (Martin 1988). These properties are more in line with a priori expectations about turbulent fields. This proposal bypasses the question what causes the strong fields observed as spots and active regions: it only looks at the weak field component and assumes it has a different origin. It is not clear, however, whether the weak fields are really an independent phenomenon: they might just be a part of the solar cycle as seen in active regions. The weak fields might represent either a small scale tail in the distribution of emerging flux units, or some kind of 'debris' from the fragmentation of larger units during the decay phase of active regions. Finally, they might be related to the 'annealing step' by which old flux disappears again from the convection zone (section 4.4). In the latter case, they would be part of the decay of magnetic fields produced in the cycle, rather than an amplification process. Numerical simulations Convincing 3-D numerical simulations of a solar cycle 'from scratch' are likely to remain out reach for the indefinite future. This is because of the well-known problems of dynamical range in length and time scale intrinsic to a convective stellar envelope. A brute-force simulation of the entire convection zone would have to resolve time scales as short as a few seconds in the photospheric layers and above, as well as years to cover the duration of a cycle. Corresponding length scales range from a kilometer at the surface to a solar radius. Extrapolating Moore's law with a constant doubling time of 1.5 years, the computing resources needed for a simulation at this resolution would become available 100 years from now (Schüssler 2008). Existing 'global' simulations of the convection zone or its magnetic cycle are possible only by leaving out key parts of the physics. Usually, the top layers where most of the dynamic range in length-and time scales is located, are left out. Conclusions drawn from such simulations are unlikely to be very meaningful, since the easier case of a convective envelope without magnetic fields is already known to produce results that bear no relation to observations when these surface layers are left out from the simulation. Individual aspects of the cycle still provide interesting unsolved conceptual problems, however, that may be addressed in isolation before realistic numerical simulations are attempted on more global scales. Two such problems are discussed in the next subsections. The annealing step, 'turbulent diffusion' The most challenging problem may well be finding a satisfactory description for the process by which the mass of buoyant vertical flux tubes resulting from a cycle's worth of eruptions gets 'annealed' back into a simpler configuration. As the eruption of active regions from the toroidal field proceeds during the cycle, an ever increasing number of magnetic strands develops connecting the surface with the base of the convection zone (cf. Fig. 5). The sections of field remaining at the base are sufficient to provide the toroidal field of the next cycle, but this picture does not explain how the clutter of strands connecting to the surface gets simplified from one cycle to the next. In dynamo parlance, this is the 'turbulent diffusion' step. The difficulty here is that appeal to traditional convective 'turbulent diffusion' will not work (even if the concept itself is accepted), since the fields are now much stronger than equipartition with convection (at least near the base of the convection zone where this annealing has to take place). At the moment, it is not clear how long this annealing process takes, or even by which mechanism. For a discussion relating to this problem, see Parker (2009). Perhaps the 'residence time' of dispersed active regions is substantially longer than the cycle length? In that case a large amount of small scale mixed-polarity magnetic flux should be present distributed over the solar surface. Such a component could remain undetected except at very high spatial resolution. In fact, recent observations of mixed-polarity fields in quiet regions of the solar surface with Hinode and the Swedish 1-m solar telescope appear to show evidence in this direction (cf Pietarila et al. 2009, but see also Deforest et al. 2008) . Extrapolating this thought further, it might even be that such a component is relevant for total solar irradiance (TSI, the solar energy flux received at earth), since small scale magnetic fields are known to produce a net brightening of the solar surface (Spruit 1977). If this is the case, the small but systematic decrease of TSI during the current extended minimum, below the previous shorter minima, might be indicative of the decrease of surface magnetic flux in the course of the annealing process. Thermodynamics A central question concerns the thermodynamics of fields of ∼ 10 5 G at the base of the convection zone. For the field to be wound up quietly over several years before becoming unstable, it has to reside in a stable buoyant equilibrium near the base of the convection zone. In the absence of such a stable equilibrium, the field would rise to the surface on a time scale of weeks (as it actually does in the emergence of a new active region). Magnetic pressure produces buoyancy, and this has to be compensated for equilibrium to hold. Neutral buoyancy through density equilibrium requires significantly lower temperatures in the field than in its environment. For a field strength of 10 5 G, the required amount of temperature reduction is some 100 times larger than canonical temperature fluctuations in a mixing length model of convection. To solve this equilibrium problem, it is often postulated that flux tubes erupt from stably stratified layers below the base of the convection zone. Though this recognizes the buoyancy problem, it does not actually help much since it begs the question how the magnetic field got to this location in the first place (in particular: on a time scale less than the solar cycle). Conclusions Observations of active region phenomenology, most of them already old and wellestablished, show that the solar cycle operates on buoyant instability of the magnetic field itself rather than the conventional view based on interaction with convection. This puts us back to ideas developed half a century ago. Significant steps forward, however are the direct 3-D, radiative numerical MHD simulations which are now beginning to make contact with some of the classical observations. Though these simulations cannot deal with the cycle as a whole, their success in reproducing limited aspects such as the emergence of magnetic flux discussed above, or the observed structure of sunspots (Heinemann et al. 2007, Scharmer et al. 2008, Rempel et al. 2009 give confidence for the future. At the same time they clean the table by eliminating a number of dead-end views on the solar cycle, some of which considered well-established thus far. At the same time, a number of unsolved questions appear that are specific for the picture of a magnetic cycle operating on buoyant instability. Some of these questions are unlikely to be answered from first principles or numerical simulations. Clues taken from observations may well play an important role in making progress in figuring out the physics relevant for these questions. As the history of the subject shows, however, taking observational clues serious will require one to jettison the turbulent mean field baggage that has impeded the development of a sensible theory of the solar cycle for so long. This process would be assisted by healthy skepticism on the part of the observational community. In fact, it is rather surprising how easily observers have acquiesced in the past to the treatment of their data by mean field theories ('sorry but your observations are just turbulence, they have to be averaged out'). Figure 3 : 3Sequence (time from left to right) showing the emergence of an active region at the solar surface observed with the Hinode satellite. The opposite magnetic polarities (vertical component of the field) are shown in black and white. 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We introduce normalized exponential Yang-Mills energy functional YM 0 e , stress-energy tensor S e,YM 0 associated with the normalized exponential Yang-Mills energy functional YM 0 e , e-conservation law. We also introduce the notion of the e-degree de which connects two separate parts in the associated normalize exponential stress-energy tensor S e,YM 0 (cf. (3.10) and (4.15)), derive monotonicity formula for exponential Yang-Mills fields, and prove a vanishing theorem for exponential Yang-Mills fields. These monotonicity formula and vanishing theorem for exponential Yang-Mills fields augment and extend monotonicity formula and vanishing theorem for F -Yang-Mills fields in [DW] and [W11, 9.2]. We also discuss an average principle (cf. Proposition 8.1), isoperimetric and Sobolev inequalities, convexity and Jensen's inequality, p-Yang-Mills fields, an extrinsic average variational method in the calculus of variation and Φ (3) -harmonic maps, from varied, coupled, generalized viewpoints and perspectives (cf. Theorems 6.

2205.03016

8343f841a19fe78166656afa8976d552b4e2355d

6 May 2022 Shihshu Walter Wei 6 May 2022On exponential Yang-Mills fields and p-Yang-Mills fields1719192101102111311141115) 11 An extrinsic average variation method and Φ (3) -harmonic maps 2010 Mathematics Subject Classification Primary 58E2053C2181T13; Secondary 26D1553C20 Key words and phrases Normalized exponential Yang-Mills-energy functionalstress-energy tensore-conservation lawexponential Yang-Mills connectionmonotonicity formulavanishing theoremexponential Yang-Mills fieldp-Yang-Mills field We introduce normalized exponential Yang-Mills energy functional YM 0 e , stress-energy tensor S e,YM 0 associated with the normalized exponential Yang-Mills energy functional YM 0 e , e-conservation law. We also introduce the notion of the e-degree de which connects two separate parts in the associated normalize exponential stress-energy tensor S e,YM 0 (cf. (3.10) and (4.15)), derive monotonicity formula for exponential Yang-Mills fields, and prove a vanishing theorem for exponential Yang-Mills fields. These monotonicity formula and vanishing theorem for exponential Yang-Mills fields augment and extend monotonicity formula and vanishing theorem for F -Yang-Mills fields in [DW] and [W11, 9.2]. We also discuss an average principle (cf. Proposition 8.1), isoperimetric and Sobolev inequalities, convexity and Jensen's inequality, p-Yang-Mills fields, an extrinsic average variational method in the calculus of variation and Φ (3) -harmonic maps, from varied, coupled, generalized viewpoints and perspectives (cf. Theorems 6. Introduction The Yang-Mills functional, brought to mathematics by physics is broadly analogous to functionals such as the length functional in geodesic theory, the area functional in minimal surface, or minimal submanifold theory, the energy (resp. p-energy) functional in harmonic (resp. p-harmonic) map theory, or the mass functional in stationary or minimal current, geometric measure theory (cf.,e.g., [FF, L, HoW]). A critical point of the Yang-Mills functional with respect to any compactly supported variation in the space of smooth connections ∇ on the adjoint bundle is called aYang-Mills connection. Its associated curvature field R ∇ is known asYang-Mills field and is "harmonic", i.e., a harmonic 2-form with values in the vector bundle. The Euler-Lagrange equation for the Yang-Mills functional isYang-Mills equation. Whereas Hodge theory of harmonic forms is motivated in part byMaxwell's equations of unifying magnetism with electricity in a physics world, and harmonic forms are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian, harmonic maps can be viewed as a nonlinear generalization of harmonic 1-form and Yang-Mills field can be viewed as a nonlinear generalization of harmonic 2-form. On the other hand, Yang-Mills equation which can be viewed as a non-abelian generalization of Maxwell's equations, has had wide-ranging consequences, and influenced developments in other fields such as low-dimensional topology, particularly the topology of smooth 4-manifolds. For example, M. Freedman and R. Kirby first observed the startling fact that there exists an exotic R 4 , i.e., a manifold homeomorphic to, but not diffeomorphic to, R 4 (cf. [K, p. 95], [D, FK, Go]). This is in stunning contrast to a phenomenal theorem of J. Milnor in compact high-dimensional topology which shows that there exist exotic seven-spheres S 7 , i.e., manifolds that are homeomorphic to, but not diffeomorphic to, the stadard Euclidean S 7 (cf. [M]). In [DW], we unify the concept of minimal hypersurfaces in Euclidean space R n+1 , maximal spacelike hypersurfaces in Minkowski space R n,1 , harmonic maps, p-harmonic maps, F -harmonic maps, Yang-Mills fields, and introduce F -Yang-Mills fields, F -degree, and generalized Yang-Mills-Born-Infeld fields (with the plus sign or with the minus sign) on manifolds, where F : [0, ∞) → [0, ∞) is a strictly increasing C 2 function with F (0) = 0. (1.1) When F (t) = t , p −1 (2t) p 2 , √ 1 + 2t − 1 , and 1 − √ 1 − 2t, F -Yang-Mills field becomes an ordinary Yang-Mills field, p-Yang-Mills field, a generalized Yang-Mills-Born-Infeld field with the plus sign, and a generalized Yang-Mills-Born-Infeld field with the minus sign on a manifold respectively (cf. [BI,BL,BLS,CCW,D,La,LWW,LY,SiSiYa,W12,Ya]). When F (t) = t , e t , p −1 (2t) p 2 , √ 1 + 2t − 1 , and 1 − √ 1 − 2t , F -harmonic map or the graph of F -harmonic map becomes an ordinary harmonic map, exponentially harmonic map, p-harmonic map, minimal hypersurface in Euclidean space R n+1 , and maximal spacelike hypersurface in Minkowski space R n,1 respectively (cf. [ES, WY, EL, Ar, WWZ]). We use ideas from physics -stress-energy tensors and conservation laws to simplify and unify various properties in F -Yang-Mills fields, F -harmonic maps, and more generally differential k-forms, k ≥ 0 with values in vector bundles. In this paper, we introduce normalized exponential Yang-Mills energy functional YM 0 e , resp. exponential Yang-Mills energy functional YM e , stress-energy tensor S e,YM 0 associated with the normalized exponential Yang-Mills energy functional YM 0 e , resp. stress-energy tensor S e,YM associated with the exponential Yang-Mills energy functional YM e , ( A critical point of YM 0 e , i.e. a normalized exponential Yang-Mills connection, and its associated normalized exponential Yang-Mills field are just the same as the Yang-Mills connection and its associated exponential Yang-Mills field ). We also introduce the notion of the e-degree d e which connects two separate parts in the associated normalized exponential stress-energy tensors S e,YM 0 (cf. (4.15)). These stress-energy tensors arise from calculating the rate of change of various functionals when the metric of the domain or base manifold is changed and are naturally linked to various conservation laws. For example, we prove that every normalized exponential Yang-Mills field or every exponential Yang-Mills field R ∇ satisfies an e-conservation law (cf. Theorem 3.11). Every normalized exponential Yang-Mills connection or exponential Yang-Mills connection satisfies the exponential Yang-Mills equation (cf. Corollary 3.7). We then prove monotonicity formulae, via the coarea formula and comparison theorems in Riemannian geometry (cf. [GW,DW,HLRW,W11]). Whereas a "microscopic" approach to some of these monotonicity formulae leads to celebrated blow-up techniques due to E. de-Giorgi ( [Gi]) and W. L. Fleming ([F1]), and regularity theory in geometric measure theory(cf. [A, Al, FF, HL, Lu, PS, SU], for example, the regularity results of Allard ([A]) depend on the monotonicity formulae for varifolds. Monotonicity properties are also dealt with by Price and Simon ( [PS]), Price ([P]) for Yang-Mills fields, and by Hardt-Lin ([HL]) and Luckhaus ([Lu]) for p-harmonic maps), a "macroscopic" version of these monotonicity formulae enable us to derive some vanishing theorems under suitable growth conditions on Cartan-Hadamard manifolds or manifolds which possess a pole with appropriate curvature assumptions. In particular, we have Theorem 5.1 -the monotonicity formula for exponential Yang-Mills fields and Theorem 6.1 -the vanishing theorem for exponential Yang-Mills fields. These monotonicity formula and vanishing theorem for exponential Yang-Mills fields augment and extend vanishing theorems for F -Yang-Mills fields in [DW] and [W11]. We note that even when F (t) = e t or F (t) = e t − 1 for t = ||R ∇ || 2 2 , F -Yang-Mills field becomes exponential Yang-Mills field, the following vanishing theorem for F -Yang-Mills fields are not applicable to exponential Yang-Mills fields, This is due to the fact that for F (t) = e t , the degree of F , d F := sup t≥0 tF ′ (t) F (t) = ∞, and the F -Yang-Mills energy functional growth condition (1.3) is not satisfied for λ = −∞ in (1.4). To overcome this difficulty in getting estimates, we introduce the notion of e-degree d e , for a given curvature tensor R ∇ cf. (4.15) . Theorem A (Vanishing theorem for F -Yang-Mills fields ( [DW,W11])). Suppose that the radial curvature K(r) of M satisfies one of the seven conditions (P ) is an F -Yang-Mills field and satisfies (i) − α 2 ≤ K(r) ≤ −β 2 with α > 0, β > 0 and (n − 1)β − 4αd F ≥ 0; (ii) K(r) = 0 with n − 4d F > 0; (iii) − A (1 + r 2 ) 1+ǫ ≤ K(r) ≤ B (1 + r 2 ) 1+ǫ with ǫ > 0 , A ≥ 0 , 0 < B < 2ǫ , and n − (n − 1) B 2ǫ − 4e A 2ǫ d F > 0; (iv) − A r 2 ≤ K(r) ≤ − A 1 r 2 with 0 ≤ A 1 ≤ A , and 1 + (n − 1) 1 + √ 1 + 4A 1 2 − 2(1 + √ 1 + 4A)d F > 0; (v) − A(A − 1) r 2 ≤ K(r) ≤ − A 1 (A 1 − 1) r 2 and A ≥ A 1 ≥ 1 , and 1 + (n − 1)A 1 − 4Ad F > 0; (vi) B 1 (1 − B 1 ) r 2 ≤ K(r) ≤ B(1 − B) r 2 , with 0 ≤ B, B 1 ≤ 1 , and 1 + (n − 1)(|B − 1 2 | + 1 2 ) − 2 1 + 1 + 4B 1 (1 − B 1 ) d F > 0; (vii) B 1 r 2 ≤ K(r) ≤ B r 2 with 0 ≤ B 1 ≤ B ≤ 1 4 , and 1 + (n − 1) 1 + √ 1 − 4B 2 − (1 + 1 + 4B 1 ) R ∇ 2 ∞ > 0. (1.2) If R ∇ ∈ A 2 AdBρ(x0) F ( ||R ∇ || 2 2 ) dv = o(ρ λ ) as ρ → ∞, (1.3) where λ is given by λ ≤                          n − 4 α β d F if K(r) obeys (i) n − 4d F if K(r) obeys (ii) n − (n − 1) B 2ǫ − 4e A 2ǫ d F if K(r) obeys (iii) 1 + (n − 1) 1+ √ 1+4A1 2 − 2(1 + √ 1 + 4A)d F if K(r) obeys (iv) 1 + (n − 1)A 1 − 4Ad F if K(r) obeys (v) 1 + n−1 (|B− 1 2 |+2 −1 ) −1 − 2 1 + 1 + 4B 1 (1 − B 1 ) d F if K(r) obeys (vi) 1 + (n − 1) 1+ √ 1−4B 2 − 2(1 + √ 1 + 4B 1 )d F if K(r) obeys (vii). (1.4) Then R ∇ ≡ 0 on M . In particular, every F -Yang-Mills field R ∇ with finite F -Yang-Mills energy functional vanishes on M . We also discuss An Average Principle (cf. Proposition 8.1) and Jensen's inequality from varied, generalized viewpoints and perspectives of exponential Yang-Mills fields, p-Yang-Mills fields, and Yang-Mills fields. (Theorems 7.1,9.1,9.2,10.1,10.2). In the context of harmonic maps, the stress-energy tensor was introduced and studied in detail by Baird and Eells ([BE]). Following Baird-Eells ([BE], Sealey [Se2] introduced the stress-energy tensor for vector bundle valued p-forms and established some vanishing theorems for L 2 harmonic p-forms (cf. [DLW,Se1,Xi1]). In a more general frame, Dong and Wei use a unified method to study the stressenergy tensors and yields monotonicity inequalities, and vanishing theorems for vector bundle valued p-forms ( [DW]). The idea and methods can be extended and unified in σ 2 -version of harmonic maps -Φ-Harmonic maps (cf. [HW]). These are the second elementary symmetric function of a pull-back tensor, whereas harmonic maps are the first elementary symmetric function of a pull-back tensor. More recently, Feng-Han-Li-Wei use stress-energy tensors to unify properties in Φ S -harmonic maps (cf. [FHLW]), Feng-Han-Wei extend and unify results in Φ S,pharmonic maps (cf. [FHW]), and Feng-Han-Jiang-Wei further extend and unify results in Φ (3) -harmonic maps (cf. [FHJW]). Whereas we can view harmonic maps as Φ (1) -harmonic maps (involving σ 1 ) and Φ-harmonic maps as Φ (2) -harmonic maps (involving σ 2 ), Φ (3) -harmonic maps involve σ 3 , the third elementary symmetric function of the pullback tensor. In fact, an extrinsic average variational method in the calculus of variation can be carried over to more general settings by which we introduce a notion of Φ (3) -harmonic map and find a large class of manifolds, Φ (3) -superstrongly unstable (Φ (3) -SSU) manifolds, introduce notions of a stable Φ (3) -harmonic map, and Φ (3) -strongly unstable (Φ (3) -SU) manifolds (cf. Theorems 11.8, 11.9, 11.10, and 11.11). By an extrinsic average variational method in the calculus variations proposed in [W3], we find multiple large classes of manifolds with geometric and topological properties in the setting of varied, coupled, generalized type of harmonic maps, and summarize some of the results in Table 1. For some details, related ideas, techniques, we refer to [CW3], [W1]- [W12], [WLW]. = π 2 = 0 Φ (1) − harmonic map E Φ (1) Φ (1) − SSU manifolds Φ (1) − SU π 1 = π 2 = 0 p − harmonic map Ep p − SSU manifolds p − SU π 1 = · · · = π [p] = 0 Φ − harmonic map or Φ − energy functional E Φ or Φ − SSU manifolds or Φ − SU or π 1 = · · · = π 4 = 0 Φ (2) − harmonic map E Φ (2) Φ (2) − SSU manifolds Φ (2) − SU π 1 = · · · = π 4 = 0 Φ S − harmonic map E Φ S Φ S − SSU manifolds Φ S − SU π 1 = · · · = π 4 = 0 Φ S,p − harmonic map E Φ S,p Φ S,p − SSU manifolds Φ S,p − SU π 1 = · · · = π [2p] = 0 Φ (3) − harmonic map Φ (3) − energy functional E Φ (3) Φ (3) − SSU manifolds Φ (3) − SU π 1 = · · · = π 6 = 0 Fundamentals in vector bundles and principal G-bundle This section is devoted to a brief discussion of the fundamental notions in vector bundles and principal G-bundle. Definition 2.1. A (differentiable) vector bundle of rank n consists of a total space E, a base M , and a projection π : E → M , where E and M are differentiable manifolds, π is differentiable, each fiber E x := π −1 (x) for x ∈ M , carries the structure of an n-dimensional (real) vector space, with the following local triviality: For each x ∈ M , there exist a neighborhood U and a diffeomorphism ϕ : π −1 (U ) → U × R n such that for every y ∈ U ϕ y := ϕ |Ey : E y → {y} × R n is a vector space isomorphism. Such a pair (ϕ, U ) is called a bundle chart. Note that local trivializations ϕ α , ϕ β with U α ∩ U β = ∅ determines transition maps ϕ βα : U α ∩ U β → Gl(n, R) by ϕ β • ϕ −1 α (x, v) = (x, ϕ βα (x)v) for x ∈ M, v ∈ R n , where Gl(n, R) is the general linear group of bijective linear self maps of R n . As direct consequences, the transition maps satisfy: ϕ αα (x) = id R n for x ∈ U α ; ϕ αβ (x)ϕ βα (x) = id R n for x ∈ U α ∩ U β ; ϕ αγ (x)ϕ γβ (x)ϕ βα (x) = id R n for x ∈ U α ∩ U β ∩ U γ . (cf. [J]) A vector bundle can be reconstructed from its transition maps E = α U α × R n / ∼ , where denotes disjoint union, and the equivalence relation ∼ is defined by (x, v) ∼ (y, w) :⇐⇒ x = y and w = ϕ βα (x)v (x ∈ U α , y ∈ U β , v, w ∈ R n ) . (2.1) Definition 2.2. Let G be a subgroup of Gl(n, R), for example the orthogonal group O(n) or special orthogonal group SO(n) . By a vector bundle has the structure group G, we mean there exists an atlas of bundle charts for which all transition maps have their values in G . Definition 2.3. Let G be a Lie group. A principal G-bundle consists of a base M , the total space P of the bundle, and a differentiable projection π : P → M , where P and M are differentiable manifolds, with an action of G on P satisfying (i) G acts freely on P from the right: (q, p) ∈ P × G is mapped to qp ∈ P , and qp = q for q = e . The G action then defines an equivalence relation on P : p ∼ q :⇐⇒ ∃g ∈ G such that p = qg . (ii) M is the quotient of P by this equivalence relation, and π : P → M maps q ∈ M to its equivalence class. By (i), each fiber π −1 (x) can then be identified with G. (iii) P is locally trivial in the following sense: For each x ∈ M , there exists a neighborhood U of x and a diffeomorphism ϕ : π −1 (U ) → U × G of the form ϕ(p) = (π(p), ψ(g)) which is G-equivariant, i.e. ϕ(pg) = (π(p), ψ(p)g) for all g ∈ G. Example 2.4. We have the following results. (i) The projection S n → P n (R) of the n-sphere to the real projective space is a principal bundle with group G = O(1) = Z 2 (ii) The Hopf map S 2n+1 → P n (C) of the 2n + 1-sphere to the complex projective space is a principal bundle with group G = U (1) = S 1 (iii) The Hopf map S 4n+1 → P n (Q) of the 4n + 1-sphere to the quaternionic projective space is a principal bundle with group G = Sp(1) = S 3 (iv) Hopf fibrations: S 1 → S 1 , S 3 → S 2 , S 7 → S 4 , and S 15 → S 8 For k = 1, 2, 4, 8 , the Hopf construction is defined by (z, w) → u(z, w) = (|z| 2 − |w| 2 , 2z · w) : R k × R k → R k+1 . In fact, Hopf fibrations are p-harmonic maps and p-harmonic morphisms for every p > 1 (c.f., e.g., [W8,CW2]). We recall a C 2 map u : M → N is said to be a p-harmonic morphism if for any p-harmonic function f defined on an open set V of N , the composition f • u is p-harmonic on u −1 (V ). Example 2.5. If E → M is a vector bundle with fiber V , the bundle of bases of E, B(E) → M is a principle bundle with group Gl(V ) . 2.1. Reversibility of principal and vector bundles. (=⇒) Given a principal G-bundle P → M and a vector space V on which G acts from the left, we construct the associated vector bundle E → M with fiber V as follows: We have a free action of G on P × V from the right: P × V × G → P × V (p, v) · g = (p · g, g −1 v) . If we divide out this G-action, i.e. identify (p, v) and (p, v) · g, the fibers of (P × V )/G → P/G becomes vector spaces isomorphic to V , and E := P × G V := (P × V ) /G → M is a vector bundle with fiber G × G V := (G × V ) /G = V and structure group G . The transition functions for P also give transition functions for E via the left action of G on V. (⇐=) Conversely, given a vector bundle E with structure group G, we construct a principal G-bundle as α U α × G / ∼ with (x α , g α ) ∼ (x β , g β ) :⇐⇒ x α = x β ∈ U α ∩ U β and g β = ϕ βα (x)g α where {U α } is a local trivialization of E with transition functions ϕ βα as in (2.1). Example 2.6. We have the following assertions. (i) The canonical line bundles (real, complex and quaternionic) over the projective spaces P n (R) , P n (C) and of the P n (Q) are the associated bundles of the principal bundles in Example 2.4 (i) − (iii) via the canonical actions of O(1), U (1) and Sp(1) on R, C and Q respectively. (ii) Let E → M be a bundle with fiber F and structure group G and f : N → M be a map between manifolds N and M . Then the pull-back of E → M is a bundle f −1 E → M with fiber F , structure group G, and bundle charts (ϕ • f, f −1 (U )), where ϕ(U ) are bundle charts of E . The pull-back f −1 E → M is called the pull-back bundle. Normalized exponential Yang-Mills functionals and e-conservation laws Our basic set-up is the following: We consider a Riemannian manifold M , and a principal bundle P with compact structure Lie group G over M . Let Ad(P ) be the adjoint bundle Ad(P ) = P × Ad G , (3.1) where G is the Lie algebra of G. Every connection ρ on P induces a connection ∇ on Ad(P ). A connection ∇ on the vector bundle Ad(P ) is a rule that equips us to take derivatives of smooth cross sections of Ad (P ). We also have the Riemannian connection ∇ M on the tangent bundle T M , and the induced connection on the tensor product Λ 2 T * M ⊗ Ad(P ), where Λ 2 T * M is the second exterior power of the cotangent bundle T * M . An Ad G invariant inner product on G induces a fiber metric on Ad(P ) and makes Ad(P ) and Λ 2 T * M ⊗ Ad(P ) into Riemannian vector bundles. Denote by Γ Λ 2 T * M ⊗ Ad(P ) the (infinite-dimensional) vector space of smooth sections of Λ 2 T * M ⊗ Ad(P ) . For k ≥ 0 set A k Ad(P ) = Γ(Λ k T * M ⊗ Ad(P )) be the space of smooth k-forms on M with values in the vector bundle Ad(P ). Although ρ is not a section of A 1 Ad(P ) , via its induced connection ∇, the associated curvature tensor R ∇ , given by R ∇ X,Y = [∇ X , ∇ Y ] − ∇ [X,Y ] , is in A 2 (Ad(P )). Let C be the space of smooth connections ∇ on Ad(P ) , and dv be the volume element of M . Recall the Yang-Mills functional is the mapping YM : C → R + given by YM(∇) = M 1 2 ||R ∇ || 2 dv , (3.2) the p-Yang-Mills functional, for p ≥ 2 (resp. the F -Yang-Mills functional ) is the mapping YM p : C → R + given by YM p (∇) = M 1 p ||R ∇ || p dv resp. YM F (∇) = M F ( 1 2 ||R ∇ || 2 ) dv , (3.3) where the norm is defined in terms of the Riemannian metric on M and a fixed Ad G -invariant inner product on the Lie algebra G of G . That is, at each point x ∈ M , its norm ||R ∇ || 2 x = i<j ||R ∇ ei,ej || 2 x (3.4) where {e 1 , · · · , e n } is an orthonormal basis of T x (M ) and the norm of R ∇ ei,ej is the standard one on Hom(Ad (P ), Ad(P ))-namely, S, U ≡ trace (S T • U ) . A connection ∇ on the adjoint bundle Ad(P ) is said to be a Yang-Mills connection (resp. p-Yang-Mills connection, p ≥ 2, F -Yang-Mills connection) and its associated curvature tensor R ∇ is said to be a Yang-Mills field (resp. p-Yang-Mills field, p ≥ 2, F -Yang-Mills field ), if ∇ is a critical point of YM (resp. YM p , YM F ) with respect to any compactly supported variation in the space of smooth connections on Ad (P ) . We now introduce Definition 3.1. The normalized exponential Yang-Mills energy functional is the mapping YM 0 e : C → R + given by YM 0 e (∇) = M exp( 1 2 ||R ∇ || 2 ) − 1 dv , (3.5) the exponential Yang-Mills energy functional is the mapping YM e : C → R + given by YM e (∇) = M exp( 1 2 ||R ∇ || 2 ) dv , (3.6) on M , the uniform norm ||R ∇ || ∞ is given by ||R ∇ || 2 ∞ = sup x∈M ||R ∇ || 2 x . (3.7) The normalized exponential Yang-Mills energy functional YM 0 e has the following simple and useful advantage. Proposition 3.2. YM 0 e (∇) ≥ 0 and YM 0 e (∇) = 0 ⇐⇒ R ∇ ≡ 0 . (3.8) This is an analog of p-Yang-Mills functional, for p ≥ 2 , YM p (∇) ≥ 0 and YM p (∇) = 0 ⇐⇒ R ∇ ≡ 0 . (3.9) Definition 3.3. The stress-energy tensor S e,YM 0 associated with the normalized exponential Yang-Mills energy functional YM 0 e and the stress-energy tensor S e,YM associated with the exponential Yang-Mills energy functional YM e are defined respectively as follows: S e,YM 0 (X, Y ) = exp( ||R ∇ || 2 2 ) − 1 g(X, Y ) − exp( ||R ∇ || 2 2 ) i X R ∇ , i Y R ∇ , (3.10) S e,YM (X, Y ) = exp( ||R ∇ || 2 2 ) g(X, Y ) − i X R ∇ , i Y R ∇ (3.11) where , is the induced inner product on A 1 Ad(P ) , and i X R ∇ is the interior multiplication by the vector field X given by (i X R ∇ )(Y 1 ) = R ∇ (X, Y 1 ) ,(3.12) for any vector fields Y 1 on M . We calculate the rate of change of the normalized exponential Yang-Mills energy functional YM 0 e,g and exponential Yang-Mills energy functional YM e,g when the metric g on the domain or base manifold is changed. To this end, we consider a compactly supported smooth one-parameter variation of the metric g , i.e. a smooth family of metrics g s such that g 0 = g . Set δg = ∂gs ∂s s=0 . Then δg is a smooth 2covariant symmetric tensor field on M with compact support. These give birth to their associated stress-energy tensors. Lemma 3.4. With the same notations as above, we have d ds YM 0 e,gs (∇) s=0 = 1 2 M S e,YM 0 , δg dv g (3.13) d ds YM e,gs (∇) s=0 = 1 2 M S e,YM , δg dv g (3.14) where S e,YM 0 and S e,YM are as in (3.8) and (3.9) respectively. Proof. From ( [Ba]), we obtain d||R ∇ || 2 gs ds s=0 = − i,j i ei R ∇ , i ej R ∇ δg(e i , e j ) (3.15) and d ds dv gs s=0 = 1 2 g, δg dv g . (3.16) Then by the chain rule, (3.15), (3.16), and (3.10), we have d ds YM 0 e,gs (∇) s=0 = M d ds exp( ||R ∇ || 2 gs 2 − 1 dv gs s=0 = M exp( ||R ∇ || 2 2 ) d ds ||R ∇ || 2 gs 2 s=0 dv g + M exp( 1 2 ||R ∇ || 2 ) − 1 d ds dv gs s=0 = 1 2 M exp( 1 2 ||R ∇ || 2 ) − 1 g, δg − exp( ||R ∇ || 2 2 ) i,j i ei R ∇ , i ej R ∇ δg(e i , e j ) dv g = 1 2 M S e,YM 0 , δg dv g . ( 3.17) Similarly, we can calculate d ds YM e,gs (∇) s=0 and obtain the desired (3.14). The exterior differential operator d ∇ : A 1 Ad(P ) → A 2 Ad(P ) relative to the connection ∇ is given by (d ∇ σ)(X 1 , X 2 ) = (∇ X1 σ)(X 2 ) − (∇ X2 σ)(X 1 ) . (3.18) Relative to the Riemannian structures of Ad(P ) and T M , the codifferential operator δ ∇ : A 2 Ad(P ) → A 1 Ad(P ) is characterized as the adjoint of d via the formula M d ∇ σ, ρ dv g = M σ, δ ∇ ρ dv g ,(3.19) where σ ∈ A 1 Ad(P ) , ρ ∈ A 2 Ad(P ) , one of which has compact support and dv g is the volume element associated with the metric g on T M . Then (δ ∇ ρ)(X 1 ) = − i (∇ ei ρ)(e i , X 1 ) . (3.20) Definition 3.5. A connection ∇ on the adjoint bundle Ad(P ) is said to be an exponential Yang-Mills connection and its associated curvature tensor R ∇ is said to be an exponential Yang-Mills field, if ∇ is a critical point of YM e with respect to any compactly supported variation in the space of connections on Ad(P ) . Lemma 3.6 (The first variation formula for normalized exponential Yang-Mills functional YM 0 e or YM e ). Let A ∈ A 1 Ad(P ) and ∇ t = ∇ + tA be a family of connections on Ad(P ). Then d dt YM 0 e (∇ t ) s=0 = d dt YM 0 e (∇ t ) s=0 = M δ ∇ exp( 1 2 ||R ∇ || 2 )R ∇ , A dv . (3.21) Furthermore, The Euler-Lagrangian equation for YM 0 e or YM e is exp( 1 2 ||R ∇ || 2 )δ ∇ R ∇ − i grad exp( 1 2 ||R ∇ || 2 ) R ∇ = 0 , (3.22) or δ ∇ exp( 1 2 ||R ∇ || 2 )R ∇ = 0 . (3.23) Proof. By assumption, the curvature of ∇ t is given by R ∇ t = R ∇ + t(d ∇ A) + t 2 [A, A] , (3.24) where [A, A] ∈ A 2 (Ad(P )) is given by [A, A] X,Y = [A X , A Y ] . Indeed, for any local vector fields X, Y on M . with [X, Y ] = 0 , we have via (3.18) R ∇ t X,Y = (∇ X + tA X )(∇ Y + tA Y ) − (∇ Y + tA Y )(∇ X + tA X ) = R ∇ X,Y + t[∇ X , A Y ] − t[∇ Y , A X ] + t 2 [A X , A Y ] = R ∇ X,Y + t∇ X (A Y ) − t∇ Y (A X ) + t 2 [A, A] X,Y = R ∇ X,Y + t(d ∇ A) X,Y + t 2 [A, A] X,Y . (3.25) Thus, exp ( 1 2 ||R ∇ t || 2 ) = exp ( 1 2 ||R ∇ || 2 + t R ∇ , d ∇ A + ε(t 2 )) , (3.26) where ε(t 2 ) = o(t 2 ) as t → 0 . Therefore, YM e (∇ t ) = M exp ( 1 2 ||R ∇ || 2 + t R ∇ , d ∇ A + ε(t 2 )) dv (3.27) and via (3.19), we have d dt YM 0 e (∇ t ) s=0 = d dt YM e (∇ t ) s=0 = M exp ( 1 2 ||R ∇ || 2 ) R ∇ , d ∇ A dv = M δ ∇ exp ( 1 2 ||R ∇ || 2 )R ∇ , A dv . (3.28) This derives the Euler-Lagrange equation for YM 0 e or YM e by (3.20) as follows 0 = δ ∇ exp ( 1 2 ||R ∇ || 2 )R ∇ = − m i=1 ∇ ei exp ( 1 2 ||R ∇ || 2 )R ∇ (e i , ·) = exp ( 1 2 ||R ∇ || 2 )δ ∇ R ∇ − i grad exp ( 1 2 ||R ∇ || 2 ) R ∇ . (3.29) Corollary 3.7. Every normalized exponential Yang-Mills connection or every exponential Yang-Mills connecton ∇ satisfies (3.29). Dong and Wei derive Theorem B ([DW]) (i) The Euler-Lagrangian equation for F -Yang-Mills func- tional YM F is F ′ ( 1 2 ||R ∇ || 2 )δ ∇ R ∇ − i grad F ′ ( 1 2 ||R ∇ || 2 ) R ∇ = 0 (3.30) or δ ∇ F ′ ( 1 2 ||R ∇ || 2 )R ∇ = 0 . (ii) The Euler-Lagrangian equation for p-Yang-Mills functional YM p , p ≥ 2 is δ ∇ (||R ∇ || p−2 R ∇ ) = 0 (3.31) or ||R ∇ || p−2 δ ∇ R ∇ − i grad(||R ∇ || p−2 ) R ∇ = 0 . (3.30) is also due to C. Gherghe ([G]). Corollary 3.8. Let ||R ∇ || = constant. Then the following are equivalent: (i) A curvature tensor R ∇ is a normalized exponential Yang-Mills field . (ii) A curvature tensor R ∇ is a Yang-Mills field . (iii) A curvature tensor R ∇ is a p − Yang-Mills field, p ≥ 2 . (iv) A curvature tensor R ∇ is an exponential Yang-Mills field . (v) A curvature tensor R ∇ is an F -Yang-Mills field . (3.32) Proof. This follows at once from (3.29)-(3.31). Lemma 3.9. Let S e,YM 0 and S e,YM be the stress-energy tensors defined by (3.9) and (3.10) respectively, then for any vector field X on M , we have (div S e,YM 0 )(X) = (div S e,YM )(X) = exp( ||R ∇ || 2 2 ) δ ∇ R ∇ , i X R ∇ + exp( ||R ∇ || 2 2 ) i X d ∇ R ∇ , R ∇ − i grad(exp( ||R ∇ || 2 2 )) R ∇ , i X R ∇ ,(3. 33) where grad ( • ) is the gradient vector field of • . Definition 3.10. A curvature tensor R ∇ ∈ A 2 Ad(P ) is said to satisfy an e-conservation law if S e,YM 0 is divergence free, i.e., divS e,YM 0 = divS e,YM = 0 . (3.34) Theorem 3.11. Every normalized exponential Yang-Mills field or every exponential Yang-Mills field R ∇ satisfies an e-conservation law. Proof. It is known that R ∇ satisfies the Bianchi identity d ∇ R ∇ = 0 . (3.35) Therefore, by Corollary 3.7, Lemma 3.9 and (3.35), we immediately derive the desired (3.34). Comparison theorems in Riemannian geometry In this section, we will discuss comparison theorems with applications on Cartan-Hadamard manifolds or more generally on complete manifolds with a pole. We recall a Cartan-Hadamard manifold is a complete simply-connected Riemannian manifold of nonpositive sectional curvature. A pole is a point x 0 ∈ M such that the exponential map from the tangent space to M at x 0 into M is a diffeomorphism. By the radial curvature K of a manifold with a pole, we mean the restriction of the sectional curvature function to all the planes which contain the unit vector ∂(x) in T x M tangent to the unique geodesic joining x 0 to x and pointing away from x 0 . Let the tensor g − dr dr = 0 on the radial direction ∂, and is just the metric tensor g on the orthogonal complement ∂ ⊥ . ⇒ β coth(βr) g − dr ⊗ dr ≤ Hess(r) ≤ α coth(αr) g − dr ⊗ dr ; (4.1) K(r) = 0 (ii [GW]) ⇒ 1 r g − dr ⊗ dr = Hess(r); (4.2) − A (1 + r 2 ) 1+ǫ ≤ K(r) ≤ B (1 + r 2 ) 1+ǫ with ǫ > 0, A ≥ 0, and 0 ≤ B < 2ǫ (iii [GW], [DW,Lemma 4 .1.(iii)]) ⇒ 1 − B 2ǫ r g − dr ⊗ dr ≤ Hess(r) ≤ e A 2ǫ r g − dr ⊗ dr ; (4.3) − A r 2 ≤ K(r) ≤ − A 1 r 2 with 0 ≤ A 1 ≤ A (iv [HLRW], [W11, Theorem A]) ⇒ 1 + √ 1 + 4A 1 2r g − dr ⊗ dr ≤ Hess(r) ≤ 1 + √ 1 + 4A 2r g − dr ⊗ dr ; (4.4) − A(A − 1) r 2 ≤ K(r) ≤ − A 1 (A 1 − 1) r 2 with A ≥ A 1 ≥ 1 (v [W11, Corollary 3.1]) ⇒ A 1 r g − dr ⊗ dr ≤ Hessr ≤ A r g − dr ⊗ dr ; (4.5) B 1 (1 − B 1 ) r 2 ≤ K(r) ≤ B(1 − B) r 2 , with 0 ≤ B, B 1 ≤ 1 (vi [W11, Corollary 3.5]) ⇒ |B − 1 2 | + 1 2 r g − dr ⊗ dr ≤ Hessr ≤ 1 + 1 + 4B 1 (1 − B 1 ) 2r g − dr ⊗ dr ; (4.6) B 1 r 2 ≤ K(r) ≤ B r 2 with 0 ≤ B 1 ≤ B ≤ 1 4 (vii [W11, Theorem 3.5]) ⇒ 1 + √ 1 − 4B 2r g − dr ⊗ dr ≤ Hessr ≤ 1 + √ 1 + 4B 1 2r g − dr ⊗ dr ; (4.7) −Ar 2q ≤ K(r) ≤ −Br 2q with A ≥ B > 0 , q > 0 (viii [GW]) ⇒ B 0 r q g − dr ⊗ dr ≤ Hess(r) ≤ ( √ A coth √ A)r q g − dr ⊗ dr , for r ≥ 1 , (4.8) where B 0 = min{1, − q + 1 2 + B + ( q + 1 2 ) 2 1 2 } . (4.9) Proof. (i), (ii) and (viii) are treated in section 2 of [GW], (iii) is proved in [DW], (iv) is derived in [HLRW,W11], (v) -(vii) are proved in [W11]. We note there are many applications of this Theorem (cf., e.g., [WW]), (iv) extends the asymptotic comparison theorem in ( [GW], [PRS], p.39), and (vii) generalizes ( [EF], Lemma 1.2 (b)). Let ♭ denote the bundle isomorphism that identifies the vector field X with the differential one-form X ♭ , and let ∇ be the Riemannian connection of M . Then the covariant derivative ∇X ♭ of X ♭ is a (0, 2)-type tensor, given by ∇X ♭ (Y, Z) = ∇ Y X ♭ Z = ∇ Y X, Z , ∀ X, Y ∈ Γ(M ) . (4.10) If X is conservative, then X = ∇f, X ♭ = df and ∇X ♭ = Hess(f ) . (4.11) for some scalar potential f (cf. [CW], p. 1527). A direct computation yields (cf., e.g., [DW]) div(i X S e,YM 0 ) = S e,YM 0 , ∇X ♭ + (divS e,YM 0 )(X) , ∀ X ∈ Γ(M ) . (4.12) By Theorem 3.11, every normalized exponential Yang-Mills field R ∇ satisfies an e-conservation law. It follows from the divergence theorem that for every bounded domain D in M with C 1 boundary ∂D , ∂D S e,YM 0 (X, ν)ds g = D S e,YM 0 , ∇X ♭ dv g ,(4.13) where ν is unit outward normal vector field along ∂D with (n − 1)-dimensional volume element ds g . When we choose scalar potential f (x) = 1 2 r 2 (x), (4.11) becomes X = r∇r, X ♭ = rdr and ∇X ♭ = Hess( 1 2 r 2 ) = dr ⊗ dr + rHess(r) . (4.14) The conservative vector field X and e-conservation law will illuminate that the curvature of the base manifold M via Hessian Comparison Theorems 4.1 influences the behavior of the stress energy tensor S e,YM 0 and the behavior of the underlying criticality -curvature field R ∇ ∈ A 2 (Ad(P )) with the help from the following concept (4.15) and estimate (4.20). Analogous to F -degree, we introduce Definition 4.2. For a given curvature field R ∇ , the e-degree d e is the quantity, given by d e = sup x∈M exp ||R ∇ || 2 2 (x) exp ||R ∇ || 2 2 (x) − 1 . (4.15) The e-degree d e will play a role in connecting two separated parts of the normalized stress-energy tensor S e,YM 0 . Since e t e t −1 is a decreasing function, with 1 ≤ e t e t −1 ≤ ∞, we have Proposition 4.3. Suppose ||R ∇ || 2 2 (x) ≤ c ∀ x ∈ M , (4.16) where c > 0 is a constant. Then d e ≥ e c e c − 1 . (4.17) Lemma 4.4. Let M be a complete n-manifold with a pole x 0 . Assume that there exist two positive functions h 1 (r) and h 2 (r) such that h 1 (r)(g − dr ⊗ dr) ≤ Hess(r) ≤ h 2 (r)(g − dr ⊗ dr) (4.18) on M \{x 0 }. If h 2 (r) satisfies rh 2 (r) ≥ 1 , (4.19) and ||R ∇ || > 0 on M , then S e,YM 0 , ∇X ♭ ≥ 1 + (n − 1)rh 1 (r) − 2d e ||R ∇ || 2 ∞ rh 2 (r) exp( ||R ∇ || 2 2 ) − 1 , (4.20) where X = r∇r. Proof. Choose an orthonormal frame {e i , ∂ ∂r } i=1,...,n−1 around x ∈ M \{x 0 }. Take X = r∇r. Then ∇ ∂ ∂r X = ∂ ∂r , (4.21) ∇ ei X = r∇ ei ∂ ∂r = rHess(r)(e i , e j )e j .(4.22) Using (3.10), (4.14), or (4.21), (4.22) , we have S e,YM 0 , ∇X ♭ = exp( ||R ∇ || 2 2 ) − 1 (1 + n−1 i=1 rHess(r)(e i , e i )) − n−1 i,j=1 exp( ||R ∇ || 2 2 ) i ei R ∇ , i ej R ∇ rHess(r)(e i , e j ) − exp( ||R ∇ 2 2 ) i ∂ ∂r R ∇ , i ∂ ∂r R ∇ . (4.23) By (4.18) and (4.15), (4.23) implies that S e,YM 0 , ∇X ♭ ≥ exp( ||R ∇ || 2 2 ) − 1 1 + (n − 1)rh 1 (r) − exp( ||R ∇ || 2 2 ) − 1 n−1 i=1 i ei R ∇ , i ei R ∇ rh 2 (r) exp( ||R ∇ || 2 2 ) exp( ||R ∇ || 2 2 ) − 1 − exp( ||R ∇ || 2 2 ) − 1 i ∂ ∂r R ∇ , i ∂ ∂r R ∇ exp( ||R ∇ || 2 2 ) exp( ||R ∇ || 2 2 ) − 1 ≥ exp( ||R ∇ || 2 2 ) − 1 1 + (n − 1)rh 1 (r) − 2||R ∇ || 2 rh 2 (r) exp( ||R ∇ || 2 2 ) exp( ||R ∇ || 2 2 ) − 1 + exp( ||R ∇ || 2 2 ) − 1 (rh 2 (r) − 1) i ∂ ∂r R ∇ , i ∂ ∂r R ∇ exp( ||R ∇ || 2 2 ) exp( ||R ∇ || 2 2 ) − 1 ≥ 1 + (n − 1)rh 1 (r) − 2||R ∇ || 2 rh 2 (r)d e exp( ||R ∇ || 2 2 ) − 1 . (4.24) The last two steps follow from (4.19) and the fact that where e n = ∂ ∂r . Now the Lemma follows immediately from (4.24) and (3.7). n−1 i=1 i ei R ∇ , i ei R ∇ + i ∂ ∂r R ∇ , i ∂ ∂r R ∇ = 1≤j1≤n n i=1 R ∇ (e i , e j1 ), R ∇ (e i , e j1 ) = 2||R ∇ || 2 , Monotonicity formulae In this section, we will establish monotonicity formulae on complete manifolds with a pole. Theorem 5.1 (Monotonicity formulae). Let (M, g) be an n−dimensional complete Riemannian manifold with a pole x 0 , Ad(P ) be the adjoint bundle and the curvature tensor R ∇ ∈ A 2 Ad(P ) be an exponential Yang-Mills field. Assume that the radial curvature K(r) of M and the curvature tensor R ∇ satisfy one of the following seven conditions: (i) − α 2 ≤ K(r) ≤ −β 2 with α > 0, β > 0 and (n − 1)β − 2d e α R ∇ 2 ∞ ≥ 0; (ii) K(r) = 0 with n − 2d e R ∇ 2 ∞ > 0; (iii) − A (1 + r 2 ) 1+ǫ ≤ K(r) ≤ B (1 + r 2 ) 1+ǫ with ǫ > 0 , A ≥ 0 , 0 < B < 2ǫ , and n − (n − 1) B 2ǫ − 2d e e A 2ǫ R ∇ 2 ∞ > 0; (iv) − A r 2 ≤ K(r) ≤ − A 1 r 2 with 0 ≤ A 1 ≤ A , and 1 + (n − 1) 1 + √ 1 + 4A 1 2 − d e (1 + √ 1 + 4A) R ∇ 2 ∞ > 0; (v) − A(A − 1) r 2 ≤ K(r) ≤ − A 1 (A 1 − 1) r 2 and A ≥ A 1 ≥ 1 , and 1 + (n − 1)A 1 − 2d e A R ∇ 2 ∞ > 0; (vi) B 1 (1 − B 1 ) r 2 ≤ K(r) ≤ B(1 − B) r 2 , with 0 ≤ B, B 1 ≤ 1 , and 1 + (n − 1)(|B − 1 2 | + 1 2 ) − d e 1 + 1 + 4B 1 (1 − B 1 ) R ∇ 2 ∞ > 0; (vii) B 1 r 2 ≤ K(r) ≤ B r 2 with 0 ≤ B 1 ≤ B ≤ 1 4 , and 1 + (n − 1) 1 + √ 1 − 4B 2 − d e (1 + 1 + 4B 1 ) R ∇ 2 ∞ > 0. (5.1) Then 1 ρ λ 1 Bρ 1 (x0) exp( ||R ∇ || 2 2 ) − 1 dv ≤ 1 ρ λ 2 Bρ 2 (x0) exp( ||R ∇ || 2 2 ) − 1 dv , (5.2) for any 0 < ρ 1 ≤ ρ 2 , where λ ≤                          n − 2d e α β R ∇ 2 ∞ if K(r) obeys (i) , n − 2d e R ∇ 2 ∞ if K(r) obeys (ii) , n − (n − 1) B 2ǫ − 2d e e A 2ǫ R ∇ 2 ∞ if K(r) obeys (iii) , 1 + (n − 1) 1+ √ 1+4A1 2 − d e (1 + √ 1 + 4A) R ∇ 2 ∞ if K(r) obeys (iv) , 1 + (n − 1)A 1 − 2d e A R ∇ 2 ∞ if K(r) obeys (v) , 1 + n−1 (|B− 1 2 |+ 1 2 ) −1 − d e 1 + 1 + 4B 1 (1 − B 1 ) R ∇ 2 ∞ if K(r) obeys (vi) , 1 + (n − 1) 1+ √ 1−4B 2 − d e (1 + √ 1 + 4B 1 ) R ∇ 2 ∞ if K(r) obeys (vii) . (5.3) Proof. Take a smooth vector field X = r∇r on M . If K(r) satisfies (i), then by Theorem 4.1 and the increasing function αr coth(αr) → 1 as r → 0 , (4.19) holds. Now Lemma 4.1 is applicable and by (4.20), we have on B ρ (x 0 )\{x 0 } , for every ρ > 0, S e,YM 0 , ∇X ♭ ≥ 1 + (n − 1)βr coth(βr) − 2d e αr coth(αr) R ∇ 2 ∞ exp( ||R ∇ || 2 2 ) − 1 = 1 + βr coth(βr)(n − 1 − 2 · d e · αr coth(αr) βr coth(βr) R ∇ 2 ∞ ) exp( ||R ∇ || 2 2 ) − 1 > 1 + 1 · (n − 1 − 2 · d e · α β · 1 R ∇ 2 ∞ ) exp( ||R ∇ || 2 2 ) − 1 ≥ λ exp( ||R ∇ || 2 2 ) − 1 , (5.4) provided that n − 1 − 2 · d e · α β R ∇ 2 ∞ ≥ 0 , since βr coth(βr) > 1 for r > 0 , and coth(αr) coth(βr) < 1 , for 0 < β < α , and coth is a decreasing function. Similarly, from Theorem 4.1 and Lemma 4.4, the above inequality holds for the cases (ii) -(vii) on B ρ (x 0 )\{x 0 } . Thus, by the continuity of S e,YM , ∇X ♭ and exp( ||R ∇ || 2 2 ) , and (3.10), we have for every ρ > 0, S e,YM 0 , ∇X ♭ ≥ λ exp( ||R ∇ || 2 2 ) − 1 in B ρ (x 0 ) ρ exp( ||R ∇ || 2 2 ) − 1 ) ≥ S e,YM 0 (X, ∂ ∂r ) on ∂B ρ (x 0 ) . (5.5) It follows from (4.13) and (5.5) that ρ ∂Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 ds ≥ λ Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 dv . (5.6) Hence, we get from (5.6) the following ∂Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 ds Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 dv ≥ λ ρ . (5.7) The coarea formula implies that d dρ Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 dv = ∂Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 ds . (5.8) Thus we have d dρ Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 dv Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 dv ≥ λ ρ (5.9) for a.e. ρ > 0 . By integration (5.9) over [ρ 1 , ρ 2 ], we have ln Bρ 2 (x0) exp( ||R ∇ || 2 2 )− 1 dv − ln Bρ 1 (x0) exp( ||R ∇ || 2 2 )− 1 dv ≥ ln ρ λ 2 − ln ρ λ 1 . (5.10) This proves (5.2). Corollary 5.2. Suppose that M has constant sectional curvature −α 2 ≤ 0 and n − 1 − 2d e R ∇ 2 ∞ ≥ 0 if α = 0; n − 2d e R ∇ 2 ∞ > 0 if α = 0. Let R ∇ ∈ A 2 Ad(P ) be an exponential Yang-Mills field. Then 1 ρ n−2de R ∇ 2 ∞ 1 Bρ 1 (x0) exp( ||R ∇ || 2 2 ) − 1 dv ≤ 1 ρ n−2de R ∇ 2 ∞ 2 Bρ 2 (x0) exp( ||R ∇ || 2 2 ) − 1 dv , (5.11) for any x 0 ∈ M and 0 < ρ 1 ≤ ρ 2 . Proof. In Theorem 5.1, if we take α = β = 0 for the case (i) or α = 0 for the case (ii), this corollary follows immediately. Proposition 5.3. Let (M, g) be an n−dimensional complete Riemannian manifold whose radial curvature satisfies (viii) − Ar 2q ≤ K(r) ≤ −Br 2q with A ≥ B > 0 and q > 0. (5.12) Let R ∇ be an exponential Yang-Mills field, and δ := (n − 1)B 0 − 2d e R ∇ 2 ∞ √ A coth √ A ≥ 0 , (5.13) where B 0 is as in (4.9). Suppose that (5.18) holds. Then 1 ρ 1+δ 1 Bρ 1 (x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv ≤ 1 ρ 1+δ 2 Bρ 2 (x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv , (5.14) for any 1 ≤ ρ 1 ≤ ρ 2 . Proof. Take X = r∇r. Applying Theorem 4.1, (4.19), and (4.20), we have S e,YM 0 , ∇X ♭ ≥ exp( ||R ∇ || 2 2 ) − 1 (1 + δr q+1 (5.15) and S e,YM 0 (X, ∂ ∂r ) = exp( ||R ∇ || 2 2 ) 1 − i ∂ ∂r R ∇ , i ∂ ∂r R ∇ − 1 on ∂B 1 (x 0 ) S e,YM 0 (X, ∂ ∂r ) = ρ(exp( ||R ∇ || 2 2 ) 1 − i ∂ ∂r R ∇ , i ∂ ∂r R ∇ − ρ on ∂B ρ (x 0 ) . (5.16) It follows from (4.13) that ρ ∂Bρ(x0) exp( ||R ∇ || 2 2 ) 1 − i ∂ ∂r R ∇ , i ∂ ∂r R ∇ − 1 ds − ∂B1(x0) exp( ||R ∇ || 2 2 ) 1 − i ∂ ∂r R ∇ , i ∂ ∂r R ∇ − 1 ds ≥ Bρ(x0)−B1(x0) (1 + δr q+1 ) exp( ||R ∇ || 2 2 ) − 1 . (5.17) Whence, if ∂B1(x0) exp( ||R ∇ || 2 2 ) 1 − i ∂ ∂r R ∇ , i ∂ ∂r R ∇ − 1 ds ≥ 0 , (5.18) then ρ ∂Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 ds ≥ (1 + δ) Bρ(x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv , (5.19) for any ρ > 1 . Coarea formula then implies d Bρ(x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv Bρ(x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv ≥ 1 + δ ρ dρ (5.20) for a.e. ρ ≥ 1. Integrating (5.20) over [ρ 1 , ρ 2 ], we get ln Bρ 2 (x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv − ln Bρ 1 (x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv ≥ (1 + δ) ln ρ 2 − (1 + δ) ln ρ 1 . (5.21) Hence we prove the proposition. Corollary 5.4. Let K(r) and δ be as in Proposition 5.3, satisfying (5.12) and (5.13) respectively, R ∇ be an exponential Yang-Mills field. Suppose exp( ||R ∇ || 2 2 ) 1 − i ∂ ∂r R ∇ , i ∂ ∂r R ∇ ≥ 1 (5.22) on ∂B 1 . Then (5.14) holds. Proof. The assumption (5.22) implies that (5.18) holds, and the assertion follows from Proposition 5.3. Vanishing theorems for exponential Yang-Mills fields Theorem 6.1 (Vanishing Theorem). Suppose that the radial curvature K(r) of M satisfies one of the seven growth conditions in (5.1) (i)-(vii), Theorem 5.1. Let R ∇ be an exponential Yang-Mills field satisfying the YM 0 e -energy functional growth condition Bρ(x0) exp( ||R ∇ || 2 2 ) − 1 dv = o(ρ λ ) as ρ → ∞ , (6.1) where λ is given by (5.3). Then exp( ||R ∇ || 2 2 ) ≡ 1 , and hence R ∇ ≡ 0. In particular, every exponential Yang-Mills field R ∇ with finite normalized exponential Yang-Mills YM 0 e -energy functional vanishes on M . Proof. This follows at once from Theorem 5.1. Proposition 6.2. Let (M, g) be an n−dimensional complete Riemannian manifold whose radial curvature satisfies (5.12) (viii) , Proposition 5.1. Let δ be as in (5.13) in which B 0 is as in (4.9). Suppose (5.18) holds. Then every exponential Yang-Mills field R ∇ with the growth condition Bρ(x0)−B1(x0) exp( ||R ∇ || 2 2 ) − 1 dv = o(ρ 1+δ ) as ρ → ∞ (6.2) vanishes on M − B 1 (x 0 ) , In particular, if R ∇ has finite normlalized exponential Yang-Mills energy on M − B 1 (x 0 ), then R ∇ ≡ 0 on M − B 1 (x 0 ). Proof. This follows at once from Proposition 5.3. Vanishing theorems from exponential Yang-Mills fields to F -Yang-Mills fields Theorem 7.1. Suppose that the radial curvature K(r) of M satisfies one of the seven growth conditions in (1.1) ( i) − ( vii), Theorem A , in which d F = 1. Let R ∇ be an exponential Yang-Mills field with ||R ∇ || =constant and Volume B ρ (x 0 ) = o(ρ λ ) as ρ → ∞ , (7.1) where λ is given by (1.4), in which d F = 1. Then R ∇ ≡ 0. In particular, every exponential Yang-Mills field R ∇ with constant ||R ∇ || over manifold which has finite volume, Volume(M ) < ∞ vanishes. Proof. By Corollary 3.8, this exponential Yang-Mills filed R ∇ is a Yang-Mills field which is a special case of F -Yang-Mills field, where F is the identity map. Thus the F-degree of the identity map d F = 1 . Now we apply F -Yang-Mills Vanishing Theorem A in which F (t) = t, d F = 1, the F -Yang-Mills functional YM F growth condition, (1.3) is transformed to the volume of the base manifold growth condition, (7.1), and the conclusion R ∇ ≡ 0 follows. An average principle, isoparametric and sobelov inequalities In this section, we state, interpret, and apply an average principe in a simple discrete version, then extend it to a dual (or continuous) version: Proposition 8.1 (An average principle of concavity (resp. convexity, linearity)). Let f be a concave function (resp. convex function, linear function ) . Then f (average) ≥ average (f ) , resp. f (average) ≤ average (f ) , f (average) = average (f ) . (8.1) Applying (8.1), where a convex function f = exp and "average" is taken over two positive numbers with respect to the sum, yields one of the simplest inequalities that has far-reaching impacts √ a · b = "f (average)"ւ exp( A + B 2 ) (Average Principle) ≤ "average(f )"ւ exp A + exp B 2 = a + b 2 . (8.2) That is, Among all rectangles on the Euclidean plane with a given perimeter L, the square has the largest area A. By duality, this means parallelly Among all rectangles on the Euclidean plane with a given area A, the square has the least perimeter L. Indeed, 16 A = 16 a · b ≤ (2a + 2b) 2 = L 2 (8.4) Equality holds if and only if the rectangles are squares, i.e., a = b. A dual approach from discreteness to continuity yields A sharp isoperimetric inequlality for plane curves: Among all simple closed smooth curves on the Euclidean plane with a given length L, the circle encloses the largest area A . 4πA ≤ L 2 (8.5) Equality holds if and only if the curve encloses a disk. This is equivalent to The Sobolev inequality on R 2 with optimal constant: If u ∈ W 1,1 (R 2 ), then 4π R 2 |u| 2 dx ≤ R 2 |∇u| dx 2 . (8.6) Similarly, applying (8.1), where f = exp and "average" is averaging the sum of n positive numbers, n ≥ 2, yields n √ a 1 · a n = "f (average)"ւ exp(n −1 n j=1 A j ) (Average Principle) ≤ "average(f )"ւ n −1 n j=1 exp A j = n −1 n j=1 a j . (8.7) That is, Example 8.3 (The geometric mean of the numbers is no greater than the arithmetic mean of n positive numbers). n √ a 1 · a n ≤ a 1 + · · · + a n n , for a 1 , · · · a n > 0 , with " = " holds if and only if a 1 = · · · = a n . whenever the right side is defined. Isoperimetric and Sobolev inequlalities can be generalized to higher dimensional Euclidean spaces. As in dimension two, the n-dimensional sharp isoperimetric inequality is equivalent (for sufficiently smooth domains) to : The Sobolev inequality on R n with optimal constant If u ∈ W 1,1 (R n ) and ω n is the volume of the unit ball in R n , then Isoperimetric and Sobolev inequlalities are extended to Riemannian manifolds M with sharp constants and applications to optimal sphere theorems (cf., e.g., Wei-Zhu [WZ]). Theorem 8.5 ( A sharp isoperimetric inequality [Du, WZ]). For every domain Ω (in M ), there exists a constant C(M ) depending on M such that P n ≥ n n ω n V n−1 (1 − C(M )V 2 n ), (8.11) where P = vol(∂Ω), V = vol(Ω) , and ω n is the volume of the unit ball in R n . Furthermore, on simply connected Riemannian manifolds of dimension n with Ricci curvature bounded from below by n − 1, the best C(M ) one can take in the above inequality (8.10) is greater than or equal to C 0 = n(n − 1) 2(n + 2)ω 2 n n . (8.12) It is then by a standard technique, via coarea formula and Cavalieri's principle, that (8.11) is equivalent to the following: Theorem 8.6 (A sharp Sobolev inequality [WZ]). There exists a constant A = A(M ) such that ∀ϕ ∈ W 1,1 (M ), This isoperimetric inequality (8.11) certainly has its roots in global analysis and partial differential equations (see, e.g., [AuL]). Furthermore, the optimal constants in (8.11) will have some geometric and even topological applications. An immediate example is that sharp estimate on C(M ) recaptures Theorem 8.7 (Bernstein isoperimetric inequality [Ber]). On the 2-sphere S 2 , L 2 ≥ 4πA(1 − 1 4π A) " = " holds if and only if the domain in question is a disk. (8.14) Remark 8.8. For a generalization of isoperimetric inequality to n-dimensional integer multiplicity rectifiable current in R n+k , which follows from the deformation theorem in geometric measure theory, we refer to Federer and Fleming ([FF]). Convexity and Jensen's inequalities We note by Proposition 8.1, every convex function f enjoys an Average Principle of Convexity and Jensen's inequality in an average sense. From the duality between discreteness and continuity, we consider Jensen's inequality involving normalized exponential Yang-Mills energy functional YM 0 e . Let M be a compact manifold and E be a vector bundle over M . Denote L p 1 (E) the Sobolev space of connections of E which are p-integrable and so are their first derivatives. Set W(E) = p≥1 L p 1 (E) ∩ {∇ : YM 0 e (∇) < ∞} . (9.1) Theorem 9.1. (Jensen's inequality involving normalized exponential Yang-Mills energy functional YM 0 e ) Let ∇ be a connection in W(E). Then applying (8.1) yields exp 1 Volume(M ) M ||R ∇ || 2 2 dv − 1 ≤ 1 Volume(M ) M exp( ||R ∇ || 2 2 ) − 1 dv . (9.2) That is, exp 1 Volume(M ) YM(∇) − 1 ≤ 1 Volume(M ) YM 0 e (∇) . (9.3) Equality is valid if and only if ||R ∇ || is constant almost everywhere. Proof. This is a form of Jensen's inequality for the convex function e t − 1(c.f. [Mo,p.21]). Theorem 9.2. Let ∇ be a minimizer in W(E) of the Yang-Mills functional MY, and the norm ||R ∇ || be constant almost everywhere. Then the same connection ∇ is a minimizer of the normalized exponential Yang-Mills functional YM 0 e , and for any minimizer∇ of the normalized exponential Yang-Mills functional YM 0 e in W(E), the norm ||R∇|| is almost everywhere constant. Proof. By the definition of minimizer ∇, the monotone of t → e t − 1 , and Jensen's inequality (9.3), we have for each∇ in W(E), exp 1 Volume(M ) YM(∇) − 1 ≤ exp 1 Volume(M ) YM(∇) − 1 ≤ 1 Volume(M ) YM 0 e (∇) . (9.4) so that exp 1 Volume(M ) YM(∇) − 1 ≤ inf ∇∈W(E) 1 Volume (M ) YM 0 e (∇) . (9.5) On the other hand, since ||R ∇ || = constant a.e., 1 Volume (M ) YM 0 e (∇) = exp( ||R ∇ || 2 2 ) − 1 = exp 1 Volume (M ) YM(∇) − 1 (9.6) so that ∇ is also a minimizer of the normalized exponential Yang-Mills functional YM 0 e . Now we assume that∇ is any minimizer of the normalized exponential Yang-Mills functional YM 0 e in W(E). Then 1 Volume (M ) YM 0 e (∇) ≤ 1 Volume (M ) YM 0 e (∇) (9.7) and combining (9.7), (9.6) and (9.4), allows us to improve all inequalities in (9.4) to equalities, so that we are ready to apply Theorem 9.1 and conclude that ||R∇|| is constant almost everywhere. p-Yang-Mills fields Similarly, we set W p (E) = L p 1 (E) ∩ L 2 1 (E) , p ≥ 2 and obtain via (8.1) Theorem 10.1 (Jensen's inequality involving p-Yang-Mills energy functional YM p , p ≥ 2). Let ∇ be a connection in W p (E). Then 1 p 2 Volume(M ) M ||R ∇ || 2 2 dv p 2 ≤ 1 Volume(M ) M ( ||R ∇ || p p ) dv . (10.1) That is, 1 p 2 Volume(M ) YM(∇) p 2 ≤ 1 Volume(M ) YM p (∇) . (10.2) Equality is valid if and only if ||R ∇ || is constant almost everywhere. Proof. This is a form of Jensen's inequality for the convex function t → 1 p (2t) p 2 , p ≥ 2 (c.f. [Mo,p.21]). Theorem 10.2. Let ∇ be a minimizer in W p (E) of the Yang-Mills functional MY, and the norm ||R ∇ || be constant almost everywhere. Then the same connection ∇ is a minimizer of the p-Yang-Mills functional YM p , and for any minimizer ∇ of the p-Yang-Mills functional YM p in W p (E), the norm ||R∇|| is almost everywhere constant. Proof. By the definition of minimizer ∇, and Jensen's inequality (10.2), we have for each∇ in W p (E), 1 p 2 Volume(M ) YM(∇) p 2 ≤ 1 p 2 Volume(M ) YM(∇) p 2 ≤ 1 Volume(M ) YM p (∇). (10.3) so that 1 p 2 Volume(M ) YM(∇) p 2 ≤ inf ∇∈W p (E) 1 Volume(M ) YM p (∇). (10.4) On the other hand, since ||R ∇ || = constant a.e., 1 Volume (M ) YM p (∇) = ||R ∇ || p p = 1 p 2 Volume (M ) YM(∇) p 2 (10.5) so that ∇ is also a minimizer of the p-Yang-Mills functional YM p . Now we assume∇ is any minimizer of the p-Yang-Mills functional YM p in W p (E). Then 1 Volume (M ) YM p (∇) ≤ 1 Volume (M ) YM p (∇) (10.6) and combining (10.6), (10.5) and (10.3) allows us to improve all inequalities in (10.3) to equalities, so that we are ready to apply Theorem 9.1 and conclude that ||R∇|| is constant almost everywhere. An extrinsic average variation method and Φ (3) -harmonic maps We propose an extrinsic, average variational method as an approach to confront and resolve problems in global, nonlinear analysis and geometry (cf. [W1, W3]). In contrast to an average method in PDE that we applied in [CW3] to obtain sharp growth estimates for warping functions in multiply warped product manifolds, we employ an extrinsic average variational method in the calculus of variations ( [W3]), find a large class of manifolds of positive Ricci curvature that enjoy rich properties, and introduce the notions of superstrongly unstable (SSU) manifolds and p-superstrongly unstable (p-SSU) manifolds ( [W5,W2,W4,WY]). Definition 11.1. A Riemannian manifold M with its Riemannian metric , M is said to be superstrongly unstable (SSU) , if there exists an isometric immersion of M in (R q , · R q ) with its second fundamental form B, such that for every unit tangent vector v to M at every point x ∈ M , the following symmetric linear operator Q M x is negative definite. Q M x (v), v M = m i=1 2 B(v, e i ), B(v, e i ) R q − B(v, v), B(e i , e i ) R q (11.1) and M is said to be p-superstrongly unstable (p-SSU) for p ≥ 2 if the following functional is negative valued. F p,x (v) = (p − 2) B(v, v), B(v, v) R q + Q M x (v), v M ,(11.2) where {e 1 , . . . , e m } is a local orthonormal frame on M . We prove, in particular that every compact SSU manifold must be strongly unstable (SU), i.e., (a) A compact SSU manifold cannot be the target of any nonconstant stable harmonic maps from any manifold, (b) The homotopic class of any map from any manifold into a compact SSU manifold contains elements of arbitrarily small energy E, (c) A compact SSU manifold cannot be the domain of any nonconstant stable harmonic map into any manifold, and (d) The homotopic class of any map from a compact SSU manifold into any manifold contains elements of arbitrarily small energy E (cf. [HoW2,Theorem 2.2,p.321]). (i) the simply connected simple Lie groups (A l ) l≥1 , B 2 = C 2 and (C l ) l≥3 ; (ii) SU (2n)/Sp(n), n ≥ 3; (iii) Spheres S k , k > 2; (iv) Quaternionic Grassmannians Sp(m + n)/Sp(m) × Sp(n), m ≥ n ≥ 1; (v) E 6 /F 4 ; (vi) Cayley Plane F 4 /Spin(9) . (11.5) Theorem 11.5 (Topological Vanishing Theorem). Suppose that M is a compact SSU(resp. p-SSU ) manifold. Then M is SU and π 1 (M ) = π 2 (M ) = 0 resp. π 1 (M ) = · · · = π [p] = 0 . (11.6) Furthermore, the following three statements are equivalent: (a) π 1 (M ) = π 2 (M ) = 0 . (b) the infimum of the energy E is 0 among maps homotopic to the identity on M . (c) the infimum of the energy E is 0 among maps homotopic to a map from M . (11.7) That is, π 1 (M ) = π 2 (M ) = 0 [Wh] ⇐⇒ inf{E(u ′ ) : u ′ is homotopic to Id on M }, [EL2] ⇐⇒ inf{E(u ′ ) : u ′ is homotopic to u : M → •} 11.2. Φ-harmonic maps, from a viewpoint of the second elementary symmetric function σ 2 ( [HW]). We introduce the notion of Φ-harmonic map which is the second symmetric function σ 2 of the pullback metric tensor u * h, an analogue of σ 1 in the above subsection 11.1. In [HW], Han and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of harmonic maps, p-harmonic maps, F -harmonic maps and Yang-Mills fields can be extended to the study of Φharmonic maps. In fact, we find a large class of manifolds with rich properties, Φsuperstrongly unstable (Φ-SSU) manifolds, establish their links to p-SSU manifolds and topology, and apply the theory of p-harmonic maps, minimal varieties and Yang-Mills fields to study such manifolds. With the same notations as above, we introduce the following notions: Definition 11.6. A Riemannian manifold (M m , g) with a Riemannian metric g is said to be Φ-superstrongly unstable (Φ-SSU) if there exists an isometric immersion R q such that, for all unit tangent vectors v to at every point x ∈ M m , the following functional is always negative: F Φx (v) = m i=1 4 B(v, e i ), B(v, e i ) − B(v, v), B(e i , e i ) ,(11.9) where B is the second fundamental form of M m in R q , and {e 1 , · · · , e m } is a local orthonormal frame on M near x. Definition 11.7. A Riemannian manifold M is Φ-strongly unstable (Φ-SU) if it is neither the domain nor the target of any nonconstant smooth Φ-stable stationary map, and the homotopic class of maps from or into M contains a map of arbitrarily small energy. Theorem 11.8. Every compact Φ-superstrongly unstable (Φ-SSU) manifold is Φ-strongly unstable (Φ-SU) . 11.3. Φ S -harmonic maps, from a viewpoint of an extended second symmetric function σ 2 ( [FHLW]). We introduce the notion of Φ S -harmonic maps, which is a σ 2 version of the stress energy tensor S. In [FHLW], Feng, Han, Li, and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of σ 1 and σ 2 versions of the pullback metric u * h on M can be extended to the study of a σ 2 version of the stress energy tensor S. In fact, we find a large class of manifolds, Φ S -superstrongly unstable (Φ S -SSU) manifolds, introduce the notions of a stable Φ S -harmonic map, Φ S -strongly unstable (Φ S -SU) manifolds, and prove Theorem 11.9. Every compact Φ S -superstrongly unstable (Φ S -SSU) manifold is Φ S -strongly unstable (Φ S -SU) . 11.4. Φ S,p -harmonic maps, from a viewpoint of a combined extended second symmetric function σ 2 ( [FHW]). We introduce the notion of Φ S,p -harmonic maps, which is a combined generalized σ 2 version of the stress energy tensor S, and a σ 1 version of the pullback u * . In [FHLW], Feng, Han, Li, and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of σ 1 and σ 2 versions of the pullback metric u * h on M and stress-energy tensor can be extended to the study of a combined extended second symmetric function σ 2 version. In fact, we find a large class of manifolds, Φ S,p -superstrongly unstable (Φ S,p -SSU) manifolds, introduce the notions of a stable Φ S,p -harmonic map, Φ S,p -strongly unstable (Φ S,p -SU) manifolds, and prove Theorem 11.10. Every compact Φ S,p -superstrongly unstable (Φ S,p -SSU) manifold is Φ S,p -strongly unstable (Φ S,p -SU) . 11.5. Φ (3) -harmonic maps, from a viewpoint of the third elementary symmetric function σ 3 ( [FHJW]). We introduce the notion of of Φ (3) -harmonic maps, which is a σ 3 version of the pullback u * . In fact, Feng, Han, Jiang, and Wei show that the extrinsic average variational method in the calculus of variations employed in the study of σ 1 and σ 2 versions of the pullback metric u * h on M can be extended to the study of the third symmetric function σ 3 version. Whereas we can view harmonic maps as Φ (1) -harmonic maps (involving σ 1 ) and Φ-harmonic maps as Φ (2) -harmonic maps (involving σ 2 ) , we introduce the notion of a Φ (3) -harmonic map and find a large class of manifolds, Φ (3) -superstrongly unstable (Φ (3) -SSU) manifolds, introduce the notions of a stable Φ (3) -harmonic map, Φ (3) -strongly unstable (Φ (3) -SU) manifolds, and prove Theorem 11.11 ( [FHJW]). Every compact Φ (3) -superstrongly unstable (Φ (3) -SSU) manifold is Φ (3) -strongly unstable (Φ (3) -SU) . Definition 11.12 ( [FHJW]). A Riemannian manifold M m is said to be Φ (3)superstrongly unstable (Φ (3) -SSU) if there exists an isometric immersion of M m in R q with its second fundamental form B such that for all unit tangent vectors v to M m at every point x ∈ M m , the following functional is negative valued. (11.10) where {e 1 , · · · , e m } is a local orthonormal frame field on M m near x. F Φ (3) x (v) = m i=1 6 B(v, e i ), B(v, e i ) R q − B(v, v), B(e i , e i ) R q , Theorem 11.13. Every Φ (3) -SSU manifold M is p-SSU for any 2 ≤ p ≤ 6. Proof. By Definition 11.12, Φ (3) -SSU manifold enjoys (11.11) for all unit tanget vector v ∈ T x (M ). It follows from (11.2) and (11.10) that for 2 ≤ p ≤ 6. So by Definition 11.1, M is p-SSU for any 2 ≤ p ≤ 6. F Φ (3) x (v) = m i=1 6 B(v, e i ), B(v, e i ) R q − B(v, v), B(e i , e i ) R q < 0F p,x (v) = (p − 2) B(v, v), B(v, v) R q + Q M x (v), v M ≤ (p − 2) Theorem 11.14. Every compact Φ (3) -SSU manifold M is 6-connected , i.e., π 1 (M ) = · · · = π 6 (M ) = 0. (11.13) Proof. Since every compact p-SSU is [p]-connected (cf. [W5, Theorem 3.10 , p. 645]), and p = 6 by the previous Theorem, the result follows. Theorem 11.15 (Sphere Theorem). Every compact Φ (3) -SSU manifold M of dimension m ≤ 13 is homeomorphic to an m-sphere. Proof. In view of Theorem 11.13, M is 6-connected. By the Hurewicz isomorphism theorem, the 6-connectedness of M implies homology groups H 1 (M ) = · · · = H 6 (M ) = 0. It follows from Proincare Duality Theorem and the Hurewicz Isomorphism Theorem ( [SP]) again, H m−6 (M ) = · · · = H m−1 (M ) = 0, H m (M ) = 0 and M is (m − 1)-connected. Hence N is a homotopy m-sphere. Since M is Φ (2) -SSU, m ≥ 7. Consequently, a homotopy m-sphere M for m ≥ 5 is homeomorphic to an m-sphere by a Theorem of Smale ([Sm]). We summarize some of new manifolds found and these results obtained by an extrinsic average method in Table 1 in Section 1. Theorem 4.1. (Hessian comparison theorem [GW, DW, HLRW, W11]) Let (M, g) be a complete Riemannian manifold with a pole x 0 . Denote by K(r) the radial curvature of M . Then − α 2 ≤ K(r) ≤ −β 2 with α > 0, β > 0 (i [GW]) Let a = exp(A) and b = exp(B) . Then applying An Average Principle of Convexity (8.1), where f = exp yields A geometric interpretation of this inequality: dual version, let a concave function f = log, An Average Principle, Proposition 8.1 yields Example 8.4. Let g be a nonnegative measurable function on [ Remark 10.3. J. Eells and L. Lemaire first derive Jensen's inequality and establish its optimality in the setting of exponentially harmonic maps ([EL]). F. Matsuura and H. Urakawa show exp YM(∇) Volume(M )≤ YM e (∇) Volume(M ) for any ∇ ∈ W(E) , and the validity of equality ([MU]). ( Cf. [W3, the diagram on p.58].) 26 B(v, e i ), B(v, e i ) R q − B(v, v), B(e i , e i ) v, e i ), B(v, e i ) − B(v, v), B(e i , e i ) B(v, e i ), B(v, e i ) − B(v, v), B(e i , e i ) < 0,(11.12) Table 1 . 1An Extrinsic Average Variational MethodMappings Functionals New manifolds found Geometry Topology harmonic map or energy functional E or SSU manifolds or SU or π 1 11.1. Harmonic maps and p-harmonic maps, from a viewpoint of the first elementary symmetric function σ 1 .We recall at any fixed point x 0 ∈ M , a symmetric 2-covariant tensor field α on(M, g)in general, or the pullback metric u * in particular, has the eigenvalues λ relative to the metric g of M ; i.e., the m real roots of the equationand {e 1 , · · · e m } is a basis for T x0 (M ) (cf.,e.g.,[HW]). A harmonic map u : (M, g) → (N, h) can be viewed as a critical point of the energy functional, given by the integral of a half of first elementary symmetric function σ 1 , of engenvalues relative to the metric g, or the trace of the pulback metric tensor u * h, with respect to g, where {e 1 , · · · , e m } is an local orthonormal frame field on M . That is,A p-harmonic map can be viewed as a critical point of the p-energy functional E p (u), given by the integral of 1 p times σ 1 or the trace of the pullback metric tensor to the power p 2 , i.e.,For the study of the stability of harmonic maps ( resp. p-harmonic maps ),Howard and Wei ( [HoW2]) resp. Wei and Yau ([WY]) introduce the following notions:Definition 11.2. A Riemannian manifold M is said to be strongly unstable (SU) resp. p-strongly unstable (p-SU) if M is neither the domain nor the target of any nonconstant smooth stable harmonic map, (resp. stable p-harmonic map), and the homotopic class of maps from or into M contains a map of arbitrarily small energy E (resp. p-energy E p ).This definition leads toTheorem 11.3. Every compact superstrongly unstable (SSU)-manifold resp. p-superstrongly unstable (p-SSU) manifold is strongly unstable(SU). resp. pstrongly unstable (p-SU) .And, we make the following classification.Theorem 11.4 ([O, HoW]). 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We analyze stationary self-gravitating disks around spinning black holes that satisfy the recently found general-relativistic Keplerian rotation law. There is a numerical evidence that the angular velocity, circumferential radius and angular momenta yield a bound onto the asymptotic mass of the system. This bound is proven analytically in the special case of massless disks of dust in the Kerr spacetime.

10.1103/physrevd.99.024004

1810.12134

beeb72f032953585ab34bd2a7e69ededaf3d6847

Two mass conjectures on axially symmetric black hole-disk systems Wojciech Kulczycki Instytut Fizyki im. Mariana Smoluchowskiego Uniwersytet Jagielloński Lojasiewicza 1130-348KrakówPoland Patryk Mach Instytut Fizyki im. Mariana Smoluchowskiego Uniwersytet Jagielloński Lojasiewicza 1130-348KrakówPoland Edward Malec Instytut Fizyki im. Mariana Smoluchowskiego Uniwersytet Jagielloński Lojasiewicza 1130-348KrakówPoland Two mass conjectures on axially symmetric black hole-disk systems PACS numbers: 04.20.-q, 04.25.Nx, 04.40.Nr, 95.30.Sf We analyze stationary self-gravitating disks around spinning black holes that satisfy the recently found general-relativistic Keplerian rotation law. There is a numerical evidence that the angular velocity, circumferential radius and angular momenta yield a bound onto the asymptotic mass of the system. This bound is proven analytically in the special case of massless disks of dust in the Kerr spacetime. I. INTRODUCTION Angular velocities Ω of test bodies in a nonrelativistic Keplerian circular motion encode information about central masses. The central mass is inferred to be Ω 2 r 3 C /G, where G is the gravitational coupling constant and r C denotes the circumferential radius of the orbit of the body. Recent numerical analysis suggests that one can estimate the total mass m also for a system with selfgravitating fluids in Keplerian rotation around a spherical centre, Ω 2 r 3 C /G ≤ m [1,2]. In this paper we shall investigate general-relativistic selfgravitating disks or toroids around spinning central black holes. We shall propose two modified inequalities for selfgravitating rotating systems in the general-relativistic Keplerian motion [3,4]; they imply Ωr 3/2 C ≤ √ GM ADM + 2|a| √ rC , where a and M ADM are the spin parameter of the black hole and the asymptotic mass of the whole configuration, respectively. There is a rich numerical evidence, some reported in what follows, that supports our conjecture. The question of estimating masses or determining rotation curves in toroidal systems has been addressed, within Newtonian physics, in the astrophysical literature [5][6][7][8][9]. We would like to stress that our methods are different and we work within the general relativistic context. For simplicity, we assume only polytropic fluids rotating around a (spinning or spinless) black hole, but we take into account the selfgravity of the fluid. Stationary axially symmetric systems of rotating barotropic fluids need the so-called rotation curve, that tells particles of fluid how to rotate. They are described by the freeboundary elliptic systems of partial differential equations -the shape of rotating fluids cannot be set a priori, but comes with the solution. There exists a numerical technology to deal with such problems, developed by many authors (a sample of sources: [10][11][12][13][14]), but the number of analytical results on the existence of solutions is small. There is a paper concerning self-gravitating fluids in Newtonian gravity [15] and a research on rigid rotations of self-gravitating gases in general relativity [16][17][18]. In Newtonian gravity the Poincaré-Wavre theorem restricts allowed rotations curves to Ω = Ω(r C ), where the circumferential radius r C is equal to the distance from the rotation axis. They obviously include the Keplerian rotation law. The general-relativistic rotation has been investigated since early 1970's -primarily a rigid rotation [19,20] and its modifications [21][22][23]. New general-relativistic differential rotation laws j = j(Ω), where j denotes the specific angular momentum, have recently been found for stationary systems consisting of self-gravitating toroids around spinless [24,25] and spinning black holes [3,4]. They describe, in particular, the motion of a massless disks of dust, with Ω = √ GM ADM /r 3/2 C in the case of the Schwarzschild geometry. In the nonrelativistic limit one has the monomial rotation law Ω = w/r λ C (0 ≤ λ ≤ 2); that obviously includes the Keplerian rotation law. The order of the main part of this paper is as follows. Section II describes formalism. Section III contains two main conjectures concerning masses of axially symmetric stationary systems with black holes. In Section IV we show that test-like disks of dust satisfy the two mass conjectures in Schwarzschild and Kerr geometries. The proof concerning the Kerr spacetime is relegated to the Appendix. Section V presents a sample of numerical examples that confirm our two conjectures. II. EQUATIONS We assume a stationary metric of the form ds 2 = −α 2 dt 2 + r 2 sin 2 θψ 4 (dϕ + βdt) 2 + ψ 4 e 2q dr 2 + r 2 dθ 2 . (1) Here t is the time coordinate, and r, θ, ϕ are spherical coordinates. In the general-relativistic part of this paper the gravitational constant G = 1 and the speed of light c = 1. We assume axial symmetry and employ the stressmomentum tensor T αβ = ρhu α u β + pg αβ , where ρ is the baryonic rest-mass density, h is the specific enthalpy, and p is the pressure. Metric functions α, ψ, q and β in (1) depend on r and θ only. The following method can be applied to any barotropic equation of state, but we will deal with polytropes p(ρ) = Kρ γ . Then one has the specific enthalpy h(ρ) = 1 + γp (γ − 1)ρ . arXiv:1810.12134v1 [gr-qc] 29 Oct 2018 2 The 4-velocity (u α ) = (u t , 0, 0, u ϕ ) is normalized, g αβ u α u β = −1. The coordinate angular velocity reads Ω = u ϕ u t .(2) We define the angular momentum per unit inertial mass ρh [26] j ≡ u ϕ u t . It is known since early 1970's that general-relativistic Euler equations are solvable under the condition that j ≡ j(Ω) [19,20]. Within the fluid region, the Euler equations ∇ µ T µν = 0 can be integrated, j(Ω)dΩ + ln h u t = C.(4) In [24] we have had the rotation law j(Ω) ≡w 1−δ Ω δ 1 − κw 1−δ Ω 1+δ + Ψ ;(5) here Ψ is of the order of the binding energy per unit baryonic mass and w, δ, and κ = (1 − 3δ)/(1 + δ) are parameters. This law was obtained in a procedure involving an "educated guess-work" [24]. Equation (5) can be transformed (through a rescaling ofw) into j(Ω) ≡ w 1−δ Ω δ 1 − κw 1−δ Ω 1+δ = −κ Ω + w δ−1 Ω −δ −1 , The Keplerian rotation corresponds to the parameter δ = −1/3 and κ = 3; it is interpreted as a rotation law for the fluid around a spin-less black hole. The rotation law describing motion of gaseous tori around spinning black holes, j (Ω) = − 1 2 d dΩ ln 1 − ã 2 Ω 2 + 3w 4 3 Ω 2 3 (1 −ãΩ) 4 3 ,(6) has been found in [3,4]. Hereã is a kind of a bare spin parameter of a black hole (see a comment below); it coincides with the Kerr spin parameter for massless disks of dust. The rotation curves-angular velocities as functions of spatial coordinates Ω(r, θ)-can be recovered from Eq. (3), j(Ω) = V 2 (Ω + β) (1 − V 2 ) .(7) Here the squared linear velocity is given by V 2 = r 2 sin 2 θ (Ω + β) 2 ψ 4 α 2 . Following [13] we introduce the central black hole using the puncture method [27]. The black hole is surrounded by a minimal two-surface S BH (the horizon) embedded in a fixed hypersurface of constant time, and located at r = r s = √ m 2 −ã 2 /2, where m is a mass parameter. Its area defines the irreducible mass M irr = AH 16π and its angular momentum J BH follows from the Komar expression J BH = 1 4 π/2 0 r 4 ψ 6 α ∂ r β sin 3 θdθ.(8) In this construction the angular momentum is given rigidly on the event horizon S BH , in terms of data taken from the Kerr solution and independently of the content of mass in a torus, J BH = mã [13]. The mass of the black hole is then defined in terms of the irreducible mass and the angular momentum, M BH = M irr 1 + J 2 BH 4M 4 irr .(9) We define the black hole spin parameter as a = J BH /M BH . If the disk is sufficiently massive (selfgravitating), we have in general a =ã [3,4]. If the selfgravity of the torus can be neglected, then M BH = m, a =ã, and the metric of the spacetime coincides (by construction) with the Kerr solution. Asymptotic (total) mass M ADM and angular momentum J ADM can be defined as apropriate Arnowitt-Deser-Misner charges, and they can be computed by means of corresponding volume integrals [13]. A circumferential radius corresponding to the circle r = const on the symmetry plane θ = π/2 is given by r C = rψ 2 .(10) The numerical part of this paper is based on the scheme described in [3,4]. In the rest of the paper we always assume that Ω > 0. Corotating disks have a > 0, while counterrotating disks have negative spins: a < 0. III. TWO MASS CONJECTURES General-relativistic Keplerian systems with tori are characterized by their asymptotic masses M ADM and angular momenta J ADM , and the quasilocal characteristics of central black holes-the mass M BH and the angular momentum J BH . It is clear that in the Newtonian limit we should get Ωr 2]. From this, and from the dimensional analysis, one may guess that Ωr 3/2 C ≤ √ m [1,3/2 C ≤ √ M ADM + 2 |JX| mY √ rC , where X, Y = BH, ADM. The asymptotic mass is larger than the mass of the central black hole, but the angular momentum is not monotonic -the asymptotic angular momentum might be smaller than J BH . That means that there are two independent possibilities. Conjecture 1. Ωr 3/2 C ≤ M ADM + 2|J ADM | M ADM √ r C .(11) Conjecture 2. Ωr 3/2 C ≤ M ADM + 2 |J BH | M ADM √ r C .(12) Notice that |JBH| MADM ≤ |a|, where a is the black hole spin parameter. Thus Conjecture 2 implies Ωr 3/2 C ≤ √ M ADM + 2|a| √ rC . Numerical calculations reported in Sec. VI suggest the validity of both Conjectures. They coincide for test disks of dust in the Kerr geometry, since then M BH = M ADM and J ADM = J BH . The inequality reduces to the equality in the Schwarzschild spacetime and it is satisfied in Kerr spacetimes (see the Appendix for an algebraic proof). The inspection of both inequalities (11) and (12) shows that far from the center the corresponding expressions on the right-hand sides are expected to be constant; if a toroid is light and large, then Ωr 3/2 C is expected to be close to √ M ADM (see Sec. V). The inspection of these inequalities in the interior might give some information about angular momentum. IV. ANGULAR MOMENTUM AND MASS ESTIMATES IN KERR AND SCHWARZSCHILD SPACETIMES A. Kerr geometry in conformal coordinates Define r K = r 1 + m r + m 2 − a 2 4r 2 ,(13)∆ K = r 2 K − 2r K + a 2 , and Σ K = r 2 K + a 2 cos 2 θ. The Kerr metric can be written in form (1) as follows [13,27]. The conformal factor ψ K reads ψ K = 1 √ r r 2 K + a 2 + 2ma 2 r K sin 2 θ Σ K 1/4 .(14) The only component β K of the shift vector is given by β K = − 2mar K (r 2 K + a 2 )Σ K + 2ma 2 r K sin 2 θ .(15) Finally, the functions α K and q K are defined as α K = Σ K ∆ K (r 2 K + a 2 )Σ K + 2ma 2 r K sin 2 θ 1/2 , e qK = Σ K (r 2 K + a 2 )Σ K + 2ma 2 r K sin 2 θ .(16) The surface r = r s ≡ 1 2 √ m 2 − a 2 is an apparent horizon, that coincides with the event horizon. Test particles can rotate along circular orbits r = const in the Kerr geometry. That implies the existence of a test-like disk made of dust, that moves circularly and lies in the plane θ = π/2. Its angular velocity reads Ω(r) = 8r 3/2 ((2r + 1) 2 − a 2 ) 3/2 + 8ar 3/2 .(17) The dragging angular velocity of the Kerr space-time is given by Ω d = −β K , that is Ω d = 2ma r K (r 2 K + a 2 ) + 2ma 2 .(18) The circumferential radius of this circular orbit is equal to r C = rψ 2 = r 2 K + a 2 + 2ma 2 r K(19) It is clear that Ω d = 2ma r K r 2 C .(20) It appears that the product Ωr 3/2 C exceeds the value √ m for all strictly negative spin parameters a and for a > 0.9525. There exists, however, a modified inequality that takes into account the dragging effects and that is always true: Ωr 3/2 C ≤ √ m + |Ω d |r 3/2 C = √ m + 2m|a| r K r 1/2 C .(21) The proof of (21) is relegated to the Appendix. The quantity m|a| is the absolute value |J BH | of the angular momentum, while r K ≥ m outside the region encircled by the trapped surface r = r s . Thus for a disk in Keplerian motion around a Kerr black hole we have Ωr 3/2 C ≤ √ m + 2 J BH m √ r C = √ m + 2 |a| √ r C .(22) This agrees with both inequalities (11) and (12), since they coincide in this case. B. Schwarzschild geometry Schwarzschild space-time in conformal coordinates is given by metric functions of the preceding subsection assuming the spin parameter a = 0. Test bodies in a Schwarzschild space-time can move on circular orbits with the angular velocity Ω = (11) and (12) coincide, and they are saturated. V. NUMERICAL RESULTS In this section we deal with numerical solutions describing self-gravitating fluids around black holes. The numerical method was described in detail in [3,4]. It is convenient to rewrite inequalities (11) and (12) in the following form. i) Conjecture 1. max (Ω − Ω d,1 )r 3/2 C ≤ M ADM ;(23) here Ω d,1 = 2|JADM| MADMr 2 C . ii) Conjecture 2. max (Ω − Ω d,2 )r 3/2 C ≤ M ADM ;(24) here Ω d,2 = 2|JBH| MADMr 2 C . We performed a large number of numerical calculations, with equations of state p = Kρ 4/3 and p = Kρ 5/3 , for a range of values of the spin parameter a, the radius of the inner boundary r 1 and for several values of the maximal mass density ρ max . In this paper we report only results concerning the polytrope with the polytropic index γ = 4/3, but the other case gives similar results. In numerical solutions of Figs. 1 and 2 we fix the coordinate radius of the disk's outer boundary r 2 = 20, but the inner radius r 1 is changed, within limits shown in captions of Figures. Figures 3-5 are dedicated to the analysis of Ωr 3/2 C in specific solutions. In all cases we calculated values of the left hand sides of (23) and (24) in the symmetry plane θ = π/2. Figure 1 summarizes results of more than four hundred (counter-rotating) numerical solutions. The diagram shows maximal values of the left hand sides of (23) and (24) in units of the square root of the asymptotic mass. There is a sharp spike in the diagram corresponding to Conjecture 1 (see (23)); around r 1 = 11.7 the angular momentum of the disk cancels the black hole spin. The asymptotic angular momentum J ADM decreases, vanishes at the top of the spike and becomes more and more negative. Conjecture 2 is also satisfied, but the expression max((Ω−Ω d,2 )r 3/2 C )/ √ M ADM changes only moderately. Figure 2 presents results of more than seven hundred (co-rotating) numerical solutions. It is clear that both proposed conjectures, 1 and 2, are valid in our numerical calculations, for co-and counter-rotating systems. on the symmetry plane of the disk. The disk is quite extended and somewhat heavier -its outer circumferential radius is larger than 100 and the asymptotic mass M ADM = 2 while the mass of the black hole is M BH = 1.02. We display two curves, for a = 0.49 and a = −0.49. The dependence on the spin is seen in disk's interior and becomes negligible in disk's peripherals. Figure 5 shows the same as Fig. 4, but the spin is higher. Again the disk is extended but it is light-its outer circumferential radius is larger than 100 and the asymptotic mass M ADM = 1.1 while the mass of the black hole is M BH = 1.00. We display two curves, for a = 0.99 and a = −0.99. A strong dependence on the spin is seen in disk's interior, but it is noticeable even in disk's peripherals. Notice that sup(Ωr 3/2 C ) > 1.1 for the two counter-rotating branches, but only the massless co-rotating branch has Ωr 3/2 C exceeding 1. It is instructive to compare the two self-gravitating solutions in Fig. 3 with the two massless disks of dust in the Kerr geometry. It is clear that the self-gravity merely pushes down the value of Ωr co-rotating one, but they almost coincide at the outer boundary. A similar picture is seen in Fig. 5. We see the same effect in Fig. 4, but with a more pronounced shift downwards. The co-rotating and counter-rotating branches cross around r C ≈ 30 and then their positions do reverse. Thus the more massive is the rotating toroid, the stronger the self-gravity impacts the quantity Ωr 3/2 C . Figure 5 demonstrates yet another influence of the self-gravity; the inner boundary of the co-rotating toroid shifts upwards from r C = 2.108 (in the Kerr spacetime) to r C = 2.32 and of the counter-rotating toroid moves downwards from r C = 9.04 (in the Kerr spacetime) to r C = 4.68. VI. SUMMARY The two conjectures on masses of rotating systems are based partly on analytic arguments and partly on extensive numerical data. Their proof would pose a serious analytic challenge. We envisage two further applications. The first concerns astrophysics. Rotating axially symmetric systems are quite common. They are known to exist in some active galactic nuclei. Particularly interesting are those containing supermasers. Our inequalities might be useful in extracting information about masses and angular momenta, provided that more information on modelling of AGN's with toroids becomes available. The two statements of (11) and (12) can be useful in estimating the amount of angular momentum within a fixed volume. There is already a formidable work done in this direction [29,30], although results are far from being precise [14]. We think that (11) and (12) are true and they would lead to substantial improvements of present estimates. VII. APPENDIX. PROOF OF THE MASS INEQUALITY IN THE KERR GEOMETRY We shall do this proof in Boyer-Lindquist coordinates (r K , θ, φ), where the Boyer-Lindquist radius r K relates to the conformal radius r via formula (13) [27]. The metric takes on the standard form ds 2 = − 1 − 2mr K Σ K dt 2 + Σ K ∆ K dr 2 K + Σ K dθ 2 + ∆ K Σ K + 2mr K (r 2 K + a 2 ) Σ K sin 2 θdφ 2 − 4mr K a sin 2 θ Σ K dtdφ,(25) where Σ K = r 2 K + a 2 cos 2 θ, ∆ K = r 2 K − 2mr K + a 2 . The inequalities of Section V are expressed in terms of geometric quantities, hence they are independent of the choice of coordinates within a fixed Cauchy slice. We shall consider a massless disk, that is located in the symmetry plane θ = π/2. A. Geometric quantities The angular velocity Ω reads Ω = u φ u t = √ m r 3/2 K + a √ m .(26) The circular radius r C can be expressed as follows, in the symmetry plane: r C = r 2 K + a 2 + 2ma 2 r K .(27) The angular velocity due to dragging plays a significant role in the calculation. It is given by Ω d = 2ma r 2 C r K .(28) The irreducible mass of the Kerr black hole reads M irr = m 2 2 1 + 1 − a 2 m 2 .(29) The quantity r ISCO denotes the coordinate radius of the innermost stable circular orbit (ISCO), that depends on a and m. In the case of co-rotation r ISCO ≥ m (with the equality when a = 1) while in the case of counter-rotation 6m ≤ r ISCO ≤ 9m (with the upper bound saturated for a = −1). B. Proofs We shall prove analytically the validity of the following inequality, provided that the areal radius is not smaller than r ISCO : r K ≥ r ISCO : Ωr 3/2 C ≤ √ m + |Ω d |r 3/2 C .(30) For the simplicity, but without the loss of generality, we shall put m = 1. We divide both sides of (30) by r 3/2 C ; that leads to Ω − |Ω d | ≤ 1 r 3 C .(31) Using (13,(26)(27)(28)(29), one arrives at 1 r 3/2 K + a − 2|a| r K (r 2 K + a 2 ) + 2a 2 ≤ ≤ a 2 + 2a 2 r K + r 2 K −3/4 .(32) a. The case a = 0 corresponds to the Schwarzschild geometry and it has been already discussed. FIG. 1 .FIG. 2 . 12Maximal values of left hand-sides of Conjectures 1 (conj. 1) and 2 (conj. 2) for γ = 4/3,ã = −0.99 and ρmax = 3.0 × 10 −4 . Here r2 = 20 and r1 varies from 6.01 to 18.97. Asymptotic masses are in the range (1.005, 1.46). Each point on the diagram corresponds to a solution. Maximal values of left hand-sides of Conjectures 1 and 2 for γ = 4/3,ã = 0.99 and ρmax = 3.0 × 10 −4 . Here r2 = 20 and r1 varies from 0.805 to 19.05. Asymptotic masses are in the range (1.005, 1.46). Each point on the diagram corresponds to a solution. Figure 3 M 3ADM − 1 as function of rC. Here γ = 4/3 and ρmax = 3.0 × 10 −4 . i) Blue line: a = 0.49, rC =∈ (7.35, 21.57), MADM = 1.45 and MBH = 1.02. ii) Green line: a = −0.485, rC ∈ (7.36, 21.58), MADM = 1.47 MBH = 1.03. iii) For the comparison, the plot of massless disk in the Kerr geometry, with m = 1. Figure 4 4Figure 4 shows the behaviour of the product Ωr 3/2 C CM /M ADM − 1 obtained for dust solutions in Kerr space-times. The branch corresponding to the counter-rotation remains above the Kerr: a = −ADM − 1 as function of rC. Here γ = 4/3, rC ∈ (7.27, 102.32), MADM = 2 and MBH = 1.02. i) Blue line: a = 0.49, ρmax = 7.47 × 10 −6 . ii) Green line: a = −0.49, ρmax = 7.74 × 10 −6 . iii) For the comparison, a massless disk in the Kerr geometry, with m = 1. M ADM − 1 as function of rC. Here γ = 4/3, MADM = 1.1 and MBH = 1.00. i) Green line: a = −0.99, ρmax = 1.92 × 10 −6 , rC ∈ (4.68, 101.36). ii) Blue line: a = 0.99, ρmax = 2.5 × 10 −6 , rC ∈ (2.32, 101.36). iii) For the comparison, the plot of 1 for a massless disk in the Kerr geometry (m = 1). For a = −0.99: rC ∈ (9.04, 101.00), while for a = 0.99: rC ∈ (2.108, 101.00). ACKNOWLEDGMENTSThis research was carried out with the supercomputer "Deszno" purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (Contract no. POIG. 02.01.00-12-023/08).b. In the corotating case, a > 0, inequality (32) takes the formIt is easy to show that the left hand side of this inequality is non-negative, since (as we show below) for r K ≥ r ISCO ,Indeed, notice that r K ≥ √ r K ; this is so because r ISCO ≥, which is obviously non-negative. That in turn implies the non-negativity of (34). Therefore the sign of the inequality (33) does not reverse when we calculate the quartic power of both sides,The denominator of (35) is positive for r K ≥ r ISCO ≥ 1. One can find out, using the REDUCE function of Mathematica[28], that the numerator of (35) is also nonnegative, if r K is not smaller than 1.c. The counter-rotating case a < 0. Changing a → −|a| in the inequality (32), one arrives atIt is easy to show that the left hand side of this inequality is nonpositive, if and only ifHowever for r K ≥ 0:which is manifestly non-negative. Thus the left-hand side of (36) is non-negative for r ≥ r K . Again we can calculate the quartic power of both sides,The denominator of (38) is positive for r K ≥ 1. One can find out, using the REDUCE function of Mathematica[28], that the numerator is also non-negative, if r K is not smaller than 6. In conclusion, in the interval of interest for counter-rotation (r K > r ISCO ≥ 6) inequality (36) holds true. 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We present a nonequilibrium strong-coupling approach to inhomogeneous systems of ultracold atoms in optical lattices. We demonstrate its application to the Mott-insulating phase of a twodimensional Fermi-Hubbard model in the presence of a trap potential. Since the theory is formulated self-consistently, the numerical implementation relies on a massively parallel evaluation of the selfenergy and the Green's function at each lattice site, employing thousands of CPUs. While the computation of the self-energy is straightforward to parallelize, the evaluation of the Green's function requires the inversion of a large sparse 10 d × 10 d matrix, with d > 6. As a crucial ingredient, our solution heavily relies on the smallness of the hopping as compared to the interaction strength and yields a widely scalable realization of a rapidly converging iterative algorithm which evaluates all elements of the Green's function. Results are validated by comparing with the homogeneous case via the local-density approximation. These calculations also show that the local-density approximation is valid in non-equilibrium setups without mass transport.

10.1103/physreve.89.023306

1309.5994

0ab946c828136f1617b59a06e3ba80e199192796

Simulation of inhomogeneous distributions of ultracold atoms in an optical lattice via a massively parallel implementation of nonequilibrium strong-coupling perturbation theory (Dated: May 22, 2014) Andreas Dirks Department of Physics Georgetown University 20057WashingtonDCUSA Karlis Mikelsons Department of Physics Georgetown University 20057WashingtonDCUSA H R Krishnamurthy Centre for Condensed Matter Theory Department of Physics Indian Institute of Science 560012BangaloreIndia Jawaharlal Nehru Centre for Advanced Scientific Research 560064BangaloreIndia James K Freericks Department of Physics Georgetown University 20057WashingtonDCUSA Simulation of inhomogeneous distributions of ultracold atoms in an optical lattice via a massively parallel implementation of nonequilibrium strong-coupling perturbation theory (Dated: May 22, 2014)numbers: 0260Nm0375-b0375Ss7110Fd We present a nonequilibrium strong-coupling approach to inhomogeneous systems of ultracold atoms in optical lattices. We demonstrate its application to the Mott-insulating phase of a twodimensional Fermi-Hubbard model in the presence of a trap potential. Since the theory is formulated self-consistently, the numerical implementation relies on a massively parallel evaluation of the selfenergy and the Green's function at each lattice site, employing thousands of CPUs. While the computation of the self-energy is straightforward to parallelize, the evaluation of the Green's function requires the inversion of a large sparse 10 d × 10 d matrix, with d > 6. As a crucial ingredient, our solution heavily relies on the smallness of the hopping as compared to the interaction strength and yields a widely scalable realization of a rapidly converging iterative algorithm which evaluates all elements of the Green's function. Results are validated by comparing with the homogeneous case via the local-density approximation. These calculations also show that the local-density approximation is valid in non-equilibrium setups without mass transport. I. INTRODUCTION The field of ultracold atoms in optical lattices has been a promising new opportunity for studying many-body effects which are important for condensed-matter physics in controlled environments [1,2]. In particular, fermionic atoms such as 40 K may provide a direct path towards a "quantum simulation" of the Hubbard model which itself has a paradigmatic role in condensed matter physics and is a key in understanding phenomena such as hightemperature superconductivity and strongly correlated magnetism. In these experiments, some novel possibilities to study physical correlations between constituents of the model are being explored. In many cases, such experiments [3][4][5] drive the systems substantially beyond thermal equilibrium, so that they are inaccessible to methods of conventional equilibrium or linear-response theory. Usually, one also encounters spatially inhomogeneous situations, since the atoms in the optical lattice are being held in a trap potential which co-exists with the lattice potential. In one-dimensional systems, many opportunities to provide computational benchmarks for such experiments exist, such as via the density-matrix renormalization group [6][7][8][9][10]. However, it is a challenging problem to describe two-and threedimensional systems out of thermal equilibrium, especially when they are also inhomogeneous. In this paper we present a nonequilibrium strongcoupling approach to inhomogeneous systems of ultra- * Electronic address: [email protected] cold atoms in optical lattices. The paper is structured as follows. Section II discusses the Hubbard model for an optical lattice in a trap. In Section III, we outline the strong-coupling approach which enables us to simulate inhomogeneous higher-dimensional Hubbard systems out of equilibrium. In Section IV, we develop the massively parallel algorithm which is used to solve the resulting equations on a supercomputer. Section V presents results of the algorithm for the example of a modulated lattice depth and validates them by comparing to the previously introduced strong-coupling method for homogeneous systems [12] within the local-density approximation (LDA). Conclusions are given in Section VI. II. MODEL We consider a Fermi Hubbard model in the presence of a trap potential, i.e. H(t) = H 0 (t) − i,j,σ J ij (t)c † i,σ c j,σ ,(1) with H 0 (t) = i H (i) 0 (t) = i,σ ε i (t)n i,σ + i U i (t)n i↑ n i↓ ,(2) where the on-site single-particle energy levels Between the dashed blue lines, the system is driven out of equilibrium by a timedependent Hamiltonian. In the numerical part of the paper, we refer to the two points in time as t = 0 and t = t mod for simplicity. The real times between the Matsubara branch and the point at which the system is driven out of equilibrium can be used to check for numerical convergence, since expectation values of observables have to be constant here. The time discretization is shown in red -in the implementation itself, the point t = 0 is identical to the point to which the Matsubara branch of the contour is attached. The total number of time slices is N t = 2N t,real + N t,imag . ε i (t) = V trap ( r i ; t) − U i (0) 2 − µ(3 are determined by the trap potential V trap ( r i ; t) and a global chemical potential µ which characterizes the initial equilibrium state at time t = 0. The initial state is assumed to have a temperature k B T = β −1 . The timedependent interaction U i (t) and the time-dependent hopping J ij (t) are chosen to be results of a tight-binding calculation for maximally localized Wannier functions computed from the translationally invariant case. In the future, we plan to include corrections to the tight-binding parameters which result from generalized Wannier functions for the inhomogeneous problem. We assume the system to be in thermal equilibrium at time t = 0 and to be driven out of equilibrium subsequently by the time-dependence of the model parameters. III. FORMALISM We employ a second-order self-consistent expression for the self-energy [12] around the atomic limit which is described by H 0 . The self-consistency takes advantage of a resummation of diagrams which yields a better approximation. It can be extended to an inhomogeneous system in the following way: Σ lm,σ (t, t ) = − δ lm j1,σ1 dt dt 3 dt 4 dt × G −1 lσ (t,t)G II l;σσ1 (t, t 4 ; t 3 ,t ) × J lj1 (t 3 ) G (loc) j1σ1 (t 3 , t 4 ) × J j1l (t 4 ) G −1 lσ (t , t ) =: δ lm Σ l,σ (t, t ).(4) Here, G lσ (t, t ) = −i T C c lσ (t)c † lσ (t ) H (l) 0 (5) is the contour-ordered on-site no-hopping Green's function at site l with spin σ (in the paramagnetic phase, the Green's function is independent of the spin σ). Times t and t are located on the Kadanoff-Baym-Keldysh contour C depicted in Fig. 1. G II m;σσ (t 1 , t 2 ; t 1 , t 2 ) = G II m;σσ (t 1 , t 2 ; t 1 , t 2 ) + G mσ (t 1 , t 1 )G mσ (t 2 , t 2 )δ σσ − G mσ (t 1 , t 2 )G mσ (t 2 , t 1 )(6) is the second-order cumulant for the on-site no-hopping two-particle propagator G II m;σσ (t 1 , t 2 ; t 1 , t 2 ) = (−i) 2 T C c mσ (t 1 )c mσ (t 2 )c † mσ (t 1 )c † mσ (t 2 ) H (m) 0(7) at site m. In a typical numerical implementation, one cannot store this tensor for a reasonable grid of time slices. However, it is easy to compute it on the fly in the particle-number basis {|E 1 (t) , |E (m) 2 (t) , |E (m) 3 (t) } = {|0 m , | ↓ m , | ↑ m , | ↓↑ m } by inserting the time evolutions c ( †) mσ (t) = e i t 0 H (m) 0 (t ) dt c ( †) mσ e −i t 0 H (m) 0 (t ) dt(8) for each possible T C time ordering of the creation and annihilation operators. Note that it is crucial to the applicability of the approach that the expression in Eq. (8) does not involve any time-ordered products, because [H ζ (m) ν (t) = e i t 0 E (m) ν (t ) dt(9) values. For example, when t 1 > t 2 > t 1 > t 2 , G II m;↑↓ (t 1 , t 2 ; t 1 , t 2 ) =(−i) 2 e −βE (m) 0 (0) Z (m) × ζ (m) 0 (t 1 )ζ (m) 2 (t 2 ) ζ (m) 2 (t 1 )ζ (m) 3 (t 2 ) × ζ (m) 3 (t 1 )ζ (m) 2 (t 2 ) ζ (m) 2 (t 1 )ζ (m) 0 (t 2 ) ,(10) with Z (m) = 3 ν=0 e −βE (m) ν (0) .(11) In the calculation, one requires the expressions for all possible time orderings. The relation in Eq. (4) is solved iteratively. At a given step of this procedure, the selfenergy at site i is given by Σ i,σ (t, t ) = tt G −1 i,σ (t,t)Σ i,σ (t,t )G −1 i,σ (t , t ),(12) wherẽ Σ i,σ (t,t ) = − j1,σ1 t3,t4G II i,σσ1 (t, t 4 ; t 3 ,t )J i,j1 (t 3 ) × G (loc) j1,σ1 (t 3 , t 4 )J j1,i (t 4 ).(13) The local Green's function at site j 1 , G j1,σ1 is given by the j 1 -th block-diagonal element of the lattice Green's function G (latt) σ ( r i , t; r i , t ) = − i T C c riσ (t)c † r i σ (t ) = Ĝ −1 σ −Ĵ −Σ σ −1 ( r i , t; r i , t ),(14) i.e. G (loc) i,σ (t, t ) = G (latt) σ ( r i , t; r i , t ).(15) Here, the hatted quantitiesĜ σ ,Ĵ, andΣ σ denote the non-interacting Green's function, the hopping, and the self-energy as operators acting on both, time and space coordinates. A. Observables We comment now on the measurement of some important physical observables within our method. The spin-σ occupancy of site i may be evaluated by n iσ (t) = − i · (−i) · lim δ→0 + T C c iσ (t)c † iσ (t + δ) = − iG (loc) iσ (t, t + 0 + )(16) from the local Green's function. The kinetic energy contribution of spin σ at lattice site i can be deduced from the lattice Green's function e kin iσ (t) := +i · j is NN of i J ij,σ (t)G (latt) σ ( r j , t; r i , t).(17) It is thus required to measure the equal-time hopping Green's functions from site i to its nearest neighbors j. A quantity of particular interest is the double occupancy D i (t) = n i↑ n i↓ (t) as a function of time t and lattice site i. We can derive it from some elements of the lattice Green's function and its equal-time derivative, using the following relation (see Appendix B): ∂ ∂t G (loc) i,σ (t, t ) t =t + = U i (t)D i (t) + ε i (t) n iσ (t) + e kin iσ (t).(18) IV. ALGORITHM In this section, we outline the massively parallel algorithm required to solve Eqs. (12), (13), and (14) iteratively on a supercomputer. A. Representation of the Green's functions and self-energies In order to represent the contour-ordered Green's function on a computer, the Kadanoff-Baym-Keldysh contour is discretized to N t time slices, as shown in Fig. 1. In the implementation used here, we chose N t = 2N t,real + N t,imag , where N t,imag = N t /32. Here, N t,real is the number of time steps on each real branch of the Kadanoff-Baym-Keldysh contour and N t,imag the number of time steps on the imaginary time axis. Furthermore, the system consists of a finite number N r of lattice sites. Assuming that spatial symmetries such as reflection and rotation symmetries are given for not only the Hamiltonian but also the quantum-statistical states, we can reduce the actual number of lattice sites N r within the algorithm by symmetry maps to the numberÑ r which is the number of sites in the irreducible wedge of the lattice. Many sites can then be represented by the equivalence class with respect to the symmetry. Note that the full number of lattice sites still plays a role in computational complexity when the propagation of excitations is considered. Since Dyson's equation performs an infinite resummation of such processes, it is relevant for the calculation of the Green's function. Appendix A describes the exploitation of symmetries in more detail. B. Global Layout Computationally, the self-consistency condition in Eq. (4) is solved iteratively using an alternating sequence u Σ,0 u Σ,1 u Σ,2 u Σ,3 u Σ,4 u Σ,5 G,0 u G,1 u G,2 u G,3 u processor index Σ G C C . . . . . . of self-energy and Green's function evaluations. The selfenergy evaluations are performed via Eqs. (12) and (13) and will be described in detail in Sec. IV C. The Green's functions required for self-consistency and measurements of observables are evaluated via Eq. (14) which will be described in Sec. IV D. Due to the structure of Eq. (13), the computation of Σ i (t, t ) at lattice site i requires only the Green's functions of neighboring lattice sites. However, the second step, i.e. the Dyson's equation evaluation in Eq. (14) of the lattice Green's requires self-energy information from all lattice sites. From a computational perspective, these demands are quite different in terms of optimal memory arrangement and distribution of tasks over a large set of compute nodes. As a consequence, we define two different configurations of the simulation. The Green's function evaluation in Eq. (14) is performed in what we call the C G configuration of the simulation. The self-energy computation is done in the C Σ configuration. Fig. 2 shows a schematic flowchart for the computation. The two different global machine states are sketched in Fig. 3. All processors are thought to be arranged along the direction of the abscissa in Fig. 3. Each configuration defines groups of processors which share tasks, we call them self-energy units u Σ,i and Green's function units u G,i . The self-energy unit u Σ,i evaluates self-energies for the i-th range of representative sites. The Green's function units evaluate relevant information for the i-th range of symmetry-representative site indices for blocked rows of the lattice Green's function. There is an optimal value for the size of the units which is usually different for u Σ,i and u G,i . The determining factors for the optimal size will be elaborated below. C. Self-energy evaluation for unit uΣ,i By a given self-energy unit u Σ,i , the self-energy Σ n (t, t ) is computed for a certain range of (representatives of) sites n for all t, t using Eqs. (12) and (13). The expressions involve an on-the-fly evaluation of the cumu-lantG II using the tabulated values of Eq. (9), as well as local Green's functions which do not require any permanent storage on u Σ,i . Memory-wise, the unit is required to have access to the local Green's functions of neighboring sites, as well as to the value of the self-energy from the previous iteration. The latter is necessary, because in our implementation, Eq. (12) is regularized as follows: Σ = λ mixΣnew + (1 − λ mix )Σ old ,(19) where λ mix ≈ 0.7 is a linear mixing parameter which stabilizes the convergence by averaging between the current and the previous iteration. Convergence is reached once N −1 bd Σ new −Σ old 2 + Σ new −Σ old ∞ ≤ δ sc ,(20) where · 2 is the Frobenius norm of the self-energy matrix with N bd entries on the block diagonal, and · ∞ is element-wise maximum norm of the matrix. A reasonable value for the accuracy of the self-consistency condition is δ sc ≈ 10 −8 U 0 ,(21) where U 0 is some typical value of the interaction strength (which often serves as the energy unit). Convergence is usually reached after approximately ten iterations. In all relevant cases we have encountered so far, the self-energy evaluation step is computationally more costly than the Green's function evaluation, if the latter has been optimized appropriately. This is due to the fact, that the contraction of cumulant indices in Eq. (13) scales with the fourth power of N t . The evaluation of the cumulant contraction is however massively parallelizable, i.e. no significant communication even within the u Σ,i is required during the computation. Thus, the size of the cumulant units can be chosen rather freely regarding the aspect of communication between CPUs within the unit. A small size is preferable, due to the typically small memory consumption. For convenience, when switching between configurations C Σ and C G , the self-energies just remain within the compute nodes of the respective processors of the self-energy units. However, it is better to ensure that each self-energy unit deals with more than one lattice site, because we encounter the situation that the time to evaluate the selfenergy varies substantially from lattice site to lattice site. If the site indices are assigned randomly, this effect averages out for a sufficient number of sites. As a rule of thumb, it is reasonable to assign self-energies for sixteen random lattice sites to each u Σ,i . D. Green's Function evaluation for unit uG,i Let us have a closer look at the Dyson's equation in Eq. (14). It contains a block-diagonal part G −1 σ −Σ σ = diag i ∈ lattice (G −1 i,σ − Σ i,σ ),(22) and a sparse off-diagonal part given by the hopping ma-trixĴ. The latter is composed of a contour-δ-function in time and a tight-binding structure in real space. In particular,Ĵ is a small quantity, due to the very nature of the strong-coupling expansion. This fact is exploited algorithmically. It is insightful to change notation in Eq. (14). We can write the lattice Green's function in a Dirac type notation, as a matrix element G (latt) σ ( r, t; r , t ) = r, t| 1 G −1 σ −Ĵ −Σ σ | r , t ,(23) where {| r, t } r,t is an orthonormal basis associated with space and time variables. Each u G,i computes G (latt) σ ( r, t; r , t ) at the sites r assigned to it for all r , t, t and stores the local and hopping Green's functions as required. Determining the lattice Green's function corresponds to solving N r · N t linear equations in N r · N t variables. That is, the (N r N t ) 2 matrix elements of the Green's function are determined by solving such linear equations N r N t times. Typically, in our application, N t is at least 512 or 1024 and N r is around 10000. Note that since the simulation discretizes the points on the Kadanoff-Baym-Keldysh contour with N t time slices, there is a finite size of time slices ∆t. The effect of a finite ∆t usually affects the accuracy of the calculations, and thus we perform a quadratic extrapolation of the simulation results from three finite values of ∆t to ∆t → 0. Cross-checks with linear extrapolations show the quadratic extrapolation is superior in most instances. The matrix to be inverted in Eq. (23), Ĝ (latt) σ −1 , is typically around 5 · 10 6 × 5 · 10 6 dimensional. Its block structure can be written as Ĝ (latt) σ −1 =                   B 1 0 · · · 0 −J 1,N N (1)i 0 · · · 0 0 B 2 0 · · · −J 2,N N (2)i 0 · · · . . . 0 B 3 0 · · · . . . . . . . . . 0 · · · −J Nr,N N (Nr)i · · · 0 B Nr                   ,(24) where each block represents an N t × N t matrix and B i = G −1 i,σ − Σ i,σ ,(25) as given in Eq. (22). Each row and each column in Eq. (24) contains at most 4 hopping matrices J n,N N (n)i to the nearest neighbors N N (n) i , i = 1, . . . , 4 (in two dimensions). The hopping matrices J n,N N (n)i are essentially diagonal matrices whose diagonal elements are the time-dependent hopping amplitudes. The matrix in Eq. (24) is extremely sparse, while its inverseĜ (latt) σ is dense. However, due to the small numerical value of the hopping in the strong-coupling expansion, matrix elements ofĜ (latt) σ fall off as a function of spatial distance and eventually become irrelevant; in other words, the lattice Green's function is typically block diagonal dominant. This fact can be exploited in a numerically controlled way by utilizing an iterative procedure. We choose the Generalized Minimal Residue (GMRES) method as a solver [15]. The GMRES method considers an equation system y =Âx,(26) as well as an "almost" correct solutioñ x = P(y),(27) where P is the so-called preconditioner. P is an arbitrary operator whose action on y is cheap to compute but is a good approximation to −1 y. In our case, due to the small numerical value of the hopping, a natural choice for the preconditioner is the inverse block diagonal matrix P(y) :=B −1 y,(28) whereB :=Ĝ −1 σ −Σ σ(29) as defined earlier in Eqs. (22) and (25). Further details of the GMRES method as applied to the Dyson's equation in the inhomogeneous strong-coupling expansion are discussed in Appendix C. We distribute the task of applying GMRES to each unit vector |e j of the vector space C NtNr within the set of Green's function units {u G,i } i=1,...,N G . Each Green's function unit must store all blocks of the block diagonal B. Memory allocated to each CPU within a unit thus defines a lower bound to the size of a Green's function unit through this requirement in the following way: |u G,i | ≥ Ñ r · N 2 t · 16 Bytes per complex memory per CPU ,(30) assuming a double precision representation of complex numbers. If one also decides to store the preconditioner, this value doubles. We store the blocks ofB as depicted in Fig. 4. Practically, within each u G,i , an efficient application of B andB −1 to a given vector |x has to be achieved. The performance of these operations crucially relies on the small size of u G,i and on a good load-balancing within u G,i . The former is due to an increased number of communication events through relatively slow connections between CPUs and also due to its role in the latter. In order to avoid an unnecessarily large number of CPUs in u G,i , a minimization of memory consumption plays a key role. We construct the representation ofB within Nx FIG. 4: Distribution of the non-zero entries (lightly shaded boxes) B xν (t, t ) of the block-diagonal matrixB within each unit u G,i (darkly shaded area). If symmetries apply, only the blocks associated with representatives of lattice sites need to be stored, and N r is then taken as N r . a Green's function unit by distributing respective selfenergies to the CPUs dealing with specific parts ofB. A practical example is shown in table I. A further important optimization when applying GM-RES takes advantage of both cache optimizations in Basic Linear Algebra Subprograms (BLAS) [16] level 3 routines and a reduction of network latency effects -rather than letting u G,i apply GMRES individually to each unit vector |e j of its concern serially, we choose a parallel implementation. It applies the algorithm to blocks rather than vectors. Thus, highly optimized BLAS level 3 rather than multiple calls of BLAS level 2 are used. In addition, less communication processes between CPUs occur. More specifically, we can write this blockwise procedure as solving the blocked equation system u G,i B x 3n B x 2n B x n B x 2 B x 1 CPU m CPU m−1 CPU 3 CPU 2 CPU 1 B xÊ 1Ê2 · · ·Ê N b = (B−Ĵ)· Γ 1Γ2 · · ·Γ N b , (31) whereÊ i = iNrNt/N b j=(i−1)NrNt/N b |e j e j |,(32) with appropriately defined unit vectors |e j which belong to a single lattice site in the notation introduced above with Eq. (23). The blocksΓ i are the respective blocks of G latt σ . Typical block sizes are listed in Table I. In the GMRES procedure, this definition of blocked equations which are solved one after another has the following advantage. The method operates iteratively, where the start value is the preconditioner solution, which is still localized at some lattice site:B −1 |e j (or B −1Ê i ). The computation ofB −1 is done with LAPACK calls [16] for the diagonal blocks by the CPUs assigned to them according to Fig. 4. By means of the spatial structure of this then iteratively refined approximation to the solution, only the application ofB −Ĵ introduces further lattice sites to the problem. As a consequence, if GMRES converges quickly for a given numerical accuracy of the GMRES method, non-zero elements only occur within the spatial vicinity associated to the consideredÊ i block in the blocked representation of the converging GMRES solutionX = x 1 x 2 · · · x NrNt/N b (see App. C for details). Indeed, the GMRES procedure converges very rapidly, because the hoppingĴ between adjacent lattice sites is required to be small in the strong-coupling expansion. For the block representationsX of iteratively refined GMRES solutions, we thus choose to only store non-zero subblocks and distribute them within the u G,i units using the storage scheme forB (see above and Fig. 4). In the case thatX contains non-zero contributions for reducible lattice sites (this happens whenÊ i is located near the boundary of the range of site representatives, see Appendix A), the respective block is handled by the CPU associated to its equivalence class. This minimizes both communication and memory consumption. The former is because applying the preconditioner is only a local operation and applying requires only communications with units storing neighbors of the respective lattice sites. The size of the blocks inX is to be chosen as large as possible. However, memory in u G,i is usually very limited, so a tradeoff in unit size and block length has to be made. Let us also comment on reasonable values for the GM-RES convergence parameter. The GMRES method for our matrix inversion runs until a certain accuracy for the result is achieved, i.e. e j −Âx 2 ≤ δ GMRES ,(33) where δ GMRES is the desired numerical precision and · 2 is the Euclidian norm. For all practical purposes we encountered so far, a value δ GMRES = 10 −2(34) has been sufficient. This surprisingly large value was verified by comparing to simulations with higher accuracy, that is δ GMRES = 10 −3 , for the physical systems studied in this paper at several parameter values. The plots of numerical results of interest are identical to the eye. Similar tests were done for completely homogeneous systems by comparing to a numerically exact implementation in momentum space. It may be that for different applications than the one presented here a smaller value of δ GMRES is required. In order to understand the meaning of δ GMRES better it may be useful to compare it to the dimension of the vector it constrains. In our case, the dimension of e j −Âx is N r N t ≥ 1 × 10 6 . Thus, if one chooses to normalize the convergence criterion in Eqs. (33) and (34) by the dimension, the constraint reads 10 −8 . In this context it may also be worthwhile to consider the fact that the GMRES procedure only involves transformations withB −1 ,B, andĴ. That is, it applies only transformations which comply with the causal structure of the Green's function and do not introduce artificial discontinuities with respect to the time variables in the end result for the Green's function which are in principle part of the vector space which is being searched by the algorithm. In other words, the physical choice of the preconditioner already constrains the solution space so drastically that even a relatively large value of δ GMRES might be sufficient. With all these optimizations, the Green's function evaluation typically consumes no more than five to ten percent of the time required for the self-energy evaluation on a Cray XE6 machine in the application to lattice depth modulation spectroscopy described below. The optimizations are necessary to speed up the Green's function evaluation appropriately, because we encountered increases in speed by a factor of at least 10 and up to 1000, for each blockwise application of GMRES, distributed storage of the GMRES vector blocks, and random assignment of site indices. In other applications than lattice depth modulation, with a large value of the hopping applied for a longer period of time, the requirement of computer time for the Green's function evaluation may still exceed the time to evaluate the self-energy. However, we find these requirements to be within reasonable bounds. E. Switching between the global configurations The only time when global communication and synchronization across all processors is required is when either the self-energy or the Green's function evaluations are finished. Then, convergence has to be checked, and the global configurations C Σ and C G have to be replaced by each other. This requires point-to-point communications of individual processors across the machine and broadcasts within smaller groups of processors which all require the same data. The occuring communication events are displayed in Fig. 5. During the switch C G → C Σ (Fig. 5a), the Green's function contents on the nearest neighbors of the sites assigned to a given u Σ,i have to be sent from the Green's function unit which computed them. Due to the possibly random assignment of site indices m to spatial coordinates r m , the input to u Σ,i is collected from various u G,j . The storage scheme used for the results within u G,j determines the actual processors which send the data. When switching from C Σ to C G , spatial ranges tasked to specific parts of each Green's function unit have to be sent from the self-energy units which contain this information (Fig. 5b). The storage pattern for u G,i is already G C Σ C Σ n m G u G,N−1 u G,1 u G,N u G,2 u Σ,i Σ i (a) Σ C G C u Σ,1 u Σ,2 u Σ,N−1 u Σ,N CPU 1 CPU 2 u G,i all Σ n site index of computed self−energy (b) FIG. 5: (color online) Global data transfers during the configuration switches C Σ ↔ C G . (a): Transfers from the u G,i units to a given u Σ,i unit during the configuration switch C G → C Σ . Only the local Green's functions G m (t, t ) on the nearest neighbors of the assigned self-energies Σ n (t, t ) are required by each u Σ,i due to Eq. (13). (b): Transfers of the full self-energy data Σ n (t, t ) from the u Σ,i units to all u G,i units. The process can be improved by sending the data only to u G,1 and then broadcasting the data internally to the set of equivalent member processors CPU j , j = const., of each u G,i , i > 1. The memory structure of the Green's function unit is identical to the block-diagonal storage scheme introduced in Fig. 4. the one specified for the block diagonal matrix denominator of the Dyson's equation (Fig. 4). To finish the change of configurations, an in-place substitution of the self-energies by the respective blocks B xi is performed by computing the respective atomic Green's functions and inserting B xi = G −1 i − Σ i , as defined in Eq. (29). The process C Σ → C G is typically more time-consuming than C G → C Σ . However, it can be optimized by using broadcasts between groups of processors with the same data requirements: each j-th CPU in u G,i requires the same data set to operate. The switching processes cost no more than five percent of the total computation time and are thus negligible. However, the implementation involves a considerable amount of book-keeping. F. Summary and notes on the implementation Let us summarize the algorithm and also provide some implementation details on the way. As the method implements the self-consistent solution of Eq. (4), the algorithm is split in two steps: the self-energy evaluation (Eqs. (12) and (13)) and the Green's function evaluation (Eq. (14)). Since the computation of the self-energy and the evaluation of the Green's function have different requirements in terms of computational resources on the supercomputer, they use different data structures and collaboration patterns amongst the CPUs. We refer to the data structures of the algorithm in the self-energy evaluation state as the configuration C Σ and to the data structures of the algorithm in the Green's function evaluation state as the configuration C G . The respective configurations are subdivided into mutually independent units u Σ,i and u G,i spanning several CPUs and the memory associated with them, respectively. This approach is tailored to a cluster, rather than a shared-memory architecture. If one chooses to employ the Message Passing Interface (MPI) standard in order to implement the algorithm, it is advantageous to use the MPI Group feature to ensure an efficient communication within the units [18]. It turns out to be useful to define internal communicators for the u G,i and u Σ,i , respectively, as well as for sets of processors with shared requirements, such as the n-th processor of each u G,i , since they all require the same self-energies. Within these shared-interest communicators, data can be broadcasted efficiently. It may also be reasonable to use advanced MPI features to perform an optimization with respect to the network topology of the supercomputer, such that communication within the u G,i units is optimal and equally fast for all i. In contrast, the communication within u Σ,i is not time-critical, because the main effort, computing the integrals in Eq. (13), is done by each processor in u Σ,i independently. Let us now comment on the implementation of Eq. (13). Each u Σ,i computesΣ for a certain range of sites. Here, the main effort is the contraction of the time indices. The atomic-limit cumulant Green's function G II is computed on the fly using tabulated values of the exponentials in Eq. (9). The computational effort of Eq. (13) scales with N 4 t and is the computationally most costly operation. However, it may also be implemented on GPUs, due to little memory and bandwidth requirements. After having evaluated Eq. (13), the units u Σ,i compute the updated self-energy using Eqs. (12) and (19). Then, as described in section IV E, the resulting local self-energies are sent to the u G,i units which may or may not overlap with the respective u Σ,i . Within each u G,i , the self-energy for all sites has to be available and is thus equally distributed over the CPUs according to the storage pattern depicted in Fig. 4. It is advised to keep the self-energy results in u Σ,i for the next update as described by Eq. (19), even though the machine changes to configuration C G in the meantime. This is because theΣ old in Eq. (19) has to be available in the self-energy computation of the next iteration of the self-consistency loop. In order to minimize the memory consumption of the relevant range ofΣ old , it can be distributed equally within each u Σ,i . In order to establish the configuration C G to compute Eq. (14), the self-energies in u G,i are then replaced by the blocks of the matrixB according to Eq. (29). Optionally, the elements of the preconditionerB −1 can also be computed and stored at this point in time, also using the previous storage pattern. However, this competes with the requirement to keep u G,i small, because the action of the preconditioner can also be computed on the fly from B with a smaller memory requirement. Having fully set up the C G configuration, Eq. (14) is written as the vectorized linear Eq. (31) as described in section IV D. The key variable is the bundle of GM-RES vectorsX whose initial valueX 0 is the preconditioner P =B −1 applied to a bundle of unit vectors, as in Eq. (32).X 0 has only non-zero entries at a single site index.X is also stored according to the scheme in Fig. 4. A good optimization here is to store only nonzero components ofX emerging fromX 0 due to the application of the hopping matrix. For this purpose, each processor in u G,i can keep track of the sites with nonzero elements inX based on the site index ofX 0 and the number of hopping events applied toX during the GM-RES procedure elaborated in App. C. Once a hopping occurs due to the application ofB −Ĵ, a given processor may have contributions to be stored and/or added to a value assigned to another processor within the storage scheme (compare to the block structure ofB −Ĵ in Eq. (31)). Such transmissions are the major communication events within u G,i . The required communication bandwidth within u G,i can only be minimized by assigning connected spatial domains with a minimal surface to single processors within u G,i . However, doing so is strongly disadvantageous in the case that the GMRES is converging very rapidly, i.e. if the hopping is small and t max is small. In this case, all but one of the CPUs in u G,i will remain idle, because the non-zero elements in X do not leave the CPU storing the non-zero elements ofX 0 . Thus, for a rapidly converging GMRES, a random assignment of the site leading to largely scattered domains is more appropriate. Once each u G,i has computed the Green's functions on the spatial range assigned to it, the different spatial components of G loc are distributed to the u Σ,i units which require them for evaluating the Green's function sum in Eq. (13) in order to start the next iteration. V. RESULTS In this section, we present results of the algorithm for trapped atoms in a two-dimensional optical lattice which is subject to a periodic modulation of the lattice depth. We compare this method to a homogeneous version of the algorithm which was previously successfully applied to a lattice depth modulation experiment within the LDA [13]. The LDA is generally expected to yield good results for systems without mass transport. This is a wellestablished observation in equilibrium [17]. We show that also in a nonequilibrium scenario without mass transport, the high accuracy of the LDA can be explicitly demonstrated. At the same time we validate our direct computational approach. A. Lattice depth modulation In lattice-depth modulation spectroscopy, the atoms are subject to a time-dependent optical lattice potential V ( r, t) = V trap ( r) + V lattice ( r, t).(35) The trap potential does not depend on time and has the parabolic shape V trap ( r) ∝ | r| 2 .(36) The lattice potential satisfies V lattice ( r, t) = V (t) 2 i=1 sin 2 (kx i ),(37) which contains the time-dependent lattice depth V (t) = V 0 + χ [0,t mod ] (t) · ∆V sin ωt.(38) We assume that the lattice is modulated over a finite time interval [0, t mod ] and that the system is in an initial thermal state at time t = 0. Numerically, we start the simulation at an earlier point in time, in order to be able to check for convergence, as discussed in Fig. 1. The lattice constant k = 2π/λ is defined by the laser wavelength λ. The single-particle Hamiltonian H single (t) = − 2 2m ∇ 2 + V ( r, t)(39) yields the recoil energy E R = k 2 /2m as a natural choice for an energy unit. In order to compute the coefficients of the many-body Hamiltonian in Eq. (1) from the singleparticle Hamiltonian, we insert the constant hoppings J and interactions U of a translationally invariant lattice. These can be computed easily with maximally localized Wannier functions [14]. Due to the time-dependence of the lattice depth V (t), we also obtain a time-dependent interaction U (t) and hopping J(t). We write the initial values of the interaction and the hopping as U 0 and J 0 , respectively. In these units, the trap potential can be written as V trap ( r) = J 0 | r/ρ trap | 2 .(40) Hence, ρ trap can be interpreted as the length scale on which the trap potential reaches the strength of the initial hopping amplitude. It is important to keep ρ trap larger than a couple of lattice spacings, since otherwise the trap potential interferes drastically with the hopping between neighboring sites and the density changes too fast for the LDA to be accurate. B. Numerical Results As a test system, we set the lattice depth to V 0 = 10E R and modulate it with an amplitude ∆V V0 = 20% at the resonant frequency ω = U 0 . The interaction strength is chosen to be U 0 /6J 0 = 7.77, and we assume an initial temperature k B T = 0.15U 0 . We choose to study two cycles of the modulation, that is t mod = 2h/U 0 . For the simulations, we use up to 1024 time slices and a lattice with up to 1024 symmetry-irreducible lattice sites, that is up to 7844 actual lattice sites. The computational effort for a system with 512 symmetry-irreducible sites and a maximum of 1024 time slices is approximately 5 × 10 5 CPU-hours on a Cray XE6. For instance, this involves 32768 CPUs for approximately 12 hours by the main simulation and some further CPU time for the cheaper simulations at larger ∆t which are required for the extrapolation ∆t → 0. Figure 6 shows simulation results for distribution of the double occupancy in a trapped system with ρ trap = 4 sites and the global chemical potential µ = 0. Each subfigure displays the distribution at a different point in time. Due to the lattice depth modulation, the hopping in units of the interaction J(t)/U (t) drives the system. The increases in the double occupancy occur as J(t)/U (t) is decreasing. To provide a better picture of the time dependence, Fig. 7 shows the fraction of atoms on doubly occupied sites,D (t) = 2 i n i↑ n i↓ (t) N ,(41) where N = i n i (t) = const. as a function of time for several values of the trap curvature. In the cases ρ trap = 4 sites and ρ trap = 5.5 sites, the results lie on top of each other, whereas for ρ trap = 2 sites a slight deviation occurs. This agrees with the results found in our previous publication Ref. [13] for homogeneous systems. Comparison to LDA In order to perform the comparison to the LDA, we solve numerous mutually independent homogeneous versions of the problem at several chemical potentials µ = −V trap ( r) and compare to local observables obtained from the full trap simulation at a position r. We consider a set of test systems with three different trap curvatures, that is different values of the Figure 8 shows a comparison of the double occupancy D(µ, t) as a function of the initial chemical potential µ at times before (t = t 0 ) and after (t = t max ) it has been driven out of equilibrium by the lattice depth modulation. As we see for both, the equilibrium (t = t 0 ) and nonequilibrium (t = t max ) situations, the numerical results for the inhomogeneous system agree well with the LDA, even for the rather steep trap potential with ρ trap = 2 sites. The slight deviation of the solution for 2 sites from the other nonequilibrium curves may still be due to numerical imperfections. The agreement with the LDA indicates that the creation of doubly occupied sites in a Mott insulator subject to a modulated lattice depth is caused by strongly local excitation processes. VI. CONCLUSION We presented a computational approach to an inhomogeneous Mott-insulating system of ultracold atoms. A major challenge is to compute a large matrix inverse in the Dyson equation. We show that a GMRES-based inversion approach exploiting the small numerical value of the hopping as compared to the many-body interaction yields a feasible implementation on supercomputers. A comparison to the LDA shows that both methods are well-suited for the problem of lattice-depth modulation spectroscopy. This hints towards mainly local processes being involved in the coherent excitations between lower and upper Hubbard bands in this particular setting, as might have been anticipated. In the future, we will apply the inhomogeneous method to problems with mass transport where the LDA is expected to fail. At present, the computational complexity of the algorithm is proportional to N 4 t . It may be worthwhile to investigate the possibility to extend the time range by truncating certain parts of the self-energy at a given threshold for t − t . This measure could increase the applicability of the algorithm greatly but requires further efforts. The circles denote lattice sites. Solid circles are representatives of an equivalence class of lattice sites with respect to the symmetry. The symmetry partners of the red/gray solid circle are displayed in red/gray. Green's function and self-energy transform as the identity representation of the point group, the computations only need to be performed for those representatives. The representative of a given site can be retrieved by reflections with respect to the coordinate axes and the diagonals which are shown as dashed lines in Fig. 9. online) Kadanoff-Baym-Keldysh contour C for the simulation time. t )] = 0 for any combination of t, t . The on-the-fly evaluation of the action of the operators in the basis can be realized by a fast multiplication with and division by tabulated FIG. 2 : 2The self-consistent strong-coupling algorithm as a flow chart. The configurations C Σ and C G are introduced in Sec. IV and correspond to evaluating Eqs.(12) and(13)(both C Σ ), and Eq. (14) (C G ). FIG. 3 : 3Global configurations for the parallelization scheme. FIG . 6: (color online) Direct numerical results for the double occupancy distribution D( x, t) as a function of time for a uniformly modulated lattice depth in a trap. The insets of each panel show the hopping in units of the interaction J(t)/U (t). The trap curvature is specified by ρ trap = 4 sites. FIG. 8 : 8(color online) Comparison of the double occupancies computed for the full trap simulation of traps with different curvatures as compared to the LDA result. characteristic length ρ trap of the trap potential, namely ρ trap = 2 sites, ρ trap = 4 sites, and ρ trap = 5.5 sites. In the simulations, the real part of the Kadanoff-Baym-Keldysh contour extends over an interval [t 0 , t max ] = [−2 /U 0 , 14 /U 0 ], whereas the modulation acts over the interval [0, t mod ]. online) C 4v point-group symmetry imposed by the circular trap potential on the 2D lattice. TABLE I : ITable oftypical parallelization parameters on a Cray XE6 (2GB memory per CPU) for the application presented in Sec. V B. The number n b = N r N t /N b denotes the block size in the parallel GMRES implementation. Appendix A: Utilization of SymmetriesWe illustrate the exploitation of symmetries for the example of spherical symmetry on a two-dimensional square lattice with an s-orbital basis. In this case, a C 4v point group symmetry[19]is imposed on the lattice.Fig. 9displays the lattice sites of a lattice. For the C 4v symmetry, all sites can be represented by the sites in the irreducible wedge x ≥ 0 ∧ y ≤ x. These representatives of the equivalence classes are denoted by solid black circles. If the Appendix B: Formula for the double occupancyWe provide a brief derivation of Eq.(18). Let us start by assuming an equidistant discretization {t 0 , . . . , t N } (∆t = t i+1 − t i ) of the forward part of the Kadanoff-Baym-Keldysh contour. We obtainThe operator lσ simplifies as follows:Taking the limit ∆t → 0 results in Eq.(18). Numerically, this limit must be performed via linear and/or quadratic extrapolation of multiple simulations for different ∆t values.Appendix C: GMRESThe Generalized Minimal Residue Method was introduced by Saad and Schultz[15]to solve a linear equationA good introduction to the method can be found in Ref.[20], and a useful C++ implementation is provided by the NIST IML++ template library[21]. In order to solve the equation (C1), GMRES operates in d-dimensional Krylov subspaceswhich are successively built up, starting with d = 1.is the residue of the initial guess u (0) for the solution. The GMRES method forms a refined approximate solution u (d) ∈ u (0) + K d (Â, r (0) ) defined by the minimization requirementwhere the expression "x = ! min" demands x to be minimal. The Krylov space dimension d is increased until the convergence criterion (33) is reached or d approaches the threshold d max at which the minimization is considered too expensive. If d = d max , GMRES is restarted with a Krylov space size d = 1, where the initial guess u (0) is taken to be u (dmax) from the previous GMRES iteration before restarting. With a preconditioner P, one applies GMRES to the systemrather than Eq. (C1). If P ≈Î this system is better conditioned than Eq. (C1). In the case that is a sparse matrix with large entries on the diagonal and some randomly occuring off-diagonal entries, the diagonal matrixB defined by the diagonal entries of yields a good preconditioner B −1 , because the condition numberis not as close to 1 as the regularizedHere, σ max and σ min represent the respective maximal and minimal singular values. Also, the application of the diagonal matrixB to a vector is a cheap operation. The same argument can be made in the case that has large entries on the block diagonalB, as it is the case in the nonequilibrium inhomogeneous strong-coupling expansion, where =B −Ĵ. The block diagonalB is defined inEq. (29), and the hopping matrixĴ is a sparse matrix with small numerical values (see also Eq. (24)). Due to the latter, in factB −1 ≈Î, so the equation system is well-conditioned. The actual GMRES algorithm with preconditioner in pseudo code reads [20, 21] 1. For the initial guess u (0) =B −1 b compute the preconditioned residue z (0) =B −1 (b −Âu (0) ), as well as q (1) = z (0) / z (0) 2 . Initialize the Hessenberg matrixIf h d+1,d = 0 then proceed with step 3 to compute the result.Otherwise use step 3 to check for convergence, Eq. (33). Continue if not converged.Solve the d-dimensional linear minimization problemwhere H (d) is the upper left d-dimensional square of H, to obtain the result y (d) .Practically, the GMRES with preconditioner scans for solutions in the modified affine Krylov spacesHere, the basis vectors areIn the case of the inhomogeneous nonequilibrium strongcoupling expansion, b is a unit vector localized at a given lattice site. The n-th basis vector v n of the Krylov spacẽ K d (B −1 , z (0) ) corresponds to n hopping processes, be-cause =B −Ĵ, and in v n , is applied n times to b. That is, the GMRES method includes exactly d iterated hopping processes in iteration d until convergence is reached. If GMRES is restarted, the d'th iteration includes md max + d iterated hopping processes, where m is the number of restarts. This is crucial for the computational optimizations used in the implementation. The procedure can be seen as a numerically controlled analogue to a partial summation of the Dyson series . I Bloch, J Dalibard, W Zwerger, Rev. Mod. Phys. 80885I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008) . T Esslinger, Ann. Rev. Condens. Mat. Phys. 1129T. Esslinger, Ann. Rev. Condens. Mat. Phys. 1, 129 (2010) . Th, H Stöferle, C Moritz, M Schori, T Köhl, Esslinger, Phys. Rev. Lett. 92130403Th. Stöferle, H. Moritz, C. Schori, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 92, 130403 (2004) . R Jördens, N Strohmaier, K Günter, H Moritz, T Esslinger, Nature. 455204R. Jördens, N. Strohmaier, K. Günter, H. Moritz, and T. Esslinger, Nature 455, 204 (2008) . D Greif, L Tarruell, Th, R Uehlinger, T Jördens, Esslinger, Phys. Rev. Lett. 106145302D. Greif, L. Tarruell, Th. Uehlinger, R. Jördens, and T. Esslinger, Phys. Rev. Lett. 106, 145302 (2011) . S R Clark, D Jaksch, Phys. Rev. A. 7043612S.R. Clark and D. Jaksch, Phys. Rev. A 70, 043612 (2004) . C Kollath, U Schollwöck, W Zwerger, Phys. Rev. Lett. 95176401C. Kollath, U. Schollwöck, W. Zwerger, Phys. Rev. Lett. 95, 176401 (2005) . C Kollath, A Iucci, T Giamarchi, W Hofstetter, U Schollwöck, Phys. Rev. Lett. 9750402C. Kollath, A. Iucci, T. Giamarchi, W. Hofstetter, and U. Schollwöck, Phys. Rev. Lett. 97, 050402 (2006); . C Kollath, A Iucci, I P Mcculloch, T Giamarchi, Phys. Rev. A. 7441604C. Kol- lath, A. Iucci, I.P. McCulloch, and T. Giamarchi, Phys. Rev. A 74, 041604(R) (2006) . K Winkler, G Thalhammer, F Lang, R Grimm, J Denschlag, A J Daley, A Kantian, H P Büchler, P Zoller, Nature. 441853K. Winkler, G. Thalhammer, F. Lang, R. Grimm, J. Hecker Denschlag, A. J. Daley, A. Kantian, H. P.Büchler, P. Zoller, Nature 441, 853 (2006) . A J Daley, P Zoller, B Trauzettel, Phys. Rev. Lett. 100110404A. J. Daley, P. Zoller, and B. Trauzettel, Phys. Rev. Lett. 100, 110404 (2008) . A J Daley, J M Taylor, S Diehl, M Baranov, P Zoller, Phys. Rev. Lett. 10240402A. J. Daley, J. M. Taylor, S. Diehl, M. Baranov, and P. Zoller, Phys. Rev. Lett. 102, 040402 (2009) . K Mikelsons, J Freericks, H R Krishnamurthy, Phys. Rev. Lett. 109260402K. Mikelsons, J. Freericks, and H. R. Krishnamurthy, Phys. Rev. Lett. 109, 260402 (2012) . A Dirks, K Mikelsons, H R Krishnamurthy, J Freericks, arXiv:1304.7802A. Dirks, K. Mikelsons, H. R. Krishnamurthy, and J. Freericks, arXiv:1304.7802 . W Kohn, Phys. Rev. 115809W. Kohn, Phys. Rev. 115, 809 (1959) . Y Saad, M H Schultz, SIAM J. Sci. Stat. Comput. 7856Y. Saad and M.H. Schultz, SIAM J. Sci. Stat. Comput. 7, 856 (1986) . E Anderson, Z Bai, C Bischof, S Blackford, J Demmel, J Dongarra, J Croz, A Greenbaum, S Hammarling, A Mckenney, D Sorensen, LAPACK Users' Guide (Society for Industrial and Applied MathematicsPhiladelphiaE. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammar- ling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, Philadelphia 1999) . E Assmann, S Chiesa, G G Batrouni, H G Evertz, R T Scalettar, Phys. Rev. B. 8514509E. Assmann, S. Chiesa, G. G. Batrouni, H. G. Evertz, and R. T. Scalettar, Phys. Rev. B 85, 014509 (2012) Using MPI. W Gropp, E Lusk, A Skjellum, MPI Press9780262571326W. Gropp, E. Lusk, and A. Skjellum, Using MPI, MPI Press, 1999, ISBN 9780262571326 Chemical Applications of Group Theory. F A Cotton, Wiley and Sons Ltd997151267F.A. Cotton, Chemical Applications of Group Theory, Wiley and Sons Ltd., 1990, ISBN 997151267X

Multiple-input multiple-output (MIMO) systems are well suited for millimeter-wave (mmWave) wireless communications where large antenna arrays can be integrated in small form factors due to tiny wavelengths, thereby providing high array gains while supporting spatial multiplexing, beamforming, or antenna diversity. It has been shown that mmWave channels exhibit sparsity due to the limited number of dominant propagation paths, thus compressed sensing techniques can be leveraged to conduct channel estimation at mmWave frequencies. This paper presents a novel approach of constructing beamforming dictionary matrices for sparse channel estimation using the continuous basis pursuit (CBP) concept, and proposes two novel low-complexity algorithms to exploit channel sparsity for adaptively estimating multipath channel parameters in mmWave channels. We verify the performance of the proposed CBP-based beamforming dictionary and the two algorithms using a simulator built upon a three-dimensional mmWave statistical spatial channel model, NYUSIM, that is based on real-world propagation measurements. Simulation results show that the CBPbased dictionary offers substantially higher estimation accuracy and greater spectral efficiency than the grid-based counterpart introduced by previous researchers, and the algorithms proposed here render better performance but require less computational effort compared with existing algorithms. Approach in [13], N = 162, K = 3, SNR = 20 dB Algo 2, N = 162, K = 3, SNR = 20 dB Algo 3, N = 162, K = 3, SNR = 20 dB Approach in [13], N = 128, K = 4, SNR = 20 dB Algo 2, N = 128, K = 4, SNR = 20 dB Algo 3, N = 128, K = 4, SNR = 20 dB

10.1109/iccw.2017.7962632

1703.08227

3c54a71c4c1ff5fd1010df533738972bdee72961

Millimeter wave MIMO channel estimation based on adaptive compressed sensing May 2017 S Sun T S Rappaport Millimeter wave MIMO channel estimation based on adaptive compressed sensing 2017 IEEE International Conference on Communications Workshop (ICCW) ParisMay 2017 Multiple-input multiple-output (MIMO) systems are well suited for millimeter-wave (mmWave) wireless communications where large antenna arrays can be integrated in small form factors due to tiny wavelengths, thereby providing high array gains while supporting spatial multiplexing, beamforming, or antenna diversity. It has been shown that mmWave channels exhibit sparsity due to the limited number of dominant propagation paths, thus compressed sensing techniques can be leveraged to conduct channel estimation at mmWave frequencies. This paper presents a novel approach of constructing beamforming dictionary matrices for sparse channel estimation using the continuous basis pursuit (CBP) concept, and proposes two novel low-complexity algorithms to exploit channel sparsity for adaptively estimating multipath channel parameters in mmWave channels. We verify the performance of the proposed CBP-based beamforming dictionary and the two algorithms using a simulator built upon a three-dimensional mmWave statistical spatial channel model, NYUSIM, that is based on real-world propagation measurements. Simulation results show that the CBPbased dictionary offers substantially higher estimation accuracy and greater spectral efficiency than the grid-based counterpart introduced by previous researchers, and the algorithms proposed here render better performance but require less computational effort compared with existing algorithms. Approach in [13], N = 162, K = 3, SNR = 20 dB Algo 2, N = 162, K = 3, SNR = 20 dB Algo 3, N = 162, K = 3, SNR = 20 dB Approach in [13], N = 128, K = 4, SNR = 20 dB Algo 2, N = 128, K = 4, SNR = 20 dB Algo 3, N = 128, K = 4, SNR = 20 dB I. INTRODUCTION Multiple-input multiple-output (MIMO) is a key enabling technology for current and future wireless communication systems [1]- [5], as it offers great spectral efficiency and robustness due to many spatial degrees of freedom. Millimeter wave (mmWave) frequencies have been envisioned as a promising candidate for the fifth-generation (5G) wireless communications [6]. Two prominent advantages of the mmWave spectrum are the massive bandwidth available and the tiny wavelengths compared to conventional UHF (Ultra-High Frequency)/microwave bands, thus enabling dozens or even hundreds of antenna elements to be implemented at communication link ends within a reasonable physical form factor. This suggests that MIMO and mmWave technologies should be combined to provide higher data rates, higher spectrum efficiency, thus resulting in lower latency [7]. Channel state information (CSI) is needed to design precoding and combining procedures at transmitters and receivers, and it can be obtained through channel estimation. Conventional MIMO channel estimation methods may not be Sponsorship for this work was provided by the NYU WIRELESS Industrial Affiliates program and NSF research grants 1320472, 1302336, and 1555332. applicable in mmWave systems because of the substantially greater number of antennas, hence new channel estimation methods are required [8]. Due to the sparsity feature of mmWave channels observed in [6], [9], compressed sensing (CS) techniques [10] can be leveraged to effectively estimate mmWave channels [11], [12]. Adaptive CS, as a branch of CS, yields better performance at low signal-to-noise ratios (SNRs) compared to standard CS techniques, and low SNRs are typical for mmWave systems before implementing beamforming gain [13]. Adaptive CS algorithms for mmWave antenna arrays were derived in [13] to estimate channel parameters for both single-path and multipath scenarios, and it was shown that the proposed channel estimation approaches could achieve comparable precoding gains compared with exhaustive training algorithms. Additionally, Destino et al. proposed an adaptiveleast absolute shrinkage and selection operator (A-LASSO) algorithm to estimate sparse massive MIMO channels [14]. In [15], reweighted l 1 minimization was employed to realize sparsity enhancement based on basis pursuit denoising. The authors of [16] demonstrated a CS-based channel estimation algorithm for mmWave massive MIMO channels in ultradense networks, in conjunction with non-orthogonal pilots transmitted by small-cell base stations (BSs). In this paper, we propose an enhanced approach of creating the beamforming dictionary matrices for mmWave MIMO channel estimation in comparison with the one introduced in [13], based on adaptive CS concepts. The main novelty of the proposed method here is the adoption of the continuous basis pursuit (CBP) method instead of the conventional gridbased approach to build beamforming dictionary matrices. This paper shows that the proposed dictionary can significantly improve the estimation accuracy, i.e., reduce the probability of estimation error, of angles of departure (AoDs) and angles of arrival (AoAs). Furthermore, built on the CBP-based dictionary, two new multipath channel estimation algorithms are proposed that have lower computational complexity compared to the one introduced in [13], while offering better estimation accuracy for various signal sparsities. A three-dimensional (3D) statistical spatial channel model (SSCM)-based simulator, NYUSIM [9], [17], developed for mmWave systems from extensive real-world propagation measurements, was used in the simulation to investigate the performance of the proposed algorithms. The following notations are used throughout this paper. The boldface capital letter X and the boldface small letter x denote a matrix and a vector, respectively; A small italic letter x denotes a scalar; C represents the set of complex numbers; N denotes the set of natural numbers; ||X|| F denotes the Frobenius norm of X; The conjugate, transpose, Hermitian, and Moore-Penrose pseudo inverse of X are represented by X * , X T , X H , and X † , respectively; Tr(X) and vec(X) indicate the trace and vectorization of X, respectively; The Hadamard, Kronecker and Khatri-Rao products between two matrices are denoted by •, ⊗, and * , respectively. II. SYSTEM MODEL Let us consider a BS equipped with N BS antennas and N RF RF chains communicating with a mobile station (MS) with N MS antennas and N RF RF chains, where N RF ≤ N MS ≤ N BS . We intentionally do not consider system interference issues, such as co-channel interference from other BSs and MSs, because of the limited interference found in directional mmWave channels [8], and also, the focus of this paper is to quantify and compare the performance of channel estimation methods in a single link. System aspects are ongoing research topics. In the channel estimation stage, the BS employs M BS beamforming vectors to transmit M BS symbols, while the MS utilizes M M S combining vectors to combine the received signal. The BS is assumed to implement analog/digital hybrid precoding with a precoding matrix F = F RF F BB , where F RF ∈ C NBS×NRF and F BB ∈ C NRF×MBS denote the RF and baseband precoding matrices, respectively. Similarly, at the MS, the combiner W also consists of RF and baseband combiners represented by W RF ∈ C NMS×NRF and W BB ∈ C NRF×MMS , respectively. The received signal at the MS is given by Y = W H HFS + Q(1) where H ∈ C NMS×NBS denotes the channel matrix, S ∈ C MBS×MBS is a diagonal matrix containing the M BS transmitted symbols, and Q ∈ C MMS×MBS represents the complex Gaussian noise. The design of analog/digital hybrid precoding and combining matrices have been extensively investigated [18], [19], and we defer this topic to future work and focus on channel estimation in this paper. Additionally, although CSI can also be obtained by uplink training and channel reciprocity in time-division duplexing (TDD) systems, we focus on the downlink training in this paper since channel reciprocity usually does not hold for frequency-division duplexing (FDD) systems, and even in TDD systems if there exist non-linear devices that are not self-calibrated so as to incur non-reciprocal effects. The mmWave channel can be approximated by a geometric channel model with L scatterers due to its limited scattering feature [6], [20], [21], and the channel matrix can be written as H = N BS N MS L L l=1 α l a MS (ϕ l , ϑ l )a H BS (φ l , θ l )(2) where α l is the complex gain of the l th path between the BS and MS including the path loss, where a path refers to a cluster of multipath components traveling closely in time and/or spatial domains, φ l , ϕ l ∈ [0, 2π) are the azimuth AoD and AoA of the l th path, θ l , ϑ l ∈ [−π/2, π/2] are the elevation AoD and AoA. a BS (φ l , θ l ) and a MS (ϕ l , ϑ l ) are the antenna array response vectors at the BS and MS, respectively. The NYUSIM simulator produces a wide range of sample ensembles for (2) and incorporates multiple antenna elements and physical arrays including uniform linear arrays (ULAs) [17]. Using a ULA, the array response vector can be expressed as (take the BS for example) a BS (φ l ) = 1 √ N BS [1, e j 2π λ dcos(φ l ) , · · · , e j(NBS−1) 2π λ dcos(φ l ) ](3) where the incident angle is defined as 0 if the beam is parallel with the array direction, λ denotes the carrier wavelength, and d is the spacing between adjacent antenna elements. III. FORMULATION OF THE MMWAVE CHANNEL ESTIMATION PROBLEM Considering the mmWave channel matrix given by (2), estimating the channel is equivalent to estimating the AoD, AoA, and path gain of each path, and training precoders and combiners are necessary to conduct the channel estimation. The mmWave channel estimation can be formulated as a sparse problem due to its limited dominant paths, e.g., on average 1 to 6 time clusters and 2 to 3 spatial lobes were found from real-world measurements using a 10 dB down threshold, as presented in [9]. Therefore, some insights can be extracted from the CS theory. Assuming all transmitted symbols are equal for the estimation phase, i.e., S = √ P I MBS (P is the average power per transmission) and by vectorizing the received signal Y in (1) to y, we can approximate the received signal with a sparse formulation as follows [13] y = √ P vec(W H HF) + vec(Q) = √ P (F T ⊗ W H )vec(H) + n Q = √ P (F T ⊗ W H )(A * BS,D * A MS,D )z + n Q = √ P (F T A * BS,D ⊗ W H A MS,D )z + n Q = √ P F T A * BS,D z BS ⊗ W H A MS,D z MS + n Q(4) where A BS,D and A MS,D denote the beamforming dictionary matrices at the BS and MS, respectively. z BS ∈ C N ×1 and z MS ∈ C N ×1 are two sparse vectors that have non-zero elements in the locations associated with the dominant paths, with N denoting the number of measurements in the channel estimation stage, and z = z BS * z MS . A beamforming dictionary based on angle quantization was proposed in [13], where the AoDs and AoAs were assumed to be taken from a uniform grid of N points with N L where L denotes the number of paths, and the resulting dictionary matrix is expressed as (take the BS side for example, the MS dictionary matrix can be derived similarly) A BS,D = [a BS (φ 1 ), · · · , a BS (φ N )](5) where a BS (φ n ) (n = 1, ..., N ) denotes the BS array response vector for the grid pointφ n . Given that the true continuous-domain AoDs and AoAs may lie off the center of the grid bins, the grid representation in this case will destroy the sparsity of the signal and result in the so-called basis mismatch [22]. This can be mitigated to a certain extent by finer discretization of the grid, but that may lead to higher computation time and higher mutual coherence of the sensing matrix, thus becoming less effective for sparse signal recovery [10]. There are several approaches to mitigate the basis mismatch problem. One promising approach, named continuous basis pursuit (CBP), is proposed in [22], where one type of CBP is implemented with first-order Taylor interpolator, which will be demonstrated shortly. Since the antenna array factor a(φ) is a continuous and smooth function of φ, it can be approximated by linearly combining a(φ k ) and the derivative of a(φ) at the point φ k via a first-order Taylor expansion: a(φ) = a(φ k ) + (φ − φ k ) ∂a(φ) ∂φ φ k + O((φ − φ k ) 2 ) (6) where φ k = 2π(k − 1)/N is the grid-point with minimal distance from φ. This motivates a dictionary consisting of the original discretized array factors a(φ) and its derivatives ∂a(φ) ∂φ , i.e., a(φ) and ∂a(φ) ∂φ can be regarded as two sets of basis for the dictionary. Therefore, the entire basis for the proposed dictionary matrix can be formulated as B BS = [a BS (φ 1 ), · · · , a BS (φ N ), b BS (φ 1 ), · · · , b BS (φ N )] (7) where b BS (φ n ) = ∂aBS(φ) ∂φ φn , and the corresponding interpolator is given by t BS = [1, · · · , 1 N , ∆φ, · · · , ∆φ N ] T(8) where ∆φ denotes the angle offset from the angles on the grid, and |∆φ| ≤ π N . The proposed dictionary is hence written as A BS,D = B BS t BS =[a BS (φ 1 ), · · · , a BS (φ N ), ∆φb BS (φ 1 ), · · · , ∆φb BS (φ N )](9) IV. MULTI-RESOLUTION HIERARCHICAL CODEBOOK The proposed hierarchical beamforming codebook is composed of S levels, where each level contains beamforming vectors with a certain beamwidth that covers certain angular regions. Due to the symmetry of the antenna pattern of a ULA, if a beam covers an azimuth angle range of [φ a , φ b ], then it also covers 2π − [φ a , φ b ]. In each codebook level s, the beamforming vectors are divided into K s−1 subsets, each of which contains K beamforming vectors. Each of these K beamforming vectors is designed such that it has an almost equal projection on the vectors a BS (φ), whereφ denotes the angle range covered by this beamforming vector, and zero projection on the array response vectors corresponding to other angles. Note that there is no strict constraint on the number of sectors K at each stage, yet considering practical anglesearching time, K = 3 or 4 is a reasonable choice. Once the value of K is defined, the total number of estimation measurements N is 2K S . The value of N should be minimized while guaranteeing the successful estimation of angles, thus S should be neither too large nor too small. Through simulations, it is found that S = 3, K = 4 (N = 128) and S = 4, K = 3 (N = 162) are two sensible combinations. In each codebook level s and subset k, the m th column of the beamforming vector [F (s,k) ] :,m , m = 1, ..., K in the codebook F is designed such that [F (s,k) ] H :,m a BS (φ u ) =    C, forφ u ∈ ⊕ BS s,k,m 0, otherwise [F (s,k) ] H :,m b BS (φ u ) = 0, ∀φ u (10) with ⊕ BS s,k,m = π K s K(k BS s − 1) + m BS − 1 , π K s (K(k BS s − 1) + m BS ) ∪ 2π − π K s (K(k BS s − 1) + m BS ), 2π − π K s K(k BS s − 1) + m BS − 1(11) where C is a constant such that each F (s,k) has a Frobenius norm of K. The fact that the product of [F (s,k) ] H :,m and b BS (φ u ) is zero in (10) can be derived from (6) to (9). The matrix F (s,k) hence equals the product of the pseudo-inverse ofà BS,D and the (k × K − (K − 1)) th to (k × K) th columns of the angle coverage matrix G (s) with its m th column given by (12): where C's are in the locations ⊕ BS s,k,m . The combining matrix W (s,k) in the codebook W at the receiver can be designed in a similar manner. It is noteworthy that the difference between the angle coverage matrix G (s) in [13] and the one proposed here is that the m th column of the former contains only the first N rows without the last N 0's in (12), i.e., the former did not force [F (s,k) ] H :,m b BS (φ u ) to be zero, hence failing to alleviate the leakage incurred by angle quantization. Fig. 1 illustrates the beam patterns of the beamforming vectors in the first codebook level of an example hierarchical codebook introduced in [13] and the hierarchical codebook proposed in this paper with N = 162 and K = 3. Comparing the two beam patterns, we can see that the codebook generated using the CBP-based dictionaryà BS,D in (9) produces a smoother pattern contour in contrast to that yielded by the codebook introduced in [13], namely, the beams associated withà BS,D are able to cover the intended angle ranges more evenly. Due to the more uniform projection on the targeted angle region, the beamforming vectors generated usingà BS,D can mitigate the leakage induced by angle quantization, thus improving the angle estimation accuracy, as will be shown later. V. ADAPTIVE ESTIMATION ALGORITHMS FOR MMWAVE MIMO CHANNELS For single-path channels, there is only one non-zero element in the vector z in (4). To effectively estimate the location of this non-zero element, and consequently the corresponding AoD, AoA, and path gain, the following algorithm, which is an improved version of Algorithm 1 in [13], is used in conjunction with the innovative CBP-based dictionary matrices. Algorithm 1 operates as follows. In the initial stage, the BS uses the training precoding vectors of the first level of the codebook F. For each of those vectors, the MS uses the measurement vectors of the first level of W to combine the received signal. After the precoding-measurement steps of this stage, the MS compares the power of the received signals to determine the one with the maximum received power. As each one of the precoding/measurement vectors is associated with a certain range of the quantized AoA/AoD, the operation of the first stage divides the entire angle range [0, 2π) into K partitions, and compares the power of the sum of each of them. Hence, the selection of the strongest received signal implies the selection of the range of the quantized AoA/AoD that is highly likely to contain the single path of the channel. The output of the maximum power is then used to determine the subsets of the beamforming vectors of level s + 1 (1 ≤ s ≤ S − 1) of F and W to be used in the next stage. Since N must be even multiples of K in order to construct the precoding and measurement codebooks, there are two possible ranges of AoD/AoA selected out after Step 15 of Algorithm 1, which are denoted asφ can andφ can . Step 16 is aimed at "filtering" out the AoD/AoA from these two ranges. The MS then feeds back the selected subset of the BS precoders to the BS to use it in the next stage, which needs only log 2 K bits. Based on Algorithm 1, two low-complexity algorithms for estimating multipath channels are established, as explained below. In Algorithm 2, I BS (i,s) and I MS (i,s) contain the precoding and measurement matrix indexes of the i th path in the s th stage, respectively. Algorithm 2 operates as follows: A procedure Output:φ,φ,α similar to Algorithm 1 is utilized to detect the first strongest path. The indexes of the beamforming matrices corresponding to the previous detected l (1 ≤ l ≤ L − 1) paths are stored and used in later iterations. Note that in each stage s from the second iteration on, the contribution of the paths that have already been estimated in previous iterations are projected out one path by one path before determining the new promising AoD/AoA ranges. In the next stage s+1, two AoD/AoA ranges are selected for further refinement, i.e., the one selected at stage s of this iteration, and the one selected by the preceding path at stage s + 1 of the previous iteration. The algorithm makes L outer iterations to estimate L paths. Thanks to the sparse nature of mmWave channels, the number of dominant paths is usually limited, which means the total number of precoding-measurement steps will not be dramatically larger compared to the single-path case. Y (s) = √ P s [W (s,k MS s ) ] H H[F (s,k BS s ) ] + Q 12: {m * BS , m * MS } = argmax ∀mBS,mMS=1,...,K [Y (s) • Y *(k BS s+1 = K(k BS s − 1) + m * BS , k MS s+1 = K(k MS s − 1) + m * Algorithm 3 is similar to Algorithm 2, but with an even lower complexity. The major difference between Algorithm 3 and Algorithm 2 stems from the way of projecting out Step 11 in Algorithm 1 4: For the s th stage in the i th (2 ≤ i ≤ L) iteration, project out previous path contributions one path by one path Y (s) = √ P s [W (s,k MS s ) ] H H[F (s,k BS s ) ] + Q y (s) = vec(Y (s) ) V (i,s) = F TY (s) = √ P s [W (s,k MS s ) ] H H[F (s,k BS s ) ] + Q y (s) = vec(Y (s) ) V (i,s) = [W (s,k MS s ) ] H MS H BS [F (s,k BS s ) ] % Calculating the contribution of previous paths in the form of Eq. (1) v (i,s) = vec(V (i,s) ) y (s) = y (s) − v (i,s) v † (i,s) y (s) 5: Convert y (s) to the matrix form Y (s) 6: Repeat Algorithm 1 from Step 12 to obtain the AoD, AoA, and path gain for the i th strongest path until all the L paths are estimated Output: AoDs, AoAs, and path gains for the L dominant paths previous path contributions: Algorithm 3 does not require storing the beamforming matrix indexes, but instead, it utilizes the antenna array response vectors associated with the estimated AoDs/AoAs to subtract out the contributions of previously detected paths simultaneously. Therefore, compared with Algorithm 2, Algorithm 3 results in less computation and storage cost, and a higher estimation speed (i.e., lower latency). When compared with the multipath channel estimation presented in [13], the most prominent advantages of both Algorithm 2 and Algorithm 3 are that they do not require the re-design of multi-resolution beamforming codebooks for each stage when the number of dominant paths vary, and only a single path is selected in each stage instead of L paths in [13], thus substantially reducing the calculation and memory overhead. VI. SIMULATION RESULTS In this section, the performance of the proposed CBP-based dictionary and Algorithms 1, 2, and 3 are evaluated in terms of average probability of estimation error of AoDs and AoAs, and spectral efficiency, via numerical Monte Carlo simulations. The channel matrix takes the form of (2), where the path powers, phases, AoDs, and AoAs are generated using the 5G open-source wideband simulator, NYUSIM, as demonstrated in [9], [17]. The channel model in NYUSIM utilizes the time-cluster-spatial-lobe (TCSL) concept, where for an RF bandwidth of 800 MHz, the number of TCs varies from 1 to 6 in a uniform manner, the number of subpaths per TC is uniformly distributed between 1 and 30, and the number of SLs follows the Poisson distribution with an upper bound of 5. The NYUSIM channel model is applicable to arbitrary frequencies from 500 MHz to 100 GHz, RF bandwidths between 0 and 800 MHz, various scenarios (urban microcell (UMi), urban macrocell (UMa), and rural macrocell (RMa) [23]), and a vast range of antenna beamwidths [9], [17]. ULAs are assumed at both the BS and MS with 64 and 32 antenna elements, respectively. All simulation results are averaged over 10,000 random channel realizations, with a carrier frequency of 28 GHz and an RF bandwidth of 800 MHz. In calculating spectral efficiency, eigen-beamforming is assumed at both the transmitter (with equal power allocation) and receiver. Other beamforming techniques can also be employed, and we found the performance of the beamforming dictionaries and algorithms are similar. The simulated probabilities of estimation errors of AoDs and AoAs as a function of the average receive SNR, using Algorithm 1 and both grid-based and CBP-based dictionaries for single-path channels, are depicted in Fig. 2 for the cases of N = 162, K = 3, and N = 128, K = 4, which are found to yield the best performance via numerous trials. As shown by Fig. 2, the CBP-based approach renders much smaller estimation errors, by up to two orders of magnitude. For the two cases considered in Fig. 2, the grid-based method generates huge estimation error probability that is over 80% even at an SNR of 20 dB; on the other hand, the estimation error probability of the CBP-based counterpart decreases rapidly with SNR, and is less than 0.5% for N = 128, K = 4 and a 20 dB SNR. These results imply that the CBP-based approach is able to provide much better channel estimation accuracy with a small number of measurements compared to the conventional grid-based fashion, hence is worth using in mmWave MIMO systems for sparse channel estimation and signal recovery. To explicitly show the effect of estimation error on channel spectral efficiency using different beamforming dictionaries, we plot and compare the achievable spectral efficiency as a function of the average receive SNR for both the gridbased and CBP-based dictionaries for single-path channels, as well as the spectral efficiency with perfect CSI at the transmitter, for the case of N = 162, K = 3, and N = 128, K = 4, as described in Fig. 3. It is evident from Fig. 3 that for both cases considered, the CBP-based dictionary yields much higher spectral efficiency, by about 2.7 bits/s/Hz to 13 bits/s/Hz, compared with the grid-based one over the entire SNR range of -20 dB to 20 dB. Furthermore, the CBP-based method achieves near-optimal performance over the SNRs spanning from 0 dB to 20 dB, with a gap of less than 0.7 bits/s/Hz. Fig. 4 illustrates the average probability of error in estimating AoDs/AoAs for multipath channels with N = 162, K = 3, and N = 128, K = 4 for an average receive SNR of 20 dB, using proposed Algorithms 2 and 3 for two to six dominant paths, as well as Algorithm 2 in [13]. For the approach in [13], since all L paths have to be estimated simultaneously in a multipath channel, it does not work for L < K, thus no results are available for L = 2 when K = 3 or 4. The SNR denotes the ratio of the total received power from all paths to the noise power. As shown in Fig. 4, both Algorithm 2 (Algo 2) and Algorithm 3 (Algo 3) produce lower estimation errors than the approach in [13] in both multipath-channel cases; for the case of N = 128, K = 4, Algorithm 3 yields the lowest estimation error, i.e., highest accuracy, and meanwhile enjoys the lowest computation expense among the three algorithms. In addition, the estimation error tends to increase more slowly and converge to a certain value as the number of dominant paths increase for all of the three algorithms. The spectral efficiency performance of the three algorithms above, with N = 162, K = 3, and L = 3, is displayed in Fig. 5, which reveals the superiority of Algorithm 3 pertaining to spectral efficiency, followed by Algorithm 2, compared with the approach in [13]. For instance, at an SNR of 10 dB, Algorithms 2 and 3 yield around 5 and 8 more bits/s/Hz than the approach in [13], respectively, and the discrepancies expand as the SNR ascends. The proposed algorithms work well for single-path channels, and significantly outperforms the approach in [13] method for multipath channels, although there is still a noticeable spectral efficiency gap compared to the perfect CSI case, due to the non-negligible angle estimation errors shown in Fig. 4. Further work is needed to improve Algo . Average spectral efficiency for multipath channels with the CBP-based dictionary using the approach in [13], and Algorithms 2 and 3 proposed in this paper, with N = 162, K = 3, and L = 3. 2 and Algo 3 to more effectively estimate multipath channels. VII. CONCLUSION Based on the concept of adaptive compressed sensing and by exploiting the sparsity of mmWave channels, in this paper, we presented an innovative approach for designing the precoding/measurement dictionary matrices, and two new lowcomplexity algorithms for estimating multipath channels. In contrast to the conventional grid-based method, the principle of CBP was leveraged in devising the beamforming dictionary matrices, which had lower mutual coherence due to the first-order Taylor interpolation, and was shown to be more beneficial for sparse signal reconstruction. Simulations were performed based on the open-source 5G channel simulator NYUSIM for broadband mmWave systems. Results show that the CBP-based dictionary renders up to over two orders of magnitude higher estimation accuracy (i.e., lower probability of estimation error) of AoDs and AoAs, and more than 12 bits/s/Hz higher spectral efficiency, with a small number of estimation measurements for single-path channels, as opposed to the grid-based approach, as shown in Figs. 2 and 3. Moreover, the newly proposed two algorithms, Algorithm 2 and Algorithm 3, can offer better estimation and spectral efficiency performance with lower computational complexity and time consumption for multipath channels, when compared with existing algorithms, as shown in Figs. 4 and 5. Interesting extensions to this work will be to improve the multipath estimation algorithms to make them more effective, and to extend the multipath estimation algorithms to the case where the number of dominant paths is unknown, as well as to implement the proposed dictionary matrices and algorithms to other types of antenna arrays such as 2D arrays. Fig. 1 . 1Beam patterns of the beamforming vectors in the first codebook level of an example hierarchical codebook using the grid-based and CBP-based dictionaries with N = 162, K = 3. m BS ≤ K 9: end for s ≤ S 10: for s ≤ S do 11: (φ,φ) = argmax Z • Z * % Finding the optimal AoD and AoA that maximize the Hadamard product of the received signal matrix α = Z (φ,φ) • Z * (φ,φ) /(N BS * N MS ) % Estimated path gain magnitude associated with the estimated AoD and AoA Algorithm 2 2Adaptive Estimation Algorithm for Multipath mmWave MIMO channels Input: K, S, codebooks F and W, N = 2K S 1: Initialization: I BS (:,1) = [1, ..., 1] T , I MS (:,1) = [1, ..., 1] T , where I BS ∈ N L×S , I MS ∈ N L×S 2: Use Algorithm 1 to detect the AoD, AoA, and path gain for the first strongest path 3: Repeat Algorithm 1 for the l th (2 ≤ l ≤ L) path until s) ) [à MS,D ] :,I MS (i,s) % Calculating the contribution of previous paths in the form of Eq. (4) y (s) = y (s) − V (i,s) V † (i,s) y (s) 5: Convert y (s) to the matrix form Y (s) 6: Repeat Algorithm 1 from Step 12 to obtain the AoD, AoA, and path gain for the i th strongest path until all the L paths are estimated Output: AoDs, AoAs, and path gains for the L dominant paths Algorithm 3 Adaptive Estimation Algorithm for Multipath mmWave MIMO channels Input: K, S, codebooks F and W, N = 2K S 1: Initialization: I BS (:,1) = [1, ..., 1] T , I MS (:,1) = [1, ..., 1] T , where I BS ∈ N L×S , I MS ∈ N L×S 2: Use Algorithm 1 to detect the AoD, AoA, and path gain for the first strongest path 3: Repeat Algorithm 1 for the l th (2 ≤ l ≤ L) path until Step 11 in Algorithm 1 4: For the s th stage in the i th (2 ≤ i ≤ L) iteration, project out previous path contributions simultaneouslŷ A BS = [a BS (φ)], MS = [a MS (φ)] %φ andφ are the AoDs and AoAs of all the previously detected paths, respectively Fig. 2 . 2Average probability of error in estimating AoD/AoA for single-path channels, using both the grid-based dictionary and CBP-based dictionary. Fig. 3 . 3Average spectral efficiency for single-path channels for the cases of perfect CSI, grid-based dictionary and CBP-based dictionary. Fig. 4 . 4Average probability of error in estimating AoD/AoA for multipath channels using the CBP-based dictionary. Fig. 5. Average spectral efficiency for multipath channels with the CBP-based dictionary using the approach in [13], and Algorithms 2 and 3 proposed in this paper, with N = 162, K = 3, and L = 3. MS , 15: end for s ≤ S 16: A can,BS = [a BS (φ can )] % Antenna array matrix for the candidate AoDs A can,MS = [a MS (φ can )] % Antenna array matrix for the candidate AoAs Z = A H can,MS HA can,BS + Q % Received signal matrix corresponding to the candidate AoDs and AoAs What will 5G be?. J G Andrews, IEEE Journal on Selected Areas in Communications. 326J. G. Andrews et al., "What will 5G be?" IEEE Journal on Selected Areas in Communications, vol. 32, no. 6, pp. 1065-1082, Jun. 2014. 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We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.This implies again by [40, Lemma 3.1] that ϕ (n,m) (z) ≤ 1. Since this is valid for an arbitrary element of K n,m , the proof is concluded.Remark 3.9. The proof of Theorem 3.8 shows something more precise: if K is a rectangular convex set, then for every n, m ∈ N, K n,m is equal to the closed rectangular convex hull of K r,s for r ≤ n and s ≤ m.The following is an immediate corollary of the rectangular Krein-Milman theorem as formulated in Remark 3.9.Corollary 3.10. Suppose that X is an operator space, and K = CBall(X ′ ) is the corresponding compact rectangular matrix convex set. If n, m ∈ N and x ∈ M n,m (X), then x is the supremum of ϕ (n,m) (x) where ϕ ranges among all the rectangular extreme points of K r,s for r ≤ n and s ≤ m.

10.14760/owp-2016-24

1610.05828

e569ebecbaa9be7e1c0e6a4f4f93430d6f425985

BOUNDARY REPRESENTATIONS OF OPERATOR SPACES, AND COMPACT RECTANGULAR MATRIX CONVEX SETS 10 Dec 2017 Adam H Fuller Michael Hartz Martino Lupini BOUNDARY REPRESENTATIONS OF OPERATOR SPACES, AND COMPACT RECTANGULAR MATRIX CONVEX SETS 10 Dec 2017arXiv:1610.05828v2 [math.OA] We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.This implies again by [40, Lemma 3.1] that ϕ (n,m) (z) ≤ 1. Since this is valid for an arbitrary element of K n,m , the proof is concluded.Remark 3.9. The proof of Theorem 3.8 shows something more precise: if K is a rectangular convex set, then for every n, m ∈ N, K n,m is equal to the closed rectangular convex hull of K r,s for r ≤ n and s ≤ m.The following is an immediate corollary of the rectangular Krein-Milman theorem as formulated in Remark 3.9.Corollary 3.10. Suppose that X is an operator space, and K = CBall(X ′ ) is the corresponding compact rectangular matrix convex set. If n, m ∈ N and x ∈ M n,m (X), then x is the supremum of ϕ (n,m) (x) where ϕ ranges among all the rectangular extreme points of K r,s for r ≤ n and s ≤ m. Introduction It is hard to overstate the importance of the theory of convexity in analysis. This is all the more true in the study of operator systems, which can be seen as the noncommutative analog of compact convex sets. Indeed, given any operator system S, the space of matrix-valued unital completely positive maps on S is endowed with a natural notion of convex combinations with matrix coefficients (matrix convex combination), and a topology which is compact as long as one restricts the target to a fixed matrix algebra. The compact matrix convex sets that arise in this way have been initially studied by Effros and Wittstock in [49,20]. The program of developing the theory of compact matrix convex sets as the noncommutative analog of compact convex sets has been proposed by Effros in [20]. This program has been pursued in [22,47], where matricial generalizations of the classical Krein-Milman and bipolar theorems are proved. Compact matrix convex sets and the corresponding notion of matrix extreme points have been subsequently studied in a number of papers. This line of research has recently found outstanding applications. These include the matrix convexity proof of Arveson's conjecture on boundary representations due to Davidson and Kennedy [17] building on previous work of Farenick [24,25], and the work of Helton, Klep, and McCullogh in free real algebraic geometry [31,32]. The main goal of this paper is to provide the nonselfadjoint analog of the results above in the setting of operator spaces. Precisely, we introduce the notion of compact rectangular matrix convex set, which is the natural analog of the notion of compact matrix convex set where convex combinations with rectangular matrices are considered. We then prove generalizations of the Krein-Milman and bipolar theorems in the setting of compact rectangular matrix convex sets. We then deduce that compact rectangular matrix convex sets are in canonical functorial one-to-one correspondence with operator spaces. It follows from this that any operator space is completely normed by the matrix-valued completely contractive maps that are rectangular matrix extreme points. We also introduce the notion of boundary representation for operator spaces. The natural operator space analog of Arveson's conjecture is then established: any operator space is completely normed by its boundary representations. This gives an explicit description of the triple envelope of an operator space in terms of boundary representations. We also obtain in this setting an analog of Arveson's boundary theorem. As an application, we compute boundary representations for multiplier spaces associated with pairs of reproducing kernel Hilbert spaces. The results of this paper can be seen as the beginning of a convexity theory approach to the study of operator spaces. Convexity theory has played a crucial role in the setting of Banach spaces, such as in the groundbreaking work of Alfsen and Effros on M-ideals in Banach spaces [2,3] or the work of Lazar and Lindenstrauss on L 1predual spaces [36,37]. This work can be seen as a first step towards establishing noncommutative analogs of the results of Alfsen-Effros and Lazar-Lindenstrauss mentioned above. We will see below that the crucial notion of collinearity for bounded linear functionals on a Banach space, which is of key importance for the work of Alfsen and Effros on facial cones and M-ideals, has a natural interpretation in the setting of rectangular matrix convexity. The results of the present paper have already found application in [38]. The fact established here that an operator space is completely normed by its matrix-valued completely contractive maps that are rectangular matrix extreme is used there to prove that the noncommutative Gurarij space introduced by Oikhberg in [39] is the unique separable nuclear operator space with the property that the canonical map from the maximal TRO to the triple envelope is injective. This paper is divided into three sections, besides the introduction. In Section 2 we introduce the notion of boundary representation for operator spaces, and prove that any operator space is completely normed by its boundary representations. The boundary theorem for operator spaces and applications to multiplier spaces for pairs of reproducing kernel Hilbert spaces are also considered in this section. In Section 3 we introduce the notion of compact rectangular matrix convex set and rectangular matrix extreme point. We prove the Krein-Milman and bipolar theorem for compact rectangular matrix convex sets, and deduce the correspondence between compact rectangular matrix convex sets and operator spaces. Finally in Section 4 we consider the notion of matrix-gauged space. Such a concept has recently been introduced by Russell in [42] in order to provide an abstract characterization of selfadjoint subspaces of C*-algebras (selfadjoint operator spaces), and to capture the injectivity property of B(H) in such a category. We prove in Section 4 that the construction of the injective envelope and the C*-envelope of an operator system can be naturally generalized to the setting of matrix-gauged spaces. A geometric approach to the study of matrix-gauged spaces and selfadjoint operator spaces is also possible, as we show that matrix-gauged spaces are in functorial one-to-one correspondence with compact matrix convex sets with a distinguished matrix extreme point. Boundary representations and the Shilov boundary of an operator space 2.1. Notation and preliminaries. Recall that a ternary ring of operators (TRO) T is a subspace of the C*-algebra B(H) of bounded linear operators on a Hilbert space H that is closed under the triple product (x, y, z) → xy * z. An important example of a TRO is the space B(H, K), where H, K are Hilbert spaces. A TRO has a canonical operator space structure coming from the inclusion T ⊂ B(H), which does not depend on the concrete representation of T as a ternary ring of operators on H. A triple morphism between TROs is a linear map that preserves the triple product. Any TRO can be seen as the 1-2 corner of a canonical C*algebra L (T ) called its linking algebra. A triple morphism between TROs can be seen as the 1-2 corner of a *-homomorphism between the corresponding linking algebras [30]; see also [10,Corollary 8.3.5]. The notions of (nondegenerate, irreducible, faithful) representations admit natural generalizations from C*algebras to TROs. A representation of a TRO T is a triple morphism θ : T → B(H, K) for some Hilbert spaces H, K. A linear map ψ : T → B(H, K) is nondegenerate if, whenever p, q are projections in B(H) and B(K), respectively, such that qθ(x) = θ(x)p = 0 for every x ∈ T , one has p = 0 and q = 0. Similarly a representation θ of T is irreducible if , whenever p, q are projections in B(H) and B(K), respectively, such that qθ(x)p + (1 − q) θ(x) (1 − p) = θ(x) for every x ∈ T (equivalently, qθ(x) = θ(x)p for every x ∈ T ), one has p = 1 and q = 1, or p = 0 and q = 0. Finally, θ is called faithful if it is injective or, equivalently, completely isometric. Various characterizations of nondegenerate and irreducible representations are obtained in [11, Lemma 3.1.4 and Lemma 3.1.5]. A concrete TRO T ⊂ B(H, K) is said to act nondegenerately or irreducibly if the corresponding inclusion representation is nondegenerate or irreducible, respectively. In the following we will use frequently without mention the Haagerup-Paulsen-Wittstock extension theorem [40,Theorem 8.2], asserting that, if H, K are Hilbert spaces, then the space B(H, K) of bounded linear operators from H to K is injective in the category of operator spaces and completely contractive maps. We will also often use the canonical way, due to Paulsen, to assign to an operator space X ⊂ B(H, K) an operator system S(X) ⊂ B(K ⊕ H). This operator system, called the Paulsen system, is defined to be the space of operators λI K x y * µI H : x, y ∈ X, λ, µ ∈ C where I H and I K denote the identity operator on H and K, respectively. Any completely contractive map φ : X → Y between operator spaces extends canonically to a unital completely positive map S(φ) : S(X) → S(Y ) defined by λI K x y * µI H → λI K φ(x) φ(y) * µI H ,ψ : X → B( H, K) is a dilation of φ if there exist linear isometries v : H → H and w : K → K such that w * ψ(x)v = φ(x) for every x ∈ X. The same proof as [40,Theorem 8.4] gives the following. In order to prove Proposition 2.1 one can proceed as in [40,Theorem 8.4], by replacing M 2 (A) for a given C*-algebra A with the C*-algebra generated by S (T ) inside B(K 0 ⊕ H 0 ). It is clear that in Proposition 2.1 one can choose θ and the linear isometries v : H → H and w : K → K in such a way that K is the linear span of θ(T )θ(T ) * wK ∪ θ (T ) vH, and H is the linear span of θ(T ) * θ(T )vH ∪ θ (T ) * wK. In this case, we call such a dilation θ a minimal dilation of φ. In the sequel, it will often be convenient to identify H with a subspace of H and K with a subspace of K. Definition 2.2. Let φ : X → B(H, K) be a rectangular operator state and let ψ : X → B( H, K) be a dilation of φ. We can assume that H ⊂ H and K ⊂ K. Let p be the orthogonal projection from H onto H and let q be the orthogonal projection from K onto K. The dilation ψ is trivial if ψ (x) = qψ (x) p + (1 − q)ψ (x) (1 − p) for every x ∈ X. The operator state φ on an operator space X is maximal if it has no nontrivial dilation. It is clear that, when X is an operator system, H = K, q = p, and ψ is a unital completely positive map, the notion of trivial dilation as above recovers the usual notion of trivial dilation. Definition 2.3. Suppose that X is a subspace of a TRO T such that T is generated as a TRO by X. The operator state φ on X has the unique extension property if any rectangular operator state φ of T whose restriction to X coincides with φ is automatically a triple morphism. We now observe, that a rectangular operator state on an operator space is maximal if and only if it has the unique extension property. The analogous fact for unital completely positive maps on operator systems is well known; see [6]. Lemma 2.4. For a TRO T , the set of positive elements of the C * -algebra T T * is the closed convex cone generated by {xx * : x ∈ T }. Proof. Let C ⊂ T T * denote the closed convex cone generated by {xx * : x ∈ T }. It is clear that C is contained in the set of positive elements of T T * . Conversely, suppose that a ∈ T T * is positive. By the remarks at the beginning of Section 2.2 of [23], the C * -algebra T T * admits a contractive approximate identity (e i ) of elements of the form j x j x * j where x j ∈ T . For such an element of T T * one has that a 1/2   j x j x * j   a 1/2 = j (a 1/2 x j )(a 1/2 x j ) * ∈ C, as T is a left T T * -module. Thus, a 1/2 e i a 1/2 ∈ C for every i. It follows that a = lim i a 1/2 e i a 1/2 ∈ C. Lemma 2.5. Let T be a TRO, let ψ : T → B( H, K) be a completely contractive linear map and suppose that H ⊂ H and K ⊂ K are closed subspaces with corresponding orthogonal projections p ∈ B( H) and q ∈ B( K). If the map θ : T → B(H, K), x → qψ(x)p, is a non-degenerate triple morphism, then ψ is a trivial dilation of θ. Proof. By dilating ψ if necessary, we may assume without loss of generality that ψ is a triple morphism. Thus, there exists a unique * -homomorphism σ : T T * → B( K) such that σ(xy * ) = ψ(x)ψ(y) * (x, y ∈ T ). The assumption further implies that there exists a * -homomorphism π : T T * → B(K) such that π(xy * ) = qψ(x)pψ(y) * q. Since θ is non-degenerate, π is non-degenerate as well. Consider now the map ϕ : T T * → B(K), a → qσ(a)q − π(a). We claim that ϕ = 0. To this end, observe that if x ∈ T , then ϕ(xx * ) = qψ(x)(1 − p)ψ(x) * q ≥ 0, so ϕ is a positive map by Lemma 2.4. Let (e i ) be an approximate identity for T T * . Since π is non-degenerate, π(e i ) tends to q in the strong operator topology. Since qσ(e i )q ≤ q, it follows that ϕ(e i ) = qσ(e i )q − π(e i ) tends to zero in the strong operator topology. Combining this with positivity of ϕ, it follows that ϕ = 0 (see, e.g. [16,Lemma I.9.5]). In particular, we see that for x ∈ T , 0 = ϕ(xx * ) = qψ(x)(1 − p)ψ(x) * q = [qψ(x)(1 − p)][qψ(x)(1 − p)] * , so that qψ(x)(1 − p) = 0. A similar argument, replacing T T * with T * T , shows that pψ(x) * (1 − q) = 0. Thus, ψ is a trivial dilation of θ. Proposition 2.6. Suppose that φ : X → B(H, K) is a rectangular operator state of X, and T is a TRO containing X as a generating subspace. Then φ is maximal if and only if it has the unique extension property. Proof. Suppose initially that φ is maximal. Let φ : T → B(H, K) be an extension of φ. Let θ : T → B( H, K) be a dilation of φ to a triple morphism. We can identify H with a subspace of H and K with a subspace of K. Let p and q be the orthogonal projections of H and K onto H and K, respectively. We have that φ(x) = qθ(x)| H for every x ∈ T . The restriction of θ to X is a rectangular operator state that dilates φ. By maximality of φ, we can conclude that θ(x) = qθ(x)p + (1 − q) θ(x) (1 − p) for every x ∈ X. Since X generates T as a TRO and θ is a triple morphism, it follows that this identity holds for every x ∈ T . It follows that φ is a triple morphism as well. Suppose now that φ has the unique extension property. Let ψ : X → B( H, K) be a dilation of φ. As above, we will identify H and K as subspaces of H and K, respectively, and denote by p and q the corresponding orthogonal projections. We can extend ψ to a rectangular operator state ψ : T → B( H, K). Observe that x → qψ(x)| H is a rectangular operator state extending φ. Since φ has the unique extension property, x → qψ(x)| H is a triple morphism. Hence from Lemma 2.5 we can conclude that ψ is a trivial dilation of φ. Since ψ was arbitrary, we can conclude that φ is maximal. Simple examples show that the implication "unique extension property implies maximal" of Proposititon 2.6 may fail if φ is a degenerate completely contractive map. Indeed, there are degenerate representations of TROs which have non-trivial dilations. Boundary representations. Suppose that X is an operator space, and T is a TRO containing X as a generating subspace. Definition 2.7. A boundary representation for X is a rectangular operator state φ : X → B(H, K) with the property that any rectangular operator state on T extending X is an irreducible representation of T . In other words, a rectangular operator state φ : X → B(H, K) is a boundary representation for X if and only if it has the unique extension property, and the unique extension of φ to T is an irreducible representation of T . In the following we will identify a boundary representation of X with its unique extension to an irreducible representation of T . It follows from Proposition 2.6 that the notion of boundary representation does not depend on the concrete realization of X as a space of operators. We remark that our terminology differs slightly from Arveson's original use of the term boundary representation in the context of operator systems [7]. Indeed, for Arveson, a boundary representation is a representation of the C * -algebra generated by the operator system. More precisely, if S is an operator system that generates the C * -algebra A, then according to Arveson, a boundary representation for S is an irreducible representation π of A such that π is the unique completely positive extension of π| S . We follow the convention, which is for example used in [17], that a boundary representation of S is a unital completely positive map φ : S → B(H) such that every extension of φ to a completely positive map on A is an irreducible representation of A. Since this notion does not depend on the concrete representation of S, these two points of view are equivalent. In the rest of this section, we will observe that the boundary representations of X completely norm X. This will be deduced from the corresponding fact about operator systems, proved in [6] in the separable case and in [17] in full generality. Proposition 2.8. Suppose ω : S(X) → B(L ω ) is a boundary representation of the Paulsen system S(X) associated with X. Then one can decompose L ω as an orthogonal direct sum K ω ⊕ H ω in such a way that ω = S(ψ) for some boundary representation ψ : X → B(H ω , K ω ) of X. Proof. Suppose that T ⊂ B(H, K) is a TRO containing X as a generating subspace. Let A be the C*-algebra generated by S(X) inside B(K ⊕ H). Observe that A = x 11 + λI K x 12 x 21 x 22 + µI H : x 11 ∈ T T * , x 12 ∈ T , x 21 ∈ T * , x 22 ∈ T * T , λ, µ ∈ C . Since ω is a boundary representation of S(X), it extends to an irreducible representation ω : A → B(L ω ). Set q ω := ω I K 0 0 0 , and p ω := ω 0 0 0 I H . Observe that p ω , q ω ∈ B(L ω ) are orthogonal projections such that p ω + q ω = I Lω . Denote by K ω the range of q ω and by H ω the range of p ω . The fact that ω is a unital *-homomorphism implies that, with respect to the decomposition L ω = K ω ⊕ H ω one has that ω = σ θ θ * π where θ : T → B(H ω , K ω ) is a triple morphism. We claim that θ is irreducible. Suppose that p ∈ B(H ω ) and q ∈ B(K ω ) are projections such that qθ(x) = θ(x)p for every x ∈ T . Since σ (ab * ) = θ(a)θ(b) * and π (a * b) = θ(a) * θ(b) for every a, b ∈ T , we can conclude that (q ⊕ p) ω(x) = ω(x)(q ⊕ p) for every x ∈ A. Since ω is an irreducible representation of A, it follows that q ⊕ p = 1 or q ⊕ p = 0. This concludes the proof that θ is irreducible. Denote by ψ the restriction of θ to X. Observe that ω = S(ψ). We claim that ψ is maximal. Indeed, let φ : X → B( H, K) be a dilation of ψ. We can identify H and K as subspaces of H and K, with corresponding orthogonal projections p and q. Then S(φ) : S(X) → B( K ⊕ H) is a dilation of ω. By maximality of ω, we have that q 0 0 p ω(x) = ω(x) q 0 0 p for every x ∈ S(X). It follows that qφ(x) = φ(x)p for every x ∈ X. This shows that φ is a trivial dilation of ψ, concluding the proof that ψ is maximal. The following result is now an immediate consequence of Proposition 2.8 and [17, Theorem 3.4]. Theorem 2.9. Suppose that X is an operator space. Then X is completely normed by its boundary representations. 2.4. Rectangular extreme points and pure unital completely positive maps. Suppose that X is an operator space, and φ : X → B(H, K) is a completely contractive linear map. A rectangular operator convex combination is an expression φ = α * 1 φ 1 β 1 + · · · + α * n φ n β n , where β i : H → H i and α i : K → K i are linear maps, and φ i : X → B(H i , K i ) are completely contractive linear maps for i = 1, 2, . . . , ℓ such that α * 1 α 1 +· · ·+α * n α n = 1, and β * 1 β 1 + · · · + β * n β n = 1. Such a rectangular convex combination is proper if α i , β i are surjective, and trivial if α * i α i = λ i 1, β * i β i = λ i 1, and α * i φ i β i = λ i φ for some λ i ∈ [0, 1]. Definition 2.10. A completely contractive map φ : X → B(H, K) is a rectangular operator extreme point if any proper rectangular operator convex combination φ = α * 1 φ 1 β 1 + · · · + α * n φ n β n is trivial. Suppose now that X is an operator system. An operator state on X is a unital completely positive map φ : X → B(H). An operator convex combination is an expression φ = α * 1 φ 1 α 1 +· · ·+α * n φ n α n , where α i : H → H i are linear maps, and φ i : X → B(H i ) are operator states for i = 1, 2, . . . , ℓ such that α * 1 α 1 + · · · + α * n α n = 1. Such an operator convex combination is proper if α i is right invertible for i = 1, 2, . . . , ℓ, and trivial if α * i α i = λ i 1 and α * i φ i α i = λ i φ for some λ i ∈ [0, 1]. We say that φ is an operator extreme point if any proper operator convex combination φ = α * 1 φ 1 α 1 + · · · + α * n φ n α n is trivial. The proof of [24,Theorem B] shows that an operator state is an operator extreme point if and only if it is a pure element in the cone of completely positive maps. When H is finite-dimensional, the notion of proper operator convex combination coincides with the notion of proper matrix convex combination from [47]. In this case, the notion of operator extreme point coincides with the notion of matrix extreme point from [47, Definition 2.1]. Lemma 2.11. Suppose that φ : X → B(H, K) is a completely contractive linear map, Ψ : S(X) → B(K ⊕ H) is a completely positive map such that S(φ)−Ψ is completely positive. Suppose that Ψ(1) is an invertible element of B(K ⊕ H). Then there exist positive invertible elements a ∈ B(K) and b ∈ B(H), and a completely contractive map ψ : X → B(H, K) such that Ψ λ x y * µ = a 0 0 b λI K ψ(x) ψ(y) * µI H a 0 0 b . Proof. Fix a concrete representation X ⊂ B(L 0 , L 1 ) of X. In this case S(X) ⊂ B(L 1 ⊕ L 0 ). Set Φ := S(φ) and let T ⊂ B(L 0 , L 1 ) denote the TRO generated by X. By Arveson's extension theorem, we may extend Φ and Ψ to the C*-algebra A generated by S(X) inside B(L 1 ⊕ L 0 ) in such a way that Φ − Ψ is still completely positive. In the following we regard Φ, Ψ as maps from A to B(K ⊕ H). Since Φ − Ψ is completely positive, the argument in the proof of [40,Theorem 8.3] shows that there exist linear maps ϕ 1 : T T * + CI L1 → B(K) and ϕ 0 : T * T + CI L0 → B(H) such that Ψ x 0 0 0 = ϕ 1 (x) 0 0 0 . and Ψ 0 0 0 y = 0 0 0 ϕ 0 (y) . In particular, w = Ψ(1) is a diagonal element, and the unital completely positive map Ψ 0 = w −1/2 Ψw −1/2 satisfies Ψ 0 I L1 0 0 0 = I K 0 0 0 and Ψ 0 0 0 0 I L0 = 0 0 0 I H . It follows that the two projections I L1 0 0 0 and 0 0 0 I L0 belong to the multiplicative domain of Ψ 0 [10, Proposition 1.3.11], so that there exists a completely contractive map ψ : X → B(H, K) such that Ψ 0 λI L1 x y * µI L0 = λI K ψ(x) ψ(y) * µI H , which finishes the proof. (2) S(φ) is an operator extreme point; (3) φ is a rectangular operator extreme point. Proof. We have already observed that the equivalence of (1) and (2) holds, as the argument in the proof of [24,Theorem B] shows. (2) =⇒ (3) Suppose that φ = α * 1 φ 1 β 1 + · · · + α * ℓ φ ℓ β ℓ is a proper rectangular matrix convex combination. Define γ i = α i ⊕ β i for i = 1, 2, . . . , ℓ. Then we have that S(φ) = γ * 1 S (φ 1 ) γ 1 + · · · + γ * ℓ S (φ ℓ ) γ ℓ is a proper matrix convex combination. Since by assumption S(φ) is an operator extreme point in the state space of S(X), we can conclude that the proper matrix convex combination γ * 1 S (φ 1 ) γ 1 + · · · + γ * ℓ S (φ ℓ ) γ ℓ is trivial. This implies that the proper rectangular matrix convex combination α * 1 φ 1 β 1 + · · · + α * ℓ φ ℓ β ℓ is trivial as well. (3) =⇒ (1) Suppose that S(φ) = Ψ 1 + Ψ 2 for some completely positive maps Ψ 1 , Ψ 2 : S(X) → B(K ⊕ H). Fix ε > 0 and define Ξ i = (1 − ε) Ψ i + (ε/2) S(φ) for i = 1, 2. Then Ξ 1 , Ξ 2 : S(X) → B(K ⊕ H) are completely positive maps such that Ξ 1 + Ξ 2 = S(φ) and Ξ i (1) is invertible for i = 1, 2; cf. the proof of [17, Lemma 2.3]. By Lemma 2.11 we have that, for i = 1, 2, Ξ i λ x y * µ = a i 0 0 b i λ1 ψ i (x) ψ i (y) * µ1 a i 0 0 b i for some positive invertible elements a i ∈ B(K), b i ∈ B(H) , and completely contractive ψ i : X → B(H, K). Thus we have that φ = a 1 ψ 1 b 1 + a 2 ψ 2 b 2 is a proper rectangular operator convex combination. By assumption, we have that a 2 i = t i 1, b 2 i = t i 1, and a i ψ i b i = t i φ for some t i ∈ [0, 1] and i = 1, 2. It follows that Ξ i = t i S(φ) for i = 1, 2. Since this is true for every ε, it follows that the Ψ i are also scalar multiples of S(φ). This concludes the proof that S(φ) is pure. The following corollary is an immediate consequence of Proposition 2.12, Proposition 2.8, and [17, Theorem 2.4]. Corollary 2.13. Suppose that φ : X → B(H, K) is a rectangular operator state. If φ is rectangular operator extreme, then φ admits a dilation to a boundary representation of X. Suppose that X is a Banach space. We regard X as an operator space endowed with its canonical minimal operator space structure obtained from the canonical inclusion of X in the C*-algebra C (Ball (X ′ )), where X ′ denotes the dual space of X. In [2] Alfsen and Effros considered the following notion. Suppose that φ 0 , φ 1 are nonzero contractive linear functionals on X. Then φ 0 , φ 1 are codirectional if φ 0 + φ 1 = φ 0 + φ 1 . This is equivalent to the assertion that ||φ 0 + φ 1 ||S( φ0+φ1 φ0+φ1 ) = φ 0 S( φ0 φ0 ) + φ 1 S( φ1 φ1 ). The relation ≺ on nonzero contractive linear functionals is defined by setting φ ≺ ψ if and only if either φ = ψ or φ and ψ − φ are codirectional. This is equivalent to the assertion that φ S( φ φ ) ≤ ψ S( ψ ψ ). Proposition 2.14. Suppose that X is a Banach space and φ is a bounded linear functional on X of norm 1. The following statements are equivalent: (1) φ is an extreme point of Ball(X ′ ); (2) if ψ ∈ Ball(X ′ ) is a nonzero linear functional such that ψ ≺ φ, then ψ is a scalar multiple of φ; (3) φ is a rectangular extreme point. Proof. The implications (3)⇒(1)⇒(2) are straightforward. (2)⇒(3) Suppose that, for every nonzero ψ ∈ Ball(X ′ ), ψ ≺ φ implies that ψ is a scalar multiple of φ. The proof of Proposition 2.12 shows that it suffices to show that every proper rectangular convex combination of two elements is trivial. Thus, consider a rectangular convex combination φ = s 0 t 0 φ 0 + s 1 t 1 φ 1 for some φ 0 , φ 1 ∈ Ball(X ′ ) and non-zero s 0 , s 1 , t 0 , t 1 ∈ C such that |s 0 | 2 + |s 1 | 2 = 1 and |t 0 | 2 + |t 1 | 2 = 1. Observe that 1 = s 0 t 0 φ 0 + s 1 t 1 φ 1 ≤ s 0 t 0 φ 0 + s 1 t 1 φ 1 ≤ s 0 t 0 φ 0 + s 1 t 1 φ 1 ≤ s 0 t 0 + s 1 t 1 ≤ 1. Hence s 0 t 0 φ 0 + s 1 t 1 φ 1 = φ 0 = φ 1 = 1 and |s 0 | = |t 0 | and |s 1 | = |t 1 |. By hypothesis we have that 1], so that the rectangular convex combination was trivial. s 0 t 0 φ 0 = ρ 0 φ and s 1 t 1 φ 1 = ρ 1 φ for some ρ 0 , ρ 1 ∈ C. In particular, |ρ i | = |s i | 2 = |t i | 2 for i = 0, 1. Since φ = s 0 t 0 φ 0 + s 1 t 1 φ 1 , it follows that ρ 0 + ρ 1 = 1. Combined with |ρ 0 | + |ρ 1 | = 1, this implies that ρ 0 , ρ 1 ∈ [0, 2.5. TRO-extreme points. Suppose that X is an operator space and ϕ : X → B(H, K) is a completely contractive map. We say that ϕ is a TRO-extreme point if whenever ϕ = α * 1 ϕ 1 β 1 + · · · + α * ℓ ϕ ℓ β ℓ is a proper rectangular matrix convex combination such that ϕ i : X → B(H, K) for i = 1, 2, . . . , ℓ, then α * i α i = t i 1, β * i β i = t i 1, and α * i ϕ i β i = t i ϕ for some t i ∈ [0, 1]. When H, K are finite-dimensional, this is equivalent to requiring that there exist unitaries u i ∈ M n (C) and w i ∈ M m (C) such that ϕ i = u * i ϕw i . This can be seen arguing as in the proof of Lemma 3.7 below. The notion of TRO-extreme point can be seen as the operator space analog of the notion of C*-extreme point considered in [33,27,26]. A similar proof as [24, Theorem B] gives the following lemma. Using this lemma, one can prove similarly as [24, Theorem C] the following fact: Proposition 2.16. Suppose that T is a TRO, and ϕ : T → B(H, K) is a TRO-extreme point. Then there exist pairwise orthogonal projections (p i ) i∈I in B(H) and (q i ) i∈I in B(K) such that p i ϕq i is rectangular operator extreme for every i, and p i ϕq j = 0 for every i = j. 2.6. The Shilov boundary of an operator space. Suppose that X is an operator space. A triple cover of X is a pair (ι, T ) where T is a TRO and ι : X → T is a completely isometric linear map whose range is a subspace of T that generates T as a TRO. Among the triple covers there exists a canonical one: the triple envelope. This is the (unique) triple cover (ι e , T e (X)) with the property that for any other triple cover (ι, T ) of X, there exists a triple morphism θ : T → T e (X) such that θ • ι = ι e . The existence of the triple envelope was established by Hamana in [29] using the construction of the injective envelope of an operator space; see also [10,Section 4.4]. The triple envelope of an operator space is referred to as the (noncommutative) Shilov boundary in [9]. It is remarked in [9] at the beginning of Section 4, referring to the Shilov boundary of an operator space, that "the spaces above are not at the present time defined canonically", and "this lack of canonicity is always a potential source of blunders in this area, if one is not careful about various identifications." We remark here that the theory of boundary representations provides a canonical construction of the Shilov boundary of an operator space X. Indeed one can consider ι e : X → B(H, K) to be the direct sum of all the boundary representations for X, and then let (ι e , T e (X)) be the subTRO of B(H, K) generated by the image of ι e . Proposition 2.6 implies that ι e is maximal, so we may argue as in the proof of [18,Theorem 4.1] that T e (X) is indeed the triple envelope of X. It also follows that if X has a completely isometric boundary representation θ : X → B(H), then the triple envelope of X is the TRO generated by the range of θ inside B(H). The Shilov boundary of a Banach space. A TRO T is commutative if xy * z = zy * x for every x, y, z ∈ T . Several equivalent characterizations of commutative TROs are provided in [10, Proposition 8.6.5]. Suppose that E is a locally trivial line bundle over a locally compact Hausdorff space U . Then the space Γ 0 (E) of continuous sections of E that vanish at infinity is a commutative TRO such that Γ 0 (E) * Γ 0 (E) = C 0 (E). Conversely, it is observed in [9, Section 4]-see also [19]-that any commutative TRO is of this form. One can also describe the commutative TROs as the C σ -spaces from the Banach space literature. Suppose that E is a locally trivial line bundle over a locally compact Hausdorff space U with point at infinity ∞ and X ⊂ Γ 0 (E) be a closed subspace. Assume that the set of elements { x, y : x, y ∈ X} of C 0 (U ) separates the points of U and does not identitically vanish at any point of U . This is equivalent to the assertion that X generates Γ 0 (E) as a TRO, as proved in [9,Theorem 4.20]. An irreducible representation of Γ 0 (E) is of the form x → x (ω 0 ) for some ω 0 ∈ U . A linear map from Γ 0 (E) to C of norm 1 has the form x → x (ω) dµ (ω) for some Borel probability measure µ on U . We say that µ is a representing measure for ω 0 ∈ U if x (ω) dµ (ω) = x (ω 0 ) for every x ∈ X. A point ω 0 ∈ U is a Choquet boundary point if the point mass at ω 0 is the unique representing measure for ω 0 . It follows from the observations above that ω 0 is a Choquet boundary point for X if and only if the map x → x (ω 0 ) is a boundary representation for X. The Choquet boundary Ch(X) of X is the set of Choquet boundary points of X. Suppose that ∂ S X ∪ {∞} is the closure of Ch(X) ∪ {∞} inside U ∪ {∞}. Then it follows from Theorem 2.9 that E| ∂S X is the Shilov boundary of X in the sense of [9,Theorem 4.25]. This means that the linear map X → Γ 0 (E| ∂S X ), x → x| ∂S X is isometric, and for any locally trivial line bundle over a locally compact Hausdorff space V and linear isometry J : X → Γ 0 (V ) with the property that the set {J(x) * J(y) : x, y ∈ X} separates the points of V and does not identically vanish at any point of V , there exists a proper continuous injection ϕ : ∂ S X → V with the property that J(x) • ϕ = x| ∂S X for every x ∈ X. This gives a canonical construction of the Shilov boundary of a Banach space, analogous to the canonical construction of a Shilov boundary of a unital function space; see [ Proof. Suppose first that the quotient map π is completely isometric on X. Then π, regarded as a map from X → π(X), admits a completely isometric inverse, which extends to a complete contraction ψ : B(H, K)/K(H, K) → B(H, K). Clearly, ψ • π is a completely contractive map which extends the inclusion of X into B(H, K), but it does not extend the inclusion of T into B(H, K), since it annihilates the compact operators. Conversely, suppose that the quotient map is not completely isometric on X. Let S(X) ⊂ B(K ⊕ H) denote the Paulsen system associated with X. We will verify that S(X) satisfies the assumptions of Arveson's boundary theorem. To see that S(X) acts irreducibly, suppose that p is an orthogonal projection on K ⊕ H which commutes with S(X). In particular, p commutes with I K ⊕ 0 and 0 ⊕ I H , from which we deduce that p = p 1 ⊕ p 2 , where p 1 ∈ B(K) and p 2 ∈ B(H) are orthogonal projections. For x ∈ X, we therefore have 0 p 1 x 0 0 = p 1 0 0 p 2 0 x 0 0 = 0 x 0 0 p 1 0 0 p 2 = 0 xp 2 0 0 , hence p 1 x = xp 2 for all x ∈ X. Since X acts irreducibly, it follows that either p 1 ⊕ p 2 = 0 or p 1 ⊕ p 2 = I K⊕H , so that S(X) acts irreducibly. The assumption that T contains a non-zero compact operator implies that S(X) contains a non-zero compact operator, hence by irreducibility of S(X), we see that K(K ⊕ H) ⊂ C * (S(X)). Since the quotient map B(H, K) → B(H, K)/K(H, K) is not completely isometric on X, there exists x ∈ M n (X) and k ∈ M n (K(H, K)) such that ||x − k|| < ||x||. Regarding x as an element of M n (S(X)) in the canonical way and correspondingly k as an element of M n (K(K ⊕H)), we see that the quotient map B(K ⊕H) → B(K ⊕ H)/K(K ⊕ H) is not completely isometric on S(X). Thus, Arveson's boundary theorem implies that the identity representation is a boundary representation of S(X). According to Proposition 2.8, there exists a boundary representation ψ : X → B(L 1 , L 2 ) of X such that S(ψ) is the inclusion of S(X) into B(K ⊕ H). It easily follows now that L 1 = H, L 2 = K and that ψ is the inclusion of X into B(H, K), which finishes the proof. 2.9. Rectangular multipliers. A reproducing kernel Hilbert space H on a set X is a Hilbert space of functions on X such that for every x ∈ X, the functional H → C, f → f (x), is bounded. The unique function k : X × X → C which satisfies k(·, x) ∈ H for all x ∈ X and f, k(·, x) = f (x) for all x ∈ X and f ∈ H is called the reproducing kernel of H. We will always assume that H has no common zeros, meaning that there does not exist x ∈ X such that f (x) = 0 for all x ∈ X. Equivalently, k(x, x) = 0 for all x ∈ X. We refer the reader to the books [41] and [1] for background material on reproducing kernel Hilbert spaces. If H and K are reproducing kernel Hilbert spaces on the same set X, we define the multiplier space Nevertheless, the rectangular case of two different reproducing kernel Hilbert spaces has been studied as well, see for example [46] and [45], where multipliers between weighted Dirichlet spaces are investigated. Mult(H, K) = {ϕ : X → C : ϕ · f ∈ K for all f ∈ H}, where (ϕ · f ) (x) = ϕ (x) f (x) We say that a reproducing kernel Hilbert space K on X with reproducing kernel k is irreducible if X cannot be partitioned into two non-empty sets X 1 and X 2 such that k(x, y) = 0 for all x ∈ X 1 and y ∈ X 2 . This definition is more general than the definition of irreducibility in [1, Definition 7.1], but it suffices for our purposes. For the next Lemma, we observe that if H contains the constant function 1, then Mult(H, K) is contained in K. holds. Since Mult(H, K) is dense in K, we deduce that ψ ∈ Mult(K) and that q = M ψ . We claim that ψ is necessarily constant. To this end, let k denote the reproducing kernel of K. Note that M ψ is in particular selfadjoint, so that ψ(x)k(x, y) = M ψ k(·, y), k(·, x) = k(·, y), M ψ k(·, x) = ψ(y)k(x, y) for all x, y ∈ X. Hence ψ(x) is real for all x ∈ X and ψ(x) = ψ(y) if k(x, y) = 0. Fix x 0 ∈ X, and suppose for a contradiction that X 1 = {x ∈ X : ψ(x) = ψ(x 0 )} is a proper subset of X and let X 2 = X \ X 1 . If x ∈ X 1 and y ∈ X 2 , then ψ(y) = ψ(x 0 ) = ψ(x), hence k(x, y) = 0. This contradicts irreducibility of K, so that ψ is constant. Moreover, since M ψ is a projection, we necessarily have ψ = 1 or ψ = 0. If ψ = 1, then q = M ψ = I K and M ϕ (I H − p) = 0 for all ϕ ∈ Mult(H, K). Similarly, if ψ = 0, then q = 0 and M ϕ p = 0 for all ϕ ∈ Mult(H, K). We may thus finish the proof by showing that ϕ∈Mult(H,K) ker(M ϕ ) = {0}. To this end, note that if x ∈ X, then {f ∈ K : f (x) = 0} is a proper closed subspace of K, as K has no common zeros. Since Mult(H, K) is dense in K, it cannot be contained in such a subspace, thus for every x ∈ X, there exists ϕ ∈ Mult(H, K) such that ϕ(x) = 0. If f ∈ H satisfies M ϕ f = 0 for all ϕ ∈ Mult(H, K), it therefore follows that f (x) = 0 for all x ∈ X, that is, f = 0, as desired. We can now use the rectangular boundary theorem to show that for many multiplier spaces, the identity representation is always a boundary representation. This scale of spaces is a frequent object of study in the theory of reproducing kernel Hilbert spaces. The space H 0 is the classical Hardy space H 2 , the space H −1 is the Dirichlet space, and the space H 1 is the Bergman space. The elements of Mult(H s , H t ) were characterized in [46] and [45]. We remark that the spaces D α of [46] are related to the spaces above via the formula D α = H −α . In [45], a slightly different convention is used. There, D α = H −2α , at least with equivalent norms. Theorem 4 of [46] In the square case s = t, boundary representations of operator spaces related to the algebras Mult(H s ), and their analogs on higher dimensional domains, were studied in [28,34,14]; see in particular [28, Section 2] and [34, Section 5.2]. It is well known that if s ≥ 0, then Mult(H s ) = H ∞ , the algebra of all bounded analytic functions on the unit disc, endowed with the supremum norm. This can be deduced, for example, from [44,Proposition 26 (ii)]. In particular, the C*-envelope of Mult(H s ) is commutative, so that the identity representation of Mult(H s ) on H s is not a boundary representation. On the other hand, if s < 0, then the identity representation of Mult(H s ) on H s is a boundary representation. This follows, for instance, from Corollary 2 in Section 2 of [5] and its proof. We use the results above to prove that, in the rectangular case, the identity representation is always a boundary representation. Corollary 2.20. If s < t, then the identity representation is a boundary representation of Mult(H s , H t ). Proof. We verify that the pair (H s , H t ) satisfies the assumptions of Proposition 2.19. It is clear that H s contains the constant function 1. By the remark above, Mult(H s , H t ) contains the polynomials, and is therefore dense in K. Moreover, since k t (0, w) = 1 for all w ∈ D, the space H t is irreducible. Finally, since ||z n || 2 Ht ||z n || 2 Hs = (n + 1) s−t , which tends to zero as n → ∞, the inclusion H s ⊂ H t is compact, so that M 1 ∈ Mult(H s , H t ) is a compact operator. Therefore, the result follows from Proposition 2.19. Operator spaces and rectangular matrix convex sets In the following we will use notation from [22] and [47]. In particular, if V and V ′ are vector spaces in duality via a bilinear map ·, · , x = [x ij ] ∈ M n,m (V ), and ψ = [ψ αβ ] ∈ M r,s (V ′ ), then we let x, ψ be the element [ x ij , ψ αβ ] of M nr,ms (C), where the rows of x, ψ are indexed by (i, α) and the columns of x, ψ are indexed by (j, β). We also let ψ (n,m) be the map ·, ψ : M n,m (V ) → M nr,ms (C). Rectangular matrix convex sets. Definition 3.1. A rectangular matrix convex set in a vector space V is a collection K = (K n,m ) of subsets of M n,m (V ) with the property that for any α i ∈ M ni,n (C) and β i ∈ M mi,m (C) and v i ∈ K ni,mi for 1 ≤ i ≤ ℓ such that α * 1 α 1 + · · · + α * ℓ α ℓ β * 1 β 1 + · · · + β * ℓ β ℓ ≤ 1 one has that α * 1 v 1 β 1 + · · · + α * ℓ v ℓ β ℓ ∈ K n,m . When V is a topological vector space, we say that K is compact if K n,m is compact for every n, m. The following characterization of rectangular matrix convex sets can be easily verified using Proposition 2.1 and the fact that any finite-dimensional representation of M n,m (C) as a TRO is unitarily conjugate to a finite direct sum of copies of the identity representation [11, Lemma 3.2.3]. Lemma 3.2. Suppose that K = (K n,m ) where K n,m ⊂ M n,m (V ). The following assertions are equivalent: (1) K is a rectangular convex set; (2) x ⊕ y ∈ K n+m,r+s for any x ∈ K n,r and y ∈ K m,s , and α * xβ ∈ K r,s for any x ∈ K n,m , α ∈ M n,r (C) and β ∈ M m,s (C) with α * α β * β ≤ 1; (3) x ⊕ y ∈ K n+m,r+s for any x ∈ K n,r and y ∈ K m,s , and (σ ⊗ id V ) [K n,m ] ⊂ K r,s for any completely contractive map σ : M n,m (C) → M r,s (C). It is clear that, if K is a rectangular matrix convex set, then (K n,n ) is a matrix convex set in the sense of [49]. Furthermore if K and T are rectangular matrix convex sets such that T n = K n for every n ∈ N then T n,m = K n,m for every n, m ∈ N. If S = (S n,m ) is a collection of subsets of a (topological) vector space V , the (closed) rectangular matrix convex hull of S is the smallest (closed) rectangular matrix convex set containing S. Example 3.3. Suppose that X is an operator space. Set K n,m to be space of completely contractive maps from X to M n,m (C). Then CBall(X) = (K n,m ) is a rectangular matrix convex set. 3.2. The rectangular polar theorem. Suppose that V and V ′ are vector spaces in duality. We endow both V and V ′ with the weak topology induced from such a duality. Let S = (S n,m ) be a collection of subsets S n,m ⊂ M n,m (V ). We define the rectangular matrix polar S ρ to be the closed rectangular matrix convex subset of V ′ such that f ∈ S ρ n,m if and only if v, f ≤ 1 for every r, s ∈ N and every v ∈ S r,s . The same proof as [22, Lemma 5.1] shows that f ∈ S ρ n,m if and only if v, f ≤ 1 for every v ∈ S n,m . If A ⊂ V , then its absolute polar A • is the set of f ∈ V ′ such that | v, f | ≤ 1 for every v ∈ A. The classical bipolar theorem asserts that the absolute bipolar A •• is the closed absolutely convex hull of A [15, Theorem 8.1.12]. We will prove below the rectangular analog of this fact. The proof is analogous to the one of [22,Theorem 5.4]. Theorem 3.4. If S = (S n,m ) is a collection of subsets S n,m ⊂ M n,m (V ), then the rectangular matrix bipolar S ρρ is the closed rectangular matrix convex hull of S. The proof of [21,Theorem C] shows that if K is a rectangular matrix convex set in a vector space V , and F is a linear functional on M n,m (V ) satisfying F | Kn,m ≤ 1, then (1) there exist states p on M n (C) and q on M m (C) such that |F (α * vβ)| 2 ≤ p (α * α) q (β * β) for every r, s ∈ N, α ∈ M n,r (C), β ∈ M m,s (C), and v ∈ M r,s (V ), and (2) there exist matrices γ ∈ M n 2 ,1 (C), δ ∈ M m 2 ,1 (C), and a map ϕ : V → M n,m (C), such that F (w) = γ * w, ϕ δ for every w ∈ M n,m (W ) and w, ϕ ≤ 1 for every r, s ∈ N and w ∈ K r,s . From this one can easily deduce the following proposition, which gives the rectangular matrix bipolar theorem as an easy consequence. Proposition 3.5. Suppose that V and V ′ are vector spaces in duality, and K is a compact rectangular convex space in V . If v 0 ∈ M n,m (V ) \K n,m , then there exists ϕ ∈ K ρ n,m such that v 0 , ϕ > 1. Proof. By the classical bipolar theorem there exists a continuous linear functional F on M n,m (V ) such that F | Kn,m ≤ 1 and |F (v 0 )| > 1. By the remarks above there exists ϕ ∈ K ρ n,m and contractive γ ∈ M n 2 ×1 (C) and δ ∈ M m 2 ×1 (C) such that F (v) = γ * v, ϕ δ. Thus we have v 0 , ϕ ≥ γ * v 0 , ϕ δ = F (v 0 ) > 1. 3.3. Representation of rectangular convex sets. Suppose that K is a rectangular matrix convex set in a vector space V . A rectangular matrix convex combination in a rectangular convex set K is an expression of the form α * 1 v 1 β 1 +· · ·+α * ℓ v ℓ β ℓ for v i ∈ K ni,mi , α i ∈ M ni,n (C), and β i ∈ M mi,m (C) such that α * 1 α 1 +· · ·+α * ℓ α ℓ = 1, and β * 1 β 1 + · · ·+ β * ℓ β ℓ = 1. A proper rectangular matrix convex combination is a rectangular convex combination α * 1 v 1 β 1 + · · · + α * ℓ v ℓ β ℓ where furthermore α 1 , . . . , α ℓ and β 1 , . . . , β ℓ are right invertible. Observe that these notions are a particular instance of the notions of (proper) rectangular operator convex combination introduced in Subsection 2.4. Definition 3.6. A rectangular matrix affine mapping from a rectangular convex set K to a rectangular convex set T is a sequence θ of maps θ n,m : K n,m → T n,m that preserves rectangular matrix convex combinations. When K and T are compact rectangular convex sets, we say that θ is continuous (respectively, a homeomorphism) when θ n,m is continuous (respectively, a homeomorphism) for every n, m ∈ N. Given a compact rectangular matrix convex set K we let A ρ (K) be the complex vector space of continuous rectangular matrix affine mappings from K to CBall (C). Here CBall (C) is the compact rectangular matrix convex set defined as in Example 3.3, where C is endowed with its canonical operator space structure. The space A ρ (K) has a natural operator space structure where M n,m (A ρ (K)) is identified isometrically with a subspace of C (K n,m , M n,m (C)) endowed with the supremum norm. More generally if Y is any operator space, then we define A ρ (K, Y ) to be the operator space of continuous rectangular affine mappings from K to CBall(Y ). Observe that M n,m (A ρ (K)) is completely isometric to A ρ (K). Starting from the operator space A ρ (K) one can consider the compact rectangular matrix convex set CBall(A ρ (K) ′ ) as in Example 3.3. Here A ρ (K) ′ denotes the dual space of the operator space A ρ (K), endowed with its canonical operator space structure. There is a canonical rectangular matrix affine mapping θ from K to CBall(A ρ (K) ′ ) given by point evaluations. It is clear that this map is injective. It is furthermore surjective in view of the rectangular bipolar theorem. The argument is similar to the one of the proof of [47,Proposition 3.5]. This shows that the map θ is indeed a rectangular matrix affine homeomorphism from K onto CBall(A ρ (K) ′ ). This implies that the assignment X → CBall(X ′ ) is a 1:1 correspondence between operator spaces and rectangular convex sets. It is also not difficult to verify that this correspondence is in fact an equivalence of categories, where morphisms between operator spaces are completely contractive linear maps, and morphisms between rectangular convex sets are continuous rectangular matrix affine mappings. 3.4. The rectangular Krein-Milman theorem. The notion of (proper) rectangular convex combination yields a natural notion of extreme point in a rectangular convex set. An element v of a rectangular convex set K is a rectangular matrix extreme point if for any proper rectangular convex combination α * 1 v 1 β 1 +· · ·+α * ℓ v ℓ β ℓ = v for v i ∈ K ni,mi one has that, for every 1 ≤ i ≤ ℓ, n i = n, m i = m, and v i = u * i vw i for some unitaries u i ∈ M n (C) and w i ∈ K m . We now observe that the notion of rectangular extreme point coincides with the notion of rectangular operator extreme operator state from Definition 2.10. The argument is borrowed from the proof of [24,Theorem B]. Lemma 3.7. Suppose that X is an operator space, K = CBall(X), and φ ∈ K n,m . Then φ is a rectangular matrix extreme point of K if and only if it is a rectangular operator extreme operator state of X. Proof. It is clear that a rectangular operator extreme point is a rectangular matrix extreme point. We prove the converse implication. Suppose that φ is a rectangular matrix extreme point. Let φ = α * 1 φ 1 β 1 + · · · + α * ℓ φ ℓ β ℓ be a proper rectangular matrix convex combination, where φ i ∈ K ni,mi for i = 1, 2, . . . , ℓ. By assumption, we have that n i = n and m i = m for i = 1, 2, . . . , m, and there exist unitaries u i ∈ M n (C) and w i ∈ M m (C) such that φ i = u * i φw i for i = 1, 2, . . . , ℓ. Therefore we have that φ = (u 1 α 1 ) * φ (w 1 β 1 ) + · · · + (u ℓ α ℓ ) * φ ℓ (w ℓ β ℓ ) .(1) Define R ⊂ M m+m (C) to be the range of S(φ). Observe that it follows from the fact that φ is a rectangular extreme point that the commutant of R is one-dimensional. Set A i = u i α i 0 0 w i β i for i = 1, 2, . . . , n. Define the unital completely positive map Ψ : M n+m (C) → M n+m (C), z → A * 1 zA 1 + · · · + A * ℓ zA ℓ . By Equation (1) we have that Ψ(z) = z for every z ∈ R. It follows from this and [4, Theorem 2.11] that Ψ(z) = z for every z ∈ M n+m (C). By the uniqueness statement in the Choi's representation of a unital completely positive map [13], we deduce that there exist λ i ∈ C such that A i = λ i 1 for i = 1, 2, . . . , ℓ. Therefore α * i α i = (u i α i ) * (u i α i ) = |λ i | 2 1, β * i β i = (w i β i ) * (w i β i ) = |λ i | 2 1, and α * i φ i β i = (u i α i ) * φ (w i β i ) = |λ i | 2 φ. This concludes the proof that φ is a rectangular operator extreme point. We denote by ∂ ρ K = (∂ ρ K n,m ) set of rectangular matrix extreme points of K. Recall that the Krein-Milman theorem asserts that, if K ⊂ V is a compact convex subset of a topological vector space V , then K is the closed convex hull of the set of its extreme points. The following is the natural analog of the Krein-Milman theorem for compact rectangular matrix convex sets. The proof is analogous to the proof of the Krein-Milan theorem for compact matrix convex sets [47,Theorem 4.3]. Theorem 3.8. Suppose that K is a compact rectangular convex set. Then K is the closed rectangular matrix convex hull of ∂ ρ K. Proof. Suppose that K is a compact rectangular convex set. In view of the representation theorem from Subsection 3.3, we can assume without loss of generality that K = CBall(X ′ ) for some operator space X. We will assume that X is concretely represented as a subspace of B(H) for some Hilbert space H. We will also canonically identify M n,r (X ′ ) with the space of bounded linear functionals on M n,r (X). Fix n, m ∈ N. LetX be the space of operators of the form λI ⊕n x y * µI ⊕m for λ, µ ∈ C and x, y ∈ M n,m (X), where I ⊕n and I ⊕m are the identity operator on, respectively, the n-fold and m-fold Hilbertian sum of H by itself. If ϕ ∈ M r,s (X ′ ), then we denote byφ ∈ M nr+ms (X ′ ) the element defined by λI ⊕n x y * µI ⊕n → λI rn ϕ (n,m) (x) ϕ (n,m) (y) * µI ms where I rn and I ms denote the identity rn × rn and ms × ms matrices. If ξ ∈ M r,n (C) and η ∈ M s,m (C) we also set ξ ⊙ η := I n ⊗ ξ 0 0 I m ⊗ η . We let ∆ n,m be the set of elements of M n 2 +m 2 (X ′ ) of the form (ξ ⊙ η) * φ (ξ ⊙ η) for r, s ∈ N, ϕ ∈ K r,s , ξ ∈ M r,n (C) and η ∈ M s,m (C) such that ξ 2 = η 2 = 1. It is not difficult to verify as in [47, §4] that one can assume without loss of generality that r ≤ n, s ≤ m, and ξ, η are right invertible. The computation below shows that ∆ n,m is convex. If t 1 , t 2 ∈ [0, 1] are such that t 1 + t 2 = 1 then t 1 (ξ 1 ⊙ η 1 ) * φ 1 (ξ 1 ⊙ η 1 ) + t 2 (ξ 1 ⊙ η 1 ) * φ 2 (ξ 2 ⊙ η 2 ) = (ξ ⊙ η) * ϕ (ξ ⊙ η) where ξ = t 1 ξ 1 t 2 ξ 2 , η = t 1 η 1 t 2 η 2 , and ϕ = ϕ 1 0 0 ϕ 2 . Thus ∆ n,m is a compact convex subset of the space of unital completely positive maps fromX to M n 2 +m 2 (C). Consider now an element (ξ ⊙ η) * φ (ξ ⊙ η) of ∆ n,m , where ξ ∈ M r,n (C) and η ∈ M s,m (C) are right invertible and ϕ ∈ K n,m . Assume that (ξ ⊙ η) * φ (ξ ⊙ η) is an extreme point of ∆ n,m . We claim that this implies that ϕ is a rectangular extreme point of K. Indeed suppose that, for some s k , r k ∈ N, ϕ k ∈ K r k ,s k , δ k ∈ M s k ,s (C), and γ k ∈ M r k ,r (C), γ * 4. Boundary representations and the C*-envelope of a matrix-gauged space 4.1. Selfadjoint operator spaces. By a concrete selfadjoint operator space we mean a closed selfadjoint subspace X of B(H). Any selfadjoint operator space is endowed with a canonical involution, matrix norms, and matrix positive cones inherited from B(H). An (abstract) matrix-ordered matrix-normed * -vector space-see [42,Subsection 3 .1]-is a vector space V endowed with • a conjugate-linear involution v → v * , • a complete norm in M n (V ) for every n ∈ N, • a distinguished positive cone M n (V ) + ⊂ M n (V ) such that, for every n, k ∈ N, x ∈ M n (X), and a, b ∈ M n,k (C), (1) M n (V ) + is proper, i.e. M n (V ) + ∩ (−M n (V ) + ) = {0},(2) M n (V ) + is closed in the topology induced by the norm, (3) a * xb ≤ a x b , and (4) when a * xa ∈ M k (V ) + , when x ∈ M n (V ) + . A matrix-ordered matrix-normed * -vector space V is normal if, for every n ∈ N and x, y, z ∈ M n (V ), x ≤ y ≤ z implies that y ≤ max { x , z }. It is essentially proved in [48]-see also [ +,ν = {x ∈ V : ν(x) = 0}. The following notion is considered in [42] under the name of L ∞ -matricially ordered vector space. Definition 4.1. A matrix-gauged space is a * -vector space V endowed with a sequence of proper gauges ν n : M n (V ) sa → [0, +∞) for n ∈ N with the property that, for every n, k ∈ N, x ∈ M n (V ), y ∈ M k (V ), and a ∈ M n,k (C), one has that ν k (a * xa) ≤ a 2 ν n (x) and ν n+k (x ⊕ y) = max {ν n (x), ν k (y)} . A linear map φ : V → W between matrix-gauged spaces is completely gauge-contractive if it is selfadjoint and ν φ (n) (x) ≤ ν(x) for every n ∈ N and x ∈ M n (X) sa , and completely gauge-isometric if it is selfadjoint and ν φ (n) (x) = ν(x) for every n ∈ N and x ∈ M n (X) sa ; see [42,Definition 3.11]. Matrix-gauged spaces naturally form a category, where the morphisms are the completely gauge-contractive maps, and isomorphism are completely gauge-isometric surjective maps. In the following we will consider matrix-gauged spaces as objects in this category. By [42, Corollary 3.10], every matrix-gauged space is completely gauge-isometrically isomorphic to a concrete selfadjoint operator space. Any matrix-gauged space V has a canonical normal matrix-ordered matrix-normed * -vector space structure, obtained by considering the gauge norms and the gauge cones associated with the given matrix-gauges. Conversely, suppose that V is a normal matrix-ordered matrix-normed * -vector space. Letting ν n (x) be the distance of x from −M n (V ) + for every x ∈ M n (V ) sa defines a canonical matrix-gauged structure on X. This matrix-gauge structure induces the original matrix-order and matrix-norms on V that one started from; see [42,Proposition 3.5]. Furthermore a selfadjoint linear map φ : V → B(H) is completely positive and completely contractive if and only if it is completely gauge-contractive with respect to these specific matrix-gauges. However, there might be different matrix-gauges on V that induce the same matrix-order and matrix-norms on V . Suppose now that S is an operator system with order unit 1. Then, in particular, S is a normal matrixordered matrix-normed * -vector space. Furthermore, it admits a unique matrix-gauge structure compatible with its matrix-order and matrix-norms. These matrix-gauges are defined by ν n (x) = inf {t > 0 : x ≤ t1} for a selfadjoint x ∈ S. Uniqueness can be deduced from Arveson's extension theorem [12, Theorem 1.6.1], as proved in [43, Theorem 6.9]. In the following we will regard an operator system as a matrix-gauge space with such canonical matrix-gauges. A unital selfadjoint linear map between operator systems is completely positive if and only if it is completely gauge-contractive, and completely isometric if and only if it is completely gauge-isometric. It is proved in [42,Subsection 3.3] that any matrix-gauged space W admits a completely gauge isometric embedding as a subspace of codimension 1 into an operator system W † , called the unitization of W , that satisfies the following universal property: any completely gauge-contractive map from W to an operator system V admits a unique extension to a unital completely positive map from W † to V . The unitization W † of W is uniquely characterized by the above. If W is a normal matrix-ordered matrix-normed space, then we define the unitization of W to be the unitization of W endowed with the canonical matrix-gauges described above. Suppose that A is a (not necessarily unital) C*-algebra. Then A is endowed with canonical matrix-gauges, obtained by setting ν n (x) = x + for a selfadjoint x ∈ A, where x + denotes the positive part of x. In the following we will consider a C*-algebra as a matrix-gauged space with these canonical matrix-gauges. It follows from the unitization construction that any matrix-gauged space admits a completely gauge-isometric embedding into B(H). The following result can also be found in [12, Proposition 2.2.1] with a different proof. if z = x + α1 ∈ M n (Z) for α ∈ M n (C) is positive, then φ (n) (x) + α1 ∈ M n (Y ) is positive. Since 1 / ∈ A, we have that α is positive. Without loss of generality, we can assume that α is invertible. After replacing z with α − 1 2 zα − 1 2 we can assume that α = 1. By Stinespring's theorem [8, Theorem II.6.9.7], there exist a Hilbert space K, a *-homomorphism π : A → B(K), and a linear map v : L → K such that v = 1 and φ(x) = v * π(x)v for every x ∈ A. Observe that π extends to a unital * -homomorphism from Z into B(L), which we still denote by π. Let v (n) : H (n) → K (n) be the map v ⊕ · · · ⊕ v. Then we have that φ (n) (x) + 1 = v (n) * π (n) (x)v (n) + 1 ≥ v (n) * π (n) (x)v (n) + v (n) * π (n) (1)v (n) = v (n) * π (n) (x + 1) v (n) ≥ 0. This concludes the proof. It follows from the previous lemma that the unitization of a C*-algebra A as a matrix-gauged space coincides with the unitization of A as a C*-algebra; see also [48,Corollary 4.17]. Furthermore, it follows from Lemma 4.2, [43, Theorem 6.9], and Arveson's extension theorem that a C*-algebra admits unique compatible matrix-gauges. One can then deduce from [42,Theorem 3.16] that a linear map between C*-algebras or operator systems is completely gauge-contractive if and only if it is completely positive contractive. 4.3. The injective envelope of a matrix-gauged space. We say that a matrix-gauged space is injective if it is injective in the category of matrix-gauged spaces and completely gauge-contractive maps. Theorem 3.14 of [42] shows that B(H) is an injective matrix-gauged space when endowed with its canonical matrix-gauges. It follows from this that the unitization functor W → W † is an injective functor from the category of matrixgauged spaces and gauge-contractive maps to the category of operator systems and unital completely positive maps. Our goal now is to show that any injective matrix-gauged space is (completely gauge-isometrically isomorphic to) a unital C*-algebra. This is a generalization of a theorem of Choi and Effros see [40,Theorem 15.2]. Proposition 4.3. Let X be an injective matrix-gauged space. Then X is completely gauge-isometrically isomorphic to a unital C*-algebra. Proof. We may assume that X ⊂ B(H) is concretely represented as a selfadjoint operator space. Since X is injective, there exists a gauge-contractive and hence completely contractive and completely positive projection Φ : B(H) → X. We define the Choi-Effros product on X by x · Φ y = Φ(xy). As in the proof of [40,Theorem 15.2], one shows that Φ(Φ(a)x) = Φ(ax) and Φ(xΦ(a)) = Φ(xa) holds for all x ∈ X and all a ∈ B(H). Indeed, the proof only requires the Schwarz inequality for unital completely positive maps, which remains valid for completely positive completely contractive maps. In particular, we see that e := Φ(I H ) ∈ X is a unit for the Choi-Effros product. Moreover, the proof of [40,Theorem 15.2] shows that (X, · Φ ), endowed with the norm and involution of B(H), is a C*-algebra. It is clear from the above that the identity map from X onto (X, · Φ ) is an isometry. To see that it is an order isomorphism, suppose that x ∈ X is positive with respect to the order on B(H). Then ||cI H − x|| ≤ c for all c ≥ ||x||, so since Φ is contractive ||ce − x|| ≤ c for all c ≥ ||x||, thus x is a positive element of the C*-algebra (X, · Φ ). Conversely, if x is positive in the C*-algebra (X, · Φ ), then there exists y ∈ X such that x = y * · Φ y, hence x = Φ(y * y) is positive in B(H). Moreover, the argument at the end of the proof of [40,Theorem 15.2] shows that M n (X), endowed with the Choi-Effros product · Φ (n) , is the C*-tensor product of (X, · Φ ) with M n (C). By the above, the identity map is an isometry and an order isomorphism between M n (X) ⊂ M n (B(H)) and (M n (X), · Φ ). Therefore, the identity map from X onto (X, · Φ (n) ) is a selfadjoint complete isometry and complete order isomorphism. To see that the identity map from X onto (X, · Φ ) is in fact a complete gauge isometry, observe that Φ is a unital completely positive map from B(H) onto (X, · Φ ) by the preceding paragraph, so it is completely gauge contractive. Conversely, if x ∈ M n (X) is self-adjoint and satisfies ||x + || ≤ 1, where the positive part is taken in the C*-algebra (M n (X), · Φ ), then x ≤ e (n) in (M n (X), · Φ ), hence x ≤ I C n ⊗ I H in M n (B(H)) by the preceding paragraph, so that the identity map from (X, · Φ ) to X is completely gauge contractive as well. In particular, we see that every injective matrix-gauged space is (completely gauge-isometrically isomorphic to) an injective operator system. Conversely, since the unitization functor is injective, every operator system that is injective in the category of operator systems and unital completely positive maps is also injective as a matrix-gauged space, when endowed with the unique compatible matrix-gauge structure. The usual proof of the existence of the injective envelope of an operator system yields the existence of a gauge analog of Hamana's injective envelope of operator spaces. Let us say that a gauge-extension of a matrix-gauged space X is a pair (Y, i) where Y is a matrix-gauged space and i : X → Y is a completely gauge-isometric map. As in the case of operator systems, we say that such a gauge-extension is: (1) rigid if the identity map of Y is the unique gauge-contractive map φ : Y → Y such that φ • i = i; (2) essential if whenever u : Y → Z is a gauge-contractive map to a matrix-gauged space Z such that u • i is a completely gauge-isometric, then u is a completely gauge-isometric; (3) an injective envelope if Y is injective, and there is no proper injective subspace of Y that contains X. The same proof as [10,Lemma 4.2.4] shows that if X is a matrix-gauged space, and (Y, i) is a gauge-extension of X such that Y is injective, then the following assertions are equivalent: 1) (Y, i) is an injective envelope of X; 2) (Y, i) is essential; 3) (Y, i) is rigid. To this purpose one can consider the gauge analog of the notion of projections and seminorms from [10, Subsection 4.2.1]. Suppose that W is a matrix-gauged space, and X is a selfadjoint subspace of W . A completely gaugecontractive X-projection on W is an idempotent completely gauge-contractive map u : W → W that restricts to the identity on X. A gauge X-seminorm on W is a seminorm of the form p(x) = u(x) for some completely gauge-contractive X-projection u on W . One can define an order on completely gauge-contractive X-projections by u ≤ v if and only if u • v = v • u = u, while gauge X-seminorms are ordered by pointwise comparison. The same proof as [10,Lemma 4.2.2] shows that any gauge X-seminorm majorizes a minimal gauge X-seminorm, and if p is a minimal gauge X-seminorm and u : W → W is a completely gauge-contractive map that restricts to the identity on X, then u is a minimal gauge X-projection. To this purpose, it is enough to observe that the set of completely gauge-contractive selfadjoint maps from W to B(H) is closed in the weak* topology of the space of CB (W, B(H)) of completely bounded maps from W to B(H). Indeed φ : W → B(H) is completely gauge-contractive if and only if it is selfadjoint and φ (n) (x)ξ, ξ ≤ ν(x) for every n ∈ N, ξ ∈ H ⊕n , and x ∈ M n (W ) sa . The proof of [10, Lemma 4.2.4] can now be easily adapted to prove the claim above, by replacing X-projections with gauge X-projections and X-seminorms with gauge X-seminorms. Similarly the same proof as [10, Theorem 4.2.6] shows that if a matrix-gauged space X is contained in an injective matrix-gauged space W , then there exists an injective envelope X ⊂ Z ⊂ W . Furthermore the injective envelope of X is essentially unique. We denote by I(X) the injective envelope of a matrix-gauged space X, and we identify X with a selfadjoint subspace of I(X). It is clear that, when X is an operator system endowed with its canonical matrix-gauges, the injective envelope of X as a matrix-gauged space coincides with the injective envelope of X as an operator system (endowed with the canonical matrix-gauges). Furthermore, it is a consequence of Proposition 4.3 that the unitization of a matrix-gauged space X is span {X, 1} ⊂ I(X), where 1 denotes the identity of the unital C*-algebra I(X). Boundary representations. Most fundamental notions in dilation theory admit straightforward versions in the setting of matrix-gauged spaces. Suppose that X is a matrix-gauged space. An operator state on X is a completely gauge-contractive map φ : X → B(H). We say that an operator state ψ : X → B( H) is a dilation of φ if there exists a linear isometry v : H → H such that v * ψ(x)v = φ(x) for every x ∈ X. It follows from Stinespring's dilation theorem [8, Theorem II.6.9.7] that if A is a C*-algebra, then an operator state on A admits a dilation which is a *-homomorphism. A dilation ψ of an operator state φ : x → v * ψ(x)v on X is trivial if ψ(x) = vv * ψ(x)vv * + (1 − vv * )ψ(x) (1 − vv * ). We say that φ is maximal if it has no nontrivial dilation. As in the case of operator systems, one can prove that an operator state φ : X → B(H) is maximal if and only if for any dilation ψ of φ one has that ψ(x)ξ = φ(x)ξ for every x ∈ X, and ξ ∈ H. Suppose that X is a selfadjoint subspace of a C*-algebra A such that A is generated as a C*-algebra by X. An operator state φ on X has the unique extension property if any completely positive contractive map φ : A → B(H) whose restriction to X coincides with φ is automatically a *-homomorphism. The same argument as in the operator systems setting shows that an operator state is maximal if and only if it has the unique extension property; see [6]. In the following we will identify a boundary representation of X with its unique extension to an irreducible representation of C * (X). It follows from the remarks above that the notion of boundary representation does not depend on the concrete realization of X as a selfadjoint space of operators. In the following we will assume that A is a C*-algebra, and X ⊂ A is a selfadjoint subspace that generates A as a C*-algebra. We regard X as a matrix-gauged space endowed with the matrix-gauges induced by A. Proposition 4.5. Suppose that φ : X → B(H) is an operator state of X, and φ † : X † → B(H) is its canonical unital completely positive extension to the unitization of X. If φ † is a boundary representation for X † , then φ is a boundary representation for X. Proof. Let Φ : A → B(H) be a completely positive contractive map extending φ. Extend Φ to a unital completely positive Φ † : A † → B(H). Then since by assumption φ † is a boundary representation, we conclude that Φ † is an irreducible representation for A † . Therefore Φ| A is an irreducible representation of A. This concludes the proof. The following result is then a consequence of Proposition 4.5 and [17, Theorem 3.1]. Theorem 4.6. Suppose that X is a matrix-gauged space. Then the matrix-gauges of X are completely determined by the boundary representations of X. Precisely, if x ∈ M n (X), then ν n (x) is the supremum of φ (n) (x) + where φ ranges among all the boundary representations of X. Suppose that X is a matrix-gauged space, and φ : X → B(H) is an operator state. An operator convex combination is an expression φ = α * 1 φ 1 α 1 + · · · + α * n φ n α n , where α i : H → H i are linear maps, and φ i : X → B(H i ) are operator states for i = 1, 2, . . . , ℓ. Such a rectangular convex combination is proper if the α i 's are right invertible and α * 1 α 1 + · · · + α * n α n = 1 and trivial if α * i α i = λ i 1 and α * i φ i α i = λ i φ for some λ i ∈ [0, 1]. Definition 4.7. An operator state φ : X → B(H) is an operator extreme point if for any proper operator convex combination φ = α * 1 φ 1 α 1 + · · · + α * n φ n α n is trivial. Observe that the map φ → φ † establishes a 1:1 correspondence between operator states on X and operator states on the operator system X † . Furthermore this correspondence is operator affine in the sense that it preserves operator convex combinations. The following proposition is then an immediate consequence of this observation and (the proof of) [24, Theorem B]. Proposition 4.8. Suppose that φ : X → B(H) is an operator state, and let φ † : X † → B(H) be its unital extension to the unitization of X. The following assertions are equivalent: (1) φ is a pure element in the cone of completely gauge-contractive maps from X to B(H); (2) φ is an operator extreme point; (3) φ † is an operator extreme point. The following corollary is an immediate consequence of Proposition 4.8, Proposition 4.5, and [17]. Corollary 4.9. Suppose that X is a matrix-gauged space, and φ is an operator state on X. If φ is operator extreme, then φ admits a dilation to a boundary representation of X. 4.5. The C*-envelope of a matrix-gauged space . Suppose that X is a matrix-gauged space. A pair (A, i) is a C*-cover if A is a C*-algebra and i : X → A is a completely gauge-isometric map whose range generates A as a C*-algebra. Definition 4.10. A C*-envelope (C * e (X), i) of X is a C*-cover of X if it has the following universal property: for any C*-cover (B, j) of X, there exists a *-homomorphism θ : B → C * e (X) such that θ • j = i. It is clear that the C*-envelope of a matrix-gauged space, if it exists, it is essentially unique. We will prove below that any matrix-gauged space has a C*-envelope. The proof is essentially the same as the one for the existence of the C*-envelope of an operator system. Suppose that X is an matrix-gauged space. Let X ⊂ I(X) be the injective envelope of X. By Proposition 4.3, I(X) is a unital C*-algebra. Let A be the C*-subalgebra of I(X) generated by X. As in the proof of [40,Theorem 15.16], one sees that (i, A) where i : X → A is the inclusion map, is the C*-envelope of X. It follows from the construction that the C*-envelope (as defined above) of an operator system regarded as matrix-gauged space with its unique compatible matrix-gauges coincides with the usual notion of C*-envelope of an operator system. Alternatively, one can construct the C*-envelope of a matrix-gauged space using boundary representations, as for the C*-envelope of an operator system. Indeed let X be a matrix-gauged space. Define ι e : X → B(H) to be the direct sum of all the boundary representations for X, and then let A be the C*-subalgebra of B(H) generated by the image of ι e . It follows from the unique extension property of boundary representations that (ι e , A) is indeed the C*-envelope of X. In particular, this construction shows that, for any C*-algebra B, C * e (B) = B. As in the case of operator systems and operator spaces, one can define a maximal or universal C*-algebra that contains a given ordered operator space as a generating subset. Explicitly, the maximal C*-algebra C * max (X) of an ordered operator space is a C*-cover (i, A) of X that has the following universal property: given any other completely gauge-contractive map f : X → B, where B is a C*-algebra, there exists a *-homomorphism θ : A → B such that θ • i = f . In order to see that such a maximal C*-algebra exists, one can consider the collection F of all completely gauge-contractive maps from X to M n (C) for n ∈ N. Then let i be the direct sum of the elements s : X → M ns (C) of F , and then A to be the C*-subalgebra of ∞ s∈F M ns (C) generated by the image of i. The same proof as [35,Proposition 8] shows that such a C*-cover satisfies the required universal property. 4.6. Selfadjoint ordered operator spaces and compact matrix convex sets. We want to conclude by observing that selfadjoint operator spaces are in canonical 1:1 correspondence with compact matrix convex sets with a distinguished extreme point. Suppose that K = (K n ) is a compact matrix convex set, and e ∈ K 1 is a matrix extreme point. Define A 0 (K, e) to be the set of continuous matrix-affine functions from K to (M n (C)) n∈N that vanish at e. Then A 0 (K, e) is a selfadjoint subspace of codimension 1 of the operator system A(K). Conversely suppose that X ⊂ B(H) is a selfadjoint operator space. Consider X as a normal matrix-ordered and matrix-normed space with respect to the induced matrix-cones and matrix-norms, and let X † be the unitization of X. Let, for n ∈ N, K n be the space of completely positive completely contractive selfadjoint maps from X to M n (C), endowed with the topology of pointwise convergence. Observe that K = (K n ) is a compact matrix-convex set, and K n can be identified with the space of unital completely positive maps from X † to M n (C). Let e ∈ K 1 be the zero functional on X. We have a canonical unital complete order isomorphism X † ∼ = A(K). Under this isomorphism X is mapped into A 0 (K, e). Since X has codimension 1 in X † , such an isomorphism in fact maps X onto A 0 (K, e). The above construction shows that one can identify the unitization of A 0 (K, e) with the operator system A(K). Furthermore it is easy to see that the correspondence (K, e) → A 0 (K, e) is a contravariant equivalence of categories from the category of compact matrix convex sets K with a distinguished matrix extreme point e ∈ K 1 , where morphisms are continuous matrix-affine maps that preserve the distinguished point, to the category of selfadjoint operator spaces and completely positive completely contractive selfadjoint maps. The commutative analog of the argument above establishes a correspondence between compact convex sets with a distinguished extreme point and selfadjoint operator spaces that can be represented inside an abelian C*-algebra (selfadjoint function spaces). Proposition 2 . 12 . 212Suppose that φ : X → B(H, K) is a completely contractive map and S(φ) : S(X) → B(K ⊕ H) is the associated unital completely positive map defined on the Paulsen system. The following assertions are equivalent:(1) S(φ) is a pure completely positive map; Lemma 2 . 15 . 215Let X be an operator space, and A be the C*-algebra generated by S(X). If ϕ : X → B(H, K) is a TRO-extreme point such that the range of ϕ is an irreducible subspace of B(H, K), then there exists a pure unital completely positive map Φ : A → B(K ⊕ H) that extends S(ϕ). The rectangular boundary theorem. Arveson's boundary theorem [5, Theorem 2.1.1] asserts that if S ⊂ B(H) is an operator system which acts irreducibly on H such that the C*-algebra C * (S) contains the algebra of compact operators K(H), then the identity representation of C * (S) is a boundary representation for S if and only if the quotient map B(H) → B(H)/K(H) is not completely isometric on S. The following result is a rectangular generalization of Arveson's boundary theorem. Theorem 2.17. Let X ⊂ B(H, K) be an operator space such that the TRO T generated by X acts irreducibly and such that T ∩ K(H, K) = {0}. Then the identity representation of T is a boundary representation for X if and only if the quotient map B(H, K) → B(H, K)/K(H, K) is not completely isometric on X. for x ∈ X; see[41, Section 5.7]. By[41, Theorem 5.21], every ϕ ∈ Mult(H, K) induces a bounded multiplication operator M ϕ : H → K. Moreover, since K has no common zeros, every multiplier ϕ is uniquely determined by its associated multiplication operator M ϕ . We may thus regard Mult(H, K) as a subspace of B(H, K).The best studied case occurs when H = K, in which case Mult(H) = Mult(H, H) is an algebra, called the multiplier algebra of H; [1, Section 2.3]. Lemma 2 . 18 . 218Let H and K be reproducing kernel Hilbert spaces on the same set. Suppose that • H contains the constant function 1, • Mult(H, K) is dense in K, and • K is irreducible. Then Mult(H, K) ⊂ B(H, K) acts irreducibly. Proof. Suppose that p ∈ B(H) and q ∈ B(K) are orthogonal projections which satisfy qM ϕ = M ϕ p for all ϕ ∈ Mult(H, K). Define ψ = p1 ∈ H. Then for all ϕ ∈ Mult(H, K), the identity qϕ = qM ϕ 1 = M ϕ p1 = ψϕ Proposition 2 . 19 . 219Let H and K be reproducing kernel Hilbert spaces on the same set and let M = Mult(H, K). Suppose that • H contains the constant function 1, • M is dense in K, • K is irreducible, and • M contains a non-zero compact operator. Then the identity representation is a boundary representation of M . In particular, the triple envelope of M is the TRO generated by M . Proof. Lemma 2.18 shows that M acts irreducibly. Moreover, the quotient map by the compacts is not isometric on M since M contains a non-zero compact operator. An application of the rectangular boundary theorem (Theorem 2.17) now finishes the proof. For s ∈ R, let H s = f (z) = ∞ n=0 a n z n : ||f || H 2 s = ∞ n=0 |a n | 2 (n + 1) −s < ∞ . This is a reproducing kernel Hilbert space on the open unit disc D with reproducing kernel k s (z, w) = ∞ n=0(n + 1) s (zw) n . 42 , 42Theorem 3.2] and [43, Theorem 5.6]-that a matrix-ordered matrix-normed * -vector space V is normal if and only if there exists a completely positive completely isometric selfadjoint linear map φ : V → B(H), where H is a Hilbert space and the space B(H) of bounded linear operators on H is endowed with its canonical matrix-ordered matrix-normed * -vector space structure. 4.2. Matrix-gauged spaces. Suppose that V is a real vector space. A gauge over V is a subadditive and positively-homogeneous function ν : V → [0, +∞). The conjugate gauge ν is defined by ν(x) = ν(−x). The seminorm x ν corresponding to a gauge is given by x ν = max {ν(x), ν(x)}. A gauge is proper if the seminorm · ν is a norm. The positive cone associated with a gauge ν is the set V Lemma 4 . 2 . 42Suppose that A ⊂ B(H) is a C*-algebra such that the identity 1 of B(H) does not belong to A. Let Y be an operator system. Then for any completely positive completely contractive map φ : A → Y there exists a unital completely positive map ψ : span {A, 1} → Y extending φ. Proof. Let Z := span {A, 1} ⊂ B(H) and assume that Y ⊂ B(L) for some Hilbert space L. We have to prove that Definition 4.4. A boundary representation for a matrix-gauged space X ⊂ B(H) is an operator state φ : X → B(H) with the property that any completely positive contractive map ψ : C * (X) → B(H) extending X is an irreducible representation of C * (X). Dilations of rectangular operator states. Suppose that X is an operator space. A rectangular operator state on X is a nondegenerate linear map φ : X → B(H, K) such that φ cb = 1. We say that a rectangular operator statesee [40, Lemma 8.1]. (The Paulsen system is defined in [40, Chapter 8] and [10, Section 1.3] only in the case when H = K. The same proofs from [40, Chapter 8] and [10, Section 1.3] apply with no change to this more general situation.) 2.2. Proposition 2.1. Any rectangular operator state φ : T → B(H, K) on a TRO T ⊂ B (H 0 , K 0 ) can be dilated to a nondegenerate triple morphism θ : T → B( H, K). If H 0 , K 0 , H, K are finite-dimensional, then one can take H and K to be finite-dimensional. shows that Mult(H s , H t ) = {0} if s > t. On the other hand, if s ≤ t, then H s ⊂ H t , hence Mult(H s ) ⊂ Mult(H s , H t ). Since Mult(H s ) at least contains the polynomials, the same is true for Mult(H s , H t ). 1 ϕ 1 δ 1 + · · · + γ * ℓ ϕ ℓ δ ℓ is a proper rectangular convex combination in K that equals ϕ. Then we have that (ξ ⊙ η) * φ (ξ ⊙ η) = (γ 1 ξ ⊙ δ 1 η) * φ 1 (γ 1 ξ ⊙ δ 1 η) + · · · + (γ1Therefore, if we set ψ k := t −2 k (ξγ 1 ⊙ ηδ 1 ) * φ k (ξγ 1 ⊙ ηδ 1 ) for k = 1, 2, . . . , ℓ, we obtain elements ψ 1 , . . . , ψ ℓ of ∆ n,m such that t 2The fact that ξ and η are right invertible now easily implies thatThis conclude the proof that ϕ is a rectangular extreme point of K. We are now ready to conclude the proof that K is the rectangular convex hull of ∂ ρ K. In view of the rectangular bipolar theorem, it is enough to prove that if n, m ∈ N and z ∈ M n,m (X) are such that ϕ (n,m) (z) ≤ 1 for every r ≤ n, s ≤ m, and ϕ ∈ ∂ ρ K s,t , then ψ (n,m) (z) ≤ 1 for every ψ ∈ K n,m . If x ∈ M n,m (X) then we letx be the element I ⊕n x x * I ⊕m ofX. 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The discrete Efimov scaling behavior, well-known in the low-energy spectrum of three-body bound systems for large scattering lengths (unitary limit), is identified in the energy dependence of atommolecule elastic cross-section in mass imbalanced systems. That happens in the collision of a heavy atom with mass mH with a weakly-bound dimer formed by the heavy atom and a lighter one with mass mL mH . Approaching the heavy-light unitary limit the s−wave elastic cross-section σ will present a sequence of zeros/minima at collision energies following closely the Efimov geometrical law. Our results open a new perspective to detect the discrete scaling behavior from low-energy scattering data, which is timely in view of the ongoing experiments with ultra-cold binary mixtures having strong mass asymmetries, such as Lithium and Caesium or Lithium and Ytterbium.

10.1103/physreva.97.012701

1708.00034

df4270e968e5598a95eeeecbc0f7100b174dbb74

Probing the Efimov discrete scaling in atom-molecule collision 31 Jul 2017 M A Shalchi Instituto de Física Teórica Universidade Estadual Paulista 01140-070São PauloSPBrazil M T Yamashita Instituto de Física Teórica Universidade Estadual Paulista 01140-070São PauloSPBrazil M R Hadizadeh Institute for Nuclear and Particle Physics Department of Physics and Astronomy Ohio University 45701AthensOHUSA College of Science and Engineering Central State University 45384WilberforceOHUSA E Garrido Instituto de Estructura de la Materia IEM-CSIC Serrano 123E-28006MadridSpain Lauro Tomio Instituto de Física Teórica Universidade Estadual Paulista 01140-070São PauloSPBrazil DCTA Instituto Tecnológico de Aeronáutica 12228-900São José dos CamposSPBrazil T Frederico DCTA Instituto Tecnológico de Aeronáutica 12228-900São José dos CamposSPBrazil Probing the Efimov discrete scaling in atom-molecule collision 31 Jul 2017(Dated: August 2, 2017) The discrete Efimov scaling behavior, well-known in the low-energy spectrum of three-body bound systems for large scattering lengths (unitary limit), is identified in the energy dependence of atommolecule elastic cross-section in mass imbalanced systems. That happens in the collision of a heavy atom with mass mH with a weakly-bound dimer formed by the heavy atom and a lighter one with mass mL mH . Approaching the heavy-light unitary limit the s−wave elastic cross-section σ will present a sequence of zeros/minima at collision energies following closely the Efimov geometrical law. Our results open a new perspective to detect the discrete scaling behavior from low-energy scattering data, which is timely in view of the ongoing experiments with ultra-cold binary mixtures having strong mass asymmetries, such as Lithium and Caesium or Lithium and Ytterbium. The Efimov effect [1] refers to a discrete scaling symmetry, which emerges in the quantum three-body system at the unitary limit (when the two-body scattering lengths diverge). The optimal condition to observe this discrete scaling symmetry in cold atomic laboratories is now found for heteronuclear three-atom systems with large mass asymmetry and large interspecies scattering lengths. In the Efimov (unitary) limit, the shallow three-body levels are geometrically spaced, namely the ratio between the binding energies of the n and n + 1 levels is given by B = exp (2π/s 0 ), where s 0 is a universal constant, which depends only on the mass ratio and not on the details of the interaction. The energy ratio for three identical bosons is exp (2π/s 0 ) ≈ 515, decreasing for the case of two heavy particles and light one. When m L /m H = 0.01, for example, the value of this energy ratio goes to exp (2π/s 0 ) = 4.698 [2]. The Efimov geometric scaling factor has been measured in a cold-atom experiment with mass-imbalance mixtures of Caesium ( 133 Cs) and Lithium ( 6 Li) gases by different groups [3,4]. The ratio between the positions of two successive peaks in the three-body recombination rate, obtained by varying the large negative scattering lengths (a HL ), was found in close agreement with the theory. Complementary to this finding, a fingerprint of the Efimov scaling can be found in the s−wave ultracold atom-molecule cross-section by varying the incident momentum energy k instead of the scattering lengths. Natural, but not yet evidenced experimentally or theoretically. What we expect is beyond the trimer crossing the corresponding continuum, which creates the resonant enhancement of the inelastic collisions of Caesium atoms with Caesium dimers, as observed by Knoop et al. [5]. Furthermore, there is an evident strong interest in ultra-cold heteronuclear atom-molecule collisions by experimental groups [6][7][8]. Trap setups with ultra-cold de-generated mixtures of alkali-metal-rare-earth molecules with strong mass-imbalanced systems as Ytterbium and Lithium ( 174,173 Yb− 6 Li) have also been reported in Refs. [9,10]. We should mention that on the theory side [11], reactions at ultra-cold temperatures with threebody systems such as 6 Li + 174 Yb 6 Li were also investigated. Therefore, the present possibilities to manipulate collisions with Lithium(Li)-Caesium(Cs) [12] and Ytterbium-Lithium [9,10], as well as the molecules of LiCs and LiYb in ultra-cold experimental setups [13], open new opportunities to probe the discrete Efimov scaling with the large mass asymmetries. This can be achieved by using low-energy collisions of a heavy atom, such as Caesium or Ytterbium, in the weakly-bound molecules as LiCs or LiYb, with m L /m H =0.045 and 0.034, respectively. Going back in time, what was known theoretically from the pioneer works for the tri-nucleon systems [14][15][16][17], was the existence of a pole in the spin doublet s−wave neutron-deuteron k cot δ 0 , which was associated with a virtual state in the tri-nucleon system. Furthermore, such pole is also present in the neutron-19 C scattering [18][19][20], with a corresponding pronounced minimum of the s−wave elastic cross-section. If the Efimov geometrical factor decreases, which is possible with atomic systems, our expectation for H − (HL) collision is that several minima in the s−wave elastic cross-sections or poles in the k cot δ 0 would emerge from the characteristic log-periodic behavior carried by the wave-function in the region where the Efimov long-range potential is dominant and being reflected in a geometrical law for the spacing of the energies of cross-section minima corresponding to these poles. In this letter, we show that the s−wave elastic crosssection for the H − (HL) collision has minima at geometrically spaced incident energies, for large values of a HL near the unitary limit. We compute the s−wave phase-shift using the three-body Faddeev formalism with zeroand short-ranged interactions, as well as by considering the Born-Oppenheimer (BO) approximation [21]. The real part of the s−wave phase shift (δ 0 ) shows zeros and k cot δ 0 has a sequence of poles at colliding energies which tend to follow the Efimov geometric scaling. The BO approximation applied to the H−(HL) system provides a universal long-range attractive 1/R 2 effective potential (R is the relative H − H distance) close to the unitary limit, which acts up to distances ∼ |a HL |, as shown in Ref. [21]. At short distances, the BO potential brings the details of the finite range pairwise potentials expressed as a boundary condition at R 0 << |a HL | that determines the reference energy B 3 . The eigenstates of the H − H effective hamiltonian has the characteristic log-periodic solutions for R 0 R |a HL |, which leads to the geometrical ratio between the binding energies and also to the log-periodic properties of s−wave scattering observables. We extend the procedure used in [21] to the scattering region, considering the collision of a heavy particle in the weakly-bound subsystem of the remaining ones. This approach was used to interpret the results obtained with the renormalized zero-range model [22], as well as with the Gaussian finite-range interactions. To simplify our study, we assume no interaction between the heavy particles and that the heavy-light molecule (HL) has a weakly-bound energy B 2 . When B 2 → 0 the three-body Efimov levels are given by B (n) 3 → e −(2nπ/s0) B 3 , where B 3 ≡ B (0) 3 is the ground state binding energy of the models we use in our approaches to obtain the s−wave cross-sections. We start our analysis by introducing a scaling function for the dimensionless product of the s−wave crosssection and energy. With B 3 and B 2 as the scales of the HHL system and E the colliding energy at the threebody center-of-mass, this function can be written as σ B 3 = S (E/B 3 , B 2 /B 3 , A) ,(1) where A ≡ m L /m H . This is strictly valid at the zerorange limit where B 2 = 1/(2µ HL a 2 HL ), with µ HL being the reduced mass for the HL subsystem. Here and in the next, the units are such that = 1 and m L = 1. The scaling function for A = 0.01 is shown in Fig. 1 for the renormalized zero-range model [19] and for the Gaussian potential model calculated with the method developed in Ref. [23], which was extended to energies above the breakup threshold in Ref. [24]. The Gaussian potential with r 0 being the interaction range is given by V (r) = V 0 e −r 2 /r 2 0 ,(2) where we have used a HL /r 0 = 50 and B 2 /B 3 = 0.0012. Noticeable, are the minima of the s−wave crosssection, where k cot δ 0 has poles. We observe that positions of such poles tend to obey the Efimov law for (k a HL ) −1 → 0. Between the zeros, there is a sequence of maxima for the cross-section where the phase-shift passes through (2n + 1)π/2, as seen in Fig. 1. It is tempting to associate the maxima obtained for the cross-section with resonances; however, a calculation by using the complex scaling method [25] for the Gaussian potential, excludes that. These results are also corroborating the conclusions of [18,20] for the neutron- 19 C system, where no resonance is found when changing the neutron separation energy in 19 C. By considering different mass-ratios A, varying from 0.01 till 0.08, our results for the cross-sections σ (in arbitrary units) are presented in Fig. 2 for three fixed weaklybound two-body energies B 2 /B 3 =0.01, 0.03 and 0.05. In the given eight panels we are presenting σ as a function of E/B 3 . From these panels, one can notice a sequence of zeros (or minima) which can appear for σ as we decrease the mass ratio, by considering a fixed interval for the collision energy, such that E/B 3 < 1. Within this interval, when the mass-imbalance is less pronounced, e.g. for A = 0.08, we can verify the occurrence of only one zero for σ within the given energy range, whereas for A = 0.01 it is possible to verify the existence of up to six zeros. Therefore, the large mass asymmetry (A << 1) is more favorable for the occurrence of several zeros/minima in σ. In order to verify the emergence of a possible scaling factor between the position of successive zeros/minima in the s−wave cross-section, in correspondence with the Efimov bound-state spectrum, one should be able to extrapolate the two-body bound-state energies to the unitary limit (i.e., to B 2 = 0). Corresponding to the upper-left panel of Fig. 2, when B 2 /B 3 = 0.01 and A=0.01, we also have Fig. 1 where E/B 3 was extended up to 1, which showed that it is possible to observe another minimum in σ for a collision energies much larger than the breakup threshold. As we can observe, in this case, the value of the minimum in σ is affected by absorption, an expected behavior for energies above the break-up threshold. Therefore, σ is not being reduced to zero, but have just a minimum, with the value of the energy E also being deviated slightly to the right as B 2 is increased in Fig. 2. The ratio between the energy position of the successive zeros is about the Efimov geometric factor as one can easily check (we will explore such feature in a systematic way later on), and as one could expect it should be distorted by absorption effects, but far away from the breakup threshold. It is noticeable to find minima of the cross-section for E >> B 2 and quite deeply immersed in the three-body continuum, where still the s−wave inelasticity parameter is very close to unity. This astonishing suppression of the breakup channel for energies of about two orders of magnitude the two-body binding is a manifestation of the long-range coherence between the heavy and light particles and the associated diluteness of the target, making it hard to destroy the system, where the light particle binds with any one of the heavy particles and the dynamics is dominated only by the exchange of the light particle between the two heavy ones. The HL molecule becomes invisible to the collision of the heavy one. Semiclassically, the possibility of the destructive interference between the direct trajectory and the one from the exchange process gives the zeros of the phase-shift. The fact that the breakup channel is suppressed is closely related to the non-existence of resonances. In the adiabatic hyper-spherical representation of this mass im-balanced three-body system, it happens that the coupling between the lowest adiabatic channel, which asymptotically goes to the atom-dimer channel, with the breakup channels is weak (see e.g. [24]). In addition, asymptotically the lowest adiabatic hyper-spherical potential is attractive, while the breakup channels have a barrier around ρ ∼ |a HL |. Indeed, in the case of Borromean systems, such barrier makes the Efimov turn to a continuum resonance when |a HL | is decreased [26]. We summarize the findings presented in figures 1 and 2 as: (i) the number of minima of the s−wave cross-section decreases significantly when A and the Efimov ratio increases, and (ii) more minima are seen when B 2 /B 3 decreases. Particularly, with respect to the second point, we found that the zeros of the cross-section are coming out from the scattering threshold and the H −(HL) scattering length passes through zero values when B 2 /B 3 is driven towards the more favorable condition for the Efimov effect. That is the counterpart of the unitary limit where virtual states come from the second energy sheet to become bound states. In the continuum region, zeros and maxima of the cross-section come one by one as B 2 /B 3 → 0, which completes the final picture of the Efimov limit including the scattering region. The manifestation of the Efimov discrete scaling in the atom-molecule collision can be systematically studied by the ratio between the energies of successive zeros/minima as a function of the mass ratio and a dimensionless ratio between two and three-body scales as follows. For that, a scaling function is introduced relating the energies of two adjacent minima obtained for the cross-section σ. Within a convention that E n+1 > E n , this function is given by E n+1 /E n = R 1/(E 1/2 n+1 a HL ), A ,(3) where R (0, A) = e 2π/s0 is the unitary limit. The universal scaling function (3) is shown in Fig. 3 for the extreme case A = 0.01, calculated with the Gaussian and zero range potentials. The curious behavior of the scaling function around the Efimov ratio, indicated by the horizontal dashed line, by departing from the unitary limit decreases, has a minimum and then increases, namely the zeros more distant. Note that we have plotted results for the renormalized zero-range potential with different B 2 and scattering lengths, ranging from 0.001 ≤ B 2 /B 3 ≤ 0.05. With the Gaussian potentials, within our numerical accuracy, we were able to approach more closely the Efimov limit. However, when going to smaller values of 1/(E 1/2 n+1 a HL ), we stand above the breakup threshold and evidently the coupling to the breakup channel affects the ratio, as the figure suggests. The curious behavior of the ratio, namely when starting from smaller to larger collision energies it is first above the Efimov geometrical factor then it decreases and increases again towards it, can be qualitatively understood by considering the collision within the BO ap- proximation. In this case, the effective H − H long-range potential is supplemented by a boundary condition at some short distance R, with the continuity of the logarithmic derivative of the wave-function u(R) imposed at R = a HL . In our illustration, the elastic scattering S-matrix is found from the boundary condition at R = a HL . To make our point clear, we assume no twobody H − H potential; and, we expand the effective BO potential [21], where the leading-order term is ∼ 1/R 2 , and we also consider the effect of the next order term, implying in the inclusion of a Coulomb-like 1/R interaction. Therefore, as one can extract from the expansion of the potential presented in [21], we have the following effective two-body equation for the collision of the heavy particle H with relative momentum k with respect to the HL dimer: − d 2 dR 2 − s 2 0 + 1 4 R 2 g R a HL u(R) = k 2 u(R),(4) where g(y) ≡ 1 + 2y + 2.07y 2 , such that the leading term in the interaction, − s 2 0 + 1/4 /R 2 , provides the Efimov limit. The wave number is related to the collision energy by k = 2µ H,HL E, where µ H,HL ≡ m H (1 + A)/(2 + A). The expansion for g(y) is found by requiring an approximation of the BO potential valid not only for R a HL , but also for R/a HL ∼ 1. With this approximation, the Coulomb-like correction −2 s 2 0 + 1/4 /(a HL R) is added to the Efimov term, as well as a constant which is negligible for larger scattering lengths. As shown by [21], in case of negative-energies we can obtain exact solutions for the Eq. (4), given by Bessel functions in case we consider the leading term 1/R 2 for the interaction. In the present extension to scattering energies, we can also verify analytical solutions for the Eq. (4), which are given by Whittaker functions. This eigenvalue equation has no lower bound energy, namely, the Thomas collapse is present, which requires a short-range scale imposed by a boundary condition at R = R 0 a HL . In what follows, a hard wall will be used, and from the boundary condition at R = a HL the phase-shift is finally obtained. In this way, the log-periodicity of the s−wave phase-shift with the energy is only deformed by the presence of the 1/R contribution. As a result, if the BO potential in Eq. (4) is given only by the Efimov term, the ratio (not plotted in Fig. 3) would approach the Efimov limit monotonically from above when decreasing 1/(E 1/2 n+1 a HL ). The minimum observed in the BO results (dashed-blue curve in Fig. 3) comes from the Coulomb-like correction. As shown by using different values for the position of the hard wall at short distances, there are no significative range corrections. Therefore, we note that the first two terms of the BO potential are quite relevant to provide a qualitative description of the scaling function. This approximation is working surprisingly well in particular for large values of E 1/2 n+1 a HL , when approaching the Efimov limit, considering that in this limit the coupling to the breakup channel (which is not being taken into account) is expected to be relevant. For smaller values of E 1/2 n+1 a HL the expansion of the BO potential starts to breakdown due to its poor efficacy when decreasing the collision energy, with the wavelength being of the order of the scattering length. Practical implications. The poles of k cot δ 0 , which correspond to the zeros/minima of the s−wave cross section, are directly connected with the Efimov spectrum of the heavy-heavy-light (HHL) system near the unitary limit. This is shown by considering a mass-imbalanced system A << 1 with no interaction between the two-heavy particles and with the heavy-light sub-system bound with energy close to zero (near unitary limit). By considering the mass ratio between Li and Yb, A =0.034, the cross-section for the Yb + LiYb collision can in principle present a couple of zeros. We can imagine a situation where a YB−Li is adjusted at some large positive values, with the colliding energy being varied slowly. In this case, σ should present minima at some specific colliding energies, whose positions are approximately geometrically spaced. In conclusion, we suggest as the best possible situation to probe the Efimov discrete scaling in the continuum to consider the atom-molecule scattering with large mass asymmetry through cold collisions, which are now feasible [12]. The challenge in these experiments would be to control the scattering length towards the large values and then observe the cross-section minima at geometrically spaced colliding energies. FIG. 1 : 1The s−wave cross-section is shown as a function of the energy-collision E, for zero-ranged (ZR) (blue-solid lines) and finite-ranged Gaussian (G) (red-dashed lines) potentials, for fixed mass ratio A = 0.01 and given two-body energies (B G 2 is a factor smaller than B ZR 2 to keep both results close to the unitary limit). Results are in units of B3. FIG. 2 : 2The zero-ranged results for σ (in arbitrary units) as functions of E/B3 are given in eight panels, with mass-ratios A as shown inside the frames. In all the panels the two-body energies are fixed such that B2/B3 =0.01 (solid-blue lines), 0.03 (dot-dashed-red lines) and 0.05 (dashed-black lines). FIG. 3 : 3Ratio between the energy positions of the successive zeros (n + 1, n) of the cross-section σ aHL) for mass-ratio A = 0.01. The results obtained with renormalized zero-range (ZR) model, for given two-body energies, are indicated inside the frame. The straight dashed line indicates the Efimov limit for A = 0.01. The solid-line with dots shows the results obtained with the Gaussian potential (2). The Born-Oppenheimer (B-O) results (connected by a dashed-blue line) are shown for three boundary conditions. Acknowledgements:The authors acknowledge partial support from Conselho Nacional de Desenvolvimento Científico 2016/01816-2(MTY) and 2013/26258-4(TF)], Coordenação de Aperfeiçoamento de Pessoal de Nível Superior. Tecnológico, Proc. 8888.1.030363/2013-01LT)], Fundação de Amparoà Pesquisa do Estado de São Paulo ; LT and TF ; Central State University (MRH)] and the Institute of Nuclear and Particle Physics at Ohio University (MRH). Spanish Ministerio de Economia y CompetitividadMAS and MTY) and Senior visitor program in ITA-DCTA (LT)]. National Science Foundation. 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Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel's theorem, about continuous solutions of Fréchet's functional equation ∆ m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel's type theorem for the case of complex valued functions defined over the discrete group Z d . Furthermore, we also state and demonstrate an improved version of Montel's Theorem for complex functions of several real variables and complex functions of several complex variables.

1310.3378

18f1983f3279232befcc83b54e071de03a132b8d

ON MONTEL'S THEOREM IN SEVERAL VARIABLES J M Almira Kh F Abu-Helaiel ON MONTEL'S THEOREM IN SEVERAL VARIABLES Functional EquationsMontel's TheoremInvariant subspaces 2010 Mathematics Subject Classification: 47A1546F0546F1039B2239B32 Recently, the first author of this paper, used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of classical Montel's theorem, about continuous solutions of Fréchet's functional equation ∆ m h f = 0, for real functions (and complex functions) of one real variable. In this paper we use similar ideas to prove a Montel's type theorem for the case of complex valued functions defined over the discrete group Z d . Furthermore, we also state and demonstrate an improved version of Montel's Theorem for complex functions of several real variables and complex functions of several complex variables. Motivation A famous result proved by Jacobi in 1834 claims that if f : C → C is a non constant meromorphic function defined on the complex numbers, then P 0 (f ) = {w ∈ C : f (z + w) = f (z) for all z ∈ C}, the set of periods of f , is a discrete subgroup of (C, +). This reduces the possibilities to the following three cases: P 0 (f ) = {0}, or P 0 (f ) = {nw 1 : n ∈ Z} for a certain complex number w 1 = 0, or P 0 (f ) = {n 1 w 1 + n 2 w 2 : (n 1 , n 2 ) ∈ Z 2 } for certain complex numbers w 1 , w 2 satisfying w 1 w 2 = 0 and w 1 /w 2 ∈ R. In particular, these functions cannot have three independent periods and there exist meromorphic functions f : C → C with two independent periods w 1 , w 2 as soon as w 1 /w 2 ∈ R. These functions are called doubly periodic (or elliptic) and have an important role in complex function theory [9]. Analogously, if the function f : R → R is continuous and non constant, it does not admit two Q-linearly independent periods. Obviously, Jacobi's theorem can be formulated as a result which characterizes the constant functions as those meromorphic functions f : C → C which solve a system of functional equations of the form (1) ∆ h1 f (z) = ∆ h2 f (z) = ∆ h3 f (z) = 0 (z ∈ C) for three independent periods {h 1 , h 2 , h 3 } (i.e., h 1 Z + h 2 Z + h 3 Z is a dense subset of C). For the real case, the result states that, if h 1 , h 2 ∈ R \ {0} are two nonzero real numbers and h 1 /h 2 ∈ Q, the continuous function f : R → R is a constant function if and only if it solves the system of functional equations (2) ∆ h1 f (x) = ∆ h2 f (x) = 0 (x ∈ R). In 1937 Montel [15] proved an interesting nontrivial generalization of Jacobi's theorem. Concretely, he substituted in the equations (1), (2) above the first difference operator ∆ h by the higher differences operator ∆ m+1 h (which is inductively defined by ∆ n+1 h f (x) = ∆ h (∆ n h f )(x), n = 1, 2, · · · ) and proved that these equations are appropriate for the characterization of ordinary polynomials. In particular, he proved the following result: Theorem 1.1 (Montel). Assume that f : C → C is an analytic function which solves a system of functional equations of the form (3) ∆ m+1 h1 f (z) = ∆ m+1 h2 f (z) = ∆ m+1 h3 f (z) = 0 (z ∈ C) for three independent periods {h 1 , h 2 , h 3 }. Then f (z) = a 0 + a 1 z + · · · + a m z m is an ordinary polynomial with complex coefficients and degree ≤ m. Furthermore, if {h 1 , h 2 } ⊂ R \ {0} satisfy h 1 /h 2 ∈ Q, the continuous function f : R → R is an ordinary polynomial with real coefficients and degree ≤ m if and only if it solves the system of functional equations (4) ∆ m+1 h1 f (x) = ∆ m+1 h2 f (x) = 0 (x ∈ R). The functional equation ∆ m+1 h f (x) = 0 had already been introduced in the literature by M. Fréchet in 1909 as a particular case of the functional equation (5) ∆ h1h2···hm+1 f (x) = 0 (x, h 1 , h 2 , . . . , h m+1 ∈ R), where f : R → R and ∆ h1h2···hs f (x) = ∆ h1 (∆ h2···hs f ) (x), s = 2, 3, · · · . In particular, after Fréchet's seminal paper [6], the solutions of (5) are named "polynomials" by the Functional Equations community, since it is known that, under very mild regularity conditions on f , if f : R → R satisfies (5), then f (x) = a 0 + a 1 x + · · · a m x m for all x ∈ R and certain constants a i ∈ R. For example, in order to have this property, it is enough for f being locally bounded [6], [2], but there are stronger results [7], [10], [14], [16]. The equation (5) can be studied for functions f : X → Y whenever X, Y are two Q-vector spaces and the variables x, h 1 , · · · , h m+1 are assumed to be elements of X: (6) ∆ h1h2···hm+1 f (x) = 0 (x, h 1 , h 2 , . . . , h m+1 ∈ X). In this context, the general solutions of (6) are characterized as functions of the form (6) if and only if it satisfies f (x) = A 0 + A 1 (x) + · · · + A m (x), where A 0 is a constant and A k (x) = A k (x, x, · · · , x) for a certain k-additive symmetric function A k : X k → Y (we say that A k is the diagonalization of A k ). In particular, if x ∈ X and r ∈ Q, then f (rx) = A 0 + rA 1 (x) + · · · + r m A m (x). Furthermore, it is known that f : X → Y satisfies(7) ∆ m+1 h f (x) := m+1 k=0 m + 1 k (−1) m+1−k f (x + kh) = 0 (x, h ∈ X). A proof of this fact follows directly from Djoković's Theorem [5] (see also [8,Theorem 7 (−1) 1+···+ s ∆ s α ( 1 ,..., s ) (h1,··· ,hs) f (x + β ( 1,..., s ) (h 1 , · · · , h s )), where α ( 1 ,..., s) (h 1 , · · · , h s ) = (−1) s r=1 r h r r and β ( 1,..., s) (h 1 , · · · , h s ) = s r=1 r h r . In his seminal paper [15], Montel also studied the equation (7) for X = R d , with d > 1, and f : R d → C continuous, and for X = C d and f : C d → C analytic. Concretely, he stated (and gave a proof for d = 2) the following result. Theorem 1.2 (Montel's Theorem in several variables). Let {h 1 , · · · , h } ⊂ R d be such that (9) h 1 Z + h 2 Z + · · · + h Z is a dense subset of R d , and let f ∈ C(R d , C) be such that ∆ m h k (f ) = 0, k = 1, · · · , . Then f (x) = |α|<N a α x α for some N ∈ N, some complex numbers a α , and all x ∈ R d . Thus, f is an ordinary complex valued polynomial in d real variables. Consequently, if d = 2k, {h i } i=1 satisfies (9), the function f : C k → C is holomorphic and ∆ m h k (f ) = 0, k = 1, · · · , , then f (z) = |α|<N a α z α is an ordinary complex valued polynomial in k complex variables. Remark 1.3. The finitely generated subgroups of (R d , +) which are dense in R d have been deeply studied, and many characterizations of them are known [20,Proposition 4.3]. For example, a theorem by Kronecker guarantees that, given θ 1 , θ 2 , · · · , θ d ∈ R, the group Z d + (θ 1 , θ 2 , · · · , θ d )Z (which is generated by d + 1 elements) is dense in R d if and only if, n 0 + n 1 θ 1 + · · · + n d θ d = 0, for every (n 0 , n 1 , · · · , n d ) ∈ Z d+1 (see, e.g., [20,Theorem 4.1], for the proof of this result). In [1] the author used the structure of finite dimensional translation invariant subspaces of C(R, C) to give a new proof of Theorem 1.1. In this paper we use similar ideas to prove a Montel's type theorem for the case of complex valued functions defined over the discrete group Z d . Furthermore, we also state and demonstrate an improved version of Theorem 1.2. In all the paper we use the following standard notation: If α = (α 1 , · · · , α d ) ∈ N d , x = (x 1 , · · · , x d ) ∈ C d , λ = (λ 1 , · · · , λ d ) ∈ (C \ {0}) d , and n = (n 1 , · · · , n d ) ∈ Z d , then n α = n α1 1 n α2 2 · · · n α d d , x α = x α1 1 x α2 2 · · · x α d d , λ n = λ n1 1 · · · λ n d d , |α| = d k=1 α k . Π d m denotes the set of complex polynomials in d vari- ables with total degree ≤ m (when d = 1 we write Π m instead of Π 1 m ). Finally, f ∈ C(Z d , C) is named an "exponential monomial" if there exists a polynomial in the variables n 1 , · · · , n d ∈ Z, p(n) = |α|≤N a α n α , and a vector λ ∈ (C \ {0}) d , such that f (n) = p(n)λ n , for all n ∈ Z d . Main resuts For the discrete case, we will need to use the following well known result by M. Lefranc, whose proof is based on algebraic geometry arguments (see, e.g., [4], [12], [18]). Theorem 2.1 (Lefranc, 1958). Assume that V is a closed vector subspace of C(Z d , C) which is invariant by translations, and let Γ V denote the set of exponential monomials which belong to V . Then V = span(Γ V ). An immediate consequence of Lefranc's Theorem is the following Corollary 2.2. If V is a finite dimensional vector subspace of C(Z d , C) which is invariant by translations, then V ⊆ W = span( s k=0 {n α λ n k : |α| ≤ m k − 1}) for certain s ∈ N, {λ k } s k=0 ⊆ (C \ {0}) d and {m k } s k=0 ⊆ N. Remark 2.3. Here and, in all what follows, we assume that λ 0 = (1, · · · , 1) and that m 0 = 0 means that there are no elements of the form n α in the basis. We always assume that m k ≥ 1 for k = 1, · · · , s. Let us now state two technical results, which are extremely important for our arguments in this section. Lemma 2.4. Let E be a vector space and L : E → E be a linear operator defined on E. If V ⊂ E is an L m -invariant subspace of E, then the space 2 m L (V ) = V + L(V ) + L 2 (V ) + · · · + L m (V ) is L-invariant. Furthermore, 2 m L (V ) is the smallest L-invariant subspace of E containing V . Proof. The linearity of L implies that L(2 m L (V )) = L(V ) + L 2 (V ) + L 3 (V ) + · · · + L m (V ) + L m+1 (V ). Now, L m+1 (V ) = L(L m (V )) ⊆ L(V ) and L(V ) + L(V ) = L(V ), so that L(2 m L (V )) ⊆ 2 m L (V ). On the other hand, let us assume that V ⊆ F ⊆ E and F is an L-invariant subspace of E. If {v k } m k=0 ⊆ V , then L k (v k ) ∈ F for all k ∈ {0, 1, · · · , m}, so that v 0 + L(v 1 ) + · · · + L m (v m ) ∈ F . This proves that 2 m L (V ) ⊆ F . Lemma 2.5. Let E be a vector space and L 1 , L 2 , · · · , L t : E → E be linear operators defined on E. Assume that L i L j = L j L i for all i = j. If V ⊂ E is a vector subspace of E which satisfies t i=1 L m i (V ) ⊆ V , then m L1,L2,··· ,Lt (V ) = 2 m Lt (2 m Lt−1 (· · · (2 m L1 (V )) · · · ) ) is L i -invariant for i = 1, 2, · · · , t, and contains V . L m t ( m L1,L2,··· ,Lt−1 (V )) = L m t (2 m Lt−1 ( m L1,L2,··· ,Lt−2 (V ))) = L m t ( m L1,L2,··· ,Lt−2 (V ) + L t−1 ( m L1,L2,··· ,Lt−2 (V )) + · · · + L m t−1 ( m L1,L2,··· ,Lt−2 (V ))) = L m t ( m L1,L2,··· ,Lt−2 (V )) + L m t (L t−1 ( m L1,L2,··· ,Lt−2 (V ))) + · · · + L m t (L m t−1 ( m L1,L2,··· ,Lt−2 (V ))) = L m t ( m L1,L2,··· ,Lt−2 (V )) + L t−1 (L m t ( m L1,L2,··· ,Lt−2 (V ))) + · · · + L m t−1 (L m t ( m L1,L2,··· ,Lt−2 (V ))) = 2 m Lt−1 (L m t ( m L1,L2,··· ,Lt−2 (V ))), since L t , L t−1 commute. Repeating this process, we get L m t ( m L1,L2,··· ,Lt−1 (V )) = 2 m Lt−1 (L m t ( m L1,L2,··· ,Lt−2 (V ))) = 2 m Lt−1 (2 m Lt−2 (L m t ( m L1,L2,··· ,Lt−3 (V )))) . . . = 2 m Lt−1 (2 m Lt−2 (· · · (2 m L1 (L m t (V ))) · · · )) ⊆ 2 m Lt−1 (2 m Lt−2 (· · · (2 m L1 (V )) · · · )) = m L1,L2,··· ,Lt−1 (V ), since L t L i = L i L t for all i < t and L m t (V ) ⊆ V . This proves that m L1,L2,··· ,Lt−1 (V ) is L m t -invariant, and Lemma 2.4 implies that m L1,L2,··· ,Lt (V ) = 2 m Lt ( m L1,L2,··· ,Lt−1 (V )) is L t -invariant. On the other hand, given i < t, the identity L t L i = L i L t , and the fact that L i ( m L1,L2,··· ,Lt−1 (V )) ⊆ m L1,L2,··· ,Lt−1 (V ) (which follows from the hypothesis of induction) imply that L i ( m L1,L2,··· ,Lt (V )) = L i (2 m Lt ( m L1,L2,··· ,Lt−1 (V ))) = L i ( m L1,L2,··· ,Lt−1 (V ) + L t ( m L1,L2,··· ,Lt−1 (V )) + · · · + L m t ( m L1,L2,··· ,Lt−1 (V ))) = L i ( m L1,L2,··· ,Lt−1 (V )) + L t (L i ( m L1,L2,··· ,Lt−1 (V ))) + · · · + L m t (L i ( m L1,L2,··· ,Lt−1 (V ))) ⊆ m L1,L2,··· ,Lt−1 (V ) + L t ( m L1,L2,··· ,Lt−1 (V )) + · · · + L m t ( m L1,L2,··· ,Lt−1 (V )) = 2 m Lt ( m L1,L2,··· ,Lt−1 (V )) = m L1,L2,··· ,Lt (V ), so that m L1,L2,··· ,Lt (V ) is L i -invariant for all i ≤ t. Finally, it is clear that V ⊆ 2 m L1 (V ) ⊆ · · · ⊆ m L1,L2,··· ,Lt (V ). It is important to note that there are many examples of linear transformations T : E → E such that T and T m have different sets of invariant subspaces. For example, if T is not of the form T = λI for any scalar λ and satisfies T m = I or T m = 0, then all subspaces of E are invariant subspaces of T m and, on the other hand, there exists v ∈ E such that T v ∈ span{v}, so that span{v} is not an invariant subspace of T . On the other hand, as the following lemma proves, sometimes it is possible to show that T and T m share the same set of invariant subspaces. Lemma 2.6. Let K be a field and E be a K-vector space with basis β = {v k } n k=1 and let m ∈ N, m ≥ 1. Assume that T : E → E is such that A = M β (T ) is of the form A = λI + B, where λ ∈ C and B is strictly upper triangular with nonzero entries in the first superdiagonal. Then the full list of T -invariant subspaces of E is given by V 0 = {0} and V k = span{v 1 , · · · , v k }, k = 1, 2, · · · , n. Furthermore, if λ = 0, then T m has the same invariant subspaces as T . Proof. Assume that A = M β (T ) is of the form A = λI + B, where λ = 0 and B is strictly upper triangular with nonzero entries in the first superdiagonal, and let V = {0} be a T -invariant subspace. Let v ∈ V , v = a 1 v 1 + · · · + a s v s , a s = 0. Then w = T v − λv ∈ V and a simple computation shows that w = α 1 v 1 + · · · + α s−1 v s−1 with α s−1 = b s−1,s a s = 0, where B = (b ij ) n i,j=1 . It follows that, if V is T -invariant and v = a 1 v 1 + · · · + a s v s ∈ V with a s = 0, then span{v 1 , v 2 , · · · , v s } ⊆ V . Take k 0 = max{k : exists v ∈ V, v = a 1 v 1 + · · · + a s v s and a s = 0}. Then V = span{v 1 , · · · , v k0 }. Finally, it is clear that all the spaces V k = span{v 1 , · · · , v k }, k = 1, 2, · · · , n are T -invariant. Let us now assume that λ = 0. To compute the invariant subspaces of T m we take into account that A m = M β (T m ) and A m = (λI + B) m = m k=0 m k (−1) m−k B k = λ m I + mλ m−1 B + m k=2 m k (−1) m−k B k . This shows that A m = λ m I + C, with C strictly upper triangular with nonzero entries in the first superdiagonal, since the only contribution to the first superdiagonal of C = mλ m−1 B + m k=2 m k (−1) m−k B k is got from mλ m−1 B, and λ = 0 . It follows that we can apply the first part of the lemma to the linear transformation T m , which concludes the proof. Proposition 2.7. Assume that V is a finite dimensional subspace of C(Z d , C), {h 1 , · · · , h t } ⊂ Z d and h 1 Z + h 2 Z + · · · + h t Z = Z d . If ∆ m h k (V ) ⊆ V , k = 1, · · · , t, then there exists a finite dimensional subspace W of C(Z d , C) which is invariant by translations and contains V . Consequently, all elements of V are exponential polynomials, f (n) = s k=0 ( |α|≤m k a k,α n α )λ n k Proof. We apply Lemma 2.5 with E = C(Z d , C), L i = ∆ hi , i = 1, · · · , t, to conclude that V ⊆ W = m ∆ h 1 ,∆ h 2 ,··· ,∆ h t (V ) and W is a finite dimensional subspace of C(Z d ) satisfying ∆ hi (W ) ⊆ W , i = 1, 2, · · · , t. Hence W is invariant by translations, since h 1 Z + h 2 Z + · · · + h t Z = Z d . Applying Corollary 2.2, we conclude that all elements of W (hence, also all elements of V ) are exponential polynomials. Theorem 2.8 (Discrete Montel's Theorem). Let {h 1 , · · · , h t } ⊂ Z d be such that h 1 Z + h 2 Z + · · · + h t Z = Z d , and let f ∈ C(Z d , C). If ∆ m h k (f ) = 0, k = 1, · · · , t, then f (n) = |α|<N a α n α for some N ∈ N, some complex numbers a α , and all n ∈ Z d . In other words: f is an ordinary polynomial on Z d . Furthermore, if d = 1, then f (n) = a 0 + a 1 n + · · · + a m−1 n m−1 is an ordinary polynomial on Z, of degree ≤ m − 1. Proof. Assume that ∆ m h k (f ) = 0, k = 1, · · · , t. Then V = span{f } is a one dimensional subspace of C(Z d , C) which satisfies the hypotheses of Proposition 2.7. Hence all elements of V are exponential polynomials. In particular, f is an exponential polynomial, (10) f (n) = s k=0 ( |α|≤m k a k,α n α )λ n k and we can assume, with no loss of generality, that λ 0 = (1, 1, · · · , 1), m 0 ≥ m − 1, and λ i = λ j for all i = j. Let (11) β = {n α λ n k , 0 ≤ |α| ≤ m k and k = 0, 1, 2, · · · , s} and E = span{β} be an space with a basis of the form (11) which contains V . Let us consider the linear map ∆ h : S → S induced by the operator ∆ h when restricted to E. Obviously, E = P ⊕E 1 ⊕E 2 ⊕· · ·⊕E s , where P = span{n α } 0≤|α|≤m0 and E k = span{n α λ n k } 0≤|α|≤m k , k = 1, 2, · · · , s. Furthermore, ∆ h (P ) ⊆ P and ∆ h (E k ) ⊆ E k for k = 1, 2, · · · , s, since q α (n) = ∆ h n α = (n + h) α − n α is a polynomial of degree ≤ |α| − 1 and ∆ h (n α λ n k ) = ((n + h) α λ h k − n α )λ n k = (n α (λ h k − 1) + q α (n))λ n k . It follows that, for any h ∈ Z d , the operator ∆ m h also satisfies ∆ h (P ) ⊆ P and ∆ h (E k ) ⊆ E k for k = 1, 2, · · · , s, so that, if g ∈ E, then ∆ m h g = 0 if and only if ∆ m h p = 0, ∆ m h b k = 0, k = 1, · · · , s, where g = p + b 1 + · · · + b s , p ∈ P , b k ∈ E k , k = 1, · · · , s. Let us fix k ∈ {1, · · · , s} and let us consider the operator (∆ h ) |E k : E k → E k . The matrix A k associated to this operator with respect to the basis β k = {n α λ n k } 0≤|α|≤m k , which we consider ordered by the graduated lexicographic order, n α λ n k ≤ grlex n γ λ n k ⇔ (|α| ≤ |γ| or (|α| = |γ| and α ≤ lex γ)) , is upper triangular and the terms in the diagonal are all equal to d k (h) = λ h k − 1 (Recall that α ≤ lex γ if and only if α k0 − γ k0 < 0, where k 0 = max{k ∈ {1, 2, · · · , d} : α k = γ k }). Obviously, these computations imply that, if d k (h) = 0, then (∆ h ) |E k is invertible and, in particular, ∆ h b k = 0 and b k ∈ E k imply b k = 0. Let us now study the equations d k (h j ) = 0, with h j = (h j,1 , · · · , h j,d ), j = 1, · · · , t, being the vectors fixed by the hypotheses of the theorem. We know that λ k = (ρ k,1 e 2πiθ k,1 , ρ 2,k e 2πiθ 2,k , · · · , ρ k,d e 2πiθ k,d ) ∈ (C \ {0}) d with ρ k,d = 1 for at least one of the values k, since λ k = (1, 1, · · · , 1). Hence d k (h j ) = 0 if and only if (ρ k,1 e 2πiθ k,1 ) hj,1 (ρ k,2 e 2πiθ k,2 ) hj,2 · · · (ρ k,d e 2πiθ k,d ) h j,d = 1, which is equivalent to the system of relations h j,1 log ρ k,1 + h j,2 log ρ k,2 + · · · + h j,d log ρ k,d = 0 θ k,1 h j,1 + θ k,2 h j,2 + · · · + θ k,d h j,d ∈ Z . Now, h 1 Z + h 2 Z + · · · + h t Z = Z d , so that {h 1 , h 2 , · · · , h t } contains a basis of R d . Furthermore, w k = (log ρ k,1 , log ρ k,2 , · · · , log ρ k,d ) ∈ R d \ {(0, 0, · · · , 0)}. This implies that there exists j k ∈ {1, · · · , t} such that h j k is not orthogonal to w k . In particular, d k (h j k ) = 0 and (∆ hj k ) |E k is invertible. Consider the function f given by (10). Then f = p 0 + b 1 + · · · + b s ∈ E (with p 0 = |α|≤m0 a 0,α n α ∈ P and b k = ( |α|≤m k a k,α n α )λ n k ∈ E k , k = 1, · · · , s) and ∆ m hj f = 0 for all j. For every k ∈ 1, · · · , s we have that ∆ m hj k b k = 0, which implies b k = 0, since (∆ hj k ) |E k is invertible. Hence f = p 0 , which proves the first part of the theorem. Let us now assume that d = 1. We know that f (n) = a 0 +a 1 n+· · ·+a m0 n m0 is an ordinary polynomial (with m 0 ≥ m − 1) and we want to demonstrate that deg(f ) ≤ m − 1. To prove this assertion we fix our attention on the matrix A associated to ∆ h : Π m0 → Π m0 with respect to the basis β 0 = {n k } m0 k=0 . A simple computation shows that (12) A =        0 h h 2 · · · h m0 0 0 2h · · · m0 1 h m0−1 . . . . . . . . . · · · . . . 0 0 0 · · · m0 m0−1 h 0 0 0 · · · 0        , so that the matrix associated to (∆ m h ) |Πm 0 with respect to the basis β 0 is given by A m . Now, ker(A m ) = span{(0, 0, · · · , 0, 1 (i-th position) , 0, · · · , 0) : i = 1, 2, · · · , m}. Hence rank(A m ) = m 0 + 1 − m = dim C (Π m0 ) − m and dim C ker(A m ) = m. On the other hand, another simple computation shows that the space of ordinary polynomials of degree ≤ m − 1, Π m−1 , is contained into ker(∆ m h ). Hence ker(∆ m h ) = Π m−1 , since both spaces have the same dimension. This, in conjunction with f ∈ ker(∆ m h ), ends the proof. Corollary 2.9. Let us assume that h 1 , h 2 ∈ Z are coprime numbers. If f ∈ C(Z, C) satisfies ∆ m h k (f ) = 0, k = 1, 2, then f (n) = a 0 + a 1 n + · · · + a m−1 n m−1 is an ordinary polynomial of degree ≤ m − 1. Proof. If h 1 , h 2 are coprime then, by Bézout's identity, there exists a, b ∈ Z such that 1 = ah 1 + bh 2 . Hence h 1 Z + h 2 Z = Z and the result follows from Theorem 2.8. The estimation of the degree of f in Theorem 2.8 when d > 1 can be achieved in certain special cases. To prove a result of this type, we introduce the following technical result. Lemma 2.10. Let p(x) = |α|≤N a α x α ∈ C[x 1 , x 2 , · · · , x d ] be a complex polynomial of d complex variables with total degree ≤ N . Assume that, for all a = (a 1 , · · · , a i−1 , a i+1 , · · · , a d ) ∈ C d−1 and all i ∈ {1, · · · , d}, the polynomial g a,i (t) = p(a 1 , · · · , a i−1 , t, a i+1 , · · · , a d ) satisfies g a,i ∈ Π m . Then N ≤ md. Furthermore, the extremal case is attained for p(x) = x m 1 x m 2 · · · x m d . Proof. Let i ∈ {1, · · · , d} be fixed and let us assume that α = (α 1 , · · · , α d ) satisfies a α = 0 and α i ≥ m+1. Then p(x) = N k=0 b k (x 1 , x 2 , · · · , x i−1 , x i+1 , · · · , x d )x k i ; and b αi (x 1 , · · · , x i−1 , x i+1 , · · · , x d ) = 0. In particular, there exists a = (a 1 , · · · , a i−1 , a i+1 , · · · , a d ) ∈ C d−1 such that b αi (a) = 0 since b αi is a nonzero polynomial. It follows that g a,i (t) = p(a 1 , · · · , a i−1 , t, a i+1 , · · · , a d ) = N k=0 b k (a 1 , a 2 , · · · , a i−1 , a i+1 , · · · , a d )t k is a polynomial of degree bigger than m, which contradicts our hypotheses. Hence a α = 0 implies max{α 1 , α 2 , · · · , α d } ≤ m and N ≤ md. Corollary 2.11. Let e i = (0, · · · , 0, 1 i−th position , 0, · · · , 0) ∈ Z d , i = 1, · · · , d and let us assume that f ∈ C(Z d , C) satisfies ∆ m ei f = 0 for i = 1, · · · , d. Then f (n) = |α|≤(m−1)d a α n α for all n ∈ Z d and certain complex values a α . Furthermore, the extremal case is attained for f (n) = n m−1 1 n m−1 2 · · · n m−1 d . Proof. It follows from e 1 Z + · · · + e d Z = Z d and Theorem 2.8 that f (n) = |α|<N a α n α for some N ∈ N, some complex numbers a α , and all n ∈ Z d . Let us introduce the polynomial p(x) = |α|<N a α x α ∈ C[x 1 , x 2 , · · · , x d ], and let us take i ∈ {1, · · · , d} and a = (a 1 , · · · , a i−1 , a i+1 , · · · , a d ) ∈ C d−1 . Then q a,i (t) = p(a 1 , · · · , a i−1 , t, a i+1 , · · · , a d ) is a polynomial in the complex variable t and φ a,i (x) = ∆ m ei q a,i (t) ∈ C[t] satisfies (φ a,i ) |Z = 0, so that φ a,i = 0 and q a,i is a polynomial of degree ≤ m − 1. The result follows by applying Lemma 2.10 to p. To prove the last claim of the corollary it is enough to check that f (n) = n m−1 1 n m−1 2 · · · n m−1 d satisfies ∆ m ei f = 0 for i = 1, · · · , d, which is an easy exercise. In [11,Lemma 15.9.4.] it is proved that, if f : R d → R is an ordinary polynomial separately in each one of its variables, and the partial degrees of f are uniformly bounded by a certain natural number m, independently of the concrete variable and independently of the values of the other variables, then f is itself an ordinary polynomial. Corollary 2.11 proves that an analogous result holds for functions f : Z d → C and gives a concrete upper bound for the total degree of f . In [17,Theorem 14] the authors proved that, if K is a field and f : K d → K is an ordinary polynomial separately in each variable (but without any other assumption about the degrees of the polynomials appearing in this way), then f is an ordinary polynomial jointly in all its variables, provided that K is finite or uncountable. Furthermore, for the case of K with infinite countable cardinal, they constructed a function χ : K d → K which is not a polynomial function on K d , but it is an ordinary polynomial separately in each variable. This construction can be translated to the case of functions Z d → K. Indeed, if K has characteristic zero, the result follows just by considering the function χ |Z d . An interesting open question is if these functions can also be constructed from Z d into Z. Let X d denote either C(R d , C) or the space of complex valued distributions defined on R d . The same kind of arguments we have used for the proof of Theorem 2.8, with small variations, jointly with Anselone-Korevaar's characterization of translation invariant finite dimensional subspaces of X d [3] as the spaces admitting a basis of the form (13) β = {x (α k,1 ,··· ,α k,d ) e <x,λ k > , 0 ≤ α k,i ≤ m k,i − 1 for i = 1, · · · , d and k = 0, 1, 2, · · · , s}, (where λ 0 = 1 and m k,i = 0 means that the variable x i does not appear in the polynomial part of the exponential monomials of the form x α e <x,λ k > ) lead to a proof of the following result, which is an improvement of classical Montel's theorem in several variables, since it is formulated for distributions: Theorem 2.12 (Montel's Theorem for distributions). Let {h 1 , · · · , h } ⊂ R d be such that h 1 Z+· · ·+h Z is a dense subset of R d , and let f ∈ X d . If ∆ m h k (f ) = 0, k = 1, · · · , , then f (x) = |α|<N a α x α for some N ∈ N, some complex numbers a α , and all x ∈ R d . Thus, f is an ordinary complex valued polynomial in d real variables. Furthermore, if d = 1, then f (x) = a 1 + a 1 x + · · · + a m−1 x m−1 is an ordinary polynomial of degree ≤ m − 1. Proof. A repetition of the proof of Proposition 2.7, but considering the operators ∆ hi defined for distributions f ∈ X d , shows that every finite dimensional subspace V of X d which satisfies (14) ∆ m hi (V ) ⊆ V for i = 1, · · · , , is included into an space W which admits a basis of the form (13), since we can obtain a subspace W ⊆ X d which contains V and it is invariant under the translations τ hi : X d → X d , i = 1, · · · , , and, which implies that W is invariant under all translations τ h with h ∈ h 1 Z + · · · + h Z, which is a dense subset of R d . Hence W is translation invariant and, thanks to Anselone-Korevaar's theorem, admits a basis of the form (13). Now we include the space W into another space which admits a basis of the form (15) β = {x α } 0≤|α|≤m0 ∪ {x α e <x,λ k > , 0 ≤ |α| ≤ m k and k = 1, 2, · · · , s}, since these bases are better suited for our computations. The proof ends taking V = span{f } with ∆ m h k (f ) = 0, k = 1, · · · , and repeating the arguments of Theorem 2.8 with small modifications. Remark 2.13. In [19, Page 78, Theorem 10.2], Anselone-Korevaar's theorem was generalized to the context of measurable complex valued functions defined on locally compact abelian groups. This produces the expectative of proving a new version of Theorem 2.12, by changing the condition f ∈ X d by the hypothesis that f : R d → C is a measurable function. In other words, the question if Montel's theorem holds true for measurable functions arises in a natural way. Unfortunately, the answer is negative: Montel's theorem fails for measurable functions, for all d ≥ 1. Indeed, let us assume that f is measurable, ∆ m h k (f ) = 0, k = 1, · · · , , with {h 1 , · · · , h } ⊂ R d such that h 1 Z+· · ·+h Z is a dense subset of R d , and let us apply the arguments in Lemma 2.5 to V = span{f } with L i = ∆ hi . Then V is contained into a finite dimensional space W which is invariant under all translations of the form τ h with h ∈ h 1 Z + · · · + h d+1 Z. But this does not imply that W is translation invariant! Obviously, this explains why our argument fails. To prove that Montel's theorem fails for measurable functions we give a counterexample. Assume d = 1 and h 1 , h 2 ∈ R \ {0}, h 1 /h 2 ∈ Q. Define f (ph 1 + qh 2 ) = pq for all p, q ∈ Z, and f (x) = 0 for all x ∈ h 1 Z + h 2 Z. Then f is measurable and a simple computation shows that ∆ n h1 f = ∆ n h2 f = 0 for all n ≥ 2. On the other hand, f is not a polynomial. In fact f is also not a solution of Fréchet's functional equation. Similar examples can be constructed for all d > 1. Proof. We proceed by induction on t. For t = 1 the result follows from Lemma 2.4. Assume the result holds true for t − 1. Obviously, m L1,L2,··· ,Lt (V ) = 2 m Lt ( m L1,L2,··· ,Lt−1 (V )). By definition, Montel's Theorem and subspaces of distributions which are ∆ m -invariant, to appear in Numer. J M Almira, 10.1080/01630563.2013.813537Funct. Anal. Optimization. J. M. Almira, Montel's Theorem and subspaces of distributions which are ∆ m -invariant, to appear in Numer. Funct. Anal. Optimization (2013) DOI:10.1080/01630563.2013.813537. On solutions of the Fréchet functional equation. J M Almira, A J López-Moreno, J. Math. Anal. Appl. 332J. M. Almira, A. J. López-Moreno, On solutions of the Fréchet functional equation, J. Math. Anal. Appl. 332 (2007), 1119-1133. Translation invariant subspaces of finite dimension. P M Anselone, J Korevaar, Proc. Amer. Math. Soc. 15P. M. Anselone, J. Korevaar, Translation invariant subspaces of finite dimension, Proc. Amer. Math. Soc. 15 (1964), 747-752. Ron Polynomial ideals and multivariate splines. C De Boor, A , Multivariate Approximation Theory IV, ISNM 90. C. Chui, W. Schempp, and K. ZellerBirkhäuser VerlagC. de Boor, A. Ron Polynomial ideals and multivariate splines, in Multivariate Approximation Theory IV, ISNM 90, C. Chui, W. Schempp, and K. Zeller (eds.), Birkhäuser Verlag (1989) 31-40. A representation theorem for (X 1 − 1)(X 2 − 1) · · · (Xn − 1) and its applications. D Z Djoković, Ann. Polon. Math. 22D. Z. Djoković, A representation theorem for (X 1 − 1)(X 2 − 1) · · · (Xn − 1) and its applications, Ann. Polon. Math. 22 (1969/1970) 189-198. Une definition fonctionelle des polynomes. M Fréchet, Nouv. Ann. 9M. Fréchet, Une definition fonctionelle des polynomes, Nouv. Ann. 9 (1909) 145-162. On some properties of polynomial functions. R Ger, Ann. Pol. Math. 25R. Ger, On some properties of polynomial functions, Ann. Pol. Math. 25 (1971) 195-203. Stability of functional equations in several variables. D H Hyers, G Isac, T M Rassias, BirkhäuserD. H. Hyers, G. Isac, T. M. Rassias, Stability of functional equations in several variables, Birkhäuser, 1998. Complex functions. An algebraic and geometric viewpoint. G A Jones, D Singerman, Cambridge Univ. PressG. A. Jones, D. Singerman, Complex functions. An algebraic and geometric viewpoint, Cambridge Univ. Press, 1987. On measurable functions with vanishing differences. M Kuczma, Ann. Math. Sil. 6M. Kuczma, On measurable functions with vanishing differences, Ann. Math. Sil. 6 (1992) 42-60. An introduction to the theory of functional equations and inequalities, Second Edition. M Kuczma, Birkhäuser VerlagM. Kuczma, An introduction to the theory of functional equations and inequalities, Second Edition, Birkhäuser Verlag, 2009. Analyse spectrale sur Z n. M Lefranc, C. R. Acad. Sci. París. 246M. Lefranc, Analyse spectrale sur Z n , C. R. Acad. Sci. París 246 (1958) 1951-1953. Finite dimensional translation invariant spaces. K O Leland, Amer. Math. Monthly. 75K. O. Leland, Finite dimensional translation invariant spaces, Amer. Math. Monthly 75 (1968) 757-758. On vanishing n-th ordered differences and Hamel bases. M A Mckiernan, Ann. Pol. Math. 19M. A. Mckiernan, On vanishing n-th ordered differences and Hamel bases, Ann. Pol. Math. 19 (1967) 331-336. Sur quelques extensions d'un théorème de Jacobi. P Montel, Prace Matematyczno-Fizyczne. 441P. Montel, Sur quelques extensions d'un théorème de Jacobi, Prace Matematyczno-Fizyczne 44 (1) (1937) 315-329. The Fréchet functional equation with application to the stability of certain operators. D Popa, I Rasa, Journal of Approximation Theory. 1641D. Popa, I. Rasa, The Fréchet functional equation with application to the stability of certain operators, Journal of Approximation Theory 164 (1) (2012) 138-144. Generalized polynomials in one and in several variables. W Prager, J Schwaiger, Mathematica Pannonica. 202W. Prager, J. Schwaiger, Generalized polynomials in one and in several variables, Mathematica Pannonica 20 (2) (2009) 189-208. L Székelyhidi, Discrete Spectral Synthesis and its Applications. SpringerL. Székelyhidi, Discrete Spectral Synthesis and its Applications, Springer Monographs on Mathematics, Springer, 2006. L Székelyhidi, Convolution type functional equations on topological abelian groups. World ScientificL. Székelyhidi, Convolution type functional equations on topological abelian groups, World Scientific, 1991. Topologie des Points Rationnels, Cours de Troisième Cycle. M Waldschmidt, 95M. Waldschmidt, Topologie des Points Rationnels, Cours de Troisième Cycle 1994/95 . P Université, M Et, Curie, Paris VIUniversité P. et M. Curie (Paris VI), 1995.

Pairwise key establishment is one of the fundamental security services in sensor networks which enables sensor nodes in a sensor network to communicate securely with each other using cryptographic techniques. It is not feasible to apply traditional public key management techniques in resource-constrained sensor nodes, and also because the sensor nodes are vulnerable to physical capture. In this paper, we introduce a new scheme called the identity based key pre-distribution using a pseudo random function (IBPRF), which has better trade-off between communication overhead, network connectivity and resilience against node capture compared to the other key pre-distribution schemes. Our scheme can be easily adapted in mobile sensor networks. This scheme supports the addition of new sensor nodes after the initial deployment and also works for any deployment topology. In addition, we propose an improved version of our scheme to support large sensor networks.

1103.4676

adff1ab0c2536c93d7f97d235f4843fdcd635372

An Identity Based Key Management Scheme in Wireless Sensor Networks 24 Mar 2011 Ashok Kumar Das [email protected] Department of Computer Science and Engineering Department of Mathematics Indian Institute of Technology Indian Institute of Technology 721 302, 721, 302Kharagpur, KharagpurIndia, India Debasis Giri [email protected] Department of Computer Science and Engineering Department of Mathematics Indian Institute of Technology Indian Institute of Technology 721 302, 721, 302Kharagpur, KharagpurIndia, India An Identity Based Key Management Scheme in Wireless Sensor Networks 24 Mar 2011arXiv:1103.4676v1 [cs.CR] Pairwise key establishment is one of the fundamental security services in sensor networks which enables sensor nodes in a sensor network to communicate securely with each other using cryptographic techniques. It is not feasible to apply traditional public key management techniques in resource-constrained sensor nodes, and also because the sensor nodes are vulnerable to physical capture. In this paper, we introduce a new scheme called the identity based key pre-distribution using a pseudo random function (IBPRF), which has better trade-off between communication overhead, network connectivity and resilience against node capture compared to the other key pre-distribution schemes. Our scheme can be easily adapted in mobile sensor networks. This scheme supports the addition of new sensor nodes after the initial deployment and also works for any deployment topology. In addition, we propose an improved version of our scheme to support large sensor networks. Introduction In a sensor network, many tiny computing nodes called sensors, are scattered in an area for the purpose of sensing some data and transmitting the data to nearby base stations for further processing. The transmission between the sensors is done by short range radio communications. The base station is assumed to be computationally well-equipped whereas the sensor nodes are resource-starved. Such networks are used in many applications including tracking of objects in an enemy's area for military purposes, distributed seismic measurements, pollution tracking, monitoring fire and nuclear power plants, tacking patients, engineering and medical explorations like wildlife monitoring, etc. Mostly for military purposes, data collected by sensor nodes need be encrypted before transmitting to neighboring nodes and base stations. The following issues make secure communication between sensor networks different from usual (traditional) networks: • Limited resources in sensor nodes: Each sensor node contains a primitive processor featuring very low computing speed and only small amount of programmable memory. An example is the popular Atmel ATmega 128L processor. • Limited life-time of sensor nodes: Each sensor node is battery-powered and is expected to operate for only few days. Therefore, once the deployed sensor nodes expire, it is necessary to add some fresh nodes for continuing the data collection operation. This is referred to as the dynamic management of security objects (like keys). • Limited communication abilities of sensor nodes: Sensor nodes have the ability to communicate each other and the base stations by the short range wireless radio transmission at low bandwidth and over small communication ranges (typical example is 30 meters (100 feet)). • Lack of knowledge about deployment configuration: Most of cases, the post deployment network configuration is not known a priori. As a result, it is unreasonable to use security algorithms that have strong dependence on locations of sensor nodes in a sensor network. • Mobility of sensor nodes: Sensor nodes may be mobile or static. If sensor nodes are mobile then they can change the network configuration at any time. • Issue of node capture: A part of the network may be captured by the adversary/enemy. The resilience measurement against node capture is computed by comparing the number of nodes captured, with the fraction of total network communications that are exposed to the adversary not including the communications in which the compromised nodes are directly involved. Thus, it is not feasible to use public-key cryptosystems in resource constrained sensor networks. Hence, only the symmetric cipher such as DES/IDEA/RC5 [12,11] is the viable option for encryption/decryption of secret data. But setting up symmetric keys among communication nodes is a challenging task in a sensor network. A survey on sensor networks can be found in [2,1]. The topology of sensor networks changes due to the following three phases: • Pre-deployment and deployment phase: Sensor nodes can be deployed from the truck or the plane in the sensor field. • Post-deployment phase: Topology can change after deployment because of irregularities in the sensor field like obstacles or due to jamming, noise, available energy of the nodes, malfunctioning, etc., or due to the mobile sensor nodes in the network. • Redeployment of additional nodes phase: Additional sensor nodes can be redeployed at any time to replace the faulty or compromised sensor nodes. A protocol that establishes cryptographically secure communication links among the sensor nodes is called the bootstrapping protocol. Several methods [6,3,8,5] are already proposed in order to solve the bootstrapping problem. All these techniques are based on random deployment models, that is, they do not use the pre-deployment knowledge of the deployed sensor nodes. Eschenauer and Gligor [6] proposed the basic random key predistribution called the EG scheme, in which each sensor is assigned a set of keys randomly selected from a big key pool of the keys of the sensor nodes. Chan et al. [3] proposed the q-composite key predistribution and the random pairwise keys schemes. For both the EG and the q-composite schemes, if a small number of sensors are compromised, it may reveal to compromise a large fraction of pairwise keys shared between non-compromised sensors. However, the random pairwise keys predistribution is perfectly secure against node captures, but there is a problem in supporting the large network. Liu and Ning's polynomial-pool based key predistribution scheme [8] and the matrix-based key predistribution proposed by Du et al. [4] improve security considerably. Liu and Ning [10] proposed the extended version of the closest pairwise keys scheme [9] for static sensor networks. Their scheme is based on the pre-deployment locations of the deployed sensor nodes and a pseudo random function (PRF) proposed by Goldreich et al. [7]. There is no communication overhead for establishing direct pairwise keys between neighbor nodes and the scheme is perfectly secure against node capture. The rest of the paper is organized as follows. Section 2 describes our proposed scheme called the identity based key predistribution using a pseudo random function (IBPRF). In Section 3, we provide a theoretical analysis for this scheme. In Section 4, we discuss the security issues with respect to our scheme. In Section 5, we provide an improved version of our scheme for distributed sensor networks. In Section 6, we compare our scheme with the previous schemes [6,3,8] with respect to communication overhead, network connectivity, and resilience against node captures. Finally, Section 7 concludes the paper. Identity Based Key Pre-Distribution using a Pseudo Random Function (IBPRF) The bootstrapping protocol for the random key predistribution schemes [6,3,8] incurs much more communication overhead for establishing direct pairwise keys between sensor nodes in a sensor network. Our goal is to design a protocol which basically reduces the communication overhead for establishing direct pairwise keys between sensors during direct key establishment phase of the bootstrapping. We propose a new scheme called the identity based key predistribution using a pseudo random function (IBPRF), which serves our above desired purpose. IBPRF has the following interesting properties: • There is no communication overhead during direct key establishment phase for establishing direct pairwise keys between sensors. • There is no communication overhead during the addition of new sensor nodes. • When the sensor nodes are mobile, our scheme easily establish direct pairwise keys between the mobile sensor nodes and their physical neighbors with which they do not share keys currently with some desired probability. • It works for any deployment topology. IBPRF is based on the following two ingredients: • A pseudo random function (PRF) proposed by Goldreich et al. in 1986 [7]. • A master key (MK) shared between each sensor node and the key setup server. The different phases for this scheme are as follows. Key Pre-Distribution Let N be a pool of the ids of n sensor nodes in a sensor network. Assume that each sensor node u is capable of holding a total of m + 1 cryptographic keys in its key ring K u . The key predistribution has the following steps: • Step-1: For each sensor node u, the key setup server randomly generates a master-key M K u . • Step-2: For each sensor node u, the key setup server selects a set S of m randomly generated ids of sensor nodes from the pool N which are considered as the probable physical neighbors' ids. Let S = {v 1 , v 2 , . . . , v m }. For each node id v i ∈ S (i = 1, 2, . . . , m), the key setup server generates a symmetric key SK u,vi = PRF MKv i (u) as the pairwise key shared between the nodes u and v i , where M K vi is the master key for v i and u is the id of the node u. For each v i ∈ S, the key-plus-id combination (SK u,vi , v i ) is stored in u's key ring K u . We note that each node v i can easily compute the same key SK u,vi with its master key and the id of node u. The sensor node v is called a master sensor node of u if the shared key between them is calculated by SK u,v = PRF MKv (u). In other words, node u is called a slave sensor node of v if v is a master sensor node of u. Direct Key Establishment After deployment of sensor nodes in a deployment area (i.e., target field), sensor nodes will establish direct pairwise keys between them. Direct key establishment phase has the following steps: • Step-1: Each sensor node first locates its all physical neighbors. Nodes u and v are called physical neighbors if they are within the communication range of one another. They are called key neighbors if they share a pairwise key. They are said to be direct neighbors if they are both physical as well as key neighbors. Now, after identifying the physical neighbors by a sensor node u, it can easily verify which ids of the physical neighbors exist in its key ring K u . If u finds that it has the predistributed pairwise key SK u,v = PRF MKv (u) with node v then it informs sensor v that it has such a key. This notification is done by sending a short message containing the id of node u that u has such a key. We note that this message never contains the exact value of the key SK u,v . • Step-2: Upon receiving such a message by node v, it can easily calculate the shared pairwise key SK u,v = PRF MKv (u) by using its own master key and the id of node u. Thus, nodes u and v can establish a direct pairwise key shared between them very easily and use this key for their future communication. Path Key Establishment This is an optional stage, if requires, adds the connectivity of the network. After direct key establishment, if the connectivity is still poor, nodes u and v which are physical neighbors not sharing a pairwise key, can establish a direct key between them as follows. • Step-1: u first finds a path u = u 0 , u 1 , u 2 , . . . , u h−1 , u h = v such that each (u i , u i+1 ) (i = 0, 1, 2, . . . , h − 1) is a secure link. • Step-2: u generates a random number k ′ as the shared pairwise key between u and v and encrypts it using the shared key SK u,u1 and sends to node u 1 . • Step-3: u 1 retrieves k ′ by decrypting the encrypted key using SK u,u1 and encrypts it using the shared key SK u1,u2 between u 1 and u 2 and sends to u 2 . • Step-4: This process is continued until the key k ′ reaches to the desired destination node v. As a result, nodes u and v use k ′ as the direct pairwise key shared between them for future communication. Since this process involves more communication overhead to establish a pairwise key between nodes, in practice h = 2 or 3 is recommended. Mobility of Sensor Nodes Suppose that a sensor node u moves from one location to another. Due to location updation of u, the connectivity of u with the new neighbors may also change. In the new location, assume that u finds the ids of its some new physical neighbors with which it does not currently share any keys. If v be one such physical neighbor, u informs to v that it has a pairwise key with v. This notification takes place by sending a request message to v containing the id of sensor node u excluding the exact value of the key. Upon receiving this message, v can immediately compute the pairwise key shared between them by executing one efficient PRF operation and by using the master key M K v for v and the id of sensor node u. Thus, u and v use this key for their future communication. After performing this stage, if sensor node u finds still poor connectivity, it may opt for at most 1-hop path key establishment because path key establishment involves more communication overhead. Of course we assume that mobility of sensor nodes are infrequent. Addition of Sensor Nodes In order to add a new sensor node u, the key setup server selects a set S of m randomly generated ids of sensor nodes from the pool N . The key setup server randomly generates a master key M K u for node u. For each sensor node id v ∈ S, the key setup server takes the master key M K v and compute the secret key SK u,v = PRF MKv (u) as the shared pairwise key between nodes u and v, and distributes the key-plus-id combination (SK u,v , v) to u. After deployment of sensor node u, it establishes direct pairwise keys using direct key establishment phase of IBPRF with the physical neighbors for which the ids are in u's key ring K u . Now, if u finds still poor connectivity after direct key establishment, it can perform path key establishment stage with 2 or 3 hops. Analysis In this section, we shall now compute the probability of establishing direct keys between two sensor nodes during direct key establishment, and the probability of establishing a pairwise key between two sensor nodes during path key establishment. We shall also analyze the storage overhead and the communication overhead required by our scheme. Probability of Establishing Direct Keys Let p be the probability that two physical neighbors can establish a direct pairwise key. For the derivation of p, we first observe that two physical neighbors u and v can establish a pairwise key only if the key ring K u of node u contains the shared secret key SK u,v = PRF MKv (u) and the id of node v, or the key ring K v of node v contains the shared secret key SK v,u = PRF MKu (v) and the id of node u because of the fact that any of nodes u and v can initiate for establishing a pairwise key between them. We then have, p = 1− (probability that both u and v do not establish a pairwise key). The total number of ways to select m ids from the pool N of size n is n m . For a fixed key ring K u of node u, the total number of ways to select K v of a node v such that K v does not have the id of u is n − 1 m . Thus, we have p = 1 − n − 1 m n m = m n .(1) We note that p strictly depends on the network size n and the key ring size. It is clear from Figure 1 that when the network size is small, our scheme provides better connectivity. Therefore, our scheme can not support a large network. In section 5, we have proposed an improved version of our scheme to support large networks. Probability of Establishing Keys using 1hop Path Key Establishment If d be the average number of neighbor nodes that each sensor node can contact, it follows from the similar analysis in [8] that the probability of two sensor nodes establishing a pairwise key (directly or indirectly) is p s = 1 − (1 − p)(1 − p 2 ) d .(2) The network connectivity probabilities for path key establishment with 1-hop are plotted in Figure 2. From this figure it is also clear that we are able to achieve better connectivity after executing this stage even if the network is almost disconnected initially. Calculation of Storage Overhead Each sensor node has to store a master key which is shared with the key setup server and m predistributed key-plus-id combinations. Hence, our scheme requires a storage overhead of maximum m + 1 keys for each sensor node. Calculation of Communication Overhead For establishing a pairwise key between two sensor nodes u and v, one of them, say u, initiates a request message to node v that its key ring contains the shared key between them. Then, after receiving such a request, node v computes the shared key between u and v by performing only one efficient PRF operation. Hence, the communication overhead involves only one short message for informing the other node that it has a pairwise key and the computational overhead due to single efficient PRF operation. Security Considerations The security of IBPRF depends on the following facts: • The security of PRF [7]. • A node's master key MK which is shared with the key setup server. It is observed that if a node's master key is not disclosed, no matter how many pairwise keys generated by this master key are disclosed, the task is still computationally difficult for an adversary to recover the master key MK as well as the non-disclosed pairwise keys generated with different ids of sensor nodes. Again, each pre-distributed pairwise key between two sensor nodes is generated by using PRF function randomly. Thus, no matter how many sensor nodes are compromised, the direct pairwise keys between noncompromised nodes are still secure. In other word, node compromise does not eventually lead to compromise of the direct pairwise keys between the other non-compromised nodes. In this way, our scheme provides perfect security against node captures. The Improved Scheme We note that our basic scheme (IBPRF) provides better connectivity if the network size is small, whereas it provides perfect security against node captures. In fact, there is no communication overhead during establishment of the direct pairwise keys between sensors and also during the addition of nodes after their initial deployment. To support a large sensor network, we wish to apply our basic scheme in distributed sensor networks. The deployment region is divided into c number of sub-regions called the cells such that each cell can communicate with the base stations comfortably. Let the i-th cell be denoted by cell i . Assume that each cell i contains n i number of sensor nodes. In practical situation, it is not always possible to deploy each node to a pre-determined location in the deployment region. We further assume that the key setup server only knows the nodes containing to a particular cell which will be deployed in that region randomly. In practice, this assumption is appropriate. Under this configuration, we now apply our basic scheme to each cell as follows. Let N i be the pool of the ids of n i sensor nodes in a cell cell i . Assume that each sensor node u is capable of holding a total of m + 1 cryptographic keys. In key pre-distribution phase, for each node u ∈ cell i , the key setup server randomly generates a master key M K u . For each node u ∈ cell i , the key setup server also selects a set S of m randomly generated ids of the sensor nodes from the pool N i . For each v ∈ S, the key setup server generates a symmetric key SK u,v = PRF MKv (u) as the pairwise key shared between nodes u and v, where M K v is the master key for node v and u is the id for node u. The key-plus-id combination (SK u,v , v) is stored in u's key ring K u . After deployment of the sensor nodes, they establish direct pairwise keys using direct key establishment phase of our basic scheme (IBPRF). The other phases like path key establishment, mobility of sensor nodes, and addition of sensor nodes remain same as our basic scheme. Thus, sensor nodes in each cell establish pairwise keys between them and communicate with each other in that cell securely. For mobility of the sensor nodes, we restrict the sensor nodes to move in a particular cell only. Let p i denote the probability that two sensor nodes in the i-th cell cell i can establish a direct pairwise key between them. Similar to analysis in 3.1, we have p i = 1 − n i − 1 m n i m = m n i .(3) The (average) probability that two sensor nodes in a network of size n = c i=1 n i , establish a direct pairwise key between them is given by p = c i=1 p i c .(4) Hence, we are able to achieve better connectivity for the entire network by using our improved version and selecting the appropriate size of the cells. However, the communication overhead as well as resilience measurement against node captures remain same as our basic scheme (IBPRF). We note that this improved scheme may not always work for ad hoc mode sensor networks. Comparison with Previous Schemes In this section, we compare both our basic scheme (IBPRF) and the improved scheme with the EG [6], the q-composite (qC) [3], and the polynomial-pool based [8] schemes with respect to the communication overhead, network connectivity and resilience against node captures. (1) Communication overhead For the EG and the q-composite schemes, when a node wishes to establish pairwise keys with its physical neighbor nodes, it needs to send a list of some messages encrypted by keys in its key ring. In case of the polynomial-pool based scheme, a sensor node also needs to send a list of some messages encrypted by potential pairwise keys based on its polynomial shares for establishing a direct pairwise key with a physical neighbor. Thus, the communication overhead is on the order of the key ring size for these schemes. But, for our schemes, the communication overhead is only due to one short message sent by a node to inform its physical neighbor that it has a pairwise key in its key ring and a single efficient PRF operation for computing the shared key SK by the physical neighbor. Hence, both our basic scheme (IBPRF) and the improved scheme have much less communication overhead than the EG, the q-composite, and the polynomial-pool based schemes. (2) Resilience against node capture From the analysis of the EG scheme [6] and the qcomposite scheme [3], it follows that even if the number of nodes captured is small, these schemes may reveal a large fraction of pairwise keys shared between non-compromised sensors. The analysis of the polynomial-pool based scheme [8] shows that this scheme is unconditionally secure and t-collusion resistant. Thus, it has better resilience against node captures than the EG and the q-composite schemes. However, both our basic scheme (IBPRF) and the improved scheme provide perfect security against node captures. (3) Network connectivity For the EG scheme [6], the probability of establishing a direct pairwise key between two sensor nodes is p EG = 1 − M − m m M m = 1 − m−1 i=0 M − m − i M − i(5) where M and m are the key pool size and key ring size of a sensor node. For the q-composite scheme [3], the probability of establishing a direct pairwise key between two sensor nodes is p qC = 1 − q−1 i=0 p i(6) where p i = M i M − i 2(m − i) 2(m − i) m − i M m 2 , M is the key pool size and m is the key ring size of a sensor node. For the polynomial-pool based scheme [8], the probability of establishing a direct pairwise key between two sensor nodes is p poly−pool = 1− s − s ′ s ′ s s ′ = 1− s ′ −1 i=0 s − s ′ − i s − i(7) where s is the polynomial-pool size and s ′ is the number of shares given to a sensor node. Thus, we see that the EG and the q-composite schemes depend on M and m. The polynomial-pool based scheme depends on s and s ′ and the maximum supported network size is bounded by (t+1)s s ′ , where t is the degree of the symmetric bivariate polynomial, whereas our scheme depends on the network size n and the key ring size m. Probability of sharing a common key Maximum supported network size polynomial-pool(2 shares, t=99) polynomial-pool(4 shares, t=49) polynomial-pool(6 shares, t=32) IBPRF(m=200) Figure 3: The probability p of establishing a common key v.s. the maximum supported network size n in order to be resilient against node compromise. Assume that each sensor node is capable of holding 200 keys. For comparison of network connectivity, we only consider the polynomial-pool based scheme because it is more resilient against node compromise than the EG scheme and the q-composite scheme. However, both the EG scheme and the q-composite scheme support networks of arbitarily big sizes. The relationship between the probability of establishing direct keys and the maximum supported network size for the polynomial-pool based scheme and our basic scheme (IBPRF) is shown in Figure 3. We assume that each sensor is capable of storing 200 keys in its key ring. From this figure, it is very clear that our scheme provides better connectivity than the polynomial-pool based scheme in order to be resilient against node compromise. Conclusion Our basic scheme (IBPRF) is an alternative to direct key establishment of the bootstrapping protocol. Both IBPRF and the improved scheme guarantee that they have better trade-off between communication overhead, network connectivity and also resilience against node captures compared to the EG, the q-composite, and the polynomial-pool based schemes. Both schemes can also be adapted for mobile sensor networks by initiating direct key establishment phase and one can achieve reasonable connectivity by applying these schemes. Figure 1 : 1The probability p that two sensors establish a direct pairwise key v.s. the network size n, with m = 100, 150, 200. Figure 2 : 2The probability p s of establishing a pairwise key v.s. the probability p that two sensor nodes establish a direct pairwise key, with d = 20, 60, 100. A survey on sensor networks. I F Akyildiz, W Su, Y Sankarasubramaniam, E Cayirci, IEEE Communications Magazine. 408I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. A survey on sensor networks. IEEE Com- munications Magazine, 40(8):102-114, August 2002. Wireless sensor networks : A survey. I F Akyildiz, W Su, Y Sankarasubramaniam, E Cayirci, Computer Networks38I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. Wireless sensor networks : A survey. Com- puter Networks, 38(4):393-422, 2002. Random key predistribution schemes for sensor networks. H Chan, A Perrig, D Song, IEEE Symposium on Security and Privacy. Berkeley, CaliforniaH. Chan, A. Perrig, and D. Song. Random key pre- distribution schemes for sensor networks. In IEEE Symposium on Security and Privacy, pages 197-213, Berkeley, California, 2003. Varshney. A key management scheme for wireless sensor networks using deployment knowledge. W Du, J Deng, Y S Han, S Chen, P , 23rd Conference of the IEEE Communications Society. Hong Kong, ChinaW. Du, J. Deng, Y. S. Han, S. Chen, and P. K. Varsh- ney. A key management scheme for wireless sensor networks using deployment knowledge. In 23rd Con- ference of the IEEE Communications Society (Info- com'04), Hong Kong, China, March 21-25 2004. A pairwise key pre-distribution scheme for wireless sensor networks. W Du, J Deng, Y S Han, P K Varshney, ACM Conference on Computer and Communications Security (CCS'03). USAW. Du, J. Deng, Y. S. Han, and P. K. Varshney. A pair- wise key pre-distribution scheme for wireless sensor networks. In ACM Conference on Computer and Com- munications Security (CCS'03), pages 42-51, Wash- ington DC, USA, October 27-31 2003. A key management scheme for distributed sensor networks. L Eschenauer, V D Gligor, 9th ACM Conference on Computer and Communication Security. L. Eschenauer and V. D. Gligor. A key management scheme for distributed sensor networks. In 9th ACM Conference on Computer and Communication Secu- rity, pages 41-47, November 2002. How to construct random functions. O Goldreich, S Goldwasser, S Micali, Journal of the ACM. 334O. Goldreich, S. Goldwasser, and S. Micali. How to construct random functions. Journal of the ACM, 33(4):792-807, October 1986. Establishing pairwise keys in distributed sensor networks. D Liu, P Ning, Proceedings of 10th ACM Conference on Computer and Communications Security (CCS). 10th ACM Conference on Computer and Communications Security (CCS)Washington DCD. Liu and P. Ning. Establishing pairwise keys in dis- tributed sensor networks. In Proceedings of 10th ACM Conference on Computer and Communications Secu- rity (CCS), pages 52-61, Washington DC, Oct 27-31 2003. Location-based pairwise key establishments for static sensor networks. D Liu, P Ning, ACM Workshop on Security in Ad Hoc and Sensor Networks (SASN '03). D. Liu and P. Ning. Location-based pairwise key establishments for static sensor networks. In ACM Workshop on Security in Ad Hoc and Sensor Networks (SASN '03), pages 72-82, October 2003. Improving key pre-distribution with deployment knowledge in static sensor networks. D Liu, P Ning, ACM Transactions on Sensor Networks. 12D. Liu and P. Ning. Improving key pre-distribution with deployment knowledge in static sensor networks. ACM Transactions on Sensor Networks, 1(2):204- 239, 2005. R L Rivest, The RC5 Algorithm. Dr. Dobb's Journal. R. L. Rivest. The RC5 Algorithm. Dr. Dobb's Journal, January 1995. Cryptography and Network Security: Principles and Practices. W Stallings, Prentice Hall3rd editionW. Stallings. Cryptography and Network Security: Principles and Practices. Prentice Hall, 3rd edition, 2003.

In the studies of dynamics of pathogens and their interactions with a host immune system, an important role is played by the structure of antigenic variants associated with a pathogen. Using the example of a model of antigenic variation in malaria, we show how many of the observed dynamical regimes can be explained in terms of the symmetry of interactions between different antigenic variants. The results of this analysis are quite generic, and have wider implications for understanding the dynamics of immune escape of other parasites, as well as for the dynamics of multi-strain diseases.

10.1007/s00285-012-0508-y

1202.4290

1ff77c389da0c5ffdc0301f9436e47ad45dca211

The effects of symmetry on the dynamics of antigenic variation February 21, 2012 K B Blyuss Department of Mathematics University of Sussex BN1 9QHBrightonUnited Kingdom The effects of symmetry on the dynamics of antigenic variation February 21, 2012 In the studies of dynamics of pathogens and their interactions with a host immune system, an important role is played by the structure of antigenic variants associated with a pathogen. Using the example of a model of antigenic variation in malaria, we show how many of the observed dynamical regimes can be explained in terms of the symmetry of interactions between different antigenic variants. The results of this analysis are quite generic, and have wider implications for understanding the dynamics of immune escape of other parasites, as well as for the dynamics of multi-strain diseases. Introduction In the course of evolution, pathogens have developed various methods of evading the immune system of their hosts. Whilst there are many contributing factors that determine individuals aspects of host-parasite interactions, from a more general perspectives strategies of immune escape can be divided into two major classes. In the first class, parasites remain largely invisible to the immune system of their host through an extended period latency when production of new viruses or bacteria inside the host organism is very small or absent. In this case, the pathogens do not trigger immune response, and thus are able to remain undetected in their hosts for long periods of time. Notable examples of such pathogens include various viruses, such as members of herpes virus family [39] and retroviruses [8,42]. Whilst this method of immune escape is largely unavailable to bacteria due to fundamental differences in replication strategies, several types of bacteria have shown persistence for long periods of time with little evident replication; examples include mycobacteria and T. Pallidum causing syphilis [40]. Another possible strategy of immune escape is that, in which pathogen is actively replicating in its host, but this replication is dynamically regulated by the host immune system. There are two ways how this regulation can be achieved: the pathogen can either reach a certain chronic state (equilibrium) through the balance of its proliferation and destruction (see [43] for Trypanosoma cruzi example), or it can go through the process of antigenic variation, whereby it keeps escaping immune response by constantly changing its surface proteins, thus going through a large number of antigenic variants. Perhaps, the best studied pathogens relying on this strategy for immune escape are Plasmodium falciparum and African Trypanosoma exemplified by Trypanosoma brucei, with other examples including several families of viruses, bacteria and even fungi [9,17,33,36,38,63]. In the case of T. brucei, the organism that causes sleeping sickness, parasite covers itself with a dense homogeneous coat of variant surface glycoprotein (VSG). Genome of T. brucei has over 1000 genes that control the expression of VSG protein, and switching between them provides the mechanism of antigenic variation [41]. What makes T. brucei unique is the fact that unlike other pathogens, whose antigenic variation is typically mediated by DNA rearrangements or transcriptional regulation, activation of VSGs requires recombination of VSG genes into an expression site (ES), which consists of a single vsg gene flanked by an upstream array of 70 base pair repeats and expression site associated genes (ESAGs). T. brucei expresses one VSG at any given time, and the active VSG can either be selected by activation of a previously silent ES (and there are up to 20 ES sites), or by recombination of a VSG sequence into the active ES. The precise mechanism of VSG switching has not been completely identified yet, but it has been suggested that the ordered appearance of different VSG variants is controlled by differential activation rates and density-dependent parasite differentiation [41,60]. For the malaria agent P. falciparum, the main target of immune response is Plasmodium falciparum erythrocyte membrane protein-1 (PfEMP1), which is expressed from a diverse family of var genes, and each parasite genome contains approximately 60 var genes encoding different PfEMP1 variants [25]. The var genes are expressed sequentially in a mutually exclusive manner, and this switching between expression of different var gene leads to the presentation of different variant surface antigens (VSA) on the surface of infected erythrocyte, thus providing a mechanism of antigenic variation [11,49]. In all cases of antigenic variation, host immune system has to go through a large repertoire of antigenic variants, and this provides parasites with enough time to get transmitted to another host or cause a subsequent infection with a different antigenic variant in the same host. Despite individual differences in the molecular implementation of antigenic variation, such as, gene conversion, site-specific DNA inversions, hypermutation etc., there are several features common to the dynamics of antigenic variation in all pathogens. These include ordered and often sequential appearance of parasitemia peaks corresponding to different antigenic variants, as well as certain degree of cross-reactivity. Several mathematical models have been put forward that aim to explain various aspects of antigenic variation. Agur et al. [3] have studied a model of antigenic variation of African trypanosomes which suggests that sequential appearance of different antigenic variants can be explained by fitness differences between single-and double-expressors -antigenic variants that express one or two VSGs. However, this idea is not supported by the experimental evidence arising from normal in vivo growth and reduced immunogenicity of artificially created double expressors [47]. Frank [23,24] has suggested a model that highlights the importance of cross-reactivity between antigenic variants in facilitating optimal switching pattern that provides sequential dominance and extended infection. Antia et al. [4] have considered variant-transcending immunity as a basis for competition between variants, which can promote oscillatory behaviour, but this failed to induce sequential expression. Many other mathematical models of antigenic variation have been proposed and studied in the literature, but the discussion of their individuals merits and limitations is beyond the scope of this work. In this paper we concentrate a model proposed by Recker et al. [52] (to be referred to as Recker model), which postulates that in addition to a highly variant-specific immune response, the dynamics of each variant is also affected by cross-reactive immune responses against a set of epitopes not unique to this variant. This assumption implies that each antigenic variant experiences two types of immune responses: a long-lasting immune response against epitopes unique to it, and a transient immune response against epitopes that it shares with other variants. The main impact of this model lies in its ability to explain a sequential appearance of antigenic variants purely on the basis of cross-reactive inhibitory immune responses between variants sharing some of their epitopes, without the need to resort to variable switch rates or growth rates (see Gupta [33] for a discussion of several clinical studies in Ghana, Kenya and India, which support this theory). From mathematical perspective, certain understanding has been achieved of various types of dynamics that can be obtained in the Recker model. In the case when long-lasting immune responses do not decay, numerical simulations in the original paper [52] showed that eventually all antigenic variants will be cleared by the immune system, with specific immune responses reaching protective levels preventing each of the variants from showing up again. Blyuss and Gupta [10] have demonstrated that the sequential appearance of parasitemia peaks during such immune clearance can be explained by the existence of a hypersurface of equilibria in the phase space of the system, with individual trajectories approaching this hypersurface and then being pushed away along stable/unstable manifolds of the saddle-centres lying on the hypersurface. They also numerically analysed robustness of synchronization between individual variants. Under assumption of perfect synchrony, when all variants are identical to each other, Recker and Gupta [54] have analysed peak dynamics and threshold for chronicity, while Mitchell and Carr [45] have investigated the additional effect of time delay in the development of immune response. De Leenheer and Pilyugin [18] have replaced linear growth of antigenic variants in the original model by the logistic growth, and have studied the effects of various types of cross-reactivity on the dynamics, ranging from no cross-reactivity to partial and complete cross-immunty. Mitchell and Carr [46] have studied the appearance of synchronous and asynchronous oscillations in the case of global coupling between variants (referred to as "perfect cross immunity" in [18]). So far, mathematical analyses of the Recker model have concentrated primarily on identifying and studying different types of behaviour in the model. Whilst this has given a certain headway in the understanding of possible dynamics, symmetric properties of the model have remained largely unstudied, and yet they can provide important insights into the dynamics of the model allowing one to distinguish the interactions between variants and the immune system from the effects of topology of coupling between antigenic variants. The importance of this topology has been highlighted in recent works [12,13,56]. In this paper we study Recker model from the perspective of symmetric dynamical systems. This allows us to perform a systematic analysis of steady states and their stability, as well as to classify various periodic behaviours in terms of their symmetries. The outline of the paper is as follows. In the next section we formalize the Recker model and discuss some of its properties. Section 3 reviews some concepts and techniques from equivariant bifurcation theory. In Section 4 we analyse steady states and their stability with account for symmetry properties of the system. Section 5 is devoted to symmetry-based classification of different dynamical regimes. The paper concludes in Section 6 with discussion of results. Mathematical model Following Recker et al. [52], we assume that within a human host, the parasite population of P. falciparum consists of N distinct antigenic variants, with each antigenic variant i, 1 ≤ i ≤ N , containing a single unique major epitope that elicits a long-lived (specific) immune response, and also several minor epitopes that are not unique to the variant. Assuming that all variants have the same net growth rate φ, their temporal dynamics is described by the equation dy i dt = y i (φ − αz i − α w i ),(1) where α and α denote the rates of variant destruction by the long-lasting immune response z i (specific to variant i) and by the transient immune response w i , respectively. The dynamics of the variant-specific immune response can be written in its simplest form as dz i dt = βy i − µz i ,(2) with β being the proliferation rate and µ being the decay rate of the specific immune response. Finally, the transient (cross-reactive) immune response can be described by the minor modification of the above equation (2): dw i dt = β j∼i y j − µ w i ,(3) where the sum is taken over all variants j sharing the epitopes with the variant i. We shall use the terms long-lasting and specific immune response interchangeably, likewise for transient and cross-reactive. The above system can be formalized with the help of adjacency or connectivity matrix A whose entries A ij are equal to one if the variants i and j share some of their minor epitopes and equal to zero otherwise [10,18]. Obviously, the matrix A is always a symmetric matrix. Prior to constructing this matrix it is important to introduce an ordering of the variants according to their epitopes. Whilst this choice is pretty arbitrary, it has to be fixed before the analysis can be done. Consider a system of antigenic variants with just two minor epitopes and two variants in each epitope. In this case, the total number of variants is four, and we enumerate them as follows After the ordering of variants has been fixed, it is straightforward to construct the connectivity matrix A of variant interactions. For the particular system of variants (4) illustrated in Fig. 1, this matrix has the form A =     1 1 0 1 1 1 1 0 0 1 1 1 1 0 1 1     .(5) With the help of lexicographic ordering, one can systematically construct matrix A for an arbitrary number of minor epitopes [10]. For the rest of the paper we will concentrate on the case of two minor epitopes, but the results can be generalized to larger systems of antigenic variants. Using the connectivity matrix one can rewrite the system (1) -(3) in a vector form d dt   y z w   = F (y, z, w) =    y(φ1 N − αz − α w), βy − µz, β Ay − µ w,(6) where y = (y 1 , y 2 , ..., y N ), z = (z 1 , z 2 , ..., z N ), w = (w 1 , w 2 , ..., w N ), 1 N denotes a vector of the length N with all components equal to one, and in the right-hand side of the first equation multiplication is taken to be entry-wise, so that the output is a vector again. The above system has to be augmented by appropriate initial conditions, which are taken to be y(0) ≥ 0, z(0) ≥ 0, w ≥ 0. As it has been shown in [10], with these initial conditions the system (6) is well-posed, so its solutions remain non-negative for all time. We will assume that cross-reactive immune responses develop at a slower rate than specific immune responses, have a shorter life time, and are less efficient in destroying the infection. This implies the following biologically realistic relations between the system parameters α ≤ α, µ ≤ µ , β ≤ β.(7) Elements of equivariant bifurcation theory Before proceeding with the analysis of symmetry effects on the dynamics of systems of antigenic variants, we recall some concepts and results from equivariant bifurcation theory [30,31]. Let Γ ⊆ GL(R N ) be a compact Lie group acting on R N . We say that a system of ODEṡ x = F (x), x ∈ R N , F : R N → R N ,(8) is equivariant with respect to a symmetry group Γ if the vector field F commutes with the action of Γ, i.e. if it satisfies an equivariance condition F (γx) = γF (x) for all x ∈ R N , γ ∈ Γ. The main examples of the symmetry groups we are interested in are Γ = S n , Z n and D n (of order n!, n, and 2n, respectively). Here, S n denotes the symmetric group of all permutations in a network with an all-to-all coupling, i.e. all permutations on n symbols. The cyclic group Z n describes the symmetry of a unidirectional ring (rotations only), while the dihedral group D n corresponds to a bidirectional ring of n coupled units (rotations and reflections in the plane that preserve a regular n-gon). One can define the group orbit of a point x ∈ R N under the action of Γ is defined as Γx = γx ∈ R N : γ ∈ Γ . Note that the equivariance of the system (8) implies that its equilibria come in group orbits, since if F (x) = 0 for some x ∈ R N , then F (γx) = γ0 = 0 for all γ ∈ Γ. The isotropy subgroup Σ(x) ⊂ Γ of a point x ∈ R N is defined as the subgroup of Γ which fixes the point x, i.e. Σ(x) = {γ ∈ Γ : γx = x} . Associated with each isotropy subgroup Σ ⊂ Γ is the fixed-point subspace denoted by Fix(Σ), which is the set of points x ∈ R N invariant under the action of Σ: Fix(Σ) = x ∈ R N : σx = x for all σ ∈ Σ . An important property of fixed-point subspaces is that they are flow-invariant, since if x ∈ Fix(Σ), then σF (x) = F (σx) = F (x), and therefore, F (x) ∈ Fix(Σ). The equivariant branching lemma states that provided certain (generic) conditions are satisfied by the bifurcation, there exists a branch of equilibrium solutions with symmetry Σ for each isotropy subgroup Σ ⊂ Γ with dim(Fix(Σ)) = 1. Such isotropy subgroups with one-dimensional fixed-point subspaces are called axial. Similarly, one can define C-axial subgroups as those subgroups Σ ⊂ Γ × S 1 , for which Σ is an isotropy subgroup of the action of Γ × S 1 on the centre subspace of the equilibrium, and dim(Fix(Σ)) = 2. Next, we recall that a subspace V of R N is called Γ-invariant if γV ⊂ V, for all γ ∈ Γ. Two Γ-invariant subspaces V and W are called Γ-isomorphic if there exists a linear isomorphism T : V → W , such that T (γv) = γ(T v), for all v ∈ V, γ ∈ Γ. A Γ-invariant subspace V is Γ-irreducible if the only Γ-invariant subspaces of V are {0} and V . Γ-irreducible subspaces can be used to perform an efficient decomposition of the phase space that would allow block-diagonalization of the linearization matrix, thus simplifying the analysis of stability of steady states. The isotypic decomposition proceeds by decomposing R N into Γ- irreducible subspaces V j so that R N = V 0 ⊕ V 1 ⊕ · · · ⊕ V m . The isotypic components are then formed by combining the irreducible subspaces that are Γ-isomorphic. The isotypic decomposition is R N = W 0 ⊕ W 1 ⊕ · · · ⊕ W k , k ≤ m, where the W j are uniquely defined [31]. Now, if x(t) is a T -periodic solution of a Γ-equivariant system (8), then γx(t) is another Tperiodic solution of (8) for any γ ∈ Γ. Uniqueness of solutions implies that either x(t) and γx(t) are identical, or there exists a phase shift θ ∈ S 1 ≡ R/Z ≡ [0, T ), such that γx(t) = x(t − θ). The pair (γ, θ) is called a spatio-temporal symmetry of the solution x(t), and the collection of all spatio-temporal symmetries of x(t) forms a subgroup ∆ ⊂ Γ × S 1 . One can identify ∆ with a pair of subgroups, H and K, such that K ⊂ H ⊂ Γ. Define H = {γ ∈ Γ : γ{x(t)} = {x(t)}} spatio-temporal symmetries, K = {γ ∈ Γ : γx(t) = x(t) ∀t} spatial symmetries. Here, K consists of the symmetries that fix x(t) at each point in time, while H consists of the symmetries that fix the entire trajectory. Let L K = ∪ γ∈H\K Fix(γ),(9) and let N Γ (K) denote the normalizer of K in Γ: N Γ (K) = {γ ∈ Γ : γKγ −1 = K}. The following theorem gives the necessary and sufficient conditions for H and K to characterize spatio-temporal symmetries of a periodic orbit. Theorem (H/K Theorem [14,30]). Let Γ be a finite group acting on R N . There is a periodic solution to some Γ-equivariant systems of ODEs on R N with spatial symmetries K and spatio-temporal symmetries H if and only if (a) H/K is cyclic. (b) K is an isotropy subgroup. (c) dim Fix(K) ≥ 2. If dim Fix(K) = 2, then either H = K or H = N Γ (K). (d) H fixes a connected component of Fix(K)\L K . Moreover, when these conditions hold, there exists a smooth Γ-equivariant vector field with an asymptotically stable limit cycle with the desired symmetries. The H/K theorem was originally derived in the context of equivariant dynamical systems by Buono and Golubitsky [14], and it has subsequently been used to classify various types of periodic behaviours in systems with symmetry that arise in a number of contexts, from speciation [59] to animal gaits [51] and vestibular system of vertebrates [28]. Now we can proceed with the analysis of symmetry properties of the system (6) as represented by its adjacency matrix A. In the case of two minor epitopes with m variants in the first epitope and n variants in the second, the system (6) is equivariant with respect to the following symmetry group [10] Γ = S m × S n , m = n, S m × S m × Z 2 , m = n.(10) This construction can be generalized in a straightforward way to a larger number of minor epitopes. It is noteworthy that while within each stratum we have an all-to-all coupling, the full system does not possess this symmetry. System (6) provides an interesting example of a linear coupling, which does not reduce to known symmetric configurations, such as diffusive, star or all-to-all [50]. A really important aspect is that two antigenic systems with the same total number of variants N may have different symmetry properties as described by the group Γ depending on m and n, such that N = mn. The simplest example of different kinds of splitting is given by N = 12, which can be represented as N = 2 × 6 or as N = 3 × 4. Symmetry analysis of steady states In the particular case of non-decaying specific immune response (µ = 0), equilibria of the system (6) are not isolated but rather form an N -dimensional hypersurface H 0 = {(y, z, w) ∈ R N : y = w = 0 N } in the phase space [10]. This hypersurface consists of saddles and stable nodes, and in addition to the original symmetry of the system it possesses an additional translational symmetry along the z axes. The existence of this hypersurface of equilibria in the phase space leads to a particular behaviour of phase trajectories, which mimics the occurrence of sequential parasitimea peaks in the immune dynamics of malaria [10,52]. When µ > 0, the structure of the phase space of the system (6) and its steady states is drastically different. Now, the only symmetry present is the original symmetry Γ, and the hypersurface of equilibria H 0 disintegrates into just two distinct points: the origin O, which is always a saddle, and the fully symmetric equilibrium E = (Y 1 N , Z1 N , W 1 N ), where Y = φµµ αβµ + α n c β µ , Z = φβµ αβµ + α n c β µ , W = φµn c β αβµ + α n c β µ .(11) Here n c is the total number of connections for each antigenic variant. It has been previously found by means of numerically computing eigenvalues of the Jacobian that the fully symmetric equilibrium E may undergo Hopf bifurcation as the parameters are varied [10]. It is worth noting that if one assumes all variants to be exactly the same, the original system (6) collapses to a system with just 3 dimensions, and in this case it is possible to show analytically that the fully symmetric equilibrium is always stable [54]. This implies that Hopf bifurcation of the fully symmetric equilibrium takes place outside the hyperplane of complete synchrony. In order to find analytically the boundary of Hopf bifurcation in terms of system parameters, as well as to understand the structure of the bifurcating solution, we concentrate on a specific connectivity matrix A that corresponds to a particular case of two epitopes with two variants in each epitope, as described by the system (4). Despite its simplicity, this case is still important for the following two reasons. Firstly, this is the simplest non-trivial combination of antigenic variants which can produce interesting dynamics of interactions with the immune system. Secondly, it can be used as a paradigm for many two-locus two-allele models [34,55]. In the case of two epitopes with two variants in each epitope, we have m = n = 2 and N = 4, and the system (6) is equivariant under the action of a dihedral group D 4 , which is an 8-dimensional symmetry group of a square. This group can be written as D 4 = {1, ζ, ζ 2 , ζ 3 , κ, κζ, κζ 2 , κζ 3 }, and it is generated by a four-cycle ζ corresponding to counterclockwise rotation by π/2, and a flip κ, whose line of reflection connects diagonally opposite corners of the square, see Fig. 2. The group D 4 has eight different subgroups (up to conjugacy): 1, Z 4 , and D 4 , as well as D p 1 = {1, κ} generated by a reflection across a diagonal, D s 1 = {1, κζ} generated by a reflection across a vertical, D p 2 = {1, ζ 2 , κ, κζ 2 } generated by reflections across both diagonals, and D s 2 = {1, ζ 2 , κζ, κζ 3 } generated by the horizontal and vertical reflections. Finally, the group Z 2 is generated by rotation by π. The lattice of these subgroups is shown in Fig. 2(b). D 4 has two other subgroups Z 2 (κζ 2 ) = {1, κζ 2 } and Z 2 (κζ 3 ) = {1, κζ 3 }, which will be omitted as they are conjugate to D p 1 and D s 1 , respectively. There is a certain variation in the literature regarding the notation for subgroups of D 4 , and we are using the convention adopted in Golubitsky and Stewart [30], c.f. [6,14,31]. The group D 4 has four one-dimensional irreducible representations [20,29]. Equivariant Hopf Theorem [30,31] states that under certain genericity hypotheses, there exists a branch of smallamplitude periodic solutions corresponding to each C-axial subgroup Γ × S 1 acting on the centre subspace of the equilibrium. To find out what type of periodic solution the fully symmetric steady state will actually bifurcate to, we can use the subspaces associated with the above-mentioned one-dimensional irreducible representations to perform an isotypic decomposition of the full phase space R 12 as follows [19,62]: R 12 = R 3 (1, 1, 1, 1) ⊕ R 3 (1, −1, 1, −1) ⊕ R 3 (1, 0, −1, 0) ⊕ R 3 (0, 1, 0, −1).(12) Jacobian of the linearization of system (6) near any steady state S = (y 1 , y 2 , y 3 , y 4 , z 1 , z 2 , z 3 , z 4 , w 1 , w 2 , w 3 , w 4 ) T , is given by J(S) =                      P 1 0 0 0 −αy 1 0 0 0 −α y 1 0 0 0 0 P 2 0 0 0 −αy 2 0 0 0 −α y 2 0 0 0 0 P 3 0 0 0 −αy 3 0 0 0 −α y 3 0 0 0 0 P 4 0 0 0 −αy 4 0 0 0 −α y 4 β 0 0 0 −µ 0 0 0 0 0 0 0 0 β 0 0 0 −µ 0 0 0 0 0 0 0 0 β 0 0 0 −µ 0 0 0 0 0 0 0 0 β 0 0 0 −µ 0 0 0 0 β β 0 β 0 0 0 0 −µ 0 0 0 β β β 0 0 0 0 0 0 −µ 0 0 0 β β β 0 0 0 0 0 0 −µ 0 β 0 β β 0 0 0 0 0 0 0 −µ                      (13) where P i = φ − αz i − α w i , i = 1, 2, 3, 4. For the fully symmetric steady state E, this Jacobian takes the block form J(E) =   0 4 −α1 4 −α 1 4 β1 4 −µ1 4 0 4 β A 0 4 −µ 1 4   , where 0 4 and 1 4 are 4 × 4 zero and unit matrices, and A is the connectivity matrix (5). Rather than compute stability eigenvalues from this 12 × 12 matrix, we use the isotypic decomposition (12) to rewrite this Jacobian in the block-diagonal form [29,62] J (E) =     C + 2D 0 3 0 3 0 3 0 3 C − 2D 0 3 0 3 0 3 0 3 C 0 3 0 3 0 3 0 3 C     ,(14) where C =   0 −α −α β −µ 0 β 0 −µ   , D =   0 0 0 0 0 0 β 0 0   .(15) Here, matrix C is associated with self-coupling, and D is associated with nearest-neighbour coupling. Stability changes in the C + 2D, C − 2D and C matrices describe a bifurcation of the fully symmetric steady state E in the even, odd, and V 4 subspaces, respectively [62]. Prior to performing stability analysis, we recall the Routh-Hurwitz criterion, which states that all roots of the equation λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0, are contained in the left complex half-plane (i.e. have negative real part), provided the following conditions hold [48] a i > 0, i = 1, 2, 3, a 1 a 2 > a 3 .(16) If the last condition is violated, then the above cubic equation has a pair of purely imaginary complex conjugate eigenvalues when a i > 0, i = 1, 2, 3, a 1 a 2 = a 3 ,(17) as discussed in Farkas and Simon [21]. Proposition 1. The fully symmetric steady state E is stable for α < α H , unstable for α > α H , and undergoes a Hopf bifurcation in the odd subspace at α = α H , where α H = αβµ + µµ (µ + µ ) β µ .(18) Proof. Stability of the fully symmetric steady state E changes when one of the eigenvalues of the Jacobian (14) goes through zero along the real axis or a pair of complex conjugate eigenvalues crosses the imaginary axis. Due to the block-diagonal form of the Jacobian it suffices to consider separately possible bifurcations in the matrices C, C ± 2D. For the matrix C given in (15), the characteristic equation takes the form λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0, with a 1 = µ + µ , a 2 = αβ + α β + µµ , a 3 = αβµ + α β µ. Clearly, in this case a 1,2,3 > 0, and also a 1 a 2 − a 3 = αβµ + α β µ + µµ (µ + µ ) > 0, which according to the Routh-Hurwitz conditions (16) implies that the eigenvalues of the matrix C are contained in the left complex half-plane for any values of system parameters. This means that the steady state E is stable in the V 4 subspace. Matrix C + 2D is equivalent to C upon replacing β with 3β , which allows one to conclude that the steady state E is also stable in the even subspace for any values of the system parameters. Finally, for the matrix C − 2D, the coefficients of the characteristic equation are a 1 = µ + µ , a 2 = αβ − α β + µµ , a 3 = αβµ − α β µ. Due to biological restrictions on parameters (7), it follows that a 1,2,3 > 0. Computing a 1 a 2 − a 3 gives a 1 a 2 − a 3 = αβµ + µµ (µ + µ ) − α β µ . When α < α H , where α H = αβµ + µµ (µ + µ ) β µ ,(19) we have a 1 a 2 − a 3 > 0, which according to (16) implies that the steady state E is stable in the odd subspace. When α = α H , one has a 1 a 2 = a 3 , which coincides with the condition (17). Hence, we conclude that at α = α H , the steady state E undergoes a Hopf bifurcation in the odd subspace. For α > α H , the steady state E is unstable in the odd subspace. The implication of the fact that the Hopf bifurcation can only occur in the odd subspace of the phase space [62] is that in the system (6) the fully symmetric state E can only bifurcate to an odd periodic orbit, for which variants 1 and 3 are synchronized and half a period out-of-phase with variants 2 and 4, i.e. each variant is π out of phase with its nearest neighbours. Besides the origin O and the fully symmetric equilibrium E, the system (6) possesses 14 more steady states characterized by a different number of non-zero variants y (for a general system with N antigenic variants there would be 2 N − 2 of such steady states). To comprehensively analyse these steady states and their stability, let us first introduce auxiliary quantities Y 1 = φµµ β α µ + αβµ , Y 2 = φµµ 2β α µ + αβµ , and Z 1,2 = βY 1,2 /µ, and W 1,2 = β Y 1,2 /µ . There are four distinct steady states with a single non-zero variant y i , which all have the isotropy subgroup D p 1 or its conjugate. A representative steady state of this kind is E 1 = (Y 1 , 0, 0, 0, Z 1 , 0, 0, 0, W 1 , W 1 , 0, W 1 ).(20) Other steady states E 2 , E 3 and E 4 are related to E 1 through elements of a subgroup of rotations Z 4 . Proposition 2. All steady states E 1 , E 2 , E 3 , E 4 with one non-zero variant are unstable. Proof. As it has already been explained, the steady states E 1,2,3,4 all lie on the same group orbit. In the light of equivariance of the system, this implies that all these states have the same stability type, and therefore it is sufficient to consider just one of them, for example, E 1 . Substituting the values of variables in E 1 into the Jacobian (13) gives the characteristic equation for eigenvalues that can be factorized as follows (λ − φ) · (λ + µ) 3 (λ + µ ) 3 (λ + α W 1 − φ) 2 × λ 3 + λ 2 (µ + µ ) + λ µµ + Y 1 (αβ + α β ) + Y 1 (αβµ + α β µ) = 0. It follows from this characteristic equation that one of the eigenvalues is λ = φ > 0 for any values of system parameters, which implies that the steady state E 1 is unstable, and the same conclusion holds for E 2 , E 3 and E 4 . Before moving to the case of two non-zero variants, it is worth noting that the symmetry group D 4 is an example of a more general dihedral group D n of order 2n, for which there are two distinct options in terms of conjugacies of reflections. If n is odd, all reflections are conjugate to each other by a rotation. However, when n is even (as is the case for D 4 ), reflections split into two different conjugacy classes: one class contains reflections along axes connecting vertices, and another class contains reflections connecting the sides. These two conjugacy classes are related by an outer automorphism, which can be represented as a rotation through π/n, which is a half of the minimal rotation in the dihedral group D n [31]. For the D 4 group, these two conjugacy classes are reflections along the diagonals, and reflections along horizontal/vertical axes. Now we consider the case of two non-zero variants, for which there are exactly six different steady states. The steady states with non-zero variants being nearest neighbours on the diagramme (1), i.e. (1,2), (2,3), (3,4) and (1,4), form one cluster: E 12 = (Y 2 , Y 2 , 0, 0, Z 2 , Z 2 , 0, 0, 2W 2 , 2W 2 , W 2 , W 2 ), E 23 = (0, Y 2 , Y 2 , 0, 0, Z 2 , Z 2 , 0, W 2 , 2W 2 , 2W 2 , W 2 ), E 34 = (0, 0, Y 2 , Y 2 , 0, 0, Z 2 , Z 2 , W 2 , W 2 , 2W 2 , 2W 2 ), E 14 = (Y 2 , 0, 0, Y 2 , Z 2 , 0, 0, Z 2 , 2W 2 , W 2 , W 2 , 2W 2 ), while the steady states with non-zero variants lying across each other on the diagonals, i.e. (1,3) and (2,3) are in another cluster E 13 = (Y 1 , 0, Y 1 , 0, Z 1 , 0, Z 1 , 0, W 1 , 2W 1 , W 1 , 2W 1 ), E 24 = (0, Y 1 , 0, Y 1 , 0, Z 1 , 0, Z 1 , 2W 1 , W 1 , 2W 1 , W 1 ), The difference between these two clusters of steady states is in the above-mentioned conjugacy classes of their isotropy subgroups: the isotropy subgroup of the first cluster belongs to a conjugacy class of reflections along the horizontal/vertical axes, with a centralizer given by D s 2 , and the isotropy subgroup of the second cluster belongs to a conjugacy class of reflections along the diagonals, with a centralizer given by D p 2 . Proposition 3. All steady states E 12 , E 23 , E 34 , E 14 , and also E 13 and E 24 , with two non-zero variants are unstable. Proof. Using the same approach as in Proposition 2, due to equivariance of the system and the fact that within each cluster all the steady states lie on the same group orbit, it follows that for the analysis of stability of these steady states it is sufficient to consider one representative from each cluster, for instance, E 12 and E 13 . Substituting the values of variables in E 12 into the Jacobian (13), one can find the characteristic equation for eigenvalues in the form (λ + α W 2 − φ) 2 (λ + µ ) 3 (λ + µ) 2 (λ 2 + µλ + αβY 2 )× λ 3 + λ 2 (µ + µ ) + λ (µµ + Y 2 (αβ + 2α β )) + Y 2 (αβµ + 2α β µ) = 0. It follows that this characteristic equation has among its roots an eigenvalue λ = φ − α W 2 = φ(αβµ + α β µ) αβµ + 2α β µ > 0. Since this eigenvalue is positive for any values of parameters, we conclude that the steady state E 12 (and also the steady states E 23 , E 34 , E 14 ) is unstable. In a similar way, the characteristic equation for the steady state E 13 can be written as follows (λ + 2α W 1 − φ) 2 (λ + µ) 2 (λ + µ ) 2 × λ 3 + λ 2 (µ + µ ) + λ (µµ + Y 1 (αβ + α β )) + Y 1 (αβµ + α β µ) 2 = 0. Hence, one of the eigenvalues is λ = φ − 2α W 1 = φ(αβµ − α β µ) αβµ + α β µ , which is always positive for biologically realistic restrictions on parameters (7). Therefore, we conclude that E 13 are E 24 are always unstable. For three non-zero variants, we again have four different steady states, which have an isotropy subgroup D p 1 or its conjugate. Introducing relative efficacies of the specific and cross-reactive immune responses E z and E w [45,46]: E z = αβ µ , E w = α β µ , one can write the values of state variables for these states in terms of E z and E w as follows: y 1 = E z − E w E z Y, y 2 = y 4 = Y, y 3 = 0, Y = φE z E 2 z + 2E z E w − E 2 w , z 1 = β µ E z − E w E z i Y, z 2 = z 4 = β µ Y, z 3 = 0, w 1 = β µ 2 + E z − E w φ Y, w 2 = w 4 = β µ 1 + E z − E w E z Y, w 3 = 2 β µ Y.(21) Here, we have chosen the three non-zero variants to be variants 1, 2 and 4. The other three steady states can be obtained from this one by rotation of variants. Proposition 4. All steady states with three non-zero variants are unstable. Proof. Similar to the proofs of Proposition 2 and 3, we employ system equivariance and the fact that the steady states with three non-zero variants all lie on the same group orbit to conclude that they all have the same stability type. Substituting the values of variables in the steady state (21) into the Jacobian (13) yields the characteristic equation λ − φ + 2 α β µ Y (λ + µ)(λ + µ )P 9 (λ) = 0, where P 9 (λ) is a 9-th degree polynomial in λ. It follows that one of the characteristic eigenvalues is λ = φ − 2 α β µ Y = φ α 2 β 2 µ 2 − α 2 β 2 µ 2 α 2 β 2 µ 2 − α 2 β 2 µ 2 + 2αβµα β µ , which is always positive due to biological restrictions on parameters (7). Hence, the steady state (21) and the other three steady states with three non-zero variants are all unstable. This completes the analysis of the steady states of system (6) with D 4 symmetry for generic values of parameters. It is worth noting, however, that in a particular case when E z = E w , the determinant of the system of linear equations that describes the steady states is equal to zero, and this results in the existence of a line of equilibria y 1 = y 3 = φµ 2αβ − s, y 2 = y 4 = s, 0 < s < φµ 2αβ . parameterized by an additional free parameter s. This line is reminiscent of the hypersurface of equilibria in the case µ = 0 in that every point on each such line is a steady state of the system (6), and as one moves along the line of equilibria, stability of the steady states can change, as illustrated in Fig. 3. One can observe that as the decay rate of the specific immune responses increases, this leads to stabilization of such steady states, and provided this rate is sufficiently high, all steady states on lines of equilibria are stable. Dynamical behaviour of the model In the previous section we established that a fully symmetric steady state E can undergo Hopf bifurcation, giving rise to a stable anti-phase periodic solution. Now we look at the evolution of this solution and its symmetries under changes in system parameters. For convenience, we fix all the parameters except for the rate of variant destruction by transient immune response α , which is taken to be a control parameter. The results of numerical simulations are presented in Fig. 4. When α is sufficiently small to satisfy the condition α < α H , the fully symmetric steady state is stable, as shown in plot (a). As it crosses the boundary E w = E z , the fully symmetric steady state loses stability through a Hopf bifurcation giving rise to a periodic solution with the spatio-temporal symmetry H = D 4 and a spatial symmetry K = D p 2 , which ensures that the variants 1 and 3 have the same behaviour, as do variants 2 and 4, as is illustrated in figure (b). Figure (c) indicates that as α increases, the periodic solution retains its symmetry but changes temporal profile, acquiring two peaks during a single period. When analysing other types of periodic behaviour in the model, one can note that according to the H/K theorem (see Section 3), periodic states can have spatio-temporal symmetry group pairs (H, K) only if H/K is cyclic, and K is an isotropy subgroup [10,30]. In the case of D 4 symmetry group acting on four elements, there are eleven pairs of subgroup H and K satisfying these requirements [30]. Now we look at actual solutions of the model and identify these spatial and spatio-temporal symmetries. For the same value of α as in Fig. (c), the system exhibits two more solutions with quite different spatial and spatio-temporal symmetries, as demonstrated in Figs. (d) and (e). The solution shown in Fig. (d) has the symmetry (H, K) = (Z 4 , 1) and is a discrete travelling wave, also known as a "splay state" [61], "periodic travelling (or rotating) wave" [7], or "ponies on a merry-go-round" or POMs [5] in the studies of systems of coupled oscillators. In this dynamical regime all variants appear sequentially one after another along the diagramme (1) with quarter of a period difference between two neighbouring variants. From the perspective of equvariant bifurcation theory, this solution is generic since the group Z n is always one of the subgroups of the D n group for the ring coupling, or the S n group for an all-to-all coupling, and its existence has already been extensively studied [5,29,31]. From the immunological point of view, this is an extremely important observation that effectively such solution, which immunologically represents sequential appearance of parasitemia peaks corresponding to different antigenic variants, owes its existence not to the individual dynamics of antigenic variants, but rather to the particular symmetric nature of cross-reactive interactions between them. This immunological genericity ensures that the same conclusions hold for a wide variety of immune interactions between human host and parasites, which use antigenic variation as a mechanism of immune escape, as illustrated, for instance, by malaria, African Trypanosomes, several members of Neisseria family (N. meningitidis and N. gonorrhoeae), Borrelia hermsii etc. [33,63]. Another co-existing solution for the same value of α is a state shown in Fig. (e). This solutions is characterized by the symmetry (H, K) = (D p 2 , D p 1 ) and corresponds to a situation, in which variants 1 and 3 are oscillating half a period out-of-phase with each other, and the variants 2 and 4 coincide and oscillate at twice the frequency of the pair (1,3). As the value of α is increased further, the system demonstrates a state of deterministic chaos, when variants appear in arbitrary order and magnitude. These kinds of periodic and chaotic solutions have been previously identified as most relevant for the clinical analysis of blood-stage malarial infection [57]. Discussion In this paper we have used techniques of equivariant bifurcation theory to perform a comprehensive analysis of steady states and periodic solutions in a model of antigenic variation in malaria. In the simplest case of two epitopes with two variants in each epitope, the system is equivariant with respect to a D 4 symmetry group of the square. Using isotypic decomposition of the phase space based on the irreducible representations of this symmetry group has allowed us to find a closed form expression for the boundary of Hopf bifurcation of the fully symmetric steady state in terms of system parameters, as well as to show that for any parameter values this bifurcation leads to an anti-phase periodic solution. All other steady states have been classified in terms of their isotropy subgroups, and this has provided insights into their origin and relatedness. With the help of the H/K theorem, we have classified various periodic states of the model in terms of their spatial and temporal symmetries. One of the important issues that have to be taken into account when applying methods of equivariant bifurcation theory to the models of realistic systems is the fact that these systems do not always fully preserve the assumed symmetry. In the context of modelling immune interactions between distinct antigenic variants, this means that not all variants cross-react in exactly the same quantitative manner. Despite this limitation, due to the normal hyperbolicity, which is a generic property in such models, main phenomena associated with the symmetric model survive under perturbations, including symmetry-breaking perturbations. The discussion of this issue in the context of modelling sympatric speciation using a symmetric model can be found in [30]. Whilst the analysis in this paper was constrained to the models of within-host dynamics antigenic variation, a similar approach can also be used for the population level mathematical models of multi-strain diseases. These models usually have a similar structure in that all strains/serotypes are assumed to be identical and have a certain degree of cross-protection or cross-enhancement based on antigenic distance between them [1,2,15,16,26,32,35,44,53,55]. The fact that multiple strains are antigenically related introduces symmetry into a model, and this then transpires in different types of periodic behaviours observed in these models. To give one example, almost all of these models support a solution in the form of discrete travelling wave having a symmetry (H, K) = (Z n , 1), which represents sequential appearance of different strains [35,44,55]. In a model studied by Calvez et al. [15] (based on an earlier model of Gupta et al. [35]), the authors study a multi-strain model having symmetry of a cube, and they find that the fully symmetric steady state can undergo Hopf bifurcation and give rise to an antiphase solution with a tetrahedral symmetry, which was not expected a priori. At the same time, if one looks at this system from the symmetry perspective, such periodic solution is to be expected as it has a D 4 symmetry, which corresponds to one of the three maximal isotropy subgroups and is therefore generically expected to arise at a Hopf bifurcation [22,37]. Observations of this kind illustrate that taking into account symmetries of the underlying models of multi-strain diseases can help make significant inroads in understanding and classifying possible periodic behaviours in such systems. The results presented in this paper are quite generic, and the conclusions we obtained are valid for a wide range of mathematical models of antigenic variation. In fact, they are applicable to the analysis of within-host dynamics of any parasite, which exhibits similar qualitative features of immune interactions based on the degree of relatedness between its antigenic variants. The significance of this lies in the possibility to classify expected dynamical regimes of behaviour using very generic assumption regarding immune interactions, and they will still hold true provided the actual system preserves the underlying symmetries. In the model studied in this paper the degree of cross-reactivity between antigenic variants does not vary with the number of epitopes they share. It is straightforward, however, to introduce antigenic distance between antigenic variants in a manner similar to the Hamming distance [2,53,58]. Such a modification would not alter the topology of the network of immune interactions but rather assign different weights to connections between different antigenic variants in such a network. Symmetry analysis of the effects of antigenic distance on possible dynamics are the subject of further study. Figure 1 : 1Interaction of malaria variants in the case of two minor epitopes with two variants in each epitope. Figure 2 : 2(a) Symmetries of the square. (b) Lattice of subgroups of D 4 symmetry group. Figure 3 : 3Lines of equilibria in the system (6) for E z = E w . Each vertical line is parameterized by s. Parameter values are φ = 7.5, α = 15, β = β = 10, µ = 0.5 Figure 4 : 4Temporal dynamics of the system (6). Parameter values are φ = 7.5, β = 10, β = 9, α = 15, µ = 0.01, µ = 0.02. (a) Stable fully symmetric equilibrium (α = 8). (b) Anti-phase periodic solution (α = 8.5), spatio-temporal symmetry (H, K) = (D 4 , D p 2 ). 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In this paper, we prove that the fluctuation of the extreme process of the maxima of all the largest eigenvalues of m× m principal minors (with fixed m) of the classical Gaussian orthogonal ensemble (GOE) of size n × n is given by the Gumbel distribution as n tends to infinity. We also derive the joint distribution of such maximal eigenvalue and the corresponding eigenvector in the large n limit, which will imply that these two random variables are asymptotically independent.

2205.05732

759f35e601bef91a54bc35be9bbca6ca1d7b51bb

PRINCIPAL MINORS OF GAUSSIAN ORTHOGONAL ENSEMBLE 11 May 2022 GANG TIANRenjie Feng ANDDongyi Wei Dong Yao PRINCIPAL MINORS OF GAUSSIAN ORTHOGONAL ENSEMBLE 11 May 2022arXiv:2205.05732v1 [math.PR] In this paper, we prove that the fluctuation of the extreme process of the maxima of all the largest eigenvalues of m× m principal minors (with fixed m) of the classical Gaussian orthogonal ensemble (GOE) of size n × n is given by the Gumbel distribution as n tends to infinity. We also derive the joint distribution of such maximal eigenvalue and the corresponding eigenvector in the large n limit, which will imply that these two random variables are asymptotically independent. Introduction Random matrix theory is a classical topic in probability which has applications to a variety of fields, such as statistics [4], high energy physics [7], wireless communication networks [8], deep neutral network [17], compressed sensing [19] and so forth. Motivated by high-dimensional statistics and signal processing, the authors in [9] derived the growth order of the maxima of all the largest eigenvalues of the principal minors of the classical random matrices of GOE and Wishart matrices, where the results have applications for the construction of the compressed sensing matrices as in [19]. In this paper, we further study the fluctuation of such maxima for the GOE case. Our main result is that the fluctuation is given by the Gumbel distribution with some Poisson structure involved in the limit, and we also derive the limiting joint distribution of such maxima and its corresponding eigenvector which indicates that these two random variables are asymptotically independent. The Gaussian orthogonal ensemble (GOE) is the Gaussian measure defined on the space of real symmetric matrices, i.e., G = (g ij ) 1≤i,j≤n is a symmetric matrix whose upper triangular entries are independent real Gaussian variables with the following distribution g ij d = N R (0, 2) if i = j; N R (0, 1) if i < j. Let λ 1 (G) > λ 2 (G) > · · · > λ n (G) be eigenvalues of GOE, then the distribution of these eigenvalues is invariant under the orthogonal group action and the joint density is (1) 1 Z n n k=1 e − 1 4 λ 2 k i<j λ i − λ j , where (2) Z n = 2 n(n+1)/4 (2π) n/2 (n!) −1 n j=1 Γ(1 + j/2) Γ(3/2) is the partition function. And the limit of the empirical measure of these eigenvalues is given by the classical semicircle law [1]. Let's first introduce some notations in order to present our main results. Given symmetric matrices G = (g ij ) 1≤i≤j≤n sampled from GOE, for α ⊂ {1, · · · , n} with cardinality |α| = m ∈ Z + , we denote G α = (g ij ) i,j∈α as the principal minors of G of size m × m, then G α is also symmetric. Let λ 1 (G α ) > λ 2 (G α ) > · · · > λ m (G α ) be the ordered eigenvalues of G α . Now we define the extreme process of the maxima of all the largest eigenvalues of the principal minors as T m,n = max α⊂{1,··· ,n},|α|=m λ 1 (G α ). In [9], the authors studied the asymptotic properties of T m,n and proved that under the assumption that m fixed or m → +∞ with m = o (ln n) 1/3 ln ln n , it holds T m,n − 2 √ m ln n → 0 in probability as n → +∞. In this paper, we further derive the fluctuation of T m,n when m is fixed as n tends to infinity. Our first result is the following Theorem 1. For GOE, we have the following convergence in distribution Our proof for Theorem 1 can imply the joint distribution of the maxima of the largest eigenvalues of principal minors and its corresponding eigenvector. To be more precise, given n, let v * ∈ S m−1 be the unit eigenvector corresponding to the largest eigenvalue of the principal minor that attains the maxima T m,n . By symmetry, −v * is also the corresponding eigenvector. Now we have the following limit for the joint distribution of (T m,n , v * ). Theorem 2. Given any y ∈ R and symmetric Borel set Q ⊂ S m−1 such that −Q = Q, let y 2 m = 4m ln n + 2(m − 2) ln ln n + y, then the joint distribution satisfies (4) P(T m,n > y m , v * ∈ Q) → (1 − F Y (y))ν(Q) as n → +∞, which implies that T m,n and v * are asymptotically independent. Here, ν is the uniform distribution on S m , i.e., ν(Q) = µ(Q ∩ S m )/µ(S m ). Let a 1 , ..., a n be i.i.d. Gaussian random variables N R (0, 2), for the extreme process M n := max{a 1 , ..., a n }, let a n = 2 √ ln n and b n = 2 √ ln n − ln ln n + ln(4π) 2 √ ln n . Then for any y ∈ R, the following classical result holds (Theorem 1.5.3 in [14]) lim n→+∞ P [a n (M n − b n ) ≤ y] = e −e −y/2 . One can check that Theorem 1 for T 2 1,n when m = 1 is equivalent to this classical result for M n . In this sense, our result is fundamental which can be considered as a natural generalization of such classical result for the extreme process of the scalar-valued random variables to the matrix-valued random variables (with correlations). One motivation to study the maxima of the largest eigenvalues of principle minors is from the study of compressed sensing, where one has to recover an input vector f from the corrupted measurements y = Af + e. Here, A is a coding matrix and e is an arbitrary and unknown vector of errors. The famous result by Candès-Tao [19] is that if the coding matrix A satisfies the restricted isometry property (Definition 1.1 in [19]), then the input f is the unique solution to some ℓ 1 -minimization problem provided that the support S (the number of nonzero entries) of errors e is not too large. Therefore, one of the major goals in compressed sensing is to construct the coding matrix A that satisfies the restricted isometry property. In Section 3 of [19], Candès-Tao proved that the Gaussian random matrices A can satisfy such property with overwhelming probability, and they can derive the estimate about the support S via the probabilistic estimate on the maxima of the largest eigenvalues of principle minors of Wishart matrices A T A. A simple proof based on the concentration measure theory is present in [5]. In this article, we only deal with the GOE case, but it seems that the method can be applied to the Wishart case and it's expected that some Gumbel fluctuation will be observed as well, which can imply better estimates on S. We would like to postpone the Wishart case for further investigate. Another motivation is that the extreme process of the maxima of the largest eigenvalues of principal minors may provide a model that interpolates between the Gumbel distribution in the Poisson regime and the Tracy-Widom law in the random matrix regime. It's well-known that for GOE, the largest eigenvalue T n,n when m = n in our setting is asymptotic to 2 √ n, and its fluctuation is given by the Tracy-Widom law, F 1 (y) = lim n→+∞ P (T n,n − 2 √ n)n 1/6 ≤ y , where F 1 (y) can be expressed in term of the Painlevé equation [1]. This together with our main result Theorem 1 indicate that there may exist several transitions from the Gumbel distribution to the Tracy-Widom law when m is increasing with n. There are some other models that have such phenomena. In [13], Johansson studied a family of determinantal processes whose edge behavior interpolates between a Poisson process with density e −x and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of the random MNS-models [16]. Therefore, it provides a model that the largest eigenvalue has a density transition from the Gumbel distribution to the Tracy-Widom law. Another important model is provided by the (Gaussian) random band matrices. It's also conjectured that there is a transition from the Poisson regime to the random matrix regime while the band width has different critical growth orders, the results in [18] almost confirm this conjecture at the spectral edge of some random band matrices. Our proof of Theorem 1 is based on Lemma 1, which roughly states that if some random variables are weakly correlated, then the point processes constructed via these random variables have a chance to converge to the Poisson processes. In our case when m is fixed or a very slowly varying function of n, the principal minors of size m× m are weakly correlated with each other as n large enough, therefore, one can expect that the point process of the largest eigenvalues of these principal minors converges to some Poisson point process, and thus some Gumbel distribution for the extreme process of the maxima of these largest eigenvalues will be observed in the limit. But this is not the case if m is a rapid varying function of n such as m = √ n, and the arguments in this article will not work, especially, the Poisson limit will not hold any more. Here, we would like to propose some natural questions such as the descriptions of the intermediate phases and the critical growth orders of m corresponding to these phases. It's worth mentioning that there are many other contexts about the (principal) minors of random matrices, and we list few of them as follows. In [10], Diaconis conjectured that the size of minors of the random matrices sampled from the orthogonal group O(n) with the Haar measure such that the minors can be approximated by independent standard normals is of order o( √ n), which can be considered as a generalization of the classical Poincaré-Borel Lemma. The conjecture is solved in [12] and we refer to [15] for more details and other relevant results. In [6,20] and the reference therein, the authors studied the principal minor assignment problems of the determinantal point processes with applications in graph theory and machine learning theory. In statistical physics, the matrix minor process constructed via eigenvalues of minors of random matrices will form an interlacing particle system. For example, the minor process of the Gaussian unitary ensemble is a determinantal point process [11]. One can find some other random matrix minor processes in [2]. Notation. In this paper, c, C and C ′ stand for positive constants, but their values may change from line to line. For simplicity, the notation a n ∼ b n means lim n→+∞ a n /b n = 1. Proof of Theorem 1 In this section, we will prove Theorem 1 by assuming some technical lemmas where the proofs of these lemmas are postponed to §3. The proof of Theorem 1 is based on Lemma 1 with the proof given in [3] by the Stein-Chen method. It provides a criteria to prove the convergence of the total number of occurrences of the point process to the Poisson distribution, and thus it provides a method to derive the distribution for some extreme processes. Lemma 1. Let I be an index set, and for α ∈ I, let X α be a Bernoulli random variable with p α = P(X α = 1) = 1 − P(X α = 0). For each α ∈ I, let N α be a subset of I with α ∈ N α , that is, α ∈ N α ⊂ I. Let S = α∈I X α , λ = ES = α∈I p α ∈ (0, +∞), let Z be the Poisson random variable with intensity EZ = λ, then it holds that L(S) − L(Z) ≤ 2(b 1 + b 2 + b 3 ), and the probability of no occurrence has the estimate |P(S = 0) − e −λ | =|P(X α = 0, ∀ α ∈ I) − e −λ | ≤ min(1, λ −1 )(b 1 + b 2 + b 3 ). Here, L(S) − L(Z) is the total variation distance between the distributions S and Z, and b 1 = α∈I β∈Nα p α p β , b 2 = α∈I α =β∈Nα E[X α X β ], b 3 = α∈I E|E[X α |σ(X β , β ∈ N α )] − p α |, and σ(X β , β ∈ N α ) is the σ-algebra generated by {X β , β ∈ N α }. In particular, if X α is independent of {X β , β ∈ N α } for each α, then b 3 = 0. One may think of N α as a 'neighborhood of dependence' for α such that X α is independent or nearly independent of all of X β for β / ∈ N α . And Lemma 1 indicates that when b 1 , b 2 and b 3 are all small enough, then S which is the total number of occurrences tends to a Poisson distribution. 2.1. Proof of Theorem 1. Given any ℓ × ℓ symmetric matrix S, we rearrange the eigenvalues of S in descending order λ 1 (S) ≥ · · · ≥ λ ℓ (S), and we denote N α = {β ∈ I m : α ∩ β = ∅} for α ∈ I m . Throughout the article, for any fixed real number y, we define y 2 k := 4k ln n + 2k ln ln n, y k > 0, 1 ≤ k < m and y 2 m := 4m ln n + 2(m − 2) ln ln n + y, y m > 0. For symmetric matrices G = (g ij ) 1≤i≤j≤n sampled from GOE, for α ∈ I m where |α| = m, we denote G α = (g ij ) i,j∈α as the principal minor of size m × m and we define the event A α = λ 1 (G α ) > y m ; λ * 1 (G β ) ≤ y k , ∀ 1 ≤ k < m, β ⊂ α, |β| = k .(7) Recall the definition of T m,n in §1, we first have 0 ≤ P(∩ α∈Im A c α ) − P(T m,n ≤ y m ) ≤ P(λ * 1 (G β ) > y k for some β ∈ I k , 1 ≤ k < m) ≤ m−1 k=1 β∈I k P(λ * 1 (G β ) > y k ) = m−1 k=1 n k P(λ * 1 (G {1,··· ,k} ) > y k ).(8) Now we need the following lemma and we postpone its proof to the next section. Lemma 2. For fixed k ≥1, there exists a constant C > 0 (depending on k) so that for all x > 1, P(|G {1,··· ,k} | 2 > x 2 ) ≤ Cx k(k+1)/2−2 e −x 2 /4 ,(9)P(λ 1 (G {1,··· ,k} ) > x) ≤ Cx k−2 e −x 2 /4 ,(10)P(λ * 1 (G {1,··· ,k} ) > x) ≤ Cx k−2 e −x 2 /4 .(11) Using (11) we have (12) m−1 k=1 n k P(λ * 1 (G {1,··· ,k} ) > y k ) ≤ C m−1 k=1 n k y k−2 k e −y 2 k /4 ≤ C/ ln n. Combining this with (8) we get (13) 0 ≤ P(∩ α∈Im A c α ) − P(T m,n ≤ y m ) ≤ C/ ln n, and thus we have (14) lim n→+∞ P(T m,n ≤ y m ) = lim n→+∞ P(∩ α∈Im A c α ). Therefore, it's enough to derive the limit of P(∩ α∈Im A c α ) to prove Theorem 1. By Lemma 1, we have (15) |P(∩ α∈Im A c α ) − e −tn | ≤ b n,1 + b n,2 , where t n = n m P(A {1,··· ,m} ), b n,1 ≤ n m n m − n − m m P(A {1,··· ,m} ) 2 , b n,2 ≤ m−1 k=1 n k n − k m − k n − m m − k P(A {1,··· ,k,··· ,m} ∩ A {1,··· ,k,m+1,··· ,2m−k} ). Using (10) we have P(A {1,··· ,m} ) ≤ P(λ 1 (G {1,··· ,m} ) > y m ) ≤ Cy m−2 m e −y 2 m /4 ≤ Cn −m . And thus we have b n,1 ≤ Cn 2m−1 P(A {1,··· ,m} ) 2 ≤ C ′ n −1 , which tends to 0 in the limit. It remains to find the limit of t n and show that b n,2 tends to 0 in order to complete the proof of Theorem 1. Let α = {1, · · · , m}, γ = {m − k + 1, · · · , m}, ζ = {m − k + 1, · · · , 2m − k}, then α ∩ ζ = γ, |α| = |ζ| = m and |γ| = k. By rearranging the indices, we have P(A {1,··· ,k,··· ,m} ∩ A {1,··· ,k,m+1,··· ,2m−k} ) =P(A {1,··· ,m} ∩ A {m−k+1,··· ,2m−k} ) ≤P(λ 1 (G α ) > y m , λ 1 (G ζ ) > y m , λ * 1 (G α\γ ) ≤ y m−k , λ * 1 (G ζ\γ ) ≤ y m−k , λ * 1 (G γ ) ≤ y k ) =E[P(λ 1 (G α ) > y m , λ 1 (G ζ ) > y m , λ * 1 (G α\γ ) ≤ y m−k , λ * 1 (G ζ\γ ) ≤ y m−k |G γ ) × 1 {λ * 1 (Gγ )≤y k } ] =E[P(λ 1 (G α ) > y m , λ * 1 (G α\γ ) ≤ y m−k |G γ ) 2 1 {λ * 1 (Gγ )≤y k } ]. The following lemma will imply that b n,2 tends to 0 as n → +∞. Lemma 3. For α ∈ I m , γ ⊂ α, |γ| = k, 1 ≤ k < m, β = α \ γ, x > 1, δ, δ ′ ∈ (0, 1) , then there are some constants C and C ′ (depending on m, δ and δ ′ ) such that P(λ 1 (G α ) > x|G β , G γ )1 {λ * 1 (G β )≤(1−δ)x,λ * 1 (Gγ )≤(1−δ ′ )x}(16)≤Cx −1 (x/(λ * 1 (G β ) + 1) + x/(λ * 1 (G γ ) + 1)) k(m−k)−1 e −(x−λ * By assuming Lemma 3, we have E[P(λ 1 (G α ) > y m , λ * 1 (G α\γ ) ≤ y m−k |G γ ) 2 1 {λ * 1 (Gγ )≤y k } ] ≤CE[y 2m−2k−4 m (y m /(λ * 1 (G γ ) + 1)) 2k(m−k)−2 e ((λ * 1 (Gγ )) 2 −y 2 m )/2 1 {λ * 1 (Gγ )≤y k } ] = : CE[f (λ * 1 (G γ )))1 {λ * 1 (Gγ )≤y k } ](18) where we define (19) f (t) = y 2m−2k−4 m (y m /(t + 1)) 2k(m−k)−2 e (t 2 −y 2 m )/2 , t ≥ 0. Integration by parts, we have E(f (λ * 1 (G γ ))1 λ * 1 (Gγ )≤y k ) = y k 0 f ′ (t)P(λ * 1 (G γ ) > t)dt − f (y k )P(λ * 1 (G γ ) > y k ) + f (0).(20) Note that f ′ (t) = −(2k(m − k) − 2)(t + 1) −1 f (t) + tf (t) ≤ tf (t).(21) Hence, using (11) for t > 1 and (21), we have y k 0 f ′ (t)P(λ * 1 (G γ ) > t)dt ≤ 1 0 tf (t)dt + y k 1 f ′ (t)P(λ * 1 (G γ ) > t)dt ≤ max 0≤t≤1 f (t) + C y k 1 ty 2m−2k−4 m (y m /(t + 1)) 2k(m−k)−2 e (t 2 −y 2 m )/2 t k−2 e −t 2 /4 dt.(22) Therefore, by (20) we further have E(f (λ * 1 (G γ ))1 λ * 1 (Gγ )≤y k ) ≤ 2 max 0≤t≤1 f (t) + C y k 1 ty 2m−2k−4 m (y m /(t + 1)) 2k(m−k)−2 e (t 2 −y 2 m )/2 t k−2 e −t 2 /4 dt.(23) We now separate this integration into 1 < t < y k /2 and y k /2 < t < y k . For 1 < t < y k /2 the integrand is bounded by Cy 2m−k−6 m y 2k(m−k)−1 m e y 2 k /16−y 2 m /2 , where we used the fact that y k ≤ y m for n large enough. For y k /2 < t < y k , we can bound the integrand by Cy 2m−k−6 m te t 2 /4−y 2 m /2 , where we used the fact that y m /y k ≤ 2 m/k as n large enough. Therefore, as n large enough, we have y k 1 f ′ (t)P(λ * 1 (G β ) > t)dt ≤Cy 2m−k−6 m e −y 2 m /2 y k /2 1 y 2k(m−k)−1 m e y 2 k /16 dt + y k y k /2 te t 2 /4 dt ≤Cy 2m−k−6 m e −y 2 m /2 y 2k(m−k) m e y 2 k /16 + e y 2 k /4 ≤Cy 2m−k−6 m e y 2 k /4−y 2 m /2 ,(24) where in the last inequality we used the fact that y 2k(m−k) m e −3y 2 k /16 can be bounded from above uniformly for all n. The definition of f (t) in (19) and the fact that y m /y k ∼ m/k imply (25) max 0≤t≤1 f (t) ≤ y 2m−2k−4 m y 2k(m−k)−2 m e (1−y 2 m )/2 ≤ Cy 2m−k−6 m e y 2 k /4−y 2 m /2 as n large enough. It follows from (18), (23), (24) and (25) that E[P(λ 1 (G α ) > y m , λ * 1 (G α\γ ) ≤ y m−k |G γ ) 2 1 {λ * 1 (Gγ )≤y k } ] ≤Cy 2m−k−6 m e y 2 k /4−y 2 m /2(26) as n large enough. Therefore, we have b n,2 ≤ m−1 k=1 n k n − k m − k n − m m − k P(A {1,··· ,k,··· ,m} ∩ A {1,··· ,k,m+1,··· ,2m−k} ) ≤ m−1 k=1 n 2m−k P(A {1,··· ,m} ∩ A {1,··· ,k,m+1,··· ,2m−k} ) ≤ C m−1 k=1 n 2m−k y 2m−k−6 m e y 2 k /4−y 2 m /2 ≤ C m−1 k=1 y 2m−k−6 m (ln n) k/2−m+2 ≤ C(ln n) −1 ,(27) which tends to 0 as n → +∞. The following lemma gives the limit of t n . Lemma 4. We have the limit (28) lim n→+∞ t n = c m e −y/4 , where c m = (2m) (m−2)/2 K m (m − 1)!2 3/2 Γ(1 + m/2) , where K m is the constant defined in Theorem 1. By assuming Lemma 4, by (15) together with the facts that b n,1 → 0 and b n,2 → 0, we can conclude that lim n→+∞ P(∩ α∈I A c α ) = exp(−c m e −y/4 ). By (14), this further implies lim n→+∞ P(T m,n ≤ y m ) = exp(−c m e −y/4 ), which proves Theorem 1. Proofs of lemmas In this section, we will prove Lemma 2, Lemma 3 and Lemma 4, and thus we complete the proof of Theorem 1. Proof of Lemma 2. Proof. We simply have the following estimates. For s ∈ R, there are some constants C, C ′ > 0 depending on s such that for all x > 1 we have (29) +∞ x r s exp(−r)dr ≤ Cx s exp(−x) and (30) +∞ x r s exp(−r 2 /2)dr ≤ C ′ x s−1 exp(−x 2 /2). Now let g 1 , . . . , g ℓ be ℓ independent N R (0, 1) random variables, then for all t ≥ 1, we have (31) P ℓ i=1 g 2 i ≥ t ≤ Ct ℓ/2−1 exp(−t/2), where C > 0 only depends on ℓ. The proof of (31) follows if we combine the estimate (29) and the fact that the probability density of the chi-squared distribution χ 2 (ℓ) := ℓ i=1 g 2 i with ℓ degrees of freedom is given by 1 2 ℓ/2 Γ(ℓ/2) x ℓ/2−1 e −x/2 . Lemma 2 holds obviously when k = 1 (note that g ii d = N R (0, 2)), and thus in the followings we consider the case when k ≥ 2. By the definition of the principal minor, (9), we note that by definition, G 1,...,k = (g ij ) 1≤i,j≤k is also sampled from GOE. Now letḡ ij = g ij if j = i and g ij / √ 2 otherwise, thenḡ ij , 1 ≤ i ≤ j ≤ k are i.i.d. N R (0, 1) random variables. To prove|G 1,...,k | 2 = Tr(G 2 1,...,k ) = k i,j=1 g 2 ij = 2 1≤i≤j≤kḡ 2 ij , therefore, by (29) and (31) we have P(|G 1,...,k | 2 > x 2 ) = P 1≤i≤j≤kḡ 2 ij > x 2 /2 ≤ Cx k(k+1)/2−2 e −x 2 /4 . This proves (9). To prove (10), by formula (1), the joint density of eigenvalues λ 1 ≥ · · · ≥ λ k of G {1,··· ,k} is J k (λ 1 , . . . , λ k ) = 1 Z k 1≤i<j≤k (λ i − λ j ) exp − k i=1 λ 2 i 4 . Note that if λ 1 > x > 1, we have λ 1 − λ j ≤ (λ 1 + 1)(|λ j | + 1) ≤ 2λ 1 (|λ j | + 1), and thus we have (32) 1≤i<j≤k (λ i − λ j ) ≤ Cλ k−1 1 2≤i≤k (|λ i | + 1) 2≤i<j≤k (λ i − λ j ), which further implies λ1>x J k (λ 1 , . . . , λ k )dλ 1 · · · dλ k ≤ C ∞ x λ k−1 1 exp(−λ 2 1 /4)dλ 1 × +∞>λ2>···>λ k >−∞ 2≤i≤k (|λ i | + 1) 2≤i<j≤k (λ i − λ j ) exp − k i=2 λ 2 i 4 dλ 2 · · · dλ k , which is bounded from above by C ′ x k−2 exp(−x 2 /4) by (30), thereby proving (10). Now we prove (11). By definition of λ * 1 (G 1,...,k ), we have P(λ * 1 (G 1,...,k ) > x) =P k i=2 λ 2 i + 2λ 2 1 > 2x 2 ≤ ⌊ √ 2x⌋+1 y=0 P 2λ 2 1 > (2x 2 − (y + 1) 2 ) + , y 2 ≤ k i=2 λ 2 i ≤ (y + 1) 2 + P k i=2 λ 2 i > 2x 2 :=I 1 + I 2 . By the fact λ 1 − λ j ≤ (|λ 1 | + 1)(|λ j | + 1) for all 2 ≤ j ≤ k, we first have (33) 1≤i<j≤k (λ i − λ j ) ≤ C(|λ 1 | + 1) k−1 2≤i≤k (|λ i | + 1) 2≤i<j≤k (λ i − λ j ). Then by the inequality of arithmetic and geometric means, for k i=2 λ 2 k ≥ 1, we have 2≤i≤k (|λ i | + 1) 2≤i<j≤k (λ i − λ j ) ≤ 2≤i≤k (|λ i | + 1) 2≤i<j≤k (|λ i | + |λ j |) ≤ 1 + k i=2 |λ i | k/2 k(k−1)/2 ≤   1 + √ k − 1 k i=2 λ 2 i k/2   k(k−1)/2 ≤ C k i=2 λ 2 i k(k−1)/4 .(34) Therefore, for x > 1, combining (33) and (34), we can bound I 2 as follows k i=2 λ 2 i >2x 2 >2 J k (λ 1 , . . . , λ k )dλ 1 · · · dλ k ≤ C ∞ 0 (|λ 1 | + 1) k−1 exp(−λ 2 1 /4)dλ 1 × k i=2 λ 2 i >2x 2 k i=2 λ 2 i k(k−1)/4 exp − k i=2 λ 2 i /4 dλ 2 · · · dλ k . The first integral is bounded. Using the polar coordinate to the second integral, and by (30) we have the bound (35) I 2 ≤ C r> √ 2x r k(k−1)/2 exp(−r 2 /4)r k−2 dr ≤ Cx k(k+1)/2−3 exp(−x 2 /2), which can be further bounded by C ′ x k−2 exp(−x 2 /4) for x > 1 by choosing C ′ large enough. Now we estimate I 1 for x > 1. For the case x 2 − (y + 1) 2 /2 > 1, by (30) and (32), we have P 2λ 2 1 > (2x 2 − (y + 1) 2 ) + , y 2 ≤ k i=2 λ 2 i ≤ (y + 1) 2 ≤C λ 2 1 >x 2 −(y+1) 2 /2>1 |λ 1 | k−1 exp(−λ 2 1 /4)dλ 1 × λ2>···>λ k :(y+1) 2 ≥ k i=2 λ 2 i ≥y 2 2≤i≤k (|λ i | + 1) 2≤i<j≤k (λ i − λ j ) exp − k i=2 λ 2 i /4 dλ 2 · · · dλ k ≤Cx k−2 exp(−(x 2 − (y + 1) 2 /2)/4) × R k−1 :(y+1) 2 ≥ k i=2 λ 2 i ≥y 2 2≤i≤k (|λ i | + 1) 2≤i<j≤k (|λ i | + |λ j |) exp(− k i=2 λ 2 i /4)dλ 2 · · · dλ k . We denote the last integral as A k (y) := R k−1 :(y+1) 2 ≥ k i=2 λ 2 i ≥y 2 2≤i≤k (|λ i | + 1) 2≤i<j≤k (|λ i | + |λ j |) exp(− k i=2 λ 2 i /4)dλ 2 · · · dλ k . For the case x 2 − (y + 1) 2 /2 ≤ 1, by (33) and the arguments as above, we simply have P 2λ 2 1 > (2x 2 − (y + 1) 2 ) + , y 2 ≤ k i=2 λ 2 i ≤ (y + 1) 2 ≤C R (|λ 1 | + 1) k−1 exp(−λ 2 1 /4)dλ 1 A k (y) ≤CA k (y) ≤Cx k−2 exp(−(x 2 − (y + 1) 2 /2)/4)A k (y) for x > 1. Therefore, in both cases, by the polar coordinate, we further have the upper bound, P 2λ 2 1 > (2x 2 − (y + 1) 2 ) + , y 2 ≤ k i=2 λ 2 i ≤ (y + 1) 2 ≤Cx k−2 e −x 2 /4 y+1>r>y (1 + r) k−1 r (k−1)(k−2)/2 r k−2 exp((r + 1) 2 /8 − r 2 /4)dr. By taking the summation, I 1 can be bounded from above by I 1 ≤Cx k−2 exp(−x 2 /4) ⌊ √ 2x⌋+1 0 (1 + r) k−1 r (k−1)(k−2)/2 r k−2 exp((r + 1) 2 /8 − r 2 /4)dr ≤Cx k−2 exp(−x 2 /4) +∞ 0 · · · = C ′ x k−2 exp(−x 2 /4). This will complete the proof of (11) by the estimates of I 1 and I 2 . Proof of Lemma 3. Proof. We first prove (16). We claim that (16) is equivalent to the following statement: for any δ > 0, x > 1, there exists a constant C depending on δ, such that P(λ 1 (G α ) > x|G β , G γ )1 {λ * 1 (G β )≤(1−δ)x,λ * 1 (Gγ )≤(1−δ)x} (36) ≤Cx −1 (x/(λ * 1 (G β ) + 1) + x/(λ * 1 (G γ ) + 1)) k(m−k)−1 e −(x−λ * The implication that (16) ⇒ (36) is trivial. We now show that (36) implies (16). For any δ, δ ′ ∈ (0, 1), we defineδ = min{δ, δ ′ }. (36) implies that there exists a constant C(δ) such that P(λ 1 (G α ) > x|G β , G γ )1 {λ * 1 (G β )≤(1−δ)x,λ * 1 (Gγ )≤(1−δ)x} (37) ≤Cx −1 (x/(λ * 1 (G β ) + 1) + x/(λ * 1 (G γ ) + 1)) k(m−k)−1 e −(x−λ * 1 (G β ))(x−λ * 1 (Gγ ))/2 . (37) implies (16) since λ * 1 (G β ) ≤ (1 − δ)x, λ * 1 (G γ ) ≤ (1 − δ ′ )x ⊂ λ * 1 (G β ) ≤ (1 −δ)x, λ * 1 (G γ ) ≤ (1 −δ)x . This completes the proof of the equivalence between (16) and (36). We now prove (36). Without loss of generality, we may assume α = {1, . . . , m}, γ = {1, . . . , k}, β = {k + 1, . . . , m}. Since G β and G γ are both symmetric matrices sampled from GOE (independently), we can find orthogonal matrices U and U ′ such that U G β U t = X, U ′ G γ U ′ t = Z, where the diagonal matrices X =      λ 1 (G β ) λ 2 (G β ) . . . λ ℓ (G β )      , Z =      λ 1 (G γ ) λ 2 (G γ ) . . . λ k (G γ )      , where ℓ := m − k. It follows that U U ′ G α U t U ′ t = X V V t Z , where V is an ℓ × k matrix with i.i.d. N R (0, 1) entries. Given any 1 × m vector v, we decompose it as v = (p, q), p = (p 1 , . . . , p ℓ ), q = (q 1 , . . . , q k ), then we have v X V V t Z v t = ℓ i=1 λ i (G β )p 2 i + k j=1 λ j (G γ )q 2 j + 2 ℓ i=1 k j=1 p i v ij q j . For simplicity, we use P * to denote the conditional probability (conditional on G β and G γ ). By Rayleigh quotient, for x > 1, we have P * (λ 1 (G α ) > x) =P *   ∃(p, q) = 0, ℓ i=1 λ i (G β )p 2 i + k j=1 λ j (G γ )q 2 j + 2 ℓ i=1 k j=1 p i v ij q j ≥ x ℓ i=1 p 2 i + x k j=1 q 2 j   =P *   ∃(p, q) = 0, 2 ℓ i=1 k j=1 p i v ij q j ≥ ℓ i=1 (x − λ i (G β ))p 2 i + k j=1 (x − λ j (G γ )q 2 j   . We define the event Ω := λ * 1 (G β ) ≤ (1 − δ)x, λ * 1 (G γ ) ≤ (1 − δ)x . For the rest of the proof, all the arguments are restricted on the event Ω and x > 1. On Ω, by the definition λ * (A) for any symmetric matrix A in (6), we simply have x + √ 2(1 − δ)x ≥ x−λ i (G β ) ≥ x−λ * 1 (G β ) ≥ δx > 0 and x+ √ 2(1−δ)x ≥ x−λ j (G γ ) ≥ x−λ * 1 (G γ ) ≥ δx > 0. Therefore, on Ω it holds that (38) z ij := (x − λ i (G β ))(x − λ j (G γ )) ∈ [δ 2 x 2 , ( √ 2 − √ 2δ + 1) 2 x 2 ]. We denotep i = x − λ i (G β )p i ,q j = x − λ j (G γ )q j ,v ij = v ij √ z ij . Using Cauchy-Schwartz inequality we have 2 ℓ i=1 k j=1p ivijqj ≤ 2 ℓ i=1p 2 i    ℓ i=1   k j=1v ijqj   2    ≤ 2 ℓ i=1p 2 i   k j=1q 2 j     ℓ i=1 k j=1v 2 ij   ≤ ℓ i=1 k j=1v 2 ij   ℓ i=1p 2 i + k j=1q 2 j   . Therefore, we further have the probabilistic estimate of λ 1 (G α ) as P * (λ 1 (G α ) > x) =P *   ∃(p,q) = 0, 2 ℓ i=1 k j=1p ivijqj ≥ ℓ i=1p 2 i + k j=1q 2 j   ≤P *   ℓ i=1 k j=1v 2 ij ≥ 1   . If k = ℓ = 1, on the event Ω, by (30) for Gaussian random variable v 11 we have P * (v 2 11 ≥ 1) = P * (v 2 11 > z 11 ) ≤ C √ z 11 exp(−z 11 /2) ≤ C x exp(−(x − λ 1 (G β ))(x − λ 1 (G γ ))/2) ≤ C x exp(−(x − λ * 1 (G β ))(x − λ * 1 (G γ ))/2),(39) where we have used the fact that z 11 ≥ δ 2 x 2 . This will imply (36) in the case m = 2, k = ℓ = 1. Now we consider the case k ≥ 2 or ℓ ≥ 2. We define κ =        min λ1(G β )−λ2(G β ) x−λ2(G β ) , λ1(Gγ )−λ2(Gγ ) x−λ2(Gγ ) if ℓ, k ≥ 2; λ1(G β )−λ2(G β ) x−λ2(G β ) if k = 1, ℓ ≥ 2; λ1(Gγ )−λ2(Gγ ) x−λ2(Gγ ) if ℓ = 1, k ≥ 2. On the event Ω, one can show that (40) 0 ≤ κ ≤ (1 − δ)x − λ 2 (G β ) x − λ 2 (G β ) ≤ (1 − δ)x − (− √ 2(1 − δ)x) x − (− √ 2(1 − δ)x) = (1 + √ 2)(1 − δ) 1 + √ 2(1 − δ) := c δ < 1. Note that ℓ i=1 k j=1v 2 ij = ℓ i=1 k j=1 v 2 ij (x − λ i (G β ))(x − λ j (G γ )) = 1 (x − λ 1 (G β ))(x − λ 1 (G γ ))   v 2 11 + (i,j) =(1,1) (x − λ 1 (G β ))(x − λ 1 (G γ )) (x − λ i (G β ))(x − λ j (G γ )) v 2 ij   ≤ 1 (x − λ 1 (G β ))(x − λ 1 (G γ ))   v 2 11 + (1 − κ) (i,j) =(1,1) v 2 ij   , where we used the fact that (x − λ 1 (G β ))(x − λ 1 (G γ )) (x − λ i (G β ))(x − λ j (G γ )) ≤ max x − λ 1 (G β ) x − λ 2 (G β ) , x − λ 1 (G γ ) x − λ 2 (G γ ) = 1 − min λ 1 (G β ) − λ 2 (G β ) x − λ 2 (G β ) , λ 1 (G γ ) − λ 2 (G γ ) x − λ 2 (G γ ) for (i, j) = (1, 1). Hence, by (31) we can conclude the following bounds on the event Ω P * (λ 1 (G α ) > x) ≤P *   ℓ i=1 k j=1v 2 ij ≥ 1   ≤P *   v 2 11 + (1 − κ) (i,j) =(1,1) v 2 ij ≥ z 11   ≤P *   ℓ i=1 k j=1 v 2 ij ≥ z 11   ≤Cz kℓ/2−1 11 exp(−z 11 /2) ≤Cx kℓ−2 exp(−(x − λ 1 (G β ))(x − λ 1 (G γ ))/2),(41) where in the last step we used (38), and C is a constant depending on δ and m. We now consider the case when one of the following two conditions holds. Condition 1: λ * 1 (G β ) ≤ 1 or λ * 1 (G γ ) ≤ 1. Under this condition, since λ 1 (G β ) ≤ λ * 1 (G β ) and λ 1 (G γ ) ≤ λ * 1 (G γ ), we have P *   ℓ i=1 k j=1v 2 i,j ≥ 1   ≤ Cx kℓ−2 exp(−(x − λ * 1 (G β ))(x − λ * 1 (G γ ))/2), which further yields the bound (36) by the assumption λ * 1 (G β ) ≤ 1 or λ * 1 (G γ ) ≤ 1. Condition 2: max{|λ i (G β )| , 2 ≤ i ≤ ℓ} > |λ 1 (G β )| /2 or max{|λ i (G γ )| , 2 ≤ i ≤ k} > |λ 1 (G γ )| /2. If condition 1 fails (i.e., λ * 1 (G β ) > 1 and λ * 1 (G γ ) > 1) and condition 2 holds (say, max{|λ i (G β )| , 2 ≤ i ≤ ℓ} > |λ 1 (G β )| /2), then we have λ * 1 (G β ) − λ 1 (G β ) = ℓ i=2 λ 2 i (G β ) 2(λ * 1 (G β ) + λ 1 (G β )) ≥ C(λ * 1 (G β )) 2 4λ * 1 (G β ) = C ′ λ * 1 (G β ). In this case, on Ω we have (x − λ * 1 (G β ))(x − λ * 1 (G γ )) ≤ (x − λ 1 (G β ))(x − λ * 1 (G γ )) − (λ * 1 (G β ) − λ 1 (G β ))δx ≤ (x − λ 1 (G β ))(x − λ 1 (G γ )) − C ′ λ * 1 (G β )δx ≤ (x − λ 1 (G β ))(x − λ 1 (G γ )) − C ′ δx. (42) Using (41), we have P * (λ 1 (G α ) > x) ≤ Cx kℓ−2 exp(−(x − λ * 1 (G β ))(x − λ * 1 (G γ ))/2) exp(−C ′ δx) . This can be further bounded from above by Cx −1 x 1 + λ * 1 (G β ) + x 1 + λ * 1 (G γ ) kℓ−1 exp(−(x − λ * 1 (G β ))(x − λ * 1 (G γ ))/2), this is because on Ω it holds exp(−C ′ δx) ≤ C 1 1 + (1 − δ)x kℓ−1 ≤ C 1 1 + λ * 1 (G β ) + 1 1 + λ * 1 (G γ ) kℓ−1 for x > 1 by choosing C large enough. Note that the constant C > 0 only depends on m and δ, and does not depend on (the conditional) λ * 1 (G β ) and λ * 1 (G γ ). As a summary, we have verified (36) when either of the two conditions is satisfied. Now we assume that both conditions fail so that λ * 1 (G β ) > 1, λ * 1 (G γ ) > 1, max{|λ i (G β )| , 2 ≤ i ≤ ℓ} ≤ |λ 1 (G β )| /2 and max{|λ i (G γ )| , 2 ≤ i ≤ k} ≤ |λ 1 (G γ )| /2. Then there exists a constant c > 0 such that λ 1 (G β ) − λ 2 (G β ) ≥ cλ * 1 (G β ) ≥ c and λ 1 (G γ ) − λ 2 (G γ ) ≥ cλ * 1 (G γ ) ≥ c. Therefore, by definition of κ, (40) and the assumptions λ * 1 (G β ) > 1, λ * 1 (G γ ) > 1, we have (43) 1 > c δ ≥ κ ≥ c min λ * 1 (G β ) x − λ 2 (G β ) , λ * 1 (G γ ) x − λ 2 (G γ ) ≥ c 2 min λ * 1 (G β ) + 1 x − λ 2 (G β ) , λ * 1 (G γ ) + 1 x − λ 2 (G γ ) . By the fact that 0 < δx ≤ x − λ 2 (G β ), x − λ 2 (G γ ) ≤ x + √ 2(1 − δ)x on Ω, we further have (44) c δ ≥ κ ≥ C δ min λ * 1 (G β ) + 1 x , λ * 1 (G γ ) + 1 x , which implies (45) 1 κ ≤ Cx 1 λ * 1 (G β ) + 1 + 1 λ * 1 (G γ ) + 1 for some constant C > 0 that depends on m and δ. Note that the probability density function p(t) for v 2 11 is exp(−t/2)/ √ 2πt, recall (41), we have P *   ℓ i=1 k j=1v 2 ij ≥ 1   ≤P *   v 2 11 + (1 − κ) (i,j) =(1,1) v 2 ij > z 11   = z11 0 exp(−(z 11 − t)/2) 2π(z 11 − t) P *   (1 − κ) (i,j) =(1,1) v 2 ij > t   dt + P * (v 2 11 > z 11 ) ≤C exp(−z 11 /2)   1 √ z 11 + z11 0 exp(t/2) √ z 11 − t P *   (1 − κ) (i,j) =(1,1) v 2 ij > t   dt   , where we have used P * (v 2 11 > z 11 ) ≤ Ce −z 11 /2 √ z11 by (31). By (31) again, we have P *   (1 − κ) (i,j) =(1,1) v 2 i,j > t   ≤ C t 1 − κ (kℓ−1)/2−1 exp − t 2(1 − κ) ≤ Ct (kℓ−1)/2−1 exp − t 2(1 − κ) , where in the last inequality we used the fact that 1 − κ ≥ 1 − c δ > 0 by (40). It follows that P *   ℓ i=1 k j=1v 2 ij ≥ 1   ≤C exp(−z 11 /2) 1 √ z 11 + z11 0 t (kℓ−3)/2 √ z 11 − t exp t 2 − t 2(1 − κ) dt ≤C ′ exp(−z 11 /2) 1 √ z 11 + 1 √ z 11 z11/2 0 t (kℓ−3)/2 exp − κt 2(1 − κ) dt +z (kℓ−3)/2 11 exp − κz 11 4(1 − κ) z11 z11/2 1 √ z 11 − t dt ≤C ′ exp(−z 11 /2) √ z 11 1 + 1 κ (kℓ−1)/2 ∞ 0 s (kℓ−3)/2 exp(−s)ds + z (kℓ−1)/2 11 exp − κz 11 4 . Note that the integration of s (kℓ−3)/2 exp(−s) converges since kℓ ≥ 2, and thus the second term can be bounded from above by c(1/κ) (kℓ−1)/2 . For the third term, the global maxima of the function x (kℓ−1)/2 e −κx/4 is obtained at the point x = c ′ /κ, here c ′ depends on k, ℓ, and thus the third term can be bounded from above by C(1/κ) (kℓ−1)/2 , where C depends on δ and m. Therefore, on Ω we have P *   ℓ i=1 k j=1v 2 ij ≥ 1   ≤C ′ exp(−z 11 /2) √ z 11 1 + c(1/κ) (kℓ−1)/2 + C(1/κ) (kℓ−1)/2 ≤C ′ exp(−z 11 /2) √ z 11 (1/κ) (kℓ−1)/2 + c(1/κ) (kℓ−1)/2 + C(1/κ) (kℓ−1)/2 [since 1 < 1/κ] =C exp(−z 11 /2) √ z 11 (1/κ) (kℓ−1)/2 ≤C exp(−z 11 /2) √ z 11 (1/κ) kℓ−1 ≤Cx −1 x λ * 1 (G β ) + 1 + x λ * 1 (G γ ) + 1 kℓ−1 exp(−(x − λ * 1 (G β ))(x − λ * 1 (G γ ))/2)) [by (38), (45)]. This proves (36), and thus we complete the proof of (16). We now prove (17) using (11) and (16). By convexity of the function s → s k(m−k)−1 for s > 0, we have x −1 (x/(λ * 1 (G β ) + 1) + x/(λ * 1 (G γ ) + 1)) k(m−k)−1 e −(x−λ * 1 (G β ))(x−λ * 1 (Gγ ))/2 ≤Cx −1 (x/(λ * 1 (G β ) + 1)) k(m−k)−1 e −(x−λ * 1 (G β ))(x−λ * 1 (Gγ ))/2 + Cx −1 (x/(λ * 1 (G γ ) + 1)) k(m−k)−1 e −(x−λ * 1 (G β ))(x−λ * 1 (Gγ ))/2 :=I 3 + I 4 .(46) LetĒ be the conditional expectation with respect to G γ and ℓ := m − k as before, and we define (16) and (46), we have f (t) = x −1 (x/(t + 1)) kℓ−1 exp(−(x − t)(x − λ * 1 (G γ ))/2) and h(t) = x −1 (x/(λ * 1 (G γ ) + 1)) kℓ−1 exp(−(x − t)(x − λ * 1 (G γ ))/2). We define the event Ω ′ = {λ * 1 (G γ ) ≤ (1 − δ ′ )x}. UsingP(λ 1 (G α ) > x, λ * 1 (G β ) ≤ (1 − δ)x|G γ )1 {λ * 1 (Gγ )≤(1−δ ′ )x} =E[E 1 {λ1(Gα)>x, λ * 1 (G β )<(1−δ)x} |G β , G γ |G γ ]1 {λ * 1 (Gγ )≤(1−δ ′ )x} =E[E 1 {λ1(Gα)>x} |G β , G γ 1 {λ * 1 (G β )<(1−δ)x} 1 {λ * 1 (Gγ )≤(1−δ ′ )x} |G γ ]1 {λ * 1 (Gγ )≤(1−δ ′ )x} ≤CĒ x −1 (x/(λ * 1 (G β ) + 1) + x/(λ * 1 (G γ ) + 1)) kℓ−1 × exp − (x − λ * 1 (G β ))(x − λ * 1 (G γ ))/2 1 {λ * 1 (G β )≤(1−δ)x} 1 Ω ′ ≤CĒ(I 3 1 {λ * 1 (G β )≤(1−δ)x} )1 Ω ′ + CĒ(I 4 1 {λ * 1 (G β )≤(1−δ)x} )1 Ω ′ ≤C Ē (f (λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} )1 Ω ′ +Ē(h(λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} )1 Ω ′ .(47) Note that G β and G γ are independent, therefore, λ * 1 (G β ) and λ * 1 (G γ ) are independent as well. Hence for any t > 0,P(λ * 1 (G β ) > t) is equal to unconditional probability P(λ * 1 (G β ) > t). Then using integration by parts, we havē E(f (λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} ) = (1−δ)x 0 f ′ (t)P(λ * 1 (G β ) > t)dt − f ((1 − δ)x)P(λ * 1 (G β ) > (1 − δ)x) + f (0).(48) On the event Ω ′ = {λ * 1 (G γ ) ≤ (1 − δ ′ )x}, one has f ′ (t) ≤ 1 2 (x/(t + 1)) kℓ−1 exp(−(x − t)(x − λ * 1 (G γ ))/2). Therefore, we have 1 0 f ′ (t)P(λ * 1 (G β ) > t)dt ≤ 1 2 x kℓ−1 exp(−(x − 1)(x − λ * 1 (G γ ))/2) = 1 2 x kℓ−1 e 1/4 exp((λ * 1 (G γ ) 2 − x 2 )/4) exp − ((x − λ * 1 (G γ )) − 1) 2 4 .(49) On the event Ω ′ , we have x − λ * 1 (G γ ) ≥ δ ′ x. Therefore, for C large enough, one has (50) exp − ((x − λ * 1 (G γ )) − 1) 2 4 ≤ C exp(−δ ′2 x 2 /8). It further follows from (49) and (50) that for x > 1, there exists C large enough such that (51) 1 0 f ′ (t)P(λ * 1 (G β ) > t)dt ≤ Cx ℓ−2 exp(((λ * 1 (G γ )) 2 − x 2 )/4). If (1 − δ)x ≤ 1, then we trivially have (52) (1−δ)x 0 f ′ (t)P(λ * 1 (G β ) > t)dt ≤ 1 0 · · · ≤ Cx ℓ−2 exp(((λ * 1 (G γ )) 2 − x 2 )/4). Now we consider the case when (1 − δ)x ≥ 1. Recall (11) in Lemma 2, we have (53) P(λ * 1 (G β ) > t) ≤ Ct ℓ−2 exp(−t 2 /4), t ≥ 1, this implies that (1−δ)x 1 f ′ (t)P(λ * 1 (G β ) > t)dt ≤ Cx ℓ−2 exp(((λ * 1 (G γ )) 2 − x 2 )/4) × (1−δ)x 1 (x/(t + 1)) kℓ−1 exp(−(t − (x − λ * 1 (G γ ))) 2 /4)dt,(54) where we have used the identity − (x − t)(x − λ * 1 (G γ ))/2 − t 2 /4 = − (t − (x − λ * 1 (G γ ))) 2 4 + (x − λ * 1 (G γ )) 2 4 − x(x − λ * 1 (G γ )) 2 = − (t − (x − λ * 1 (G γ ))) 2 4 + (λ * 1 (G γ )) 2 − x 2 4 . On Ω ′ , we have, (1−δ)x 1 (x/(t + 1)) kℓ−1 exp(−(t − (x − λ * 1 (G γ ))) 2 /4)dt ≤ min{δ ′ x/2,(1−δ)x} 0 x kℓ−1 exp(−(δ ′ ) 2 x 2 /16)dt + (1−δ)x min{δ ′ x/2,(1−δ)x} 2 δ ′ + 1 1 − δ kℓ−1 exp(−(t − (x − λ * 1 (G γ ))) 2 /4)dt ≤Cx kℓ exp(−(δ ′ ) 2 x 2 /16) + C ≤ C ′ .(55) Combining (51), (54) and (55), for (1 − δ)x > 1, we have (56) (1−δ)x 0 f ′ (t)P(λ * 1 (G β ) > t)dt = 1 0 · · · + (1−δ)x 1 · · · ≤ Cx ℓ−2 exp(((λ * 1 (G γ )) 2 − x 2 )/4). By (52), the above estimate is actually true for all x > 1 on Ω ′ . We also have (56) and (57), on Ω ′ we get (57) f (0) = x kℓ−2 exp(−x(x − λ * 1 (G γ ))/2) ≤ Cx ℓ−2 exp(((λ * 1 (G γ )) 2 − x 2 )/4) for C large enough, this is because on Ω ′ we have −x(x − λ * 1 (G γ ))/2 − ((λ * 1 (G γ )) 2 − x 2 )/4 = −(x − λ * 1 (G γ )) 2 /4 ≤ −δ ′2 x 2 . Combining (48),(58)Ē(f (λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} ) ≤ C ′ x ℓ−2 exp(((λ * 1 (G γ )) 2 − x 2 )/4). On Ω ′ , for x > 1, it holds that x/(λ * 1 (G γ ) + 1) ≥ x/((1 − δ ′ )x + 1) ≥ 1/(2 − δ ′ ) , therefore, we further have the upper bound, (59)Ē(f (λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} ) ≤ C ′ x ℓ−2 (x/(λ * 1 (G γ ) + 1)) kℓ−1 exp(((λ * 1 (G γ )) 2 − x 2 )/4). We can similarly control the conditional expectation of h(λ * 1 (G β ))1 {λ * 1 (G β )≤(1−δ)x} . Analogously to (48), we havē E(h(λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} ) = (1−δ)x 0 h ′ (t)P(λ * 1 (G β ) > t)dt − h((1 − δ)x)P(λ * 1 (G β ) > (1 − δ)x) + h(0).(60) On the event Ω ′ = {λ * 1 (G γ ) ≤ (1 − δ ′ )x}, one has h ′ (t) ≤ 1 2 (x/(λ * 1 (G γ ) + 1)) kℓ−1 exp(−(x − t)(x − λ * 1 (G γ ))/2). We basically repeat the proofs of (56) and (57) to get E(h(λ * 1 (G β ))1 {λ * 1 (G β )<(1−δ)x} ) ≤ Cx ℓ−2 (x/(λ * 1 (G γ ) + 1)) kℓ−1 exp(((λ * 1 (G γ )) 2 − x 2 )/4).(61) Then (17) follows from (47), (59) and (61) (recall that ℓ = m − k). Proof of Lemma 4. Proof. By the definition of A α in (7) with α = {1, 2, ..., m}, we have P(A {1,2,...,m} ) =P(λ 1 (G α ) > y m ; λ * 1 (G β ) ≤ y k , ∀ 1 ≤ k < m, β ⊂ α, |β| = k) =P(λ 1 (G α ) > y m )P(λ * 1 (G β ) ≤ y k , ∀ 1 ≤ k < m, β ⊂ α, |β| = k|λ 1 (G α ) > y m ).(62) Lemma 4 follows from the following two limits, We first prove (63). By formula (1) for G α where |α| = m, we have lim n→+∞ P(λ 1 (G α ) > y m )/[m2 −(1+m)/2 /Γ(1 + m/2) · y mP(λ 1 (G α ) > y m ) = 1 Z m λ1>···>λm;λ1>ym e − m i=1 λ 2 i /4 1≤i<j≤m |λ i − λ j |dλ 1 · · · dλ m = 1 Z m λ1>ym e −λ 2 1 /4 g(λ 1 )dλ 1 ,(65) where we denote g(λ 1 ) = λ1>···>λm e − m i=2 λ 2 i /4 1≤i<j≤m |λ i − λ j |dλ 2 · · · dλ m . We claim that lim λ1→+∞ g(λ 1 )/(λ m−1 1 Z m−1 ) = 1.(66) Dividing into two cases λ m > − √ λ 1 and λ m < − √ λ 1 , g(λ 1 ) is bounded from above by λ2>···>λm>− √ λ1 (λ 1 + λ 1 ) m−1 2≤i<j≤m (λ i − λ j )e − m i=2 λ 2 i /4 dλ 2 · · · dλ m + λ2>···>λm,λm<− √ λ1 1≤i<j≤m (λ i − λ j )e − m i=2 λ 2 i /4 dλ 2 · · · dλ m := I 5 + I 6 . (67) Note that I 5 is further bounded from above by ( λ 1 + √ λ 1 ) m−1 Z m−1 . For I 6 , we note that 1≤i<j≤m (λ i − λ j ) ≤ 1≤i<j≤m (|λ i | + 1)(|λ j | + 1) ≤ m i=1 (|λ i | + 1) m−1 . Therefore, we can bound I 6 from above by (|λ 1 | + 1) m−1 λm<− √ λ1 (|λ m | + 1) m−1 exp(−λ 2 m /4)dλ m × R (|λ 2 | + 1) m−1 exp(−λ 2 2 /4)dλ 2 m−2 .(68) For λ 1 large enough, (68) can be further bounded from above by (using (30)) Cλ m−1 1 λ 1 m−2 exp(−λ 1 /4) ≤ C ′ λ m−2 1 . Combining the estimates for I 5 and I 6 we get g(λ 1 ) ≤ Z m−1 (λ 1 + λ 1 ) m−1 + Cλ m−2 1 for λ 1 large enough. It follows that lim sup λ1→+∞ g(λ 1 ) λ m−1 1 ≤ Z m−1 . For the lower bound, for all λ 1 > 1 we have g(λ 1 ) ≥ (λ 1 − λ 1 ) m−1 √ λ1≥λ2>···>λm e − m i=2 λ 2 i /4 2≤i<j≤m |λ i − λ j |dλ 2 · · · dλ m . It follows that lim inf λ1→+∞ g(λ 1 ) λ m−1 1 ≥ lim inf λ1→+∞ √ λ1≥λ2>···>λm e − m i=2 λ 2 i /4 2≤i<j≤m |λ i − λ j |dλ 2 · · · dλ m = Z m−1 . Then (66) follows from the upper and lower bounds. Therefore, as n → +∞, i.e., y m → +∞, we have This completes the proof of (63). Now we prove (64). We define the following two auxiliary events: B α := y m < λ 1 (G α ) < y m + 1, λ 2 (G α ) < log log n, λ m (G α ) > − log log n and D α := y m < λ 1 (G α ) < y m + 1 . Then it holds that B α ⊂ D α ⊂ λ 1 (G α ) > y m . We first show that (72) lim n→+∞ P(B α |λ 1 (G α ) > y m ) = 1. Note that (72) follows if we can prove the following two limits, By (63), we find that lim n→+∞ P(λ 1 (G α ) > y m + 1) P(λ 1 (G α ) > y m ) = 0, which implies (73). We now prove (74) by finding a sequence of numbers ǫ n → 0 such that for n large enough it holds that (75) P(λ 2 (G α ) > √ log log n or λ m (G α ) < − √ log log n|D α ) P(λ 2 (G α ) < 1, λ m (G α ) > 0|D α ) ≤ ǫ n . Indeed, (75) implies that (76) P(λ 2 (G α ) > log log n or λ m (G α ) < − log log n|D α ) ≤ ǫ n ǫ n + 1 , which converges to 0 as n → +∞, and thus this yields (74). To prove (75), we divide the set λ 2 (G α ) > √ log log n or λ m (G α ) < − √ log log n as the union of two disjoint subsets S 1 ∪ S 2 as follows. On the subset S 1 := λ m (G α ) ≤ − √ log log n , if we condition on D α , we have 1≤i<j≤m (λ i − λ j ) ≤C(λ 1 + |λ m |) m−1 (|λ 2 | + |λ m |) (m−1)(m−2)/2 ≤Cλ m−1 1 |λ m | m−1 (|λ 2 | + 1) (m−1)(m−2)/2 |λ m | (m−1)(m−2)/2 =Cλ m−1 1 |λ m | (m−1)m/2 (|λ 2 | + 1) (m−1)(m−2)/2 .(77) On the subset S 2 := λ 2 (G α ) > √ log log n, λ m (G α ) > − √ log log n , if we condition on D α where λ 1 (G α ) ∼ 2 √ m log n, then we easily have the upper bound 1≤i<j≤m (λ i − λ j ) ≤ Cλ m−1 1 λ (m−1)(m−2)/2 2 . It follows that the left hand side of (75) can be bounded from above by the summation of C[ ym+1 ym λ m−1 1 e −λ 2 1 /4 dλ 1 ] λm<− √ log log n |λ m | (m−1)m/2 (|λ 2 | + 1) (m−1)(m−2)/2 e − m i=2 λ 2 i /4 dλ 2 · · · dλ m [ ym+1 ym (λ 1 − 1) m−1 e −λ 2 1 /4 dλ 1 ] 1>λ2>···>λm>0 2≤i<j≤m (λ i − λ j )e − m i=2 λ 2 i /4 dλ 2 · · · dλ m and C[ ym+1 ym λ m−1 1 e −λ 2 1 /4 dλ 1 ] λ2> √ log log n,λm>− √ log log n λ (m−2)(m−1)/2 2 e − m i=2 λ 2 i /4 dλ 2 · · · dλ m [ ym+1 ym (λ 1 − 1) m−1 e −λ 2 1 /4 dλ 1 ] 1>λ2>···>λm>0 2≤i<j≤m (λ i − λ j )e − m i=2 λ 2 i /4 dλ 2 · · · dλ m . The above summation can be further bounded from above by := ǫ n as n large enough. Clearly it holds that ǫ n → 0 since √ log log n → +∞. This completes the proof of (75) and thus the proof of (72). Now we are ready to prove (64). We define the event H α = λ * 1 (G β ) ≤ y k , ∀ 1 ≤ k < m, β ⊂ α, |β| = k . Using (72), we see that (64) is equivalent to (78) lim n→+∞ P(H α |B α ) = K m . To prove (78) we need to use the fact that G α d = U T diag(λ 1 , · · · , λ m )U where |α| = m and U d = U(O(m)) is sampled from the uniform measure on the orthogonal group O(m) which is independent of (λ 1 , · · · , λ m ) and (λ 1 , · · · , λ m ) d = (λ 1 (G α ), · · · , λ m (G α )). Given any X := U T diag(λ 1 , · · · , λ m )U, U ∈ O(m), λ 1 ≥ · · · ≥ λ m , by definition it holds λ 1 (X) = λ 1 , λ * 1 (X) 2 = λ 2 1 + m k=2 λ 2 k /2. Let λ − 1 (X) := m k=2 λ 2 k = |X| 2 − λ 2 1 (X), X 1 := U T diag(λ 1 , 0, · · · , 0)U, X 2 := X − X 1 , then we have |X 2 |(= Tr(X 2 2 )) = λ − 1 (X). For β ⊂ α = {1, · · · , m} we have (79) |λ * 1 (X β ) − λ * 1 ((X 1 ) β )| ≤ |(X 2 ) β | ≤ |X 2 | = λ − 1 (X). Here, by the Lipschitz continuity of eigenvalues for symmetric matrices (see Corollary A.6 in [1]), we can further derive the fact that |λ * 1 (A) − λ * 1 (B)| ≤ |A − B|. Since (X 1 ) β is a matrix of rank at most 1, we have (80) λ * 1 ((X 1 ) β ) = (Tr((X 1 ) 2 β )) 1/2 = |λ 1 | k∈β u 2 k , here u = (u 1 , · · · , u m ) is the first row of U . Note that u has the uniform distribution on the unit sphere S m−1 . By (79) and (80), we have (81) λ * 1 (X β ) − |λ 1 | k∈β u 2 k ≤ λ − 1 (X). This together with the facts that y k ∼ 2 √ k log n and y m ∼ 2 √ m log n as n → +∞ imply lim n→+∞ P(H α |B α ) = P   k∈β u 2 k ≤ k/m, ∀β ∈ α, |β| = k   , We note that P (∪ α∈Im A α ,v ∈ Q) = α∈Im P(A α , v 1 (G α ) ∈ Q; ∀α ′ = α, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )) = α∈Im k 1 (α). By the definition of k 1 (α) and k 2 (α) together with the union bound we have (93) α∈Im |k 1 (α) − k 2 (α)| ≤ α∈Im P A α ∩ ∪ α ′ ∩α =∅ A α ′ ≤ α,α ′ ∈Im;α∩α ′ =∅ P(A α ∩ A α ′ ). Combining (91), (92) and (93) The advantage of introducing k 2 (α) is that, we have the conditional probability P(v 1 (G α ) ∈ Q|A α ; ∀α ′ ∩ α = ∅, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )) =P(v 1 (G α ) ∈ Q|A α ), since G α is independent of {G α ′ : α ′ ∩ α = ∅}. Consequently, we have k 2 (α) =P(A α ; ∀α ′ ∩ α = ∅, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )) × P(v 1 (G α ) ∈ Q|A α ; ∀α ′ ∩ α = ∅, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )) =P(v 1 (G α ) ∈ Q|A α )k 4 (α). Clearly the term P(v 1 (G α ) ∈ Q|A α ) is the same for all α ∈ I m . Let α 1 = {1, . . . , m}. Then we have (100) α∈Im k 2 (α) = α∈Im P(v 1 (G α ) ∈ Q|A α )k 4 (α) = P(v 1 (G α1 ) ∈ Q|A α1 ) α∈Im k 4 (α). Let u be sampled from the uniform distribution on the unit sphere S m−1 . Inspecting the proof of (64) in §3.3, especially (72) and (83), we have (101) lim n→+∞ P(H α , v 1 (G α ) ∈ Q|λ 1 (G α ) > y m ) = P(u ∈ S m ∩ Q), where S m has been defined in (3). By the fact A α = H α ∩ {λ 1 (G α ) > y m }, we have lim n→+∞ P(v 1 (G α1 ) ∈ Q|A α1 ) = lim n→+∞ P(H α , v 1 (G α ) ∈ Q|λ 1 (G α ) > y m ) P(H α |λ 1 (G α ) > y m ) = P(u ∈ S m ∩ Q) P(u ∈ S m ) =ν(Q),(102) where ν is the uniform distribution on the set S m . Combining (94) This proves (89), and thus the proof of Theorem 2. n − 4m ln n − 2(m − 2) ln ln n d −→ Y as n → +∞ for fixed m, where the random variable Y has the Gumbel distribution functionF Y (y) = exp(−c m e −y/4 ), y ∈ R. Here, the constant c m = (2m) (m−2)/2 Km (m−1)!2 3/2 Γ(1+m/2) , where K m = µ(S m ) is the probability of the event (3) S m := x ∈ S m−1 : j∈β x 2 j ≤ k/m, ∀ β ⊂ {1, · · · , m} with ∀ 1 ≤ |β| = k < munder the uniform distribution µ on the unit sphere S m−1 . In particular, S) := [(λ 1 (S) 2 + |S| 2 )/2] For fixed m ≥ 2, we define the index set I m = {α ⊂ {1, · · · , n}, |α| = m} and the neighborhood set G β ) ≤ y k , ∀ 1 ≤ k < m, β ⊂ α, |β| = k|λ 1 (G α ) > y m ) = K m ,where K m is the constant defined in Theorem 1. Indeed, (62), (63), (64) and the definition of y lim n→+∞ P(D α |λ 1 (G α ) > y m B α |D α ) = 1. Now we replace X by G α . On the event B α we have λ − 1 (G α ) ≤ m log log n and y m < λ 1 < y m − m log log n ≤ λ * 1 (G β ) ≤ (y m + 1) k∈β u 2 k + m log log n for all β ⊂ α. By (82) and the fact that the first row u of the orthogonal group are independent of the eigenvalues λ i (G α ), 1 ≤ i ≤ m, k 1 (α), . . . , k 4 (α) don't depend on the specific choice of α. Recall the definition of b α∈Im A α ) → 1 − F Y (y)in the proof of Theorem 1 as n → +∞. Hence, combining (95) and (96) we have (97) lim n→+∞ α∈Im k 4 (α) = 1 − F Y (y). 1 (G α1 ) ∈ Q|A α1 ) α∈Im k 4 (α) =ν(Q)(1 − F Y (y)). (G β ))(x−λ * 1 (Gγ ))/2 . RENJIE FENG, GANG TIAN, DONGYI WEI, AND DONG YAO ≤2P ((∩ α∈Im H α ) c ) ,(88)which converges to 0 as n → +∞. Hence, (4) is equivalent to the limit(89) P(∪ α∈Im A α ,v ∈ Q) → (1 − F Y (y))ν(Q).All of the rest is to prove this convergence. Now we define four quantitiesk 1 (α) = P(A α , v 1 (G α ) ∈ Q; ∀α ′ = α, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )), k 2 (α) = P(A α , v 1 (G α ) ∈ Q; ∀α ′ ∩ α = ∅, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )), k 3 (α) = P(A α ; ∀α ′ = α, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )), k 4 (α) = P(A α ; ∀α ′ ∩ α = ∅, A α ′ fails or λ 1 (G α ′ ) < λ 1 (G α )).(90) which is exactly the definition of K m . This proves (78), and hence (64). Therefore, we complete the proof of Lemma 4.Proof of Theorem 2Now we prove Theorem 2. As before, for G = (g ij ) 1≤i≤j≤n sampled from GOE, we denote G α = (g ij ) i,j∈α as the principal minor of size m × m for α ⊂ {1, ..., n} with |α| = m and I m is the collection of all such α. Let v 1 (G α ) be the eigenvector of the largest eigenvalue of G α and v * ∈ S m−1 be the eigenvector of the largest eigenvalue of the principal sub-matrix that attains the maximal eigenvalue T m,n , i.e., we haveAs in §3.3, we recall the definition of the eventWe define the random variableα as follows. If the event ∪ α∈Im H α holds, then we set α := argmax α∈Im, Hα holds λ 1 (G α ).Otherwise, we setα to be {1, . . . , m}. We now set (86)v := λ 1 (Gα).In other words,v is the eigenvector of the largest eigenvalue of the principal sub-matrix G α that achieves the maximal eigenvalue under the constraint that H α is true. By(8)and(12), we have lim n→+∞ P(∩ α∈Im H α ) = 1.On the event ∩ α∈Im H α we clearly haveα = α * andv = v * since the constraint forα doesn't have any effect. Recall the definition of A α in(7)where A α = H α ∩ {λ 1 (G α ) > y m }, on ∩ α∈Im H α , the two events {T m,n ≥ y m } and ∪ α∈Im A α coincide. In other words, for symmetric Q, we haveIt follows that |P(T m,n ≥ y m , v * ∈ Q) − P(∪ α∈Im A α ,v ∈ Q)| = |P ((∩ α∈Im H α ) c ∩ {T m,n ≥ y m , v * ∈ Q}) − P ((∩ α∈Im H α ) c ∩ (∪ α∈Im A α ,v ∈ Q))| An introduction to random matrices. G W Anderson, A Guionnet, O Zeitouni, Cambridge Studies in Advanced Mathematics. 118Cambridge University PressG. W. Anderson, A. Guionnet and O. Zeitouni, An introduction to random matrices. Cambridge Studies in Advanced Mathematics, 118. Cambridge University Press, Cambridge, 2010. Random matrix minor processes related to percolation theory. Mark Adler, Pierre Van Moerbeke, Dong Wang, Random Matrices: Theory and Applications. 021350008Mark Adler, Pierre van Moerbeke and Dong Wang, Random matrix minor processes related to percolation theory, Random Matrices: Theory and Applications, Vol. 02, No. 04, 1350008 (2013). Two moments suffice for Poisson approximations: The Chen-Stein method. R Arratia, L Goldstein, L Gordon, Ann. Probab. 17Arratia R., Goldstein L., Gordon L. Two moments suffice for Poisson approximations: The Chen-Stein method. Ann. Probab. 17(1989):9-25. Spectral analysis of large dimensional random matrices. Z D Bai, J W Silverstein, Springer20Bai, Z. D. and Silverstein, J. W. (2010). Spectral analysis of large dimensional random matrices, volume 20. Springer. A simple proof of the restricted isometry property for random matrices. R Baraniuk, M Davenport, R Devore, M Wakin, Constructive Approximation. 283Baraniuk, R., Davenport, M., DeVore, R., and Wakin, M. (2008). A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28(3):253-263. Learning signed determinantal point processes through the principal minor assignment problem. Victor-Emmanuel Brunel, Advances in Neural Information Processing Systems. S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. GarnettCurran Associates, Inc31Victor-Emmanuel Brunel, Learning signed determinantal point processes through the principal minor as- signment problem. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. Black Holes and Random Matrices. J S Cotler, G Gur-Ari, M Hanada, J Polchinski, P Saad, S H Shenker, D Stanford, A Streicher, M Tezuka, J. High Energ. Phys. 118J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher and M. Tezuka, Black Holes and Random Matrices, J. High Energ. Phys. (2017) 2017: 118. R Couillet, M Debbah, Random Matrix Methods for Wireless Communications. Cambridge University PressR. Couillet and M. Debbah, Random Matrix Methods for Wireless Communications, Cambridge University Press, 2011. Asymptotic Analysis for Extreme Eigenvalues of Principal Minors of Random Matrices. T.-T Cai, T Jiang, X Li, Ann. Appl. Probab. 316Cai T.-T., Jiang T., Li X. Asymptotic Analysis for Extreme Eigenvalues of Principal Minors of Random Matrices, Ann. Appl. Probab. 31 (6) 2953 -2990, December 2021. P Diaconis, Patterns in eigenvalues: The 70th Josiah Willard Gibbs Lecture. Bulletin of the. American Mathematical SocietyDiaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs Lecture. Bulletin of the American Mathematical Society. Published as part of Dimer Models and Random Tilings. P Ferrari, Dimers and orthogonal polynomials: connections with random matrices. C. Boutillier, N. Enriquez45Extended lecture notes of the minicourse at IHPP. Ferrari, Dimers and orthogonal polynomials: connections with random matrices, Extended lecture notes of the minicourse at IHP (5-7 Oct. 2009). Published as part of Dimer Models and Random Tilings, Panoramas et Synthèses 45 (2015). B. de Tilière, P. Ferrari, edited by C. Boutillier, N. Enriquez. How Many Entries of A Typical Orthogonal Matrix Can Be Approximated By Independent Normals?. T Jiang, Ann. Probab. 344T. Jiang, How Many Entries of A Typical Orthogonal Matrix Can Be Approximated By Independent Nor- mals? Ann. Probab. 34(4), 1497-1529 (2006). K Johansson, From Gumbel to Tracy-Widom. Probab. Theory Relat. Fields. 138K. Johansson, From Gumbel to Tracy-Widom. Probab. Theory Relat. Fields 138, 75-112 (2007). Extremes and related properties of random sequences and processes. M R Leadbetter, G Lindgren, H Rootzen, Springer Series in Statistics. SpringerLeadbetter, M.R., Lindgren, G., Rootzen, H.: Extremes and related properties of random sequences and processes. Springer Series in Statistics, Springer (1983). The Random Matrix Theory of the Classical Compact Groups. Elizabeth S Meckes, Cambridge University PressElizabeth S. Meckes, The Random Matrix Theory of the Classical Compact Groups, Cambridge University Press, 2019. Generalized ensemble of random matrices. 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To make better use of file diversity provided by random caching and improve the successful transmission probability (STP) of a file, we consider retransmissions with random discontinuous transmission (DTX) in a large-scale cache-enabled heterogeneous wireless network (HetNet) employing random caching. We analyze and optimize the STP in two mobility scenarios, i.e., the high mobility scenario and the static scenario. First, in each scenario, by using tools from stochastic geometry, we obtain a closed-form expression for the STP in the general signal-to-interference ratio (SIR) threshold regime.The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios; random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. In each scenario, we also derive a closed-form expression for the asymptotic outage probability in the low SIR threshold regime. The asymptotic analysis shows that the diversity gain is jointly affected by random caching and random DTX in both scenarios. Then, in each scenario, we consider the maximization of the STP with respect to the caching probability and the BS activity probability, which is a challenging non-convex optimization problem. In particular, in the high mobility scenario, we obtain a globally optimal solution using interior point method. In the static scenario, we develop a low-complexity iterative algorithm to obtain a stationary point using alternating optimization.Finally, numerical results show that the proposed solutions achieve significant gains over existing baseline schemes and can well adapt to the changes of the system parameters to wisely utilize storage resources and transmission opportunities.Index TermsRandom caching, retransmission, random discontinuous transmission (DTX), heterogeneous wireless networks, optimization, stochastic geometry.Y. Cui is with the

10.1109/tcomm.2018.2863367

1802.03271

292f93647820e6dcfc68c1ece003761a19aa194f

Enhancing Performance of Random Caching in Large-Scale Heterogeneous Wireless Networks with Random Discontinuous Transmission February 13, 2018 12 Feb 2018 Wanli Wen Department of Electronic Engineering are with the National Mobile Communications Research Laboratory Shanghai Jiao Tong University China Ying Cui Department of Electronic Engineering are with the National Mobile Communications Research Laboratory Shanghai Jiao Tong University China Fu-Chun Zheng Department of Electronic Engineering are with the National Mobile Communications Research Laboratory Shanghai Jiao Tong University China Shi Jin Department of Electronic Engineering are with the National Mobile Communications Research Laboratory Shanghai Jiao Tong University China Yanxiang Jiang Department of Electronic Engineering are with the National Mobile Communications Research Laboratory Shanghai Jiao Tong University China W Wen Southeast University NanjingChina F.-C Zheng Southeast University NanjingChina S Jin Southeast University NanjingChina Y Jiang Southeast University NanjingChina Enhancing Performance of Random Caching in Large-Scale Heterogeneous Wireless Networks with Random Discontinuous Transmission February 13, 2018 12 Feb 20181 This paper will be presented in part at IEEE WCNC 2018. DRAFT 2 To make better use of file diversity provided by random caching and improve the successful transmission probability (STP) of a file, we consider retransmissions with random discontinuous transmission (DTX) in a large-scale cache-enabled heterogeneous wireless network (HetNet) employing random caching. We analyze and optimize the STP in two mobility scenarios, i.e., the high mobility scenario and the static scenario. First, in each scenario, by using tools from stochastic geometry, we obtain a closed-form expression for the STP in the general signal-to-interference ratio (SIR) threshold regime.The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios; random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. In each scenario, we also derive a closed-form expression for the asymptotic outage probability in the low SIR threshold regime. The asymptotic analysis shows that the diversity gain is jointly affected by random caching and random DTX in both scenarios. Then, in each scenario, we consider the maximization of the STP with respect to the caching probability and the BS activity probability, which is a challenging non-convex optimization problem. In particular, in the high mobility scenario, we obtain a globally optimal solution using interior point method. In the static scenario, we develop a low-complexity iterative algorithm to obtain a stationary point using alternating optimization.Finally, numerical results show that the proposed solutions achieve significant gains over existing baseline schemes and can well adapt to the changes of the system parameters to wisely utilize storage resources and transmission opportunities.Index TermsRandom caching, retransmission, random discontinuous transmission (DTX), heterogeneous wireless networks, optimization, stochastic geometry.Y. Cui is with the I. INTRODUCTION With the proliferation of smart mobile devices and multimedia services, the global mobile data traffic is expected to increase exponentially in the coming years. However, the majority of such traffic is asynchronously but repeatedly requested by many users at different times and thus a tremendous amount of mobile data traffic have actually been redundantly generated over networks [1]. Motivated by this, caching at base stations (BSs) has been proposed as a promising approach for reducing delay and backhaul load [2]. When the coverage regions of different BSs overlap, a user can fetch the desired file from multiple adjacent BSs, and hence the performance can be increased by caching different files among BSs, i.e., providing file diversity. In [3]- [16], the authors analyze the performance of various caching designs in large-scale cache-enabled wireless networks. In particular, in [3] and [4], the authors study the most popular caching design, where each BS only stores the most popular files. As the most popular caching design can not provide any spatial file diversity, it may not yield the optimal network performance. To provide more spatial file diversity, the authors in [5]- [16] consider random caching. Specifically, in [5], the authors consider uniform caching where each BS randomly stores a file according to the uniform distribution. In [6], the authors consider i.i.d. caching where each BS stores a file in an i.i.d. manner. In [7]- [16], besides analysis, the authors also consider the optimization of random caching to either maximize the cache hit probability [7], [8], the successful offloading probability [9] and the successful transmission probability (STP) [10]- [14], [16] or minimize the average caching failure probability [15]. Note that, in [5]- [16], a user may associate with a relatively farther BS when nearer BSs do not cache the requested file [16]. In this case, the signal is usually weak compared with the interference, and the user may not successfully receive the requested file and benefit from the file diversity offered by random caching. Increasing the number of transmissions of a file can increase the probability that eventually the file is successfully transmitted, at the cost of delay increase. Enabling retransmissions at BSs is an effectively way to improve the STP for some applications without strict delay requirements, e.g., elastic services. The interferences experienced by a user at different slots are usually correlated as they come from the same set of BSs [17]. In [17]- [21], the authors study retransmissions and show that the interference correlation significantly degrades the performance, e.g., the diversity gain [18] or the transmission delay [19]- [21]. In [17]- [21], the authors adopt random discontinuous transmission (DTX) together with retransmissions to effectively manage such interference correlation, by creating randomness for interferers, and analyze the performance in large-scale wireless networks. Note that, [17]- [21] do not consider caching at BSs. Recently, [22] studies the effect of retransmissions on the performance of random caching, and analyzes and optimizes the STP in a large-scale single tier network. Note that [22] does not consider DTX, and hence the gain of retransmissions is limited due to the strong interference correlation across multiple retransmissions. Therefore, it is still not clear how retransmissions with random DTX can maximally improve the performance of random caching. Heterogeneous wireless networks (HetNets) can further improve the network capacity by deploying small BSs together with traditional macro BSs, to provide better time or frequency reuse. Caching at small BSs can effectively alleviate the backhaul capacity requirement in HetNets. For cache-enabled HetNets, it is also not known how to jointly design random caching and random DTX across different tiers. In this paper, we would like to address the above issues. We consider a large-scale cacheenabled HetNet. We adopt random caching and a simple retransmission protocol with random DTX to improve the STP, which is defined as the probability that a file can be successfully transmitted to a user. Our focus is on the analysis and optimization of joint random caching and random DTX in two scenarios of user mobility, i.e., the high mobility scenario and the static scenario. The main contributions of the paper are summarized below. • First, we analyze the STP in both scenarios. The random caching and retransmission with random DTX make the analysis very challenging. In each scenario, by carefully considering the joint impacts of random caching and random DTX on the distribution of the signal-tointerference ratio (SIR) in each slot, we derive closed-form expression for the STP in the general SIR threshold regime, utilizing tools from stochastic geometry. The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios, which reveals the advantage of caching. In addition, the analysis reveals that random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. We also derive a closed-form expression for the asymptotic outage probability in the low SIR threshold regime, utilizing series expansion of some special functions. The asymptotic analysis shows that the diversity gain is jointly affected by random caching and random DTX. • Next, we consider the maximization of the STP with respect to the caching probability and the BS activity probability in both scenarios, which is a challenging non-convex optimization February 13, 2018 DRAFT problem. In the high mobility scenario, we obtain a globally optimal solution using interior point method. In the static scenario, we develop a low-complexity iterative algorithm to obtain a stationary point using alternating optimization. • Finally, numerical results show that the proposed solutions achieve significant gains over existing baseline schemes and can well adapt to the changes of the system parameters to wisely utilize storage resources and transmission opportunities. As the maximum number of transmissions increases, more files are stored and more BSs are silenced for improving the STP. II. SYSTEM MODEL A. Network Model We consider a large-scale HetNet consisting of K independent network tiers. We denote the set of K tiers by K = {1, 2, · · · , K}. All network tiers are co-channel deployed. The BS locations in tier k are modeled by an independent homogeneous Poisson point process (PPP) Φ k with density λ k . Let Φ be the superposition of Φ k , k ∈ K, i.e., Φ k∈K Φ k , which denotes the locations of all tiers of BSs in the network. Each BS in tier k has one transmit antenna with transmission power P k . For the propagation model, we consider a general power-law path-loss model in which a transmitted signal from a BS with distance r, is attenuated by a factor r −α , where α > 2 denotes the path-loss exponent. For the small-scale fading model, we assume Rayleigh fading. Since a HetNet is primarily interference-limited, we ignore the thermal noise for simplicity. Each user has one receive antenna. We consider a discrete-time system with time being slotted. Let t = 1, 2, · · · denote the slot index. Two scenarios of user mobility are considered: the high mobility scenario and the static scenario [19]. In the high mobility scenario, in each slot, the user locations follow an independent homogeneous PPP, i.e., a new realization of the PPP for the user locations is drawn in each slot, and the user locations are independent over time. Mathematically, the high mobility scenario is equivalent to the case where the user locations follow an independent homogeneous PPP and stay fixed over time, and in each slot, new realizations of the K independent PPPs for the K tiers of BSs are drawn [19]. In the static scenario, the user locations stay fixed over time and follow an independent PPP. Note that, the mobility scenario in a practical network is between the two scenarios, and hence the results in this paper provide some theoretical performance bounds for a practical network. In both scenarios, February 13, 2018 DRAFT without loss of generality (w.l.o.g.), we can study the performance of a typical user u 0 , which is located at the origin o, according to Slivnyak's Theorem [23]. Let N = {1, 2, · · · , N } denote the set of N files in the HetNet. For ease of analysis, as in [10], [13], we assume that all files have the same size, and file popularity distribution is identical among all users. 1 The probability that file n ∈ N is requested by each user is a n ∈ (0, 1), where n∈N a n = 1. Thus, the file popularity distribution is given by a (a n ) n∈N , which is assumed to be known a priori. 2 In addition, w.l.o.g., we assume that a 1 ≥ a 2 ≥ · · · ≥ a N , i.e., the popularity rank of file n is n. Assume that at the beginning of slot 1, each user randomly requests a file according to the file popularity distribution a. We shall consider the delivery of each requested file over M consecutive slots. The network consists of cache-enabled BSs. In particular, each BS in tier k is equipped with a cache of size C k ≤ N to store C k different files out of N . B. Random Caching and Retransmissions with Random DTX To provide high spatial file diversity, we consider a random caching design similar to the one in [12], where file n is stored at each BS in tier k according to a certain probability T n,k ∈ [0, 1], called the caching probability of file n in tier k. Denote T (T n ) n∈N ∈ [0, 1] N K×1 , where T n (T n,k ) k∈K ∈ [0, 1] K×1 , as the caching distribution of the N files in the K-tier HetNet. Note that, the random caching design is parameterized by T. We have [12], [13] : 3 0 ≤ T n,k ≤ 1, n ∈ N , k ∈ K,(1)n∈N T n,k = C k , k ∈ K.(2) Let Φ n,k denote the point process of the BSs in tier k which store file n. Note that, Φ n,k ⊆ Φ k , n ∈ N . Under the random caching design, Φ n,k , n ∈ N are independent PPPs with densities λ k T n,k , n ∈ N . 1 Note that, the results in this paper can be easily extended to the case of different file sizes. To be specific, we can consider file combinations of the same total size, but formed by files of possibly different sizes. Consider a user requesting file n at the beginning of slot 1. If file n is not stored in any tier, the user will not be served. Otherwise, the user is associated with the BS which not only stores file n but also provides the maximum average received signal strength (RSS) [8] among all BSs in the K-tier HetNet, referred to as its serving BS. Note that, in the high mobility scenario, the user association changes from slot to slot, while in the static scenario, the user association does not change over slots. Under this content-based user association, in each slot, a user may not be associated with the BS which provides the maximum average RSS if it has not stored file n. As a result, a user may suffer from more severe inter-cell interference under this content-based user association than under the traditional connection-based user association. The transmission of a file in one slot is more likely to fail, if u 0 is not associated with the BS providing the maximum average RSS. Increasing the number of transmissions of a file can increase the probability that eventually the file is successfully transmitted, at the cost of delay increase. In addition, there are some applications without strict delay requirements, e.g., elastic services. Therefore, for those applications, we consider a simple retransmission protocol in which a file is repeatedly transmitted until it is successfully received or the number of transmissions exceeds M . 4 However, in a practical HetNet, the interference suffered by a user is temporally correlated since it comes from the same set of interferers in different time slots [20]. Such correlation makes the SIRs temporally correlated and thus dramatically decreases the performance gain of retransmission. In order to manage such correlation, we consider random DTX at the BSs [20], [21], where each BS has two possible transmission states in each slot, i.e., the active state and the inactive state. Specifically, in each slot, a BS is active with probability β ∈ (0, 1], called the activity probability, and is inactive with probability 1 − β, independent of the BS location and slot. 5 Note that, the random DTX design is parameterized by β. The density of active BSs in tier k is βλ k . C. Performance Metric Suppose that the typical user u 0 requests file n at the beginning of slot 1. Let k 0 denote the index of the tier with which u 0 is associated and l 0 ∈ Φ k 0 denote the index of the serving BS of u 0 . We denote X k,l,0 (t) and h k,l,0 (t) as the distance and the fading power coefficient between BS l ∈ Φ k and u 0 in slot t, respectively. Assume h k,l,0 (t), l ∈ Φ, t = 1, 2, · · · , M are i.i.d., according to the exponential distribution with unit mean. Let B a k (t) be the set of active BSs in tier k in slot t. When u 0 requests file n and file n is transmitted by BS l 0 , the signal-to-interference ratio (SIR) of u 0 in slot t is given by SIR n,0 (t) = P k0 h k0,l0,0 (t)X k0,l0,0 (t) −α (t)1(l 0 ∈ B a k0 (t)) l∈Φk 0 \{l0} P k0 h k0,l,0 (t)X −α k0,l,0 (t)1(l ∈ B a k0 (t)) + j∈K\{k0} l∈Φj P j h j,l,0 (t)X −α j,l,0 (t)1(l ∈ B a j (t)) ,(3) where 1(·) denotes the indicator function. We say that file n is successfully transmitted to u 0 in slot t if SIR n,0 (t) is greater than or equal to a given threshold θ, i.e., SIR n,0 (t) ≥ θ. Let S n (t) denote the event that file n is successfully transmitted to u 0 in slot t and S c n (t) denote the complementary event of S n (t), i.e., the event that file n is not successfully transmitted to u 0 in slot t. The probability that file n is successfully transmitted to u 0 in M consecutive slots, referred to as the successful transmission probability (STP) of file n, under the adopted simple retransmission protocol in the high mobility and static scenarios, is given by q n,i (T n , β) = 1 − Pr (S c n (1), S c n (2), · · · , S c n (M )) , i ∈ {hm, st}.(4) In the high mobility scenario, since the events S n (t) (or S c n (t)), t = 1, 2, · · · , M , are i.i.d., the STP of file n in (4) can be expressed as q n,hm (T n , β)=1 − M t=1 Pr (S c n (t)) = 1 − (1 − Pr (S n )) M ,(5) where Pr (S n ) = Pr (SIR n,0 (t) ≥ θ) is the STP of file n in one slot. Here, we have dropped the index t in S n (t), as S n (t), t = 1, 2, · · · , M , are i.i.d. In the static scenario, as the locations of BSs and u 0 do not change, the events S c n (t), t = 1, 2, · · · , M , are correlated. Let S n|Φ (t) denote the event that file n is successfully transmitted to u 0 in slot t, conditioned on Φ. Similarly, S c n|Φ (t) denotes the complementary event of S n|Φ (t). Note that, the events S n|Φ (t) (or S c n|Φ (t)), t = 1, 2, · · · , M , are i.i.d. due to the fact that the fading power coefficients are i.i.d. with respect to (w.r.t.) t. Thus, in the static scenario, the STP of file n in (4) can be expressed as q n,st (T n , β)=E Φ 1 − Pr S c n|Φ (1), S c n|Φ (2), · · · , S c n|Φ (M ) =E Φ 1 − M t=1 Pr S c n|Φ (t) = E Φ 1 − 1 − Pr S n|Φ M ,(6) where E(·) is the expectation operation and Pr S n|Φ = Pr (SIR n,0 (t) ≥ θ|Φ). Note that, E Φ Pr S n|Φ = Pr (S n ). Here, we have dropped the index t in S n|Φ (t), as S n|Φ (t), t = 1, 2, · · · , M , are i.i.d.. Note that, f (x) = 1 − (1 − x) M , x ∈ [0, 1] , is a linear function when M = 1, and a concave function when M = 2, 3, · · ·. Thus, by Jensen's inequality, we have q n,st (T n , β) ≤ q n,hm (T n , β), where the equality holds when M = 1, implying that mobility has a positive effect on the STP. The intuitions are given as follows. In the static scenario, the locations of BSs during M consecutive slots stay fixed, leading to temporal SIR correlation. That is, if the transmission in one slot fails, there is a higher chance that the transmission in another slot also fails. In contrast, in the high mobility scenario, the locations of BSs during M consecutive slots are independent, and hence there is no correlation among SIRs during M consecutive slots. Consequently, a user has a higher chance to experience a favorable transmission channel with high SIR within M transmissions. Therefore, mobility increases temporal diversity, leading to the STP increase. Users are mostly concerned about whether their requested files can be successfully received. Therefore, in this paper, we adopt the probability that a randomly requested file by the typical user is successfully transmitted in M consecutive slots, referred to as the STP, as the network performance metric. By total probability theorem, the STP in the high mobility and static scenarios is given by q i (T, β)= n∈N a n q n,i (T n , β), i ∈ {hm, st},(7) where T and β are the design parameters of random caching and random DTX, respectively. III. HIGH MOBILITY SCENARIO In this section, we consider the high mobility scenario. We first analyze the STP and then maximize the STP by optimizing the design parameters of random caching and random DTX. A. Performance Analysis In this part, we analyze the STP in the general SIR threshold regime and the low SIR threshold regime, respectively. To be specific, we only need to analyze the STP of file n, i.e., q n,hm (T n , β), in the two regimes. Then, by (7), we can directly obtain the STP, i.e., q hm (T, β). 1) Performance Analysis in General SIR Threshold Regime: In this part, we analyze q n,hm (T n , β) in the general SIR threshold regime, using tools from stochastic geometry. To calculate q n,hm (T n , β), based on (5), we first need to analyze the distribution of the SIR, SIR n,0 (t). Under random caching, there are three types of interferers for u 0 : i) all the other BSs in the same tier as the serving BS of u 0 which have stored the desired file of u 0 (apart from the serving BS of u 0 ), ii) all the BSs in the same tier as the serving BS of u 0 which have not stored the desired file of u 0 , and iii) all the BSs in other tiers. In addition, under random DTX, the serving BS of u 0 is active with probability β, and the number of interferers of u 0 is β times that for the case where the BSs are always active. By jointly considering the impacts of random caching and random DTX on SIR n,0 (t), we can derive the distribution of SIR n,0 (t) and then q n,hm (T n , β), as summarized in the following theorem. Theorem 1 (STP in High Mobility Scenario): The STP of file n in the high mobility scenario is given by q n,hm (T n , β) = 1 − 1 − β k∈K z k T n,k W (β) k∈K z k T n,k + V (β)(1 − k∈K z k T n,k ) M , n ∈ N , (8) where z k λ k P 2/α k j∈K λ j P 2/α j , W (β) 1−β+β 2 F 1 (− 2 α , 1; 1− 2 α ; −θ), V (β) βΓ 1 + 2 α Γ 1 − 2 α θ 2 α , and 2 F 1 (a, b; c; z) and Γ(x, y) denote the Gauss hypergeometric function and Gamma function, respectively. Proof : See Appendix A. Theorem 1 provides a closed-form expression for q n,hm (T n , β) in the general SIR threshold regime. From Theorem 1, we can see that the system parameters K, M , α, λ k , k ∈ K, P k , k ∈ K, θ, β and T n jointly affect q n,hm (T n , β) in a complex manner. Based on Theorem 1, we characterize how q n,hm (T n , β) changes with T n,k and β, as summarized blow. Lemma 1 (Effects of Random Caching and Random DTX): • q n,hm (T n , β) is an increasing and concave function of T n,k , for all k ∈ K. • q n,hm (T n , β) is an increasing and concave function of β. The first result in Lemma 1 shows that a larger T n,k corresponds to a larger q n,hm (T n , β), which reveals the advantage of caching. This is because the average distance between a user requesting file n and its serving BS decreases with T n,k . The second result in Lemma 1 shows that a larger β corresponds to a larger q n,hm (T n , β), which reveals that it is not beneficial to W, λ 1 = 1 250 2 π , λ 2 = 1 50 2 π , α = 3.5, and M = 3. The power parameters are chosen according to [25]. In the Monte Carlo simulations, we choose a large spatial window, which is a square of 10 4 × 10 4 m 2 , and the final simulation results are obtained by averaging over 10 5 independent realizations. apply random DTX in the high mobility scenario. To understand this result, we first study the STP of file n in one slot, i.e., Pr (S n ). By setting M = 1 in (8), we have Pr (S n ) = β k∈K z k T n,k W (β) k∈K z k T n,k + V (β)(1 − k∈K z k T n,k ) , n ∈ N .(9) It is easy to verify that Pr (S n ) is an increasing function of β, implying that the penalty of random DTX in signal reduction overtakes its advantage in interference reduction in one slot. In the high mobility scenario, S n (t), t = 1, 2, · · · , M are i.i.d., implying that random DTX has no further benefit of reducing interference correlation. Therefore, random DTX cannot improve the STP in the high mobility scenario. Fig. 1 plots q n,hm (T n , β) versus T n,k and β, respectively, verifying Theorem 1 and Lemma 1. 2) Performance Analysis in Low SIR Threshold Regime: To further obtain insights, in this part, we analyze the outage probability of file n which is defined asq n,hm (T n , β) 1 − q n,hm (T n , β), in the low SIR threshold regime, i.e., θ → 0, where the (normalized) target bit rate τ log 2 (1 + θ) → 0. 6 Let 1 (1, 1, · · · , 1) T ∈ R K denote the K-dimensional all-one vector. Here, (·) T denotes the transpose operation. Denote T n {T n |T n,k ∈ [0, 1], k ∈ K}. For ease of illustration, in the following, we consider four cases. • Case i): File n is stored at each BS and random DTX is not applied, i.e., T n = 1 and β = 1. • Case ii): File n is not stored at any BS and random DTX is not applied, i.e., T n ∈ T n \{1} and β = 1. • Case iii): File n is stored at each BS and random DTX is applied, i.e., T n = 1 and β < 1. • Case iv): File n is not stored at any BS and random DTX is applied, i.e., T n ∈ T n \ {1} and β < 1. By analyzing the four cases, we have the following result. 7 Lemma 2 (Outage Probability in High Mobility Scenario When θ → 0): In the high mobility scenario, when θ → 0, we havē q n,hm (T n , β) ∼ (1 − β) M + c hm (T n , β), n ∈ N ,(10) where c hm (T n , β)                      θ M 2 α−2 M , case i), θ 2M α 1 k∈K zkTn,k − 1 Γ 1 + 2 α Γ 1 − 2 α M , case ii), θ (1 − β) M −1 M β 2 2 α−2 , case iii), θ 2 α (1 − β) M −1 M β 2 1 k∈K zkTn,k − 1 Γ 1 + 2 α Γ 1 − 2 α , case iv).(11) Proof : See Appendix B. From Lemma 2, we can see that both T n and β significantly affect the asymptotic behaviours ofq n,hm (T n , β) when θ → 0, but in different manners. Fig. 2 plotsq n,hm (T n , β) versus θ and indicates that Lemma 2 provides a good approximation forq n,hm (T n , β) when θ is small. In addition, from Fig. 2, we observe that the rates of decay to zero ofq n,hm (T n , β) when θ → 0 in the four cases are different. In the following, we further characterize such rate in each case, referred to as the diversity gain [28], i.e., d hm = lim θ→0 log(q n,hm (T n , β)) log θ .(12) Note that the definition in (12) is similar to the usual definition of diversity gain as the rate of decay to zero of the error probability in the high SNR regime [29]. A larger diversity gain implies a faster decay to zero ofq n,hm (T n , β) with decreasing θ. From Lemma 2, we have the following result. Lemma 3 (Diversity Gain in High Mobility Scenario): The diversity gain in the high mobility scenario is given by d hm =            M, case i), 2M α , case ii), 0, cases iii) and iv). Lemma 3 tells us that as long as random DTX is applied, there is no diversity gain. Without random DTX, caching a file at every BS can achieve the full diversity gain M , and caching a file only at some BSs, irrespective of the caching probability, achieves the same smaller diversity gain 2M α ∈ (0, M ) (as α > 2 and M ≥ 1). Fig. 2 verifies Lemma 3. B. Performance Optimization By substituting (8) into (7), the STP in the high mobility scenario is calculated as q hm (T, β) = n∈N a n 1 − 1 − β k∈K z k T n,k W (β) k∈K z k T n,k + V (β)(1 − k∈K z k T n,k ) M .(14) The caching distribution T and BS activity probability β significantly affect the STP in the high mobility scenario. We would like to maximize q hm (T, β) in (14) by jointly optimizing T and β. Specifically, we have the following optimization problem. where q * hm = q hm (T * , β * ) denotes an optimal value and (T * , β * ) denotes an optimal solution. Problem 1 maximizes a non-concave function over a convex set, and hence is non-convex. In general, it is difficult to obtain a globally optimal solution of a non-convex problem. By exploring properties of the objective function q hm (T, β), in the following, we can obtain a globally optimal solution of Problem 1. Recall that Lemma 1 shows that q n,hm (T n , β) increases with β for all T n . Thus, we know that β * = 1. It remains to obtain T * by maximizing q n,hm (T n , 1) w.r.t. T, i.e., solving the following problem. Problem 2 (Optimization of Random Caching in High Mobility Scenario): (2). q * hm = max T q hm (T, 1) s.t. (1), Recall that Lemma 1 shows that q n,hm (T n , β) is a concave function of T n , implying that (14) is a concave function of T. Thus, Problem 2 is a convex optimization problem and can be efficiently solved by the interior point method. Consider a special case of C k = C for all k ∈ K, i.e., equal cache size across all tiers. Using KKT conditions, the optimal solution of Problem 2 can be characterized as follows. q hm (T, β) in Lemma 4 (Optimal Solution of Problem 2 When C k = C, k ∈ K): When C k = C for all k ∈ K, an optimal solution T * of Problem 2 is given by 8 T * n,k =            0, if η * ≥ g(0), 1, if η * ≤ g(1), g −1 (η * ), otherwise, n ∈ N , k ∈ K, where g −1 (·) denotes the inverse function of function g(·), given by g(x) a n M 1 − x xW (1) + (1 − x)V (1) M −1 V (1) (xW (1) + (1 − x)V (1)) 2 , and η * satisfies n∈N T * n,k = C. Based on β * = 1 and an optimal solution of Problem 2, we can obtain a globally optimal solution of Problem 1. 8 Note that, for all k ∈ K, T n,k = g −1 (η * ) can be obtained by solving g(T * n,k ) = η using the bisection method, and η * can be obtained by solving n∈N T * n,k = C using the bisection method. IV. STATIC SCENARIO In this section, we consider the static scenario. We first analyze the STP and then maximize the STP by optimizing the design parameters of random caching and random DTX. A. Performance Analysis In this part, we analyze the STP in the general SIR threshold regime and the low SIR threshold regime, respectively. To be specific, we only need to analyze the STP of file n, i.e., q n,st (T n , β), in the two regimes. Then, by (7), we can directly obtain the STP, i.e., q st (T, β). 1) Performance Analysis in General SIR Threshold Regime: In this part, we analyze q n,st (T n , β) in the general SIR threshold regime, using tools from stochastic geometry. It is challenging to calculate q n,st (T n , β), as SIR n,0 (t), t = 1, 2, · · · , M are correlated. To address this challenge, by using the binomial expansion theorem, we first rewrite q n,st (T n , β) in (6) as q n,st (T n , β) = M m=1 M m (−1) m+1 E Φ ((Pr (SIR n,0 (t) > θ|Φ)) m ) . Note that, conditioned on Φ, SIR n,0 (t), t = 1, 2, · · · , M are i.i.d.. Thus, we can first analyze the distribution of SIR n,0 (t), conditioned on Φ, and then derive q n,st (T n , β) by deconditioning on Φ. Thus, we have the following theorem. Theorem 2 (STP in Static Scenario): The STP of file n in the static scenario is given by, q n,st (T n , β) = M m=1 M m (−1) m+1 β m k∈K z k T n,k F m (β) k∈K z k T n,k + G m (β)(1 − k∈K z k T n,k ) , n ∈ N ,(15) where z k is given by Theorem 1, F m (β) and G m (β) are given, respectively, by F m (β)= m i=0 m i β i (1 − β) m−i 2 F 1 − 2 α , i; 1 − 2 α ; −θ ,(16)G m (β)= m i=0 m i β i (1 − β) m−i Γ i + 2 α Γ(i) Γ 1 − 2 α θ 2 α .(17) Proof : See Appendix C. Similarly to Theorem 1, Theorem 2 provides a closed-form expression for q n,st (T n , β) in the general SIR threshold regime. However, the expression in Theorem 2 is more complex than that in Theorem 1, due to the correlations across the M consecutive slots. Fig. 3 plots q n,st (T n , β) versus T n,k and β, respectively, in the static scenario, verifying Theorem 2. It is worth noting that unlike the high mobility scenario, it is hard to analytically characterize how q n,st (T n , β) changes with T n,k and β. However, from Fig. 3, we can observe some properties in the static scenario for the considered setup. Specifically, Fig. 3(a) shows that q n,st (T n , β) is an increasing function of T n,k , for all k ∈ K, which reveals the advantage of caching. Fig. 3(b) shows that in the static scenario, for some T n , there exists an optimal BS activity probability β * ≤ 1 that maximizes q n,st (T n , β), in contrast to the case in the high mobility scenario, where β * = 1 for any given T n . Although the penalty of random DTX in signal reduction may overtake its advantage in interference reduction in one particular slot (see (9)), random DTX is also able to reduce interference correlation across different slots in the static scenario [20]. When β > β * , the advantages of random DTX outweigh its penalty and when β < β * , its penalty outweighs its advantages. 2) Performance Analysis in Low SIR Threshold Regime: To further obtain insights, in this part, we analyze the outage probability of file n which is defined asq n,st (T n , β) 1 − q n,st (T n , β), in the low SIR threshold regime, i.e., θ → 0. By considering the same four cases as in Lemma 2, we have the following result. Lemma 5 (Outage Probability in Static Scenario When θ → 0): In the static scenario, when θ → 0, we haveq n,st (T n , β) ∼ (1 − β) M + c st (T n , β), n ∈ N ,(18) where c st (T n , β)                    θ M ∂ M ∂x M 1 F 1 (− 2 α ; 1 − 2 α ; x) −1 | x=0 , case i), θ 2 α 1 k∈K z k T n,k − 1 Γ 1 − 2 α M m=1 M m ×(−1) m+1 β m m i=0 m i β i (1 − β) m−i Γ(i+ 2 α ) Γ(i) , cases ii and iv), θ(1 − β) M −1 M β 2 2 α−2 , case iii).(19) Here, 1 F 1 (a; b; x) is the confluent hypergeometric function of the first kind. Proof : See Appendix D. From Lemma 5, we can see that both the caching probability T n and the BS activity probability β significantly affect the asymptotic behaviours ofq n,st (T n , β) when θ → 0, but in different manners. Fig. 4 plotsq n,st (T n , β) versus θ and indicates that Lemma 5 provides a good approximation forq n,st (T n , β) when θ is small. Similarly, we further characterise the diversity gain in the static scenario, i.e., d st = lim θ→0 log (q n,st (T n , β)) log θ .(20) From Lemma 5, we have the following result. Lemma 6 (Diversity Gain in Static Scenario): The diversity gain in the static scenario is given by d st =            M, case i), 2 α , case ii), 0, cases iii) and iv). Lemma 6 can be interpreted in the same way as Lemma 3. Comparing the diversity gains for case ii) in Lemma 6 and Lemma 3, we know that for case ii), the diversity gain in the high mobility scenario is M times of that in the static scenario due to the fact that there is no interference correlation in the high mobility scenario. B. Performance Optimization By substituting (15) into (6), the STP in the static scenario is calculated as q st (T, β) = n∈N a n M m=1 M m (−1) m+1 β m k∈K z k T n,k F m (β) k∈K z k T n,k + G m (β)(1 − k∈K z k T n,k ) .(22) The caching distribution T and BS activity probability β significantly affects the STP in the static scenario. We would like to maximize q st (T, β) in (22) by jointly optimizing T and β. Specifically, we have the following optimization problem. Problem 3 (Optimization of Random Caching and Random DTX in Static Scenario) : q * st max T,β q st (T, β) s.t. (1), (2), β ∈ (0, 1],(23) where q * st = q st (T * , β * ) denotes the optimal value and (T * , β * ) denotes the optimal solution. Problem 3 maximizes a non-concave function over a convex set, and hence is non-convex in general. Recall that in Section II-C, when M = 1, we have q n,hm = q n,st , implying that when M = 1, Problem 3 can be solved by using the same method as for Problem 1. Thus, in the following, we focus on solving Problem 3 when M ≥ 2. Note that, as q st (T, β) is differentiable, in general, we can obtain a stationary point of Problem 3 when M ≥ 2, using the gradient projection method with a diminishing stepsize. 9 However, the rate of convergence of the gradient projection method is strongly dependent on the choices of stepsize. If it is chosen improperly, it may take a large number of iterations to meet some convergence criterion, especially when the number of variables in Problem 3 is large. To address this issue, we propose a more efficient algorithm to obtain a stationary point of Problem 3, based on alternating optimization. Specifically, we partition the variables in Problem 3 into two blocks, i.e., T and β, and separate the constraint sets of these two blocks. Then, we solve a random caching optimization problem and a random DTX optimization problem alternatively. To solve Problem 4, we first analyze its structural properties. Let M 1 and M 2 denote the sets of all the odd and even numbers in the set {1, 2, · · · , M }, respectively. We rewrite q st (T, β) in (22) as q st (T, β) = q 1 (T, β) − q 2 (T, β), where q i (T, β) is given by q i (T, β) = n∈N a n m∈M i M m β m k∈K z k T n,k F m (β) k∈K z k T n,k + G m (β)(1 − k∈K z k T n,k ) , i = 1, 2. It can be easily verified that q i (T, β) is a concave function of T. Thus, Problem 4 is a differenceof-convex (DC) programming problem and can be solved based on the convex-concave procedure (CCP) [30]. The basic idea of the CCP is to linearize the convex terms of the objective function (i.e., −q 2 (T, β) in q st (T, β)) to obtain a concave objective for a maximization problem, and then solve a sequence of convex problems successively. Specifically, at iteration j, we solve the following problem: T (j) arg max T q 1 (T, β) −q 2 (T, β; T (j−1) )(24)s.t. (1),(2) , −1) , β), and T q 2 (T (j−1) , β) denotes the gradient of q 2 (T, β) at T = T (j−1) . whereq 2 (T, β; T (j−1) ) q 2 (T (j−1) , β) + (T − T (j−1) ) T T q 2 (T (j Since q 1 (T, β) andq 2 (T, β; T (j−1) ) are concave and linear w.r.t. T, respectively, the optimization in (24) is a convex problem which can be efficiently solved by the interior point method. The details of the proposed iterative algorithm are summarized in Algorithm 1. Note that, it has been shown in [30] that the sequence {T (j) } ∞ j=1 generated by Algorithm 1 converges to a stationary point of Problem 4. Next, we consider a special case of C k = C for all k ∈ K, i.e., equal cache size across all tiers. In this case, the optimization in (24) is convex. Similarly, using KKT conditions, we can obtain an optimal solution of the problem in (24) as follows. Lemma 7 (Optimal Solution of Problem in (24) When C k = C for all k ∈ K): When C k = C for all k ∈ K, an optimal solution T * of problem in (24) is given by T (j) n,k =            0, if η * ≥ f (0), 1, if η * ≤ f (1), f −1 (η * ), otherwise, n ∈ N , k ∈ K, where f −1 (·) denotes the inverse function of function f (·), given by f (x) a n m∈M1 M m β m G m (β) (F m (β)x + G m (β)(1 − x)) 2 − a n m∈M2 M m β m G m (β) F m (β)T (j−1) n,k + G m (β)(1 − T (j−1) n,k ) 2 , and η * satisfies n∈N T * n,k = C. 2) Random DTX Optimization: Next, we consider the optimization of the BS activity probability β while fixing T. When M = 2, it can be easily verified that Problem 5 is convex and thus can be efficiently solved by the interior point method. When M ≥ 3, it is hard to determine the convexity of Problem 5. Since q st (T, β) is a continuously differentiable function of β, a stationary point of Problem 5 can be efficiently obtained by the gradient projection method. 3) Alternating Optimization Procedure: Based on the results in Section IV-B1 and Section IV-B2, we develop an alternating optimization procedure for Problem 3, as summarized in Algorithm 2. If the sequence {(T (i) , β (i) )} ∞ i=1 generated by Algorithm 2 is convergent, then every limit point of {(T (i) , β (i) )} ∞ i=1 is a stationary point of Problem 3. Fig. 5 compares Algorithm 2 and the gradient projection method for solving Problem 3 in terms of the convergence rate and computational complexity. From Fig. 5(a), we can see that Algorithm 2 is convergent and the rate of convergence of the gradient projection method is strongly dependent on the choices of stepsize. In contrast, Algorithm 2 has robust convergence performance. In addition, from Fig. 5(b), we can see that the computing time of Algorithm 2 is shorter than the gradient projection method. These demonstrate the advantage of Algorithm 2 over the gradient projection method. V. NUMERICAL RESULTS In this section, we first illustrate the proposed solutions in the high mobility and static scenarios. Then, we compare the performance of the proposed solutions with some baselines in both scenarios. In the simulations, we consider a two-tier HetNet, i.e., K = 2, consisting of a macrocell network as the 1st tier overlaid with a picocell network as the 2nd tier. Unless otherwise stated, the simulation settings are as follows: P 1 = 20 W, P 2 = 0.13 W, λ 1 = 1 250 2 π , λ 2 = 1 50 2 π , α = 4, θ = 3 dB, C 1 = 25, C 2 = 15, N = 50, and a n = n −γ n∈N n −γ , where γ = 0.8 is the Zipf exponent. We can see that a more popular file corresponds to a larger caching probability, which is consistent with intuition. In addition, as the maximum number of transmissions M increases, more files can be stored at BSs, implying that retransmissions have a positive effect on the spatial file diversity. Fig. 6(b) plots the proposed BS activity probability versus M in the two scenarios. We can see that in the high mobility scenario, the optimal BS activity probability is β * = 1, verifying Lemma 1. In the static scenario, we can see that β * decreases with M , which means that the larger the number of transmissions the more BSs should be silenced in one slot. This is because with M increasing, smaller interference correlation can compensate lower BS availability. A. Proposed Solutions B. Comparisons between Proposed Solutions and Baselines In this part, we compare the proposed solutions with three baselines in the two scenarios. Baseline 1 adopts the most popular caching design where each BS in tier k selects the C k most popular files to store [3]. Baseline 2 adopts the uniform caching design, where each BS in tier k randomly selects C k files to store, according to the uniform distribution [5]. Baseline 3 adopts the i.i.d. caching design, where each BS in tier k randomly selects C k files to store, in an i.i.d. manner with file n being selected with probability a n [6]. In addition, for each baseline, in the high mobility scenario, the BS activity probability is chosen as β = 1 and in the static scenario, the BS activity probability is obtained by solving Problem 5 for the corresponding caching probability T using the method proposed in Section IV-B2. Fig. 7 and Fig. 8 plot the STP versus the SIR threshold θ and the Zipf exponent γ, respectively, in the two scenarios. Clearly, we see that the proposed solutions outperform all the baselines in both scenarios. In addition, we can see that when θ is high or γ is large, the most popular caching can achieve almost the same performance as the proposed random caching; when θ is low or γ is small, the uniform caching can achieve almost the same performance as the proposed random caching. The reasons are given as follows. The typical user can be successfully served by only the strongest BS when θ is high (the tail of the popularity distribution becomes small when γ is large), and thus caching the most popular files at each BS is almost optimal; the typical user can be successfully served by more BSs when θ is low (the tail of the popularity distribution becomes flat when γ is small), and thus caching more files in the network is better. VI. CONCLUSIONS In this paper, we consider retransmissions with random DTX in a large-scale cache-enabled HetNet employing random caching. We analyze and optimize the STP in the high mobility and static scenarios, and show that mobility increases temporal diversity, leading to the STP increase. First, in each scenario, we obtain closed-form expressions for the STP in the general and low SIR threshold regimes. The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios, which reveals the advantage of caching. In addition, the analysis reveals that random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. Next, in each scenario, we consider the maximization of the STP with respect to the caching probability and the BS activity probability, which is a challenging nonconvex optimization problem. We obtain a globally optimal solution in the high mobility scenario and a stationary point in the static scenario. Numerical results show that the proposed solutions achieve significant gains over existing baseline schemes and can well adapt to the changes of the system parameters to wisely utilize storage resources and transmission opportunities. The practical situations are most likely to be somewhere between these two scenarios. The results here therefore have provided some theoretical performance bounds and insights for a practical large-scale cache-enabled HetNet. More general practical scenarios will be explored in our future work. APPENDIX A PROOF OF THEOREM 1 For ease of illustration, we rewrite the SIR in (3) as SIR n,0 (t) = P k0 h k0,l0,0 (t)X −α k0,l0,0 (t)1(l 0 ∈ B a k0 (t)) I , where I I n,k 0 + I −n,k 0 + K j=1, =k 0 (I n,j + I −n,j ) with I n,k 0 l∈Φ n,k 0 \{k 0 } P k 0 h k 0 ,l,0 (t)X −α k 0 ,l,0 (t) 1(l ∈ B a k 0 (t)), I −n,k 0 l∈Φ −n,k 0 P k 0 h k 0 ,l,0 (t)X −α k 0 ,l,0 (t)1(l ∈ B a k 0 (t)), I n,j l∈Φ n,j P j h j,l,0 (t) X −α j,l,0 (t)1(l ∈ B a j (t)) and I −n,j l∈Φ −n,j P j h j,l,0 (t)X −α j,l,0 (t)1(l ∈ B a j (t)). Note that Φ −n,j Φ j \ Φ n,j for all j ∈ K. Due to independent thinning induced by random caching, we know that Φ n,j is a homogeneous PPP with density λ j T n,j and Φ −n,j is a homogeneous PPP with density λ j (1 − T n,j ). By (5), we have q n,hm (T n , β)=1 − (1 − Pr (SIR n,0 ≥ θ)) M .(26) Thus, to calculate q n,hm (T n , β), we only need to calculate Pr (SIR n,0 ≥ θ). First, we calculate Pr (SIR n,0 ≥ θ|X k 0 ,l 0 ,0 = x). By (25), we have Pr (SIR n,0 ≥ θ|X k0,l0,0 = x) = E In,k 0 exp − θx α P k0 I n,k0 LI n,k 0 (s)| s= θx α P k 0 E I−n,k 0 exp − θx α P k0 I −n,k0 LI −n,k 0 (s)| s= θx α P k 0 × K j=1, =k0 E In,j exp − θx α P k0 I n,j LI n,j (s)| s= θx α P k 0 E I−n,j exp − θx α P k0 I −n,j LI −n,j (s)| s= θx α P k 0 ,(27) where (a) is obtained by noting that h k 0 ,l 0 ,0 is exponentially distributed with unit mean; (b) is due to the independence of the Rayleigh fading channels and the independence of the PPPs. L I n,k 0 (s), L I −n,k 0 (s), L I n,j (s) and L I −n,j (s) represent the Laplace transforms of the interference I n,k 0 , I −n,k 0 , I n,j and I −n,j , respectively, which are calculated as follows. L In,k 0 (s)=E Φn,k 0   l∈Φn,k 0 \{k0} E hk 0 ,l,0 exp −sP k0 h k0,l,0 X −α k0,l,0 1(l ∈ B a j (t))   (c) =exp −πλ k0 T n,k0 x 2 βθ 2 α ∞ θ − 2 α 1 − 1 1 + v − α =exp −πλ k0 T n,k0 x 2 β 2 F 1 − 2 α , 1; 1 − 2 α ; −θ − 1 ,(28) where (c) is obtained by first noting that h k 0 ,l 0 ,0 is exponentially distributed with unit mean and each interfering BS is active with probability β, and then utilizing the probability generating functional of PPP. Similarly, L I −n,k 0 (s), L I n,j (s) and L I −n,j (s) can be calculated as follows. L I−n,k 0 (s)=exp −πλ k0 (1 − T n,k0 ) x 2 βθ 2 α Γ 1 + 2 α Γ 1 − 2 α ,(29) L In,j (s)=exp −πλ j T n,j P j P k0 2 α x 2 β 2 F 1 − 2 α , 1; 1 − 2 α ; −θ − 1 ,(30) L I−n,j (s)=exp −πλ j (1 − T n,j ) P j P k0 2 α x 2 βθ 2 α Γ 1 + 2 α Γ 1 − 2 α .(31) Substituting (28)-(31) into (27), we can obtain Pr (SIR n,0 ≥ θ|X k 0 ,l 0 ,0 = x). Next, we calculate Pr (SIR n,0 ≥ θ) by removing the condition X k 0 ,l 0 ,0 = x. Note that, we have the probability density function (p.d.f.) of the distance x, which is given by f Xk 0 ,l 0 ,0 (x) = 2πλ k0 T n,k0 A k0 x exp   − K j=1 πλ j T n,j P j P k0 2 α x 2   ,(32) where A k 0 is the probability that the typical user u 0 is associated with tier k 0 . Thus, we have Pr (SIR n,0 ≥ θ) = β ∞ 0 f Xk 0 ,l 0 ,0 (x) (Pr (SIR n,0 ≥ θ|X k0,l0,0 = x)) dx (d) = β K k0=1 A k0 ∞ 0 exp − K j=1 πλ j P j P k0 2 α x 2 T n,j β 2 F 1 − 2 α , 1; 1 − 2 α ; −θ − 1 +(1 − T n,j )βθ 2 α Γ 1 + 2 α Γ 1 − 2 α 2πλ k0 T n,k0 A k0 x exp   − K j=1 πλ j T n,j P j P k0 2 α x 2   dx (e) = β K k=1 πλ k P 2 α k T n,k K k=1 πλ k P 2 α k (T n,k W (β) + (1 − T n,k )V (β)) ,(33) where (d) follows from the law of total probability and the serving BS is active with probability β; (e) follows from the definitions of W (β) and V (β) in Theorem 1. Finally, substituting (33) into (26), we complete the proof of Theorem 1. APPENDIX B PROOF OF LEMMA 2 In case i), q n,hm (T n , β) in (8) can be rewritten as where SIR n,0 is given by (25). Thus, to calculate q n,st (T n , β) in (38), it remains to calculate E Φ ((Pr (SIR n,0 ≥ θ|Φ)) m ). As in (27) Next, we calculate E Φ ((Pr (SIR n,0 ≥ θ|Φ)) m ) by removing the condition X k 0 ,l 0 ,0 = x in E Φ ((Pr (SIR n,0 ≥ θ|Φ, X k 0 ,l 0 ,0 = x)) m ). By using the p.d.f. of X k 0 ,l 0 ,0 in (32) and applying some similar algebraic manipulations as used in (33), we have E Φ ((Pr (SIR n,0 ≥ θ|Φ)) m ) = β m ∞ 0 f Xk 0 ,l 0 ,0 (x)E Φ ((Pr (SIR n,0 ≥ θ|Φ, X k0,l0,0 = x)) m ) dx Fig. 1 . 1q n,hm (T n , β) versus T n,k and β, respectively, in the high mobility scenario. K = 2, P 1 = 20 W, P 2 = 0.13 Fig . 2.q n,hm (T n , β) versus θ in the high mobility scenario in different cases. P 1 = 20 W, P 2 = 0.13 W, λ 1 = 1 250 2 π , λ 2 = 1 50 2 π , α = 4, and M = 3. Problem 1 ( 1Optimization of Random Caching and Random DTX in High Mobility Scenario): Fig. 3 . 3q n,st (T n , β) versus T n,k and β, respectively, in the static scenario. The simulation parameters are given inFig. 1. Fig . 4.q n,st (T n , β) versus θ in the static scenario in different cases. P 1 = 20 W, P 2 = 0.13 W, λ 1 = 1 250 2 π , λ 2 = 1 50 2 π , α = 4, and M = 3. 1 ) 1Random Caching Optimization: First, we consider the optimization of the random caching probability T while fixing β. Fix T (i) , and obtain an optimal solution β (i+1) of Problem 5 when M = 2 using the interior point method or a stationary point β (i+1) of Problem 5 when M ≥ 3 using the gradient projection method.5: i ← i + 1.6: until convergence criterion is met. Computing time versus C2 (C1), C1 = C2 + 5. Fig. 5 . 5Convergence rate and computing complexity of Algorithm 2 at M = 3. For the gradient projection method, we choose the stepsize at iteration i as (i) = c 2+i 0.55 . Note that, in (b), each point corresponds to the minimum computing time by choosing the optimal parameter c ∈ {5, 10, 15, 20, 25}. P 1 = 20 W, P 2 = 0.13 W, λ 1 = 1 250 2 π , λ 2 = 1 50 2 π , α = 4, θ = 3 dB, C 1 = 25, C 2 = 15, N = 50, and a n = n −γ n∈N n −γ , where γ = 0.8 is the Zipf exponent. caching probability versus n. BS activity probability versus M . Fig. 6 . 6Proposed caching probability in tier 2 and BS activity probability in the high mobility and static scenarios at θ = 3 dB. Fig. 6 6(a) plots the proposed caching probability in tier 2 versus file index n in the two scenarios. Fig. 7 . 7STP versus θ at γ = 0.8 and M = 5. Fig. 8 . 8STP versus γ at θ = −3 dB and M = 5. q n,hm (T n , β) = 1 − 1 − 1 2 F 1 (− 2 α , 1; 1 − 2 α ; −θ) m+1 E Φ ((Pr (SIR n,0 ≥ θ|Φ)) m ) , , conditioning on X k 0 ,l 0 ,0 = x, we haveE Φ ((Pr (SIR n,0 ≥ θ|Φ, X k0,l0,0 = x)) m ) = E Φ E {In,k 0 |Φ ,I−n,k 0 |Φ ,In,j|Φ,I−n,j|Φ} exp − θx α P k0 I m = E Φ E In,k 0 |Φ exp − θx α P k0 I n,k0|ΦLI n,k 0 |Φ (s)| I n,k 0 |Φ (s), L I −n,k 0 |Φ (s), L I n,j|Φ (s) and L I −n,j|Φ (s) represent the Laplace transforms of the interference I n,k 0 |Φ , I −n,k 0 |Φ , I n,j|Φ and I −n,j|Φ , conditioned on Φ, respectively, which can be calculated as L I n,k 0 |Φ (s February 13, 2018 DRAFT Note that, the file popularity evolves at a slower timescale and various learning methodologies can be employed to estimate the file popularity over time[24].3 To implement the random caching design, we randomly place a file combination of C k different files at each BS in tier k according to a corresponding caching probability for file combinations. The detailed relationship between T and the caching probability for file combinations can be found in[12].February 13, 2018 DRAFT Note that, we consider the case that a user will not request any new file until the current file request is served or expires (i.e., the number of transmissions exceeds M ).5 Note that, a user cannot be served in a slot if its serving BS is inactive in this slot.February 13, 2018 DRAFT Different types of files may have different target bit rates. For instance, some video files such as MPEG 1, MPEG 4 and H.323[26] and audio files such as CD and MP3[27] require relatively low target bit rates.February 13, 2018 DRAFT Note that, f (x) ∼ g(x) when x → 0 means limx→0 f (x)/g(x) = 1.February 13, 2018 DRAFT Note that a stationary point is a point that satisfies the necessary optimality conditions of a non-convex optimization problem, and it is the classic goal in the design of iterative algorithms for non-convex optimization problems.February 13, 2018 DRAFT dvFebruary 13, 2018 DRAFT Then, when θ → 0, we havē q n,hm (T n , β) (a)where (a) is due to2In case ii), q n,hm (T n , β) in (8) can be rewritten asThen, when θ → 0, we havēwhere (c) follows from 2 F 1 (− 2 α , 1; 1 − 2 α ; −θ) ∼ 1 + 2θ α−2 as θ → 0; (d) uses the fact that the dominant term of the polynomial cθ + dθ 2 α , c, d > 0, α > 2, is dθ 2 α when θ → 0; (e) is due toIn case iii), q n,hm (T n , β) in(8)can be rewritten asThen, when θ → 0, we havēwhere the last step follows from the binomial expansion.In case iv), based on q n,hm (T n , β) in (8), by using a similar method to case i) -iii), whenCombining (34)-(37) and using the definition of c hm (T n , β) in (11), we complete the proof of Lemma 2. 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We describe an approach to learn, in a term-rewriting setting, function definitions from input/output equations. By confining ourselves to structurally recursive definitions we obtain a fairly fast learning algorithm that often yields definitions close to intuitive expectations. We provide a Prolog prototype implementation of our approach, and indicate open issues of further investigation.

1802.01177

822959d14c790e4ec3a79637444de3fbe4e0f440

A Scheme-Driven Approach to Learning Programs from Input/Output Equations Feb 2018 Jochen Burghardt A Scheme-Driven Approach to Learning Programs from Input/Output Equations Feb 2018inductive functional programming We describe an approach to learn, in a term-rewriting setting, function definitions from input/output equations. By confining ourselves to structurally recursive definitions we obtain a fairly fast learning algorithm that often yields definitions close to intuitive expectations. We provide a Prolog prototype implementation of our approach, and indicate open issues of further investigation. Introduction This paper describes an approach to learn function definitions from input/output equations. 1 In trivial cases, a definition is obtained by syntactical anti-unification of the given i/o equations. In non-trivial cases, we assume a structurally recursive function definition, and transform the given i/o equations into equations for the employed auxiliary functions. The latter are learned from their i/o equations in turn, until a trivial case is reached. We came up with this approach in 1994 but didn't publish it until today. In this paper, we explain it mainly along some learning examples, leaving a theoretical elaboration to be done. Also, we indicate several issues of improvement that should be investigated further. However, we provide at least a Prolog prototype implementation of our approach. In the rest of this section, we introduce the term-rewriting setting our approach works in. In Sect. 2, we define the task of function learning. In Sect. 3 and 4, we explain the base case and the inductive case of our approach, that is, how to learn trivial functions, and how to reduce learning sophisticated functions to learning easier functions, respectively. Section 5 sketches some ideas for possible extensions to our approach; it also shows its limitations. Some runs of our Prolog prototype are shown in Appendix A. We use a term-rewriting setting that is well-known from functional programming: A sort can be defined recursively by giving its constructors. For example, sort definition 1, shown in Fig. 1, defines the sort nat of all natural numbers in 0-s notation. In this example, we use 0 as a nullary, and s as a unary constructor. add : blist × blist −→ blist addition of binary numbers (lists) Figure 2. Employed function signatures A sort is understood as representing a possibly infinite set of ground constructor terms, 2 e.g. the 1 We will use henceforth "i/o equations" for brevity. We avoid calling them "examples" as this could cause confusion when we explain our approach along example sort definitions, example signatures, and example functions. size(nd(x, y, z)) = s(size(x)+size(z)) 0)), s(s(s(0))), . . .}. A function has a fixed signature; Fig. 2 gives some examples. The signature of a constructor can be inferred from the sort definition it occurs in, e.g. 0 :−→ nat and s : nat −→ nat. We don't allow non-trivial equations between constructor terms, therefore, we have T 1 = T 2 iff T 1 syntactically equals T 2 , for all ground constructor terms T 1 , T 2 . A Fig. 3. Given some functions f 1 , . . . , f m defined by such a term rewriting system, for each i and each ground constructor terms T 1 , . . . , T n we can find a unique ground constructor term T such that f i (T 1 , . . . , T n ) = T . We then say that f i (T 1 , . . . , T n ) evaluates to T . Given a term T , we denote by vars(T ) the set of variables occurring in T . The task of learning functions The problem our approach shall solve is the following. Given a set of sort definitions, a non-constructor function symbol f , its signature, and a set of input/output equations for f , construct a term rewriting system defining f such that it behaves as prescribed by the i/o equations. We say that we want to learn a definition for f , or sloppily, that we want to learn f , from the given i/o equations. For example, given sort definition 1, signature 10, and the following input/output ground equa- dup(s(s(0))) = s(s(s(s(0)))) 25: dup(s(s(s(0)))) = s(s(s(s(s(s(0)))))) we are looking for a definition of dup such that equations 22, 23, 24, and 25 hold. One such definition is 26: dup(0) = 0 27: dup(s(x)) = s(s(dup(x))) We say that this definition covers the i/o equations 22, 23, 24, and 25. In contrast, a definition 28: dup(0) = 0 29: dup(s(x)) = s(s(x)) would cover i/o equations 22 and 23, but neither 24 nor 25. We wouldn't accept this definition, since we are interested only in function definitions that cover all given i/o equations. It is well-known that there isn't a unique solution to our problem. In fact, given i/o equations f (L 1 ) = R 1 , . . . , f (L n ) = R n and an arbitrary function g of appropriate domain and range, e.g. the function defined by 3 f (x) = ( if x = L 1 then R 1 elif . . . elif x = L n then R n else g(x) fi ) trivially covers all i/o equations. Usually, the "simplest" function definitions are preferred, with "simplicity" being some user-defined measure loosely corresponding to term size and/or case-distinction count, like e.g. in [Bur05,p.8] and [Kit10,p.77]. However, the notion of simplicity depends on the language of available basic operations. 4 In the end, the notion of a "good" definition can hardly be defined more precisely than being one that meets common human prejudice. From our prototype runs we got the feeling that our approach often yields "good" definition in that sense. Learning functions by anti-unification One of the simplest ways to obtain a function definition is to syntactically anti-unify the given i/o equations. Given i/o equations f (L 11 , . . . ,L m1 ) = R 1 . . . . . . . . . f (L 1n , . . . ,L mn ) = R n , let f ( L 1 , . . . , L m ) = R be their least general generalization (lgg for short, see [Plo70,Plo71,Rey70]). If the variable condition vars(R) ⊆ vars(L 1 ) ∪ . . . ∪ vars(L m ) holds, then the lgg will cover all n given i/o equations. 3 We use common imperative notation here for sake of readability. 4 The "invariance theorem" in Kolmogorov complexity theory (e.g. [LV08, p.105, Thm.2.1.1]) implies that ∀L 1 , L 2 ∃c ∀x : |C L 1 (x) − C L 2 (x)| c, where the L i range over Turing-complete algorithm description languages, c is a natural number, x ranges over i/o equation sets, and C L (x) denotes the length of the shortest function definition, written in L, that covers x. This theorem is sometimes misunderstood to enable a language-independent notion of simplicity; however, it does not, at least for small i/o example sets. For example, assume we are to generate a definition for a unary function called g 2 from the i/o equations 30: g 2 ( 0 ) = s(s( 0 )) 31: g 2 ( s(s(0)) ) = s(s( s(s(0)) )) 32: g 2 (s(s(s(s(0))))) = s(s(s(s(s(s(0)))))) . We obtain the lgg 33: g 2 ( x 024 ) = s(s( x 024 )) As another example, we can generate a definition for a binary function called g 4 from the i/o equations 34: g 4 (a, 0 ) = s( 0 ) 35: g 4 (a, s(0) ) = s( s(0) ) 36: g 4 (a,s(s(0))) = s(s(s(0))) . We obtain the lgg dup( s(s(0)) ) = s(s(s(s(0)))) 25 dup(s(s(s(0)))) = s(s(s(s(s(s(0)))))) is computed as 38: dup( x 0123 ) = x 0246 which violates the variable condition, and thus cannot be used to reduce a term dup(T ) to a ground constructor term, i.e. to evaluate dup(T ). The above anti-unification approach can be extended in several ways, they are sketched in Sect. 5.1. However, in all but trivial cases, an lgg will violate the variable condition, and we need another approach to learn a function definition. Learning functions by structural recursion For a function f that can't be learned by Sect. 3, we assume a defining term rewriting system that follows a structural recursion scheme obtained from f 's signature and a guessed argument position. For example, for the function dup with the signature given in 10 and the only possible argument position, 1, we obtain the schematic equations dup(s(x)) = g 2 (dup(x)) where g 1 and g 2 are fresh names of non-constructor functions. If we could learn appropriate definitions for g 1 and g 2 , we could obtain a definition for dup just by adding equations 39 and 40. The choice of g 1 is obvious: 41: 0 22 = dup(0) 39 = g 1 In order to learn a definition for g 2 , we need to obtain appropriate i Altogether, we obtain the rewriting system 39 dup(0) = g 1 40 dup(s(x)) = g 2 (dup(x)) 41 g 1 = 0 33 g 2 (x 024 ) = s(s(x 024 )) as a definition for dup that covers its i/o equations 22, 23, 24, and 25. Subsequently, this system may be simplified, by inlining, to 39 dup(0) = 0 40 dup(s(x)) = s(s(dup(x))) which is the usual definition of the dup function. Returning to the computation of i/o equations for g 2 from those for dup, note that g 2 's derived i/o equations 30, 31, and 32 were necessary in the sense that they must be satisfied by each possible definition of g 2 that leads to dup covering its i/o equations (23, 24, and 25). Conversely, g 2 's i/o equations were also sufficient in the sense that each possible definition of g 2 covering them ensures that dup covers 23, 24, and 25, provided it covers 22: Proof of 23: dup(s(0)) 40 = g 2 (dup(0)) 22 = g 2 (0) 30 = s(s(0)) Proof of 24: dup(s(s(0))) 40 = g 2 (dup(s(0))) 23 = g 2 (s(s(0))) 31 = s(s(s(s(0)))) Proof of 25: dup(s(s(s(0)))) 40 = g 2 (dup(s(s(0)))) 24 = g 2 (s(s(s(s(0))))) 32 = s(s(s(s(s(s(0)))))) Observe that the above proofs are based just on permutations of the equation chains from 30, 31, and 32. Moreover, note that the coverage proof for dup(s(T )) relies on the coverage for dup(T ) already being proven. That is, the coverage proofs follow the employed structural recursion scheme. As for the base case, g 1 's coverage of 41 is of course necessary and sufficient for dup's coverage of 22. Non-ground i/o equations As an example that uses i/o equations containing variables, consider the function lgth, with the signature given in 7. Usually, i/o equations for this functions are given in a way that indicates that the particular values of the list elements don't matter. For example, an i/o equation like lgth(a::b::nil) = s(s(0)) is seen much more often than lgth(s(0)::0::nil) = s(s(0)). Our approach allows for variables in i/o equations, and treats them as universally quantified. That is, a non-ground i/o equation is covered by a function definition iff all its ground instances are. Assume for example we are given the i/o equations 42: lgth(x::y) = g 4 (x, lgth(y)) . lgth(nil) = 0 . Similar to the dup example, we get Functions of higher arity For functions with more than one argument, we have several choices of the argument on which to do the recursion. In these cases, we currently systematically try all argument positions 7 in succession. This is feasible since • our approach is quite simple, and hence fast to compute, and • we have a sharp and easy to compute criterion (viz. coverage 8 of all i/o examples) to decide whether recursion on a given argument was successful. For the function +, with the signature given in 5, and argument position 2, we obtain the structural recursion scheme 51: x+0 = g 5 (x) 52: x+s(y) = g 6 (x, x+y) . Appendix A.1 shows a run of our Prolog prototype implementation that obtains a definition for +. In Sect. 5.2, we discuss possible extensions of the structural recursion scheme, like simultaneous recursion. Constructors with more than one recursion argument When computing a structural recursion scheme, we may encounter a sort s with a constructor that takes more than one argument of sort s. A common example is the sort of all binary trees (of natural numbers), as given in 3. The function size, with the signature given in 9, computes the size of such a tree, i.e. the total number of nd nodes. A recursion scheme for the size and argument position 1 looks like: 53: size(null) = g 9 54: size(nd(x, y, z)) = g 10 (y, size(x), size(z)) 7 In particular, the recursive argument's sort and the function's result sort needn't be related in any way, as the lgth example above demonstrates. 8 Checking if an i/o equation is covered by a definition requires executing the latter on the lhs arguments of the former. Our structural recursion approach ensures the termination of such computations, and establishes an upper bound for the number of rewrite steps. For example, g 2 and g 4 , defined in 33 and 37, respectively, need one such step, while their callers dup and lgth, defined in 39,40 and 46,47, respectively, need a linear amount of steps. An upper-bound expression for learned functions' time complexity remains to be defined and proven. In App. A.2, we show a prototype run to obtain a definition for size. General approach In the previous sections, we have introduced our approach using particular examples. In this section, we sketch a more abstract and algorithmic description. Given a function and its signature f : s 1 × . . . × s n −→ s, and given one of its argument positions 1 i n, we can easily obtain a term rewriting system to define f by structural recursion on its ith argument. Assume in the definition of f 's ith domain sort s i we have an alternative s i ::= . . . | c(s 1 , . . . , s l ) | . . . , assume {s ν(1) , . . . , s ν(m) } s i is the set of non-recursive arguments of the constructor c, and s ρ(1) = . . . = s ρ(k) = s i are the recursive arguments of c. Let g be a new function symbol. We build an equation f (x 1 , . . . , x i−1 , c(y 1 , . . . , y l ), x i+1 , . . . , x n ) = g(x 1 , . . . , x i−1 , x i+1 , . . . , x n , y ν(1) , . . . , y ν(m) , f (x 1 , . . . , x i−1 , y ρ(1) , x i+1 , . . . , x n ) . . . f (x 1 , . . . , x i−1 , y ρ(k) , x i+1 , . . . , x n ) ) In a somewhat simplified presentation, we build the equation This way, we can reduce the problem of synthesizing a definition for f that reproduces the given i/o equations to the problem of synthesizing a definition for g from its i/o equations. As a base case for this process, we may synthesize non-recursive function definitions by anti-unification of the i/o equations. It should be possible to prove that f covers all its i/o equations iff g covers its, under some appropriate conditions. We expect that a sufficient condition is that all recursive calls to f could be evaluated. At least, we could demonstrate this in the above dup and lgth example. Fct Eqn Lf Rg Fct Eqn Lf Rg Termination In order to establish the termination of our approach, it is necessary to define a criterion by which g is easier to learn from it i/o equations than f is from its. Term size or height cannot be used in a termination ordering; when proceeding from f to g they may remain equal, or may even increase, as shown in Fig. 4 for the dup vs. g 2 example. However, the number of i/o equations decreases in this example, and in all other ones we dealt with. A sufficient criterion for this is that f 's i/o equations don't all have the same left-hand side top-most constructor. However, the same criterion would have to be ensured in turn for g, and it is not obvious how to achieve this. In any case, by construction of g's i/o example from f 's, no new terms can arise. 10 Even more, each term appearing in an i/o example for g originates from a right-hand side of an i/o example for f . Therefore, our approach can't continue generating new auxiliary functions forever, without eventually repeating the set of i/o equations. Our prototype implementation doesn't check for such repetitions, however. Possible extensions In this section, we briefly sketch some possible extensions of our approach. Their investigation in detail still remains to be done. Extension of anti-unification In Sect. 3 we used syntactical anti-unification to obtain a function definition, as a base case of our approach. Several way to extend this technique can be thought of. Set anti-unification It can be tried to split the set of i/o equations into disjoint subsets such that from each one an lgg satisfying the variable condition is obtained. This results in several defining equations. An additional constraint might be that each subset corresponds to another constructor symbol, observed at some given fixed position in the left-hand side terms. sq( s(s(0)) ) = s(s(s(s(0)))) 59: sq(s(s(s(0)))) = s 9 (0) 60: sq( x 0123 ) = x 0123 * x 0123 Depth-bounded anti-unification In many cases, defining equations obtained by syntactical anti-unification appear to be too particular. For example, s 4 (0) and s 9 (0) are generalized to s 4 (x 05 ), while being by 4 greater than something wouldn't be the first choice for a common property of both numbers for most humans. As a possible remedy, a maximal depth d may be introduced for the anti-unification algorithm. Beyond this depth, terms are generalized by a variable even if all their root function symbols agree. Denoting by lgg d (t 1 , t 2 ) the result of an appropriately modified algorithm, it should be easy to prove that lgg d (t 1 , t 2 ) can be instantiated to both t 1 and t 2 , and is the most special term with that property among all terms of depth up to d. If d is chosen as ∞, lgg d and lgg coincide. In our prototype implementation, we meanwhile built in such a depth boundary. size( x null null ) = 0 size( nd(x, y, z) ) = f 1 ( y, size(x), size(z)) f 1 (x, y 0 0 , z ) = s( z ) f 1 (x, s(y) , z ) = f 2 ( x, z, f 1 (x, y, z)) f 2 (x, y 0 0 , z s(z) s(0) ) = s( z s(z) s(0) ) f 2 (x, Extension of structural recursion Some functions are best defined by simultaneous recursion on several arguments. As an example, consider the sort definition 4 with nl, o, and i denoting an empty list, a 0 digit, and a 1 digit, respectively. For technical reasons, such a list is interpreted in reversed order, e.g. o(i(i(nl))) denotes the number 6. The sum function add, its signature shown in 11, may then be defined by the following rewrite system: inc : blist −→ blist is a function to increment a binary digit list. This corresponds to the usual hardware implementation, with inc being used for the carry. It is obvious that this definition cannot be obtained from our simple structural recursion scheme from Sect. 4, neither by recurring over argument position 1 nor over 2. Instead, we would need recursion over both positions simultaneously, i.e. a scheme like add( i(x) , i(y) ) = g 23 (add(x, y)) An extension of our approach could provide such a scheme, additionally to the simple structural recursion scheme. If we could prove that each function definition obtainable by the simple recursion scheme can also be obtained by a simultaneous recursion scheme, we needed only to employ the latter. This way, we would no longer need to guess an appropriate argument position to recur over; instead we could always recur simultaneously over all arguments of a given sort. Unfortunately, simultaneous recursion is not stronger than simple structural recursion. For example, the function app to concatenate two given lists can be obtained by simple recursion over the first argument (see 18,19 in Fig. 3), but not by simultaneous recursion: app(w::x, y::z) = g 24 (w, y, app(x, z)) doesn't lead to a sensible definition, for any choice of g 24 . One possible remedy is to try simple structural recursion first, on any appropriate argument position, and simultaneous recursion next, on any appropriate set of argument positions. Alternatively, user commands may be required about which recursion to try on which argument position(s). Another possibility might be to employ a fully general structural recursion scheme, like 77: app(w::x, y::z) = g 24 (w, y, app(w::x, z) , app(x, y::z) , app(x, z) ) and 78: add(o(x), o(y)) = g 25 ( add(o(x), y) , add(x, o(y)) , add(x, y) ) . In this scheme, calls for simple recursion over each position are provided, as well as for simultaneous recursion over each position set. A new symbol Ω, intended to denote an undefined term, could be added to the term language. When e.g. i/o equations are missing to compute add(o(x), y) for some particular instance, the first argument of g 25 would be set to Ω in the respective i/o equation. In syntactical anti-unification and coverage test, Ω needed to be handled appropriately. This way, only one recursion scheme would be needed, and no choice of appropriate argument position(s) would be necessary. However, arities of auxiliary functions might grow exponentially. Limitations of our approach In this section, we demonstrate an example where our approach fails. Consider again the squaring function, its signature shown in 55, and consider again its i/o equations 56, 57, 58, and 59. Since syntactical anti-unification as in Sect. 3 (i.e. not considering an equational background theory E) doesn't lead to a valid function definition, we build a structural recursion scheme as in Sect. 4: 79: sq(0) = g 11 80: sq(s(x)) = g 12 (sq(x)) We get g 11 = 0, and the following i/o equations for g 12 : (s(0)))) 80 = g 12 (sq(s(s(0)))) 58 = g 12 (s(s(s(s(0))))) Observe that we are able to obtain i/o equations for g 12 only on square numbers. For example, there is no obvious way to determine the value of g 12 (s(s(s(0)))). Syntactically anti-unifying g 12 's i/o equation still doesn't yield a valid function definition. So we set up a recursion scheme for g 12 , in turn: 84: g 12 (0) = g 13 85: g 12 (s(x)) = g 14 (g 12 (x)) Again, g 13 = s(0) is obvious. Trying to obtain i/o equations for g 14 , we get stuck, since we don't know how g 12 should behave on non-square numbers: ?? ?? = g 12 (s(s(0))) 85 = g 14 (g 12 (s(0))) 82 = g 14 (s(s(s(s(0))))) 88: ?? ?? = g 12 (s(s(s(0)))) 85 = g 14 (g 12 (s(s(0)))) ?? = g 14 (??) 89: s 9 (0) 83 = g 12 (s(s(s(s(0))))) 85 = g 14 (g 12 (s(s(s(0))))) ?? = g 14 (??) As an alternative, by applying 85 sufficiently often rather than just once, we can obtain: 90: s 9 (0) 83 = g 12 (s(s(s(s(0))))) 85 = g 14 (g 12 (s(s(s(0))))) 85 = g 14 (g 14 (g 12 (s(s(0))))) 85 = g 14 (g 14 (g 14 (g 12 (s(0))))) 82 = g 14 (g 14 (g 14 (s(s(s(s(0))))))) However, no approach is known to learn g 14 from an extended i/o equation like 90, which determines g 14 • g 14 • g 14 rather than g 14 itself. In such cases, we resort to the excuse that the original function, sq isn't definable by structural recursion. A precise criterion for the class that our approach can handle is still to be found. It is not even clear that such a criterion can be computable. If not, it should still be possible to give computable necessary and sufficient approximations. size(nd(nd(nl,va,nl),vb,nl)) = s(s(0)), | size(nd(nl,va,nd(nl,vb,nl))) = s(s(0)), | size(nd(nd(nl,va,nl),vb,nd(nl,vc,nl))) = s(s(s(0))), | size(nd(nl,va,nd(nd(nl,vb,nl),vc,nl))) = s(s(s(0))), | size(nd(nl,va,nd(nl,vb,nd(nl,vc,nl)))) = s(s(s(0))), | size(nd(nd(nl,va,nl),vb,nd(nd(nl,vc,nl),vd,nl))) = s(s(s(s(0)))), | size(nd(nd(nd(nl,va,nl),vb,nl),vc,nd(nl,vd,nl))) = s(s(s(s(0)))) | ], | run(size,SgI,SD,ExI size(nl) = 0 . inducePos(size,1,nl) . inducePos(size,1,nd(tree,nat,tree)) . . matching examples: [size(nd(nl,va,nl))=s(0),size(nd(nd(nl,va,nl),vb,nl))=s(s(0)),size(nd(nl,va,nd(nl,vb,nl)))=s(s(0)),size(nd(nd(nl,va,nl),vb,n... s(s(s(0))) = size(nd(nl,va,nd(nd(nl,vb,nl),vc,nl))) = f12(va,0,s(s(0))) . . derive new equation: s(s(s(0))) = size(nd(nl,va,nd(nl,vb,nd(nl,vc,nl)))) = f12(va,0,s(s(0))) . . derive new equation: s(s(s(s(0)))) = size(nd(nd(nl,va,nl),vb,nd(nd(nl,vc,nl),vd,nl))) = f12(vb,s(0),s(s(0))) . . derive new equation: s(s(s(s(0)))) = size(nd(nd(nd(nl,va,nl),vb,nl),vc,nd(nl,vd,nl))) = f12(vc,s(s(0)),s (0) [f12(va,0,0)=s(0),f12(va,0,s(0))=s(s(0)),f12(va,0,s(s(0)))=s(s(s(0))),f12(vb,s(0),0)=s(s (0) s(s(s(s(0)))) = f12(vb,s(0),s(s(0))) = f46(vb,s(s(0)),s(s(s(0)))) . . . . derive new equation: s(s(s(s(0)))) = f12(vc,s(s(0)),s(0)) = f46(vc,s(0),s(s(s(0)))) . . . . induce(f46 [f46(vb,0,s(0))=s(s(0)),f46(vb,s(0),s(s(0)))=s(s(s(0))),f46(vb,s(s(0)),s(s(s(0))))=s(s(s(s(0)))),f46(vc,s(0),s(s(s(0... 0)))=s(s(s(0))),f46(vb,s(s(0)),s(s(s(0))))=s(s(s(s(0)))),f46(vc,s(0),s(s(s(0))))=s(s(s(s(0))))] . . . . . . anti-unifier: f46(v63,s(v64),s(s(v65))) = s(s(s(v65))) . . . . . inducePos(f46,2,s(nat) nd(nl,va,nl),vb,nl))=s(s(0)) size(nd(nl,va,nd(nl,vb,nl)))=s(s(0)) size(nd(nd(nl,va,nl),vb,nd(nl,vc,nl)))=s(s(s(0))) size(nd(nl,va,nd(nd(nl,vb,nl),vc,nl)))=s(s(s(0))) size(nd(nl,va,nd(nl,vb,nd(nl,vc,nl))))=s(s(s(0))) size(nd(nd(nl,va,nl),vb,nd(nd(nl,vc,nl),vd,nl)))=s(s(s(s(0)))) size(nd(nd(nd(nl,va,nl),vb,nl),vc,nd(nl,vd,nl)))=s(s(s(s(0)))) FUNCTION DEFINITIONS: size(nl)=0 size(nd(v66,v67,v68))=f12(size(v66),size(v68)) f12(0,v69)=s(v69) f12(s(v70),v71)=f46(v71,f12(v70,v71)) f46(0,s(0))=s(s(0)) f46(s(v72),s(s(v73)))=s(s(s(v73))) ?- A.3 Reversing a list ::= null | nd(tree, nat, tree) binary trees of natural numbers 4: blist ::= nl | o(blist) | i(blist) list of binary digits Figure 1 . 1Employed sort definitions nat × nat −→ nat addition of natural numbers 6: * : nat × nat −→ nat multiplication of natural numbers 7: lgth : list −→ nat number of elements of a list 8: app : list × list −→ list concatenation of lists 9: size : tree −→ nat number of elements of a binary tree 10: dup : nat −→ nat duplicating a natural number 11: 2 i.e. terms without variables, built only from constructor symbols Figure 3 . 3Example function definitions sort nat represents the set {0, s(0), s(s( (a, y 012 ) = s( y 012 ) which satisfies the variable condition. 5 Hence when g 4 is defined by equation 37, it covers i/o equations 36, 35, and 34.As a counter-example, the lgg of the above /o examples for g 2 from those for dup. Joining equation 40 with dup's relevant i/o equations yields three i/o equations for g 2 : for g 2 covering its i/o examples 30, 31, and 32 has already been derived by antiunification in Sect. 3 as 33 g 2 (x 024 ) = s(s(x 024 )) . Given the signature of lgth (see 7) and argument position 1, we obtain a = g 4 4can obtain i/o equations for g 4 from those for lgth: (a, lgth(b::c::nil)) 44 = g 4 (a, s(s(0))) Again, a function definition covering these i/o equation happens to have been derived by antiunification in Sect. 3: 37 g 4 (a, y 012 ) = s(y 012 )Altogether, equations 46, 47, 48, and 37 build a rewriting system for lgth that covers all its given i/o equations. By subsequently inlining g 3 's and g 4 's definition, we obtain a simplified definition for lgth: x::y) = s(lgth(y)) which agrees with the usual one found in textbooks.Similar to the ground case, g 4 's derived i/o equations 34, 35, and 36 were necessary for lgth renaming substitutions were used in the application of 43 and 44. f (. . . , c(y 1 , . . . , y l ), . . .) = g(. . . , f (..., y ρ(1) , ...), . . . , f (..., y ρ(k) , ...)). From the i/o equations for f , we often 9 can construct i/o equations for g: If we have an i/o equation that matches the above equation's left-hand side, and we have all i/o equations needed to evaluate the recursive calls to f on its right-hand side, we can build an i/o equation equation for g. Figure 4 . 4Left-and right-hand term sizes of i/o equations for dup and g 2 10 except for the fresh left-hand side top function symbols Figure 5 . 5Application of E-anti-unification to learn squaring Anti-unification modulo equational theory Another extension consists in considering an equational background theory E in anti-unification; it wasn't readily investigated in 1994. See[Hei94b,Hei94a,Hei95] for the earliest publications, and [Bur05,Bur17] for the latest.As of today, the main application of E-anti-unification turned out to be the synthesis of nonrecursive function definitions from input/ output equations [Bur17, p.3]. To sketch an example, let E consist just of definitions 12, 13, 14, and 15. Assume the signature 55: sq : nat −→ nat and the i/o equations 56, 57, 58, and 59 of the squaring function. Applying syntactical antiunification to the left-hand sides yields a variable x 0123 , and four corresponding substitutions.Applying constrained E-generalization [Bur05, p.5, Def.2] to the right-hand sides yields a term set that contains x 0123 * x 0123 as a minimal-size member, seeFig. 5. Figure 6 6compares the learned function definitions for size for d = 2, 3, 4 (top to bottom). For example, for d = 2, the -nonsensical-equation size(x) = 0 is learned, while for d 3 the respective equation reads size(null) = 0. Not surprisingly, for d = 2 only one of the given 9 i/o equations is covered. For d 1, the attempt to learn defining equations for size fails. For d = 4 , 4the learned equations agree with those for d = ∞, and hence also with those for all intermediate depths. The prototype run for d = ∞ is shown in App. A.2. Note that the prototype simplifies equations by removing irrelevant function arguments. For this reason, f12 there has only two arguments, while the corresponding function f 1 inFig. 6has three. Figure 6 . 6Learned tree size definition for anti-unification depth 2, 3, and 4 . . induce(f10) . . . trying argument position: 1 . . . inducePos(f10,1,0) . . . . matching examples: [f10(0,0)=s(0),f10(0,s(0))=s(s(0))] . . . . anti-unifier: f10(0,v13) = s(v13) . . . inducePos(f10,1,0) . . . inducePos(f10,1,s(nat)) . . . . matching examples:[f10(s(0),s(0))=s(s(0)),f10(s(0),s(s(0)))=s(s(s(0))),f10(s(s(0)),s(s(0)))=s(s(s(0)))] . . . . anti-unifier: f10(s(v15),s(v16)) = s(s(v16)) . . . inducePos(f10,1,s(nat)) . . . all examples covered . . induce(f10) . inducePos(+,1,s(nat)) . all examples covered induce(+) +++++ Examples output check: +++++ Examples output check done FUNCTION SIGNATURES: f10 signature [nat,nat]-->nat (+)signature[nat,nat]-SgI = [ size signature [tree] --> nat], | SD = [ tree sortdef nl ! nd(tree,nat,tree), | nat sortdef 0 ! s(nat)], | ExI = [ size(nl) . . new recursion scheme: size(nd(v10,v9,v11)) = f12(v9,size(v10),size(v11)) . . derive new equation: s(0) = size(nd(nl,va,nl)) = f12(va,0,0) . . derive new equation: s(s(0)) = size(nd(nd(nl,va,nl),vb,nl)) = f12(vb,s(0),0) . . derive new equation: s(s(0)) = size(nd(nl,va,nd(nl,vb,nl))) = f12(va,0,s(0)) . . derive new equation:s(s(s(0))) = size(nd(nd(nl,va,nl),vb,nd(nl,vc,nl))) = f12(vb,s(0),s(0)) . . derive new equation: ),f12(vb,s(0),s(0))=s(s(s... . . . trying argument position: 2 . . . inducePos(f12,2,0) . . . . matching examples: [f12(va,0,0)=s(0),f12(va,0,s(0))=s(s(0)),f12(va,0,s(s(0)))=s(s(s(0)))] . . . . anti-unifier: f12(va,0,v37) = s(v37) . . . inducePos(f12,2,0) . . . inducePos(f12,2,s(nat)) . . . . matching examples: [f12(vb,s(0),0)=s(s(0)),f12(vb,s(0),s(0))=s(s(s(0))),f12(vb,s(0),s(s(0)))=s(s(s(s(0)))),f12(vc,s(s(0)),s(0))=s(s(s(s... . . . . new recursion scheme: f12(v43,s(v45),v44) = f46(v43,v44,f12(v43,v45,v44)) . . . . derive new equation: s(s(0)) = f12(vb,s(0),0) = f46(vb,0,s(0)) . . . . derive new equation: s(s(s(0))) = f12(vb,s(0),s(0)) = f46(vb,s(0),s(s(0))) . . . . derive new equation: . . . . . trying argument position: 2 . . . . . inducePos(f46,2,0) . . . . . . matching examples: [f46(vb,0,s(0))=s(s(0))] . . . . . . anti-unifier: f46(vb,0,s(0)) = s(s(0)) . . . . . inducePos(f46,2,0) . . . . . inducePos(f46,2,s(nat)) . . . . . . matching examples: [f46(vb,s(0),s(s( non-constructor function can be defined by giving a terminating ([DJ90, Sect.5.1, p.270]) term rewriting system for it such that its left-hand sides are sufficiently complete ([Gut77], [Com86], [DJ90, Sect.3.2, p.264]). Examples for function definitions are shown in ) . . induce(f12) . . . trying argument position: 1 . . . inducePos(f12,1,0) . . . . matching examples: [] . . . . no examples . . . inducePos(f12,1,0) . . . inducePos(f12,1,s(nat)) . . . . matching examples: [] . . . . no examples . . . inducePos(f12,1,s(nat)) . . . uncovered examples: ) . . . . . trying argument position: 1 . . . . . inducePos(f46,1,0) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f46,1,0) . . . . . inducePos(f46,1,s(nat)) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f46,1,s(nat)) . . . . . uncovered examples: ) . . . . . all examples covered . . . . induce(f46) . . . inducePos(f12,2,s(nat)) . . . all examples covered . . induce(f12) . inducePos(size,1,nd(tree,nat,tree)) . all examples covered induce(size) +++++ Examples output check: +++++ Examples output check done FUNCTION SIGNATURES: f46 signature [nat,nat,nat]-->nat f12 signature [nat,nat,nat]-->nat size signature [tree]-->nat FUNCTION EXAMPLES: size(nl)=0 size(nd(nl,va,nl))=s(0) size(nd( ? - ?SgI = [rev signature [list] --> list], Variable sorts: [vc:nat,vb:nat,va:nat] +++++ Examples input check done induce(rev) . trying argument position: 1 . inducePos(rev,1,[]) . . matching examples: [rev([])=[]] . . anti-unifier: rev([]) = [] . inducePos(rev,1,[]) . inducePos(rev,1,[nat|list]) . . matching examples: [rev([va])=[va],rev([vb,va])=[va,vb],rev([vc,vb,va])=[va,vb,vc]] . . new recursion scheme: rev([v7|v8]) = f9(v7,rev(v8)) . . derive new equation: [va] = rev([va]) = f9(va,[]) . . derive new equation: [va,vb] = rev([vb,va]) = f9(vb,[va]) . . derive new equation: [va,vb,vc] = rev([vc,vb,va]) = f9(vc,[va,vb]) . . induce(f9) . . . trying argument position: 1 . . . inducePos(f9,1,0) . . . . matching examples: [] . . . . no examples . . . inducePos(f9,1,0) . . . inducePos(f9,1,s(nat)) . . . . matching examples: [] . . . . no examples . . . inducePos(f9,1,s(nat)) . . . uncovered examples: [f9(va,[])=[va],f9(vb,[va])=[va,vb],f9(vc,[va,vb])=[va,vb,vc]] . . . trying argument position: 2 . . . inducePos(f9,2,[]) . . . . matching examples: [f9(va,[])=[va]] . . . . anti-unifier: f9(va,[]) = [va] . . . inducePos(f9,2,[]) . . . inducePos(f9,2,[nat|list]) . . . . matching examples: [f9(vb,[va])=[va,vb],f9(vc,[va,vb])=[va,vb,vc]] . . . . new recursion scheme: f9(v22,[v23|v24]) = f25(v22,v23,f9(v22,v24)) . . . . derive new equation: [va,vb] = f9(vb,[va]) = f25(vb,va,[vb]) . . . . derive new equation: [va,vb,vc] = f9(vc,[va,vb]) = f25(vc,va,[vb,vc]) . . . . induce(f25) . . . . . trying argument position: 1 . . . . . inducePos(f25,1,0) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f25,1,0) . . . . . inducePos(f25,1,s(nat)) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f25,1,s(nat)) . . . . . uncovered examples: [f25(vb,va,[vb])=[va,vb],f25(vc,va,[vb,vc])=[va,vb,vc]] . . . . . trying argument position: 2 . . . . . inducePos(f25,2,0) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f25,2,0) . . . . . inducePos(f25,2,s(nat)) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f25,2,s(nat)) . . . . . uncovered examples: [f25(vb,va,[vb])=[va,vb],f25(vc,va,[vb,vc])=[va,vb,vc]] . . . . . trying argument position: 3 . . . . . inducePos(f25,3,[]) . . . . . . matching examples: [] . . . . . . no examples . . . . . inducePos(f25,3,[]) . . . . . inducePos(f25,3,[nat|list]) . . . . . . matching examples: [f25(vb,va,[vb])=[va,vb],f25(vc,va,[vb,vc])=[va,vb,vc]] . . . . . . anti-unifier: f25(v37,va,[vb|v38]) = [va,vb|v38] . . . . . inducePos(f25,3,[nat|list]) . . . . . all examples covered . . . . induce(f25) . . . inducePos(f9,2,[nat|list]) . . . all examples covered . . induce(f9) . inducePos(rev,1,[nat|list]) . all examples covered induce(rev) +++++ Examples output check: +++++ Examples output check done FUNCTION SIGNATURES: f25 signature [nat,nat,list]-->list f9 signature [nat,list]-->list rev signature [list]-->list FUNCTION EXAMPLES: rev([])=[] rev([va])=[va] rev([vb,va])=[va,vb] rev([vc,vb,va])=[va,vb,vc] FUNCTION DEFINITIONS: rev([])=[] rev([v39|v40])=f9(v39,rev(v40)) f9(v41,[])=[v41] f9(v42,[v43|v44])=f25(v43,f9(v42,v44)) f25(v41,[v45|v46])=[v41,v45|v46]| SD = [ list sortdef [] ! [nat|list], | nat sortdef 0 ! s(nat)], | ExI = [ rev([]) = [], | rev([va]) = [va], | rev([vb,va]) = [va,vb], | rev([vc,vb,va]) = [va,vb,vc]], | run(rev,SgI,SD,ExI). +++++ Examples input check: +++++ Example 1: +++++ Example 2: +++++ Example 3: +++++ Example 4: ?- Whenever applied to terms T 1 , . . . , T m that don't start all with the same function symbol, Plotkin's lgg algorithm returns a variable that uniquely depends on T 1 , . . . , T m . We indicate the originating terms by an index sequence; e.g. y 012 was obtained as lgg(0, s(0), s(s(0))). In the rightmost equation of each line, we employ a renaming substitution. For example, we apply {a → b, b → c} to i/o equation 44 in line 36. For this reason, our approach wouldn't work if a, b, c were considered non-constructor constants rather than universally quantified variables. Our construction isn't successful in all cases. We give a counter-example in Sect. 5.3 E-generalization using grammars. Jochen Burghardt, Artificial Intelligence Journal. 1651Jochen Burghardt. E-generalization using grammars. Artificial Intelligence Journal, 165(1):1-35, 2005. An improved algorithm for E-generalization. Jochen Burghardt, BerlinTechnical reportJochen Burghardt. An improved algorithm for E-generalization. Technical report, Berlin, Sep 2017. Sufficient completeness, term rewriting systems and "anti-unification. Hubert Comon, Proc. 8th International Conference on Automated Deduction. 8th International Conference on Automated DeductionSpringer230Hubert Comon. Sufficient completeness, term rewriting systems and "anti-unification". In Proc. 8th International Conference on Automated Deduction, volume 230 of LNCS, pages 128-140. Springer, 1986. Rewrite Systems, volume B of Handbook of Theoretical Computer Science. N Dershowitz, J.-P Jouannaud, ElsevierN. Dershowitz and J.-P. Jouannaud. Rewrite Systems, volume B of Handbook of Theoretical Computer Science, pages 243-320. Elsevier, 1990. Abstract data types and the development of data structures. John V Guttag, Communications of the ACM. 206John V. Guttag. Abstract data types and the development of data structures. Communications of the ACM, 20(6):396-404, Jun 1977. Anti-unification and its application to lemma discovery. Birgit Heinz, Workshop Talk given in CADE-12Birgit Heinz. Anti-unification and its application to lemma discovery, 1994. Workshop Talk given in CADE-12. Lemma discovery by anti-unification of regular sorts. Birgit Heinz, 94-21TU BerlinTechnical ReportBirgit Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94-21, TU Berlin, 1994. Anti-Unifikation modulo Gleichungstheorie und deren Anwendung zur Lemmagenerierung. Birgit Heinz, TU BerlinPhD thesisBirgit Heinz. Anti-Unifikation modulo Gleichungstheorie und deren Anwendung zur Lemmagenerierung. PhD thesis, TU Berlin, Dec 1995. A Combined Analytical and Search-Based Approach to the Inductive Synthesis of Functional Programs. Emanuel Kitzelmann, Univ. BambergPhD thesisEmanuel Kitzelmann. A Combined Analytical and Search-Based Approach to the Inductive Synthesis of Functional Programs. PhD thesis, Univ. Bamberg, May 2010. An Introduction to Kolmogorov Complexity and Its Applications. texts in computer science. Ming Li, Paul Vitányi, SpringerNew York3rd editionMing Li and Paul Vitányi. An Introduction to Kolmogorov Complexity and Its Applications. texts in computer science. Springer, New York, 3rd edition, 2008. A note on inductive generalization. Gordon D Plotkin, Machine Intelligence. 5Gordon D. Plotkin. A note on inductive generalization. Machine Intelligence, 5:153-163, 1970. A further note on inductive generalization. Gordon D Plotkin, Machine Intelligence. 6Gordon D. Plotkin. A further note on inductive generalization. Machine Intelligence, 6:101- 124, 1971. Transformational systems and the algebraic structure of atomic formulas. John C Reynolds, Machine Intelligence. 5John C. Reynolds. Transformational systems and the algebraic structure of atomic formulas. Machine Intelligence, 5:135-151, 1970.

We develop a criterion for a point of global function field to be a unique wild point of some self-equivalence of this field. We show that this happens if and only if the class of the point in the Picard group of the field is 2-divisible. Moreover, given a finite set of points, whose classes are 2-divisible in the Picard group, we show that there is always a self-equivalence of the field for which this is precisely the set of wild points. Unfortunately, for more than one point this condition is no longer a necessary one.

10.4064/cm6979-1-2018

1501.06168

580ef59635d65dd41d09b2597c0d5401ea4102cc

WILD AND EVEN POINTS IN GLOBAL FUNCTION FIELDS 25 Jan 2015 Alfred Czogała ANDPrzemysław Koprowski Beata Rothkegel WILD AND EVEN POINTS IN GLOBAL FUNCTION FIELDS 25 Jan 2015 We develop a criterion for a point of global function field to be a unique wild point of some self-equivalence of this field. We show that this happens if and only if the class of the point in the Picard group of the field is 2-divisible. Moreover, given a finite set of points, whose classes are 2-divisible in the Picard group, we show that there is always a self-equivalence of the field for which this is precisely the set of wild points. Unfortunately, for more than one point this condition is no longer a necessary one. Introduction and related works Hilbert-symbol equivalence (formerly known under the name reciprocity equivalence) appeared for the first time in early 90's in papers by J. Carpenter, P.E. Conner, R. Litherland, R. Perlis, K. Szymiczek and the first author of this paper (see e.g. [PSCL94]). It was originally introduced as a tool for investigating Witt equivalence of global fields (two fields are said to be Witt equivalent, when their Witt rings of similarity classes of non-degenerate quadratic forms are isomorphic-roughly speaking, Witt equivalent fields admit "equivalent" classes of orthogonal geometries). Later, however, Hibert-symbol equivalence developed into a research subject by itself. It was generalized to higher-degree symbols (see e.g. [CS97], [CS98]), to quaternion-symbol equivalence of real function fields (see e.g. [Kop02]), as well to a ring setting (see e.g. [RC07]). One of the subjects considered in this theory is the problem of describing self-equivalences of a given field. Let K be a global field of characteristic = 2 and let X denotes the set of all primes of K (i.e. classes of non-trivial places on K). The self-equivalence of K is a pair of maps (T, t), consisting of a bijection T : X ∼ − → X and an automorphism t :K/K 2 ∼ − →K/K 2 of the square-class group of K, satisfying the condition: (λ, µ) p = (tλ, tµ) T p , for all p ∈ X and λ, µ ∈K/K 2 . Here, (·, ·) p denotes the Hilbert symbolKp/K 2 p ×Kp/K 2 p → {±1}. Every selfequivalence of a global field induces an automorphism of its Witt ring. Given a self-equivalence of a global field K, a prime p of K is called tame if ord p λ ≡ ord T p tλ (mod 2) for all λ ∈ K. Otherwise p is called wild. Few years ago, M. Somodi gave a full characterization all finite sets of wild primes in Q (see [Som06]) and in Q(i) (see [Som08]). His results were recently generalized to a broad class of number fields by two of the authors of this article (for details see [CR14]). In this paper, we consider the same question for function fields. Hence from now on, K is a global function field of characteristic = 2 and k is the full field of constants of K. We may think of K as of a field of rational functions on some smooth, irreducible complete curve X. The closed points of X are identified with non-trivial places of K. We shall never explicitly refer to the generic point of X. Thus, in what follows, we use the word "point" meaning actually "closed point". We denote the set of closed points again by X. We show (cf. Theorem 4.7) that a point p ∈ X is a unique wild point for some self-equivalence of K if and only if its class in the Picard group of K is 2-divisible (i.e. belongs to the subgroup 2 Pic K). On implication of this theorem, still holds even when we increase the number of points, this way we obtain a complete counterpart (Theorem 4.8) for function fields of the results from [Som06,Som08,CR14]. These two results establish a direct link between the property of being wild (for some-self equivalence) and 2-divisibility in the Picard group of K. For this reason, we develop in Section 3 some criteria for the class of a point p ∈ X to be 2-divisible in the group Pic K. In particular, we show (cf. Theorem 3.6), that a point of a hyperelliptic curve (of an odd degree) is 2-divisible in Pic K (hence is a unique wild point of some self-equivalence) if and only if its norm over the rational function field is represented by the norm of the field extension K /k(x). This in turn implies that for such curves, wild points always exist (see Proposition 3.10). In the paper we use the following notation (beside the one already explained above). Given a function field K and a point p ∈ X, by O p we denote the associated valuation ring, by K p the completion of K and by K(p) the residue field. Given a non-empty, open subset Y ⊆ X, we write O Y := p∈Y O p and E Y := λ ∈K | ∀ p∈Y ord p λ ≡ 0 (mod 2) This set is a union of cosets ofK 2 and we denote its image in the square-class group of K by E Y := E Y /K 2 . Further, when Y is a proper subset, we consider the subset of E Y consisting of all those function that are local squares everywhere outside Y , namely: ∆ Y := E Y ∩ p / ∈YK 2 p = E X ∩ p / ∈YK 2 p . This set again contains full square classes of K and so we write ∆ Y := ∆ Y /K 2 . In the special case, when Y is of the form X \ {p}, i.e. is obtained from X by excluding just a single point, we abbreviate the notation writing E p , E p , ∆ p and ∆ p for E X\{p} , E X\{p} , ∆ X\{p} and ∆ X\{p} , respectively. The square-class groupk/k 2 has order two. We write ζ ∈ k ⊂ K for a fixed generator of this group, with the convention that ζ = −1, whenever −1 is not a square in K (i.e. when card(k) ≡ 3 (mod 4)). Abusing the notation slightly, we tend to use the same symbols λ, µ, . . . to denote elements of the field as well as their classes in the square-class group of this field. Likewise, the fraktur letters p, q, . . . denote, depending on the context, either points of K or their classes in the Picard group Pic K or Pic O Y . Divisors, as well as their classes in the Picard group, are always written additively. Preliminaries Recall that if K p is a local field, then the square class group of K p consists of four elements: 1, u p , π p and u p π p , where π p is the class of a uniformizer and u p is the class of a unit which is not a square (see. e.g. [Lam05, Theorem Vi.2.2]). We call u p the p-primary unit. If (T, t) is a self-equivalence of K, then t factors over all the local square-class groups by [PSCL94,Lemma 4]. In particular it maps 1 ∈Kp/K 2 p onto 1 ∈KTp/K 2 T p . If it also maps u p to u T p , then it is necessarily tame by the pigeonhole principle. Thus we proved: Observation 2.1. The self-equivalence (T, t) is wild at a point p ∈ X if and only if ord T p tu p ≡ 1 (mod 2). The primary unit u p may also be characterized using Hilbert symbols as follows: (u p , λ) p = (−1) ord p λ for every λ ∈K p . The Hilbert symbol (·, ·) p can be viewed as a non-degenerate F 2 -inner product oṅ K p/K 2 p , provided the additive group F 2 is identified with the multiplicative group {±1}. The following observation is now immediate: Observation 2.2. Let p, q ∈ X be two points of K such that −1 ∈ K 2 p ∩ K 2 q , then the isomorphism τ :Kp/K 2 p →Kq/K 2 q defined by the formula: τ (u p ) = u q π q , τ (π p ) = π q is an isometry of the inner product spaces K p/K 2 p , (·, ·) p and K q/K 2 q , (·, ·) q . Below we gather some results concerning 2-ranks of the class groups: either the Picard group Pic K of the complete curve X or the Picard group Pic O Y for some fixed open subset ∅ = Y X. Recall that the latter group can be identified with the ideal class group Cl O Y of the coordinate ring O Y of Y , as O Y is a Dedekind domain. We begin with a proposition that is not new. The first assertion was proved in [Czo01,p. 607] and the second in [Czo01, Lemma 2.1]. The third assertion is a simple consequence of the previous two. We state the result explicitly only to simplify further references. Proposition 2.3. Let ∅ = Y X be a proper open subset of X, then (1) rk 2 E Y = rk 2 Pic O Y + card(X \ Y ); (2) rk 2 ∆ Y = rk 2 Pic O Y ; (3) rk 2 E Y /∆ Y = card(X \ Y ); An identity similar to (1) above can be also proved for a complete curve. Lemma 2.4. rk 2 E X = 1 + rk 2 Pic 0 K. Proof. Let H be the subgroup of Pic 0 K consisting of elements of order 2. The map E X → H, λ → 1 2 div K λ = p∈X 1 2 ord p λ · p is a surjective homomorphism with the kernelk·K 2 . Thus, rk 2 E X /kK 2 = rk 2 Pic 0 K. The groupskK 2 /K 2 andk/k 2 are isomorphic and the 2-rank ofk/k 2 equals one. This proves the lemma. Now, we consider a case, when we have two open subsets Z ⊂ Y ⊂ X. Lemma 2.5. If ∅ = Z ⊂ Y X are two proper open subsets of X, then (1) rk 2 Pic O Z = rk 2 Pic O Y − rk 2 {p + 2 Pic O Y | p ∈ Y \ Z} ; (2) rk 2 E Z = rk 2 Pic O Y − rk 2 {p + 2 Pic O Y | p ∈ Y \ Z} + card(X \ Z). Proof. Since Z ⊂ Y , hence O Z ⊃ O Y and by functoriality, there is a natural morphism Pic O Y → Pic O Z . It is clearly an epimorphism, since a class of a divisor p∈Z n p p is the image of the class of any divisor of a form p∈Z n p p + q∈Y \Z n q q . This epimorphism induces an epimorphism of the quotient groups Pic O Y /2 Pic O Y ։ Pic O Z /2 Pic O Z , whose kernel is generated by the set {p+2 Pic O Y | p ∈ Y \ Z}. It proves the first assertion of the lemma and the second follows immediately from the first one and Proposition 2.3. It is natural to compare the 2-rank of Pic 0 K with the 2-rank of the class group Pic O Y of a proper open subset Y X. Below we formulate two relevant results for the case when Y = X \ {p}. Lemma 2.6. Let ζ ∈ k be a fixed generator of the square class groupk/k 2 of the full field of constants of K. If p ∈ X is a point of an odd degree, then (1) E X = E p = ζ ⊕ ∆ p ; (2) rk 2 Pic 0 K = rk 2 Pic O p . Proof. Take an arbitrary element λ ∈ E p . The degree of the principal divisor div K λ is 0. Thus, we have ord p λ · deg p = − q =p ord q λ · deg q. Now, ord q λ is even for every q = p, since λ ∈ E p . On the other hand deg p is odd by the assumption. It follows that ord p λ is even, too. Hence λ ∈ E X and so we proved that E p ⊆ E X . The other inclusion is trivial and the equality E p = ζ ⊕∆ p follows from Proposition 2.3 and the fact that ζ is a local square at a given point if and only if this point has an odd degree. This proves (1). Now, the other assertion follows immediately from Lemma 2.4 and Proposition 2.3.(1). Proposition 2.7. If p ∈ X is any point, then rk 2 Pic O p = rk 2 Pic 0 K, if p / ∈ 2 Pic K 1 + rk 2 Pic 0 K, if p ∈ 2 Pic K. The proof of this proposition will be postponed till the next section. 2-divisibility of classes of prime divisors This section is fully devoted to the following problem: let p ∈ X be a point, when does the class of p in Pic K is divisible by 2 (i.e. lies in 2 Pic K)? Points having this property will be subsequently called 2-divisible or shorty, albeit less formally, even. The results of this section, not only have direct applications in the rest of this paper, but (at least some of them) are somehow interesting by themselves. Let us begin with the following basic observation. Observation 3.1. If p ∈ X is an even point, then its degree deg p is an even integer. The assertion of this observation follows immediately from the fact (see e.g. [Lor96, Corollary VII.7.10]), that the epimorphism deg : Div K ։ Z factors over the subgroup of principal divisors, inducing a well defined group epimorphism deg : Pic K ։ Z. It is well known (see e.g. [Lor96, Proposition VII.7.12]), that for a field of rational functions this map is actually an isomorphism. Hence, in such a field, even points are precisely the points of even degrees. Of course, it is not so in general. For example, if K is the function field of an elliptic curve over F 3 given in Weierstrass normal form by the polynomial y 2 − x 3 + x, then there are exactly 6 points of degree 2 and 12 points of degree 4 in K but none (!) of them is 2-divisible in Pic K (verified using computer algebra system 1 Magma [BCP97]). Thus, we are forced to search for some other criteria of 2-divisibility. Proposition 3.2. A point p ∈ X is 2-divisible in Pic K if and only if there exists an element λ ∈ E p such that ord p λ ≡ 1 (mod 2). Proof. Assume that p is an even point, this means that p + div K λ = q∈X 2n q · q for some n q ∈ Z almost all equal 0 and some element λ ∈ K. It is clear that λ satisfies the assertion. Conversely, assume the existence of λ ∈ E p of an odd order at p, say ord p λ = 2k + 1. Write the divisor of λ div K λ = (2k + 1)p + q∈X q =p 2n q q, for some k ∈ Z and n q ∈ Z almost all equal 0. Therefore the following equality holds in the Picard group of K p = 2 kp + q∈X q =p n q q − div K λ. In particular p ∈ 2 Pic K, as claimed We are now ready to prove Proposition 2.7. Proof of Proposition 2.7. Let d := deg p be the degree of p. It follows from [Ros02,Proposition 14.1] that the following sequence is exact 0 → Pic 0 K → Pic O p → Z d → 0. Therefore the following sequence is exact, as well: 0 → Pic 0 K/2 Pic 0 K → Pic O p /2 Pic O p → Z d /2Z d → 0. Consequently (1) rk 2 Pic O p ≤ 1 + rk 2 Pic 0 K. Lemma 2.4 asserts that rk 2 E X = 1 + rk 2 Pic 0 K, while Proposition 2.3 states that rk 2 Pic O p = rk 2 E p − 1. Clearly E X ⊆ E p . If p / ∈ 2 Pic K, then E X = E p by Proposition 3.2, hence rk 2 Pic O p = rk 2 Pic 0 K. On the other hand, if p ∈ 2 Pic K, then E X E p , again by Proposition 3.2. Thus rk 2 Pic O p > rk 2 Pic 0 K and the assertion follows from Eq. (1). One immediate consequence of Proposition 2.7 is the following criterion for 2divisibility. Proposition 3.3. Let p ∈ X be any point, then p is 2-divisible in Pic K if and only if every function having even order everywhere on X is a local square at p (i.e. if E X = ∆ p ). Proof. Think of ∆ p as of a subspace of a F 2 -linear space E X . Lemma 2.4 asserts that rk 2 E X = 1 + rk 2 Pic 0 K, while rk 2 ∆ p = rk 2 Pic O p by Proposition 2.3. Now, it follows from Proposition 2.7 that rk 2 Pic O p = 1 + rk 2 Pic 0 K = rk 2 E X if and only if p ∈ 2 Pic K. Consequently, dim F2 ∆ p = dim F2 E X , and so ∆ p is the full space E X , if and only if p is even. Proof. By the assumption there exists an element λ ∈K such that div OY λ = p+2D for some O Y -divisor D ∈ Div O Y . Fix an element µ ∈ ∆ Y . Then, for every q ∈ X \ Y , the element µ is a local square at q, hence the quaternion algebra λ,µ Kq splits. On the other hand, when q ∈ Y \ {p}, then both µ and λ are q-adic units moduloK 2 q and so again λ,µ Kq splits. Consequently, all quaternion algebras λ,µ Kq split for q ∈ X, except possibly p. It follows from Hilbert's reciprocity formula, that in such a case also λ,µ Kp splits. But µ is arbitrary, which implies that λ must be a local square at p. Conversely, let Z = Y \ {p}. Since µ ∈K 2 p for every µ ∈ ∆ Y by the assumption, hence ∆ Y = ∆ Z and it follows from Proposition 2.3.(2) that rk 2 Pic O Y = rk 2 Pic O Z . Consequently p ∈ 2 Pic O Y , by the means of Lemma 2.5. Finally, we present a proposition connecting 2-divisibility in the Picard group of a complete curve with 2-divisibility over its open subset. Proposition 3.5. Let p, q be two points of X with deg p even and deg q odd, then p ∈ 2 Pic K ⇐⇒ p ∈ 2 Pic O X\{q} . Proof. Let Y := X \ {q}. If p is 2-divisible in Pic K, then p = div K λ + 2D for some λ ∈ K and D ∈ Div K. Drop any occurrences of q in D and the principal divisor div K λ, to get O Y -divisors D ′ and div OY λ. Therefore, over O Y , we have p = div OY λ + 2D ′ ∈ Div O Y and so p ∈ 2 Pic O Y . Conversely, assume that p ∈ 2 Pic O Y , this means that there are: an element λ ∈ K and O Y -divisor D ∈ Div O Y such that div OY λ = p + 2D ∈ Div O Y . Passing from Y to the complete curve X, write div K λ = p + 2D + ord q λ · q. Compute the degrees of both sides to get 0 = deg p + 2 deg D + ord q λ · deg q. We assumed that deg q is odd, while deg p is even, hence ord q λ must be even, too. Say ord q λ = 2k for some k ∈ Z. Thus, div K λ = p + 2(D + kq), which means that p is even, as desired. All the above results are of rather general nature and are valid for any global function field. It should not come as a big surprise, that, if we concentrate on function fields of a special type, more can be proved. Recall that a smooth curve X, whose affine part X aff is defined by the polynomial y 2 − f (x), is called hyperelliptic when deg f ≥ 4, elliptic when deg f = 3 and conic when deg f ≤ 2. In what follows, we will deal with elliptic and hyperelliptic curves in a uniform fashion, therefore, stretching the term a little, we shall call all curves of this form "hyperelliptic", treating elliptic curves as special case of hyperelliptic ones. We warn the reader, however, that this is not a standard terminology. Let K /L be an extension of function fields and π : X ։ Y be the corresponding morphism of their associated (smooth) curves. Recall (cf. [Lor96, Ch. VII, §7]) that a norm is a function Norm K/L : Div K → Div L given by the formula (2) Norm K/L i a i p i := i a i f p i /π(p i ) π(p i ), where f p /π(p) is the inertia degree of p over π(p). If Y aff is an affine part of Y , O L = k[Y aff ] is the ring of functions regular on Y aff and O K = int. cl. K O L is the integral closure of O L in K, then Norm K/L Div OK restricted to Div O K is a morphism Norm K/L : Div O K → Div O L . If additionally L = k(x) is a field of rational functions, then to every point p of Y = P 1 k one may unambiguously assign either a monic polynomial p ∈ k[x] with a single zero at p and no other zeros or a function 1 /x, when p is the point at infinity. This constitutes a morphism Div L →L from the group of divisors to the multiplicative group of the field L. Composing it over Norm K/L , we arrive at the map norm K/L : Div K →L, which (harmlessly abusing the notation) we shall again call a norm. In what follows, we shall prefer norm K/L over Norm K/L since the former allows us to compare the norm of a divisor with values of the standard norm of the field extension norm K/L :K →L. Theorem 3.6. If K is a function field of a smooth hyperelliptic curve X of an odd degree and p ∈ X is a point of an even degree. Then p is 2-divisible in Pic K if and only if norm K/L p is representable by the norm K/L :K →L, where L is a field of rational functions. In other words p ∈ 2 Pic K ⇐⇒ ∃ λ∈K norm K/L p = norm K/L λ. The proof of this theorem will be divided into three lemmas. In Lemmas 3.7-3.9, K = qf k[x, y] /y 2 − f (x) is always a function field of a hyperelliptic curve X with its affine part defined by the polynomial y 2 − f (x), further L = k(x) is a field of rational functions in x and O K = int. cl. k[x]. We denote by : K → K the unique non-trivial L-automorphism of K. The ring O K is a Dedekind domain, hence its Picard group can be identified with its ideal class group Cl O K . The first lemma is basically a recap of [BS66, Theorem III.8.7]. Unfortunately in [BS66] it is proved only for number fields, hence for the sake of completeness we feel obliged to explicitly state and prove its function field counterpart. D = m i=1 a i p i + b i p i + n j=1 c j q j , where the points q j = q j are fixed under the action of and p i = p i are not. Then, norm K/L p i = norm K/L p i = p i and norm K/L q j = q fj j for some monic polynomials We are now in a position to prove the first implication of Theorem 3.6. p i , q j ∈ k[x], f j ∈ {1, 2}, i ≤ m, j ≤ n. Therefore 1 = norm K/L D = m i=1 p ai+bi i · n j=1 q cj j . Lemma 3.8. If deg p ∈ 2Z and norm K/L p ∈ norm K/LK , then p is even. Proof. By the assumption of the theorem, the degree of X is odd and it follows from [Lor96, Lemma V.10.15] that X has a unique point at infinity (denote it ∞ K ) and this point is ramified. In particular, deg ∞ K = 1 / ∈ 2Z and so p and ∞ K are distinct. If the inertia degree of p (in the field extension K /L) equals 2, then norm K/L p = p 2 , for some monic polynomial p ∈ k[x]. This means that div K p = p − 2∞ K . Therefore p = div K p + 2∞ K ∈ 2 Pic K. From now on, we assume that p = ∞ K and the inertia degree of p equals 1. Hence, norm K/L p = p and by the assumption there exists such an element λ ∈ K, that p = norm K/L λ = λλ. Take a divisor D := p − div OK λ ∈ Div O K . Clearly norm K/L D = norm K/L p norm K/L λ = 1 and so the previous lemma asserts that D ∈ 2 Pic O K . Since ∞ K is the unique point at infinity and deg ∞ K = 1, therefore [Lor96, Proposition VIII.9.2] implies that the group Pic O K is isomorphic to Pic 0 K. Hence, passing with D to Pic K, we have p − div K λ + 2k∞ K ∈ 2 Pic K for some k ∈ Z. In particular p ∈ 2 Pic K, as desired. We are now ready to prove the opposite implication of Theorem 3.6. Lemma 3.9. The norm norm K/L p of every even point lies in norm K/LK . Proof. Take a point p ∈ X and assume that it is 2-divisible in Pic K. Thus, there is a divisor D ∈ Div K and an element λ ∈ K such that p = 2D + div K λ. Compute the norms of both sides to get norm K/L p = norm K/L (2D + div K λ) = (norm K/L D) 2 · norm K/L λ. If λ = a + by for some a, b ∈ L, then norm K/L λ = a 2 − b 2 f , therefore norm K/L p = (ac) 2 − (bc) 2 f, where c = norm K/L D ∈ L. In particular norm K/L p ∈ norm K/LK . The proof of Theorem 3.6 is now complete. Remark 1. One should note that the condition deg f / ∈ 2Z occurs only in the proof of Lemma 3.8. Therefore, the implication p ∈ 2 Pic K =⇒ norm K/L p ∈ norm K/LK of the theorem holds even without this assumption. Nevertheless, for the other implication this condition is indispensable. Indeed, take K = qf F 5 [x, y] /y 2 − x 4 + x + 1 . Using computer algebra system Magma one checks that there are a total of 8 points of K of degree 2 that are not 2-divisible in Pic K, but their norms lie in norm K/LK . Remark 2. The assumption, that deg p is even, is also essential. Take a field K = qf F 13 [x, y] /y 2 + 12x 3 + x 2 + 3x + 10 . As it was mentioned in the proof of Lemma 3.8, the field K has the unique point at infinity ∞ K and deg ∞ K = 1. On the other hand, norm K/L ∞ K = 1 /x ∈ norm K/LK . Again this example was checked using Magma. The criterion in the above theorem, let us show that even points do exist. Proposition 3.10. Let K be a function field of a (smooth) hyperelliptic curve given by a polynomial y 2 − f (x). If f ∈ k[x] is monic of an odd degree, then there are infinitely many points of K that are 2-divisible in Pic K. Proof. As observed in the proof of Lemma 3.8, K has the unique point at infinity (denote it ∞ K ). This point is ramified and the Picard group Pic O K of O K = int. cl. K k[x] is isomorphic to Pic 0 K. Let f = f 1 · · · f n be the decomposition of f into irreducible monic factors. Fix a non-zero integer M ∈ N and take an irreducible polynomial q ∈ k[x] of an even degree strictly greater than M and prime to char k. Let k 0 := k(α 0 ), where α 0 is a root of q. Clearly, k 0 = k since the degree of q 0 is even and greater than M = 0. Denote λ 1 := f 1 (α), . . . , λ i := f n (α) and take a field k 1 := k 0 √ λ 1 , . . . , √ λ n . Further, let β be a primitive element of the extension k 1 /k and p ∈ k[x] be its minimal polynomial. Take p ∈ X to be a point of K dominating p. Clearly the degree of p is even and we have (3) f 1 p = · · · = f n p = 1. If the inertia degree of p equals 2, then p = div OK p in Div O K , hence p = 0 in Pic O K ∼ = Pic 0 K. It follows that the class of p in the Picard group Pic K ∼ = Pic 0 K ⊕ Z can be written as (0, deg p) and so clearly belong to 2 Pic K. Thus, assume that the inertia degree f ( p /p) of p is one. We claim that norm K/L p ∈ norm K/L K or in other words, that p = norm K/L p is represented over L by the quadratic form 1, −f . This is equivalent to saying that the form ϕ := 1, −f, −p is isotropic over L. By the local-global principle, it suffices to show that the form is locally isotropic in every completion of L. First, take the completion at infinity L ∞ . By the assumption, − ord ∞ f = deg f / ∈ 2Z, while − ord ∞ p = deg p ∈ 2Z. Decompose the form ϕ ⊗ L ∞ into the sum 1, −p ⊗L ∞ with coefficients of even order and −f ⊗L ∞ of odd order. A well known consequence of Springer's theorem (see e.g. [Lam05, Proposition VI.1.9]) asserts that ϕ⊗ L ∞ is isotropic if and only if the residue form of 1, −p is isotropic. But the latter is just 1, −1 , hence trivially isotropic, since p is monic. Take now a completion L s of L at the place associated to some irreducible polynomial s different from p and not dividing f . Using [Lam05, Proposition VI.1.9], we see that ϕ ⊗ L s is again isotropic, because its residue form has dimension three (over a finite field) and therefore is isotropic. Next, consider the completion L p of L at the place associated to p. We know, that all f i 's are squares modulo p and so is f itself. Consequently 1, −f ⊗ L p is isotropic, hence ϕ ⊗ L p is isotropic, too. Finally, take the f i -adic completion L fi for some monic irreducible factor f i of f . We have ( fi p ) = 1 by (3) and Dedekind's quadratic reciprocity law says that p f i · f i p = (−1) (card(k) − 1)(deg f · deg p) /2 , but deg p is even and it follows that ( p fi ) = 1, too. Thus, ϕ ⊗ L fi is again isotropic. All in all, ϕ is isotropic over L, which proves our claim. Theorem 3.6 asserts now that p is even. It is immediate that taking subsequently M := deg p and repeating the above construction, we ultimately produce an infinite sequence of 2-divisible points in K. It seems interesting to investigate, how classes of prime divisors of K are distributed among the cosets of 2 Pic K in the Picard group of K. If we fix a degree, not all the cosets of 2 Pic K must contain classes of points having this degreee. It s trivial for an odd degree, since already the subgroup 2 Pic K itself cannot contain a point of an odd degree by Observation 3.1. The example in the paragraph following Observation 3.1 shows that this may happen even for some fixed even degrees. We do not know, how general this phenomenon is, but computer-verified examples make us believe that this happens only for low enough degrees and justify the following conjecture: Conjecture. If K is a function field of a hyperelliptic curve X of an odd degree, then there is such an integer N 0 ∈ N that for every D ∈ Pic 0 K /2 Pic 0 K and every n > N 0 , there exists a point p ∈ X with deg p = n and such that the class of p in Pic K /2 Pic K = Pic 0 K /2 Pic 0 K ⊕ Z 2 is (D, deg p mod 2). In a nutshell, the above conjecture says, that from certain degree onward every coset contains a class of a point of a given degree, with the only exception forced by the necessary condition of Observation 3.1. Main results In this section, we prove our two main results, namely: Theorem 4.7 showing that a point is even if and only it is a unique wild point for some self-equivalence and its partial generalization Theorem 4.8. First, however, we need the following lemma, generalizing Proposition 3.4. ≤ i ≤ n λ i / ∈ K 2 pi and λ i ∈ j =i K 2 pj . Proof. We proceed by an induction on the number of points. For n = 1 the assertion follows from Proposition 3.4. Suppose that n > 1 and the assertion holds true for n − 1. Classes of p 1 , . . . , p n are linearly independent in Pic O Y /2 Pic O Y and so, in particular, p 1 is not be 2-divisible in Pic O Y . Proposition 3.4 asserts that there exists µ ∈ ∆ Y such that µ / ∈ K 2 p1 . Take a subset Z := Y \ {p 1 } of Y . By the means of Lemma 2.5, we have rk 2 Pic O Z = rk 2 Pic O Y − 1. Clearly, ∆ Z ⊂ ∆ Y with µ ∈ ∆ Y \ ∆ Z . Moreover, p 2 , . . . , p n remains linearly independent in Pic O Z /2 Pic O Z . It follows from the inductive hypothesis, that there are elements λ 2 , . . . , λ n ∈ ∆ Z linearly independent in ∆ Z and such that for every 2 ≤ i ≤ n λ i / ∈ K 2 pi and λ i ∈ j =i j≥2 K 2 pj . By the very definition of ∆ Z , all λ i 's for i ≥ 2 lie in K 2 p1 . Let λ 1 := µ · i>1 λ εi i , where ε i = 0, if µ ∈ K 2 pi 1, if µ / ∈ K 2 pi . It is now immediate, that λ 1 ∈ j =1 K 2 pj while at the same time λ 1 / ∈ K 2 p1 . This proves one implication. The other one follows from [Czo01, Lemma 2.1]. Lemma 4.2. Let p ∈ 2 Pic K be an even point, then for any other even point q ∈ 2 Pic K, the set E p \E X is fully contained in a square-class of the completion K q . Proof. Since ∆ p = E X by Proposition 3.3, thus E X is a subgroup of E p of index (E p : E X ) = 2 by Proposition 2.3. Take any λ, µ ∈ E p \ E X , then λ · E X = µ · E X and so λ · µ ∈ E X = ∆ q ⊂K 2 q . ord p u p ≡ 0 (mod 2) and ord p π p ≡ 1 (mod 2). For two square classes λ, µ ∈Kp/K 2 p , λ, µ = 1, the Hilbert symbol can be computed with the formula (λ, µ) p = 1 ⇐⇒ λ = µ. Therefore, every bijection of the local square-class groups mapping squares to squares is an isomorphism and preserves the Hilbert symbols. Consequently, the condition (SE3) is always satisfied for this type of fields. Proposition 4.5. Let K be a global function field and X an associated smooth curve. Let p, p 1 , . . . , p l be 2-divisible points such that p i ⌣ p j for every i = j Then, there is a self-equivalence (T, t) of K such that: • p is the unique wild point of (T, t), i.e. W(T, t) = {p}; • T preserves the selected points in the sense that T p = p and T p i = p i for i = 1, . . . , l; • for every p i ⌣ p, the isomorphism t restricted to the local square-class grouṗ K pi/K 2 pi is an identity; • for every p i ⌣ p, the isomorphism t restricted to the local square-class grouṗ rk 2 ∆ Y = rk 2 Pic O Y = = rk 2 Pic O p − rk 2 p 1 + 2 Pic O p , . . . , p l + 2 Pic O p = = rk 2 Pic O p = rk 2 ∆ p , where the first and the last equalities follows from Proposition 2.3, the second follows from Lemma 2.5, while the third one is due to the fact that every p i is 2-divisible in Pic K so consequently also in Pic O p . Therefore, the F 2 -linear spaces ∆ p and ∆ Y are equal, but the former one is just E X by Proposition 3.3. All in all, ∆ Y = E X . Take a basis q 1 , . . . , q m of Pic O Y /2 Pic O Y . Lemma 4.1 asserts that there are elements µ 1 , . . . , µ m ∈ ∆ Y linearly independent in ∆ Y and such that µ i ∈K 2 qj if and only if i = j. Clearly, they form a basis of ∆ Y = E X . Now, rk 2 E p/E X = 1 by Proposition 3.3 and Proposition 2.3. Likewise, rk 2 E pi /E X = 1 for every i = 1, . . . , l. Therefore, there are square-classes λ ∈ E p \ E X , λ 1 ∈ E p1 \ E X , . . . , λ l ∈ E p l \ E X . By the assumption p i ⌣ p j for every 1 ≤ i = j ≤ l, thus every λ i is a local square at every p j for j = i. Multiplying, if necessary by appropriate µ j 's we may assume without loss of generality, that λ, λ 1 , . . . , λ l are local squares at q j for every j = 1, . . . , m. Denote S := {p; p 1 , . . . , p l ; q 1 , . . . , q m } and let Z := X \ S ⊂ Y . It follows from Lemma 2.5 that rk 2 Pic O Z = 0 and so S is a sufficiently large set. We claim that the set B := {λ; λ 1 , . . . , λ l ; µ 1 , . . . , µ m } forms a basis of the F 2 -linear space E Z . First, we show that it is linearly independent. Suppose, a contrario, that it is not. Thus ν := λ a · l i=1 λ bi i · m j=1 µ cj j is a square in K, for some a, b 1 , . . . , b l , c 1 , . . . , c m ∈ F 2 . This means that 0 ≡ ord p ν ≡ a (mod 2), since all the other elements have even order at p, consequently a = 0. Similarly, for every 1 ≤ i ≤ l, 0 ≡ ord pi ν ≡ b i (mod 2) so also b 1 = · · · = b l = 0. Finally, c 1 = · · · = c m = 0, because µ 1 , . . . , µ m are linearly independent in ∆ Y , a subspace of E X . Further, Proposition 2.3 asserts that dim F2 E Z = rk 2 Pic O Z + card(S) = card(B), proving that B is a basis of E Z . Observe that, if p is related to every point p i , i = 1, . . . , l, then a p-primary unit u does not belong to E Z . On the other hand, if p ⌣ p i for some i ∈ {1, . . . , l}, then the obtained above element λ i is a p-primary unit (and symmetrically λ is a p i -primary unit). Construct a triple T S , t S , (t r | r ∈ S) in the following way: • let T S : S → S be the identity; • define the automorphism t S : E Z → E Z fixing its values on the basis B: -t S (λ) := λ; -t S (λ i ) := λ i , if p ⌣ p i , λλ i , if p ⌣ p i ; -t S (µ j ) := µ j for j = 1, . . . , m. • finally, the automorphisms of the local square-class groups are given as follows: t p is the transposition (u, uλ) onKp/K 2 p = {1, u, λ, uλ} (recall that u = λ i (modK 2 p ) whenever p ⌣ p i ); -for a point p i related to p, take t pi to be an identity ofKp i /K 2 pi ; -for a point p i not related to p, let t pi be a "tame transposition" (λ i , λλ i ) on the groupKp i /K 2 pi = {1, λ, λ i , λλ i }; -eventually, for the remaining points q 1 , . . . , q m , let t qj be the identity on the corresponding square class group. The commutativity of the diagram (4) is now immediate. It follows that the triple T S , t S , (t r | r ∈ S) is a small equivalence and Theorem 4.4 asserts that it can be extended to a self-equivalence (T, t) of K tame on Z. Since, only t p is wild, thus p is the unique wild point of (T, t). Lemma 4.6. Let K be a global function field and X an associated smooth curve and (T, t) be a self-equivalence of K. If (T, t) has a unique wild point p, then p ∈ 2 Pic K. Proof. By the assumption W(T, t) = {p}. Denote q := T p. Suppose, a contrario, that p is not 2-divisible. Thus, Proposition 3.2 asserts that every element of E p has an even order at p, in particular E p = E X . Now, it follows from Proposition 2.3(3), that there is an element λ ∈ K such that E X = E p = λ ⊕ ∆ p . Clearly, ord p λ ≡ 0 (mod 2) and λ is not a local square at p, that is λ is a p-primary unit. Now, p is a wild point of (T, t), hence ord q tλ ≡ 1 (mod 2) by Observation 2.1. It follows from Proposition 3.2 that q is an even point of K. It is straightforward to show that tE p = E T p = E q . In particular, the 2-ranks must agree: rk 2 E p = rk 2 E q . Use Proposition 2.3 to express these 2-ranks as: rk 2 Pic O p + 1 = rk 2 Pic O q + 1. Now, q is 2-divisible in Pic K, while p is not. Proposition 2.7 asserts that the lefthand side equals rk 2 Pic 0 K + 1, while the right-hand side is rk 2 Pic 0 K + 2. This is clearly a contradiction. Combining now Proposition 4.5 with the above lemma, we arrive at our first main result. Theorem 4.7. Let K be a global function field and X an associated smooth curve. Given a point p ∈ X, the following two conditions are equivalent: • p is 2-divisible in Pic K; • p is the unique wild point of some self-equivalence of K. Looking at Proposition 4.5 it is obvious that if we have a set of even points and each of them is related to every other, then we can we can build a number of self-equivalences, each wild at precisely one of these points and preserving the rest. Then the wild set of the composition of all these self-equivalences consists of all our (related) even points. It turns out that this is still true, even when not all the points are related. Theorem 4.8 below, not only generalizes one implication of Theorem 4.7, but also constitutes a direct counterpart of [CR14, Theorem 1.1] for the case of global function fields. Theorem 4.8. Let K be a global function field and X be its associated smooth curve. Given a finitely many points p 1 , . . . , p n ∈ X, that are 2-divisible in Pic K, there is a self-equivalence (T, t) of K such that p 1 , . . . , p n are precisely its wild points, i.e. W(T, t) = {p 1 , . . . , p n }. Proof. We proceed by an induction on the number n of the points. The case n = 1 simply boils down to the assertion of Theorem 4.7. Hence, suppose that the thesis holds for all sets of cardinality n − 1 and consider a set of n even points {p 1 , . . . , p n } ⊂ X. Since p 1 is even, thus Proposition 4.5 asserts that there exists a self-equivalence (T 1 , t 1 ) of K such that p 1 is the unique wild point of (T 1 , t 1 ) and T 1 p 1 = p 1 . Denote the images of the remaining points by q 2 := T 1 p 2 , . . . , q n := T 1 p n . We claim that the points q 2 , . . . , q n are all 2-divisible in Pic K. In order to prove the claim, observe first, that since p 1 is even, thus ∆ p1 = E X by Proposition 3.3. Moreover (T 1 , t 1 ) is tame on X \ {p 1 }, therefore t 1 E p1 = E T1p1 = E p1 . It follows that also t 1 ∆ p1 = t 1 (E p1 ∩K 2 p1 ) = E p1 ∩K 2 p1 = ∆ p1 , as every self-equivalence preserves local squares. Consequently we obtain: tE X = t 1 ∆ p1 = ∆ p1 = E X . Take now any point p i with i > 1 and write E X = t 1 E X = t 1 ∆ pi = t 1 E X ∩K 2 pi = t 1 E X ∩ t 1K 2 pi = E X ∩K 2 qi = ∆ qi . It follows from Proposition 3.3 that q i ∈ 2 Pic K, as claimed. By the inductive hypothesis, there exists a self-equivalence (T 2 , t 2 ) of K with the wild set W(T 2 , t 2 ) = {q 2 , . . . , q n }. The composition (T, t) = (T 2 • T 1 , t 2 • t 1 ) is now the desired self-equivalence of K with the wild set W(T, t) = {p 1 , . . . , p n }. Remark 5. The above theorem generalizes only one of the implications of Theorem 4.7 to sets having more than one point. This is all we can do, since the opposite implication no longer holds for larger sets. The simplest counterexample, we are aware of, is probably the following one: let K be the function field of an elliptic curve over F 5 given in Weierstrass normal form by the polynomial y 2 + x 3 + x + 2. Take two points: p ∼ (1, 1) and q ∼ (1, 4). Then neither of them is even, since both are rational. Nevertheless, there exists a self-equivalence of K that is wild precisely at these two points. We will discuss the structure of bigger wild set in another paper. 1 The source codes for Magma of all the counter examples are available from the second author's web page at http://z2.math.us.edu.pl/perry/papers. So far we have been considering 2-divisibility in the Picard group of the complete curve. The next proposition deals with 2-divisibility in Pic O Y (or equivalently in Cl O Y ), that is over some proper open subset Y of X. Proposition 3 . 4 . 34Let ∅ = Y X be a proper open subset and p ∈ Y . Then p is 2-divisible in Pic O Y if and only if ∆ Y ⊂K 2 p . Lemma 3. 7 . 7If the norm K/L D of a divisor D ∈ Div O K equals 1, then the class of D lies in 2 Pic O K . Proof. We closely follow the lines of the proof of [BS66, Theorem III.8.7]. Write the divisor D in the form Now, all the polynomials are irreducible and pairwise distinct and k[x] is a u.f.d., hence all the exponents must vanish. In particular c j = 0 for every j and a i = −b i for every i. p i − p i , but p i + p i = div OK p, hence p i = −p i in the class group Pic O K . All in all, we write the class of D asm i=1 2a i p i ∈ 2 Pic O K . Lemma 4 . 1 . 41Let ∅ = Y X be a proper open subset and p 1 , . . . , p n ∈ Y . Then p 1 , . . . , p n are linearly independent (over F 2 ) in Pic O Y /2 Pic O Y if and only if there are λ 1 , . . . , λ n ∈ ∆ Y linearly independent in ∆ Y and such that for every 1 is a transposition of the square-classes of odd orders. Proof. Take an open subset Y := X \ {p; p 1 , . . . , p l } of X and let m := rk 2 Pic O Y . Observe that We define a relation on the set of 2-divisible points, saying that p ∈ 2 Pic K is related to q ∈ 2 Pic K, when E p \ E X ⊂K 2 q . We write p ⌣ q, when p is related to q. Unfortunately this relation-although symmetric-is neither reflexive nor transitive (see Remark 3).Lemma 4.3. The relation ⌣ is symmetric.Proof. Take λ ∈ E p \ E X and µ ∈ E q \ E X . Assume that p is related to q, so that λ ∈K 2 q . Take any point r distinct from both p and q, then a local quaternion algebra λ,µ Kr splits, since ord r λ ≡ ord r µ ≡ 0 (mod 2). Next, also λ,µ Kq splits, because λ is a square in K q . It follows from Hilbert's reciprocity law that λ,µ Kp splits, as well. But ord p λ ≡ 1 (mod 2), hence µ must be a local square at p. Consequently E q \ E X is contained inK 2 p and so q is related to p. Remark 3. While it is obvious (and harmless) that ⌣ is not reflexive, but it is less obvious that in general it is not transitive. Take the function field of an elliptic curve X over F 3 given by the equation y 2 = x 3 + x − 1. Consider the points p, q, r ∈ X, where p is the common zero of x and x 3 + x; q is the common zero of x 4 + x 2 + 2x + 1 and y + x 2 + 2x and finally r is the common zero of x 4 + x 2 + 2x + 1 and y + 2x 2 + x. Then, using computer algebra system Magma one can check that, p ⌣ q and p ⌣ r, nevertheless the points q and r are not related.Let us now recall the notion of a small equivalence. Let ∅ = S ⊂ X be a finite (hence closed) subset of X. We say that S is sufficiently large if rk 2 Pic O X\S = 0. If S ⊂ X be a sufficiently large set of points of K, then a triple T S , t S , (t p | p ∈ S) is called (cf. [PSCL94, §6]) a small S-equivalence of the field K if (SE1) T S : S → X is injective, (SE2) t S : E X\S → E X\T S is a group isomorphism, (SE3) for every p ∈ S the map t p :Kp/K 2 p →KTp/K 2 T p is an isomorphism of local square class groups preserving Hilbert symbols, in the sense thatfor all x, y ∈Kp/K 2 p ; (SE4) the following diagram commuteswhere the maps i S = p∈S i p and i T S = q∈T S i q are the diagonal homomorphisms, i p : E X\S →Kp/K 2 p , i q : E X\T S →Kq/K 2 q . We say that the local isomorphism t p :Kp/K 2 p →KTp/K 2 T p is tame, when ord p λ ≡ ord T p t p λ (mod 2) for every λ ∈Kp/K 2 p . The following result follows from [PSCL94, Theorem 2 and Lemma 4]:Theorem 4.4. Every small S-equivalence T S , t S , (t p | p ∈ S) of the field K can be extended to a self-equivalence (T, t) of K tame on X \ S. Moreover, the self-equivalence (T, t) is tame at p ∈ S if and only if the local isomorphism t p is tame.Remark 4. In the case consider in this paper (that is over global function fields) any local square-class groupKp/K 2 p consists of just four elements {1, u p , π p , u p π p }, with The Magma algebra system. I. The user language. Wieb Bosma, John Cannon, Catherine Playoust, Computational algebra and number theory. London24Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235-265, 1997. Computational algebra and number theory (London, 1993). Number theory. Translated from the Russian by Newcomb Greenleaf. A I Borevich, I R Shafarevich, Pure and Applied Mathematics. 20Academic PressA. I. Borevich and I. R. Shafarevich. Number theory. Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20. Academic Press, New York-London, 1966. Wild primes of a self-equivalence of a global function field. Alfred Czogała, Beata Rothkegel, Acta Arith. 1664Alfred Czogała and Beata Rothkegel. Wild primes of a self-equivalence of a global function field. Acta Arith., 166(4):335-348, 2014. Higher degree Hilbert-symbol equivalence of number fields. Alfred Czogała, Andrzej Sładek, Number theory. Liptovský Ján11Alfred Czogała and Andrzej Sładek. Higher degree Hilbert-symbol equivalence of num- ber fields. Tatra Mt. Math. Publ., 11:77-88, 1997. Number theory (Liptovský Ján, 1995). Higher degree Hilbert symbol equivalence of algebraic number fields. Alfred Czogała, Andrzej Sładek, II. J. Number Theory. 722Alfred Czogała and Andrzej Sładek. Higher degree Hilbert symbol equivalence of alge- braic number fields. II. J. Number Theory, 72(2):363-376, 1998. Witt rings of Hasse domains of global fields. Alfred Czogała, J. Algebra. 2442Alfred Czogała. Witt rings of Hasse domains of global fields. J. Algebra, 244(2):604-630, 2001. Witt equivalence of algebraic function fields over real closed fields. Przemysław Koprowski, Math. Z. 2422Przemysław Koprowski. Witt equivalence of algebraic function fields over real closed fields. Math. Z., 242(2):323-345, 2002. Introduction to quadratic forms over fields. T Y Lam, Graduate Studies in Mathematics. 67American Mathematical SocietyT. Y. Lam. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2005. An invitation to arithmetic geometry. Dino Lorenzini, Graduate Studies in Mathematics. 9American Mathematical SocietyDino Lorenzini. An invitation to arithmetic geometry, volume 9 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1996. Matching Witts with global fields. R Perlis, K Szymiczek, P E Conner, R Litherland, Recent advances in real algebraic geometry and quadratic forms. Berkeley, CA; San Francisco, CA; Providence, RIAmer. Math. Soc155R. Perlis, K. Szymiczek, P. E. Conner, and R. Litherland. Matching Witts with global fields. In Recent advances in real algebraic geometry and quadratic forms (Berkeley, CA, 1990/1991; San Francisco, CA, 1991), volume 155 of Contemp. Math., pages 365-387. Amer. Math. Soc., Providence, RI, 1994. Witt equivalence of semilocal Dedekind domains in global fields. B Rothkegel, A Czogała, Abh. Math. Sem. Univ. Hamburg. 77B. Rothkegel and A. Czogała. Witt equivalence of semilocal Dedekind domains in global fields. Abh. Math. Sem. Univ. Hamburg, 77:1-24, 2007. Number theory in function fields. Michael Rosen, Graduate Texts in Mathematics. 210Springer-VerlagMichael Rosen. Number theory in function fields, volume 210 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2002. A characterization of the finite wild sets of rational self-equivalences. Marius Somodi, Acta Arith. 1214Marius Somodi. A characterization of the finite wild sets of rational self-equivalences. Acta Arith., 121(4):327-334, 2006. Self-equivalences of the Gaussian field. Marius Somodi, Rocky Mountain J. Math. 386Marius Somodi. Self-equivalences of the Gaussian field. Rocky Mountain J. Math., 38(6):2077-2089, 2008. . Bankowa. 14Institute of Mathematics, University of SilesiaPoland E-mail address: [email protected] of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland E-mail address: [email protected]

The Noether Symmetry Approach can be used to construct spherically symmetric solutions in f (R) gravity. Specifically, the Noether conserved quantity is related to the gravitational mass and a gravitational radius that reduces to the Schwarzschild radius in the limit f (R) → R. We show that it is possible to construct the M − R relation for neutron stars depending on the Noether conserved quantity and the associated gravitational radius. This approach enables the recovery of extreme massive stars that could not be stable in the standard Tolman-Oppenheimer-Volkoff based on General Relativity. Examples are given for some power law f (R) gravity models.

10.1016/j.physletb.2018.03.001

1802.09073

671201b1e76fada4e0b8218eb94e5b5c7afbcd81

25 Feb 2018 Mariafelicia De Laurentis Institute for Theoretical Physics Dipartimento di Fisica "E. Pancini" Goethe University Max-von-Laue-Str. 160438FrankfurtGermany Federico II?, Compl. Univ. di Monte S. Angelo Lab.Theor.Cosmology Universitá di Napoli Edificio G, Via CinthiaI-80126NapoliItaly Systems and Radioelectronics (TUSUR) Tomsk State University of Control 634050TomskRussia Compl. Univ. di Monte S. Angelo INFN Sezione di Napoli Edificio G, Via CinthiaI-80126NapoliItaly 25 Feb 2018(Dated: August 28, 2018)Noether's stars in f (R) gravitynumbers: 0450Kd0420Cv Keywords: Modified gravityNoether symmetriesstellar structures The Noether Symmetry Approach can be used to construct spherically symmetric solutions in f (R) gravity. Specifically, the Noether conserved quantity is related to the gravitational mass and a gravitational radius that reduces to the Schwarzschild radius in the limit f (R) → R. We show that it is possible to construct the M − R relation for neutron stars depending on the Noether conserved quantity and the associated gravitational radius. This approach enables the recovery of extreme massive stars that could not be stable in the standard Tolman-Oppenheimer-Volkoff based on General Relativity. Examples are given for some power law f (R) gravity models. I. INTRODUCTION Compact stars are natural laboratories to test strong gravity effects or, in general, alternative theories of gravity. In particular, some neutron stars present properties, as the Mass-Radius (M − R) relation, that can be hardly explained in the context of General Relativity adopting simple equations of state. For examples, PSR J0348 + 0432 [1] and PSR J1614 − 2230 [2] represent a challenge for standard theory and could be a possible testbed for modified gravity [3][4][5][6][7][8][9][10][11][12]. On the other hand, understanding the structure of neutron stars allows to constrain the parameters of any given gravitational theory in the strong field regime [15][16][17][18]. However, the most important problem in this research concerns the choice of equation of state for matter, that, up to now, are not known with certainty. In order to explain observations, one can either ask for exotic (unknown) equations of state or for modifying gravity in the strong field regime inside the star [19][20][21]. To constrain the observational parameters in modified theories of gravity, one can use the M − R relation as discussed in [22]. A drawback in the study of neutron stars models is the fact that one cannot always perform selfconsistent matching of internal and external solutions. This is because, in modified gravity, the exterior space-time geometry is not described exclusively by the mass of the star. This point needs to be clarified. According to the stellar structure, if a theory of gravity is viable and can describe, for example, a neutron star, a unique solution should be achieved and internal and external solutions should be consistently matched. This fact strictly depends on the well formulation and the well position of the Cauchy problem. In a modified theory of gravity, assigning the mass M and the radius R could not be sufficient to obtain self-consistent boundary conditions. The problem gets worse if the field equations are higher than second order in derivatives because one needs initial data up to (n − 1) order, being n the derivative order of the field equations 1 . This means that it could result extremely difficult to get a unique solution matching internal and external ones. This lack of effective mathematical tools to achieve unique solutions can be partially circumvented considering in detail the Cauchy problem. As discussed in [9,24], a choice of source fluid and suitable coordinates in the gravitational field equations can lead to a well position and well formulation of the problem. However, a general recipe, working for any modified theory of gravity, does not exist at the moment. Furthermore, the Birkhoff Theorem [23] is not always valid in modified gravity and the consistency of solutions must be carefully verified according to the boundary conditions [24]. This means that other information concerning the mass distribution is necessary in order to obtain a unique solution for both the interior and exterior regions of stars. In general, the external solution is imposed by hand to be coincident with the internal Schwarzschild or Tolman-Oppenheimer-Volkoff (TOV) solution: the method is equivalent to freezing-out the further degrees of freedom emerging from Modified Gravity with respect to those of General Relativity outside the star. This approach is controversial because it means that the full field equations are not considered, and hence the self-consistency of the whole problem is strongly violated. Consequently, artificial effects on the structure of the star can arise. A self-consistent analysis of compact objects, in particular of neutron stars and their properties in Modified Theories, in particular in f (R) gravity 2 , is a fundamental challenge which needs to be addressed. It is worth stressing that modified theories of gravity were introduced to explain the accelerated expansion of the Universe, the presence of dark matter and, finally, the impossibility to renormalize gravity [3][4][5][6][7][8][9][10][11][12]. All the fundamental inter-actions have already been described at fundamental level by quantum field theory, except gravity. In other words, a selfconsistent theory of quantum gravity is not at hand until now. This means that General Relativity is not the final theory of gravitation, but only an approximation of it working very well at local and infrared scales. The simplest generalization of General Relativity is assuming that the Hilbert -Einstein action of gravity, linear in the Ricci curvature scalar R, can be generalized as f (R) where f is an analytic function of R not necessarily linear. The fundamental reason for this approach lies on the fact that the formulation of quantum field theory on curved space-times gives rise to higher order corrections to the gravitational action like R + αR 2 [3]. Furthermore, the effective action of any unified theory, involving gravity, implies corrections to the Hilbert-Einstein Lagrangian, then f (R) gravity is a natural approach to be pursued. On the other hand, the form of f (R) can be constrained assuming a sort of "inverse scattering procedure" considering fine experiments and observations that can fix the parameters of gravitational interaction [13]. It is interesting to see that a wide range of astrophysical phenomena can be addressed by f (R) gravity ranging from Solar System scales up to cosmological scales without assuming the dark energy and dark matter hypotheses [14]. The investigation predicts the existence of new stable neutron star branches with respect to General Relativity [15]. In particular, techniques related to the existence of symmetries and conserved quantities can aid in the construction of self-consistent neutron star models. The so-called Noether Symmetry Approach [25] is one these techniques suitable for these purposes. In fact, identifying Noether symmetries enables one to "reduce" dynamics by finding out first integrals and, if a complete set of first integrals is identified, to solve this one through a suitable changes of variables. In other words, if the number of conserved quantities coincides with the dimension of the configuration space, the resulting system is fully integrable. On the other hand, such conserved quantities are always related to the physical parameters of dynamical systems. In general, the technique has been successfully applied to dark energy and inflationary cosmology [25,26] and to dynamical systems in spherical and axial symmetry [27]. In this paper, the Noether Symmetry Approach is adopted to fix the radius R and the mass M of neutron stars. As it can be shown, both quantities can be related to the Noether conserved quantity emerging in f (R) gravity. In this case we say that we are in the presence of a Noether Star. Specifically, because the existence of a Noether symmetry is related to the identification of a vector field in the configuration space whose Lie derivative is conserved, it is possible to perform a change of variables where one (or more than one) cyclic variable appears in the dynamics. A conserved quantity is related to this variable and then a first integral is derived. We will show that such a conserved quantity coincides with the gravitational mass and therefore the gravitational radius of the stellar system. In particular, the Noether vector allows to fix a power-law form f (R) = f 0 R 1+ , where the deviations with respect to General Relativity can be easily identified. The mass and the radius of the system are functions of . The standard Schwarzschild radius and mass of General Relativity are recovered for → 0. A power law Lagrangian, like that we are using here, has been largely tested at different scales. Several works have been done on the study of deviations on the apsidal motion of eccentric eclipsing binary systems [28], as well as tests on the geodesic motions of a massive particles [29]. Primordial gravitational waves in the early universe have been widely studied [30]. As discussed in [26], powerlaw f (R) models have several application in cosmology and can partially alleviate the problem of today observed accelerated expansion also if they have to be improved in order to address the whole cosmic evolution (see [4,6] for details). The outline of the paper is as follows. In Sec. II, the field equations for f (R) gravity are derived. Sec. III is devoted to the Noether Symmetry Approach. The power-law form of f (R), associated conserved quantities and the spherically symmetric solutions are derived. The modified TOV solution related to f (R) = f 0 R 1+ is discussed in Sec. IV. Herein the M − R diagram, considering values of = 0 and then demonstrating the deviation of the diagram with respect to General Relativity case ( = 0), is also discussed. The conclusions are drawn in Sec. V. II. FIELD EQUATIONS AND SPHERICAL SYMMETRY IN f (R) GRAVITY Let us start from the following action A = 1 16π d 4 x √ −g [f (R) + L m ] ,(1) where g is the determinant of the metric tensor and L m is the standard fluid matter Lagrangian. We adopt for the moment the physical units G = c = 1. The field equations, in the metric formalism, for action (1) are obtained by the variational principle f R G µν − 1 2 [f − f R R] g µν − (∇ µ ∇ ν − g µν )f R = 8πT µν . (2) Here G µν = R µν − 1 2 Rg µν is the Einstein tensor, f = f (R), f R (R) = f R = df (R)/dR is the derivative of f (R) with respect to the Ricci scalar and T µν is the energy-momentum tensor of matter. Spherically-symmetric solutions can be looked for, computing a point-like Lagrangian in which the spherically symmetry is placed in the action (1). It is worth noting that a given symmetry can be imposed whether in the Lagrangian formalism, from which the Euler-Lagrange equations are subsequently derived, or directly into the field equations. The results are entirely equivalent. We will adopt the first strategy in order to define the space configuration where the Noether vector acts on the point-like Lagrangian. A generic spherically-symmetric metric is: ds 2 = −A(r)dt 2 + B(r)dr 2 + C(r)dΩ ,(3) where dΩ = dθ 2 +sin θ 2 dφ 2 is the angular element. Imposing (3) in the action (1), in principle, a canonical form with a finite number of degrees of freedom may be assumed, that is A = drL(A, A , B, B , C, C , R, R ) ,(4) where the Ricci scalar R and the metric coefficients A, B, C are the set of independent variables defining the space configuration (see also [27] for details). The prime indicates the derivative with respect to the radial coordinate r. In order to obtain the point-like Lagrangian in the above coordinates, we write the action as A = d 4 x √ −g f − λ(R −R) ,(5) where λ is a Lagrangian multiplier andR is the Ricci scalar expressed in terms of the metric (3), i.e. in more compact form, asR = R * + A AB + 2 C BC ,(6) where R * collects first order derivative terms R * = A C ABC − A 2 2A 2 B − C 2 2BC 2 − A B 2AB 2 − B C B 2 C − 2 C .(7) Varying the action (5) with respect to R we obtain that λ = f R . Then, the action (1) becomes A = drC √ A √ B f − f R R − R * − A AB − 2 C BC = dr C √ A √ B f − f R (R − R * ) − f R C A ( √ A) ( √ B) − 2 ( √ A) ( √ B) f R C . Then the canonical point-like Lagrangian is L = − √ Af R 2C √ B C 2 − f R √ AB A C − Cf RR √ AB A R + − 2 √ Af RR √ B R C − √ AB[(2 + CR)f R − Cf ] .(8) The above Lagrangian can be recast in a suitable form introducing the matrix formalism: The general form of the Euler -Lagrange equations is L = q TT q + V ,(9)d dr ∇ q L − ∇ q L = 2 d dr T q − ∇ q V − q T ∇ qT q = = 2T q + 2 q · ∇ qT q − ∇ q V − q T ∇ qT q = 0 ,(10) which gives the equations of motion in terms of A, B, C and R, respectively. After some manipulations, it is possible to demonstrate that the variable B can be expressed as a combination of A and C, that is B = 2C 2 f RR A R + 2Cf R A C + 4ACf RR M R +Af R C 2 × (2AC[(2 + CR)f R − Cf ]) −1 .(11) By inserting Eq. (11) into the Lagrangian (8), we obtain a nonvanishing Hessian matrix which removes the singular dynamics, and then the Lagrangian (8) may be recast in the more manageable form L = [(2 + CR)f R − f C] C [2C 2 f RR A R +2CC (f R A + 2Af RR R ) + Af R C 2 ] .(12) Since ∂L ∂r = 0, L is canonical (L is the quadratic form of generalized velocities, A , C and R and then coincides with the Hamiltonian), so that we can consider L as a Lagrangian with three degrees of freedom. III. SPHERICALLY SYMMETRIC SOLUTIONS VIA NOETHER SYMMETRY APPROACH We now search for symmetries for the Lagrangian (12) in order to obtain exact solutions. It is known that if the following relation holds L X L = 0 , → XL = 0 ,(13) then Noether symmetries exist. Here L X is the Lie derivative with respect to the Noether vector X ≡ α∇ q + α ∇ q ,(14) α are functions of configuration variables and α their derivatives. The second part of Equation (13) means that the vector derivative X is applied to the Lagrangian L. Being, for ex- ample, X = α ∂ ∂q i +α ∂ ∂q i , it is XL = α ∂L ∂q i +α ∂L ∂q i , That is the contraction of X on L. In general, Equation (13) is the contraction of the Noether vector X on the tangent space T Q = {A, A , C, C , R, R } with the space of the configuration given by Q = {A, C, R}. Explicitly, we have: L X L = α·∇ q L+α ·∇ q L = q T α·∇ qL +2 ∇ q α TL q ,(15) where, in the matrix formalism, it is L = q TL q . Equation (15) vanishes if the functions α satisfy the following system α · ∇ qL + 2(∇ q α) TL = 0 −→ α i ∂L km ∂q i + 2 ∂α i ∂q kL im = 0 .(16) The functions α i , which fix the Noether vector, are obtained by solving the system (16). The system of equations (16) is related to the form of f (R)-Lagrangian. In particular, classes of f (R) models, consistent with the spherical symmetry, are determined by solving the above system [27]. Conversely, by choosing the f (R) form, we can explicitly solve (16). We find that the system (16) is satisfied for f (R) = f 0 R 1+ ,(17) and α = (α 1 , α 2 , α 3 ) = (1 − 2 )kA, −kC, kR ,(18) where is any real number, k an integration constant and f 0 a dimensional coupling constant. Eq. (17) is not the unique possible f (R) solution that can be derived from the Noether Symmetry Approach, however it is the only analytic and available in explicit form [9]. This means that for any f (R) = f 0 R 1+ , a Noether symmetry exists and it is related to a constant of motion Σ 0 given by the equations of motion, that is Σ 0 = α · ∇ q L = 2(1 + )kCR 2 −1 [2(1 + ) + CR] × ×[( − 1)RA − (2(1 + ) 2 − 3 − 2)AR ] .(19) A physical interpretation of Σ 0 is possible by starting from General Relativity, i.e. = 0. In this case, the Noether symmetry yields the solution α GR = (−kA, kC) , f (R) = f 0 R .(20) The functions A and C give the Schwarzschild solution and then, upon restoration of standard units, the constant of motion is Σ 0 = 2GM c 2 ,(21) where M is the gravitational mass of the system. In other words, in the case of Einstein gravity, the Noether symmetry gives the Schwarzschild radius (and the gravitational mass) as a conserved quantity. In the general case (17), the Lagrangian (12) becomes, L = (1 + )R 2 −1 [2(1 + ) + CR] C 2 C 2 A R + +2CRC A + 4 ACC R + ARC 2 ,(22) and exact solutions, using the constant of motion, can be given in the form B = 1 + 2ACR[2(1 + ) + CR] 2 C 2 A R + +2CRC A + 4 ACC R + ARC 2 ,(23)A = R (2 +1) −1 k 1 + Σ 0 R (4 −1) 1− dr 2k( 2 − 1)C[2( + 1) + CR](24) where k 1 an integration constant. General Relativity is clearly recovered for = 0. Such solutions can be used to obtain TOV solutions and M − R relations parameterized by . Reversing the problem, the M − R relation fixes the underlying theory of gravity, corrected with respect to General Relativity. IV. NOETHER'S STARS The above relations enable general solutions for the field equations to be determined, giving the dependence of the scalar curvature R vs the radial coordinate r. The first step is to calculate the interior metric solution that must be matched with the corresponding exterior solution. In order to restore the TOV standard notation, let us set A(r) = e 2ψ , B(r) = e 2λ , C(r) = r 2 , where ψ and λ are functions of the radial coordinate r only 3 . The metric (3) can then be recast in the standard form: ds 2 = −e 2ψ dt 2 + e 2λ dr 2 + r 2 dΩ 2 .(25) The energy-momentum tensor is T µν = diag e 2ψ ρ, e 2λ p, r 2 p, r 2 sin 2 θ p , where ρ is the matter density and p is the pressure [31]. The nontrivial components of the field equations (2) give the TOV equations for f (R) gravity [15], which in our case, for f (R) = f 0 R 1+ , are: R r 2 r 1 − e −2λ = 8πρ + 1 2 R 1+ + +e −2λ 2 r − dλ dr (1 + )R −1 R + + (1 + )R −2 RR + ( − 1)R 2 ,(27)R r 2e −2λ dψ dr − 1 r 1 − e −2λ = 8πp + + 1 2 R 1+ + e −2λ 2 r + dψ dr (1 + )R −1 R .(28) 3 C(r) is the function that rules how 2D surfaces, embedded in spacetime, are measured. Choosing C(r) = r 2 implies that the length of a circle, centered in the origin of the coordinates, is 2πr (i.e. in such a way we preserve the spherical symmetry). If C(r) = r 2 , the circle is deformed. Furthermore, the system can present singularities if C(r) is not continuos and derivable. These cases can be interesting in the cases of anisotropic and/or inhomogeneous collapses. Here, now the prime indicate the derivative with respect R. Adopting physical units, we may set f 0 = 1. For = 0, the standard TOV equations of General Relativity are recovered. The stellar configuration is a solution of the field equations and the conservation equations for the energy-momentum tensor, ∇ µ T µν = 0, from which the hydrostatic equilibrium condition follows: dp dr = −(ρ + p) dψ dr .(29) In f (R) gravity, the scalar curvature is a dynamical variable and the equation for R can be obtained by taking into account the trace of the field equations (2). We have 3 f R + Rf R − 2f = −8π(ρ − 3p) ,(30) that explicitly becomes ( − 1)R +1 + 3 (1 + )e 2λ ( − 1)R −2 R 2 + +R −1 R 2 r − dλ dr + dψ dr + R −1 R = = −8π(ρ − 3p) .(31) The above equation give us a further constraint to solve the TOV equations [15]. These equations (27)- (30) can be solved by numerical integration from r = 0, but we require a set of boundary conditions to fix the integration constants, and an equation of state that gives a relation between the density and pressure (see e.g. [22] for details on the numerical method). In Figs. 1-4, the M − R diagram for various values of is represented. Herein some popular equations of state are used, namely Sly, BSK19, BSK20 and BSK21 respectively [32,33]. It is clear to see that for | | > 0.01 there is a significant deviation with respect to General Relativity. Noteworthy is the fact that, for increasingly large values of | |, the M − R diagrams assume a self-similar behaviour. Larger radii and masses are achieved for negative values of the scaling parameter while, in the case of positive values, the traces are bent with usual General Relativity TOV equations. It is straightforward to see that we can reach masses about (2.8−3)M using the BSK20 and BSK21 equations of state. A final comment is in order at this point. The radius in the figures has not to be identified with the constant of motion. The constant of motion fixes the functional relation between the mass M and the radius R saying that there is a characteristic gravitational radius which coincides with the Schwarzschild radius of General Relativity, i.e. for = 0. Clearly, for any the gravitational radius changes. The integration constant k can be chosen equal to 1 without affecting the system. The sign of is related to the (M − R) relation. If < 0 larger stars can be achieved with respect to General Relativity. For > 0, we obtains smaller stars. V. CONCLUSIONS The mass of a self-gravitating system can be considered as a Noether charge according to the existence of the Noether sym- (17). Here different curves for different values of (yellow color scale) using the BSk21 EoS are shown. The classical TOV corresponding to = 0 is also shown as a red line. metries. In this paper, we derived both the conserved quantities and the functional form of f (R) gravity according to the so-called Noether Symmetry Approach [25]. The final output is that a power-law form of f (R) gravity is determined by the Noether vector. The power can be any real number. Such a parameter is useful in order to study deviations with respect to General Relativity. In particular, spherically-symmetric solutions are considered and we derived the field equations parameterized by . Starting from this scheme, modified TOV equations are obtained and, assuming reliable equations of state discussed in the literature, the M −R relation is achieved. According to the value and the sign of , it is possible to show that radii and masses of compact neutron stars change with respect to General Relativity. This fact allows, in principle, that larger/smaller objects can be obtained by varying the gravitational sector with respect to those provided by the standard theory. In particular, extremely large objects could be framed depending on modified gravity [15]. Some considerations are in order at this point. The first is related to the Noether symmetries. The associated conserved quantity leads the M − R relation. In other words, the existence of the symmetry is capable of ruling the stellar parameters and then the position of the star on the Hertzsprung-Russell diagram. In a general sense, the whole diagram could depend on the given theory of gravity and compact objects, where strong field effects are effective, could be a useful testbed to retain or rule out alternative models. Another consideration is related to the role of gravity in this framework. It seems that the parameter can really point out deviations with respect General Relativity emerging at given interaction lengths. Such lengths, depending on , has a similar role of the Schwarzschild radius (derived for = 0). The paradigm is that any theory of gravity has its own characteristic gravitational radius that can be something else with respect to the standard one of General Relativity. It is worth noticing that for small deviation with respect to General Relativity we can write R 1+ R + R ln R + O( 2 ) ,(32) and then control the magnitude of the corrections with respect to the standard Hilbert-Einstein action. Such deviation could come out in the strong field regimes inside compact objects that could be very similar to some situations present in the early universe where logarithmic corrections emerge from quantization of curved space time [15,34]. Finally, neutron stars, achieved in such a framework, could really discriminate between modified gravity and dark matter scenarios: in fact no exotic particle is requested in this context. The only natural assumption is that a symmetry breaking of gravitational interaction can happen at a given scale and energy, exactly like in the case of Starobinsky model of early universe where higher order curvature terms like R 2 give rise to inflation [9,34]. The Noether Symmetry Approach deserves some further general considerations. As firstly discussed in [25], the utility of the method is twofold. From one hand, it allows to find out exact solutions since the presence of Noether symmetries reduces the related dynamical systems. Clearly, if the number of symmetries coincides with the number of dimensions of configuration space, the system is completely integrable. On the other hand, as shown here, the approach allows to select the class of models, in this case the power-law form of f (R) gravity. This means that the further degrees of freedom of any modified theory of gravity (scalar tensor, vector tensor, and so on) can be linked to the symmetries that rule the dynamics (see [25] for scalar tensor gravity). In this perspective, the Noether Symmetry Approach is a general criterion to select viable theories of gravity. where q = (A, B, C, R) and q = (A , B , C , R ) are the generalized positions and velocities associated with L. The index T indicates the transposed column vector. The kinetic tensor is given byT ij = ∂ 2 L ∂q i ∂q j . V = V (q) is the potential depending only on the configuration variables. Figure 1 : 1M − R diagram for f (R) = R 1+ for the Sly EoS with different values of (purple color scales). The classical TOV solution corresponding to = 0 is also shown as a red line. Figure 2 : 2M − R diagrams for(17) using the Bsk19 EoS with different values of represented as a blue color scale. The classical TOV solution corresponding to = 0 is also shown as a red line. Figure 3 : 3M − R diagrams for(17) using EoS BSk20. The green color scale represent the different values of . The classical TOV corresponding to = 0 is also shown as a red line. Figure 4 : 4M − R diagrams for f (R) given in equation In the case of f (R) gravity, being the field equations of order 4, we need initial data up to the third derivative.2 To avoid confusion between the radius R of the star and the Ricci scalar curvature R, we adopt a different notation. 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We discuss black hole solutions in six-dimensional gravity with a Gauss-Bonnet term in the bulk and an induced gravity term on a thin 3-brane of codimension-2. We show that these black holes can be localized on the brane, and they can further be extended into the bulk by a warp function. These solutions have regular horizons and no other curvature singularities appear apart from the string-like ones. The projection of the Gauss-Bonnet term on the brane imposes a constraint relation which requires the presence of matter in the extra dimensions. * of the stability and thermodynamics of these solutions were worked out in[32].A lower dimensional version of a black hole living on a (2+1)-dimensional braneworld was considered in [33] by Emparan, Horowitz, and Myers. They based their analysis on the so-called C-metric [34] modified by a cosmological constant term. They found a BTZ black hole [38] on the brane which can be extended as a BTZ black string in a fourdimensional AdS bulk. Their thermodynamical stability analysis showed that the black string remains a stable configuration when its transverse size is comparable to the fourdimensional AdS radius, being destabilized by the Gregory-Laflamme instability[25]above that scale, breaking up to a BTZ black hole on a 2-brane.Three-dimensional gravity, because of its simplicity, is widely recognized as a useful laboratory to study important issues of general relativity. Earlier work on (2+1)-gravity[35,36]has been followed by many authors studying various aspects of classical and quantum gravity (for a review see[37]). In spite of the fact that general relativity in (2+1) dimensions has neither Newtonian limit nor propagating degrees of freedom, a black hole solution was found (BTZ black hole[38]). The BTZ black hole differs from the Schwarzschild and Kerr solutions in some important aspects: it has a conical-like axially symmetric metric, it is asymptotically anti-de Sitter rather than asymptotically flat, and it has no curvature singularity at the origin. Nonetheless, it is clearly a black hole: it has an event horizon and (in the rotating case) an inner horizon, it appears as the final state of collapsing matter, and it has thermodynamic properties much like those of a (3+1)-dimensional black hole. A singular solution at the origin was presented in [41] as a result of the coupling of BTZ black hole to a conformal matter field, and it was further extended in[42].In our previous work [39] we studied black holes on an infinitely thin conical 2-brane and their extension into a five-dimensional bulk with a Gauss-Bonnet term. We had found two classes of solutions. The first class consists of the familiar BTZ black hole which solves the junction conditions on a conical 2-brane in vacuum. These solutions in the bulk are BTZ string-like objects with regular horizons and no pathologies. The warping to fivedimensions depends on the length √ α where α is the Gauss-Bonnet coupling, and this length scale defines the shape of the horizon. Consistency of the bulk solutions requires a fine-tuned relation between the Gauss-Bonnet coupling and the five-dimensional cosmological constant. The second class of solutions consists of BTZ black holes with short distance corrections. These solutions correspond to a BTZ black hole conformally dressed with a scalar field[41,42]. Localization of these black holes on the 2-brane leads to the interesting result that the energy-momentum tensor required to support such solutions on the brane corresponds to the energy-momentum tensor of a scalar field in the limit r/L 3 << 1, where L 3 is the length scale of the three-dimensional AdS space and r the radial distance on the brane. Also these solutions have black string-like extensions into the bulk.In this work we generalize our previous work to black objects in six-dimensional braneworlds of codimension-2. We find solutions of four-dimensional Schwarzschild-AdS black holes on the brane which in the six-dimensional spacetime look like black string-like objects with regular horizons. The warping to extra dimensions depends on the Gauss-Bonnet coupling which is fine-tuned to the six-dimensional cosmological constant. In the case of constant deficit angle the localization of the four-dimensional black hole requires matter in the two extra dimensions. The energy-momentum tensor corresponding to this matter

10.1016/j.nuclphysb.2008.11.003

0804.4459

f45ef3f4df820705a3f2b268b33cda77d315fea7

Black Holes on Thin 3-branes of Codimension-2 and their Extension into the Bulk 5 Nov 2008 Bertha Cuadros-Melgar [email protected]**e-mailaddress:[email protected]♭ Eleftherios Papantonopoulos Minas Tsoukalas [email protected]♮ ♭ Vassilios Zamarias [email protected] Departamento de Física Department of Physics Universidad de Santiago de Chile Casilla 307SantiagoChile National Technical University of Athens Zografou Campus GR 157 73AthensGreece Black Holes on Thin 3-branes of Codimension-2 and their Extension into the Bulk 5 Nov 2008 We discuss black hole solutions in six-dimensional gravity with a Gauss-Bonnet term in the bulk and an induced gravity term on a thin 3-brane of codimension-2. We show that these black holes can be localized on the brane, and they can further be extended into the bulk by a warp function. These solutions have regular horizons and no other curvature singularities appear apart from the string-like ones. The projection of the Gauss-Bonnet term on the brane imposes a constraint relation which requires the presence of matter in the extra dimensions. * of the stability and thermodynamics of these solutions were worked out in[32].A lower dimensional version of a black hole living on a (2+1)-dimensional braneworld was considered in [33] by Emparan, Horowitz, and Myers. They based their analysis on the so-called C-metric [34] modified by a cosmological constant term. They found a BTZ black hole [38] on the brane which can be extended as a BTZ black string in a fourdimensional AdS bulk. Their thermodynamical stability analysis showed that the black string remains a stable configuration when its transverse size is comparable to the fourdimensional AdS radius, being destabilized by the Gregory-Laflamme instability[25]above that scale, breaking up to a BTZ black hole on a 2-brane.Three-dimensional gravity, because of its simplicity, is widely recognized as a useful laboratory to study important issues of general relativity. Earlier work on (2+1)-gravity[35,36]has been followed by many authors studying various aspects of classical and quantum gravity (for a review see[37]). In spite of the fact that general relativity in (2+1) dimensions has neither Newtonian limit nor propagating degrees of freedom, a black hole solution was found (BTZ black hole[38]). The BTZ black hole differs from the Schwarzschild and Kerr solutions in some important aspects: it has a conical-like axially symmetric metric, it is asymptotically anti-de Sitter rather than asymptotically flat, and it has no curvature singularity at the origin. Nonetheless, it is clearly a black hole: it has an event horizon and (in the rotating case) an inner horizon, it appears as the final state of collapsing matter, and it has thermodynamic properties much like those of a (3+1)-dimensional black hole. A singular solution at the origin was presented in [41] as a result of the coupling of BTZ black hole to a conformal matter field, and it was further extended in[42].In our previous work [39] we studied black holes on an infinitely thin conical 2-brane and their extension into a five-dimensional bulk with a Gauss-Bonnet term. We had found two classes of solutions. The first class consists of the familiar BTZ black hole which solves the junction conditions on a conical 2-brane in vacuum. These solutions in the bulk are BTZ string-like objects with regular horizons and no pathologies. The warping to fivedimensions depends on the length √ α where α is the Gauss-Bonnet coupling, and this length scale defines the shape of the horizon. Consistency of the bulk solutions requires a fine-tuned relation between the Gauss-Bonnet coupling and the five-dimensional cosmological constant. The second class of solutions consists of BTZ black holes with short distance corrections. These solutions correspond to a BTZ black hole conformally dressed with a scalar field[41,42]. Localization of these black holes on the 2-brane leads to the interesting result that the energy-momentum tensor required to support such solutions on the brane corresponds to the energy-momentum tensor of a scalar field in the limit r/L 3 << 1, where L 3 is the length scale of the three-dimensional AdS space and r the radial distance on the brane. Also these solutions have black string-like extensions into the bulk.In this work we generalize our previous work to black objects in six-dimensional braneworlds of codimension-2. We find solutions of four-dimensional Schwarzschild-AdS black holes on the brane which in the six-dimensional spacetime look like black string-like objects with regular horizons. The warping to extra dimensions depends on the Gauss-Bonnet coupling which is fine-tuned to the six-dimensional cosmological constant. In the case of constant deficit angle the localization of the four-dimensional black hole requires matter in the two extra dimensions. The energy-momentum tensor corresponding to this matter Introduction Recently there has been a growing interest in codimension-2 braneworlds. The most attractive feature of these models is that the vacuum energy (tension) of the brane instead of curving the brane world-volume, merely induces a deficit angle in the bulk solution around the brane [1]. This observation led several people to utilize this property in order to selftune the effective cosmological constant to zero and provide a solution to the cosmological constant problem [2]. However, soon it was realized [3] that one can only find nonsingular solutions if the brane energy momentum tensor is proportional to its induced metric. To reproduce an effective four-dimensional Einstein equation on the brane one has to introduce a cut-off (brane thickness) [4,5,6] with the price of loosing the predictability of the theory. Alternatively, in the thin brane limit four dimensional gravity is recovered as the dynamics of the induced metric on the brane if the gravitational action is modified by the inclusion of either a Gauss-Bonnet term [7] or an induced gravity term on the brane [8]. We are still lacking an understanding of time dependent cosmological solutions in codimension-2 braneworlds. In the thin brane limit, because the energy momentum tensors on the brane and in the bulk are related, the brane equation of state and energy density are tuned and we cannot get the standard cosmology on the brane [9,10]. One then has to regularize the codimension-2 branes by introducing some thickness and then consider matter on them [11,12,13,14]. To have a cosmological evolution on the regularized branes the brane world-volume should be expanding and in general the bulk space should also evolve in time. This is a formidable task, so an alternatively approach was followed in [15,16] by considering a codimension-1 brane moving in the regularized static background. The resulting cosmology, however, was unrealistic having a negative Newton's constant (for a review on the cosmology in six dimensions see [17]). We do not either fully understand black hole solutions on codimension-2 braneworlds. Recently a six-dimensional black hole localized on a 3-brane of codimension-2 [18] was proposed. These solutions are generalization of the 4D Aryal, Ford, Vilenkin [19,20] black hole pierced by a cosmic string adjusted to the codimension-2 branes with a conical structure in the bulk and deformations accommodating the deficit angle. However, it is not clear how to realize these solutions in the thin brane limit where high curvature terms are needed to accommodate matter on the brane. Generalizations to include rotations were presented in [21] and perturbative analysis of these black holes were carried out in [22,23]. The localization of a black hole on the brane and its extension to the bulk is a difficult task. In codimension-1 braneworlds the first attempt was to consider the Schwarzschild metric and study its black string extension into the bulk [24]. Unfortunately, as suspected by the authors, this string is unstable to classical linear perturbations [25] (for a recent review see [26]). Since then, several authors have attempted to find the full metric using numerical techniques [27]. Analytically, the brane metric equations of motion were considered with the only bulk input coming from the projection of the Weyl tensor [28] onto the brane. Since this system is not closed because it contains an unknown bulk dependent term, assumptions have to be made either in the form of the metric or in the Weyl term [29]. So far there is no clear evidence of what the brane black hole metric is, however, some interesting features which do occur are wormholes and singular horizons [30,31]. Analysis scales as 1/r 6 . This fact defines a length scale in the six-dimensional spacetime above which we recover the standard four-dimensional General Relativity (GR), while at small distances GR is strongly modified. There are also solutions with variable deficit angle, in which case matter is also necessary in the other directions. However, consistency of the bulk equations requires the deficit angle to be constant. The presence of the Gauss-Bonnet term in codimension-2 braneworlds has important consequences in our solutions. Its projection on the brane gives a consistency relation [8] that dictates the form of the solutions. It allows black string solutions in five dimensions and in six dimensions it specifies the kind of matter which is needed in the bulk in order to support a black hole solution on the brane. The paper is organized as follows. In section 2 we present in a self-contained way the BTZ string-like solutions of the five-dimensional case. In section 3 we discuss the black string-like solutions of the six-dimensional Einstein equations for constant and variable deficit angles. To complete our solutions we introduce branes, and solving the junction equations we find the conditions to localize the black holes on these branes. In section 4 we discuss the special rôle played by the Gauss-Bonnet term, and finally, in section 5 we conclude. where H N M = 1 2 g N M (R (5) 2 − 4R (5) 2 KL + R (5) 2 ABKL ) − 2R (5) R (5)N M +4R (5) M P R N P (5) +4R (5) N KM P R KP (5) − 2R (5) M KLP R N KLP(5) . (2.4) To obtain the braneworld equations we expand the metric around the brane as L(x, ρ) = β(x)ρ + O(ρ 2 ) . (2.5) At the boundary of the internal two-dimensional space where the 2-brane is situated the function L behaves as L ′ (x, 0) = β(x), where a prime denotes derivative with respect to ρ. We also demand that the space in the vicinity of the conical singularity is regular which imposes the supplementary conditions that ∂ µ β = 0 and ∂ ρ g µν (x, 0) = 0 [7]. The extrinsic curvature in the particular gauge g ρρ = 1 that we are considering is given by K µν = g ′ µν . The above decomposition will be helpful in the following for finding the induced dynamics on the brane. We will now use the fact that the second derivatives of the metric functions contain δ-function singularities at the position of the brane. The nature of the singularity then gives the following relations [7] L ′′ L = −(1 − L ′ ) δ(ρ) L + non − singular terms , (2.6) K ′ µν L = K µν δ(ρ) L + non − singular terms . (2.7) From the above singularity expressions and using the Gauss-Codazzi equations, we can match the singular parts of the Einstein equations (2.3) and get the following "boundary" Einstein equations G (3) µν = 1 M 3 (5) (r 2 c + 8π(1 − β)α) T (br) µν + 2π(1 − β) r 2 c + 8π(1 − β)α g µν . (2.8) Note that in the above boundary Einstein equations, as a result of the Gauss-Codazzi reduction procedure, there will also appear terms proportional to the extrinsic curvature and terms coming from the GB term in the bulk. However, if we allow only conical singularities there is no contribution from these terms [7] (see next section for the most general case). Also observe, that the presence of the induced gravity on the brane or the GB term in the bulk is necessary in order to have a non zero energy momentum tensor on the brane. We assume that there is a localized (2+1) black hole on the brane. The brane metric is ds 2 3 = −n(r) 2 dt 2 + n(r) −2 dr 2 + r 2 dφ 2 ,(2.9) where 0 ≤ r < ∞ is the radial coordinate, and φ has the usual periodicity (0, 2π). We will look for black string solutions of the Einstein equations (2.3) using the five-dimensional metric (2.2) in the form ds 2 5 = f 2 (ρ) −n(r) 2 dt 2 + n(r) −2 dr 2 + r 2 dφ 2 + a 2 (r, ρ)dρ 2 + L 2 (r, ρ)dθ 2 . (2.10) The space outside the conical singularity is regular, therefore, we demand that the warp function f (ρ) is also regular everywhere. We assume that there is only a cosmological constant Λ 5 in the bulk and we take a(r, ρ) = 1. Then, from the bulk Einstein equations G (5) M N − αH M N = − Λ 5 M 3 5 g M N ,(2.11) combining the (rr, φφ) equations we get ṅ 2 + nn − nṅ r 1 − 4α L ′′ L = 0 , (2.12) while a combination of the (ρρ, θθ) equations gives f ′′ − f ′ L ′ L 3 − 4 α f 2 ṅ 2 + nn + 2 nṅ r + 3f ′2 = 0 , (2.13) where a dot denotes derivatives with respect to r. The solutions of the equations (2.12) and (2.13) are summarized in the following table [39] n(r) In the above table L 3 is the length scale of AdS 3 space. Note that in all solutions there is a fine-tuned relation between the Gauss-Bonnet coupling α and the five-dimensional cosmological constant Λ 5 , except for the solution in the fourth row. Also observe that the solution in the third row is a kind of combination of the solutions in the first and second row. This is a result of the way we solve the factorized equations (2.12) and (2.13) [39]. f (ρ) L(ρ) −Λ 5 Constraints BTZ cosh ρ 2 √ α ∀L(ρ) 3 4α L 2 3 = 4 α BTZ cosh ρ 2 √ α 2 β √ α sinh ρ 2 √ α 3 4α - BTZ cosh ρ 2 √ α 2 β √ α sinh ρ 2 √ α 3 4α L 2 3 = 4 α BTZ ±1 1 γ sinh (γ ρ) 3 l 2 γ = − 2Λ 5 3+4αΛ 5 ∀n(r) cosh ρ 2 √ α 2 β √ α sinh ρ 2 √ α 3 4α - −M + r 2 L 2 3 − ζ r cosh ρ 2 √ α 2 β √ α sinh ρ 2 √ α 3 4α L 2 3 = 4 α −M + r 2 L 2 3 − ζ r ±1 2 β √ α sinh ρ 2 √ α 1 4α Λ 5 = − 1 4α = − 3 L 2 3 To introduce a brane we must solve the corresponding junction conditions given by the Einstein equations on the brane (2.8) using the induced metric on the brane given by (2.9). For the case when n(r) corresponds to the BTZ black hole n 2 (r) = −M + r 2 L 2 3 , and the brane cosmological constant is given by Λ 3 = −1/L 2 3 , we found that the energy-momentum tensor is null. Therefore, the BTZ black hole is localized on the brane in vacuum. When n(r) is of the form given by n(r) = −M + r 2 L 2 3 − ζ r , (2.14) which is the BTZ black hole solution with a short distance correction term, we can go back to (2.8) and solve for T br µν . Then we find that the matter source necessary to sustain such a solution on the brane is given by T β α = diag ζ 2r 3 , ζ 2r 3 , − ζ r 3 , (2.15) which is conserved on the brane [40]. Interesting enough, for a scalar field conformally coupled to BTZ [41,42], the energy-momentum tensor needed to support such a solution at a certain limit reduces to (2.15) which is necessary to localize this black hole on the conical 2-brane. These solutions extend the brane BTZ black hole into the bulk. Calculating the square of the Riemann tensor we find that at the AdS horizon (ρ → ∞) all solutions give finite result and hence the only singularity is the BTZ-corrected black hole singularity extended into the bulk. The warp function f 2 (ρ) gives the shape of a 'throat' to the horizon of the BTZ string-like solution. The size of the horizon is defined by the scale √ α and this scale is fine-tuned to the length scale of the five-dimensional AdS space. Black String-Like solutions in Six-Dimensional Braneworlds of Codimension-2 In this section we will look for black string solutions in six-dimensions with conical singularities. We consider the gravitational action (2.1) in six dimensions S grav = M 4 6 2 d 6 x −g (6) R (6) + α R (6)2 − 4R (6) M N R (6)M N + R (6) M N KL R (6)M N KL + r 2 c d 4 x −g (4) R (4) + d 6 xL bulk + d 4 xL brane . (3.1) Here r c = M 4 /M 2 6 is the induced gravity "cross-over" scale (marking the transition from 4D to 6D gravity), M 6 is the six-dimensional Planck mass and M 4 is the four-dimensional one. The metric as in the five-dimensional case is ds 2 6 = g µν (r, χ)dx µ dx ν + a 2 (r, χ)dχ 2 + L 2 (r, χ)dξ 2 ,(3.2) now with µ = 0, 1, 2, 3 whereas χ, ξ denote the radial and angular coordinates of the two extra dimensions (the χ direction may or may not be compact and the ξ coordinate ranges form 0 to 2π). Note that we have assumed that there exists an azimuthal symmetry in the system, so that both the induced four-dimensional metric and the functions a and L do not depend on ξ. The corresponding Einstein equations are G (6)N M + r 2 c G (4)ν µ g µ M g N ν δ(χ) 2πL − αH N M = 1 M 4 6 −Λ 6 + T (B)N M + T (br)ν µ g µ M g N ν δ(χ) 2πL , (3.3) where H N M is the corresponding six-dimensional term of (2.4) To obtain the braneworld equations we expand the metric around the 3-brane as L(r, χ) = β(r)χ + O(χ 2 ) ,(3.4) and as in the five-dimensional case the function L behaves as L ′ (r, 0) = β(r), where a prime now denotes derivative with respect to χ. The "boundary" Einstein equations are G (4) µν r 2 c + 8π(1 − β)α | 0 = 1 M 4 6 T (br) µν | 0 + 2π(1 − β)g µν | 0 + πL(r, χ) E µν | 0 − 2πβα W µν | 0 , (3.5) where the term E µν | 0 = (K µν − g µν K) | 0 , (3.6) appears because of the presence of the induced gravity term in the gravitational action, while the term W µν | 0 = g λσ ∂ χ g µλ ∂ χ g νσ | 0 − g λσ ∂ χ g λσ ∂ χ g µν | 0 + 1 2 g µν g λσ ∂ χ g λσ 2 − g λσ g δρ ∂ χ g λδ ∂ χ g σρ 0 , (3.7) is the Weyl term due to the presence of the Gauss-Bonnet term in the bulk [7]. The effective four-dimensional mass and cosmological constant are M 2 P l = M 4 6 (r 2 c + 8π(1 − β)α) , (3.8) Λ 4 = λ − 2πM 4 6 (1 − β) , (3.9) where λ is the brane tension. If we demand that the space in the vicinity of the conical singularity is regular (∂ µ β = 0) then (3.5) simply becomes [7,8] G (4) µν r 2 c + 8π(1 − β)α | 0 = 1 M 4 6 T (br) µν | 0 + 2π(1 − β)g µν | 0 . (3.10) 3.1 Black String-Like Solutions with pure conical singularity: Case ∂ µ β = 0 In this subsection we make the assumption that the singularity is purely conical. Thus, we will solve the bulk equations with a constant deficit angle β. We assume that the brane metric is ds 2 4 = −A(r) 2 dt 2 + A(r) −2 dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 , (3.11) where 0 ≤ r < ∞ is the radial coordinate, and φ has the usual periodicity (0, 2π). We will look for black string solutions of the Einstein equations (3.3) using the six-dimensional metric (3.2) in the form ds 2 6 = F 2 (χ) −A(r) 2 dt 2 + A(r) −2 dr 2 + r 2 dθ 2 + r 2 sin 2 θ dφ 2 + a 2 (r, χ)dχ 2 + L 2 (r, χ)dξ 2 . (3.12) As we discussed in the previous section, the space outside the conical singularity is regular, therefore we demand that the warp function F (χ) is also regular everywhere. We have split the general bulk energy momentum tensorT . Moreover, we take a(r, χ) = 1. Then from the bulk equations G (6) M N − αH M N = 1 M 4 6 −Λ 6 g M N + T (B) M N , (3.13) by taking the combination rr − θθ and χχ − ξξ of the Einstein equations we respectively get Ȧ 2 + AÄ − A 2 r 2 + 1 r 2 1 − 4α L ′′ L + F ′′ F + F ′ L ′ F L = 0 , (3.14) F ′′ − F ′ L ′ L 1 − 2α F 2 Ȧ 2 + AÄ + A 2 r 2 + 4 AȦ r − 1 r 2 + 6F ′2 = 0 . (3.15) The χχ and ξξ components of the Einstein equations are G χ χ = T χ χ , G ξ ξ = T ξ ξ ,(3.16) and therefore to get (3.15) we must take the difference G χ χ − G ξ ξ = T χ χ − T ξ ξ , and as the only remaining energy contribution in the bulk is a cosmological constant, the matter in the extra two dimensions must satisfy the relation T χ χ = T ξ ξ . Black String-Like Solutions of the Bulk Equations: Case 1 We will consider firstȦ 2 + AÄ − A 2 r 2 + 1 r 2 = 0 ,(3.17) which has as a solution A 2 (r) = 1 + r 2 L 2 4 − ζ r , (3.18) where L 4 is the length scale of the AdS 4 space, and ζ is an integration constant. Then equation (3.15) becomes F ′′ − F ′ L ′ L 1 − 12 α F 2 1 L 2 4 + F ′2 = 0 . (3.19) From the above equation we have two cases: • Case 1a: The first case is F ′2 − F 2 12α + 1 L 2 4 = 0 . (3.20) This equation has the following solution F (χ) = C 1 e χ 2 √ 3α + C 2 e −χ 2 √ 3α ,(3.21) where C 1 and C 2 are integration constants which satisfy the relation C 1 C 2 = 3α/L 2 4 . The function F (χ) is regular and if we require on the position of the brane the boundary condition F 2 (χ = 0) = 1, the integration constants can be expressed in terms of α and L 4 as C 1 = ± 1 + ε 1 − 12 α L 2 4 2 , C 2 = ± 1 − ε 1 − 12 α L 2 4 2 ,(3.22) where ε = ±1 independently of the ± sign in C 1 and C 2 . Moreover, we need ∂ χ g µν | (χ=0) = 0, i.e., ∂ χ F | (χ=0) = 0, therefore C 1 = C 2 = 1 2 and α = T (B) χ χ = T (B) ξ ξ = − 6 α ζ 2 r 6 1 F (χ) 4 ,(3.24) with the other components to be zero. Notice that except for the warp function F (χ) these components of the energy-momentum tensor do not depend explicitly on χ but only on the radial coordinate on the brane. We will give a detailed account of this dependence in the next section. • Case 1b: The second case is to consider 25) which means that L(χ) = L 0 F ′ (χ). In any subcase we recover in the same way the results of case 1a. However L(χ) is no more arbitrary and is given by L(χ) = β L 4 sinh χ L 4 , where we used the boundary conditions L(χ = 0) = 0 and L ′ (χ = 0) = β. F ′′ − F ′ L ′ L = 0 ,(3. • There are also two constant solutions for F (χ) which read F (χ) = ±1, (3.26) A(r) = 1 + r 2 L 2 4 − ζ r , (3.27) L(χ) = β sinh (γ χ) γ , (3.28) with γ = 1 L 4 1 − L 2 4 4α 1 − L 2 4 12α Λ 6 = − 6 L 2 4 1 − 2α L 2 4 (3.29) for ζ = 0 T (B) χ χ = T (B) ξ ξ = − 6αζ 2 r 6 . (3.30) The second one has the same F (χ) and A(r) functions, as well as T (B) χ χ = T (B) ξ ξ , but L(χ) = β χ sinh γ γ , (3.31) T (B) t t = T (B) r r = T (B) θ θ = T (B) φ φ = 3 (4α − L 2 4 ) L 4 4 . (3.32) Black String-Like Solutions of the Bulk Equations: Case 2 In this section we chose from (3.14) the equation 1 − 4α L ′′ L + F ′′ F + F ′ L ′ F L = 0 . (3.33) • Case 2a: The first one is to consider from (3.15) F ′′ − F ′ L ′ L = 0 ,(3.34) which means that L(χ) = L 0 F ′ (χ). Although this is not enough to solve (3.33), there is an exponential solution for both functions given by F (χ) = C 1 e χ 2 √ 3α + C 2 e −χ 2 √ 3α , (3.35) L(χ) = L 0 F ′ (χ) . (3.36) Substituting the above solution into the tt, rr, θθ, and φφ components of the Einstein equations we get a fine-tuned relation between α and Λ 6 Λ 6 = − 5 12α . (3.37) If we choose as in the first case C 1 C 2 = 3α L 2 4 then we get a differential equation for A(r), which has the following solution , therefore imposing the boundary conditions F 2 (χ = 0) = 1 (∂ χ F | (χ=0) = 0 is already satisfied), L(χ = 0) = 0 and A 2 (r) = 1 + r 2 L 2 4 ± 1 + C 3 L 2 4 + C 4 L 4 4 r ,(3.L ′ (χ = 0) = β we have F (χ) = cosh χ 2 √ 3α and L(χ) = 2 √ 3α β sinh χ 2 √ 3α which satisfy all Einstein equations. • Case 2b: The second case is to consider from (3.15) F 2 − 12α F ′2 − 2α Ȧ 2 + AÄ + A 2 r 2 + 4 AȦ r − 1 r 2 = 0 , (3.39) The first term is a function of χ while the second one is a function of r. Therefore, each term should be, in general, equal to a constant κ. We then have F 2 − 12α F ′2 = κ , (3.40) 2α Ȧ 2 + AÄ + A 2 r 2 + 4 AȦ r − 1 r 2 = κ ,(3.41) which give F (χ) = C 1 e χ 2 √ 3α + C 2 e −χ 2 √ 3α ,(3. 42) A 2 (r) = 1 + 2 C 3 r 2 + C 4 r + κ r 2 12 α ,(3.43) with C 1 C 2 = κ 4 . No solution can be found unless we set Λ 6 = − 5 12α . Then we need to solve the following differential equation −L(χ) + √ 3α   1 − κ 4 C 2 1 e −χ √ 3α 1 + κ 4 C 2 1 e −χ √ 3α   L ′ (χ) + 6α L ′′ (χ) = 0,(3.44) which has the following solution L(χ) = 4 C 2 1 C 5 e χ √ 3α κ 2 F 1 1 2 , 2, 5 2 , − 4 C 2 1 κ e χ √ 3α ,(3.45) being 2 F 1 the hypergeometric function of the second kind. The χχ and ξξ components of the Einstein equations impose us C 1 = 0, but then we cannot have ∂ χ g µν = 0. Therefore we must have C 1 = 0 and we have to consider a specific r and χ dependent χχ and ξξ components of the energy momentum tensor given by T (B) χ χ = T (B) ξ ξ = − 2α (40 C 2 3 + 24 C 3 C 4 r + 3 C 2 4 r 2 ) r 8 1 F (χ) 4 . (3.46) Imposing the boundary conditions F 2 (χ = 0) = 1, L(χ = 0) = 0 and L ′ (χ = 0) = β we get C 1 = ± 1 + ε √ 1 − κ 2 , (3.47) C 2 = ± 1 − ε √ 1 − κ 2 , (3.48) κ = β, (3.49) C 5 = 5 √ 3α β 3 η 2 5 β 2 F 1 1 2 , 2, 5 2 , − η 2 β − 2 η 2 2 F 1 3 2 , 3, 7 2 , − η 2 β , (3.50) with η = 1 + 1 − β. Moreover, we must have ∂ χ F | (χ=0) = 0 therefore C 1 = C 2 = 1 2 i.e. κ = 1 which imposes C 5 = 0. Therefore there is no solution. • There is also a constant solution for F (χ) which gives F (χ) = ±1, (3.51) A(r) 2 = 1 + r 2 4α − √ 3 12α 2 r 4 − 3 C 4 r + 48 α (α − C 3 ), (3.52) L(χ) = 2 √ α β sinh χ 2 √ α , (3.53) Λ 6 = − 1 4α . (3.54) Localization of the Bulk Black Hole on the Brane In order to complete our solution with the introduction of the brane we must solve the corresponding junction conditions given by the Einstein equations on the brane (3.10) using the induced metric on the brane given by (3.11). Equation (3.10) can be written as T (br) ν µ | 0 M 6 4 = r 2 c + 8π (1 − β) α G (4) ν µ | 0 − 2π (1 − β) g ν µ | 0 ,(3. 55) Moreover, the (χχ) component of the six-dimensional Einstein tensor evaluated at χ = 0 is − 1 2 R (4) | 0 − α 2 R (4) 2 − 4R (4) 2 µν + R (4) 2 µνκλ 0 = 1 M 4 6 T (B) χ χ | 0 − Λ 6 M 4 6 | 0 , (3.56) which gives the form of the (χχ) component of the bulk energy momentum tensor in terms of brane quantities 1 M 4 6 T (B) χ χ | 0 = − 1 2 R (4) | 0 − α 2 R (4) 2 − 4R (4) 2 µν + R (4) 2 µνκλ 0 + Λ 6 M 4 6 | 0 . (3.57) • For case 1a we have: A 2 (r) = 1 + r 2 L 2 4 − ζ r , F (χ) = cosh χ 2 √ 3α , In this case L(χ) is arbitrary, and we have the constraint α = 12 . In addition, • For case 1b we have: • For case 2a we have: Λ 6 = − 5 12α = − 5 L 2 4 , (3.58) T (B) χ χ = T (B) ξ ξ = − 6 α ζ 2 r 6 1 F (χ) 4 .A 2 (r) = 1 + r 2 L 2 4 − ζ r , F (χ) = cosh χ L 4 , (3.61) L(χ) = β L 4 sinh χ L 4 ,(3.A 2 (r) = 1 + r 2 L 2 4 ± 1 + C 3 L 2 4 + C 4 L 4 4 r, F (χ) = cosh χ 2 √ 3α , L(χ) = 2 √ 3α β sinh χ 2 √ 3α , Λ 6 = − 5 12α = − 5 L 2 4 , In this case (3.57) and (3.55) give respectively some complicated r dependent expressions for T χ χ and T µ µ , as well as for the solution (3.51)-(3.54). • For case 2b we have no solution. Our results are summarized in Table 2. 3.2 Curvature singularity: Case ∂ µ β = 0 A 2 (r) F (χ) L(χ) −Λ 6 Constraints & T (B) 1 + r 2 L 2 4 − ζ r cosh χ 2 √ 3α ∀L(χ) 5 12α α = L 2 4 12 , T χ χ = T ξ ξ = − 6αζ 2 r 6 F (χ) 4 1 + r 2 L 2 4 − ζ r cosh χ 2 √ 3α 2 √ 3αβ sinh χ 2 √ 3α 5 12α α = L 2 4 12 , T χ χ = T ξ ξ = − 6αζ 2 r 6 F (χ) 4 1 + r 2 L 2 4 − ζ r ±1 β γ sinh (γ χ) 6 L 2 4 1 − 2α L 2 4 γ = 1 L 4 1− L 2 4 4α 1− L 2 4 12α , T χ χ = T ξ ξ = − 6αζ 2 r 6 1 + r 2 L 2 4 − ζ r ±1 β γ χ sinh γ 6 L 2 4 1 − 2α L 2 4 γ = 1 L 4 1− L 2 4 4α 1− L 2 4 12α , T χ χ = T ξ ξ = − 6αζ 2 r 6 , T t t = T r r = T θ θ = T φ φ = 3(4α−L 2 4 ) L 2 4 (3.38) cosh χ 2 √ 3α 2 √ 3α β sinh χ 2 √ 3α 5 12α α = L 2 4 12 (3.52) ±1 2 √ α β sinh χ 2 √ α 1 4α α = L 2 4 4 In this section we relax the assumption of the purely conical singularity. Therefore, in general ∂ χ g µν = 0 and β(r) is a function of r. Black String-Like Solutions of the Bulk Equations In this case the combination of the rr − tt components of the bulk equations (3.13) give rr − tt: − A 2L r 2 F 4 L 4α − 4αA 2 + r 2 F 2 − 4αF ′2 − 8αF F ′′ . (3.64) If we want to keep this factorized form, we can chooseL = 0 which will not simplify our task. Therefore, we consider in generalL = 0. The other possibility is to consider the term in square brackets equal to zero. • Case 1: In this case the term in square brackets of (3.64) is equal to zero. Thus we will have T (B) r r = T (B) t t and we need to solve the following equations 4l 2 α − 4αA 2 = −κr 2 , (3.65) F 2 − 4αF ′2 − 8αF F ′′ = κ , (3.66) where κ is a constant. Then the solutions are A 2 (r) = 1 + κ 4α r 2 , (3.67) F (χ) = C 1 e χ 2 √ 3α + C 2 e −χ 2 √ 3α , (3.68) with κ = 4 C 1 C 2 3 . If we redefine 4α κ = L 2 4 we get A 2 (r) = 1 + r 2 L 2 4 , (3.69) F (χ) = C 1 e χ 2 √ 3α + C 2 e −χ 2 √ 3α with C 1 C 2 = 3α L 2 4 . (3.70) Then all the bulk Einstein equations are satisfied for Λ (6) = − 5 12α and with no matter in the bulk. Furthermore, if we require on the position of the brane the boundary condition F 2 (χ = 0) = 1, the integration constants can be expressed as in the case 1a with constant deficit angle, in terms of α and L 4 C 1 = ± 1 + ε 1 − 12 α L 2 4 2 , C 2 = ± 1 − ε 1 − 12 α L 2 4 2 , (3.71) where ε = ±1 independently of the ± sign in C 1 and C 2 . Moreover, L(r, χ) is arbitrary. we consider that A(r) has the same form as in the previous subsection for cases 1a and 1b, A 2 (r) = 1 + r 2 L 2 4 − ζ r , (3.72) then the combination θθ −tt and χχ−ξξ of the bulk equations (3.13) can respectively be factorized as θθ − tt : (2l 2 r − 3ζ)L [8αζL 2 4 + 4αr 3 + r 3 L 2 4 (4αF ′2 + 8αF F ′′ − F 2 )] 2r 5 L 2 4 F 4 L − 12αζrL (r 3 + l 2 L 2 4 r − L 2 4 ζ) 2r 5 L 2 4 F 4 L , (3.73) χχ − ξξ : 12α − L 2 4 F 2 + 12αL 2 4 F ′2 × × L (4r 3 + 2l 2 L 2 4 r − ζL 2 4 ) + rL (r 3 + L 2 4 r l 2 − ζ L 2 4 ) + 4r 2 L 2 4 F (F ′ L ′ − F ′′ L) r 2 L 4 4 F 4 L , (3.74) where a dot denotes derivatives with respect to r. We note here that in the θθ − tt combination the first term in brackets can never be zero while the second one cannot be solved analytically. Therefore here T r r = T θ θ . In the χχ − ξξ combination the second term in brackets can not also be solved analytically. Therefore the only term which can be equal to zero in order to keep a factorized form is the first bracket in (3.74). Then we get that F (χ) = C 1 e χ 2 √ 3α + C 2 e −χ 2 √ 3α with C 1 C 2 = 3α L 2 4 . Finally, the bulk Einstein equations are satisfied for Λ (6) = − 5 12α and for ζ = 0, unless we have the following bulk energy momentum tensor T (B) t t = −T (B) θ θ = −T (B) φ φ = 2αζ r 5 LF 4 × 3ζ − 2l 2 r L + 2r 2L l 2 + r 2 L 2 4 − ζ r , (3.75) T (B) r r = 2αζ r 5 LF 4 3ζ − 2l 2 r L , (3.76) T (B) χ χ = T (B) ξ ξ = − 6 α ζ 2 r 6 1 F (χ) 4 . (3.77) Furthermore, if we require on the position of the brane the boundary condition F 2 (χ = 0) = 1, the integration constants can be expressed as in (3.71). Moreover, L(r, χ) is arbitrary. Localization of the Bulk Black Hole on the Brane In order to complete our solution with the introduction of the brane we must solve the corresponding junction conditions given by the Einstein equations on the brane (3.5) using the induced metric on the brane given by (3.11). Equation (3.5) can be written as T (br) ν µ | 0 M 6 4 = r 2 c + 8π(1 − β)α G (4) µ ν | 0 − 2π(1 − β)δ µ ν | 0 − πL(r, χ) E µ ν | 0 + 2πβα W µ ν | 0 . ,(3.78) Moreover, the (χχ) component of the six-dimensional Einstein tensor evaluated at χ = 0 is given in terms of brane quantities as 1 M 4 6 T (B) χ χ 0 = − 1 2 R (4) 0 − α 2 R (4) 2 − 4R (4) 2 µν + R (4) 2 µνκλ 0 − K ′ 4 0 − 1 8 K ν µ K µ ν | 0 + g ′ L ′ 4gL 0 + ▽ (4) µ ∂ µ L L 0 + Λ 6 M 4 6 0 . (3.79) In the above relation we have K ′ 4 0 = 2 F ′′ F − F ′2 F 2 0 (3.80) 1 8 K ν µ K µ ν | 0 = 2 F ′2 F 2 0 , (3.81) g ′ L ′ 4gL 0 = F ′ L ′ 2F L 0 , (3.82) ▽ (4) µ ∂ µ L L 0 = 1 F 2 L 2L A 2 r + AȦ + A 2L 0 ,(3. The rôle of the Gauss-Bonnet Term It is known that from a Ricci flat (D-1)-dimensional solution a D-dimensional solution can be generated which satisfies the Ricci flat D-dimensional Einstein equations [43]. This procedure can also be applied if there is a D-dimensional negative cosmological constant in the bulk. This result was used in [24] to construct the five-dimensional black string in codimension-1 branes. If there is a Gauss-Bonnet term in the bulk there is a drastic change in this result [44,45]. In the five-dimensional case consistency of the four-dimensional Einstein equations forces the four-dimensional Gauss-Bonnet term projected on the brane to be constant [44] 1 . This implies that there could not exist black string solutions of the type in [24] with a Gauss-Bonnet term in the bulk in codimension-1 braneworlds. In codimension-2 braneworlds there is a relation connecting the Gauss-Bonnet term projected on the brane with the components of the bulk energy-momentum tensor corresponding to the extra dimensions [8]. In six dimensions it reads while the Ricci scalar and Ricci tensor are constants. Therefore, for the relation (4.1) to be satisfied the bulk energy-momentum tensor T (B) χ χ | 0 has to scale as 1/r 6 with the right coefficients. This is actually what happens considering the result (3.24). Moreover, it is easy to verify that relation (4.1) is satisfied substituting the relevant quantities. Thus, the presence of the Gauss-Bonnet term in the bulk, which acts as a source term because of its divergenceless nature, dictates the form of matter that must be introduced in the bulk in order to sustain a black hole on the brane 2 . − 1 2 R (4) | 0 − 1 2 α R (4) 2 − 4R (4) 2 µν + R (4) 2 µνκλ 0 = 1 M 4 6 T (B) χ χ | 0 − Λ 6 M 4 6 | 0 . For the physically, most interesting solutions of the Schwarzschild-AdS black hole on the brane, we found that there must be non-trivial matter in the extra two dimensions given by (3.24). These components of the energy-momentum tensor depend on the radial distance on the brane r and on one of the extra dimensions χ through the warp function F (χ). Therefore, if we go far away from the brane (large χ) because of the form of the warp function (see Table 2) the energy momentum tensor coming from the bulk decouples. This means that on the brane we have standard four-dimensional gravity without any corrections from the bulk. On the contrary, near the brane the 1/r 6 term dominates (the warp function goes to a constant) giving a strong modification of the four-dimensional gravity on the brane. In five-dimensions a similar relation to (4.1) holds. Then, if we use the BTZ solution of Table 1 of section 2, the corresponding relation in five-dimensions is automatically satisfied. The reason is that the BTZ black hole does not have an r = 0 curvature singularity [48] and, therefore, all the curvature invariants appearing in the relation are constants. Also the BTZ solution does not require matter in the bulk [39]. Thus, the corresponding relation to (4.1) in five dimensions is trivially satisfied, allowing the existence of a black string-like solution in five-dimensional braneworlds of codimensionality two. The situation is more subtle for the sort distance BTZ-corrected solution of section 2. This black hole has 1/r curvature singularity giving, therefore, a non-constant Krestschmann scalar proportional to 1/r 6 . This implies that for the relation to hold the combination of the three-dimensional squared Ricci scalar and the squared Ricci tensor should also be proportional to 1/r 6 with the appropriate coefficients. These curvature invariants can be obtained solving the three-dimensional Einstein equations on the brane (2.8). In order to get a non-trivial solution matter should be introduced on the brane, and this is actually what happens as it was shown in [39] (see relation (2.15)). In the five-dimensional case we have found that the matter necessary to sustain the BTZcorrected black hole solution on the brane is provided by a scalar field conformally coupled to the BTZ black hole. In six dimensions it is not clear to what system the "holographic matter" necessary to sustain the Schwarzschild-AdS black hole on the brane, corresponds. Considering the similarities between the five and six-dimensional cases it might correspond also to a scalar field coupled to the six-dimensional gravitational action. Conclusions We discussed black hole localization on an infinitely thin 3-brane of codimension-2 and its extension into a six-dimensional AdS bulk. To have a four-dimensional gravity on the brane we introduced a six-dimensional Gauss-Bonnet term in the bulk and an induced gravity term on the brane. We showed that a Schwarzschild-AdS black hole can be localized on the brane which is extended into the bulk with a warp function. Consistency of the six-dimensional bulk equations requires a fine-tuned relation between the Gauss-Bonnet coupling constant and the length of the six-dimensional AdS space. The use of this finetuning gives to the non-singular horizon the shape of a throat up to the horizon of the AdS space with no other curvature singularities except the Schwarzschild string-like singularity. If the deficit angle is constant, independent of the radial coordinate of the brane, there is a consistency relation between the Gauss-Bonnet term projected on the brane and the energy-momentum tensor of the two extra dimensions. This relation for the Schwarzschild-AdS black hole solution on the brane requires the presence of a form of "holographic matter" in the extra dimensions which scales as 1/r 6 . This gives a strong modification of gravity at short distances while standard GR is obtained only at large distances. If the deficit angle is variable, the effective four-dimensional Einstein equations on the brane acquire extra terms related to the projection of the Weyl tensor on the brane. Also, the constraint relation connecting the Gauss-Bonnet term projected on the brane and the bulk energy-momentum tensor is more involved, and in spite of the fact that the Schwarzschild-AdS black hole solution on the brane is still a solution of the bulk equations, it gives an inconsistency forcing the deficit angle to be constant. The presence of the Gauss-Bonnet term is important in our considerations. It allows the existence of black string solutions in five-dimensions and in six dimensions it specifies the form of matter which is needed in the bulk in order to sustain a black hole on the brane. It would have been interesting to find out what modifications the gravitational action is needed in order to obtain bulk solutions without the need of matter in the extra dimensions. F (χ) = cosh χ L 4 . Substituting the above solutions into the tt, rr, θθ, and φφ components of the Einstein equations we get a fine-tuned relation between α and Λ 6 the positivity of α the six-dimensional bulk space is Anti-de-Sitter. In addition we have a relation between the six-dimensional cosmological constant Λ 6 and the AdS 4 length scale L 4 . If we require the bulk equations to have as a solution the Schwarzschild-AdS black hole (3.18) with ζ = 0 then consistency of the bulk equations requires the χχ and ξξ components of the energy-momentum tensor to have the form relates the 6-dimensional cosmological constant Λ 6 and the AdS 4 length scale L 4 as Λ 6 µ whereas (3.57) and (3.63) are consistent. • Case 2: In this case the factorized equation (3.64) is not equal to zero (i.e. T (B) r r = T (B) t t ) but then the bulk Einstein equations cannot be solved in general. However, if 83)and we can see that requiring L(r, χ = 0) = 0 all terms are regular except (3.82) which has a 1 χ contribution which is singular. This can only be avoided if F ′ | 0 = 0, thus α = i.e., β constant. Another way to make the relation (3.79) regular is to take the pure Gauss-Bonnet case, where we do not take under consideration the induced gravity term in the action. In this case the bulk solutions are the same and we do not have the contributions(3.80)-(3.83) in (3.79) which becomes as (3.57). Then for both cases 1 and 2 if we want to match the T (B) χ χ component of the bulk solution with the one derived in (3.79) we must have the relation α = λ 2 12 which gives the constant β case. Thus, the relation (3.79) between bulk and brane quantities in order to be regular in the vicinity of the conical singularity requires the deficit angle to be constant. solutions have to satisfy this relation which acts as a consistency relation. In spite of the fact that in four dimensions the Gauss-Bonnet term is a topological invariant, when it is projected on the brane, it leaves its traces through this relation. 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This work makes use of the VST observations to select variable sources. We use also the IR photometry, SED fitting and X-ray information where available to confirm the nature of the AGN candidates. The IR data, available over the full survey area, allow to confirm the consistency of the variability selection with the IR color selection method, while the detection of variability may prove useful to detect the presence of an AGN in IR selected starburst galaxies.

10.1007/978-3-319-19330-4_43

1507.04923

a99b5642f53841b566ac90ef14199bd6361d3f96

A new search for variability-selected active galaxies within the VST SUDARE-VOICE survey: the Chandra Deep Field South and the SERVS-SWIRE area 17 Jul 2015 Falocco S Physics Department of University Federico II Via Cintia CAP80126 De Cicco Physics Department of University Federico II Via Cintia CAP80126 D Physics Department of University Federico II Via Cintia CAP80126 Paolillo M Physics Department of University Federico II Via Cintia CAP80126 Covone G Physics Department of University Federico II Via Cintia CAP80126 Longo G Physics Department of University Federico II Via Cintia CAP80126 Grado A Physics Department of University Federico II Via Cintia CAP80126 Limatola L Physics Department of University Federico II Via Cintia CAP80126 Vaccari M Physics Department of University Federico II Via Cintia CAP80126 Botticella M T Physics Department of University Federico II Via Cintia CAP80126 Pignata G Physics Department of University Federico II Via Cintia CAP80126 Cappellaro E Physics Department of University Federico II Via Cintia CAP80126 Trevese D Physics Department of University Federico II Via Cintia CAP80126 Vagnetti F Physics Department of University Federico II Via Cintia CAP80126 Salvato M Physics Department of University Federico II Via Cintia CAP80126 Radovich M Physics Department of University Federico II Via Cintia CAP80126 Hsu L Physics Department of University Federico II Via Cintia CAP80126 Brandt W N Physics Department of University Federico II Via Cintia CAP80126 Capaccioli M Physics Department of University Federico II Via Cintia CAP80126 Napolitano N Physics Department of University Federico II Via Cintia CAP80126 Baruffolo A Physics Department of University Federico II Via Cintia CAP80126 Cascone E Physics Department of University Federico II Via Cintia CAP80126 Schipani P Physics Department of University Federico II Via Cintia CAP80126 S Falocco Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples De Cicco Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples D Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples Paolillo M Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples Covone G Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples Longo G Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples Capaccioli M Italy Vaccari M. Astrophysics Group, Physics Department INAF Osservatorio Di Capodimonte Naples Grado A Max Plank Institute fur Extraterrestrische Physik Departamento de Ciencias Fisicas University of the Western Cape Cape Town, GarchingSouth Africa, Germany Pignata G Limatola L Max Plank Institute fur Extraterrestrische Physik Departamento de Ciencias Fisicas University of the Western Cape Cape Town, GarchingSouth Africa, Germany Pignata G Botticella M T Max Plank Institute fur Extraterrestrische Physik Departamento de Ciencias Fisicas University of the Western Cape Cape Town, GarchingSouth Africa, Germany Pignata G Napolitano N Max Plank Institute fur Extraterrestrische Physik Departamento de Ciencias Fisicas University of the Western Cape Cape Town, GarchingSouth Africa, Germany Pignata G Cascone E Max Plank Institute fur Extraterrestrische Physik Departamento de Ciencias Fisicas University of the Western Cape Cape Town, GarchingSouth Africa, Germany Pignata G P Schipani Max Plank Institute fur Extraterrestrische Physik Departamento de Ciencias Fisicas University of the Western Cape Cape Town, GarchingSouth Africa, Germany Pignata G Radovich M Trevese D. Department of Physics University Department of Physics University Tor Vergata Roma Department of Astronomy and Astrophysics Universidad Andres Bello La Sapienza RomaSantiagoChile, Italy Vagnetti F., Italy Cappellaro E Trevese D. Department of Physics University Department of Physics University Tor Vergata Roma Department of Astronomy and Astrophysics Universidad Andres Bello La Sapienza RomaSantiagoChile, Italy Vagnetti F., Italy Baruffolo A Inaf-Osservatorio Di Padova Trevese D. Department of Physics University Department of Physics University Tor Vergata Roma Department of Astronomy and Astrophysics Universidad Andres Bello La Sapienza RomaSantiagoChile, Italy Vagnetti F., Italy Italy Hsu Trevese D. Department of Physics University Department of Physics University Tor Vergata Roma Department of Astronomy and Astrophysics Universidad Andres Bello La Sapienza RomaSantiagoChile, Italy Vagnetti F., Italy L Trevese D. Department of Physics University Department of Physics University Tor Vergata Roma Department of Astronomy and Astrophysics Universidad Andres Bello La Sapienza RomaSantiagoChile, Italy Vagnetti F., Italy Salvato M Trevese D. Department of Physics University Department of Physics University Tor Vergata Roma Department of Astronomy and Astrophysics Universidad Andres Bello La Sapienza RomaSantiagoChile, Italy Vagnetti F., Italy Brandt W N The Pennsylvania State University 16802University ParkPAUSA A new search for variability-selected active galaxies within the VST SUDARE-VOICE survey: the Chandra Deep Field South and the SERVS-SWIRE area 17 Jul 2015 This work makes use of the VST observations to select variable sources. We use also the IR photometry, SED fitting and X-ray information where available to confirm the nature of the AGN candidates. The IR data, available over the full survey area, allow to confirm the consistency of the variability selection with the IR color selection method, while the detection of variability may prove useful to detect the presence of an AGN in IR selected starburst galaxies. Aims and method The luminosity of virtually all AGN varies at every wavelength (see e.g. [10,14,7,19] and references therein), thus making variability one of the most distinctive properties of these sources. The variability selection method assumes that all AGN vary intrinsically in the observed band, without requiring assumptions on the spectral shape, colours, and/or spectral line ratios. In the work described in this proceeding we aim at constructing a new variabilityselected AGN sample exploiting the data from the ongoing survey performed with the VST (VLT Survey Telescope), see [6] for details. We make use of data in the r band from the SUDARE-VOICE survey performed with the VST telescope ( [2], Cappellaro et al. in prep.). In a companion contribution by De Cicco et al. in this volume (also see [4], hereafter Paper I) we focused on the COSMOS region. Here we examine two fields around the CDFS region, that we label CDFS1 and CDFS2. We examined a total of 27 VST epochs for the CDFS1 and 22 epochs for the CDFS2 spanning five and four months respectively and covering an area of two square degrees. The data reduction and the analysis were performed using the procedure explained in Paper I (and in De Cicco et al., this volume) consisting in identifying as candidate AGNs all sources whose lightcurve showed an excess variability of 3 r.m.s. from the average variability of all sources with similar magnitude. To validate our catalogue of variable objects we exploited SWIRE by [12] and SERVS by [13]. We also used SED (Spectral Energy Distribution) classification given in [9] and [15]. Results and discussion We obtained a sample of 175 sources that we investigated further in detail analysing the diagnostics described below. 12% of the selected sample are classified as SN, based on both visual inspection of the light-curves and template fitting by the SUDARE-I collaboration (Botticella et al., this volume and Cappellaro et al. in prep.). We used the information contained in [9], to extract X-ray and SED data for our variable candidates located within the ECDFS area. There are only 15 sources in common between the sample presented in [9] and our selected sample. The 15 common sources belong to the CDFS1 which encloses the ECDFS. Twelve of the 15 common sources are detected in the X-rays and their SEDs require a strong AGN contribution (in particular in the NIR part of the spectrum). All these sources have also been identified as non-SN on the basis of the inspection of their lightcurves. The remaining three sources are non-detected in the X-rays and their best-fit SED template shows no evidence for a significant AGN contribution. These three sources were identified as SN according to their lightcurves. Therefore, we conclude that they are SN explosions in normal galaxies. We validate our catalogue of variable objects with the overlapping surveys SWIRE [12] and SERVS [13] which provide data in the 3.6, 4.5, 5.6, 8, 24, 70, 160 µm bands, and in the U, g, r, i, z filters. In Fig. 1 we compare the r − i versus the 3.6 µm to r band flux ratio of our variable candidates with the SERVS+SWIRE source catalog. This diagram has been proposed by [15] to separate stars from galaxies. The populations represented in the plot are segregated into two regions: stars and extragalactic objects. Figure 1 shows that six of 57 variable objects are along the stellar sequence, likely variable stars. We further make use of the mid-IR colors in order to confirm the identification of our AGN candidates. Figure 2 shows the diagnostic developed by [11]. Due to the different dust content and temperature, normal galaxies, starforming galaxies and AGNs occupy different regions of this diagram. This allows us, as shown in [11], to define an empirical wedge (solid line in Fig. 2) which encloses a large fraction of the AGN population. Out of the 115 sources of the selected sample represented in the plot, 103 lie within the Lacy wedge, supporting their AGN nature. We also note from Fig. 2 that the average stellarity index of the variable candidates inside the Lacy region decreases towards the left side of the diagram, where the contamination is more severe. According to [18], many of such sources are low ionisation narrow emission regions (LINERs). To improve the purity 1 of the IR selected AGN sample and to reduce the starburst contamination to IR-selected AGN samples, [5] defined a more restrictive criterion, which is shown in Figure 2 as a dashed line. The majority of pointlike sources lie within the Donley wedge, strengthening the view that the Donley region is occupied prevalently by AGN-dominated galaxies. Conclusions We identified 175 candidates selected through variability using VST observations in the CDFS. To validate the sample, we used information available both within the VST-SUDARE consortium and in the literature. The total number of candidates for which we could employ the diagnostics discussed in the previous section is 137 out of a total 175 candidates in the selected sample. We found 103 confirmed AGN (by at least one diagnostic of those explored in the previous section), that is 75% of the 137 candidates with ancillary data and 59% of the selected sample. As expected, contaminants are mainly stars and SN: the stars constitutes 3 % of the selected sample of 175 candidates, while the SN constitute 12 %. In conclusion the purity of our sample of optically variable sources is 75%, close to the 80% obtained for the COSMOS field in Paper I. The completeness of the variability-selected survey presented in this work is 22 % (computed with respect to the IR selection of [5]). In Paper I, the completeness (computed with respect to X-ray samples) has been estimated to be 15 % for a 5 months baseline; the two results are thus in broad agreement considering that they are estimated with respect 1 We define the purity as the number of confirmed AGN divided by the number of AGN candidates to different reference populations. These completeness levels can improve extending the monitoring baseline, as pointed out in previous papers (e.g. [16] and Paper I). The new observations which are currently being acquired for the COSMOS field (P.I: G. Pignata) with VST will allow us to directly compare these results using a 3-year long monitoring baseline. Fig. 1 1Flux (F λ ) ratios between the r band and 3.6 µm versus r − i colour. Small points: SERVS+SWIRE 82254 sources. Triangles: 57 sources in common with the selected sample. Diamonds: SN. Crosses: X-ray detected sources. The colours indicate the increasing stellarity, from red (extended sources) to blue (pointlike). The solid line separates the stellar sequence and the non-stellar region. . M A Bershady, D Trevese, R G Kron, APJ. 496103M. A. Bershady, D. Trevese, and R. G. Kron., APJ, 496, 103 (1998) . M T Botticella, E Cappellaro, G Pignata, The Messenger, 1512932M. T. Botticella, E. Cappellaro, G. Pignata, The Messenger, 151:2932 (2013) . C N Cardamone, P G Van Dokkum, C M Urry, APJs. 189270285C. N. Cardamone, P. G. van Dokkum, C. M. Urry, APJs, 189:270285 (2010) . D De Cicco, M Paolillo, G Covone, A&A. 574112De Cicco, D., Paolillo, M., Covone, G., et al., A&A, 574, A112 (2015) . J L Donley, A M Koekemoer, M Brusa, APJ. 748142J. L. Donley, A. M. Koekemoer, M. Brusa et al., APJ, 748:142 (2012) . S Falocco, M Paolillo, G Covone, arXiv:1505.02668eprintFalocco S., Paolillo M., Covone G., et al., eprint arXiv:1505.02668 (2015) . J García-González, A Alonso-Herrero, P G Pérez-González, MNRAS. 446J. García-González, A. Alonso-Herrero, P. G. Pérez-González, et al., MNRAS, 446, I 3, 3199- 3223 (2015) . E Gawiser, P G Van Dokkum, D Herrera, APJs. 162119E. Gawiser, P. G. van Dokkum, D. Herrera et al., APJs, 162:119 (2006) . L.-T Hsu, M Salvato, K Nandra, ApJ. 79660L.-T. Hsu, M. Salvato, K. Nandra, et al., ApJ, 796, 60 (2014) . T Kawaguchi, S Mineshige, M Umemura, APJ. 504671679T. Kawaguchi, S. Mineshige, M. Umemura, APJ, 504:671679 (1998) . M Lacy, L J Storrie-Lombardi, A Sajina, Apjs , 154166169M. Lacy, L. J. Storrie-Lombardi, A. Sajina, APJs, 154:166169 (2004) . C Lonsdale, M D C Polletta, J Surace, Apjs , 1545459C. Lonsdale, M. d. C. Polletta, J. Surace, APJs, 154:5459 (2004) . J.-C Mauduit, M Lacy, D Farrah, Pasp, 12411351136J.-C. Mauduit, M. Lacy, D. Farrah,PASP, 124:11351136 (2012). . M Paolillo, E J Schreier, R Giacconi, APJ. 61193106M. Paolillo, E. J. Schreier, R. Giacconi, et al., APJ, 611:93106 (2004) . M Rowan-Robinson, E Gonzalez-Solares, M Vaccari, MNRAS. 42819581967M. Rowan-Robinson, E. Gonzalez-Solares, M. Vaccari,et al., MNRAS, 428:19581967 (2013) Flux (F λ ) ratio (logarithmic) at 5.8 and 3.6 µm versus flux ratio at 8 and 4.5 µm. Small points: SERVS+SWIRE 18436 sources; Triangles (enclosed in yellow edges): 115 sources in common with the selected sample. Cyan stars: stars; Diamonds: SN. Crosses: X-ray detected sources. Colour code as in Fig. 1. The solid line is the Lacy region and the dashed line the Donley region (see textFig. 2 Flux (F λ ) ratio (logarithmic) at 5.8 and 3.6 µm versus flux ratio at 8 and 4.5 µm. Small points: SERVS+SWIRE 18436 sources; Triangles (enclosed in yellow edges): 115 sources in com- mon with the selected sample. Cyan stars: stars; Diamonds: SN. Crosses: X-ray detected sources. Colour code as in Fig. 1. The solid line is the Lacy region and the dashed line the Donley region (see text). . B Sesar, AJ. 13422362251B. Sesar, et al., AJ, 134:22362251 (2007) . D Trevese, R G Kron, S R Majewski, APJ. 433494509D. Trevese, R. G. Kron, S. R. Majewski, et al., APJ, 433:494509 (1994) . D Trevese, K Boutsia, F Vagnetti, A&A. 4887381D. Trevese, K. Boutsia, F. Vagnetti, et al, A&A, 488:7381 (2008) . M.-H Ulrich, L Maraschi, C M Urry, ARAA. 35445502M.-H. Ulrich, L. Maraschi, and C. M. Urry, ARAA, 35:445502 (1997)

Thermal and isospin composition effects over the heat capacity of infinite nuclear matter are studied within the binodal coexistence region of the nuclear phase diagram. Assuming the independent conservation of both proton and neutron densities, a second order phase transition is expected, leading to a discontinuous behavior of the heat capacity. This discontinuity is analyzed for the full range of the thermodynamical variables coherent with the equilibrium coexistence of phases. Two different effective models of the nuclear interaction are examined in the mean field approximation, the non-relativistic Skyrme force and the covariant QHD formulation. We found qualitative agreement between both descriptions. The discontinuity in the specific heat per particle is finite and decreases with both the density of particles and the isospin asymmetry.As a byproduct the latent heat for isospin-symmetric matter is considered. * Electronic address: [email protected]

10.1103/physrevc.85.064314

1306.3202

d918187911dbc5de02b597ea0d30a612213cb323

Specific heat capacity in the low density regime of asymmetric nuclear matter 13 Jun 2013 R Aguirre Departamento de Fisica Facultad de Ciencias Exactas IFLP Universidad Nacional de La Plata CCT-La PlataCONICET. Argentina Specific heat capacity in the low density regime of asymmetric nuclear matter 13 Jun 2013 Thermal and isospin composition effects over the heat capacity of infinite nuclear matter are studied within the binodal coexistence region of the nuclear phase diagram. Assuming the independent conservation of both proton and neutron densities, a second order phase transition is expected, leading to a discontinuous behavior of the heat capacity. This discontinuity is analyzed for the full range of the thermodynamical variables coherent with the equilibrium coexistence of phases. Two different effective models of the nuclear interaction are examined in the mean field approximation, the non-relativistic Skyrme force and the covariant QHD formulation. We found qualitative agreement between both descriptions. The discontinuity in the specific heat per particle is finite and decreases with both the density of particles and the isospin asymmetry.As a byproduct the latent heat for isospin-symmetric matter is considered. * Electronic address: [email protected] I. INTRODUCTION The in-medium nuclear interaction gives rise to a complex thermodynamical phase diagram. Several phases are expected to take place as temperature and density are varied, as for instance superfluid, superconducting, boson condensed, Bose-Einstein condensed deuterons, and quark deconfined phases. Along these thermodynamical changes some constraints must be fulfilled, such as conservation of baryonic number, electric charge, etc., which have severe consequences on the evolution of the state of matter. As a typical situation we can mention the liquid-gas phase transition (LGPT), expected to occur in the low density-low temperature regime of the nuclear matter. It is a subject of study since long time ago, and it has received renewed attention in recent years [1][2][3][4][5]. The theoretical treatment requires an interesting combination of quantum statistical approaches with models of the nuclear interaction. The contrast with empirical results could be made in the field of ion collision experiences, see for example [6], as well as with observational data concerning the thermal equilibration of proto-neutron stars [7,9]. Different situations prevent a direct application of the theoretical predictions, for instance, finite size effects are significant in heavy ion collisions. Furthermore, the very short characteristic times of the reactions, turn it dubious the applicability of equilibrium thermodynamics. On the other hand, the possible frustration of the binodal transition and the coming up of a non-homogeneous phase near the surface layer of neutron stars complicate the interpretation of the transport properties of the star matter. It is clear that detailed calculations should take into account a multitude of specific effects. However, with the purpose of highlight the basic features, some simplified calculations are admissible and they serve as a useful reference for more complex statements [5]. The existence of more than one conserved charge is a feature of the above mentioned situations. A phase transition taking place under such requirements has distinctive consequences, such as the fact that the conserved charges do not distribute uniformly among the coexisting phases [10]. Certainly, the isospin fractionation observed in multifragmentation experiences [11] could be the fingerprint of a LGPT occurring after the collision. A variety of models and approximations have been used to describe the nuclear equation of state. The combination of mean field approaches with effective models, adjusted to reproduce the nuclear phenomenology, offers the advantages of simple calculations and reliable results. Among the most used representations of the nuclear force, we can mention the non-relativistic Skyrme model and the covariant formulation known as Quantum Hadro-Dynamics (QHD). They have dissimilar foundations, a density dependent nucleon-nucleon potential and a covariant exchange of mesons are respectively used, but eventually both give rise to an energy density functional. Within this formulation, they can be fairly compared. The fact that nuclear systems could have more than one conserved charge, has noticeable consequences on the evolution of the thermodynamical instabilities. Indeed, there is a change in the order of the transition, allowing a continuous variation of the thermodynamic potentials. Discontinuities are relegated to the first derivatives, corresponding, for instance, to the compressibility and heat capacity of the system. It is worthwhile to mention that the inclusion of other effects, such as the Coulomb and surface tension forces, could change dramatically this description [8]. In the present work we intend to study the behavior of the heat capacity of infinite nuclear matter within the coexistence region. The heat capacity is of great significance, for instance, in the evaluation of the rate of change of temperature through the outer shell of young neutron stars [9]. It has also been studied in relation to the nuclear multifragmentation, where it is considered as an indicator of the LGPT [12,13]. A statistical model of multinucleon clusters was frequently used for this purpose, focusing the calculations in the low isospin asymmetry regime. The upper limit for the temperature is determined by a characteristic value, of the order T ∼ 10 MeV, for which clusters start to dissolve. In this work we explore a wide range of temperature, particle density and isospin composition, which can be easily combined in a mean field calculation. We examine two effective models of the nuclear interaction: the Skyrme potential and the QHD relativistic formulation. This article is organized as follows. The general features of the models are presented in the next section, the results are shown and discussed in Section III. A final summary is given in Section IV. II. THE NUCLEAR LIQUID-GAS PHASE TRANSITION IN DIFFERENT MOD-ELS In order to check the generality of the results found, two phenomenological models of the nuclear force have been used. In first place the well known Skyrme model, where medium effects are included through density dependent parameters for the nucleon-nucleon potential. As an alternative formulation, we choose the QHD model. Here the interaction is mediated by meson fields, which are evaluated in a self-consistent way. In both cases the isospin composition can be easily handled. It can be parameterized by the asymmetry fraction w = (n 2 − n 1 )/n, with n 1 , n 2 standing for the particle number density of protons and neutrons respectively, and n = n 1 + n 2 is the total nucleon density. We assume both proton and neutron numbers are conserved independently. Hence, different chemical potentials µ a can be assigned to each isospin component. The statistical distribution function can be written f a (T, p) = [1 + exp β(ε a (p) − µ a )] −1 , where the particle energy spectrum ε a (p) is provided by the proposed model. Throughout this article we use units such that c = 1, = 1, k B = 1. A. The Skyrme model The Skyrme model is a well known effective formulation of the nuclear interaction [14]. It consists of a basic Hamiltonian with contact potentials and density dependent coefficients v Sky (r 1 , r 2 ) = t 0 (1 + x 0 P σ )δ(r 1 − r 2 ) + 1 2 t 1 (1 + x 1 P σ ) ← q 2 δ(r 1 − r 2 )+ → q 2 δ(r 1 − r 2 ) + t 2 (1 + x 2 P σ ) ← q ·δ(r 1 − r 2 ) → q + 1 6 t 3 (1 + x 3 P σ )δ(r 1 − r 2 )ρ γ ((r 1 + r 2 )/2) + iW 0 (σ 1 + σ 2 )· ← q ×δ(r 1 − r 2 ) → q where σ k represent the Pauli matrices for spin, P σ = (1 + σ 1 · σ 2 )/2 is the spin exchange operator, and q = −i(∇ 1 − ∇ 2 )/2 is the relative momentum operator. Several parameterizations have been given, according to the applications planned. They cover from exotic nuclei to stellar matter. By taking the Hartree-Fock expectation value from this force, an energy density functional is obtained, which for infinite homogeneous nuclear matter is given by E Skm = δ s j K j 2m * j + 1 16 (a 0 + a 2 w 2 ) n 2 .(1) the factor δ s = 2 takes account of the spin degeneracy, and the kinetic density of particles with isospin j (j=1,2 for protons and neutrons, respectively) is given by K j = 1 (2π) 3 d 3 p p 2 f j (T, p)(2) Here, f j (T, p) is the Fermi occupation number for the isospin component j at temperature T . The effective nucleon mass m * j for this state is given by 1 m * j = 1 m + 1 4 n (b 0 − b 2 wI j )(3) m represents the in vacuum degenerate nucleon mass, and I j = (−1) 1+j . The density dependent coefficients a 0 , a 2 and b 0 , b 2 can be expressed in terms of the standard parameters of the Skyrme model a 0 = 6t 0 + t 3 n γ , b 0 = [3t 1 + t 2 (5 + 4x 2 )]/2 a 2 = −2t 0 (1 + 2x 0 ) − t 3 (1 + 2x 3 )n γ /3, b 2 = [t 2 (1 + 2x 2 ) − t 1 (1 + 2x 1 )]/2 Within a Landau-Fermi liquid scheme, the particle spectra can be obtained by the functional derivatives ε as (p) = δE Skm /δf as (T, p). In such a way, the following result is obtained [15] ε j (p) = p 2 2m * j + 1 8 v j + ∆ε, v j = (a 0 − a 2 wI j ) n + δ s k (b 0 + I j I k b 2 )K k ∆ε = 3n 2 − (1 + 2x 3 )n 2 w 2 σt 3 n σ−1 /48 The set of self-consistent equations is completed with the relation between the conserved particle numbers and the corresponding chemical potentials n j = δ s (2π) 3 d 3 p f j (T, p)(4) B. QHD model This is a model of the covariant field theory, proposed to deal with in-medium nuclear properties [16]. The interaction is mediated by iso-scalar mesons σ and ω µ , the first one can be considered as a resonant state. In the present case the isovector mesons φ and ρ µ are also included, as well as polynomial terms in the σ field. The lagrangian density is L =Ψ (i ∂ − M + g s σ + g c τ · φ − g w ω − g r τ · ρ) Ψ + 1 2 (∂ µ σ∂ µ σ − m 2 s σ 2 ) − A 3 σ 3 − B 4 σ 4 + 1 2 (∂ µ φ · ∂ µ φ − m 2 c φ 2 ) − 1 4 F µν F µν + 1 2 m 2 w ω 2 − 1 4 R µν · R µν + 1 2 m 2 r ρ 2 where Ψ(x) is the isospin multiplet nucleon field, F µν = ∂ µ ω ν −∂ ν ω µ , R µν = ∂ µ ρ ν −∂ ν ρ µ , and g s , g c , g w , g r , A, and B are coupling constants. The nonlinear self-interaction of the σ meson is necessary to obtain an adequate behavior for the incompressibility around the saturation density. Within a mean field approximation, the equations of motion are (i ∂ − M + g s σ + g c τ 3 φ − g w γ 0 ω − g r γ 0 τ 3 ρ) Ψ = 0,(5)m 2 s σ + Aσ 2 + Bσ 3 = g s j n sj , m 2 w ω = g w j n j m 2 c φ = g c j I j n sj m 2 r ρ = g r j I j n j . As in the previous section, the density of particles with isospin projection j is represented by n j =<Ψ j γ 0 Ψ j >, whereas n sj =<Ψ j Ψ j > was used for the scalar density. They can be explicitly written as The statistical distribution function f j (p, T ) depends on the quasi-particle energies ε j = E pj + g w ω + g r I j ρ. n j = δ s d 3 p (2π) 3 f j (p, T ), n sj = δ s d 3 p (2π) 3 f j (p, T ) m j E pj where m j = m − g s σ − g c I j φ is The energy density can be obtained by first evaluating the energy-momentum tensor and then taking mean values. In such a way, it is obtained E = δ s j d 3 p (2π) 3 ε j (p)f j (p) + 1 2 m 2 s σ 2 + 1 2 m 2 c φ 2 + 1 2 m 2 w ω 2 + 1 2 m 2 r ρ 2 + A 3 σ 3 + B 4 σ 4 The entropy density for a system in thermodynamical equilibrium is given by S = −δ s j d 3 p (2π) 3 [f j log f j + (1 − f j ) log (1 − f j )] It can be used, together with the corresponding energy density E, to evaluate the free energy density F = E − T S and the pressure P = j µ j n j − F of the system. A homogeneous system at temperature T and isospin composition n 1 , n 2 remains thermodynamically stable if the free energy per unit volume F is lower than any linear combination of energies corresponding to independent thermodynamical states, say a and b, satisfying the conservation laws [10], i. e. F (T, n 1 , n 2 ) < λ F (T, n (a) 1 , n (a) 2 ) + (1 − λ) F (T, n (b) 1 , n (b) 2 )(6) Otherwise, the system becomes unstable and a change of phase is feasible. In such a case, two states can coexist if they verify the thermodynamical equilibrium conditions P (T, n (a) 1 , n (a) 2 ) = P (T, n (b) 1 , n (b) 2 ),(7) µ (a) 1 = µ (b) 1 , µ (a) 2 = µ (b) 2(8) Within the coexistence region, the total density of particles is a combination of contributions coming from each phase n k = λ n (a) k + (1 − λ) n (b) k , 0 < λ < 1, k = 1, 2,(9) where the λ parameter stands for the partial volume fraction of the phase a. As a consequence, if the system has global densities n 1 , n 2 , within the binodal region it could be composed of phases with densities differing considerably from the global values. Any extensive thermodynamical quantity can be evaluated in a similar way, for instance the free energy can be written [10], F (T, n 1 , n 2 ) = λ F (T, n (a) 1 , n (a) 2 ) + (1 − λ) F (T, n (b) 1 , n (b) 2 ). In practice, we proceed as follows. For a given temperature T , we fix the pressure P 0 within a reasonable range. We explore this isobar and find a set of values 2 ) which fulfill Eq. (8). For these pairs of states we find all the solutions (n 1 , n 2 , λ) consistent with the requirement of Eq. (9). Exploring the range of temperatures and pressures for which there exist solutions of Eqs. (7)-(9), we obtain the binodal region immersed in a 3-dimensional space, say (T, P, w). An interesting situation occurs for certain values of the variables (T, P, w), for which the parameter λ does not exhaust the full range [0, 1]. In such situations λ = 0 at low density, then it grows with the density, reaches a maximum value and then comes back to zero. This phenomenon is known as the retrograde transition [10], because the system starts and ends at the same phase but in between develops a germ of the other phase. Within these prescriptions the thermodynamical potentials remain continuous throughout the phase transition. Discontinuities are relegated to their derivatives. Of special meaning are the heat capacity, compressibility and thermal expansion coefficient. All of them are evaluated at constant particle number, for instance the heat capacity at constant volume is defined as c v = (∂S/∂T ) N 1 ,N 2 ,V . Within the binodal this derivative must be done carefully, since the total number of protons is distributed between the two coexisting phases. Furthermore, the parameter λ of Eq. (9) has also a temperature dependence which is not explicitly written. Taken these facts into account, and using E a = E(T, n (a) 1 , n (a) 2 ), E b = E(T, n (b) 1 , n (b) 2 ), the heat capacity per unit volume in the binodal can be written c v (T, n 1 , n 2 ) = λ ∂E a ∂T n 1 ,n 2 + (1 − λ) ∂E b ∂T n 1 ,n 2 + (E a − E b ) ∂λ ∂T n 1 ,n 2 (10) ∂E c ∂T n 1 ,n 2 = R c + ∂E c ∂T n (c) 1 ,n (c) 2(11)R c = k=1,2 ∂E c ∂n (c) k T,n (c) j ∂n (c) k ∂T n 1 ,n 2 , j = k(12) The last term in Eq. (11) can be recognized as the heat capacity for a homogeneous system composed of only one phase c (c) v = c v (T, n (c) 1 , n(c) 2 ). Therefore, Eq. (10) can be summarized as c v (T, n 1 , n 2 ) = λc (a) v + (1 − λ)c (b) v + ∆c v(13)∆c v = λR a + (1 − λ)R b + (E a − E b ) ∂λ ∂T n 1 ,n 2(14) When the system approaches to the binodal boundary from inside, λ → 0, or λ → 1. For instance, c v (T, n 1 , n 2 ) → c (a) v + R a + (E a − E b ) (∂λ/∂T ) n 1 ,n 2 when λ → 1. Approaching the same point, but from outside the binodal, yields c v (T, n 1 , n 2 ) → c (a) v . Hence we have a discontinuity R a + (E a − E b ) (∂λ/∂T ) n 1 ,n 2 , where the first term contains several derivatives evaluated at the binodal boundary, while the second contribution is proportional to the energy difference between the coexisting phases. Expressions for the several derivatives appearing in Eqs. (10)- (12), can be found in the Appendix. III. RESULTS AND DISCUSSION In this section we show and discuss the results obtained for the binodal region and its thermodynamical properties as described by the selected models of the nuclear interaction. In particular we analyze the specific heat at constant volume throughout the phase transition, and in the case of symmetric nuclear matter we consider the definition of a latent heat and evaluate its temperature dependence. respectively. It is worthwhile to mention that for high isospin asymmetries w (low y) the transition is of retrograde character. This situation can be appreciated more clearly in a plot of the pressure as a function of the global particle density, and several isospin asymmetries, as shown for T = 10 MeV in Fig. 2. Continuous lines correspond to the physical results, whereas dashed lines represent non-equilibrium states previous to the Gibbs construction. For high neutron excess (w = 0.6) the retrograde transition differs only slightly from the noncorrected pressure. In this circumstance, the mechanical stability condition (∂P/∂n > 0) is verified, but some of the matter diffusion conditions ∂µ 1 /∂n > 0, ∂µ 2 /∂n < 0 is not fulfilled. Furthermore, from the same figure it can be appreciated that within the non-equilibrium region of highly asymmetric matter the Skyrme model presents higher incompressibility than the QHD case. This is a consequence of the fast increase of the pressure, but keeping almost unchanged densities, chemical potentials, and specially the derivatives of the chemical potentials. Hence, larger pressures are obtained in the former case within the range of densities that do not satisfy the equilibrium conditions. This fact explains why the binodal region extends up to larger pressures in the Skyrme than in the QHD calculations, as shown in Fig. 1. The effects of the Gibbs construction on the free energy are shown in Fig. 3. In order to ease the comparison, the rest mass contribution has been removed from the QHD results. It can be corroborated that the coexistence of phases effectively minimizes the free energy, and changes its convexity also. For the temperature shown, T = 5 MeV, there is a retrograde transition for w = 0.9. As a special case we consider matter globally iso-symmetric, in such a case it behaves as a one component system [10]. Along the phase separation the coexisting states have w = 0, and the pressure remains almost constant. In this sense the LGPT resembles a first order transition. This circumstance can be appreciated in Fig. 4, where the density dependence of the pressure in the Skyrme model is shown for several temperatures. The dashed line encloses the coexistence area. As it was pointed out, there is no discontinuity in the thermodynamical potentials and it is particularly true for the entropy. Notwithstanding, the difference T (S(T, n (a) ) − S(T, n (b) )) represents the heat transferred as the LGPT is accomplished isothermically. It is interesting to compare this quantity with the latent heat corresponding to a first order phase transition. It must be noticed that some calculations find a first order LGPT in the nuclear medium, even for two components systems. See for instance [13], where particle correlations beyond the mean field approximation are included. Therefore the thermal dependence of this variation could be used to characterize the order of the change of phases. In a recent paper [3] the latent heat for the LGPT in symmetric nuclear matter was studied for several parameterizations of the Skyrme model. In order to compare results, we consider the quantity L = T (S(T, n (a) )/n (a) − S(T, n (b) )/n (b) ), (15) along an isothermal within the binodal, which coincides with the definition of the specific latent heat for a first order transition. For a given temperature, the coexisting states with nucleon densities n (a) and n (b) are determined by the conditions of equal chemical potentials and pressures, so that L depends only on the temperature. In Fig. 5 There are discontinuities at the thresholds of the binodal, as discussed at the end of section II C. For a given temperature, the discontinuity decreases with the asymmetry w, as expected from the fact that pure neutron matter does not exhibit instabilities of the LGPT type. Within the binodal, c v is a decreasing function of the density, in contrast to its behavior for which a retrograde transition takes place. The comparison with the results obtained using the QHD model can be made by examining Fig. 7. There is a general agreement with the previous description, with slightly greater values of c v corresponding to the QHD case. As a particular situation, it can be mentioned that for T = 10 MeV, w = 0.6 the curve for c v does not show an appreciable discontinuity at the higher transition density n = 0.47 n 0 , because it is at the limit, in the isospin asymmetry variable, separating full and retrograde evolution. The thermal dependence of C v is shown in Fig. 8, for some selected values of the global density and asymmetry. For this purpose we have chosen the Skyrme model, because the QHD results do not differ qualitatively. The heat capacity per particle exhibits an evident discontinuity of a few units at the threshold of the binodal. An increasing behavior is obtained for the full range of temperatures examined. As expected, this quantity approaches to zero as the temperatures vanishes, according to the Nernst principle [19]. On the other hand for high enough temperatures, the specific heat approaches asymptotically to the noninteracting limit 3k B /2. The degree of convergence to this limit depends essentially on the global density n, with a negligible influence of the isospin asymmetry. The transition temperature is both a decreasing function of w (for fixed n), and n (for fixed w). It can be observed that the magnitude of the discontinuity diminishes with both n and w. For the lowest density shown n/n 0 = 0.2 the greatest jump correspond to the lower isospin asymmetry w = 0.2, at T ≃ 14 MeV, and for the neutron rich state w = 0.8 the discontinuity at T ≃ 10.5 MeV decreases a 60 %. The results shown in Fig. 8(a) can be contrasted with Fig. 6 of Ref. [13]. The comparison must be cautious since the latter used a canonical description for a high (but finite) number of particles in a inhomogeneous probe of nucleons. For A=1000 the heat capacity increases softly with temperature, up to a characteristic temperature T ≃ 10 MeV, where reaches a peaked maximum. For higher T the heat capacity remains almost constant. The difference ∆C v /N between the maximum and the plateau varies within the range 5-25, decreasing with the asymmetry w. In contrast we obtain a high rate of growth before the critical temperature, which is of the same order T ≃ 10 MeV. Furthermore, the drop from peak to the plateau is lesser than 4. In our calculations the precise location of the temperature corresponding to the maximum decreases with w, in opposition to the behavior shown in [13]. IV. CONCLUSIONS In this work we have examined the behavior of the heat capacity of infinite homogeneous nuclear matter in the region of low particle density and low temperature, taking the isospin composition as a relevant parameter. For this purpose we have selected two different descriptions of the nuclear interaction. Both, the non-relativistic Skyrme potential and the field theoretical QHD model have been extensively used in the literature with remarkable success. Although these effective models have very different foundations, they describe appropriately the nuclear matter phenomenology for sub-saturation densities. We have examined the region of thermodynamical instability, where nuclear matter separates spontaneously in different phases. As we consider conservation of both neutron n 2 and proton n 1 densities, it is found a three dimensional region of coexistence of phases. As a consequence the thermodynamical potentials are continuous, leading to a second order phase transition. Under these conditions, the heat capacity at constant volume has been particularly analyzed. As a first approximation we neglected the electromagnetic interaction, which could have significant influence on the thermodynamical fluctuations leading to a change of phase, see for example [8]. The description obtained corresponds to a region of the phase space wider than in previous calculations, where more complex states of matter were considered [12,13]. The Gibbs construction allows the conservation of the global densities of each isospin component through the coexistence of two phases with local densities differing appreciably from the global values n 1 , n 2 . The relative abundance of each of these two phases can be represented through a parameter 0 < λ < 1. The energy of the system is expressed as a linear combination of the energy of each phase, with coefficients λ and 1 − λ. The evaluation of the heat capacity requires some care, since its definition prescribes derivatives at constant global densities, which does not imply fixed local densities. The temperature dependence of λ must also be considered. We have found qualitative agreement between the predictions of both models. Only small differences can be found, for instance in the extension of the binodal region, the critical temperature and the maximum value of the specific heat capacity. As expected in a second order phase transition, the heat capacity exhibits a discontinuity at the boundary of the binodal. A detailed characterization of this discontinuity has been presented in terms of the thermodynamical variables for the full range of the coexistence of phases. For a fixed temperature we found a discontinuity at very low density and other one at a relatively greater value, corresponding to the passage to pure gas and pure liquid, respectively. As the isospin asymmetry is increased, the full transition is replaced by a retrograde one. The high density discontinuity in c v has opposite behavior for each of these situations. For instance, in the retrograde evolution, c v decreases suddenly when the matter leaves the coexistence region towards the pure liquid phase. The thermal variation of the specific heat, at fixed density or isospin asymmetry, also shows a sharp but finite jump at a characteristic value. The location of this critical temperature decreases with both n and w. The magnitude of the discontinuity in the heat capacity per particle is lesser than 4, diminishing for increasing density and asymmetry. As a special situation, we examined the change of phase for symmetric nuclear matter, which develops at very low pressures. In such a case we examined the entropy variation between the final states of an isothermal process within the binodal, and we have compared it with the latent heat L, defined for a first order phase transition. As a function of temperature L(T ) has a maximum value and vanishes for the critical temperature. We have found small differences between Skyrme and QHD predictions, and a general agreement with recently published results [3]. Appendix The derivatives appearing in Eqs. (10), (11) are obtained as solutions of a set of algebraic linear equations. We start taking derivatives of Eqs. (9) keeping constants n 1 , n 2 , for k = 1, 2. After rearranging terms we obtain 0 = (n (a) 1 − n (b) 1 ) λ ∂n (a) 2 ∂T + (1 − λ) ∂n (b) 2 ∂T + (n (a) 2 − n (b) 2 ) λ ∂n (a) 1 ∂T + (1 − λ) ∂n (b) 1 ∂T (A.1) ∂λ ∂T = λ ∂n (a) 1 ∂T + (1 − λ) ∂n (b) 1 ∂T /(n (a) 1 − n (b) 1 ) (A.2) In the next step, derivatives of Eqs. ∂T = ∂µ (b) j ∂T n (b) 1 , n (b) 2 + k=1,2 ∂µ (b) j ∂n (b) k n (b) l , T ∂n (b) k ∂T (A.3) where j = 1, 2 and l = k. When there is no explicit statement, partial derivatives respect to T are evaluated holding the global densities n 1 , n 2 fixed. the in-medium effective mass and E pj = p 2 + m 2 j . Due to the assumed isotropy, only the zero component of the vector fields survives. Furthermore as there are not decaying channels between nucleons, only the third component of the isovectors contributes. For the Skyrme model the SLy4 parametrization is used, for which t 0 = −2488.91 MeV fm 3 ,t 1 = 486.82 MeV fm 5 , t 2 = −546.39 MeV fm 5 , t 3 = 13777 MeV fm 7/2 , x 0 = 0.834, x 1 = −0.344, x 2 = −1, x 3 = 1.354, γ = 1/6[17].For the QHD model with iso-vector mesons the parametrization given by Ref.[18] is used,for which (g s /m s ) 2 = 10.33 fm 2 , (g w /m w ) 2 = 5.42 fm 2 , (g c /m c ) 2 = 2.5 fm 2 , (g r /m r ) 2 = 3.15 fm 2 , A/g 3 s = 0.033 fm −1 , B/g 4 s = −0.0048. The saturation density, binding energy, incompressibility and symmetry energy obtained are n 0 = 0.159 fm −3 , E B = −15.97 MeV, K = 229.9 MeV, E S = 32 MeV in the Skyrme model, and n 0 = 0.16 fm −3 , E B = −16 MeV, K = 240 MeV, E S = 30.5 MeV for the QHD model. Another significative quantity is the in-medium nucleon mass m * at the saturation density, the values m * /m = 0.694, and m * /m = 0.75 are obtained for the Skyrme and QHD models, respectively. In first place the binodal region is constructed for both models. Some results corresponding to the temperatures T =5 and 10 MeV are shown in Fig. 1, in a plot of pressure versus proton abundance y = (1 − w)/2. It can be seen that for the Skyrme interaction the coexistence of phases extends up to larger pressures. The range of temperatures, instead, is smaller. The critical temperatures are 14.5 MeV and 15.9 MeV for Skyrme and QHD, the results obtained for the SLy4 parametrization and nonlinear QHD model are shown. The behavior is similar in both cases, with maximum values L max = 29.7 MeV and L max = 30.8 MeV for Skyrme and QHD forces, respectively. The value L = 0 is reached at the critical temperature. We corroborate some of the conclusions enunciated in [3], a) when T → 0, L approaches to the binding energy at the saturation density; b) for low temperatures L grows almost linearly, c) for the greater L max corresponds the larger critical temperature. Now we consider the specific heat capacity at constant volume, evaluated according to Eqs. (10)-(12). In first place we examine the results for the Skyrme model, as shown in Fig. 6, where the density dependence of the heat capacity is displayed for several isospin asymmetries. Dashed lines represent the results obtained without the Gibbs construction. outside. Comparing the upper and lower panels of this figure, a general increment around 60% is observed in the specific heat at T = 10 MeV respect to the T = 5 MeV outcome. For all the curves shown, there are jumps toward greater values of c v as the system reach the pure liquid state. The only exception occurs for the greater value of w shown in each figure, FIG. 1 : 1Isothermal sections of the binodal corresponding to T=5 MeV (a) and T=10 MeV (b), for the selected models. The line convention specified in (a) is used for both cases. FIG. 2 : 2The pressure as a function of the global density of particles corresponding to T=10 MeV and several isospin asymmetries w, for the Skyrme (a) and QHD (b) models. Continuous lines correspond to physical states in thermodynamical equilibrium, dashed lines represent predictions of the models without the Gibbs construction. FIG. 3 :FIG. 4 : 34The free energy density as a function of the global density of particles corresponding to T=5 MeV and several isospin asymmetries w, for the Skyrme (a) and QHD (b) models. Continuous lines correspond to physical states in thermodynamical equilibrium, dashed lines represent predictions of the models without the Gibbs construction. The pressure for isospin symmetric nuclear matter as a function of the global density of particles, for a set of temperatures ranging from T =0 to the critical temperature T c = 14.5 MeV within the Skyrme model. The dashed line represent the boundary of the binodal. FIG. 5 :FIG. 6 :FIG. 7 : 567The thermal dependence of L, see Eq. 15, corresponding to the LGPT for symmetric nuclear matter, as given by the selected models. The heat capacity per unit volume in terms of the global density of particles for several asymmetries w, corresponding to T=5 MeV (a) and T=10 MeV (b), within the Skyrme model. Continuous lines correspond to physical states in thermodynamical equilibrium, dashed lines represent predictions of the model without the Gibbs construction. The heat capacity per unit volume in terms of the global density of particles for several asymmetries w, corresponding to T=5 MeV (a) and T=10 MeV (b), within the QHD model. Continuous lines correspond to physical states in thermodynamical equilibrium, dashed lines represent predictions of the model without the Gibbs construction. 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We consider the family of irreducible crystalline representations of dimension 2 of Gal(Q p /Q p ) given by the V k,ap for a fixed weight integer k ≥ 2. We study the locus of the parameter a p where these representations have a given reduction modulo p. We give qualitative results on this locus and show that for a fixed p and k it can be computed by determining the reduction modulo p of V k,ap for a finite number of values of the parameter a p . We also generalize these results to other Galois types.2010 Mathematics Subject Classification. 11F80,14G22.1 Lemma 1.2.1. Let K be a finite extension of Q p . Let D be a disk defined over K. Suppose that D contains an a ∈ Q p of degree n over K. Then D contains an element b ∈ Q p of degree ≤ p s over K where s = v p (n). Corollary 1.2.2. Let K be a finite extension of Q p . Let D be a disk defined over K. Suppose that D contains an element a such that [K(a) : K] = n. Then the minimal degree over K of an element of D is of the form p t for some t ≤ v p (n).Proof. It follows from Lemma 1.2.1 that the minimal degree over K of an element of D is a power of p. On the other hand, applying Lemma 1.2.1 to a, we get an element of degree at most p s for s = v p (n). Hence the minimal degree is of the form p t for some t ≤ s.Then the minimal ramification degree over E of an element of D is a power of p, and it can be reached for an element a such that [E(a) : E] is a power of p.Proof. We first apply Corollary 1.2.2 with K = E nr to see that the minimal ramification degree is a power of p. Let b ∈ D be such that e E(b)/E = p t is the minimal ramification degree.LetWe apply Corollary 1.2.2 to K = F , and we get an element a ∈ D of degree at most p t over F .By minimality of t, we get that in fact [E(a) : F ] = p t , and E(a)/F is totally ramified. Finally, [E(a) : E] is a power of p and e E(a)/E = p t .Let π E be a uniformizer of E, and let F be a finite unramified extension of E. For x ∈ F , we define the E-part of x, which we denote by x 0 , as follows: we write x as x = n≥N a n π n E where the a n are Teichmueller lifts of elements of the residue field of F . Let x 0 = m n=N a n π n E with a n ∈ E for all n ≤ m and a m+1 ∈ E (or m = ∞ if a ∈ E) so that x 0 ∈ E. We have that v E (x − x 0 ) = m + 1. This definition depends on the choice of π E . Proposition 1.2.4. Let D be a disk defined over E, and suppose that F ∩ D = ∅ for some unramified extension F of E. Then E ∩ D = ∅.Proof. Let a ∈ F ∩ D. We fix π E a uniformizer of E, and let a 0 be the E-part of a. Let σ be the Frobenius of Gal(F/E). Then v E (a − σ(a)) = v E (a − a 0 ). So any disk containing a and σ(a) also contains a 0 .We also recall the well-known result: Lemma 1.2.5. Let f be a rational function. Then for any disk D, if f does not have a pole in D then f (D) is also a disk. Moreovoer, if D is defined over E and f ∈ E(X) then f (D) is defined over E.1.3. Proofs. The part that states that e is a power of p in Theorems 1.1.1 and 1.1.2 is a consequence of Corollary 1.2.3.We start with the rest of the proof of Theorem 1.1.2 which is actually easier.Proof of Theorem 1.1.2. By applying Corollary 1.2.3, we get an element a ∈ D that generates a totally ramified extension F of K of degree e = p s , where K is an unramified extension of E of degree a power of p, and we take [K : E] minimal. If K = E, let K ′ ⊂ K with [K : K ′ ] = p. We will show that we can find b ∈ D of degree e over K ′ , which gives a contradiction by minimality of K so in fact K = E.

10.2140/ant.2020.14.655

1705.01060

8428f5d79f3c51171004e99d40c177d530fb90cc

ON THE LOCUS OF 2-DIMENSIONAL CRYSTALLINE REPRESENTATIONS WITH A GIVEN REDUCTION MODULO p 9 Feb 2018 Sandra Rozensztajn ON THE LOCUS OF 2-DIMENSIONAL CRYSTALLINE REPRESENTATIONS WITH A GIVEN REDUCTION MODULO p 9 Feb 2018 We consider the family of irreducible crystalline representations of dimension 2 of Gal(Q p /Q p ) given by the V k,ap for a fixed weight integer k ≥ 2. We study the locus of the parameter a p where these representations have a given reduction modulo p. We give qualitative results on this locus and show that for a fixed p and k it can be computed by determining the reduction modulo p of V k,ap for a finite number of values of the parameter a p . We also generalize these results to other Galois types.2010 Mathematics Subject Classification. 11F80,14G22.1 Lemma 1.2.1. Let K be a finite extension of Q p . Let D be a disk defined over K. Suppose that D contains an a ∈ Q p of degree n over K. Then D contains an element b ∈ Q p of degree ≤ p s over K where s = v p (n). Corollary 1.2.2. Let K be a finite extension of Q p . Let D be a disk defined over K. Suppose that D contains an element a such that [K(a) : K] = n. Then the minimal degree over K of an element of D is of the form p t for some t ≤ v p (n).Proof. It follows from Lemma 1.2.1 that the minimal degree over K of an element of D is a power of p. On the other hand, applying Lemma 1.2.1 to a, we get an element of degree at most p s for s = v p (n). Hence the minimal degree is of the form p t for some t ≤ s.Then the minimal ramification degree over E of an element of D is a power of p, and it can be reached for an element a such that [E(a) : E] is a power of p.Proof. We first apply Corollary 1.2.2 with K = E nr to see that the minimal ramification degree is a power of p. Let b ∈ D be such that e E(b)/E = p t is the minimal ramification degree.LetWe apply Corollary 1.2.2 to K = F , and we get an element a ∈ D of degree at most p t over F .By minimality of t, we get that in fact [E(a) : F ] = p t , and E(a)/F is totally ramified. Finally, [E(a) : E] is a power of p and e E(a)/E = p t .Let π E be a uniformizer of E, and let F be a finite unramified extension of E. For x ∈ F , we define the E-part of x, which we denote by x 0 , as follows: we write x as x = n≥N a n π n E where the a n are Teichmueller lifts of elements of the residue field of F . Let x 0 = m n=N a n π n E with a n ∈ E for all n ≤ m and a m+1 ∈ E (or m = ∞ if a ∈ E) so that x 0 ∈ E. We have that v E (x − x 0 ) = m + 1. This definition depends on the choice of π E . Proposition 1.2.4. Let D be a disk defined over E, and suppose that F ∩ D = ∅ for some unramified extension F of E. Then E ∩ D = ∅.Proof. Let a ∈ F ∩ D. We fix π E a uniformizer of E, and let a 0 be the E-part of a. Let σ be the Frobenius of Gal(F/E). Then v E (a − σ(a)) = v E (a − a 0 ). So any disk containing a and σ(a) also contains a 0 .We also recall the well-known result: Lemma 1.2.5. Let f be a rational function. Then for any disk D, if f does not have a pole in D then f (D) is also a disk. Moreovoer, if D is defined over E and f ∈ E(X) then f (D) is defined over E.1.3. Proofs. The part that states that e is a power of p in Theorems 1.1.1 and 1.1.2 is a consequence of Corollary 1.2.3.We start with the rest of the proof of Theorem 1.1.2 which is actually easier.Proof of Theorem 1.1.2. By applying Corollary 1.2.3, we get an element a ∈ D that generates a totally ramified extension F of K of degree e = p s , where K is an unramified extension of E of degree a power of p, and we take [K : E] minimal. If K = E, let K ′ ⊂ K with [K : K ′ ] = p. We will show that we can find b ∈ D of degree e over K ′ , which gives a contradiction by minimality of K so in fact K = E. Introduction Let p be a prime number. Fix a continuous representation ρ of G Qp = Gal(Q p /Q p ) with values in GL 2 (F p ). In [Kis08], Kisin has defined local rings R ψ (k, ρ) that parametrize the deformations of ρ to characteristic 0 representations that are crystalline with Hodge-Tate weights (0, k − 1) and determinant ψ. These rings are very hard to compute, even for relatively small values of k. We are interested in this paper in the rings R ψ (k, ρ)[1/p]. These rings lose some information from R ψ (k, ρ), but still retain all the information about the parametrization of deformations of ρ in characteristic 0. We can relate the study of the rings R ψ (k, ρ)[1/p] to another problem: When we fix an integer k ≥ 2 and set the character ψ to be χ k−1 cycl , the set of isomorphism classes of irreducible crystalline representations of dimension 2, determinant ψ and Hodge-Tate weights (0, k − 1) is in bijection with the set D = {x ∈ Q p , v p (x) > 0} via a parameter a p , and we call V k,ap the representation corresponding to a p . So given a residual representation ρ we can consider the set X(k, ρ) of a p ∈ D such that the semi-simplified reduction modulo p of V k,ap is equal to ρ ss . It turns out that X(k, ρ) has a special form. We say that a subset of Q p is a standard subset if it is a finite union of rational open disks from which we have removed a finite union of rational closed disks. Then we show that under some hypotheses on ρ (including in particular the fact that is has trivial endomorphisms, so that the rings R ψ (k, ρ) are well-defined): Theorem A. The set X(k, ρ) is a standard subset of Q p , and R ψ (k, ρ)[1/p] is the ring of bounded analytic functions on X(k, ρ). This tells us that we can recover R ψ (k, ρ)[1/p] from X(k, ρ). But we need to be able to understand X(k, ρ) better. We can define a notion of complexity for a standard subset X which invariant under the absolute Galois group of E for some finite extension E of Q p . This complexity is a positive integer c E (X), which mostly counts the number of disks involved in the definition of X, but with some arithmetic multiplicity that measures how hard it is to define the disk on the field E. A consequence of this definition is that if an upper bound for c E (X) is given, then X can be recovered from the sets X ∩ F for some finite extensions F of E, and even from the intersection of X with some finite set of points under an additional hypothesis (Theorems 4.5.1 and 4.5.2). A key point is that this complexity, which is defined in a combinatorial way, is actually related to the Hilbert-Samuel multiplicity of the special fiber of the rings of analytic functions bounded by 1 on the set X (Theorem 4.4.1). This is especially interesting in the case where the set X is X(k, ρ) as in this case this Hilbert-Samuel multiplicity can be bounded explicity using the Breuil-Mézard conjecture. So, under some hypotheses on ρ, we have: Theorem B (Proposition 5.4.9). There is an explicit upper bound for the complexity of X(k, ρ). As a consequence we get: Theorem C (Theorem 5.4.10). The set X(k, ρ) can be determined by computing the reduction modulo p of V k,ap for a p in some finite set. In particular, it is possible to compute the set X(k, ρ), and also the ring R ψ (k, ρ)[1/p], by a finite number of numerical computations. We give some examples of this in Section 6. One interesting outcome of these computations is that when ρ is irreducible, in every example that we computed we observed that the upper bound for the complexity given by Theorem B is actually an equality. It would be interesting to have an interpretation for this fact and to know if it is true in general. Finally, we could ask the same questions about more general rings parametrizing potentially semi-stable deformations of a given Galois type, instead of only rings parametrizing crystalline deformations. Our method relies on the fact that we work with rings that have relative dimension 1 over Z p , so we cannot use it beyond the case of 2-dimensional representations of G Qp . But in this case we can actually generalize our results to all Galois types. In order to do this, we need to introduce a parameter classifying the representations that plays a role similar to the role the function a p plays for crystalline representations, and to show that it defines an analytic function on the rigid space attached to the deformation ring. This is the result of Theorem 5.3.1. Once we have this parameter, we show that an analogue of Theorem A holds, and an analogue of Theorem B (Theorem 5.3.3). However we get only a weaker analogue of Theorem C (Theorem 5.3.6). The main ingredient of this theorem that is known in the crystalline case, but missing the case of more general Galois types, is the fact that the reduction of the representation is locally constant with respect to the parameter a p , with an explicit radius for local constancy. Plan of the article. The first three sections contain some preliminaries. In Section 1 we prove some results on the smallest degree of an extension generated by a point of a disk in C p . These results may be of independant interest. In Section 2 we prove some results on Hilbert-Samuel multiplicities and how to compute them for some special rings of dimension 1. In Section 3 we introduce the notion of standard subset of P 1 (Q p ) and prove some results about some special rigid subspaces of the affine line. Section 4 contains the main technical results. This is where we introduce the complexity of so-called standard subsets of P 1 (Q p ), and show that it can be defined in either a combinatorial or an algebraic way. We apply these results in Section 5 to the locus of points parametrizing potentially semi-stable representations of a fixed Galois type with a given reduction. We also explain some particularities of the case of parameter rings for crystalline representations. In Section 6 we report on some numerical computations that were made using the results of Section 5 in the case of crystalline representations, and mention some questions inspired by these computations. Finally in Section 7 we explain the construction of a parameter classifying the representations on the potentially semi-stable deformation rings. Notation. If E is a finite extension of Q p , we denote its ring of integers by O E , with maximal ideal m E , and its residue field by k E . We write π E for a uniformizer of E, and v E for the normalized valuation on E and its extension to C p . We write also write v p for v Qp . Finally, G E denotes the absolute Galois group of E. If R is a ring and n a positive integer, we denote by R[X] <n the subspace of R[X] of polynomials of degree at most n − 1. If a ∈ C p and r ∈ R, we write D(a, r) + for the set {x ∈ C p , |x − a| ≤ r} (closed disk) and D(a, r) − for the set {x ∈ C p , |x − a| < r} (open disk). We denote by χ cycl the p-adic cyclotomic character, and ω its reduction modulo p. We denote by unr(x) the unramified character that sends a geometric Frobenius to x. Points in disks in extensions of the base field Let D ⊂ C p be a disk (open or closed). It can happen that D is defined over a finite extension E of Q p (that is, invariant by G E ), but E ∩ D is empty. For example, let π be a p-th root of p and let D be the disk {x, v p (x − π) > 1/p}. Then D is defined over Q p , as it contains all the conjugates of π, that is the ζ i p π for a primitive p-th root ζ p of 1. On the other hand, D does not contain any element of Q p . The goal of this section is to understand the relationship between the smallest ramification degree over E of a field F such that F ∩ D = ∅, and the smallest degree over E of such a field. In this Section a disk will mean either a closed or an open disk. The results of this Section are used in the proofs of Propositions 4.5.8 and 4.5.10. We can in fact do better in the case where p = 2. Note that this result proves Conjecture 2 of [Ben15] in this case. Preliminaries. We recall the following result, which is [Ben15, Lemma 2.6] (it is stated only for closed disks, but applies also to open disks). Let µ be the minimal polynomial of a over K, so µ ∈ K[X] is monic of degree e. Now we use that p = 2: let (1, u) be a basis of K over K ′ , and write µ = µ 0 + uµ 1 with µ 0 , µ 1 in K ′ [X]. If µ 0 has a root in D we are finished, so we can assume that µ 0 has no zero in D, and let f = µ 1 /µ 0 ∈ K ′ (X). Let D ′ = f (D). It is a disk defined over K ′ , containing −u ∈ K, so by Lemma 1.2.4, D ′ contains an element c ∈ K ′ . This means that µ 0 − cµ 1 has a root b in D. Then b is of degree at most e over K ′ . By minimality of e, it means that b is of degree exactly e over K ′ , and K ′ (b)/K ′ is totally ramified. So this gives the contradiction we were looking for. Now we turn to the proof of Theorem 1.1.1. We start with a special case. Proof. Let K = E(a) ∩ E nr . Let µ be the minimal polynomial of a over K, so that µ has degree e. We write µ = b i X i , b i ∈ K. Define µ 0 = b 0 i X i where b 0 i ∈ E is the E-part of b i . Let x 1 , . . . , x e be the roots of µ 0 . Then v E (µ 0 (a)) = e i=1 v E (a − x i ). On the other hand, µ 0 (a) = µ 0 (a) − µ(a) = e−1 i=0 (b 0 i − b i )a i . By the condition on v E (a), we get that v E (µ 0 (a)) = min 0≤i<e (v E (b 0 i − b i ) + in/e). Let σ be an element of G E that induces the Frobenius on K. Let y 1 , . . . , y e be the roots of σ(µ) = σ(b i )X i . Then as before, v E (σ(µ)(a)) = v E (a − y i ), and v E (σ(µ)(a)) = min 0≤i<e (v E (σ(b i ) − b i ) + in/e). As v E (b 0 i − b i ) = v E (σ(b i ) − b i ) for all i, we get that v E (µ 0 (a)) = v E (σ(µ)(a) ). Suppose first that D is closed. Write D as the set {z, v E (z − a) ≥ λ} for some λ, then we get that v E (σ(µ)(a)) ≥ eλ as the y i are among the conjugates of a over E and hence are in D, so v E (µ 0 (a)) ≥ eλ and so there exists an i with x i ∈ D. Let F = E(x i ) then F is an extension of E of degree at most e. The case of an open disk is similar. Note that if we take e to be minimal, then necessarily F/E is totally ramified and of degree e. K] ≤ p vp([E(b):K]) , that is [K(a ′ ) : K] ≤ p s . So finally a ′ ∈ D and [E(a ′ ) : E] ≤ p 2s−1 . We prove now the existence of such a polynomial f . Fix a uniformizer π F of F , and let E be the set of pairs of e-uples (α, P ) where α = α 1 , . . . , α e are elements of K, P = P 1 , . . . , P e are elements of E[X] <e , and i α i P i (a) = π F . Then E is not empty: we can write π F = Q(a) for some Q ∈ K[X] <e ; now let α 1 , . . . , α e be a basis of F over K, and write Q = α i P i with P i ∈ E[X] <e . For each (α, P ) ∈ E let m (α,P ) = inf i v E (α i P i (a)), so m (α,P ) ≤ 1/e. It is enough to show that there is an (α, P ) with m (α, P ) = 1/e. Indeed, if v E (α i P i (a)) = 1/e, let β i ∈ E with v E (α i ) = v E (β i ) then β i P i is the f we are looking for. So choose a (α, P ) ∈ E with m = m (α,P ) minimal, and with minimal number of indices i such that v E (α i P i (a)) = m. Suppose that m < 1/e. Then there are at least two indices i with v E (α i P i (a)) = m. Say for simplicity that v E (α 1 P 1 (a)) = v E (α 2 P 2 (a)) = m. By minimality of e, P 1 and P 2 have no root in D. Let f = P 1 /P 2 , and D ′ = f (D). Then D ′ is defined over E, and contains an element f (a) of valuation r = v E (P 1 (a)/P 2 (a)) ∈ Z, (P 1 (a)). We define an element (α ′ , P ′ ) of E by setting P ′ 1 = P 1 − xP 2 and α ′ 2 = α 2 + xα 1 , and We give some results that will enable us to compute e(A) for some special cases of rings A of dimension 1. Proof. For n large enough, we have m n+1 ⊂ (z). So the surjective map A → A/(z) factors through A/m n+1 (and in particular len A (A/(z)) is finite). We have an exact sequence: as r = v E (α 2 /α 1 ). Consider π −r E D ′ . It does not contain 0, so it is contained in a disk {z, v E (z − c) > 0} for some element c that is the Teichmueller lift of an element of F × p . So v E (π −r E P 1 (a)/P 2 (a)−c) > 0. As π −r E D ′ is defined over E, we have that c ∈ E. Let x = cπ r E , then v E (P 1 (a) − xP 2 (a)) > r + v E (P 2 (a)) = v Eα ′ i = α i and P ′ i = P i for all other indices. We observe that v E (α ′ 1 P ′ 1 (a)) > m, v E (α ′ 2 P ′ 2 (a)) ≥ m,A z → A/m n+1 → A/(z) → 0 For n large enough, the kernel of the first map is m n by the assumptions on z: it contains m n , and as multiplication by z is injective, it is exactly equal to m n . So we have an exact sequence: 0 → A/m n z → A/m n+1 → A/(z) → 0 This gives len A (m n /m n+1 ) = len A (A/(z)) = dim k A/(z) as stated. Corollary 2.1.2. Let k be a field, and (A, m) be a local noetherian k-algebra of dimension 1, with A/m = k. Suppose that there exist an element z ∈ m such that A has no z-torsion and a nilpotent ideal I such that m = (z, I). Then e(A) = dim k A/(z). Proof. We need only show that zm n = m n+1 for all n large enough, as we can then apply Lemma 2.1.1. Let m be an integer such that I m = 0. Then for n > m we have m n = m i=0 I i z n−i , which gives the result. Let k be a field. Let A 1 , . . . , A s be a family of local noetherian complete k-algebras of dimension 1 with maximal ideals V i and A i /V i = k. Let A be a local noetherian complete k-algebra with A/m = k. We say that A is nearly the sum of the family (A i ) if A = k ⊕ (⊕ s i=1 V i ) as a k-vector space and m = ⊕ s i=1 V i , and for all i, A i ⊂ A with image k ⊕ V i . In this case, for α = (α 1 , . . . , α s ) ∈ Z s ≥0 , we denote by V α the closure of the vector space generated by elements of the form x 1 . . . x s , where x i is an element of the image V α i i of the ideal V α i i of A i . Note that this is not in general an ideal of A. We also denote by V α V n i the set V β where β j = α j , except for β i = α i + n. Lemma 2.1.3. Let k be a field. Let A 1 , . . . , A s be a family of local noetherian complete k-algebras of dimension 1 with maximal ideals V i and A i /V i = k. Suppose that for all i, there is an element z i ∈ A i such that A i has no z i -torsion and that for all n large enough, z i V n i = V n+1 i . Let A be a k-algebra with maximal ideal m that is nearly the sum of the family (A i ). Moreovoer, suppose that there exist integers N 0 ≥ t 0 such that for all i and j, V j V n i ⊂ V n−t 0 i for all n ≥ N 0 . Then e(A) = s i=1 e(A i ) . Note that if we had the stronger property that V i V j = 0 for all i = j the result would be trivial. Proof. Observe first that there exist integers N ≥ t such that for all α, for all i, V α V n i ⊂ V n−t i for all n ≥ N. Indeed, V α ⊂ V j 1 . . . V jr where {j 1 , . . . , j r } ⊂ {1, . . . , s} is the set of indices with α j > 0. Then if n ≥ rN 0 , then V j 1 . . . V jr V n i ⊂ V n−rt 0 i . So we can take N = sN 0 and t = st 0 . If α j > N then V α ⊂ V α j −t j ⊂ V j . So if there are two different indices i, j with α i > N and α j > N then V α = 0 as it is contained in V i ∩ V j . If |α| > sN then there exists at least one i with α i > N so V α = (V α ∩ V j ). Fix some index i. Let n > 0. Then m n = |α|=n V α . So if n > Ns then (m n ∩ V i ) = α (V α ∩ V i ) and the only contributing terms are those with α j ≤ N for all j = i, and α i > N. For such an α, we have V α ⊂ V n−sN i as α i ≥ n − (s − 1)N. Let r = sN, so that V n−r i ⊂ V i for all n > r. So for all n > r and all such α we have V α ⊂ V i , so finally for n > r we have: (1) (m n ∩ V i ) = |α|=n,α j ≤N if j =i V α We see that V n i ⊂ (m n ∩ V i ) ⊂ V n−r i for all n > r. Note that (m n ∩ V i ) is an ideal of A i , which we denote by W i,n . We know that z i V n i = V n+1 i for all n large enough, so by the formula (1) for W i,n we see that z i W i,n = W i,n+1 for all n large enough. In A i , multiplication by z i induces an isomorphism from V n i to V n+1 i and from W i,n to W i,n+1 , so it also induces an isomorphism from V n−r i /W i,n to V n+1−r i /W i,n+1 for all n large enough. Note that these vector spaces are finite-dimensional, so they have the same dimension, as dim k V n−r i /V n i is finite for all n. We consider the inclusions V n i ⊂ W i,n ⊂ V n−r i ⊂ W i,n−r ⊂ V n−2r i We know that, for all n ≫ 0, dim k V n−r i /V n i = dim k V n−2r i /V n−r i = re(A i ) and dim k V n−r i /W i,n = dim k V n−2r i /W i,n−r , which gives that dim k W i,n−r /W i,n = re(A i ). We now go back to A. For all n ≫ 0 we have that dim k (m n−r /m n ) = re(A). On the other hand, we have seen that for all n ≫ 0, m n = ⊕ i (m n ∩ V i ), so m n−r /m n is isomorphic to ⊕ i (m n−r ∩ V i )/(m n ∩ V i ) = ⊕ i (W i,n−r /W i,n ). So re(A) = s i=1 re(A i ), and so e(A) = i e(A i ). 2.2. Hilbert-Samuel multiplicity of the special fiber. Let R be a discrete valuation ring with uniformizer π and residue field k. Let A be a local R-algebra with maximal ideal m, and let M be an A-module of finite type. We denote by e R (A, M) the Hilbert-Samuel multiplicity of M ⊗ R k as an A ⊗ R kmodule, with respect to the ideal m ⊗ R k. When M = A we just write e R (A) instead of e R (A, A), and we omit the subscript R when the choice of the ring is clear from the context. Proof. Let n ≥ 0 be an integer. Then S/m n S is a quotient of S/(m T S) n , so len T (S/m n S ) ≤ len T (S/(m T S) n ). Morevoer, Proof. Note that B is also a local complete noetherian local R-algebra which is a domain. Indeed, A is henselian and B is a finite A-algebra, so B is a finite product of local rings, and so it is a local ring as it is a domain. len T (S/m n S ) = n−1 i=0 dim k T m i S /m i+1 S = [k S : k T ] n−1 i=0 dim k S m i S /m i+1 S = [k S : k T ] len S (S/m n S ) so finally len S (S/m n S ) ≤ [k S : k T ] len T (S/(m T S) n ) which It is enough to prove the result when πB ⊂ A, as B is generated over A by a finite number of elements of the form x/π n for x ∈ A. We have an exact sequence of R-modules: 0 → A → B → B/A → 0 After tensoring by k over R we get the exact sequence: 0 → B/A → A ⊗ R k → B ⊗ R k → B/A → 0 Indeed, (B/A) ⊗ R k = BR = Z p , C = R[[X]], A n = R[[pX, X n ]] ⊂ C for n ≥ 1, B n = R[[pX, pX 2 , . . . , pX n−1 , X n ]] ⊂ C for n ≥ 1. We check easily that A n ⊂ B n ⊂ C and that C is finite over A n , and A n is not equal to B n if n > 2. We compute that e(A n ) = e(B n ) = n, and e(C) = 1. So we see that in Proposition 2. 2.3. Change of ring. We suppose now that R is the ring of integers of a finite extension K of Q p . If K ′ is a finite extension of K, we denote by R ′ its ring of integers. Proposition 2.3.1. Let K ′ be a finite extension of K, with ramification degree e K ′ /K . Let A be a local noetherian R ′ -algebra. Then e R (A) = e K ′ /K e R ′ (A). Proof. Suppose first that K ′ is an unramified extension of K, and let k and k ′ be the residue fields of K and K ′ respectively, and let π be a uniformizer of R and R ′ . Then A ⊗ R ′ k ′ = A ⊗ R k = A/πA. So e R (A) = e(A/πA) = e R ′ (A). Suppose now that K ′ is a totally ramified extension of K. Let u be an Eisenstein polynomial defining the extension, so that R ′ = R[X]/u(X), and u(X) = X s where s = [K ′ : K]. Then A ⊗ R k = A ⊗ R ′ (R ′ ⊗ R k) = A ⊗ ′ R (k[X]/X s ) = (A ⊗ R ′ k) ⊗ k k[X]/X s . So e R (A) = se R ′ (A) = [K ′ : K]e R ′ (A). For the general case, let R 0 be the ring of integers of the maximal unramified extension K 0 of K in K ′ , then e R (A) = e R 0 (A) and e R 0 (A) = [K ′ : K 0 ]e R ′ (A) which gives the result. We recall the following result, which is [BM02, Lemme 2.2.2.6]: Lemma 2.3.2. Let A be a local noetherian R-algebra, with the same residue field as R and A is complete and topologically of finite type over R. Let K ′ be a finite extension of K, and A ′ = R ′ ⊗ R A. Suppose that A ′ is still a local ring. Then e R (A) = e R ′ (A ′ ). 3. Rigid geometry and standard subsets of the affine line 3.1. Quasi-affinoids. We recall some definitions and results from [LR00]. Let F be a finite extension of Q p , with ring of integers R. We denote by R n,m = R[[x 1 , . . . , x n ]] y 1 . . . , y m ′ the F -algebra R[[x 1 , . . . , x n ]] y 1 . . . , y m ⊗ R F , and we say that such an algebra is quasi-affinoid. We say that it is of closed type if n = 0, and of open type if m = 0. In general, we call a F -algebra quasi-affinoid if it is a quotient of an R n,m for some n and m. We can attach canonically to such an algebra A a rigid space X A , which has the property that A is the ring of bounded functions on X. Such a rigid space is called quasi-affinoid. If X is a quasi-affinoid rigid space, we denote by A(X) the set of bounded functions on X, and by A 0 (X) the set of functions on X bounded by 1. 3.1.1. R-subdomains. As in the case of affinoid algebras and rigid spaces, we define some special subsets. Let X be a quasi-affinoid rigid space. Let h, f 1 , . . . , f n , g 1 , . . . , g m be elements of A(X) that generate the unit ideal of A(X). A quasi-rational subdomain of X is a subset U of the form {x, |f i (x)| ≤ |h(x)| ∀i and |g i (x)| < |h(x)| ∀i} (see [LR00,Definition 5.3.3]). In contrast to the case of affinoid rigid spaces, it is not necessarily true that a quasirational subdomain of a quasi-rational subdomain of X is itself a rational subdomain of X, see [LR00, Example 5.3.7]. We recall the definition of a R-subdomain of X ([LR00, Definition 5.3.3]): the set of R-subdomains of X is defined as the smallest set of subsets of X that contains X and is closed by the operation of taking a quasi-rational subdomain of an element of this set . An important result is the following: {x, |x − a| < |b|} with a, b ∈ Q p , b = 0 (open disk), or of the form {x, |x − a| ≤ |b|} with a, b ∈ Q p , b = 0 (closed disk). From now on, when we write "disk" we always mean "rational disk". Definition 3.2.2. We say that a subset X of P 1 (Q p ) is a connected standard subset if it is of one of the following forms: (2) P 1 (Q p ) \ ∪ n i=1 D i where the D i are rational disks, and D i and D j are disjoint if i = j (unbounded connected standard subset). (1) D 0 \ ∪ n i=1 D i One checks easily that a bounded connected standard subset is an R-subdomain of the rigid affine line, and even a quasi-rational subset. Note that if X is a connected standard subset, then the disks D i , their radii, and the integer n are entirely determined. A (1) {x, |b| < |x − a| < |c|} for some a, b, c ∈ Q p with b = 0 (2) {x, |x−a| ≤ |b| and for all i ∈ {1, . . . , N}, |x−α i | ≥ |β i |} for some a, b, α i , β i ∈ Q p with b = 0 (that is, a connected standard subset of closed type). Then we have the following result: Theorem 3.2.5 (Theorem 4.5 of [LR96]). An R-subdomain of Q p is a finite union of special sets. Proof. We recall the following property of quasi-affinoid spaces of open type ([LR00, Proposition 5.3.9]), which we will use repeatedly: let f ∈ A(X), then the set {x, |f (x)| = f X } is a union of Zariski components of X. The set U is an R-subdomain of a disk by Theorem 3.1.1, hence a finite union of special sets by Theorem 3.2.5. We write U as (∪ n i=1 Y i ) ∪ (∪ m i=1 C i ) where the Y i are as in (2) of Definition 3.2.4 and the C i as in (1) of this definition. Note that the Y i are connected standard subsets of closed type, hence we can assume that they are pairwise disjoint by Lemma 3.2.3, and that each of them is non-empty. So there exists an ε > 0 such that for all x ∈ Y i , y ∈ Y j with i = j, we have |x − y| ≥ ε. Fix an i, and write Y i = D(a, r 0 ) + \ ∪ N j=1 D(α j , r j ) − . Suppose that there is an η > 0, η < ε such that U ∩ {x, r 0 < |x − a| < r 0 + η} is empty. Then X ′ = φ −1 (D(a, r 0 + η) − ) is union of Zariski components of X, hence a quasi-affinoid space of open type. Consider f (x) = x − a ′ for any a ′ ∈ D(a, r 0 ) + , then φ * f X ′ = r 0 (where φ * f = f • φ). Then Y i ⊂ {x, |x − a ′ | = r 0 } for all such a ′ , as Y i is Zariski connected. So Y i = ∅ as we can take a ′ ∈ Y i , a contradiction. So for all η > 0 with η < ε, the set U ∩ {x, r 0 < |x − a| < r 0 + η} is not empty. It does not meet any of the Y j for j = i. This means that for some ℓ 0 , the set C ℓ 0 is of the form {x, ρ 1 < |x − a ′ | < ρ 2 } for some ρ 1 ≤ r 0 < ρ 2 and a ′ ∈ D(a, r 0 ) + . Similarly, we also have for all j that for some ℓ j , the set C ℓ j is of the form {x, ρ 1,j < |x − α ′ j | < ρ 2,j } for some α ′ j ∈ D(α j , r j ) − and ρ 1,j < r j ≤ ρ 2,j . We set Y ′ i = D(a ′ , ρ 2 ) − \ ∪ N j=1 D(α ′ j , ρ 1,j ) + . Then Y i ⊂ Y ′ i ⊂ U,∪ i Y ′ i ) ∪ (∪ i C i ) which is finite union of standard subsets of open type, as stated. Assume now that X is Zariski connected. We can write U uniquely as a disjoint union of connected standard subsets of open type. Let Y be one of these subsets, which we write as D(a 0 , 3.3.1. Rings of functions of standard subsets. Let X ⊂ Q p be a bounded connected standard subset. Then X is the set of points of a well-defined quasi-affinoid space X which is a rational subdomain of an open disk (defined by strict inequalities). We define by A(X) the set of bounded analytic functions on X and A 0 (X) the set of analytic functions on X bounded by 1. r 0 ) − \ ∪ m i=1 D(a i , r i ) + . It is enough to show that U ⊂ D(a 0 , r 0 ) − . The set {x, |a − x| = r 0 } is) − , does not meet U. Let f (x) = 1/(x − b). Then φ * f X = 1/r 0 , and |φ * f (x)| = 1/r 0 for all x ∈ φ −1 (D(a, r 0 ) + ), and |φ * f (x)| < r 0 if |φ(x) − a| > r 0 . So X = φ −1 (D(a, r 0 ) + ) as X is Zariski connected. Now let f (x) = x − a. Then φ * f X = r 0 . For all x ∈ φ −1 (D(a 0 , r 0 ) − ), we have φ * f (x)| < φ * f X . As X is Zariski connected, this means that X = φ −1 (D(a 0 , r 0 ) − ), what we wanted. Let X ⊂ P 1 (Q p ) be an unbounded connected standard subset, which is not equal to all of P 1 (Q p ). Let f be a homography with Q p -coefficients with its pole outside of X, then Y = f (X) is a bounded connected standard subset of Q p , so A(Y ) and A 0 (Y ) are well-defined. We define A(X) and A 0 (X) to be the functions of X of the form u • f for u ∈ A(Y ) and A 0 (Y ) respectively. It is clear that this does not depend on the choice of f , as different choices of f give rise to bounded connected standard subsets coming from isomorphic quasi-affinoids. Let now X be a standard subset. It can be written uniquely as X = ∪ n i=1 X i where the X i are disjoint connected standard subsets. Then we set A(X) = ⊕ n i=1 A(X i ) and A 0 (X) = ⊕ n i=1 A 0 (X i ). 3.3.2. Subsets defined over a field. If E is a finite extension of Q p , denote by G E its absolute Galois group. We say that X ⊂ P 1 (Q p ) is defined over F if σ(X) = X for all σ ∈ G F . Let E be a finite extension of Q p . The field of definition of X over E is the fixed field of {σ ∈ G E , σ(X) = X}. The field of definition of X is the field of definition of X over Q p . Then X is defined over F if and only if F contains the field of definition of X. 3.3.3. Standard subsets defined over a field. Let X be a standard subset. Suppose that X is defined over E. In this case G E acts on A(X) and A 0 (X) by (σf )(x) = σ(f (σ −1 x)). If F is a finite extension of E, we write A F (X) and A 0 F (X) for A(X) G F and A 0 (X) G F . So for example, if X = D(0, 1) − , then X is defined over Q p , and A 0 (X) is O Cp [[x]] with G Qp acting on the coefficients. So A 0 F (X) = O F [[x]] for any finite extension F of Q p . Proposition 3.3.1. Let X = D(a 0 , r 0 ) − \ ∪ n i=1 D(a i , r i ) + or X = P 1 (Q p ) \ ∪ n i=1 D(a i , r i ) + be a connected standard subset of P 1 (Q p ) , with a i ∈ Q p for all i, and the sets D(a i , r i ) + are pairwise disjoint for i > 0. For each i, let t i ∈ Q p be such that |t i | = r i . Let E be the finite extension of Q p generated by the elements a i and t i . Then X is defined over E, and for any finite extension F/E, we have: A F (X) = {f, f (x) = i≥0 c i,0 x − a 0 t 0 i + n j=1 i>0 c i,j t j x − a j i with c i,j ∈ F for all i, j and {c i,j , 0 ≤ j ≤ n, i ≥ 0} bounded } if X is bounded and A F (X) = {f, f (x) = c 0 + n j=1 i>0 c i,j t j x − a j i with c i,j ∈ F for all i, j and {c i,j , 0 ≤ j ≤ n, i ≥ 0} bounded } if X is unbounded. Moreover, f X = sup i,j |c i,j | if f is written as above. If we write f 0 = i≥0 c i,0 x−a 0 t 0 i (or f 0 = c 0 in the unbounded case), and f j = i>0 c i,j t j x−a j i for j > 0 so that f = n i=0 f i then f X = max 0≤i≤n f i X . In particular, f ∈ A 0 F (X) if and only if c i,j ∈ O F for all i, j. Proof. The fact that any element of A F (X) can be written this way is a consequence of the description of the ring of functions of a quasi-rational subsets, as described in [LR00, Proposition 5.3.2]. Let f be as in the statement of the Proposition, and let M = sup i,j |c i,j |. Then it is clear from the formula that for all x ∈ X, the series defining f (x) converges and that |f (x)| ≤ M. As F has discrete valuation, the sup defining M is in fact a maximum. Let us show that f X = M. Fix first a j such that there exists an i with |c i,j | = M. For simplicity of notation we will assume that j = 0, the other cases being similar. Let i 0 be the smallest index such that |c i 0 ,0 | = M. Let ρ ′ < r 0 be such that |a j | < ρ ′ for all j > 0. Note that for all j > 0, we have r j < ρ ′ and D(a j , r j ) + ⊂ D(0, ρ ′ ). Fix ρ with ρ ′ < ρ < r 0 . Let x ∈ Q p with ρ < |x| < r 0 , so that x ∈ X. For all j > 0 and all i we have Remark 3.3.2. The description of A F (X) is similar to the result given by the Mittag-Leffler theorem (see [Kra83]) in the situations studied by Krasner. Our situation is slightly different as are we are considering subspaces of P 1 (Q p ) that are "open", and simpler as we have only a finite number of "holes". |c i,j (t j /(x − a j )) i | ≤ M(ρ ′ /ρ) i ≤ M(ρ ′ /ρ) < M. For i < i 0 , we have |c i,0 (x/t 0 ) i | < M ′ for some M ′ < M as |c i,0 | < M. For i < i 0 we have |c i,j (x/t 0 ) i | < M(|x|/r 0 ) i 0 as |c i,0 | ≤ M and |(x/t 0 ) i | < |(x/t 0 ) i 0 |. Finally |c i 0 ,0 (x/t 0 ) i 0 | = M(|x|/r 0 ) i 0 . By taking |x| close enough to r 0 , we get that |c i 0 ,0 (x/t 0 ) i 0 | > |c i,0 (x/t 0 ) i | for all i = i 0 , and |c i 0 ,0 (x/t 0 ) i 0 | > |c i,j (t j /(x − a j )) i | for all j > 0 and all i. So |f (x)| = |c i 0 ,0 (x/t 0 ) i 0 | = M(|x|/r 0 ) i Proposition 3.3.3. Let X be a standard subset defined over E. Let F be a finite extension of E. Then A F (X) = F ⊗ E A E (X), and O F ⊗ O E A 0 E (X) ⊂ A 0 F (X), with A 0 F (X) finite over O F ⊗ O E A 0 E (X). If F/E is unramified, then this inclusion is an isomorphism. Note that we do note assume that the conditions of Proposition 3.3.1 are satisfied. Proof. We define a map φ : F ⊗ E A E (X) → A F (X) by φ(a ⊗ f ) = af . Let us describe the inverse ψ of φ. Let Q = G E /G F . If a is in F and f ∈ A F (X), σ(a) and σ(f ) are well-defined for σ ∈ Q as a and f are invariant by G F . Moreover, for a ∈ F , we have that tr F/E (a) = σ∈Q σ(a). Let (e 1 , . . . , e n ) be a basis of F over E, and (u 1 , . . . , u n ) be the dual basis with respect to tr F/E , that is, tr F/E (e i u j ) = δ i,j . One checks easily that for σ ∈ G E , we have n i=1 e i σ(u i ) = 1 if σ ∈ G F , and 0 otherwise. For f ∈ A F (X), we set t i (f ) = σ∈Q σ(u i f ). Let ψ(f ) = n i=1 e i ⊗ t i (f ). Let us check that ψ is the inverse of φ. Let f ∈ A F (X), and f ′ = φ(ψ(f )). Then f ′ = i e i Q σ(u i )σ(f ) = Q σ(f ) ( i e i σ(u i )), so f ′ = f . Let f ∈ A E (X), and a ∈ F . Let g = φ(a ⊗ f ). Then t i (g) = tr F/E (au i )f , as σ(f ) = f for all σ ∈ Q. So ψ(g) = i e i ⊗ tr F/E (au i )f = i e i tr F/E (au i ) ⊗ f as tr F/E (au i ) ∈ E. Then we check that i e i tr F/E (au i ) = a, so ψ(φ(a ⊗ f )) = a ⊗ f . So we see that ψ is the inverse map of φ, so φ is an isomorphism. We see that φ induces a map φ 0 from O F ⊗ O E A 0 E (X) to A 0 F (X). When F/E is unramified, we can choose (e i ) and (u i ) to be in O F , and in this case the restriction ψ 0 of ψ to A 0 F (X) maps into O F ⊗ O E A 0 E (X) , and so ψ 0 is the inverse map of φ 0 , and so φ 0 is an isomorphism. 3.3.4. Some algebraic results. Let X be a standard subset of P 1 (Q p ) that is defined over E for some finite extension E of Q p . Let F be a finite extension of E. We say that X is irreducible over F if it can not be written as a finite disjoint union of standard subsets of P 1 (Q p ) that are defined over F . There exists a unique decomposition of X as a finite disjoint union of standard subsets of P 1 (Q p ) that are irreducible over F . A standard subset is connected if and only if it is irreducible over any field of definition. Lemma 3.3.4. Let X be a connected standard subset of P 1 (Q p ). Then A(X) is a domain, and A 0 (X) is a local ring. Suppose that moreover X is defined over E. Then A 0 E (X) is a local ring which has the same residue field as E. Proof. Let m be the ideal in A 0 (X) of functions f such that |f (x)| < 1 for all x in X. Then m is a closed ideal, and it is maximal. Indeed, consider the description of A(X) given in Proposition 3.3.1. We see that m contains the constant functions with values in m Cp , (x − a 0 )/t 0 and the t j /(x − a j ). So A 0 (X) \ m is the set of non-zero constant functions with values in O × Cp so all functions that are not in m are units. For A 0 E (X), note that the set of constant functions on X that are in A 0 E (X) is O E . Lemma 3.3.5. Let X be defined and irreducible over E, and let X = ∪ r i=1 X i its decompo- sition in a finite union of connected standard subsets. Let F be the field of definition of X 1 over E. Then the restriction map A 0 (X) → A 0 (X 1 ) induces an O E -linear isomorphism A 0 E (X) → A 0 F (X 1 ) . Note in particular that: [F : E] is the number of connected components of X, and the isomorphism class of A 0 F (X 1 ) as an O E -algebra does not depend on the choice of X 1 . Proof. The group G E acts transitively on the set of the (X i ) as X is irreducible, and G F is the stabilizer of X 1 . We fix a system (σ i ) of representatives of G E /G F , numbered so that σ i (X 1 ) = X i for all i. Let f be an element of A 0 E (X). First note that f is invariant under the action of G F , so f |X 1 is in A 0 F (X 1 ). Moreover, we have that for all x ∈ X i , f (x) = σ i ((σ −1 i f )(σ −1 i (x)) = σ i (f |X 1 (σ −1 i x)) So f |X i is entirely determined by f |X 1 , so the restriction map is injective, and moreover for any f ∈ A 0 F (X 1 ) the formula above defines an element of A 0 E (F ), so the restriction map is bijective. Corollary 3.3.6. If X is defined and irreducible over E then A E (X) is a domain, and A 0 E (X) is a local ring. Proof. We apply Lemma 3.3.5: A 0 E (X) is isomorphic as a ring to A 0 F (X 1 ), which is local. Definition 3.3.7. If X is defined and irreducible over E, we denote by k X,E the residue field of A 0 E (X). By construction, k X,E is a finite extension of k E . In the notation of Lemma 3.3.5, we have k X,E = k F (which does not depend on the choice of X 1 ). Complexity of standard subsets Algebraic complexity of a standard subset over a field of definition. 4.1.1. Definition. Recall that we defined e in Section 2.2. Definition 4.1.1. Let X be a standard subset of P 1 (Q p ) that is defined over E. If X is irreducible over E, we define the complexity of X over E to be: c E (X) = [k X,E : k E ]e O E (A 0 E (X)) In general, let X = ∪ r i=1 X i be(X) = r i=1 c E (X i ). The above definition makes sense as A 0 E (X) is a complete noetherian local O E -algebra if X is irreducible over E by Corollary 3.3.6. Note that in particular if X is connected then c E (X) = e O E (A 0 E (X)) as k X,E = k E in this case. 4.1.2. Some general results on algebraic complexity. We now give explicit formulas for the complexity. It is enough to give such formulas for subsets X that are irreducible over E. Proposition 4.1.2. In the situation of Proposition 3.3.5, we have c E (X) = [F : E]c F (X 1 ). Note that c F (X 1 ) does not depend on the choice of X 1 among the connected components. Proof. Let e F/E be the ramification degree of F/E. We have that A 0 F (X 1 ) = A 0 E (X) as O E -algebras, and k X,E = k X 1 ,F = k F . So c E (X) = [k F : k E ]e O E (A 0 E (X)) = [k F : k E ]e O E (A 0 F (X 1 )) which is equal to [k F : k E ]e F/E e O F (A 0 F (X 1 )) = [F : E]c F (X 1 ) by Propo- sition 2.3.1. Proposition 4.1.3. Let X be a connected standard subset defined over E, and F a finite extension of E. Then c E (X) ≥ c F (X) with equality when F/E is unramified. Proof. From Proposition 2.3.2 we see that e(O F ⊗ O E A 0 E (X)) = e(A 0 E (X)) = c E (X),e(O F ⊗ O E A 0 E (X)) ≥ e(A 0 F (X)) with equality when F/E is unramified. Proposition 4.1.4. Let X be a standard subset defined over E, and F a finite extension of E. Then c E (X) ≥ c F (X) with equality when F/E is unramified. Proof. By additivity of the complexity we can assume that X is irreducible over E. Write X = ∪ n i=1 X i where each X i is connected. Let E i be the field of definition of X i over E, so that c E (X) = nc E 1 (X 1 ). Then F E i is the field of definition of X i over F . Suppose that the action of G F on the set of the irreducible components of X has r orbits, with representatives say X 1 , . . . , X r . Then c F (X) = r j=1 [F E j : F ]c F E j (X j ). We have that c F E j (X j ) ≤ c E j (X j ) by Proposition 4.1.3, and c E j (X j ) is independent of j, and equal to (1/n)c E (X). Moreover, [F E j : F ] is the cardinality of the orbit of X j , so r j=1 [F E j : F ] = n. Finally we get that c F (X) ≤ c E (X), with equality if and only if c F E j (X j ) = c E j (X j ) for all j, which happens in particular if F/E is unramified. 4.1.3. Does c E (X) characterize A 0 E (X)? We ask the following question: let X be defined and irreducible over E. Let R ⊂ A 0 E (X) be a local, noetherian, complete, O E -flat O E - subalgebra of A 0 E (X), such that R[1/p] = A E (X). Suppose moreover that R and A 0 E (X) both have residue field k E , and e(R) = e(A 0 E (X)), that is e(R) = c E (X). Do we have R = A 0 E (X) ? It follows from [BM02, Lemme 5.1.8] that the equality holds if c E (X) = 1, and in this case both rings are isomorphic to O E [[x]], and X is a disk of the form {x, |x − a| < |b|} for some a, b ∈ E. But as soon as c E (X) > 1 there are counterexamples. We give a few, with E = Q p . (1) Let X = {x, 0 < v p (x) < 1}. Then A 0 Qp (X) is isomorphic to Z p [[x, y]]/(xy − p). Let R be the closure of the subring generated by px, py and x − y. Here e(R) = c Qp (X) = 2. (2) Let X = {x, v p (x) > 1/2}. Then A 0 Qp (X) is isomorphic to Z p [[x, y]]/(x 2 − py) . Let R be the closure of the subring generated by y and px. Here e(R) = c Qp (X) = 2. (3) Let X = {x, |x − π| < |π|} where π p = p. Then A 0 Qp (X) is isomorphic to Z p [[x, y]]/(x p − p(y + 1) ). Let R be the closure of the subring generated by y and px. Here e(R) = c Qp (X) = p. 4.2. Computations of the algebraic complexity in some special cases. 4.2.1. Preliminaries. If P ∈ E[X], and a ∈ C p , let P a (X) = P (X + a) ∈ C p [X].E. Let λ ∈ R be such that D = {x, v E (x − a) > λ}. Let P ∈ E[X] <s , and write P a (X) = s−1 i=0 b i X i . Then: v E (b i ) ≥ v E (b 0 ) − iλ for all i. In particular, if v E (b 0 ) ≥ 0, then v E (b i ) ≥ −iλ for all i > 0, and if v E (b 0 ) > 0, then v E (b i ) > −iλ for all i > 0. Proof. Consider the Newton polygon of P a : if the conclusion of the Lemma is not satisfied, then it has at least one slope µ which is < −λ. So P a has a root y of valuation −µ > λ. Let b = a + y, then b is a root of P , so of degree < s over E. On the other hand, v E (b − a) = v E (y) > λ so b is in D, which contradicts the definition of s. A similar proof shows: Lemma 4.2.2. Let D be a closed disk defined over E, let s be the smallest degree over E of an element in D. Let a be in D of degree s over E. Let λ ∈ R be such that Proof. Let a ∈ D be as in the statement. As the complexity does not change by unramified extensions by Proposition 4.1.4, we can enlarge E so that E(a)/E is totally ramified. Let µ be the minimal polynomial of a over E, so that µ has degree s. D = {x, v E (x − a) ≥ λ}. Let P ∈ E[X] <s , and write P a (X) = s−1 i=0 b i X i . Then: v E (b i ) > v E (b 0 ) − iλ for all i > 0. In particular, if v E (b 0 ) ≥ 0, then v E (b i ) > −iλ for all i > 0. Let L/Q p be a finite extension. Let f ∈ O L [[T ]], f = i≥0 f i T i . We say that f is regular of degree n if f n ∈ O × L and f m ∈ m LWrite F = E(a). For ν ∈ Q, let F ν be the set {x ∈ F, v E (x) ≥ ν} (so that F 0 = O F ). Let λ be such that D = {x, v E (x − a) > λ}. Let also ρ ∈ F such that v E (ρ) = stλ, which is possible by the condition on r. Let L be a Galois extension of E containing F and an element u such that v E (u) = λ. Then A 0 L (D) is isomorphic to O L [[T ]], with T corresponding to (x − a)/u. Let E n be the subset of E[X] <s of polynomials that can be written as s−1 i=0 b i (X − a) i with v E (b i ) ≥ −(i + ns)λ. Note that by Lemma 4.2.1, E n is the set of polynomials in E[X] <s with v E (b 0 ) ≥ −nsλ. In fact E n is in bijection with the set F −nsλ by P → P (a), as any element of F can be written uniquely as P (a) for some P ∈ E[X] <s . Note that ρ −1 ∈ F −stλ . We fix R ∈ E t the unique polynomial such that R(a) = ρ −1 . We set α = Rµ t . We check that α is regular of degree st when seen as an element of A 0 L (D) = O L [[T ]]. Let also E ′ be the subset of E[X] <st of polynomials that can be written as st−1 i=0 b i (X − a) i with v E (b i ) ≥ −iλ. Then A 0 E (D) = { n≥0 P n α n , P n ∈ E ′ } and any element of A 0 E (D) can be written uniquely in such a way. Indeed: Let f ∈ A 0 E (D), which we see as an element of A 0 L (D) = O L [[T ] ]. Applying repeatedly the Weierstrass Division Theorem, f can be written uniquely as n≥0 P n α n with P n ∈ O L [T ] <st . The fact that f is in A 0 E (D) means that f is invariant under Gal(L/E). As α itself is invariant under this group, this means that each P n is invariant, and so P n ∈ E ′ (where we see E ′ ⊂ O L [T ] <st by T = (X − a)/u). We observe that E ′ = t−1 j=0 µ j E j . For 0 ≤ i < t, let (U i,j ) 1≤j≤s be a basis of E i as an O E -module, where we take U 0,1 = 1, and v E (U 0,j (a)) > 0 for j > 1. We can satisfy this condition as taking a basis of E 0 is the same as taking a basis of O F over O E , and F is totally ramified over E. Write Y i,j = U i,j µ j and Z = α (note that Y 1,0 = 1). Then A 0 E (D) is a quotient of O E [[Y i,j , Z]], hence the ring A = A 0 E (D)/π E is a quotient of k E [[Y i,j , Z]]. Let y i,j , z be the images of Y i,j , Z in A, so that the maximal ideal m of A is generated by z and the y i,j for (i, j) = (1, 0). Let f ∈ E ′ , and suppose that when we write f (X) = st−1 i=0 b i (X − a) i , we have for all i, that v E (b i ) > −iλ. The condition implies that f = π L g for some g ∈ A 0 L (D), where π L is a uniformizer of L. Let n = e L/E , so that nv E (π L ) ≥ 1, then f n /π E ∈ A 0 L (D) ∩ A E (D) = A 0 E (D). So the image of f n in A is zero, hence the image of f in A is nilpotent. We see that the Y i,j for (i, j) = (1, 0) satisfy this condition, as t is the smallest integer such that there exists an element of F of valuation stλ, hence y i,j is nilpotent for all (i, j) = (1, 0). Let I be the ideal generated by the y i,j for (i, j) = (1, 0). Then I is nilpotent. We deduce that the conditions of Lemma 2.1.1 are satisfied. So e(A) = dim k A/(z), and we see easily that the y i,j , 1 ≤ i ≤ s and 0 ≤ i < t form a k-basis of A/(z). Holes. Proposition 4.2.5. Let X = P 1 (Q p ) \ T where T = ∪ N i=1 D i is a G E -orbit of closed disks of positive radius r ∈ p Q , with each disk defined over a totally ramified extension of E. Let K be the field of definition of D 1 . Let s be the smallest ramification degree of K(a)/K for a ∈ D 1 . Let t be the smallest positive integer such that r st ∈ |E(a) × |. Then c E (X) = Nst. Proof. Write X ′ = P 1 (Q p ) \ D 1 , so that X ′ is defined over K, and let a ∈ D 1 as in the statement of the Proposition. Note that [K : E] = N. Let F = E(a). Note that K ⊂ F so E(a) = K(a). As the complexity does not change by unramified extensions, we can assume that F/E is totally ramified. We write [F : K] = s. Write D 1 as the set {x, v E (x−a) ≥ λ} for some λ ∈ Q. Let µ be the minimal polynomial of a over K, so that µ has degree s. Let also ρ ∈ F be such that v E (ρ) = stλ, which is possible by the condition on r. Let L be an extension of E containing a and an element u such that v E (u) = λ, and which is Galois over E. Let Q = {σ 1 , . . . , σ N } be a system of representatives in G E of G E /G K , numbered so that σ i D 1 = D i (so we take σ 1 = id). For f ∈ K(x), we denote by tr f ∈ E(x) the element N i=1 σ i f . Note that A 0 E (X) = {a + tr f, a ∈ O E , f ∈ A 0 K (X ′ )}. So we begin first by describing A 0 K (X ′ ). Let R be the unique element of F [x] <s such that R(a) = ρ. Note that when we write R(X) = b i (x − a) i , we have v E (b i ) > (st − i)λ for all i > 0 by Lemma 4.2.2. For n > 1, set α n = ρ µ t R µ t n−1 . Note that A 0 L (X ′ ) is isomorphic to O L [[Y ]], with Y corresponding to the function u/(x − a). In this isomorphism, observe that α n is regular of degree nst and is divisible by Y st . Let f = Y g ∈ Y A 0 L (X ′ ). Then I can write Y st−1 f = Y st g as n≥1 P n (Y )α n for P n ∈ O L [Y ] <st (there is no remainder as Y st and α 1 differ by a unit). So f = n≥1 Y 1−st P n (Y )α n . Write Y 1−st P n (Y ) = Q n (1/Y ), Q n (1/Y ) ∈ O L [1/Y ] <st . Finally, any element of A 0 L (X ′ ) can we written uniquely f = a 0 + n≥1 Q n (1/Y )α n . Note that β n = ρ −1 α n is in fact in A K (X ′ ). So the elements of A 0 L (X ′ ) that are in A 0 K (X ′ ) are those for which a 0 ∈ O K and ρQ n (1/Y ) (which is a polynomial in x of degree < st) is in K[x]. Let E ′ the set of elements P ∈ K[x] <st such that when we write P (x) = i≥0 b i (x − a) i , we have v E (b i ) ≥ (st − i)λ. Then we have shown that: A 0 K (X ′ ) =    a 0 + n≥1 P n (x) R(x) n−1 µ(x) tn , a 0 ∈ O K , P n ∈ E ′    For 0 ≤ j < t, let E j be the subset of K[x] <s of polynomials that can be written as s−1 i=0 b i (x − a) i with b i ∈ F , v E (b i ) ≥ (s(t − j) − i)λ. Note that by Lemma 4.2.2, E j is the subset of elements of K[x] <s with v E (b 0 ) ≥ s(t − j)λ, and if P ∈ E j then for all i > 0, v E (b i ) > (s(t − j) − i)λ. Moreover, E j is in bijection with the set F s(t−j)λ = {b ∈ F, v E (b) ≥ s(t − j)λ} by P → P (a). Indeed, if b ∈ F , it can be written uniquely as b = P (a) for some P ∈ K[x] <s as F = K(a). Note that by definition, for 0 < j < t, F s(t−j)λ does not contain an element of valuation s(t − j)λ. We note that E ′ = ⊕ t−1 j=0 µ j E j . We define bases for the E j as O K -modules as follows: fix δ j in F s(t−j)λ of minimal valuation (take δ 0 = 1, and note that v E (δ j ) > s(t − j)λ if j = 0). Let ̟ be a uniformizer of F , so that (1, ̟, . . . , ̟ s−1 ) is a basis of O F as an O K -module. Then let Q i,j ∈ E j be the polynomial such that Q i,j (a) = δ j ̟ i−1 for 1 ≤ i ≤ s. So we deduce a basis (P i,j ) 0≤j<t,1≤i≤s of E ′ as an O K -module by taking P i,j = Q i,j µ j . Finally let U i,j = P i,j /µ t ∈ A 0 K (X ′ ), and V = R/µ t = U 1,0 , so that the elements of A 0 K (X ′ ) can be written uniquely as a 0 + n≥0 i,j a i,j,n U i,j V n , with a 0 and the a i,j,n in O K . Consider such a function f with a 0 = 0 as an element of A 0 L (X ′ ) = O L [[Y ]], its image f ∈ k L [[Y ]]. If (i, j) = (1, 0), then U i,j goes to zero in k L [[Y ] ]. So f is equal to n a 1,0,n V n+1 , and V has valuation st as a series in Y . So the image is non-zero if and only if there exists an n such that a 1,0,n is in O × K , and then f has valuation st(n + 1) for the smallest such n. Let α be a uniformizer of K, so that O K = O E [α]. Let f i,j,ℓ,n = α ℓ U i,j V n , for 0 ≤ ℓ < N, so that elements of A 0 K (X ′ ) can be written uniquely as a 0 + n≥0 i,j,ℓ, a i,j,ℓ,n f i,j,ℓ,n ,f i,j,ℓ,n is ℓv E (α) + (i − 1)v E (̟) + v E (δ j ) and the leading term is Y s(t(n+1)−j) . So we can determine j and n from the leading term. Note also that v E (α) = 1/N, v E (̟) = 1/sN. As 0 ≤ ℓ < N and 0 ≤ i − 1 < s, we see that for a given j, the valuations of f i,j,ℓ,n and f i ′ ,j,ℓ ′ ,n are not equal modulo Z except if i = i ′ and ℓ = ℓ ′ . Using the description of A 0 E (X) from A 0 K (X ′ ) we see that: A 0 E (X) =    a 0 + n≥0 i,j N −1 ℓ=0 b i,j,ℓ,n tr α ℓ U i,j V n , a 0 ∈ O E , b i,j,ℓ,n ∈ O E    and elements can be written uniquely in such a way. We deduce this from the previous description by setting a i,j,n = ℓ b i,j,ℓ,n α ℓ . We write S i,j,ℓ,n = tr α ℓ U i,j V n . We also set Y i,j,ℓ = S i,j,ℓ,0 , and Z = S 1,0,0,0 = Y 1,0,0 . We denote by lowercase letters their images in A = A 0 E (X)/π E . Recall that L is an extension of E containing a, an element u such that v E (r) = λ, and Galois over E. Also, note that if i = j then σ i a and σ j a are not in the same disk, so v E (σ i a − σ j a) < λ. We also assume that the uniformizer π L of L satisfies v E (π L ) ≤ λ − sup v E (σ i a − σ j a). Let I be the ideal of A generated by the s i,j,ℓ,m for (i, j, ℓ) = (1, 0, 0). Then I is a nilpotent ideal. Indeed, consider f one of the elements S i,j,ℓ,m , that is, f = tr α ℓ U i,j V m . We see f as an element of A 0 L (X). When we write α ℓ U i,j V m as an element of O L [[Y ]], with Y = u/(x − a) as before, we see that in fact it is in π L O L [[Y ]], as either ℓ > 0 or (i, j) = (1, 0). So f is in π L A 0 L (X). As in the proof of Proposition 4.2.4, this means that the image of f in A is nilpotent. As I is generated by nilpotent elements, and A is noetherian, we see that I is nilpotent. Let us show that s 1,0,0,m − z m ∈ I for all m > 0. We write Z m − S 1,0,0,m as tr f for some f ∈ A 0 K (X ′ ) (up to a constant, which goes to zero in A anyway). To study f we work in A 0 L (X), then f is the part with poles in D = D 1 . Consider a product u i x−a i u j x−a j (with u i = σ i u, a i = σ i a). We see that if i = j it can be written as ε i u i x−a i + ε j u j x−a j with v E (ε i ) = v E (ε j ) = v E (u/(a i − a j )) ≥ v E (π L ) . So when we compute Z m = (tr V ) m , all the parts coming from the product of terms with poles in differents disks D i s are in π L A 0 L (X). So Z m − S 1,0,0,m = tr f with f ∈ π L A 0 L (X). We see f as an element of O L [[Y ]] as before, then f ∈ π L O L [[Y ]] , which means that when we write f = i,j,n a i,j,n U i,j V n , we have a 1,0,n ∈ π K O K for all n, and so the image of tr f in A is indeed in I. From this we deduce that the maximal ideal m of A is generated by z and I. Let us show that A has no z-torsion. Let f ∈ A 0 K (X ′ ) which we write as b i,j,ℓ,n S i,j,ℓ,n , where we can assume that each coefficient is either 0 or in O × E , and at least one coefficient is not zero. Let g = b i,j,ℓ,n f i,j,ℓ,n . Let Y s((n 0 +1)t−j 0 ) be the leading term. Then the leading coefficient comes from f i 0 ,j 0 ,ℓ 0 ,n 0 for a well-determined (i 0 , j 0 , ℓ 0 , n 0 ). Consider now Zf = tr h for some h ∈ A 0 K (X ′ ) (up to a constant in m E ). We write h = h 1 + h 2 where h 1 is the part coming from (σ 1 h)(σ 1 V ), and h 2 the part coming from the (σ i h)(σ j V ) where either i or j is not 1. From the previous computations, we see that the valuation of h 2 is strictly smaller than the valuation of g. On the other hand, the valuation of h 1 is the same as the valuation of g and its leading term is Y s((n 0 +2)t−j 0 ) , and as the valuation is the same it means that it comes from f i 0 ,j 0 ,ℓ 0 ,n 0 +1 which appears with a coefficient of the same valuation as the coefficient of f i 0 ,j 0 ,ℓ 0 ,n 0 in g, that is 0. So we see that when we write Zf = b ′ i,j,ℓ,n S i,j,. Write X = D \ T , where D is an open disk, T = ∪ m i=1 T i where each T i is a disjoint union of closed disks D i,j such that the T i are pairwise disjoint, with each defined and irreducible over E, and the field of definition of each D i,j is totally ramified over E. Then c E (X) = c E (D) + m i=1 c E (P 1 (Q p ) \ T i ). Proof. For simplicity we treat only the case where X = D 0 \ (D 1 ∪ D 2 ), with D 1 and D 2 being disjoint disks defined over E. The general case needs no new ideas but requires more complicated notation. Write X i = P 1 \ D i for i = 1, 2. In this case, each X i satisfies the conditions of Proposition 4.2.5. Also, denote D 0 by X 0 , it satisfies the conditions of Proposition 4.2.4. We fix a finite Galois extension L of E such that each of the disks that appear in the definition of X is defined over L and contains a point of L, and each radius that appears is in |L × |. So we write D i = D(a i , |u i |) ± , with a i and u i in L. Note that |u i /(a i − a j )| < 1 if {i, j} = {1, 2}, so v L (u i /(a i − a j )) ≥ 1. Let Y i = u i /(x − a i ) for i = 1, 2, and Y 0 = (x − a 0 )/u 0 . Then A 0 E (X i ) ⊂ O L [[Y i ]]. Let t 0 = e L/E . If h ∈ A 0 E (X i ) ∩ π t 0 L O L [[Y i ]], then h is in π E A 0 E (X i ). For 0 ≤ i ≤ 2, denote by A i the ring A 0 E (X i ),write A 0 E (X i ) = O E ⊕ W i for some O E -module W i , such that m i is the ideal generated by π E and W i . We have then that A 0 E (X) = O E ⊕ (⊕ 2 i=0 W i ) , and the maximal ideal of A 0 E (X) is the ideal generated by π E and the submodules W i , 0 ≤ i ≤ 2. Denote by We want to apply Proposition 2.1.3, which will give the result we want. Note first that the existence of the elements z i ∈ A i was established in the course of the proofs of Propositions 4.2.4 and 4.2.5. So we only need to find integers N, t such that V j V n i ⊂ V n−t i for all n > N and all i, j. Fix some f ∈ W i such that its image in V i is in V n i , and g ∈ W j for j = i. What we want to do is look at α k (f g), and show that it goes to zero in V k if k = i, and to an element of V n−t i in V i for k = i. For simplicity we do the proof only for i = 1 and j = 2, but there is no added difficulty when one of the indices is 0. α i : A 0 E (X) → W i the projection with respect to O E ⊕ W j ⊕ W k where {i, j, k} = {0, 1, 2}. Let A i = A i /π E , and V i ⊂ A i its maximal ideal. Note that A i = k ⊕ V i , V i is the image in A i of m i , hence also of W i . Let A = A 0 E (X)/π E . Then we get that A = k ⊕ (⊕ 2 i=0 V i ), and m = ⊕ 2 i=0 V i is Denote by Z 1 the element that was called Z in the proof of Proposition 4.2.5 applied to X 1 (which is also the element that was called V , as we are in the case where N = 1), and denote by τ the integer that was denoted by st. Then in O L [[Y 1 ]], Z 1 is equal to π L P +Y τ 1 U for some P ∈ O L [Y 1 ] <τ and U ∈ O L [[Y 1 ]] × . For m ≥ 0, write Z m 1 = j≥0 u m,j Y j 1 with u m,j ∈ O L . Then we have that v L (u m,j ) ≥ m − j/τ . On the other hand, we can write Y mτ 1 = i≥0 Q i Z i 1 with Q i ∈ π max(0,m−i) L O L [[Y 1 ]] . Let z 1 be the image of Z 1 in A 1 . Then as in the proof of Proposition 4.2.5, V 1 is generated by z 1 and a nilpotent ideal I of A 1 . Let t 1 be an integer such that I t 1 = 0. Then any element of V n 1 for n large enough is a multiple of z n−t 1 1 . Let f ∈ W 1 such that its image in V 1 is in V n 1 , then we can assume that f is divisible by Z n−t 1 1 . So when we write f as j f j Y j , we have v L (f j ) ≥ n − t 1 − j/τ . We see easily that for all integers a, b, we can write Y a 1 Y b 2 = a i=1 λ a,b,i Y 1 i + b i=1 µ a,b,i Y i 2 with λ a,b,i and µ a,b,i in O L , and v L (λ a,b,i ) ≥ a + b − i and v L (µ a,b,i ) ≥ a + b − i. Let g ∈ W 2 , which we see as an element of O L [[Y 2 ]]. We study first α 1 (f g). We have α 1 (f g) = j≥0 f j α 1 (Y j 1 g). As v L (f j ) ≥ n − t 1 − j/τ , all terms f j α 1 (Y j 1 g) for j ≤ (n − t 0 − t 1 )τ contribute elements that are in π t 0 L O L [[Y 1 ]]. Consider now α 1 (Y j 1 g) for j > (n − t 0 − t 1 )τ . It contributes to Y i 1 with a coefficent of valuation ≥ j − i. So all terms in Y i 1 with i ≤ (n − t 0 − t 1 )τ − t 0 are in π t 0 L O L [[Y 1 ]]. So we see that α 1 (f g) is in (π t 0 L O L [[Y 1 ]] + Y (n−t 2 )τ 1 O L [[Y 1 ]]) ∩ A 0 E (X 1 ) for t 2 = t 1 + 2t 0 . We have that Y (n−t 2 )τ 1 = i Q i Z i 1 with Q i ∈ π max(0,(n−t 2 −i)τ ) L O L [[Y 1 ]]. So finally, α 1 (f g) ∈ (π t 0 L O L [[Y 1 ]] + Z (n−t 3 ) 1 O L [[Y 1 ]]) ∩ A 0 E (X 1 ) for t 3 = t 2 + t 0 . From this we deduce that the image We see also that if n ≥ 2t 0 +t 1 , then α 2 (f g) goes to 0 in V 2 (and also clearly α 0 (f g) = 0). So we get the result we wanted by taking t = t 3 and N = t. 4.3. Combinatorial complexity of a standard subset with respect to a field. We give another definition of complexity of a standard subset. It is defined in more cases than the algebraic complexity, as we do not require X to be defined over E to define the complexity of X with respect to E. 4.3.1. Definition. Let X be a standard subset of Q p , and E be a finite extension of Q p . We define an integer γ E (X) which we call combinatorial complexity of X. Let D be a disk (open or closed). Let F be the field of definition of D over E. Let s be the smallest integer such that there exists an extension K of F , with e K/F = s, and K ∩ D = ∅. Let t be the smallest positive integer such that D can be written as {x, stv E (x − a) ≥ v E (b)} or as {x, stv E (x − a) > v E (b)} for elements a, b in K. Then we set γ E (D) = st. We also set γ E (P 1 (Q p )) = 0. If X is a connected standard subset, it can be written uniquely as D 0 \ ∪ n j=1 D j with D 0 an open disk or D 0 = P 1 (Q p ), D j a closed disk for j > 0, and the D j are disjoint for j > 0. We set γ E (X) = n j=0 γ E (D j ). Now let X be a standard subset. We can write uniquely X = ∪ s i=1 X i where X i is a connected standard subset and the X i are disjoint. Then we set γ E (X) = s i=1 γ E (X i ). We also define γ E (X) when X = ∪ s i=1 D i is a disjoint union of closed disks: in this case we set γ E (X) = i γ E (D i ). Some properties of the combinatorial complexity. Lemma 4.3.1. Let X be a standard subset. Let F/E be a finite extension. Then γ E (X) ≥ γ F (X), with equality when F/E is unramified, or when F is contained in the field of definition of X. Proof. It suffices to show that γ E (D) ≥ γ F (D), with equality when F/E is unramified, for any disk D (open or closed), and then it is clear from the definition. Proposition 4.3.2. Let X be a standard subset defined and irreducible over E, and write X = ∪ s i=1 X i its decomposition in connected standard subsets. Let E 1 be the field of definition of X 1 over E. Proof. We can assume that X is irreducible over E, as both multiplicities are additive with respect to irreducible standard subsets. Then γ E (X) = [E 1 : E]γ E 1 (X 1 ). Proof. We have γ E (X) = s i=1 γ E (X i ) = s i=1 γ E i (X i ). Observe first that γ E i (X i ) does not depend on i. Indeed, for all i there exists σ ∈ G E such that σ(X 1 ) = X i and σ(E 1 ) = E i . Such a σ transforms an equation {x, v E (x − a) ≥ v E (b)} (or {x, v E (x − a) > v E (b)}) of a disk Write now X = ∪X i where the X i are connected standard subsets, and let E i be the field of definition of X i . Then c E (X) = [E : E 1 ]c E 1 (X 1 ) by Proposition 4.1.2, and γ E (X) = [E : E 1 ]γ E 1 (X 1 ) by Proposition 4.3.2. So we can assume that X is a connected standard subset defined over E. Note that c E (X) = c E ′ (X) and γ E (X) = γ E ′ (X) for any finite unramified extension E ′ /E by Propositions 4.1.4 and 4.3.1. So we can enlarge E if needed to an unramified extension, and we can assume that we have written X = D \ ∪Y i satisfying the hypotheses of Proposition 4.2.6. So we have c E (X) = c E (D) + i c E (P 1 (Q p ) \ Y i ) by Proposition 4.2.6, and the analogous result for γ E follows from the definition. So we need only prove the equality for these standard subsets. Let Let now X = P 1 (Q p ) \ T , where T is defined and irreducible over E, and T = ∪ N i=1 D i where the D i are disjoint closed disks defined over a totally ramified extension of E. We have γ E (X) = γ E (D i ) = Nγ E (D 1 ) as the D i are G E -conjugates. Let F be the field of definition of D 1 . Then γ E (X) = Nγ F (D 1 ) = Nγ F (P 1 (Q p ) \ D 1 ). On the other hand, it follows from Propostion 4.2.5 that c E (X) = Nc F (P 1 (Q p ) \ D 1 ). Now the proof that γ F (P 1 (Q p ) \ D 1 ) = c F (P 1 (Q p ) \ D 1 ) is the same as in the case of a disk. So finally c E (X) = γ E (X). From now on we only write c E to denote eithe c E or γ E (so we can consider c E (X) even for X that is not defined over E, or for X a disjoint union of closed disks). For a connected standard subset Y of D(0, 1) − , written as D(a, r) − \ ∆ for some finite union of closed disks ∆, we define its outer part as D(a, r) − . If Y is any standard subset, we define its outer part as the union of the outer parts of its connected components. Note that if Y is defined over a field E, then so is its outer part Y ′ , and Y ′ is an approximation of Y . Let Y be a connected standard subset. If the outer part of Y contains 0, we define its circular part as follows: write Y as D(0, r) − \ ∪ n i=1 D i where the D i are disjoint closed disks. If none of the D i contains 0, we define the circular part of Y as D(0, r) − . If 0 is contained in one of the D i , say D 1 , then we define the circular part of Y as D(0, r) − \ D 1 . Note that the circular part of Y is defined over Q p , hence over the definition field of Y . The circular part of Y is an approximation of Y . Main results. Theorem 4.5.1. Let X be a standard subset of Q p defined over E. Let m be an integer such that c E (X) ≤ m. Then there exists a finite set E of finite extensions of E, depending only on E and m, such that X is entirely determined by the sets X ∩ F for all extensions F ∈ E. We can actually take the set E to be the set of all extensions of E of degree at most N for N depending only on E and m. Corollary 4.5.2. Let X be a standard subset of D(0, 1) − defined over E. Let m be an integer such that c E (X) ≤ m. Moreover suppose that there exists an ε > 0 such that for all x ∈ X, D(x, ε) − ⊂ X, and for all x ∈ X, D(x, ε) − ∩ X = ∅. Then there exists a finite subset P of D(0, 1) − , depending only on E, m, and ε, such that X is entirely determined by X ∩ P. Proof of Corollary 4.5.2. Let N be the integer as in Theorem 4.5.1. For each extension F of E of degree at most N, F ∩ D(0, 1) − can be covered by a finite number of open disks of radius ε, and we define a finite set P F by taking an element in each of these disks. Then we set P to be the union of the sets P F , which is finite as there is only a finite number of extensions of E of degree at most N. Remark 4.5.3. As is clear from the proof, the set P can be huge. However in practice for a given X we need only test points in a very small proportion of this subset. We give the proof of Theorem 4.5.1 in Section 4.5.5. We work by constructing a sequence (X i ) of approximations of X, such that each X i is defined over E and is an approximation of X i+1 and c E (X i+1 ) > c E (X i ), so that at some point we get X i = X. A(a, b) the annulus {x, a < v E (x) < b}. If c is a rational number, denote by C(c) the circle {x, v E (x) = c}. Sometimes we also write C(r) to denote the circle {x, |x| = r} when no confusion can arise. Notation. If a < b are rational numbers, denote by If t ∈ Q, we introduce denom(t) the denominator of t, which is the smallest integer d such that t ∈ (1/d)Z. Let v be the valuation on Q p that extends the normalized valuation on E. If x ∈ Q p , we write denom(x) for denom(v E (x)). Note that [E(x) : E] ≥ denom(x). Preliminaries. Lemma 4.5.4. Let Z be an irreducible standard subset defined over E, which is contained in the set C(λ) for λ ∈ (1/d)Z for d minimal. Then c E (Z) ≥ d. Proof. We can replace Z by the union of the outer parts of its connected components, as in can only lower the multiplicity. Write Z = ∪ n i=1 Z i with each Z i connected, so Z i is a disk. Let F be the field of definition of Z 1 , Then c E (Z) = [F : E]c F (Z 1 ) by Corollary 4.1.2. As Z ⊂ C(λ), we see that for all y ∈ Z 1 , e F (y)/E ≥ e E(y)/E ≥ d, so e F (y)/F ≥ d/e F/E . We have that c F (Z 1 ) ≥ e F (y)/F ≥ d/e F/E , and so [F : E]c F (Z 1 ) ≥ d, that is, c E (Z) ≥ d. Lemma 4.5.5. Let X = P 1 (Q p ) \ T where T is a disjoint union of closed disks defined over E and contained in C(λ) for λ ∈ (1/d)Z for d minimal. Then c E (X) ≥ d. Proof. Write T = ∪ n i=1 T i with each T i a closed disk. Let F be the field of definition of T 1 , then c E (X) = [F : E]c F (T 1 ) by Proposition 4.3.2. As T ⊂ C(λ), we see that for all y ∈ T 1 , e F (y)/E ≥ e E(y)/E ≥ d, so e F (y)/F ≥ d/e F/E . By definition of γ E , we have that c F (T 1 ) ≥ e F (y)/F ≥ d/e F/E , and so [F : E]c F (T 1 ) ≥ d, that is, c E (T ) ≥ d. Lemma 4.5.6. Let X be a standard open subset defined over E and contained in C(r) for some r > 0, and suppose that c E (X) ≤ m. Then X is contained in a union of at most m open disks of radius r contained in C(r). Proof. Let Y be the union of the outer parts of the connected components of X, so that X ⊂ Y , Y is defined over E and is a disjoint union of open disks, and c E (Y ) ≤ c E (X) ≤ m. So it is enough to prove the result for Y , but it is clear in this case. Proof. Let Y be the connected component of X containing P 1 (Q p ) \ C(r), so that X ⊂ Y , Y is defined over E and is of the form P 1 (Q p ) \ T where T is a disjoint union of closed disks, and c E (Y ) ≤ c E (X) ≤ m. So it is enough to prove the result for Y , but it is clear in this case. Note that s is a power of p by Theorem 1. Proof of Proposition 4.5.8. We show first that there exists a function ψ 1 E such that for all X a standard connected subset defined over E with c E (X) ≤ m, there exists an extension F of E with [F : E] ≤ ψ 1 E (m) and X ∩ F = ∅. We can write X as D \ Y for some open disk D. By Lemma 4.5.9, there exists an extension K of E of degree at most ψ 0 E (m) such that D contains a point in K and has a radius in |K × |. Moreover, c K (X) ≤ c E (X) ≤ m. By doing an affine transformation in K, we can assume that X is of the form D(0, 1) − \ Y . Then: either 0 ∈ Y , in which case K ∩X = ∅, or 0 ∈ Y . In the latter case, m > 1 and X is contained in a standard subset X ′ of the form D(0, 1) − \(D(0, r) + ∪Z) for some r ∈ p Q , r < 1, with c K (X ′ ) ≤ c K (X) (we take D(0, r) + to be the outer part of the irreducible component of Y containing 0). So we have c K (P 1 (Q p ) \ D(0, r) + ) + c K (P 1 (Q p ) \ Z) ≤ m − 1. Let s be the smallest integer such that r s ∈ |K × |, then c K (P 1 (Q p ) \ D(0, r) + ) = s and s ≤ m − 1. There exists an extension L of K of degree 2s such that L contains an element of norm ρ = √ r. The circle C = C(v E (ρ)) is contained in L ∩ D(0, 1) − \ D(0, r) + . Then Z ∩ C meets at most c K (P 1 \ Z) ≤ m − 2 open disks of radius ρ in C by Lemma 4.5.6. Let q be the cardinality of the residue field of E, so that q L ≥ q E . By replacing if necessary L by an unramified extension of L of degree 1+max(0, ⌊log q (m−1)⌋), we can assume that q L −1 > m−2, so that L∩C is not contained in the union of these open disks, and so (C ∩ L) \ (C ∩ Z) is not empty, hence L ∩ X is not empty. Finally, we notice that [L : E] ≤ 2(m − 1)ψ 0 E (m)(1 + max(0, ⌊log q (m − 1)⌋)). So we can take ψ 1 E (m) = 2(m − 1)ψ 0 E (m)(1 + max(0, ⌊log q (m − 1)⌋)) if m > 1, and ψ 1 E (1) = 1. Now we go back to the general case. Write X as a disjoint union of irreducible componenents over E. Each of them has complexity at most m, and it is enough to find a point in one of them. So we can assume that X is irreducible over E. Suppose now that X is irreducible over E: write X = ∪ s i=1 X i where the X i form a G E -orbit. Let F be the field of definition of X 1 , and s = [F : E]. Then c E (X) = sc F (X 1 ), so c F (X 1 ) ≤ m ′ = ⌊m/s⌋. There exists an extension K of F of degree at most ψ 1 F (m ′ ) such that K ∩ X 1 = ∅. As K is an extension of E of degree at most sψ 1 F (m ′ ), we see that we can take ψ E (m) = sup 1≤s≤m sup [F :E]=s sψ 1 F (⌊m/s⌋), which is finite as E has only a finite number of extensions of a given degree. Proof of Proposition 4.5.10. The proof is very similar to the proof of 4.5.8. We first define φ 1 E for X of the form P 1 \ Y for Y connected, which we can take to be φ 1 E (m) = φ 0 E (m)(1 + max(0, ⌊log q (m)⌋)) if m > 1, and φ 1 E (1) = 1. Indeed, after introducing K of degree at most φ 0 E (m) as before, and transforming Y to D(0, 1) + \ Z for some standard open subset Z, we can look for points of Y that are in the circle of radius 1 so we do not need to introduce the ramified extension L. We then take as before φ E (m) = sup 1≤s≤m sup [F :E]=s sφ 1 F (⌊m/s⌋). 4.5.5. Proof of Theorem 4.5.1. We work by constructing a sequence (X i ) of approximations of X, such that each X i is defined over E and is an approximation of X i+1 and c E (X i+1 ) > c E (X i ), so that at some point X i = X and we stop. We divide X in two parts Y and Z, each being defined over E. The first part Y is the union of connected components such that their outer part contains 0. The other part Z is the union of the other connected components. We have that c E (X) = c E (Y ) + c E (Z), and the outer part of Z does not contain 0. Let Y 0 be the circular part of Y , so that Y ⊂ X 0 . It is clear from the definition that Y 0 is an approximation of X (and of Y ). We write Y = Y 0 \ T , so that T is a union of closed disks that do not contain 0. Our first approximation of X will be X 0 = Y 0 . We now explain how to compute Y 0 . Observe first: Lemma 4.5.12. The set Z is contained in ∪ λ∈Q,denom(λ)≤m C(λ). Proof. By definition of Z, it is equal to the union of the Z λ = Z ∩ C(λ) for λ ∈ Q, each Z λ being a standard open subset. Suppose that there exists a λ ∈ Q with denom(λ) > m and Z λ is not empty. By Lemma 4.5.4, we see that c E (Z) ≥ c E (Z λ ) ≥ denom(λ) > m, which is not possible. Similarly to Lemma 4.5.12, but using Lemma 4.5.5 instead of Lemma 4.5.4, we see that: Lemma 4.5.13. The set T is contained in ∪ λ∈Q,denom(λ)≤m C(λ). As a consequence of Lemmas 4.5.12 and 4.5.13, we have: A(a, b) Lemma 4.5.14. Let x ∈ D(0, 1) − such that denom(x) > m. Then x ∈ Y 0 if and only if x ∈ X. Lemma 4.5.15. Write Y 0 = ∪ n i=1 A(a i , b i ) or Y 0 = D(0, b 0 ) − (∪ n i=1 A(a i , b i )), with b i−1 ≤ a i < b i for all i. Then i denom(a i ) + i denom(b i ) = c E (Y 0 ). Corollary 4.5.16. Let a < b be two rational numbers such that for all rational numbers c strictly between a and b, we have that denom(c) > m. Then either ⊂ X or A(a, b) ∩ X = ∅. Proof. By Lemma 4.5.14, A(a, b) ∩ X = A(a, b) ∩ Y 0 . So we can work with Y 0 . The result then follows from Lemma 4.5.15. Fix a sequence of rationals 0 = t 0 < t 1 · · · < t n = 1 such that for any rational number c strictly between t i and t i+1 , we have denom(c) > m. Extend this sequence to (t i ) i∈Z by setting t i+n = t i + 1. Choose for each i ∈ Z an element x i with t i < v E (x i ) < t i+1 . We can do this by taking the elements x i in some totally ramified extension L m of E, of degree bounded in terms of m. Then for each annulus A(t i , t i+1 ), we know whether it is contained in X (if x i ∈ X), or if it does not meet X (if x i ∈ X) by considering only X ∩ L m . Note that X being a standard subset, then if 0 ∈ X then there is an open disk around 0 contained in X, and otherwise there is an open disk around 0 that does not meet X; and likewise with ∞ instead of 0. Moreover, we only need to understand additionally whether C(t i ) ⊂ Y 0 for i ∈ Z in order to understand Y 0 . Let I be the set of indices such that both A(t i−1 , t i ) and A(t i , t i+1 ) are contained in X. If C(t i ) ⊂ Y 0 , then t i ∈ I, but the converse is not necessarily true. Let Y 1 = Y 0 (∪ i∈I C(t i )) (so Y 1 is entirely known at this step). Then Y 1 is an approx- i ∈ I, and let x 1 , . . . , x m imation of Y 0 and c E (Y 1 ) ≤ c E (Y 0 ). Let m 1 = m − c E (Y 1 ), then c E (T ) + c E (Z) ≤ m 1 , as c E (X) = c E (Y 0 ) + c E (T ) + c E (Z).1 +1 be such that v E (x j ) = t i for all j, and v E (x j − x j ′ ) = t i if j = j ′ . Then C(t i ) ⊂ Y 0 if x j ∈ X for all j, and C(t i ) does not meet Y 0 if none of the x j are in X. Proof. Suppose that x j ∈ X for all j, but C(t i ) is not contained in Y 0 . Then it means that x j ∈ Z for all j. But this is a contradiction by Lemma 4.5.6. Suppose that none of the x j are in X, but that C(t i ) ⊂ Y 0 . This means that x j is in T for all j. But this is a contradiction by Lemma 4.5.7. So we see how to determine whether C(t i ) ⊂ Y 0 for i ∈ I: choose an element x of valuation t i , compute if x is in X or not. After a finite number of such computations, one of the hypotheses Lemma 4.5.17 is satisfied, so we can conclude. Moreover, we can speed this up by noting that if denom(t i ) ≥ m/2 and i ∈ I, then C(t i−1 , t i+1 ) ⊂ Y 0 , by Lemma 4.5.15. So for such t i we do not have to do the computations. So finally we have computed Y 0 = X 0 our first approximation of X. From the method we used to compute X 0 , we see that for each E there is a non-decreasing function f E such that if c E (X) ≤ m, then we can compute X 0 by testing only if x ∈ X for elements x with [E(x) : E] ≤ f E (m). We now assume that we have computed an approximation X i of X defined over E, and we explain how to compute another approximation X i+1 of X such that X i is an approximation of X i+1 . Note that if c E (X i ) = m then X i = X so we are finished. We can write uniquely X = (X i \ T i ) ∪Z i where T i is a disjoint union of closed disks and Z i is a disjoint union of connected standard subsets that do not meet X i \ T i , and T i and Z i are defined over E. Let m i = m−c E (X i ). Note that c E (X) = c E (X i ) + c E (Z i ) + c E (T i ), so that c E (Z i ) + c E (T i ) ≤ m i . If there exists a point that is in X but not in X i , then Z i is not empty. By Proposition 4.5.8, it means that there exists an extension F/E with [F : E] ≤ ψ E (m i ) such that Z i ∩ F = ∅. Let Y i = (P 1 (Q p ) \ T i ) ∪ Z i . If there exists a point x that is in X i but not in X, then x is in T i but not in Z i , so x is not in Y i and so Y i is not P 1 (Q p ). We see that c E (Y i ) ≤ c E (Z i ) + c E (T i ) ≤ m i . So if Y i is not P 1 (Q p ),E] ≤ φ E (m i ) and F ⊂ Y i . So we see that we can determine whether X = X i by doing computations only in extensions of E of degree at most max(ψ E (m i ), φ E (m i )). If X = X i we explain how to compute an X i+1 . Suppose first that we have found some a ∈ Y i , and let F = E(a). We have that (X \ Y i ) ∩ D(a, |a|) − ⊂ D(a, r) + for some r < |a|, as T i ∩ D(a, |a|) − is a closed disk. Consider X ′ = Y i ∩D(a, |a|) − . Then it is a standard subset defined over F , with c F (X ′ ) ≤ c F (Y i ) + 1 ≤ m i + 1. We can compute an approximation X ′ 0 of X ′ defined over F in the same way that we computed the approximation X 0 of X. Then we define a standard subset X i+1 as follows: X i+1 coincides with X i outside of the G E -orbit of D(a, |a|) − ; D(a, |a|) − ∩ X i+1 = D(a, |a|) − ∩ X ′ 0 ; and X i+1 is defined over E. We check that X i+1 is an approximation of X, X i is an approximation of X i+1 and c E (X i+1 ) > c E (X i ). Suppose now that we have found some a ∈ Z i , and let F = E(a). Let X ′ = Z i ∩ D(a, |a|) − , it is an approximation of Z i and defined over F so c F (X ′ ) ≤ m i . We can compute an approximation X ′ 0 of X ′ defined over F in the same way that we computed the approximation X 0 of X. Then we define a standard subset X i+1 as follows: X i+1 coincides with X i outside of the G E -orbit of D(a, |a|) − ; D(a, |a|) − ∩X i+1 = D(a, |a|) − ∩X ′ 0 ; and X i+1 is defined over E. We check that X i+1 is an approximation of X, X i is an approximation of X i+1 and c E (X i+1 ) > c E (X i ). In both cases, we see that in order to compute X i+1 we needed only to test if x ∈ X for elements x with [E(x) : E] ≤ [F : E]f F (m i ) ≤ [F : E]f F (m), where F = E(a) satisfies [F : E] ≤ max(ψ E (m), φ E (m)). So we see how to compute the sequence of approximations of X. From the construction, we see that we need only to test if x ∈ X for elements x such that [E( x) : E] ≤ max F [F : E]f F (m), where the max is taken over extensions F such that [F : E] ≤ max(ψ E (m), φ E (m)). 5. Application to potentially semi-stable deformation rings 5.1. Definition of the potentially semi-stable deformation rings. We recall the definition and some properties of the rings defined by Kisin in [Kis08] (see also [Kis10]). Let ρ : G Qp → GL 2 (Q p ) be a potentially semi-stable representation. Then we know from [Fon94] that we can attach to ρ a Weil-Deligne representation WD(ρ), that is, a smooth representation σ : W Qp → GL 2 (Q p ), and an endomorphism N of Q 2 p such that Nσ(x) = p deg x σ(x)N for all x ∈ W Qp . We say that σ is the extended type of ρ, and σ |I Qp the inertial type of ρ, where I Qp is the inertia subgroup of W Qp . We make the following definition: Definition 5.1.1. A Galois type of dimension 2 is one of the following representations with values in GL 2 (Q p ): (1) a scalar smooth representation τ = χ⊕χ of I Qp , such that χ extends to a character of W Qp . (2) a smooth representation τ = χ 1 ⊕ χ 2 of I Qp , where both χ 1 and χ 2 extend to characters of W Qp . (3) if p > 2, a smooth representation τ = χ 1 ⊕ χ 2 of W Qp , such that χ 1 and χ 2 have the same restriction to inertia, and χ 1 (F ) = pχ 2 (F ) for any Frobenius element F in W Qp . (4) if p > 2, a smooth irreducible representation τ of W Qp . We call Galois types of the form (1) and (2) inertial types, and those of the forms (3) and (4) discrete series extended types. If ρ is a potentially semi-stable representation of G Qp of dimension 2 and p > 2, then we know from the classification of 2-dimension smooth representations of W Qp that either its inertial type is isomorphic to a Galois type of the form (1) or (2), or its extended type is isomorphic to a Galois type of the form (3) or (4) (if p = 2 there are other possibilities). Note that if the Galois type of ρ is of the form (2) and (4) then it is potentially crystalline (that is, the endomorphism N of the Weil-Deligne representation is zero), and that if ρ is potentially semi-stable but not potentially crystalline (that is, N = 0) then its Galois type is of the form (3). Definition 5.1.2. A deformation data (k, τ, ρ, ψ) is the data of: (1) an integer k ≥ 2. (2) a Galois type τ . (3) an continuous representation ρ of G Qp of dimension 2, with trivial endomorphisms, over some finite extension F of F p . (4) a continuous character ψ : G Qp → Q × p lifting det ρ such that ψ and χ k−1 cycl det τ coincide. If the type τ is a discrete series extended type, we will assume that p > 2. Let (k, τ, ρ, ψ) be a deformation data, and let E be a finite extension of Q p over which τ and ψ are defined, and such that its residue field contains F. Let R(ρ) be the universal deformation ring of ρ over O E , it is a local noetherian complete O E -algebra. Let R ψ (ρ) the quotient of R(ρ) that parametrizes deformations of determinant ψ. Then Kisin in [Kis08] defines deformation rings R ψ (k, τ, ρ) that are quotients of R ψ (ρ). We will use a refinement of these rings introduced in [Roz15], which are better for our purposes in view of Theorem 5.3.1. If the Galois type τ is an inertial type, we denote by R ψ (k, τ, ρ) the ring classifying potentially crystalline representations with Hodge-Tate weights (0, k−1), inertial type τ , determinant ψ with reduction isomorphic to ρ, as defined by Kisin in [Kis08]. If the Galois type τ is a discrete series extended type, we denote by R ψ (k, τ, ρ) the complete local noetherian O E -algebra which is a quotient of R ψ (ρ), classifying potentially semi-stable representations with Hodge-Tate weights (0, k − 1), extended type τ , determinant ψ with reduction isomorphic to ρ defined in [Roz15,2.3.3]. We know that R ψ (k, τ, ρ) is a complete flat O E -algebra, such that Spec R ψ (k, τ, ρ)[1/p] is formally smooth of dimension 1. A consequence of the properties of these potentially semi-stable deformation rings is the following: There is a bijection between the maximal ideals of R ψ (k, τ, ρ)[1/p] and the set of isomorphism classes of lifts ρ of ρ of determinant ψ, potentially crystalline of inertial type τ (resp. potentially semi-stable of extended type τ ), and Hodge-Tate weights 0 and k − 1. In this bijection, a maximal ideal x, corresponding to a finite extension E x of E, corresponds to ρ x : G Qp → GL 2 (E x ). The Breuil-Mézard conjecture gives us some information about these rings ( [BM02], proved in [Kis09], [Paš15], [Paš16]; and [Roz15] for the cases of discrete series extended type): Theorem 5.1.3. Let ρ be a continuous representation of G Qp of dimension 2, with trivial endomorphisms. If p = 3, assume that ρ is not a twist of an extension of 1 by ω, and let (k, τ, ρ, ψ) be a deformation data. Then there is an explicit integer µ aut (k, τ, ρ) such that e(R ψ (k, τ, ρ)/π E ) = µ aut (k, τ, ρ). For our purposes, what is important to know about µ aut (k, τ, ρ) is that it can be easily computed in a combinatorial way. For more details on the formula for this integer see the introduction of [BM02]. Definition 5.1.4. We will say that a representation ρ with trivial endomorphisms is good if it satisfies the hypothesis of Theorem 5.1.3, that is, if p = 3 then ρ is not a twist of an extension of 1 by ω. Note that the condition of trivial endomorphisms implies that ρ is not reducible with scalar semi-simplification. Rigid spaces attached to deformation rings. We denote by X ψ (k, τ, ρ) the rigid space attached to R ψ (k, τ, ρ)[1/p] by the construction of Berthelot (see [dJ95, Section 7]). Let p 1 , . . . , p n the minimal prime ideals of R ψ (k, τ, ρ), and let R i = R ψ (k, τ, ρ)/p i . As R ψ (k, τ, ρ) has no p-torsion, the set of ideals (p i ) is in bijection with the set of minimal prime ideals (p Let (k, τ, ρ, ψ) be a deformation data. There exists a finite extension E = E(k, τ, ρ, ψ) of Q p , such that if X is the rigid space attached to a deformation ring over E, then there exists an analytic function λ : X → P 1,rig E defined over E that is injective on X (Q p ). ′ i ) of R ψ (k, τ, ρ)[1/p], with R i [1/p] = R ψ (k, τ, ρ)[1/p]/p ′ i . Let X i be the rigid space attached to R i [1/p], then X ψ (k, τ, ρ) = ∪ n i=1 X i . Let R 0 i be the integral closure of R i in R i [1/p], so that R i ⊂ R 0 i ⊂ R i [ This will be proved as Propositions 7.4.1, 7.5.3, 7.6.1, and 7.7.4, with an explanation of the choice of the field E(k, τ, ρ, ψ). Proof. Let X be a rigid analytic space that is smooth of dimension 1, and f : X → P 1,rig a rigid map that induces an injective map X (Q p ) → P 1 (Q p ). Then f is an open immersion. Indeed, this follows from the well-know fact that an analytic function f from some open disk D to Q p that is injective satisfies f ′ (x) = 0 for all x ∈ D. Now we apply this to X = X ψ (k, τ, ρ) and f = λ. Let X = X ψ (k, τ, ρ) be the image of X (Q p ) by λ. It is clear that X is defined over E. Assume first that X is contained in some bounded subset of Q p (this is automatic when τ is an inertial type, see Paragraphs 7.4 and 7.5). Then λ is an analytic open immersion from the quasi-affinoid space X to some quasi-affinoid space D attached to an open disk in A 1,rig . By Corollary 3.2.6, X is a bounded standard subset of P 1 (Q p ). We do not assume anymore that X is contained in some bounded subset of Q p . By the Breuil-Mézard conjecture, there is an infinite number of ρ ′ with trivial endomorphisms such that X ′ = X ψ (k, τ, ρ ′ ) is non-empty. For such a ρ ′ , X ′ contains a disk D(a, r) − for some r > 0 as it is open. For any ρ ′ with trivial endormophisms such that is semisimplification is not the same as the semi-simplification of ρ, we have that the intersection of X and X ′ is empty. So there exists some a ∈ P 1 (Q p ) and r > 0 such that D(a, r) − ∩X = ∅. Let u be an homography sending a to ∞, then u(X) is a bounded subset of P 1 (Q p ). This means that u • λ is a bounded analytic function on X . So we can apply the same reasoning as before to show that u(X) is a bounded standard subset of P 1 (Q p ), and so X is a standard subset of P 1 (Q p ). We denote by X ψ (k, τ, ρ) the subset λ(X ψ (k, τ, ρ)(Q p )) of P 1 (Q p ). 5.3.2. Complexity bounds. Now we give more information on the sets X ψ (k, τ, ρ). Theorem 5.3.3. Let (k, τ, ρ, ψ) be a deformation data. Then X ψ (k, τ, ρ) is a standard subset of P 1 (Q p ), defined over E = E(k, τ, ρ, ψ), with c E (X ψ (k, τ, ρ)) ≤ e(R ψ (k, τ, ρ)/π E ). In particular, c E (X ψ (k, τ, ρ)) ≤ µ aut (k, τ, ρ) if ρ is good. Remark 5.3.4. Note that the right-hand side of the inequality does not depend on the choice of E, whereas the left-hand side can get smaller when E has more ramification. In particular, to get a statement as strong as possible we want to take E with as little ramification as possible. Proof. Let p 1 , . . . , p n be the minimal prime ideals of R ψ (k, τ, ρ), R i = R ψ (k, τ, ρ)/p i and R 0 i be the integral closure of R i in R i [1/p] as in Section 5.2. Let X i be the rigid space attached to R i [1/p], then X ψ (k, τ, ρ) is the disjoint union of the X i = λ(X i (Q p )), and each of the X i is a standard subset of P 1 (Q p ) which is defined over E. (k, τ, ρ)) by [BM02, Lemme 5.1.6]. Then A 0 E (X i ) = R 0 i , so c E (X i ) = [k X i ,E : k E ]e(R 0 i ) by definition. Note that k X i ,E is the residue field of R 0 i , while k E is the residue field of R i . So by Proposition 2.2.2, we have c E (X i ) ≤ e(R i ). So we get c E (X ψ (k, τ, ρ)) ≤ n i=1 e(R i ). Finally, n i=1 e(R i ) = e(R ψ Note that in the proof above, the decomposition X ψ (k, τ, ρ) = ∪ i X i is the decomposition of X ψ (k, τ, ρ) in standard subsets that are defined and irreducible over E. So we also have the following result: Proposition 5.3.5. Let X ψ (k, τ, ρ) = ∪ i X i the decomposition of X ψ (k, τ, ρ) in standard subsets that are defined and irreducible over E. Then R ψ (k, τ, ρ) [1/p] = ⊕ i A E (X i ). Finally, we have the following result: Theorem 5.3.6. Let (k, τ, ρ, ψ) be a deformation data, and assume that ρ is good. There exists a finite set E of finite extensions of E = E(k, τ, ρ, ψ), depending only on µ aut (k, τ, ρ), such that X ψ (k, τ, ρ) is determined by the sets X ψ (k, τ, ρ) ∩ F for F ∈ E. Proof. This is a consequence of Theorem 5.1.3 and Corollary 4.5.1, where we take m = µ aut (k, τ, ρ). The case of crystalline deformation rings. We are interested here in the case of the deformation ring of crystalline representations, that, we take τ to be the trivial representation. This case is of particular interest as we are able to deduce additional information. In this case R ψ (k, triv, ρ) is zero unless ψ is a twist of χ k−1 cycl by an unramified character. Note that R ψ (k, triv, ρ) and R ψ ′ (k, triv, ρ) are isomorphic as long as ψ/ψ ′ is an unramified character with trivial reduction modulo p. So without loss of generality we will assume from now on that ψ = χ k−1 cycl and det ρ = ω k−1 . We denote by R(k, ρ) the ring R χ k−1 cycl (k, triv, ρ). It parametrizes the set of crystalline lifts of ρ with determinant χ k−1 cycl and Hodge-Tate weights 0 and k − 1. We also write µ aut (k, ρ) for µ aut (k, triv, ρ) Let F be the extension of F p over which ρ is defined (so F = F p when ρ is irreducible), and E the unramified extension of Q p with residue field F (so E = Q p when ρ is irreducible). Then R(k, ρ) is an O E -algebra with residue field F. Classification of filtered φ-modules. For a p ∈ Z p and F a finite extension of Q p containing a p , we define a filtered φ-module D k,ap as follows: D k,ap = F e 1 ⊕ F e 2 φ(e 1 ) = p k−1 e 2 , φ(e 2 ) = −e 1 + a p e 2 Fil i D k,ap = D k,ap if i ≤ 0 Fil i D k,ap = F e 1 if 1 ≤ i ≤ k − 1 Fil i D k,ap = 0 if i ≥ k Denote by V k,ap the crystalline representation such that D cris (V * k,ap ) = D k,ap . Then: V k,ap has Hodge-Tate weights (0, k−1) and determinant χ k−1 cycl . Moreover, V k,ap is irreducible if v p (a p ) > 0, and a reducible non-split extension of an unramified character by the product of an unramified character by χ k−1 cycl if v p (a p ) = 0. We have the following well-know result: Lemma 5.4.1. Let V be a crystalline representation with Hodge-Tate weights (0, k − 1) and determinant χ k−1 cycl . If V is irreducible there exists a unique a p ∈ m Zp such that V is isomorphic to V k,ap . If V is reducible non-split there exists a unique a p ∈ Z × p such that V is isomorphic to V k,ap . 5.4.2. The parameter a p . We show in Proposition 7.4.1 that the parameter a p actually defines a rigid analytic function. This is the function that plays the role of λ of Theorem 5.3.1 for crystalline representations. From Theorem 5.3.1 we can already deduce some results. It is a well-know conjecture (see [BG16, Conjecture 4.1.1]) that if p > 2, k is even, and v(a p ) ∈ Z then V ss k,ap is irreducible. From this we get: Proposition 5.4.2. Let p > 2, k even, n ∈ Z ≥0 . If the conjecture above is true, then there is an irreducible representation ρ such that the set {x, n < v p (x) < n + 1} is contained in X(k, ρ). Proof. If the conjecture holds, then the set C = {x, n < v p (x) < n + 1} is the union of the C ∩ X(k, ρ) for ρ irreducible. So we have written C as a finite disjoint union of standard subsets, which means that one of these subsets is equal to C. Reduction and semi-simplification. We know want to show that the case of crystalline deformation rings is accessible to numerical computations. However we must change slightly our setting: indeed, we can compute numerically only the semi-simplified reduction of V k,ap . So we need to express the result of Theorem 5.3.3 in terms of semisimple representations instead of in terms of representations with trivial endomorphisms. Let r be a semi-simple representation of G Qp with values in GL 2 (F p ). We define Y (k, r) to be the set {a p ∈ D(0, 1) − , V k,ap = r}. Let ρ be a representation of G Qp with trivial endomorphisms with semi-simplification isomorphic to r. Let X ′ (k, ρ) = X(k, ρ) ∩ D(0, 1) − . This means we are only interested in elements in X(k, ρ) that correspond to irreducible representations V k,x . Then we have that X ′ (k, ρ) ⊂ Y (k, r). Y (k, r). There exists a G Qp -invariant lattice T ⊂ V k,x such that T is a non-split extension of α by β, and so isomorphic to ρ. This means that x ∈ X ′ (k, ρ). Definition 5.4.4. We say that ρ is nice if it has trivial endomorphisms and either ρ is irreducible, or ρ is a non-split extension of α by β where β/α ∈ {1, ω}. We say that a semi-simple representation r is nice if r is not scalar, and in addition when p = 3 if r is not of the form α ⊕ β with α/β ∈ {ω, ω −1 }. Note that any ρ with trivial endomorphisms that is nice is also good, hence satisfies the hypotheses of Theorem 5.1.3. If r is semi-simple and nice, then there exists a nice ρ with trivial endomorphisms such that ρ ss = r, so we have Y (k, r) = X ′ (k, ρ). Note that we can choose such a ρ so that in addition, E(ρ) = E(r). We know some information about the difference between X(k, ρ) and X ′ (k, ρ): Proposition 5.4.5. Let ρ be a representation of G Qp with trivial endomorphisms. If ρ is not an extension unr(u) by unr(u −1 )ω n for some n which is equal to k −1 modulo p − 1, and u ∈ F × p , then X(k, ρ) ⊂ D(0, 1) − . If ρ is an extension of unr(u) by unr(u −1 )ω n for some u ∈ F × p and 0 ≤ n < p − 1, and n = k − 1 modulo p − 1, and u ∈ {±1} if n = 0 or n = 1, then X(k, ρ) ∩ {x, |x| = 1} is the disk {x, x = u}. Proof. For a p ∈ Z × p , the representation V k,ap is the unique crystalline non-split extension of unr(u) by unr(u −1 )χ k−1 cycl , where u ∈ Z × p and u and u −1 p k−1 are the roots of X 2 −a p X +p k−1 . In particular, for any invariant lattice T ⊂ V k,ap such that T is non-split, we get that T is an extension of unr(u) by unr(u −1 )ω k−1 . So X(k, ρ) does not meet {x, |x| = 1} unless ρ has the specific form given. Moreover, u = a p . So X(k, ρ) ∩ {x, |x| = 1} ⊂ {x, x = u}. If ρ is an extension of unr(u) by unr(u −1 )ω n for some u ∈ F p and 0 ≤ n < p − 1, the conditions on (n, u) imply there is a unique non-split extension of unr(u) by unr(u −1 )ω n , and so X(k, ρ) ∩ {x, |x| = 1} = {x, x = u} Remark 5.4.6. We could actually also determine X(k, ρ) ∩ {x, |x| = 1} when n = 1 and u ∈ {±1}. However, we will have to exclude this case later (see Proposition 5.4.3), so we do not need it. 5.4.4. Local constancy results. We recall the following results: Proposition 5.4.7. Let a p ∈ m Zp . If a p = 0, then for all a ′ p such that v p (a p − a ′ p ) > 2v p (a p ) + ⌊p(k − 1)/(p − 1) 2 ⌋, we have V ss k,ap ≃ V ss k,a ′ p . Moreovoer, V ss k,ap ≃ V ss k,0 for all a p with v p (a p ) > ⌊(k − 2)/(p − 1)⌋. Proof. The result for a p = 0 is Theorem A of [Ber12]. The result for a p = 0 is the main result of [BLZ04]. Corollary 5.4.8. Let X ′ (k, ρ) = X(k, ρ) ∩ D(0, 1) − . If ρ is not an extension unr(u) by unr(u −1 )ω n for some n which is equal to k − 1 modulo p − 1, then X ′ (k, ρ) = X(k, ρ) and c E (X ′ (k, ρ)) ≤ e (R(k, ρ)). If ρ is good and is an extension unr(u) by unr(u −1 )ω n for some n which is equal to k − 1 modulo p − 1, and u ∈ {±1} if n = 0 or n = 1, then c E (X ′ (k, ρ)) ≤ e (R(k, ρ)) − 1. Proof. The first part is clear by Proposition 5.4.5. For the second part, we can write X(k, ρ) as a disjoint union of X ′ (k, ρ) and X + (k, ρ) = X(k, ρ) ∩ {x, |x| = 1}, and both are standard subsets defined over E, so c E (X(k, ρ)) = c E (X ′ (k, ρ))+c E (X + (k, ρ)). By Proposition 5.4.5, c E (X + (k, ρ)) = 1 under the hypotheses, hence the result. 5.4.5. Computation of Y (k, r). We explain now how we can compute numerically the sets Y (k, r) for r nice (and hence the sets X(k, ρ) for ρ with nice semi-simplification). From Corollary 5.4.8 we deduce: Proposition 5.4.9. Suppose that r is nice, and let ρ be nice with ρ ss = r. Then Y (k, r) is a standard subset of D(0, 1) − defined over E = E(r), with c E (Y (k, r)) ≤ µ aut (k, ρ). Moreover if ρ is an extension of an unramified character by another character then c E (Y (k, r)) ≤ µ aut (k, ρ) − 1. that Y (k, r) is determined by the sets Y (k, r) ∩ F for F ∈ E. Proof. This is Corollary 4.5.1, where we take for E the field E(r), and for m the bound given by Proposition 5.4.9, that is m = µ aut (k, ρ) or µ aut (k, ρ) − 1 where ρ is some nice representation with ρ ss = r. Theorem 5.4.11. Suppose that the semi-simple representation r is nice. Then there exists a finite set of points P ⊂ D(0, 1) − , depending only on k and r, such that Y (k, r) is determined by Y (k, r) ∩ P. Proof. This is Corollary 4.5.2, where we take for E the field E(r), for m the bound given by Proposition 5.4.9, and for ε we can take the norm of an element of valuation ⌊3p(k − 1)/(p − 1) 2 ⌋ by Proposition 5.4.7. As a consequence, we see that if we are able to compute V ss k,ap for given p, k, a p , then we can compute Y (k, r) for r nice in a finite number of such computations, bounded in terms of E(r) and k. We give some examples of such computations in Section 6. We give a last application of these results: It follows from the formula giving µ aut (k, ρ) that there exists an integer m(k), depending only on k, such that µ aut (k, ρ) ≤ m(k) for all ρ. The optimal value for m(k) is of the order of 4k/p 2 when k is large. In general, the value of V ss k,ap depends on more information than just the valuation of a p . But there are some cases where it depends only on v p (a p ): Corollary 5.4.12. Fix k, and let m be an integer such that m ≥ e (R(k, ρ)) for all nice ρ with trivial endomorphisms. Let a and b be rational numbers such that for all rational c between a and b, the denominator of c is strictly larger than m. Then either for all a p with a < v p (a p ) < b, V ss k,ap is not nice, or V ss k,ap is constant on the annulus A(a, b). In particular, let c ∈ Q with denominator strictly larger than m. Then either for all a p with v p (a p ) = c, V ss k,ap is not nice, or V ss k,ap is constant on the circle C(c). Note that if p > 3 and k is even, V ss k,ap is always nice. Proof. Suppose that there exists at least an a p in A(a, b) such that r = V ss k,ap is nice. Then c E (Y (k, r)) ≤ m for E = E(ρ) which is an unramifed extension of Q p . So we can apply Corollary 4.5.16: the annulus A(a, b) is a subset of Y (k, r). Numerical examples We give some numerical examples for the deformations rings of crystalline representations. We have computed some examples of X(k, ρ) using Theorem 5.4.11 and a computer program written in SAGE ([SAGE]) that implements the algorithm described in [Roz]. We also used the fact that V ss k,ap is known for v p (a p ) < 2 in almost all cases, by the results of [BG09,BG13,GG15,BG15,BGR15], which reduces the number of computations that are necessary to determine X(k, ρ). We make the following remark: let ρ be a representation such that ρ ⊗ unr(−1) is isomorphic to ρ. Then X(k, ρ) is invariant by x → −x. Indeed, V k,−ap is isomorphic to V k,ap ⊗ unr(−1). This applies in particular when ρ is irreducible. 6.1. Observations for p = 5. We have computed X(k, ρ) for p = 5, k even, k ≤ 102, or k odd and k ≤ 47, and ρ irreducible (so in this case we have E(ρ) = Q p ). We summarize here some observations from these computations: (1) in each case, we have V ss k,ap = V ss k,0 for all a p with v p (a p ) > ⌊(k − 2)/(p + 1)⌋, and not only v p (a p ) > ⌊(k − 2)/(p − 1)⌋ which is the value predicted by [BLZ04]. (2) in each case, we have c Qp (X(k, ρ)) = e (R(k, ρ)), that is, the inequality of Proposition 5.4.9 is an equality. (3) each disk D appearing in the description of a X(k, ρ) has γ Qp (D) = 1. (4) each disk D appearing in a X(k, ρ) is defined over an extension of Q p of degree at most 2, which is unramified if k is even and totally ramified if k is odd. (5) for each disk D appearing in a X(k, ρ), either 0 ∈ D, or D is included in the set {x, v p (x) = n} for some n ∈ Z ≥0 if k is even, and in the set {x, v p (x) = n + 1/2} for some n ∈ Z ≥0 if k is odd. It would be interesting to know which of these properties hold in general. Property (1) is expected to be in fact true for all p and k, but nothing is known about the other properties. We comment further on Property (2) in Section 6.4. Some detailed examples. Let p = 5. Let r 0 = ind ω 2 and r 1 = ind ω 3 2 , and for all n, r(n) = r ⊗ ω n . We describe a few examples of sets X(k, r). In each case, the sets given contain all the values of a p for which V ss k,ap is irreducible. We also give the generic fibers of the deformation rings. 6.2.1. The case k = 26. We get that: • X(26, r 0 ) = {x, v p (x) < 2} ∪ {x, v p (x) > 2}, with c Qp (X(26, r 0 )) = 3, and R(26, r 0 )[1/p] = (Z p [[X]] ⊗ Q p ) × (Z p [[X, Y ]]/(XY − p) ⊗ Q p ). • X(26, r 0 (2)) = {x, v p (x − a) > 3} ∪ {x, v p (x + a) > 3}, where a = 4 · 5 2 , with c Qp (X(26, r 0 (2))) = 2, and R(26, r 0 (2))[ (X(26, r 1 (1))) = 4, and R(26, 1/p] = (Z p [[X]] ⊗ Q p ) 2 . • X(26, r 1 (1)) = {x, 2 < v p (x−a) < 3}∪{x, 2 < v p (x+a) < 3}, with c Qpr 1 (1))[1/p] = (Z p [[X, Y ]]/(XY − p) ⊗ Q p ) 2 . Here we see an example where the geometry begins to be a little complicated, with annuli that do not have 0 as a center. 6.2.2. The case k = 28. We get that: • X(28, r 1 ) = {x, 0 < v p (x) < 1} ∪ {x, v p (x) > 2 and v p (x − a) < 4 and v p (x + a) < 4}, where a = 4 · 5 3 + 5 4 , with c Qp (X(28, r 1 )) = 5, and R(28, r 1 )[1/p] = (Z p [[X, Y ]]/(XY − p) ⊗ Q p )×(Z p [[X, Y, Z]]/(XY − p 2 − (a/p 2 )Y, XZ − p 2 + (a/p 2 )Z, Y Z − (p 4 • X(28, r 0 (1)) = {x, 1 < v p (x) < 2}, with c Qp (X(28, r 0 (1))) = 2, and R(28, r 0 (1))[1/p] = (Z p [[X, Y ]]/(XY − p) ⊗ Q p ). • X(28, r 0 (3)) = {x, v p (x − a) > 4} ∪ {x, v p (x + a) > 4}, with c Qp (X(28, r 0 (3))) = 2, and R(28, r 0 (3))[1/p] = (Z p [[X]]) ⊗ Q p ) 2 . Here we see an example with an irreducible component that has complexity 3. 6.2.3. The case k = 30. We get that: • X(30, r 0 ) = {x, 0 < v p (x) < 1} ∪ {x, v p (x) > 4} , with c Qp (X(30, r 0 )) = 3, and R(30, r 0 )[1/p] = (Z p [[X]] ⊗ Q p ) × (Z p [[X, Y ]]/(XY − p) ⊗ Q p ). • X(30, r 0 (2)) = {x, v p (x − a) > 3} ∪ {x, v p (x + a) > 3}, where a = 5 3 · √ 3 , with c Qp (X(30, r 0 (2))) = 2, and R(30, r 0 (2))[1/p] = (Z p 2 [[X]] ⊗ Q p 2 ). • X(30, r 1 (1)) = {x, 1 < v p (x) < 3}∪{x, 3 < v p (x) < 4}, with c Qp (X(30, r 1 (1))) = 4, and R(30, r 1 (1)) [1/p] = (Z p [[X, Y ]]/(XY − p) ⊗ Q p )×(Z p [[X, Y ]]/(XY − p 2 ) ⊗ Q p ) . The interesting part here is X(30, r 0 (2)): we see that A 0 Qp (X(30, r 0 (2))), which is a domain, has residue field F p 2 , whereas R(30, r 0 (2)) has residue field F p . So R(30, r 0 (2)) = A 0 Qp (X(30, r 0 (2))). 6.3. Criteria for non-normality. Recall the notation of Section 5.2. Then we see, by Proposition 5.3.5, that if we know X(k, ρ) then we know R(k, ρ) X(k, ρ) gives no indication about how the R i glue together so we can not hope for complete information on R(k, ρ) anyway if it is not irreducible). We do not expect this to hold, as this would mean that each of the R i is a normal ring. So we can ask instead, how can we recognize when R i is not R 0 i ? A first criterion is when they have different residue fields, as in the example of R(30, r 0 (2)) in Paragraph 6.2.3. Another criterion is when R i and R 0 i have the same residue field (a situation that we can always obtain by replacing E by an unramified extension, which does not change the complexities), but e(R 0 i ) < e(R i ). This is a situation that does not seem to arise often, see Section 6.4. [1/p] = ⊕ i R i [1/p] = ⊕ i A E (X i ). We can ask whether we can recover each R i , that is, if R i = A 0 E (X i ), or equivalently if R i = R 0 i for all i (the description of We give a last, more subtle criterion. Let X i be one of the components of X(k, ρ), and assume that each of the disks that appears in the description of X i is defined over Q p , and has complexity 1. In this case, a closer look at the proof of Proposition 4.2.6 show that Spec(A 0 Qp (X i )/p) has exactly c Qp (X i ) distinct irreducible components. On the other hand, the geometric version of the Breuil-Mézard conjecture, proved in [EG14], shows that if ρ is irreducible then Spec(R(k, ρ)/p) has at most two irreducible components (which can have large multiplicity), and so Spec(R i /p) also has at most two irreducible components. So if c Qp (X i ) > 2 then we certainly have that R i = R 0 i . This happens for example for the second irreducible component of X (28, r 1 ). It would be interesting in this case to understand how the irreducible components of Spec(R 0 i /p) map to the irreducible components of Spec(R i /p). 6.4. Complexity and multiplicity. An interesting result coming from our computations is the following: for p = 5, for all irreducible representation ρ, for all k ≤ 47 and all even k ≤ 102, we have that c Qp (X(k, ρ)) = e (R(k, ρ)), instead of simply the inequality c Qp (X(k, ρ)) ≤ e (R(k, ρ)). Given this, it is tempting to make the following conjecture: For all p > 2, for all k ≥ 2 and for all irreducible ρ, we have that c Qp (X(k, ρ)) = e (R(k, ρ)). Note that this equality between complexity and multiplicity does not necessarily hold when ρ is reducible. However, it may be true that for all p > 2, for all k ≥ 2, there is only a finite number of reducible (nice) representations ρ for which the equality does not hold. We can also reformulate this equality in a different way: recall the notation of Section 5.2. So R(k, ρ) has a family of quotients R i that are integral domains, and e (R(k, ρ) ) = i e(R i ). On the other hand, c Qp (X(k, ρ)) = i [k R 0 i : F p ]e(R 0 i ) where k R 0 i is the residue field of R 0 i . The equality between complexity and multiplicity can be reformulated as saying that for all i, e(R i ) = [k R 0 i : F p ]e(R 0 i ). Written in this way without any reference to the sets X(k, ρ), the equality can be generalized to any potentially semi-stable deformation ring, including those that are of dimension larger than 1, such as the deformation rings classifying representations of dimension > 2 or representations of G K for some finite extension K/Q p . Let τ be a Weil representation. The field of definition of τ , denoted by E(τ ), is the subfield of Q p generated by the tr τ (x), x ∈ W Qp . This is a finite extension of Q p , as a Weil representation factors through a finitely generated group. Let E be a finite extension of Q p . We say that τ is realizable over E if there is a representation τ ′ : W Qp → GL n (E) that is isomorphic to τ . Then we have: 7.1.2. (φ, Gal(F/Q p ))-modules. We fix a finite Galois extension F of Q p , and denote by F 0 the maximal subextension of F that is unramified over Q p . Let A be a Q p -algebra. Then a (φ, Gal(F/Q p ))-module M over F 0 ⊗ Qp A is a free F 0 ⊗ Qp A-module of finite rank, endowed with commuting actions of an automorphism φ and the group Gal(F/Q p ). The action of φ is A-linear and F 0 -semi-linear (with respect to the Frobenius automorphism of F 0 ), and the action of Gal(F/Q p ) is F 0 -semi-linear (with respect to the action of Gal(F/Q p ) on F 0 ) and A-linear. Then: Proposition 7.1.2. Let A be an F 0 -algebra. Then there is an equivalence of categories between (φ, Gal(F/Q p ))-modules over F 0 ⊗ Qp A and Weil representations over a free Amodule that are trivial on I F , and this equivalence preserves rank. Moreover this construction is functorial in A (in the category of F 0 -algebras). Proof. For a given A, the construction of the Weil representation from the (φ, Gal(F/Q p ))module is explained in [BM02], and the converse construction is immediate. We will make use of this equivalence as some things are more naturally expressed in terms of (φ, Gal(F/Q p ))-modules, whereas others are more easily proved in terms of representations of the Weil group (for example Proposition 7.3.2). In the same situation, we also define a (φ, N, Gal(F/Q p ))-module over F 0 ⊗ Qp A to be a (φ, Gal(F/Q p ))-module over F 0 ⊗ Qp A that is additionally endowed with a F 0 ⊗ Qp -linear endomorphism N satisfying Nφ = pφN that commutes with the action of Gal(F/Q p ). 7.2. Universal (filtered) (φ, N)-modules with descent data. We recall a few definitions concerning objects attached to p-adic representations of G Qp . If F/Q p is a finite extension, we denote by F 0 be maximal unramified extension of Q p contained in F . Let V be a continuous representation of G Qp over an E-vector space for some finite E/Q p . Let F be a finite Galois extension of Q p . We denote by D F crys (V ) the F 0 ⊗ Qp E- module (B crys ⊗ Qp (V )) G F . It is a (φ, Gal(F/Q p ))-module over F 0 ⊗ Qp E. If V becomes crystalline over F then D F crys (V ) is a free F 0 ⊗ Qp E-module of rank dim E (V ). We de- note by D F st (V ) the F 0 ⊗ Qp E-module (B st ⊗ Qp V ) G F . It is endowed with a structure of (φ, N, Gal(F/Q p ))-module over F 0 ⊗ Qp E. If V becomes semi-stable over F then is it a free F 0 ⊗ Qp E-module of rank dim E (V ). If V becomes crystalline over F then D F st (V ) and D F crys (V ) coincide as (φ, Gal(F/Q p ))-modules, and N = 0. We denote by D F dR (V ) the F ⊗ Qp E-module (B dR ⊗ Qp V ) G F . It is a F ⊗ Qp E-module with a semi-linear action of Gal(F/Q p ), and is endowed with a separated exhaustive decreasing filtration by sub-F ⊗ Qp E-modules that is stable under the action of Gal(F/Q p ), and satisfies an additional condition called admissibility. If V is potentially semi-stable, then D Qp dR (V ) is an E-vector space of dimension dim E V . Moreover, we have that D F dR (V ) = F ⊗ F 0 D F st (V ) as F ⊗ Qp E, so this endows F ⊗ F 0 D F st (V ) with a filtration as above, that is, a structure of filtered (φ, N, Gal(F/Q p ))-module. Theorem 7.2.1. Let F be a finite Galois extension of Q p . Let X be a reduced rigid analytic space, let V be a locally free O X -module of rank n with a continuous action of G Qp . Assume that for all x ∈ X, V x is potentially semi-stable with weights independent of x, and becomes semi-stable on F . Then there exists a projective F 0 ⊗ Qp O X -module D of rank n, endowed with a structure of (φ, N, Gal(F/Q p ))-module over F 0 ⊗ Qp O X , such that for all x, D x is isomorphic, as a (φ, N, Gal(F/Q p ))-module, to D F st (V x ). Proof. This follows immediately from [Bel15, Theorem 5.1.2]: we take the module D to be the module called D Bst (V) there, considering V as a representation of G F (see also [BC08,Théorème C] B dR (V x ) are actually free (then their rank is independent of x by the condition on the weights). This comes from [Sav05, Lemma 2.1], and here we use the fact that we start from a representation of G Qp . Let now (k, τ, ρ, ψ) be a deformation data, as defined in Definition 5.1.2. Let E be a finite extension of Q p satisfying the following conditions: (1) the residual representation ρ can be realized on the residue field of E (2) the type τ can be realized on E (3) the character ψ takes its values in E × Let R ψ (k, τ, ρ)[1/p] be the ring defined by Kisin attached to this data, as recalled in Section 5.1. It is an O E -algebra. We can apply Theorems 7.2.1 and 7.2.2 to the rigid analytic space X = X ψ (k, τ, ρ) attached to the Kisin ring R ψ (k, τ, ρ)[1/p]. Indeed, we know that these rings are reduced, and the hypotheses come from the definition of the rings. Working in families. Reduction of an endomorphism. Proposition 7.3.1. Let K be a field and A be a K-algebra. Let φ be an A-linear endomorphism of A 2 , and assume that the characteristic polynomial of φ is in fact in K [X], and that it is split over K with distinct eigenvalues. Then, Zariski-locally on A, φ is diagonalizable. Proof. Let λ and µ be the roots of the characteristic polynomial of φ, and let ( a b c d ) be the matrix of φ in the canonical basis of A 2 (so that a + d = λ + µ and ad − bc = λµ). We are looking for a basis (f 1 , f 2 ) of A 2 , with f 1 = xe 1 + ye 2 , f 2 = e 2 , such that the matrix of φ in this basis is upper triangular. The new basis is as wanted if x, y satisfy one of the following systems of equations: (a − λ)x + by = 0 and cx + (d − λ)y = 0 or (a − µ)x + by = 0 and cx + (d − µ)y = 0 Assume that u = d − λ is invertible. We solve the first system by setting x = 1, y = −c/(d − λ). In the first case, in our new basis φ has a matrix of the form ( λ b 0 d ), and actually d = µ by the trace condition. As λ − µ is invertible, we can change the basis again so that in the new basis, φ has matrix λ 0 0 µ . Assume now that v = a − λ is invertible. Then so is d − µ = −v. We solve the second system by setting x = 1, y = −c/(d − µ). In this case we do the same thing after exchanging λ and µ. Note that u + v = µ − λ is invertible by assumption. We set f = (d − λ)/(µ − λ), A 1 = A[f −1 ], A 2 = A[(1 − f ) −1 ] . Then as we just saw in A 1 and A 2 there is a basis in which the matrix of φ is λ 0 0 µ , which gives the result. Isomorphism of group representations. Theorem 7.3.2. Let K be a field of characteristic zero, and A a K-algebra. Let G be a group. Let ρ : G → GL n (K) be a representation that is absolutely irreducible. Let ρ ′ : G → GL n (A) be a representation. Assume that for all g in G, we have tr ρ(g) = tr ρ ′ (g). Then, Zariski-locally on A, there is an M ∈ GL n (A) such that ρ ′ (g) = Mρ(g)M −1 for all g ∈ G. Proof. By [Rou96, Théorème 5.1], there is an A-algebra automorphism τ of M n (A) such that for all g ∈ G, ρ ′ (g) = τ ρ(g). By Let now A be as in point (2) and let c ∈ H 1 (Gal(L/K), (L ⊗ K A) × ). For an extension H 1 (Gal(L/K), (L ⊗ K A[f −1 i ]) × ) is zero for all i. Proof. Let M be a K-algebra, and c ∈ H 1 (Gal(L/K), (L ⊗ K M) × ). Let x ∈ L. We set φ(c, x) = γ∈Gal(L/K) γ(x)c(γ) ∈ L⊗ K M. We have for all g ∈ Gal(L/K), c(g)g(φ(c, x)) = φ(c,A ′ of A, denote by c A ′ the image of c in H 1 (Gal(L/K), (L ⊗ K A ′ ) × ). Let m be a maximal ideal of A, and K m = A/m. Then K m is a finite extension of K. So there exists an In particular, we can see a p as an analytic map from X (k, ρ) to A 1,rig . Moreover, a p induces an injective map from X (k, ρ)(Q p ) to D(0, 1) + . x ∈ L such that φ(c Km , x) is invertible in L ⊗ K K m . Let f = N A (φ(c, x)) ∈ A. Then D f is a neighborhood of m in Spec A. Moreover the image of φ(c, x) in L ⊗ K A[f −1 ] is invertible, so c A[f −1 ] = 0. So Proof. Consider the φ-module D which is obtained from applying Theorem 7.2.1 to the rigid space X (k, ρ) attached to the ring R(k, ρ)[1/p]. It is a projective module of rank 2 over R(k, ρ)[1/p] and is such that for all x : R(k, ρ)[1/p] → E x corresponding to a representation ρ x , D ⊗ R(k,ρ)[1/p] E x is the φ-module D x attached to ρ x (forgetting the filtration). Now observe that a p , as defined in Lemma 5.4.1, is the trace of φ on the dual of D, so it is an element of R(k, ρ)[1/p], and a p (x) is the evaluation at x of the trace of φ on the dual of D. 7.5. The crystabelline case. We suppose here that τ = χ 1 ⊕ χ 2 , where χ 1 and χ 2 are distinct characters of I Qp with finite image that extend to characters of W Qp , so that the representations classified by R ψ (k, τ, ρ) become crystalline on an abelian extension of Q p . In this case we show the existence of a function λ as in Proposition 5.3.1 when χ 1 = χ 2 . We make use of the results of [GM09], which classifies the filtered φ-modules with descent data that give rise to a Galois representation of inertial type τ and Hodge-Tate weights (0, k − 1). We summarize their results for such a τ . The characters χ i factor through F = Q p (ζ p m ) for some m ≥ 1, so the Galois representations we are interested in become crystalline on F , and so are given by filtered (φ, Gal(F/Q p ))-modules. Note that here F 0 = Q p . Let E be a finite extension of Q p containing the values of χ 1 and χ 2 . Let α, β be in O E with v p (α) + v p (β) = k − 1. We define a (φ, Gal(F/Q p ))-module ∆ α,β as follows: let ∆ α,β = Ee 1 ⊕ Ee 2 , with g(e 1 ) = χ 1 (g)e 1 and g(e 2 ) = χ 2 (g)e 2 for all g ∈ Gal(F/Q p ). The action of φ is given by: φ(e 1 ) = α −1 e 1 and φ(e 2 ) = β −1 e 2 . We are looking at filtrations on ∆ α,β,F = F ⊗ Qp ∆ α,β satisfying Fil i ∆ α,β,F = 0 if i ≤ 1 − k, Fil i ∆ α,β,F = ∆ α,β if i > 0, and Fil i ∆ α,β,F = Fil 0 ∆ α,β,F for 1 − k < i ≤ 0 is a F ⊗ Qp E-line. We summarize now the results that are given in [GM09, Section 3]. Proposition 7.5.1. Fix α, β in O E with v p (α) + v p (β) = k − 1. Then there exists a way to choose Fil 0 (∆ α,β,F ) ⊂ ∆ α,β,F = ∆ α,β ⊗ F that makes it an admissible filtered (φ, Gal(F/Q p ))-module. If neither α nor β is a unit, then all such choices give rise to isomorphic filtered (φ, Gal(F/Q p ))-modules, which are irreducible. If α or β is a unit, the choices give rise to two isomorphism classes of filtered (φ, Gal(F/Q p ))modules, one being reducible split and the other reducible non-split. We denote by D α,β the isomorphism class of admissible filtered (φ, Gal(F/Q p ))-module given by a choice of filtration that makes it into either an irreducible module (if neither α nor β is a unit) or a reducible non-split module (if α or β is a unit). Then it follows from the computations of [GM09, Section 3] that: Proposition 7.5.2. Let V be a potentially crystalline representation with coefficients in E, of inertial type τ and Hodge-Tate weights (0, k − 1) that is not reducible split. Then there exists a unique pair (α, β) ∈ O E with v p (α) + v p (β) = k − 1 such that D F crys (V ) is isomorphic to D α,β as a filtered (φ, Gal(F/Q p ))-module. Let E = E(k, τ, ρ, ψ) be a finite extension of Q p such that ρ can be defined over the residue field of E, E contains the images of χ 1 and χ 2 and of the character ψ. Then the ring R ψ (k, τ, ρ) can be defined over E. Moreover: Proposition 7.5.3. Let ρ be a representation with trivial endomorphisms. There are elements α, β ∈ R ψ (k, τ, ρ)[1/p] such that for each closed point x of Spec R ψ (k, τ, ρ)[1/p] corresponding to a representation ρ x , D F crys (ρ x ) is isomorphic to ∆ α(x),β(x) as a (φ, Gal(F/Q p ))module. Proof. By Theorem 7.2.1 applied to the rigid analytic space X ψ (k, τ, ρ) attached to R ψ (k, τ, ρ)[1/p], there exists a φ-module D with descent data by Gal(F/Q p ), where D is a projective module of rank 2 over R ψ (k, τ, ρ)[1/p], such that for each closed point x of Spec R ψ (k, τ, ρ)[1/p], D F crys (ρ x ) is isomorphic to D ⊗ R E x (where E x is the field of coefficients of ρ x ) as a (φ, Gal(F/Q p ))-module. Applying Proposition 7.3.1, we see that the action of Gal(F/Q p ) on D is given as the action of Gal(F/Q p ) on each ∆ α,β : that is, Zariski-locally on Spec R ψ (k, τ, ρ)[1/p], we can write D = Re 1 ⊕ Re 2 , with g(e 1 ) = χ 1 (g)e 1 and g(e 2 ) = χ 2 (g)e 2 . As the action of φ on D commutes with the action of Gal(F/Q p ), this shows that the eigenvalues of φ acting on D are in fact in R ψ (k, τ, ρ)[1/p], that is, α and β are elements of R ψ (k, τ, ρ)[1/p]. Moreover, if we fix the determinant of the Galois representation corresponding to D α,β then we fix αβ. So the function α is injective on points, so it can play the role of the function λ of Theorem 5.3.1. Let X ψ (k, τ, ρ) be the image of X ψ (k, τ, ρ)(Q p ) in Q p , then we see that X ψ (k, τ, ρ) is contained in the set {x, 0 ≤ v p (x) ≤ k − 1}, with the irreducible representations corresponding the subset of elements that are in {x, 0 < v p (x) < k − 1}. 7.6. Semi-stable representations. We now assume p > 2 and we study the case of the deformation rings attached to a discrete series extended type of the form τ = χ 1 ⊕χ 2 , where χ 1 and χ 2 are characters of W Qp that have the same reduction to inertia, and such that χ 1 (F ) = pχ 2 (F ) for any Frobenius element F . As in the case of crystalline representations, we can twist by a smooth character of W Qp and reduce to the case where χ 1 and χ 2 are trivial on inertia. Then the deformation rings R ψ (k, τ, ρ) classify representations that are semi-stable, and only a finite number of the representations that appear can be crystalline. Let ρ be a semi-stable, non-crystalline representation of dimension 2 of G Qp , with Hodge-Tate weights (0, k − 1) for some k ≥ 2. Then we know (see for example [GM09, Section 3.1], that the filtered (φ, N)-module D st (ρ) is isomorphic to exactly one D α,L for some α with v(α) = k/2, some L ∈ Q p and some finite extension E containing α and L, for (φ, N)-modules D α,L defined as follows: D α,L = Ee 1 ⊕ Ee 2 , φ(e 1 ) = pα −1 e 1 , φ(e 2 ) = α −1 e 2 , Ne 1 = e 2 , Fil 0 D α,L = E(e 1 − Le 2 ). Then L is the L-invariant of Fontaine, as defined in [Maz94,§9]. Let ρ be a crystalline representation of dimension 2 of G Qp , we set its L-invariant to be ∞. Proposition 7.6.1. Let X be a rigid analytic space defined over some finite extension E of Q p . Assume that X is endowed with a 2-dimensional representation ρ of G Qp such that for all x ∈ X , ρ x is semi-stable with Hodge-Tate weights (0, k −1), the Weil representation attached to ρ x is independent of x, there exists at least one x such that ρ x is not crystalline, and none of the ρ x is reducible split. Then there exists a rigid analytic map L : X → P 1 E , defined over E, such that for all x, L(x) is the L-invariant of ρ x . Note that under these conditions, the α of D α,L is independent of x, and is in E. This proposition applies in the following situation: let p > 2, let X = X ψ (k, τ, ρ) be the deformation space for the extended type τ , and ρ is not reducible split. Then the function L can play the role of λ of Proposition 5.3.1. Proof. Let D = D dR,0 ⊗ E 0 E, with its action of Gal(F/Q p ), which is isomorphic to the φmodule D F dR (ρ) with its action of Gal(F/Q p ) for some potentially crystalline representation ρ. Then D F dR (ρ) Gal(F/Qp) = D Qp dR (ρ) is an E-vector space of dimension 2, as ρ is de Rham as a G Qp -representation. The action of Gal(F/Q p ) on D dR,0 is E 0 -linear. So the dimension of its subspace of fixed elements is invariant by extension of scalars. Hence the result. Remark 7.7.2. We could also make use of the results of [GM09], which give an explicit basis of the E-vector space (D dR,0 ⊗ E 0 E) Gal(F/Qp) for some extension E of E 0 . We denote by V τ the E 0 -vector space of dimension 2 given by Lemma 7.7.1. Any potentially semi-stable representation of extended type τ becomes crystalline when restricted to G F . For any such representation ρ, with coefficients in an extension E of E 0 , D F crys (ρ) is a (φ, Gal(F/Q p ))-module over F 0 ⊗ Qp E. We have that D F dR (ρ) is canonically isomorphic to F ⊗ F 0 D F crys (ρ), and is endowed with an admissible filtration. Moreover, D F dR (ρ) Gal(F/Qp) = D Qp dR (ρ) is an E-vector space of dimension 2. We also fix an integer k ≥ 2, a continuous character ψ : G Qp → E × 0 . Note that there is no loss of generality in considering only characters with values in E 0 , as the compatibility condition between type and determinant shows that if R ψ (k, τ, ρ) is non-zero then ψ takes its values in E 0 . Let E τ be the set of Galois representations ρ : G Qp → GL 2 (Q p ) that are potentially crystalline of extended type τ , Hodge-Tate weights (0, k − 1), and determinant ψ. Then: Theorem 7.7.3. There exists a map L τ : E τ → P(V τ ⊗ E 0 Q p ) such that two elements ρ, ρ ′ of E τ are isomorphic if and only if L τ (ρ) = L τ (ρ ′ ). Proof. We can assume that E τ is not empty, otherwise the statement is trivially true. Let ρ : G Qp → GL 2 (Q p ) be an element of E τ . Then WD(ρ), the Weil-Deligne representation attached to ρ, is actually a Weil representation as ρ is potentially crystalline. By definition, WD(ρ) is isomorphic to τ ⊗ E 0 Q p as a representation of W Qp . We fix such an isomorphism u, it is unique up to a scalar by the irreduciblity of τ . Then u gives us an isomorphism between D F crys (ρ) and D crys,0 ⊗ E 0 Q p as φ-modules with an action of Gal(F/Q p ), by Proposition 7.1.2. This also gives us an isomorphism, that we still call u, between D F dR (ρ) and D dR,0 ⊗ E 0 Q p . The isomorphism class of ρ is entirely determined by the filtration on D F dR (ρ). As the Hodge-Tate weights of ρ are known, the only necessary information is the F ⊗ Qp Q p -line corresponding to the non-trivial steps of the filtration. This line is invariant by the action of Gal(F/Q p ). By the isomorphism u, this gives rise to a Gal(F/Q p )-invariant F ⊗ Qp Q pline in D dR,0 ⊗ E 0 Q p . This line is generated by an element of D dR,0 ⊗ E 0 Q p that is invariant by Gal(F/Q p ) by (1) of Proposition 7.3.3, hence by an element of D Gal(F/Qp) dR,0 ⊗ E 0 Q p . We define L τ (ρ) ∈ P(D Gal(F/Qp) dR,0 ⊗ E 0 Q p ) to be the line generated by this element in D Gal(F/Qp) dR,0 ⊗ E 0 Q p . This does not depend on the choices made, as u is unique up to multiplication by a scalar, and the invariant element generating the line is well-defined up to multiplication by a scalar. 7.7.2. Making it into an analytic function. Let X be the rigid analytic space corresponding to the deformation ring R ψ (k, τ, ρ) for some representation ρ with trivial endomorphisms and some supersingular extended type τ . Let E = E(k, τ, ρ, ψ) be the field E 0 defined above. Proposition 7.7.4. There exists a rigid analytic map L τ : X → P(V τ ), defined over E, such that for all x, L τ (x) is the L τ -invariant of ρ x as defined in 7.7.3. By fixing a basis of the 2-dimensional E-vector space V τ , we then get a map L τ : X → P 1 E , which plays the role of λ in Theorem 5.3.1. Proof. It is enough to do this on an admissible covering of X by affinoid subspaces. So we can assume that X = Max(A) for some affinoid algebra A, and replace X by an admissible covering by affinoid subspaces as needed. Let D F crys (A) be the (φ, Gal(F/Q p ))-module corresponding to the representation ρ. We can assume that D crys (A) is a free A-module of rank 2. Using the correspondence between (φ, Gal(F/Q p ))-modules and representations of the Weil group as in Section 7.1.2, and Theorem 7.3.2, we can assume that D F crys (A) = D F crys,0 ⊗ E A as a (φ, Gal(F/Q p ))-module over F 0 ⊗ Qp A. Consider now D F dR (A). It is isomorphic to F ⊗ F 0 D F crys (A), so to D F dR,0 ⊗ E A as a φ-module with action of Gal(F/Q p ). In particular, it is trivial as an F ⊗ Qp A-module with an action of Gal(F/Q p ). Also, it has a basis as an A-module given by the chosen basis of D F dR,0 . D F dR (A) contains a locally free sub-F ⊗ Qp A-module F of rank 1, such that D F dR (A)/F is also locally free of rank 1, that gives at each point x the filtration on D F dR (ρ x ). We can assume that F and D F dR (A) are free of rank 1 over F ⊗ Qp A. Moreover, this submodule is invariant by the action of Gal(F/Q p ). Consider a basis f of F . Then the action of Gal(F/Q p ) on f gives rise to an element c ∈ H 1 (Gal(F/Q p ), (F ⊗ Qp A) × ). Using Theorem 7.3.3 and replacing Max(A) by an admissible covering if necessary, we can assume that f itself is fixed by the action of Gal(F/Q p ). So we get that f is in D F dR (A) Gal(F/Qp) , which is canonically isomorphic to D Theorem 1.1.2. Let p = 2. Let D be a disk defined over E. Let e be the smallest integer such that there exists a finite extension F of E with e F/E = e and F ∩ D = ∅. Then e = p s for some s, and there exists a totally ramified extension F of E with [F : E] = p s such that F ∩ D = ∅. Proposition 1 .3. 1 . 11Let D be a disk defined over E and a ∈ D. Suppose that v E (a) = n/e where e = e E(a)/E and n is prime to e. Then there exists an extension F of E of degree at most e such that F ∩ D = ∅. Proof of Theorem 1.1.1. The case e = 1 is a consequence of Proposition 1.2.4. Assume now that e > 1. Let a ∈ D, F = E(a) with e F/E = e, K = E(a) ∩ E nr . If a is a uniformizer of F , the result follows from Proposition 1.3.1. Otherwise, let f ∈ E[X] <e be a polynomial such that f (a) is a uniformizer of F . Assume first that such a f exists. Let D ′ = f (D). Then D ′ is a disk defined over E by Lemma 1.2.5, containing an element ̟ = f (a) with e E(̟)/E = e and v E (̟) = 1/e, so it satisfies the hypotheses of Proposition 1.3.1. Hence there exists a c ∈ D ′ with [E(c) : E] ≤ e. Let b ∈ D such that f (b) = c, then [E(b) : E] ≤ e(e − 1) as b is a root of f (X) − c, which is a polynomial of degree at most e − 1 with coefficients in an extension of degree e of E. Now we apply again Lemma 1.2.1, but with K the maximal subextension of E(b) ∩ E nr with degree a power of p. Then [K : E] ≤ e − 1 so [K : E] ≤ p s−1 where e = p s , and D contains a point a ′ with [K(a ′ ) : and all other valuations are unchanged. This contradicts the choice we made for (α, P ) at the beginning. So in fact m = 1/e. 2. Some results on Hilbert-Samuel multiplicities 2.1. Hilbert-Samuel multiplicity. Let A be a noetherian local ring with maximal ideal m, and d be the dimension of A. Let M be a finite-type module over A. We recall the definition of the Hilbert-Samuel multiplicity e(A, M) (see [Mat86, Chapter 13]). For n large enough, len A (M/m n M) is a polynomial in n of degree at most d. We can write its term of degree d as e(A, M)n d /d! for an integer e(A, M), which is the Hilbert-Samuel multiplicity of M (relative to (A, m)). We also write e(A) for e(A, A). If dim A = 1, it follows from the definition that e(A, M) = len A (M/m n+1 M)−len A (M/m n M) = len A (m n M/m n+1 M) for n large enough. Lemma 2 .1. 1 . 21Let k be a field, and (A, m) be a local noetherian k-algebra of dimension 1, with A/m = k. Suppose that there exists an element z ∈ m such that A has no z-torsion and for all n large enough, zm n = m n+1 . Then e(A) = dim k A/(z). Lemma 2 .2. 1 . 21Let (T, m T ) → (S, m S ) be a local morphism of local noetherian rings of the same dimension, with residue fields k T and k S respectively, then e(T, S) ≥ [k S : k T ]e(S). 2.2, both possibilities e(B) < e(A) and e(B) = e(A) can happen for A = B. See also Paragraph 4.1.3 for more examples. where the D i are rational disks, ∞ ∈ D 0 , D 0 = D i for all i > 0, D i ⊂ D 0 , and D i and D j are disjoint if i = j and i, j > 0 (bounded connected standard subset). a disjoint union of an infinite number of open disks of radius r 0 . Hence one of these disks, say D(b, r 0 3. 3 . 3Rings of functions on standard subsets of open type. From now on, we will be only interested in standard subsets that are of open type. So we will simply write standard subset and connected standard subsets for standard subsets of open type and connected standard subsets of open type. 0 can get arbitrarily close to M. So finally f X = M, and the rest follows. Lemma 4.2.1. Let D be an open disk defined over E, let s be the smallest degree over E of an element in D. Let a be an element of D of degree s over for all m < n. We recall the following result (see for example [Was97, Proposition 7.2]: Lemma 4.2.3 (Weierstrass Division Theorem). Let f ∈ O L [[T ]] that is regular of degre n, and g ∈ O L [[T ]]. Then there exists a unique pair (q, r) with q ∈ O L [[T ]], r ∈ O L [T ] <n and g = qf + r. 4.2.2. Open disks. Proposition 4.2.4. Let D be an open disc of radius r ∈ p Q defined over E. Let s be the smallest ramification degree of E(a)/E for a ∈ D. Let t be the smallest positive integer such that r st ∈ |E(a) × |. Then c E (D) = st. with a 0 0and the a i,j,ℓ,n in O E . Let f be such a function with a 0 = 0 and consider f as an element of O L [[Y ]]. We define the valuation of f as the smallest valuation of the coefficients of f , and the leading term of f as the smallest power of Y where this valuation occurs. We compute easily that the valuation of the maximal ideal of A. Indeed, m is the image of the maximal ideal of A 0 E (X), hence also the image of i W i . Moreover we have k-algebra inclusions A i ⊂ A. So A is nearly the sum of the family (A i ) (see definition before Proposition 2.1.3). appearing in the definition of X 1 to an equation defining the corresponding disk in X i . Moreover, s = [E 1 : E], as G E acts transitively on the set of X i because we have assumed X to be irreducible over E.4.4. Comparison of complexities.The important result is that the two definitions of complexity actually coincide when both are defined. Theorem 4.4. 1 . 1Let X be a standard subset defined over E. Then c E (X) = γ E (X). D be a disk defined over E, of the form {x, v E (x − a) > λ}. Let s be the minimal ramification degree of an extension F of E such that F ∩ D = ∅, and t > 0 be the smallest integer such that stλ ∈ (1/s)Z. Then c E (D) = γ E (D) = st. For c E (D) it follows from Proposition 4.2.4, and for γ E (D) it is the definition. So we get that c E (D) = γ E (D). Corollary 4.4.2. The complexity of X is at least equal to the number of connected components of X. Corollary 4.4. 3 . 3Let X = P 1 (Q p ) \ Z, where Z is defined over E is a disjoint union of d disks. Then c E (X) ≥ d.4.5. Finding a standard subset from a finite set of points.4.5.1. Approximations of a standard subset. Let X = ∪ N n=1 (D n,0 \ ∪ mn i=1 D n,i ) be a standard subset, where the D n,0 \ ∪ mn i=1 D n,i form the decomposition of X as a disjoint union of connected standard subset. For J ⊂ {1, . . . , N} and I n ⊂ {1, . . . , m n } for n ∈ J, we set Y J,I = ∪ n∈J (D n,0 \ ∪ i∈In D n,i ). This is a standard subset with c E (Y J,I ) ≤ c E (X) and equality if and only if X = Y J,I . Such standard subsets are called approximations of X. Lemma 4 .5. 7 . 47Let X be a standard open subset defined over E and of the form P 1 (Q p )\Z with Z ⊂ C(r) for some r > 0, and suppose that c E (X) ≤ m. Then Z is contained in a union of at most m open disks of radius r contained in C(r). Proposition 4.5. 8 . 8Let E be a finite extension of Q p . There exists a function ψ E such that for any standard subset X of Q p defined over E, if c E (X) ≤ m then there exists an extension F of E with [F : E] ≤ ψ E (m) and X ∩ F = ∅. Lemma 4 .5. 9 . 49Let E be a finite extension of Q p . There exists a function ψ 0 E such that for any open disk D of Q p defined over E, if c E (D) ≤ m then there exists an extension F of E with [F : E] ≤ ψ 0 E (m) and D ∩ F = ∅ and the radius of D is in |F × |. For m < p 2 or p = 2 we can take ψ 0 E (m) = m and consider only extensions F/E that are totally ramified. Proof. Let s be the minimal ramification degree of an extension K of E with K ∩ D = ∅, and let t be the smallest positive integer such that D can be written as {x, stv E (x − a) > v E (b)} for a b ∈ K. So by definition c E (D) = st. By Theorem 1.1.1, there exists an extension K of E with e K/E = s and [K : E] ≤ s 2 and K ∩ D = ∅. Then if F is a totally ramified extension of degree t of K, then F satisfies the conditions, and we have [F : E] ≤ s 2 t. As st ≤ m, this means that we can take ψ 0 E (m) = m 2 . Proposition 4.5. 10 . 10Let E be a finite extension of Q p . There exists a function φ E such that for any standard subset X of Q p defined over E and different from P 1 (Q p ), if c E (X) ≤ m then there exists an extension F of E with [F : E] ≤ φ E (m) and F ⊂ X. Lemma 4.5. 11 . 11Let E be a finite extension of Q p . There exists a function φ 0 E such that for any closed disk D of Q p defined over E, if c E (D) ≤ m then there exists an extension F of E with [F : E] ≤ φ 0 E (m)and D ∩ F = ∅ and the radius of D is in |F × |. For m < p 2 or p = 2 we can take φ 0 E (m) = m and consider only extensions F/E that are totally ramified. The proof is the same as the proof of Lemma 4.5.9. Corollary 5 .3. 2 . 52In the conditions of Theorem 5.3.1, the map λ defines an open immersion of analytic spaces. The image of X (Q p ) by λ is a standard subset of P 1 (Q p ) that is defined over E. Proposition 5 .4. 3 . 53Suppose that either ρ is irreducible, or ρ is an extension of α by β where β/α ∈ {1, ω}. Then X ′ (k, ρ) = Y (k, r). Proof. The result is clear when ρ is irreducible. Recall that dim Ext 1 (α, β) > 1 if and only if β/α ∈ {1, ω}. Suppose that ρ is an extension of α by β where β/α ∈ {1, ω}. Let x ∈ 7 . 7Parameters classifying potentially semi-stable representations 7.1. Results on Weil representations. 7.1.1. Field of definition. Let W Qp be the Weil group of Q p . A Weil representation is a representation of W Qp with coefficients in Q p that is trivial on an open subgroup of I Qp . [KO74, IV. Proposition 1.3], there is a family (f i ) in A generating the unit ideal such that for all i, the automorphism of M n (A[1/f i ]) induced by τ is inner. Hence the result. 7.3.3. Variations on Hilbert 90. Proposition 7.3.3. Let K be an infinite field, and L/K be a finite Galois extension of fields. (1) Let M be a finite K-algebra. Then H 1 (Gal(L/K), (L ⊗ K M) × ) = 0. (2) Let A be a K-algebra. Assume that for every maximal ideal m of A, A/m is a finite extension of K. Let c ∈ H 1 (Gal(L/K), (L ⊗ K A) × ). There exists a family of elements (f i ) in A that generate the unit ideal such that the image of c in x), so c = 0 as soon as we can find an x such that φ(c, x) is invertible in L ⊗ K M. Point (1) is well-known, and is proved by showing that if M is finite over K then such an x exists, with a proof similar to the case where M = M n (K) (here we do not need M to be commutative). For any commutative K-algebra M, the M-algebra L ⊗ K M is finite. We denote by N M the norm map L ⊗ K M → M, so that for all x ∈ L ⊗ K M, we have x ∈ (L ⊗ K M) × if and only if N M (x) ∈ M × . Moreover the norm map commutes with base change: let u : M → M ′ be a map of K-algebras, then N M ′ (1 ⊗ u)(x) = u(N M (x)) for all x ∈ L ⊗ K M. dR,0 ⊗ E A.So f defines an analytic map over Max(A) with values in P(D Qp dR,0 ) = P(V τ ), which is what we wanted. 1.1. Statements. Theorem 1.1.1. Let D be a disk defined over E. Let e be the smallest integer such that there exists a finite extension F of E with e F/E = e and F ∩ D = ∅. Then e = p s for some s, and there exists an extension F of E with [F : E] ≤ max(1, p 2s−1 ) such that F ∩ D = ∅. For s ≤ 1 any such F/E is totally ramified. gives the result. Proposition 2.2.2. Let A be a local complete noetherian local R-algebra which is a domain. Let B ⊂ A[1/π] be a finite A-algebra. Let k A and k B be the residue fields of A and B respectively. Then e(A) ≥ [k B : k A ]e(B). /A, and (B/A)[π] = B/A and B is π-torsion free so B[π] = 0. Hence we get that e(A, B) = e(A, A). So we only need to show that e(A, B) ≥ [k B : k A ]e(B), which follows from Lemma 2.2.1 applied to T = A ⊗ R k and S = B ⊗ R k. Remark 2.2.3. We give some examples: Let connected standard subset is called of closed type if D 0 is a closed disk (in the bounded case), and the D i are open disks for i > 0. A connected standard subset is called of open type if D 0 is an open disk (in the bounded case), and the D i are closed disks fori > 0. A subset of P 1 (Q p ) is called standard if it is a finite union of disjoint connected standard subsets. It is called a standard subset of open type if it is a finite union of disjoint connected standard subsets of open type, and we define similarly a standard subset of closed type. If X is a standard subset of open type of P 1 (Q p ) it can be written uniquely as a disjoint finite union of connected standard subsets of open type, which we call the connected components of X. We check easily the following result: Lemma 3.2.3. Let X and Y be two connected standard subsets of closed (resp. open) type. If X ∩ Y = ∅ then X ∩ Y and X ∪ Y are connected standard subsets of closed (resp. open) type. As a consequence, any finite union of connected standard subsets of closed (resp. open) type is a standard subset of closed (resp. open) type. Following [LR96, Definition 4.1], we define: Definition 3.2.4. A special subset of Q p is a subset of one of the following form: and Y ′ i is a standard subset of open type. As we can do this for all i, we have written U as ( the decomposition of X as a disjoint union of standard subsets that are defined and irreducible over E. We define the complexity of X over E to be c E Proposition 4.2.6. Let X be a connected standard subset defined over Eℓ,n , one of the coefficients at least is in O × E , and so the image of Zf in A is not zero. So we are in the conditions of Corollary 2.1.2, and so e(A) = dim k A/(z) = Nst. 4.2.4. Additivity formula. 1.1, and s ≤ m. So if m < p 2 then s = 1 or s = p so we can take [K : E] ≤ s and K/E totally ramified instead of [K : E] ≤ s 2 , and so we can take [F : E] ≤ m. When p = 2 the result comes from applying Theorem 1.1.2 instead of Theorem 1.1.1. then by Proposition 4.5.10, there exists an extension F/E with [F : 1/p] and R 0 i is finite over R i . As R i [1/p] is formally smooth, it is normal, hence so is R 0 X i that are bounded by 1, and that R i [1/p] is equal to the ring of bounded analytic functions on X i . 5.3. Results.5.3.1. Parameters on deformation spaces.i . From [dJ95, Theorem 7.4.1], we deduce that R 0 i is equal to the ring Γ(X i , O 0 X i ) of analytic functions on Theorem 5.3.1. Lemma 7.1.1. Let τ be an irreducible Weil representation. Then there exists a finite unramified extension E of E(τ ) such that τ is realizable over E. Proof. From the results of [Kra83, 1.4], we see that the obstruction to realizing τ over E(τ ) is in the Brauer group of E(τ ). An element of the Brauer group can be killed by taking a finite unramified extension, hence the result. ) . )Theorem 7.2.2. Let F be a finite Galois extension of Q p . Let X be a reduced rigid analytic space, let V be a locally free O X -module of rank n with a continuous action of G Qp . Assume that for all x ∈ X, V x is potentially semi-stable with weights independent of x, and becomes semi-stable on F . Then F ⊗ F 0 D is endowed of a filtration by locally free sub-F ⊗ Qp O X -modules, such that the graded parts are also locally free, such that for all x, (F ⊗ F 0 D) x is isomorphic, as a filtered (φ, N, Gal(F/Q p ))-module, to D F dR (V x ). Proof. This follows from [Bel15, Theorem 5.1.7], as F ⊗ F 0 D is the F ⊗ Qp O X -module that is called D B dR (V) there, considering V as a representation of G F . Indeed the filtration, and the graded parts, are given by the modules called D[a,b]B dR (V). The point that we need to check is that for all[a, b], the F ⊗ Qp E x -modules D[a,b] we see that there is a covering of Spec A by open subsets of the form D f with c A[f −1 ] = 0, which is what we wanted.7.4. The crystalline case. We want to prove Theorem 5.3.1 for the case where the Galois type is of the form (1), that is, τ = χ ⊕ χ for some smooth character χ of I Qp that extends to W Qp . By twisting by the character χ, we can reduce to the case where τ is the trivial representation of I Qp , that is, the case of crystalline deformation rings. Recall from Section 5.4 the definition of the parameter a p .Proposition 7.4.1. There is an element a p ∈ R(k, ρ)[1/p] such that for any finite extension E x of E and x : R(k, ρ)[1/p] → E x corresponding to a representation ρ x , a p (x) is the value of a p corresponding to ρ x by the classification of Lemma 5.4.1. Proof. In order to prove this result, it is enough to prove it for an admissible covering of X . Indeed, the condition that L(x) is the L-invariant of ρ x ensures that the functions defined on each subset of the covering will glue. In particular, we can assume that X is affinoid, coming from a Tate algebra A over E. By Theorems 7.2.1 and 7.2.2, there is a projective A-module D of rank 2 over A, endowed with a structure of filtered (φ, N)-module, such that for all x ∈ Max(A), D x is D st (ρ x ). Consider the action of φ on D: it has eigenvalues pα −1 and α −1 . By Proposition 7.3.1, we can assume, after replacing A by a Zariski covering, that D is free over A, with a basis e 1 , e 2 such that φ(e 1 ) = pα −1 e 1 and φ(e 2 ) = α −1 e 2 . By the commutation relations between φ and N, there is a λ ∈ A such that Ne 1 = λe 2 . Moreover, we can assume that there is a free A-module L of rank 1 in D, with quotient that is also free of rank 1, that gives the non-trivial step of the filtration. We fix a basis f of L.Let h = det(f, φ(f )). Let us show that N and h do not vanish simultaneously. If this is the case, let x be a point where they both vanish. Then ρ x is crystalline, as N x = 0, and the filtration of the associated filtered φ-module is generated by an eigenvector of φ, as h x = 0. Then the representation ρ x is necessarily split reducible. But by hypothesis this can not happen. So by replacing Max(A) by a Zariski cover, we can assume that either N never vanishes, or h in a unit in A.Assume first that N never vanishes, that is, ρ x is never crystalline. Then the λ as defined above is actually a unit in A, so we can modify the basis (e 1 , e 2 ) so that λ = 1. Write f in this basis as ae 1 +be 2 , with a, b ∈ A. By specializing at each x ∈ Max(A), we see that a(x) = 0 for all x, as this would contradict the admissibility condition of the filtered module. So a ∈ A × . Then by definition of the L-invariant, we have L(x) = −(b/a)(x) for all x ∈ Max(A). So the function L is indeed an analytic function on Max(A).Assume now that h is a unit in A. Let (e 1 , e 2 ) be the basis of D defined above such that each e i is an eigenvector for φ. We can write f = ae 1 + be 2 for some a, b ∈ A. Then the condition on h implies that a and b are in A × , that is, (ae 1 , be 2 ) is also a basis of D over A. So we can modify the basis so that we have moreover f = e 1 + e 2 . After specializing at x ∈ Max(A) an easy computation shows that λ(x) = −1/L(x) (and in particular the condition on h implies that L does not take the value 0). So we have defined an analytic function Max(A) → P 1 by taking L = 1/λ. 7.7. Supersingular types. In this Section, assume that p > 2. We consider now the case where the type is supersingular, that is, the Weil representation is (absolutely) irreducible. 7.7.1. Defining the generalized L-invariant. We fix once and for all a supersingular extended type τ , that is, a smooth absolutely irreducible representation τ : W Qp → GL 2 (E 0 ) for some finite extension E 0 of Q p . This corresponds to cases (2) and (3) of the classification of types of [GM09, Lemma 2.1]. Note that we can take E 0 to be an unramified extension of the definition field of τ by Lemma 7.1.1.Let F be a finite Galois extension of Q p such that τ is trivial on I F , and let F 0 be the maximal unramified extension of Q p contained in F . We assume, after taking an unramified extension of E 0 if necessary, that F 0 ⊂ E 0 .Let D crys,0 be the (φ, Gal(F/Q p ))-module corresponding to τ via the correspondence of Proposition 7.1.2. Let D dR,0 = F ⊗ F 0 D crys,0 . It is endowed with an action of Gal(F/Q p ) coming from the one on D crys,0 . Then: is an E 0 -vector space of dimension 2. Familles de représentations de de Rham et monodromie p-adique. Laurent Berger, Pierre Colmez, Représentations p-adiques de groupes p-adiques. I. Représentations galoisiennes et (φ, Γ)-modules. Laurent Berger and Pierre Colmez. Familles de représentations de de Rham et monodromie p-adique. Astérisque, (319):303-337, 2008. Représentations p-adiques de groupes p-adiques. I. Représentations galoisiennes et (φ, Γ)-modules. Rebecca Bellovin. p-adic Hodge theory in rigid analytic families. Algebra Number Theory. 9Rebecca Bellovin. p-adic Hodge theory in rigid analytic families. Algebra Number Theory, 9(2):371-433, 2015. 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We evaluate the πNN , πΣΣ, πΛΣ, KΛN and KΣN coupling constants and the corresponding monopole masses in lattice QCD with two flavors of dynamical quarks. The parameters representing the SU(3)-flavor symmetry are computed at the point where the three quark flavors are degenerate at the physical s-quark mass. In particular, we obtain α ≡ F/(F + D) = 0.395(6). The quark-mass dependences of the coupling constants are obtained by changing the u-and the d-quark masses. We find that the SU(3)-flavor parameters have weak quark-mass dependence and thus the SU(3)-flavor symmetry is broken by only a few percent at each quark-mass point we consider.

10.1103/physrevd.79.074509

0805.3068

49375a4d57b0ca34445228d6b2b0c9b5c5a6e499

Pseudoscalar-meson-Octet-baryon Coupling Constants in Two-flavor Lattice QCD 14 Apr 2009 Güray Erkol Ozyegin University Kusbakisi Caddesi No:2 Altunizade Uskudar Istanbul 34662Turkey Department of Physics Tokyo Institute of Technology H-27152-8551MeguroTokyoJapan Makoto Oka Department of Physics Tokyo Institute of Technology H-27152-8551MeguroTokyoJapan Toru T Takahashi Yukawa Institute for Theoretical Physics Kyoto University 606-8502SakyoKyotoJapan Pseudoscalar-meson-Octet-baryon Coupling Constants in Two-flavor Lattice QCD 14 Apr 2009(Dated: April 14, 2009)arXiv:0805.3068v2 [hep-lat]R(t 2 , t 1p ′ , pΓµ) = G BPB ′ (t 2 , t 1p ′ , pΓ) G B ′ (t 2p ′Γ 4 ) G B (t 2 − t 1pΓ 4 ) G B ′ (t 2 − t 1p ′Γ 4 ) × G B ′ (t 1p ′Γ 4 ) G B ′ (t 2p ′Γ 4 ) G B (t 1pΓ 4 ) G B (t 2pΓ 4 ) We evaluate the πNN , πΣΣ, πΛΣ, KΛN and KΣN coupling constants and the corresponding monopole masses in lattice QCD with two flavors of dynamical quarks. The parameters representing the SU(3)-flavor symmetry are computed at the point where the three quark flavors are degenerate at the physical s-quark mass. In particular, we obtain α ≡ F/(F + D) = 0.395(6). The quark-mass dependences of the coupling constants are obtained by changing the u-and the d-quark masses. We find that the SU(3)-flavor parameters have weak quark-mass dependence and thus the SU(3)-flavor symmetry is broken by only a few percent at each quark-mass point we consider. Meson-baryon coupling constants are important ingredients for hadron physics as they provide a measure of baryon-baryon interactions in terms of One Boson Exchange (OBE) models, and production of mesons off the baryons. In phenomenological potential models, the meson-baryon coupling constants are determined so as to reproduce the nucleon-nucleon, hyperon-nucleon and the hyperon-hyperon interactions in terms of, e.g., OBE models. On the other hand, it is an important issue to determine the coupling constants at the hadronic vertices directly from QCD, the underlying theory of the strong interactions. The only method we know that provides a first-principles calculation of hadronic phenomena is lattice QCD, which serves as a valuable tool to determine the hadron couplings in a model-independent way. Among other meson-baryon coupling constants, the πNN coupling constant, g πNN , which enters as a fundamental quantity in low-energy dynamics of nucleonnucleon and pion-nucleon, has been a subject of intense investigation. It is defined as the πNN form factor, g πNN (q 2 ), at zero momentum transfer, q 2 = 0. The value of the coupling constant at the pion pole is relatively well-known from experiment: g 2 πNN (m 2 π )/4π ≃ 13.6 (see, e.g., Ref. [1,2] for a review). The value at zero momentum transfer can be extracted from the Goldberger-Treimann relation (GTR), g πNN ≡ g A m N /f π ∼ 12.8, where f π is the pion decay constant and m N and g A are the mass and the axial-vector coupling constant of the nucleon, respectively. An earlier determination of the πNN coupling constant from a quenched-lattice QCD calculation, which reports g πNN = 12.7 ± 2.4 [3], is in agreement with the phenomenological value. In the SU(3)-flavor [SU(3) F ] symmetric limit, one can classify the pseudoscalar-meson-octet-baryon coupling constants in terms of two parameters: the πNN coupling constant and the α = F/(F +D) ratio of the pseudoscalar octet [4]. This systematic classification is expected to govern all the meson-baryon couplings, however as we move from the symmetric case to the realistic one, the SU(3) F breaking occurs as a result of the s-quark mass and the physical masses of the baryons and mesons. The broken sym-metry no longer provides a pattern for the meson-baryon coupling constants, and therefore they should be individually calculated based on the underlying theory, QCD. In this work we extract the coupling constants g πNN , g πΣΣ , g πΛΣ , g KΛN and g KΣN (denoted by g MBB ′ hereafter) by employing lattice QCD with two flavors of dynamical quarks. The evaluation of the coupling constants allows us to check the following SU(3) F relations: g πNN = g, g πΣΣ = 2gα, g πΛΣ = 2 √ 3 g(1 − α), g KΛN = − 1 √ 3 g(1 + 2α), g KΣN = g(1 − 2α),(1) which phenomenologically work rather well but are not known a priori to hold. The pseudoscalar current matrix element is written as B(p)|P (0)|B ′ (p ′ ) = g P (q 2 )ū(p)iγ 5 u(p ′ ),(2) where g P (q 2 ) is the pseudoscalar form factor, q µ = p ′ µ − p µ is the transferred four-momentum and P (x) = ψ(x)iγ 5 τ3 2 ψ(x) is the pseudoscalar current. We compute this matrix element using the ratio [5,6] R(t 2 , t 1 ; p ′ , p; Γ; µ) = G BPB ′ (t 2 , t 1 ; p ′ , p; Γ) G B ′ (t 2 ; p ′ ; Γ 4 ) G B (t 2 − t 1 ; p; Γ 4 ) G B ′ (t 2 − t 1 ; p ′ ; Γ 4 ) × G B ′ (t 1 ; p ′ ; Γ 4 ) G B ′ (t 2 ; p ′ ; Γ 4 ) G B (t 1 ; p; Γ 4 ) G B (t 2 ; p; Γ 4 ) 1/2 ,(3) where the baryonic two-and three-point correlation functions are respectively defined as with Γ ≡ γ 3 γ 5 Γ 4 and Γ 4 ≡ (1 + γ 4 )/2. The baryon interpolating fields are given as G B (t; p; Γ 4 ) = x e −ip·x Γ αα ′ 4 × vac|T [η α B (x)η α ′ B ′ (0)]|vac ,(4)G BPB ′ (t 2 , t 1 ; p ′ , p; Γ) = −i x2,x1 e −ip·x2 e iq·x1 × Γ αα ′ vac|T [η α B (x 2 )P (x 1 )η α ′ B ′ (0)]|vac ,(5)η N (x) = ǫ abc [u T a (x)Cγ 5 d b (x)]u c (x), η Σ (x) = ǫ abc [s T a (x)Cγ 5 u b (x)]u c (x), η Λ (x) = 1 √ 6 ǫ abc {[u T a (x)Cγ 5 s b (x)]d c (x) − [d T a (x)C × γ 5 s b (x)]u c (x) + 2[u T a (x)Cγ 5 d b (x)]s c (x)},(6) where C = γ 4 γ 2 and a, b, c are the color indices. t 1 is the time when the meson interacts with a quark and t 2 is the time when the final baryon state is annihilated. The ratio in Eq. (3) reduces to the desired pseudoscalar form factor when t 2 − t 1 and t 1 ≫ a, viz. R(t 2 , t 1 ; 0, p; Γ; µ) t1≫a − −−−−− → t2−t1≫a g L P (q 2 ) [2E(E + m)] 1/2 q 3 ,(7) where m and E are the mass and the energy of the ini-tial baryon, respectively, and g L P (q 2 ) is the lattice pseudoscalar form factor. Since the ratio in (7) is proportional to the transfered momentum q 3 , it cannot be used directly to obtain g L P (q 2 ) at q 2 = 0. We apply a procedure (similarly to the one in Ref. [3]) of seeking plateau regions as a function of t 1 in the ratio (7) and calculating g L P (q 2 ) at the momentum transfers q 2 a 2 = n(2π/L) 2 (for the lowest nine n points), where L is the spatial extent of the lattice. We then obtain the meson-baryon form factor via the relation g L P (q 2 ) = G M g MBB ′ (q 2 ) m 2 M − q 2 ,(8) assuming that the pseudoscalar form factors are dominated by the pseudoscalar-meson poles. Here G M ≡ vac|P (0)|M is extracted from the two-point mesonic correlator P (x)P (0) . Finally we extract the mesonbaryon coupling constants g MBB ′ = g MBB ′ (0) by means of a monopole form factor: g MBB ′ (q 2 ) = g MBB ′ Λ 2 MBB ′ Λ 2 MBB ′ − q 2 .(9) We employ a 16 3 ×32 lattice with two flavors of dynamical quarks and use the gauge configurations generated by the CP-PACS collaboration [7] with the renormalization group improved gauge action and the mean-field improved clover quark action. We use the gauge configurations at β = 1.95 with the clover coefficient c SW = 1.530, which give a lattice spacing of a = 0.1555(17) fm (a −1 = 1.267 GeV), which is determined from the ρ-meson mass. The simulations are carried out with four different hopping parameters for the sea and the u,d valence quarks, κ sea , κ u,d val = 0.1375, 0.1390, 0.1400 and 0.1410, which correspond to quark masses of ∼ 150, 100, 65, and 35 MeV, and we use 490, 680, 680 and 490 such gauge configurations, respectively. The hopping parameter for the s valence quark is fixed to κ s val = 0.1393 so that the Kaon mass is reproduced [7], which corresponds to a quark mass of ∼ 90 MeV. We employ smeared source and smeared sink, which are separated by 8 lattice units in the temporal direction. Source and sink operators are smeared in a gauge-invariant manner with the root mean square radius of 0.6 fm. All the statistical errors are estimated via the jackknife analysis. In Table I, we give the fitted values of the meson and baryon masses as obtained from the two-point correlation function in Eq. (4). We extract the meson-baryon coupling constants, g MBB ′ , and the corresponding monopole masses, Λ MBB ′ , for each κ u,d val . In Fig. 1, the q 2 dependence of the form factors, g MBB ′ , for κ u,d val = 0.1393 is given. Our complete results are presented in Table II: We give the fitted value of the πNN coupling constant and the corresponding monopole mass in lattice unit, as well as the fitted values of the πΣΣ, πΛΣ, KΛN and KΣN coupling constants and the corresponding monopole masses normalized with g πNN and Λ πNN , respectively. In Table II, g R MBB ′ and Λ R MBB ′ denote g MBB ′ /g πNN and Λ MBB ′ /Λ πNN , respectively. We expect that the systematic errors cancel out to some degree in the ratios of the coupling constants and those of the monopole masses. We give a graphical representation of our results in Figs. 2, 3 and 4. In Fig. 2 we plot g πNN and Λ πNN as a function of the pion-mass squared. The ratios of the πΣΣ, πΛΣ, KΛN and KΣN coupling constants to the πNN coupling constant, and the corresponding monopole masses normalized with Λ πNN are shown in Fig. 3. g πNN is consistent with the experimental value at κ ≤ 0.1393 and Λ πNN decreases towards the chiral limit. Note that, in addition to the monopole form, we have tried fitting the form factors to dipole and exponential forms, which have produced coupling-constant ratios consistent with those given in Table II. Having discussed the results for g πNN , we proceed with the octet-meson-baryon coupling constants. We first concentrate on the SU(3)-flavor symmetric case, where κ u,d val ≡ κ s val = 0.1393 and the SU(3) F relations in Eq.(1) are exact. (Here we take κ u,d sea =0.1390 and neglect the difference in the sea-quark effects.) As expected, all the coupling ratios, g R πΣΣ , g R πΛΣ , g R KΛN , and g R KΣN are well reproduced with α = 0.395 (6), which is obtained by a global fit. The ratios of the monopole masses, Λ R πΣΣ , Λ R πΛΣ , Λ R KΛN , and Λ R KΣN , are consistent with unity. The obtained value of α is consistent with that in the SU(6) spin-flavor symmetry (α = 2/5) [8], which is the symmetry based on the nonrelativistic quark model. We have also tried fixing all the quark masses at κ u,d,s val = 0.1390. g πNN and the ratios of the coupling constants obtained in this case are as follows: g πNN = 12.769(495), g R πΣΣ = 0.785(10), g R πΛΣ = 0.704(6), g R KΛN = −1.003 (6), and g R KΣN = 0.211(10). We have found that the coupling constants again satisfy SU(3) F and the resulting α = 0.387(5) is consistent with that obtained at κ u,d,s val = 0.1393. In Fig. 5 we plot the ratio in Eq. (3) for g πNN as a function of current-insertion point, in order to show the plateau regions. We present the data at κ u,d,s val = 0.1393 for the first three momentum transfer values. We next discuss the SU(3) F broken case. The quarkmass dependences we find for g R MBB ′ and Λ R MBB ′ are not large. The ratios of the coupling constants, g R MBB ′ , are similar in value to those in the SU(3) F symmetric limit, and the monopole-mass ratios, Λ R MBB ′ , are almost unity independently of the quark masses. This suggests that the SU(3) F breaking is small at the quark masses we consider. Our data do not allow a direct determination of SU(3) F breaking in the chiral limit, as we have flavorsymmetric data only at κ u,d val ≡ κ s val = 0.1393. On the other hand, the value of g πNN at the chiral point is wellknown, which may serve as a reference point for us to obtain a measure of SU(3) F breaking. For this purpose, we construct the following three sets of relations: A 1 ≡ 1 2 √ 3g R πΛΣ + g R πΣΣ , A 2 ≡ g R KΣN + g R πΣΣ , A 3 ≡ 1 2 g R KΣN − √ 3g R KΛN , A 4 ≡ −g R πΣΣ − √ 3g R KΛN , A 5 ≡ 1 √ 3 g R πΛΣ − g R KΛN , A 6 ≡ √ 3g R πΛΣ − g R KΣN ,(10)B 1 ≡ 1 4 √ 3g R πΛΣ + 3g R πΣΣ + 2g R KΣN , B 2 ≡ 1 4 2g R πΣΣ + 3g R KΣN − √ 3g R KΛN , B 3 ≡ 1 √ 12 g R πΛΣ − 4g R KΛN − √ 3g R πΣΣ , B 4 ≡ 1 √ 12 4g R πΛΣ − √ 3g R KΣN − g R KΛN ,(11) and C 1 ≡ 1 2 √ 3g R πΛΣ − √ 3g R KΛN − g R πΣΣ − g R KΣN ,(12) which can be readily obtained from those in Eq. (1). In the SU(3) F symmetric limit, the above equations satisfy A 1 ≡ . . . ≡ A 6 ≡ B 1 ≡ . . . ≡ B 4 ≡ C 1 = 1, which can be verified by inserting the coupling constants at κ u,d val = 0.1393 in Table II. At other quark masses, the deviations from unity represent the amount of SU(3) F breaking. Inserting the values of the coupling constants corresponding to the lowest quark mass we consider in Table II which amounts to δ SU(3) =0.014(03), 0.003(02), 0.012(02), and 0.028(17) for the quark masses at ∼ 150, 100, 65, and 35 MeV, respectively. This suggests for the pseudoscalar-meson couplings of the octet baryons that SU(3) F is a good symmetry in the quark-mass range we consider, which is broken by only a few percent. We have also tried a quadratic fit of δ SU(3) and extracted δ SU(3) = 0.072(16) in the chiral limit. Fig. 6 shows the value of δ SU(3) as a function of m 2 π and the chiral extrapolations with errors. In Fig. 7, we plot the value of α as obtained from a global fit of the SU(3) F relations at each quark mass we consider. α slightly decreases toward the chiral limit. The deviation of α in the present quark-mass range is at most 10%, whereas that of δ SU(3) is less than 5%. We infer from this that the deviation in α should be small in the chiral limit, as we find that the ratios of the coupling constants have weak quark-mass dependence. The SU(3) F breaking effect seems to appear in α rather than in SU(3) F relations (δ SU (3) ). We have also repeated our analysis with local source and local sink for consistency check. In Fig. 7, we show the values of α as obtained from such a setup as well, where both analysis lead to consistent results with each other. In summary, we have evaluated the pseudoscalarmeson-octet-baryon coupling constants, g πNN , g πΣΣ , g πΛΣ , g KΛN and g KΣN , in two-flavor lattice QCD with the hopping parameters κ sea , κ u,d val = 0.1375, 0.1390, 0.1400 and 0.1410, which correspond to quark masses of ∼ 150, 100, 65, and 35 MeV. The parameters representing the SU(3) F symmetry have been computed at the point where the three flavors are degenerate at the physical strange-quark mass. In particular, we have obtained α ≡ F/(F + D) = 0.395 (6), which is consistent with the prediction from SU(6) spin-flavor symmetry (α = 2/5). The monopole mass we find leads to a πNN form factor which is softer than those typically used in the phenomenological OBE potential models. The ratios of the coupling constants, which are supposed to be less prone to systematic errors, show very weak quark-mass dependence. We have discussed to what extent the SU(3) F symmetry is broken as we approach the physical masses of the uand the d-quarks. Our results indicate for the pseudoscalar-meson couplings of the octet baryons that SU(3) F is a good symmetry, which is broken by only a few percent (at least) in the 35 MeV to 150 MeV range of the light quark masses. FIG. 1 : 1The q 2 dependence of the form factors, g M BB ′ for κ u,d val = 0.1393. The diamonds show the lattice data, and the solid curves denote the fitted form factors. FIG. 2 : 2gπNN and ΛπNN as a function of m 2 π . The empty circle denotes the SU(3)F limit and the diamond marks the experimental result. FIG. 3 : 3The πΣΣ, πΛΣ, KΛN and KΣN coupling constants normalized with gπNN as a function of m 2 π . The empty circle denotes the SU(3)F limit. FIG. 4 : 4Same as Fig. 3 but for monopole masses ΛπΣΣ, ΛπΛΣ, ΛKΛN and ΛKΣN normalized with ΛπNN . FIG. 5 :FIG. 6 : 56The ratio in Eq. (3) for gπNN as a function of currentinsertion point, t1, at κ u,d,s val = 0.1393 for the first three momentum-transfer values. The bands represent the adopted plateau regions. The value of δ SU (3) as a function of m 2 π . The curve and the shaded region denote linear chiral extrapolations with errors. FIG. 7 : 7The value of α as obtained from a global fit of the SU(3)F relations at each quark mass we consider (in black filled circles). We also show our results as obtained with local source and local sink (in red triangles). The empty circle and the triangle denote the SU(3)F limit. The line at α = 0.4 is shown for reference only. into (10)-(12), we find A 1 = 1.045(29), A 2 = 1.040(30), A 3 = 1.002(25), A 4 = 0.963(42), A 5 = 1.017(22), A 6 = 1.049(40), B 1 = 1.043(28), B 2 = 1.021(24), B 3 = 0.990(28), B 4 = 1.033(28) and C 1 = 1.006(27), which indicate a breaking in SU(3) F by less than 10%. Moreover, we define the average SU(3) F breaking as follows: TABLE I : IThe fitted values of mπ, mK, mN , mΛ and mΣ in lattice units.κ u,d val mπ mK mN mΛ mΣ 0.1375 0.899(1) 0.834(1) 1.707(06) 1.658(06) 1.648(06) 0.1390 0.737(1) 0.725(1) 1.475(05) 1.466(06) 1.464(06) 0.1393 0.713(1) 0.713(1) 1.455(06) 1.455(06) 1.455(06) 0.1400 0.603(1) 0.635(1) 1.289(05) 1.312(04) 1.318(05) 0.1410 0.440(1) 0.533(1) 1.051(08) 1.114(06) 1.134(07) TABLE II : IIThe fitted value of the πNN coupling constant and the corresponding monopole mass (given in lattice units), together with the fitted values of the πΣΣ, πΛΣ, KΛN and KΣN coupling constants and the corresponding monopole masses normalized with gπNN and ΛπNN , respectively. Here, we define g R M BB ′ = g M BB ′ /gπNN and Λ R M BB ′ = Λ M BB ′ /ΛπNN .κ u,d val gπNN ΛπNN g R πΣΣ g R πΛΣ g R KΛN g R KΣN Λ R πΣΣ Λ R πΛΣ Λ R KΛN Λ R KΣN 0.1375 13.953(412) 1.053(123) 0.759(11) 0.698(11) -1.038(07) 0.231(14) 1.074(065) 0.908(039) 1.011(27) 0.714(118) 0.1390 13.257(448) 1.228(189) 0.785(12) 0.697(07) -1.034(07) 0.209(12) 1.020(066) 0.988(042) 1.006(28) 0.978(223) 0.1393 13.236(478) 1.223(202) 0.789(13) 0.699(08) -1.033(08) 0.209(13) 1.020(068) 0.989(044) 1.009(30) 0.970(236) 0.1400 13.098(393) 1.013(111) 0.781(13) 0.723(08) -1.017(07) 0.242(15) 1.034(053) 0.970(033) 1.026(24) 0.802(124) 0.1410 12.834(1.092) 0.719(123) 0.781(38) 0.756(28) -1.007(30) 0.260(30) 1.083(106) 0.985(074) 1.032(58) 0.958(191) AcknowledgmentsAll the numerical calculations were performed on NEC SX-8R at CMC, Osaka university, on SX-8 at YITP, Kyoto University, and on TSUBAME at TITech. The unquenched gauge configurations employed in our analysis were all generated by CP-PACS collaboration[7]. This work was supported in part by the 21st Century COE 'Center for Diversity and University in Physics", Kyoto University and Yukawa International Program for Quark-Hadron Sciences (YIPQS), by the Japanese Society for the Promotion of Science under contract number P-06327 and by KAKENHI(17070002 and 19540275). . D V Bugg, Eur. Phys. J. 33505D. V. Bugg, Eur. Phys. J. C33, 505 (2004). . T E O Ericson, B Loiseau, A W Thomas, Phys. Rev. 6614005T. E. O. Ericson, B. Loiseau, and A. W. Thomas, Phys. Rev. C66, 014005 (2002). . K F Liu, S J Dong, T Draper, W Wilcox, Phys. Rev. Lett. 742172K. F. Liu, S. J. Dong, T. Draper, and W. Wilcox, Phys. Rev. Lett. 74, 2172 (1995). . J J De Swart, Rev. Mod. Phys. 35916J. J. de Swart, Rev. Mod. Phys. 35, 916 (1963). . C Alexandrou, T Leontiou, J W Negele, A Tsapalis, Phys. Rev. Lett. 9852003C. Alexandrou, T. Leontiou, J. W. Negele, and A. Tsapalis, Phys. Rev. Lett. 98, 052003 (2007). . C Alexandrou, G Koutsou, T Leontiou, J W Negele, A Tsapalis, Phys. Rev. 7694511C. Alexandrou, G. Koutsou, T. Leontiou, J. W. Negele, and A. Tsapalis, Phys. Rev. D76, 094511 (2007). . A , CP-PACSAli Khan, CP-PACSPhys. Rev. 6554505A. Ali Khan et al. (CP-PACS), Phys. Rev. D65, 054505 (2002). . A Pais, Rev. Mod. Phys. 38215A. Pais, Rev. Mod. Phys. 38, 215 (1966).

The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between 4 × 10 5 and ≤ 5 × 10 9 and Prandtl numbers of 0.025 ≤ P r ≤ 0.033. Our measurements are performed in two cylindrical samples of aspect ratios Γ = diameter/height = 0.5 and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beamlines in different planes reveals a clear breakdown of coherence of the large-scale circulation for Γ = 0.5. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number Re. The scaling exponent β of the power law N u ∝ Ra β crosses eventually over from β = 0.221 to 0.124 when the liquid metal flow at Γ = 0.5 reaches Ra 2 × 10 8 and the coherent large-scale flow is completely collapsed. arXiv:2110.15807v2 [physics.flu-dyn]

10.1103/physrevlett.128.164501

2110.15807

ee65b48c890563ff499de0706cef5c98cfdcebb1

Collapse of Coherent Large Scale Flow in Strongly Turbulent Liquid Metal Convection 21 Jan 2022 Felix Schindler Helmholtz-Zentrum Dresden-Rossendorf 01328DresdenGermany Sven Eckert Helmholtz-Zentrum Dresden-Rossendorf 01328DresdenGermany Till Zürner Technische Universität Ilmenau 98684IlmenauGermany UME ENSTA Paris Institut Polytechnique de Paris 91120PalaiseauFrance Jörg Schumacher Technische Universität Ilmenau 98684IlmenauGermany Tobias Vogt Helmholtz-Zentrum Dresden-Rossendorf 01328DresdenGermany Collapse of Coherent Large Scale Flow in Strongly Turbulent Liquid Metal Convection 21 Jan 2022(Dated: January 24, 2022) The large-scale flow structure and the turbulent transfer of heat and momentum are directly measured in highly turbulent liquid metal convection experiments for Rayleigh numbers varied between 4 × 10 5 and ≤ 5 × 10 9 and Prandtl numbers of 0.025 ≤ P r ≤ 0.033. Our measurements are performed in two cylindrical samples of aspect ratios Γ = diameter/height = 0.5 and 1 filled with the eutectic alloy GaInSn. The reconstruction of the three-dimensional flow pattern by 17 ultrasound Doppler velocimetry sensors detecting the velocity profiles along their beamlines in different planes reveals a clear breakdown of coherence of the large-scale circulation for Γ = 0.5. As a consequence, the scaling laws for heat and momentum transfer inherit a dependence on the aspect ratio. We show that this breakdown of coherence is accompanied with a reduction of the Reynolds number Re. The scaling exponent β of the power law N u ∝ Ra β crosses eventually over from β = 0.221 to 0.124 when the liquid metal flow at Γ = 0.5 reaches Ra 2 × 10 8 and the coherent large-scale flow is completely collapsed. arXiv:2110.15807v2 [physics.flu-dyn] In turbulent convection flows, heat and momentum are primarily transported by a large-scale circulation (LSC) which is built up by a successive clustering of thermal plumes, fragments of the thermal boundary layer that rise from the top and bottom plates into the interior [1][2][3]. The LSC is manifested as a single, partly twisted roll (SR) that fills a closed cuboid or cylindrical cell in case of width-to-height aspect ratios Γ ≈ 1 [4][5][6][7]; it appears as a whole coherent pattern of circulation rolls in convection layers with very large Γ [8][9][10][11][12][13][14]. Even though the basic SR is superposed by three-dimensional dynamics of fluctuations, a number of investigations demonstrated that their coherence and role as a backbone of the heat transport remains intact, even for higher Rayleigh number Ra, a dimensionless measure of the vigor of convective turbulence, see e.g. refs. [2,3,[15][16][17][18]. For Γ < 1, the LSC forms multiple rolls arranged on top of each other [4,[19][20][21]. The particular LSC configuration determines the magnitude of transferred heat which is quantified by the Nusselt number N u [17,21,22]. Furthermore, most theories of turbulent heat transfer [23,24] rely on the existence of a mean wind, another notion for the LSC, that provides the major fraction of kinetic energy dissipation close to the plates and allows to separate this region from the interior. Most of the experimental studies are conducted in air or water, the simulations are typically run for Prandtl numbers 0.1 ≤ P r ≤ 7. It is thus still open how the LSC is connected to the turbulent transfer when we study convection beyond this parameter range, such as for very low P r. Our presented laboratory experiments in liquid metal convection extend this range to very low P r. Here, we demonstrate the collapse of the LSC into a highly turbulent flow which causes a dramatic decrease of the amount of heat that is transferred across the fluid layer. Our experiment is designed for simultaneous measurement of temperature and velocity fields (see Fig. 1(a)) which allows an unprecedented structural analysis of the LSC in liquid metals. A total of 17 ultrasound Doppler velocimetry (UDV) sensors provide a volumetric reconstruction of the LSC in the opaque liquid metal flow to relate its structure to the turbulent heat transfer. Our data reflect a strong dependence of the turbulent heat transfer on the cell geometry; it also demonstrates that the transport mechanisms in convection can be altered profoundly when the Prandtl number of the fluid is taken to the lower limits that are possible in laboratory flows, all this in a parameter range that is inaccessible to long-term direct numerical simulations [25]. We show that the observed LSC collapse causes a significantly smaller scaling exponent β of the global heat transfer law N u ∝ Ra β that goes even below the value β = 1/4 of an asymptotic twodimensional theory [26]. Experimental setup. Our study is performed in two upright cylindrical vessels with aspect ratios Γ = D/H = 1 (D = H = 180 mm) and Γ = 0.5 (H = 2D = 640 mm) between two copper plates at constant temperatures and a thermally insulated sidewall. The vessels are filled with the eutectic alloy GaInSn, which is liquid at room temperature. The flow is actuated by the temperature difference ∆T = T Bot − T T op between the heated bottom and the cooled top. The thermal driving is represented by the Rayleigh number Ra = αgH 3 ∆T /νκ and the Prandtl number is given by P r = ν/κ with kinematic viscosity ν, thermal diffusivity κ, thermal expansion coefficient α, acceleration due to gravity g, cell diameter D, and height H. Respective thermophysical properties are found in [27]. We consider Ra ranges 4 × 10 5 ≤ Ra ≤ 6 × 10 7 for Γ = 1 and 2 × 10 7 ≤ Ra ≤ 5 × 10 9 for Γ = 0.5. Details of the experimental setup such as the sensor arrangement, N u measurement principle and tables with the measurement data can be found in the supplemen- tary information [28]. The high thermal conductivity of liquid metals requires special attention in the design of the thermal conditions at the plates. An assessment in this respect can be made by the Biot number [29,31] which incorporates the effects of turbulent transport using the Nusselt number and is given by Bi = N u λ λ Cu H Cu H ,(1) where λ and λ Cu are the thermal conductivity of the liquid metal and copper, respectively, and H Cu denotes the thickness of the copper plates which have been chosen sufficiently thin to minimize thermal inertia. The boundary conditions are assumed to be isothermal for Bi 1 [30]. For our experiments, the Biot-number ranges between 0.03 ≤ Bi ≤ 0.125 for Γ = 1 and 0.01 ≤ Bi ≤ 0.019 for Γ = 0.5, respectively. Heat losses through the side walls and the copper plates are minimized by insulating the experiment with closed-cell polyethylene foam. We thus conclude that the thermal boundary conditions in our experiment satisfy the isothermal and adiabatic properties at the top/bottom plates and the sidewall, respectively [28]. Linear velocity profiles are measured by UDV [32,33] with a total of 17 sensors, 16 of which are located at different azimuthal positions to determine the radial velocity distribution in five different horizontal planes (see Fig. 1(b)). The remaining sensor measures the vertical velocity component along the cell height at r/R = 2/3 with the cell radius R = D/2. When all ultrasonic sensors are scanned in series, a sampling rate of 1 Hz and an accuracy of 0.1 mm/s are achieved. Large-scale flow structure. Figure 1(b) shows a snapshot of a typical incoherent flow pattern at Ra = 5×10 9 which is interrupted rarely by short-term periods with a single roll structure. The supplemental movie [28] displays an example for the visualization of the LSC structure by simultaneous temperature and velocity measurements in the cell at Γ = 0.5 and Ra = 5 × 10 9 . Already at first glance, the volatile character of the flow becomes obvious. Drastic changes can be observed on short time scales involving frequent reorientations as well as rapid and irregular rotations. This accumulates to a disintegration of the SR structure and temporary transitions into doubleor multi-roll structures. The coherence level of the LSC is apparently quite low. A robust criterion to estimate the coherence of the LSC is to determine the phase correlation of the mean flow direction at both copper plates. We rely on two radial UDV sensors, placed at 90 • to each other, which determine the mean flow direction φ T op and φ Bot as well as the time-averaged LSC velocity magnitude u LSC = (u LSC,T op + u LSC,Bot )/2 near the viscous boundary layers at a distance of 10 mm from the top/bottom plate, as shown in Fig. 2(a) (see [18] for more details). From the angular difference ∆φ LSC = |φ T op − φ Bot | between the top and bottom flow, the coherence of the flow can be assessed in a relatively simple way [4]. In the case of a SR-LSC, the fluid at the top and the bottom is expected to move in opposite directions, causing an angular difference ∆φ LSC ∼ 180 • , while the formation of a double-roll structure is associated with a vanishing angular difference. From [2,4], the well-proven approach is known to evaluate the coherence of the LSC based on the temperature measurements at three different heights, where the position of upward (downward) flow was determined by locating the maximum (minimum) of the temperature distribution on the circumference. The main flow direction can be accessed directly by UDV and has not to rely on an indirect determination via temperature, which is affected by the large thermal diffusivity at low P r. In Fig. 2, the measured angular difference is plotted versus time normalized by the turnover time t to = L/u LSC assuming a SR-LSC, where the LSC path length L is taken as the largest ellipse that fits into the vertical midplane of the convection cell. Fig. 2(b) contains the results for the Γ = 1 cell at two different Ra. The stable average value of about 180 • indicates a coherent SR-LSC. The regular oscillations, which are visible particularly at the higher Ra and correlate with the turnover time, provide a clear indication of the torsional and sloshing modes typically observed in Γ = 1 cells [18,34]. The corresponding data for Γ = 0.5 in Fig. 2(c) display rapid fluctuations and weak correlation between the flow at both plates. The mean value deviates significantly from 180 • . However, the SR state is not just replaced by another LSC structure, e.g. a double-roll, since ∆φ LSC does not level off to a value close to zero. In addition to the results in Fig. 2, we plot in Fig. 3 cor- In this plot, a coherent single-roll LSC would exhibit a narrow peak near 180 • . The measurements show that the flow increasingly deviates from the SR state as Ra increases and Γ decreases. For the measurements at Γ = 0.5, there is even a trend towards a uniform distribution of the angular differences. In such a case the flows at the top and bottom would be fully uncorrelated; an inherent coupling between the flows at bottom and top (which exists in case of Γ = 1) no longer occurs here. Mean value ∆Φ LSC and standard deviation σ of ∆Φ LSC are drawn in Fig. 4 versus Ra. For Γ = 1 and low Ra, the mean value is quite close to the ideal value of 180 • for the SR-LSC and the standard deviation is comparatively small. With increasing Ra, we find a decrease in the mean value and a growing standard deviation. This demonstrates that the flow becomes more turbulent and, conversely, the coherence decreases. Just changing the aspect ratio to Γ = 0.5 reduces the mean angle difference and increases the standard deviation drastically. A further increase of Ra does not show a significant effect. Scalings and transport laws. Previous liquid metal convection experiments in Γ ≥ 1 and moderate Ra exhibit a robust and coherent LSC [18,35]. This is associated with intense flow velocities of u LSC /u ff = (Ra/P r) −0.5 Re ≈ 0.7 in a Γ = 2 cylinder [35] and a Γ = 5 rectangular box [36], where Re = u LSC H/ν is the Reynolds number and the free-fall velocity u ff = √ gα∆T H is the theoretical upper velocity limit in case the potential buoyancy is completely converted into momentum. In contrast, our measurements performed in Γ = 0.5 reveal an almost complete breakdown of the SR-LSC for the whole Ra range. This is associated with a drastic reduction of the LSC velocity as reflected in a ratio of u LSC /u ff < 0.2 (see inset of Fig. 5(a)). The SR-LSC is still clearly pronounced in Γ = 1 cylinders [18], although our velocity measurements there already show a smaller ratio u LSC /u ff ≈ 0.4. The corresponding Re is plotted for both aspect ratios in Fig. 5(a). For the same Ra, significantly higher Re values occur in the Γ = 1 cell with a scaling of Re ∝ Ra 0.428 . The data at Γ = 0.5 indicate a gradual change of the scaling of Re(Ra) at Ra ≈ 2 × 10 8 . The Nusselt number measurements in Fig. 5(b) give a qualitatively similar picture, although the accessible measurement range of N u is narrower than that of the UDV measurements, which reliably measure even the lowest velocities of 0.5 mm/s. The data is referred to P r = 0.029 where the deviations in the Prandtl number are accounted for by assuming a P r dependence of the heat transfer of N u ∝ P r 0.14 and correcting the N u values accordingly. This power law is based on simulations by Verzicco and Camussi [40] which provide data for P r < 1 [15,37]. The measured scaling law N u ∝ Ra 0.28±0.01 for Γ = 1 is in good agreement with existing data at similar P r [18,38,39]. In contrast, the N u values and the scaling exponent β for Γ = 0.5 are lower than those for Γ = 1 at comparable Ra. Moreover, our data indicate two regions with different power laws: (i) for 4 × 10 7 ≤ Ra 2 × 10 8 we find N u ∝ Ra 0.22±0.04 and (ii) for 2 × 10 8 Ra ≤ 5 × 10 9 the scaling N u ∝ Ra 0.124±0.005 results from the data. The transition between the two scalings is observed at approximately the same Ra at which the change in Re scaling is found. Final discussion. Our heat transfer measurements differ from those at P r ∼ 1 [41][42][43] where no significant differences were found for the heat transfer scaling for Γ = 0.5, 1. The same holds for a comparison of Γ = 0.1, 1 in direct numerical simulations for 10 8 ≤ Ra ≤ 10 10 [44]. Our findings differ also from measurements in mercury at P r ≈ 0.025 [38,39,45]. Naert et al. [45] obtained exponents of β = 0.27, 0.25 and 0.28 for aspect ratios of Γ = 2, 1, and 0.5, respectively. Glazier et al. [39] reported a scaling N u ∝ Ra 0.29 for the entire range 10 5 < Ra < 10 11 using cells with Γ = 0.5, 1, and 2. Cioni et al. [38] identified three different scalings at Γ = 1 between Ra = 7 × 10 6 and 2 × 10 9 . Interestingly, their scaling law for the heat transfer changes from N u ∝ Ra 0.26 to N u ∝ Ra 0.20 which occurs around Ra = 4.5 × 10 8 , i.e., close to our value of Ra ≈ 2 × 10 8 . The scaling for Ra < 4.5 × 10 8 agrees well with our results in Γ=1. It is also worth noting that our Γ = 0.5 measurements for Ra 2×10 8 show a comparable scaling as found by Cioni et al. in their region II [38]. Both experiments [38,39] date back more than 20 years and do not provide any direct velocity measurements to characterize the LSC and its possible breakdown. Their analysis was based on temperature measurements only. We recall also that the N u measurements of both works come to different results. A comparative analysis of our data with both pioneering works is hardly possible. Furthermore, we note that the exponent β = 0.22 for Ra ≤ 2 × 10 8 is in agreement to predictions from the Grossmann-Lohse theory for low Prandtl numbers [24]. The exponent of β = 0.124 for Ra ≥ 2 × 10 8 , however, falls even below a prediction of β = 1/4 from a two-dimensional asymptotic theory [26] Thus, the N u(Ra) scaling law in Γ = 0.5 for Ra 2×10 8 reveals a very small β, which to the best of our knowl- edge, has not been reported anywhere before. The change in scaling at Ra ∼ 2 × 10 8 might originate from the transition of a highly fluctuating, partially decoherent to a fully collapsed large-scale flow. A similar sharp decrease of the N u(Ra) scaling with increasing Ra is known only from an experimental study in a cylindrical gap [46] where a sharp transition from N u ∝ Ra 0.274 to N u ∝ Ra 0.17 is found at Ra ≈ 6.35 × 10 8 . The authors trace this transition to a change from a high-symmetry, coherent state to a low-symmetry, turbulent state. The present low-P r measurements consistently show a simillar loss of coherence in the flow at Γ = 0.5, which leads to significantly lower velocity amplitudes and reduced heat transport in comparison to the stable LSC at Γ = 1. This explanation is supported by the fact that the effect works in the opposite direction as well. In [47] and [36] it was shown that an increase of flow structure coherence can increase the heat transport. It is also mentioned that higher P r seem to cause more stable LSC configurations [48,49]. We conclude that our present knowledge regarding the LSC properties at low P r and large Ra for Γ < 1 is still incomplete. Theoretical models assume a coherent wind. Its breakdown alters heat and momentum transport in ways that ask to our view for further detailed research on this topic. This work is supported by the Deutsche Forschungsgemeinschaft with Grants No. VO 2332/1-1 and SCHU 1410/29-1. Figure 1 . 1Experimental setup. a) Photograph of the experimental setup without thermal insulation. b) Instantaneous large-scale flow visualization by means of UDV (grey sensors) in the measurement at Ra = 5 × 10 9 . The velocity magnitude u is given in units of the free-fall velocity u ff = (gα∆T H) 1/2 . Figure 2 . 2Large-scale flow analysis. a) Determination of the LSC orientation using two crossed UDV sensors (see [18] for more details). b) Angular difference ∆φLSC ∈ [0 • , 360 • [ of the flow direction at the top and bottom plate vs. time normalized by the turnover time tto for two Ra in the Γ = 1 cell showing oscillations around a stable mean value of about 180 • due to torsion and sloshing of a SR-LSC. c) Respective data for the Γ = 0.5 cell revealing non-periodic frequent changes between various flow states. responding histograms of ∆Φ LSC ∈ [0 • , 180 • ], for which the values larger 180 • are transformed from ∆φ LSC by ∆Φ LSC = 360 • − ∆φ LSC to fit into the domain. 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In the paper the so-called modified Yamaguchi function for the Bethe-Salpeter equation with a separable kernel is discussed. The type of the functions is defined by the analytic stucture of the hadron current with breakup -the reactions with interacting nucleon-nucleon pair in the final state (electro-, photo-, and nucleon-disintegration of the deuteron).

10.1051/epjconf/201817302005

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0f499bad71fe81393bed9e8c0c9ee4118add4b16

On the Relativistic Separable Functions for the Breakup Reactions Serge G Bondarenko Joint Institute for Nuclear Research 141980Dubna, Moscow regionRussia Valery V Burov Joint Institute for Nuclear Research 141980Dubna, Moscow regionRussia Elena P Rogochaya Joint Institute for Nuclear Research 141980Dubna, Moscow regionRussia On the Relativistic Separable Functions for the Breakup Reactions 10.1051/epjconf/201817302005 In the paper the so-called modified Yamaguchi function for the Bethe-Salpeter equation with a separable kernel is discussed. The type of the functions is defined by the analytic stucture of the hadron current with breakup -the reactions with interacting nucleon-nucleon pair in the final state (electro-, photo-, and nucleon-disintegration of the deuteron). Introduction One of the most consistent nucleon-nucleon (NN) interaction theories is based on the solution of the Bethe-Salpeter (BS) equation [1]. In this case, we have to deal with a nontrivial integral equation for the bound state (deuteron) or for the interacting unbound NN pair. The approximations based on a kernel with particle exchange are difficult to solve. An effective and solvable approach to the exact solution of the BS equation uses the separable Ansatz for the interaction kernel in the BS equation [2]. In this case one can transform an initial integral equation into a system of linear equations. The parameters of the kernel are obtained by fitting the phase shifts, inelasticity and low-energy parameters for the respective partial-wave states. The first separable parameterizations were worked out within nonrelativistic models. The separable functions (called form factors) of the interaction kernel used in these models had no poles on the real axis in the relative energy complex plane [3,4]. However, such poles appeared when the interaction kernel was relativistically generalized. In some cases they do not prevent to perform the calculations (for example in elastic reactions). However, at high energies, one have to deal with several thresholds corresponding to the production of one, two and more mesons of different types, which is clearly not feasible to deal with. A more practical approach is to employ a phenomenological covariant separable kernel, which do not exhibit the meson-production thresholds and can even be constructed in a singularity-free fashion, using separable form factors and Wick-rotation prescription as it is done in the present paper. Thus, an accurate description of the on-shell nucleon-nucleon data is possible up to quite high energies. One then can hope that the obtained separable interaction kernels have also a reasonable off-shell behavior, so that they can be applied to other reactions as well. e-mail: [email protected] Bethe-Salpeter formalism We start with the partial-wave decomposed Bethe-Salpeter equation for the nucleon-nucleon scattering matrix T (in the rest frame of the two-nucleon system): t L L (p 0 , p , p 0 , p; s) = v L L (p 0 , p , p 0 , p; s) (1) + i 4π 3 L dk 0 k 2 dk v L L (p 0 , p , k 0 , k; s)t L L (k 0 , k, p 0 , p; s) ( √ s/2 − E k + i ) 2 − k 2 0 . Here t is the partial-wave decomposed T matrix and v is the kernel of the NN interaction, E k = √ k 2 + m 2 . There is only one term in the sum for the singlet (uncoupled triplet) case (L = J) and there are two terms for the coupled triplet case (L = J ∓ 1). We introduce the square of the total momentum s = P 2 = (p 1 + p 2 ) 2 and the relative momentum p = (p 1 − p 2 )/2 [p = (p 1 − p 2 )/2] (for details, see reference [2]). Assuming the separable form (rank I) for the partial-wave decomposed kernels of NN interactions: v L L (p 0 , p , p 0 , p; s) = λg [L ] (p 0 , p )g [L] (p 0 , p),(2) we can solve Eq. (1) and write for the T matrix: t L L (p 0 , p , p 0 , p; s) = τ(s)g [L ] (p 0 , p )g [L] (p 0 , p),(3) with the function τ(s) being given by τ(s) = 1/(λ −1 + h(s)).(4) The function h(s) has the following form: h(s) = L h L (s) = − i 4π 3 dp 0 p 2 dp L [g [L] (p 0 , p)] 2 ( √ s/2 − E p + i ) 2 − p 2 0 .(5) The simplest separable function g(p 0 , p) which can be used is a covariant generalization of the non-relativistic Yamaguchi-type [5,6] function: g(p 0 , p) = 1 p 2 0 − p 2 − β 2 + i ,(6) where β is a parameter. Modified Yamaguchi-type functions Let us consider the integral h(s) (Eq. 5). Taking into account the pole structure of the propagators: p (1,2) 0 = ± √ s/2 ∓ E p ± i(7) and of the g functions: p (3,4) 0 = ∓E β ± i(8) and using the Cauchy theorem, the h(s) function can be written as follows: 1 2π 2 p 2 dp 1 (s/4 − √ sE p + m 2 − β 2 ) 2 1 √ s − 2E p + i .(9) To calculate this integral one should analyze the factor f = (s/4 − √ sE p + m 2 − β 2 ) in the denominator as a function of s: • if 2(m − β) < √ s< 2(m + β) then always f < 0 and the function 1/ f n is integrable for any integer n and any E p ; • for a bound state √ s= M d = (2m − D ). Since for minimal β min = 0.2 GeV always β min > D /2 then the function 1/ f n is integrable for any integer n and any E p ; • if √ s< 2(m − β) or √ s> 2(m + β) then f can be positive and negative and 1/ f n is non-integrable for even n at any E p . The critical value s c = 4(m + β) 2 corresponds to the laboratory kinetic energy of np-pair T c lab = 4β + 2β 2 /m 4β. If β min = 0.2 GeV then T min lab = 0.8 GeV. So, if we consider breakup processes of the deuteron such as photo-, electro-and nucleon-breakup Yamaguchi-functions can be used only if the laboratory kinetic energy of the NN-pair is less than T min lab . To avoid this restriction we suggest to use Yamaguchi-type functions modified in the following way: g Y (p 0 , p) = 1/(p 2 with a propagator which has poles only on the real axis in the p 0 complex plane; one of them is circled from below and the other -from above. So, the path of integration is defined by an appropriate contour for the propagator; 2. the calculation over the presented path leads to a pure real contribution from the form factor poles and, therefore, to the unitary S matrix (or the corresponding unitarity condition for the T matrix). We also obtain a correct transition to ordinary form factors of type g ∼ 1/(p 2 0 − p 2 −β 2 ) 2 in the α → 0 limit. In general, the modified Yamaguchi-type functions can be written as: g [a] i (p 0 , p) = (p ci − p 2 0 + p 2 ) n i (p 2 0 − p 2 ) m i ((p 2 0 − p 2 − β 2 1i ) 2 + α 4 1i ) k i ((p 2 0 − p 2 − β 2 1i ) 2 + α 4 1i ) l i ,(11) where the parametersn i , m i , k i , l i (integer), p ci , β 1i , β 2i , α 1i , α 2i (real) depend on the channel [a] under consideration. Such g form factors are used to describe neutron-proton scattering observables (phase shifts, inelasticities, low-energy parameters and deuteron characteristics) for the total angular momentum J = 0, 3 in a wide energy range (see [9]- [13]). − p 2 − β 2 ) −→ g MY (p 0 , p) = 1/((p 2 EPJ Web of Conferences 173, 02005 (2018) https://doi.org/10.1051/epjconf/201817302005 Mathematical Modeling and Computational Physics 2017 AcknowledgementOne of the authors (S.G. Bondarenko) would like to thank the organizers of the International Conference "Mathematical Modeling and Computational Physics, 2017" (JINR, Dubna, July 3-7, 2017) for the invitation and opportunity to present our results.This work was partially supported by the Russian Foundation for Basic Research grant N o 16-02-00898.here Y stands for the Yamaguchi and MY -for Modified Yamaguchi functions.To work with the modified Yamaguchi-type functions the procedure of p 0 integration should be modified, too. This procedure is worthy of a special discussion. The poles of the h(s) integral with the modified Yamaguchi-type functions are: All poles and the contour of integration are pictured inFig. 1(a,b). The idea how to choose the contour appeared owing to[7,8]. It is:1. the contour must envelope the poles of g form factors which will be inside the standard contour in the α → 0 limit. "Standard" means the contour used in the quantum field theory calculations . E E Salpeter, H A Bethe, Phys. Rev. 841232E.E. Salpeter and H.A. Bethe, Phys. Rev. 84, 1232 (1951) . S G Bondarenko, V V Burov, A V Molochkov, G I Smirnov, H Toki, Prog. Part. Nucl. Phys. 48449S.G. Bondarenko, V.V. Burov, A.V. Molochkov, G.I. Smirnov, and H. Toki, Prog. Part. Nucl. Phys. 48, 449 (2002) . L Mathelitsch, W Plessas, W Schweiger, Phys. Rev. 2665L. Mathelitsch, W. Plessas, and W. Schweiger, Phys. Rev. C26, 65 (1982) . J Haidenbauer, W Plessas, Phys. Rev. 301822J. Haidenbauer and W. Plessas, Phys. Rev. C30, 1822 (1984) . Y Yamaguchi, Phys. Rev. 951628Y. Yamaguchi, Phys. Rev. 95, 1628 (1954) . Y Yamaguchi, Y Yamaguchi, Phys. Rev. 951635Y. Yamaguchi, Y. Yamaguchi, Phys. Rev. 95, 1635 (1954) . R E Cutkosky, P V Landshoff, D I Olive, J C Polkinghorne, Nucl. Phys. 12281R.E. Cutkosky, P.V. Landshoff, D.I. Olive, and J.C. Polkinghorne, Nucl. Phys. B12, 281 (1969) . T D Lee, G C Wick, Nucl. Phys. 9209T.D. Lee and G.C. Wick, Nucl. Phys. B9, 209 (1969) . S G Bondarenko, V V Burov, W.-Y. Pauchy Hwang, E P Rogochaya, Nucl. Phys. 832233S.G. Bondarenko, V.V. Burov, W.-Y. Pauchy Hwang, and E.P. Rogochaya, Nucl. Phys. A832, 233 (2010) . S G Bondarenko, V V Burov, W.-Y. Pauchy Hwang, E P Rogochaya, Nucl. Phys. 84875S.G. Bondarenko, V.V. Burov, W.-Y. Pauchy Hwang, and E.P. Rogochaya, Nucl. Phys. A848, 75 (2010) . S G Bondarenko, V V Burov, E P Rogochaya, Phys. Lett. 705S.G. Bondarenko, V.V. Burov, and E.P. Rogochaya, Phys. Lett. B705, 264-268 (2011); . Nucl. Phys. Proc. Suppl. 219-220Nucl. Phys. Proc. Suppl. 219-220, 126-129 (2011) . S G Bondarenko, V V Burov, E P Rogochaya, Nucl. Phys. Proc. Suppl. 245S.G. Bondarenko, V.V. Burov, and E.P. Rogochaya, Nucl. Phys. Proc. Suppl. 245, 291-297 (2013) Proceedings of the 12th International Workshop "Relativistic Nuclear Physics from Hundreds of MeV to TeV. S G Bondarenko, V V Burov, S E Kemelzhanova, E P Rogochaya, N Sagimbaeva, Slovak Republic, Stara LesnaS.G. Bondarenko, V.V. Burov, S.E. Kemelzhanova, E.P. Rogochaya, and N. Sagimbaeva, Pro- ceedings of the 12th International Workshop "Relativistic Nuclear Physics from Hundreds of MeV to TeV" (Slovak Republic, Stara Lesna, June 16-20, 2014)

We consider determination of spin-orbit (SO) coupling constants for the two-dimensional electron gas from measurements of electric properties in rotated in-plane magnetic field. Due to the SO coupling the electron backscattering is accompanied by spin precession and spin mixing of the incident and reflected electron waves. The competition of the external and SO-related magnetic fields produces a characteristic conductance dependence on the in-plane magnetic field value and orientation which, in turn, allows for determination of the absolute value of the effective spin-orbit coupling constant as well as the ratio of the Rashba and Dresselhaus SO contributions. arXiv:1604.01270v2 [cond-mat.mes-hall] 30 Sep 2016

10.1103/physrevb.94.121304

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Conductance measurement of spin-orbit coupling in the two-dimensional electron systems with in-plane magnetic field K Kolasiński Faculty of Physics and Applied Computer Science AGH University of Science and Technology al. Mickiewicza 3030-059KrakówPoland Institut Néel CNRS Université Joseph Fourier BP 16638042GrenobleFrance H Sellier Institut Néel CNRS Université Joseph Fourier BP 16638042GrenobleFrance B Szafran Faculty of Physics and Applied Computer Science AGH University of Science and Technology al. Mickiewicza 3030-059KrakówPoland Conductance measurement of spin-orbit coupling in the two-dimensional electron systems with in-plane magnetic field We consider determination of spin-orbit (SO) coupling constants for the two-dimensional electron gas from measurements of electric properties in rotated in-plane magnetic field. Due to the SO coupling the electron backscattering is accompanied by spin precession and spin mixing of the incident and reflected electron waves. The competition of the external and SO-related magnetic fields produces a characteristic conductance dependence on the in-plane magnetic field value and orientation which, in turn, allows for determination of the absolute value of the effective spin-orbit coupling constant as well as the ratio of the Rashba and Dresselhaus SO contributions. arXiv:1604.01270v2 [cond-mat.mes-hall] 30 Sep 2016 Introduction. Charge carriers in semiconductor devices are subject to spin-orbit (SO) interactions [1] stemming from the anisotropy of the crystal lattice and/or the device structure. The SO interactions translate the carrier motion into an effective magnetic field leading to carrier spin relaxation and dephasing [2][3][4], spin Hall effects [5][6][7], formation of topological insulators [8], persistent spin helix states [9][10][11], Majorana fermions [12]. Moreover, the SO coupling paves the way to spin-active devices, including spin-filters based on quantum point contacts (QPCs) [13] or spin transistors [14][15][16][17][18], which exploit the precession of the electron spin in the effective magnetic field [19]. The most popular playground for studies of spin effects and construction of spin-active devices is the two-dimensional electron gas (2DEG) confined at an interface of an asymmetrically doped III-V heterostructure, with a strong built-in electric fields in the confinement layer giving rise to the Rashba SO coupling [20] and with the Dresselhaus coupling due to the anisotropy of the lattice which is enhanced by a strong localization of the electron gas in the growth direction [21]. The SO interaction is sample-dependent and its characterization is of a basic importance for description of spin-related phenomena and devices. The SO coupling constant are derived from the Shubnikov-de Haas [22][23][24][25][26][27][28][29] oscillations, antilocalization in the magnetotransport [30], photocurrents [31], or precession of optically polarized electron spins as a function of their drift momentum [32]. Usually both the Rashba and Dresselhaus interactions contribute to the overall SO coupling. Separation of contributions of both types of SO coupling is challenging and requires procedures based on optical polarization of the electron spins [31][32][33]. In this Letter we investigate the possibility for extraction of the Rashba and Dresselhaus constants from a purely electric measurement of the two-terminal conductance. The proposed method does not involve application of optical excitation [31,32] or a particularly complex gating [32]. The procedure given below requires rotation of the sample in an external in-plane magnetic field [34], which is straightforward as compared to application of the rotated electric field to 2DEG [32]. Also, the present approach is suitable for high mobility samples and goes without analysis of the localization effects in the magnetotransport [30]. The procedure which is proposed below bases on an idea that the effects of the SO coupling related to the wave vector component in the direction of the current flow can be excluded by a properly oriented external inplane magnetic field. The procedure exploits spin effects of backscattering -due to intentionally introduced potentials -or simply to intrinsic imperfections within the sample. In particular we show that the linear conductance of a disordered sample reveals an oscillatory behavior as a function of the magnetic field direction and amplitude. The dependence allows one to determine the strength of the SO interaction as compared to the spin Zeeman effect as well as the relative strength of both Rashba and Dresselhaus contributions. Spin-dependent scattering model. Let us start from a simple model of electron scattering (see Fig. 1). The electron is injected to the system from e.g. a QPC and comes to the potential defect from the left. The defect is taken as an infinite potential step, so that the scattering probability is 1. The incident and backscattered waves are denoted by |k + σ where k ± σ stands for the absolute value of the wave vector for the spin state σ and the superscript sign indicates the electron incoming from the left (+) or backscattered (−). Only the backscattering which returns the carriers to the QPC can alter the conductance, so we consider the scattering wave function along the line between the QPC and the defect |Ψ σ = e ik + σ r k + σ + Σ σ a σσ e −ik − σ r k − σ ,(1) where a σσ stand for the scattering amplitudes. Scattering at other angles does not decrease the conductance and is neglected for a moment. Within the 2DEG, outside the scattering center and the QPC channel the 2D electron Hamiltonian for in-plane field B = r=(0,0) Figure 1. Sketch of considered scattering process. The electron wave is incoming from the left from a source (a QPC for instance) in spin state σ, propagates to the right and is backscattered at position r = (0, 0) by the potential barrier induced by the impurity. (B x , B y , 0) reads H = E kin I+σ x (αk y −βk x +b x )+σ y (βk y −αk x +b y ), (2) where E kin = 2 k 2 2m eff , b x/y = 1 2 gµ B B x/y , m eff is the electron effective mass, α and β are the Rashba and Dresselhaus constants. The spin Zeeman effect is introduced via Pauli matrices σ x/y and the Zeeman energy will be denoted below by E B = 1 2 gµ B |B| = b 2 x + b 2 y . Note, that we use the symmetric gauge A = (B y z, −B x z, 0) then by choosing the plane of the 2DEG confinement to be located at z = 0, we get A = 0, and the magnetic field enters the Hamiltonian only via the spin Zeeman term i.e. the orbital effects do not affect the electron transport. Let us first neglect the Dresselhaus coupling (β = 0). Plane wave solution for the eigenvalues of the Schrödinger equation gives E σ = 2 k 2 2m eff + σ |p| ,(3) with σ = {+, −} denoting projections of the spin on the direction of polarization p = (αk y + b x , −αk x + b y ), and eigenvectors k ± σ = 1 √ 2 1 σ p ± x +ip ± y p ± ≡ 1 √ 2 1 σe iφ(k ± σ ,B) ,(4) for the incident (+) and backscattered (−) directions of the electron motion with p ± = |p ± |. Due to the assumed infinite scattering potential, the wave function in Eq. (1) has to vanish at r = 0 (see Fig. 1 ), Ψ σ (r = 0) = |k + σ + Σ σ a σσ k − σ = 0, hence a σσ = −σ σe iφ(k + σ ,B) + σ e iφ k − −σ ,B e iφ(k − + ,B) + e iφ(k − − ,B) .(5) In the following we use In 0.5 Ga 0.5 As material parameters with m = 0.0465m 0 , Landé factor g = 9, and the Fermi energy E F = 20meV. For the bulk Rashba [35] constant α 3D = 57.2 Å 2 , the 2D value is α = α 3D F z , where F z is the electric field in the growth-direction. The Rashba constant can be controlled by the external voltages [22] and for In 0.5 Ga 0.5 As SO coupling constants of the order of 5 to 10 meV nm [22] were recorded. In Fig. 2(a) we present the scattering amplitudes a σσ obtained from Eq. (5) as a function of the direction of the magnetic field B = (B cos (θ) , B sin (θ)), with B = 5T for scattering along the x direction, k = (k x , 0). Note that for the magnetic field oriented in the y direction θ = π/2, i.e. for B = (0, B y ) the diagonal elements of the scattering amplitudes are zero. This is a special case for which the spinor in Eq. (4) can be written in form |k ± σ = 1 iσd ± , where d ± = sign (−αk ± x + b y ) . For a weak magnetic field |αk ± x | > |b y |, we get d ± = ∓, and the orthogonality relation k d σ |k d σ = 1 2 (1 + σσ dd ), gives zero for the backscattering to states with the same spin projection on the polarization vector (p), k − σ |k + σ = 0 [see Fig. 2(a)] . On the other hand for high magnetic field |αk ± x | < |b y |, we get d ± = 1, and the spin projection on the polarization vector is conserved k − σ |k + σ = 1. In Fig. 2(b) we show the evolution of the scattering amplitudes for the orientation of magnetic field fixed at θ = π, as a function of the Rashba constant α. Then, in Eq. (2) k y = 0 and b y = 0. The scattering amplitudes cross near α ≈ 8meVnm [ Fig. 2 (b)]. In this point the Zeeman energy E B is equal to the SO coupling energy E SO = αk x . For αk x = E B the off-diagonal terms are: −b x σ x +(E B +b y )σ y , for which the scattering amplitudes (5) for eigenvectors (4) simplify to |a σσ | 2 = 1 2 for any σσ and for any in-plane orientation of B vector. The E B ≈ E SO case is presented in Fig. 2(a) where the black dashed lines show the scattering amplitudes for B = 7.2T, which shows that almost complete spin mixing |a σσ | 2 ≈ 1 2 is present for any angle. Let us now include the Dresselhaus SO coupling. The 2D Dresselhaus constant is given by β = β 3D k 2 z = β 3D π 2 d 2 , where β 3D is the bulk constant and d is the width of the 2DEG confinement in the growth direction. We consider β values from 0 to α [9,10]. The cubic Dresselhaus interaction is neglected as a small effect [10]. In the absence of the B field, for the electron incident along the x direction i.e. k y = 0, the polarization di-rection is p = (−βk x , −αk x ) with the energy eigenvalues E σ = 2 k 2 x 2m eff + σk x α 2 + β 2 . As a result the Dresselhaus interaction sets the direction of the electron spin polarization to θ = arctan α β and increases the effective SO coupling constant to γ eff = α 2 + β 2 . The above conclusions can also be reached by a direct inspection of the off-diagonal part of Hamiltonian (2) for the electron transport along the x direction (k y = 0, k x = k F ). The effective magnetic field in Eq. (2) is (−βk F +b x , −αk F +b y ). Both components of the effective magnetic field vanish for tan θ = b y b x = α β(6) and E B = 1 2 gµ B B = b 2 x + b 2 y = k F γ eff ≡ E SO .(7) For illustration we calculated the electron density at the source position -including the incident and backscattered waves using Eqs. (1,5) as ρ = σ Ψ σ |Ψ σ . The backscattering probability is roughly proportional to the electron density at the QPC [36]. The electron density -is depicted in Fig. 3 (b-d) for α = 12meVnm, β = 0; and Fig. 3(c-d) α = 9meVnm, β = 8meVnm. These values produce the same effective coupling constant γ eff ≈ 12meVnm. The results of Fig. 3(b-d) contain a distinct circular pattern in the θ, B plane. The position of the center is given by Eqs. (6,7). The angular coordinate of the center allows one to determine the ratio of the Rashba and Dresselhaus constants and the SO coupling constant γ eff can be read-out from the position of the center of the pattern on the B scale, provided that the Fermi wave vector is known. In presence of SO coupling and / or the Zeeman effect k F is spin dependent [21]. However, for the E B = E SO the off-diagonal terms of the Hamiltonian (2) vanish and the Fermi wavevector is directly related to the Fermi energy E F = 2 k 2 F 2m eff , which for the adopted parameters gives k F = 0.156/nm. For γ eff = 12 meV nm one obtains E SO = 1.875 meV, which coincides with E B for B = 7.2 T (see Fig. 3(b-d)). In Fig. 3(c) one notices a reduction of the period with respect to 3(b) with the source-impurity distance increased to 2000 nm from 1500 nm. The period of the oscillations ∆ B is ∆ (c) B ≈ 1.5T in (c), and ∆ (b) B ≈ 2.0T in (b). The ratio ∆ (c) B /∆ (b) B ≈ 3/4, is exactly an inverse of the source-impurity d s−i distance ratio. Coherent quantum transport calculations. With the intuitions gained by the simple analytical model we can pass to the calculations of the coherent transport using a standard numerical method [38], based on the quantum transmitting boundary solution of the quantum scattering equation at the Fermi level implemented in the finite difference approach, which produces the electron transfer probability used in the Landauer formula for conductance summed over the subbands of the channels far from the scattering area. Zero temperature is assumed. For the numerical calculations we consider a channel extended along the x direction, hence k x in Eq. (2) remains a quantum number characterizing the asymptotic states of the channel. Within the computational box the wave vector is replaced by an operator k = (k x , k y ) = −i∇. We consider a QPC/defect system of Fig. 1. The results presented in Fig. 3(a) indicate a reduction of the conductance below the maximal value M e 2 h for M subbands passing across the QPC. The central position of the pattern nearly coincides with the one of Fig. 3(c). The local extremum of conductance in the center of the pattern indicates the value and orientation of the external magnetic field which lifts the SO interaction effects. The angular position of Fig. 3(c) is exactly reproduced, and the amplitude of the field is B = 6.5 T instead of B = 7.2 T. The deviation in the location of the central point in the B axis in Fig. 3(a) results from the confinement in the QPC channel which is not included in our free particle model (see below). The effects described so far dealt with interference of the electron waves between the source (QPC) and the defect. In fact, the role of the source can be played by any scattering center, and the extraction of the SO coupling constant requires a presence of two or more scatterers to allow for formation of standing waves described in the previous section. For the rest of the paper we consider a channel of homogenous width W , which carries M transport modes at the Fermi level. In Fig. 4(d) we presented the conductance results for a clean channel of width W = 180nm and the computational box of length L = 1.6µm. A smooth potential barrier is introduced across the channel with height 10 meV and width 200nm. Depending on the orientation of the magnetic field the number of transport modes varies between M = 17 and M = 18. The simulation was performed for α = 9meVnm and β = 8meVnm as in Fig. 3(b,c). The conductance plot possesses an extremum precisely at the angle of θ = arctan 9 8 . The magnetic field of the extremum is slightly shifted to lower values than 7.2 T -which is a result of the reduction of k x within the potential barrier. The lack of conductance oscillations that were observed above in Fig. 3 results from a small barrier length (d s−i =200nm). The oscillations reappear when one replaces the barrier by a random disorder due to the random nonmagnetic and spin-diagonal potential fluctuations. The fluctuations simulate inhomogeneity of the doping of the potential barrier which provides the charge to the 2DEG. In 2DEG in III-V's due to the spatial separation of the impurities of the 2DEG, the defects do not introduce any significant contribution to the spin-orbit interaction (see Ref. [39] and the Supplement [40]). Figure 4(c) displays the conductance for the channel of the same width and length. The potential -displayed in Fig. 4(a) is locally varied within the range of (-0.5E F ,0.5E F ). The perturbation induces a multitude of scattering evens -the local density of states at the Fermi level for B = 0 is displayed in Fig. 4(b). In spite of the complexity of the density of states the angular shift is still arctan (9/8) ≈ π/4. The shift of the G extremum along the B scale with respect to 7.2 T is detectable -but small and of an opposite sign than in Fig. 4(d). This shift is related to the fact that for a finite width channel k y is an operator that mixes the subbands. The wave vector k y is a well-defined quantum number for electrons moving in an unconfined space. The small -but detectable -effects of a finite W disappear completely for a wider channel -which is illustrated in Fig. 4(e) for W = 0.8 µm. Here, the number of conducting bands varies between 80 and 81. The local extremum of conductance appears exactly at the positions indicated in the previous section. Note, that although the number of subbands change by 1 in Fig. 4(c,e), the variation of conductance is as large as ∼ 3e 2 /h in Fig. 4(c) and ∼ 6e 2 /h in Fig. 4(e). The conductance variation in Fig. 3(a) was very small -since the defect was far away from the QPC, for the disordered channel it is no longer the case. For completeness in Fig. 4(f) we presented calculations for a twice smaller SO coupling constants α = 4.5 meVnm, β = 4 meVnm, and γ eff = 6meVnm. The position of the maximal G along the B scale is consistently reduced from 7.2 T to 3.6 T, and the orientation of the magnetic field vector corresponding to the extremum is unchanged. Summary. We have shown that the in-plane magnetic field can lift the off-diagonal terms of the transport Hamiltonian for the two-dimensional electron gas that result from the Zeeman effect and the SO interaction. The effect appears only for a value and orientation of the external magnetic field which excludes the spin mixing effects that accompany the backscattering in presence of the SO coupling. In consequence the conductance maps for a system containing two or more scatterers -intentionally introduced -or inherently present in a disordered sample exhibit a pronounced extremum as a function of the magnetic field modulus B and orientation θ. An experimental value of B -for which the Zeeman energy is equal to the SO coupling energy -should allow one to extract the effective SO coupling constant including both the Rashba and Dresselhaus terms, and the orientation field indicates the relative contributions of both. The results indicate the ratio of the Dresselhaus and Rashba constants is exactly resolved by the procedure, and the amplitude of the magnetic field -hence the effective SO constant varies only within a 10% from the exact value depending on the channel width and disorder profile. Supplemental Material for Conductance measurement of spin-orbit coupling in the two-dimensional electron systems with in-plane magnetic field The Letter indicated a well resolved extremum of conductance for the conditions given by Eq. (6) and (7), for which the external magnetic field cancels the SO effective magnetic field that depends on the electron wave vector. The presented analysis is based on the electron transport along the x direction. The conductance extremum was found for a quantum point contact with a single defect [ Fig. 3(a)] and for a potential barrier [ Fig. 3(d)]. These two systems corresponded to a nearly 1D transport, since the potential barrier was introduced in a separable manner, and in the quantum point contact an appearance of the non-zero k y implies removal of the electron from the beam of backscattered electrons. However, for the perturbed sample [ Fig. 4(c,e,f)] the electrons undergo multiple scattering and acquire non-zero k y values between the scattering events. Results of Fig. 4(c,e,f) obtained for a disorder covering basically the entire width of the sample indicate that the effects of temporarily non-zero k y for the spin precession averages out to zero and the conductance extremum stays in the position defined by Eqs. (6,7). In order to support the discussion further we present in this document the case with a smaller number of defects, where the averaging may not be complete. We take a channel of width W =400nm and length L =1200nm. The simulation was performed for α = 9meVnm and β = 8meVnm as in Fig. 3(b,c). In the first column of Fig. S1 we show the obtained conductance images for different configurations of disorder potential V dis in the channel (second column). The third column present an image of LDOS computed for B = 0. One notices the angle shift θ = arctan α β , equal to the ratio between Rashba and Dresselhaus strengths, remains constant for different densities and configurations of disorder. Hence still for a limited number of defects the measurement of θ should provide good information about the α β . However, the position of the vertical line for which E B = E SO can be estimated roughly from the obtained images to be between 7 and 8 T which agrees with the assumed value of the SO interaction in simulated device. The exact position of the center on the B scale for the adopted parameters is 7.2 T. To conclude, in general we have found that detecting the angle shift θ is more robust against the sample variations than finding the magnetic field strength symmetry point. II. THE EFFECT OF THE IMPURITIES ON SO COUPLING CONSTANT The disorder considered in the Letter is a potential fluctuation, which results from e.g. non-ideally homogenous doping of the barrier layer under which the twodimensional electron gas is confined. In the 2DEG heterostructures the electron gas is separated from the impurities by an undoped spacer of a few to about 20 nm. The potential fluctuation affects the electric field in the growth direction, so an estimation of the resulting local change of the Rashba SO coupling constant is needed. The Rashba SO interaction strength α = α 3D F z results from the structural inversion asymmetry in the 2DEG confinement plane and is proportional to the electric field in the growth direction F z , where for the bulk In 0.5 Ga 0.5 As alloy Rashba [1] constant is α 3D = 57.2 Å 2 . The strength of the electric gradient F z can be tuned by external voltages [2] and may be varied from 50 to 100 kV/cm in In 0.5 Ga 0.5 As as reported in Ref. [2]. The value of the electric field can be also changed by random impurities in the structure or result from the inhomogeneity of the distribution of the impurities, which may locally change the value of F z → F z + ∆F z . The vertical position of the 2DEG sheet is set at z = 0. In order to estimate the strength of the electric gradient induced by arXiv:1604.01270v2 [cond-mat.mes-hall] 30 Sep 2016 single impurity we computed the electric field ∆F z (x, y) = − ∂ ∂z e 4π * |r imp − r | r =(x,y,0)(1) induced by negatively charged impurity located 10nm from the 2DEG surface (see Fig. S2(c)). The correction to the electric field due to a single impurity located is negligibly weak as compared to the nominal value of F z , hence it can be neglected in our simulations. III. FRIEDEL OSCILLATIONS AND THE EFFECTIVE POTENTIAL In strongly perturbed systems the charge density forms ripples with the periodicity of half of the Fermi wavelength. We considered the effect for the present perturbation -an impurity at a distance from the 2DEG. We simulated the effect of the single negatively charged impurity on the electron density using the DFT method described in Ref. [3]. In simulations we put α = 10meVnm and β = 0 i.e. only the Rashba term is active. The impurity is introduced 10nm from the the 2DEG computational sheet in the middle of the computational box. The average electron density in the simulation is 1×10 12 [ 1 cm 2 ]. The self-consistent DFT solution is reached for temperature T = 1K. The rest of the parameters is the same as in the transport simulations. In Fig. S2(a) we show the self consistent electron density obtained from our DFT solver. One may see that the effect of introduced impurity is relatively weak for a given configuration. The electron density for E F = 20 meV is large, hence the screening is quite efficient, resulting in small depletion in the electron density. The change of the effective potential energy due to the presence of the defect is well visible (see Fig. S2(b)) and vary from about 42meV to 46meV. In Fig. S3(a) we plot the cross section of the electron density and effective potential from Fig. S2(a-b) with peak in the V eff and deep in electron density n at x = 200nm. Due to the relatively weak perturbation of the impurity (∼ 25% of depletion) the amplitude of the Friedel oscillations are very small but still visible in the electron density plot. However, due to the smearing property of the Coulomb kernel V (r) =´dr n(r )/ |r − r | the weak Friedel oscillations in n become even weaker in the resulting potential energy, hence the lack of the Friedel oscillation in the obtained V eff plot. In order to enhance the oscillation of the charge density in Fig. S3(b) we put the impurity at 2.5nm only from the electron gas. Now, the ripples on the density are more pronounced, but their are not resolved in the self-consistent potential anyway. To conclude, the effect of Friedel oscillation on the DFT potential for the considered perturbation is weak, and the modification of the charge density by the rota- tion of the magnetic field in presence of the SO coupling should not alter the results for the electron transport. Figure 2 . 2(a) Scattering amplitudes calculated from Eq. (5) for α = 12meVnm and β = 0 as a function of angle formed by the magnetic field vector and the x axis θ. (b) Same as in (a), but for a fixed angle θ = π as a function of Rashba constant α. In (a) and (b) the solid lines show the result for B = 5T and the black dashed lines for B = 7.2T. Figure 3 . 3(a) Reduction of QPC [Fig. 1] conductancẽ G = M e 2h − G from its quantized value for M subbands [EF =20 meV] passing across the QPC for a potential defect at a distance of 1500 nm from the QPC as a function of the inplane magnetic field value and orientation. The results were calculated numerically within the Landauer approach. The QPC gate potential modeled with analytical formulas for a rectangle gate adapted from Ref.[37].(b) Charge density at the entrance to the QPC calculated with a simple model of Eq.(5) for the QPC at 1500 nm from the scatterer. (c) Same as (b) but with Rashba and Dresselhaus SO interactions present. (d) Same as (c) but with the source at 2000nm from the scatterer. The values of the coupling constants α and β are given in meV×nm units. The vertical dashed line in(b-d) indicates B = 7.2 T for which the Zeeman energy is equal to the SO coupling energy, EB = ESO, see text. The horizontal dashed line shows the angle arctan α β Figure 4 . 4a) Potential disorder in simulated quantum wire. b) Local density of states obtained for channel from (a) at B=0. c) Conductance through the wire in (a) as a function of magnetic field amplitude B and direction angle. d) Same as c) but with one potential barrier in the middle of channel instead of random disorder. e) Same as (c) but for the wire with length L = 4000nm and W = 800nm. f) Same as (e) but for SO couplings twice smaller γ eff = 6meVnm. For comparison of (c,e) seeFig. 3(b). The values M show the number of non-degenerated modes in the channel. Figure S1 . S1First column -conductance as a function of magnetic field amplitude and angle. Second column -the random disorder potential energy in the channel in units of meV. Third column -resulting LDOS for B = 0 T in arbitrary units. Figure S2 . S2(a) The self consistent electron density in the channel of width 200nm and length 400nm. (b) DFT self-consistent potential energy. (c) The bare electric field ∆Fz(x, y) at z = 0 induced by the impurity (see Eq. (1)). Figure S3 . S3(a) The cross section of normalized electron density for y = 100 nm from Fig. S2(a) (black line) and the effective potential -potential of the impurity screened by the deformation of the electron gas (red line). (b) Same as (a) only for the impurity at 2.5 nm from the electron gas. [ 1 ] 1E. A. de Andrada e Silva, G. La Rocca, and F. Bassani, Phys. Rev. B 55, 16293 (1997) AcknowledgmentsThis work was supported by National Science Centre (NCN) grant DEC-2015/17/N/ST3/02266, and by PL-Grid Infrastructure. The first author is supported by the Smoluchowski scholarship from the KNOW funding and by the NCN Etiuda stipend DEC-2015/16/T/ST3/00310. . A Manchon, H C Koo, J Nitta, S M Frolov, D R A , Nat. Mater. 14871A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and D. R. A., Nat. Mater. 14, 871 (2015) . 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B. 9335304K. Kolasiński, A. Mreńca-Kolasińska, and B. Szafran, Phys. Rev. B 93, 035304 (2016)

Enterprise resource planning (ERP) systems are commonly used in technical educational institutions(TEIs). ERP systems should continue providing services to its users irrespective of the level of failure. There could be many types of failures in the ERP systems. There are different types of measures or characteristics that can be defined for ERP systems to handle the levels of failure. Here in this paper, various types of failure levels are identified along with various characteristics which are concerned with those failures. The relation between all these is summarized. The disruptions causing vulnerabilities in TEIs are identified .A vulnerability management cycle has been suggested along with many commercial and open source vulnerability management tools. The paper also highlights the importance of resiliency in ERP systems in TEIs.

10.5120/8409-2043

1209.6484

bbe19e0f64eb98a652bf38662c6f61f65d95d7ea

Vulnerability Management for an Enterprise Resource Planning System September 2012 Assistant ProfessorShivani Goel ProfessorRavi Kiran Deepak Garg Computer Science and Engineering Department School of Behavioral Sciences and Business Studies Associate Professor Computer Science and Engineering Department Thapar University Patiala Punjab India Thapar University 147001Patiala Punjab India-147001 Thapar University Patiala Punjab India-147001 Vulnerability Management for an Enterprise Resource Planning System International Journal of Computer Applications 534September 201219Enterprise resource planningdisruptionsresiliencevulnerability management, tools Enterprise resource planning (ERP) systems are commonly used in technical educational institutions(TEIs). ERP systems should continue providing services to its users irrespective of the level of failure. There could be many types of failures in the ERP systems. There are different types of measures or characteristics that can be defined for ERP systems to handle the levels of failure. Here in this paper, various types of failure levels are identified along with various characteristics which are concerned with those failures. The relation between all these is summarized. The disruptions causing vulnerabilities in TEIs are identified .A vulnerability management cycle has been suggested along with many commercial and open source vulnerability management tools. The paper also highlights the importance of resiliency in ERP systems in TEIs. INTRODUCTION ERP systems are used for providing an approach that integrates the functioning of various departments in any technical educational institution (TEI), to conduct the operations from a central database with accuracy and convenience. Any ERP system involves an information system which has two parts: hardware and software. Hardware part can be called as infrastructure system which includes network, databases and computer peripherals including servers. The software part is the data and information which flows through hardware. There can be failures in both parts. The hardware and software resources might fail and information may get corrupted and it can lead to ERP failure. Various types of failure conditions are handled differently in systems ensuring security. These systems can have different characteristics defined for handling failure and hence security. These are discussed in next section. FAILURE CONDITIONS During the life cycle of a system, there occur certain situations which cause many threats to system security. The system is required to have certain security attributes in that situation. This relationship between conditions and attributes is depicted in figure 1. Figure 1 : Relation between failure conditions and attributes With no disruptions, the system works accurately. Whenever there is a need for change, disruptions can occur in a system which can cause harm to system operations. The probability of disruptions is judged by vulnerability of the system. The system should be least vulnerable in order to predict as many disruptions as possible. A reliable system is supposed to maintain correct functioning of the system till there is no damage. A system should be highly reliable so that it can perform without damage for a longer duration. In case there is no damage, but certain noise, the robust system will be able to perform within an acceptable range of errors. Thus, a system needs to be highly robust so that it can perform near correct functions even in the presence of noise. When there is partial damage to the system, but the system is still performing the correct function, the system is said to be fault tolerant. But only fault tolerance will not makeup for the partial damage to the system. So the system needs to recover back to stable state while performing correct function. Thus a system should be resilient so that it can recover as early as possible to a stable state after a partial damage. In other words, resilience engineering goes beyond reliability engineering and robust engineering [1]. For this, the system needs to have flexibility so that it can recover early. The capacity to predict the disruptions should be as high as possible so that it can take measures to avoid or recover fast. Recovery may need resources, so sustainability is required which can be provided at a high level with greater amount of redundancy. Sustainability is defined as the system's property that measures the balanced generation and consumption of the system resource and supply resources when needed. If a system is able to speedy recovery, it is said to be safe . In case a system gets full damage, i.e. resilience fails, the system needs to be healed. Healing is defined as Healing refers to recovery of a lost function of the system (after damage) by means of the external resource instead of a system's own resource. Thus, the main source of system failure is vulnerabilities. The earliest the system can identify the vulnerabilities, better will be the security of the system. DISRUPTIONS FOR ERP IN TEIs The main reasons for failures are disruptions caused in a system. In order to assess the security of an ERP system, the common disruptions leading to vulnerabilities are needed to be identified. A total of 25 technical users from TEIs were consulted using unstructured interviews as an information gathering tool for identifying the major disruptions in ERP in TEIs. The major technical disruptions identified in ERP system in a TEI are network failure, failure of surveillance system and interface issues. These can cause vulnerabilities like attack on the web based interface for getting the database configuration management, attack on network devices, wireless access points etc. Poor security system, inadequate training and poor error tracking system have been identified as organizational factors. Due to gap in security of the network, attackers can get access to network's critical assets. Poor error attacking system can cause serious attacks unnoticed which can consume system resources. The error by the person operating the ERP system can also cause the system to fail. The wrong intensions of the users in ERP may cause malicious attacks in the system like man in the middle attack, phishing and spoofing attacks etc . Various sources of disruptions which may cause vulnerabilities in ERP in TEIs are shown in figure 2 below: Figure 2: Disruptions for ERP in TEIs ERP can be best suited for handling vulnerabilities caused by disruptions when it supports resilience. This is because a resilient system is capable of supporting recovery mechanisms along with correct functioning when it finds vulnerabilities. Resilience of a system is measured by the level of its vulnerability to a specific risk [2, 4 to 3]. Reducing vulnerability has a positive impact on the resilience of a system [13 to 4]. Resilience depends upon the anticipating unexpected disruptive events and designing solutions to eliminate errors as early as possible [9 to 5]. The attributes of a resilient system are highlighted in next section. ATTRIBUTES OF A RESILIENT SYSTEM There are many attributes which a resilient system should have. These are identified as adaptive capacity, agility, fault tolerance , flexibility and redundancy [6]. Each of these is defined as follows: a) Adaptive capacity Adaptability demonstrates the ability to adapt to changing environments while delivering the intended functionality under changing operating conditions. b) Agility Agility has been used in conjunction with flexibility as a defining attribute of resilience [7]. According to Christopher and Peck [3], resilience involves agility and it helps a system to rapidly reorganise itself. Morello [8], on the other hand, suggests that agility may introduce new risks and vulnerabilities which result in lower resilience. c) Fault Tolerance Fault tolerance is defined as the ability to deliver service in the presence of faults. The ability to function after software component damage under attack is a kind of the fault tolerance in software and is the resilience of a software system [9]. d) Flexibility Fiksel [10] defines flexibility as a major system characteristic that contributes to resilience. Helaakoski [11] relates flexibility to agility and adaptability, which indicates that flexibility is a system's ability to rapidly adapt to its changing environment. e) Redundancy Redundancy is defined as keeping extra capacity or resources kept in reserve to be used in case of a disruption [4]. According to Haimes et al. [12], redundancy is the ability of certain components of a system to support the functions of failed components without any considerable effect on the performance of the system itself. From the analysis of all these definitions, it is found that an ERP system should support flexibility and redundancy to be a resilient system. Since vulnerability detection is the key to success for a resilient system, it is important to handle vulnerabilities in the system. A vulnerability management cycle is proposed in next section. VULNERABILITY MANAGEMENT Vulnerability management is a pro-active approach for managing security of a network because it involves continuous monitoring of system conditions for identifying vulnerabilities. Regular assessment of vulnerability is important for checking the security system of any system. Various users have studied various methods for ensuring security in ERP systems. A multi layer tree model for enterprise vulnerability management has been proposed by Wu and Wang [13] which helps in qualifying the overall score of the company. A vulnerability management cycle has been proposed which consists of following activities: Planning for known vulnerability Before starting with any management, a planning is required. There are a large number of resources and processes which can be attacked by vulnerabilities. When any part is affected by the attack, then that needs to be cured. But for a pro-active approach, it is impossible to monitor all of these. In order to identify vulnerabilities, the resources or processes which are important and are susceptible to vulnerabilities must be first identified. The type of potential threats to each identified resource must be listed. This will lead to a plan for monitoring for the threats becoming a reality. The known types of vulnerabilities can be listed with their symptoms while for the unknown types some methods can be learned only from organizational data and experience. Examples identified for ERP in TEIs are man in the middle attack, denial of service attack, Monitoring for vulnerability Regular monitoring of the system activities identified in first step need to be done continuously. This can be done by regular network scanning, firewall logging, penetration testing . There is also a tool called vulnerability scanner which can be used for the purpose. Analyzing to identify vulnerabilities This involves analyzing the results indicated during the monitoring of the vulnerability. If there are symptoms for some threat becoming a reality, then mitigation for that vulnerability should be used to minimize the consequences if an attack occurs. Mitigating the vulnerabilities The process of finding how to prevent the vulnerabilities. Patches can be applied in the affected area. The vendors of the affected software or hardware can provide the patch as early as possible to minimize the ill effects of the vulnerability attack. Updating the list of known vulnerabilities In the case of new vulnerabilities found, the information should be updated so that the planning can include for newly identified vulnerabilities for monitoring in future. Figure 3: Vulnerability management cycle The above cycle can be used in any system. In order to implement it, the vulnerabilities need to be identified first so that planning for those can be done. In this paper, the vulnerabilities are identified in an ERP system in TEIs. In any ERP system, there is a requirement for security at many levels since there are many areas where vulnerabilities can occur. The most common is in authentication and authorization procedures. This is because due to centralized systems, a number of users need to be assigned different roles and responsibilities. But in case of lack of proper security, there is a threat of wrong manipulation of information. The man in the middle attack, SQL injections, data theft, least privileges violations are common. This can be monitored by network scanner tools. The errors caused due to operator errors also cause disruptions in the system. Proper training should be provided to reduce operator errors. Another mitigating action is providing the Autocorrect facilities to handle the errors due to wrong entry by operators. In order to avoid these, it is advisable to introduce some security solutions at the nearest points of vulnerability attacks. This will also ease the vulnerability analysis for identifying the attacks as early as possible. There are many tools available in the market for various processes in vulnerability management. These are either available as open source and are free in the market or these are commercial. Some of these are shown in figure 4: Along with these tools, security metrics such as vulnerability counts, intrusion attempts, unauthorized access attempts can also be used for monitoring vulnerabilities in ERP systems for TEIs [14]. CONCLUSION For any ERP system, agility is required to be an essential feature to be successful. Allowing changes at any time increases the probability of disruptions and hence failures. So if there is flexibility in a system to allow changes, the system should be flexible enough to incorporate recovery mechanisms also [3,4,15]. The crucial condition is when there is partial damage in the system and speedy recovery is required so that system is safe. This indicates the importance of resilience. In order to enhance resilience, adaptive capacity and hence flexibility should be increased even after a disruption. The redundancy within the system can help increase sustainability of a system and hence can pave the way to faster recovery before full damage [10]. Resilience is a system's ability to bounce back from disruptions and disasters by building in redundancy and flexibility [16]. Thus, some form of redundancy is found to be important for ERP system in TEIs. In any ERP system for TEIs, failure or disruptions like network failure, operator error and malicious attacks on data are inevitable. The ability to predict these disruptions can be one step in taking preventive measures for handling these before these become a reality. This enhances the security of the system. Vulnerability management cycle can be used for monitoring and predicting the vulnerabilities in the system. Various tools can be used for handling vulnerabilities. Figure 4 : 4Tools for Vulnerability Management Examples of commercial vulnerability management tools are:1. Attachmate's NetIQ Vulnerability Manager enables users to define and maintain configuration policy templates, vulnerability bulletins, and automated checks via AutoSync technology. 2. eEye Digital Security: It provides a suite consists of the Retina Network Security Scanner which is a vulnerability assessment tool, Blink Professional which is a host-based security technology, and the REM Security Management Console. 3. Mayflower Chorizo Intranet edition , a scanner for intranet web applications 4. McAfee's Foundstone Enterprise is also solution that offers asset discovery, inventory, and vulnerability prioritization with threat intelligence, correlation, remediation tracking, and reporting and it also does not use agents. 5. Symantec BindView's Compliance Manager is a software-based solution which allows organizations to evaluate their assets against corporate standards or industry best practices, without the need for agents.1. Advchk (Advisory Check), gathers security advisories automatically and compares them to a list of known services, and gives an alert if you are vulnerable. 2. Grabber, A scanner which analyses vulnerabilities in web applications. 3. Information Resource Manager (IRM) a powerful Web- based asset tracking . 4. Mavutina Netsparker, aweb application security scanner. 5. Nmap is a free, open source utility for network exploration or security auditing. 6. OSSIM (Open Source Security Information Management) which is used to provide a network/security administrator a detailed view of the network and devices. International Journal of Computer Applications (0975 -8887) Volume 53-No.4, September 2012 Security metrics for Enterprise Information System. 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E Hollnagel, D D Woods, N Levesson, Resilience engineering: Concepts and precepts. HampshireAshgateHollnagel, E., Woods, D. D. and Levesson, N. 2006. Resilience engineering: Concepts and precepts, Hampshire: Ashgate. A framework for Enterprise Resilience Using Service Oriented Architecture Approach. IEEE SysCon 2009 -3rd Annual IEEE International Systems Conference Vancouver Canada. O Erol, M Monsouri, B Sauser, Erol, O., Monsouri, M. and Sauser, B. 2009. A framework for Enterprise Resilience Using Service Oriented Architecture Approach. IEEE SysCon 2009 - 3rd Annual IEEE International Systems Conference Vancouver Canada (March 23-26), 2009. From Metaphor to Measurement: Resilience of What to What? Ecosystems. S Carpenter, B Walker, J M Anderies, N Abel, 4Carpenter, S., Walker, B., Anderies, J.M. and Abel, N. 2001. From Metaphor to Measurement: Resilience of What to What? Ecosystems. 4, 765-781. The Blueprint for the Resilient Virtual Organization. D Morello, Morello, D. 2001. The Blueprint for the Resilient Virtual Organization, Gartner January 2001. Basic concepts and taxonomy of dependable and secure computing. J Avizienis, Laprie, B Randell, C Landwehr, IEEE Transactions on Dependable Secure Computing. 11Avizienis, J. Laprie, Randell, B. and Landwehr, C. 2004. Basic concepts and taxonomy of dependable and secure computing. IEEE Transactions on Dependable Secure Computing. 1 (1), 11-33. Sustainability and Resilience: Toward a Systems Approach. J Fiksel, Sustainability: Science, Practice, & Policy. 2Fiksel, J. 2006. Sustainability and Resilience: Toward a Systems Approach, Sustainability: Science, Practice, & Policy. 2, 14-21. Agent-based Architecture For Virtual Enterprirses To Support Agility, in Establishing the Foundation of Collaborative Networks. H Helaakoski, P Iskanius, I Peltomaa, L. Camarinha-Matos, H. Afsarmanesh, P. Novais and C. AnalideSpringer243BostonHelaakoski, H., Iskanius, P. and Peltomaa, I. 2007. 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φ meson production is studied by the NA49 Collaboration in central Pb+Pb collisions at 20A, 30A, 40A, 80A, and 158A GeV beam energy. The data are compared with measurements at lower and higher energies and with microscopic and thermal models. The energy dependence of yields and spectral distributions is compatible with the assumption that partonic degrees of freedom set in at low SPS energies.

10.1103/physrevc.78.044907

0806.1937

0113e15c013d2580390a62cc38aa5c93baad0600

Energy dependence of φ meson production in central Pb+Pb collisions at √ s N N = 6 to 17 GeV 27 Oct 2008 C Alt Fachbereich Physik Universität FrankfurtGermany T Anticic Rudjer Boskovic Institute ZagrebCroatia B Baatar Joint Institute for Nuclear Research DubnaRussia D Barna KFKI Research Institute for Particle and Nuclear Physics BudapestHungary J Bartke Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Science CracowPoland L Betev CERN GenevaSwitzerland H Bia Institute for Nuclear Studies WarsawPoland C Blume Fachbereich Physik Universität FrankfurtGermany B Boimska Institute for Nuclear Studies WarsawPoland M Botje NIKHEF AmsterdamNetherlands J Bracinik Comenius University BratislavaSlovakia R Bramm Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany P Bunčić CERN GenevaSwitzerland V Cerny Comenius University BratislavaSlovakia P Christakoglou Department of Physics University of Athens AthensGreece P Chung Department of Chemistry Stony Brook University (SUNYSB) Stony BrookNew YorkUSA O Chvala Institute of Particle and Nuclear Physics Charles University PragueCzech Republic J G Cramer Nuclear Physics Laboratory University of Washington SeattleWashingtonUSA P Csató KFKI Research Institute for Particle and Nuclear Physics BudapestHungary P Dinkelaker Fachbereich Physik Universität FrankfurtGermany V Eckardt Max-Planck-Institut für Physik MunichGermany D Flierl Fachbereich Physik Universität FrankfurtGermany Z Fodor KFKI Research Institute for Particle and Nuclear Physics BudapestHungary P Foka Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany V Friese Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany J Gál KFKI Research Institute for Particle and Nuclear Physics BudapestHungary M Gaździcki Fachbereich Physik Universität FrankfurtGermany Institute of PhysicsŚwi etokrzyska Academy KielcePoland V Genchev Institute for Nuclear Research and Nuclear Energy SofiaBulgaria G Georgopoulos Department of Physics University of Athens AthensGreece E G Ladysz Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Science CracowPoland K Grebieszkow Faculty of Physics Warsaw University of Technology WarsawPoland S Hegyi KFKI Research Institute for Particle and Nuclear Physics BudapestHungary C Höhne Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany K Kadija Rudjer Boskovic Institute ZagrebCroatia A Karev Max-Planck-Institut für Physik MunichGermany D Kikola Faculty of Physics Warsaw University of Technology WarsawPoland M Kliemant Fachbereich Physik Universität FrankfurtGermany S Kniege Fachbereich Physik Universität FrankfurtGermany V I Kolesnikov Joint Institute for Nuclear Research DubnaRussia T Kollegger Fachbereich Physik Universität FrankfurtGermany E Kornas Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Science CracowPoland R Korus Institute of PhysicsŚwi etokrzyska Academy KielcePoland M Kowalski Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Science CracowPoland I Kraus Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany M Kreps Comenius University BratislavaSlovakia D Kresan Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany A Laszlo KFKI Research Institute for Particle and Nuclear Physics BudapestHungary R Lacey Department of Chemistry Stony Brook University (SUNYSB) Stony BrookNew YorkUSA M Van Leeuwen NIKHEF AmsterdamNetherlands P Lévai KFKI Research Institute for Particle and Nuclear Physics BudapestHungary L Litov Atomic Physics Department Sofia University St. Kliment Ohridski SofiaBulgaria B Lungwitz Fachbereich Physik Universität FrankfurtGermany M Makariev Atomic Physics Department Sofia University St. Kliment Ohridski SofiaBulgaria A I Malakhov Joint Institute for Nuclear Research DubnaRussia M Mateev Atomic Physics Department Sofia University St. Kliment Ohridski SofiaBulgaria G L Melkumov Joint Institute for Nuclear Research DubnaRussia A Mischke NIKHEF AmsterdamNetherlands M Mitrovski Fachbereich Physik Universität FrankfurtGermany J Molnár KFKI Research Institute for Particle and Nuclear Physics BudapestHungary St Mrówczyński Institute of PhysicsŚwi etokrzyska Academy KielcePoland V Nicolic Rudjer Boskovic Institute ZagrebCroatia G Pálla KFKI Research Institute for Particle and Nuclear Physics BudapestHungary A D Panagiotou Department of Physics University of Athens AthensGreece D Panayotov Atomic Physics Department Sofia University St. Kliment Ohridski SofiaBulgaria A Petridis Department of Physics University of Athens AthensGreece W Peryt Faculty of Physics Warsaw University of Technology WarsawPoland M Pikna Comenius University BratislavaSlovakia J Pluta Faculty of Physics Warsaw University of Technology WarsawPoland D Prindle Nuclear Physics Laboratory University of Washington SeattleWashingtonUSA F Pühlhofer Fachbereich Physik Universität MarburgGermany R Renfordt Fachbereich Physik Universität FrankfurtGermany C Roland MIT CambridgeMassachusettsUSA G Roland MIT CambridgeMassachusettsUSA M Rybczyński Institute of PhysicsŚwi etokrzyska Academy KielcePoland A Rybicki Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Science CracowPoland A Sandoval Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany N Schmitz Max-Planck-Institut für Physik MunichGermany T Schuster Fachbereich Physik Universität FrankfurtGermany P Seyboth Max-Planck-Institut für Physik MunichGermany F Siklér KFKI Research Institute for Particle and Nuclear Physics BudapestHungary B Sitar Comenius University BratislavaSlovakia E Skrzypczak Institute for Experimental Physics University of Warsaw WarsawPoland M Slodkowski Faculty of Physics Warsaw University of Technology WarsawPoland G Stefanek Institute of PhysicsŚwi etokrzyska Academy KielcePoland R Stock Fachbereich Physik Universität FrankfurtGermany C Strabel Fachbereich Physik Universität FrankfurtGermany H Ströbele Fachbereich Physik Universität FrankfurtGermany T Susa Rudjer Boskovic Institute ZagrebCroatia I Szentpétery KFKI Research Institute for Particle and Nuclear Physics BudapestHungary J Sziklai KFKI Research Institute for Particle and Nuclear Physics BudapestHungary M Szuba Faculty of Physics Warsaw University of Technology WarsawPoland P Szymanski Institute for Nuclear Studies WarsawPoland V Trubnikov Institute for Nuclear Studies WarsawPoland D Varga KFKI Research Institute for Particle and Nuclear Physics BudapestHungary M Vassiliou Department of Physics University of Athens AthensGreece G I Veres KFKI Research Institute for Particle and Nuclear Physics BudapestHungary G Vesztergombi KFKI Research Institute for Particle and Nuclear Physics BudapestHungary D Vranić Gesellschaft für Schwerionenforschung (GSI) DarmstadtGermany A Wetzler Fachbereich Physik Universität FrankfurtGermany Z W Lodarczyk Institute of PhysicsŚwi etokrzyska Academy KielcePoland I K Yoo Department of Physics Pusan National University PusanRepublic of Korea J Zimányi KFKI Research Institute for Particle and Nuclear Physics BudapestHungary Energy dependence of φ meson production in central Pb+Pb collisions at √ s N N = 6 to 17 GeV 27 Oct 2008(NA49 Collaboration) φ meson production is studied by the NA49 Collaboration in central Pb+Pb collisions at 20A, 30A, 40A, 80A, and 158A GeV beam energy. The data are compared with measurements at lower and higher energies and with microscopic and thermal models. The energy dependence of yields and spectral distributions is compatible with the assumption that partonic degrees of freedom set in at low SPS energies. I. INTRODUCTION The production of strange particles is considered one of the key observables for understanding the reaction mechanisms in ultrarelativistic heavy-ion collisions. Enhanced strangeness production with respect to proton-proton collisions was originally proposed as a signature of the transition to a deconfined state of quarks and gluons during the initial * Deceased. stage of the reactions [1]. The enhancement was predicted to arise from gluon fragmentation into quark-antiquark pairs which is believed to have a significantly lower threshold than strange-antistrange hadron pair production channels. Indeed, it has been observed [2,3] that the ratio of the number of produced kaons to that of pions is higher by a factor of about 2 in central S + S and Pb + Pb reactions than that in p + p collisions at the top energy available at the CERN Super Proton Synchroton (SPS). Statistical hadron gas models have been successfully employed to describe the measured particle yields at various collision energies [4,5,6,7,8]. The fact that the hadronic final state of the collision resembles a hadron gas in chemical equilibrium has been interpreted as a consequence of the hadronization process [9] or as a result of a fast hadronic equilibration process involving multiparticle collisions [10]. In this hadron gas picture, enhanced production of strange particles in collisions of large nuclei arises as a consequence of the increased reaction volume, relaxing the influence of strangeness conservation [11]. Technically, this requires the application of the canonical ensemble to small collision systems, while for larger volumes such as those encountered in central collisions of heavy ions, the grand-canonical approximation is valid. It has been shown that this "canonical strangeness suppression" also applies to a partonic system [12]. In addition to this volume effect, the strange particle phase space appears to be undersaturated in elementary interactions. The deviation of the strange particle yields from a hadron gas in full equilibrium was parametrized by a strangeness undersaturation factor γ S [8,13]. The additional suppression becomes much weaker in heavy-ion collisions. However, fits to the hadron multiplicities in full phase space are still unsatisfactory when not taking into account γ S [8]. A possible interpretation is that the total amount of strangeness available for hadronization is determined in a prehadronic stage of the collision. A change in γ S between p + p and A + A would then reflect the difference in the initial conditions of the respective fireballs. The hadron gas model was extended to describe the energy dependence of produced hadron multiplicities by a smooth parametrization of the fit parameters T and µ B , determined at energies available at the BNL Alternating Gradient Synchroton (AGS), SPS, and BNL Relativistic Heavy Ion Collider (RHIC), as a function of collision energy [14]. However, this extended model failed to reproduce the detailed features of the energy dependence of relative strangeness production measured by NA49 in its energy scan program. In particular, the sharp maximum at around 30A GeV beam energy [15,16] could not be described. The same holds true for microscopic reaction models such as UrQMD [17]. On the other hand, this feature was predicted as a consequence of the onset of a phase transition to a deconfined state at the respective beam energy [18]. In this context, it is certainly interesting to investigate specific strangeness-carrying hadrons. Among these, the φ meson is of particular interest because of its ss valence quark composition. In a purely hadronic scenario, being strangeness-neutral, it should not be sensitive to hadrochemical effects related to strangeness. If on the other hand, the amount of available strange quarks is determined in a partonic stage of the collision, the φ is expected to react more sensitively than singly strange particles. In particular, one would expect the φ meson yield to be suppressed by γ 2 s with respect to equilibrium. Analogously, the canonical suppression mechanism in small systems should have a stronger effect on the φ, leading to a larger relative enhancement in Pb + Pb collisions with respect to p + p reactions than observed for kaons. In the evolution of the fireball after hadronization, φ mesons can be both formed by kaon coalescence and destroyed by rescattering. In addition, when decaying inside the fireball, the daughter particles can rescatter, leading to a loss of signal in the invariant mass peak of the respective decay channel. This is more likely to happen for slow φ mesons, which spend more time in the fireball. Thus the effect could lead to a depletion of the φ meson yield at low p t in central nucleus-nucleus collisions [19]. Theoretical investigations have suggested that the properties of the φ meson might be modified in a dense hadronic medium. In particular, a decrease of its mass of the order of 10 MeV 1 [20] and an increase of its width by a factor of 2-3 [21] were predicted. So far, there is only one experimental claim for a broadening of the width in p + Cu collisions [22]. In an earlier publication [23], we reported on φ production at top SPS energy, where we found the φ enhanced by a factor of about 3, compared to minimum bias p + p collisions at the same beam energy. Meanwhile, the φ meson was measured at the same energy by the NA50 [24], NA45 [25], and NA60 [26] experiments. At the AGS, data on φ production were obtained by the E917 Collaboration in Au + Au collisions at p beam = 11.7A GeV ( √ s N N = 4.88 GeV) in a restricted rapidity range [y c.m. − 0.4, y c.m. ] [27]. At the RHIC, the STAR Collaboration measured the φ meson at √ s N N = 130 and √ s N N = 200 GeV at midrapidity [28,29]. For the latter energy, data are also available from the PHENIX experiment [30]. In this article, we report on φ production in central Pb+Pb collisions at five different beam energies from 20A to 158A GeV. Together with the data obtained at the AGS and the RHIC, our findings enable the study of energy dependence of φ production over a large range of collision energies. II. EXPERIMENT The NA49 experiment at CERN is based on a fixed-target hadron spectrometer using heavy-ion beams from the SPS accelerator. Its main components are four large-volume time projection chambers for charged-particle tracking, two of which operate inside the magnetic field of two superconducting magnets, thus providing an excellent momentum measurement. Two larger main time projection chambers (MTPCs) are placed downstream, outside of the field, and enable particle identification by the measurement of the specific energy loss in the detector gas. The particle identification capabilities are enhanced by a time-of-flight (TOF) scintillator system behind the MTPCs, albeit in a restricted geometrical acceptance. A thin lead foil with 1% interaction probability for Pb nuclei was used as a target. For the different runs, the magnetic field was scaled proportionally to the beam energies in order to have similar acceptance in the c.m. system. The centrality of the reactions was determined from the energy deposited by the beam spectators in the zero-degree calorimeter, placed 20 m downstream of the target. By setting an upper limit on this energy, the online central trigger selected the 7.2% most central collisions at 20A-80A GeV and the 10% most central collisions at 158A GeV. The latter data set was restricted to 5% centrality in the offline analysis. The corresponding mean numbers of wounded nucleons were obtained by Glauber model calculations (see Table I). Details of the experimental apparatus can be found in Ref. [31]. III. DATA ANALYSIS A. Event and track selection Offline quality criteria were applied to the events selected by the online centrality trigger to suppress nontarget interactions, pileup, and incorrectly reconstructed events. The cut variables include the position and χ 2 of the reconstructed vertex and the track multiplicity. For the central data sets used in this analysis, however, the impact of these quality cuts is marginal; only about 1% of all events were rejected. Table I shows the event statistics used in the analysis for the five data sets. The analysis was restricted to tracks reconstructed in the MTPCs which could be assigned to the primary vertex. A minimal track length of 2 m out of the maximal 4 m in the MTPCs was required to suppress ghost or split tracks and to ensure a good resolution in dE/dx. Detailed studies including reconstruction of simulated tracks embedded into real raw data events showed that for such a selection of tracks, losses due to track reconstruction and high track density are negligible. particles in the acceptance of the time-of-flight detectors and parametrized as a function of βγ as shown in Fig. 1(a). This allowed one to extend the momentum range for the identification from the TOF acceptance to higher momenta. The lower momentum limit was given by either the MTPC acceptance or the crossing of the Bethe-Bloch curves of pions and kaons. The momentum limits for the different data sets are summarized in Table I. Fixing the mean dE/dx of kaons and protons to this parametrization, the resolution was obtained by unfolding the energy-loss spectra in momentum bins into the Gaussian contributions of the particle species (p, K, π, and e). The resolution is about 4% and has a slight momentum dependence which was again parametrized [ Fig. 1 (b)]. Kaon candidates were selected by a momentum-dependent dE/dx window around the expectation value, the size of which was chosen to optimize the φ signal quality. In addition, the window had to be symmetric and large enough to minimize the sensitivity to the errors in the determination of the dE/dx expectation value and resolution. A window of ±1.5 σ was found to be the best choice. This selection contains 87% of all kaons, giving an efficiency of 75% for the pair. The fraction of true kaons within the selected candidate track sample varies between 40% and 60%. C. Extraction of raw yields The φ signal was obtained by calculating the invariant mass of all combinations of positive and negative kaon candidates in one event. To reconstruct the combinatorial background of uncorrelated pairs, candidates from different events were combined. The mixed-event spectrum was subtracted from the same-event spectrum after normalization to the same number of pairs [32]. Figure 2 shows the background-subtracted invariant-mass spectra in the total forward acceptance for different collision energies. In all cases, clear signals are observed at the expected position. While the subtracted spectrum is flat on the right side of the signal, a depletion is observed between the peak and the threshold. As a possible source of this undershoot, the correlation of kaons stemming from different φ mesons has been discussed in Ref. [32]. In our case, it was shown by simulation that this effect is small thanks to the large acceptance of the NA49 MTPCs. Another possible source of the distortion is the reflection of other resonances, e.g., ∆ 0 → π − p, into the K + K − spectrum by misidentification of pions and protons, as discussed in detail in Ref. [33]. This effect was shown to be present in our previous analysis of another data set [23], where the dE/dx resolution was significantly worse. However, all such resonances would distort the spectrum over a broad range above threshold, which can be excluded by the observed flatness at higher masses. This conclusion is further strengthened by the observation that the depletion does not vanish when applying a stricter dE/dx cut on the kaons. Hence, the undershoot is likely to originate from a true correlation of kaon pairs. Simulations show that it can be explained by final state strong interaction of kaons [34]. This is demonstrated in Fig. 3 by showing the K + K − correlation Fig. 3(a)] and in m inv [ Fig. 3(b)]. While the repulsive interaction causes a depletion in m inv , the stronger attractive Coulomb effect is squeezed into 0.8 MeV above threshold and is thus hardly seen. In combination with the steeply rising unsubtracted m inv distribution, this depletion can easily account for the deficit observed in the subtracted spectrum. function in q inv = ( p 1 − p 2 ) 2 − (E 1 − E 2 ) 2 [ To correct this effect quantitatively by simulation is difficult and would moreover be model dependent. As the narrow signal is easily distinguished from the broad residual background, we accounted for the depletion by fitting a straight line in the vicinity of the peak. For the description of the signal itself, we used a relativistic p-wave Breit-Wigner distribution [35] of the form dN dm ∝ mΓ(m) (m 2 − m 2 0 ) 2 + m 2 0 Γ 2 (m)(1) with the mass-dependent width Γ(m) = 2 Γ 0 q q 0 3 q 2 0 q 2 + q 2 0 ,(2) where q := 1 4 m 2 − m 2 K and q 0 := 1 4 m 2 0 − m 2 K . This distribution was folded with a Gaussian representing the invariant-mass resolution σ m of the spectrometer. Since in general, mass resolution and width cannot be determined separately, we fixed the width to its book value Γ 0 = 4.26 MeV [36], leaving m 0 , σ m , a normalization and two parameters for the linear background as free parameters for the fit, which was performed in the mass range 994-1050 MeV. It was checked by simulations that this procedure gives the correct values for position, width and integral of the distribution. As Fig. 2 demonstrates, the fit gives a good description of the signal. The numerical values of the fitted parameters are listed in Table II. To obtain longitudinal and transverse spectra, the signal was extracted in rapidity and in p t bins, respectively, in the same way as in the total acceptance. Generally, the limited statistics prevented a simultaneous division into y-p t bins. Thus, transverse momentum spectra could only be derived averaged over rapidity. To reduce the number of free fit parameters, m 0 and σ m were fixed for the fits in the phase space bins to the values obtained from the signal in the total acceptance. For the 158A GeV data set, where the statistics in the signal allowed to do so, we checked that leaving these parameters free did not significantly alter the results. In particular, no significant dependence of m 0 or σ m on rapidity or p t was observed. Since the straight-line background is only an approximation for the residual background in the vicinity of the signal, the stability of the fit against the variation of the fit region was checked. The parameters m 0 and σ m show no significant dependence. The variation of the normalization constants, which determine the fit integral, is in all bins far below the statistical error returned by the fit procedure. We conclude that the latter properly takes into account the possible variations of the baseline. The raw yields in the phase-space bins were obtained by integrating the fit function from threshold up to m 0 +30Γ 0 ≈ 1.148 MeV. This mass cutoff is somehow arbitrary; the corresponding integral varies by about 3% for cutoff values from m 0 + 10Γ 0 to infinity. We take this as a systematic uncertainty due to the mass cutoff. Using alternatively a (analytically integrable) nonrelativistic Lorentz distribution for the fit does not change the integral by more than 1%. D. Geometrical acceptance The geometrical acceptance of the NA49 detector for the decay φ → K + K − was obtained double-differentially in y and p t (integrated over azimuth) by geant simulations of the φ decay including in-flight decay of the kaon daughters, assuming an azimuthally flat φ emission and isotropic decay. The resulting acceptance is shown in Fig. 4 for 20A and 158A GeV. While the upper momentum limit for the daughter candidates restricts the acceptance at forward rapidity for the top SPS energy, at lower beam energies there is lack of acceptance near midrapidity because of the lower momentum limit for the daughter tracks and the increased losses due to in-flight decay for low-momentum kaons. As the acceptance is a function of y and p t , the proper correction factor for a given extended phase-space bin (integrated either over y or p t ) as used in the analysis is the mean acceptance a S = S dydp t a(y, p t ) f (y, p t ) S dydp t f (y, p t ) ,(3) where S denotes the region in the y, p T plane, a(y, pt) the acceptance probability averaged over the azimuthal angle, and f (y, p t ) the differential φ meson yield. For the rapidity distributions, the differential yields have in addition to be extrapolated to the full p t range. The extrapolation factor, however, is small (<5%) due to the large p t range covered by the experiment. Both the acceptance correction and the extrapolation to full p t require the knowledge of the y and p t dependence of φ meson yields, which leads to an iterative procedure (see Sec. III E). E. Spectra and yields Apart from the differential acceptance correction, the raw yields obtained from the fit to the invariant-mass spectra were corrected for the branching ratio φ → K + K − (49.1%) and the efficiency of kaon dE/dx selection (75% for the pair), and normalized to the number of collisions. These global correction factors are common for all bins in phase space and for all beam energies. The transverse spectra are fitted by the thermal ansatz dn dp t ∝ p t e −mt/T ,(4) where the transverse mass m t = m 2 0 + p 2 t . The distributions in rapidity were parametrized by a single Gaussian dn dy ∝ e − y 2 2σ 2 y .(5) As the parameters T and σ y must be obtained by the analysis itself, an iterative procedure was employed. Starting from some reasonable parameter values, the acceptance correction was calculated according to Eq. (3), assuming factorization of the emission function f (y, p t ) into the transverse and longitudinal distributions (4) and (5), i.e., independence of T on rapidity. The corrected yields in the p t and y bins were then fitted with the distributions (4) and (5), respectively, obtaining new values for T and σ y which serve as input for the next iteration. Convergence of the method was reached after three to five steps. It was checked that the final results do not depend on the choice of start values for the parameters. After the final step of the iteration, the yields in full phase space were obtained by summing up the measured yields in the rapidity distributions and numerically extrapolating Eq. (5) to the full rapidity range. In a similar way, the quantities p t , m t , and σ y were determined. The midrapidity yield dn/dy was obtained directly from the fit function. As demonstrated later in Fig. 7, the Gaussian parametrization gives a satisfactory description of the rapidity distribution for all data sets. However, because of the lack of midrapidity data points at the lower beam energies, an ambiguity for the extrapolation to full phase space arises. To check the sensitivity of the results to the assumed shape of the rapidity distribution, we alternatively parametrized the latter by the sum of two Gaussian functions displaced symmetrically around midrapidity by a shift a: dn dy ∝ e − (y−a) 2 2σ 2 y + e − (y+a) 2 2σ 2 y .(6) The width of this distribution will be characterized by its rms value. Total yield φ , midrapidity yield dn φ /dy and rms y were calculated for both parametrizations (5) and (6). The final values listed in Tables V and VI were calculated as the mean of the results of the two methods; their differences enter the systematic errors. F. Statistical and systematic errors Statistical errors in the raw differential φ meson yields originate from the statistical bin-by-bin errors in the sameevent and mixed-event invariant mass spectra, which were found to be in good approximation Poissonian and uncorrelated between mass bins. Then, the statistical errors in the event-mix subtracted invariant-mass spectrum was calculated as [32] σ 2 i = n 0,i + k 2 n em,i ,(7) where n 0,i is the number of entries in mass bin i in the same-event spectrum, n em,i the same number in the mixedevent spectrum, and k the normalization constant for the event mix. These errors were propagated toward the raw differential yields by the least-squares fit of the Breit-Wigner distribution to the signal peak. The acceptance calculation was performed with sufficiently high statistics such that the relative statistical error of the differential acceptance is below 1% and thus far below the uncertainty in the raw yields over the entire y, p t region used for the analysis. Finally, the errors in the acceptance-corrected differential yields are propagated through the least-square fits to the spectra to obtain the statistical uncertainties in the spectral parameters and the integrated quantities. Systematic uncertainties in the uncorrected yields arise from the approximation of the residual background in the invariant-mass spectra as a straight line. This approximation is only valid in a limited mass range around the signal peak. Thus, the stability of the results of the Breit-Wigner fit against the variation of the fit range was checked. We found no significant dependence of the parameters m 0 and σ m ; the variation of the normalization constant, determining the fit integral, was in all y and p t bins found to be smaller than the statistical error. Another source of systematic error arises from the dE/dx selection of kaon candidates. Uncertainties in the parametrization of the mean kaon dE/dx and the resolution result in systematic deviations of the efficiency correction from its true value. To estimate this error, the analysis was repeated for different widths of the dE/dx selection window around the kaon expectation value, applying the respective efficiency correction. This error was found to be the dominating one; for most raw yields, it is comparable to or slightly larger than the statistical one. Imperfect detector description in the simulation leads to systematic uncertainties in the acceptance correction. To reduce possible errors, the analysis was restricted to phase-space regions where the acceptance is above 1%. The remaining error was estimated by repeating the analysis with varying acceptance conditions (minimal track length in the MTPCs). It was in all cases found to be much smaller than the error originating from the kaon selection by dE/dx. As the spectral parameters enter the acceptance correction through Eq. (3), their uncertainties add to the systematic errors of the corrected yields. This was accounted for by determining the range of acceptance values allowed by the errors in T and σ y . In addition, for the rapidity bins close to beam rapidity, a possible deviation of the slope parameter by 50 MeV from its averaged value was taken into account in the acceptance correction. The resulting error, however, is small thanks to the large and approximately uniform p t acceptance. The systematic errors in the corrected differential yields were assumed to be independent and added in quadrature. They were propagated to the respective errors in the spectral parameters by repeating the fit of Eqs. (4)-(6) with statistical and systematic errors added and comparing the resulting errors to those obtained from the fit with statistical errors only. For the determination of the averaged quantities φ , p r , m t , and rms y , the summation of the measured differential yields as well as extrapolation to full phase space are required. The systematic errors of these observables were determined from the errors of the differential yields and the uncertainties in the spectral shapes. II: Approximate number of detected φ mesons S, background-to-signal ratio B/S, signal-to-noise ratio SNR, position of the signal peak m0, and invariant-mass resolution σm. The latter two were obtained by a Breit-Wigner fit to the signal peak (see text). The width was fixed to its literature value 4.26 MeV. S and B were calculated in a window of ± 4 MeV around the peak. The quoted errors are statistical only. Table II summarizes the parameters obtained from the invariant-mass signals in the total acceptance. The signal quality decreases when going to lower beam energy because of both the reduced φ meson yield and the reduced acceptance due to the increased in-flight decay probability for the daughter kaons. At all five energies, the fitted peak position is slightly below the literature value of 1019.43 MeV [36]. We investigated the effect of an error in the normalization of the magnetic field used for momentum determination in the reconstruction chain and found that a bias of 1% in the magnetic field is needed to explain the observed shift. This is slightly above the momentum scale uncertainty deduced from a precision study of the K 0 s signal. We thus cannot exclude that the deviation of the peak position is due to experimental effects. The widths of the mass peaks obtained from the fits are consistent with those obtained from a full detector simulation and reconstruction. Their slight increase toward lower beam energies can be understood as the increasing influence of multiple scattering on lower momentum tracks. For the signal at 158A GeV, we fitted simultaneously width and mass resolution and obtained Γ 0 = (4.41 ± 0.61) MeV, σ m = (1.81 ± 0.26) MeV, i.e., no deviation from the free-particle width. Thus, within experimental uncertainties, we do not observe indications for a mass shift or a broadening of the φ meson. The observation that the mass and width of the φ meson agree with the Particle Data Group values is in line with the results of AGS and RHIC experiments [27,28,29,30]. It should be noted that because of the long lifetime of the φ meson (τ = 46 fm), only a fraction decays inside the fireball. Thus, only a part of the φ mesons can be expected to be influenced by the surrounding medium. B. Transverse momentum spectra The transverse momentum spectra obtained for the five beam energies are shown in Fig. 5; numerical data are given in Table III. In all cases, the thermal distribution (4) gives a good description of the data; the fit parameters are summarized in Table IV. At top SPS energy with the best signal quality, a modest deviation from the fit function is indicated by the χ 2 /ndf of 1.5. A slight curvature of the transverse mass spectrum at this energy, as expected from a hydrodynamical expansion scenario, is visible for this energy in Fig. 6(a). For the other energies, no deviations from pure exponential behavior can be seen within the experimental uncertainties. The transverse momentum spectrum can be also characterized by its first moment or the average transverse mass. These parameters were calculated from the measured data points and extrapolated to full p t using the exponential fit function. As the extrapolation contributes only marginally because of the large p t coverage, p t and m t − m 0 are largely independent of the spectral shape. Their values are also listed in Table IV. The assumption of the slope parameter being independent of y could be checked for 158A GeV, where statistics allowed us to extract transverse spectra in four different rapidity bins. The resulting slope parameters are shown in Fig. 6(b). Within the measured rapidity range, we observe no significant change of the slope parameter with y. Using the y-dependent slope parameters for correcting the rapidity distribution had no sizable effect on the results. The spectrum obtained for 158A GeV agrees with that from an earlier publication [23] of the NA49 experiment, which was based on the analysis of an older data set at the same beam energy. For comparison, the previously Table IV. The full lines show the fits of thermal distributions (4). The squared symbols denote previously published results [23]. Only statistical errors are shown. Table IV. The exponential fits indicated by the full lines correspond to the fits shown in Fig. 5. The spectra for different beam energies are scaled for better visibility. Only statistical errors are shown. The data at 158A GeV are compared with previously published results of NA49 [23] and CERES [25]. (b) Slope parameter as function of rapidity at 158A GeV. The values agree within errors with that obtained from the y-integrated pt spectrum, the latter indicated with its standard deviation by the shaded bar. published data are shown by the square symbols in Figs. 5(e) and 6(a). There is agreement with the results of the CERES experiment in both decay channels φ → K + K − and φ → e + e − [25], as also demonstrated in Fig. 6(a). The data disagree with the spectrum measured by the NA50 experiment in the di-muon decay channel φ → µ + µ − , where a significantly smaller slope was obtained [24]. IV: Rapidity range (in c.m. system), pt range, slope parameter T , χ 2 per degree of freedom, average pt, and average mt for the transverse momentum spectra. T and χ 2 are results from the fit of Eq. (4) to the spectrum; pt and mt − m0 were obtained by summation over the data points and extrapolation to full pt using the fit function. The first error is statistical, the second one systematic. (6) to the rapidity distributions. The RMS was calculated from the data points and extrapolated to the full rapidity range using the average of the two parametrizations. The first error is statistical, the second one systematic. Figure 7 shows the rapidity distributions, which for all five energies are in good agreement with both the single-Gaussian and the double-Gaussian parametrization (see curves). Numerical data are given in Table III. For the data sets at 20A and 30A GeV, due to the low number of data points, the double-Gaussian fit was constrained to a = σ y as suggested by the data at 40A and 80A GeV. p beam (A GeV) y range pt range (GeV) T (MeV) χ 2 /ndf pt (MeV) mt − m0 [MeV Only at 80A GeV is the complete forward hemisphere covered. At 158A GeV, large rapidities are not measured because of the upper momentum cut on the secondary kaons. Since kaons below 2 GeV laboratory momentum cannot be reliably identified by dE/dx because of the crossing of the Bethe-Bloch curves, no signal could be extracted at midrapidity for the lower three beam energies. The uncertainties in the extrapolation toward midrapidity is demonstrated by the difference of the two parametrizations. It adds to the systematic error of the total yield and, in particular, to that of dn/dy at midrapidity. Table V lists the parameters obtained by the two fit functions, respectively. Alternatively, the rapidity distributions can be characterized by their second moments in a model-independent fashion. The root mean square of the distributions was calculated from the measured data and extrapolated to the full rapidity range using the parametrizations (5) and (6). The average of the two results is listed in Table V. Total yields were obtained by summation of the data points in the rapidity spectra and extrapolation to the full rapidity range by the average of the fit functions. The midrapidity yield dn/dy was calculated analytically from the average of the fit functions. For the determination of statistical and systematic errors, the correlation of the spectral parameters were properly taken into account. The results for the mean multiplicity of φ mesons φ and for the midrapidity yield dn φ /dy are listed in Table VI. All results obtained at 158A GeV are consistent within statistical errors with NA49 results published earlier [23], which were obtained from a data set taken in 1995 (squared symbols in Fig. 7(e). The main difference of the two data sets is an improved dE/dx resolution, resulting in a reduced pion contamination of the kaon candidate sample. The cleaner kaon identification reduces the distortions in the background-subtracted invariant-mass spectrum induced by resonances with a pion as decay daughter [33], thus leading to a smaller systematic error of the Breit-Wigner fit to the spectrum. We thus prefer to use the newly obtained results at 158A GeV for the discussion. (5), the dashed lines that by the sum of two Gaussians (6). The data at 158A GeV are compared with previously published results of NA49 [23] and CERES [25]. Only statistical errors are shown. The enhancement of relative strangeness production in heavy-ion collisions with respect to proton-proton reactions is a well-known fact. In an earlier publication [23], the enhancement factor for the φ meson at top SPS energy was found to be 3.0 ± 0.7, thus larger than for kaons and Λ, but smaller than for multistrange hyperons. We calculate the φ enhancement by normalizing the measured φ meson yield in A + A by the number of wounded nucleon pairs and dividing by the corresponding yield in p + p. For the lower beam energies, no reference measurements in elementary collisions are available. Here, we employ a parametrisation of the φ excitation function in p + p collisions as described in Ref. [27]. For top SPS energy and RHIC, the φ meson yield measured in p + p [23,29] was used. Figure 8 shows the resulting enhancement factor E φ := 2 φ A+A N w φ p+p(8) as a function of energy per nucleon pair. The measurement of the E917 Collaboration at AGS (p beam = 11.7A GeV) was extrapolated to full phase space assuming the same rapidity distribution as for K − as suggested by the authors [27]. [27] and the SPS refer to multiplicities in full phase space, data from the RHIC [29,30] to midrapidity yields. The shaded boxes represent the systematic errors. In the AGS/SPS energy region, the value of E φ lies between 3 and 4, and within our experimental uncertainties we find no systematic variation here. At RHIC energies, the enhancement appears to be lower, significantly so, should the PHENIX result be validated. It should be noted, however, that the RHIC values were derived from midrapidity data while at lower energies phase-space integrated yields were used. In the context of statistical models, the enhancement of strangeness production can be interpreted as a result of the release of suppression due to strangeness conservation when going from small (p + p) to large (central A + A) systems. Technically, this is reflected in the application of the canonical ensemble for small systems, while large systems can be described by the grand-canonical ensemble. In this picture, a smaller enhancement at RHIC energies points to the fact that at such high energies, strangeness is produced with sufficient abundance for the canonical suppression to be relaxed even in p + p collisions. However, in a purely hadronic picture, canonical suppression does not act on the φ meson because it is a strangeness-neutral hadron. Enhanced φ production can thus be attributed either to enhanced strangeness production in a partonic stage of the collision or to the coalescence of kaons which suffer canonical suppression also in a hadronic scenario. The hadrochemical models have been extended not only to fit hadron multiplicities for a given reaction but also to describe the energy dependence of particle yield ratios by a smooth variation of the relevant parameters T and µ B with collision energy [14,37]. Here, the energy dependence of temperature and baryochemical potential is obtained by a parametrization of the values for T and µ B obtained from fits to particle yield ratios at various collision energies. The model reproduces many yield ratios of the bulk hadrons; however, this does not hold for the φ meson, as shown in Fig. 9(a), where the measured excitation function of the φ / π ratio [ π = 1.5( π + + π − )] is compared with the model prediction. The relative φ meson yields at the SPS are overpredicted by factors of up to 2. The situation remains essentially unchanged when midrapidity ratios are considered instead of integrated yields [ Fig. 9(b)]. At the RHIC, there is a large experimental ambiguity as a result of the different results on φ production obtained by the STAR and PHENIX experiments [29,30]. A better description of the data is obtained if a strangeness saturation parameter γ s is allowed. The corresponding model predictions [8] for the φ multiplicity, resulting from a fit to the hadron abundances at 11.7A, 30A, 40A, 80A, and 158A GeV, are compared with the data in Fig. 10 (solid points). Note that this model does not provide a continuous description of the energy dependence; the points are only connected to guide the eye. The agreement with the measurements at the higher SPS energies is very good. The successful application of the saturation parameter γ s on the strangeness-neutral φ meson for p beam ≥ 40A GeV again suggests that the strangeness content at chemical freeze-out is determined on a partonic level for these energies. Final state interactions after chemical freeze-out could change the equilibrium φ yield and spectra. In particular, scattering of the daughter kaons with other produced hadrons would lead to a loss of the φ signal in the experimentally . The CERES data point [25] was displaced horizontally for visibility. Note that the CERES measurement is at y ≈ yc.m. −0.5. The full line shows the predictions of the extended hadron gas model (HGM) with strangeness equilibration [37], the dashed curves those obtained with UrQMD 1.3 [17]. The shaded boxes represent the systematic errors. observed decay channel, predominantly at small rapidities and low values of p t . Such a loss is not expected in the leptonic decay modes, since electrons or muons will leave the fireball without interaction. A comparison of the measured m t spectrum via the K + K − and e + e − decay channels [see Fig. 6(a)] indicates that the effect cannot be large. To study the effect on the total yield, we used the string-hadronic transport model UrQMD [17]. It was found that only about 8% of the decayed φ mesons are lost for detection due to rescattering of their daughter particles, independent of collision energy. Similar results have been obtained with the RQMD model [19]. The effect is thus not sufficient to account for the deviation of the relative φ multiplicities from their equilibrium values. On the other hand, φ mesons can be produced by KK scattering. In fact, kaon coalescence is the dominant (≈ 70%) production mechanism for the φ in UrQMD, again for all investigated collision systems. As shown by the dotted curve in Fig. 10, the model gives a reasonable description of the φ meson yields at lower energies, whereas it starts to deviate from the measurements at intermediate SPS energies. The discrepancy with data is more pronounced when studying the φ / π ratio (Fig. 9) because UrQMD overestimates the pion yields at SPS energies by about 30%. The hypothesis that the φ meson is produced predominantly by kaon coalescence can be tested by comparing the φ and kaon distributions in phase space. Figure 11(a) shows the width of the φ rapidity distribution as a function of beam rapidity at SPS energies, together with that measured for π − , K + , and K − [38,39]. The φ meson width does not fit into the systematics observed for the other particle species but increases much faster with energy. While at 20A GeV, the φ rapidity distribution is narrower than that of K − , we find it at top SPS energy comparable to the pions. In addition, at 158A GeV it is much larger in central Pb + Pb collisions than measured in p + p collisions at the same energy [23], a feature which is not observed for other particle species. In the kaon coalescence picture, there would be a tendency for the φ rapidity distribution to be narrower than those of the kaons. In an ideal case, neglecting correlations, 1 σ 2 φ = 1 σ 2 K + + 1 σ 2 K − ,(9) where the distributions were approximated by Gaussians. As shown in Fig. 11(b), the φ data rule out kaon coalescence as dominant formation mechanism for beam energies above 30A GeV. Only at 20A GeV, the observed rapidity widths are consistent with the coalescence picture. As mentioned before, this would also explain the φ enhancement at low energies, where a transient deconfined state is not expected. The observation that models based on a purely hadronic reaction scenario have serious problems in describing relative strangeness production in the upper SPS energy range is not unique to the φ meson but holds for kaons and other strange particles, too. It has been related to the onset of deconfinement at around 30A GeV as predicted by the statistical model of the early stage [18]. A striking experimental evidence is the narrow maximum in the K + /π + ratio at this energy [38,39]. A similar structure is, within experimental errors, not observed for the φ meson (Fig. 9); instead, the energy dependence of the relative φ meson yield resembles that of the K − . This can be understood since the K + yield is in good approximation proportional to the total strangeness production, which is not the case for K − and φ because a large, energy-dependent fraction of s quarks is carried by hyperons. The data from E917 [27] were averaged over the measured rapidity interval (see Table IV). Results from NA50 [24] and RHIC [28,29,30] were obtained at midrapidity, the result from CERES at y = −0.71. Data from NA49 are integrated over rapidity. The PHENIX data point was slightly displaced horizontally for visibility. For the NA49 data, mt was calculated from the transverse momentum spectra using an exponential extrapolation to full pt. For the other data sets, it was derived analytically from the exponential fit function. The shaded boxes represent the systematic errors. The energy dependences of both the inverse slope parameter and the mean transverse mass of the φ meson are shown in Fig. 12. The transverse mass spectra of the φ are well described by exponential fits [see Fig. 6(a)]; consequently, the two parameters show a similar behavior. Over the energy range AGS-SPS-RHIC, there is an overall tendency for both parameters to increase. However, a constancy of the values in the lower SPS energy range, as has been observed for pions, kaons, and protons [39]-a fact interpreted as being consistent with a mixed partonic/hadronic phase [40]-cannot be excluded. VI. SUMMARY We have presented new data on φ production in central Pb+Pb collisions obtained by the NA49 experiment at 20A, 30A, 40A, 80A, and 158A GeV beam energies. No indications of medium modifications of the φ meson mass or width were observed. The energy dependence of the production characteristics was studied by comparing them with measurements at AGS and RHIC energies. We find that at low SPS energy, the data can be understood in a hadronic reaction scenario; while at higher energies, hadronic models fail to reproduce the data. A statistical hadron gas model with undersaturation of strangeness gives a good description of the measured yields. This suggests that φ production is ruled by partonic degrees of freedom, consistent with the previously found indications for the onset of deconfinement at lower SPS energy. B. Selection of kaon candidatesNA49 observes the φ meson through its hadronic decay into charged kaons. To reduce the large contribution of pions and protons to the combinatorial background, kaon candidates were selected based on their specific energy loss dE/dx in the MTPCs. The mean dE/dx of pions, kaons, and (anti-)protons was determined from TOF-identified online) dE/dx parametrization for the data set at 80A GeV. (a) Mean dE/dx as function of βγ determined for TOF-identified pions, kaons, and protons; (b) dE/dx resolution as function of momentum, obtained from the deconvolution of the energy loss spectra into the contributions of π + , K + , and p. FIG . 2: (Color online) K + K − invariant-mass spectra after subtraction of the combinatorial background in the forward rapidity hemisphere for the five different beam momenta. The full lines show the Breit-Wigner fits to the signals as described in the text. The bin size is 4 MeV for 20A and 30A GeV and 2 MeV for the other beam energies. FIG. 3 : 3K + K − correlation function close to threshold in (a) qinv and (b) minv[34]. FIG. 4 : 4Geometrical acceptance probability for φ → K + K − including kaon decay in flight for (a) 20A GeV and (b) 158A GeV. FIG. 5 : 5(Color online) φ transverse momentum spectra integrated over the rapidity intervals given in FIG. 6 : 6(Color online) (a) φ transverse mass spectra integrated over the rapidity intervals given in FIG. 7 : 7(Color online) φ rapidity distributions. The solid points refer to measured data, the open points are reflected at midrapidity. The full lines show the parametrization by a single Gaussian online) φ enhancement factor E φ [see Eq. (8)] as function of energy per nucleon pair. Data from the AGS FIG. 9 : 9(Color online) φ / π ratio (a) in full phase space and (b) at midrapidity as function of energy per nucleon pair [ π = 1.5( π + + π − )] √s NN [GeV] <φ> FIG. 10: (Color online) φ multiplicity in central A + A collisions as function of energy per nucleon pair. The solid points denote the results of the statistical hadronization model (SHM) which allows a deviation from strangeness equilibrium[8]. They are connected by the solid line to guide the eye. The dotted curve shows the φ yield predicted by the UrQMD 1.3 model[17]. The shaded boxes represent the systematic errors. FIG. 11 : 11(Color online) (a) Widths of the rapidity distributions of π − , K + , K − , and φ in central Pb + Pb collisions at SPS energies as function of beam rapidity [38, 39]. The dashed lines are to guide the eye. The open star denotes the φ rapidity width measured in p + p collisions [23]. (b) Widths of the φ rapidity distributions in central Pb+Pb collisions compared with the expectations in a kaon coalescence picture [Eq. (9)]. The shaded boxes represent the systematic errors (shown only for φ mesons). FIG. 12: (Color online) (a) Inverse slope parameter T and (b) average transverse mass mt − m0 of the φ meson in central A + A collisions as function of energy per nucleon pair. TABLE I : ICharacteristics of the data sets employed in the analysis. The mean numbers of wounded nucleons Nw were obtained by Glauber model calculations.E beam √ sNN y beam Year Centrality Nw Nevents Momentum range (A GeV) (GeV) (GeV) 20 6.3 1.88 2002 7.2% 349 ± 1 ± 5 352 309 2.0-23.0 30 7.6 2.08 2002 7.2% 349 ± 1 ± 5 368 662 2.0-27.0 40 8.8 2.22 1999 7.2% 349 ± 1 ± 5 586 768 2.0-27.0 80 12.3 2.57 2000 7.2% 349 ± 1 ± 5 300 992 2.0-32.0 158 17.3 2.91 1996 5.0% 362 ± 1 ± 5 345 543 3.5-35.0 TABLE TABLE III : IIIDifferential φ meson yields in the pt (left) and y (right) distributions. Data in the pt bins are integrated over the rapidity ranges given inTable IV. The errors are statistical.pt (GeV) dn/(dydpt) (GeV −1 ) y dn/dy E beam = 20A GeV 0.0-0.4 0.382 ± 0.074 0.2-0.6 1.043 ± 0.250 0.4-0.8 0.528 ± 0.097 0.6-1.0 0.536 ± 0.077 0.8-1.2 0.257 ± 0.054 1.0-1.4 0.159 ± 0.033 1.2-1.6 0.079 ± 0.030 1.4-1.8 0.032 ± 0.017 1.6-2.0 0.033 ± 0.015 E beam = 30A GeV 0.0-0.3 0.231 ± 0.051 0.3-0.6 0.735 ± 0.194 0.3-0.6 0.578 ± 0.079 0.6-0.9 0.651 ± 0.090 0.9-1.2 0.386 ± 0.079 0.9-1.2 0.456 ± 0.052 1.2-1.5 0.257 ± 0.050 1.2-1.5 0.193 ± 0.036 1.5-1.8 0.070 ± 0.019 1.5-1.8 0.097 ± 0.029 E beam = 40A GeV 0.0-0.2 0.185 ± 0.035 0.3-0.6 1.067 ± 0.108 0.2-0.4 0.668 ± 0.052 0.6-0.9 0.756 ± 0.059 0.4-0.6 0.780 ± 0.064 0.9-1.2 0.611 ± 0.038 0.6-0.8 0.625 ± 0.075 1.2-1.5 0.348 ± 0.028 0.8-1.0 0.569 ± 0.073 1.5-1.8 0.188 ± 0.023 1.0-1.2 0.413 ± 0.059 1.2-1.4 0.275 ± 0.040 1.4-1.6 0.081 ± 0.028 1.6-1.8 0.086 ± 0.019 1.8-2.0 0.057 ± 0.014 E beam = 80A GeV 0.0-0.2 0.337 ± 0.031 -0.3-0.0 1.591 ± 0.304 0.2-0.4 0.886 ± 0.051 0.0-0.3 1.474 ± 0.138 0.4-0.6 1.148 ± 0.057 0.3-0.6 1.258 ± 0.086 0.6-0.8 0.996 ± 0.056 0.6-0.9 1.351 ± 0.062 0.8-1.0 0.861 ± 0.052 0.9-1.2 1.041 ± 0.049 1.0-1.2 0.517 ± 0.048 1.2-1.5 0.718 ± 0.043 1.2-1.4 0.344 ± 0.045 1.5-1.8 0.408 ± 0.037 1.4-1.6 0.173 ± 0.040 1.8-2.1 0.197 ± 0.040 E beam = 158A GeV 0.0-0.2 0.582 ± 0.053 0.0-0.2 2.557 ± 0.166 0.2-0.4 1.275 ± 0.086 0.2-0.4 2.386 ± 0.121 0.4-0.6 1.924 ± 0.098 0.4-0.6 2.229 ± 0.098 0.6-0.8 2.016 ± 0.099 0.6-0.8 2.202 ± 0.089 0.8-1.0 1.778 ± 0.092 0.8-1.0 1.974 ± 0.090 1.0-1.2 1.339 ± 0.080 1.0-1.2 1.816 ± 0.094 1.2-1.4 0.956 ± 0.067 1.2-1.4 1.636 ± 0.105 1.4-1.6 0.567 ± 0.055 1.4-1.6 1.528 ± 0.126 1.6-1.8 0.370 ± 0.044 1.6-1.8 1.125 ± 0.171 1.8-2.0 0.200 ± 0.034 TABLE TABLE VI : VITotal φ multiplicity φ and midrapidity yield dn φ /dy calculated from the rapidity distributions ofFig. 7. 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An emergence of Local Parity Breaking (LPB) in central heavy-ion collisions (HIC) at high energies is discussed. LPB in the fireball can be produced by a difference between the number densities of right-and left-handed chiral fermions (Chiral Imbalance) which is implemented by a chiral (axial) chemical potential. The effective meson lagrangian induced by QCD is extended to the medium with Chiral Imbalance and the properties of light scalar and pseudoscalar mesons (π, a 0 ) are analyzed. It is shown that exotic decays of scalar mesons arise as a result of mixing of π and a 0 vacuum states in the presence of chiral imbalance. The pion electromagnetic formfactor obtains an unusual parity-odd supplement which generates a photon polarization asymmetry in pion polarizability. We hope that the above pointed indications of LPB can be identified in experiments on LHC, RHIC, CBM FAIR and NICA accelerators. ⋆

10.1051/epjconf/201715803012

1710.01760

7e586549bf087d8ced0965be54657f5761dd8b3e

Exotic meson decays in the environment with chiral imbalance A A Andrianov [email protected] Faculty of Physics Saint Petersburg State University Universitetskaya nab. 7/9, Saint Petersburg 199034Russia, Departament de Física Quàntica i Astrofísica and Institut de Ciéncies del Cosmos (ICCUB) Universitat de Barcelona Martí i Franqués 108028BarcelonaSpain V A Andrianov ⋆⋆ D Espriu Faculty of Physics Saint Petersburg State University Universitetskaya nab. 7/9, Saint Petersburg 199034Russia, Departament de Física Quàntica i Astrofísica and Institut de Ciéncies del Cosmos (ICCUB) Universitat de Barcelona Martí i Franqués 108028BarcelonaSpain ⋆⋆⋆ A V Iakubovich ⋆⋆⋆⋆e-mail:[email protected] Faculty of Physics Saint Petersburg State University Universitetskaya nab. 7/9, Saint Petersburg 199034Russia, ⋆⋆⋆⋆A E Putilova Faculty of Physics Saint Petersburg State University Universitetskaya nab. 7/9, Saint Petersburg 199034Russia, Exotic meson decays in the environment with chiral imbalance 10.1051/epjconf/201715803012EPJ Web of Conferences 158, 03012 (2017) An emergence of Local Parity Breaking (LPB) in central heavy-ion collisions (HIC) at high energies is discussed. LPB in the fireball can be produced by a difference between the number densities of right-and left-handed chiral fermions (Chiral Imbalance) which is implemented by a chiral (axial) chemical potential. The effective meson lagrangian induced by QCD is extended to the medium with Chiral Imbalance and the properties of light scalar and pseudoscalar mesons (π, a 0 ) are analyzed. It is shown that exotic decays of scalar mesons arise as a result of mixing of π and a 0 vacuum states in the presence of chiral imbalance. The pion electromagnetic formfactor obtains an unusual parity-odd supplement which generates a photon polarization asymmetry in pion polarizability. We hope that the above pointed indications of LPB can be identified in experiments on LHC, RHIC, CBM FAIR and NICA accelerators. ⋆ Topological charge, Chiral Imbalance and axial chemical potential The behaviour of baryonic matter under extreme conditions has got recently a lot of interest [1,2]. A medium generated in the heavy ion collisions may serve for detailed studies, both experimental and theoretical, of various phases of hadron matter. In this context new properties of QCD in the hot and dense environment are tested in current accelerator experiments on RHIC and LHC [3,4]. In heavy ion collisions, in principle, there are two distinct experimental situations for peripheral and central collisions. In the first case the so-called Chiral Magnetic Effect (CME) can be detected, details see in [5] and also [6] for a review and additional references. In the second case there are some experimental indications of an abnormal dilepton excess in the range of low invariant masses and rapidities and moderate values of the transverse momenta [7]- [11] (see the reviews in [12]), which can be thought of as a result of LPB in the medium (the details can be found in [13]). In particular, in heavy-ion collisions at high energies, with raising temperatures and baryon densities, metastable states can appear in the finite-volume fireball with a nontrivial topological axial charge (due to fluctuations of gluonic fields) T 5 , which is related to the gluon gauge field G i , T 5 (t) = 1 8π 2 vol. d 3 x ε jkl Tr G j ∂ k G l − i 2 3 G j G k G l , j, k, l = 1, 2, 3,(1) where the integration is over the fireball volume. Its jump ∆T 5 can be associated with the space-time integral of the gauge-invariant Chern-Pontryagin density, ∆T 5 = T 5 (t f ) − T 5 (0) = 1 16π 2 t f 0 dt vol. d 3 x Tr(G µν G µν ) = 1 4π 2 t f 0 dt vol. d 3 x ∂ µ K µ , K µ = 1 2 ǫ µνρσ Tr G ν ∂ ρ G σ − i 2 3 G ν G ρ G σ .(2) For the time being we adopt a static case and neglect a topological current flux through the fireball boundary during lifetime of the corresponding thermodynamic phase in a domain. It is known that the divergence of isosinglet axial quark current J 5,µ = qγ µ γ 5 q is locally constrained via the relation of partial conservation of axial current affected by the gluon anomaly, ∂ µ J 5,µ − 2i m q J 5 = N f 2π 2 ∂ µ K µ ; J 5 = qγ 5 q(3) This relation allows to find the relation of a nonzero topological charge with a non-trivial quark axial charge Q q 5 . Namely, integrating over a finite volume of fireball we come to the equality, d dt (Q q 5 − 2N f T 5 ) ≃ 2i vol. d 3 x m q qγ 5 q, Q q 5 = vol. d 3 x q † γ 5 q = �N L − N R �,(4) where �N L − N R � stands for the vacuum averaged difference between left and right chiral densities of baryon number. Therefrom it follows that in the chiral limit (when the masses of light quarks are taken zero) the axial quark charge is conserved in the presence of non-zero (metastable) topological charge. If for the lifetime of fireball and the size of hadron fireball of order L = 5 − 10 fm , the average topological charge is non-zero, �∆T 5 � 0, then it may be associated with a topological chemical potential µ T or an axial chemical potential µ 5 [14] for neglected masses of light u, d quarks. Thus we have, �∆T 5 � ≃ 1 2N f �Q q 5 � ⇐⇒ µ 5 ≃ 1 2N f µ T ,(5) Thus adding to the QCD lagrangian the term ∆L top = µ T ∆T 5 or ∆L q = µ 5 Q q 5 , we get the possibility of accounting for non-trivial fluctuations of topological charge (fluctons) in the nuclear (quark) fireball. In the general, Lorentz covariant form the field dual to the fluctons is described by means of the classical pseudoscalar field a(x), so that, ∆L a = N f 2π 2 K ν ∂ ν a(x) ≃ 1 π 2 K ν b ν ⇐⇒ b ν qγ ν γ 5 q, b ν ≃ �∂ ν a(x)� ≃ const.(6) Thus in a quasi-equilibrium situation the appearance of a nearly conserved chiral charge can be incorporated with the help of an axial (chiral) vector chemical potential b ν . The appearance of a space vector part in b ν can be associated with the non-equilibrium axial charge flow [15][16][17]. For the detection of Local Parity Breaking in the hadron fireball we implement the generalized sigma model with a background 4-vector of axial chemical potential [18], symmetric under S U L (N f ) × S U R (N f ), for u, d-quarks (N f = 2), L = 1 4 Tr (D µ H (D µ H) † ) + B 2 Tr [ m(H + H † )] + M 2 2 Tr (HH † ) − λ 1 2 Tr � (HH † ) 2 � − λ 2 4 [ Tr (HH † )] 2 + c 2 (det H + det H † ),(7) where H = ξ Σ ξ is an operator for meson fields, m is an average mass of current quarks, M is a "tachyonic" mass generating the spontaneous breaking of chiral symmetry, B, c, λ 1 , λ 2 are real constants. The matrix Σ includes the singlet scalar meson σ, its vacuum average v and the isotriplet of scalar mesons a 0 0 , a − 0 , a + 0 , Σ = � v + σ + a 0 0 √ 2 a + 0 √ 2 a − 0 v + σ − a 0 0 � .(8) The operator ξ realizes a nonlinear representation of the chiral group and is determined by the isotriplet π 0 , π − , π + of pseudoscalar mesons, ξ = exp       i � π� τ 2 f π       ≈ 1 + i � π� τ 2 f π − (� π� τ) 2 8 f 2 π , (9) � π� τ = � π 0 √ 2 π + √ 2 π − −π 0 � ,(10) where � τ are Pauli matrix, f π is a decay constant of π mesons. The covariant derivative of H contains external gauge fields R µ and L µ , D µ H = ∂ µ H − iL µ H + iHR µ(11) These fields include the photon field A µ and are supplemented also a background 4-vector of axial chemical potential (b µ ) = (b 0 , b), R µ = e Q em A µ − b µ · 1 2×2 , L µ = e Q em A µ + b µ · 1 2×2 ,(12) where Q em = 1 2 τ 3 + 1 6 1 2×2 is a matrix of electromagnetic charge. The complete effective meson lagrangian has to include a P-odd part the Wess-Zumino-Witten effective action [19] which is modified in the chirally imbalanced medium. The relevant parts of WZW action read, ∆L WZW = − ie N c b ν 6π 2 v 2 ǫ νσλρ A ρ (∂ σ π + ) (∂ λ π − ) − e 2 N c 24 π 2 v ǫ νσλρ (∂ σ A λ )(∂ ν A ρ )π 0(13) Chiral (scalar) condensate depending on chiral chemical vector The mass gap equation for the scalar condensate follows from (7), −4 (λ 1 + λ 2 ) v 3 + 2M 2 + 4b 2 + 2c v + 2B m = 0. The general solution of this equation reads v(b µ ) = 1 6 2 3 (λ 1 + λ 2 ) 9B m(λ 1 + λ 2 ) 2 + 3(λ 1 + λ 2 ) 3 27B 2 m 2 (λ 1 + λ 2 ) − 2 M 2 + 2b 2 + c 3 1/3 + M 2 + 2b 2 + c 6 1 3 9B m(λ 1 + λ 2 ) 2 + 3(λ 1 + λ 2 ) 3 27B 2 m 2 (λ 1 + λ 2 ) − 2 M 2 + 2b 2 + c 3 −1/3 . There are different regions for chiral vector covariant under Lorentz transformations of fireball frame. We stress that in a hot medium Lorentz invariance is broken by thermal bath and the physical effects depend on a particular set of components of (b µ ). The plots for condensates display the enhancement of CSB and the restoration of chiral symmetry depending on the sign of b µ b µ . Namely, in the chiral imbalance region with b 2 > 0 the increasing of chemical potentials causes the growth of chiral condensate, i.e. enhances Chiral Symmetry Breaking (CSB). Instead, in the chiral vector imbalance region with b 2 < 0 the chiral condensate is decreasing with growing |b 2 | up to |b 2 | = 1 2 (M 2 + c). At this scale in the chiral limit m → 0 the CSB parameter v → 0 (see below) and spontaneous CSB is restored. At last the CSB parameter v is insensitive to the light-cone b µ . For b 2 > 0 in the rest frame of vector (b µ ) = (µ 5 , 0, 0, 0) one can compare the predictions of our effective meson lagrangian (see Fig.1, the left plot) with the lattice estimations [20] (see Fig.1, the right plot) which clearly shows the enhancement of CSB. However non-zero real space components of (b µ ) produce non-Hermitian purely imaginary vertices after euclidization of QCD which makes it difficult to compute on the lattice their contribution to the quark determinant. Meson mass spectrum in different chiral imbalance regions Introduce the definitions for meson state masses in the chiral imbalance environment. The mass matrix for scalar and pseudoscalar mesons on the diagonal takes the following values, m 2 a = −2 � M 2 − 2 (3λ 1 + λ 2 ) v 2 − c + 2b 2 � , m 2 σ = −2 � M 2 − 6 (λ 1 + λ 2 ) v 2 + c + 2b 2 � , m 2 π = 2B m v .(14) After diagonalization we define distorted masses as m e f f + for the fieldã and m e f f − for the fieldπ m 2 e f f ± = 1 2 � m 2 a + m 2 π ± � � m 2 a − m 2 π � 2 + (8 b µ k µ ) 2 � .(15) Parameters of QCD-inspired generalized sigma model Let us normalize the vacuum parameters of our model. We take m π = 139 MeV, m a = 980 MeV, m σ = 500 MeV, m = 5.5 MeV, µ 5 = 0, M = 300 MeV, v = 92 MeV. Then from the following Eqs.                                  m 2 σ = −2 � M 2 − 6 (λ 1 + λ 2 )v 2 + c � m 2 a = −2 � M 2 − 2 (3λ 1 + λ 2 )v 2 − c � m 2 π = 2 B m v v(µ 5 = 0) = � M 2 + c 2(λ 1 + λ 2 ) + B m 2(M 2 + c) ,(16) Diagonalization matrix After diagonalization of mass matrix the states for a 0 and π mesons happen to be mixed. The eigenstates are defined as a 0 = C aãã + C aππ , π = C πãã + C πππ , C aã = iC ππ = C + , C aπ = −iC πã = −C − ,(17) with C ± = 1 √ 2 1 ± m 2 a − m 2 π (m 2 a − m 2 π ) 2 + (8 b µ k µ ) 2 .(18) We use the notationã,π, indicating that these states tend to a 0 , π when b µ = 0. One can see (Fig.2, left plot) that for b µ b µ > 0 the growing chemical potential very quickly enforces the mixing between pions and isotriplet scalars so that distorted scalars happen to be involved into typical reactions of pion decay and pion formfactor. Yet with increasing of chemical potential the content of original pion states in distorted scalars and pseudoscalars diminishes. In turn, for b µ b µ < 0 (Fig.2, right plot) the restoration of CS does not suppress the mixing of parity counterpartners and for increasing chemical potential tends to a nearly constant mixing coefficients with values in between 0 and 1. Masses in CSB region with b µ b µ > 0 This is a region where CSB is enhancing and theã 0 and σ mesons become more heavy with growing chemical potential. Meantime theπ-meson effective mass is slowly decreasing at rest and decreasing faster in flight with | � k| 0. One can see how theπ-meson reaches the massless point and further on its mass squared becomes negative, "tachyonic" which however does not cause any causality problems. One can check that the group velocity of these states remains less than the light velocity. In this region in the rest frame one can see clearly the CS restoration with merging masses of all scalars and pseudoscalars. But in flight the behavior of pion masses is more peculiar. First pion masses vanish and then mass squared become negative. Next they reappear with positive mass squared and slowly approach to scalar masses in asymptotics. Thus the in-flight effect of chiral vector imbalance on pion spectrum deviates strongly from naive expectations. 5 The decaysã ± 0 →π ± γ,π 0 → γγ,ã 0 0 → γγ After mixing π, a 0 0 →π,ã 0 0 the decayã ± 0 →π ± γ arises which breaks space parity and therefore is forbidden in vacuum. with q = |� q| being space momenta of scalars. As well in the decay process π 0 → γγ an adjacent resonance decayã 0 0 → γγ emerges after mixing. From the plots for mixing coefficients we conclude that these processes are comparable in decay widths. Conclusions and outlook In this work we described a possibility of local parity breaking (LPB) emerging in a dense hot baryon matter (hadron fireball) in heavy-ion collisions at high energies. The phenomenology of LPB in a fireball is based on introducing a topological (axial) charge and a topological (chiral) chemical potential. Topological charge fluctuations transmit their influence to hadronic physics via an axial chemical potential. We suggested QCD-motivated sigma model for the description of isotriplet pseudoscalar and isoscalar and isotriplet scalar mesons in the body of a fireball. We conclude: • Strong CP violation is quite a challenging possibility to be revealed in heavy-ion collisions both at high energy densities (temperatures) and being triggered by large baryon densities. • However the existing theoretical arguments for arising CP violation in FINITE volumes are not well sufficient to calculate the production rate of CP violating nuclear processes. • There are two ways to improve the discovery potential: firstly, to elaborate the recipes for experimentalists to detect peculiar effects generated in CP-odd background, secondly, to measure the production of the mass states without a firm CP parity. In both cases the chiral chemical potential method helps a lot in predictions. • In addition we already suggested [14] the vector meson dominance model with chiral imbalance: the spectrum of massive vector mesons splits into three components with different polarizations and with different effective masses that can be used to detect local parity breaking. schemes for revealing local parity breaking helps to (partially) explain qualitatively and quantitatively the anomalous yield of dilepton pairs in the CERES, PHENIX, STAR, NA60, and ALICE experiments. Accordingly the identification of its physical origin might serve as a base for a deeper understanding of QCD properties in a medium under extreme conditions. Experimental collaborations should definitely check this possibility. • Recently an interesting proposal was given in [21] to detect the LPB by measuring photon polarization asymmetry in the process π ± γ → π ± γ. We extend this proposal indicating the resonance enhancement at energies comparable with the mass ofã ± 0 scalars. 1 . 1Chiral charge imbalance region when b 2 > 0, in the rest frame the chiral background (b µ ) = (µ 5 , 0, 0, 0). 2. Chiral vector imbalance region with b 2 < 0, in the static frame the chiral background is taken along the beam axis (b µ ) = (0, 0, 0, b).3. Transition region with b 2 = 0, in the light-cone background (b µ ) = (b, 0, 0, ±b). Figure 1 . 1Left: the condensate in b µ b µ > 0 and b µ b µ < 0 region; right: the vacuum average v(µ 5 ) ∼ �ψψ�/B from the lattice calculations[20]. one can find λ 1 , λ 2 , c and b. We have for parameters λ 1 = 1.64850 × 10, λ 2 = −1.31313 × 10, c = −4.46874 × 10 4 MeV 2 , B = 1.61594 × 10 5 MeV 2 . Figure 2 . 2Left: mixing coefficients dependence on chemical potential µ 5 for | � k| = 1000 MeV in b µ b µ > 0 region; right: mixing coefficients dependence on chemical potential |b| for | � k| = 1000 MeV in b µ b µ < 0 region . Figure 3 . 3ã 0 -meson andπ-meson effective mass dependence as well as σ-meson mass dependence on chemical potential b = µ 5 , for different values of | � k| in b µ b µ > 0 region. Figure 4 . 4ã 0 -meson and σ-meson mass dependence on chemical potential b,π-meson effective mass dependence on b for different values of | � k| in b µ b µ < 0 phase, the angle between � k and � b is θ = 0 4.4 Masses in CSR region with b µ b µ < 0 Figure 5 . 5Decay width a ± → π ± γ, µ 5 = 100 MeV Figure 6 . 6Decay widths; left:π 0 → γγ, right:ã 0 0 → γγ, µ 5 = 100 MeV For pseudoscalars and scalars in flight the speeds of decays are considerably increasing. The effects are opposite to the Lorentz retardation. EPJ Web of Conferences 158, 03012 (2017) DOI: 10.1051/epjconf/201715803012 QFTHEP 2017 AcknowledgementsIt is a pleasure to thank the organizers of the XXIII International Workshop on High Energy Physics and Quantum Field Theory, QFTHEP 2017 for a fruitful meeting and an excellent atmosphere. 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The key parameter controlling the glass transition of colloidal suspensions is ϕ, the fraction of the sample volume occupied by the particles. Unfortunately, changing ϕ by varying an external parameter, e.g. temperature T as in molecular glass formers, is not possible, unless one uses thermosensitive colloidal particles, like the popular poly(Nisopropylacrylamide) (PNiPAM) microgels. These however have several drawbacks, including high deformability, osmotic deswelling and interpenetration, which complicate their use as a model system to study the colloidal glass transition. Here, we propose a new system consisting of a colloidal suspension of non-deformable spherical silica nanoparticles, in which PNiPAM hydrogel spheres of ∼ 100 − 200 µm size are suspended. These non-colloidal 'mesogels' allow for controlling the sample volume effectively available to the silica nanoparticles and hence their ϕ, thanks to the T -induced change in mesogels volume. Using optical microscopy, we first show that the mesogels retain their ability to change size with T when suspended in Ludox suspensions, similarly as in water. We then show that their size is independent of the sample thermal history, such that a well-defined, reversible relationship between T and ϕ may be established. Finally, we use space-resolved dynamic light scattering to demonstrate that, upon varying T , our system exhibits a broad range of dynamical behaviors across the glass transition and beyond, comparable with those exhibited by a series of distinct silica nanoparticle suspensions of various ϕ.

10.1063/5.0086822

2203.04407

1f3b3a2d55da9982ee05446666f8b0993f48a1f6

Controlling the volume fraction of glass-forming colloidal suspensions using thermosensitive host 'mesogels' J S Behra Laboratoire Charles Coulomb (L2C) Université Montpellier CNRS MontpellierFrance A Thiriez Laboratoire Charles Coulomb (L2C) Université Montpellier CNRS MontpellierFrance D Truzzolillo Laboratoire Charles Coulomb (L2C) Université Montpellier CNRS MontpellierFrance L Ramos Laboratoire Charles Coulomb (L2C) Université Montpellier CNRS MontpellierFrance L Cipelletti Laboratoire Charles Coulomb (L2C) Université Montpellier CNRS MontpellierFrance Institut Universitaire de France ParisFrance Controlling the volume fraction of glass-forming colloidal suspensions using thermosensitive host 'mesogels' (Dated: 10 March 2022)Controlling the volume fraction The key parameter controlling the glass transition of colloidal suspensions is ϕ, the fraction of the sample volume occupied by the particles. Unfortunately, changing ϕ by varying an external parameter, e.g. temperature T as in molecular glass formers, is not possible, unless one uses thermosensitive colloidal particles, like the popular poly(Nisopropylacrylamide) (PNiPAM) microgels. These however have several drawbacks, including high deformability, osmotic deswelling and interpenetration, which complicate their use as a model system to study the colloidal glass transition. Here, we propose a new system consisting of a colloidal suspension of non-deformable spherical silica nanoparticles, in which PNiPAM hydrogel spheres of ∼ 100 − 200 µm size are suspended. These non-colloidal 'mesogels' allow for controlling the sample volume effectively available to the silica nanoparticles and hence their ϕ, thanks to the T -induced change in mesogels volume. Using optical microscopy, we first show that the mesogels retain their ability to change size with T when suspended in Ludox suspensions, similarly as in water. We then show that their size is independent of the sample thermal history, such that a well-defined, reversible relationship between T and ϕ may be established. Finally, we use space-resolved dynamic light scattering to demonstrate that, upon varying T , our system exhibits a broad range of dynamical behaviors across the glass transition and beyond, comparable with those exhibited by a series of distinct silica nanoparticle suspensions of various ϕ. I. INTRODUCTION Glasses are characterized by a structure that resembles that of liquids, while microscopic dynamics are orders of magnitude slower than in fluids. 1 Typically, the control parameter in molecular glasses is temperature, T : when the sample is cooled quickly enough below the glass transition temperature, crystallization is avoided, leading to an amorphous solid. 1 Remarkably, other systems exhibit a phenomenology similar to that of molecular glass formers, e.g. granular systems, 2 dense colloidal suspensions, 3 and active 4 or biological matter. 5 On the one hand, these analogies motivate the quest for a general scenario for the glass transition. On the other hand, they pave the way for using systems such as colloids as model glass formers, because structural and dynamical quantities of interest are more readily accessible in colloidal suspensions than in molecular glasses. 3,6 Colloids are sub-micron particles dispersed in a solvent. The simplest colloidal glass former comprises colloidal hard spheres, whose relevant parameter is ϕ, the fraction of the sample volume occupied by the particles, 7,8 rather than T , as in molecular glasses. In other colloidal systems, the interparticle potential is more complex than the no-overlap hard sphere potential. Both attractive and repulsive interactions are routinely encountered in colloidal systems, and in general the colloidal glass transition depends on both ϕ and particle interactions. This results in a very rich behavior that often has no counterpart in molecular materials, see e.g. the nonmonotonic (reentrant) glass transition of colloids with shortrange attractive interactions. 9 Investigating the colloidal glass transition usually involves preparing a series of distinct samples whose composition is varied in order to explore a range of ϕ and/or of interparticle interactions. This is different from molecular systems, where a single sample may be used across the glass transition by simply varying T . In colloids, the lack of an easily tunable external parameter poses several challenges. Controlling and measuring the volume fraction with the required accuracy is difficult, even for hard spheres. 10 Manipulating concentrated colloids, e.g. to transfer them to a measuring cell, can be quite tricky, especially for suspensions of small particles, which are in general very stiff. For example, the shear modulus of glassy hard spheres scales as G ∼ k B T /a 3 , with k B Boltzmann's constant and a the colloid radius. 11 Accordingly, a (marginally) glassy suspension of hard spheres with ϕ ≈ 0.6 and a = 200 nm has G ∼ 0.5 Pa and flows easily when poured from a container, while a similar suspension with a = 20 nm has G ∼ 500 Pa. The latter is pasty and can only be transferred using a spatula, which inevitably introduces further uncertainties on ϕ. Finally, it is difficult to impose the equivalent of a well-controlled 'thermal history' to a colloidal glass former, thus preventing the in-depth study of the effect of sample history on aging. 1,3 While mechanical agitation is a popular way of initializing glassy colloidal suspensions, 12 a recent work suggests differences in aging when shear or a ϕ quench is applied. 13 In general, a protocol alternative to mechanical agitation is desirable when studying the interplay between the microscopic structure and dynamics of colloidal suspensions and their rheological properties, an increasingly active research field. [14][15][16][17] In response to these difficulties, the possibility of studying colloidal phase transitions by tuning the volume fraction or the interparticle interactions with T has been explored for several years. One approach leverages on varying interactions, e.g. using selectively wettable particles suspended in mixture of fluids close to its critical point, 18,19 or thanks to depletion forces 20 whose strength varies with T . 21,22 Another approach is based on micelles of self-assembled block-copolymers as colloidal objects. 23 Thanks to the T dependence of the affinity with the solvent of each block, it is possible to design systems where the degree of micellization, hence the colloidal volume fraction, depends on T . 24 However, increasing the micelle number density often leads to the formation of (poly)crystalline phases, rather than glasses. 23 Additionally, data interpretation is complicated by the difficulty to precisely quantify the degree of micellization as a function of T . Thermosensitive microgels are sub-micrometric, deformable particles that provide another popular way to control the colloidal volume fraction. By varying T , the affinity of the polymer chains for the solvent is changed, resulting in microgel swelling or deswelling and hence in a change of ϕ for samples at fixed microgel number density. Poly(Nisopropylacrylamide) (PNiPAM)-based materials have been extensively studied, 3,13,[25][26][27][28][29][30] unveiling intriguing aspects of the glass transition of soft colloids, distinct from those of hard spheres. Features such as the 'strong' (i.e. Arrheniuslike) increase of the relaxation time on approaching the glass transition, 25,27 supra-linear aging, 27 or the existence of highdensity states where the dynamics depend surprisingly weakly on ϕ 29,30 distinguish soft colloids from hard spheres and have become an active research field per se. Thermosensitive microgels, however, come with several complications. At high ϕ, they are subject to interpenetration with one another, shape modification, and osmotic deswelling, resulting in changes of the interparticle interactions. 27,29,31,32 It is difficult to disentangle the contribution of these phenomena from that of the variation of ϕ, hindering the understanding of the microscopic dynamics and rheological properties of microgel suspensions. Moreover, these phenomena make it difficult to compare experimental results to numerical simulations, where interactions are usually modeled by simplified central potentials and are assumed to be independent of particle density. Charge-stabilized hard particles such as silica colloids are an appealing alternative as model soft colloids, 30,33 since they have a well-defined spherical shape and are not subject to osmotic swelling, interpenetration nor compression. In this case, softness arises from the shape of the screened Coulomb repulsive potential. Unfortunately, however, neither the volume fraction nor the interparticle potential of these systems can be significantly varied by tuning T . To circumvent these difficulties, we develop a new colloidal system, comprising a dense suspension of charged-stabilized silica nanoparticles and non-colloidal, thermosensitive PNi-PAM spheres. The PNiPAM spheres have a typical diameter ∼ 100 − 200 µm, intermediate between the sub-micron scale of usual microgels and the macroscopic scale; we thus term them 'mesogels'. Since the mesogels are more than three orders of magnitude larger than the silica nanoparticles and well beyond the colloidal length scale, they do not alter the interaction potential between nanoparticles, as it would be the case if they had comparable size, e.g. due to depletion interactions. Furthermore, we do not expect the nanoparticles to penetrate in the PNiPAM mesogels, as their diameter (∼ 30 − 40 nm) is too large compared to the average mesh size expected for the mesogels ( 10 nm, see Refs. and Sec. I.A of the Supplementary Material (SM)) 34,35 ). The role of the mesogels is to control the volume of the sample effectively available to the nanoparticles, thanks to the swelling or deswelling of PNi-PAM gels upon temperature changes. Thus, the effective volume fraction of the silica nanoparticles can be simply tuned by varying T , paving the way for an easier sample manipulation (by transiently reducing ϕ), the straightforward study of ϕ-dependent properties with a single sample, and the investigation of the effect of an arbitrary ϕ history imposed to the system. The rest of the paper is organized as follows: in Sec. II we describe the synthesis of the mesogels, as well as microscopy and Dynamic Light Scattering (DLS) setups. In Sec. III we first present and discuss the T -dependent size of the mesogels, both in water and in concentrated Ludox suspensions, for various T histories. We then use DLS to show that, upon the addition of a few % vol. of mesogels, a suspension of nanoparticles can span the whole range of dynamic behaviors from marginally supercooled to fully glassy upon changing T . Finally, in Sec. IV we recapitulate our main findings and briefly discuss future research paths opened by this work. II. MATERIALS AND METHODS Sample preparation PNiPAM mesogel synthesis was carried out at T room = 20 • C with a similar protocol as that described by Kanai et al. 36 A stable water-in-silicone oil emulsion was prepared with a home-made microfluidic device (see Fig. S1 in SM), and the so-obtained aqueous drops were subsequently polymerized under UV light. Details are provided in Sec. I in SM. The mesogels were thoroughly washed with isopropanol prior to being transferred into DI water, and the remaining traces of silicone oil were removed with diethyl ether. The mesogels, suspended in DI water, were stored in the fridge prior to use. They looked relatively transparent to the naked eye, indicating that their structure is homogeneous upon the scale of the visible light wavelengths, consistent with previous investigations of the influence of the synthesis temperature on the structure and appearance of PNiPAM hydrogels, which showed that syntheses performed at T below 25 • C yield more transparent PNiPAM materials as compared to syntheses performed at higher T 37-39 (see Sec. I.A in SM). To prepare the concentrated Ludox suspensions, ∼ 20 g of the commercial suspension (Ludox TM-50, average particle diameter 35 nm and polydispersity index 0.25 as determined by DLS on a diluted suspension) were centrifuged at 10500 rpm during ∼ 4 h in a 3-15 centrifuge (Sigma) equipped with a 12158-H rotor (Sigma). The supernatant was then removed, and the suspensions were homogenized with a spatula. They were vortexed and further centrifuged at ∼ 1500 rpm for about 15 min in a 2-4 centrifuge (Sigma) to release trapped air bubbles. The Ludox volume fractions ϕ of all the Ludox suspensions used in the present study (including that of the commercial Ludox suspension) were determined by drying a small aliquote of the suspension, as described in the supplemental material of Ref. 30 . The concentrated Ludox suspensions were stored in the fridge prior to use. Two types of mesogels and Ludox mixtures were prepared: suspensions of mesogels (i) in the commercial Ludox solution (ϕ = 0.350) and (ii) in concentrated Ludox suspensions (ϕ = 0.396 and ϕ = 0.412 for the suspensions characterized with optical microscopy and DLS, respectively). The vial containing the mesogels was gently shaken to redisperse the mesogels which had sedimented over time. Mesogels were quickly sampled with a plastic pipette and immediately transferred into a 2.0 mL Eppendorf tube. They were left to sediment in the tube and as much as possible DI water was removed. The Ludox suspension was then added. For concentrated Ludox suspensions, centrifugation was carried out to bring the Ludox suspension to the bottom of the Eppendorf tube. Finally, the mixture was gently mixed with a spatula to disperse the mesogels throughout the sample. Samples obtained from the concentrated Ludox suspension were centrifuged at low speeds to remove trapped air bubbles. Optical microscopy Using a plastic pipette for samples prepared in DI water and in commercial Ludox solution, and a spatula for those prepared in concentrated Ludox suspensions, the samples were transferred in a 130 µL volume cell formed by a microscope slide and a coverslip spaced by two superimposed 65 µL gene frames (16 mm × 15 mm × 0.25 mm; ref: AB0577; Thermo Scientific), to ensure no mesogel is squeezed. To ease the microscopy observations, the mesogel volume fraction was kept low, about 0.045% at room temperature. To measure the temperature directly in the observation cell, a thermocouple connected to a temperature controller (di 32 PID; Jumo) was sandwiched between the two gene frames with the sensor being right in the middle of the cell. The microscope slide was then placed under the microscope (Laborlux 12 Pol S; Leitz) in a microscope hot stage (HS400; Instec) linked to a temperature controller (STC200; Instec) and a liquid nitrogen cooling unit (LN2-P; Instec). Micrographs were taken with a digital camera (D5200; Nikon) and processed with ImageJ (version 1.53f51; National Institutes of Health; USA). In particular, we used it to determine d, the Feret's diameter of the mesogels, 40 which for our spherical mesogels effectively corresponds to the usual diameter. Prior to any observation, samples were allowed to equilibrate in the microscope hot stage for at least 20 min. To characterize the mesogels at equilibrium at different T , the temperature was increased in small steps (no more than 1 • C at a time) using T ramps of +0.2 • C/min. After each ramp, the mesogels were allowed to thermally equilibrate for ∼ 10 min prior to taking pictures of the mesogels across the sample and subsequently measuring their size. To characterize the mesogel behavior upon successive changes in temperature, T cycles were performed, withṪ up ranging from +0.02 • C/min to +3 • C/min, andṪ down = −0.5 • C/min. After each ramp up and down, the mesogels were allowed to thermally equilibrate for ∼ 10 min and ∼ 5 min, respectively, prior to taking pictures of the mesogels across the sample and subsequently measuring their size. Dynamic light scattering The sample was prepared as described in Sec. II B 1. A mass of 1.450 g concentrated Ludox suspension (ϕ = 0.412) was added to 1.094 g mesogels (as determined after removal of the supernatant; note that this amount also accounts for the water in-between the mesogels which we assumed to be at the random packing volume fraction). Using a thin spatula, ∼ 200 mg of the sample were deposited on the inner walls of a NMR tube of diameter 4 mm, cut to a height of about 8 cm. The NMR tube was centrifuged to bring the sample down to its bottom and to remove trapped air bubbles. It was then placed in a hemolysis tube filled with DI water (diameter 1 cm), which in turn was placed in the temperature-controlled copper sample holder of the home-built set-up performing space-resolved DLS measurements described by El Masri et al. 41 . All the measurements were performed at a scattering angle of 90 • , corresponding to a scattering wave vector with modulus q = 22.3 µm −1 . The power of the laser (532 nm wavelength in vacuum) was set to 150 mW for the highest investigated T and was then decreased to 37.5 mW for all the other measurements (see discussion). The measurements were performed from high to low T , i.e. from the sample with the fastest dynamics to the slowest one. III. RESULTS AND DISCUSSION We first show in Fig. 1 pearance. As expected, 36 the size of the PNiPAM mesogels immersed in water decreases when the temperature increases. This behavior -key to our approach -is preserved when the mesogels are immersed in Ludox suspensions of various concentrations. In the latter, PNiPAM mesogels become less visible at temperatures around 31 • C. The variation of PNiPAM mesogel volume indeed induces a change in their refractive index, which matches that of Ludox suspensions at temperatures around 31 • C (see discussion of Fig. 2(b)). PNiPAM mesogel size at equilibrium is plotted as a function of temperature and the immersion medium in Fig. 2(a). For the three investigated suspension media, the size of the mesogels decreases when the temperature increases. In each case, the temperature at which the sharper change in size occurs -usually called Volume Phase Transition Temperature (VPTT) -was determined as the inflection point of d(T ), and the obtained values are shown in Table I. The VPTT of our mesogels in water is 33.6 • C and belongs to the range of 31.5 − 34 • C typically found for PNiPAM materials in wateronly environments. 36,[42][43][44][45] At low temperatures, the size of the mesogels is smaller in Ludox suspensions than in water, and the VPTT decreases as the Ludox concentration is increased. These differences in size and in VPTT are likely to be induced by the electrolytes present in the Ludox suspensions as a result of the charge stabilization performed by the manufacturer (we estimate from the data sheet that the concentration of the main electrolyte, Na + , is ca. 0.1 M 46 ). Previous studies have indeed shown that, upon salt addition, (i) the size of PNiPAM microgels decreases at a given temperature, and (ii) the volume phase transition is shifted towards lower temperatures. 42,47,48 Our data are in qualitative agreement with these works. The main mechanism responsible for these changes is the dehydration of the PNi-PAM chains by the added free ions, which leads to a smaller size at the lowest T and promotes the volume transition at lower temperatures. 42,47,48 To investigate the close index matching of PNiPAM meso- n mgel φ n mgel φ n Ludox φ T n Ludox φ T (a) (b)n mgel (T ) = x(T ) n mono + [1 − x(T )]n w (1) where n i is the refractive index of the object or fluid i (i = 'mgel', 'mono', and 'w' for the mesogel, NiPAM monomer and water, respectively), and x is the volume fraction of Ni-PAM monomers within a mesogel sphere, calculated from the monomer concentration used in the synthesis as explained in Sec. III of the SM. In Eq. 1, we use n mono = 1.52, 49 and n w = 1.325, the latter measured at T = 23.8 • C. The variation of the refractive index of both NiPAM monomers and water is considered negligible across the investigated temperature range. The so-estimated values of the PNiPAM mesogel refractive index are plotted as a function of temperature in Fig. 2 (symbols), together with the refractive index values measured for the two Ludox suspensions (dashed lines). In both cases, n mgel is lower than that of the surrounding medium at low temperatures, where the mesogels are highly swollen, such that n mgel is close to the refractive index of water. Upon increasing the temperature, n mgel increases and becomes higher than that of the surrounding medium. The crossover -indicative of index matching conditions -takes place at temperatures around 31 • C (values are provided in Table I), corresponding to the temperature range where the mesogels almost 'disappear' (see mid and bottom rows of Fig. 1). It is worth noting that, as the water refractive index is lower than that of the mesogels for all investigated temperatures, these conditions are never met in water and the balls never 'disappear' when they are suspended in water (top row of Fig. 1). Finally, we emphasize that the fact that mesogels in Ludox suspensions meet index-matching conditions at intermediate temperatures confirms that the silica particles cannot penetrate the mesogels. Indeed, if a significant amount of Ludox could penetrate the mesogels, n mgel would be larger than the refractive index of the background suspension at all T and index matching would never occur. Our final goal is to use PNiPAM mesogel ability to change size with temperature to access a wide range of Ludox volume fractions in a controlled manner using a single sample, as well as to impose well-controlled variations of the volume fraction with time (e.g. quenches and ramps 13,27,28 or cycles 50 ). To this end, it is important to asses whether a unique, welldefined relationship between temperature and Ludox volume fraction holds, i.e. to check whether the mesogel size at a given T is reproducible and independent of the thermal history imposed to the sample. (b) We show in Fig. 3 the influence of successive identical temperature cycles on PNiPAM mesogel appearance and size in both water and the commercial Ludox suspension. In both cases, the mesogels are found to recover the same size and appearance after each T ramp up and each T ramp down. One may notice that the appearance of the mesogels during the high temperature plateau (bottom row of images in Fig. 3(a),(b)) is different than that seen in Fig. 1. This is due to the fact that the way the mesogels undergo the volume phase transition upon an increase in temperature depends on the rateṪ up , as illustrated in Fig. 4 for mesogels suspended in a commercial Ludox suspension (similar results are obtained for mesogels suspended in pure water -data not shown). When the temperature is increased slowly (Fig. 4, bottom part), the mesogel appearance is very similar to that of thermally-equilibrated mesogels (see Fig. 1). Overall, the volume phase transition appears to be a smooth and continuous process. By contrast, when the temperature is increased quickly (Fig. 4, upper part), the change in mesogel size is very different. Once the mesogel temperature reaches that of the volume phase transition, its surface is deformed by the escaping water, forming some sorts of transient 'bubbles'. At the end of the transition, the mesogels look dark, indicating that their structure is heterogeneous on length scales comparable to or larger than the wavelength of visible light. If the mesogels are kept at a high temperature, their appearance keeps on evolving (from the mesogel surface towards its centre) and finally becomes much lighter, identical to that of the mesogels which have been subjected to a slow increase in temperature (see Fig. 1 and bottom part of Fig. 4), suggesting that rearrangements have occurred and led to a more homogeneous structure. The formation of 'bubbles' upon a quick increase in T has been observed for many types of PNiPAM materials (e.g. mesogels, 51 100 µm − 1.0 mm, ∼ 1.1 mm and ∼ 15 mm, respectively) prepared with a small crosslinker/monomer ratio (2%mol. in our case) and is associated with a three-stage shrinking process, 51,52 which can be seen in the video in Fig. S5 (Multimedia view) in SM and is further discussed in Sec. IV of that same document. Our videos where very fast heating is carried out using a hair-dryer ( Fig. S6 (Multimedia views) in SM) show that the three-step process becomes more and more pronounced asṪ up increases. In the videos, water release at the third stage can be visualised due to both the high speed at which water is expelled out of the mesogels and the difference in refractive index between water and the Ludox suspension. Although the appearance (and hence the microscopic structure) of the mesogels depends on the rate at which temperature is increased, Fig. 5 shows that their final size does not depend on that rate. Indeed, after T ramps up with ratesṪ up varied between 0.02 to 3 • C/min, the mesogel diameter reaches essentially the same value of ∼ 80 µm (solid symbols in Fig. 5), independent ofṪ up , and equal to that obtained after the samples stayed at high T for a couple of hours ('Equilibrated' data in Fig. 5). After the samples have been cooled back to low T (open symbols, see caption of Fig. 5), the mesogels recover their diameter of ∼ 185 µm and ∼ 165 µm in water and in the commercial Ludox suspension, respectively, similar to those they had prior to any T cycle ('Equilibrated' data). These results, together with those presented in Fig. 3, confirm the ability of the mesogels to recover their size at either low or high T , regardless of the imposed thermal history. Our microscopy observations suggest that, for fast enough ramps, the time scale of mesogel volume change is limited by the time required by the solvent to enter or leave the mesogel. By measuring the time evolution of d while imposing upwards or downwards T ramps between the fully swollen and fully shrunk states, we find that forṪ up ≥ 2.7 • C/min (resp., |Ṫ down | ≥ 1.3 • C/min), the final size is reached up to 300 s (resp., up to 130 s) after attaining the target temperature. Note that, in the case of fast shrinking, a longer time scale of the or-der of 1.5 h is needed for the mesogels to recover their transparent appearance. By contrast, for slower upwards or downwards ramps, d smoothly follows the evolution of T , with no further changes once the final T is reached. Now that we have characterized the behavior of the mesogels in Ludox suspensions upon T changes and demonstrated that the stationary state is independent from the T history, we investigate the dynamics of the Ludox nanoparticles in our mesogel-Ludox mixtures with DLS. We use mesogels from another batch than that used for optical microscopy, see Sec. I.B in SM. Note that the dynamics we probe are those of the Ludox suspension rather than those of the mesogels. Indeed, our experiments are carried out at a scattering angle of 90 • , where the mesogels do not scatter significantly due to their large size. Furthermore, our DLS set-up includes an imaging collection optics and a CMOS camera, allowing us to take space-resolved speckle pictures of the scattering volume over time, which are then processed to obtain the intensity auto-correlation (IAC) functions. 54 When selecting the regions of interest (ROIs) to process these images, we make sure that they are free from mesogels at all times. An example of an image collected during DLS measurements showing the scattering volume, the mesogels and ROI selection is available in Fig. S7 in SM. Fig. 6(a) shows the IAC functions collected for our mesogel-Ludox mixture. As T decreases, the IAC curves decay at longer lag times, indicating slower dynamics of the Ludox particles. This behavior is consistent with what is expected from the T dependence of the mesogels diameter: as temperature decreases, the mesogels become bigger and occupy more volume in the sample. Hence, the volume available to Ludox particles is smaller, and their effective volume fraction ϕ increases, leading to slower dynamics 30 . Interestingly, during the DLS measurements at the highest temperature (T = 35.9 • C, above the VPTT), movies of the speckle images showed that PNiPAM mesogels moved throughout the sample (see SM), which we explain by the relatively low background medium viscosity. Indeed, as discussed above, when T increases, ϕ decreases, which leads to a decrease in the viscosity of the Ludox suspension the mesogels are suspended in. Two types of motion were observed: creaming at first (Fig. S8(a) in SM), followed by convection ( Fig. S8(b) in SM). Creaming is due to the density mismatch between the Ludox suspension and the PNiPAM mesogels; the latter having a lower density. Convection is likely to be due to slight local heating of the sample, due to the (small) absorption of laser light by PNi-PAM. We find that convection only sets in after illuminating the sample for extended periods of time (∼ 4 h) at the maximum laser power. To avoid convection, the laser power was decreased from 150 mW to 37.5 mW for all the measurements performed at lower temperatures, starting from T = 28.4 • C, and no creaming nor convection was observed over the time of the measurements. Importantly, even at the highest temperature, we find that mesogel motion is slow enough not to perturb the dynamics of the Ludox nanoparticles (see Fig. S9 and Sec. V.B in SM for details). To quantify the T dependence of the dynamics, we fit the decay of the IAC functions with where A is the amplitude of the relaxation mode, τ c its relaxation time and β the stretching exponent. Fits using Eq. 2 are shown as dashed lines in Fig. 6(a). For IAC data collected at the lowest temperatures (T ≤ 20.7 • C), two relaxation modes were clearly observed, a distinctive feature of the slow dynamics of supercooled systems. 1,3 In this case, we fitted the slowest relaxation mode. The fitting parameters τ c and β are shown in Fig. 6(b). Across the investigated T range, τ c increases as T decreases, spanning more than four decades. Note that, for all T , τ c is significantly larger than 0.18 ms, the relaxation time obtained from Eq. 2 in the infinite dilute limit. To confirm that the wide variation of τ c is due to the variation of the mesogel size, values of τ c measured at different T on a concentrated Ludox suspension without added mesogels are also shown (ϕ = 0.395, open symbols in Fig. 6(b)). As T decreases, τ c increases only slightly, indicating a marginal slowing down of the dynamics when T is lowered. Note that this variation is orders of magnitude smaller than that observed in the Ludox sample containing the mesogels. Indeed, in the case of the pure Ludox sample, τ c is multiplied by ∼ 2.8 when T decreases from 33.2 • C to 19.4 • C, while it is multiplied by more than 800 over the same T range in the case of the mesogels/Ludox mixture. At fixed ϕ, the relaxation time of a colloidal suspension is expected to scale as D −1 0 ∝ η 0 /T , with D 0 the infinite dilution particle diffusion coefficient and η 0 the (T -dependent) solvent viscosity. The dashed-dotted line in Fig. 6(b) shows the variation of η 0 /T over the measured T range: it accounts for most of the observed change of τ c in the Ludox suspension without mesogels, confirming that the dramatic change in relaxation time of the Ludox-mesogels mixtures is due to the volume change of the mesogels upon varying temperature. g 2 (τ) − 1 = A exp[−(τ/τ c ) β ] 2(2) As a further proof that our mesogel-Ludox mixture exhibits a broad range of dynamical behaviors with T , we look into the T -dependence of the stretching exponent β . When T decreases (i.e. ϕ increases), β first decreases, reaching a value as low as ∼ 0.3 at T = 31.4 • C, and then increases again, reaching a value of 1.5 at T = 18.7 • C. These variations in β are similar to those reported by Philippe et al. 30 for a pure Ludox suspension, where the volume fraction was varied by preparing distinct samples. Values of β below 1 -characteristic of a stretched exponential relaxation -are a feature of the intermediate ϕ regime observed in Ref. 30 (termed 'regime II' therein), which corresponds to the supercooled regime and where τ c increases sharply with volume fraction. By contrast, values of β above 1 -characteristic of a compressed exponential relaxation -are a feature of high-ϕ regime ('regime III' in Ref. 30 ), where all the samples are in a glassy state and τ c is nearly independent of the volume fraction. These results fully demonstrate that varying T over a few degrees in a single Ludox-mesogels mixture allows us to access the same states as those obtained for pure Ludox suspensions prepared at different volume fractions. Using τ c = f (ϕ) data collected by Philippe et al. 30 for pure Ludox suspensions and the values of τ c of our mesogel-Ludox sample, we estimate ∆ϕ, the maximum variation in the Ludox volume fraction corresponding to the range of τ c measured when varying T in the Ludox-mesogel mixture of Fig. 6. We find ∆ϕ ≈ 3.5% of the same order of magnitude but somehow larger than ∆ϕ ≈ 1.2%, the value estimated from the volumes and volume fractions of the stock mesogel and the stock Ludox suspensions used to prepare the sample (see details in Sec. VI in SM). Three hypotheses may explain this discrepancy: (i) The estimation of ∆ϕ is based on the assumption that the behavior of our Ludox sample is exactly the same as that studied by Philippe et al., 30 which may not be the case. Indeed, mesogel addition is accompanied by a small dilution of the Ludox sample due to the simultaneous addition of a small amount of water, which may affect electrolyte concentrations and thus electrostatic screening and Ludox surface charge. Furthermore, the experiments presented here were performed using a batch of Ludox particles different from that of Ref. 30 , and commercial suspensions are known to exhibit batch-tobatch differences; (ii) As previously discussed, creaming has occurred in the sample at the highest investigated temperature, which is also the first temperature the sample was subjected to. This resulted in a greater mesogel concentration near the top of the sample, where the scattering volume is located, as compared to the average mesogel concentration used to estimate ∆ϕ; (iii) Various hypotheses and approximations (see Sec. VI in SM for details) are required to estimate both values of ∆ϕ, which probably also contribute to the observed difference. IV. CONCLUSIONS AND PERSPECTIVES We have successfully prepared a single suspension of silica nanoparticles that exhibits a broad range of dynamical behaviors upon varying T , showing characteristics similar to those obtained for a series of distinct Ludox-only samples with ϕ ≈ 0.367 − 0.403. This was achieved by adding a relatively small amount of PNiPAM mesogels (∼ 2 − 5% vol., depending on T ) to a concentrated colloidal suspension. While the same result could likely be achieved by immersing a macroscopic piece of PNiPAM gel in the sample, our ∼ 200 µm mesogels offer several advantages. First, they allow for a faster change of volume of PNiPAM, since the swelling time of a gel in a good solvent is proportional to the square of its linear size. 55 Second, any change in PNiPAM volume creates a local gradient of nanoparticle concentration, which will relax slowly in dense suspensions. Mesogels allow for splitting the overall volume over which gradients occur in many smaller regions, thereby accelerating sample equilibration. Finally, mesogels/Ludox mixtures can easily be transferred in cells of arbitrary size and shape, e.g. thin capillaries used for X-ray scattering, and may be used for rheology experiments, since the mesogel size is smaller than the gap of typical plateplate or couette geometries. The achievable change in volume fraction ∆ϕ is limited essentially by the amount of mesogels that one is willing to add to the sample. Absolute volume fraction changes larger than 20% are in principle possible (see Fig. S9 in the SM). However, this would typically come at the expense of having 10% or more of the total sample volume occupied by mesogels. Another factor limiting ∆ϕ is the extent of the volume variation of the mesogels. We have shown that the swelling of mesogels in Ludox suspensions is reduced as compared to that in water. This is likely due to the electrolytes present in Ludox suspensions, although the osmotic pressure exerted by the colloids themselves may also play a role. Finally, since ϕ is tuned by varying temperature, a potential concern in experiments is the impact of (unwanted) temperature fluctuations. In Sec. VI of the SM, we show that for typical experimental conditions and considering a relatively large temperature fluctuation δ T = 0.1 • C, the resulting change in colloid volume fraction is modest: of order 10 −3 close to the VPTT and significantly smaller at the lower and higher ends of the typical T range. A few points could still be improved to optimize our method, essentially concerning the control of the amount and distribution of mesogels in the sample. As discussed above, a poorly known amount of water is added to the nanoparticle suspension together with the mesogels. As a result, the amount of added mesogels is not known precisely, nor is the actual volume fraction of the nanoparticles. A solution that we are currently investigating consists in adding the mesogels in a freeze-dryed state and let them rehydrate in the final sample. This would have the twofold advantage of knowing precisely the amount of added mesogels (e.g. by weighting and counting them) and avoiding any dilution of the nanoparticle suspension. Another strategy involves labelling the mesogels with a fluorophore, 56,57 allowing for their visualization in the scattering volume. This approach would also address the issue of the uneven spatial distribution of mesogels due to creaming, since their number concentration would be directly determined in the very sample region where the nanoparticle dynamics are probed. The effect of creaming could also be mitigated by using a light scattering cell with a smaller height. Although we have tested the method described here only on Ludox samples, we expect it to apply quite generally to any water-based colloidal suspensions, provided that the physicochemistry of the system does not severely interferes with the swelling/deswelling capability of PNiPAM. The present work, where PNiPAM thermosensitivity was shown to still hold in the presence of electrolytes and at a basic pH ∼ 9, together with previous works, 58,59 suggest that PNiPAM swelling/deswelling is indeed preserved for the solvent conditions encountered in a wide range of suspensions. We believe that the versatility of our method opens several research paths. Among various possibilities, we mention the question of how far a local disturbance of volume fraction (due to swelling/deswelling of a mesogel) has an impact on the dynamics of the surrounding suspension, which would be an experimentally new way to study the role of spatial correlations of the dynamics. 1,60 By coupling rheology and DLS measurements, 16 the method presented here could also allow assessing whether or not rejuvenation upon shear or ϕ jumps are equivalent to each other in concentrated suspensions of soft nanoparticles, when avoiding complications potentially arising from particle interpenetration. Finally, the enhanced ease of handling of nanoparticle suspension at high T , due to the decrease in effective volume fraction, could allow for investigating the behavior of samples comprising particles smaller than those typically used so far. This should provide interesting insights about aging, as colloidal glasses made of smaller particles age quicker than those made of larger particles and should therefore lead to samples closer to equilibrium. 3,61 Additionally, this would allow testing experimentally recent numerical findings that have unveiled intriguing differences in the non-linear mechanical properties of glasses depending on equilibration. 62 SUPPLEMENTARY MATERIAL See supplementary material for details about PNiPAM mesogel production for both optical microscopy and DLS, micrographs of equilibrated PNiPAM mesogels at different temperatures and immersion media, 'bubble' appearance during PNiPAM mesogel shrinkage following a quick increase in T , DLS data analysis (where the selection of regions of interest for data processing is discussed, together with creaming and convection at T = 35.9 • C), and calculation of the effective volume fraction of the Ludox particles in the presence of mesogels. Video files showing showing mesogels suspended in Ludox suspensions undergoing shrinkage forṪ up = 3 • C/min at magnification ×20, and upon heating up with a hair dryer at magnifications ×3.5, ×10 and ×20 are also available, as well as video files showing creaming and convection observed during DLS measurements at T = 35.9 • C. ACKNOWLEDGMENTS This work was financially supported by CNES and L2C. L. Cipelletti gratefully acknowledges support from the Institut Universitaire de France. DOWSIL TM RSN-0749 resin was kindly provided free-of-charge by Univar Solutions SAS. We are grateful to T. Phou and E. Chauveau for their assistance regarding PNiPAM mesogel synthesis, and to C. Blanc for his help with optical microscopy. DATA AVAILABILITY STATEMENT The data that support the findings of this study are openly available in Zenodo at http://doi.org/10.5281/zenodo.5891491. Harvard Apparatus). A 0.45 µm pore size mixed cellulose ester syringe filter was inserted between the aqueous phase syringe and the corresponding entrance of the microfluidic device to prevent clogging. The flow rates on the syringe pumps were set to 10 mL/h (syringe diameter: 19 mm) and 0.3 mL/h (syringe diameter: 7 mm) for the oil and the aqueous phases, respectively. Once the steady-state was reached, the emulsion was collected in a container flushed with argon. Still under argon flush, the cap was removed and replaced by cling film prior to placing the container under a UV lamp (Dual Wave UV Analysis Lamp, 2 x 4 W, 254 nm and 365 nm; ref: H466.1; Herolab) for ∼ 1 h for polymerization of the aqueous drops to take place, yielding mesogels. The cap was put back (still under argon) on the vial and the sample was left to rest overnight. The mesogels were then washed and transfered in water as described in Section II B 1 of the main text. The room air conditioning temperature was set to 20 • C at all times during the synthesis, since the synthesis temperature influences the structure and appearance of PNiPAMbased materials. Previous research has shown that the structure of PNiPAM macrogels synthesized at temperatures below 25 • C is homogeneous on length scales of the order of the light wavelength and the materials appear relatively transparent in water, while the structure of PNiPAM macrogels synthesized at higher temperatures is heterogeneous upon that same scale and the materials are white. [37][38][39] We performed a synthesis at 30 • C to verify that this is also true for mesogels and indeed obtained mesogels that looked white in water at room temperature. Once suspended in Ludox, mesogels synthesized at T = 30 • C remained visible at all temperatures, contrary to the mesogels used in our study (see Figs 1 and 2(b) as well as related discussions in the main text). Finally it is worth mentioning that, assuming a homogeneous distribution of crosslinker within the mesogels, the average distance between two neighboring crosslinkers can be computed and it is equal to about 8 nm. For such a reason we expect a scarce Ludox nanoparticle diffusion, if any, into the mesogels. B. Samples for Dynamic Light Scattering (DLS) The synthesis of mesogels for the DLS study was similar to that described above, but polymerization was performed directly in a home-made microfluidic device similar to that shown in Fig. S1, using a protocol adapted from that of Chen et al. 63 A redox activator -both oil-and water-soluble -was added to the silicone oil, and subsequently diffused into the aqueous phase where it activated the initiator (see next paragraph for the composition of the two phases). The tip of the inner capillary where drops form was 18 µm wide and the collecting tubing was 3.6 m long (i.e. significantly longer than for the synthesis with the photoinitiator described in Section VI A), to allow the aqueous drops to polymerize while travelling in the tubing before reaching the collecting container. The container was left to rest overnight prior to adding 1 mL DI water, and washing the mesogels with diethyl ether, followed by DI water. No surfactant was used and no exposure to UV light was needed. The oil and aqueous phase flow rates were 7 mL/h and 1 mL/h, respectively. Air conditioning was set to 24 • C. For this synthesis, the oil phase was prepared by dispersing 1.5 mL tetramethylethylenediamine (TEMED) in 28.5 mL silicone oil. The aqueous phase was prepared by dispersing 15.5 mg N,N'-methylenebisacrylamide (BIS) crosslinker (0.05 M), 8.6 mg potassium persulfate (KPS) initiator (0.02 M) and 226 mg NiPAM monomers in 2.0 mL DI water. The so-produced mesogels were spherical and looked relatively transparent in water. The mesogel/background medium interface was slightly less well defined than that of the mesogels prepared with the photoinitiated synthesis, which we attribute to the absence of surfactant. In water, the so-produced PNiPAM mesogels had a diameter of 221 µm (SD: 11 µm) at room temperature, and in commercial Ludox (ϕ = 0.350), the diameters were 209 µm (SD: 13 µm) and 149 µm (SD: 10 µm) at room and high temperatures, respectively. The fact that the ratio between the mesogel diameter at high and low temperature, d(T high )/d(T low ), is higher in the present case as compared to that obtained for the mesogels prepared with the photoinitiated synthesis is due to the difference in the crosslinker/monomer ratios (5 mol% and 2 mol%, respectively), consistent with literature data. 45 VII. MICROGRAPHS OF EQUILIBRATED PNIPAM MESOGELS AT DIFFERENT TEMPERATURES AND IMMERSION MEDIA Micrographs of equilibrated PNiPAM mesogels synthesized with the protocol of Section VI A, taken at various temperatures and for different immersion media are shown in Figs. S2-S4. These images complement Fig. 1 of the main text. VIII. VOLUME FRACTION x OF NIPAM MONOMERS IN A MESOGEL SPHERE To calculate the volume fraction x of NiPAM monomers in a mesogel sphere introduced in Eq. 1 of the main text, we use the following expression: x(T ) = m NiPAM ρ NiPAM π 6 π[d(T )] 3 . (S1) In Eq. S1, the numerator corresponds to the volume occupied by NiPAM monomers in a mesogel. It is obtained from the mass density of NiPAM, 64 ρ NiPAM = 1.1 g.cm −3 , taken to be constant across the investigated temperature range, and the mass of NiPAM initially contained in each drop produced by the microfluidic device, m NiPAM = c NiPAM V drop , with c NiPAM the concentration of NiPAM monomers in the aqueous solution injected in the microfluidic device and V drop the drop volume. The denominator corresponds to the volume of a mesogel at temperature T , with d(T ) the diameter of the mesogel at that temperature (see Fig. 2(a) of the main paper). IX. 'BUBBLE' APPEARANCE DURING PNIPAM MESOGEL SHRINKAGE FOLLOWING A QUICK INCREASE IN T When the variation in T is quick, the shrinking process takes place in 3 steps. 51,52 (i) The quick initial shrinking leads to the formation of a dense skin layer at the surface of the mesogels. (ii) The skin layer is so dense that water can temporarily not diffuse out of the mesogels, causing their size to plateau. Meanwhile, the pressure inside the mesogels increases. (iii) When the inner pressure becomes high enough to overcome the strength of the skin layer, some areas of the skin layer are blown up like the surface of a balloon, forming bubble-like structures, and allowing water to be expelled from the mesogels as the expanded skin layer is no longer impermeable. Mesogel size further decreases. Note that, although commonly used in the literature by convenience, 45,51-53 the word 'bubbles' is a somehow abusive as those actually correspond to pockets of water. The 3-stage shrinking process described above can be seen in our videos of PNiPAM mesogels shrinking in a Ludox commercial suspension. FigS6left_Mesogel_shrinking_Hair_dryer_x20.mp4'), water release during stage (i) can also be observed. It appears as a thin light, even layer around the mesogel ∼ 6 s after the beginning of the movie and is obvious after ∼ 9 s. X. DYNAMIC LIGHT SCATTERING (DLS) DATA ANALYSIS A. Selection of Regions of Interest (ROIs) for data processing Fig. S7 shows a representative DLS image collected by the CMOS camera of our home-built set-up performing space-resolved DLS measurements. The rod-shaped region at the top corresponds to the scattering volume: it is bright because the laser beam illuminates the Ludox suspension, which scatters light collected by the camera objective lens. Some mesogels out of the scattering volume are also visible (highlighted by the orange circles in Fig. S7(b)). Although they are not directly illuminated by the laser beam, these mesogels are visible because of scattering from the Ludox suspension. Indeed, the Ludox particles scatter light in all directions. Part of this light illuminates the mesogels located out of the scattering volume, which in turn scatter light that is collected by the set-up. Mesogels superposed to the scattering volume region appear as darker zones, highlighted by the green circles in Fig. S7(b). They could be located either in between the scattering volume and the collection optics (partially blocking the light scattered by the Ludox suspension), or in the scattering volume (if the mesogels scatter less efficiently than Ludox at the detected scattering angle of 90 • ). To ensure we only probe light scattered by the Ludox suspension, we carefully checked all the images and processed selected areas of the scattering volume that were free of mesogels at all times during the measurements. Two such ROIs are shown as red rectangles in Fig. S7(b). B. Creaming and convection at T = 35.9 • C As mentioned when discussing Fig. 6 in Section III of the main text, at T = 35.9 • C mesogels moved in the sample. They initially underwent creaming, followed by convection after ∼ 4 h as shown in the movies 'NoFig_creaming.avi' and 'NoFig_convection.avi' (available as independent supplementary material files). Videos were recorded at 1 fps and are played at 10 fps. The field of view is the same as that in Fig. S7. Although we took care to process ROIs that were mesogels-free at all times, one may wonder if the motion of nearby mesogels due to creaming or convection may accelerate the Ludox dynamics, by providing some sort of 'stirring' mechanism. To address this question, intensity autocorrelation functions (IACs) were calculated during both creaming and convection. Since convection involved mesogel motion at speeds much higher than for creaming (typically, 1.0 µm/s vs 0.18 µm/s), one would expect faster relaxation times during convection, had the mesogel motion had an impact on the Ludox dynamics. Fig. S8 shows that the Ludox dynamics is essentially the same, independently of the kind of mesogel motion. Furthermore, for the creaming phase, we also processed the data using the 'drift correction' algorithm described in Ref. 65 , which corrects the IACs for any spurious contribution due to a drift of the speckle pattern. We find very little difference between data with and without correction (compare the open and solid green triangles in Fig. S8), ruling out any significant contribution of the mesogel motion to the measured Ludox dynamics. Finally, we recall that no mesogel motion was observed in all the other measurements shown in Fig. 6 of the main text. In this section we shall derive expressions for the Tdependent volume fraction of mesogels and Ludox particles, as a function of the composition of samples prepared by mixing Ludox-alone and mesogel-alone stock suspensions. We start by calculating the effective volume fraction of the Ludox particles at a reference temperature, T re f , defined as the temperature at which the two stock suspensions are mixed. To avoid any confusion, we use the subscripts 'L' and 'mgel' to designate the Ludox particles and the mesogels, respectively. Thus, ϕ L in this section correspond to ϕ in the main text; it is defined as the volume occupied by the particles divided by the sample volume effectively available to them, i.e. the total volume excepted that occupied by the mesogels: ϕ L (T re f ) = V L ϕ (0) L V L +V mgel 1 − ϕ (0) mgel (T re f ) d(T re f ) d (0) (T re f ) 3 , (S2) where V L and V mgel are the volumes of, respectively, the Ludox-alone and mesogel-alone stock suspensions mixed to obtain the final sample, and ϕ mgel are the corresponding volume fractions of Ludox and microgels in the two suspensions before mixing. The numerator of Eq. S2 represents the volume of the Ludox particles in the final sample and the denominator the sample volume accessible to them. d and d (0) are the diameters of the mesogels when suspended in the final sample and in the medium of the mesogels-alone suspension (usually, water), respectively. Their ratio appears in the denominator of Eq. S2 as a correcting factor for ϕ (0) mgel , because the size of the mesogels is somehow smaller in a water-Ludox background as compared to pure water, as discussed in the main text. The ratio is raised to the third power because the volume occupied the mesogels scales as d 3 . By introducing the ratio χ = V mgel /V L of the volumes of the two initial suspensions and by taking into account the variation of d (in the final suspension) with T , one derives the following expression for the Ludox effective volume fraction at any temperature: ϕ L (T ) = ϕ (0) L 1 + χ 1 − ϕ (0) mgel (T re f ) d(T ) d (0) (T re f ) 3 . (S3) For completeness, we also report the volume fraction ϕ mgel of the mesogels in the final sample, calculated with respect to the total volume sample (i.e including the volume of the mesogels themselves): ϕ mgel (T ) = ϕ (0) mgel (T re f ) d(T ) d (0) (T re f ) 3 1 + 1/χ .(S4) In practice, it is desirable to prepare samples that cover a broad range of dynamical behaviors upon changing T , while keeping the mesogel content as low as possible. To strike the right balance between these two conflicting requirements, we use Eq. S3 to determine the maximum achievable variation of Ludox volume fraction, defined as ∆ϕ = ϕ L (T low )−ϕ L (T high ), where T low and T high are the temperatures where the mesogel reach their largest and smallest size, respectively. We calculate ∆ϕ for a wide range of χ values, and for several values of d (0) (T ref ), to study the influence of the temperature at which the supernatant is removed from the mesogel suspension after it has been centrifuged and prior to adding the Ludox suspension. Assuming that the mesogels in that suspension are randomly packed spheres, we use ϕ (0) mgel = 0.64 in Eq. S3. The results of these calculations are shown in Fig. S9(a) for the mesogels synthesized according to the protocol of Section VI A and fully characterized with optical microscopy (see main text), using ϕ (0) L = 0.412. As one may expect, at a given preparation temperature (i.e. d (0) fixed), ∆ϕ increases when χ increases, which corresponds to the case where the number of added mesogels is increased while the volume of the Ludox suspension is kept constant or increased in lower proportion. At a given χ value, the smaller the mesogels in water (i.e. the higher the preparation temperature T ref ), the higher ∆ϕ, because more mesogels are contained in a set volume of the mesogel stock suspension if the mesogels have a smaller size (recall that we assume a constant ϕ (0) mgel = 0.64). For the same set of d (0) values, we plot in Fig. S9(b) the fraction ϕ mgel of the total sample volume occupied by the mesogels, calculated with Eq. S4 at both T low (solid lines) and T high (dotted lines). The curves shown in both panels of Fig. S9 provide guidelines for sample preparation. Importantly, they demonstrate that changes of a few % of ϕ L (sufficient to vary the microscopic dynamics by several orders of magnitude for samples in the supercooled regime) are achievable even with less than 10% of mesogels by volume. As a final remark, we note that in experiments T may slightly fluctuate. We use Eq. S3 to estimate the change δ ϕ L of Ludox volume fraction due to a small temperature fluctuation δ T , finding δ ϕ L (T ) ≈ ∂ ϕ L ∂ T δ T = = ϕ 2 L δ T ϕ (0) L χϕ (0) mgel (T re f ) 3d 2 (T ) d (0) 3 (T re f ) ∂ d ∂ T . (S5) We evaluate δ ϕ L for a temperature fluctuation δ T = 0.1 • C by inserting in Eq. S5 the same set of parameters as for FIG. 2 . 2Measured size (a) and estimated refractive index n mgel (b) of thermally-equilibrated PNiPAM mesogels as a function of the surrounding medium temperature in pure water or Ludox suspensions with ϕ = 0.350 and ϕ = 0.396. Error bars in (a) represent the standard deviation (SD). In (b), the values of n mgel (symbols) were estimated using Eq. 1. The dashed lines show the refractive index of the background Ludox suspensions. gels in Ludox suspensions around 31 • C shown in Fig. 1, their refractive index was estimated as a function of temperature (T ) using FIG. 3 . 3Appearance and size of PNiPAM mesogels upon successive identical temperature cycles when suspended in water (a) or in a commercial Ludox suspension (ϕ = 0.350) (b).Ṫ up ranges from 2.3 to 2.8 • C/min, andṪ down from −0.7 to −0.5 • C/min depending on the cycle. Error bars represent the SD. t = 0 corresponds to the time at which micrographs were taken to assess mesogel size at equilibrium prior to T cycles. Micrographs of a given mesogel are provided for each data point. Micrographs were all collected at magnification ×20 and the scale bar (100 µm) applies to all the micrographs. Slow T ramp (Ṫ up = 0.02°C/min) FIG. 4. Micrographs of two mesogels suspended in a commercial Ludox suspension (ϕ = 0.350) undergoing a fast T ramp up (top part; T up = 2.8 • C/min) and a slow T ramp up (bottom part;Ṫ up = 0.02 • C/min). Micrographs were collected at magnification ×3.5. The scale bar applies to all of them. Fig. S8 (FIG. 5 . S85Multimedia Mesogel size in water and in a commercial Ludox suspension (ϕ = 0.350) after T ramps up (full symbols; high T , corresponding to T = 35.4 ± 0.2 • C and T = 33.0 ± 0.4 • C for water and the Ludox suspension as background fluids, respectively) and down (empty symbols; low T , corresponding to T = 23.8 ± 0.3 • C and T = 23.6 ± 0.2 • C for water and the Ludox suspension as background fluids, respectively). Data are plotted as a function of the rate at which the temperature is varied during the ramp up,Ṫ up . Ramps down were all performed atṪ down ≈ −0.5 • C/min. Equilibration times prior to data collection for mesogel sizing were ∼ 15 min and ∼ 10 min at high and low T , respectively, except for the 'Equilibrated' data. The latter, on the left of the axis break, correspond to data collected prior to any temperature cycle (empty symbols; low T ) or after a 215 min T plateau (full symbol; high T ). The set of points at the highestṪ also includes the data ofFig. 3, and error bars represent SD. FIG. 6 . 6DLS data collected accross a wide range of temperatures on a concentrated Ludox suspension containing PNiPAM mesogels. Details about the mesogels used for this series of experiments are provided in the Sec. I.B of the Supplementary Material. (a) Intensity auto-correlation (IAC) functions (symbols) and their fits (lines), obtained using Eq. 2. IACs functions were normalized to the smallest available delay time to bring their intercepts to unity. (b) Fitting parameters τ c (full triangles) and β (crosses) as a function of T for the studied mesogels/Ludox mixtures. The color code is the same as in panel (a). The relaxation time τ c of a Ludox sample with no added mesogels (ϕ = 0.395) measured at three different temperatures is also shown (empty triangles), together with the expected scaling with T due to changes in thermal energy and solvent viscosity (dashed-dotted line). containing N-isopropylacrylamide (Ni-PAM) monomers, a crosslinker and a photoinitiator were produced in silicone oil and stabilized with a surfactant. The drops were then polymerized under UV-light to form PNi-PAM mesogels. To prepare the oil phase, 0.742 g silicone surfactant DOWSIL TM RSN-0749 resin (3 wt%) was added to 24 g silicone oil 47 V 100. The oil phase was degassed with argon for ∼ 4 h. To prepare the aqueous solution, 6.2 mg N,N'-methylenebisacrylamide (BIS) crosslinker (0.02 M) and 226 mg NiPAM monomers (1 M) were weighted and transferred in a brown glass vial (to avoid light-induced polymerization), prior to adding 6.2 µL 2hydroxy-2-methylpropiophenone photoniator (0.3 wt%) and 2.0 mL deionized (DI) water. Argon was bubbled in the solution for ∼ 15 min just before starting the synthesis.The microfluidic setup used to produce the water-insilicone oil emulsion is sketched inFig. S1.A 20 mL plastic syringe (Omnifix ® Luer-Lock 20 mL; ref: T550.1; B. Braun) and a 2 mL plastic syringe (Omnifix ® Luer-Lock 2 mL; ref: LY21.1 B. Braun) were loaded with the oil and the aqueous phases, respectively, connected to a home-built microfluid setup (see schematic at the bottom of Fig. S1) and placed on two distinct syringe pumps (PHD2000 Infusion; ref: 70-2000; Fig. S5 ( S5Multimedia view) shows a frame taken from movie 'FigS5_Mesogel_shrinking_3Cpmin_x20.mp4', available as an independent supplementary material file, where the mesogel is heated up atṪ up ≈ 3 • C/min, i.e. the same rate as that used in Figs 3, 4 (top part) and 5 (rightmost data points) of the main text. At stage (iii), the mesogel outer layer is subject to small local deformations and a few water pockets appear, grow and disappear over time. When the T ramp is an order of magnitude faster (Ṫ up ≈ 35 • C/min), the local deformations are significantly more pronounced, both in amplitude and in number, as seen inFig. S6 (Multimedia views), which shows mesogels shrinking in a commercial Ludox suspension recorded at various magnifications. In the associated movies (i.e. 'FigS6left_Mesogel_shrinking_Hair_dryer_x20.mp4', 'FigS6right_Mesogel_shrinking_Hair_dryer_x10.mp4' and 'FigS6bottom_Mesogel_shrinking_Hair_dryer_x3.5.mp4', all available as independent supplementary material files), water escaping the mesogels can be visualized due to both the high speed of the flow and the difference between the water refractive index and that of the Ludox suspension the mesogels are suspended in. In the movie corresponding to the top-right panel ofFig. S6(' Fig. S9, together with typical values χ = 0.05, ϕ L = 0.38 and d (0) = 90 µm. The largest fluctuation is found at intermediate temperatures, where the microgel size changes steeply with T : for T = 31.3 • C, δ ϕ L = 1.4 × 10 −3 , a rather small fluctuation. Far from the Volume Phase Transition Temperature, the impact of T fluctuations is even lower: for T = 24.5 • C, δ ϕ L = 4.5 × 10 −4 , while at high temperature, T = 35.3 • C, δ ϕ L is as low as 4.8 × 10 −7 . FIG. S1 . S1Schematic of the microfluidic setup for drop production. The bottom frame shows details about the home-made device where the drops are formed (dashed box in the upper part of the figure). 1: Stretched glass capillary with a tip diameter of 22 µm (unstretched capillary ref: cf 5 hereafter); 2: Barbed connector, I.D: 3/32", Harvard Apparatus (ref: 72-1426); 3: Glass capillary, O.D: 1.5 − 1.6 mm, I.D: 1.1 − 1.2 mm, Labbox (ref: MPC3-090-500); 4: T-connector, I.D: 1/8", Harvard Apparatus (ref: 72-9282); 5: Glass capillary, O.D: 1.0 mm, I.D: 0.5 mm, l: 10 cm, World Precision Instruments (ref: TW100-4). FIG. S4 . S4Micrographs of an equilibrated PNiPAM mesogel dispersed in a concentrated Ludox suspension (ϕ = 0.396) taken at different temperatures. Images were collected at magnification ×20. The scale bar applies to all of them. FIG. S5. Frame taken from movie 'FigS5_Mesogel_shrinking_3Cpmin_x20.mp4' -available as an independent supplementary material file -showing the shrinkage of a PNiPAM mesogel suspended in a commercial Ludox suspension (ϕ = 0.350) and heated atṪ up = 3 • C/min. Magnification: ×20. Field of view dimensions: 234 µm × 244 µm. Exposure time: 20 ms. Video recorded with a high-speed camera at 24 fps and played at recording speed. Multimedia view. FIG. S7. Images of the light scattered at 90 degrees by a Ludox-mesogels mixture at 39.5 • C. The bright rod-shaped region near the top of the images corresponds to the scattering volume, i.e. the sample region directly illuminated by the laser beam, whose thickness is ∼ 245 µm. (a) Raw image after the contrast has been enhanced. (b) Same image with annotations. The dashed circles are around PNiPAM mesogels. The mesogels appear darker than the background when they are either in the scattering volume or between the scattering volume and the camera (green circles), while they appear bright when located elsewhere (orange circles), see discussion in the text. Examples of two Regions of Interests (ROIs) selected to calculate the intensity autocorrelation function are shown as red rectangles.FIG. S8. Intensity autocorrelation functions collected at 35.9 • C, where mesogels exhibited creaming or convection. Data collected during creaming -without and with drift correction, open and solid green triangles -and convection, red circles, exhibit essentially the same behavior in spite of the difference of mesogel velocity for the two kinds of motion, supporting the argument that the Ludox dynamics are not affected by mesogel motion. FIG. S9. (a): Maximum achievable variation of the Ludox volume fraction, ∆ϕ, and (b): the fraction of the total sample volume occupied by the mesogels, ϕ mgel , as a function of the ratio χ between the volumes of the two stock suspensions used to prepare Ludox-mesogel mixtures. Curves are obtained from Eqs. S3, S4 for (a) and (b), respectively, using the parameters for the mesogels fully characterized by optical microscopy: ϕ(0) mgel = 0.64 and ϕ (0) L = 0.412. The color code is identical for both (a) and (b) and refers to the diameter d (0) of the mesogels at the temperature at which the two stock suspensions are mixed together, as shown by the labels in (a). TM-50 colloidal silica (50 wt% suspension in H 2 O; ref: 420778-1L) were acquired from Sigma-Aldrich. Silicone surfactant DOWSIL TM RSN-0749 resin (∼ 50% cyclopentasiloxane, ∼ 50% trimethylsiloxysilicate; ref: 4119565) was provided by Dow. Silicone oil 47 V 100 (100 cSt; ref: 84542.290) and isopropanol (≥ 99.7%; ref: 20842.298) were bought from VWR. Diethyl ether (≥ 99.5%; ref: D/2450/17) was acquired from Fischer Scientific. 'Ultrapure' water type I was obtained from a Milli-Q ® Synergy ® -R ultrapure water station (Merck Millipore), and is next called deionized (DI) water.A. Materials N-isopropylacrylamide (NiPAM) monomers (≥ 99%; ref: 731129-25G), N,N'-methylenebisacrylamide (BIS) crosslinker (≥ 99.5%; ref: M7279-25G), 2-hydroxy-2- methylpropiophenone photoniator (97%; ref: 405655-50ML) and LUDOX ® B. Methods the influence of temperature and the immersion medium on PNiPAM mesogels size and ap-Concentrated Ludox® suspension ( = 0.396) 25.5°C 31.5°C 33°C 34.5°C Commercial Ludox® suspension ( = 0.350) 25.4°C 30.8°C 32.8°C 34.3°C Water ( = 0) 25.5°C 30.9°C 33°C 34.3°C FIG. 1. Micrographs of equilibrated PNIPAM mesogels at different temperatures and immersion media (i.e. pure water or Ludox sus- pensions with ϕ = 0.350 and ϕ = 0.396, for the top, mid and bottom row, respectively). Images were collected at magnification ×20. The scale bar applies to all of them. TABLE I . IVolume Phase Transition Temperature (VPTT) and temperature at which near index matching occurs for equilibrated PNi-PAM mesogels dispersed in different media.Background medium VPTT (°C) T index matching (°C) Water 33.6 ± 0.1 - Commercial Ludox (ϕ = 0.350) 31.4± 0.1 31.1 ± 0.2 Concentrated Ludox (ϕ = 0.396) 31.3± 0.1 31.3 ± 0.2 ,52 cylinders 45 and disks 53 with diameters of92 s 186 s 228 s 250 s 300 s 318 s 360 s 1112 s 24.9°C 29.5°C 31.2°C 31.7°C 32.3°C 32.6°C 32.8°C 32.8°C t < t 0 9261 s 13987 s 15377 s 17045 s 17879 s 18435 s 33891 s XI. CALCULATION OF ϕ L , THE EFFECTIVE VOLUMEFRACTION OF THE LUDOX PARTICLES IN THE PRESENCE OF MESOGELS DATA AVAILABILITY STATEMENTThe data that support the findings of this study are openly available in Zenodo at http://doi.org/10.5281/zenodo.5891491. aq. sol.Filtered Oil Aq. drops in oil 2 1 4 3 5 Drop production device Filter Syringe pump for aq. sol. Syringe pump for oil Container to collect the emulsion Argon 30 cm 0.56 mm PTFE tubing 0.5 mm 22 µm FIG. S2. Micrographs of an equilibrated PNiPAM mesogel dispersed in water taken at different temperatures. Images were collected at magnification ×20. The scale bar applies to all of them. FIG. S3. Micrographs of an equilibrated PNiPAM mesogel dispersed in a commercial Ludox suspension (ϕ = 0.350) taken at different temperatures. Images were collected at magnification ×20. 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Caroff, "Simultaneous measurement of the microscopic dynamics and the mesoscopic displace- ment field in soft systems by speckle imaging," Optics Express 21, 22353- 22366 (2013). On the Structure of Poly(N-isopropylacrylamide) Microgel Particles. B R Saunders, 10.1021/la036390vLangmuir. 20B. R. Saunders, "On the Structure of Poly(N-isopropylacrylamide) Micro- gel Particles," Langmuir 20, 3925-3932 (2004). Top left: magnification ×20, field of view 277 µm × 238 µm, exposure time: 2 ms; video 'FigS6left_Mesogel_shrinking_Hair_dryer_x20.mp4' recorded at 10 fps and played at recording speed. Top right: magnification ×10, field of view 711 µm × 533 µm, exposure time: 2 ms, video 'FigS6right_Mesogel_shrinking_Hair_dryer_x10.mp4' recorded at 10 fps and played at recording speed. Bottom: magnification ×3.5, field of view 2133 µm × 917 µm, exposure time: 1 ms, video 'FigS6bottom_Mesogel_shrinking_Hair_dryer_x3.5.mp4' recorded at 10 fps and played at recording speed. Fig, S6, Shrinking of PNiPAM mesogels suspended in a commercial Ludox suspension (ϕ = 0.350) upon heating with a hair-dryer (Ṫ up ≈ 35 • C/min). Videos are available as independent supplementary material files. Multimedia viewsFIG. S6. Shrinking of PNiPAM mesogels suspended in a commercial Ludox suspension (ϕ = 0.350) upon heating with a hair-dryer (Ṫ up ≈ 35 • C/min). Top left: magnification ×20, field of view 277 µm × 238 µm, exposure time: 2 ms; video 'FigS6left_Mesogel_shrinking_Hair_dryer_x20.mp4' recorded at 10 fps and played at recording speed. Top right: magnification ×10, field of view 711 µm × 533 µm, exposure time: 2 ms, video 'FigS6right_Mesogel_shrinking_Hair_dryer_x10.mp4' recorded at 10 fps and played at recording speed. Bottom: magnification ×3.5, field of view 2133 µm × 917 µm, exposure time: 1 ms, video 'FigS6bottom_Mesogel_shrinking_Hair_dryer_x3.5.mp4' recorded at 10 fps and played at recording speed. Videos are available as inde- pendent supplementary material files. Multimedia views.

Due to the global pandemic, in March 2020 we in academia and industry were abruptly forced into working from home. Yet teaching never stopped, and neither did developing software, fixing software, and expanding into new markets. Demands for flexible ways of working, responding to new requirements, have never been so high. How did we manage to continue working, when we had to suddenly switch all communication to online and virtual forms of contact? In this short paper we describe how Ocuco Ltd., a medium-sized organization headquartered in Ireland, managed our software development teams -distributed throughout Ireland, Europe, Asia and America during the COVID-19 pandemic. We describe how we expanded, kept our customers happy, and our teams motivated. We made changes, some large, such as providing emergency financial support; others small, like implementing regular online social pizza evenings. Technology and process changes were minor, an advantage of working in globally distributed teams since 2016, when development activities were coordinated according to the Scaled Agile Framework (SAFe). The results of implementing the changes were satisfying; productivity went up, we gained new customers, and preliminary results from our wellness survey indicate that everyone feels extremely well-supported by management to achieve their goals. However, the anonymised survey responses did show some developers' anxiety levels were slightly raised, and many are working longer hours. Administering this survey is very beneficial, as now we know, so we can act.CCS CONCEPTS• Software and its engineering → Agile software development; Programming teams; • Social and professional topics → Employment issues.

10.1109/ser-ip52554.2021.00010

2103.17181

08d6f2bed6fb4a9cc6ae013e39d474f1928592b6

Globally Distributed Development during COVID-19 Clodagh NicCanna Mohammad Abdur Razzak Ocuco Ltd Dublin Ireland John Noll [email protected] Ocuco Ltd Dublin Ireland Sarah Beecham [email protected]@the Irish Software Research Centre Limerick University of Hertfordshire Hatfield HertsUK, Ireland Globally Distributed Development during COVID-19 Clodagh NicCanna 10.1145/nnnnnnn.nnnnnnnGlobally Distributed Development during COVID-19 SER&IP, June, 2021, ACM ISBN 978-x-xxxx-xxxx-x/YY/MM. . . $15.00 https://doi.org/10.1145/nnnnnnn.nnnnnnn KEYWORDS Global Software Development, virtual teams, wellness, remote work-ing practices, home-working, working-from-home, WFH, pandemic, Covid-19, change management, Scaled Agile Framework ACM Reference Format: Clodagh NicCanna, Mohammad Abdur Razzak, John Noll, and Sarah Beecham. 2021. Globally Distributed Development during COVID-19. In Proceedings of 8th International Virtual Workshop on Software Engineering Research and Industrial Practice (SER&IP). ACM, New York, NY, USA, 8 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn Due to the global pandemic, in March 2020 we in academia and industry were abruptly forced into working from home. Yet teaching never stopped, and neither did developing software, fixing software, and expanding into new markets. Demands for flexible ways of working, responding to new requirements, have never been so high. How did we manage to continue working, when we had to suddenly switch all communication to online and virtual forms of contact? In this short paper we describe how Ocuco Ltd., a medium-sized organization headquartered in Ireland, managed our software development teams -distributed throughout Ireland, Europe, Asia and America during the COVID-19 pandemic. We describe how we expanded, kept our customers happy, and our teams motivated. We made changes, some large, such as providing emergency financial support; others small, like implementing regular online social pizza evenings. Technology and process changes were minor, an advantage of working in globally distributed teams since 2016, when development activities were coordinated according to the Scaled Agile Framework (SAFe). The results of implementing the changes were satisfying; productivity went up, we gained new customers, and preliminary results from our wellness survey indicate that everyone feels extremely well-supported by management to achieve their goals. However, the anonymised survey responses did show some developers' anxiety levels were slightly raised, and many are working longer hours. Administering this survey is very beneficial, as now we know, so we can act.CCS CONCEPTS• Software and its engineering → Agile software development; Programming teams; • Social and professional topics → Employment issues. INTRODUCTION The Coronavirus pandemic global crisis has brought academics, scientists, and practitioners together as we all fight a common cause for our very survival. A key mitigation strategy to stem the viral transmission, while balancing health and economic factors, is to work from home where possible. Software development, especially distributed software development where teams collaborate across multiple sites, would seem particularly suited to this transition. However, even teams familiar with remote forms or work, would traditionally meet other members of their teams, face to face, at key points in the project, in which they would plan for the future, learn together, socialize, and thereby gain trust [9]. Lack of social contact is shown to create many additional problems, associated especially with well-being [10]. As Europe enters a second and arguably more intense wave of COVID-19 transmissions, in this paper we reflect on how software practitioners working-from-home are faring. As a medium-sized enterprise, head-quartered in Ireland, developing software in distributed teams throughout the globe (see Figure 1), we share some of our COVID-19 strategies. We recognized early that changes to our social and technical practices must be made; to ensure the wellness of all our employees, acting fast is key [8]. We heed the warning expressed by a seasoned developer, who, despite having worked remotely for many years, after only one month of quarantine noted "I'm feeling a tinge of burn out for the first time in my life" [10]. While our experience of global software development does help, in that we have both infrastructure and process for remote working in place, this in itself does not guarantee a smooth transition to working-from-home full time. Home working presents a very different rhythm and structure to the working week, and there can be additional pressures of friends and family needing support, dependents becoming ill, and the worry of reduced income. In this paper, we share our experience with the transition to working-from-home, and address the research question, "what changes does a global software development organization need to make for the wellness of their employees during a pandemic?" This paper is organized as follows: in the next section we present a brief background into transitioning from the office to working-fromhome in a software engineering context. In Section 3, we outline our method to include our company setting, followed by our results in Section 4. We conclude the paper with a discussion and a distilled set of recommendations in Section 5, and final remarks in Section 6. BACKGROUND To provide a context for this paper, we look to the literature on developing software during the COVID-19 pandemic, and how those working remotely are managing to retain a level of physical and mental health. Several surveys have been conducted to assess the impact of COVID-19 pandemic on software developers, ranging from measuring practitioner wellness [10], productivity [1], and job satisfaction and work-life balance [2]. Some shift the focus from problem to solution, proposing mitigation strategies [8]; others [3], recognize the dichotomy where developers prefer, and at the same time dislike, some aspects of working from home. Ralph et al's [10] timely and large global study on the impact of COVID-19 on the wellness of developers (with over 2,200 responses across 53 countries), identified that poor disaster preparedness, fear relating to the pandemic, and not having the right set up at home (home office ergonomics), all adversely affect well-being. They were also, through statistical tests, able to show a close relationship between wellness and productivity. Other areas of concern are indications that "women, parents and people with disabilities may be disproportionately affected." They conclude that support needs to be tailored to the needs of individuals. Ford et al [3] noted the dichotomy of the same factors having both a negative and positive effect. For example, people missed social interactions, found it hard to create a clear boundary between home life and work, suffered from poor ergonomics, had less visibility and awareness of how other people are working, and exercised less. Communication was also an issue, and some parent employees suffered from a lack of childcare. On the other hand, the same participants see benefits to working from home, such as, no commute and associated reduction in expense, flexible hours, being close to family, comfort at home, health, and more time. This dichotomy was also observed by Boa et al [1], "Many . . . agreed that [working-from-home ] can have both positive and negative impacts on developer productivity." A big question that will guide the future of working-from-home post pandemic is whether productivity is impacted. Boa et al's [1] study on the productivity before and during COVID-19 found very little difference (using a selection of productivity measures on the output of 139 developers). The literature presents mixed messages here, as these neutral findings contrast with those of Ralph et al [10] who suggest that conditions imposed by the pandemic and working-from-home had a negative impact on productivity. The other extreme comes from pre-pandemic studies on teleworking, and home working, where several studies revealed productivity improvements when working-from-home [1]. Moving to the open-source community, a special report published by the GitHub Data Science team [4] highlights trends and insights into developer activity on GitHub at the start of COVID-19. The research team investigated how a sudden shift to working from home affected developers according to three themes: productivity and activity, work cadence, and collaboration. Their findings suggest that developers have continued to contribute and show resilience in the face of uncertainty. Developers are working longer, by up to an hour a day. "The cadence of work has changed. [4, Key Findings]" The researchers suggest that these longer workdays (when working from home) may be due to non-work interruptions such as childcare, that cannot be ignored when working from home. The team warn that "patterns of developer activity have implications for burnout" [4]. Transitioning to new work routines can lead to developers spending more time online, and completing tasks on time might be taking away from "personal time and breaks to replenish, ponder, and maintain healthy separation. [4, Key Findings]" They query whether this is sustainable. On a positive side, developers are collaborating more, with many open-source projects seeing a spike in activity. Key cross cutting themes from the research on development during the COVID-19 pandemic are: maintaining a work-life balance with boundaries to avoid burnout, setting up home office environment, childcare, women being disadvantaged due to care responsibilities, productivity changes, lack of awareness of others' work, communication, and exercise. The next section looks at how we in Ocuco Ltd. responded and implemented changes to support our employees during the pandemic. METHOD To answer our question, "what changes does a global software development organization need to make for the wellness of their employees during a pandemic?" we looked at wellness according to three key concepts: People, Technology, and Process. Fig. 1 shows the timeline of Ocuco Ltd. interventions in response to the emergence of COVID-19 in Ireland. Ocuco Ltd. Setting Our company, Ocuco Ltd., is a medium-sized Irish software company that develops practice and laboratory management software for the optical industry. We employ more than 300 staff members in our software development organization (including support and management personnel). Of these, a growing team of 75 developers and 40 operations engineers work from Ocuco Ltd.'s Dublin Headquarters, working on software development projects across twelve countries. Fig. 2 shows the distribution of countries and roles. Problem identification, data collection and analysis We conducted a series of one-to-one meetings in March 2020. The first set of meetings focused on checking staff welfare and needs in transitioning to working-from-home; the second set of meetings were held straight after the announcement of pay cuts, in which financial assistance was offered where necessary. A "work location" survey was administered to 88 employees in July 2020 (to include developer, product owner, QA and project manager roles) based across all geographic locations. With many members of Ocuco Ltd. already working remotely, we wanted a picture of changes imposed by the working-from-home regime, and future wishes. The survey asked participants to select from one of seven options relating to working-from-home and/or working from the office. The data analysis involved identifying individual needs (from the one to one meetings), and aggregating the responses from future work location survey. Interventions based on this analysis were implemented with immediate effect in March, 2020 as described in Section 4. Evaluation of impact of changes made during the pandemic In order to check management perception directly as to how well the interventions were working, we administered an evaluation survey on 15th January 2021 to a stratified sample, to assess People, Technology, Process, and Wellness factors (see Appendix A). The Because some of these roles have only one team member in the sample, to preserve anonymity the survey does not ask respondents to identify their roles; nevertheless, we can deduce that at least four of the seven roles are represented in our results (Section 4). RESULTS This section presents results derived from our data collection, problem identification and interventions according to the timeline given in Fig. 1. First we present results from our 'work location preference' survey in Table 1, followed by a list of problems and interventions to support employees working-from-home (WFH) in Tables 2 to 5. Finally, Table 7 presents preliminary results from our working-fromhome evaluation survey that we administered in January 2021 (see Appendix A for questions). Table 6 provides the demographic breakdown of the seven practitioners who responded to our survey. Note, we cannot specify roles of respondents as we did not include this identifying feature in our preliminary survey, to preserve anonymity. However, we can be confident that we have at least four roles included in the responses (see breakdown of roles in Section 3). Although all Irish based staff were asked to work from home to keep safe and comply with Ireland's COVID-19 guidance, results from our world-wide work location survey (Table 1) shows 88 practitioners (comprising developers, product owners, QAs and project managers) had a mix of preferences. 20% already work from home (prior to COVID-19). 59% (32 +14+13) of office-based workers would like flexibility to work from home or office. 8% of officebased workers would like to work from home or another country full-time (with willingness to go to office/Dublin whenever required, e.g. for PI planning meetings). A small number (5%) feel the need to work in office full time in the future. The majority wanted the flexibility to work from both the office and home. Problem identification The recurrent one-to-one online meetings identified anxieties, needs and preferences, as follows: People concerns. (1) How to share work with childcare? 1 forms.office.com. (2) How to relieve financial demands such as mortgage payments, due to reduction in salary or partner's loss of job? (3) How to facilitate move from office to working-from-home, and compensate for reduced level of social and informal interaction with colleagues. (4) How to welcome new staff remotely? (5) How to support (new) staff moving to Ireland from abroad? Technology concerns. (1) Is our current technology adequate for us to adapt to working from home, and does it scale? (2) Does everyone have the right set up at home? (3) Does need for fast broadband connection from home result in any additional cost to our staff? Process concerns. (1) How can we adapt processes to keep people socially connected (beyond having the right technology)? For example, prepandemic, we held regular face-to-face program increment (PI) planning meetings that take place over two weeks at the Dublin headquarters. Attendees from 10 different countries in five time zones all travel to be physically in the same space. During the pandemic we held a virtual version of the PI planning meeting, and in the retrospective attendees reported that they "miss social contact and cross team collaboration enabled by on-site meetings." (2) How can we replicate (in a virtual setting) the informal gatherings and interactions enjoyed by our staff? While "daily stand-ups" and other Scaled Agile Framework ® (SAFe) virtual ceremonies offer opportunities for staff to meet, many employees noted that this did not satisfy the need for informal gatherings, and getting to know new members in a relaxed social setting. Interventions We introduced new initiatives in recognition of the needs expressed by employees in our one-to-one interviews. Tables 2 to 4 list our interventions to address the above problems. All interventions were implemented immediately, and where appropriate, repeated regularly. Evaluation Survey On 15 January, 2021, we administered an online "Work from home evaluation" survey (see Appendix A for questions). We received seven completed responses (out of 10); Table 6 shows the sample distribution. Information has been aggregated to ensure anonymity in this small sample. A selected set of Evaluation Survey results are shown in Table 7, Fig. 3 and Fig. 4. Even with our small sample, we can draw some conclusions from these results with a degree of confidence 3 Looking at the level of support from Ocuco (Q2.1, Table 7), there was near consensus that the support was excellent. Given the range of responses for other questions, we describe further in Fig. 3 and Fig. 4. Fig. 3 suggests that Ocuco Ltd. employees are working at least as long when working-from-home as when they were in the office. Allowed flexibility to staff with kids who are working-from-home-could adjust their day to share childcare. Unsuitable home environment for remote working (e.g. no space, too many people in house). Offered a Taxi service to office (adapted to be COVID-19 compliant) to those unable to working-fromhome comfortably. Concerns about family who are living in a different country. Facilitated staff living abroad to go home for extended periods to see their families allowing for COVID-19 restrictions. Concerns about living arrangements for those committed to moving countries. Provided accommodation to staff moving to Ireland to start new roles within Ocuco Ltd.. Loss of contact. Line Managers check in with teams at least once a month. (Teams meet regularly with their daily stand-ups). Lack of job security. Offered free and independent advice to staff with financial commitments, e.g. whether to avail of mortgage extensions offered by their banks. Anxiety over status of Ocuco Ltd. business. CEO gave key updates at least once a month in company wide meetings. Fig. 3 also shows that Ocuco Ltd. developers perceive they are at least as productive during the pandemic as they were before the pandemic. There was a more mixed response when considering the impact of working-from-home on personal responsibilities. Regarding productivity, we wanted to know whether there was a change in productivity at the beginning of the pandemic, when working-from-home was novel, and little was known about short-or long-term effects of the pandemic. The responses suggest increased productivity immediately after shifting to working-from-home. This increase in productivity seems to have been maintained, with a majority of respondents continuing to report higher productivity compared Problem Solution Anxiety and fear of burnout. Check for burn-out, and encourage setting work/home boundaries. Meet staff one-to-one meetings, or administer anonymous surveys in work routines and levels of anxiety can be reported anonymously or directly with line manager. New employee feelings of uncertainty. Set-up remote staff induction programs to ensure new staff are onboarded efficiently and quickly to the Ocuco Ltd. team. Provide accommodation to staff committed to moving to Ireland to start new roles. All work and no play. We maintain social connections globally by hosting: Pizza Fridays (with synchronized expensed lunches), knowledge and music quizzes (Fig. 5). Coffee mornings where groups of four from different countries and offices meet via video conference for casual chat. "Coffee Dock 2 " meet ups where anyone can open up a 'meet' an invite colleagues. Feeling lost and isolated. The management team has a monthly informal one-to-one chat with each staff member. to the "pre-pandemic" (Fig. 4). Finally, looking at the evolution from when we first moved to working-from-home to now, we see a further upward shift in productivity. This could be due to taking time to acclimatize to the early move to working-from-home. But we don't know whether this productivity increase is the result of longer working hours, or other factors related to working-from-home specifically. Also, these observations are based on participant's self-reported perceptions; it remains to be seen whether actual productivity has increased. DISCUSSION Politicians, organizations, economists and individuals are all consumed with the pressing economic question of the long-term cost arising from the COVID-19 pandemic [6]. Organizations need to balance keeping afloat and surviving during the pandemic, maintaining an experienced and trained workforce, and, being prepared for growth in the near future. Agile takes on a whole new meaning in the pandemic, moving way beyond the development team. SAFe paves the way for the whole company to be involved, so that decisions can be made quickly, communication channels are open with regular contact across roles and divisions, and management are flexible in terms of changing processes to meet the new needs of employees. As a people-intensive field, software engineering relies heavily on the wellness of a skilled workforce. A particular challenge is to reverse the negative impact working-from-home can have on work-life balance [2]. From our initial survey feedback, we need to be particularly aware of raised anxiety in our employees and longer working hours. Recommendations Returning to our original research question, "what changes does a global software development organization need to make for the wellness of their employees during a pandemic?" we distill the results from the previous section into five recommendations: Recommendation 1: Be flexible. Example: allow flexible working hours, and flexibility of work location. Allow employees who have relocated from abroad to return to their home country, when travel restrictions allow. Recommendation 2: Be proactive and supportive. Example: reach out to all staff to ensure they have a suitable work from home set up. Provide financial support for acquiring same. Facilitate staff living away from their families to either find a place to live (for new members), or travel home when COVID-19 restrictions allow. We are planning more remote events during 2021 to replace the larger "all hands" workshops that normally take place in person. In addition, we are coordinating the safe return to office day events when local government guidelines permit. Recommendation 4: Give financial support where needed. Allocate funds to support staff experiencing financial shortfalls as a result of temporary pay cuts. Offer impartial, professional independent advice on financial affairs, such as bank loans and mortgages. Ocuco Ltd. is a medium-size company, and so can implement recommendations such as these rapidly, on a company-wide basis. Nevertheless, we feel that larger organizations can still benefit from these recommendations, which can be adapted to the department or division level. Not doing so, on the other hand, risks depletion of the social capital created from face-to-face working [5]. CONCLUSION In this paper we presented our experience of managing remote development teams during the COVID-19 pandemic. Ocuco Ltd. has taken very naturally to remote working, having already had a strong ethos, based on agile practices, for working with colleagues in any of the 14 different Ocuco Ltd. locations around the world. But we recognized early that our previous processes and infrastructure do not guarantee a smooth transition to working-from-home full time. As a result, we implemented several new interventions, in three key areas: People, Process and Technology. Our key message when it comes to People is that management need to be quick to recognize and react to the crisis, show strong leadership, listen to employees' needs, and be informative. Technology, on the other hand, as the route to remaining connected, must be ubiquitous throughout the company, regardless of location; equipment and infrastructure costs must be met by the employer. Finally, Process changes need to be fast, flexible, and inclusive. Our results are encouraging, as during the pandemic crisis, we have kept all our current staff on the payroll, employed new people, and won new contracts. Early feedback on productivity is also positive. We still have a lot of hard work to do to prevent losing the social fabric of the company and the culture we have worked so hard to build over 25 years of trading. So, although we've been successful in working remotely in the past, we recognize there is more to do, and we do all miss the office for the human interactions. Our future plans involve administering the Ocuco Ltd. COVID-19 Working from Home Evaluation Survey to all Ocuco Ltd. employees. ACKNOWLEDGMENTS We are indebted to the many members of Ocuco Ltd. who responded to our survey and volunteered their insights about working during the COVID-19 pandemic. This work was supported, in part, by Science Figure 1 :Figure 2 : 12Coronavirus Timeline in Ireland and Ocuco Ltd.'s response Ocuco Ltd.. Figure 3 : 3Plot of survey results re working hours (Q1.7, Appendix A) and impact on personal life (Q2.7) (#7). Figure 4 : 4Plot of survey results related to productivity (Q4.5-7, Appendix A) (#7). Recommendation 5 : 5Show strong leadership. Have the CEO and Management team provide regular updates on the current and future direction of the company, and provide an open door policy for staff to come and ask questions or raise concerns. Figure 5 : 5Virtual music quiz hosted by Ocuco Ltd.'s BeNeLux team. Table 1 : 1Work location preference (#88) -Home/Office/Abroad? Work location (pre COVID-19 and future) Count No change, worked from abroad before: 18 20% No change, working-from-home (WFH) full time before: 2 2% Need to be in office full time (once conditions allow): 4 5% Change: Would like flexibility (50 WFH/50 office): 32 36% Change: Would like flexibility -mainly WFH: 14 16% Change: Would like flexibility -mainly office: 13 15% Change: Would like flexibility -other country/city: 5 6% online survey was administered using Microsoft Forms 1 in which all responses were anonymous and voluntary. The sample comprised ten team members having one of seven roles: Senior Software Engineer (x 2), Software Engineer (x 3), QA Manager, QA Engineer, Automated Test Architect, Project Manager, Development & PMO Director; we received seven responses. Table 2 : 2New strategies-PeopleProblem Solution Sharing work with child- care. Table 3 : 3New strategies -ProcessProblem Solution Feeling isolated. Established 'Remote working Initia- tives' Team to keep people connected. Simulated team lunches/nights out, coffee dock/water cooler chats, etc. Required everyone in teleconferences to share their cameras. Financial difficulties. Introduced Employee Assistance fund for those struggling during temporary pay cut period. Table 4 : 4New strategies -TechnologyProblem Solution Home office and er- gonomics. Reached out to ensure all staff had a suitable working-from-home envi- ronment; budget provided for chairs, desks, and extra screens. How to scale existing video conferencing applications. Switched to Microsoft Teams®. Retain privacy and security of code and data. Provided access to virtual build ma- chines via virtual private network (VPN). Connection costs. Paid broadband subsidies and up- grade costs where needed. Table 5 : 5New strategies -Wellness Table 6 : 6Survey (Appendix A) respondent demographics (#7) ('DNS' = "did not specify"). Category Number Gender: Female Male Other DNS 3 4 - - Age: 20-29 30-39 40-49 DNS 1 2 3 1 Location: GB Ireland DNS 1 5 1 Time at Ocuco Ltd. (years): 1-2 3-5 5-10 10+ 1 1 1 4 q Recommendation 3: Keep connected. Example: Remove technical barriers to communication. Initiate and maintain communication. Keep lines of communication open, bi-directional, and active. Balance number of work meetings with regular and varied social events to on-board new members of staff and keep existing people connected. Have fun! Due to the wide distribution of sites, Ocuco Ltd. had already invested in strong tools to collaborate and host meetings across Table 7 : 7Selected results from Ocuco Ltd.'s COVID-19 working-from-home evaluation survey (#7) (see Appendix A).distance. As such, technology interventions during COVID-19 were more about scaling, upgrading personal connections, and ensuring privacy, than introducing new applications.Working hours now vs pre-pandemic (Q1.7) Much longer Somewhat longer About the same Somewhat shorter Much shorter 1 4 2 0 0 Level of support from Ocuco (Q2.1) Excellent Adequate Neither helped nor hindered Inadequate Very inadequate 6 1 0 0 0 Working-from-home impact on personal responsibilities (Q2.7) Very positive Somewhat positive No impact Somewhat negative Very negative 2 0 3 2 0 Productivity Much more productive Somewhat more productive About the same Somewhat less productive Much less productive Early-vs pre-pandemic (Q4.5) 1 3 3 0 0 Now vs pre-pandemic (Q4.6) 1 3 3 0 0 Now vs early-pandemic (Q4.7) 3 1 3 0 0 This an authors' preprint. Please cite as: Clodagh NicCanna, Mohammad Abdur Razzak, John Noll, and Sarah Beecham (2021) "Globally Distributed Development during COVID-19" 8th International Virtual Workshop on Software Engineering Research and Industrial Practice. This an authors' preprint. Please cite as: Clodagh NicCanna, Mohammad Abdur Razzak, John Noll, and Sarah Beecham (2021) "Globally Distributed Development during COVID-19" 8th International Virtual Workshop on Software Engineering Research and Industrial Practice. SER&IP, June, 2021, NicCanna, Noll, Razzak,& Beecham According to the "rule of five [participants]"[7] there is nearly a 95% likelihood that the median response of a population is within the highest and lowest responses of a random sample of only five members of the population. So with a sample of seven we have high confidence that the median response represents the company as a whole.This an authors' preprint. Please cite as: Clodagh NicCanna, Mohammad Abdur Razzak, John Noll, and Sarah Beecham (2021) "Globally Distributed Development during COVID-19" 8th International Virtual Workshop on Software Engineering Research and Industrial Practice. This an authors' preprint. Please cite as: Clodagh NicCanna, Mohammad Abdur Razzak, John Noll, and Sarah Beecham (2021) "Globally Distributed Development during COVID-19" Foundation Ireland grant 13/RC/2094 to Lero -The SFI Software Research Centre.A OCUCO LTD. COVID-19 WORKING FROM HOME EVALUATION SURVEYThis survey was administered online to employees after 11 months of lockdown. Respondents were informed that participation is voluntary, completely anonymous, that results would be disseminated in aggregate form only, and they had the option of not answering any particular question.(1) Demographics3.4. Do these tools help you to communicate and collaborate (e.g share knowledge, resolve issues quicker etc.) more efficiently while working from home compared to in an office environment?□ Much more efficient □ Somewhat more efficient □ About the same □ Somewhat less efficient □ A lot less efficient 3.5. How has the number of meetings you attend changed compared to pre-pandemic? □ Many more meetings □ Somewhat more meetings □ About the same □ Somewhat fewer meetings □ A lot fewer meetings 3.6. How has the length of meetings you attend changed compared to pre-pandemic? □ Much longer meetings □ Somewhat longer meetings □ About the same □ Somewhat shorter meetings □ Much shorter meetings 3.7. Have you changed your work from home technology to meet changing needs in the pandemic? □ Reconfigure/make space for home office □ Increase broadband speed □ Obtain larger/multiple monitors □ Upgrade home computer speed, memory, or storage □ Better camera/microphone □ Other (please specify): □ Communities of practice 4.3. Looking back to the beginning of the pandemic (March-April 2020), how did your productivity at the beginning of the pandemic compare to pre-pandemic? □ Much more productive at beginning of pandemic than pre-pandemic □ Somewhat more productive at beginning of pandemic than pre-pandemic □ About the same □ Somewhat less productive at beginning of pandemic than prepandemic □ Much less productive at beginning of pandemic than pre-pandemic □ Don't knowjoined Ocuco Ltd. during the pandemic 4.4. How is your productivity now compared to pre-pandemic? □ Much more productive now than pre-pandemic □ Somewhat more productive now than pre-pandemic □ About the same □ Somewhat less productive now than pre-pandemic □ Much less productive now than pre-pandemic □ Don't know -joined Ocuco Ltd. during the pandemic 4.5. How is your productivity now compared to the beginning of the pandemic (March-April 2020)?□ Much more productive now than at the beginning of the pandemic □ Somewhat more productive now than at the beginning of the pandemic □ About the same now than at the beginning of the pandemic □ Somewhat less productive now than at the beginning of the pandemic □ Much less productive now than at the beginning of the pandemic □ Don't know -joined Ocuco Ltd. recently 4.6. How effective are Ocuco Ltd.'s pre-pandemic distributed development processes and practices for working from home? □ Very effective □ Somewhat effective □ Neither effective nor ineffective □ Somewhat ineffective □ Very ineffective □ Don't know -joined Ocuco Ltd. during the pandemic 4.7. Is there any practice introduced for working from home that you would like to see implemented in the office environment post-pandemic (flexibility of working hours, better communication, helpful attitude, etc.)? Please elaborate: (5) Wellness 5.1. During the pandemic, have you found it is more or less difficult to concentrate as compared to pre-pandemic? □ Much more difficult □ Somewhat more difficult □ About the same □ Somewhat less difficult □ Much less difficult 5.2. During the pandemic, have you felt more or less anxiety as compared to pre-pandemic? □ Much more anxious □ Somewhat more anxious □ About the same □ Somewhat less anxious □ Much less anxious 5.3. During the pandemic, how often do you engage in vigorous activities (like running or HIIT) compared to pre-pandemic? □ Much more often □ Somewhat more often □ About the same □ Somewhat less often □ Much less often 5.4. During the pandemic, how often do you engage in moderate activities (other than walking) compared to pre-pandemic? □ Much more often □ Somewhat more often □ About the same □ Somewhat less often □ Much less often 5.5. During the pandemic, how often do you engage in walking compared to pre-pandemic? □ Much more often □ Somewhat more often □ About the same □ Somewhat less often □ Much less often 5.6. Overall, how do you feel your wellbeing has been impacted during the pandemic? □ Very positive impact □ Somehat positive impact □ No impact □ Somewhat negative impact □ Very negative impact 5.7. Any other comment? How does Working from Home Affect Developer Productivity?-A Case Study of Baidu During COVID-19 Pandemic. Lingfeng Bao, Tao Li, Xin Xia, Kaiyu Zhu, Hui Li, Xiaohu Yang, arXiv:2005.13167arXiv preprintLingfeng Bao, Tao Li, Xin Xia, Kaiyu Zhu, Hui Li, and Xiaohu Yang. 2020. How does Working from Home Affect Developer Productivity?-A Case Study of Baidu During COVID-19 Pandemic. arXiv preprint arXiv:2005.13167 (2020). Working from home, job satisfaction and work-life balance-robust or heterogeneous links?. Lutz Bellmann, Olaf Hübler, International Journal of Manpower. Lutz Bellmann and Olaf Hübler. 2020. Working from home, job satisfaction and work-life balance-robust or heterogeneous links? International Journal of Manpower (2020). Denae Ford, Margaret-Anne Storey, Thomas Zimmermann, Christian Bird, Sonia Jaffe, Chandra Maddila, Jenna L Butler, Brian Houck, Nachiappan Nagappan, arXiv:2008.11147A tale of two cities: Software developers working from home during the covid-19 pandemic. arXiv preprintDenae Ford, Margaret-Anne Storey, Thomas Zimmermann, Christian Bird, Sonia Jaffe, Chandra Maddila, Jenna L Butler, Brian Houck, and Nachiappan Nagappan. 2020. A tale of two cities: Software developers working from home during the covid-19 pandemic. arXiv preprint arXiv:2008.11147 (2020). Octoverse spotlight: An analysis of developer productivity, work cadence, and collaboration in the early days of covid-19. N Forsgren, WWW pageN. Forsgren. 2020. Octoverse spotlight: An analysis of developer productivity, work cadence, and collaboration in the early days of covid-19. WWW page, accessed 10 January 2021. https://github.blog/2020-05-06-octoverse-spotlight- an-analysis-of-developer-productivity-work-cadence-and-collaboration-in-the- early-days-of-covid-19/ Remote working productivity will slump as firms burn up their social capital. insight online magazine. Neil Franklin, Neil Franklin. 2020. Remote working productivity will slump as firms burn up their social capital. insight online magazine, accessed 6 March 2021. workplaceinsight.net/remote-working-productivity-will-slump-as-firms- burn-up-their-social-capital Working from home across countries. Charles Gottlieb, Jan Grobovšek, Markus Poschke, Covid Economics. 1Charles Gottlieb, Jan Grobovšek, and Markus Poschke. 2020. Working from home across countries. Covid Economics 1, 8 (2020), 71-91. W Douglas, Hubbard, How to Measure Anything: Finding the Value of Intangibles in Business. Wiley3 ed.Douglas W. Hubbard. 2014. How to Measure Anything: Finding the Value of Intangibles in Business (3 ed.). Wiley. Optimal mitigation policies in a pandemic: Social distancing and working from home. J Callum, Thomas Jones, Venky Philippon, Venkateswaran, National Bureau of Economic Research. Technical ReportCallum J Jones, Thomas Philippon, and Venky Venkateswaran. 2020. Optimal mitigation policies in a pandemic: Social distancing and working from home. Technical Report. National Bureau of Economic Research. Global Software Development and Collaboration. John Noll, Sarah Beecham, Ita Richardson, Barriers and Solutions. ACM Inroads. 13John Noll, Sarah Beecham, and Ita Richardson. 2010. Global Software Develop- ment and Collaboration: Barriers and Solutions. ACM Inroads 1, 3 (September 2010). . Ralph Paul, Sebastian Baltes, Adisaputri Gianisa, Richard Torkar, Vladimir Kovalenko, Kalinowski Marcos, Novielli Nicole, Shin Yoo, Devroey Xavier, Xin Tan, Empirical Software Engineering. 25et al. 2020. Pandemic programmingRalph Paul, Sebastian Baltes, Adisaputri Gianisa, Richard Torkar, Vladimir Ko- valenko, Kalinowski Marcos, Novielli Nicole, Shin Yoo, Devroey Xavier, Xin Tan, et al. 2020. Pandemic programming. Empirical Software Engineering 25, 6 (2020), 4927-4961.

Gamma-ray induced air showers are notable for their lack of muons, compared to hadronic showers. Hence, air shower arrays with large underground muon detectors can select a sample greatly enriched in photon showers by rejecting showers containing muons. IceCube is sensitive to muons with energies above ∼500 GeV at the surface, which provides an efficient veto system for hadronic air showers with energies above 1 PeV. One year of data from the 40-string IceCube configuration was used to perform a search for point sources and a Galactic diffuse signal. No sources were found, resulting in a 90% C.L. upper limit on the ratio of gamma rays to cosmic rays of 1.2 × 10 −3 for the flux coming from the Galactic Plane region ( −80 • < ∼ l < ∼ −30 • ; −10 • < ∼ b < ∼ 5 • ) in the energy range 1.2 -6.0 PeV. In the same energy range, point source fluxes with E −2 spectra have been excluded at a level of (E/TeV) 2 dΦ/dE ∼ 10 −12 − 10 −11 cm −2 s −1 TeV −1 depending on source declination. The complete IceCube detector will have a better sensitivity, due to the larger detector size, improved reconstruction and vetoing techniques. Preliminary data from the nearly-final IceCube detector configuration has been used to estimate the 5 year sensitivity of the full detector. It is found to be more than an order of magnitude better, allowing the search for PeV extensions of known TeV gamma-ray emitters.

10.1103/physrevd.87.062002

1210.7992

8ecd705b4fd57fd8c27ca7ad79fa14ee48030d02

Search for Galactic PeV Gamma Rays with the IceCube Neutrino Observatory 30 Oct 2012 M G Aartsen School of Chemistry & Physics University of Adelaide 5005AdelaideSAAustralia R Abbasi Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA Y Abdou Dept. of Physics and Astronomy University of Gent B-9000GentBelgium M Ackermann DESY D-15735ZeuthenGermany J Adams Dept. of Physics and Astronomy University of Canterbury Private Bag 4800ChristchurchNew Zealand J A Aguilar Département de physique nucléaire et corpusculaire Université de Genève CH-1211GenèveSwitzerland M Ahlers Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA D Altmann Institut für Physik Humboldt-Universität zu Berlin D-12489BerlinGermany K Andeen Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA J Auffenberg Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA X Bai Bartol Research Institute and Department of Physics and Astronomy University of Delaware 19716NewarkDEUSA M Baker Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA S W Barwick Dept. of Physics and Astronomy University of California 92697IrvineCAUSA V Baum Institute of Physics University of Mainz Staudinger Weg 7D-55099MainzGermany R Bay Dept. of Physics University of California 94720BerkeleyCAUSA K Beattie Lawrence Berkeley National Laboratory 94720BerkeleyCAUSA J J Beatty Dept. of Physics and Center for Cosmology and Astro-Particle Physics Ohio State University 43210ColumbusOHUSA Dept. of Astronomy Ohio State University 43210ColumbusOHUSA S Bechet Université Libre de Bruxelles Science Faculty CP230B-1050BrusselsBelgium J Becker Tjus Fakultät für Physik & Astronomie Ruhr-Universität Bochum D-44780BochumGermany K.-H Becker Dept. of Physics University of Wuppertal D-42119WuppertalGermany M Bell Dept. of Physics Pennsylvania State University 16802University ParkPAUSA M L Benabderrahmane DESY D-15735ZeuthenGermany S Benzvi Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA J Berdermann DESY D-15735ZeuthenGermany P Berghaus DESY D-15735ZeuthenGermany D Berley Dept. of Physics University of Maryland 20742College ParkMDUSA E Bernardini DESY D-15735ZeuthenGermany D Bertrand Université Libre de Bruxelles Science Faculty CP230B-1050BrusselsBelgium D Z Besson Dept. of Physics and Astronomy University of Kansas 66045LawrenceKSUSA D Bindig Dept. of Physics University of Wuppertal D-42119WuppertalGermany M Bissok III. 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Physikalisches Institut RWTH Aachen University D-52056AachenGermany L Schulte Physikalisches Institut Universität Bonn Nussallee 12D-53115BonnGermany O Schulz D Seckel Bartol Research Institute and Department of Physics and Astronomy University of Delaware 19716NewarkDEUSA S H Seo Oskar Klein Centre and Dept. of Physics Stockholm University SE-10691StockholmSweden Y Sestayo S Seunarine Dept. of Physics University of Wisconsin River Falls54022WIUSA C Sheremata Dept. of Physics University of Alberta T6G 2G7EdmontonAlbertaCanada M W E Smith Dept. of Physics Pennsylvania State University 16802University ParkPAUSA M Soiron III. Physikalisches Institut RWTH Aachen University D-52056AachenGermany D Soldin Dept. of Physics University of Wuppertal D-42119WuppertalGermany G M Spiczak Dept. of Physics University of Wisconsin River Falls54022WIUSA C Spiering DESY D-15735ZeuthenGermany M Stamatikos Dept. of Physics and Center for Cosmology and Astro-Particle Physics Ohio State University 43210ColumbusOHUSA ¶ T Stanev Bartol Research Institute and Department of Physics and Astronomy University of Delaware 19716NewarkDEUSA A Stasik Physikalisches Institut Universität Bonn Nussallee 12D-53115BonnGermany T Stezelberger Lawrence Berkeley National Laboratory 94720BerkeleyCAUSA R G Stokstad Lawrence Berkeley National Laboratory 94720BerkeleyCAUSA A Stößl DESY D-15735ZeuthenGermany E A Strahler Vrije Universiteit Brussel Dienst ELEMB-1050BrusselsBelgium R Ström Dept. of Physics and Astronomy Uppsala University Box 516S-75120UppsalaSweden G W Sullivan Dept. of Physics University of Maryland 20742College ParkMDUSA H Taavola Dept. of Physics and Astronomy Uppsala University Box 516S-75120UppsalaSweden I Taboada School of Physics and Center for Relativistic Astrophysics Georgia Institute of Technology 30332AtlantaGAUSA A Tamburro Bartol Research Institute and Department of Physics and Astronomy University of Delaware 19716NewarkDEUSA S Ter-Antonyan Dept. of Physics Southern University 70813Baton RougeLAUSA S Tilav Bartol Research Institute and Department of Physics and Astronomy University of Delaware 19716NewarkDEUSA P A Toale Dept. of Physics and Astronomy University of Alabama 35487TuscaloosaALUSA S Toscano Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA M Usner Physikalisches Institut Universität Bonn Nussallee 12D-53115BonnGermany D Van Der Drift Dept. of Physics University of California 94720BerkeleyCAUSA Lawrence Berkeley National Laboratory 94720BerkeleyCAUSA N Van Eijndhoven Vrije Universiteit Brussel Dienst ELEMB-1050BrusselsBelgium A Van Overloop Dept. of Physics and Astronomy University of Gent B-9000GentBelgium J Van Santen Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA M Vehring III. Physikalisches Institut RWTH Aachen University D-52056AachenGermany M Voge Physikalisches Institut Universität Bonn Nussallee 12D-53115BonnGermany M Vraeghe Dept. of Physics and Astronomy University of Gent B-9000GentBelgium C Walck Oskar Klein Centre and Dept. of Physics Stockholm University SE-10691StockholmSweden T Waldenmaier Institut für Physik Humboldt-Universität zu Berlin D-12489BerlinGermany M Wallraff III. Physikalisches Institut RWTH Aachen University D-52056AachenGermany M Walter DESY D-15735ZeuthenGermany R Wasserman Dept. of Physics Pennsylvania State University 16802University ParkPAUSA Ch Weaver Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA C Wendt Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA S Westerhoff Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA N Whitehorn Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center University of Wisconsin 53706MadisonWIUSA K Wiebe Institute of Physics University of Mainz Staudinger Weg 7D-55099MainzGermany C H Wiebusch III. Physikalisches Institut RWTH Aachen University D-52056AachenGermany D R Williams Dept. of Physics and Astronomy University of Alabama 35487TuscaloosaALUSA H Wissing Dept. of Physics University of Maryland 20742College ParkMDUSA M Wolf Oskar Klein Centre and Dept. of Physics Stockholm University SE-10691StockholmSweden T R Wood Dept. of Physics University of Alberta T6G 2G7EdmontonAlbertaCanada K Woschnagg Dept. of Physics University of California 94720BerkeleyCAUSA C Xu Bartol Research Institute and Department of Physics and Astronomy University of Delaware 19716NewarkDEUSA D L Xu Dept. of Physics and Astronomy University of Alabama 35487TuscaloosaALUSA X W Xu Dept. of Physics Southern University 70813Baton RougeLAUSA J P Yanez DESY D-15735ZeuthenGermany G Yodh Dept. of Physics and Astronomy University of California 92697IrvineCAUSA S Yoshida Dept. of Physics Chiba University 263-8522ChibaJapan P Zarzhitsky Dept. of Physics and Astronomy University of Alabama 35487TuscaloosaALUSA J Ziemann Dept. of Physics TU Dortmund University D-44221DortmundGermany S Zierke III. Physikalisches Institut RWTH Aachen University D-52056AachenGermany A Zilles III. Physikalisches Institut RWTH Aachen University D-52056AachenGermany M Zoll Oskar Klein Centre and Dept. of Physics Stockholm University SE-10691StockholmSweden Search for Galactic PeV Gamma Rays with the IceCube Neutrino Observatory 16 M. Merck, 27 P. Mészáros 384030 Oct 2012(Dated: May 3, 2014)(IceCube Collaboration) 2 Gamma-ray induced air showers are notable for their lack of muons, compared to hadronic showers. Hence, air shower arrays with large underground muon detectors can select a sample greatly enriched in photon showers by rejecting showers containing muons. IceCube is sensitive to muons with energies above ∼500 GeV at the surface, which provides an efficient veto system for hadronic air showers with energies above 1 PeV. One year of data from the 40-string IceCube configuration was used to perform a search for point sources and a Galactic diffuse signal. No sources were found, resulting in a 90% C.L. upper limit on the ratio of gamma rays to cosmic rays of 1.2 × 10 −3 for the flux coming from the Galactic Plane region ( −80 • < ∼ l < ∼ −30 • ; −10 • < ∼ b < ∼ 5 • ) in the energy range 1.2 -6.0 PeV. In the same energy range, point source fluxes with E −2 spectra have been excluded at a level of (E/TeV) 2 dΦ/dE ∼ 10 −12 − 10 −11 cm −2 s −1 TeV −1 depending on source declination. The complete IceCube detector will have a better sensitivity, due to the larger detector size, improved reconstruction and vetoing techniques. Preliminary data from the nearly-final IceCube detector configuration has been used to estimate the 5 year sensitivity of the full detector. It is found to be more than an order of magnitude better, allowing the search for PeV extensions of known TeV gamma-ray emitters. Gamma-ray induced air showers are notable for their lack of muons, compared to hadronic showers. Hence, air shower arrays with large underground muon detectors can select a sample greatly enriched in photon showers by rejecting showers containing muons. IceCube is sensitive to muons with energies above ∼500 GeV at the surface, which provides an efficient veto system for hadronic air showers with energies above 1 PeV. One year of data from the 40-string IceCube configuration was used to perform a search for point sources and a Galactic diffuse signal. No sources were found, resulting in a 90% C.L. upper limit on the ratio of gamma rays to cosmic rays of 1.2 × 10 −3 for the flux coming from the Galactic Plane region ( −80 • < ∼ l < ∼ −30 • ; −10 • < ∼ b < ∼ 5 • ) in the energy range 1.2 -6.0 PeV. In the same energy range, point source fluxes with E −2 spectra have been excluded at a level of (E/TeV) 2 dΦ/dE ∼ 10 −12 − 10 −11 cm −2 s −1 TeV −1 depending on source declination. The complete IceCube detector will have a better sensitivity, due to the larger detector size, improved reconstruction and vetoing techniques. Preliminary data from the nearly-final IceCube detector configuration has been used to estimate the 5 year sensitivity of the full detector. It is found to be more than an order of magnitude better, allowing the search for PeV extensions of known TeV gamma-ray emitters. I. INTRODUCTION Gamma-rays are an important tool for studying the cosmos; unlike cosmic rays (CRs), they point back to their sources and can identify remote acceleration regions. Air Cherenkov telescopes have identified numerous sources of high-energy (E > 1 TeV) gamma-rays (see e.g. [1]): within our galaxy, gamma-rays have been observed coming from supernova remnants (SNRs), pulsar wind nebulae (PWNe), binary systems, and the Galactic Center. Extra-galactic sources include starburst galaxies and Active Galactic Nuclei (AGNs). Surface air-shower arrays like Milagro have performed all-sky searches for TeV gamma-rays. Although these detectors are less sensitive to point sources than Air Cherenkov telescopes, they have identified several Galactic pointlike and extended sources [2]. Interactions of CRs with interstellar matter and radiation in the Galaxy produce a diffuse flux. Hadrons interacting with matter produce neutral pions, which decay into gamma rays, while CR electrons produce gamma-rays via inverse Compton scattering on the radiation field. Milagro has measured this diffuse Galactic flux in the TeV energy range with a median energy of 15 TeV and reported an excess in the Cygnus region, which might originate from CRs from local sources interacting with interstellar dust clouds [3]. IceCube's predecessor AMANDA-II has also looked for TeV photons from a giant flare from SGR 1806-20, using 100 GeV muons. AMANDA's large muon collection area compensated for the small cross-section for photons to produce muons [4]. At higher energies, extra-galactic sources are unlikely to be visible, because more energetic photons are predicted to interact with cosmic microwave background radiation (CMBR), and with infrared starlight from early galaxies, producing e + e − pairs [5]. At 1 PeV, for example, photon propagation is limited to a range of about 10 kpc. It is unknown whether Galactic accelerators exist that can produce gamma rays of such high energy, but an expected flux results from interaction of (extragalactic) CRs with the interstellar medium (ISM) and dense molecular clouds. To date, the best statistics on photons with energies in the range from ∼300 TeV to several PeVs come from the Chicago Air Shower Array -Michigan Muon Array (CASA-MIA), built at the Dugway Proving Ground in Utah. CASA consisted of 1089 scintillation detectors placed on a square array with 15 m spacing. MIA consisted of 1024 scintillation counters buried under about 3 m of earth, covering an area of 2500 m 2 . It served as a muon veto, with a threshold of about 0.8 GeV. CASA-MIA set a limit on the fraction of photons in the cosmic-ray flux of 10 −4 at energies above 600 TeV [6]. The experiment also sets a limit of 2.4 × 10 −5 on the fraction of photons in the CR flux coming from within 5 • of the galactic disk [7] at 310 TeV. This is near the theoretical expectation due to cosmic-ray interactions with the interstellar medium. For a Northern hemisphere site like CASA-MIA, Ref. [8] predicts a gamma-ray fraction of 2 × 10 −5 for the average gas column density. In this work, we present a new approach for detecting astrophysical PeV gamma rays, based on data of the surface component, IceTop, and the in-ice array of IceCube. IceTop measures the electromagnetic component of air showers, while the in-ice array is sensitive to muons that penetrate the ice with energies above 500 GeV. While most CR showers above 1 PeV contain many muons above this threshold, only a small fraction of PeV gamma-ray showers carry muons that are energetic enough to reach the in-ice array. Therefore, gamma-ray candidates are selected among muon-poor air showers detected with IceTop and whose axis is reconstructed as passing through the in-ice array. This approach of selecting muon-poor showers as gamma-ray candidates is fundamentally different from the earlier AMANDA-II gamma-ray search described IceCube consists of a ∼km 2 surface air shower array and 86 strings holding 60 optical modules each, filling a physical volume of a km 3 . The region in the center of the buried detector is more densely instrumented. See text for details. above, which was only sensitive to gamma-ray showers that do contain high energy muons (> 100 GeV). We present a limit on the gamma-ray flux coming from the Galactic Plane, based on one year of data with half of the IceCube strings and surface stations installed. We also discuss the sensitivity of the completed detector. II. THE ICECUBE DETECTOR IceCube (see Fig. 1) is a particle detector located at the geographic South Pole. The in-ice portion consists of 86 strings that reach 2450 m deep into the ice. Most of the strings are arranged in a hexagonal grid, separated by ∼125 m. Each of these strings holds 60 digital optical modules (DOMs) separated by ∼ 17 m covering the range from 1,450 m to 2,450 m depth. Eight strings form a denser instrumented area called DeepCore. The DOMs detect Cherenkov light produced by downwardgoing muons in cosmic-ray air showers and from charged particles produced in neutrino interactions. The data used in this analysis was collected in 2008/9, when the 40 strings shown in Fig. 2 were operational. Each DOM is a complete detector system, comprising a 25 cm diameter Hamamatsu R7081-02 phototube [9], shaping and digitizing electronics [10], calibration hardware, plus control electronics and power supply. Most of the buried PMTs are run at a gain of 10 7 . Digitization is initiated by a discriminator, with a threshold set to 0.25 times the typical peak amplitude of a single photoelectron waveform. Each DOM contains two separate digitizing systems; the Analog Transient Waveform Digitizer (ATWD) records 400 ns of data at 300 Megasamples/s, with a 14 bit dynamic range (divided among 3 parallel channels), while the fast Analog-to-Digital Converter (fADC) records 6.4 µs of data at 40 Megasamples/s, with 10 bits of dynamic range. A system transmits timing signals between the surface and each DOM, providing timing calibrations across the entire array of about 2 ns [11,12]. The IceTop surface array [13] is located on the surface directly above the in-ice detectors. It consists of 81 stations, each consisting of two ice-filled tanks, about 5 m apart. For the 2008 data used here, 40 stations were operational (IC40, see Fig. 2). Each tank is 1.8 m in diameter, filled with ice to a depth of about 90 cm. The tanks are initially filled with water, and the freezing of the water is controlled to minimize air bubbles and preserve the optical clarity of the ice. Each tank is instrumented with two DOMs, a high-gain DOM run at a PMT gain of 5 × 10 6 , and a low-gain DOM, with a PMT gain of 5 × 10 5 . The two different gains were chosen to maximize dynamic range; the system is quite linear over a range from 1 to 10 5 photoelectrons. A station is considered hit when a low-gain DOM in one tank fires in coincidence with a high-gain DOM in the other; the thresholds are set to about 20 photoelectrons. When an in-ice DOM is triggered it sends a Local Coincidence (LC) message to its nearest two neighbors above and nearest two neighbors below. If the DOM also receives an LC message from one of its neighbors within 1 µs it is in Hard Local Coincidence (HLC). In that case the full waveform information of both the ATWD and fADC chip is stored. For IC40 and earlier configurations, isolated or Soft Local Coincidence (SLC) hits were discarded. In newer configurations, the SLC hits are stored albeit with limited information. Keeping the full waveforms would require too much bandwidth, since the rate of isolated hits per DOM due to noise is ∼500 Hz. Instead, only the three fADC bins with the highest values and their hit times are stored. In Sec. IV we present a gamma-ray analysis of the IC40 dataset. In Sec. VI we study the expected sensitivity of the full IceCube detector, and discuss how the inclusion of SLC hits increases the background rejection. III. DETECTION PRINCIPLE We create a sample of gamma-ray candidates by selecting air showers that have been successfully reconstructed by IceTop and have a shower axis that passes through the IceCube instrumented volume. The geometry limits this sample to showers having a maximum zenith angle of ∼30 degrees. Since IceCube is located at the geographic South Pole, the Field of View (FOV) is roughly con-layout of these strings in relation to the final 86-string IceCube configuration is shown Fig. 1. Over the entire period the detector ran with an uptime of 92%, yielding 375.5 day total exposure. Deadtime is mainly due to test runs during and after the construction sea dedicated to calibrating the additional strings and upgrading data acquisition systems. IceCube uses a simple multiplicity trigger, requiring local coincidence hits in eight DO within 5 µs. Once the trigger condition is met, local coincidence hits within a readout w dow ±10 µs are recorded, and overlapping readout windows are merged together. IceCu triggers primarily on down-going muons at a rate of about 950 Hz in this (40-string) con strained to the declination range of −60 to −90 degrees, as shown in Fig. 3. This FOV includes the Magellanic Clouds and part of the Galactic Plane. Gamma-rays at PeV energies are strongly attenuated over extra-galactic distances, thus limiting the observable sources to those localized in our galaxy. At distances of ∼ 50 kpc and ∼60 kpc, the PeV gamma-ray flux from the Large and Small Magellanic Cloud is suppressed by several orders of magnitude. The contours in the background of Fig. 3 are the integrated neutral atomic hydrogen (HI) column densities under the assumption of optical transparency based on data from the Leiden/Argentine/Bonn survey [14]. These densities are not incorporated into the analysis and are only plotted to indicate the Galactic Plane. We do expect, however, that gamma-ray sources are correlated with the HI column density. Firstly, Galactic CR accelerators are more abundant in the high density regions of the Galaxy. Secondly, the gamma-ray flux of (extra-)galactic CRs interacting with the ISM naturally correlates with the column density. However, it has to be noticed that this correlation is not linear, because of the attenuation of gamma-rays over a 10 kpc distance scale. Furthermore, the column densities do not include molecular hydrogen which can also act as a target for CRs. The gamma-ray candidate events are searched for in a background of CR showers that have exceptionally few muons or are directionally misreconstructed. In the latter case the muon bundle reaches kilometers deep into the ice but misses the instrumented volume. This background is hard to predict with Monte Carlo (MC) simulations. Cosmic-ray showers at PeV energies and with a low number of energetic muons are rare. For example, at 1 PeV Contours of integrated neutral atomic hydrogen (HI) column densities [14], in Galactic coordinates (flat projection). The blue circle indicates the gamma-ray FOV for IceCube in the present IC40 analysis. The red rectangle indicate the regions for which CASA-MIA [6] has set an upper limit on the Galactic diffuse photon flux in the 100 TeV − 1 PeV energy range. IceCube's FOV is smaller but covers a different part of the Galactic Plane, close to the Galactic Center. less than 0.1% of the simulated showers contain no muons with an energy above 500 GeV, approximately what is needed to reach the detector in the deep ice when traveling vertically. Determining how many hadronic showers produce a signal in a buried DOM would require an enormous amount of MC data to reach sufficient statistics, plus very strong control of the systematic uncertainties due to muon production, propagation of muons and Cherenkov photons through the ice, and the absolute detector efficiencies. It would also have to be able to accurately predict the errors in air shower reconstruction parameters. For example, this analysis is very sensitive to the tails of the distribution of the error on the angular reconstruction of IceTop. Even MC sets that are large enough to populate these tails are not expected to properly describe them. To avoid these issues, we determine the background directly from data. As a result, we are not able to measure a possible isotropic contribution to the gamma-ray flux, because these gamma-rays would be regarded as background. Instead, we search for localized excesses in the gamma-ray flux. We search for a correlation of the arrival directions of the candidate events with the Galactic Plane, and scan for point sources. The acceptance of IceTop-40 is a complex function of azimuth and zenith due to its elongated shape and the requirement that the axis of the detected shower passes through IC40 (with the same elongated shape). However, since the arrival time is random (there are no systematic gaps in detector uptime w.r.t. sidereal time) the reconstructed right ascension (RA) of an isotropic flux of CR showers is uniform. The correct declination distribution of the background is very sensitive to the number of background showers introduced by errors in the IceTop angular reconstruction of the air shower as a function of the zenith angle, and is taken to be unknown. However, the flat distribution of background over RA is enough to allow for a search for gamma-ray sources. Recently, IceCube found an anisotropy in the arrival direction of CRs on the southern hemisphere [15]. These deviations with RA have been established for samples of CRs with median energies of 20 TeV and 400 TeV. The two energy ranges show a very different shape of the anisotropy, but the level of the fractional variations in flux is at a part-per-mil level for both [16]. An anisotropy with comparable magnitude in the PeV energy range is too small to affect this analysis (the IC40 final sample contains 268 events). IV. IC40 ANALYSIS A. Event selection Between April 5 2008 and May 20 2009, IceCube took data with a configuration of 40 strings and 40 surface stations (IC40), using several trigger conditions based on different signal topologies. This analysis uses the 8 station surface trigger, which requires a signal above threshold in both tanks of at least 8 IceTop stations. An additional signal in IceCube is not required for this trigger, but all HLC hits in the deep detector within a time window of 10 µs before and after the surface trigger are recorded. The air shower parameters are reconstructed from the IceTop hits with a series of likelihood maximization methods. The core position is found by fitting the lateral distribution of the signal, using S(r) = S ref r R ref −β−κ log 10 r R ref (1) where S is the signal strength, r is the distance to the shower axis, S ref is the fitted signal strength at the reference distance R ref = 125 m, β is the slope parameter reflecting the shower age, and κ = 0.303 is a constant determined from simulation [13,23]. Signal times are used to find the arrival direction of the air shower. The time delay due to the shape of the shower plane is described by the sum of a Gaussian function and a parabola, both centered at the shower core, which yields a resolution of 1.5 • for IC40. The relationship between the reconstructed energy E reco and S ref is based on MC simulations for proton showers and depends on the zenith angle. IceCube data is processed in different stages: in the first two levels the raw data is calibrated and filtered, and various fitting algorithms are applied, of which only the IceTop reconstruction described above is used in this analysis. In the selection of photon shower candidates from the data sample we distinguish two more steps: level three (L3) and level four (L4). Level three includes all the conditions on reconstruction quality, geometry and energy that make no distinction between gamma-ray showers and CR showers. The L4 cut is designed to separate gamma rays from CRs. Two parameters are used to constrain the geometry and ensure the shower axis passes through the instrumented volume of IceCube. The IceTop containment parameter C IT is a measure for how centralized the core location is in IceTop. When the core is exactly in the center of the array C IT = 0, while C IT = 1 means that it is exactly on the edge of the array. More precisely, C IT = x means that the core would have been on the edge of the array if the array would be x times its actual size. The string distance parameter d str is the distance between the point where the shower reaches the depth of the first level of DOMs and the closest inner string. Inner string, in this sense, means a string which is not on the border of the detector configuration. IC40 has 17 inner strings (see Fig. 2). The L3 cuts are: • Quality cut on lateral distribution fit: 1.55 < β < 4.95 (cf. Eq. 1) • Geometry cut: C IT < C max • Geometry cut: d str < d max • Energy cut: E reco > E min The energy and geometry cuts are optimized in a later stage (Sec. IV C). The L4 stage imposes only one extra criterion to the event: there should be no HLC hits in IceCube. This removes most of the CR showers, if E min is chosen sufficiently high. The remaining background consists of CR showers with low muon content and mis-reconstructed showers, the high energy muons from which do not actually pass through IceCube. The event sample after the L4 cut might be dominated by remaining background but it can be used to set an upper limit to the number of gamma-rays in the sample. Since the event sample after L3 cuts is certainly dominated by CRs, the ratio between the number of events after L4 and L3 cuts can be used to calculate an upper limit to the ratio of gamma-ray-to-CR showers. The remaining set of candidate gamma ray events is tested for a correlation with the Galactic Plane (Sec. IV C) and the presence of point like sources (Sec. IV D). First, the results of simulations are presented, which provide several quantities needed for sensitivity calculations. B. Simulation Although we determine the background from data only, simulations are needed to investigate systematic differences in the detector response to gamma-ray showers and cosmic-ray showers. More specifically, we are interested in the energy reconstruction of gamma-ray showers, the fraction of gamma-ray showers that is rejected by the muon veto system, and a possible difference in effective detector area between both types of showers. Gamma-ray and proton showers are simulated with CORSIKA v6.900, using the interaction models FLUKA 2008 and SYBILL 2.1 for low and high energy hadronic interactions, respectively. For both primaries 10,000 showers are generated within an energy range of 800 TeV to 3 PeV with a E −1 spectrum. Because the shower axes are required to pass through IceCube, the zenith angle is restricted to a maximum of 35 degrees. The observation altitude for IceTop is 2835 m. Atmospheric model MSIS-90-E is used, which is South Pole atmosphere for July 01, 1997. Seasonal variations in the event rate are of the order of 10% [19]. The detector simulation is done with the IceTray software package. Each simulated shower is fed into the detector simulation ten times with different core positions and azimuthal arrival direction, for a total of 100,000 events for both gamma rays and protons. This resampling of showers is a useful technique for increasing the statistics when examining quantities like the resolution of the energy reconstruction of IceTop. However, it cannot be used for quantities with large shower-to-shower variations, such as the number of high energy muons. The energy of gamma-ray showers is overestimated by the reconstruction algorithm. Fig. 4 shows the distribution of the logarithm of the ratio between the reconstructed and true primary energy as function of true energy, weighted to a E −2.7 spectrum. There are two reasons for the energy offset. First, there is a selection effect of the eight-station filter, which has a bias (below a few PeV) towards showers that produce relatively large signals in the IceTop tanks. This also affects the reconstructed energy of CR showers. At higher energies, the offset decreases but the reconstructed energy of gamma- ray showers is still slightly overestimated because the energy calibration of IceTop is performed with respect to proton showers. Figure 5 shows the distribution of true energies of gamma-ray and proton showers for the energy cut E reco > 1.4 PeV (which will be adopted in Sec. IV C). After this cut, 95% of the gamma-ray showers have a true energy above 1.2 PeV, while 95% of the proton showers have an energy above 1.3 PeV. The fraction of gamma-ray showers that is falsely rejected because the showers contain muons that produce a signal in IceCube is found by applying the cuts to the MC simulation. After applying the L3 cuts (defined in Sec. IV C) to the simulated gamma-ray sample there are 737 events left in the sample, of which 121 produce a signal in IceCube. Taking into account an energy weighting of E −2.7 , this corresponds to 16%. Showers that have no energetic muons can still be rejected if an unrelated signal is detected by IceCube. This could be caused by noise hits or unrelated muon tracks that fall inside the time window. This noise rate is determined directly from the data and leads to 14% signal loss. The total fraction of gamma-ray showers that is falsely rejected is therefore 28%. Finally, because the composition, shower size and evolution of gamma-ray and CR showers are different, one might expect a difference in the number of triggered stations and the quality of the reconstruction, which could lead to different effective areas. Such an effect would be of importance when calculating the relative contribution of gamma-rays to the total received flux. We compare the effective area for gamma rays and protons by counting the number of events that are present at L3. We compensate for the energy reconstruction offset by reducing the reconstructed gamma-ray energy by a factor 1. 16. The ratio of the effective area for gamma-rays to that for protons is then found to be 0.99. It should be emphasized that we do not use the simulation to determine the number of muons and their energy distribution from CR showers. This would require a simulation set that includes heavier nuclei instead of protons only. Moreover, various hadronic interaction models generate significantly different muon fluxes [18]. Instead, this analysis estimates the rate of CR showers that do not trigger IceCube using the data itself. C. Galactic Plane test The IC40 data set consists of 368 days of combined IceCube and IceTop measurements. The data from August 2009 is used as a burn sample, which means that it is used to tune the parameters of the analysis. After this tuning the burn sample is discarded and the remaining data is used for the analysis. IceCube is sensitive to gamma rays above 1 PeV. Earlier searches by CASA-MIA in a slightly lower energy range (100 TeV -1 PeV) with better sensitivity have not established a correlation of gamma-rays with the Galactic Plane (see Fig. 14). For a Galactic diffuse flux below the CASA-MIA limit [6] no gamma rays are expected in the IC40 burn sample. However, IceCube observes a different part of the Galactic Plane (see Fig. 3), close to the Galactic Center, so the possibility exists that previously undetected sources or local enhancements in CR and dust densities create an increase in the flux from that part of the sky. In order to find a possible correlation of candidate events in our burn sample to the Galactic Plane, different sets of L3 cut parameters are applied to find a set that produces the most significant correlation. Afterwards, the cut parameters are fixed and the burn sample is discarded. The fixed cut parameters are then used in the analysis of the rest of the IC40 data set to test whether the correlation is still present. Note that these cuts are applied at L3, so they affect both the event samples after L3 and L4 cuts. This is important because the ratio between the number of events after L4 and L3 cuts is used to calculate the ratio between gamma ray and CR showers. There are three cut parameters that are optimized by using the burn sample: E min , C max and d max . This is done by scanning through all combination of parameters within the following range: 600 TeV ≤ E min ≤ 2 PeV with steps of 100 TeV, 0.5 ≤ C max ≤ 1.0 with steps of 0.1, and 50 m ≤ d max ≤ 90 m with steps of 10 m. For each combination, the number of events N S in the burn sample after the L4 cut, located in the source region, is counted. The source region is defined as within 10 degrees of the Galactic Plane. Then, the data set is scrambled multiple times by randomizing the RA of each data point. For each scrambled data set the number of events in the source region is again counted. The best combination of cut parameters is the set which has the lowest fraction of scrambled data sets for which this number is equal to or higher than N S . The result of the scan is given in the three panels of Fig. 6. For each cut parameter the fraction of scrambled data sets that has a number of events in the source region equal to or exceeding the amount in the original data set is plotted for different cut values. For each plot the values of the other two cut parameters are kept constant at their optimal value. The actual search is done in three dimensions. The ratio is lowest for E > 1.4 PeV, C IT < 0.8 and d str < 60 m. With this combination of cuts only 0.011% of the scrambled data sets produce an equal or higher number of events in the source region. Note that while this procedure of optimizing cuts should be effective in the presence of sufficient signal, the small fraction obtained here and its erratic behavior with changing cut values are consistent with fluctuations of the background CR distribution alone, given the large number of possible cut combinations which were scanned. Nonetheless, the cut parameter values found with this procedure seem very reasonable (similar values are found with an alternative method, see Sec. VI). The optimized cuts are applied to the complete IC40 data set minus the burn sample. There are 268 candidate events of which 28 are located in the source region. Figure 7 is a map of the sky showing all 268 events. The colors in the background indicate the integrated HI column densities, cf. Fig. 3 (see discussion Sec. III). These are meant to guide the eye and are not part of the analysis. The significance of the correlation with the Galactic Plane is tested by producing data sets with scrambled RA. An equal or higher number of source region events is found in 21% of the scrambled data sets, corresponding to a non-significant excess of +0.9σ. We follow the procedure of Feldman & Cousins [20] to construct an upper limit for the ratio of gamma rays to CRs. The background is determined by selecting a range of RA that does not contain the source region. Within this range the data points are scrambled multiple times and for each scrambled set the number of events in a predefined region of the same shape and size as the source region is counted. This yields a mean background of 24.13 events for the source region. Using a 90% confidence interval, the upper limit on the number of excess gamma rays from this region is 14. Since 28% of gamma-ray showers are rejected by the veto from the buried detector, the maximum number of excess gamma-rays from the Galactic Plane is 14/0.72 = 19.4. From Fig. 5, it is known that the energy cut corresponds to a threshold of 1.2 PeV for gamma-rays, and 1.3 PeV for protons. Given that at L3 the sample is dominated by CR showers, and assuming a CR and gammaray power law of γ = −2.7, a 90% C.L. upper limit of 1.2 × 10 −3 on the ratio of gamma-ray showers to CR showers in the source region can be derived in the energy range 1.2 -6.0 PeV. The upper bound of 6.0 PeV is the value for which 90% of the events are inside the energy range. This value falls outside the range for which gamma-ray showers were simulated. However, there is no indication that the energy relation plotted in Fig. 4 behaves erratically above 3 PeV. This is a limit on the average excess of the ratio of gamma-rays to CRs in the source region with respect to the rest of the sky, i.e. a limit on the Galactic component of the total gamma-ray flux. A possible isotropic component is not included. Systematic uncertainties lead to a 18% variation of the upper limit, as determined in Sec. V. D. Unbinned point source search An additional search for point-like sources tests the possibility that a single source dominates the PeV gamma-ray sky. This source does not necessarily lie close to the Galactic Plane. An unbinned point source search is performed on the sky within the declination range of −85 • to −60 • , using a method that follows [21]. The region within 5 degrees around the zenith is omitted, because the method relies on scrambled data sets that are produced by randomizing the RA of the events. Close to the zenith this randomization scheme fails due to the small number of events. The data is described by an unknown amount of signal events on top of a flat distribution of background events. In an unbinned search, a dense grid of points in the sky is scanned. For each point a maximum likelihood fit is performed for the relative contribution of source events over background events. For a particular event i the probability density function (PDF) is given by P i (n S ) = n S N S i + 1 − n S N B i (2) where n S is the number of events that is associated to the source, B i is the background PDF, and S i = 1 2πσ 2 exp − ∆Ψ 2 2σ 2(3) is the two-dimensional Gaussian source PDF, in which ∆Ψ is the space angle between the event and the source test location, and σ = 1.5 • is the angular resolution of IceTop. The background PDF B i is only dependent on the zenith angle, and is derived from the zenith distribution of the data. For each point in the sky there is a likelihood function L(n S ) = Π i P i (n S ),(4) and associated test statistic λ = −2 (log(L(0)) − log(L(n S )) ,(5) which is maximized for n S . In the optimization procedure, n S is allowed to have a negative value, which mathematically corresponds to a local flux deficit. The procedure is similar to the search method for the neutrino point sources with IceCube [22], except that the source and background PDF do not contain an energy term. Because the range of energies in the event sample is relatively small (90% of the events have an energy between 1 and 6 PeV), an energy PDF is unlikely to improve the sensitivity. Figure 8 displays a map of the sky with declination between −85 • and −60 • showing the events in this region and contours of the test statistic λ. The maximum value is λ = 2.1 at δ = −65.4 • and RA= 28.7 • , corresponding to a fit of n s = 3.5 signal events. The overall significance of this value for λ is found by producing 10, 000 scrambled data sets by randomizing the RA of each event. Figure 9 shows the distribution of λ associated to the hottest spot in each scrambled data set. The median test statistic value for the hottest spot in the scrambled data sets is λ = 2.7, so the actual data set is consistent with a flat background. Upper limits on the gamma-ray flux can be derived for each point in the sky by assuming that all events are gamma-rays. Since many events are in fact muon-poor or misreconstructed showers, this leads to a conservative upper limit. Because the acceptance of IC40 decreases as a function of declination (see Fig. 13), the limit is more constraining at lower declination. Figure 10 is a sky map of the 90% C.L. upper limit in the energy range E = 1.2 − 6.0 PeV for E −2 source spectra. Point source fluxes are excluded at a level of (E/TeV) 2 dΦ/dE ∼ 10 −12 − 10 −11 cm −2 s −1 TeV −1 depending on source declination. Corrections for signal efficiency and detector noise are taken into account. Systematic uncertainties lead to a 18% variation of the upper limit. V. SYSTEMATIC ERRORS Since this analysis derives the background from data, the systematic uncertainties due to the background estimation are small. The previously discussed cosmic-ray anisotropy measurement (see Sec. III and [16]) is too small to have an impact on this analysis. Since there are no systematic gaps in detector uptime with respect to sidereal time-of-day in our sample, the coverage of RA is homogeneous. Therefore, we focus on the systematic uncertainties in the signal efficiency, due to uncertainties in the surface detector sensitivity, and in the muon production rate for photon showers. The uncertainty in the surface detector sensitivity is studied in Ref. [23]; Table 2 there gives the uncertainties for hadronic showers as a function of shower energy and zenith angle. Although there are differences between hadronic and electromagnetic showers, most factors that contribute to this figure apply to both types of showers. Strongly contributing factors include atmospheric fluctuations, calibration stability and uncertainties in response of detector electronics (PMT saturation and droop). The contribution from the uncertainty in modeling the hadronic interaction is clearly different for electromagnetic showers, and is discussed below. For E < 10 PeV, and zenith angle less than 30 • , there is a 6.0% systematic uncertainty in energy, and a 3.5% systematic uncertainty in flux. For an E −2.7 spectrum, a 6.0% uncertainty in energy translates into a 17.0% uncertainty in flux, or, adding in quadrature, 17.4% flux uncertainty. The uncertainty in the muon production from hadronic showers emerges from theoretical uncertainties. It depends on the hadronic photoproduction and electroproduction cross-sections for energies between 10 TeV and 6 PeV. Figure 1 of [24] compares two cross-sections from two different photoproduction models, and finds (for water with a similar atomic number and mass number as air), a difference that rises from about 20% at 10 TeV to 60% at 1 PeV. The bulk of the particles in the shower are at lower energies, so we adopt a 20% uncertainty on the muon production rate via photoproduction. In addition, there is also a contribution of muon pair creation. To reach the in-ice DOMs, muons need at least 500 GeV. At 1 TeV, the fractional contribution of muon pair creation is ∼ 10% [25]. Since muon pair production is not included in SYBILL 2.1, we arrive at 30% uncertainty in total muon production rate. This uncertainty is applied to the 16% of photon showers that are lost because they contain muons for a final 4.8% uncertainty in sensitivity due to the unknown muon production cross-section. We add the uncertainties due to detector response and muon production in quadrature, and arrive at an overall 18 % uncertainty in sensitivity. VI. ICECUBE 5-YEAR SENSITIVITY The sensitivity of the full IceCube detector to a gamma-ray flux from the Galactic Plane benefits from multiple improvements that can be made with respect to the analysis presented above. In this section we use preliminary data from the IC79 configuration (79 strings, 73 surface stations, 2010/2011) to estimate the sensitivity that the full IceCube detector can reach in 5 years. Since the full detector (IC86: 86 strings and 81 surface stations) is slightly larger than the IC79 configuration, the predicted sensitivity will be slightly underestimated. Also, the new cuts proposed below are not yet optimized, as this would require the actual IC86 data set. A. Air shower reconstruction This analysis is very sensitive to the quality of the core reconstruction. If the shower core is not reconstructed accurately, a muon bundle that passes outside the in-ice array might be incorrectly assumed to be aimed at the detector. Because of the absence of a signal in the in-ice DOMs, the event is then misinterpreted as a gamma-ray candidate. A more accurate core reconstruction algorithm has been developed for IceTop and will improve the CR rejection in post-IC40 analyses. In addition, the angular resolution of the larger array is improved, increasing the sensitivity to point sources. B. Isolated hits The SLC mode (which is available since IC59, see Sec. II) increases the sensitivity to CR showers with low muon content. A muon with just enough energy to reach IceCube, might not emit enough Cherenkov light to trigger multiple neighboring DOMs. By tightening the L4 cut so that no SLC hits are allowed to be present in the data, the efficiency with which CR showers can be rejected increases. At the same time, actual gamma-ray showers may be rejected in case of a noise hit in a single DOM. To keep this chance low, SLC hits only count as veto hits if they can be associated to the shower muon bundle both spatially and temporally. Figure 11 shows the distribution of isolated hits in the complete detector as a function of time relative to the arrival time of the air shower as measured by IceTop. The plots show data at L3 level, applying the same cut values as in the IC40 analysis. The left plot shows the distribution of SLC hits for all events, while the right plot shows the same distribution but restricted to the subset of events which contain only SLC hits, i.e. events with no HLC hits. Hits associated with the muon bundle are seen throughout the detector, although the number of hits varies with depth because of variations in the optical properties of the ice due to naturally varying levels of contaminants such as dust, which attenuate Cherenkov photons The large number of isolated hits in the two bottom rows is an edge effect: the DOMs have fewer neighbors, so the chance for a hit to be isolated increases. In principle, the same effect could occur at the top two rows. However, the muon bundle deposits more energy in this region and the probability for any hit to have neighbor hits is larger here. The muon-poor showers that produce no HLC hits (right-hand plot) can still cause some isolated hits in the top of the detector. These events can be removed with an additional cut on SLC hits. Because isolated hits can also be produced by noise, only a small area is selected in which SLC hits are used as a veto. A simple additional L4 cut is that all events are removed that have an SLC hit meeting the following three criteria: • it is within 200 m from the reconstructed shower axis, • it is within a time window of 4.8-7.5 µs after the shower arrival time, and • it is in one of the six top layers of DOMs (spanning a vertical extend of 85 m). Note that the lower bound of the time window (4.8 µs) corresponds to the time it takes for a muon traveling vertically to reach the top layer of in-ice DOMs starting from the surface. Muons from an inclined shower will arrive even later. The number of background events that are discarded in the L4 cut is increased by ∼30%, while the SLC noise rate in the data implies a decrease in signal efficiency of ∼5%. With the completed detector it will be possible to optimize the SLC cuts further by making the time window dependent on zenith angle and DOM depth. The effect of this optimization was not yet studied here. C. Re-optimization of cuts For the IC40 analysis the cut parameters were optimized to increase the detection probability of a possible correlation of gamma rays with the Galactic Plane. To increase the sensitivity of future searches with the completed IC86 configuration, the cut values were reevaluated to increase the number of candidate events without losing background rejection power. This was achieved by evaluating the ratio between the number of events after L3 and L4 cuts. While the L3 event sample is completely dominated by CRs, the L4 sample is a combination of possible gamma-ray showers, muon-poor CR showers and misreconstructed CR showers. The fraction of gamma-rays and muon-poor CRs in the detected events is independent from the cuts on geometry parameters d str and C IT . The number of misreconstructed CR showers, on the other hand, will increase if the geometry cut values are chosen too loosely. Therefore, the ratio between the number of L4 and L3 events as a function of the cut parameter should be flat up to some maximum value after which it starts to increase. This maximum value is the preferred cut value since it maximizes the number of candidate events without lowering the background rejection power. It also maximizes the FOV, as looser geometry cuts imply a larger maximum zenith angle. Figure 12 shows the number of L3 (red) and L4 (blue) events together with their ratio (black dotted line; righthand axis) as a function of the three main cut parameters (with the other cut parameters kept constant at their final value). The rejection efficiency for d str is fairly stable up to 60 m. The number of events rapidly decreases above this value, while the rejection becomes worse. In this case, the alternative method of optimization yields the same result as the method used in the IC40 analysis. For the containment size C IT the ratio remains stable up to the edge of the array (C IT = 1) after which it starts to rise. It appears the cut can be relaxed with respect to the IC40 analysis. In the following we will use d str < 60 m and C IT < 1.0. The efficiency of the energy cut increases, as expected, with increasing energy, leveling off around ∼ 2.0 PeV. Since the total number of events falls off rapidly for increasing energy, the most sensitive region will be ∼ 2 − 3 PeV. However, since the spectra of possible sources in this energy regime are unknown, it is not clear what energy cut would produce the optimal sensitivity. Instead, the sensitivity is calculated for ten energy bins in the range 1-10 PeV (see Fig. 14). D. Increased acceptance With a larger array the acceptance, defined here as the effective area integrated over the solid angle of each 1 • bin in zenith angle, increases considerably. Because of the condition that the shower axis has to be inside the instrumented area of both IceCube and IceTop, the increase is especially dramatic at larger zenith angles. Fig. 13 shows the acceptance for IC40 with C IT < 0.8 and the complete IC86 array with C IT < 1.0. Not only does the acceptance increase at large zenith angles, the range of possible zenith angles is also extended (to ≈ 45 • ). This extends the FOV to cover a larger part of the Galactic Plane and probe an area closer to the Galactic Center. The Galactic Center itself is still outside the FOV at δ ≈ −29 • , corresponding to a zenith angle of 61 • . E. Sensitivity The sensitivity that can be reached with 5 years of data from the completed IceCube configuration can be estimated with preliminary data from IC79. It is assumed that the fraction of gamma-rays that are missed due to noise hits is the same as in the IC40 analysis. The full detector obviously has more noise hits, but this can be compensated by refining the in-ice cut by only allowing vetoes from DOMs that can be associated to the shower muon bundle in space and time (cf. the SLC cut described above). The sensitivity is calculated by producing scrambled data sets with randomized RA. Figure 14 shows the log(Energy/GeV) 90% C.L. sensitivity to a diffuse flux from within 10 degrees of the Galactic Plane that can be achieved with 5 years of full detector data. The blue dashed line indicates the integrated limit between 1 and 10 PeV, while the blue dots indicate the sensitivity in six smaller energy bins. The upper limits found by CASA-MIA and IC40 (present work) are also included in the plot. The KASCADE [26] results are not included since they set a limit on the all-sky gamma-ray flux. Figure 15 shows the sensitivity to point sources that is possible with 5 years of IceCube data. The sensitivity is a strong function of declination because the acceptance decreases at larger zenith angles. Point sources are expected to lie close to the Galactic Plane which reaches its lowest declination at −63 • . Within the IceCube field of view there are several PWNe and other gamma-ray sources detected by H.E.S.S. [27], listed in Table I. For these sources no significant cut-off was observed up to the maximum energy of 10 TeV, where statistics gets low. The blue dots indicate the flux that these sources would have at 1 PeV if their spectrum remains unchanged up to that energy. No correction for gamma-ray attenuation between the source and observer has been applied in this calculation. The extrapolation over two order of magnitude causes large uncertainties in the gamma-ray flux due to propagation of the errors on the spectral indices. VII. CONCLUSIONS We have presented a new method of searching for high energy gamma-rays using the IceCube detector and its surface array IceTop. One year of data from IC40 was used to perform a search for point sources and a Galactic diffuse signal. No sources were found, resulting in Table I in the absence of a cut-off. a 90% C.L. upper limit on the ratio of gamma rays to cosmic rays of 1.2 × 10 −3 for the flux coming from the Galactic Plane region ( −80 • < ∼ l < ∼ −30 • ; −10 • < ∼ b < ∼ 5 • ) in the energy range 1.2 -6.0 PeV. Point source fluxes with E −2 spectra have been excluded at a level of (E/TeV) 2 dΦ/dE ∼ 10 −12 − 10 −11 cm −2 s −1 TeV −1 depending on source declination. The full detector was shown to be much more sensitive, because of its larger size, improved reconstruction techniques and the possibility to record isolated hits. This analysis offers interesting observation possibilities. IceCube can search for a diffuse Galactic gamma-ray flux with a sensitivity comparable to CASA-MIA, but at higher energies. This sensitivity is reached, however, by studying a much smaller part of the Galactic Plane than CASA-MIA. IceCube is therefore especially sensitive to localized sources, which might be Galactic accelerators or dense targets for extragalactic CRs. The H.E.S.S. and CANGAROO-III [38] telescopes have found several high energy gamma-ray sources in IceCube's FOV. Most of these sources are identified as or correlated with PWNe. Their energy spectrum has been measured up to a couple of tens of TeV. At this energy, statistics become low and for most sources no cut-off has been established. If these spectra extend to PeV energies without a break, IceCube will be able to detect them. It is also possible that an additional spectral component in the PeV energy range is present if a nearby dense molecular cloud acts as a target for the PWN beam [39]. IceCube will be able to study these systems and place constraints on their behavior at very high energies, or possibly detect PeV gamma-rays for the first time. FIG. 1 . 1FIG. 1. IceCube consists of a ∼km 2 surface air shower array and 86 strings holding 60 optical modules each, filling a physical volume of a km 3 . The region in the center of the buried detector is more densely instrumented. See text for details. Fig. 1 . 1-Overhead view of the 40-string configuration, along with additional strings t will make up the complete IceCube detector. FIG. 2 . 2Map of location of all 86 strings of the completed IceCube detector. The blue dots represent the 40 string configuration that is used for this analysis. At surface level each of these 40 strings is complemented by an IceTop station consisting of two tanks. The large (red) circles indicate the 'inner strings' of the IC40 configuration. FIG. 3. Contours of integrated neutral atomic hydrogen (HI) column densities [14], in Galactic coordinates (flat projection). The blue circle indicates the gamma-ray FOV for IceCube in the present IC40 analysis. The red rectangle indicate the regions for which CASA-MIA [6] has set an upper limit on the Galactic diffuse photon flux in the 100 TeV − 1 PeV energy range. IceCube's FOV is smaller but covers a different part of the Galactic Plane, close to the Galactic Center. FIG. 4 . 4Ratio between reconstructed and true energy of simulated gamma-ray showers as a function of their true energy. At low energies the overestimation of the gamma-ray energy is largely due to a bias effect of the eight-station filter. At higher energies, this overestimation decreases. FIG. 5 . 5Distribution of true energy of gamma-ray (red, solid) and proton (blue, dotted) showers for an energy cut at Ereco > 1.4 PeV, weighted to a E −2.7 spectrum. FIG. 6 .FIG. 7 . 67Optimization scans for cut parameters Emin, dmax, and Cmax. The fraction of scrambled data sets that perform equal or better than the real data set is plotted against cut value. For each plot, the other 2 parameters are kept constant at their optimal value. The actual scan was done three dimensionally. For each plot, the shaded region indicates the parameter space that is excluded by the optimized cuts. Equatorial map of the 268 candidate gamma-ray events of the IC40 data set superimposed on HI column densities based on[14]. The dotted black curve encloses the source region, defined as within 10 degrees of the Galactic Plane. FIG. 8 .FIG. 9 . 89Equatorial map of the part of the sky for which an unbinned point source search is performed. The contours indicate the value of λ and the black dots are the candidate events. Distribution of the largest value of λ observed in each scrambled data set. The red dotted line indicated the value for λ that corresponds to the hottest spot in the data. FIG. 10 . 10Sky map of 90% C.L. upper limits on point source flux (E/TeV) 2 dΦ/dE in cm −2 s −1 TeV −1 for E −2 source spectra in the energy range E = 1.2 − 6.0 PeV. The limit is typically more constraining at low declinations where the effective area is largest. FIG. 11 .FIG. 12 . 1112Observed time of isolated hits (SLC) relative to the arrival time of the air shower front measured by IceTop. The left plot shows the distribution of SLC hits for all events; the right plot is the same but restricted to the subset of events which have only SLC hits and no HLC hits. There is an excess of SLC hits in the region were a muon signal associated to the shower is expected. This allows an additional cut to separate gamma-ray showers from hadronic showers. The black box indicates the region in which an SLC hit counts as a veto (see text for details). The variation in the number of hits as a function of depth in the left plot is due to variations in the optical properties of the ice. The number of L3 (red) and L4 (blue) events in the data as a function of the three main cut parameters. The ratio of the number of L4 to L3 events is given by the black dotted line, for which the corresponding axis is drawn on the right-hand side of the plot. effective area integrated over solid angle) for showers with an axis through both IceTop and Ice-Cube for IC40, CIT=0.8 (black), and IC86, CIT=1.0 (blue). FIG. 14 . 14Existing limits (red triangles for CASA-MIA and purple line for present IC40 analysis) and IceCube sensitivity to a diffuse gamma-ray flux from a region within 10 degrees from the Galactic Plane. The blue dashed line indicates the five year sensitivity of the completed detector, while the blue dots represent the sensitivity in smaller energy bins. FIG. 15 . 15IceCube 5 year sensitivity to point sources as a function of declination. The solid (dashed) black line indicates the sensitivity to an E −2 (E −2.5 ) flux. The dashed red line indicates the lowest declination reached by the Galactic Plane. The blue points indicate the flux at 1 PeV with extrapolated uncertainties of the sources listed in ACKNOWLEDGMENTSWe acknowledge the support from the following agen- sources in IceCube gamma-ray FOV. For those values that have two error margins, the first indicates the statistical error, while the second indicates the systematic error. FWV?) [36TABLE I. List of H.E.S.S. sources in IceCube gamma-ray FOV. For those values that have two error margins, the first indicates the statistical error, while the second indicates the systematic error. (FWV?) [36] FWO Odysseus programme, Flanders Institute to encourage scientific and technological research in industry (IWT), Belgian Federal Science Policy Office (Belspo); University of Oxford. for Scientific Research (FNRS-FWO). 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A set of multipartite orthogonal quantum states is strongly nonlocal if it is locally irreducible for every bipartition of the subsystems [Phys. Rev. Lett. 122, 040403 (2019)]. Although this property has been shown in three-, four-and five-partite systems, the existence of strongly nonlocal sets in N -partite systems remains unknown when N ≥ 6. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in (C d ) ⊗N for all N ≥ 3 and d ≥ 2, which for the first time reveals the strong quantum nonlocality in general N -partite systems. For N = 3 or 4 and d ≥ 3, we present a strongly nonlocal set consisting of genuinely entangled states, which has a smaller size than any known strongly nonlocal orthogonal product set. Finally, we connect strong quantum nonlocality with local hiding of information as an application. arXiv:2202.07139v1 [quant-ph]

10.1103/physreva.105.022209

2202.07139

9c0337bac66f0296dada39e047018ae71e515212

Strong quantum nonlocality in N -partite systems Fei Shi School of Cyber Security University of Science and Technology of China 230026HefeiPeople's Republic of China Zuo Ye School of Mathematical Sciences University of Science and Technology of China 230026HefeiPeople's Republic of China Lin Chen Ministry of Education School of Mathematical Sciences LMIB(Beihang University Beihang University 100191BeijingChina International Research Institute for Multidisciplinary Science Beihang University 100191BeijingChina Xiande Zhang School of Mathematical Sciences University of Science and Technology of China 230026HefeiPeople's Republic of China Strong quantum nonlocality in N -partite systems A set of multipartite orthogonal quantum states is strongly nonlocal if it is locally irreducible for every bipartition of the subsystems [Phys. Rev. Lett. 122, 040403 (2019)]. Although this property has been shown in three-, four-and five-partite systems, the existence of strongly nonlocal sets in N -partite systems remains unknown when N ≥ 6. In this paper, we successfully show that a strongly nonlocal set of orthogonal entangled states exists in (C d ) ⊗N for all N ≥ 3 and d ≥ 2, which for the first time reveals the strong quantum nonlocality in general N -partite systems. For N = 3 or 4 and d ≥ 3, we present a strongly nonlocal set consisting of genuinely entangled states, which has a smaller size than any known strongly nonlocal orthogonal product set. Finally, we connect strong quantum nonlocality with local hiding of information as an application. arXiv:2202.07139v1 [quant-ph] I. INTRODUCTION Quantum nonlocality is one of the most important properties in quantum mechanics. The entangled states show Bell nonlocality for violating Bell-type inequalities [1,2]. A set of multipartite orthogonal quantum states is locally indistinguishable if it is not possible to optimally distinguish the states by any sequence of local operations and classical communications (LOCC). This set also shows quantum nonlocality, which is different from Bell nonlocality. Bennett et al. firstly constructed a locally indistinguishable orthogonal product basis (OPB) in C 3 ⊗ C 3 [3], which shows the phenomenon of quantum nonlocality without entanglement. Later, locally indistinguishable orthogonal product sets (OPSs) and orthogonal entangled sets (OESs) have been widely studied [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. When information is encoded in a locally indistinguishable set of a composite quantum system, it cannot be completely retrieved under LOCC among the spatially separated subsystems. Consequently, local indistinguishability can be used for quantum data hiding [21][22][23][24] and quantum secret sharing [25][26][27]. Recently, Halder et al. introduced the concepts of local irreducibility and strong quantum nonlocality [28]. A set of multipartite orthogonal states is locally irreducible if it is not possible to eliminate one or more states from the set by orthogonality-preserving local measurements. A locally irreducible set must be a locally indistinguishable set, while the converse is not true in general. Further, a set of multipartite orthogonal states is strongly nonlocal if it is locally irreducible for each bipartition of the subsystems. The authors of Ref. [28] showed the phenomenon of strong quantum nonlocality without en- * shifei@mail. ustc C d ⊗C d ⊗C d , C d ⊗C d ⊗C d+1 , C 3 ⊗C 3 ⊗C 3 ⊗C 3 , and C 4 ⊗ C 4 ⊗ C 4 ⊗ C 4 . [29]. Recently, strongly nonlocal OPSs in general three-, four-, and five-partite systems, and strongly nonlocal unextendible product bases (UPBs) in general three-, and four-partite systems were constructed [30][31][32]. For OESs, based on the Rubik's cube, Shi et al. presented some strongly nonlocal OESs in C d ⊗C d ⊗C d [33] , while these OESs are not orthogonal genuinely entangled sets (OGESs). By using graph connectivity, Wang et al. successfully constructed strongly nonlocal OGESs in C d ⊗ C d ⊗ C d [34]. The concept of strong quantum nonlocality was also extended to more general settings [35][36][37][38][39]. Whether we use OPS or OES, the existence of strongly nonlocal sets are limited to three-, four-, and five-partite systems up to now. It is still unknown whether strong quantum nonlocality can be shown in general N -partite systems. So it is natural to ask the following question. Q: Can we show the strong quantum nonlocality in (C d ) ⊗N for d ≥ 2 and for all N ≥ 3? The main method of showing the strong nonlocality of a set is to show a stronger property, that is, any orthogonality-preserving local measurement that may be performed across any bipartition of the subsystems must be trivial. Although it suffices to show that the condition holds for any part consisting of N −1 subsystems [31,34], it is not easy to prove that any joint orthogonalitypreserving local measurement performed on any set of N − 1 subsystems must be trivial when N is large. To reduce the complexity, we wish to construct a set of orthogonal states which has a similar structure when restricted on any N − 1 subsystems. However, it is again not easy to give an explicit form of the states satisfying this condition, which is important in the verification of triviality of the measurement. One of the main contribution of this paper is to answer the question Q in an affirmative way. In fact, by using cyclic permutation group action, we construct a strongly nonlocal OES of size d N − (d − 1) N + 1 in (C d ) ⊗N for all d ≥ 2 and N ≥ 3. Based on this construction, we further show that when N = 3 and 4, a strongly nonlocal OGES of size d N − (d − 1) N + 1 exists in (C d ) ⊗N for all d ≥ 2. In Ref. [34], the authors asked whether we can construct a strongly nonlocal set via OGES that has a smaller size than that via OPS in the same system. The question was raised due to the intuition that OGESs show more strong nonlocality than OPSs [28], but known strongly nonlocal sets via OGESs [34] have larger sizes than that via OPSs [29]. When N = 3, the size 3d 2 − 3d + 2 of OGES in our construction is about half of the size 6(d − 1) 2 of the OPS in Ref. [29]. Finally, for applications, we show that strong quantum nonlocality can be used for local hiding of information. The rest of this paper is organized as follows. In Sec. II, we introduce the concept of strong nonlocality. In Sec. III, we construct OESs in (C d ) ⊗N for d ≥ 2 and N ≥ 3 from cyclic permutation group actions, and show the strong nonlocality in Sec. IV. Next, in Sec. V, we present strongly nonlocal OGESs when N = 3 and 4. In Sec. VI, we connect strong quantum nonlocality with local hiding of information. Finally, we conclude in Sec. VII. II. PRELIMINARIES |ψ AB = i∈Zm,j∈Zn a i,j |i A |j B .(1) Then |ψ AB corresponds to an m × n matrix M = (a i,j ) i∈Zm,j∈Zn . We denote rank(|ψ AB ) = rank(M ), which is also called the Schmidt rank of |ψ AB [40]. | GHZ N d = i∈Z d |i A1 |i A2 · · · |i A N .(3) An N -qubit W state can be expressed by |W N 2 =|1 A1 |0 A2 · · · |0 A N + |0 A1 |1 A2 · · · |0 A N + · · · + |0 A1 |0 A2 · · · |1 A N .(4) In local state discrimination, we usually perform orthogonality-preserving local measurements (OPLMs) for a set of orthogonal states, where a measurement is orthogonality preserving if the postmeasurement states keep being mutually orthogonal. Recently, Halder et al. proposed the concepts of locally irreducible sets and strong quantum nonlocality [28]. A set of orthogonal states in (C d ) ⊗N is locally irreducible if it is not possible to eliminate one or more states from the set by OPLMs. Moreover, A set of orthogonal states in (C d ) ⊗N is strongly nonlocal, if it is locally irreducible for each bipartition of the subsystems. In fact, a stronger property is often applied to show the strong nonlocality [31,34]. A set of orthogonal states in (C d ) ⊗N is said to have the property of the strongest nonlocality when the following condition holds, any OPLM that may be performed across any bipartition must be trivial. The following lemma shows that we only need to show that any N − 1 parties can perform only a trivial OPLM [31]. III. OESS FROM CYCLIC PERMUTATION GROUP ACTION In this section, we construct OESs from cyclic permutation group action. We first briefly recall the concept and properties of group action [41]. If X is a set and G is a group, then G acts on X if there is a function G × X → X, denoted by (g, x) gx, such that (i) (gh)x = g(hx) for all g, h ∈ G and x ∈ X; (ii) 1x = x for all x ∈ X, where 1 is the identity in G. For x ∈ X, the orbit of x, denoted by O x , is the subset O x = {gx | g ∈ G} ⊆ X,(5)and x is called a representative of the orbit O x . If y ∈ O x , then O x = O y ; if y / ∈ O x , then O x ∩ O y = ∅. Thus, X is the union of the mutually disjoint orbits, X = x O x ,(6) where x runs over the set of representatives of all orbits. Moreover, |O x | divides |G|. In our construction, the set we consider is the subset X N d of Z N d , which is defined by X N d := {(i 1 , i 2 , . . . , i N ) | 1≤k≤N i k = 0}.(7) That is, for any (i 1 , i 2 , . . . , i N ) ∈ X N d , there exists at least one i k = 0 for 1 ≤ k ≤ N . Then |X N d | = d N − (d − 1) N . Assume that G N = {σ k | k ∈ Z N }(8) is a cyclic permutation group of order N , where σ(i 1 , i 2 , . . . , i N ) = (i 2 , . . . , i N , i 1 )(9) for an N -tuple (i 1 , i 2 , . . . , i N ) ∈ Z N d . Then G N acts on X N d by definition, and yields a partition of X N d into disjoint orbits. For example, since G 3 acts on X 3 2 = Z 3 2 \ {(1, 1, 1)}, then X 3 2 = O (0,0,0) ∪ O (0,0,1) ∪ O (0,1,1) ,(10) In general if x = (0, 0 . . . , 0), then we can write O x = {(i (j) 1 , i (j) 2 , . . . , i (j) N ) | j ∈ Z k },(12) where k ≥ 2 is the size of O x . Note that the size may be different for different orbits, see Fig. 1 as an example. For each orbit O x of X N d , we define a set S x of states in (C d ) ⊗N as follows. If x = (0, 0 . . . , 0), let S x := { j∈Z k w sj k |i (j) 1 A1 |i (j) 2 A2 · · · |i (j) N A N : s ∈ Z k },(13)where w k = e 2π √ −1 k is a primitive kth root of unity. In fact, the coefficient matrix B = (w sj k ) s,j∈Z k(14) forms a complex Hadamard matrix of order k. = {|0 A1 |0 A2 |0 A3 |1 A4 + w s 4 |0 A1 |0 A2 |1 A3 |0 A4 + w 2s 4 |0 A1 |1 A2 |0 A3 |0 A4 + w 3s 4 |1 A1 |0 A2 |0 A3 |0 A4 : s ∈ Z 4 }. For the right cycle, the orbit is O (0,1,0,1) = {(0, 1, 0, 1), (1, 0, 1, 0)}, which yields the set of states S (0,1,0,1) = {|0 A1 |1 A2 |0 A3 |1 A4 ± |1 A1 |0 A2 |1 A3 |0 A4 }. in this case. See Fig. 1 for S (0,0,0,1) and S (0,1,0,1) . If O x = {(0, 0, . . . , 0)}, we define S (0,0,...,0) := {|0 A1 |0 A1 · · · |0 A N ± |1 A1 |1 A1 · · · |1 A N },(15) which has one more element than the orbit. Since X N d = x O x is a disjoint union, thus B N d := x S x(16) is also a disjoint union, which has size d N − (d − 1) N + 1. For example, in (C 2 ) ⊗3 , we know that B 3 2 = S (0,0,0) ∪ S (0,0,1) ∪ S (0,1,1) ,(17) by Eq. (10), where S (0,0,0) = {|0 A1 |0 A2 |0 A3 ± |1 A1 |1 A2 |1 A3 }, S (0,0,1) = {|0 A1 |0 A2 |1 A3 + w s 3 |0 A1 |1 A2 |0 A3 + w 2s 3 |1 A1 |0 A2 |0 A3 : s ∈ Z 3 }, S (0,1,1) = {|0 A1 |1 A2 |1 A3 + w s 3 |1 A1 |1 A2 |0 A3 + w 2s 3 |1 A1 |0 A2 |1 A3 : s ∈ Z 3 },(18) and |B 3 2 | = 8. One can easily check that B 3 2 is an OES in C 2 ⊗ C 2 ⊗ C 2 . Similarly, we can construct B 4 2 in C 2 ⊗ C 2 ⊗ C 2 ⊗ C 2 . Since X 4 2 =O (0,0,0,0) ∪ O (0,0,0,1) ∪ O (0,0,1,1) ∪ O (0,1,0,1) ∪ O (0,1,1,1) ,(19) we obtain that B 4 2 =S (0,0,0,0) ∪ S (0,0,0,1) ∪ S (0,0,1,1) ∪ S (0,1,0,1) ∪ S (0,1,1,1) ,(20) where S (0,0,0,0) = {|0 A1 |0 A2 |0 A3 |0 A4 ± |1 A1 |1 A2 |1 A3 |1 A4 }, S (0,0,0,1) = {|0 A1 |0 A2 |0 A3 |1 A4 + w s 4 |0 A1 |0 A2 |1 A3 |0 A4 + w 2s 4 |0 A1 |1 A2 |0 A3 |0 A4 + w 3s 4 |1 A1 |0 A2 |0 A3 |0 A4 : s ∈ Z 4 }, S (0,0,1,1) = {|0 A1 |0 A2 |1 A3 |1 A4 + w s 4 |0 A1 |1 A2 |1 A3 |0 A4 + w 2s 4 |1 A1 |1 A2 |0 A3 |0 A4 + w 3s 4 |1 A1 |0 A2 |0 A3 |1 A4 : s ∈ Z 4 }, S (0,1,0,1) = {|0 A1 |1 A2 |0 A3 |1 A4 ± |1 A1 |0 A2 |1 A3 |0 A4 }, S (0,1,1,1) = {|0 A1 |1 A2 |1 A3 |1 A4 + w s 4 |1 A1 |1 A2 |1 A3 |0 A4 + w 2s 4 |1 A1 |1 A2 |0 A3 |1 A4 + w 3s 4 |1 A1 |0 A2 |1 A3 |1 A4 : s ∈ Z 4 },(21) and |B 4 2 | = 16. We can also check that B 4 2 is an OES in C 2 ⊗ C 2 ⊗ C 2 ⊗ C 2 . In general, we have the following result. Lemma 2 The set B N d is an OES of size d N −(d−1) N +1 in (C d ) ⊗N . The proof of Lemma 2 is given in Appendix A. We will further show that B 3 d is an OGES, while B 4 d is not in Section V. However, we first show that B N d is strongly nonlocal in the next section. IV. THE STRONG NONLOCALITY FOR OESS IN N -PARTITE SYSTEMS In this section, we show that the set B N d of states in Eq. (16) } i,j∈Z2 , E =    a 00,00 a 00,01 a 00,10 a 00,11 a 01,00 a 01,01 a 01,10 a 01,11 a 10,00 a 10,01 a 10,10 a 10,11 a 11,00 a 11,01 a 11,10 a 11,11    .(22) Then the postmeasurement states {I A1 ⊗ M |ψ : |ψ ∈ B 3 2 } should be mutually orthogonal. That is ψ|I A1 ⊗ E|φ = 0(23) for |ψ = |φ ∈ B 3 2 . By using these orthogonality relations, we need to show that E ∝ I. Since E = E † , a ij,k = 0 can imply that a k ,ij = 0. Applying Lemma 8 to S (0,0,0) and S (0,0,1) , we must have a 00,01 = a 00,10 = a 00,11 = 0. Next, applying Lemma 8 to S (0,0,0) and S (0,1,1) , we obtain that a 10,11 = a 01,11 = 0. Since a 00,01 = a 00,10 = 0, we can apply Lemma 9 to S (0,0,1) . Then we obtain that a 01,10 = 0 and a 01,01 = a 10,10 = a 00,00 . Finally, applying Lemma 9 to S (0,0,0) , we have a 00,00 = a 11,11 . Thus E ∝ I, and this completes the proof. Example 4 In C 2 ⊗ C 2 ⊗ C 2 ⊗ C 2 , the set B 4 2 given by Eq. (20) is a strongly nonlocal OES of size 16. Proof. Let 2 } should be mutually orthogonal. We can show that E ∝ I from Table I. This completes this proof. In fact, we can show a general result. Theorem 5 The set B N d defined in Eq. (16) is a strongly nonlocal OES of size d N − (d − 1) N + 1 in (C d ) ⊗N for all d ≥ 2 and N ≥ 3. The proof of Theorem 5 is given in Appendix C. Note that Theorem 5 presents the first class of strongly nonlocal OESs in (C d ) ⊗N for any d ≥ 2 and N ≥ 3. V. THE STRONG NONLOCALITY FOR SMALL OGESS IN 3, 4-PARTITE SYSTEMS In this section, we consider strongly nonlocal OGESs in 3, 4-partite systems which have smaller sizes than those of previously known. Two states |ψ and |φ are called LU-equivalent if there exists a product U 1 ⊗U 2 ⊗· · ·⊗U N of unitary operators, such that |ψ = U 1 ⊗ U 2 ⊗ · · · ⊗ U N |φ .(24) It is known that LU-equivalence does not change the entanglement of a state. I: Off-diagonal elements and Diagonal elements of E = (a ijk, mn ) i,j,k, ,m,n∈Z2 . We apply Lemma 8 to two sets among S (0,0,0,0) , S (0,0,0,1) , S (0,0,1,1) , S (0,1,0,1) , S (0,1,1,1) , and apply Lemma 9 to one set among S (0,0,0,0) , S (0,0,0,1) , S (0,0,1,1) , S (0,1,0,1) , S (0,1,1,1) . Sets Elements Sets Elements S (0,0,0,0) , S (0,0,0,1) a000,001 = a000,010 = a000,100 = a000,111 = 0 S (0,0,0,0) , S (0,1,1,1) a110,111 = a101,111 = a011,111 = 0 S (0,0,0,0) , S (0,0,1,1) a000,011 = a000,110 = a100,111 = a001,111 = 0 S (0,0,0,0) , S (0,1,0,1) a000,101 = a010,111 = 0 S (0,0,1,1) , S (0,1,0,1) a011,101 = a101,110 = a010,100 = a001,010 = 0 S (0,0,0,1) , S (0,1,0,1) a001,101 = a010,101 = a100,101 = 0 S (0,0,0,1) , S (0,0,1,1) a001,011 = a001,110 = a010,011 S (0,0,1,1) a001,100 = 0 = a010,110 = a011,100 = a100,110 = 0 S (0,0,1,1) a011,110 = 0 S (0,0,0,0) a000,000 = a111,111 S (0,0,1,1) a100,100 = a011,011 = a110,110 S (0,0,0,1) a000,000 = a001,001 = a010,010 = a100,100 S (0,1,0,1) a010,010 = a101,101 First, we consider B 3 d in C d ⊗C d ⊗C d for d ≥ 2. Denote [d − 1] := {1, 2, . . . , d − 1}. Then we can explicitly write down all states in B 3 d as follows. Since X 3 d =O (0,0,0) ∪ i∈[d−1] (O (0,0,i) ∪ O (0,i,i) ) ∪ i =j∈[d−1] O (0,i,j) ,(25) we obtain that B 3 d =S (0,0,0) ∪ i∈[d−1] (S (0,0,i) ∪ S (0,i,i) ) ∪ i =j∈[d−1] S (0,i,j) ,(26) where S (0,0,0) = {|0 A1 |0 A2 |0 A3 ± |1 A1 |1 A2 |1 A3 }, S (0,0,i) = {|0 A1 |0 A2 |i A3 + w s 3 |0 A1 |i A2 |0 A3 + w 2s 3 |i A1 |0 A2 |0 A3 : s ∈ Z 3 }, S (0,i,i) = {|0 A1 |i A2 |i A3 + w s 3 |i A1 |i A2 |0 A3 + w 2s 3 |i A1 |0 A2 |i A3 : s ∈ Z 3 }, S (0,i,j) = {|0 A1 |i A2 |j A3 + w s 3 |i A1 |j A2 |0 A3 + w 2s 3 |j A1 |0 A2 |i A3 : s ∈ Z 3 },(27)for i = j ∈ [d − 1]. For i ∈ [d − 1] and s ∈ Z 3 , let |ψ s =|0 A1 |0 A2 |i A3 + w s 3 |0 A1 |i A2 |0 A3 + w 2s 3 |i A1 |0 A2 |0 A3 ∈ S(0, 0, i). Then there must exist three unitary operators P 1 : |0 A1 → |0 A1 , |i A1 → w −2s 3 |1 A1 ; P 2 : |0 A2 → |0 A2 , |i A2 → w −s 3 |1 A2 ; P 3 : |0 A3 → |0 A3 , |i A3 → |1 A3 ,(28) such that |W 3 2 = P 1 ⊗ P 2 ⊗ P 3 |ψ s . Thus |ψ s is LU-equivalent to a W-state. In the same way, we can show that any state in S(0, 0, 0) and S(0, i, j) is LU-equivalent to a GHZ-state, and any state in S(0, i, i) is LU-equivalent to a W-state. Thus, we have the following result. Lemma 6 For all d ≥ 2, the set B 3 d given by Eq. (26) is an OGES in C d ⊗ C d ⊗ C d . Combining Theorem 5 and Lemma 6, we have presented a strongly nonlocal OGES of size 3d 2 − 3d + 2 in C d ⊗ C d ⊗ C d for all d ≥ 2. When d ≥ 3, the size of the strongly nonlocal OGES in our construction is about half of the size 6(d − 1) 2 of the strongly nonlocal OPS in Ref. [29]. However, when d = 3, the authors in Ref. [31] presented a strongly nonlocal UPB of size 19, which is smaller than the size 20 of OGES in Lemma 6. For completeness, we construct a strongly nonlocal OGES of size 18 in C 3 ⊗ C 3 ⊗ C 3 , see Appendix E. See also Table II for the comparisons. It is natural to ask whether Lemma 6 can be extended to B N d with N ≥ 4. Unfortunately, the answer is no for B N d when N = 4. In fact, the following state |ψ i =|0 A1 |0 A2 |i A3 |i A4 + |0 A1 |i A2 |i A3 |0 A4 + |i A1 |i A2 |0 A3 |0 A4 + |i A1 |0 A2 |0 A3 |i A4 ∈ S (0,0,i,i) ⊂ B 4 d(29) is separable in A 1 A 3 |A 2 A 4 bipartition for each i ∈ [d−1], since it can be written as |ψ i = (|0 |i + |i |0 ) A1A3 (|0 |i + |i |0 ) A2A4 .(30) Thus B 4 d is not an OGES. However, we can modify the coefficient matrix B in Eq. (14) to get genuinely entangled states. Systems Sizes of the OPSs References Sizes of the OGESs References C 3 ⊗ C 3 ⊗ C 3 19 [31] 18 Appendix E C d ⊗ C d ⊗ C d , d ≥ 3 6(d − 1) 2 [29] d 3 − (d − 1) 3 + 1 Lemma 6 C d ⊗ C d ⊗ C d ⊗ C d , d ≥ 3 d 4 − (d − 2) 4 [30] d 4 − (d − 1) 4 + 1 Lemma 7 Let B = (b i,j ) i,j∈Z4 :=      1 1 1 2 1 −1 2 −1 5 5 −2 −4 5 −5 −4 2      ,(31) which is a row orthogonal matrix. For each i ∈ [d − 1], define S (0,0,i,i) :={b s,0 |0 A1 |0 A2 |i A3 |i A4 + b s,1 |0 A1 |i A2 |i A3 |0 A4 + b s,2 |i A1 |i A2 |0 A3 |0 A4 + b s,3 |i A1 |0 A2 |0 A3 |i A4 : s ∈ Z 4 }.(32) Let B 4 d be the set obtained from B 4 d by replacing S(0, 0, i, i) with S(0, 0, i, i) for all i ∈ [d − 1].− (d − 1) 4 + 1 in C d ⊗ C d ⊗ C d ⊗ C d . The proof of Lemma 7 is given in Appendix D. Lemma 7 provides a strongly nonlocal OGES in fourpartite systems whose size is smaller than that of the strongly nonlocal OPS in Ref. [30]. See also Table II for a comparison. Before closing this section, we remark that it is not easy to extend Lemma 6 or Lemma 7 to N -partite systems with N ≥ 5. From the proof of Lemma 2, we know that for any state |ψ ∈ B N d , |ψ is entangled in the bipartition A 1 |A 2 A 3 · · · A N . Since B N d has a similar structure under the cyclic permutation of the subsystems {A 1 , A 2 , . . . , A N }, |ψ must be entangled in the biparti- tions {A 1 A 2 . . . A N }\{A i } for all i ∈ [d]. However, this is far from enough since we need to the entangled property for all bipartitions. VI. LOCAL HIDING OF INFORMATION In this section, we indicate that strong quantum nonlocality can be used for local hiding of information [21][22][23][24]42]. Assume that information is encoded in a locally indistinguishable set of N -partite orthogonal states, and the so-called "boss" sends it to his N "subordinates". These subordinates are from different labs, and they can only communicate classically with each other and perform local measurements. Due to the local indistinguishability of the set, the information cannot be fully accessed by the subordinates, and part of it remains hidden. For example, since Bell basis is locally indistinguishable, one can encode 2 classical bits in four Bell states, but only 1 bit can be extracted locally [21,42]. Note that if some subordinates are collusive, that is some subordinates are from the same lab and they can perform joint measurements, then the information could be fully accessed by the N subordinates. For example, when the information is encoded in a three-qubit UPB, then the information could be fully accessed by the three subordinates if any two of them are collusive [4,43,44]. In order to avoid this collusion, the boss can encode the information in a genuinely nonlocal set of N -partite orthogonal states. (Note that if a set of orthogonal states is locally indistinguishable for each bipartition of the sub- encoded in a strongly nonlocal set of N -partite orthogonal states. The so called "boss" sends the information to his N "subordinates", A 1 , A 2 , . . . , A N through some quantum channels. If any k "subordinates" are collusive, k < N , and they can only perform OPLMs, then the information will not be accessible at all. systems, then this set is said to be genuinely nonlocal [36,38]). Thus the information cannot be fully accessed by the subordinates even if any k subordinates are collusive, where k < N . In the above scheme of local hiding of information, we only use genuine quantum nonlocality, and it does not require strong quantum nonlocality. These subordinates may obtain part of the information. However, if the boss encode the information in a strongly nonlocal set of N -partite orthogonal states, and these subordinates can only perform OPLMs, then the information will not be accessible at all. By Theorem 5, the new scheme of local hiding of information can be used for N -partite systems, N ≥ 3. In the local hiding scheme by using strongly nonlocal sets in Fig. 2, there are two ways to reveal the whole information by these subordinates. One way is to perform global measurements when all N subordinates are collusive, since any set of orthogonal pure states can be perfectly distinguished by using global measurements [40]. Another way is to use additional quantum resources such as entanglement [44]. It is known that if some proper Npartite genuinely entangled resource state assists all these N subordinates, then the information could be fully accessed by these subordinates [28,33,36]. Recently, k-uniform quantum information masking was proposed in [45]. When information is masked in a set of N -partite states {|ψ j }, it is required that the reductions to k parties of |ψ j are identical for all j. That is any k parties have no information about the value j. There are several differences between k-uniform quantum information masking and the above local hiding of information. The former is based on reductions of states of a set, while the latter is based on local indistinguishability of a set. In k-uniform quantum information, the masked information is completely inaccessible to each set of k parties. However, in the local hiding scheme of information, part of information remains hidden if k parties are collusive. In a sense, our scheme of local hiding of information is weaker than the scheme of k-uniform quantum information masking, which can also be seen from the range of parameters. In k-uniform quantum information masking, the parameters satisfy k ≤ N 2 [45] if the states from a whole space are masked. While in local hiding of information, k has a wide range, that is k ≤ N − 1. VII. CONCLUSION AND DISCUSSION In this paper, we constructed a set of orthogonal entangled states in (C d ) ⊗N for d ≥ 2 and N ≥ 3 from cyclic permutation group action, and showed strong quantum nonlocality in general N -partite systems for the first time. This result positively answered an open questions proposed in Refs. [33]. Further, when N = 3, 4 and d ≥ 3, we presented a strongly nonlocal orthogonal genuinely entangled set, which has a smaller size than that of all known strongly nonlocal OPS in the same systems. Finally, we connected strong quantum nonlocality with local hiding of information. There are also some open questions left. Whether we can show strong quantum nonlocality in C d1 ⊗C d2 ⊗· · ·⊗ C d N for any d i ≥ 2 and 1 ≤ i ≤ N ? How to construct a strongly nonlocal orthogonal genuinely entangled set in (C d ) ⊗N for any d ≥ 2 and N ≥ 5? Whether we can use other permutation group actions to construct a strongly nonlocal orthogonal entangled set? Can we construct a strongly nonlocal UPB for N -partite systems with N ≥ 5? What is the minimum size of a strongly nonlocal OPS in (C d ) ⊗N ? Acknowledgments We thank Mao-Sheng Li and Jun Gao for discussing this problem. F.S., Z.Y. and X. N ) | j ∈ Z k } are distinct, the states in {|i (j) 1 A1 |i (j) 2 A2 · · · |i (j) N A N | j ∈ Z k } are mutually orthogonal. Since the coefficient matrix B = (w sj k ) s,j∈Z k is a complex Hadamard matrix of order k, the states in S x must be mutually orthogonal. Next, we show that any state in S x is an entangled state. Without loss of generality, we assume that x = (i (0) 1 , i (0) 2 , . . . , i (0) N ). By definition, O x = {σ r x | r ∈ Z N }. We have the following two claims. (i) We claim that the (N − 1)-tuples {(i (j) 2 , . . . , i (j) N ) | j ∈ Z k } must be mutually distinct. If (i (j) 2 , . . . , i (j) N ) = (i (j ) 2 , . . . , i (j ) N ) for some j = j ∈ Z k , then i (j) 1 = i (j ) 1 . Otherwise, if i (j) 1 = i (j ) 1 , then (i (j) 1 , i (j) 2 , . . . , i (j) N ) / ∈ O x or (i (j ) 1 , i (j ) 2 , . . . , i (j ) N ) / ∈ O x . We obtain that (i (j) 1 , i (j) 2 , . . . , i (j) N ) = (i (j ) 1 , i (j ) 2 , . . . , i (j ) N ) , which is impossible. We obtain this claim. (ii) We also claim that there must exist j = j ∈ Z k , such that i (j) 1 = 0 and i (j ) 1 = 0. Since x = (0, 0 . . . , 0) ∈ X N d , there must exist 1 ≤ t = t ≤ N , such that i (0) t = 0 and i (0) t = 0. Further, there must exist j = j ∈ Z k , such that (i (j) 1 , i (j) 2 , . . . , i (j) N ) = σ (t−1) x, and (i (j ) 1 , i (j ) 2 , . . . , i (j ) N ) = σ (t −1) x. Then i (j) 1 = i (0) t = 0 and i (j ) 1 = i (0) t = 0. We obtain this claim. For s ∈ Z k , let |ψ s = j∈Z k w sj k |i (j) 1 A1 |i (j) 2 A2 · · · |i (j) N A N ∈ S x . By the above two claims, we know that rank(|ψ s A1|A2...A N ) ≥ 2. Thus, the set S x is an OES. Further, if O x ∩ O y = ∅, then any state in S x is orthogonal to any state in S y . Thus, B N d is an OES. Suppose that we have two orthogonal sets in C d1 ⊗ C d2 , {|ψ i = j∈Zm b i,j |p j |r j } i∈Zm , {|ϕ k = ∈Zn c k, |q |s } k∈Zn ,(B1) where {|r j } j∈Zm , {|s } ∈Zn are two nonempty subsets of B 2 , and |p j , |q ∈ B 1 for j ∈ Z m and ∈ Z n . Here we do not require that these p j 's (q l 's, respectively) are distinct. Further, assume that ψ i |I ⊗ E|ϕ k = 0, for i ∈ Z m , k ∈ Z n .(B2) If p j = q for some j ∈ Z m and ∈ Z n , then a rj ,s = a s ,rj = 0. Proof. Let B = (b i,j ) i,j∈Zm and C = (c k, ) k, ∈Zn . Since {|ψ i } i∈Zm and {|ϕ k } k∈Zn are two orthogonal sets, we obtain that B and C are both full-rank. Denote B = (b i,j ) i,j∈Zm , where b i,j is the complex conjugate of b i,j . Then B ⊗ C is also full-rank. Next, ψ i |I ⊗ E|ϕ k = j∈Zm ∈Zn b i,j c k, p j |q r j |E|s = j∈Zm ∈Zn b i,j c k, p j |q a rj ,s = u i,k · X = 0,(B4) where u i,k = (b i,0 c k,0 , b i,0 c k,1 , · · · , b i,0 c k,n−1 , · · · , b i,m−1 c k,0 , b i,m−1 c k,1 , · · · , b i,m−1 c k,n−1 ) = (b i,0 , b i,1 , · · · , b i,m−1 ) ⊗ (c k,0 , c k,1 , · · · , c k,n−1 ) and X = ( p 0 |q 0 a r0,s0 , p 0 |q 1 a r0,s1 , · · · , p 0 |q n−1 a r0,sn−1 , · · · , p m−1 |q 0 a rm−1,s0 , p m−1 |q 1 a rm−1,s1 , · · · , p m−1 |q n−1 a rm−1,sn−1 ) T . Since Eq. (B4) holds for any i ∈ Z m , and k ∈ Z n . We obtain that B ⊗ C · X = 0. Since B ⊗ C is full-rank, we have X = 0. That is p j |q a rj ,s = 0, for j ∈ Z m , ∈ Z n . If p j = q for some j ∈ Z m and ∈ Z n , then we obtain that a rj ,s = 0. Since E = E † , we also have a s ,rj = 0. Lemma 9 Assume that B i := {|0 , |1 , . . . , |d i − 1 } is the computational basis of C di for i = 1, 2. Let a d 2 × d 2 Hermitian matrix E = (a i,j ) i,j∈Z d 2 be the matrix representation of a Hermitian operator E under the basis B 2 . Suppose that we have an orthogonal set in C d1 ⊗ C d2 , {|ψ i = j∈Zm b i,j |p j |r j } i∈Zm ,(B5) where {|r j } j∈Zm is a subset of B 2 , and |p j ∈ B 1 for j ∈ Z m . Here we do not require that these p j 's are distinct. Further, assume that ψ i |I ⊗ E|ψ k = 0, for i = k ∈ Z m . (B6) If there exists t ∈ Z m , such that a rt,rj = 0 for j = t ∈ Z m , and b i,t = 0 for i ∈ Z m , then we obtain that a ri,rj = a rj ,ri = 0, for i = j and p i = p j , and a r0,r0 = a ri,ri , for i = 0 ∈ Z m . Proof. Without loss of generality, we can assume that t = 0. Let B = (b i,j ) i,j∈Zm , C = (c i,j ) i,j∈Zm , where c i,j = b i,j / j∈Zm |b i,j | 2 . First, we can normalize the states {|ψ i = j∈Zm b i,j |p j |r j } i∈Zm as {|ϕ i = j∈Zm c i,j |p j |r j } i∈Zm . Since the states {|ψ i } i∈Zm are mutually orthogonal, we obtain that the row vectors of B are mutually orthogonal. This implies that C is a unitary matrix. We can also obtain that ϕ i |I ⊗ E|ϕ k = 0, for i = k ∈ Z m . By the same discussion as Eq. (B4), we have ϕ i |I ⊗ E|ϕ k = u i,k · X = 0,(B9) where u i,k = (c i,0 , c i,1 , · · · , c i,m−1 ) ⊗ (c k,0 , c k,1 , · · · , c k,m−1 ), and X = ( p 0 |p 0 a r0,r0 , p 0 |p 1 a r0,r1 , · · · , p 0 |p m−1 a r0,rm−1 , · · · , p m−1 |p 0 a rm−1,r0 , p m−1 |p 1 a rm−1,r1 , · · · , p m−1 |p m−1 a rm−1,rm−1 ) T . Note that C ⊗ C = u T 0,0 , u T 0,1 , . . . u T 0,m−1 , . . . u T m−1,0 , u T m−1,1 , . . . u T m−1,m−1 T is still a unitary matrix. Since Eq. (B9) holds for any i = k, there exists e i ∈ C for each i ∈ Z m such that X = e 0 u † 0,0 + e 1 u † 1,1 + · · · + e m−1 u † m−1,m−1 . (B10) By the condition, we have a r0,rj = 0 for j = 0 ∈ Z m , and b i,0 = 0 for i ∈ Z m . This also implies that c i,0 = 0 for i ∈ Z m . By Eq. (B10), we have    0 . . . 0    =    p 0 |p 1 a r0,r1 . . . p 0 |p m−1 a r0,rm−1    =    c 0,1 c 1,1 · · · c m−1,1 . . . . . . . . . . . . c 0,m−1 c 1,m−1 · · · c m−1,m−1       c 0,0 e 0 . . . c m−1,0 e m−1    . (B11) Since C † is a unitary matrix, there exists k ∈ C, such that c i,0 e i = kc i,0 for i ∈ Z m . Further, since c i,0 = 0 for i ∈ Z m , we have e i = k, for i ∈ Z m . Then by Eq. (B10), we obtain that p i |p j a ri,rj = δ i,j k, for i, j ∈ Z m . Thus, if i = j, and p i = p j , then we have a ri,rj = 0. Since E = E † , we also have a rj ,ri = 0. If i = j, then we have a ri,ri = k, i.e. a r0,r0 = a ri,ri , for j = 0 ∈ Z m . (2) Next, we consider the case k = 2. Then there must exist an i = 0, where 1 ≤ ≤ N − 1. We only need to consider the case O (0,i1,i2,...,i N −1 ) = O (0,j1,j2,...,j N −1 ) . We denote i 0 = 0. Then σ (0, i 1 , i 2 , . . . , i N −1 ) = (i , i +1 , i +2 , . . . , i −1 ) ∈ O (0,i1,i2,...,i N −1 ) , where (i +1 , i +2 , . . . , i −1 ) ∈ A 1 . For any (n 0 , n 1 , . . . , n N −1 ) ∈ O (0,i1,i2,...,i N −1 ) , we must have (n 1 , . . . , n N −1 ) ∈ A 1 or A 2 . For (i 1 , i 2 , . . . , i N −1 ) = (j 1 , j 2 , . . . , j N −1 ) ∈ Z N −1 d , we have shown that a i 1 i 2 ···i N −1 ,j 1 j 2 ···j N −1 = 0 for (i 1 , i 2 , . . . , i N −1 ) ∈ A 1 , (j 1 , j 2 , . . . , j N −1 ) ∈ A 2 , and a i 1 i 2 ···i N −1 ,j 1 j 2 ···j N −1 = 0 for (i 1 , i 2 , . . . , i N −1 ), (j 1 , j 2 , . . . , j N −1 ) ∈ A 1 . This implies that a i +1 ,i +2 ··· ,i −1 ,n1n2···n N −1 = 0 for any (n 0 , n 1 , . . . , n N −1 ) = (i , i +1 , . . . , i −1 ) ∈ O (0,i1,i2,...,i N −1 ) . Then we can apply Lemma 9 to S (0,i1,i2,...,i N −1 ) , we obtain that a i1i2···i N −1 ,j1j2···j N −1 = 0 for (i 1 , i 2 , . . . , i N −1 ), (j 1 , j 2 , . . . , j N −1 ) ∈ A 2 . (3) By repeating this process N − 2 times, we obtain that a i1i2···i N −1 ,j1j2···j N −1 = 0, for (i 1 , i 2 , . . . , i N −1 ), (j 1 , j 2 , . . . , j N −1 ) ∈ A k for 1 ≤ k ≤ N − 2. (4) Finally, we consider the case k = N − 1. If (i 1 , i 2 , . . . , i N −1 ) = (j 1 , j 2 , . . . , j N −1 ) ∈ A N −1 , then O (0,i1,i2,...,i N −1 ) ∩O (0,j1,j2,...,j N −1 ) = ∅. By the same discussion as (i), we obtain that a i1i2···i N −1 ,j1j2···j N −1 = 0, for (i 1 , i 2 , . . . , i N −1 ), (j 1 , j 2 , . . . , j N −1 ) ∈ A N −1 . In a word, we know that the off-diagonal elements of E are all zeros. Next, we consider the diagonal elements of E. For any (i 1 , i 2 , . . . , i N −1 ) ∈ A k for k = 0 ∈ Z N , there must exist an (n (0,k−1) , n (1,k−1) , . . . , n (N −1,k−1) ) ∈ O (0,i1,i2,...,i N −1 ) , such that (n (1,k−1) , . . . , n (N −1,k−1) ) ∈ A k−1 . Applying Lemma 9 to S (0,i1,i2,...,i N −1 ) , we obtain that a i1i2···i N −1 ,i1i2···i N −1 = a n (1,k−1) n (2,k−1) ···n (N −1,k−1) ,n (1,k−1) n (2,k−1) ···n (N −1,k−1) . Next, there must exist an (n (0,k−2) , n (1,k−2) , . . . , n (N −1,k−2) ) ∈ O (0,n (1,k−1) ,...,n (N −1,k−1) ) , such that (n (1,k−2) , . . . , n (N −1,k−2) ) ∈ A k−2 . Applying Lemma 9 to S (0,n (1,k−1) ,...,n (N −1,k−1) ) , we obtain that a n (1,k−1) n (2,k−1) ···n (N −1,k−1) ,n (1,k−1) n (2,k−1) ···n (N −1,k−1) = a n (1,k−2) n (2,k−2) ···n (N −1,k−2) ,n (1,k−2) n (2,k−2) ···n (N −1,k−2) . By repeating this process k times, we obtain that a i1i2···i N −1 ,i1i2···i N −1 = a n (1,0) n (2,0) 00···0 . We obtain that the diagonal elements of E are all equal. Therefore, E is trivial. This completes the proof. d is strongly nonlocal. Since B 4 d has a similar structure to B 4 d . We can also obtain that B 4 d is strongly nonlocal. We only need to show that B 4 d is an OGES. It is easy to see that any state in S (0,0,0,0) and S (0,i,0,i) is LU-equivalent to a GHZ state for i ∈ [d − 1], and any state in S (0,0,0,i) is LU-equivalent to a W state for i ∈ [d − 1]. For the set S (0,0,i,j) , S (0,p,0,q) and S (0,k, ,m) , we need to use the following claim. Claim: For a state |ψ ∈ (C d ) ⊗N , if there exists a product operator P 1 ⊗ P 2 ⊗ · · · ⊗ P N such that P 1 ⊗ P 2 ⊗ · · · ⊗ P N |ψ is a genuinely entangled state, then |ψ is also a genuinely entangled state. ···n (N −1,0) ,n (1,0) n (2,0) ···n (N −1,0) , where (n (1,0) , n (2,0) , · · · , n (N −1,0) ) ∈ A 0 . That is a i1i2···i N −1 ,i1i2···i N −1 = a 00···0,S (0,0,0,0) ={|0 A1 |0 A2 |0 A3 |0 A4 ± |1 A1 |1 A2 |1 A3 |1 A4 }, S (0,0,0,i) ={|0 A1 |0 A2 |0 A3 |i A4 + w s 4 |0 A1 |0 A2 |i A3 |0 A4 + w 2s 4 |0 A1 |i A2 |0 A3 |0 A4 + w 3s 4 |i A1 |0 A2 |0 A3 |0 A4 : s ∈ Z 4 }, S (0,0,i,i) ={b s,0 |0 A1 |0 A2 |i A3 |i A4 + b s,1 |0 A1 |i A2 |i A3 |0 A4 + b s,2 |i A1 |i A2 |0 A3 |0 A4 + b s,3 |i A1 |0 A2 |0 A3 |i A4 : s ∈ Z 4 }, S (0,i,0,i) ={|0 A1 |i A2 |0 A3 |i A4 ± |i A1 |0 A2 |i A3 |0 A4 }, S (0,0,i,j) ={|0 A1 |0 A2 |i A3 |j A4 + w s 4 |0 A1 |i A2 |j A3 |0 A4 + w 2s 4 |i A1 |j A2 |0 A3 |0 A4 + w 3s 4 |j A1 |0 A2 |0 A3 |i A4 : s ∈ Z 4 }, S (0,p,0,q) ={|0 A1 |p A2 |0 A3 |q A4 + w s 4 |p A1 |0 A2 |q A3 |0 A4 + w 2s 4 |0 A1 |q A2 |0 A3 |p A4 + w 3s 4 |q A1 |0 A2 |p A3 |0 A4 : s ∈ Z 4 }, S (0,k, ,m) ={|0 A1 |k A2 | A3 |m A4 + w s 4 |k A1 | A2 |m A3 |0 A4 + w 2s 4 | A1 |m A2 |0 A3 |k A4 + w 3s 4 |m A1 |0 A2 |k A3 | A4 : s ∈ Z 4 }. The proof of the above claim is as follows. If |ψ is not a genuinely entangled state, then there exists a bipartition A|B such that |ψ = |ψ A ⊗ |ψ B . For any product operator P 1 ⊗ P 2 ⊗ · · · ⊗ P N , P 1 ⊗ P 2 ⊗ · · · ⊗ P N (|ψ A ⊗ |ψ B ) is still separable in A|B bipartition, which is not a genuinely entangled state. Contradiction. This completes the proof of this claim. For s ∈ Z 4 , and i = j ∈ [d − 1], let |ψ s = |0 A1 |0 A2 |i A3 |j A4 + w s 4 |0 A1 |i A2 |j A3 |0 A4 + w 2s 4 |i A1 |j A2 |0 A3 |0 A4 + w 3s 4 |j A1 |0 A2 |0 A3 |i A4 ∈ S (0,0,i,j) , and P = (|0 0| + |0 j| + w −2s 4 |1 i|) A1 (|0 0| + |0 j| + w −s 4 |1 i|) A2 (|0 0| + |0 j| + |1 i|) A3 (|0 0| + |0 j| + w −3s 4 |1 i|) A4 . Then P |ψ s = |0 A1 |0 A2 |1 A3 |0 A4 + |0 A1 |1 A2 |0 A3 |0 A4 + |1 A1 |0 A2 |0 A3 |0 A4 + |0 A1 |0 A2 |0 A3 |1 A4 = |W 4 2 , which is a W-state. By the above claim, |ψ s is a genuinely entangled state. In the same way, we can also show that any state in S (0,p,0,q) and S (0,k, ,m) is a genuinely entangled state for p < q ∈ [d − 1] and k, , m ∈ [d − 1]. Off-diagonal elements and Diagonal elements of E = M † M = (a ij,k ) i,j,k, ∈Z3 . We apply Lemma 8 to two sets among {A i } 6 i=1 , and apply Lemma 9 to one set among {A i } 6 i=1 . Sets Elements Sets Elements A1, A2 a00,02 = a00,20 = a00,22 = 0 A2, A5 a02,21 = a20,21 = a00,10 = 0 A1, A3 a00,11 = a10,11 = a01,11 = 0 A3, A4 a11,12 = a10,20 = a01,20 = 0 A1, A4 a00,12 = a11,20 = a01,22 = 0 A3, A5 a11,21 = a02,10 = a01,02 = 0 A1, A5 a00,21 = a02,11 = a10,22 = 0 A4, A5 a12,21 = a02,20 = a01,10 = 0 A1, A6 a11,22 = a21,22 = a12,22 = 0 A4, A6 a20,22 = a01,21 = a01,12 = 0 A2, A4 a02,12 = a12,20 = a00,01 = 0 A5, A6 a02,22 = a10,21 = a10,12 = 0 A1 a00,00 = a11,11 = a22,22 A4 a12,12 = a20,20 A2 a02,02 = a20,20 = a00,00 A5 a02,02 = a21,21 A3 a11,11 = a10,10 = a01,01 Finally, we consider S (0,0,i,i) . For s ∈ Z 4 and i ∈ [d − 1], let |λ s =b s,0 |0 A1 |0 A2 |i A3 |i A4 + b s,1 |0 A1 |i A2 |i A3 |0 A4 + b s,2 |i A1 |i A2 |0 A3 |0 A4 + b s,3 |i A1 |0 A2 |0 A3 |i A4 ∈ S (0,0,i,i) . We can calculate that rank(|λ s A1|A2A3A4 ) = rank(|λ s A2|A3A4A1 ) = rank(|λ s A3|A4A1A2 ) = rank(|λ s A4|A1A2A3 ) = rank(|λ s A1A3|A2A4 ) = 2, and rank(|λ A1A2|A3A4 ) = rank(|λ A1A4|A2A3 ) = 4. This implies that |λ s is entangled in every bipartition, and it is a genuinely entangled state. In a word, B 4 d is an OGES. This completes the proof. Lemma 1 [ 31 ] 131Let S := {|ψ j } be a set of orthogonal states in a multipartite system⊗ N i=1 H Ai . For each i = 1, 2, . . . , N , define B i = {A 1 A 2 . . . A N } \ {A i }be the joint party of all but the ith party. Then the set S has the property of the strongest nonlocality if the following condition holds for any 1 ≤ i ≤ N : if party B i performs any OPLM, then the OPLM is trivial. FIG. 1 : 1By definition, the number of states in S x equals the size of O We represent two orbits by two cycles. For the left cycle, the orbit is O A 2 , 2A 3 and A 4 come together to perform a joint OPLM {E = M † M }, where each POVM element can be written as a 8 × 8 matrix in the basis {|i A2 |j A3 |k A4 } i,j,k∈Z2 , E = (a ijk, mn ) i,j,k, ,m,n∈Z2 . Then the postmeasurement states {I A1 ⊗ M |ψ : |ψ ∈ B 4 FIG. 2 : 2Local hiding of information. The information is N Z. were supported by the NSFC under Grants No. 11771419 and No. 12171452, the Anhui Initiative in Quantum Information Technologies under Grant No. AHY150200, and the National Key Research and Development Program of China 2020YFA0713100. L.C. was supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research Funds for the Central Universities (Grant No. ZG216S2005). Appendix A: The proof of Lemma 2 Proof. We have shown that |B N d | = d N − (d − 1) N + 1. Next, we show that B N d is an OES. Obviously, S (0,0,...,0) is an OES. We only need to consider S x for x = (0, 0 . . . , 0) ∈ X N d . Assume O x = {(i ) | j ∈ Z k }, k ≥ 2. Since any two elements of O x = {(i B: Two lemmas for showing the strong nonlocality Lemma 8 Assume that B i := {|0 , |1 , . . . , |d i − 1 } is the computational basis of C di for i = 1, 2. Let a d 2 × d 2 Hermitian matrix E = (a i,j ) i,j∈Z d 2 be the matrix representation of a Hermitian operator E under the basis B 2 . for i = j ∈ [d − 1], p < q ∈ [d − 1] and k, , m ∈ [d − 1]. Note that B = (b i,j ) Since |B 4 d | = d 4 − (d − 1) 4 + 1, and |S (0,0,i,i) | = |S (0,0,i,i) | for i ∈ [d − 1], we obtain that |B 4 d | = |B 4 d | = d 4 − (d − 1) 4 + 1.By Theorem 5, we know that B 4 Throughout this paper, we only consider pure states, and we do not normalize states for simplicity. A positive operator-valued measure (POVM) on Hilbert space H is a set of semidefinite operators{E m = M † m M m } such that m E m = I H ,where each E m is called a POVM element, and I H is the identity operator on H. We only consider POVM measurements. A measurement is trivial if all its POVM elements are proportional to the identity operator. Otherwise, the measurement is called nontrivial.For an integer d ≥ 2, we denote Z d := {0, 1, . . . , d − 1},Z N d := Z d ×Z d ×· · ·×Z d and (C d ) ⊗N := C d ⊗C d ⊗· · ·⊗C d ,where Z d and C d both repeat N times. We assume that {|i } i∈Z d is a computational basis of C d . For a bipartite state |ψ AB ∈ C m ⊗ C n , it can be expressed by Then |ψ AB is an entangled state if and only if rank(|ψ AB ) ≥ 2. An N -partite state |ψ A1A2···A N is called an entangled state, if it is entangled for at least one bipartition of the subsystems {A 1 , A 2 , . . . , A N }. Moreover, an N -partite state |ψ A1A2···A N is called a genuinely entangled state, if it is entangled for each bipartition of the subsystems {A 1 , A 2 , . . . , A N }.The most well known genuinely entangled states are GHZ states and W states. An N -qudit GHZ state can be expressed by is strongly nonlocal in (C d ) ⊗N for all d ≥ 2 and N ≥ 3. Since B N d has a similar structure under the cyclic permutation of the subsystems {A 1 , A 2 , . . . , A N }, we only need to show that the party A 2 A 3 . . . A N can only perform a trivial OPLM for B N d by Lemma 1. We give Lemma 8 and Lemma 9 in Appendix B, which are useful for showing the strong nonlocality for OESs. These two lemmas can greatly reduce the complexity of proof when they are applied to {S x } of B N d . First, we give two examples to illustrate the idea of proof. Example 3 In C 2 ⊗C 2 ⊗C 2 , the set B 3 2 given by Eq. (17) is a strongly nonlocal OES of size 8. Proof. Let A 2 and A 3 come together to perform a joint OPLM {E = M † M }, where each POVM element can be written as a 4 × 4 matrix in the basis {|i A2 |j A3 TABLE TABLE II : IIComparisons of the sizes between strongly nonlocal OPSs and strongly nonlocal OGESs. Similar to Lemma 6, we can show that any state in S (0,0,0,0) and S (0,i,0,i) is LU-equivalent to a GHZ state for i ∈ [d − 1], and any state in S (0,0,0,i) is LU-equivalent to a W state for i ∈ [d − 1]. It is not clear for other states in B 4 d . However, we are still able to show the genuine entanglement for all other states in B 4 d .Lemma 7 For all d ≥ 2, the set B 4 d is a strongly nonlocal OGES of size d 4 TABLE III : III for the same discussion as (i). If O (0,i1,i2,...,i N −1 ) = O (0,j1,j2,...,j N −1 ) , then σ (0, i 1 , i 2 , . . . , i N −1 ) = (i , 0, 0, . . . , 0) ∈ O (0,i1,i2,...,i N −1 ) , where (0, 0, . . . , 0) ∈ A 0 . For any (n 0 , n 1 , . . . , n N −1 ) ∈ O (0,i1,i2,...,i N −1 ) , we must have (n 1 , . . . , n N −1 ) ∈ A 0 or A 1 . For (i 1 , i 2 , . . . , i N −1 ) = (j 1 , j 2 , . . . , j N −1 ) ∈ Z N −1 d , we have shown that a i 1 i 2 ···i N −1 ,j 1 j 2 ···j N −1 = 0 for (i 1 , i 2 , . . . , i N −1 ) ∈ A 0 , (j 1 , j 2 , . . . , j N −1 ) ∈ A 1 . This implies that a 00···0,n1n2···n N −1 = 0 for any (n 0 , n 1 , . . . , n N −1 ) = (i , 0, . . . , 0) ∈ O (0,i1,i2,...,i N −1 ) . Then we can apply Lemma 9 to S (0,i1,i2,...,i N −1 ) , and we obtain that a i1i2···i N −1 ,j1j2···j N −1 = 0 for (i 1 , i 2 , . . . , i N −1 ), (j 1 , j 2 , . . . , j N −1 ) ∈ A 1 . Appendix C: The proof ofTheorem 5Proof.Let For an (i 1 , i 2 , . . . , i N −1 ) ∈ Z N −1 d , we denote wt(i 1 , i 2 , . . . , i N −1 ) as the number of nonzero i k for 1 ≤ k ≤ N − 1. We also define N subsets of Z N −1 d ,There are two cases. 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