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anindya-hf-2002

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\begin{align*}\hat R^i_j= R_{jkl}^i \nabla z^k \wedge \nabla z^l + g{\cal F}^\Lambda \partial_j k^i_\Lambda\end{align*}

\begin{align*}(\lambda-\Delta + b \cdot \nabla)u=f, \end{align*}

G ( z _ { 1 } , z _ { 2 } ) = \left( \begin{array} { l l } { { G _ { + + } ( z _ { 1 } , z _ { 2 } ) } } & { { G _ { + - } ( z _ { 1 } , z _ { 2 } ) } } \\ { { G _ { - + } ( z _ { 1 } , z _ { 2 } ) } } & { { G _ { -- } ( z _ { 1 } , z _ { 2 } ) } } \\ \end{array} \right)

d s ^ { 2 } = d t ^ { 2 } - \left( 1 + { \frac { 4 m } { r } } \right) d \vec { x } ^ { 2 } \ .

\begin{align*} \Psi_m(z)=P_m(z)\left(\frac{M_{|m|}(z)}{M_{|m|}(i\lambda_m)}\right)^{\frac{C_1}{K_1}}\frac{\sin(\delta(z-i\lambda_m))}{\delta(z-i\lambda_m)},\end{align*}

\begin{align*}\begin{aligned}\Delta\log\frac{\rho}{e^{-\Psi}}=\mbox{div}\frac{\nabla\frac{\rho}{e^{-\Psi}}}{\frac{\rho}{e^{-\Psi}}}=-\frac{\bigg|\nabla\frac{\rho}{e^{-\Psi}}\bigg|^{2}}{\bigg(\frac{\rho}{e^{-\Psi}}\bigg)^{2}}+\frac{\Delta\frac{\rho}{e^{-\Psi}}}{\frac{\rho}{e^{-\Psi}}}=:I_1+I_2. \end{aligned} \end{align*}

\begin{align*}|(h-j)A' - jA'| = |(h-j)A - jA|.\end{align*}

\begin{align*}Bu=B\partial_x^m\delta_L^v=\partial_x^mB\delta_L^v=\partial_x^m\delta_L^{B'v}=\partial_x^m\delta_L^{Av}\;.\end{align*}

\hat { A } = A + i \psi \Lambda _ { \alpha } + \frac { 1 } { 4 } \psi ^ { 2 } { D ^ { \tau \alpha } D _ { \alpha } ^ { \tau } } A ,

\begin{align*}\tau(A(\varphi_1)A(\varphi_2)\dotsm A(\varphi_n))=\sum_{\pi\in{NC}(n)}\prod_{B\in\pi}C(B,\varphi_1,\dots,\varphi_n),\end{align*}

\sum _ { n } \left( \frac { 1 } { x _ { 1 } + i \pi n } - \frac { 1 } { x _ { 2 } + i \pi n } \right) = \coth ( x _ { 1 } ) - \coth ( x _ { 2 } ) \, ,

\begin{align*}\tau'(s) = \frac{1}{\mu_{C^0}(\dot{\xi}(\tau(s)))} , \tau(0)=0.\end{align*}

L _ { \omega } = \sum _ { \omega ^ { \prime \prime \prime } } [ \sum _ { \omega ^ { \prime } \leq \omega ^ { \prime \prime } } \sqrt { | \omega ^ { \prime } \omega ^ { \prime \prime } | } C _ { \omega ^ { \prime } \omega ^ { \prime \prime } } ^ { \omega ^ { \prime \prime \prime } } \: C _ { \omega \omega ^ { \prime \prime \prime } } ^ { 0 } \: \tilde { \alpha } _ { \omega ^ { \prime } } \: \tilde { \alpha } _ { \omega ^ { \prime \prime } } + \sum _ { \omega ^ { \prime \prime } < \omega ^ { \prime } } \sqrt { | \omega ^ { \prime } \omega ^ { \prime \prime } | } C _ { \omega ^ { \prime } \omega ^ { \prime \prime } } ^ { \omega ^ { \prime \prime \prime } } \: C _ { \omega \omega ^ { \prime \prime \prime } } ^ { 0 } \: \tilde { \alpha } _ { \omega ^ { \prime \prime } } \: \tilde { \alpha } _ { \omega ^ { \prime } } ]

\delta S [ \theta , \bar { A } _ { + } , \bar { \Pi } ^ { \perp - } ] = \frac { k } { 8 \pi } \int _ { \partial \cal M } \partial _ { + } \theta \, \partial _ { - } \theta + \partial _ { - } \theta \left( \bar { A } _ { + } - \mathrm { ~ \frac { 8 \ p i } { k } ~ } \bar { \Pi } ^ { \perp - } \right) \, .

\begin{align*} h(t_0^1, \dot{t}_0^1) = \int h (t_{T_1}^1, \dot{t}_{T_1}^1)d \mathbb P_1 \otimes d\mathbb P_2 = \int h (t_{T_1}^1, \dot{t}_{T_1}^1)d \mathbb P. \end{align*}

\begin{align*}\lim_{k\to\infty}-\frac{\ln Q(\frac{1}{k}\sum_{i=1}^k X_i\geq x)}{k}=\Lambda^*(x)\end{align*}

\begin{align*}{\bf E}_\pm(\xi_\pm)={\rm T}_\pm(\xi_\pm){\bf E}_0{\rm T}^{\dagger}_\pm(\xi_\pm),\end{align*}

G r = \frac { ( O ( 3 , 1 9 ) ) ^ { + } } { ( O ( 2 ) \times O ( 1 , 1 9 ) ) ^ { + } } ,

\begin{align*} \sum_{k=0}^n (-1)^k \binom{n}{k} B_{n-k}B_k=0 \end{align*}

\begin{align*}a > 1 \; \; S = \emptyset\end{align*}

\begin{gather*}\omega_1 = f(x,z,p)dx_1\wedge \cdots \wedge dx_n,\omega_2 =g(x,z,p)dp_1\wedge \cdots \wedge dp_n.\end{gather*}

\begin{align*}\begin{aligned}T_iT_{j}&=T_jT_i |i-j|>1, \\T_{i}T_{i+1}T_i&=T_{i+1}T_iT_{i+1} \end{aligned}\end{align*}

\begin{align*}f^{3,0}(Y,w) & = f^{0,3}(Y,w) = 1,\\ f^{1,1}(Y,w) &= f^{2,2}(Y,2) = k(Y,w),\\ f^{2,1}(Y,w) &= f^{1,2}(Y,w) = ph(Y,w) -2 + h^{2,1}(Z),\end{align*}

\begin{align*} S_{\Psi}=\psi S=e^{(\Psi ,\cdot\, )}S,\end{align*}

\begin{align*}\Delta_v^{\star} := \delta_{i_v^{\star} A} A + i_v^{\star} DA\,, \end{align*}

W _ { \alpha } \equiv Z _ { N } ^ { \alpha } ( 1 ) = \frac { M ( M + \alpha ) ( M + 2 \alpha ) \cdots ( M + ( N - 1 ) \alpha ) } { N ! } \, \, \, ,

\overline { { { \psi } } } ( \vec { x } ) \psi ( \vec { y } ) \longrightarrow \overline { { { \psi } } } ( \vec { x } ) e ^ { - i \Lambda ( \vec { x } ) } e ^ { i \Lambda ( \vec { y } ) } \psi ( \vec { y } )

\begin{align*}q_3^{\rm pole} = -i \left(-\frac{1}{2}\delta + \frac{\left|{\bf q}\right|^2}{2m} + \Sigma_-(q_3^{\rm pole},\left|{\bf q}\right|) \right), \end{align*}

\begin{align*}a(0)=0.\end{align*}

\begin{align*}0=F^n\subset F^{n-1} \subset \cdots \subset F^1 \subset F^0 \subset F^{-1} \subset \cdots \subset F^{-n}=E\,, \end{align*}

\begin{align*}a_{\ell+2,0,0}-{q}^{1/2}\sigma_1(\textbf{z})a_{\ell+1,0,0}+q\sigma_2(\textbf{z})a_{\ell,0,0}-q^{3/2}\sigma_3(\textbf{z})a_{\ell-1,0,0}+q^2\sigma_4(\textbf{z})a_{\ell-2,0,0}=0.\end{align*}

H _ { T } ( q , p ; \lambda ) = H _ { 0 } ( q , p ) \, + \, \lambda ^ { \alpha } \, \phi _ { \alpha } ( q , p ) \ \ \ ,

\begin{align*}V[\phi] = -\mu \sum_{(a)} |\phi^{(a)}|^2 + \lambda \left\{ \sum_{(a)} |\phi^{(a)}|^2 \right\}^2 + \kappa \sum_{(a) \neq (b)} |\bar{\phi}^{(a)} .\phi^{(b)}|^2,\end{align*}

\begin{align*}L_{2}(\mathbb{R} ) = \overline{ \bigoplus_{j \in \mathbb{Z}} W_{p,j}^{\star} } = \overline{ \bigoplus_{j \in \mathbb{Z}} \hat{\bigoplus}_{q = 0}^{p} W_{q,j}^{\star \star} }\end{align*}

\begin{align*}\sim\sum_{\alpha\geq0}\frac{1}{\alpha!}D^{(\alpha)}_{x}g(x)|_{x=e}\, q^{\alpha}(e,x)\end{align*}

\begin{align*}\hat g_{ij}= v^{\frac{4}{n-3}}{}^{\Phi}g_{ij},\end{align*}

\begin{align*} \Lambda_{200}=(A_{2} q+\frac{1}{2}B_{2}q^2)\Phi_0(p)+K_2(p).\end{align*}

\begin{align*}D(X,m)^2=dd^*+d^*d=-\Delta_0(X,m)-\Delta_1(X,m)\end{align*}

\begin{align*} \min_{\beta(\tau)}\sum_{i=1}^n\rho_{\tau}\bigl(y_i-x_i^{T}\beta(\tau)\bigr),\end{align*}

\begin{align*}[\tau^{-1}(D)\Delta D_0]=[D\Delta\tau(D_0)]=[D\Delta D_0]+[D_0\Delta\tau(D_0)]=[D\Delta D_0]+1\,.\end{align*}

\begin{align*}\epsilon ^A(x) = \sum_{\ell =1}^{\mathcal{N}_0} \, \rho^A_{(\ell)} \epsilon^{(\ell)}(x)~,\end{align*}

\begin{align*}L=\frac{1}{pc}\frac{1}{T}~. \end{align*}

\begin{align*} Q=\log_{2}C\end{align*}

\begin{align*}A(y_n...y_1x) - A(y_n...y_1x') = A(0^{n+k}1...) - A(0^\infty) = a_{n+k} - a \ .\end{align*}

\begin{align*} {\left| {\hat h_{\max }^{{\rm{rough}}}} \right|^{\rm{2}}} ={\sum\limits_{{{\bf{r}}_n} \in {I_1}}^{} {\max \left( {{{\left| {r_{n,1}^{}} \right|}^{\rm{2}}},{{\left| {r_{n,2}^{}} \right|}^{\rm{2}}}} \right)} }/{\left| {{I_1}} \right|} - {N_0} \end{align*}

\begin{align*}S_1(l,u,v;N)= \sum_{n=1}^{\infty}\frac{1}{n^{1/2+u+v}}\Delta_{2k,N}(l,n)\end{align*}

\begin{align*}\Delta_z \rho(u+z v)|_{z = 0} \geq 0,\end{align*}

\begin{align*}&u_{ik} := (-1)^{i-k-l}\binom{n}{k}^{-2}\frac{(i+k+l+\sigma)_{n+1-k-l}(\beta+2k+1)_{i-k-l}}{(2i+\sigma)(i-k-l)!(k+l-n)_{i-k-l}(\alpha+l+i+1-k)_{n-i}},\\begin{align*}1ex]&u_{ih} := -u_{i,h-1}\frac{h^2(l+h-n-1)(n+l+\alpha+1-h)}{(h-n-1)^2(h-k)(h+k+\beta)} (h=k+1,k+2,\ldots,n-l);\end{align*}

( \bar { F } \Gamma _ { \mu \nu } E ) ( \bar { E } \Gamma ^ { \nu } E ) + ( \bar { F } \Gamma ^ { i } E ) ( \bar { E } \Gamma _ { \mu i } E ) = 0 .

\begin{align*} \max _{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|\geq \sqrt{\frac{n-d}{d(n-1)}}= \sqrt{\frac{n-(\mathcal{X})}{(\mathcal{X})(n-1)}}. \end{align*}

\begin{align*} [D(a, b), \rho(c)]=\rho([a, b, c]), \end{align*}

\begin{align*}& 2(u+1)(4+t+u)b_1 \\& =2(u+1)(1+2t)c_1+u(u+1)(b_1+c_1) \\& +u(b_1+c_1+ub_3+uc_3)+2u(u+1)b_2\end{align*}

\begin{align*}\begin{aligned}&r_-(x)-\hat r_-(x)=\frac D2\lambda(l+1)(\gamma(\lambda)+x^2)^{\frac{l-3}2}\times\\&\Big[\frac{2c_1}{D\lambda(l+1)}(\gamma(\lambda)+x^2)^{\frac{3-l}2}(\gamma_1+x^2)^l-(l+1)\lambda x^2(\gamma(\lambda)+x^2)^{\frac{l+1}2}-\gamma(\lambda)-lx^2\Big].\end{aligned}\end{align*}

\alpha \equiv z - \theta ( t , r ^ { 1 } , r ^ { 2 } , \cdots , r ^ { d } ) ,

\theta _ { i k } \equiv \theta _ { i } - \theta _ { k }

\begin{align*} \mathcal{A}(s_1,s_2) = \sum_{\substack{ a,b \ge 1 \\ (a,b) \mid h}} \frac{\tau_{k}(a) \tau_{\ell}(b)}{[a,b] a^{s_1} b^{s_2}}. \end{align*}

\begin{align*}&F(A)=\left(a+d-2+\left(\sum_{i=1}^kx_i\right)_{a-1}\right)a-d,\\&g(A)=\sum_{r=1}^{a-1}\left(\sum_{i=1}^kx_i\right)_r+\frac{(a-1)(a+d-1)}{2},\end{align*}

\begin{align*}\psi_{p-1}(x^{(p)}, \xi^{(p)})=\big\{(\xi_p-h_p(x^{(p)}, \xi^{(p+1)}))^2+g_p(x^{(p)},\xi^{(p+1)})\big\}b_p(x^{(p)}, \xi^{(p)})\end{align*}

\begin{align*}\big(1+|J|-p_J(\mu);(a_j)_{j\in J}\big)_{[1],J}=\big(|J|;1,0^{|J|-1}\big)_{[1],J}=\frac{|J|}{24},\end{align*}

\begin{align*}U_{\delta,\xi}(x)= \log\frac{\alpha_{2m}Q\delta^{2m}}{(\delta^{2}+|x-\xi|^{2})^{2m}}\end{align*}

\begin{align*}[R_{\alpha,i},R_{\beta,j}]=s(j-i)R_{\alpha+\beta,i+j}+((i+s-1)\beta-(j+s-1)\alpha)R_{\alpha+\beta-1,i+j}\end{align*}

\begin{align*}u_t - d_u \Delta u = f(x,y,t), (x,y,t) \in \Omega \times [0,T],\end{align*}

\begin{align*}U(x):=x1_{\{x>0\}} -\frac{1}{q}[ (1-x)^{q}-1]1_{\{x\leq 0\}}.\end{align*}

\begin{align*}\nu(1+\varpi^l x+\varpi^l y\tau)\nu'(1+\varpi^l x+\varpi^l y\sigma(\tau))=\psi(\varpi^l {\tau }(x+y{\tau})+\varpi^l \sigma ({\tau}) (x+y\sigma({\tau}))), \forall x,y\in {\mathcal O}_r.\end{align*}

\begin{align*}\mathbf{ I } = \mathop{\vee} \limits_{u \in \mathcal{U}_{ac}^{\mathbf{a}} } \mathbf{s}_{u},\end{align*}

\begin{align*}\rho=\psi\psi^\dagger+[W,W^\dagger]\,.\end{align*}

\begin{align*}J = \mbox{const.} \exp \left\{ \int\!\!d^2 \xi \frac{22}{96 \pi}(\partial_a \log \sqrt{g} )^2 + \ldots \right\} .\end{align*}

\begin{align*} \mathcal{A}_{\gamma}= \begin{array}{rl} \bigoplus_{j=1}^r(\mathcal{A}_j)_{e_j},& \gamma=e\\ (\mathcal{A}_j)_{\gamma},& \gamma\in \Gamma_j\setminus {e_j},\\ 0,& . \end{array} \end{align*}

\begin{align*}c = -\gamma_{12} \gamma_{21} \;,\end{align*}

\begin{align*} v(\theta)=&v(0)+\int_0^\theta w(s) ds\le v(0)+\int_0^\theta(w(0)+\bar Ls)ds\\ =&v(0)+v'(0)\theta+\frac{\bar L}{2}\theta^2. \end{align*}

\begin{align*}W = m\Phi_1\Phi_2 + \tilde m \Sigma_1 \Sigma_2.\end{align*}

\begin{align*}\int_{C s_0 t}^{\Gamma(t)} \frac{ds}{\nu(s)} = t.\end{align*}

\begin{gather*}L_{m,n}^{II,(\alpha)}(x)=\sum_{j=0}^nc_{j,n}x^j.\end{gather*}

\begin{align*}{{\prod_t (dX^1(\xi(t))dX^2(\xi(t)))}\over{{\rm Diff}_1(t\mapsto\tau(t))}}\prod_{i=1}^k ({{dl}\over{dt_i}}dt_i)\end{align*}

V ( r \ll \Gamma _ { 0 } ^ { - 1 } ) = - \frac { G _ { N } } { r }

\begin{align*} (X+Y)^n&=(X+Y)(X+Y)^{n-1}\\ &=(X+Y)\sum\limits_{k\ge0}{n-1\brack k}_ww^{k(n-1-k)}X^kY^{n-1-k}\\ &=\sum\limits_{k\ge0}\left({n-1\brack k-1}_ww^{(k-1)(n-k)}+{n\brack k}_ww^{k(n-1-k)+2k}\right)X^kY^{n-k}\\ &=\sum\limits_{k\ge0}\left({n-1\brack k-1}_ww^{-n+k}+{n\brack k}_ww^k\right)w^{k(n-k)}X^kY^{n-k}\\ &=\sum\limits_{k\ge0}{n\brack k}_ww^{k(n-k)}X^kY^{n-k}. \end{align*}

\tilde { v } \equiv s v = \frac { 1 } { u } , \qquad \tilde { u } \equiv s u = u ( u v - \ddot { \mathrm { l n } u } ) , \qquad v _ { 0 } = 0 ,

\begin{align*}Y_{ii}=\frac{u^2_{ij}}{2\lambda^2_i}\,.\end{align*}

\begin{align*}JO_{n+2}^{(3)}+JO_{n+1}^{(3)}+JO_{n}^{(3)}=\sum_{s=0}^{7}(J_{n+s+2}^{(3)}+J_{n+s+1}^{(3)}+J_{n+s}^{(3)})e_{s}.\end{align*}

\begin{align*}\left((e^{U} R')^2\right)_{extr}|_{r=r_+}= 0 \ ,\end{align*}

\begin{align*}\partial _{t}u=\left( a+ib\right) \left[ \Delta u+Au+V\left( x,t\right)u+F\left( x,t\right) \right] ,x\in R^{n},t\in \left[ 0,1\right] , \end{align*}

\begin{align*}A_{1,2} = \frac{(4\pi)^2}{2\theta^2} \left( \gamma \pm \sqrt{\gamma^2-32 \frac{\lambda \theta^2}{(4\pi)^2}} \right),\end{align*}

\begin{align*}G_\alpha(z) = \frac{\int_0^z H'(w) m^{(\alpha-1) Q(w)}\,dw}{\int_0^1 H'(w)m^{(\alpha-1) Q(w)}\,dw}\end{align*}

\begin{align*}\frac{d}{dt}\|x_\nu^{(\frac{N-1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+3)}(t)\|^2&\leq-\alpha\nu\|x_\nu^{(\frac{N-1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+3)}(t)\|^2+\nu\|x_\nu^{(\frac{N-1}{2}-1)}(t)-x_\nu^{(\frac{N+1}{2}+4)}(t)\|^2 \\&+\nu\|x_\nu^{(\frac{N+1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+2)}(t)\|^2+\frac{1}{\nu}M_{T_1,T_2}^{\frac{N-1}{2},\frac{N+1}{2}+3,2-\alpha}(\omega)\end{align*}

\begin{align*}h_{i}(k) = \sqrt{\overline{\gamma}_{i}}\left(\sqrt{\frac{K_{i}}{K_{i}+1}} e^{j \phi_{i}} + v_{i}(k)\right)\end{align*}

\begin{align*}g = \kappa (g)\mu(g) e^{H(g)}n \in KM A N = G. \end{align*}

\begin{align*}\mathcal Z (\mathcal{H}):= \left\{ f \in L^1 (\Omega) \cap \mathcal D (\mathcal{H}) : \sup_{j \in\mathbb Z} 2^{ M |j|} \big\| \phi_j \big(\sqrt{\mathcal{H}} \big ) f \big\|_{L^1(\Omega)} < \infty M \in \mathbb N \right\}.\end{align*}

\begin{align*} \omega_1\omega_2= (\theta_1\otimes \theta_2 )(\theta_3\otimes \theta_4)= (\theta_1\wedge \theta_3 )\otimes(\theta_2\wedge \theta_4)\in {\mathcal D}^{p+r,q+s}(V).\end{align*}

\begin{align*}{\bf w}_t^{(i)}={\bf w}_{\Delta}^{(i)}\frac{t}{\Delta}+\frac{1}{2}a_{i,0}+\sum_{r=1}^{\infty}\left(a_{i,r}{\rm cos}\frac{2\pi rt}{\Delta}+b_{i,r}{\rm sin}\frac{2\pi rt}{\Delta}\right),\end{align*}

\begin{align*} \beta^n B_n(s, \alpha, \beta) &= B_n( \beta s, \alpha, 1), \end{align*}

\begin{align*}J_0 = \left( \begin{array}{cc} 0 & I \\ I & 0 \end{array} \right).\end{align*}

\begin{align*}\begin{cases}|| \delta_{x} ||_{-2,\alpha}\leq || \delta_{x} ||_{-1,\alpha} \leq C_{1} (1+ M^{\alpha}),\\|| D_{x,y} ||_{-2,\alpha}\leq || D_{x,y} ||_{-1,\alpha} \leq C_{2} (1+ M^{\alpha}).\end{cases}\end{align*}

\begin{align*} \beta_{\lambda}^{[2]}=\frac{1}{8\pi^2}\lambda\nu-\frac{25}{12\pi^2}\,e^4\lambda.\end{align*}

( F , H _ { 1 } , H _ { 2 } ) ( x ) = \frac { d } { d t } \Big | _ { t = 0 } F ( g ^ { t } x ) = d F ( ^ { * } ( d H _ { 1 } \wedge d H _ { 2 } ) ) .

\begin{align*}\phi(x,t) = \phi_c(x) + \eta (x,t),\end{align*}

\begin{align*}ac &= (1,3,9,8,5,4)(2,6,7) \\ b \cdot c^{ac} &= (1,8,6)(2,4,9)(10,11) \\ab \cdot c^{ac} &= (1,9,6,4,8,2)(3,5)(10,11)\\\end{align*}

\begin{align*} \boldsymbol{\eta}=\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 0 & \mathrm{I} & 0 \\ 1 & 0 & 0 \\ \end{array} \right] \end{align*}

\begin{align*} (x_1 -s)(x_1 - s^{-1}) = 0. \end{align*}

\begin{align*} \| \chi(p^m) \| \ll_p \max_{j=1}^r \max_{\ell=0}^{k_j-1} \left\{ |\lambda_j|^{m-\ell} {m\choose \ell} \right\}.\end{align*}

\tilde { Q } _ { - } ^ { 2 } = 2 ( \tilde { H } - M \tilde { Z } )