\begin{align*} \Phi ( s , t ) = \Phi ( \phi , s , t ) : = \sup _ { r > 0 } \left \{ \frac { s } { r } - \frac { t } { \phi ( r ) } \right \} . \end{align*}


\begin{align*} p = _ X J ^ A _ 1 x \leftrightarrow ( x - p ) \in A p \end{align*}


\begin{align*} \lim _ { k \to \infty } \frac { c _ { \langle \alpha _ { k } \rangle } ( Y , \lambda ) ^ { 2 } } { I ( \alpha _ { k } ^ { \Gamma } ) } = \lim _ { k \to \infty } \frac { A ( \alpha _ { k } ^ { \Gamma } ) ^ { 2 } } { 2 | \Lambda ( A ( \alpha _ { k } ^ { \Gamma } ) , \Gamma ) | } = \mathrm { V o l } ( Y , \lambda ) . \end{align*}


\begin{align*} \Delta : M ( A ) \rightarrow M ( A \otimes ^ I A ) = \Delta ( 1 ) M ( A \otimes A ) \Delta ( 1 ) \subseteq M ( A \otimes A ) \end{align*}


\begin{align*} \bigoplus _ { i = 1 } ^ m \left ( n _ i \delta ^ { t _ i } \odot ^ \sharp \nu \delta ^ \tau \right ) = \left ( \bigoplus _ { i = 1 } ^ j ( n _ i - \nu ) \delta ^ { t _ i } \right ) \oplus ( n _ { j + 1 } - \nu ) \delta ^ { + \infty } . \end{align*}


\begin{align*} 2 ^ { ( \mu _ * ( 1 ‐ \lambda ) ‐ \alpha _ * \lambda ) k _ * t } \stackrel { \eqref { h a t _ n u _ d e f } } { = } n ^ { ‐ \hat \nu } ; \end{align*}


\begin{align*} \left \langle \varepsilon , \phi _ { 0 , b , \beta } \left ( \frac { d + 2 } { 2 \beta } - \frac { y ^ 2 } { 2 } \right ) \right \rangle _ { L ^ 2 _ { \rho _ \beta } } = \frac { \gamma \varepsilon _ 1 } { \beta } \| \phi _ { 0 , b , \beta } \| ^ 2 _ { L ^ 2 _ { \rho _ \beta } } + O ( b ^ { \frac { \alpha } { 2 } + \delta } ) \end{align*}


\begin{align*} \int w ( x , t ) d x = 0 \end{align*}


\begin{align*} \Gamma ( \pi ) \ge \frac { 1 } { N } \sum _ { \ell = 1 } ^ N \left ( \frac { \rho _ \ell T _ { \ell , \pi } ^ { a v } } { 2 } + \frac { c _ \ell } { T _ { \ell , \pi } ^ { a v } } \right ) \implies \Gamma ( \tilde { \pi } _ { O F } ^ \star ) \ge \frac { 1 } { N } \sum _ { \ell = 1 } ^ N \left ( \frac { \rho _ \ell T _ { \ell , \tilde { \pi } _ { O F } ^ \star } ^ { a v } } { 2 } + \frac { c _ \ell } { T _ { \ell , \tilde { \pi } _ { O F } ^ \star } ^ { a v } } \right ) , \end{align*}


\begin{align*} 2 Q _ { \tau } ( Z , Z _ { \gamma } ) + 2 \lfloor p _ { i } \theta \rfloor = 0 \end{align*}


\begin{align*} \theta _ { | \Sigma ^ i | } ( x ^ i , 1 ) - \theta _ { | \Sigma ^ i | } ( x ^ i ) \leq \delta , & \ & \theta _ { | \Sigma ^ 1 | } ( x ^ 1 ) = \theta _ { | \Sigma ^ 0 | } ( x ^ 0 ) . \end{align*}


\begin{align*} \begin{array} { l l } \beta _ { 1 } ( r , \theta ) = { ( - 2 z _ { s t } \cos \theta - z _ { t t } \sin \theta ) } , & \gamma _ { 1 } ( r , \theta ) = { - z _ { t t } \cos \theta } , \\ \beta _ { 2 } ( r , \theta ) = { z _ { s s } \cos \theta } , & \gamma _ { 2 } ( r , \theta ) = { - z _ { t t } \sin \theta } , \\ \beta _ { 3 } ( r \theta ) = { z _ { s s } \sin \theta } , & \gamma _ { 3 } ( r , \theta ) = { z _ { s s } \cos \theta + 2 z _ { s t } \sin \theta } . \end{array} \end{align*}


\begin{align*} \mathbf { x } _ m [ t ] = \mathbf { D } ^ { - 1 } \tilde { \mathbf { x } } _ m [ t ] = \frac { 1 } { N } \mathbf { D } ^ { * } \tilde { \mathbf { x } } _ m [ t ] . \end{align*}


\begin{align*} \lambda ( \langle b \rangle ) = \lambda ( b ) = \lambda ( \mathfrak { c } ( \Gamma ( b ) ) ) , \end{align*}


\begin{align*} & \inf _ { \rho , v } \ ; [ \int _ 0 ^ 1 \frac 1 2 \ < v _ t , v _ t \ > _ { \theta ( \rho _ t ) } d t ] \\ & \ ; \ ; d \rho ( t ) + d i v _ G ^ { \theta } ( \rho ( t ) v ( t ) ) + d i v _ G ^ { \theta } ( \rho ( t ) \nabla _ G \Sigma ) d W _ t ^ { \delta } = 0 \\ & \ ; \ ; \rho ( 0 , \omega ) = \rho _ a , \ ; \rho ( 1 , \omega ) = \rho _ b . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \Lambda _ { ( \infty , m ) } ( M , \Gamma ) | } { | \Lambda ( M , \Gamma ) | } = 0 \end{align*}


\begin{align*} \tilde { \Phi } ( 1 \otimes X ) & = 1 \otimes X \\ \tilde { \Phi } ( X \otimes 1 - 1 \otimes X ) & = \Bar { X } \otimes 1 . \end{align*}


\begin{align*} | J ' ( \alpha , \alpha ) | ^ 2 = q . \end{align*}


\begin{align*} \begin{aligned} & - \left ( e ^ { - s } C _ p q \left | \dot { \xi } ( s ) \right | ^ { q - 2 } \dot { \xi } ( s ) \right ) ^ \prime + e ^ { - s } D f \left ( \xi ( s ) \right ) = 0 \\ \Longrightarrow & - \left ( e ^ { - s } C _ p q \left | \dot { \xi } ( s ) \right | ^ { q - 2 } \dot { \xi } ( s ) \right ) ^ \prime \cdot \dot { \xi } ( s ) e ^ s + D f \left ( \xi ( s ) \right ) \cdot \dot { \xi } ( s ) = 0 . \end{aligned} \end{align*}


\begin{align*} x _ t = x + \int _ 0 ^ t \tilde { f } ( s , x _ s ) d s - \tilde { \sigma } B _ t ^ H , 0 \le t \le 1 , x > 0 , \end{align*}


\begin{align*} M _ 6 ( p , H ) = { \pi ^ 2 \over 9 } \left ( 1 + \frac { c _ a } { p } \right ) , \hbox { w h e r e } c _ a = \begin{cases} - 2 a - 1 & \hbox { i f $ a \equiv 0 \pmod 6 $ } , \\ - 3 & \hbox { i f $ a \equiv 2 , 3 \pmod 6 $ } , \\ 2 a + 1 & \hbox { i f $ a \equiv 5 \pmod 6 $ } . \end{cases} \end{align*}


\begin{align*} h h ' & = s + ( 2 + \delta ) A s ^ { 1 + \delta } + ( 1 + \delta ) A ^ 2 s ^ { 1 + 2 \delta } \\ h h '' & = ( 1 + \delta ) \delta A s ^ \delta + ( 1 + \delta ) \delta A ^ 2 s ^ { 2 \delta } \\ ( h ' ) ^ 2 & = 1 + 2 ( 1 + \delta ) A s ^ \delta + ( 1 + \delta ) ^ 2 A ^ 2 s ^ { 2 \delta } , \end{align*}


\begin{align*} W ( M _ \eta T _ \xi f ) ( x , \omega ) = W f ( x - \xi , \omega - \eta ) . \end{align*}


\begin{align*} \lim \limits _ { H ^ 2 \rightarrow + \infty } \frac { r } { k - r } w _ 2 = \ & H ^ 2 \left ( \frac { k } { r } - \frac { 1 } { 2 } \right ) + \frac { H ^ 2 r } { 2 ( k - r ) } \left ( \frac { k } { r } - \epsilon \right ) \left ( - \frac { k } { r } - \epsilon + 1 \right ) \end{align*}


\begin{align*} \big ( \mathcal { F } ^ \# \big ) ^ { - 1 } [ \phi ] ( x ) = \int \mathcal { K } ^ \# ( x , k ) \phi ( k ) \ , d k . \end{align*}


\begin{align*} D _ L = \left \{ \begin{pmatrix} L & 0 \\ 0 & L ^ { - T } \end{pmatrix} \mid \det ( L ) \neq 0 \right \} . \end{align*}


\begin{align*} c _ { \alpha , d } = \dfrac { - \alpha ( \alpha - 1 ) \Gamma \left ( \frac { \alpha + d } { 2 } \right ) } { 4 \cos \left ( \frac { \alpha \pi } { 2 } \right ) \Gamma \left ( \frac { \alpha + 1 } { 2 } \right ) \pi ^ { \frac { d - 1 } { 2 } } \Gamma ( 2 - \alpha ) } . \end{align*}


\begin{align*} g _ { R , X _ \ast , V _ \ast } ( t , x , v ) = h _ { V _ \ast } ( t , x - X _ \ast + ( R - t ) V _ \ast , v ) . \end{align*}


\begin{align*} \Big ( \phi _ t = I d _ L + t [ \mathfrak { X } , - ] + \sum _ { i = 2 } ^ { + \infty } \phi _ i t ^ { i } , \ ; \psi _ t = I d _ V + t D ( \mathfrak { X } ) ( - ) + \sum _ { i = 2 } ^ { + \infty } \psi _ i t ^ { i } \Big ) , \end{align*}


\begin{align*} D ( 0 , A ( x ) ) = 0 x \in \mathrm { z e r } A \end{align*}


\begin{align*} b _ j = a _ 1 \Big ( q - \sum _ { k = 1 } ^ { L } q _ k \widetilde { \phi _ k } \Big ) e _ j + q _ j . \end{align*}


\begin{align*} ( \sigma u ) ( x ) = u ( \sigma ^ { - 1 } ( x ) ) , \ \forall \sigma \in G , \ u \in H ^ 1 ( M ) , \ x \in M . \end{align*}


\begin{align*} \| T _ j T ^ * _ k f \| _ { L ^ 2 ( \R ^ N , \omega ) } ^ 2 & \lesssim \sum \limits _ { \sigma \in G } ( r - 1 ) ^ 4 r ^ { - 2 | j - k | \varepsilon } \int _ { \mathbb R ^ N } \big ( M f ( \sigma ( x ) ) \big ) ^ 2 d \omega ( x ) \\ & = \sum \limits _ { \sigma \in G } ( r - 1 ) ^ 4 r ^ { - 2 | j - k | \varepsilon } \int _ { \mathbb R ^ N } \big ( M f ( x ) \big ) ^ 2 d \omega ( x ) \\ & \lesssim ( r - 1 ) ^ 4 r ^ { - 2 | j - k | \varepsilon } \| f \| _ { L ^ 2 ( \R ^ N , \omega ) } ^ 2 , \end{align*}


\begin{align*} X ( s , \varphi ) : = e ^ { i \varphi } \cdot \eta ( s ) , \forall \ , ( s , \varphi ) \in [ a , b ] \times [ 0 , 2 \pi ] , \end{align*}


\begin{align*} \mathbb { C } _ { \infty } ( ( t ) ) & \rightarrow \mathbb { C } _ { \infty } ( ( t ) ) \\ f : = \sum _ { i } a _ i t ^ i & \mapsto \sum _ { i } a _ i ^ { q ^ { n } } t ^ i = : f ^ { ( n ) } . \end{align*}


\begin{align*} \widehat { A } ( X , [ \alpha ] ) : = \int _ X c _ X \widehat { A } ( X ) \wedge \psi _ X ^ * ( \alpha ) \end{align*}


\begin{align*} g ( x ) = f ( x ) - \langle h ( x ) , \lambda ( x ) \rangle + \beta \norm { h ( x ) } ^ 2 \end{align*}


\begin{align*} \hat q _ { j i } = ( \hat q _ { i j } ) ^ * . \end{align*}


\begin{align*} \mathcal { U } ^ 1 ( h , k ) = Q \sum _ { \substack { 1 \leq c \leq C \\ ( c , h k ) = 1 } } \frac { \mu ( c ) } { c } \sum _ { \substack { 1 \leq m , n < \infty \\ ( m n , c ) = 1 \\ m h \neq n k } } \frac { \tau _ A ( m ) \tau _ B ( n ) } { \sqrt { m n } } V \left ( \frac { m } { X } \right ) V \left ( \frac { n } { X } \right ) \frac { \phi ( m n h k ) } { m n h k } \int _ 0 ^ { \infty } W ( u ) \ , d u . \end{align*}


\begin{align*} \vert ( - \Delta ) ^ s ( \zeta \circ Q _ \rho ) ( x ) \vert & = C \int _ { B _ { 3 / 4 } ( - 2 e _ 1 / \rho ) } \frac { ( \zeta \circ Q _ \rho ) ( y ) } { \vert x - y \vert ^ { n + 2 s } } \dd y \leqslant C x \in B _ { 3 / 4 } \end{align*}


\begin{align*} m ^ { ( 1 ) } ( x , t , k ) = m ( x , t , k ) G ( x , t , k ) , \end{align*}


\begin{align*} \hat { K } ^ { \lambda , \nu } ( z ; w ) : = \sum _ { w = 1 } ^ k \sum _ { i = 0 } ^ { \infty } K ^ j _ { ( i ) } ( \lambda ; \nu ) z ^ i w ^ j . \end{align*}


\begin{align*} [ \eta , \xi ] ( x ) \equiv \eta \circ \xi ( x ) - \xi \circ \eta ( x ) = o ( x ^ 3 ) x = 0 . \end{align*}


\begin{align*} H ^ { Q , j } ( x ) = \pm \mu ^ { \frac { d - 1 } { p ' } - 1 } \tilde { \Phi } ( \mu \bar { x } ) = \pm \mu ^ { \frac { d - 1 } { p } } \alpha _ i x _ { \bar { j } } \end{align*}


\begin{align*} S _ { g , \L } ^ { - 1 } f = \sum _ { \l \in \L } \langle f , \pi ( \l ) \widetilde { g } \rangle \ \pi ( \l ) \widetilde { g } . \end{align*}


\begin{align*} ( \beta _ 1 , \ldots , \beta _ n ) : = ( b _ 1 , \ldots , b _ n ) E ^ { - 1 } , \end{align*}


\begin{align*} \lambda ^ h _ k : = - \tilde \mu _ k : = - 1 - \alpha _ { 1 , k } - i ( 2 k \pi + \alpha _ { 2 , k } ) , \forall | k | \geq k _ 0 . \end{align*}


\begin{align*} u ^ { - \frac { \alpha + 1 } { \alpha } } \bar { g } ( \lambda \partial _ { r } , \nabla P ) = u ^ { - \frac { \alpha + 1 } { \alpha } } \lambda ^ { 2 } \sum _ { i } r _ { i } ^ { 2 } - \bar { g } ( \lambda \partial _ { r } , \nabla \log u ) \end{align*}


\begin{align*} ( - \Delta ) ^ s ( \tilde u - \tau \varphi ^ { ( 2 ) } ) ( a ) = - C \int _ { \R ^ n _ + } \bigg ( \frac 1 { \vert a - y \vert ^ { n + 2 s } } - \frac 1 { \vert a _ \ast - y \vert ^ { n + 2 s } } \bigg ) ( \tilde u - \tau \varphi ^ { ( 2 ) } ) ( y ) \dd y . \end{align*}


\begin{align*} \sum & p ( n ) X ^ n = \frac { 1 } { \alpha } = \frac { \alpha ( X ^ { 2 5 } ) } { \alpha ( X ^ 5 ) ^ 6 } \alpha ( \zeta X ) \alpha ( \zeta ^ 2 X ) \alpha ( \zeta ^ 3 X ) \alpha ( \zeta ^ 4 X ) \\ & = \frac { \alpha ( X ^ { 2 5 } ) } { \alpha ( X ^ 5 ) ^ 6 } ( \alpha _ 0 + \zeta \alpha _ 1 + \zeta ^ 2 \alpha _ 2 ) ( \alpha _ 0 + \zeta ^ 2 \alpha _ 1 + \zeta ^ 4 \alpha _ 2 ) ( \alpha _ 0 + \zeta ^ 3 \alpha _ 1 + \zeta \alpha _ 2 ) ( \alpha _ 0 + \zeta ^ 4 \alpha _ 1 + \zeta ^ 3 \alpha _ 2 ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { t } \frac { \alpha _ k } { \Gamma _ k } = \frac { 1 } { \Gamma _ t } ~ ~ ~ ~ ~ \implies ~ ~ ~ ~ ~ \Gamma _ t \sum _ { k = 1 } ^ t \frac { \alpha _ k } { \Gamma _ k } = 1 . \end{align*}


\begin{align*} L [ f ] = \int _ { \Omega \times \mathbb { R } ^ 3 } K ( x , y ) f ( t , y , w ) ( w - v ) \ , d w d y , \end{align*}


\begin{align*} q = g + \tau ' c \end{align*}


\begin{align*} \mu _ n ( L / A ) = \frac { [ L ' : \sigma ( L ' ) ] } { [ L : \sigma ( L ) ] _ { i n s e p } } . \end{align*}


\begin{align*} ( \varPhi _ q \ , f ) ( x , t ) = \int _ { { \mathbb R } ^ n } f ( y , t ) \wedge \phi _ q ( x , y ) \end{align*}


\begin{align*} \frac { | \nabla _ y p _ t ^ B ( x , \cdot ) | ( y ) } { p _ t ^ B ( x , y ) } = | \nabla _ y \log p _ t ^ B ( x , \cdot ) | ( y ) \leq \frac { c _ { B , T } } { \sqrt { t } } \Big ( 1 + \frac { \varrho ( x , y ) } { \sqrt { t } } \Big ) \end{align*}


\begin{align*} ( \pi ^ { s + 1 , s } ) ^ { * } \rho = f _ { 0 } \omega _ { 0 } + \sum _ { k = 0 } ^ { s } f _ { \sigma } ^ { i , j _ { 1 } \ldots j _ { k } } \omega _ { j _ { 1 } \ldots j _ { k } } ^ { \sigma } \wedge \omega _ { i } + \eta , \end{align*}


\begin{align*} \lambda ^ 1 ( m ) = \inf _ { \substack { u \in H ^ 1 _ 0 ( \Omega ) \\ \int _ { \Omega } m u ^ 2 > 0 } } \frac { \int _ { \Omega } | \nabla u | ^ 2 } { \int _ { \Omega } m u ^ 2 } . \end{align*}


\begin{align*} \lim _ { | x | + | v | \to \infty } \mathcal W ( x , v ) = \infty \end{align*}


\begin{align*} b ^ { - \frac { \gamma } { 2 } - \frac { 1 } { 2 } } \partial _ \xi \phi _ { i , i n t , \beta } \left ( \frac { y _ 0 } { \sqrt { b } } \right ) = b ^ { - \frac { \gamma } { 2 } } \frac { \phi _ { i , i n t , \beta } \left ( \frac { y _ 0 } { \sqrt { b } } \right ) } { \phi _ { i , o u t , \beta } ( y _ 0 ) } \partial _ y \phi _ { i , o u t , \beta } ( y _ 0 ) , \end{align*}


\begin{align*} _ { s + 1 } \mathcal { F } _ s ( \alpha ) ^ { q ^ d } = \sum _ { n \geq 0 } \sum _ { H \geq h \geq 1 } f _ h ( t ) \theta ^ { h q ^ { n + d } } \alpha ^ { q ^ { n + d } } = \sum _ { H \geq h \geq 1 } f _ h ( t ) \sum _ { n \geq 0 } \theta ^ { h q ^ { n + d } } \alpha ^ { q ^ { n + d } } . \end{align*}


\begin{align*} l = d _ 1 i _ 1 + \dots + d _ n i _ n 0 \leq \sum _ { t > t _ 0 } d _ t i _ t < i _ { t _ 0 } \ . \end{align*}


\begin{align*} \left \{ \begin{array} { l l l l } H _ \mu u ' \ ! + \ ! \varPhi _ q d _ q { \mathcal N } _ q u ' & = & \varPhi _ q d _ q f & ( x , t ) \in { \mathbb R } ^ n \times ( 0 , T ) , \\ u & = & u _ 0 , & ( x , t ) \in \mathbb { R } ^ n \times \{ 0 \} . \end{array} \right . \end{align*}


\begin{align*} \pi ( F ) = \pi ( F _ r ) = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \pi ( x , \omega , 0 ) \ , d ( x , \omega ) . \end{align*}


\begin{align*} U \left ( t \right ) = U \left ( t ^ S _ { j - 1 } \right ) = t ^ S _ { j - 1 } + \tau _ { \rm f } - \tau ^ { \rm R e a c } _ { V _ { j - 1 } } , t \in \mathcal { I } _ j . \end{align*}


\begin{align*} \partial _ x ^ 2 F = \sum _ j \nabla ^ 2 \phi _ j ( x ) \cdot f _ j ( x ) + 2 d \phi _ j ( x ) \otimes d f _ j ( x ) + \phi _ j ( x ) \cdot \nabla ^ 2 f _ j ( x ) . \end{align*}


\begin{align*} K _ Y + \pi _ { * } ^ { - 1 } \varepsilon D _ 1 + \pi _ { * } ^ { - 1 } \varepsilon D _ 2 + \sum _ { i = 1 } ^ n a _ i E _ i = \pi ^ { * } ( K _ X + \varepsilon D ) \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal e _ { i _ k } \frac { e _ { i _ k } ^ \intercal ( A x _ k - b ) } { \norm { A ^ \intercal e _ j } _ 2 ^ 2 } . \end{align*}


\begin{align*} \mu \left ( \bigcup _ { i \in \mathbb { N } } \bigcup _ { L \in \mathcal { F } _ { k , i } } L \right ) = 1 . \end{align*}


\begin{align*} A ( x ) \ = \ a + ( b - a ) \frac { 2 \phi ( x ) + 1 } { 2 x + 2 } , \end{align*}


\begin{align*} \lim _ { n \rightarrow + \infty } \int _ { \mathbb { R } ^ { d } } K ( x ) f ( x , w _ { n } ^ { \pm } ) w _ { n } ^ { \pm } \ d x = \int _ { \mathbb { R } ^ { d } } K ( x ) f ( x , w ^ { \pm } ) w ^ { \pm } \ d x . \end{align*}


\begin{align*} D ^ { \alpha - 1 } \left ( ( P _ t ^ { \nu _ \alpha } ) ^ * ( f ) \right ) ( x ) + \frac { 1 } { p _ \alpha ( x ) } R ^ \alpha \left ( p _ \alpha , ( P _ t ^ { \nu _ \alpha } ) ^ * ( f ) \right ) ( x ) & = \dfrac { - x e ^ { - \alpha t } } { ( 1 - e ^ { - \alpha t } ) } ( P ^ { \nu _ \alpha } _ t ) ^ { * } ( f ) ( x ) \\ & + \dfrac { e ^ { - t } } { \left ( 1 - e ^ { - \alpha t } \right ) } ( P ^ { \nu _ \alpha } _ t ) ^ { * } ( h f ) ( x ) , \end{align*}


\begin{align*} | A \backslash B | \ \leqslant \ \frac { 1 } { \inf _ n w ( n ) } \sum _ { n \in A \backslash B } w ( n ) \cdot 1 \ = \ \frac { 1 } { \inf _ n w ( n ) } \| 1 _ { A \backslash B } \| . \end{align*}


\begin{align*} \mathcal L ( \rho , m , S ) & : = \ < S ( 1 ) , \rho ^ b \ > - \ < S ( 0 ) , \rho ^ a \ > + \mathcal A ( \rho , m ) \\ & \quad - \int _ 0 ^ 1 ( \ < \dot S , \rho \ > + \ < m , \nabla _ G S \ > + \ < \nabla _ G \Sigma , \nabla _ G S \ > \dot W ^ { \delta } ( t ) ) d t . \end{align*}


\begin{align*} ( \hat y _ i ^ 1 , \dots , \hat y _ i ^ n ) : = ( - \hat f _ i ^ 1 , \dots , - \hat f _ i ^ m , \varphi _ { i , \delta } ^ { m + 1 } , \dots , \varphi _ { i , \delta } ^ n ) : B _ { R } ^ n ( 0 , \hat g _ { i , \delta } ^ * ) \to \mathbb R ^ n \end{align*}


\begin{align*} v _ n : = \frac { u _ n } { \Vert u _ n \Vert } \ \ n \in \mathbb { N } . \end{align*}


\begin{align*} \overline { \mathbb { K } } _ { \overline { \psi } _ F ^ l } ^ { r _ l ( \pi _ F ) ^ { ( l ) } } ( w ) \xi \{ \chi , 0 \} = \epsilon ( \chi ^ { - 1 } r _ l ( \pi _ F ) ^ { ( l ) } , \overline { \psi } _ F ^ l ) \xi \{ \chi , - n ( \chi ^ { - 1 } r _ l ( \pi _ F ) ^ { ( l ) } , \overline { \psi } _ F ^ l ) \} , \end{align*}


\begin{align*} \ddot { m } _ 1 ^ k = \arg \max _ { m = 1 , \ldots , M } \bigl ( \mathbf { b } _ k \bigr ) , \end{align*}


\begin{align*} \forall k \in \N \colon x _ k : = \frac 1 k , y _ k : = \frac 1 { k ^ 2 } , x _ k ^ * : = 1 , \lambda _ k : = \frac k 2 , \end{align*}


\begin{align*} C _ { n } ^ { \lambda , \beta } ( x ) = \sum _ { k = 0 } ^ { n } \binom { n } { k } \Bigg [ \sum _ { i = 0 } ^ { n - k } ( - 1 ) ^ { i } i ! B _ { n - k , i } \left ( \left ( \tilde { M } _ { \lambda , \beta } ( j ) \right ) _ { j = 1 } ^ { n - k - i + 1 } \right ) \Bigg ] ( x ) _ { k } , \end{align*}


\begin{align*} \Phi ( y , \tau ) = \frac { 1 } { 2 } \Lambda _ y Q _ b \left ( y \right ) \left [ \frac { b _ \tau } { b } - 1 \right ] \end{align*}


\begin{align*} E ' ( t ) = - \int _ U F ( u _ t ( t ) ) u _ t ( t ) \ , d x \leq 0 . \end{align*}


\begin{align*} \check { \Delta } ( \omega ) \Delta ( a ) = \sum \Delta ( a ' ) \check { \Delta } ( \omega ' ) , \end{align*}


\begin{align*} \Psi _ { a , \zeta , \mu , \sigma } ( x ) & : = \mu ^ { \frac { d - 1 } { p } } \Phi ( \mu x ) \\ \tilde { \Psi } _ { a , \zeta , \mu , \sigma } ( x ) & : = \mu ^ { \frac { d - 1 } { p ' } } \tilde { \Phi } ( \mu x ) . \end{align*}


\begin{align*} Y _ { t } ^ { x ; l ^ { \epsilon } } : = X _ { \gamma _ { t } ^ { \varepsilon } } ^ { x ; l ^ { \epsilon } } , t \geq l _ { 0 } ^ { \epsilon } . \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - W _ k ( W _ k ^ \intercal A ^ \intercal A W _ k ) ^ \dagger W _ k ^ \intercal A ^ \intercal ( A x _ k - b ) , \end{align*}


\begin{align*} \delta = Q ^ { - 9 9 } . \end{align*}


\begin{align*} \frac { e ^ { - t } u \left ( \xi ( t ) \right ) - e ^ { - ( t + h ) } u ( \xi ( t + h ) ) } { h } = \frac { 1 } { h } \int _ t ^ { t + h } e ^ { - s } C _ p \left | \dot { \xi } ( s ) \right | ^ { q } d s . \end{align*}


\begin{align*} \forall x ^ X , z ^ X \left ( z \in A x \land \norm { z } _ X = \norm { A ^ \circ _ X x } _ X \rightarrow A ^ \circ _ X x = _ X z \right ) . \end{align*}


\begin{align*} u ( x ) & = \gamma _ { n , s } \int _ { \R ^ n \setminus B _ r } \bigg ( \frac { r ^ 2 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - r ^ 2 } \bigg ) ^ s \frac { u ( y ) } { \vert x - y \vert ^ n } \dd y \end{align*}


\begin{align*} K _ { ( i j ) } ^ { ( \ell ) } : = \partial _ { x _ { i ; \alpha _ 1 } } \partial _ { x _ { j ; \alpha _ 2 } } K _ { \alpha _ 1 \alpha _ 2 } ^ { ( \ell ) } \bigg | _ { x _ { \alpha _ 1 } = x _ { \alpha _ 2 } = x _ \alpha } \end{align*}


\begin{align*} u _ t - \Delta u + a | \nabla u | ^ \alpha = 0 \end{align*}


\begin{align*} B ( z ) = B _ 0 + \dots + B _ k z ^ k = B _ k ( z - \beta _ 1 ) \cdots ( z - \beta _ k ) , \beta _ i \neq \beta _ j . \end{align*}


\begin{align*} B ( b ) = \frac { \pi } { 2 } \left | \sqrt { b } \ , \vartheta _ 3 ( 0 , i b ) \vartheta _ 3 ( 0 , i b ) \right | ^ 2 = \frac { \pi } { 2 } \left | \vartheta _ 3 ( 0 , i b ^ { - 1 } ) \vartheta _ 3 ( 0 , i b ) \right | ^ 2 = \frac { \pi } { 2 } \ , \vartheta _ 3 ( 0 , i b ^ { - 1 } ) ^ 2 \vartheta _ 3 ( 0 , i b ) ^ 2 . \end{align*}


\begin{align*} F : V \to \mathbb { R } \ , , F ( v ) = \int _ \Omega f \ , v \ , ; \end{align*}


\begin{align*} G = G _ n ( \Q ) = \{ g \in \mathrm { S L } _ n ( \mathbb { B } ) : g ^ { \ast } \phi g = \phi \} , \phi = \left [ \begin{array} { c c c } 0 & 0 & - 1 _ m \\ 0 & \zeta \cdot 1 _ r & 0 \\ 1 _ m & 0 & 0 \end{array} \right ] . \end{align*}


\begin{align*} \begin{cases} - \Delta u = \lambda m u & \Omega \ , , \\ u = 0 & \partial \Omega \end{cases} \end{align*}


\begin{align*} c ( h , q , s ) = A ( n ) \chi ( \det ( q _ { \mathbf { h } } ) ) ^ { - 1 } \det ( q _ { \infty } ) ^ s | \det ( q ) | _ { \mathbf { h } } ^ { 2 m - 1 - s } \alpha _ { \mathfrak { n } } ( q _ { \mathbf { h } } ^ { \ast } h q _ { \mathbf { h } } , s , \chi ) \xi ( q _ { \infty } q _ { \infty } ^ { \ast } , h , s + l , s - l ) . \end{align*}


\begin{align*} \Gamma _ 2 ( f ) ( x ) & = \sum _ { j , k } \left ( \partial ^ 2 _ { j , k } ( f ) ( x ) \right ) ^ 2 + \langle \nabla ( f ) ( x ) ; \operatorname { H e s s } ( V ) ( x ) \nabla ( f ) ( x ) \rangle \geq \kappa \Gamma ( f ) ( x ) . \end{align*}


\begin{align*} \phi ^ 1 = \phi ^ 0 + \tau \dot { \phi } \big | _ { t = 0 } + \frac { \tau ^ 2 } { 2 } \ddot { \phi } \big | _ { t = 0 } = \phi ^ 0 + \frac { \tau ^ 2 } { 2 } \left ( - G ( \phi ^ 0 ) + \chi ^ 0 \phi ^ 0 \right ) , \end{align*}


\begin{align*} \underset { x } { \min } f ( x ) \textrm { s u b j e c t t o } h ( x ) = 0 , \end{align*}


\begin{align*} P _ \theta [ \eta \le \eta _ \alpha ] = P _ \theta [ C ( \eta ) \le C ( \eta _ \alpha ) ] = P _ \theta [ C ( \eta ) \le \alpha ] = \alpha . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } u _ { \nu k } \ge w _ { \nu , \nu , l } , & \hat { u } _ { \nu k } ( l ) = 1 \\ u _ { \nu k } \le 1 - w _ { \nu , \nu , l } , & \end{array} \right . ~ ~ \forall l \in S _ { \nu } , k = 1 , \cdots , n _ \nu ^ 1 , \end{align*}


\begin{align*} J ( \alpha \circ \beta ) ( 0 ) = J ( \alpha ) ( 0 ) J ( \beta ) ( 0 ) \ne 0 \end{align*}


\begin{align*} w _ { * } ^ { ( \mu - 1 ) } f : = \left [ \left ( \frac { d w } { d z } \right ) ^ { \mu - 1 } \cdot f \right ] \circ w ^ { - 1 } \end{align*}


\begin{align*} \begin{aligned} \frac { 1 } { \sqrt { 2 } } = \lambda _ 1 \phi ^ 1 + \sum _ { k > 1 } \lambda _ k \phi ^ k & \leq \lambda _ 1 \phi ^ 1 + ( \lambda _ 2 + \cdots + \lambda _ k ) \sqrt { \frac { \lambda _ 2 y ^ 2 + \cdots + \lambda _ k y ^ k } { \lambda _ 2 + \cdots + \lambda _ k } } \\ & < \sqrt { \lambda _ 1 y ^ 1 + \cdots + \lambda _ k y ^ k } = \frac { 1 } { \sqrt { 2 } } , \end{aligned} \end{align*}


\begin{align*} \mathcal L ( \rho , v ) = \int _ 0 ^ 1 \bigl [ \frac 1 2 \ < v , v \ > _ { \theta ( \rho ) } - \mathcal V ( \rho ) - \mathcal W ( \rho ) + \alpha L ( \rho ) - \beta I ( \rho ) \bigr ] d t , \end{align*}


\begin{align*} \mathcal { H } ^ { \frac { \log 2 } { \log 3 } } _ { \infty } ( U _ { p , q , \delta } ) \geq \widetilde { C } q ^ { - 2 } = \mathcal { L } \Big ( B \Big ( \frac { p } { q } , q ^ { - 2 } \Big ) \Big ) . \end{align*}


\begin{align*} \begin{aligned} y _ { i , k + d _ i } & = h _ i ( \boldsymbol { f } _ O ^ { d _ i - 1 } \left ( \boldsymbol { f } ( \mathbf { x } _ k , \mathbf { u } _ k ) \right ) ) . \end{aligned} \end{align*}


\begin{align*} \eta ^ { \prime \prime } ( x ) + \lambda \xi ( x ) - \lambda \eta ( x ) = 0 , \ \ \forall x \in ( 0 , 1 ) . \end{align*}


\begin{align*} R e ( x ^ * y ) = 0 \\ \end{align*}


\begin{align*} 1 - \Phi \left ( x + \varepsilon _ n \right ) = \Big ( 1 - \Phi ( x ) \Big ) \exp \Big \{ \theta c \ , ( 1 + x ) \varepsilon _ n \Big \} , \end{align*}


\begin{align*} M \subset \R \times M _ 0 , g ( t , x ) = c ( t , x ) ( d t ^ 2 + g _ 0 ( x ) ) , \end{align*}


\begin{align*} - \Delta w = \tilde { \lambda } _ 0 \tilde { m } _ 0 w \R ^ N \ , , \end{align*}


\begin{align*} \prod _ { i \in V _ 1 } \left ( \zeta _ { 2 k } ^ { \mu _ i } - \zeta _ { 2 k } ^ { - \mu _ i } \right ) & = \prod _ { i \in V _ 1 } \sum _ { l _ i \in \{ \pm 1 \} } l _ i \zeta _ { 2 k } ^ { l _ i \mu _ i } \end{align*}


\begin{align*} x ^ \prime : y ^ \prime : z = \alpha x : \beta y : \gamma _ 1 z \ , ; \gamma _ 1 = \frac { k } { \alpha l } = \frac { \beta } { \alpha } \gamma \ , . \end{align*}


\begin{align*} f ( t ) = e ^ t \cos ( e ^ t ) . \end{align*}


\begin{align*} \begin{aligned} I & = \sum \limits _ { k = - \infty } ^ { \infty } D _ { k } = \left ( \sum \limits _ { k = - \infty } ^ { \infty } D _ { k } \right ) \left ( \sum \limits _ { j = - \infty } ^ { \infty } D _ { j } \right ) \\ & = \sum \limits _ { | k - j | \leqslant N } D _ { k } D _ { j } + \sum \limits _ { | k - j | > N } D _ { k } D _ { j } = T _ { N } + R _ { N } , \end{aligned} \end{align*}


\begin{align*} & g _ R ( x ) = 0 { g _ R ( x ) \over | x | } \to 1 \\ & \sup \{ | \nabla g _ R ( x ) | + | \nabla ^ 2 g _ R ( x ) | : x \in \R ^ n , \ R > 0 \} < \infty . \end{align*}


\begin{align*} \begin{aligned} A _ { n , n } & = a \cdot \Big ( [ 0 , 2 n + 5 ] \cup [ 2 n + 7 , 4 n + 8 ] \Big ) \\ & = A + a \cdot \Big ( [ 0 , 2 n + 4 ] \cup [ 2 n + 7 , 4 n + 7 ] \Big ) \ , , \end{aligned} \end{align*}


\begin{align*} \frac { d } { d t } \left ( e ^ { - t } C _ p q \left | \dot { \xi } ( t ) \right | ^ { q - 2 } \dot { \xi } ( t ) \right ) = e ^ { - t } D f \left ( \xi ( t ) \right ) , \end{align*}


\begin{align*} 2 \sum _ { 0 < \gamma < T } \frac { x ^ { \beta - 1 } } { \gamma } = \Sigma _ { 0 } ^ { \sigma _ 1 } + \Sigma _ { \sigma _ 1 } ^ { \sigma _ 2 } + \Sigma _ { \sigma _ 2 } ^ 1 . \end{align*}


\begin{align*} \mathcal { L i } _ { K , \mathfrak { s } } ( { \bf z } ) ^ { ( - 1 ) } = \frac { z _ r } { ( \theta - t ) ^ { s _ r } } \mathcal { L i } _ { K , ( s _ 1 , \ldots , s _ { r - 1 } ) } ( z _ 1 , \ldots , z _ { r - 1 } ) + \mathcal { L i } _ { K , \mathfrak { s } } ( { \bf z } ) \end{align*}


\begin{align*} I ( \alpha , \beta , Z ) = \epsilon ( \alpha ) - \epsilon ( \beta ) \ , \ , \ , \mathrm { m o d } \ , \ , 2 . \end{align*}


\begin{align*} f _ { R | | C } = \int f _ { R | C B } f _ B d b . \end{align*}


\begin{align*} H _ c ^ + = \left \{ x \in \mathbb { R } ^ n \middle | \left \langle c , x - z \right \rangle \leqslant 0 \right \} \end{align*}


\begin{align*} | J _ 5 | = & \Big | - \varepsilon \ ! \int _ 0 ^ t \ ! \int _ \Omega \mathcal { C } ( \mu _ { \varepsilon } - \mu _ A ) \ , \mathrm { d } x \ , \mathrm { d } \varsigma \Big | \leq C \varepsilon ^ { \frac { 7 } { 2 } } \| ( \mu _ { \varepsilon } - \mu _ A ) \| _ { L ^ 2 ( \Omega \times ( 0 , t ) ) } \\ \le & C \varepsilon ^ 7 + \eta \| ( \mu _ { \varepsilon } - \mu _ A ) \| _ { L ^ 2 ( \Omega \times ( 0 , t ) ) } ^ 2 . \end{align*}


\begin{align*} \theta R = R ^ 2 + R ^ \# . \end{align*}


\begin{align*} \sigma _ 1 \sigma _ 3 < 0 , \sigma _ j : = \frac { 1 } { j ! } \frac { d ^ j } { d t ^ j } \bigg | _ { t = 0 } \sigma ( t ) . \end{align*}


\begin{align*} \left | \frac { \beta _ k } { z - \beta _ k } \right | \le \frac { 1 } { 2 M - 1 } + 1 = \frac { 2 M } { 2 M - 1 } \le 2 . \end{align*}


\begin{align*} \tilde u _ { \bar M + \varepsilon } = \tilde u + h ^ { ( \varepsilon ) } \geqslant a - \varepsilon > 0 B _ 1 ( 2 e _ 1 ) \end{align*}


\begin{align*} J X _ \varepsilon ' = \lambda H _ { \varepsilon } ( \theta ) X _ \varepsilon \end{align*}


\begin{align*} \mathcal E ( \rho , m ) & : = \sup _ { S \in H _ 1 ^ 1 } \{ \ < S ( 1 ) , \rho ^ b \ > - \ < S ( 0 ) , \rho ^ a \ > \\ & - \int _ 0 ^ 1 ( \ < \dot S , \rho \ > + \ < m , \nabla _ G S \ > + \ < \nabla _ G \Sigma , \nabla _ G S \ > _ { \theta ( \rho ) } \dot W ^ { \delta } ( t ) ) d t \} . \end{align*}


\begin{align*} 2 e + R _ { n + 1 } - 1 = 2 e > 2 e - 1 = 2 \left ( e - \left \lfloor \dfrac { 1 - 1 } { 2 } \right \rfloor \right ) - 1 = G _ { n } \ , . \end{align*}


\begin{align*} i _ { V } \omega = \sqrt { - 1 } \overline { \partial } \theta _ { V } ^ { ( \omega ) } \quad \int _ { X } \theta _ { V } ^ { ( \omega ) } \omega ^ { n } = 0 , \end{align*}


\begin{align*} S & = { \sqrt { \Delta ( { \mathbb K } ) } } \sum _ { d \vert \frac { m } { p } } \left ( \mu ( d ) \lambda ^ { ( n / d ) ^ s } + \mu ( p d ) \lambda ^ { n / ( d p ) } \right ) \\ & = { \sqrt { \Delta ( { \mathbb K } ) } } \sum _ { d \vert \frac { m } { p } } \pm \lambda ^ { ( n / ( p d ) ) ^ s } \left ( \lambda ^ { ( n / p d ) ^ s ( p ^ s - 1 ) } - 1 \right ) . \end{align*}


\begin{align*} 0 \leq u ( x ) \leq 1 , E ( u ) : = \int _ M | \nabla u | ^ 2 d x \leq C _ 1 \ , . \end{align*}


\begin{align*} \mathbb { P } \Big ( \sup _ { t \le T \wedge \theta _ { K } } \sup _ { i } | M ^ K _ i ( t ) | \geq A \Big ) & \leq \sum _ { i = 0 } ^ { 1 / \delta _ K - 1 } \mathbb { P } \Big ( \sup _ { t \le T \wedge \theta _ { K } } | M ^ K _ i ( t ) | \geq A \Big ) \\ \leq & \frac { 1 } { A ^ 2 } \sum _ { i = 0 } ^ { 1 / \delta _ K - 1 } \mathbb { E } \Big ( \sup _ { t \le T \wedge \theta _ { K } } | M ^ K _ i ( t ) | ^ 2 \Big ) , \end{align*}


\begin{gather*} | a _ n | = \frac { f _ n } { c _ E - n ^ 2 } \lesssim \frac { 1 } { n ^ 2 } \sqrt { \delta E ( 0 ) } , \ ; \forall n \in \mathbb { Z } \setminus \{ 0 , \pm 1 \} , \\ | a _ 0 | = \left | \frac { f _ 0 } { c _ E } \right | \lesssim \sqrt { \frac { \delta } { E ( 0 ) } } \ ; \textrm { a n d } \ ; | a _ { \pm 1 } | = \left | \frac { 2 \pi f _ { \pm 1 } } { E ( 0 ) - 2 \pi } \right | \lesssim \frac { \sqrt { \delta E ( 0 ) } } { 2 \pi - E ( 0 ) } . \end{gather*}


\begin{align*} d _ { C _ 1 C n } ( M , \ , Y _ q ( \Omega ) ) \underset { \mathfrak { Z } _ 0 } { \lesssim } \sum \limits _ { t = 0 } ^ { \hat t ( n ) } \sum \limits _ { m = 0 } ^ \infty d _ { k _ { t , m } } ( W _ { t , m } , \ , l _ q ^ { \nu _ { t , m } } ) + \sup _ { f \in M } \| f \| _ { Y _ q ( \tilde \Omega _ { \hat t ( n ) + 1 } ) } . \end{align*}


\begin{align*} \Im \left \{ \log \Gamma \left ( \frac { \sigma + i T } { 2 } \right ) \right \} & = \frac { T } { 4 } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) + \left ( \frac { \sigma - 1 } { 2 } \right ) \arg \left ( \frac { \sigma + i T } { 2 } \right ) - \frac { T } { 2 } + o ( 1 ) \end{align*}


\begin{align*} h ( \mathcal { I } _ { \beta } ^ { j } ) \cdot h ( \mathcal { I } _ { \alpha \beta } ^ { i j } ) = h ( \mathcal { J } _ { j } ^ { \beta } ) \cdot h ( \mathcal { J } _ { \alpha j } ^ { i \beta } ) = 0 , \quad ( j , \beta ) \in ( \mathcal { I } \setminus \mathcal { J } ) \times ( \mathcal { J } \setminus \mathcal { I } ) . \end{align*}


\begin{align*} \frac { d } { d t } E _ 0 ( \rho , \phi ) ( t ) & = - \int _ { \Omega } \sum _ { i = 1 } ^ s D _ i ( x ) \rho _ i | \nabla ( \log \rho _ i + z _ i \phi ) | ^ 2 d x + \frac { 1 } { 2 } \int _ { \partial \Omega } \epsilon ( x ) \left [ \phi ( \partial _ n \phi ) _ t - ( \partial _ n \phi ) \phi _ t \right ] d s . \end{align*}


\begin{align*} w _ r \left ( \rho , s \right ) = e ^ { \rho \omega _ r s } \left [ 1 + f _ r \left ( s \right ) + \frac { f _ { r 1 } \left ( s \right ) } { \rho } + \frac { H _ r \left ( \rho , s \right ) } { \rho ^ 2 } \right ] \end{align*}


\begin{align*} E ( X ) ^ g = \{ \{ u ^ g , v ^ g \} \mid \{ u , v \} \in E ( X ) \} . \end{align*}


\begin{align*} d x ( t ) = ( \alpha _ { - 1 } x ( t ) ^ { - 1 } - \alpha _ { 0 } + \alpha _ { 1 } x ( t ) - \alpha _ { 2 } t ^ { 2 H - 1 } x ( t ) ^ { \rho } ) d t + \sigma x ( t ) ^ { \theta } d B _ t ^ H \end{align*}


\begin{align*} \Big ( \mathbb { E } \big [ \sup _ { 1 \leq m \leq M } \| Y _ m ^ { M , N } \| ^ { 2 p } \big ] \Big ) ^ { \frac 1 p } + \Big \| k \sum _ { j = 1 } ^ M \| A Y _ j ^ { M , N } \| ^ 2 \Big \| _ { L ^ p ( \Omega ; \mathbb { R } ) } + \Big \| k \sum _ { j = 1 } ^ M \| \nabla ( Y _ j ^ { M , N } ) ^ 2 \| ^ 2 \Big \| _ { L ^ p ( \Omega ; \mathbb { R } ) } < \infty . \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma _ { 1 0 } ( H G ) } c ( g ( \gamma ) ) = 1 8 \cdot 7 + 1 7 \cdot 7 = 2 4 5 \equiv 1 \pmod 2 . \end{align*}


\begin{align*} T ( t ) f : = \sigma \lim _ { \mathclap { n \to \infty } } T _ n ( t ) f \end{align*}


\begin{align*} u ^ i \star w ^ j = u ^ i w ^ j + \mathrm { i } \nu ( \delta ^ i _ - - \delta ^ i _ + ) u ^ i \bigg ( \frac { 1 } { \sqrt { a } } \delta ^ j _ 0 w ^ + - 2 \sqrt { a } \delta ^ j _ - w ^ 0 \bigg ) + \delta ^ i _ + \delta ^ j _ - 2 \nu ^ 2 u ^ + w ^ + . \end{align*}


\begin{align*} \mathcal { A } _ { s ' } = \mathcal { A } ^ 1 _ m + \mathcal { A } ^ { 2 , f i n } _ m = \mathcal { A } ^ { 1 , i n i t } _ m + \mathcal { C } _ m + \mathcal { D } _ m + \mathcal { A } ^ { 2 , f i n } _ m , \end{align*}


\begin{align*} \lim _ { t \rightarrow \infty } ( s ( r , t ) - t ) = \lim _ { t \rightarrow } F _ \sigma ( i t ) - i t = 0 . \end{align*}


\begin{align*} \tau _ n = \inf \Big \{ 0 \le t \le 1 ; x _ t \notin [ \frac { 1 } { n } , n ] \Big \} ( \inf \emptyset = \infty ) \end{align*}


\begin{align*} M _ t = 1 + t h + ( h \triangleleft h ) \frac { t ^ 2 } { 2 } + ( ( h \triangleleft h ) \triangleleft h ) \frac { t ^ 3 } { 6 } + \dotsb = 1 + \sum _ { n = 1 } ^ \infty R ^ { ( n - 1 ) } _ { \triangleleft h } ( h ) \frac { t ^ n } { n ! } . \end{align*}


\begin{align*} 0 \le u _ n ( x ) = u _ n ( x ) - u ( x ) + u ( x ) \leq \varepsilon + u ( x ) \forall \ , n \ge N _ \varepsilon , \ , \ , x \in \bar \Omega . \end{align*}


\begin{align*} \Omega _ { t _ n } = \{ ( z _ 1 , \dots , z _ n ) \in \Omega \ , : \ , z _ n = t _ n \} \end{align*}


\begin{align*} & m _ { i j } = \theta _ { i j } ( \rho ) ( \nabla _ G S ) _ { i j } , \ ; \forall ( i , j ) \in E , \ ; \\ & \ < \dot S , \rho \ > + \frac 1 4 \sum _ { i j } ( S _ i - S _ j ) \theta _ { i j } ( \rho ) + \sum _ { i j } ( \Sigma _ i - \Sigma _ j ) ( S _ { i } - S _ { j } ) \theta _ { i j } ( \rho ) d W ^ { \delta } ( t ) = 0 . \end{align*}


\begin{align*} \begin{aligned} \Big ( { \| x - x ' \| \over \| x - y \| } \Big ) ^ \varepsilon \frac { 1 } { \omega ( B ( x , \| x - y \| ) ) } & \leqslant C \Big ( \frac { \| x - x ' \| } { \| x - y \| } \Big ) ^ \varepsilon \frac 1 { \omega ( B ( x , d ( x , y ) ) ) } \Big ( \frac { d ( x , y ) } { \| x - y \| } \Big ) ^ { N } \\ & \leqslant C \Big ( \frac { \| x - x ' \| } { \| x - y \| } \Big ) ^ \varepsilon \frac 1 { \omega ( B ( x , d ( x , y ) ) ) } . \end{aligned} \end{align*}


\begin{align*} \Upsilon _ { \omega } ^ { + } = \left \{ \tau ^ { - 1 } - 1 , \tau \theta ^ { - 1 } \cdot \Omega \right \} , \quad \Upsilon _ { \omega } ^ { - } = \left \{ \tau - 1 , \Omega \right \} , \Omega \in \{ - \vartheta \tau ^ { - 1 } - 1 \} \cup \mathbb { C } . \end{align*}


\begin{align*} [ D _ h \Psi ] ( t ) : = \frac { \Psi ( t + h ) - \Psi ( t ) } { h } . \end{align*}


\begin{align*} \begin{aligned} u \left ( \xi ( 0 ) \right ) & = \int _ 0 ^ { t } e ^ { - s } \left ( C _ p \left | \dot { \xi } ( s ) \right | ^ q + f \left ( \xi ( s ) \right ) \right ) d s + e ^ { - t } u ( \xi ( t ) ) \\ & = \int _ 0 ^ { t } e ^ { - s } \left ( C _ p \left | \dot { \xi } ( s ) \right | ^ q + f \left ( \xi ( s ) \right ) \right ) d s + e ^ { - t } u ( y ) \end{aligned} \end{align*}


\begin{align*} \langle X g , M _ l T _ k g \rangle & = \langle g , X M _ l T _ k g \rangle \\ & = k \langle g , M _ l T _ k g \rangle + \langle g , M _ l T _ k X g \rangle \\ & = 0 + \langle T _ { - k } M _ { - l } g , X g \rangle , \end{align*}


\begin{align*} \pi _ 2 ( S ( \underline { P } , B ) ) = \{ x \in B \ ; \vert \ ; \bigvee _ { j = 1 } ^ { q } ( P _ j ( x ) \leq 0 ) \} . \end{align*}


\begin{align*} [ J ^ { \chi _ A } ] _ \mathcal { M } : = \lambda \alpha \in \mathbb { N } ^ \mathbb { N } , x \in X . \begin{cases} J ^ A _ { r _ \alpha } x & r _ \alpha > 0 x \in \mathrm { d o m } ( J ^ A _ { r _ \alpha } ) , \\ 0 & , \end{cases} \end{align*}


\begin{align*} \sqrt { - i \tau } \ , \vartheta _ 2 ( 0 , \tau ) = \vartheta _ 4 ( 0 , - 1 / \tau ) , \tau \in \mathbb { H } . \end{align*}


\begin{align*} h _ p ( r ) : = \Big ( 1 + \log _ + \frac { 1 } { r } \Big ) ^ { 1 - p } , r > 0 , \end{align*}


\begin{align*} | \mathcal { B C } _ { ( s , t ) } | = | \mathcal { S C } _ { ( s , t ) } | = \binom { \lfloor s / 2 \rfloor + \lfloor t / 2 \rfloor } { \lfloor s / 2 \rfloor } . \end{align*}


\begin{align*} \Delta _ { g _ M } f = \Delta _ B f . \end{align*}


\begin{align*} S ( z ) = | \mathbb { S } ^ 1 | \int _ { 0 } ^ { \frac \pi 2 } \sin \theta \left [ \frac 1 { \cos ^ 3 ( \theta / 2 ) } B \left ( \frac { | z | } { \cos ( \theta / 2 ) } , \cos \theta \right ) - B ( | z | , c o s \theta ) \right ] . \end{align*}


\begin{align*} X = \frac { 3 \sigma \log ( 1 / \delta ) \pm \sqrt { ( 3 \sigma \log ( 1 / \delta ) ) ^ 2 + 4 A } } { 2 } \end{align*}


\begin{align*} \mu _ \omega ( 2 B _ \rho ) \le \mu _ \omega ( B _ \rho ( \infty , 3 r ) ) = C _ { \beta , a } \ , \mu ( Z ) \ , r ^ { \tfrac { 2 \beta - a - 1 } { \beta - 1 } } & = C _ { \beta , a } \ , \mu _ \omega ( B _ \rho ( \infty , r / 2 ) ) \\ & \le C _ { \beta , a } \ , \mu _ \omega ( B _ \rho ) , \end{align*}


\begin{align*} r ( x , y ) = ( \sigma _ x ( y ) , \tau _ y ( x ) ) \end{align*}


\begin{align*} \forall R \ge R _ 0 + | X ( t ) | = R _ 0 + | x ( t ) | \Rightarrow | A _ R ( u ( t ) ) | \le 4 s _ c ( p - 1 ) \delta ( t ) , \ ; \forall t \in D _ { \delta _ 0 } . \end{align*}


\begin{align*} \mathbf { F } _ N = \{ \left ( \cos ( 2 \pi k / N ) , \sin ( 2 \pi k / N ) \right ) \mid k = 0 , \ldots N - 1 \} \end{align*}


\begin{align*} h _ \lambda ( t ) = h ( s ( | \lambda | , t ) ) , t > 0 . \end{align*}


\begin{align*} \Gamma ^ k _ { i j } = \partial ^ k _ \gamma \Gamma ^ \gamma _ { \alpha \beta } \partial ^ \alpha _ i \partial ^ \beta _ j + \partial ^ k _ \gamma \partial ^ \gamma _ { i j } \end{align*}


\begin{align*} \langle \nabla _ V T , S \rangle = \langle \nabla T , V \otimes S \rangle \end{align*}


\begin{align*} \frac { \partial ^ 2 \varphi _ R } { \partial x _ i \partial x _ j } ( x ) = ( \Delta ^ 2 \varphi _ R ) ( x ) = \frac { \partial ^ 2 \varphi _ R } { \partial x _ i ^ 2 } ( x ) - 2 = 2 N - \Delta \varphi _ R ( x ) = 0 , 2 x - \nabla \varphi _ R ( x ) = 0 , \end{align*}


\begin{align*} ( F _ H ) _ { H \in \mathrm { H a d } ( \mathcal { M } ) } \star _ { \ell } ( G _ H ) _ { H \in \mathrm { H a d } ( \mathcal { M } ) } = ( F _ H \star _ { H , \ell } G _ H ) _ { H \in \mathrm { H a d } ( \mathcal { M } ) } . \end{align*}


\begin{align*} H P ^ i ( \mathcal { C } ( \mathbb { R } ^ n ) ) = \left \{ \begin{array} { l l } 0 , & i \not \equiv n \ ( m o d \ 2 ) , \\ \mathbb { C } , & i \equiv n \ ( m o d \ 2 ) . \end{array} \right . \end{align*}


\begin{align*} T _ t = \sum _ { i = 0 } ^ { + \infty } T _ i t ^ { i } , \ ; T _ i \in H o m _ { \mathbb { K } } ( V ; L ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { E _ { n , k } Y ^ { ( n , k ) } _ i } { n k } = \frac i { k + 1 } . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta \Phi & = \sigma \varepsilon ^ { - 2 } ( 1 - ( x ^ { 2 } + y ^ { 2 } ) ) & \Omega , \\ \Phi & = \sigma g & \partial \Omega . \end{aligned} \right . \end{align*}


\begin{align*} [ z ^ n x _ h ^ { \ell } ] ( N _ 1 + N _ 2 + \cdots + N _ k ) = [ z ^ n x _ h ^ { \ell } ] \left ( \sqrt { \frac { z F _ { k , 1 } ( B ) } { B } } - \frac { z F _ { k , 1 } ( B ) } { B } \right ) . \end{align*}


\begin{align*} \partial _ t [ \nabla S _ t ( \varphi ) w ] = - \nabla ^ 2 u _ t ( \varphi ) [ \nabla S _ t ( \varphi ) w ] . \end{align*}


\begin{align*} p _ { t + s } ( x , y ) = \int _ M p _ t ( x , z ) p _ s ( z , y ) \mu ( d z ) , s , t > 0 , x , y \in M . \end{align*}


\begin{align*} = & - \frac { 1 } { 4 } \lambda ^ { n - 1 } \left ( \lambda ^ 2 - 4 \right ) ( 3 ) + \lambda ^ 3 P _ { n - 3 } ( \lambda ) \\ & \vdots \\ = & - \frac { 1 } { 4 } \lambda ^ { n - 1 } \left ( \lambda ^ 2 - 4 \right ) ( n ) + \lambda ^ n P _ { 0 } ( \lambda ) \end{align*}


\begin{align*} \pi ( \l ) \pi ( \l ' ) = e ^ { - 2 \pi i \omega ' \cdot x } \pi ( \l + \l ' ) . \end{align*}


\begin{align*} \tilde S _ 0 ( \bar x ) - \bar x \cdot \bar c - \sum _ { t = 0 } ^ { T - 1 } x _ t \Delta s _ { t + 1 } \le 0 . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \varphi _ t + | \varphi _ r | \pi \min \{ R ^ 2 , r ^ 2 \} = 0 & \ r > 0 , t > 0 , \\ \varphi ( 0 , t ) = 1 & \ t > 0 , \\ \varphi ( r , 0 ) = r + 1 & \ r \ge 0 . \end{array} \right . \end{align*}


\begin{align*} E = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } ( f + \sum _ { i = 1 } ^ s z _ i \rho _ i ) \phi \bigg ) d x + B , \end{align*}


\begin{align*} L ^ * \big ( * ^ V ( v _ i \wedge v _ j ) \big ) \otimes 1 + 1 \otimes * ^ V ( v _ i \wedge v _ j ) & = \tfrac 1 2 \left ( \mu _ { i j } \ , \omega ^ W _ \C w _ i w _ j \otimes 1 + 1 \otimes \omega ^ V _ \C v _ i v _ j \right ) \\ & = \tfrac 1 2 \left ( \mu _ { i j } \ , w _ i w _ j \omega ^ W _ \C \otimes 1 + 1 \otimes v _ i v _ j \omega ^ V _ \C \right ) \\ & = \tfrac 1 2 \left ( \mu _ { i j } \ , w _ i w _ j \otimes 1 + 1 \otimes v _ i v _ j \right ) , \end{align*}


\begin{align*} \frac { \mathbf { P } \left ( W _ n > x \right ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) , \ \ \ \ \ \ n \rightarrow \infty . \end{align*}


\begin{align*} \exists k \in \mathbb { N } \forall p \in A F _ k \forall l \leq m \left ( H ( d ( x _ { n + l } , p ) ) < G ( d ( x _ n , p ) ) + \sum _ { i = n } ^ { n + l - 1 } \varepsilon _ i + \frac { 1 } { r + 1 } \right ) . \end{align*}


\begin{align*} 0 < T ( F ) = \frac { t ( t - 1 ) } { 2 } \tau \Big ( | \alpha | ^ 2 \ F \wedge F \Big ) . \end{align*}


\begin{align*} { m \left ( \bigcup _ { r > 0 } A ( r ) \right ) = m ( A ) . } \end{align*}


\begin{align*} \rho _ \kappa ( \mathbf { h } ) = \rho \circ \delta _ \kappa ( \mathbf { h } ) = \rho ( \kappa x , \omega , \kappa \tau ) = e ^ { 2 \pi i \kappa \tau } e ^ { \pi i \kappa x \cdot \omega } T _ { \kappa x } M _ { \omega } . \end{align*}


\begin{align*} A B ^ T = B A ^ T , \ C D ^ T = D C ^ T A D ^ T - B C ^ T = I . \end{align*}


\begin{align*} d _ \rho ( ( x , y ) , \infty ) = \begin{cases} \frac { 1 } { ( \beta - 1 ) \ , y ^ { \beta - 1 } } & y \ge 1 , \\ \frac { \beta } { \beta - 1 } - y & 0 \le y \le 1 . \end{cases} \end{align*}


\begin{align*} \partial _ t h + \mathcal { L } h = R ( h ) \end{align*}


\begin{align*} \overline { \dim } _ B \big ( G ( { \textbf { f } + \textbf { g } } ) \big ) \le \max \{ \overline { \dim } _ B \big ( G ( { \textbf { f } } ) \big ) , \overline { \dim } _ B \big ( G ( { \textbf { g } } ) \big ) \} = 1 . 5 , \end{align*}


\begin{align*} \pi ^ { - \beta } \mathbf { T } ( g , h ) = \sum _ { j = 1 } ^ e \overline { \mathbf { g } _ j ( g ) } \mathbf { h } _ j ( h ) , \end{align*}


\begin{align*} P ( \boldsymbol { \varphi } ) = \max _ { \sigma \in \mathcal { A } } \ , \sup _ { \mathbf { P } _ { \ ! \sigma } \in \mathcal { Q } ^ { \sigma } } \ , \sum _ { n = 1 } ^ { d } C _ { n } ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \cdot \bigg ( H ( \mathbf { p } _ { \sigma _ { n } } ) + \int \ ! \varphi _ n ^ { \sigma } \ , \mathrm { d } \mathbf { p } _ { \sigma _ n } \bigg ) . \end{align*}


\begin{align*} M \mathbf { \mathrm { y } } _ u = \sum _ { \substack { w \sim u \\ w \in E } } \mathbf { \mathrm { v } } _ w \geq \frac { 3 } { 4 k } \lambda \mathbf { \mathrm { v } } _ u = \frac { 3 } { 4 k } \lambda \mathbf { \mathrm { y } } _ u . \end{align*}


\begin{align*} V _ { \widetilde S } = \{ ( z _ 1 , \dots , z _ n , \Pi _ { i = 1 } ^ n z _ i ) \ , : \ , ( z _ 1 , \dots , z _ n ) \in V _ S \} . \end{align*}


\begin{align*} \mathcal { Q } _ v = \{ \pi _ v ( G ^ + ) \cap \pi _ v ( G ^ - ) \mid G ^ + \in Q _ v ^ + , G ^ - \in Q _ v ^ - \} . \end{align*}


\begin{align*} \partial _ t \Phi ( h ) = 2 B ( \partial _ t h , h ) = 2 B ( - \mathcal { L } h + R , h ) = 2 B ( R , h ) . \end{align*}


\begin{align*} \int _ \mathcal { Q } | Z f ( x , \omega ) | ^ 2 \ , d \omega = \sum _ { k \in \Z ^ d } | f ( x + k ) | ^ 2 . \end{align*}


\begin{align*} \begin{aligned} \| Z _ { j } ^ { T } V _ j \| _ F ^ 2 \geq j - \frac { 1 0 \epsilon \| C \| _ F ^ 2 } { \eta _ j } . \end{aligned} \end{align*}


\begin{align*} m _ 0 = J ( \widehat { u } ) = \inf \limits _ { \mathcal { N } } J = \inf \limits _ { \mathcal { M } } J = J ( \widehat { w } ) = m _ 1 . \end{align*}


\begin{align*} \begin{aligned} ( X _ u - X _ s ) \cdot \mu ( X _ u ) + \frac { 1 } { 2 } ( p - 1 ) \cdot \sigma ^ 2 ( X _ u ) & = X _ u \mu ( X _ u ) + \frac { 1 } { 2 } ( p - 1 ) \cdot \sigma ^ 2 ( X _ u ) - X _ s \mu ( X _ u ) \\ & \le X _ u \mu ( X _ u ) + \frac { 1 } { 2 } ( p _ 0 - 1 ) \cdot \sigma ^ 2 ( X _ u ) + | X _ s | \cdot | \mu ( X _ u ) | \\ & \le c \cdot \bigl ( 1 + X _ u ^ 2 + | X _ s | \cdot ( 1 + | X _ u | ^ { \ell _ \mu + 1 } ) \bigr ) . \end{aligned} \end{align*}


\begin{align*} D : \ell ^ 2 ( \Gamma ) \to \mathcal { H } , D ( c _ \gamma ) _ { \gamma \in \Gamma } = \sum _ { \gamma \in \Gamma } c _ \gamma e _ \gamma \end{align*}


\begin{align*} \begin{aligned} x _ i - x ^ L _ i & \geq 0 & x ^ U _ i - x _ i & \geq 0 , i = 1 , \ldots , d , \\ f _ { i } ( x ) - a _ { i 0 } ( x ) & \geq 0 & a _ { i n } ( x ) - f _ { i } ( x ) & \geq 0 , i = 1 , \ldots , d , \\ f _ { i } ( x ) - u _ { i j } ( x ) & \geq 0 & a _ { i j } ( x ) - u _ { i j } ( x ) & \geq 0 , i = 1 , \ldots , d , \ ; j = 0 , \ldots , n . \end{aligned} \end{align*}


\begin{align*} g \notin H _ i ^ + . \end{align*}


\begin{align*} E ( \rho , \phi ) = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } \epsilon ( x ) | \nabla \phi | ^ 2 \bigg ) d x + \frac { \alpha } { 2 \beta } \int _ { \partial \Omega } \epsilon ( x ) | \phi | ^ 2 d s , \end{align*}


\begin{align*} \alpha ^ * A _ Y = \alpha ^ * \circ \rho _ Y ^ * A _ \ast = ( \rho _ Y \circ \alpha ) ^ * A _ \ast = \rho _ X ^ * A _ \ast = A _ X . \end{align*}


\begin{align*} \Pi _ \star ^ \mathcal { F } : = \mathrm { p r } _ { \perp \star } \circ \nabla ^ \mathcal { F } | _ { \Xi _ { t \star } \otimes _ { \mathcal { X } _ \star } \Xi _ { t \star } } \colon \Xi _ { t \star } \otimes _ { \mathcal { X } _ \star } \Xi _ { t \star } \rightarrow \Xi _ { \perp \star } \end{align*}


\begin{align*} R ( x ) & = \left \{ \theta \in \Theta : \P ( \theta | x ) \geq \frac { \lambda _ { B N } } { ( \lambda _ { B N } - \lambda _ { B P } ) + \lambda _ { N P } } \right \} . \end{align*}


\begin{align*} F ( u ) \in \mathcal { D } ( \mathcal { L } ^ { ( 1 ) } ) \mathcal { L } ^ { ( 1 ) } F ( u ) = F ' ( u ) \mathcal { L } u + F '' ( u ) \Gamma ( u ) ; \end{align*}


\begin{align*} \Phi _ t ( x , u ) = ( x + t u , u ) . \end{align*}


\begin{align*} \dim _ { \C } V _ k = \sum _ { \pi \ , \atop W ( \pi ) = - 1 } \dim V _ { \pi } \ ; . \end{align*}


\begin{align*} D _ { U } F _ { t } ( W ) = \lim _ { \epsilon \to 0 } \frac { F _ { t } ( W + \epsilon U ) - F _ { t } ( W ) } { \epsilon } \end{align*}


\begin{align*} \N ^ \perp : = \{ v \in L ^ 1 \mid \langle x , v \rangle = 0 \forall x \in \N \cap L ^ \infty \} . \end{align*}


\begin{align*} \sup _ { K \geq 1 } \ , \sum _ { j = 0 } ^ { { 1 \over \delta _ K } - 1 } h _ K G \left ( \overline { ( i _ K - j ) \delta _ K } \ , \log K \right ) e ^ { L \ , \log K \ , \rho ( j \delta _ K , i _ K \delta _ K ) } = \ : \ \overline G ( \alpha ) < + \infty . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \bigcup _ { p _ { i } \in S _ { \theta } } \Lambda _ { ( p _ { i } , \infty ) } ( M , \Gamma ) | } { | \Lambda ( M , \Gamma ) | } = 0 \end{align*}


\begin{align*} \left ( \frac { B } { A } \right ) ' = 0 \Longleftrightarrow \frac { A ' } { A } = \frac { B ' } { B } . \end{align*}


\begin{align*} u ( x ) = \inf \left \{ \left . I [ \gamma ] : = \int _ 0 ^ \infty e ^ { - s } \left ( \frac { \left ( \dot { \gamma } ( s ) \right ) ^ 2 } { 2 } + f ( \gamma ( s ) ) \right ) d s \right | \gamma \in \mathrm { A C } ( [ 0 , \infty ) ; [ - 1 , 1 ] ) , \gamma ( 0 ) = x \right \} . \end{align*}


\begin{align*} W _ { i } \upharpoonright t '' = \Phi _ i ^ { \Gamma _ A } \upharpoonright t '' , 1 = \Gamma _ A ( \langle x _ m , y _ m \rangle ) = \Psi _ i ^ { W _ i \upharpoonright t '' } ( \langle x _ m , y _ m \rangle ) . \end{align*}


\begin{align*} & \sqrt { 2 \pi } \mathcal { K } ^ { \# } ( x , k ) = \mathcal { K } ^ { \# } _ S ( x , k ) + \mathcal { K } ^ { \# } _ R ( x , k ) , \end{align*}


\begin{align*} \lim _ { x \rightarrow \infty } \left | e ^ { - i k x } \psi _ + ( x , k ) - 1 \right | = 0 , \lim _ { x \rightarrow - \infty } \left | e ^ { i k x } \psi _ - ( x , k ) - 1 \right | = 0 . \end{align*}


\begin{align*} \check { A } = \{ \varphi ( a - ) \mid a \in A \} . \end{align*}


\begin{align*} \begin{cases} \lambda _ 1 & = \sqrt { \frac { s _ { \pm } ( a ) s _ { \pm } ( c ) } { s _ { \pm } ( b ) } } \\ \lambda _ 2 & = \sqrt { \frac { s _ { \pm } ( a ) s _ { \pm } ( b ) } { s _ { \pm } ( c ) } } \\ \lambda _ 3 & = \sqrt { \frac { s _ { \pm } ( b ) s _ { \pm } ( c ) } { s _ { \pm } ( a ) } } \end{cases} \end{align*}


\begin{align*} \rho _ { \pi _ { \lambda , \beta } } ( - z , \cdot ) = \sum _ { k = 0 } ^ { \infty } \frac { ( z ) _ { k } } { k ! } Q _ { k } ^ { \pi _ { \lambda , \beta } } ( \cdot ) . \end{align*}


\begin{align*} v _ + : = \inf \big \{ v \in \mathbb { R } \colon \liminf _ { H \to \infty } p _ H ( v ) = 0 \big \} , \end{align*}


\begin{align*} f ( z ) = \sum _ { n = 1 } ^ { \infty } \lambda _ f ( n ) n ^ { ( \kappa - 1 ) / 2 } e ( n z ) . \end{align*}


\begin{align*} \phi _ t ( \mu _ n , Q , y ) = \int _ { Q _ { \alpha } ( y ) } \rho ( x , y ) ^ { - t } d \mu _ n ( x ) + \int _ { Q \setminus Q _ { \alpha } ( y ) } \rho ( x , y ) ^ { - t } d \mu _ n ( x ) . \end{align*}


\begin{align*} \kappa _ { 2 k ; \alpha } ^ { ( \ell + 1 ) } = \kappa _ { k } \big ( \underbrace { \Delta _ { \alpha \alpha } ^ { ( \ell ) } , \ldots , \Delta _ { \alpha \alpha } ^ { ( \ell ) } } _ { k } \big ) , \end{align*}


\begin{align*} \inf _ { ( \rho , m ) } \sup _ { S \in H ^ 1 } \mathcal L ( \rho , m , S ) = \sup _ { S \in H ^ 1 } \inf _ { ( \rho , m ) } \mathcal L ( \rho , m , S ) \end{align*}


\begin{align*} ( g \circ f ) ( \tau ( a , b ) ) & = ( g \circ f ) \left ( - a , a + b + \frac { 1 } { 4 k } \right ) = g \left ( - a , - k a + 2 k \left ( a + b + \frac { 1 } { 4 k } \right ) \right ) \\ & = g \left ( - a , k a + 2 k b + \frac { 1 } { 2 } \right ) = ( g \circ \tau _ 1 ) \left ( a , k a + 2 k b \right ) = g ( a , k a + 2 k b ) = ( g \circ f ) ( a , b ) . \end{align*}


\begin{align*} \begin{array} { l c l } \bar R ( X , Y ) Z & = & \langle Y , Z \rangle X - \langle X , Z \rangle Y + 2 \langle Z , E _ 3 \rangle ( \langle X , E _ 3 \rangle Y - \langle Y , E _ 3 \rangle X ) \\ \\ & & + 2 ( \langle X , Z \rangle \langle Y , E _ 3 \rangle - \langle Y , Z \rangle \langle X , E _ 3 \rangle ) E _ 3 , \end{array} \end{align*}


\begin{align*} H = \{ b _ 1 a _ 1 + \cdots + b _ m a _ m : \ ; b _ i \in \mathbb { N } i = 1 , \ldots , m \} \end{align*}


\begin{align*} \mathcal { E } _ 1 : = \left \{ \exists \ , y \in I _ { H } ( w ) \cap \mathbb { L } : X ^ { y } _ { H ^ \prime } - \pi _ 1 ( y ) \geq v ^ { \prime } H \right \} , \end{align*}


\begin{align*} \hat { f } ( \xi ) : = \mathcal { F } [ f ] ( \xi ) = \frac { 1 } { \sqrt { 2 \pi } } \int _ { \R } e ^ { - i x \xi } f ( x ) d x , \check { g } ( x ) : = \mathcal { F } ^ { - 1 } [ g ] ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \int _ { \R } e ^ { i x \xi } g ( \xi ) d \xi . \end{align*}


\begin{align*} b & : C _ k ( \mathfrak { A } ) \to C _ { k - 1 } ( \mathfrak { A } ) , \\ b & ( a _ 0 \otimes a _ 1 \otimes \cdots \otimes a _ k ) : = \sum _ { j = 0 } ^ { k - 1 } ( - 1 ) ^ { j } a _ 0 \otimes \cdots \otimes a _ j a _ { j + 1 } \otimes \cdots \otimes \cdots \otimes a _ k + \\ & \qquad \qquad \qquad \qquad + ( - 1 ) ^ k a _ k a _ 0 \otimes a _ 1 \otimes \cdots \otimes a _ { k - 1 } . \end{align*}


\begin{align*} u ( s ) - u ( t ) \rightarrow 0 | s | - | t | = c , \quad | s | , | t | \rightarrow \infty \end{align*}


\begin{align*} & \ , \iint _ { \R ^ { 2 d } } W ( f _ 1 , g _ 1 ) ( x , \omega ) \overline { W ( f _ 2 , g _ 2 ) ( x , \omega ) } \ , d x d \omega \\ = & \ , 2 ^ { 2 d } \iint _ { \R ^ { 2 d } } V _ { g _ 1 ^ \vee } f _ 1 ( 2 x , 2 \omega ) \overline { V _ { g _ 2 ^ \vee } f _ 2 ( 2 x , 2 \omega ) } \ , d x d \omega \\ = & \ , \langle f _ 1 , f _ 2 \rangle \overline { \langle g ^ \vee _ 1 , g ^ \vee _ 2 \rangle } \\ = & \ , \langle f _ 1 , f _ 2 \rangle \overline { \langle g _ 1 , g _ 2 \rangle } \end{align*}


\begin{align*} \limsup _ { \varepsilon _ 1 \rightarrow 0 ^ { + } } \limsup _ { n \rightarrow \infty } \mathbb { E } _ { \mu _ n } \Big [ \int _ 0 ^ T n ^ { \gamma - 1 } \sum _ { x , y : | x - y | \geq \varepsilon n } & [ \overleftarrow { \eta } _ t ^ { \varepsilon _ 1 n } ( y ) \overrightarrow { \eta } _ t ^ { \varepsilon _ 1 n } ( y + 1 ) \\ - & \eta _ t ^ n ( y ) \eta _ t ^ n ( y + 1 ) ] F ( t , \tfrac { x } { n } , \tfrac { y } { n } ) ( c _ { \gamma } ) ^ { - 1 } p ( x - y ) d t \Big ] = 0 . \end{align*}


\begin{align*} f = \frac { 1 } { \overline { \langle g , \widetilde { g } \rangle } } \iint _ { \R ^ { 2 d } } V _ g f ( x , \omega ) M _ \omega T _ x \widetilde { g } \ , d ( x , \omega ) . \end{align*}


\begin{align*} \sum _ { \substack { 1 \leq e < \infty \\ ( e , g ) = 1 } } \mu ( e ) \sum _ { a | g } a \mu ( a ) \sum _ { \substack { 1 \leq b < \infty \\ e a b | \frac { m h } { g } \pm \frac { n k } { g } } } b W \left ( \frac { c e a b } { Q } \right ) . \end{align*}


\begin{align*} V _ t = \{ \beta \times \gamma \in P _ { n } ^ t \times P _ { n } ^ t : \kappa _ t \pi _ t ( \beta ) = \pi _ t ( \gamma ) \kappa _ t \} , \kappa _ t = \left [ \begin{array} { c c c } 0 & 0 & - 1 _ t \\ 0 & 1 _ r & 0 \\ 1 _ t & 0 & 0 \end{array} \right ] . \end{align*}


\begin{align*} X ^ 1 = \lim _ S \left ( X _ S ^ 1 \times \prod _ { s \in S \setminus V } \{ o _ s \} \right ) . \end{align*}


\begin{align*} \| S _ { c w } ( T _ M f ) \| _ { L ^ p ( \omega ) } & = \bigg \| \bigg \{ \sum \limits _ { k \in \Bbb Z } \sum \limits _ { Q ' \in Q ^ k } | D _ k T _ { M } f ( x _ { Q ' } ) | ^ 2 \chi _ { Q ' } ( x ) \bigg \} ^ { \frac 1 2 } \bigg \| _ { L ^ p } \\ & \lesssim \bigg \| \bigg \{ \sum \limits _ { j \in \Bbb Z } \sum \limits _ { Q \in Q ^ j } | q _ Q f ( x _ { Q } ) | ^ 2 \chi _ { Q } ( x ) \bigg \} ^ { \frac 1 2 } \bigg \| _ { L ^ p } \\ & \lesssim \| S ( f ) \| _ p . \end{align*}


\begin{align*} \lambda ( L ) : = \prod _ { i = 1 } ^ n \prod _ { j \in J _ i } \left ( \frac { a _ { i j } } { - b _ { i j } } \right ) ^ { d _ { i j } } . \end{align*}


\begin{align*} \bigg \| \sum _ { i = 1 } ^ n \mathbf { E } ( \eta _ i ^ 2 | \mathcal { F } _ { i - 1 } ) - n \sigma ^ 2 \bigg \| _ { \infty } \leq N ^ 2 , \end{align*}


\begin{align*} & \sigma _ { 0 } ( t ) : = - t \qquad ( - \infty < t < + \infty ) , \\ & \sigma _ { 1 } ( t ) : = - \log ( t + 1 ) \qquad ( - 1 < t < + \infty ) , \end{align*}


\begin{align*} \tilde { x } _ { m , n } = \sqrt { P _ d } \sum _ { k = 1 } ^ { K } \sqrt { \eta _ { m k } } \hat { g } _ { m k , n } ^ * s _ { k , n } , \end{align*}


\begin{align*} \frac { \alpha _ i } { 4 } = \sum _ { \substack { j = 1 \\ j \not = i } } ^ n \frac { a _ { i j } } { ( \alpha _ i - \alpha _ j ) ^ 3 } , \end{align*}


\begin{align*} \partial _ t ^ 2 \phi - \partial _ x ^ 2 \phi + m ^ 2 \phi + V ( x ) \phi = \phi ^ 3 , ( \phi , \phi _ t ) ( 0 ) = ( \phi _ 0 , \phi _ 1 ) , \end{align*}


\begin{align*} \sum _ { ( p , 2 ) = 1 } ( \log p ) \chi _ { 8 p } ( m \ell ) V \left ( \frac { m } { p } \right ) \Phi \left ( \frac { p } { X } \right ) = & \sum ^ { \infty } _ { n = 1 } \chi _ { 8 n } ( m \ell ) \Lambda ( n ) V \left ( \frac { m } { n } \right ) \Phi \left ( \frac { n } X \right ) + O \left ( \sum _ { \substack { p ^ j \leq X ^ { 1 + \varepsilon } , j \geq 2 } } ( \log p ) \Phi \left ( \frac { p ^ j } X \right ) \right ) . \end{align*}


\begin{align*} \sum _ { \rho \in R _ \epsilon } ( \rho ) = \left ( \frac { 1 } { 2 } - \left ( \frac { 1 } { 2 } - \epsilon \right ) \right ) N _ 0 ( T ) \end{align*}


\begin{align*} J ( w ) & = J ( w ^ + ) + J ( w ^ - ) \\ & + \int _ { ( w ^ - ) } \int _ { ( w ^ + ) } G \left ( \frac { w ^ + ( x ) - w ^ - ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { d x d y } { \vert x - y \vert ^ { d } } \\ & + \int _ { ( w ^ + ) } \int _ { ( w ^ - ) } G \left ( \frac { w ^ - ( x ) - w ^ + ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { d x d y } { \vert x - y \vert ^ { d } } . \end{align*}


\begin{align*} R ^ { ( n - 1 ) } _ { \triangleleft h } ( h ) & = \sum _ { k = n } ^ \infty \bigg ( \sum _ { i _ 1 + \dotsm + i _ n = k } ( i _ 1 + 1 ) ( i _ 1 + i _ 2 + 1 ) \dotsm ( i _ 1 + \dotsb + i _ { n - 1 } + 1 ) h _ { i _ 1 } \dotsm h _ { i _ n } \bigg ) x ^ k . \end{align*}


\begin{align*} \bar M : = \sup \{ M \geqslant 0 \tilde u _ M > 0 B _ 1 ( 2 e _ 1 ) \} \end{align*}


\begin{align*} ( M ) \subseteq \bigcup _ { i = 1 } ^ k \left [ \lambda _ i - c _ i , \lambda _ i + c _ i \right ] \end{align*}


\begin{align*} L \otimes \Z _ v & = L _ { v , 0 } , \\ K _ { \ell } \otimes \Z _ v & = K _ { \ell , v , 0 } . \end{align*}


\begin{align*} ( S S ' ) ^ T J S S ' = S '^ T ( S ^ T J S ) S ' = S '^ T J S ' = J . \end{align*}


\begin{align*} f ( x , y ) = \sum _ { i = 0 } ^ { \infty } { \vphantom { \sum } } ' \sum _ { j = 0 } ^ { \infty } { \vphantom { \sum } } ' c _ { i , j } T _ i ( x ) T _ j ( y ) , \end{align*}


\begin{align*} T ( x ; k , r ) = B ( r , k ) x + O ( ( r x ) ^ { 1 / 2 + \varepsilon } ) , \ B ( r , k ) = O ( 1 / r ) . \end{align*}


\begin{align*} \Phi ( a _ k , a _ 0 , \dots , a _ { k - 1 } ) = ( - 1 ) ^ k \Phi ( a _ 0 , a _ 1 , \dots , a _ k ) , \ \ \ \forall a _ 0 , \dots , a _ k \in A . \end{align*}


\begin{align*} 0 \in \begin{bmatrix} H \mathbf { x } + \mathbf { g } - A ^ T \mathbf { y } + \partial _ \mathbf { x } I _ D ( \mathbf { x } , \mathbf { y } ) + \rho ( \mathbf { x } - \mathbf { x } _ k ) \\ A \mathbf { x } - \mathbf { b } + \partial _ \mathbf { y } I _ D ( \mathbf { x } , \mathbf { y } ) + \delta ( \mathbf { y } - \mathbf { y } _ k ) \end{bmatrix} \end{align*}


\begin{align*} & \theta _ k = \sum _ { I \in \mathcal { I } _ k } c ^ k _ I \biggl ( \sum _ { e \in E } \Bigl ( \prod _ { i \in I } a _ { i e _ i } \Bigr ) w _ e \biggr ) , & & k = 1 , \ldots , \kappa , \\ & w \geq 0 , \ \sum _ { e \in E } w _ e = 1 , \ \lambda _ { i j } = \sum _ { e \in E : e _ i = j } w _ e , & & i = 1 , \ldots , d , \ ; j = 0 , \ldots , n , \\ & f = A ( \lambda ) , \ ( \lambda _ i , \delta _ i ) \in ( \ref { e q : S O S 2 - l o g } ) , & & i = 1 , \ldots , d . \end{align*}


\begin{align*} \mu _ { T } \{ \lambda \in \mathbb { C } : | \lambda | \leq r \} = \frac { s ( r , 0 ) ^ 2 } { s ( r , 0 ) ^ 2 + r ^ 2 } , \end{align*}


\begin{align*} \gamma = \frac { \log \left ( n + \sqrt { n ^ 2 + 4 } \right ) - \log ( 2 ) } { a \ , n } . \end{align*}


\begin{align*} | \mathcal { F } ( a + b , - b \ , ; \ , \{ U \} , \{ D \} ) | = \sum _ { i = 0 } ^ { a - 1 } \binom { a + b - 2 } { \lfloor i / 2 \rfloor } \binom { a + b - 1 - \lfloor i / 2 \rfloor } { a - i - 1 } . \end{align*}


\begin{align*} \overline { u } _ n ( x ) = \frac { ( - 1 ) ^ n } { n ! } \frac { d ^ n } { d x ^ n } \lim _ { \epsilon \searrow 0 } \frac { 1 } { x \mp i \epsilon } . \end{align*}


\begin{align*} D ( \varphi ) \leq \frac C { n ^ 2 } \sum _ { i = 1 } ^ n \alpha \log \Big ( \max \Big \{ \frac 1 { \alpha \ell _ i } , e \Big \} \Big ) . \end{align*}


\begin{align*} E [ S _ n ] = \beta a _ n \mbox { f o r a n y $ n \in \mathbb { N } $ } . \end{align*}


\begin{align*} \sum _ { p \le x } \frac { ( \log p ) ^ j } { p } = \frac { ( \log x ) ^ j } { j } + O ( ( \log x ) ^ { j - 1 } ) . \end{align*}


\begin{gather*} T = \begin{pmatrix} 0 & a & 0 & 0 \\ 0 & 0 & 0 & 0 \\ b & c & d & e \\ f & g & h & k \end{pmatrix} \end{gather*}


\begin{align*} A - \lambda = 0 \end{align*}


\begin{align*} \delta ^ k _ { - 1 } = 1 , ~ \delta _ j ^ k = \prod \limits _ { i = k - j } ^ k \delta _ i , ~ \gamma ^ k _ j = \gamma \sum \limits ^ { j - 1 } _ { i = - 1 } \delta ^ k _ i . \end{align*}


\begin{align*} | \mathcal { F } ( a + b - 1 , - b \ , ; \ , \{ U \} , \emptyset ) | & = \sum _ { i = 0 } ^ { a - 1 } \binom { a + b - 2 } { \lfloor i / 2 \rfloor , b + \lfloor ( i - 1 ) / 2 \rfloor , a - i - 1 } , \\ | \mathcal { F } ( a + b - 1 , - b - 1 \ , ; \ , \{ U \} , \emptyset ) | & = \sum _ { i = 0 } ^ { a - 2 } \binom { a + b - 2 } { \lfloor i / 2 \rfloor , b + \lfloor ( i + 1 ) / 2 \rfloor , a - i - 2 } . \end{align*}


\begin{align*} \prod _ { i \in I _ \kappa ^ * } c _ i = ( - 1 ) ^ { \frac { | I _ \kappa ^ * | - \sum _ { i \in I ^ * } c _ i } { 2 } } . \end{align*}


\begin{align*} | z \rangle = A _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { z ^ { n } } { \sqrt { n ! } } | n \rangle . \end{align*}


\begin{align*} \alpha _ { i } = \sum _ { j = i } ^ k \theta _ { 0 , j } , \alpha _ { n + 1 - i } = - \alpha _ { i } , \forall i = 1 , \ldots , k . \end{align*}


\begin{align*} | B | \leq | B _ { \ell + k _ \delta } | \leq \gamma | D _ \ell \setminus D _ { \ell + k _ \delta } | = \gamma | D _ \ell | - \gamma | D _ { \ell + k _ \delta } | \leq \gamma | D _ \ell | , \end{align*}


\begin{align*} \tilde { Y } _ { k + 1 } = \tilde { Y } _ { k } - \eta \tilde { Y } _ { k } + \frac { \eta ^ { 1 / \alpha } } { \sigma } \widetilde { Z } _ { k + 1 } , k = 0 , 1 , 2 , \dots , \end{align*}


\begin{align*} \check { \Delta } ( \omega ) \Delta ( a ) ( \chi \otimes b ) = \sum \Delta ( a ' ) \check { \Delta } ( \omega ' ) ( \chi \otimes b ) . \end{align*}


\begin{align*} f ^ \infty ( x , u , \omega ) = V ^ \infty \left ( u - \sum _ { t = 0 } ^ { T - 1 } x _ t \cdot \Delta s _ { t + 1 } ( \omega ) - \bar c ( \omega ) \cdot \bar x + S _ 0 ^ \infty ( \bar x ) , \omega \right ) + \sum _ { t = 0 } ^ { T - 1 } \delta _ { D _ t ^ \infty ( \omega ) } ( x _ t , \omega ) . \end{align*}


\begin{align*} 0 & = \ < \rho , \dot S + \nabla _ { \rho } \mathcal H ( \rho , S ) + \nabla _ { \rho } \mathcal H _ 1 ( \rho , S ) \dot W ^ { \delta } \ > \\ & = \ < \rho , \dot S \ > + \mathcal H ( \rho , S ) + \mathcal H _ 1 ( \rho , S ) \dot W ^ { \delta } \le 0 , \end{align*}


\begin{align*} \frac { q + 1 } { | G | } \sum _ { \eta \in S _ { \ell } ^ \wedge } R ( \alpha \eta ) ( \mathbf { d } ( a ) ) & = \frac { q + 1 } { | G | } \left ( \alpha ( a ) \sum _ { \eta \in S _ { \ell } ^ \wedge } \eta ( a ) + \alpha ( a ^ { - 1 } ) \sum _ { \eta \in S _ { \ell } ^ \wedge } \eta ( a ^ { - 1 } ) \right ) . \end{align*}


\begin{align*} \int _ { ( - \epsilon ) } & \int _ { ( \epsilon / 2 ) } \left | \zeta ( 1 + w ) \widetilde { W } ( 1 - w ) \mathcal { H } ( z , - w ) \frac { e ^ { \delta z } - e ^ { - \delta z } } { 2 \delta z } \right | \ , | d z | \ , | d w | \\ & \ll \int _ { ( - \epsilon ) } \int _ { ( \epsilon / 2 ) } | w | ^ { - 9 8 } | z | ^ { \varepsilon - 1 } \min \left \{ 1 , \frac { 1 } { \delta | z | } \right \} \ , | d z | \ , | d w | \ll \left ( \frac { 1 } { \delta } \right ) ^ { \varepsilon } . \end{align*}


\begin{align*} \hat \xi ^ b = \omega ^ b + \epsilon ( \dot p ^ b + \omega ^ b \times p ^ b ) \end{align*}


\begin{align*} R _ 1 = J _ { 2 3 } + O \big ( ( X h k ) ^ { \varepsilon } k ^ { 1 / 2 } Q ^ { - 9 6 } \big ) . \end{align*}


\begin{align*} ( x , y ) \ = \ x y . \end{align*}


\begin{align*} \langle f , \varphi \rangle = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \langle M _ \omega T _ x g , \varphi \rangle \ , d ( x , \omega ) . \end{align*}


\begin{align*} \mathcal { L } ( f ) ( x ) & = \frac { 1 } { \alpha } \bigg ( - \langle x ; \nabla ( f ) ( x ) \rangle + \sum _ { k = 1 } ^ d \partial _ k D ^ { \alpha - 1 } _ { k } ( f ) ( x ) + \frac { 1 } { p _ \alpha ( x ) } \sum _ { k = 1 } ^ d \partial _ k R _ k ^ \alpha ( p _ \alpha , f ) ( x ) \\ & \quad + \sum _ { k = 1 } ^ d \left ( \frac { \partial _ k ( p _ \alpha ) ( x ) } { p _ \alpha ( x ) } D _ k ^ { \alpha - 1 } ( f ) ( x ) + \partial _ k D _ k ^ { \alpha - 1 } ( f ) ( x ) \right ) \bigg ) , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 4 f ( v _ i ) = 1 2 , \end{align*}


\begin{align*} \pi ( M _ p ) = \sum _ { q \in Q } \Big ( \sum _ { y \leq q , \ , g ( y ) = p } \mu _ { Q } ( y , q ) \Big ) F _ q . \end{align*}


\begin{align*} & \min _ { t \in [ 0 , T ] } \min _ { i = 1 } ^ N \rho _ i ( t ) > 0 , a . s . \\ & \max _ { i j \in E } | S _ i ( t ) - S _ j ( t ) | \le \sqrt { \frac { C _ { K i n } } { \min \limits _ { t \in [ 0 , T ] } \min \limits _ { i = 1 } ^ N \rho _ i ( t ) } } < \infty , a . s . \end{align*}


\begin{align*} \lim _ { t \to s + } \int _ \Omega u ^ { - p } ( t , x ; s , u _ 0 ) d x = \int _ \Omega u ^ { - p } _ 0 ( x ) d x . \end{align*}


\begin{align*} M _ n ( k ) & = \frac { 1 } { n } \sum _ { \mathbf { p } _ k \in \mathcal { P } ( d , k ) } \sum _ { i \neq j } ^ n \left \{ \prod _ { \ell = 1 } ^ k I ^ { ( p _ \ell ) } _ { i , j } - \left ( \frac { - 1 } { 6 n } \right ) ^ { k } \right \} . \end{align*}


\begin{align*} \tilde N _ { n , \mu } ( k ) = \sum \limits _ { ( \mathbf { p } _ { k , 1 } , \mathbf { p } _ { k , 2 } ) \in \mathcal { Z } _ k ( \mu ) } \tilde N _ { n , \mathbf { p } _ { k , 1 } } ( k ) \tilde N _ { n , \mathbf { p } _ { k , 2 } } ( k ) . \end{align*}


\begin{align*} r _ l \big ( \epsilon ( X , { \rm i n d } _ { \mathcal { W } _ E } ^ { \mathcal { W } _ F } ( \Pi _ { \widetilde { \chi _ 0 } } \otimes \Pi _ { \pi _ E } ) , \psi _ F ) \big ) = r _ l \Big ( \epsilon \big ( X , ( \Pi _ { \chi _ 0 } \otimes \Pi _ { \pi _ F } ) \otimes { \rm i n d } _ { \mathcal { W } _ E } ^ { \mathcal { W } _ F } ( 1 _ E ) , \psi _ F \big ) \Big ) = \Big ( r _ l \big ( \epsilon ( X , \Pi _ { \chi _ 0 } \otimes \Pi _ { \pi _ F } , \psi _ F ) \big ) \Big ) ^ l . \end{align*}


\begin{align*} \frac { c _ k ( \mu ) L ( \mu , \mathbf { f } , \chi ) } { \pi ^ { \beta } } \mathbf { f } ( g ) = \sum _ { j = 1 } ^ e \langle \mathbf { h } _ j , \mathbf { f } \rangle \cdot \mathbf { g } _ j ( g ) . \end{align*}


\begin{align*} \Lambda ( \phi ) : = \phi ( \mathbf { 1 } ) + \sum _ { w \in \N ^ * } \phi ( w ) x _ w , \end{align*}


\begin{align*} \mathcal { A } ^ \infty _ G ( X , E ) : = \big ( \mathcal { C } ( G ) \hat { \otimes } \Psi ^ { - \infty } ( S , E | _ S ) \big ) ^ { K \times K } . \end{align*}


\begin{align*} a + b = c + d \end{align*}


\begin{align*} R _ k ( x ) = \left \{ 0 \leq i \leq N : \bigg | \frac { ( N - n ) } { 2 } + x - i \bigg | < k \right \} & \end{align*}


\begin{align*} \lim _ { n \to \infty } \binom { c _ n k n } l \frac { ( k ( n - s - 1 ) ) _ { c _ n k n - l } } { ( k n ) _ { c _ n k n } } = \frac 1 { l ! } ( \frac c { 1 - c } ) ^ l ( 1 - c ) ^ { k ( s + 1 ) } . \end{align*}


\begin{align*} P _ t ( x , A ) : = P _ t \mathbf { 1 } _ A ( x ) , t > 0 , x \in M , A \subset M \ , \end{align*}


\begin{align*} z ^ m = ( k ^ + ) ^ m \cdot ( g _ 0 ^ { - 1 } \cdot \phi ( g ^ + ) ^ m \cdot g _ 0 ) . \end{align*}


\begin{align*} \sum _ { \substack { p \leq x ^ { 1 / 2 } \\ p \nmid d } } \frac { ( \lambda ^ 2 _ f ( p ) - 2 ) } { p ^ { 1 + 2 / \log x } } \frac { \log ( x / p ^ 2 ) } { \log x } = \sum _ { \substack { p \leq x ^ { 1 / 2 } \\ p \nmid d } } \frac { \lambda ^ 2 _ f ( p ) - 2 } { p ^ { 1 + 2 / \log x } } - 2 \sum _ { \substack { p \leq x ^ { 1 / 2 } \\ p \nmid d } } \frac { ( \lambda ^ 2 _ f ( p ) - 2 ) } { p ^ { 1 + 2 / \log x } } \frac { \log p } { \log x } . \end{align*}


\begin{align*} \mathcal T _ Q ( \bar x ) : = \left \{ d \in \mathbb X \ , \middle | \ , \begin{aligned} & \exists \{ d _ k \} _ { k \in \N } \subset \mathbb X , \ , \exists \{ t _ k \} _ { k \in \N } \subset \R _ + \colon \\ & d _ k \to d , \ , t _ k \searrow 0 , \ , \bar x + t _ k d _ k \in Q \ , \forall k \in \N \end{aligned} \right \} \end{align*}


\begin{align*} \partial ^ i f = \partial ^ i f ( z ) = \frac { \partial ^ i } { \partial z ^ i } f ( z ) . \end{align*}


\begin{align*} \Delta _ \prec ( a _ 1 \dotsm a _ n ) : = a _ 1 \dotsm a _ n \otimes \ 1 + \sum _ { 1 \in S \subsetneq [ n ] } w _ S \otimes w _ { J ^ S _ 1 } \vert \dotsm \vert w _ { J ^ S _ m } \end{align*}


\begin{align*} \Phi _ { \lambda ^ h _ k } ( x ) = \begin{pmatrix} \xi _ { \lambda ^ h _ k } ( x ) \\ \eta _ { \lambda ^ h _ k } ( x ) \end{pmatrix} , \forall | k | \geq k _ 0 . \end{align*}


\begin{align*} \boldsymbol { b } ^ n = \boldsymbol { b } ^ n + \mathcal { M } ^ n _ { 1 D } \boldsymbol { q } , \boldsymbol { q } = [ q _ 1 , . . . , q _ { N _ { e p } ^ { 1 D } } ] ^ T , \end{align*}


\begin{align*} \tau _ { j + 1 } = \tau _ j + \nu ( \tau _ j ) + 1 , \end{align*}


\begin{align*} m _ { \pm } ( x , k ) = e ^ { \mp i k x } \psi _ { \pm } ( x , k ) . \end{align*}


\begin{align*} \langle \tfrac { 1 } { x } , f \rangle = - \lim _ { \varepsilon \to 0 } \int _ { | x | > \varepsilon } \log | x | \ , \overline { f ' ( x ) } \ , d x + ( \overline { f ( - \varepsilon ) } - \overline { f ( \varepsilon ) } ) \log ( \varepsilon ) . \end{align*}


\begin{align*} \kappa ( k , l ) = \kappa ( 0 , l ) + \kappa ( k , 0 ) + k \cdot l = \kappa ( 0 , l ) + \kappa ( k , 0 ) . \end{align*}


\begin{align*} \prod _ { \iota \in 2 ^ { V ' } } h _ { x y } ^ \iota ( c ) = \sum _ { \xi \in \Xi } \prod _ { \iota \in 2 ^ { V ' } } g ^ { \iota } ( \xi ( \iota ) ) . \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ r ( \widetilde { W _ v } , \widetilde { U _ v } ) ^ l q _ F ^ { - l r / 2 } X ^ { - l r } = \omega _ \sigma ( - 1 ) ^ { n - 2 } \gamma ( X , \pi , \sigma , \psi ) ^ l \sum _ { r \in \mathbb { Z } } c ^ F _ r ( W _ v , U _ v ) ^ l q _ F ^ { l r / 2 } X ^ { l r } \end{align*}


\begin{align*} H _ i = I - 2 \frac { a _ { i } a _ { i } ^ T } { a _ { i } ^ T a _ { i } } , i = 1 , 2 , \ldots , m . \end{align*}


\begin{align*} \alpha _ + ( t ) = B ( h ( t ) , \mathcal { Y } _ - ) , \alpha _ - ( t ) = B ( h ( t ) , \mathcal { Y } _ + ) , \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal w _ k \frac { w _ k ^ \intercal ( A x _ k - b ) } { \norm { A ^ \intercal w _ k } _ 2 ^ 2 } . \end{align*}


\begin{align*} \dim _ K H ^ 2 _ { \mathrm { c r i s } } ( Z / W ) _ 0 = 1 . \end{align*}


\begin{align*} \mathfrak { F } _ { ( \alpha , \beta ) } ( e ^ { - | t | } ) = \{ \alpha \Z \times \beta \Z \mid \alpha \beta < 1 \} \end{align*}


\begin{align*} k \rho \mu ( u ) = \rho \mu ( u ^ k ) = \omega ( u ^ k ) = k \omega ( u ) , \end{align*}


\begin{align*} \int _ 0 ^ T ( z - \hat { \eta } _ i ) \dd ( \hat { \eta } _ i - h _ i ) & = \int _ 0 ^ t ( z - \hat { \eta } _ i ) \dd ( \hat { \eta } _ i - h _ i ) + \int _ t ^ T ( z - \hat { \eta } _ i ) \dd ( \hat { \eta } _ i - h _ i ) \\ & = \int _ 0 ^ t ( z - \hat { \eta } _ i ) \dd ( \hat { \eta } _ i - h _ i ) \\ & = \int _ 0 ^ t ( z - \eta _ i ) \dd ( \eta _ i - h _ i ) \ge 0 , \end{align*}


\begin{align*} & h - k = \frac { \lambda } { ( T - \tilde { t } ) ^ 2 } , \ p = \frac { 4 | \tilde { x } - \tilde { y } | ^ 2 ( \tilde { x } - \tilde { y } ) } { \varepsilon ^ 4 } , \\ & \begin{pmatrix} X & 0 \\ 0 & - Y \end{pmatrix} \le \begin{pmatrix} Z & - Z \\ - Z & Z \end{pmatrix} + \rho \begin{pmatrix} Z & - Z \\ - Z & Z \end{pmatrix} ^ 2 , \end{align*}


\begin{align*} f ^ { \omega } = \sum _ { j \in \N } g _ { j } ( \omega ) \Box _ j f . \end{align*}


\begin{align*} ( g \dot { \gamma } ^ { - 1 } ) ^ { - 1 } b \phi ( g \dot { \gamma } ^ { - 1 } ) = \dot { \gamma } g ^ { - 1 } b \phi ( g ) \phi ( \dot { \gamma } ^ { - 1 } ) = \dot { \gamma } k _ 1 ^ o \omega k \phi ( \dot { \gamma } ^ { - 1 } ) = \end{align*}


\begin{align*} | \dot { X } _ j | = O ( \delta + \delta \delta ^ * ) | \dot { \alpha } | = O ( \delta + \delta \delta ^ * ) . \end{align*}


\begin{align*} \mathbb P ( S _ n ^ c ) = O ( n ^ { - \infty } ) . \end{align*}


\begin{align*} \frac { 1 } { t } d _ K ( a + t \beta a ) = \frac { 1 } { t } | ( a + t \beta a ) - a | = | \beta | a . \end{align*}


\begin{align*} \chi _ + ( x ) = \int _ { - \infty } ^ { x } \Phi ( y ) \ , d y , \chi _ + ( x ) + \chi _ - ( x ) = 1 . \end{align*}


\begin{align*} \sum _ { \rho \in R _ \epsilon } ( \rho ) & = \frac { 1 } { 2 \pi } \int _ { 0 } ^ { T } \left [ \log \left | - \frac { 1 } { 2 } - \epsilon + i t \right | - \log \left | - \frac { 1 } { 2 } + \epsilon + i t \right | \right ] \ d t + \\ & \frac { 1 } { 2 \pi } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \left [ \arg \left ( \sigma - 1 + i T \right ) - \arg ( \sigma - 1 ) \right ] \ d \sigma \end{align*}


\begin{align*} - \triangle _ { g _ { \textnormal { p o i n } , j } } ( u _ { j } ) = e ^ { 2 u _ { j } } \ , K _ { \tilde F _ { t _ { j } } ^ * ( g _ { \textnormal { e u c } } ) } \textnormal { o n } \ , \ , \ , \Sigma \end{align*}


\begin{align*} d _ K ( \xi ( t ; s , x ) ) = d _ K ( \eta _ s ( t - s ; x ) ) \leq \Phi _ { \beta + L , 0 } ( d _ K ( x ) , \epsilon , t - s ) . \end{align*}


\begin{align*} \widehat { \Phi ^ { - 1 } } ( x ) = \frac { 1 } { 1 + \widehat { \kappa } ( x ) } . \end{align*}


\begin{align*} \sum _ { \gamma \in \Gamma _ { 6 } ( H G ) } c ( g ( \gamma ) ) = 2 \cdot 7 + 7 = 2 1 \equiv 1 \pmod 2 . \end{align*}


\begin{align*} & \ < S ( 1 ) , \rho ^ b \ > - \ < S ( 0 ) , \rho ^ a \ > \\ = & \int _ 0 ^ 1 ( \ < m , \nabla _ G S \ > + \ < \rho , \dot S \ > ) d t + \int _ 0 ^ 1 \ < \nabla _ G \Sigma , \nabla _ G S \ > _ { \theta ( \rho ) } d W ^ { \delta } ( t ) \\ \le & \mathcal A ( \rho , m ) + \int _ 0 ^ 1 \Big ( \ < \rho , \dot S \ > + \frac 1 2 \| \nabla _ G S \| _ { \theta ( \rho ) } ^ 2 + \ < \nabla _ G \Sigma , \nabla _ G S \ > _ { \theta ( \rho ) } \dot { W ^ { \delta } ( t ) } \Big ) d t \le \mathcal A ( \rho , m ) . \end{align*}


\begin{align*} c _ { \sigma } ( Y , \lambda ) = \inf \{ L > 0 | \ , \sigma \in \mathrm { I m } ( i _ { L } : \mathrm { E C H } ^ { L } ( Y , \lambda , \Gamma ) \to \mathrm { E C H } ( Y , \lambda , \Gamma ) ) \ , \} \end{align*}


\begin{align*} \left \langle E , ( \partial _ { \Sigma _ 0 , \epsilon } ^ { * } F ) ^ { ( 1 ) } [ \phi ] \otimes ( \partial _ { \Sigma _ 0 , \epsilon } ^ { * } G ) ^ { ( 1 ) } [ \phi ] \right \rangle = \left \langle E _ { \Sigma _ 0 } , F ^ { ( 1 ) } [ \partial _ { \Sigma _ 0 , \epsilon } \phi ] \otimes G ^ { ( 1 ) } [ \partial _ { \Sigma _ 0 , \epsilon } \phi ] \right \rangle . \end{align*}


\begin{align*} Y = Z _ { 2 h ( Y ) } \circ \cdots \circ Z _ { 1 } , \end{align*}


\begin{align*} \displaystyle | \mathcal { S C } _ { ( s , s + d , s + 2 d ) } | = \begin{cases} & \displaystyle \sum _ { i = 0 } ^ { \lfloor s / 4 \rfloor } \binom { ( s + d - 1 ) / 2 } { i , d / 2 + i , ( s - 1 ) / 2 - 2 i } \\ & \\ & \displaystyle \sum _ { i = 0 } ^ { \lfloor s / 2 \rfloor } \binom { \lfloor ( s + d - 1 ) / 2 \rfloor } { \lfloor i / 2 \rfloor , \lfloor ( d + i ) / 2 \rfloor , \lfloor s / 2 \rfloor - i } \end{cases} \end{align*}


\begin{align*} \phi \psi & = ( \sigma \tau , ( \phi _ { \tau ^ { - 1 } 1 } \psi _ 1 , \phi _ { \tau ^ { - 1 } 2 } , \phi _ { \tau ^ { - 1 } 3 } \psi _ 3 ) ) \\ & = ( \sigma \tau , ( \phi _ { \tau ^ { - 1 } 1 } \psi _ 1 , \varepsilon , \phi _ { \tau ^ { - 1 } 3 } \psi _ 3 ) ) \in \prescript { \sigma \tau } { } N _ l ^ A \end{align*}


\begin{align*} \norm { \phi } _ { H ^ s _ { { p e r } } ( I ) } = \left ( \sum _ { m \in \mathbb { Z } } \left ( 1 + \frac { 4 \pi ^ 2 m ^ { 2 } } { \mod { I } ^ 2 } \right ) ^ s \mod { c _ m } ^ 2 \right ) ^ \frac { 1 } { 2 } . \end{align*}


\begin{align*} \Psi ^ { A i } ( \zeta ) \sim \zeta ^ { - \frac { \sigma _ 3 } { 4 } } \Psi ^ { A i } _ { 0 } \left ( I + \sum _ { j = 1 } ^ { \infty } \frac { \mathcal { K } _ j } { \zeta ^ { 3 j / 2 } } \right ) , \end{align*}


\begin{align*} & \widetilde { f } ( \widetilde { t } ) \geqq \widetilde { f } ( 0 ) = f ( \alpha ) = 0 , \\ & \widetilde { f } ( \widetilde { t } ) \geqq { A _ { 0 } \widetilde { t } } . \end{align*}


\begin{align*} \sum _ { p \in \mathcal { S } ( j ) } ( \log p ) \left | L ( \tfrac { 1 } { 2 } , f \otimes \chi _ { 8 p } ) \right | ^ 2 \left | \mathcal { M } ( p ) \right | ^ 2 \ll X \exp \Big ( \frac { 4 } { \alpha _ j } \Big ) ( \mathcal { I } - j ) e ^ { - 1 2 2 / \alpha _ { j + 1 } } \ll ( \mathcal { I } - j ) e ^ { - 4 2 / \alpha _ { j + 1 } } X \ll e ^ { - 1 / ( 1 0 \alpha _ { j } ) } X . \end{align*}


\begin{align*} \mathrm { d i v } _ x \mathbf { S } ( u ) = \mu ^ S \Delta _ x u + \Big ( \mu ^ B + \frac { \mu ^ S } { 3 } \Big ) \nabla _ x \mathrm { d i v } _ x u = \mu _ 1 \Delta _ x u + \mu _ 2 \nabla _ x \mathrm { d i v } _ x u , \end{align*}


\begin{align*} ( f \bullet g ) \bullet h ( x ) = h ( x ) ( f \bullet g ) ( x h ( x ) ) = h ( x ) g ( x h ( x ) ) f ( x h ( x ) g ( x h ( x ) ) ) . \end{align*}


\begin{align*} E ( t , x ) = - ( \nabla \phi * _ x \rho [ f ( t , \cdot ) ] ) ( x ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \frac { \mathrm { d } } { \mathrm { d } t } ( X _ 1 - X _ 2 ) + A \big ( A ( X _ 1 - X _ 2 ) + F ( X _ 1 ) - F ( X _ 2 ) \big ) = 0 , & \\ ( X _ 1 - X _ 2 ) ( 0 ) = 0 . & \end{array} \right . \end{align*}


\begin{align*} \mathcal { F } ( t ) = E ( \phi ( \cdot , t ) ) + \| \dot { \phi } ( \cdot , t ) \| ^ 2 . \end{align*}


\begin{align*} \mu _ d ( \phi ( x , b ) ) = \nu ( \phi ) \mu _ d ( c l _ B ( \phi ( x , b ) ) ) . \end{align*}


\begin{align*} \mathcal { K } _ + ( x , k ) & = \begin{cases} T ( k ) e ^ { i k x } & k \geq 0 \\ e ^ { i k x } + R _ + ( - k ) e ^ { - i k x } & k < 0 , \end{cases} \\ \mathcal { K } _ - ( x , k ) & = \begin{cases} e ^ { i k x } + R _ - ( k ) e ^ { - i k x } & k \geq 0 \\ T ( - k ) e ^ { i k x } & k < 0 , \end{cases} \end{align*}


\begin{align*} \min _ u \mathcal { F } _ h ( u ) , s . t . A u = b , S u \geq \delta , \end{align*}


\begin{align*} ( a ^ { \dagger } a ) _ { k , \lambda } | m \rangle = ( a ^ { \dagger } a ) ( a ^ { \dagger } a - \lambda ) \cdots ( a ^ { \dagger } a - ( k - 1 ) \lambda | m \rangle = ( m ) _ { k , \lambda } | m \rangle , \end{align*}


\begin{align*} U : = \bigg \{ x \in B _ \rho ^ + \frac { u ( x ) } { \zeta ( x ) } > \frac { \partial _ 1 u ( a ) } { 2 \partial _ 1 \zeta ( a ) } \bigg \} . \end{align*}


\begin{align*} { } ^ b { \rm T r } _ { c _ Y } ( A ) = { } ^ b { \rm T r } _ { Y } \left ( \Phi _ A ( e ) \right ) . \end{align*}


\begin{align*} \mathcal { S } ( h , k ) = \mathcal { I } _ 0 ( h , k ) + \mathcal { I } _ 1 ( h , k ) + \mathcal { E } ( h , k ) , \end{align*}


\begin{align*} 2 ^ n E _ n ( u , v ) = A _ n ( x , y ) , \end{align*}


\begin{align*} \rho ( \mathbf { h } ^ { - 1 } ) = \rho ( - x , - \omega - \tau ) = e ^ { - 2 \pi i \tau } M _ { - \omega / 2 } T _ { - x } M _ { - \omega / 2 } = \rho ( \mathbf { h } ) ^ * . \end{align*}


\begin{align*} b ^ { - \frac { 1 } { 2 } } \partial _ \xi \phi _ { i , i n t , \beta } \left ( \frac { y _ 0 } { \sqrt { b } } \right ) \phi _ { i , o u t , \beta } ( y _ 0 ) - \phi _ { i , i n t , \beta } \left ( \frac { y _ 0 } { \sqrt { b } } \right ) \partial _ y \phi _ { i , o u t , \beta } ( y _ 0 ) = 0 \end{align*}


\begin{align*} \Psi ^ 0 _ { \alpha , \nu , \tilde \nu } : = \mathbf { L } _ { - \nu } ^ 0 \tilde { \mathbf { L } } _ { - \tilde \nu } ^ 0 \ : \Psi ^ 0 _ \alpha , \end{align*}


\begin{align*} | V _ g \left ( M _ \eta T _ \xi f \right ) ( x , \omega ) | = | V _ g f ( x - \xi , \omega - \eta ) | . \end{align*}


\begin{align*} e ^ { - m _ i ( x ) s + f _ { m _ i ( x ) } ( x ) } = \mu _ i ( \overline { B } ( x , \gamma _ i ) ) \leq \hat { \mu } ( \overline { B } ( x , \gamma _ i ) ) \leq C \mu _ i ( \overline { B } ( x , \gamma _ i ) ) = C e ^ { - m _ i ( x ) s + f _ { m _ i ( x ) } ( x ) } , ~ ~ \forall x \in K _ i . \end{align*}


\begin{align*} f ^ * ( v , y , \omega ) & = \sup \{ x \cdot v - l ( x , y , \omega ) \} \\ & = \sum _ { t = 0 } ^ T K _ t ^ * ( v _ { t } + \Delta y _ { t + 1 } , y _ t , \omega ) . \end{align*}


\begin{align*} v = \frac { 1 } { m } \sum _ { i = 1 } ^ { r } a _ j \ ; \rho _ { i _ j } \end{align*}


\begin{align*} S _ 0 : = \left ( \frac { 1 } { 2 \widetilde C } \right ) ^ 2 . \end{align*}


\begin{align*} d ( D , { \sf S N R } ) = \sqrt { \eta \lambda D / N } , \end{align*}


\begin{align*} \norm { Y f _ \ell - \prod \limits _ { s = 1 } ^ j \left ( 1 - \delta _ { k - 2 s + i _ s - \ell } ^ { k - 2 s + i _ s } \right ) f _ \ell } \leq \norm { g _ \ell } \sum \limits _ { s = 1 } ^ j \gamma ^ { k - 2 s + i _ s } _ { k - 2 s + i _ s - \ell } \prod \limits _ { t = 1 } ^ { s - 1 } \left ( 1 - \delta _ { k - 2 t + i _ t - \ell } ^ { k - 2 t + i _ t } \right ) \leq ( j + k ) j \gamma \norm { g _ \ell } . \end{align*}


\begin{align*} n _ { k } = \boldsymbol { n } , k = \{ 1 , 3 \} , n _ { k } = \boldsymbol { r } \times \boldsymbol { n } , k = 5 , \Gamma ^ { b o d y } , \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - A ^ \intercal e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } \frac { e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } ^ \intercal ( A x _ k - b ) } { \norm { A ^ \intercal e _ { \mathrm { r e m } ( \zeta _ { k - 1 } , n ) + 1 } } _ 2 ^ 2 } , \end{align*}


\begin{align*} Q = \begin{pmatrix} 0 & q \\ q & 0 \end{pmatrix} , V = 4 k Q + 2 i k \sigma _ 3 ( Q _ x - Q ^ 2 ) + 2 Q ^ 3 - Q _ { x x } . \end{align*}


\begin{align*} \phi _ { n , \lambda } ( x ) = e ^ { - x } \bigg ( x \frac { d } { d x } \bigg ) _ { n , \lambda } e ^ { x } = \sum _ { k = 1 } ^ { n } S _ { 2 , \lambda } ( n , k ) x ^ { k } , ( n \ge 1 ) . \end{align*}


\begin{align*} H = u _ { e e } = u _ { x x } + \tau ^ 2 u _ { y y } + 2 \tau u _ { x y } . \end{align*}


\begin{align*} \mathcal { A } ( G ( u ) ) = u . \end{align*}


\begin{align*} E ^ \alpha ( \rho ) & = \frac 1 2 \int _ \R \int _ \R K _ \alpha ( x - y ) \ , d \rho ( y ) d \rho ( x ) + \int _ \R Q _ \alpha ( x ) \ , d \rho ( x ) , \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } - i t \cdot \omega _ 2 ( i t ) & = \lim _ { t \rightarrow 0 ^ + } \frac { t | \lambda | ^ 2 } { s ( | \lambda | , t ) - t } \\ & = \lim _ { t \rightarrow 0 ^ + } \frac { | \lambda | ^ 2 } { s ( | \lambda | , t ) / t - 1 } \\ & = \lambda _ 1 ( \mu ) ^ 2 - | \lambda | ^ 2 . \end{align*}


\begin{align*} \frac { \mathcal { H } _ { d - 1 } ( W ' ( \omega ) \cap O ) } { | W ' ( \omega ) \cap O | ^ \frac { d - 1 } { d } } \geq \frac { \sum _ { l = 1 } ^ N \mathcal { H } _ { d - 1 } ( B ( x _ { j _ l } , \rho ' ) \cap O ) } { \sum _ { l = 1 } ^ N | B ( x _ { j _ l } , \rho ' ) \cap O | ^ \frac { d - 1 } { d } } \geq C _ . \end{align*}


\begin{align*} | \mathcal { D D } _ { ( s , s + d , s + 2 d ) } | = \sum _ { i = 0 } ^ { \lfloor ( s - 1 ) / 2 \rfloor } \binom { \lfloor ( s + d - 2 ) / 2 \rfloor } { \lfloor i / 2 \rfloor , \lfloor ( d + i ) / 2 \rfloor , \lfloor ( s - 1 ) / 2 \rfloor - i } . \end{align*}


\begin{align*} A = \sum _ { j = 1 } ^ { 2 d } P _ { j } A , \end{align*}


\begin{align*} \Box \phi _ k = \left ( | \phi _ { k , t } | ^ 2 - | \phi _ { k , x } | ^ 2 \right ) \phi _ { k } + \mathbf { 1 } _ { \omega } f _ { k - 1 } ^ { \phi ^ { \perp } _ k } , \ ; \phi _ k [ 0 ] = ( a , b ) , \end{align*}


\begin{align*} \inf _ { 0 \not = u \in H ^ 1 ( \mathbb { R } ^ N ) } \frac { \| u \| _ 2 ^ { ( N + \gamma ) - ( N - 2 ) p } \| \nabla u \| _ 2 ^ { N p - ( N + \gamma ) } } { \int _ { \mathbb { R } ^ N } \left ( | \cdot | ^ { - ( N - \gamma ) } * | u | ^ p \right ) | u | ^ p d x } \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d \tau } \| \varepsilon _ - \| _ { L ^ 2 _ { \rho } } ^ 2 = \left \langle \partial _ \tau \varepsilon _ - , \varepsilon _ - \right \rangle _ { L ^ 2 _ { \rho _ \beta } } + \beta ' \left \langle \varepsilon _ - , \varepsilon _ - \left ( \frac { d + 2 } { 2 \beta } - \frac { y ^ 2 } { 2 } \right ) \right \rangle _ { L ^ 2 _ { \rho _ \beta } } . \end{align*}


\begin{align*} \sum _ { u \sim w } \mathbf { \mathrm { v } } _ w = \lambda \mathbf { \mathrm { v } } _ u = \sum _ { \substack { w \sim u \\ w \in L ' \cup R } } \mathbf { \mathrm { v } } _ w + \sum _ { \substack { w \sim u \\ w \in E } } \mathbf { \mathrm { v } } _ w . \end{align*}


\begin{align*} ( \phi | \psi ) = \frac { 1 } { | A | } \sum _ { a \in A } \bar \phi ( a ) \psi ( a ) . \end{align*}


\begin{align*} \lambda \rho + \rho _ x + u _ x = f , \\ \lambda u + \rho _ x - u _ { x x } + u _ x = g . \end{align*}


\begin{align*} \frac { \omega _ { p } ( T ) } { 2 ^ { 1 - \frac { 1 } { p } } } \leq \omega _ { p } \left ( \begin{bmatrix} 0 & A \\ B & 0 \end{bmatrix} \right ) \leq \frac { \omega _ { p } ( T ) } { 2 ^ { - \frac { 1 } { p } } } . \end{align*}


\begin{align*} \tilde { S } : = \Bigl \{ ( x , s , \delta ) \Bigm | \bigl ( Z ( s ) , \delta \bigr ) \in ( \ref { e q : I n c - 1 } ) , \ ( x , s _ { \cdot n } ) \in W , \ s _ { i j } = \min \bigl \{ b _ { i j } ( \delta ) , s _ { i n } \bigr \} \ ; \forall i \ ; \forall j \Bigr \} . \end{align*}


\begin{align*} R A + \sum _ { j = 1 } ^ L \bar y _ j S B _ j \equiv 1 . \end{align*}


\begin{align*} N _ k ^ j f _ \ell = \lambda _ \ell f _ \ell , \end{align*}


\begin{align*} \underline { m } \int _ { B ^ 1 ( \mathbf { 0 } ) } \tilde { u } _ n ^ 2 + \underline { m } \int _ { B ^ 1 ( \mathbf { 0 } ) ^ c } \tilde { u } _ n ^ 2 = \underline { m } \ ; . \end{align*}


\begin{align*} | s _ \delta | = | d _ { n } ^ \delta ( C _ { n } ^ \delta , J _ { n } ^ \delta ) | \geq k _ \delta + 1 , n \in \Lambda , \end{align*}


\begin{align*} \alpha ( x \cdot y ) \cdot \beta x = \gamma y \end{align*}


\begin{align*} \sup _ { m \geq n } \| \widetilde { v } ^ { n , m } ( t ) \| _ { L ^ p ( \Omega ; H ^ \gamma ) } \leq & \| E ( t ) P _ n P X _ 0 \| _ { L ^ p ( \Omega ; H ^ \gamma ) } + \sup _ { m \geq n } \Big \| \int _ 0 ^ t E ( t - s ) P _ n A P F ( X ^ m ( s ) ) \ , \dd s \Big \| _ { L ^ p ( \Omega ; H ^ \gamma ) } \\ \leq & C \| X _ 0 \| _ { L ^ p ( \Omega ; H ^ \gamma ) } + C \leq C ( X _ 0 , \gamma , T ) . \end{align*}


\begin{align*} c _ { M a p } = \left ( c _ G + c _ { F _ k } \right ) m , \end{align*}


\begin{align*} ( \phi - 1 ) \kappa ( g ) + \kappa ( b ) = \kappa ( k ) + \kappa ( \mu ) . \end{align*}


\begin{align*} P _ n ( x ) = ( x - i ) ^ { n + 1 } A _ n \left ( { x + i \over x - i } \right ) . \end{align*}


\begin{align*} \max _ { \substack { x , y \in X \\ x \ne y } } \frac { \beta _ { v _ i } ( x ) - \beta _ { v _ i } ( y ) } { d ( x , y ) } = \max _ { \substack { x ' , y ' \in X \\ x ' \ne y ' } } \frac { \beta _ { v _ i ' } ( x ' ) - \beta _ { v _ i ' } ( y ' ) } { d ( x ' , y ' ) } \end{align*}


\begin{align*} A ( x , n ) \ = \ A _ n ( x ) \end{align*}


\begin{align*} X = \bigl \{ x \in \R ^ k \bigm | g _ i ( x ) \geq 0 , \ i = 1 , \ldots , m \bigr \} , \end{align*}


\begin{align*} \begin{aligned} P _ { n , k } ( W _ { \mathcal { S } ^ + ( n , k ; M ) } ) = \sum _ { j = 1 } ^ { n - 1 } \sum _ { l = 1 } ^ k \frac { \binom M l ( k ) _ l ( k ( j - 1 ) ) _ { M - l } } { ( k n ) _ M } \frac 1 { n - j } . \end{aligned} \end{align*}


\begin{align*} u = 1 + \dot u + r , \end{align*}


\begin{align*} \Delta _ { \mathbf { M } } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { M } } ( \mathcal { I } _ { \alpha \beta } ^ { i j } ) = c _ { 1 } \Delta _ { \mathbf { M } } ( \mathcal { I } _ { \alpha } ^ { i } ) \cdot \Delta _ { \mathbf { M } } ( \mathcal { I } _ { \beta } ^ { j } ) + c _ { 2 } \Delta _ { \mathbf { M } } ( \mathcal { I } _ { \beta } ^ { i } ) \cdot \Delta _ { \mathbf { M } } ( \mathcal { I } _ { \alpha } ^ { j } ) \end{align*}


\begin{align*} ( w - v ) ( x ) ( x ) & = \begin{cases} 0 & x \in \R ^ n _ + \cap \{ v \geqslant 0 \} , \\ u ( x ) - \tau d ^ { - n - 2 } \big ( 1 - \frac \theta 2 \big ) ^ { - n - 2 } \zeta ( x ) , & x \in \R ^ n _ + \cap \{ v < 0 \} . \end{cases} \end{align*}


\begin{align*} S _ { N _ \alpha } : = \Big [ z _ 1 + \frac { N _ \alpha - 1 } \alpha , z _ 2 \Big ) ^ 2 . \end{align*}


\begin{align*} u ( \cdot , t ; 0 , u ^ * ) = u ^ * ( \cdot ) \ , \ , t \ge 0 . \end{align*}


\begin{align*} \tilde { \phi } ( x ) : = \int _ { \R } \phi ( t , x ) \psi ( t ) \ , d t . \end{align*}


\begin{align*} \begin{cases} - k _ { \varepsilon } \Delta \tilde { \Psi } _ { \varepsilon } + | \nabla \tilde { \Psi } _ { \varepsilon } | ^ 2 - \tilde { \lambda } _ 0 \underline { m } = 0 & \Omega \ ; , \\ \tilde { \Psi } _ { \varepsilon } = - k _ { \varepsilon } \log ( w ) ( \frac { \mathbf { x } - \mathbf { x } _ { \varepsilon } } { k _ { \varepsilon } } ) & \partial \Omega \ ; , \end{cases} \end{align*}


\begin{align*} ( x g ( x ) ) _ i : = x _ i g ( x ) = \sum _ { w \in \N ^ * \cup \{ \mathbf { 1 } \} } g _ { w } x _ { i w } = x _ i + \sum _ { w \in \N ^ * } g _ { w } x _ i x _ w . \end{align*}


\begin{align*} \Xi _ t : = \{ X \in \Xi ~ | ~ X ( f ^ a ) = 0 a = 1 , \ldots , k \} \end{align*}


\begin{align*} \hat { \mu } _ { \varepsilon } \left ( \xi \right ) : = \hat { \mu } ( \xi ) \exp \left ( - \dfrac { \varepsilon ^ 2 \langle \xi ; \Sigma ( \xi ) \rangle } { 2 } \right ) , \end{align*}


\begin{align*} \begin{aligned} U ( \mathcal { O } f ) & = U \bigg [ \Big ( - \frac { d } { d x } + 2 x \Big ) \frac { d } { d x } f \bigg ] \\ & = \Big ( - \frac { d } { d x } + x \Big ) \Big ( \frac { d } { d x } + x \Big ) U f \\ & = \frac 1 2 \Big ( \Big ( \frac { d } { d x } + x \Big ) \Big ( - \frac { d } { d x } + x \Big ) + \Big ( - \frac { d } { d x } + x \Big ) \Big ( \frac { d } { d x } + x \Big ) \Big ) U f - U f \\ & = \mathcal { H } ( U f ) - U f . \end{aligned} \end{align*}


\begin{align*} \theta = \frac { v _ { + } - v _ { - } } { 6 } \in \left ( 0 , \frac { \lambda } { 6 } \right ) \end{align*}


\begin{align*} \langle v , w \rangle = v \cup \ast w \ ; v , w \in V _ k \ ; . \end{align*}


\begin{align*} \det ( 1 _ n + D A ) = \sum _ { I \subseteq N } \det ( A _ I ) X _ I , \end{align*}


\begin{align*} \begin{aligned} & H _ L ( w ^ d ) = [ \ , U _ { 1 } \ , U _ { 2 } \ , ] \begin{bmatrix} S & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} V _ 1 \\ V _ 2 \end{bmatrix} , \\ & H _ L ( \tilde { w } ^ d ) = [ \ , \tilde { U } _ { 1 } \ , \tilde { U } _ { 2 } \ , ] \begin{bmatrix} \tilde { S } & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \tilde { V } _ 1 \\ \tilde { V } _ 2 \end{bmatrix} \end{aligned} \end{align*}


\begin{align*} | \mathcal { A } | _ { | \mathcal { B } ( 1 ) } = \sum _ { k _ { \Omega } \in \mathcal { B } ( 1 ) } \phi ( k _ { \Omega } ) . \end{align*}


\begin{align*} r _ l \big ( \epsilon ( \widetilde { \chi _ 0 } \pi _ E , \psi _ F ) \big ) = \epsilon \big ( \chi r _ l ( \pi _ F ) ^ { ( l ) } , \overline { \psi } _ F ^ l \big ) . \end{align*}


\begin{align*} r _ I ( \underline { \lambda } , \sigma ) = \frac { \chi ( ( V _ f ^ I ) ^ { \langle ( \underline { \lambda } , \sigma ) \rangle } ) } { m _ I | { \rm K e r } \ , A _ \sigma \cap G | } \ , . \end{align*}


\begin{align*} u _ n = \sum _ { j = 1 } ^ { J } [ T _ n ^ j ] \phi ^ j + \omega _ n ^ { J } , \end{align*}


\begin{align*} f _ 1 L i _ { K , \mathfrak { s } _ 1 } ( \alpha ) + \cdots + f _ r L i _ { K , \mathfrak { s } _ r } ( \alpha ) = 0 \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { E \tau ^ { ( N , A ) } } N = \frac 1 { A + 1 } . \end{align*}


\begin{align*} \bigg ( x \frac { d } { d x } \bigg ) _ { n , \lambda } f ( x ) = \sum _ { k = 0 } ^ { n } S _ { 2 , \lambda } ( n , k ) x ^ { k } \bigg ( \frac { d } { d x } \bigg ) ^ { k } f ( x ) , ( \mathrm { s e e } \ [ 7 ] ) , \end{align*}


\begin{align*} \mathcal { D } ( h , k ) = \mathcal { I } ^ * _ 0 ( h , k ) + O \left ( \left ( Q + \frac { Q ^ 2 } { C } \right ) \frac { ( X H K ) ^ \varepsilon } { \sqrt { H K } } \right ) . \end{align*}


\begin{align*} \Omega = V _ S \cap \mathbb D ^ n = \{ ( z _ 1 \dots , z _ n ) \in \mathbb D ^ n \ , : \ , f _ i ( z _ 1 , \dots , z _ n ) = 0 \ , , \ , 1 \leq i \leq n \} . \end{align*}


\begin{align*} \begin{aligned} \min \ & f ( x ) \\ \enspace & h ( x ) = 0 \\ & g ( x ) \le 0 , \end{aligned} \end{align*}


\begin{align*} p _ 0 q _ 0 + \cdots + p _ n q _ n = m d _ i , \end{align*}


\begin{align*} \begin{aligned} & \left \{ \varsigma _ { Q _ j , i } = 1 \right \} \subseteq \left \{ \varsigma _ { Q _ j , m } = 1 \right \} , i \le m , \\ & \left \{ \varsigma _ { Q _ j , i } = 0 \right \} \subseteq \left \{ \varsigma _ { Q _ j , r } = 0 \right \} , r \le i . \end{aligned} \end{align*}


\begin{align*} \mathcal { S } ( h , k ) : = \sum _ { q = 1 } ^ { \infty } W \left ( \frac { q } { Q } \right ) \sideset { } { ^ \flat } \sum _ { \chi \bmod q } \chi ( h ) \overline { \chi } ( k ) \sum _ { m = 1 } ^ { \infty } \frac { \tau _ A ( m ) \chi ( m ) } { \sqrt { m } } V \left ( \frac { m } { X } \right ) \sum _ { n = 1 } ^ { \infty } \frac { \tau _ B ( n ) \overline { \chi } ( n ) } { \sqrt { n } } V \left ( \frac { n } { X } \right ) , \end{align*}


\begin{align*} \mbox { m i n i m i z e } \ ; \ ; f ( x ) : = \varphi ( x ) - ( g \circ F ) ( x ) \mbox { s u b j e c t t o } \ ; \ ; x \in \R ^ n . \end{align*}


\begin{align*} \widetilde { c } _ k ( ( s - k ) / 2 ) j ( g _ { \infty } , z _ 0 ) ^ { - k } f ( z ) = c _ k ( s ) \mathbf { f } ( h _ { \mathbf { h } } \cdot g _ { \infty } ) , \end{align*}


\begin{align*} \| f ( x ) \| _ p ^ p \leqslant C \sum \limits _ { l = - \infty } ^ \infty 2 ^ { l p } \mu ( { \Omega } _ l ) ^ { p } \leq C \| S _ { c w } ( f ) \| ^ p _ p \leqslant C \| f \| ^ p _ { H _ { c w } ^ p } \leqslant C \| f \| _ { H _ d ^ p } ^ p \end{align*}


\begin{align*} \Pr _ { V ' \subset W } [ \mathcal { E } _ i ( W ) ~ | ~ W ] = q ^ { i ^ 2 } \frac { { j \choose i } _ q { k \choose k - i } _ q } { { k + j \choose k } _ q } , \end{align*}


\begin{align*} \hat { p } _ \ell = \frac { ( p _ \ell / \mu _ \ell ) } { \sum _ { i = 1 } ^ { N } ( p _ i / \mu _ i ) } , \end{align*}


\begin{align*} F _ { ( \xi , \eta ) } ( x , \omega ) = e ^ { 2 \pi i x \cdot \omega } \ , V _ g ( M _ \eta T _ \xi f ) ( x , \omega ) \ , V _ g ( M _ \eta T _ \xi f ) ( - x , - \omega ) . \end{align*}


\begin{align*} \forall u \in \mathbb S _ { \mathbb X } \colon \ker D ^ * _ \gamma \Phi ( ( \bar x , \bar y ) ; ( u , 0 ) ) = \{ 0 \} . \end{align*}


\begin{align*} \overline { \{ g _ { \alpha } : \alpha \in S _ 2 \} } = A ^ 2 ( D , e ^ { - \varphi } ) . \end{align*}


\begin{align*} \hat { I } _ { ( n , \infty ) } ( k ) : = \bigcup _ { m = 0 } ^ { \infty } \hat { I } _ { ( n , m ) } ( k ) \end{align*}


\begin{align*} 2 ^ { \mu _ * k _ * t } \cdot 2 ^ { ‐ m _ t ( s _ * + 1 / q ‐ 1 / p _ 1 ) } = 2 ^ { ‐ \alpha _ * k _ * t } \cdot 2 ^ { ‐ m _ t ( 1 / q ‐ 1 / p _ 0 ) } \stackrel { \eqref { h a t _ t h e t a _ d e f } } { = } n ^ { ‐ q \hat \theta / 2 } ; \end{align*}


\begin{align*} \sigma _ \varepsilon : = \Big ( \frac { \varepsilon ^ d } { a _ \varepsilon ^ { d - 2 } } \Big ) ^ { \frac { 1 } { 2 } } , \ d \geqslant 3 ; \quad { \sigma _ \varepsilon } : = \varepsilon \left | \log \frac { { a _ \varepsilon } } { \varepsilon } \right | ^ { \frac { 1 } { 2 } } , \ d = 2 , \end{align*}


\begin{align*} \textbf { a } _ m ( \theta ) = \left [ e ^ { - j 2 \pi f _ 0 \tau _ { 1 m } ( \theta ) } , \ldots , e ^ { - j 2 \pi f _ 0 \tau _ { N _ s m } ( \theta ) } \right ] ^ T , \end{align*}


\begin{align*} x ^ \prime : y ^ \prime : z ^ \prime = ( a _ 1 x + b _ 1 y ) ( a _ 2 x + b _ 2 y ) : ( a _ 2 x + b _ 2 y ) ^ 2 : z y \ , . \end{align*}


\begin{align*} m ^ * ( \tau , s , u _ 0 ) = { \rm m a x } \Big \{ \int _ { \Omega } u ( \tau , x ; s , u _ 0 ) d x , \frac { a _ { \sup } } { b _ { \inf } } | \Omega | \Big \} \end{align*}


\begin{align*} \mathcal { X } ^ { M _ c } : = \mathcal { X } / \mathcal { C } ^ c = \{ [ g ] : = g + \mathcal { C } ^ c ~ | ~ g \in \mathcal { X } \} , \end{align*}


\begin{align*} ( G ' \cdot \mu + \tfrac { 1 } { 2 } G '' \cdot \sigma ^ 2 ) ( \xi _ i - ) & = \mu ( \xi _ i - ) - \alpha _ i \cdot \sigma ^ 2 ( \xi _ i ) = ( \mu ( \xi _ i - ) + \mu ( \xi _ i + ) ) / 2 = ( G ' \cdot \mu + \tfrac { 1 } { 2 } G '' \cdot \sigma ^ 2 ) ( \xi _ i ) \\ & = \mu ( \xi _ i + ) + \alpha _ i \cdot \sigma ^ 2 ( \xi _ i ) = ( G ' \cdot \mu + \tfrac { 1 } { 2 } G '' \cdot \sigma ^ 2 ) ( \xi _ i + ) . \end{align*}


\begin{align*} \overline { \cdot } : \mathbb { B } \to \mathbb { B } : a + b \zeta + c \xi + d \zeta \xi \mapsto \overline { a + b \zeta + c \xi + d \zeta \xi } = a - b \zeta - c \xi - d \zeta \xi . \end{align*}


\begin{align*} \int _ t ^ \infty \delta ( s ) d s = - \frac { 1 } { 8 s _ c ( p - 1 ) } \int _ t ^ \infty \ddot { y } ( s ) d s \lesssim e ^ { - c t } , \ ; \forall t \ge 0 . \end{align*}


\begin{align*} T _ k ( f ) ( x ) & = \sum \limits _ { i = 1 } ^ { \infty } \lambda _ i B f , \end{align*}


\begin{align*} | \mathcal { F } ( a + b , - b \ , ; \ , \{ U \} , \emptyset ) | = \sum _ { i = 0 } ^ { a } \binom { a + b - 1 } { \lfloor i / 2 \rfloor , b + \lfloor ( i - 1 ) / 2 \rfloor , a - i } . \end{align*}


\begin{align*} z y '' ( z ) = y ( z ) ; \end{align*}


\begin{align*} Z ( S _ g f ) = | Z g | ^ 2 Z f . \end{align*}


\begin{align*} w ( \gamma _ { 1 } \cup { \gamma _ { 3 } } ) = w ( \gamma _ { 2 } \cup { \gamma _ { 3 } } ) = w ( \gamma _ { 1 } \cup { \gamma _ { 4 } } ) = w ( \gamma _ { 2 } \cup { \gamma _ { 4 } } ) = 3 . \end{align*}


\begin{align*} \lambda ^ { 2 } F \sum _ { i } r _ { i } ^ { 2 } - \lambda F ^ { i j } r _ { j } \nabla _ { i } u = \lambda ^ { 2 } \sum _ { i } ( f - f ^ { i } \kappa _ { i } ) r _ { i } ^ { 2 } . \end{align*}


\begin{align*} g \rhd x ^ 0 = \epsilon ( g ) x ^ 0 , ~ g \rhd x ^ i = x ^ \mu \tau ^ { \mu i } ( g ) , ~ g \rhd \xi ^ i = \xi ^ j \tau ^ { j i } ( g ) , ~ g \rhd \partial _ i = \tau ^ { i j } ( S ( g ) ) \partial _ j . \end{align*}


\begin{align*} \Psi ^ { ( - 1 ) } _ { \mathfrak { s } , \boldsymbol { \alpha } } = \Phi _ { \mathfrak { s } , \boldsymbol { \alpha } } \Psi _ { \mathfrak { s } , \boldsymbol { \alpha } } \end{align*}


\begin{align*} B ( \mathcal { Y } _ - , h ) = B ( \mathcal { Y } _ + , h ) = 0 \Leftrightarrow \int \mathcal { Y } _ 1 h _ 2 = \int \mathcal { Y } _ 2 h _ 1 = 0 , \end{align*}


\begin{align*} \kappa ^ { * } _ { n e w } = \kappa ^ { * } - \frac { ( f ' ( \tilde { \infty } ) - 1 . ) } { g } \end{align*}


\begin{align*} \phi _ { k , \lambda } = \frac { 1 } { e } \sum _ { n = 0 } ^ { \infty } \frac { 1 } { n ! } ( n ) _ { k , \lambda } = \frac { 1 } { e } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { ( n - 1 ) ! } ( n - \lambda ) _ { k - 1 , \lambda } , ( k \in \mathbb { N } ) . \end{align*}


\begin{align*} \tau _ { \infty } : = \lim _ { n \rightarrow \infty } \tau _ n \le 1 . \end{align*}


\begin{align*} A ' ( e _ 1 ) = \begin{bmatrix} 1 & e _ 1 \cdot w _ 2 & \cdots & e _ 1 \cdot w _ n \\ 0 & & & \\ \vdots & & \bigg [ w _ { t _ 1 } \cdot w _ { t _ 2 } - ( e _ 1 \cdot w _ { t _ 1 } ) ( e _ 1 \cdot w _ { t _ 2 } ) \bigg ] _ { ( m - 1 ) \times ( m - 1 ) } & \\ 0 & & & \end{bmatrix} . \end{align*}


\begin{align*} \lambda \star ( x _ 0 , \dots , x _ n ) = \left ( \lambda ^ { q _ 0 } x _ 0 , \dots , \lambda ^ { q _ n } x _ n \right ) , \ \ \lambda \in k ^ \ast . \end{align*}


\begin{align*} 0 = \int _ { M ^ { n } } \mathrm { d i v } ( H X ^ { T } - \nabla \langle X , \nu \rangle ) \geq - ( n - 1 ) \int _ { M ^ { n } } \langle X , \nabla H \rangle . \end{align*}


\begin{align*} \Lambda ( M ) : = \bigcup _ { \Gamma \in H _ { 1 } ( M ) } \Lambda ( M , \Gamma ) \end{align*}


\begin{align*} \hat { V } = X \Sigma _ { \hat { V } } Y ^ { T } , \end{align*}


\begin{align*} \gamma _ { m a x } = \frac { N _ s P _ t \Bigl | \sum _ { n = 1 } ^ N | g _ n | | h _ n | + | h _ d | \Bigr | ^ 2 } { \sigma _ n ^ 2 } , \end{align*}


\begin{align*} [ \pi ( \l ' ) , \pi ( \l ) ] = \left ( 1 - e ^ { 2 \pi i ( \omega \cdot x ' - x \cdot \omega ' ) } \right ) \pi ( \l ' ) \pi ( \l ) , \l = ( x , \omega ) , \ \l ' = ( x ' , \omega ' ) . \end{align*}


\begin{align*} \lambda ( x ) & = \min \Big \{ h \in { \bigcup } _ { z = 0 } ^ { e } \Lambda _ z \colon h \equiv - x \ ( \bmod { \ p ^ e } ) \Big \} , \\ \overline { \lambda } ( x ) & = \frac { 1 } { 2 } ( n p ^ e + \lambda ( x ) ) . \end{align*}


\begin{align*} \nu _ { V , W , m } ( f ) : = \sup _ { g \in G } \lVert | 1 + L ( g ) | ^ m \Theta ( g ) ^ { - 1 } L _ V R _ W \big ( f \big ) ( g ) \rVert , \ f \in C _ c ^ \infty ( G ) . \end{align*}