\begin{align*} & { _ { 5 } F _ { 4 } } \left [ \begin{array} { c c c c c c c c } a , 1 + \frac { a } { 2 } , b , c , d \\ \frac { a } { 2 } , 1 + a - b , 1 + a - c , 1 + a - d \end{array} ; 1 \right ] \\ [ 1 m m ] & \ : \ : = \frac { \Gamma ( 1 + a - b ) \Gamma ( 1 + a - c ) \Gamma ( 1 + a - d ) \Gamma ( 1 + a - b - c - d ) } { \Gamma ( 1 + a ) \Gamma ( 1 + a - b - c ) \Gamma ( 1 + a - b - d ) \Gamma ( 1 + a - c - d ) } , \end{align*}


\begin{align*} & \int _ 0 ^ { \infty } e ^ { - \lambda s } p _ j ( s ) d s = \frac { q } { \lambda } \int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } ( 1 - y ) y ^ { j - 1 } d y . \end{align*}


\begin{align*} \bigl \{ \lim \langle A x _ n , y _ n \rangle & \colon x _ n , y _ n \in S _ H \ \forall n \in \N , \ \lim \langle T x _ n , y _ n \rangle = \| T \| \bigr \} \\ & = \{ \lim \langle A x _ n , T x _ n \rangle \colon x _ n \in S _ H \ \forall n \in \N , \ \lim \| T x _ n \| = \| T \| \bigr \} \end{align*}


\begin{align*} ( \gamma \ast \mu ) ( f ) ( \tau ) = \mu ( f \mid \gamma ) ( \gamma ^ { - 1 } \ast \tau ) . \end{align*}


\begin{align*} \Delta \Psi = f \ , , \lim _ { x _ { 2 } \to + \infty } \partial _ { 2 } \Psi = - \lim _ { x _ { 2 } \to - \infty } \partial _ { 2 } \Psi \ , , | \Psi | \leq C ( | x _ { 2 } | + 1 ) \end{align*}


\begin{align*} \Pr ( \widetilde { X } _ t = k + 1 ) = \sum _ { j = k + 1 } ^ n \Pr ( X _ t = j ) \overset { } { = } \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) , \end{align*}


\begin{align*} r '' _ { d - 1 } : = \sqrt { \frac { d - 1 } { d } } r ' _ d , \mbox { a n d } \gamma '' _ { d - 1 } : = \left ( \frac { 3 3 r } { \sqrt { d - 1 } } \right ) ^ { d - 1 } r \gamma _ 1 \log ^ { ( d - 1 ) / 2 } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) . \end{align*}


\begin{align*} \nu ( x ) = \langle y _ 1 ' x \Omega _ { \rho } , \Omega _ { \rho } \rangle _ { \rho } x \in M . \end{align*}


\begin{align*} \int _ { \R ^ d } \left ( \exp \left | \frac { e ^ { - t H ^ { \beta } } g } { \lambda } \right | ^ p - 1 \right ) \ , d x & \leq \sum _ { k = 1 } ^ { \infty } \frac { { C ^ { p k } } \| g \| _ { L ^ { p k } } ^ { p k } } { k ! \lambda ^ { p k } } = \int _ { \R ^ d } \left ( \exp \left | \frac { { C } g } { \lambda } \right | ^ p - 1 \right ) \ , d x . \end{align*}


\begin{align*} \langle ( f \ast - , - \ast f ) , ( g \ast - , - \ast g ) \rangle = ( h \ast - , - \ast h ) , \end{align*}


\begin{align*} \Big \| \sum _ { j = 1 } ^ \infty \alpha _ j \tilde { u } _ j \Big \| _ { p ^ * , q , \mu } ^ q \geq \sum _ { j = 1 } ^ \infty | \alpha _ j | ^ q \int _ { \delta _ { j + 1 } } ^ { \delta _ j } t ^ { \frac { q } { p ^ * } - 1 } ( \tilde { u } _ j ) _ \mu ^ * ( t ) ^ q \dd t \geq \sum _ { j = 1 } ^ \infty \frac { \| u _ j \| _ { p ^ * , q , \mu } ^ q } { ( 1 + \varepsilon _ 1 ) ^ q } | \alpha _ j | ^ q . \end{align*}


\begin{align*} \mu \left ( \tilde { \textbf { Q } } \right ) : = \left \{ \mu ( q ) \ \left | \ q \in \tilde { \textbf { Q } } \right . \right \} \subseteq \{ \gamma _ 1 , \ldots , \gamma _ \epsilon \} . \end{align*}


\begin{align*} \lambda _ n = t _ n \lambda ^ + ( q ) , \lambda _ n ' = t _ n ' \lambda ^ + ( q ) , \end{align*}


\begin{align*} { \rm I I } _ { x } = \sum _ { s \in G } \mu ( s ) \frac { d s \nu } { d \nu } ( \pi ( x ) ) \sum _ { \xi _ { n } ^ { x } \in L _ { x } \backslash G } \log \left ( P _ { \mu , x } ^ { n - 1 } ( L _ { x } s , \xi _ { n } ^ { x } ) \right ) P _ { \mu , x } ^ { n - 1 } ( L _ { x } s , \xi _ { n } ^ { x } ) \end{align*}


\begin{align*} \det \Phi _ t ( x , y ) = \det \left ( \begin{array} { c c } & - \\ t & ( 1 - t ) \end{array} \right ) = 1 \end{align*}


\begin{align*} \int _ { X } f ( x ) d \lambda ( x ) & = \int _ { X _ { 0 } } \mathcal { E } _ { s } f ( e , e , x _ { 0 } ) d \lambda ( x _ { 0 } ) = \int _ { X ' } \phi d \lambda ' \\ & = \int _ { X ' } \phi \left ( x ' \right ) d u . \lambda ' ( x ' ) = \int _ { X _ { 0 } } \mathcal { E } _ { s } f ( u , e , x _ { 0 } ) d \lambda ( x _ { 0 } ) = \int _ { X } f ( x ) d u . \lambda ( x ) , \end{align*}


\begin{align*} \mbox { i f $ \tau = \{ i \} $ a n d $ \mathfrak { u } = \{ j \} $ t h e n $ \eta ( \sigma ) ^ { \ast } ( y _ { i , j } ) = \displaystyle \frac { c ^ { \sigma } _ { i } } { c ^ { \{ i \} } _ { i } } x _ { i , j } $ } . \end{align*}


\begin{align*} \mathcal { A } \mathcal { = } \left \{ x \in X \mid 0 \in \sum _ { i = 1 } ^ { m } \partial f _ { i } \left ( x \right ) \right \} = \arg \min \sum _ { i = 1 } ^ { m } f _ { i } . \end{align*}


\begin{align*} \widetilde { A } : = A / K _ 1 A . \end{align*}


\begin{align*} \inf _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \sup _ { \lambda \in \mathbb { R } ^ 2 _ + } F ( \pi , \lambda ) = \sup _ { \lambda \in \mathbb { R } ^ 2 _ + } \inf _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } F ( \pi , \lambda ) . \end{align*}


\begin{align*} \underset { t \to \infty } { \lim } P ^ \omega ( \mathcal { R } ( \alpha t ) \cap \widehat { \Phi } _ { \alpha t } ^ \omega = \emptyset \mid S _ { \alpha t } ) = \underset { t \to \infty } { \lim } P ^ \omega ( E _ t ^ c \mid S _ { \alpha t } ) = 0 . \end{align*}


\begin{align*} \begin{aligned} \boldsymbol { K } _ \ell \left ( \mu _ \ell , \gamma ^ { \prime \prime } _ { \ell , 3 } \right ) & \leq \lambda \boldsymbol { K } _ \ell ( \mu _ \ell , \gamma _ \ell ) + ( 1 - \lambda ) \boldsymbol { K } _ \ell \left ( \mu _ \ell , \gamma _ \ell ^ \prime \right ) \leq \lambda \delta _ \ell + ( 1 - \lambda ) \delta _ \ell ^ \prime . \end{aligned} \end{align*}


\begin{align*} \nabla _ { \xi _ i } \ , \xi _ j = 0 , 1 \le i , j \le p ; \end{align*}


\begin{align*} & \int _ { \mathbb { T } ^ d \times [ 0 , T ) } \bar { f } \left ( \frac { \partial \phi } { \partial t } + u \cdot \nabla \phi \right ) \ ; d x d t \\ & = \lim _ { i \to \infty } \int _ { \mathbb { T } ^ d \times [ 0 , T ) } f ^ { \nu _ i } \left ( \frac { \partial \phi } { \partial t } + u \cdot \nabla \phi + \nu _ i \Delta \phi \right ) \ ; d x d t \\ & = - \int _ { \mathbb { T } ^ d } f _ 0 \phi _ 0 \ ; d x , \end{align*}


\begin{align*} f \left ( \sum _ { l = 1 } ^ n \| w _ l - w \| _ \infty \right ) & \leq f \left ( \sum _ { l = 1 } ^ n k \| x _ l - x \| _ \infty + k \| y _ l - y \| _ \infty + k \| z _ l - z \| _ \infty \right ) \\ & \leq k f \left ( \sum _ { l = 1 } ^ n \| x _ l - x \| _ \infty \right ) + k f \left ( \sum _ { l = 1 } ^ n \| y _ l - y \| _ \infty \right ) \\ & + k f \left ( \sum _ { l = 1 } ^ n \| z _ l - z \| _ \infty \right ) , \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( \frac { 1 } { 1 - x ^ l y ^ m z ^ n } \right ) ^ { \frac { l m ^ 2 } { n ^ 4 } } \end{align*}


\begin{align*} Y : = \left \{ y _ \ast \in [ 2 , 3 ] \ , \big | \ \exists \ , y \in ( s ( \tilde y _ \ast ) , y _ \ast ) \ \omega ( y ; \tilde y _ \ast ) = \frac 1 3 \ \ \tilde y _ \ast \in [ y _ \ast , 3 ] \right \} . \end{align*}


\begin{align*} S _ { t } ( t ) : = \left ( \begin{array} { c c } a ^ { \prime } ( t ) & 0 \\ 0 & 0 \end{array} \right ) \left ( S Q - Q ^ { * } S \right ) ( t ) : = \left ( \begin{array} { c c } 0 & a ( t ) - q ( t ) \\ q ( t ) - a ( t ) & 0 \end{array} \right ) , \end{align*}


\begin{align*} \Pr [ \hat { M } = M ] \geq \frac { \Pr [ d ( \hat { M } ' , M ) \leq n ^ { 1 - \beta } ] } { | \{ m : d ( m , \hat { M } ' ) \leq n ^ { 1 - \beta } \} | } . \end{align*}


\begin{align*} \begin{aligned} | \beta \sum \limits _ { i = n } ^ { k - 1 } \langle r ^ { i + 1 } , r ^ k \rangle | \leq \frac { \beta } { 2 } \sum \limits _ { i = n } ^ { k - 1 } ( \| r ^ { i + 1 } \| ^ 2 + \| r ^ k \| ^ 2 ) \leq \beta \sum \limits _ { i = n } ^ { k } \| r ^ { i } \| ^ 2 . \end{aligned} \end{align*}


\begin{align*} f _ { j , \flat } & = \sum _ { m \in \mathbb { Z } ^ 2 } \widetilde { \eta } _ m f _ { j , m , \flat } , \\ f _ { j , \sharp } & = \sum _ { m \in \mathbb { Z } ^ 2 } \ ! \widetilde { \eta } _ m f _ { j , m , \sharp } , \\ f _ { j , } & = \sum _ { m \in \mathbb { Z } ^ 2 } \widetilde { \eta } _ m \left [ \eta _ m \left ( \psi ^ { ( j ) } \ast _ j f _ j \right ) - \psi ^ { ( j ) } \ast _ j \left ( \eta _ m f _ j \right ) \right ] . \end{align*}


\begin{align*} \Big | f _ \epsilon ( V ) - f _ 0 ( V ) \Big | _ { \frac { 2 n } { n + 2 } } = O \bigg ( \epsilon \ln \Big | \ln \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big | \bigg ) . \end{align*}


\begin{align*} \begin{array} { r l } ( \prod \limits _ { i = 1 } ^ m a ( i ) \ast g _ { w ( l ) } ( t ( i ) ) ) \ast a ( m + 1 ) = & ( \prod \limits _ { i = 1 } ^ m a ( i ) \ast ( \prod \limits _ { p = 1 } ^ r c _ { t ( i ) } ( p ) \ast h _ l ( z _ { t ( i ) } ( p ) ) ) \ast c _ { t ( i ) } ( r + 1 ) ) ) \ast a ( m + 1 ) \\ = & ( \prod \limits _ { i = 1 } ^ u d ( i ) \ast h _ l ( q ( i ) ) ) \ast d ( u + 1 ) . \end{array} \end{align*}


\begin{align*} \ddot \l ( t ) \l ( t ) ^ 2 & = \delta , \ \ \delta \in \mathbb R , \\ \l ( 0 ) = \l _ 0 , \ \ \dot \l ( 0 ) & = \l _ 1 \end{align*}


\begin{align*} & \chi _ I ( H _ \omega ) u = \left ( \int _ B P _ \omega ( E ) d \mu _ \omega ( E ) \right ) u , \\ & f ( H _ \omega ) \chi _ I ( H _ \omega ) u = \left ( \int _ { B } f ( E ) P _ \omega ( E ) d \mu _ \omega ( E ) \right ) u , \end{align*}


\begin{align*} ( \textsf { E } \vert g _ { i j } g _ { i j } ^ { \prime } \vert ^ { p } ) ^ { 1 / p } = ( \textsf { E } ( \textsf { E } _ { ( g _ { i j } ) } \vert g _ { i j } g _ { i j } ^ { \prime } \vert ^ { p } ) ) ^ { 1 / p } \asymp p , \end{align*}


\begin{align*} H ( a ) H ( \tilde { b } ) & = H ( a ) T ( \tilde { f } ) H ( \tilde { b } ) + H ( a ) T ( \tilde { g } ) H ( \tilde { b } ) \\ & = \Big ( T ( a ) H ( \tilde { f } ) - H ( a f ) \Big ) H ( \tilde { b } ) + H ( a ) \Big ( H ( \tilde { g } ) T ( b ) - H ( \tilde { g } \tilde { b } ) \Big ) . \end{align*}


\begin{align*} \sum _ { i + j = n } \pi _ i \circ \pi _ j = 0 , f o r n \geq 0 . \end{align*}


\begin{align*} N \coloneqq d \cdot \min \left \{ \prod _ { i = 1 } ^ n ( h _ i + 1 ) , \binom { n + h } { n } \right \} \end{align*}


\begin{align*} R _ 2 = \left ( \begin{array} { l l l } x = 0 : & 0 & a _ 1 \\ x = 1 : & 0 & a _ 2 \\ x = \infty : & a _ 3 & a _ 4 \end{array} \right ) = \left ( \begin{array} { l l c } x = 0 : & 0 & 1 - c \\ x = 1 : & 0 & c - a - b \\ \xi = \infty : & a & b \end{array} \right ) = R _ { a b c } , \end{align*}


\begin{align*} g ( P \otimes \lbrace \alpha \rbrace ) = ( g P ) \otimes \lbrace g ( \alpha ) \rbrace \ , \ \ , \ P \lbrace \alpha \rbrace \in \mathbb { B } _ k , \ , g \in \Gamma . \end{align*}


\begin{align*} \det ( u _ { j \bar k } ( 0 ) ) = a u _ { n \bar n } ( 0 ) + b , \end{align*}


\begin{align*} Z = \left ( \begin{array} { c c } A & 0 \\ 0 & A \end{array} \right ) , T = \left ( \begin{array} { c c } B & 0 \\ 0 & B \end{array} \right ) , X = \left ( \begin{array} { c c } 0 & C \\ C & 0 \end{array} \right ) , Y = \left ( \begin{array} { c c } 0 & i C \\ - i C & 0 \end{array} \right ) , \end{align*}


\begin{align*} \| u _ m \| ^ 2 = \int _ \Omega f _ \epsilon ^ { ' } ( V _ m ) u _ m ^ 2 d x + \int _ \Omega f _ \epsilon ^ { ' } ( V _ m ) ( h _ m + \omega _ m ) u _ m d x . \end{align*}


\begin{align*} \begin{aligned} \sum _ { j = 1 } ^ { m } X _ { j } ^ { * } \left ( \left | X u \right | ^ { p - 2 } X _ { j } u \right ) & = - \lambda | u | ^ { p - 2 } u & & \Omega , \\ u & = 0 & & \partial \Omega , \end{aligned} \end{align*}


\begin{align*} d ( m S , \N ^ 2 \setminus m S ) = m d ( S , ( 1 / m ) \N ^ 2 \setminus S ) \leq m ( 1 - f ( 1 / m ) ) . \end{align*}


\begin{align*} d \overline { \Pi } _ { \psi } ( e ^ - ) f ( [ \epsilon , x ] ) & = \frac { d } { d t } \Big \{ \Pi _ { \psi } ( \exp ( t e ^ - ) ) f ( [ \epsilon , x ] ) \Big \} \Big | _ { t = 0 } \\ & = \frac { \epsilon i } { 4 \pi } \frac { d ^ 2 } { d x ^ 2 } f ( [ \epsilon , x ] ) . \end{align*}


\begin{align*} \mathbf { P } ( U ) : = \mathbf { p } ( u _ 1 ) + \mathbf { p } ( u _ 2 ) + \mathbf { p } ( u _ 3 ) , \end{align*}


\begin{align*} P _ { x \to y } w = \sinh ( \| v \| ) x + \cosh ( \| v \| ) w , P _ { x \to y } u = u . \end{align*}


\begin{align*} y ( q x ) = m ( x ) y ( x ) \end{align*}


\begin{align*} \zeta _ { 0 , p ^ { \prime } } ( s , c ) : = \sum _ { m , n \neq 0 } \frac { p ^ { \prime 2 } + \lambda _ { n } ^ { \prime 2 } } { p ^ { \prime } \left ( p ^ { \prime } + \frac { 1 } { \pi } \right ) + \lambda _ { n } ^ { \prime 2 } } \ , \frac { 1 } { \left ( \left ( m - \frac { 1 } { 2 } \right ) ^ { 2 } + c \ , \lambda _ { n } ^ { \prime 2 } \right ) ^ { s } } , \ , \ , \ , \ , ( s ) > 1 . \end{align*}


\begin{align*} \mathcal { M } _ { \omega , \mathbf { c } } : = \{ \Psi \in \mathcal { H } ^ 1 \backslash \{ ( \mathbf { 0 } , \mathbf { 0 } , \mathbf { 0 } ) \} \ | \ S _ { \omega , \mathbf { c } } ( \Psi ) = \mu _ { \omega , \mathbf { c } } , \ K _ { \omega , \mathbf { c } } ( \Psi ) = 0 \} . \end{align*}


\begin{align*} | S ' | \leq | N b d ( S ' ) | \leq | N b d ( S ) | - 1 < | S | - 1 = | S ' | . \end{align*}


\begin{align*} \langle S _ { \xi \chi } f , g \rangle & = \langle \int _ { \Theta } v ( w ) s ( w ) \pi _ { G ( w ) } \xi _ { w } ^ { \ast } \chi _ { w } \pi _ { F ( w ) } f d \mu ( w ) , g \rangle \\ & = \int _ { \Theta } v ( w ) s ( w ) \langle f , \pi _ { F ( w ) } \chi _ { w } ^ { \ast } \xi _ { w } \pi _ { G ( w ) } g \rangle d \mu ( w ) \\ & = \langle f , \int _ { \Theta } v ( w ) s ( w ) \pi _ { F ( w ) } \chi _ { w } ^ { \ast } \xi _ { w } \pi _ { G ( w ) } g d \mu ( w ) \rangle \\ & = \langle f , S _ { \chi \xi } g \rangle . \end{align*}


\begin{align*} \mathbb { Q } ^ \times \cap ( \widehat { \mathbb { Z } } ^ p ) ^ \times \ni \mu ( a ) = \mu ( b ) ^ { - 1 } \in \mathbb { Z } _ p ^ \times , \end{align*}


\begin{align*} M ( g a , g b ) = M ( g a _ 1 , g a _ 2 , g b _ 1 , g b _ 2 ) = M ( a , b ) \ , . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ D c _ k L p _ { k , \sigma ( j ) } ( y _ j ) = f ( y _ j ) , j = 1 , \ldots , D . \end{align*}


\begin{align*} \gamma _ 0 : = \gamma ( 0 , v ) , \gamma _ h : = \gamma ( h , v + h ) , T _ 0 : = T ( 0 , v ) , T _ h : = T ( h , v + h ) . \end{align*}


\begin{align*} A \prod _ { i = 1 } ^ n x _ i ^ { \alpha ' _ i } = B \prod _ { i = 1 } ^ n x _ i ^ { \gamma ' _ i } , \end{align*}


\begin{align*} x \ast _ { n , n } x & = ( a _ { 1 } \ast a _ { 2 } \ast \cdots \ast a _ { n } ) \ast ( a _ { 1 } \ast a _ { 2 } \ast \cdots \ast a _ { n } ) \\ & = ( a _ { 1 } \ast a _ { 1 } ) \ast ( a _ 2 \ast a _ 2 ) \ast \cdots ( a _ n \ast a _ { n } ) = ( \top _ 1 \ast a _ 1 ) \ast ( \top _ 1 \ast a _ 2 ) \ast \cdots \ast ( \top _ 1 \ast a _ n ) \\ & = ( \top _ n ) \ast ( a _ { 1 } \ast a _ { 2 } \ast \cdots \ast a _ { n } ) ; \end{align*}


\begin{align*} u = h \rho , \end{align*}


\begin{align*} E _ { \varepsilon } & = \big \{ ( t , x , z ) \in E : t - | x - z | < \varepsilon \sqrt { x z } \ ; \ ; \textrm { o r } \ ; \ ; x + z - t < \varepsilon \sqrt { x z } \big \} , \\ F _ { \varepsilon } & = \big \{ ( t , x , z ) \in F : t - ( x + z ) < \varepsilon \sqrt { x z } \ ; \ ; \textrm { o r } \ ; \ ; t > \varepsilon ^ { - 1 } \sqrt { x z } \big \} . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { 1 } { a ^ s b ^ t c ^ u } } = \exp \left \{ \left ( \sum _ { i = 1 } ^ { \infty } \frac { x ^ i } { i ^ s } \right ) \left ( \sum _ { j = 1 } ^ { \infty } \frac { y ^ j } { j ^ t } \right ) \left ( \sum _ { k = 1 } ^ { \infty } \frac { z ^ k } { k ^ u } \right ) \right \} . \end{align*}


\begin{align*} \lim _ { K \to \infty } \overline { d } _ { ( \ell _ { i } ) } ( \{ n \in \N : x _ K ( n ) \neq x ( n ) \} ) = 0 . \end{align*}


\begin{align*} M = O ( \log q / \log \log q ) \end{align*}


\begin{align*} Q = \begin{pmatrix} X & Y \\ - Y & X \end{pmatrix} , \end{align*}


\begin{align*} \lim _ { h \to + \infty } { ( _ h ) } = 0 , \end{align*}


\begin{align*} \left \{ \begin{array} { l c } u _ { t t } ( { x } , t ) = \nabla ( a ( { x } ) \nabla u ( { x } , t ) ) + f ( { x } , t ) , & ( x , t ) \in \Omega \times ( 0 , T ] , \\ u = 0 , & ( { x } , t ) \in \partial \Omega \times ( 0 , T ] , \\ u ( { x } , 0 ) = \psi _ 0 , ~ u _ t ( { x } , 0 ) = \psi _ 1 , & x \in \Omega \subset \mathbb { R } ^ { d } . \end{array} \right . \ , \end{align*}


\begin{align*} \mathcal { G } _ w \subset \bigcup _ { j = 1 } ^ { K } \mathcal { S } _ j ( \mathcal { U } ) . \end{align*}


\begin{align*} \beta ^ { ( N ) } _ n = \frac { ( b _ 1 q , \ldots , b _ N q ) _ n } { ( b _ 1 , \ldots , b _ N ) _ n } \beta _ n . \end{align*}


\begin{align*} L = B C _ E ^ { - 1 } B ^ T C _ V , B = S - T , \end{align*}


\begin{align*} \sum _ { p _ i \in \Z } \int _ { [ 0 , N ] ^ n } \chi _ { \frac { \vartheta _ i } { \| N \bar { q } + v '' \| ^ { w _ i } } } ' ( N p _ i + v _ { m + i } + \langle \bar { u } _ i , N \bar { q } + v '' \rangle ) \ , d \bar { u } _ i = 2 N ^ { n - 1 } \vartheta _ i \| N \bar { q } + v '' \| ^ { - w _ i } . \end{align*}


\begin{align*} \mathcal V _ \infty ^ { p - 1 } = \mathcal V _ \infty ^ { k _ 2 } \cdot \mathcal W _ \infty ^ { k _ 3 } = \mathcal V _ \infty ^ { k _ 2 } \cdot \mathcal U _ \infty ^ { k _ 3 } \end{align*}


\begin{align*} \begin{aligned} & | \langle \lambda ^ { k + 1 } , \zeta ^ * - \zeta ^ { k + 1 } \rangle + \langle D _ u \theta ( u ^ { k + 1 } ) , u ^ * - u ^ { k + 1 } \rangle _ \mathcal { U } | \\ & \leq \beta \sum \limits _ { i = n } ^ { k + 1 } \| r ^ { i } \| ^ 2 + | \langle \lambda ^ n , \zeta ^ * - \zeta ^ { k + 1 } \rangle + \langle D _ u \theta ( u ^ n ) , u ^ * - u ^ { k + 1 } \rangle _ \mathcal { U } | . \end{aligned} \end{align*}


\begin{align*} { \overline g = ( \alpha _ 1 / r , \dots , \alpha _ N / r ) } { \overline g = \frac { 1 } { r } ( \alpha _ 1 , \dots , \alpha _ N ) } \end{align*}


\begin{align*} \int _ { \mathbb { S } ^ n } x _ l \mathrm { d } A = 0 , \ \ l = 1 , 2 , \cdots , n + 1 . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle \tilde { \lambda } , \bar { \zeta } _ 0 \rangle > 0 , \\ & \langle \lambda ^ * , \bar { \zeta } _ 0 \rangle > 0 . \end{aligned} \right . \quad ( \mbox { C o n s i s t e n c y \ C o n d i t i o n } ) \end{align*}


\begin{align*} \mathrm { K e r } ( \tilde { \lambda } ) \cap \overline { R ( S ) } = \mathrm { K e r } ( \lambda ^ * ) \cap \overline { R ( S ) } \quad ( \mbox { C o m p a t i b i l i t y \ C o n d i t i o n } ) \end{align*}


\begin{align*} \mathcal { D } _ { \eta } ( \hat { \eta } , \eta ; \lambda ) : = \theta ( \hat { u } ) - \theta ( u ) - \langle D _ u \theta ( u ) , \hat { u } - u \rangle _ \mathcal { U } + I _ K ( \hat { \zeta } ) - I _ K ( \zeta ) - \langle \lambda , \hat { \zeta } - \zeta \rangle \geq 0 , \end{align*}


\begin{align*} \delta _ { t } ^ { \Lambda } : = \exp \circ \exp ( \ln ( t ) \Lambda ) \circ \exp ^ { - 1 } , t > 0 , \end{align*}


\begin{align*} f ^ { ( k ) } + A ( z ) f = 0 \end{align*}


\begin{align*} \mathcal { M } _ { \omega , \mathbf { c } } ^ * ( \eta ) : = \{ \Psi \in \mathcal { M } _ { \omega , \mathbf { c } } \ | \ G ( \Psi ) \ge \eta \} , \end{align*}


\begin{align*} f ' ( \lambda ) = \lambda \left \{ 2 L ( U ) + \left ( \frac { d } { 2 } + 1 \right ) \lambda ^ { \frac { d } { 2 } - 1 } N ( U ) \right \} = : \lambda g ( \lambda ) . \end{align*}


\begin{align*} \| f \| _ { L ^ 1 _ N } = \int _ 0 ^ { N T } | f ( x ) | d x \quad \mbox { a n d } \left \langle f , g \right \rangle _ { L ^ 2 _ N } = \int _ 0 ^ { N T } f ( x ) g ( x ) d x , \end{align*}


\begin{align*} \tilde { S } _ { \alpha _ i \alpha _ i } + \iota \tilde { S } _ { \alpha _ i \beta _ i } = \left ( x _ 1 + \iota y _ 1 , \ldots , x _ { | \alpha _ i | } + \iota y _ { | \alpha _ i | } \right ) . \end{align*}


\begin{align*} \begin{aligned} \ 1 ( x \in [ 0 , \Delta ^ { - 1 } ] ) & \geq | \mathcal D ^ - _ { \Delta , A } ( x ) | ^ 2 - C e ^ { - \Delta ^ { A - 1 } } \\ | \mathcal D ^ - _ { \Delta , A } ( x ) | ^ 2 & \geq ( 1 - C e ^ { - \Delta ^ { A - 1 } } ) \mathbf { 1 } ( x \in [ \Delta ^ { - A / 2 } , \Delta ^ { - 1 } - \Delta ^ { - A / 2 } ] ) , \end{aligned} \end{align*}


\begin{align*} \frac { d g . \lambda } { d \lambda } ( \psi ( x , \omega ) ) = \frac { d g . \eta } { d \eta } ( x ) \frac { d \left ( g . \lambda \right ) _ { x } } { d \lambda _ { x } } \left ( \psi ( x , \omega ) \right ) . \end{align*}


\begin{align*} \mathcal { S } _ { j k } ( A ) : = \sum \limits _ { \ell = p _ 1 + p _ 2 + \ldots + p _ { k - 1 } + 1 } ^ { p _ 1 + p _ 2 + \ldots + p _ k } a _ { p _ 1 + p _ 2 + \ldots + p _ j , \ell } \ , , j , k = 1 , \ldots , m . \end{align*}


\begin{align*} c ( \beta _ m \otimes \beta _ n ) = b \sum _ { k = 0 } ^ m \binom { m } { k } ( 1 - a ) ^ k a ^ n \beta _ { n + k } \otimes \beta _ { m - k } . \end{align*}


\begin{align*} \aligned m _ \nu ( a , b ) \leq J _ \nu ( t _ \nu \star ( u _ 0 , v _ 0 ) ) = \bar { f } \Bigl ( \int _ { \mathbb { R } ^ N } ( I _ \mu * | t _ \nu \star { u _ 0 } | ^ { p } ) | t _ \nu \star { v _ 0 } | ^ { q } \Bigr ) . \endaligned \end{align*}


\begin{align*} V _ { 2 } V _ 1 M _ { \tilde { c } \circ \kappa } V _ 1 ^ { - 1 } V _ { 2 } ^ { - 1 } = M _ { \tilde { c } \circ \kappa \circ \psi _ 1 \circ \psi _ 2 } . \end{align*}


\begin{align*} C = \begin{pmatrix} 0 _ n & A \\ B & 0 _ m \end{pmatrix} . \end{align*}


\begin{align*} & M _ { n m } = b _ { n m } \times \underset { \mathcal { R } ( \mathbf { F } , \mathbf { M } ) \ , \cap \ , \mathcal { X } ^ 0 _ { n m } } { \max } ( \theta _ n - \theta _ m ) \\ & M _ { m n } = b _ { n m } \times \underset { \mathcal { R } ( \mathbf { F } , \mathbf { M } ) \ , \cap \ , \mathcal { X } ^ 0 _ { n m } } { \max } ( \theta _ m - \theta _ n ) \end{align*}


\begin{align*} \lim _ { N \rightarrow \infty } \limsup _ { n \rightarrow \infty } n ^ { - 1 } \big \Vert \sum _ { i = 1 } ^ n Y _ { i , N } \big \Vert ^ 2 _ 2 = 0 \ , . \end{align*}


\begin{align*} \rho _ 1 ( \varpi ) = | \varpi | ^ { n - 2 } + O \left ( | \varpi | ^ { ( n - 1 ) / 2 } \right ) , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c \underline u ) ^ n \geq ( 1 - ( \delta + \mu _ 1 ) \underline u ) ^ n f ^ n \omega ^ n & \textnormal { i n } & \Omega , \\ \underline u = 0 & \textnormal { o n } & \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} J _ { 2 | \alpha _ i | } \tilde { S } _ { \gamma _ i \gamma _ i } J _ { 2 | \alpha _ i | } ^ T = \tilde { S } _ { \gamma _ i \gamma _ i } + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} J _ { \overrightarrow 0 } ( a ) = J _ { \overrightarrow 0 } ( a ^ { ( 1 ) } ) ( z _ 0 ) \cdot J _ { z _ 0 } ( a ^ { ( 2 ) } ) . \end{align*}


\begin{align*} f ' = F _ 0 + F _ 1 Q _ i + \ldots + F _ r Q _ i ^ r \end{align*}


\begin{align*} F ( D ^ { 2 } u , D u , u ) = \int _ { \Omega } \left ( G \left ( \det ( u _ { i j } ) \right ) - g u \right ) d x \end{align*}


\begin{align*} \gamma ( r ) = \begin{cases} 1 & \textrm { i f } \ , \ , 0 \leq r \leq \frac 1 2 \\ 0 & \textrm { i f \ \ } r \geq 1 \ , \end{cases} ; \quad \quad \gamma ' ( r ) < 0 . \end{align*}


\begin{align*} I = \int _ { 0 } ^ { \infty } f ( x ) \ , d x = \int _ { 0 } ^ { \infty } \sum _ { n = 0 } ^ { \infty } C ( n ) x ^ { \alpha n + \beta - 1 } = \sum _ { n = 0 } ^ { \infty } C ( n ) \langle \alpha n + \beta \rangle . \end{align*}


\begin{align*} \widetilde { w } : = \left ( \mathcal { P } _ a \vert _ { \mathcal { M } } \right ) ^ { - 1 } \circ \mathcal { P } _ a \circ w \end{align*}


\begin{align*} P _ \varepsilon ( p _ i , s ) = \{ ( p _ j , F ( \tau _ j ) ) : ( p _ j , \tau _ j ) \in P ( p _ i , s ) \} . \end{align*}


\begin{align*} \frac { d } { d \lambda } S _ { \omega , \mathbf { c } } ( \Phi ^ { \lambda } ) \big | _ { \lambda = 1 } = 0 . \end{align*}


\begin{align*} g ( v ) - g ( v ^ { \prime } ) \leq \Psi \left ( \boldsymbol { d } _ { \mathcal { V } } ( v , v ^ \prime ) , \boldsymbol { d } _ { \mathcal { V } } ( v , v ^ \prime ) \right ) = \omega ( \boldsymbol { d } _ { \mathcal { V } } ( v , v ^ \prime ) ) . \end{align*}


\begin{align*} T ( t ) = \begin{bmatrix} - t I & 0 \\ 0 & 0 \end{bmatrix} . \end{align*}


\begin{align*} 2 \binom { n - k } { k } \mbox { a n d } - 2 \binom { n - k - 1 } { k - 1 } . \end{align*}


\begin{align*} \int _ { \Omega } \sqrt { \det g } g ^ { i j } \delta ^ { k l } u _ { i k } \eta _ { j l } d x = 0 \forall \eta \in C _ { 0 } ^ { \infty } ( \Omega ) \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ u \ge 0 , & \ \ B _ R , \\ u = 0 , \ \ & \ \ \partial B _ R . \end{cases} \end{align*}


\begin{align*} Y & = ( x _ j , j \in J _ 1 , x _ j , j \in J _ 2 , \ldots , x _ j , j \in J _ m ) , \\ ( \sigma ( 1 ) , \ldots , \sigma ( | Y | ) & = ( \underbrace { 1 , \ldots , 1 } _ { | J _ 1 | } , \underbrace { 2 , \ldots , 2 } _ { | J _ 2 | } , \ldots , \underbrace { m , \ldots , m } _ { | J _ m | } ) . \end{align*}


\begin{align*} K \simeq \mathrm { E n d } _ { \overline { F } } ^ \circ ( A ) : = \mathrm { E n d } _ { \overline { F } } ( A ) \otimes _ { \mathbb { Z } } \mathbb { Q } \end{align*}


\begin{align*} = e x p \left \{ \frac { z ( 1 + 4 z + z ^ 2 ) L i _ 2 ( x ) L i _ 2 ( y ) } { ( 1 - z ) ^ 4 } \right \} , \end{align*}


\begin{align*} \Phi ( \nu , \mu ) ( y ) = \frac { 1 } { Z ' ( \nu , \mu ) } \exp \left ( { \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \mu } ( \nu , \mu , y ) - U ^ { \rho } ( y ) } \right ) , \end{align*}


\begin{align*} \int _ { U ^ - } \theta ^ - = \int _ { U ^ 0 } \theta ^ 0 = 1 \end{align*}


\begin{align*} \Sigma ( \delta ) : = \Sigma ( \mu _ { 1 3 } , \mu _ { 2 3 } , \delta ) = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \ \boldsymbol { K } _ 1 ( \mu _ { 1 3 } , \gamma _ { 1 3 } ) \leq \delta _ 1 , \ \boldsymbol { K } _ 2 ( \mu _ { 2 3 } , \gamma _ { 2 3 } ) \leq \delta _ 2 \right \} \end{align*}


\begin{align*} p ( z ) = 1 + \sum _ { n = 1 } ^ { \infty } c _ n z ^ n \end{align*}


\begin{align*} \Delta _ { \bar { \xi } _ i } u ^ { ( i ) } _ { \alpha - 1 } = 0 \ \ \ \ \Upsilon _ i , \partial _ { \bar { \nu } _ { \bar { \xi } _ i } } u ^ { ( i ) } _ { \alpha - 1 } = 0 \ \ \ \ \partial \Upsilon _ i , \langle u _ { \alpha - 1 } ^ { ( i ) } \rangle _ { \Upsilon _ i } = 0 , \end{align*}


\begin{align*} a _ 1 = \int _ { \mathbb { S } ^ n } u Y _ 1 \mathrm { d } A = \dfrac { \sqrt { n + 1 } } { \sqrt { \mathrm { A r e a } ( \mathbb { S } ^ n ) } } \sum \limits _ { l = 1 } ^ { n + 1 } v _ l \int _ { \mathbb { S } ^ n } u x _ l \mathrm { d } A = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } . \end{align*}


\begin{align*} \alpha _ { 1 } ^ { \frac { 1 } { p - 1 } } \left ( x - P _ { 1 } \left ( x \right ) \right ) + \alpha _ { 2 } ^ { \frac { 1 } { p - 1 } } \left ( x - P _ { 2 } \left ( x \right ) \right ) = 0 , \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( \frac { 1 } { 1 - x ^ l y ^ m z ^ n } \right ) ^ { \frac { l m ^ 2 } { n ^ 4 } } \end{align*}


\begin{align*} f ^ T = \sum _ { k = 1 } ^ N c _ k x _ 1 ^ { E _ { 1 k } } \cdots x _ N ^ { E _ { N k } } . \end{align*}


\begin{align*} \widetilde { d } ( \theta , w ) = d ( \theta , w ) - \chi _ 0 ( \theta ) d _ 0 ( w ) - \chi _ { \pi } ( \theta ) d _ { \pi } ( w ) \end{align*}


\begin{align*} m _ { K } = \int _ { G / Q } \tau ( y ) . m _ { K \cap Q } d \bar { m } _ { K } ( y ) . \end{align*}


\begin{align*} \left ( X _ T , X _ { T + 1 } , \ldots , X _ { T + t } \right ) = \left ( X ' _ { T ' } , X ' _ { T ' + 1 } , \ldots , X ' _ { T ' + t } \right ) . \end{align*}


\begin{align*} \lambda ( r + s ) = a r s + \lambda ( r ) + \lambda ( s ) \ , \ , r , s \in R . \end{align*}


\begin{align*} \int \bigl ( \prod _ { i = 1 } ^ n V f _ i \bigr ) \otimes g \ , d \eta = \int \bigl ( \prod _ { i = 1 } ^ n V f _ i \bigr ) d \mu \int g \ , d \nu . \end{align*}


\begin{align*} \left [ { N + 1 \atop j } \right ] = \left [ { N \atop j } \right ] + q ^ { N + 1 - j } \left [ { N \atop j - 1 } \right ] , \end{align*}


\begin{gather*} \dim W ^ { \mathrm { u } } ( p ^ - ) = 2 n - k ^ - , \dim W ^ { \mathrm { s } } ( p ^ + ) = k ^ + + 1 , \end{gather*}


\begin{align*} \zeta ( 2 , 1 ) = \zeta ( 3 ) . \end{align*}


\begin{align*} ( A \otimes B ^ { - 1 } ) ^ r = \int ^ { \infty } _ 0 ( s + ( A \otimes B ^ { - 1 } ) ) ^ { - 1 } d \mu ( s ) . \end{align*}


\begin{align*} s _ \lambda ( X , Y ) = \frac { X ^ { \lambda _ 1 + 1 } Y ^ { \lambda _ 2 } - X ^ { \lambda _ 2 } Y ^ { \lambda _ 1 + 1 } } { X - Y } . \end{align*}


\begin{align*} H _ \ell = - \alpha z _ { \bar \ell } H ; ~ ~ ~ \ ; \ ; ~ H _ { \bar \ell } = - \alpha z _ { \ell } H , ~ ~ \ell = 1 , \cdots , n , \end{align*}


\begin{align*} [ \N ^ \ell , b _ { k } ( t ) ] = \sum _ { n = 0 } ^ { \ell - 1 } \binom { \ell } { n } ( - 2 ) ^ { \ell - n } \N ^ n b _ k ( t ) \ , \forall k \in \Lambda ^ * , \ t \in \R \ . \end{align*}


\begin{align*} \big \langle z ^ m \lbrace 0 , \infty \rbrace , f \big \rangle = \int _ 0 ^ { \infty } f ( z ) z ^ m d z 0 \leq m \leq k - 2 . \end{align*}


\begin{align*} C = Y _ 0 \subset Y _ 1 \subset \dots \subset Y _ k = Y \end{align*}


\begin{align*} R _ 2 ( \hat { x } , \bar { y } ) : = \sum _ { | \beta | = 2 } \frac { 2 } { \beta ! } \left ( \int _ { 0 } ^ 1 ( 1 - t ) ^ \beta c ( \bar { y } + t ( \hat { x } - \bar { y } ) ) t \right ) ( \hat { x } - \bar { y } ) ^ \beta . \end{align*}


\begin{align*} u _ { \leq N } = \frac { 1 } { N ^ d } \check { \psi } \left ( \frac { \cdot } { N } \right ) \ast u . \end{align*}


\begin{align*} \lambda ^ * = \bar { \lambda } | _ { \overline { R ( S ) } } , \end{align*}


\begin{align*} u ' ( t ) + \mathcal { A } ( u ( t ) , t ) = f ( t ) , \mathcal { V } ^ * , \ , u ( t _ 0 ) = u _ 0 H , \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( V ^ { \lambda } ) & = 2 L ( V ^ { \lambda } ) + 3 N ( V ^ { \lambda } ) + 2 \omega M ( V ^ { \lambda } ) + 2 \mathbf { c } \cdot \mathbf { P } ( V ^ { \lambda } ) \\ & = 2 \lambda ^ 2 L ( V ) + 3 \lambda ^ { \frac { d } { 2 } + 1 } N ( V ) + 2 \omega M ( V ) + 2 \lambda \mathbf { c } \cdot \mathbf { P } ( V ) , \end{align*}


\begin{align*} \alpha = \pi \widehat { \phi ^ { ( p ) } } \psi \phi \end{align*}


\begin{align*} U = \sum _ { u \in Z _ + ^ d } a ( u ) T _ 1 ^ { u _ 1 } \cdots T _ d ^ { u _ d } \end{align*}


\begin{align*} 2 t s \cos ( \gamma + \beta ) = \frac { 2 s ^ 2 - 1 } { \sqrt { 1 - s ^ 2 } } . \end{align*}


\begin{align*} \varphi _ j ( p ) : = \frac { \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s } { \int _ 0 ^ \infty e ^ { - \lambda s } \big ( \sum _ { k = j } ^ \infty p _ k ( s ) \big ) d s } = \begin{cases} \dfrac { 1 } { j + 1 } , & p = 0 , \\ 1 - \dfrac { \int _ 0 ^ { 1 } ( 1 - p y ) ^ { - 1 } y ^ { j } d y } { \int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y } , & 0 < p < 1 , \end{cases} \end{align*}


\begin{align*} f ( \underline { k } , t , \mathbf { p } _ { n + 1 } ) = \left ( \prod _ { j = 0 } ^ { k - 1 } p _ j ^ { } \right ) \sum _ { j = 0 } ^ { k } q _ j ^ { t - k } \prod _ { \substack { i = 0 \\ i \neq j } } ^ { k } \frac { 1 } { q _ j ^ { } - q _ i ^ { } } \ , . \end{align*}


\begin{align*} E \left [ \left ( Y e ^ { \lambda t } - Z _ 0 ( t ) \right ) ^ 2 \right ] = c _ 2 e ^ { \lambda t } , \end{align*}


\begin{align*} \mathcal { E } ^ { 0 , \gamma } _ { \tilde { \alpha } - \gamma ; I } : = C ( I ; \mathcal { B } _ { \tilde { \alpha } - \gamma } ) \cap C ^ { \gamma } ( I ; \mathcal { B } _ { \tilde { \alpha } - 2 \gamma } ) , \ \ \ \mathcal { E } ^ { \gamma , 2 \gamma } _ { \tilde { \alpha } ; I } : = C ^ { \gamma } _ { 2 } ( I ; \mathcal { B } _ { \tilde { \alpha } - \gamma } ) \cap C ^ { 2 \gamma } _ { 2 } ( I ; \mathcal { B } _ { \tilde { \alpha } - 2 \gamma } ) . \end{align*}


\begin{align*} F ^ n ( P ) = \sum _ { i = 0 } ^ { m - 1 } a ^ { ( i ) } _ n \cdot F ^ i ( P ) . \end{align*}


\begin{align*} T ( z ) = \sum _ { n = 1 } ^ { \infty } n ^ { n - 1 } \frac { z ^ n } { n ! } . \end{align*}


\begin{align*} \mathbb { P } _ { \mu } \left ( B \cap \left [ H _ { 0 } \omega _ { n } ; \omega _ { n } ^ { - 1 } \omega _ { n + 1 } \right ] \neq \emptyset \mbox { i n f i n i t e l y o f t e n } \right ) = 0 . \end{align*}


\begin{align*} \begin{aligned} \int _ { A _ 1 \cap A _ 2 } | g - g _ { \rho } | \ , d \pi & \leq \int _ { A _ 1 \cap A _ 2 } M \left [ \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s ^ \star _ 1 , s _ 1 ) ^ { p _ 1 ^ { \prime } } + \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 , s ^ \star _ 1 ) ^ { p _ 2 ^ { \prime } } \right ] d \pi ( s _ 1 , s _ 2 ) \\ & \leq B ( \rho ^ { p _ 1 ^ \prime - p _ 1 } + \rho ^ { p _ 2 ^ \prime - p _ 2 } ) . \end{aligned} \end{align*}


\begin{align*} C _ n ^ \lambda ( x ) & = \frac { \Gamma ( n + 2 \lambda ) } { \Gamma ( n + 1 ) \Gamma ( 2 \lambda ) } { } _ 2 F _ 1 \left ( - n , n + 2 \lambda ; \lambda + \frac { 1 } { 2 } ; \frac { 1 } { 2 } - \frac { x } { 2 } \right ) \\ & = \frac { \Gamma ( n + 2 \lambda ) } { \Gamma ( n + 1 ) \Gamma ( 2 \lambda ) } { } _ 2 F _ 1 \left ( - \frac { n } { 2 } , \frac { n } { 2 } + \lambda ; \lambda + \frac 1 2 ; 1 - x ^ 2 \right ) \end{align*}


\begin{align*} \begin{aligned} \lim _ { N \to \infty } \ , \frac { \theta _ N ^ 2 } { N ^ 4 } \ , \Xi ( 1 ) = 0 \qquad \lim _ { N \to \infty } \ , \frac { \theta _ N ^ 2 } { N ^ 4 } \ , \sum _ { r = 3 } ^ \infty \Xi ( r ) = 0 \ , . \end{aligned} \end{align*}


\begin{align*} h = \psi ( z ^ { 1 } , \ldots , z ^ { l } , g ^ { l + 1 } ( z ) , \ldots , g ^ { k } ( z ) ) \end{align*}


\begin{align*} - ( a , b ) & = ( - a , - b ) , \\ ( a , b ) \cdot ( c , d ) & = ( a \cdot c , b \cdot d ) , \\ ( a , b ) + ( c , d ) & = \{ ( e , f ) \in F _ 1 \times F _ 2 : e \in a + c \mbox { a n d } f \in b + d \} \cap ( F _ 1 \times _ h F _ 2 ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ N q _ k = ( 1 , \dots , 1 ) E _ f ^ { - 1 } \boldsymbol { 1 } = ( 1 , \dots , 1 ) ( E _ f ^ T ) ^ { - 1 } \boldsymbol { 1 } = \sum _ { k = 1 } ^ N q _ k ^ T . \end{align*}


\begin{align*} h ( \widetilde { X } _ \eta ) = 0 \iff \mathcal { P } ( \widetilde { X } _ \eta ) = \{ \delta _ { \boldsymbol { 0 } } \} \iff \widetilde { X } _ \eta \end{align*}


\begin{align*} \prod _ { x : X } \exists _ { k _ x : \N , m _ x : M _ x } . \varphi ( x ) = \frac { m _ x } { f ( x ) ^ { k _ x } } \rlap { . } \end{align*}


\begin{align*} \gamma = \sup _ { n \in \N } \{ \nu ( x - a _ n ) \} = \sup _ { a \in K } \{ \nu ( x - a ) \} = \sup { \rm d i s t } ( \eta , K ) . \end{align*}


\begin{align*} \Psi ( [ p ] \xi _ g ) : = [ p ] \xi _ { g ^ { - 1 } } . \end{align*}


\begin{align*} L _ S [ f ] = \sum _ { T \subseteq S } ( - 1 ) ^ { | T | } E _ T [ f ] . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } \big ( { S } _ { j , + } ( t ) - { S } _ { j , - } ( t ) \big ) = \nu Y \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s . \end{align*}


\begin{align*} 0 = \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( i ) \lambda _ f ( j ) \big | \langle ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , f \rangle \big | ^ 2 . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m \mathbf { u } ( E _ i ) = 0 \end{align*}


\begin{align*} f ' = \frac { d } { d x } \left ( \sum _ { j = 1 } ^ r \tilde b _ j \tilde { \textbf { Q } } ^ { \lambda _ j } \right ) = \sum _ { j = 1 } ^ r \sum _ { l \leq i } \tilde b _ j \lambda _ j ( Q _ { l } ) \tilde { \textbf { Q } } ^ { \lambda _ j } \frac { Q ' _ { l } } { Q _ { l } } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle D _ u \theta ( u ^ * ) , v \rangle _ \mathcal { U } + \langle \tilde { \lambda } , S v \rangle _ * = 0 \forall \ v \in \mathcal { U } ; \\ & - \langle \tilde { \lambda } , \zeta - S u ^ * \rangle _ * \geq 0 \forall \ \zeta \in K , \end{aligned} \right . \end{align*}


\begin{align*} \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) = \langle \lambda , \delta \rangle + \frac { V _ { 1 , Y Y } } { 4 \lambda _ 1 } + \frac { V _ { 2 , Y Y } } { 4 \lambda _ 2 } . \end{align*}


\begin{align*} h ( v ) = \begin{cases} 0 & v \in [ - 1 , 0 ) , \\ 2 a v & v \in [ 0 , 1 ] . \end{cases} \end{align*}


\begin{align*} A _ n ( \mu ) ( x ) \leq \frac { \chi _ d } { n _ d } \sum _ { j = 0 } ^ { n _ d - 1 } U ^ j ( \mu ) ( x ) \leq \epsilon \rho ( x ) x \in ( e M e ) _ + . \end{align*}


\begin{align*} h _ 2 ( z ) : = A _ 2 z ^ { n + m } + B _ 2 \overline { z } ^ m + C _ 2 \textrm { f o r a l l } z \in \mathbb { C } , \end{align*}


\begin{align*} H ( f ) ( \theta ) = \lim _ { \epsilon \to 0 } \frac { 1 } { \pi } \int _ { \epsilon \leq | t | \leq \pi } \frac { f ( \theta - t ) } { 2 \tan ( t / 2 ) } \ , d t . \end{align*}


\begin{align*} \Theta _ j : = a _ 0 + \sum _ { S \in \mathcal { Q } _ m } \left ( \prod _ { i \in S } \alpha _ i ^ { ( j ) } \right ) f _ S ( u _ { j , S } ) \end{align*}


\begin{align*} \begin{cases} \frac { C _ 2 ( \lambda _ 2 \rho _ 0 ) } { 2 } + \frac { C _ 3 ( \lambda _ 2 \lambda _ { 2 , 3 } \rho _ 0 ) } { 2 } = C _ 1 ( \lambda _ 1 \rho _ 0 ) \\ \frac { C _ 2 ( \lambda _ 2 \rho _ 0 ) } { 2 } + \frac { C _ 4 ( \lambda _ 2 \lambda _ { 2 , 4 } \rho _ 0 ) } { 2 } = C _ 1 ( \lambda _ 1 \rho _ 0 ) \\ \lambda _ 1 + \lambda _ 2 = \lambda _ { 2 , 3 } + \lambda _ { 2 , 4 } = 1 \end{cases} . \end{align*}


\begin{align*} \textup { v a r } ( U ^ \top A U ) = 2 \textup { t r } ( A ^ 2 ) + \sum _ { i = 1 } ^ { q } \delta _ i a _ { i i } ^ 2 , \end{align*}


\begin{align*} L _ 1 ( d \underline { c } , k ) \coloneqq \{ \underline { x } \in K _ \infty ^ 2 \colon d \underline { c } \cdot \underline { x } = k \} \end{align*}


\begin{align*} \varepsilon ^ { \alpha - 1 } \ , w _ { \alpha - 1 } ^ { ( i ) } ( x _ i , t ) + \varepsilon ^ { \alpha } \ , w _ { \alpha } ^ { ( i ) } ( x _ i , t ) + \sum \limits _ { k = 2 } ^ { - \lfloor \alpha \rfloor } \varepsilon ^ { \alpha + k - 1 } \ , \Big ( w _ { \alpha + k - 1 } ^ { ( i ) } ( x _ i ) + u _ { \alpha + k - 1 } ^ { ( i ) } \big ( x _ i , \tfrac { \overline { x } _ i } { \varepsilon } , t \big ) \Big ) . \end{align*}


\begin{align*} \mathcal { I } ( \lambda ) : = \left \{ i \in \{ 1 , 2 , \ldots , t \} \mid \lambda _ { i - 1 } > \lambda _ { i } \right \} . \end{align*}


\begin{align*} \mathbb { E } ( X ) = \sum _ { \substack { 0 \leq i < j \leq h \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) + \sum _ { \substack { 0 \leq i \leq h < j \leq k \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) + \sum _ { \substack { h < i < j \leq k \\ j - i \leq t } } \mathbb { P } ( s _ i = s _ j ) . \end{align*}


\begin{align*} H ^ { m } ( \R ^ d ) \subset \left \{ f \in C ^ r ( \R ^ d ) \left | \ \displaystyle \lim _ { | x | \rightarrow \infty } \partial ^ { \alpha } f ( x ) = 0 , \ \alpha \in \Z ^ d : \ | \alpha | \le r \right . \right \} \end{align*}


\begin{align*} z _ { \mathbf { n } } : = \tfrac { 1 } { \mathbf { n } ! } \partial ^ \mathbf { n } p ( 0 ) . \end{align*}


\begin{align*} [ ( B _ n + i ) ^ { - 1 } - ( B + i ) ^ { - 1 } ] \psi = ( B _ n + i ) ^ { - 1 } [ ( B + i ) \phi - ( B _ n + i ) \phi _ n ] - \phi + \phi _ n \longrightarrow 0 \end{align*}


\begin{gather*} \{ 0 \} = E _ { 0 } ( \pi , \lambda ) \subset E _ { - 1 } ( \pi , \lambda ) \subset \ldots \subset E _ { - g } ( \pi , \lambda ) \subset E _ { c s } ( \pi , \lambda ) \\ = E _ { g + 1 } ( \pi , \lambda ) \subset E _ { g } ( \pi , \lambda ) \subset \ldots \subset E _ { 1 } ( \pi , \lambda ) = \Gamma : = \R ^ { \mathcal { A } } \end{gather*}


\begin{align*} g _ { \gamma , 2 } ( x ) = \tfrac { 1 } { 2 } ( x ^ { \frac { 1 } { \beta } } + 1 ) . \end{align*}


\begin{align*} \mu _ m ^ i \int _ { \Omega } \nabla P \psi _ { \mu ^ i _ m , \xi ^ i _ m } ^ h ( x ) \nabla u _ m ( x ) d x = o ( 1 ) . \end{align*}


\begin{align*} A ( f + T ) = T \cdot Q ( T ) \ ; \ ; \ ; \mbox { a n d } \ ; \ ; \ ; Q ( 0 ) \not = 0 \ , . \end{align*}


\begin{align*} ( x \bar { \otimes } y ) ( x ' \bar { \otimes } y ' ) = ( x x ' ) \bar { \otimes } ( y y ' ) \end{align*}


\begin{align*} g _ i v _ j = - v _ { 2 i - j \bmod 3 } , i , j \in \{ 1 , 2 , 3 \} . \end{align*}


\begin{align*} \mathcal { I } ( \delta ) \leq \mathbb { E } [ Y _ 2 ] - \mathbb { E } [ Y _ 1 ] + \mathrm { R } ( \lambda _ { \mathrm { D } } ^ \star , \delta ) \leq \mathbb { E } [ Y _ 2 ] - \mathbb { E } [ Y _ 1 ] + \mathrm { R } _ { \mathrm { D } } ( \lambda _ { \mathrm { D } } ^ { \star } , \delta ) = \mathcal { I } _ { \mathrm { D } } ( \delta ) . \end{align*}


\begin{align*} H _ k = h _ 0 L _ k . \end{align*}


\begin{align*} d _ { \lambda _ t } H = d _ { \lambda _ t } h _ { \bar u ( t ) } , \qquad \forall \ , t \in [ 0 , 1 ] . \end{align*}


\begin{align*} ( 1 + w ) R ^ \varphi _ { k j , i } = w _ { k j , i } - w _ i R ^ \varphi _ { j k } + \frac { S ^ \varphi } { m - 1 } w _ i \delta _ { j k } \ , . \end{align*}


\begin{align*} ( \Delta + \lambda - q ) u = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , u _ { | \partial \mathrm { M } } = \varphi . \end{align*}


\begin{align*} \| g \| _ { L ( \log L ) , Q } & = \inf \Big \{ \lambda > 0 : \frac { 1 } { | Q | } \int _ { Q } \frac { | g ( x ) | } { \lambda } \log ( e + | f | / \lambda ) \mathrm { d } x \le 1 \Big \} . \end{align*}


\begin{align*} ( \phi ' ) ^ 2 + K \phi ^ 2 = 1 , \ \phi '' = - K \phi . \end{align*}


\begin{align*} A = \displaystyle { \frac { \frac { 1 } { Q } - \frac { 1 } { P } } { \frac { 1 } { Q } + \frac { 1 } { n } - \frac { 1 } { R } } } . \end{align*}


\begin{align*} \lim _ { t \to + \infty } \Big ( \left \vert u ( t ) - \phi _ { 1 } ( \lambda ) \right \vert + \left \vert u _ { t } ( t ) \right \vert \Big ) = 0 \ . \end{align*}


\begin{align*} 0 = \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \ , \bigg | \sum _ { i = 1 } ^ D \alpha _ i \ , \langle T _ i ( z ^ n \otimes \lbrace 0 , \infty \rbrace ) , f \rangle \bigg | ^ 2 \end{align*}


\begin{align*} E [ Z _ 0 ( \ell \Delta ) W ^ k _ { \ell \Delta , t } ( j ) ] = \Delta \nu p _ { j , + } ^ k ( t - \ell \Delta ) E [ Z _ 0 ( \ell \Delta ) ^ 2 ] , \end{align*}


\begin{align*} s = \tfrac { 1 } { \sqrt { L } } \ , . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } M _ + ( t ) = \nu Y \int _ 0 ^ { \infty } e ^ { - \lambda s } d s = \nu Y / \lambda , \end{align*}


\begin{align*} G ( \theta ) = & \mathbb { E } _ { \Phi | \Phi ( \mathcal { A } ) > 0 } \left \{ \left ( \sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \right . \right . \\ & \left . \left . \prod \limits _ { x _ { i } \in \Phi \backslash \{ x _ 1 \} \cap { \mathcal { A } } } \frac { 1 } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M r _ i ^ { \alpha } } \right ) ^ M } \right ) ^ b \right \} , \end{align*}


\begin{align*} g ( ( \pounds _ { \xi _ i } { f } ) X , \xi _ j ) = - g ( \nabla _ { { f } X } \ , \xi _ i , \xi _ j ) = 0 . \end{align*}


\begin{align*} \rho e _ t + \rho u e _ x + p u _ x + q _ x = \frac { 1 } { \mu } S ^ 2 . \end{align*}


\begin{align*} \mathcal { P } ( X _ \varphi ) = \mathcal { P } ( [ \underline { \varphi } , \varphi ] ) . \end{align*}


\begin{align*} \min \left \{ P ( G ) + 2 \lambda f ( | E \Delta G | \big ) \ , : \ , | G | = | B | \right \} , \end{align*}


\begin{align*} L ^ \sigma _ { h _ { s _ 1 } , \ldots , h _ { s _ k } } \ , = \ , I _ { z _ 0 } ( [ \alpha _ { s _ 1 } | \cdots | \alpha _ { s _ k } ] ) , k \geq 0 , ( s _ 1 , \ldots , s _ k ) \in S _ \star ^ k , \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } \log \mathbb { P } ^ * _ N ( \lambda _ N / N \geq \beta ) = - \widehat { \psi } _ r ( \beta ) , \beta \in [ C _ r , 1 ] , \end{align*}


\begin{align*} \begin{array} { l l l } V _ { n + 1 } ( x ) & = [ K + \min _ { x \leq y \leq x + B } & ( v y + L _ { n + 1 } ( y ) + \int _ 0 ^ \infty C _ { n } ( y - \xi ) f _ { n + 1 } ( \xi ) \mathrm { d } \xi ) \\ & & - ( v x + L _ { n + 1 } ( x ) + \int _ 0 ^ \infty C _ { n } ( x - \xi ) f _ { n + 1 } ( \xi ) \mathrm { d } \xi ) ] ^ - . \end{array} \end{align*}


\begin{align*} \Psi ^ \tau _ \mu ( ( u , a ) , ( v , b ) ) = \mu ( \sigma _ v = b \mid \sigma _ u = a , \sigma _ \Lambda = \tau ) - \mu ( \sigma _ v = b \mid \sigma _ \Lambda = \tau ) \end{align*}


\begin{align*} \delta = x d _ 2 \cdots d _ l \quad \quad \tilde { \delta } = ( x , d _ l , d _ { l - 1 } \dots , d _ 2 ) . \end{align*}


\begin{align*} f \mapsto \tilde f : = ( f ( 0 ) , f ( x _ 1 ) , f ( x _ 1 , x _ 2 ) , \ldots ) \end{align*}


\begin{align*} J _ 1 : = ( ( \nabla \cdot \varphi _ 3 ) \varphi _ 2 , \theta _ { \epsilon } \varphi _ 1 ) , \ \ J _ 2 : = ( ( \nabla \cdot \overline { \varphi _ 3 } ) \varphi _ 1 , \theta _ { \epsilon } \varphi _ 2 ) , \ \ J _ 3 : = - ( \nabla ( \varphi _ 1 \cdot \overline { \varphi _ 2 } ) , \theta _ { \epsilon } \varphi _ 3 ) . \end{align*}


\begin{align*} x _ 1 \bar { \otimes } x _ 2 = ( h _ { \varphi _ 1 } ^ { \eta / q } \bar { \otimes } h _ { \varphi _ 2 } ^ { \eta / q } ) ( x _ 1 ' \bar { \otimes } x _ 2 ' ) ( h _ { \varphi _ 1 } ^ { ( 1 - \eta ) / q } \bar { \otimes } h _ { \varphi _ 2 } ^ { ( 1 - \eta ) / q } ) = h _ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } ^ { \eta / q } ( x _ 1 ' \bar { \otimes } x _ 2 ' ) h _ { \varphi _ 1 \bar { \otimes } \varphi _ 2 } ^ { ( 1 - \eta ) / q } , \end{align*}


\begin{align*} c ( t ) = \frac { \det \phi ( t ) } { \prod _ { k = 1 } ^ R u _ { \beta _ k , \tau _ k } ( t ) } \end{align*}


\begin{align*} \frac { 1 } { T _ { * } } = \frac { A \varepsilon } { k T } . \end{align*}


\begin{align*} \begin{cases} P ( ( - \Delta ) ^ s ) _ j u _ j + q _ j u _ j : = \sum _ { i = 1 } ^ M \alpha _ i ^ j ( - \Delta ) ^ { s _ i } u _ j + q _ j u _ j = 0 & \Omega , \\ u _ j = f _ j & \Omega ^ c \end{cases} \end{align*}


\begin{align*} \phi ( A \otimes B ) = U ( \varphi _ 1 ( A ) \otimes \varphi _ 2 ( B ) ) V \hbox { f o r a l l $ A \in M _ m $ a n d $ B \in M _ n $ , } \end{align*}


\begin{align*} \sum _ { j = 0 } ^ k \alpha _ j y _ { n + j } = \sum _ { j = 0 } ^ k h \beta _ j f _ { n + j } \end{align*}


\begin{align*} u _ { i , k } = \begin{bmatrix} u _ { i , k } ^ 1 \\ u _ { i , k } ^ 2 \end{bmatrix} , y _ { i , k } = \begin{bmatrix} y _ { i , k } ^ 1 \\ y _ { i , k } ^ 2 \end{bmatrix} , B _ i = [ B _ i ^ 1 , B _ i ^ 2 ] , C _ i = \begin{bmatrix} C _ i ^ 1 \\ C _ i ^ 2 \end{bmatrix} , D _ i = \begin{bmatrix} D _ i ^ { 1 1 } & D _ i ^ { 1 2 } \\ D _ i ^ { 2 1 } & D _ i ^ { 2 2 } \end{bmatrix} , i = 1 , 2 , \end{align*}


\begin{align*} | \{ x \in \mathbb { R } ^ { n } : | T _ { \alpha } ( \vec { f } ) ( x ) | > \lambda \} | & \le \sum _ { s = 1 } ^ { 2 ^ { m } } | E _ { s } | + C | \Omega ^ { * } | \\ & \le \sum _ { s = 2 } ^ { 2 ^ { m } } | E _ { s } | + C B ^ { q } \lambda ^ { - \frac { n } { m n - \alpha } } \gamma ^ { q - \frac { n } { m n - \alpha } } + C ( \lambda \gamma ) ^ { \frac { - n } { m n - \alpha } } . \end{align*}


\begin{align*} \| f ^ { = d } \| _ q \le \sqrt { \gamma } ( 6 \sqrt { 2 } r q ) ^ d \le \gamma ( 6 \sqrt { 2 } e ^ 2 r q ) ^ { d } \le \gamma \left ( 6 4 r \sqrt { q } \sqrt { \max \{ \log ( 1 / \gamma ) , \ , q \} } \right ) ^ d , \end{align*}


\begin{align*} \partial _ { x _ n } u ( \hat { x } ) = \partial _ { x _ n } u ( \hat { x } ) - \partial _ { x _ n } u ( \hat { x } ^ \prime , 0 ) = \Big ( \int _ { 0 } ^ 1 \partial _ { x _ n } ^ 2 u ( \hat { x } ^ \prime , t x _ n ) \ : t \Big ) x _ n . \end{align*}


\begin{align*} \underline { C } : = \underline { \varphi } ^ { - 1 } ( \underline { A } ) = \Gamma _ { H , H ^ * } ^ { - 1 } ( ( \varphi ^ * ) ^ { - 1 } ( \underline { A } ) ) = \Gamma _ { H , H ^ * } ^ { - 1 } \bigcap _ { s \in \underline { S } } ( R ^ * ) ^ { - s } ( W ^ * ) ^ c = \bigcap _ { s \in \underline { S } } \Gamma _ { H , H ^ * } ^ { - 1 } ( R ^ * ) ^ { - s } ( W ^ * ) ^ c . \end{align*}


\begin{align*} \lim _ { \alpha \to 0 } s _ \alpha [ Y ] ( x ) = l [ Y ] ( x ) \lim _ { \alpha \to 0 } t _ \alpha [ Y ] ( x ) = l [ Y ] ( x ) \end{align*}


\begin{align*} f = f _ 0 + f _ 1 + \dots + f _ p + f _ { a d d } , \end{align*}


\begin{align*} t _ N : = \inf \{ t \geq 0 : Y e ^ { \lambda t } = N \} . \end{align*}


\begin{align*} \nu _ i ( f ) = \min _ { 1 \leq j \leq r } \left \{ v \left ( \tilde b _ j \right ) \right \} . \end{align*}


\begin{align*} \inf \limits _ { t \in [ 0 , T ] } a ( t ) = a _ { 0 } \sup \limits _ { t \in [ 0 , T ] } a ( t ) = a _ { 1 } . \end{align*}


\begin{align*} \mu _ p ( g ) = \mu _ p ( f _ { S \to 0 } ) \leq 8 \exp \left ( - \frac { 0 . 0 0 0 1 } { p } \right ) \leq \frac { 1 } { 4 } \mu _ p ( f ) . \end{align*}


\begin{align*} \tilde { C } = ( n , \Lambda , p _ 0 , \| D ^ 2 u \| _ { L ^ { \infty } ( B _ 1 ) } ) \omega ^ { p _ 0 } . \end{align*}


\begin{align*} P _ k : = \frac { \int _ 0 ^ 1 \phi \circ T ^ k _ { \gamma + \delta } \ , d m - \int _ 0 ^ 1 \phi \circ T ^ k _ \gamma \ , d m - D \cdot \delta } { \Vert \delta \Vert } . \end{align*}


\begin{align*} \int _ \Omega f ^ { ' } _ 0 ( U _ { \mu _ i , \xi _ i } ) \phi \psi ^ h _ { \mu _ j , \xi _ j } d x = 0 . \end{align*}


\begin{align*} ( A _ q u | v ) = \mathfrak { a } _ q ( u , v ) , u \in D ( A ) , \ ; v \in H _ 0 ^ 1 ( \mathrm { M } ) . \end{align*}


\begin{align*} B = B ( T ) = - 2 \pi \int _ { 0 } ^ { \infty } \left [ e ^ { - u ( r ) / k T } - 1 \right ] r ^ { 2 } \ , d r , \end{align*}


\begin{align*} \begin{aligned} J _ 1 \geq C s ^ 3 \lambda ^ 4 \iint _ Q \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d y d t - C s ^ 3 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega _ 0 } \xi ^ 3 | u | ^ 2 d x d y d t , \end{aligned} \end{align*}


\begin{align*} Y _ 1 = 0 , \ldots , Y _ d = 0 , \ , x _ i - x _ j = 0 , \ , \ , \{ i , j \} \in E \setminus E ( H ) , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c u ) ^ n = \psi ( \cdot , u ) \ , \omega ^ n & \textnormal { o n } & \Omega \\ u = 0 & \textnormal { o n } & \partial \Omega . \end{array} \right . \end{align*}


\begin{align*} | \hat { \varphi } _ { l , d } ( \zeta ) | = | l \hat { \varphi } ( l \zeta ) e ^ { - i d \zeta } | \leq C l e ^ { ( d - l ) \Im \zeta } ( 1 + | l \zeta | ) ^ { - N } . \end{align*}


\begin{align*} n m ' - k m ' - l k ' = ( n - k ) \frac { m - l } { ( k , m - l ) } - l \frac { k } { ( k , m - l ) } = \frac { n m - n l - k m } { ( k , m - l ) } > 0 , \end{align*}


\begin{align*} D _ { i } ^ { p } ( x ) : = \alpha _ { i } \nabla d _ { S _ { i } } ^ { p } \left ( x \right ) . \end{align*}


\begin{align*} S ^ * T T ^ * S = ( I \otimes T ) Z ^ * Z ( I \otimes T ^ * ) \leq I \otimes T T ^ * . \end{align*}


\begin{align*} ( k , \sigma ) \cdot f = \psi ( k ) \cdot ( \sigma \ast f ) \ . \end{align*}


\begin{align*} V \left ( t , \xi \right ) = \inf _ { u \in \Lambda _ { t , \xi } } J \left ( t , \xi ; u \right ) \leq \inf _ { u ^ { \prime } \in \Lambda _ { t , \xi } } J \left ( t , \xi ^ { \prime } ; u ^ { \prime } \right ) = V \left ( t , \xi ^ { \prime } \right ) . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) = \sup _ { \pi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \int _ { \mathcal { V } } g _ \lambda \ , d \pi \right ] = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \sup _ { \pi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \left [ \langle \lambda , \delta \rangle + \int _ { \mathcal { V } } g _ \lambda \ , d \pi \right ] . \end{align*}


\begin{align*} \left \{ \sum _ { i = 1 } ^ r \sum _ { j = 1 } ^ s F ^ { k _ { i , j } n _ j } ( Q _ j ) \colon n _ j \ge 0 j = 1 , \dots , s \right \} , \end{align*}


\begin{align*} U ( \vec { a } , \Lambda ) \phi ^ { \alpha , n } & ( f ) U ( \vec { a } , \Lambda ) ^ { - 1 } 1 \\ & = ( \Lambda S _ 0 + \vec { a } , e ^ { - i [ \vec { a } \cdot ( \hat { H } ( \rho _ n ) \hat { g } _ 0 + \hat { P } ( \rho _ n ) \hat { g } _ 1 ) ] } f ^ { S _ 0 ^ \flat } ( \Lambda ^ { - 1 } ( \cdot - \vec { a } ) ) \otimes \rho _ n ( E ^ \alpha ) ) . \end{align*}


\begin{align*} g ' ( t ) = f ' \bigg ( \alpha ( s _ 2 ) + \frac { d ( s _ 2 ) } { d ( s _ 1 ) } ( t - \alpha ( s _ 1 ) ) \bigg ) \leq f ' ( t ) . \end{align*}


\begin{align*} \mathcal { A } ( \alpha , p ) : = \arg \min \sum \limits _ { i = 1 } ^ { m } \alpha _ { i } d _ { S _ { i } } ^ { p } , \end{align*}


\begin{align*} \begin{aligned} y = & \sqrt { P _ s } \zeta \sum _ { l = 1 } ^ L \hat h _ l e ^ { j \phi _ { l , n _ t } } g _ { l , n _ t } + \\ & \sqrt { P _ s ( 1 - \zeta ^ 2 ) } \sum _ { l = 1 } ^ L \Delta { h } e ^ { j \phi _ { l , n _ t } } g _ { l , n _ t } + n _ 0 , \end{aligned} \end{align*}


\begin{align*} F _ \lambda ( t ) : = \frac { f ' ( u ( t ) ) } { 1 - \lambda f ' ( u ( t ) ) } = \frac { F ( t ) } { 1 - \lambda F ( t ) } , \lambda > 0 , \end{align*}


\begin{align*} E \big | S _ { j , + } ^ k ( t ) - \hat { S } _ { j , + } ^ k ( t ) \big | = O ( \theta ^ { k / 2 } t e ^ { \lambda t / 2 } ) . \end{align*}


\begin{align*} I _ { m , p } ( s , x ) : = \frac { 1 } { 2 \pi i } \ , \intop _ { \mu - i \infty } ^ { \mu + i \infty } \left ( 2 z , 2 \pi p m \right ) _ { m } \ , \Gamma \left ( z \right ) \ , \Gamma \left ( s + z - \frac { 1 } { 2 } \right ) \ , ( \pi \ , x \ , m ) ^ { - 2 z } \ , d z . \end{align*}


\begin{align*} U _ m ( z ) : = \frac { U ( z _ 0 + \lambda _ m z ) } { \lambda _ m ^ { k _ 0 + s } } , \lambda _ m : = 1 - r _ m \to 0 ^ + , \end{align*}


\begin{align*} { \rm R e } ( \overline { a ( \xi ) } f ( \xi ) ) = 0 { \rm \ f o r \ } \xi \in L \end{align*}


\begin{align*} \psi ^ \ell _ { i _ 1 , \ldots , i _ n } ( f _ { i _ 1 , \ldots , i _ n } ) = f _ { i _ 1 , \ldots , i _ \ell , i _ \ell , \ldots , i _ n } \end{align*}


\begin{align*} \sup _ { \gamma \in \bar { \mathcal { P } } } \sup _ { \pi \in \Pi \left ( \mu _ 1 , \mu _ 2 , \gamma \right ) } \int _ { \mathcal { V } \times \mathcal { S } } \phi _ \lambda \ , d \pi = \sup _ { \pi \in \mathcal { G } _ { \lambda } } \int _ { \mathcal { V } \times \mathcal { S } } \phi _ \lambda \ , d \pi . \end{align*}


\begin{align*} ( a t + b ) ^ { p \theta + p - 1 } & = ( a t + b ) ^ { p \theta } ( a t + b ) ^ { p - 1 } \\ & = \begin{cases} a ^ { p \theta } t ^ { p \theta + p - 1 } b = 0 , \\ \sum _ { m = 0 } ^ { p - 1 } \binom { p - 1 } { m } \left ( a ^ { p \theta - m } b ^ m t ^ { p \theta + p - 1 - m } + a ^ { - m } b ^ { p \theta + m } t ^ { p - 1 - m } \right ) . \end{cases} \\ & = a ^ { p \theta } t ^ { p \theta + p - 1 } + r _ { \theta } ( t ) , \end{align*}


\begin{align*} C _ T : = { \rm c o n v } \left ( \cup B \left ( x ' , \alpha \delta _ h \right ) \right ) , \end{align*}


\begin{align*} Z _ { k \ell } [ a ] : = \left ( \begin{array} { c c c c c } a _ 1 & a _ 2 & \ldots & a _ { \ell - 1 } & - a _ 1 - a _ 2 - \ldots - a _ { \ell - 1 } \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a _ 1 & a _ 2 & \ldots & a _ { \ell - 1 } & - a _ 1 - a _ 2 - \ldots - a _ { \ell - 1 } \\ \end{array} \right ) , { \rm i f \ } \ell \geq 2 , \end{align*}


\begin{align*} & x \oslash y = \frac { x } { y } ( 1 + \eta _ 1 ) , & | \eta _ 1 | \le 2 ^ { - d } , \end{align*}


\begin{align*} S A - A ^ { * } S = 0 . \end{align*}


\begin{align*} \mathsf { M } ( \sigma ( A ) , i ) : = \min _ { i \leq j \leq k } \left \{ \sigma ( a _ j ) : \ \sigma ( a _ j ) < \sigma ( a _ i ) \right \} . \end{align*}


\begin{align*} Y _ { 0 } = \mathbb { E } \left [ \Psi _ { T } \left \langle G X _ { T } , X _ { T } \right \rangle + \int _ { 0 } ^ { T } \Psi _ { s } ( \left \langle Q _ { s } X _ { s } , X _ { s } \right \rangle + \left \langle R _ { s } u _ { s } , u _ { s } \right \rangle ) \mathrm { d } s \right ] . \end{align*}


\begin{align*} \overline { \overline { \Pi } } _ { \psi } [ \omega ] f ( [ \epsilon , x ] ) = - i \nu ( \epsilon , \omega ) \int _ { \R } e ^ { 2 \pi i \epsilon x y } f ( [ \epsilon , - y ] ) d y ; \end{align*}


\begin{align*} \mathcal { R } _ { \chi , \varepsilon } ( z ) = \left ( \mathcal { A } _ { 0 , \chi , \varepsilon } - z I \right ) ^ { - 1 } - S _ { \chi , \varepsilon } ( z ) M _ { \chi , \varepsilon } ( z ) ^ { - 1 } S _ { \chi , \varepsilon } ( \overline { z } ) ^ * , \end{align*}


\begin{align*} z ( s ) & = h ^ 2 \beta _ h ( s ) + \frac { h ^ 2 } { 2 } \overline { \beta _ h ( s ) } + \frac { \sigma ^ 2 } { 2 } \overline { \beta _ \sigma ( s ) } , \\ \zeta ( s ) & = i \beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \overline { \beta _ h ( s ) } K _ 2 - i \overline { \beta _ \sigma ( s ) } K , \\ \widetilde \gamma ( s ) & = - i s ( \beta _ h ( s ) - 1 ) ( | K _ 1 | ^ 2 + | K _ 3 | ^ 2 ) + i s ( \overline { \beta _ h ( s ) } - 1 ) | K _ 2 | ^ 2 + i s ( \overline { \beta _ \sigma ( s ) } - 1 ) | K | ^ 2 . \end{align*}


\begin{align*} \mathrm { f } ( k ) : = \sum _ { j = 1 } ^ { d } 2 ( 1 - \cos k _ j ) . \end{align*}


\begin{gather*} I _ { \alpha } : = [ l _ \alpha , r _ \alpha ) , l _ \alpha = \sum _ { \pi _ 0 ( \beta ) < \pi _ 0 ( \alpha ) } \lambda _ \beta , \ ; \ ; \ ; r _ \alpha = \sum _ { \pi _ 0 ( \beta ) \leq \pi _ 0 ( \alpha ) } \lambda _ \beta . \end{gather*}


\begin{align*} \sqrt { \alpha } \left ( \frac { 1 } { 4 } \gamma - \frac { 1 } { 4 } \log ( 4 \beta ) + \sum _ { m , n = 1 } ^ { \infty } K _ { 0 } ( 2 m n \ , \alpha ) \right ) = \sqrt { \beta } \left ( \frac { 1 } { 4 } \gamma - \frac { 1 } { 4 } \log ( 4 \alpha ) + \sum _ { m , n = 1 } ^ { \infty } K _ { 0 } ( 2 m n \ , \beta ) \right ) , \end{align*}


\begin{align*} \mathcal { G } _ { X ^ { \ast } X } ( f ) ( x ) & = \int _ { X ^ { \ast } } f ( [ y ^ { \ast } , 0 ] [ x , 0 ] ) d y ^ { \ast } \\ & = \int _ { X ^ { \ast } } \psi ( \langle y ^ { \ast } , x \rangle ) f ( y ^ { \ast } ) d y ^ { \ast } \\ & = \mathcal { F } ' _ { X ^ { \ast } X } ( f ) ( x ) . \end{align*}


\begin{align*} \phi ( t ) = \begin{cases} t - \frac { \delta } { 2 } , & \mbox { i f } t \in \left [ \frac { \delta } { 2 } , \frac { \delta } { 2 } + 1 \right ] , \\ 1 , & \mbox { i f } t \in \left [ \frac { \delta } { 2 } + 1 , \delta - 1 \right ] , \\ \delta - t , & \mbox { i f } t \in \left [ \delta - 1 , \delta \right ] , \\ 0 , & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} V ( p , t ) = \inf _ { t ' \in ] t , T ] } \{ V ( \bar p , t ' ) + C ( p , \bar p , t , t ' ) \} = \inf _ { t ' \in ] t , T ] } [ C ( p , \bar p , t , t ' ) + \tilde C ( \bar p , t ' ) ] \end{align*}


\begin{align*} \Pi _ { \psi } [ \begin{pmatrix} a & 0 \\ 0 & a ^ { \ast - 1 } \end{pmatrix} ] f ( [ \epsilon , y ] ) = | \det ( a ) | ^ { 1 / 2 } ( \det a , \epsilon ) _ F f ( [ \epsilon , y a ] ) . \end{align*}


\begin{align*} H ^ { - s } ( \mathbb { R } ^ d ) : = \{ \xi \in \mathcal { S } ' ( \mathbb { R } ^ d ) : \{ 1 + | \xi | ^ { - s } \hat { \xi } \} \in L ^ 2 ( \mathbb { R } ^ d ) \} , \end{align*}


\begin{align*} f _ { \beta + \delta , 2 } ' ( y ) = f _ { \beta , 1 } ' ( y ) + \partial _ 2 f ' _ { \beta , 2 } ( y ) \cdot \delta + O ( \delta ^ 2 ) , \end{align*}


\begin{align*} \frac { d } { d \lambda } S _ { \omega , \mathbf { c } } ( \Phi ^ { \lambda } ) \big | _ { \lambda = 1 } = \sum _ { j = 1 } ^ 3 \left . \left \langle ( D _ j S _ { \omega , \mathbf { c } } ) ( \Phi ^ { \lambda } ) , \frac { d } { d \lambda } \Phi ^ { \lambda } \right \rangle \right | _ { \lambda = 1 } = \sum _ { j = 1 } ^ 3 \left \langle D _ j S _ { \omega , \mathbf { c } } ( \Phi ) , \frac { d } { 2 } \Phi + \nabla \cdot \Phi \right \rangle . \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - z ^ n } \right ) ^ { \frac { m ^ 4 } { n ^ 5 } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { - n } { 3 0 } + \frac { n ^ 3 } { 3 } + \frac { n ^ 4 } { 2 } + \frac { n ^ 5 } { 5 } \right ) \frac { z ^ n } { n ^ 5 } \right \} \end{align*}


\begin{align*} g _ { u , v } ( z ) : = u | A | z ^ { n + m } + v | B | \overline { z } ^ m + | C | , z \in \mathbb { C } , \end{align*}


\begin{align*} F ( p ) = \frac { 1 } { 2 } | | p | | ^ 2 + \sum _ { i = 1 } ^ { n } \frac { m _ i } { \vert \vert p - x _ i \vert \vert } . \end{align*}


\begin{align*} t ^ n = \sum _ { k = 0 } ^ { n } S [ n , k ] ( t ) _ { k , q } , \end{align*}


\begin{align*} c _ m \Delta v = - S ^ \varphi u + \tilde S ^ { \tilde \varphi } u ^ { \frac { m + 2 } { m - 2 } } \ge - S ^ \varphi u \left ( 1 - u ^ { \frac { 4 } { m - 2 } } \right ) = - c _ m \Lambda ( x ) v \end{align*}


\begin{align*} \alpha ' _ n = \alpha _ n ( a ^ 2 , q ^ 2 ) \quad \mbox { a n d } \quad \beta ' _ n = \sum _ { j \leq n } \frac { ( - a q ) _ { 2 j } } { ( q ^ 2 ; q ^ 2 ) _ { n - j } } q ^ { n - j } \beta _ j ( a ^ 2 , q ^ 2 ) . \end{align*}


\begin{align*} \beta \phi = \lambda g ( \phi ) . \end{align*}


\begin{align*} P ( W ^ k _ { \ell \Delta , t } ( j ) = 1 ) = P ( X _ { \ell , \Delta } = 1 , D ^ j _ { \ell \Delta } ( t ) \geq k ) ( 1 + O ( \Delta ) ) . \end{align*}


\begin{align*} \frac { { \partial { J _ { G M E E F P } } } } { { \partial { { \boldsymbol { x } } _ k } } } = { \boldsymbol { W } } _ k ^ { } { { \boldsymbol { \Lambda } } _ k } { { \boldsymbol { d } } _ k } - { \boldsymbol { W } } _ k ^ T { { \boldsymbol { \Lambda } } _ k } { { \boldsymbol { W } } _ k } { { \boldsymbol { x } } _ k } \end{align*}


\begin{align*} \left [ P \right ] _ { i , j } ^ { } = \begin{dcases} \frac { i } { n } & \mbox { i f } \ , j = i ; \\ \frac { n - i } { n } & \mbox { i f } \ , j = i + 1 ; \\ 0 & \mbox { o t h e r w i s e } ; \end{dcases} \end{align*}


\begin{align*} 0 < \mathcal { H } ^ 1 \left ( \Omega \cap \left ( \overline { J _ u } \setminus J _ u \right ) \right ) \leq \sum _ { h = 1 } ^ \infty \mathcal { H } ^ 1 \left ( \Omega _ { \delta _ h } \cap \left ( \overline { J _ u } \setminus J _ u \right ) \right ) = 0 , \end{align*}


\begin{align*} u = h \rho , \end{align*}


\begin{align*} \sum _ { j \neq i _ 0 } v _ { i _ 0 j } = x _ { 0 , i _ 0 } \quad v _ { i _ 0 j } = 0 , \ , \forall j \neq 0 , \end{align*}


\begin{align*} f ( y _ 1 , y _ 2 , x ; \theta ) = \log ( 1 + \exp ( - y _ 1 \langle \theta , ( y _ 2 , x ) \rangle ) ) , \end{align*}


\begin{align*} \rho _ \sigma ^ { a b } = & \frac { 1 } { \sqrt { \det \left [ 1 + W _ { a b } ^ { c d , T } W _ { a b } ^ { c d } \right ] } } \equiv \frac { | J _ { a b } ^ \sigma | } { \sqrt { \det \left [ J _ { a b } ^ { \sigma , T } J _ { a b } ^ \sigma + J _ { c d } ^ { \sigma , T } J _ { c d } ^ \sigma \right ] } } . \end{align*}


\begin{align*} { k ( n - k ) \Pr ( X _ t = k \ , \vert \ , n ) = k n ( 1 - \frac { 1 } { n } ) ^ t \Pr ( \widetilde { X } _ t = k \ , \vert \ , n - 1 ) } , \end{align*}


\begin{align*} & a _ 1 = i - \lambda _ 1 , a _ 2 = \lambda _ 1 - \lambda _ 2 , \ldots , a _ { r - i } = \lambda _ { r - i - 1 } - \lambda _ { r - i } . \end{align*}


\begin{align*} \begin{array} { l l } \underset { x \in X } { } & \sum \limits _ { i = 1 } ^ { m } \alpha _ { i } d \left ( x , S _ { i } \right ) ^ { p } , \end{array} \end{align*}


\begin{align*} \partial _ t v = - i ( P _ 0 - i M h W - z ) v , v | _ { t = 0 } = u . \end{align*}


\begin{align*} \mathrm { E n d } _ { \mathbb { C } } ( A ) \otimes _ { \mathbb { Z } } \mathbb { Z } _ { \ell } = \mathcal { O } _ K \otimes _ { \mathbb { Z } } \mathbb { Z } _ { \ell } = : \mathcal { O } _ { K , \ell } . \end{align*}


\begin{align*} S _ { r } ( f \circ _ { } g ) = \sum _ { \substack { 0 \leq s , r \\ s + t \leq r } } \frac { r ! } { s ! t ! ( r - s - t ) ! } \alpha ^ { s / p } \beta ^ { t / p } \gamma ^ { ( r - s - t ) / p } S _ { r - t } ( f ) \circ _ { } S _ { r - s } ( g ) \ , , \end{align*}


\begin{align*} \mathcal { S } = \left \{ \{ B , \underline { B } \} : \ 2 k - 1 , 2 k \in B \right \} . \end{align*}


\begin{align*} \mathcal { F } ^ X _ t = \sigma \big ( ( X _ i ( s ) ) _ { 0 \leq s \leq t _ i } , i \in V \big ) . \end{align*}


\begin{align*} \mathcal { S } _ { \ell , M } = \bigl ( ( p _ 1 , \nu _ 1 ) , \ldots , ( p _ s , \nu _ s ) \bigr ) , \end{align*}


\begin{align*} \zeta _ { p } ( 1 - s ) = \frac { 2 \ , \cos \left ( \frac { \pi s } { 2 } \right ) \Gamma ( s ) } { ( 2 \pi ) ^ { s } } \eta _ { p } ( s ) . \end{align*}


\begin{align*} ( \underline { X } , \underline { A } ) ^ { K } = \bigcup _ { \sigma \subseteq K } ( \underline { X } , \underline { A } ) ^ { \sigma } \subseteq \prod ^ m _ { i = 1 } X _ i . \end{align*}


\begin{align*} \lim _ { y \to - 1 ^ + } \phi ( y ) = { 1 \over 2 } \sum _ j E _ j d _ j . \end{align*}


\begin{align*} \sigma _ a ^ + = \sum _ { I } \varphi ^ + ( M _ I ) S ^ I , \end{align*}


\begin{align*} [ t ( f _ 1 ' , \ldots , f _ k ' ) ] ( x _ 1 , \ldots , x _ m ) & = t ( f _ 1 ' ( x _ 1 , \ldots , x _ m ) , \ldots , f _ k ' ( x _ 1 , \ldots , x _ m ) ) \\ & = t ( f _ 1 ( x _ 1 / \alpha , \ldots , x _ m / \alpha ) , \ldots , f _ k ( x _ 1 / \alpha , \ldots , x _ m / \alpha ) ) \\ & = [ t ( f _ 1 , \ldots , f _ k ) ] ' ( x _ 1 , \ldots , x _ m ) \in \varphi ( C ) . \end{align*}


\begin{align*} \texttt { L H S } \eqref { i d : 2 r e f 2 } & \simeq \int _ a ^ b x ^ { \xi + \lambda } \ , d x \simeq \begin{cases} ( b - a ) b ^ { \xi + \lambda } & \xi + \lambda > - 1 , \\ \log \Big ( 1 + \frac { b - a } { a } \Big ) & \xi + \lambda = - 1 , \\ ( b - a ) b ^ { - 1 } a ^ { \xi + \lambda + 1 } & \xi + \lambda < - 1 \end{cases} \\ & \simeq \texttt { R H S } \eqref { i d : 2 r e f 2 } . \end{align*}


\begin{align*} & x ^ { \gamma } y ^ { \alpha } z ^ { \beta } - x ^ { \alpha } y ^ { \gamma } z ^ { \beta } \\ = & - ( x - y ) x ^ { \gamma } y ^ { \gamma } z ^ { \beta } \sum _ { i = 0 } ^ { \alpha - \gamma - 1 } x ^ i y ^ { \alpha - \gamma - 1 - i } \\ = & - ( x - y ) \bigg ( x ^ { \gamma } z ^ { \beta } \sum _ { i = 0 } ^ { \alpha - \beta - 1 } x ^ i y ^ { \alpha - 1 - i } + x ^ { \alpha - \beta + \gamma } z ^ { \beta } \sum _ { i = 0 } ^ { \beta - \gamma - 1 } x ^ i y ^ { \beta - 1 - i } \bigg ) . \end{align*}


\begin{align*} f _ { L _ 1 } \left ( M \tau ; M \begin{pmatrix} r \\ t \end{pmatrix} \right ) = \phi ( \tau ) ^ { 2 \nu } \rho _ { L _ 1 } ( M , \phi ) f _ { L _ 1 } \left ( \tau ; \begin{pmatrix} r \\ t \end{pmatrix} \right ) \end{align*}


\begin{align*} m = m _ - - m _ + + R ( \chi _ - \circ \pi - \chi _ + \circ \pi ) \end{align*}


\begin{align*} L _ N \pi _ { s } ^ N ( \phi ) = \pi _ { s } ^ N ( A \phi ) + R _ 3 ( \phi , N , s ) . \end{align*}


\begin{align*} \mathrm { V o l } ( \Omega ) = \mathrm { V o l } ( B ) , \ \mathrm { b a r } ( \Omega ) = O , \end{align*}


\begin{align*} v _ 0 ( t , x ) : = ( \tau , z ) \mapsto K _ { t - \tau } ( x , z ) \ , \sigma ( u ( \tau , z ) ) \ , 1 _ { [ 0 , t ] } ( \tau ) . \end{align*}


\begin{align*} \frac { 2 \sqrt { \pi } \ , c ^ { \frac { 1 } { 2 } - s } \Gamma \left ( s - \frac { 1 } { 2 } \right ) } { \Gamma ( s ) } \zeta _ { p ^ { \prime } } ( 2 s - 1 ) = \frac { \pi } { \sqrt { c } } \ , \frac { 1 } { s - 1 } + \frac { \pi } { \sqrt { c } } \left ( 2 C _ { p ^ { \prime } } ^ { ( 1 ) } - \log \left ( 4 c \right ) \right ) + O \left ( s - 1 \right ) . \end{align*}


\begin{align*} \phi = \phi ( r ) , \ \phi ' = \phi ' ( r ) , \ r = \rho ( 1 + u ) , \end{align*}


\begin{align*} \overline { F } ( k , t + 1 ; p , n ) = \begin{dcases} \frac { ( n - k ) p } { n } \overline { F } ( k - 1 , t ; p , n ) + \frac { n - ( n - k ) p } { n } \overline { F } ( k , t ; p , n ) & \textnormal { i f } \ ; k \le t ; \\ 0 & \textnormal { i f } \ ; k > t . \end{dcases} \end{align*}


\begin{align*} p _ { \# } = p \left ( 1 + \frac { 2 s } { n } \right ) . \end{align*}


\begin{align*} [ \varphi _ \lambda ( [ x _ \mu y _ 1 ] , y _ 2 ) _ { \lambda + \mu } [ w _ \gamma v ] ] + ( - 1 ) ^ { x y _ 1 } [ \varphi _ \lambda ( y _ 1 , [ x _ \mu y _ 2 ] ) _ { \lambda + \mu } [ w _ \gamma v ] ] = [ ( \varphi _ \lambda ( x , [ { y _ 1 } _ \eta y _ 2 ] ) ) _ { \lambda + \mu } [ w _ \gamma v ] ] . \end{align*}


\begin{align*} \eta = \frac { \Theta ( n ^ { - 1 } ) } { 1 - \left ( 1 - \Omega ( n ^ { - 1 } ) \right ) } = O ( 1 ) , \end{align*}


\begin{align*} X _ { h G } : = E G \times _ G X \end{align*}


\begin{align*} d ^ { ( 2 ) * } _ i = \arg \min \limits _ { d \in \mathcal { D } ^ { ( 2 ) } _ i \cup \{ - \beta _ i , 1 - \beta _ i \} } f ^ { ( 2 ) } _ { \boldsymbol \beta , i } ( d ) , \end{align*}


\begin{align*} E V S I = \frac { 1 } { T } \sum _ { t = 1 } ^ T \varphi ( y ^ t , d _ 0 ^ t ) - \frac { 1 } { T } \sum _ { t = 1 } ^ T \psi ( y ^ t , d _ 0 ^ t ) . \end{align*}


\begin{align*} \langle f , g \rangle _ { \mathrm { p e t } } = \int _ { \Gamma \backslash \mathbb { H } } f ( z ) \overline { g ( z ) } \ , ( \mathrm { I m } ( z ) ) ^ { k } \ , \mu _ { \mathrm { h y p } } ( z ) \ , \ \ , \ f , g \in S _ { k } ( \Gamma ) . \end{align*}


\begin{align*} | V _ e | = ( n - 2 ) | V _ i | + 2 . \end{align*}


\begin{align*} L ( \Phi ^ { \lambda } ) = \lambda ^ 2 L ( \Phi ) , \ \ N ( \Phi ^ { \lambda } ) = \lambda ^ { \frac { d } { 2 } + 1 } N ( \Phi ) , \ \ Q ( \Phi ^ { \lambda } ) = Q ( \Phi ) , \ \ \mathbf { P } ( \Phi ^ { \lambda } ) = \lambda \mathbf { P } ( \Phi ) . \end{align*}


\begin{align*} \rho _ a ( e ^ { t i } \wedge x ) = e ^ { a t i } \wedge x . \end{align*}


\begin{align*} h ^ { - 1 } g ^ { n } h = g ^ { - n } . \end{align*}


\begin{align*} C _ n ^ \lambda ( \cos \theta ) : = \frac { \Gamma ( \lambda + \frac 1 2 ) \Gamma ( 2 \lambda + n ) } { \Gamma ( 2 \lambda ) \Gamma ( \lambda + \frac 1 2 + n ) } P _ n ^ { ( \lambda - \frac 1 2 , \lambda - \frac 1 2 ) } ( \cos \theta ) , \end{align*}


\begin{align*} \frac { \varphi _ 0 ( x ) ^ k } { k ! } = \sum _ { n \geq k } B _ { n , k } ( f _ { 1 , 0 } , \ldots , f _ { n - k + 1 , 0 } ) \frac { x ^ n } { n ! } , \end{align*}


\begin{align*} L w ( x ) = P V \int _ { \R ^ n } \frac { w ( x ) - w ( x + y ) } { | y | ^ { n + 2 s } } a \left ( \frac { y } { | y | } \right ) d y , \end{align*}


\begin{align*} x _ { n + 1 } = \frac { 1 } { \| x _ n + \alpha _ n u \| } ( x _ n + \alpha _ n u ) , \end{align*}


\begin{align*} \mu _ { \omega , \mathbf { c } } = \omega ^ { 2 - \frac { d } { 2 } } \mu _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle \dfrac { N } { 4 } \Lambda a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } \eta \xi ^ 2 d x = 0 . \end{array} \right . \end{align*}


\begin{align*} \pi _ { \mu } ( f ) \bar { x } \begin{pmatrix} f ( T ^ t ) & 0 \\ \star & \star \end{pmatrix} \begin{pmatrix} \bar { x } \\ 0 \end{pmatrix} = \begin{pmatrix} \bar { x } \\ \star \end{pmatrix} . \end{align*}


\begin{align*} \| f \| _ { \mathrm { H } _ { \mathcal { H } _ { \hbar , V } } ^ { s } } : = \left \| ( I + \mathcal { H } _ { \hbar , V } ) ^ { s / 2 } f \right \| _ { \ell ^ { 2 } ( \hbar \mathbb { Z } ^ { n } ) } = \left ( \sum \limits _ { \xi \in \mathcal { I } _ { \hbar } } \langle \xi \rangle ^ { 2 s } \widehat { f } ( \xi ) | ^ { 2 } \right ) ^ { \frac { 1 } { 2 } } = \left ( \sum \limits _ { \xi \in \mathcal { I } _ { \hbar } } \left ( 1 + \lambda _ { \xi } \right ) ^ { s } \widehat { f } ( \xi ) | ^ { 2 } \right ) ^ { \frac { 1 } { 2 } } . \end{align*}


\begin{align*} \Vert B A ^ { \prime } \Vert ^ { 2 } = \Vert ( B A ^ { \prime } ) ^ { * } \Vert ^ { 2 } \le 4 \max _ { x \in \mathcal { N } } \Vert ( B A ^ { \prime } ) ^ { * } x \Vert _ { 2 } ^ { 2 } = 4 \max _ { x \in \mathcal { N } } \sum _ { j \le n } \langle W _ { j } ^ { \prime } , x \rangle ^ { 2 } . \end{align*}


\begin{align*} u \mid _ { S _ { T } \diagdown \left ( \Gamma _ { 1 T } ^ { + } { } \right ) } = 0 . \end{align*}


\begin{align*} J _ r ( x , \beta ) = \int _ { [ 0 , 1 ] } \d y \ , \mathcal { R } \big ( \widehat { r } _ \beta ( x , y ) \mid r ( x , y ) \big ) , x \in [ 0 , 1 ] , \end{align*}


\begin{align*} \boldsymbol { \Omega } ( \mathbf { x } ) = \theta ( \mathbf { x } ) \mathbf { A } + \phi ( \mathbf { x } ) \mathbf { B } + \chi ( \mathbf { x } ) \mathbf { C } . \end{align*}


\begin{align*} \mathrm { I } ( \xi _ { 1 } ^ { x } , \mathcal { T } _ { x } ) = \int _ { G ^ { \mathbb { N } } } \log \frac { d ( \omega _ { 1 } \lambda ) _ { x } } { d \lambda _ { x } } \left ( \psi ( x , \omega ) \right ) d \mathbb { P } _ { \mu } ^ { \pi ( x ) } ( \omega ) . \end{align*}


\begin{align*} \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \leq x \right ) - \Phi ( x ) & = - \left [ \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } > x \right ) - ( 1 - \Phi ( x ) ) \right ] \\ & \geq - \left [ ( 1 - \Phi ( x ) ) \exp \left \{ C \frac { 1 + x ^ { 3 } } { \sqrt { n } } \right \} - ( 1 - \Phi ( x ) ) \right ] \\ & = - ( 1 - \Phi ( x ) ) \left ( \exp \left \{ C \frac { 1 + x ^ { 3 } } { \sqrt { n } } \right \} - 1 \right ) . \end{align*}


\begin{align*} K _ \varepsilon ( | A _ \gamma ( \hat { x } , \hat { x } ) ( \hat { x } - \bar { y } ) | ) : = J _ \varepsilon ( | A _ \gamma ( \hat { x } , \hat { y } ) ( \hat { x } - \bar { y } ) | ) \big | \det F _ \gamma ( \bar { y } ) \big | - J _ \varepsilon ( | A _ \gamma ( \hat { x } , \hat { x } ) ( \hat { x } - \bar { y } ) | ) \big | \det F _ \gamma ( \hat { x } ) \big | \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - z ^ n } \right ) ^ { \frac { m ^ 1 } { n ^ 2 } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { n } { 2 } + \frac { n ^ 2 } { 2 } \right ) \frac { z ^ n } { n ^ 2 } \right \} \end{align*}


\begin{align*} \mathcal { P } _ { \textrm { o u t } } = 1 - \sum ^ M _ { m = 1 } \mathcal { V } ^ { ( M , m ) } \xi ^ m \mathcal { P } _ { \textrm { o u t } } ^ { ( m ) } . \end{align*}


\begin{align*} \frac { \partial x } { \partial r } = O ( t ) \qquad \qquad \frac { \partial y } { \partial r } = O ( t ) . \end{align*}


\begin{align*} J _ { 5 1 } = & s \lambda ^ 3 \iint _ Q \xi u A \nabla u \cdot \nabla \eta \left ( A \nabla \eta \cdot \nabla \eta \right ) d x d y d t \\ \geq & - C s ^ 2 \lambda ^ 4 \iint _ Q \xi \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d y d t - C \lambda ^ 2 \iint _ Q \xi | A \nabla u \cdot \nabla \eta | ^ 2 d x d y d t , \end{align*}


\begin{align*} H ( A _ 1 . . . A _ k - B _ 1 . . . B _ k ) \leq H ( A _ 1 ) + \sum _ { j = 3 } ^ k ( 2 j - 3 ) \log L + k \log K . \end{align*}


\begin{align*} \| y \| = \| x _ 1 \cdot x _ 1 ^ { - 1 } \cdot y \| \geq \big ( c \| x _ 1 \| - \| x _ 1 ^ { - 1 } y \| ^ { \gamma } \big ) ^ { 1 / \gamma } \geq \big ( c - \epsilon ) ^ { 1 / \gamma } > \theta , \end{align*}


\begin{align*} \nu ( f \circ T . g ) = \nu ( f . K g ) , \ ; \textrm { f o r a n y } f , g \in \mathbb L ^ 2 ( \nu ) . \end{align*}


\begin{align*} \mathcal U _ \infty ^ { p - 1 - \alpha } = \mathcal V _ \infty ^ { k _ 1 } . \end{align*}


\begin{align*} F ( p ) : = \begin{cases} \max _ { v } & \sum _ { i = 1 } ^ n c p _ i \mathbb { E } _ { \xi } \left [ \min ( x _ i + y _ i , d _ i ) \right ] - \sum _ { i \neq j } \alpha _ { i j } v _ { i j } \\ s . t \quad & \begin{cases} v \geq 0 \\ \sum _ { k \neq j } v _ { j k } \leq x _ { 0 , j } , \forall j \in I \\ \end{cases} \end{cases} , \end{align*}


\begin{align*} \cal R : = { \rm I } _ { N _ { S } } - 2 \tilde { \cal A } _ { - 1 } \cal A _ { D , N } \ , , \end{align*}


\begin{align*} \lambda _ s ( b ) \left ( \sum _ { t \in G } b _ t \delta _ t \right ) = \sum _ { t \in G } b b _ t \delta _ { s t } . \end{align*}


\begin{align*} P ^ \omega ( F _ { j , t } ^ c ) = P ^ \omega ( M _ { j h ( t ) } > k j h ( t ) ) & \leq P ( M _ { j h ( t ) } > k j h ( t ) ) \\ & \leq E [ N _ { j h ( t ) } ] \ : \mathbf { P } _ 0 \bigg ( \sup _ { 0 \le s \le j h ( t ) } | X _ s | > k j h ( t ) \bigg ) \\ & = \exp [ j h ( t ) ( \beta - k ^ 2 / 2 ) ( 1 + o ( 1 ) ) ] . \end{align*}


\begin{align*} u ^ i = g ^ { i j } \frac { \partial u } { \partial x ^ j } , \end{align*}


\begin{align*} F _ 1 = \lambda _ 0 G _ 1 + \lambda _ 1 x _ 1 G _ 2 + \cdots + \lambda _ n x _ n G _ n \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \left \{ \langle \lambda , \delta \rangle + \sup _ { \pi \in \Pi ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ { \mathcal { S } _ 1 \times \mathcal { S } _ 2 } ( f _ \mathcal { S } ) _ \lambda \ , d \pi \right \} , \end{align*}


\begin{align*} p \le s \le t \le q , \ p + q = s + t \Rightarrow f ( s ) + f ( t ) \le f ( p ) + f ( q ) . \end{align*}


\begin{align*} | A \tilde { Z } N \tilde { Z } ^ * A | & = \left | A \tilde { Z } | N | ^ { 1 / 2 } U | N | ^ { 1 / 2 } \tilde { Z } ^ * A \right | \\ & \prec _ { \log } A \tilde { Z } | N | \tilde { Z } ^ * A \end{align*}


\begin{align*} \mathcal { E } _ P ( f , g ) = \langle f , ( I - P ) g \rangle _ { \mu } = \frac { 1 } { 2 } \sum _ { x , y \in \Omega } \mu ( x ) P ( x , y ) ( f ( x ) - f ( y ) ) ( g ( x ) - g ( y ) ) , \end{align*}


\begin{align*} \theta _ s ^ M \circ \pi _ { \varphi } = \pi _ { \varphi } , \theta _ s ^ M ( \lambda ^ { \varphi } ( t ) ) = e ^ { - i t s } \lambda ^ { \varphi } ( t ) \end{align*}


\begin{align*} 2 ^ k k ! \ , S _ B ( n , k ) = \sum _ { \ell = 0 } ^ k B _ { n , \ell } \binom { n - \ell } { k - \ell } \end{align*}


\begin{align*} \left ( \overline { \nabla } _ { X } F \right ) Y = 0 , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } ( | \nabla \tilde { u } _ n | ^ 2 _ 2 + | \nabla \tilde { v } _ n | ^ 2 _ 2 ) = 0 , \ \ \lim _ { n \rightarrow \infty } ( | \nabla \tilde { u } _ n | ^ 2 _ 2 + | \nabla \tilde { v } _ n | ^ 2 _ 2 ) \geq S ^ { \frac { 2 ^ * _ \mu } { 2 ^ * _ \mu - 1 } } _ { H , L } . \end{align*}


\begin{align*} \Rightarrow \ | \langle h ^ { = d } , g \rangle | \leq e \ , \mu _ { p } ( h ) \mu _ { p } ( g ) \left ( e ^ 6 \ , r \ , \sqrt { q } \cdot \sqrt { A } \right ) ^ { d } . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } M ( t ) = \nu Y \int _ 0 ^ \infty e ^ { - \lambda s } ( 1 - p _ 0 ( s ) ) d s , \end{align*}


\begin{align*} X _ { \mathbf { K } } ( \mathbb { C } ) = \mathbf { G } ( \mathbb { Q } ) \backslash \left ( \mathbf { X } \times \mathbf { G } ( \mathbb { A } _ f ) / \mathbf { K } \right ) \end{align*}


\begin{align*} \cos \alpha _ { 2 0 3 } = \cos \alpha _ { 1 0 4 } , \end{align*}


\begin{align*} \Omega ^ { - 1 } ( \epsilon ( \sigma ) ) C \in \mathrm { C e n t } _ { \mathcal { G } ( \mathbb { Q } _ p ) } ( \mathcal { G } ^ 1 ( \mathbb { Q } _ p ) ) = Z ( \mathcal { G } ( \mathbb { Q } _ p ) ) . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { \lfloor 8 t ^ 2 \rfloor } \exp \left ( j - \frac { j ^ 2 } { 4 t ^ 2 } \right ) \leq \sum _ { j = 0 } ^ { \lfloor 8 t ^ 2 \rfloor } \exp \left ( j \right ) \leq 8 t ^ 2 \exp ( 8 t ^ 2 ) \leq L \exp ( L ) \end{align*}


\begin{align*} \psi ^ { ( n ) } ( x ) = \frac { d ^ { n + 1 } } { d x ^ { n + 1 } } \big \{ \log \Gamma ( x ) \big \} = \frac { d ^ { n } } { d x ^ { n } } \psi ( x ) . \end{align*}


\begin{align*} = ( c _ 0 + 1 ) T ( r , M ) + T ( r , h ) + S ( r , Y ) . \end{align*}


\begin{align*} ( \gamma \ast \mu ) ( f ) = \mu ( f \mid \gamma ) \mid \gamma ^ { - 1 } . \end{align*}


\begin{align*} \langle f _ b , \varphi \rangle : = \lim _ { y \searrow 0 } \int _ { - \infty } ^ { \infty } f ( x + i y ) \ , \varphi ( x ) \ , d x \end{align*}


\begin{align*} M _ X ( s ) = \frac { 1 } { \sqrt { 1 - 2 s \sigma ^ 2 } } \exp \left ( \frac { \mu ^ 2 s } { 1 - 2 s \sigma ^ 2 } \right ) . \end{align*}


\begin{align*} q = 0 \Gamma _ 1 \times ( 0 , T ) . \end{align*}


\begin{align*} \mathrm { o s c } ( v , T ^ N ( I _ \alpha ^ { ( k ) } ) ) \leq \mathrm { o s c } ( v , I _ \alpha ^ { ( k ) } ) + \int _ { I _ \alpha ^ { ( k ) } } \Big | \sum _ { i = 0 } ^ { N - 1 } D \varphi ( T ^ i x ) \Big | \ , d x . \end{align*}


\begin{align*} \begin{aligned} | K _ { \alpha } ( x , y _ { 1 } , \dots , y _ { j } , \dots , y _ { m } ) & - K _ { \alpha } ( x , y _ { 1 } , \dots , y _ { j } ' , \dots , y _ { m } ) | \\ & \le \dfrac { A } { \Big ( \sum \limits _ { j = 1 } ^ { m } | x - y _ { j } | \Big ) ^ { m n - \alpha } } \omega \Big ( \frac { | y _ { j } - y _ { j } ' | } { | x - y _ { 1 } | + \cdots + | x - y _ { m } | } \Big ) \end{aligned} \end{align*}


\begin{align*} \left ( \frac { m - 2 } { m } + w \right ) \nabla S ^ \varphi = 0 \ , M \ , . \end{align*}


\begin{align*} G _ 1 ^ { \mathrm { r e g } } ( \xi , \sigma , \alpha , \beta ) = H _ 1 ( \xi , \sigma , \alpha - \tau \beta , \beta ) \ , + \ , \phi _ 1 ( \sigma , \alpha , \beta ) \ , . \end{align*}


\begin{align*} \partial _ t \left ( \begin{array} { c } \tilde { v } _ r \\ \tilde { v } _ i \end{array} \right ) = \mathcal { A } [ \phi ] \left ( \begin{array} { c } \tilde { v } _ r \\ \tilde { v } _ i \end{array} \right ) + \widetilde { \mathcal { N } } [ \phi ] ( \tilde { v } ) , \end{align*}


\begin{align*} \Gamma ' _ { s , t } : = R ( \Sigma ' _ s , \phi ( s , t ) ) \cup R ( \Sigma ' _ t , \phi ( s , t ) ) \cup N ( s , t , \phi ( s , t ) ) . \end{align*}


\begin{align*} y ^ { \prime } \left ( x \right ) = f \left ( x , y \left ( x \right ) \right ) + \int \limits _ { x _ { 0 } } ^ { x } K \left ( x , y \left ( t \right ) , t \right ) d t \end{align*}


\begin{align*} 2 \omega Q ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) = ( 4 - d ) \omega ^ { 2 - \frac { d } { 2 } } \mu _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } . \end{align*}


\begin{align*} Q _ { x _ i } + Q _ { x _ i } ^ T = E ^ T P _ { \partial \Omega } N _ { i } E , \end{align*}


\begin{align*} i _ * I C _ N [ - 1 ] = i _ * i ^ ! p _ N ^ * ( I C _ N [ 1 ] ) \to p _ N ^ * ( I C _ N [ 1 ] ) \to R j _ * j ^ * p _ N ^ * ( I C _ N [ 1 ] ) = R j _ * I C _ { N _ \eta } [ 1 ] \rightsquigarrow \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta _ 1 , 0 ) = \sup _ { \pi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \inf _ { \lambda _ 1 \in \mathbb { R } _ { + } } \left [ \lambda _ 1 \delta _ 1 + \int _ { \mathcal { V } } g _ { \lambda , 1 } \ , d \pi \right ] . \end{align*}


\begin{align*} \mathcal U ( r ) = \frac { u ( r ) } { u _ 0 ( r ) } , \mathcal V ( r ) = \frac { v ( r ) } { v _ 0 ( r ) } , \mathcal W ( r ) = \frac { u ' ( r ) } { u ' _ 0 ( r ) } , \mathcal Y ( r ) = \frac { v ' ( r ) } { v ' _ 0 ( r ) } \end{align*}


\begin{align*} \mathcal { L } _ M ( \varepsilon ) = \sum \limits _ { k = 0 } ^ { - \lfloor \alpha \rfloor } \varepsilon ^ { \alpha + k - 1 } \ , \mathfrak { b } _ { \alpha + k - 1 } + \sum \limits _ { k = 1 } ^ { M + \lfloor \alpha \rfloor } \varepsilon ^ { k - 1 } \ , \mathfrak { b } _ { k - 1 } + \sum \limits _ { k = - \lfloor \alpha \rfloor + 1 } ^ { M } \varepsilon ^ { \alpha + k - 1 } \ , \mathfrak { b } _ { \alpha + k - 1 } , \end{align*}


\begin{align*} I _ g \leq C a ^ d \log b / a , I _ \gamma \leq C \zeta ^ { d / 2 } = C ( 1 + \abs { \log z } ) ^ { d / 2 } , I _ { | x | ^ n g } \leq C a ^ d b ^ n . \end{align*}


\begin{align*} \sum _ { n \ne 0 } \widehat { \Delta } ( a n ) ^ { 2 } \le \sum _ { n \in \mathbb { Z } } \widehat { \Delta } ( a n ) - 1 = \frac { 1 } { a } \sum _ { m \in \mathbb { Z } } \Delta \left ( \frac { m } { a } \right ) - 1 = \frac { 1 } { a } - 1 < \frac { 1 } { a } . \end{align*}


\begin{align*} \left ( \sum _ { j \in M ^ c } | a _ j | ^ 2 \right ) ^ \frac { 1 } { 2 } \leq \varepsilon \left ( \sum _ { j = 1 } ^ d | a _ j | ^ 2 \right ) ^ \frac { 1 } { 2 } . \end{align*}


\begin{align*} \det ( a _ { i j } ) = \sum _ { \pi \in S _ n } \operatorname { W e i g h t } ( \pi ) \ ; . \end{align*}


\begin{align*} \overline { A } = \{ p \in \beta S : A \in p \} . \end{align*}


\begin{align*} J _ 1 = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix} , J _ n = \mathrm { d i a g } ( J _ 1 , \cdots , J _ 1 ) \in \mathrm { G L } _ { 2 n } ( \mathbb { Z } ) . \end{align*}


\begin{align*} \omega _ 0 ( t ) = Z _ 0 ( t ) - Y e ^ { \lambda t } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \left [ u ' ( r ) ^ { p - 1 - \alpha } \right ] ' + \frac { \delta } { r } \ , u ' ( r ) ^ { p - 1 - \alpha } = \frac { \delta } { n - 1 } \ , f _ 1 ( r ) g _ 1 ( v ( r ) ) & & \\ & \left [ v ' ( r ) ^ { p - 1 } \right ] ' + \frac { n - 1 } { r } \ , v ' ( r ) ^ { p - 1 } = f _ 2 ( r ) g _ 2 ( v ( r ) ) \cdot g _ 3 ( u ' ( r ) ) & & \end{aligned} \right . \end{align*}


\begin{align*} J ( \phi , \omega , r ; t ) = C ( 1 + O ( \varepsilon ) ) | t | ^ 5 , \forall \ , t \in [ - \rho , \rho ] . \end{align*}


\begin{align*} 0 < \mu _ { 1 , \mathbf { c } _ 0 } = E ( \Phi ) + Q ( \Phi ) + \mathbf { c } _ 0 \cdot \mathbf { P } ( \Phi ) = Q ( \Phi ) - E ( \Phi ) \end{align*}


\begin{align*} \min \limits _ { x \in D _ { I } } \to f ( x ) = \sum \limits ^ { m } _ { i = 1 } f _ { i } ( x _ { i } ) , \end{align*}


\begin{align*} \max _ { A \in \mathcal { P } _ { x , n } } \frac { q _ { x , n , 2 } ^ { t } ( A ) } { q _ { x , n , 1 } ^ { t } ( A ) } & \le \frac { 1 } { \left ( 1 - \varepsilon _ { x , n , 1 } ( t ) \right ) \left ( 1 - \varepsilon _ { x , n , 2 } ( t ) \right ) } \max _ { A \in \mathcal { P } _ { x , n } } \frac { \beta _ { x , 2 } ( A ) } { \beta _ { x , 1 } ( A ) } \\ & \le \frac { C } { \left ( 1 - \varepsilon _ { x , n , 1 } ( t ) \right ) \left ( 1 - \varepsilon _ { x , n , 2 } ( t ) \right ) } = : C _ { n , t } . \end{align*}


\begin{align*} \begin{aligned} \theta ( u ^ k ) + I _ K ( \zeta ^ k ) & \leq \theta ( \hat { u } ) + I _ K ( \hat { \zeta } ) - [ \langle \lambda ^ k , \hat { \zeta } - \zeta ^ k \rangle + \langle D _ u \theta ( u ^ k ) , \hat { u } - u ^ k \rangle _ \mathcal { U } ] \\ & \leq \theta ( \hat { u } ) + I _ K ( \hat { \zeta } ) + \beta \sum \limits _ { i = n } ^ { k } \| r ^ { i } \| ^ 2 + | \langle \lambda ^ n , r ^ k \rangle | . \end{aligned} \end{align*}


\begin{align*} \left ( \mathcal { F } _ { \mathcal { H } _ { \hbar , V } } ^ { - 1 } g \right ) ( k ) : = \sum \limits _ { \xi \in \mathcal { I } _ { \hbar } } g ( \xi ) u _ { \xi } ( k ) , g \in \mathcal { S } ( \mathcal { I } _ { \hbar } ) , \end{align*}


\begin{align*} H \circ \varphi _ L = \varphi _ { K } \circ H . \end{align*}


\begin{align*} \partial _ t \big ( x ^ 2 w ( t , x ) \big ) = \partial _ x \ , \Big ( \frac { 1 } { 2 } \partial _ x \big ( x ^ 2 w ( t , x ) \big ) - \frac { 1 } { x } \big ( x ^ 2 w ( t , x ) \big ) \ ! \Big ) = \Big ( \frac { 1 } { 2 } \partial _ { x x } + \frac { 1 } { x } \partial _ x \Big ) ^ * \big ( x ^ 2 w ( t , x ) \big ) , \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } ( g _ 1 ) \overline { \Pi } _ { \psi } ( g _ 2 ) & = \overline { \Pi } _ { \psi } ( g _ 1 g _ 2 ) \overline { C } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) . \end{align*}


\begin{align*} \widehat { \vect u } = \sum _ { i = 1 } ^ 3 \alpha _ i \widehat { \Pi } _ \chi ^ { \rm s t i f f } \psi _ i ^ \chi . \end{align*}


\begin{align*} \frac { 1 - q t | } { | 1 - q } = ( 1 - q t ) \times \frac 1 { 1 - q } \end{align*}


\begin{align*} [ [ x _ \mu y ] _ { \mu + \gamma } \varphi _ { \lambda } ( w , v ) ] = [ ( \varphi _ \lambda ( x , y ) ) _ { \lambda + \mu } [ w _ \gamma v ] ] . \end{align*}


\begin{align*} \psi _ \mu ( x ) = \int g ( x - y ) \ , d \mu ( y ) \ , , \end{align*}


\begin{align*} \alpha _ { x , g } = ( \zeta _ { H } ) _ { \ast } \left ( r ^ { - 1 } \left ( \bar { p } _ { k } \right ) _ { \ast } \left ( \tau ( y ) ^ { - 1 } ( g \nu _ { P } ) ^ { y } \right ) \right ) , \ \mbox { w h e r e } x = \left ( y , r , H \right ) . \end{align*}


\begin{align*} \abs { \frac { 1 } { n } \sum _ { j = 0 } ^ { n - 1 } U ^ j ( M _ 1 ( \mu ) ) ( x ) } \leq \frac { \epsilon } { \chi _ d } \rho ( x ) x \in ( e M e ) _ + n \in \{ 1 , \ldots , ( N + 1 ) _ d \} . \end{align*}


\begin{align*} \Tilde { \theta } ( u ^ - u ^ 0 ) : = \theta ^ - ( u ^ - ) \theta ^ 0 ( u ^ 0 ) \Delta ( u ^ 0 ) ^ { - 1 } . \end{align*}


\begin{align*} \partial _ t v = - h ^ { - 1 } e ^ { h ^ { - 1 } \int _ 0 ^ t q _ 1 ( \varphi ^ s ( x ) ) d s } W w ( t , \varphi ^ t ( x ) ) . \end{align*}


\begin{align*} \theta _ { 3 / 2 } ( p _ g , \epsilon ) & = \theta _ { L , X ^ { \ast } } \Big ( J ( p _ g , i ) ^ 3 \overline { \Pi } _ { \psi } ( p _ g ) B \Big ) [ \epsilon , 0 ] \\ & = \sum _ { n \in \Z } J ( p _ g , i ) ^ 3 \overline { \Pi } _ { \psi } ( p _ g ) B ( \epsilon , n ) \\ & = \sum _ { n \in \Z } n e ^ { i \epsilon \pi n ^ 2 z _ g } . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } S _ j ( t ) = \nu Y \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s , j \geq 1 , \end{align*}


\begin{align*} u _ \mathrm { l i n } ( t , r ) = \begin{cases*} 2 & i f $ 0 < r < 1 - t $ , \\ \frac { 1 + r - t } { r } & i f $ 1 - t \leq r < t $ , \\ \frac { 1 - r + t } { r } & i f $ t \leq r < 1 + t $ , \\ 0 & o t h e r w i s e . \end{cases*} \end{align*}


\begin{align*} \mathbf { S } _ { 6 k + i + j + ( i > 0 ) } : = \xi _ k \left ( \mathbf { e } _ i \otimes \mathbf { e } _ j + \mathbf { e } _ j \otimes \mathbf { e } _ i \right ) , \end{align*}


\begin{align*} c _ 0 \cdot \Delta _ G ( f ) + c _ 1 \cdot \Delta _ G ( f ) ^ p + \cdots + c _ N \cdot \Delta _ G ( f ) ^ { p ^ N } = 0 \ , , \end{align*}


\begin{align*} \frac { \partial z } { \partial \omega } ( \phi , \omega , r ; t ) = ( 1 + O ( \varepsilon ) ) ( 1 + \varepsilon ) ^ { n } \frac { r ^ 2 t ^ 3 } { 6 } , \qquad \forall \ , t \in ( 0 , \rho ] , \end{align*}


\begin{align*} \mathcal { P } _ p ( \mathcal { X } ) = \left \{ \mu \in \mathcal { P } ( \mathcal { X } ) : \int _ { \mathcal { X } } \boldsymbol { d } ( x _ 0 , x ) ^ p d \mu ( x ) < \infty \right \} , \end{align*}


\begin{align*} \mbox { $ \mathcal { S } _ + = \left \{ A ^ + \in \mathcal { S } : \ A \in \binom { [ n ] } { k } \right \} $ a n d $ \mathcal { S } _ - = \left \{ A ^ - \in \mathcal { S } : \ A \in \binom { [ n ] } { k } \right \} $ . } \end{align*}


\begin{align*} \phi _ n ( x _ n ^ * x _ n ) = \psi _ n ( x _ n ^ * x _ n ) = 1 \end{align*}


\begin{align*} \begin{cases} \mathrm { t r } \big [ \big ( h _ \varphi ^ { ( 1 - \alpha ) / 2 z } h _ \psi ^ { \alpha / z } h _ \varphi ^ { ( 1 - \alpha ) / 2 z } \big ) ^ z \big ] & ( ) , \\ \Vert x \Vert _ z ^ z & ( ) , \\ + \infty & ( ) , \end{cases} \end{align*}


\begin{align*} \left ( \frac { \pi } { \sqrt { c } } \right ) ^ { - s } \Gamma ( s ) \ , \zeta _ { \infty , p ^ { \prime } } \left ( s , c \right ) = \left ( \frac { \pi } { \sqrt { c } } \right ) ^ { - ( 1 - s ) } \Gamma \left ( 1 - s \right ) \ , \tilde { \zeta } _ { p ^ { \prime } , \infty } \left ( 1 - s , c \right ) . \end{align*}


\begin{align*} \exp ( y f ) . x = \frac { x } { 1 + x y } . \end{align*}


\begin{align*} M _ t : = \inf \{ r \geq 0 : \mathcal { R } ( t ) \subseteq B ( 0 , r ) \} \end{align*}


\begin{align*} \sum _ i d ( k ) _ i ^ 2 + d ( k ) _ i = 2 k . \end{align*}


\begin{align*} \phi ( X E _ { i i } X ^ * \otimes B ) = W _ X ( E _ { i i } \otimes \varphi _ { i , X } ( B ) ) { W } _ X ^ * \hbox { f o r a l l \quad } B \in M _ n , \end{align*}


\begin{align*} | c _ { i _ 1 , \ldots , i _ m } | < s _ { i _ 1 + \cdots + i _ m } : = \frac { 1 } { \binom { i _ 1 + \cdots + i _ m - 1 } { m - 1 } ( i _ 1 + \cdots + i _ m ) ! } , \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \left \{ \langle \lambda , \delta \rangle + \sup _ { \pi \in \widetilde { \mathit { \Pi } } } \int _ { \mathbb { R } ^ { 2 d + 2 } } \left ( f _ { \mathcal { S } } \right ) _ \lambda \ , d \pi \right \} \end{align*}


\begin{align*} ( J \star T ) \times \Delta [ 1 ] & = \big ( ( J \times \{ 0 \} ) \star ( T \times \Delta [ 1 ] ) \big ) \sqcup _ { ( J \times \{ 0 \} ) \star ( T \times \{ 1 \} ) } \big ( ( J \times \Delta [ 1 ] ) \star ( T \times \{ 1 \} ) \big ) \end{align*}


\begin{align*} N ( P ) = c \widehat { P } ^ { n - 5 } + O \left ( \widehat { P } ^ { n - 5 - \delta ' } \right ) \end{align*}


\begin{align*} \left | \sum _ { i = 1 } ^ n B ^ * _ i A _ i \right | \le \left ( \sum _ { i = 1 } ^ n A ^ * _ i A _ i \right ) \# V ^ * \left ( \sum _ { i = 1 } ^ n B _ i ^ * B _ i \right ) V , \end{align*}


\begin{align*} ( x - a ) ^ N \prod _ { k = 1 } ^ { \lfloor { T \over 2 } \rfloor } ( f ( x ) - v _ { 2 k - 1 } ^ 2 ( x ) ) \sim \prod _ { k = 0 } ^ { \lfloor { T \over 2 } \rfloor } ( f ( x ) - v _ { 2 k } ^ 2 ( x ) ) \end{align*}


\begin{align*} d ( \overline { v } _ \lambda - \overline { v } _ \mu ) + A ( \overline { v } _ \lambda - \overline { v } _ \mu ) \ , d t = ( F _ \lambda \overline { v } _ \lambda - F _ \mu \overline { v } _ \mu ) \ , d t + \sigma ' ( u ) ( \overline { v } _ \lambda - \overline { v } _ \mu ) B \ , d W \end{align*}


\begin{align*} C _ { 2 } ( \varepsilon ) ^ { p } ( \sqrt { n p } ) ^ { p } + \sum _ { k = 1 } ^ { \infty } C _ { 2 } ( \varepsilon ) ^ { p } ( \sqrt { n p } ) ^ { p } ( 2 ^ { k } ) ^ { p / 2 } \exp ( - c 2 ^ { \frac { 2 k } { 4 + \varepsilon } } ) \le C _ { 3 } ( \varepsilon ) ^ { p } ( \sqrt { n p } ) ^ { p } . \end{align*}


\begin{align*} \int _ \Omega f \ , d x = 0 . \end{align*}


\begin{align*} ( \underbar { x } , \underbar { y } ' ) : = ( x _ 1 , \ldots , x _ N , y _ 1 , \ldots , y _ m + c ' , \dots , y _ N ) . \end{align*}


\begin{align*} \Delta _ \delta ( \tau ) : = \prod _ { D \mid N } \Delta ( D \tau ) ^ { n _ D } \end{align*}


\begin{align*} \lim _ { t \to 0 } \frac { 1 } { t } \left ( \int _ { B _ R } | \nabla g ^ t | ^ 2 \d x - \int _ { B _ R } | \nabla g ^ 0 | ^ 2 \d x \right ) = 0 . \end{align*}


\begin{align*} Q _ { \Sigma } [ \phi ] = - \int _ { \Sigma } \phi L _ { \Sigma } \phi + \int _ { \partial \Sigma } \phi ( \nabla _ n \phi - h ( \nu , \nu ) \phi ) , \end{align*}


\begin{align*} & [ u , v , w ] _ T = \rho ( T u , T v ) w , \end{align*}


\begin{align*} \rho \phi _ t + \Delta \phi - ( - \Delta ) ^ s \phi = 0 \textrm { i n } \ ; \ ; S _ T \ , , \end{align*}


\begin{align*} \sigma \cdot [ E , \lambda ] = [ E _ { \sigma ^ { - 1 } } , \lambda _ { \sigma ^ { - 1 } } ] \end{align*}


\begin{align*} b _ { j + 1 } - b _ j & = ( x - y ) ^ 2 \cdot ( z - w ) \cdot z ^ j \cdot w ^ { n - 2 - j } \\ & = ( x - y ) \cdot ( z - w ) \cdot x \cdot z ^ j \cdot w ^ { n - 2 - j } - ( x - y ) \cdot ( z - w ) \cdot y \cdot z ^ j \cdot w ^ { n - 2 - j } . \end{align*}


\begin{align*} c ^ { \wedge \sigma / \tau } = \bigwedge _ { i \in \tau } \left ( \frac { c ^ { \sigma } _ i } { c ^ { \tau } _ i } \right ) : \bigwedge _ { i \in \tau } X _ i \to \bigwedge _ { i \in \tau } X _ i . \end{align*}


\begin{align*} \widehat { ( - \Delta ) ^ \gamma u } = | \xi | ^ { 2 \gamma } \hat { u } , \hat { u } ( \xi ) = \int _ { \mathbb { R } ^ n } e ^ { - i \xi \cdot x } u ( x ) \ , d x . \end{align*}


\begin{align*} \left | \frac { N ( y ) } { N ( 1 ) } \right | = \left | \frac { c _ 5 } { b _ 3 } \right | , \left | \frac { N ( 1 ) } { N ( x ) } \right | = \left | \frac { \widetilde { c _ 5 } } { \widetilde { b _ 3 } } \right | , \left | \frac { N ( x ) } { N ( y ) } \right | = \left | \frac { \widehat { c _ 5 } } { \widehat { b _ 3 } } \right | . \end{align*}


\begin{align*} \cal A _ { D , N } = \frac 1 2 \tilde { \cal A } ( { \rm I } _ { N _ { S } } - \cal R ) \ , , \end{align*}


\begin{align*} f ' ( 0 ) = \mathbf { c } \cdot \mathbf { P } ( U ) \ge 0 , \ \ f ' ( 1 ) = I _ { \omega , \mathbf { c } } ( U ) < 0 \end{align*}


\begin{align*} \liminf _ { r \to \infty } \frac { 1 } { r ^ 2 } \int _ { B _ r } { \bar v } ^ 2 \le \liminf _ { r \to \infty } \frac { | \Omega \cap B _ r | } { r ^ 2 } = 0 . \end{align*}


\begin{align*} \mp _ s = \begin{bmatrix} \mp _ { s , 1 } & \mp _ { s , 2 } & \cdots & \mp _ { s , 2 ^ { t - s } } \end{bmatrix} . \end{align*}


\begin{align*} y _ { 1 , m } : = \left ( \frac { 1 } { 1 - r _ m } \right ) \left [ \frac { 1 + r _ m } { 2 } \sqrt { 1 - \left ( \frac { 2 | x ' | ( 1 - r _ m ) } { 1 + r _ m } \right ) ^ 2 } - 1 \right ] . \end{align*}


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


\begin{align*} \mathcal { O } _ i = \left \{ ( A ^ + , B ^ + ) \in \binom { [ n ] } { k } ^ \pm \times \binom { [ n ] } { k } ^ \pm : \ | A \cap B | = k - i \right \} , \end{align*}


\begin{align*} u : \R ^ k \times B \to G & & u ( t _ 1 , \ldots , t _ k , b ) = e ^ { t _ 1 X _ 1 } \cdots e ^ { t _ k X _ k } b \end{align*}


\begin{align*} t ^ { n } = \sum _ { k = 0 } ^ { n } S _ r [ n , k ] ( t ) _ { k , q } ^ r , \end{align*}


\begin{align*} g ( u , v ) = u ^ N \phi ( v / u ) , \end{align*}


\begin{align*} E _ N ( \mathcal { A } ) = \# \{ ( i , j , k , l ) \in [ 1 , N ] ^ 4 : a _ i + a _ j = a _ k + a _ l \} ; \end{align*}


\begin{align*} \begin{array} { l l } & \left \Vert u \right \Vert _ { \alpha , p } \\ & \cap _ { i = 1 } ^ { m } \left ( S _ { i } + u _ { i } \right ) \neq \emptyset , \\ & u = \left ( u _ { 1 } , . . . , u _ { m } \right ) \in X ^ { m } , \end{array} \end{align*}


\begin{align*} \ < X _ { T _ 1 \cdots T _ r } , X _ T \ > _ \triangleright = \sum _ { T ' } X _ { T ' } \end{align*}


\begin{align*} t _ { j } ( x ) = \dfrac { \tanh ( \alpha x _ { j } ) - \tanh ( \alpha x ) } { \tanh ( \alpha x _ { j } ) - \tanh ( \alpha x _ { j - 1 } ) } y _ { j - 1 } + \dfrac { \tanh ( \alpha x ) - \tanh ( \alpha x _ { j - 1 } ) } { \tanh ( \alpha x _ { j } ) - \tanh ( \alpha x _ { j - 1 } ) } y _ { j } . \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } ( [ h ( - 1 ) , i ] ) B ( [ \epsilon , x ] ) & = i e ^ { \tfrac { \pi i } { 4 } [ 1 + 1 ] } ( - 1 , \epsilon ) _ \R B ( [ \epsilon , - x ] ) \\ & = ( - 1 , - \epsilon ) _ \R B ( [ \epsilon , - x ] ) \\ & = ( - 1 , \epsilon ) _ \R B ( [ \epsilon , x ] ) . \end{align*}


\begin{align*} z _ \ell ' = ( \lambda _ \ell Q _ { \ell , Y Y } ) ^ { - 1 } \left [ \frac { a _ \ell } { 2 } - \lambda _ \ell Q _ { \ell , Y X } ( x ' - x _ \ell ) \right ] . \end{align*}


\begin{align*} \mathbf r _ m = & \sum _ { n \in \mathcal N } \alpha _ n g _ n ^ { \frac { 1 } { 2 } } \mathbf H _ { n , m } \mathbf s _ n + \mathbf n _ m \\ = & \sum _ { n \in \mathcal N } \alpha _ n g _ n ^ { \frac { 1 } { 2 } } \mathbf H _ { n , m } \mathbf { F } ^ { H } \tilde { \mathbf s } _ n + \mathbf n _ m , m \in \mathcal { M } , \end{align*}


\begin{align*} S _ t = \{ N _ t \geq 1 \} , S = \bigcap _ { t \geq 0 } S _ t \end{align*}


\begin{align*} f = f _ 0 + f _ 1 q + \ldots + f _ n q ^ n . \end{align*}


\begin{align*} ( x _ 1 , x _ 2 , y _ 1 , y _ 2 , w ) = \left ( r \cos \vartheta , r \sin \vartheta , p _ r \cos \vartheta - \frac { \sin \vartheta } { r } p _ \vartheta , p _ r \sin \vartheta + \frac { \cos \vartheta } { r } p _ \vartheta , w \right ) . \end{align*}


\begin{gather*} \mathcal { H } ^ 1 ( J \cap \partial B _ { r _ x } ( x ) ) = 0 , \\ \eta r _ x \leq \mathcal { H } ^ 1 \left ( J \cap B _ { r _ x } ( x ) \right ) \leq \mathcal { H } ^ 1 \left ( J \cap B _ { 2 r _ x ( x ) } \right ) < 2 \eta r _ x . \end{gather*}


\begin{align*} \sum _ { p = 1 } ^ n u ^ { p \bar { p } } \frac { \beta _ { p \bar { p } } } { \beta } \leq \sum _ { p = 1 } ^ n u ^ { p \bar { p } } \left ( \vert A u _ p + B \bar z _ { p } \vert ^ 2 - u u _ { p \bar p } - \vert u _ p \vert ^ 2 - B \right ) \end{align*}


\begin{align*} 1 + \eta _ k = \prod _ { j = 1 } ^ k ( 1 + \xi _ j ) | \xi _ j | \le 2 ^ { - d } . \end{align*}


\begin{align*} | \sigma ( r , x _ 3 , t ) | = r | v _ \theta ( r , x _ 3 , t ) | \leq C _ 1 e ^ { - c \ln ^ \frac 1 4 ( 1 / r ) } \end{align*}


\begin{align*} H _ 0 ( x ) = \sup _ { \xi \neq 0 } \frac { \xi \cdot x } { H ( \xi ) } \end{align*}


\begin{align*} Q _ { q } ( y ) : = \begin{cases} \frac { | D f ( x ) | ^ { q } } { | J ( x , f ) | } , \ , \ , & x = f ^ { - 1 } ( y ) \in \Omega \setminus ( S \cup Z ) , \\ \ , \ , 0 , \ , \ , & x = f ^ { - 1 } ( y ) \in S \cup Z . \end{cases} \end{align*}


\begin{align*} Y _ i ( t ) = X _ i ( T _ i + t ) \end{align*}


\begin{align*} \norm { u } _ { L ^ 2 } ^ 2 = \sum _ { N \in 2 ^ { \mathbb Z ^ d } } \norm { u _ N } _ { L ^ 2 } ^ 2 \end{align*}


\begin{align*} d _ i = \begin{cases} b _ i , & \mbox { i f } i \in \bigcup _ { p = 1 } ^ { r + 1 } L ( p ) \\ l , & \mbox { i f } i \in \{ s ( 1 ) , s ( 2 ) , \cdots , s ( r ) \} . \end{cases} \end{align*}