\begin{align*} \liminf _ { k \to \infty } \frac { f ( b ^ { n _ k } x _ 0 ) } { \overline { f } ( b ^ { n _ k } x _ 0 ) } \ge \liminf _ { k \to \infty } c n _ k ( e - 1 ) ^ { - 1 } n _ k ^ { - 1 / 2 } = \infty , \end{align*}


\begin{align*} r _ { \ell } ^ - \leq e ^ { 5 0 \varepsilon q _ n } \frac { e ^ { - q _ n L } } { \max ( | \ell | , 1 ) } \max ( r _ { \ell - 1 } ^ - , r _ { \ell - 1 } ^ + , r _ { \ell } ^ + , r _ { \ell + 1 } ^ - , c _ { n , \ell + 1 } r _ { \ell + 1 } ^ + ) \times \begin{cases} \max ( | \ell | , e ^ { \delta _ n q _ n } , 1 ) , & \beta _ n \geq \delta _ n + 2 0 0 \varepsilon \\ e ^ { \beta _ n q _ n } , & \beta _ n < \delta _ n + 2 0 0 \varepsilon \end{cases} , \end{align*}


\begin{align*} Q _ { r } \ge 0 , c _ { r } = r ^ { 1 - d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n c \ , d \rho \ge 0 , r < R , t < T _ { \rm m a x } . \end{align*}


\begin{align*} \delta _ H ( k ) = 2 \phi _ H ( k + 1 ) - ( \phi _ H ( 2 k ) + \phi _ H ( 2 ) ) \end{align*}


\begin{align*} ( \alpha - \alpha ^ 2 ) ( 1 - K _ { 2 } ) ^ 2 \sum \limits _ { k = 0 } ^ { \infty } \norm { \theta ^ k - \tilde { \theta } ^ k } _ { \mathcal { L } ^ 2 , G } ^ 2 \leq \norm { \theta ^ 0 - \theta ^ * } _ { \mathcal { L } ^ 2 , G } ^ 2 . \end{align*}


\begin{align*} - \sum _ { j = 0 } ^ { p } B _ j n ^ { p - j } \sum _ { k = j } ^ p ( - 1 ) ^ k \binom { p } { k } \binom { k } { j } . \end{align*}


\begin{align*} \mathcal { U } _ { \lambda } ( \tau ) = : \Theta _ { \lambda } ( t ) + \frac { 1 } { 2 } \varrho _ { 0 } \lambda ^ { \beta - 1 } \int _ { \tau } ^ { t } \Theta _ { \lambda } ( s ) \ d s \leq \Theta _ { \lambda } ( \tau ) , 0 \leq t < \tau \leq t _ { 0 } < T _ { \mathrm { g e l } } . \end{align*}


\begin{align*} \alpha \vee \beta = \begin{cases} \{ \alpha _ { i _ 1 , j _ 2 } , \alpha _ { i _ 1 , \overline { j } _ 1 } \} & i _ 2 \geq j _ 2 \\ \{ \alpha _ { i _ 1 , j _ 2 } \} & \end{cases} \end{align*}


\begin{align*} \eta ( t ) = \begin{cases} R , t \leq \delta \\ R ( 1 - ( \frac { t - \delta } { R - \delta } ) ^ 2 ) ^ 2 , t > \delta \end{cases} \end{align*}


\begin{align*} d x _ { t , j } = - a _ j x _ { t , j } d t + B _ j ( x _ t , x _ t ) d t + \sigma _ j d W _ t ^ { ( j ) } , \end{align*}


\begin{align*} \max _ { v \in \Lambda } e ( \alpha ^ { ( v ) } , G ^ { ( v ) } ) = e ( 0 , G ) + 2 H - 3 / 2 \end{align*}


\begin{align*} \partial _ { j , ( 1 ) } = \sum _ { k = 1 } ^ d a _ { j , k } \partial _ { k , ( 2 ) } \end{align*}


\begin{align*} \lim \limits _ { j \to + \infty } \frac { G ( t _ 0 ) - G ( t _ 0 + B _ j ) } { \int _ { t _ 0 } ^ { t _ 0 + B _ j } c ( t ) e ^ { - t } d t } = \liminf \limits _ { B \to 0 + 0 } \frac { G ( t _ 0 ) - G ( t _ 0 + B ) } { \int _ { t _ 0 } ^ { t _ 0 + B } c ( t ) e ^ { - t } d t } \end{align*}


\begin{align*} A = \frac { 2 \pi } { N } / \sqrt { N } \end{align*}


\begin{align*} S _ k ^ { i j } ( T _ \alpha u ) _ { i j } = T _ \alpha f . \end{align*}


\begin{align*} \displaystyle \sum _ { i = 1 } ^ { n } d _ i ^ 2 \leq \frac { m ^ 2 \Big ( \triangle ( G ) + \delta ( G ) \Big ) ^ 2 } { n \triangle ( G ) \delta ( G ) } . \end{align*}


\begin{align*} \mathcal S ( f _ 2 \circ f _ 1 ) = \left ( \mathcal S ( f _ 2 ) \circ f _ 1 \right ) ( f _ 1 ' ) ^ 2 + \mathcal S ( f _ 1 ) \end{align*}


\begin{align*} \mathbb { E } \left [ \varepsilon \right ] = & { \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { s } ^ { \left ( \right ) } \right ] } - { \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { y } ' \right ] } - { \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { s } ^ { \left ( \right ) } \right ] } + { \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { y } ' \right ] } . \end{align*}


\begin{align*} V _ 1 ^ { - 1 } ( k ) R ( x ; z ) V _ 1 ( k ) = V _ 2 ^ { - 1 } ( k ) R ( x ; z ) V _ 2 ( k ) = J ( x ; k ) , z \in \mathbb { R } , k \in \mathbb { R } \cup i \mathbb { R } . \end{align*}


\begin{align*} \varphi _ n ( t ) : = \exp \left ( - \int _ 0 ^ t \big [ f ( r ) + \rho ( X ( r , x _ n ) ) + \eta ( X ( r , x ) ) \big ] d r \right ) . \end{align*}


\begin{align*} \nu = \min \{ 1 , \varepsilon _ * , \varepsilon _ * / \lambda _ 0 , \varepsilon _ * / \lambda ^ * , \nu _ * \} . \end{align*}


\begin{align*} \sigma = \prod _ p \left ( 1 - \frac { \lambda ( p ^ 2 ) } { p ^ 2 } \right ) \ , . \end{align*}


\begin{align*} \lim _ { i \rightarrow \infty } g _ { n _ i } ( t , \omega ) = 0 , ( t , \omega ) . \end{align*}


\begin{align*} \mathbb { E } \left [ \lvert y _ k \rvert ^ 2 \right ] = a _ c ^ 2 \lvert \mathbf { h } _ k ^ { \textrm { T } } \mathbf { p } _ c \rvert ^ 2 + \sum _ { i = 1 } ^ K a _ i ^ 2 \lvert \mathbf { h } _ k ^ { \textrm { T } } \mathbf { p } _ i \rvert ^ 2 + \sigma _ w ^ 2 . \end{align*}


\begin{align*} \phi ( \sigma _ 1 \sigma _ 2 , \tau ) = \phi ( \tau , \sigma _ 1 \sigma _ 2 ) = \phi ( \tau , \sigma _ 1 ) \phi ( \tau , \sigma _ 2 ) = \phi ( \sigma _ 1 , \tau ) \phi ( \sigma _ 2 , \tau ) . \end{align*}


\begin{align*} \underline { u } = \frac { 1 } { 4 } \left ( \sum _ { j = 1 } ^ p e _ j \cdot u ( e _ j ) + \sum _ { j = p + 1 } ^ n e _ j \cdot ( - u ^ * ( e _ j ) ) \right ) . \end{align*}


\begin{align*} \omega : = \{ x \in \Omega : | x _ \beta | < \delta \frac { b _ \alpha } { \sqrt { b _ \beta } } , \beta = \alpha + 1 , \ldots , n - 1 , x _ n < \delta ^ 2 b _ \alpha \} , \end{align*}


\begin{align*} X _ { r } = \xi _ r \quad { X } _ t = \xi _ 0 + \int _ { 0 } ^ { t } \mu ( s , X ) \ , d s + \int _ { 0 } ^ { t } \sigma ( s , X ) \ , d W _ s , \end{align*}


\begin{align*} \phi : C ^ n _ c ( G , V ) ^ G \to C ^ { n - 1 } _ c ( G , V ) , \phi ( c ) ( x _ 1 , \ldots , x _ n ) : = c ( 1 , x _ 1 , x _ 1 x _ 2 , \ldots , x _ 1 \cdots x _ n ) , n \geq 1 \end{align*}


\begin{align*} g _ { \theta _ 1 } ( \omega ) = g _ { \theta _ 2 } ( \omega ) , \omega \in \Omega _ { \theta _ 1 } \cap \Omega _ { \theta _ 2 } . \end{align*}


\begin{align*} & \int _ { J _ U ^ V } \left ( A _ { \vartheta \varphi } { } ^ { ( \vartheta , \varphi ) } \mathcal I _ { a } ^ { \vec { \alpha } } [ F ] ( W , Z ) + B _ { \vartheta \varphi } { \bf i } { } ^ { ( \vartheta , \varphi ) } \mathcal I _ { a } ^ { \vec { \alpha } } [ F ] ( W , Z ) \right ) d Z \wedge d Z ^ { * } \\ = & \int _ { \partial J _ U ^ V } { } ^ { ( \vartheta , \varphi ) } \mathcal I _ { a } ^ { \vec { \alpha } } [ F ] ( W , Z ) d \sigma _ { ( \vartheta , \varphi ) } ( Z ) , \end{align*}


\begin{align*} [ P , Q ] = P \ , Q - Q \ , P \ ; . \end{align*}


\begin{align*} \log \det u _ { , i j } = - v _ p x ^ p + u _ { , q } \xi ^ q + c . \end{align*}


\begin{align*} c _ 2 ^ 2 ( \mu ) : = \int _ J \int _ J \int _ J \frac { d \mu ( z _ 1 ) s \mu ( z _ 2 ) d \mu ( z _ 3 ) } { R ^ 2 ( z _ 1 , z _ 2 , z _ 3 ) } < \infty , \end{align*}


\begin{align*} \lim _ { N \to \infty } \sup _ { n \in \mathbb N } \mathbb E \left [ \sup _ { t \in [ 0 , T ] } \left \| P _ N ^ 2 ( I ) _ t ^ n \right \| _ { \mathcal H } \right ] = 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} \mathcal H _ X ^ D ( \alpha ) _ x & = \mathcal H _ { X , x } & \\ \mathcal H _ X ^ D ( \alpha ) _ x & = \alpha _ i \cap R ^ 0 _ { y _ i , n _ i } & \end{aligned} \right . \end{align*}


\begin{align*} \frac { N - 2 } { 2 } \int | \nabla u | ^ 2 d x - \frac { N } { 2 } \lambda \int | u | ^ 2 d x - \frac { N + \alpha } { 2 p } \int ( I _ \alpha \ast | u | ^ { p } ) | u | ^ p d x = 0 . \end{align*}


\begin{align*} \hat \Phi ^ \dagger ( a , s ) : = \langle z _ 1 ^ * , \Phi ^ \dagger ( a u _ \ell , p ( s ) ) \rangle , \end{align*}


\begin{align*} \mathcal { C } : = \{ q \in \Omega : d _ H ( q , [ 0 , \xi ] ) < R , 0 < \langle q , \xi \rangle < | \xi | ^ 2 \} \end{align*}


\begin{align*} b _ t + \nabla \times ( B \times u ) + \mu \nabla \times ( \nabla \times b ) = 0 \textrm { o n } \ ( L ^ { 2 } _ { \sigma } \cap H ^ { 1 } ) ( \mathbb { R } ^ { 3 } ) ^ { * } , \end{align*}


\begin{align*} n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } - \left ( \frac { \beta } { 3 } \left ( n + 2 \right ) \right ) ^ { 3 } + \left ( \frac { \beta } { 3 } \left ( n - 1 \right ) \right ) ^ { 3 } = 0 \end{align*}


\begin{align*} H = \frac { \alpha } { n } \frac { \nu _ { n + 1 } } { x _ { n + 1 } } + \frac { \varpi } { n } , \end{align*}


\begin{align*} V = g ( c ) H \end{align*}


\begin{align*} \prescript L { } { } \ell { } ^ R ( x , w ^ { - 1 } \alpha ) = - \Phi ^ + _ R ( w ^ { - 1 } \alpha ) + \Phi _ L ^ + ( \alpha ) = 1 - \Phi ^ + _ R ( w ^ { - 1 } \alpha ) . \end{align*}


\begin{align*} \partial _ z ^ * = M _ z , \end{align*}


\begin{align*} ( x ) _ k = \frac { \Gamma ( x + k ) } { \Gamma ( x ) } = x ( x + 1 ) \dots ( x + k - 1 ) \end{align*}


\begin{align*} W ( t ) = \sum _ { l = 1 } ^ { n _ 1 } \Psi _ { k l } \beta ^ c _ l ( t ) , k = 1 , \dots , n . \end{align*}


\begin{align*} \sigma \geq v + \sum _ { k = 1 } ^ l a _ k \hat C _ { q _ k , N } . \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 + } \sup _ { f \in \Upsilon } \int _ 0 ^ { T - \delta } \Vert f ( t + \delta ) - f ( t ) \Vert _ Y ^ p d t = 0 , \end{align*}


\begin{align*} \mathbf { U } _ { x _ w ^ { ( i ) } } - \omega \mathbf { U } _ { x _ o ^ { ( i ) } } = \begin{pmatrix} 0 & 0 \\ 0 & \tilde { \rho } _ i \mathbf { P } _ { x _ o ^ { ( i ) } } \end{pmatrix} , \end{align*}


\begin{align*} ( \mu _ T ( y , z ) | \mu _ T ( x , w ) ) = ( \mu _ T ( x , y ) | \mu _ T ( z , w ) ) . \end{align*}


\begin{align*} M _ { x , y } ( m ) = \sum _ { j , j ' = 0 } ^ n \Theta _ m ( \mu _ j , \mu _ { j ' } ) \psi _ j ( x ) \psi _ { j ' } ( x ) \psi _ j ( y ) \psi _ { j ' } ( y ) \end{align*}


\begin{align*} - \nabla \cdot \big ( \nabla \chi _ i + e _ i \big ) & = 0 & \mbox { i n } & Y ^ { \ast } , \\ - \big ( \nabla \chi _ i + e _ i \big ) \cdot \nu & = 0 & \mbox { o n } & \Gamma , \\ \chi _ i \mbox { i s } Y \mbox { - p e r i o d i c , } \int _ { Y ^ { \ast } } \chi _ i d y = 0 . \end{align*}


\begin{align*} a ( u , v ) = ( f , v ) ~ v \in H ^ 1 _ 0 ( D ) , \end{align*}


\begin{align*} f ( p , s ) = s \log a + \log A _ k - \left ( \frac { p } { 2 } + 2 \right ) \log s + \log \Gamma \left ( \frac { p } { 2 } + 2 \right ) . \end{align*}


\begin{align*} \beta _ 2 ( u ) = 0 \Longrightarrow \beta _ 2 ( u _ { \varepsilon } ) = 0 . \end{align*}


\begin{align*} D _ t ^ { ( 1 ) } u _ k = b _ { k 1 } + 2 b _ { k 2 } t + \cdots , \end{align*}


\begin{align*} T _ { ( \Sigma , c ) } M ^ m = \left \{ \big ( V , w \big ) \ , \big | \ , V , w : \Sigma \rightarrow \R \int _ \Sigma w - c H V \ , \mathrm { d } \mathcal { H } ^ d = 0 \right \} . \end{align*}


\begin{align*} { \mathcal { I } } _ h \leq \int _ { \Omega } | B | ^ { 2 } \dd x \leq \liminf _ { n \to \infty } \int _ { \Omega } | B _ n | ^ { 2 } \dd x = { \mathcal { I } } _ h , \end{align*}


\begin{align*} F _ { + } ( x ) = G ( x ) + \frac { 1 } { 2 } \left [ - L , G ( x ) \right ] + \frac { 1 } { 1 2 } \left [ - L , \left [ - L , G ( x ) \right ] \right ] + \cdots . \end{align*}


\begin{align*} \underline { a } : = \begin{pmatrix} a \\ & a \\ & & \ddots \\ & & & a \\ & & & & a \end{pmatrix} , \end{align*}


\begin{align*} N ( d ) = \frac { p + 1 } { 2 d } + \xi ( d ) , \end{align*}


\begin{align*} C ( a _ k ) = \max ( \{ C ( b _ i ) \mid b _ i < _ P a _ k \} \cup \{ C ( a _ i ) \mid 1 \le i < k \} ) + 1 . \end{align*}


\begin{align*} \kappa ( t ) = ( \sqrt { t } ) ^ { \alpha - 1 } J _ { \alpha - 1 } ( 2 \sqrt { t } ) , \ k ( t ) = ( \sqrt { t } ) ^ { - \alpha } I _ { - \alpha } ( 2 \sqrt { t } ) , \ 0 < \alpha < 1 , \end{align*}


\begin{align*} a \cos ( x ) + b \sin ( x ) = r \cos ( x + \theta ) . \end{align*}


\begin{align*} c _ k = \left \{ \begin{array} { l l } { k + 2 \choose 3 } & k \in [ 0 , N ] \\ { 2 N - k + 2 \choose 3 } & k \in [ N + 1 , 2 N ] \end{array} \right . . \end{align*}


\begin{align*} & G _ x ( \theta ) : = \sum _ { n \in \Z } g \left ( \pi _ N ( x ) ( \theta + n ) \right ) \\ & = \sum _ { | n | \leq \pi _ N ( x ) } \widehat { G _ { x } } ( n ) e ( n \theta ) \\ & = \widehat { G _ { x } } ( 0 ) + \sum _ { 1 \leq n \leq \pi _ N ( x ) } \widehat { G _ { x } } ( n ) 2 \cos ( 2 \pi n \theta ) , \end{align*}


\begin{align*} P _ { \min } ^ { { \rm { N } } } \left ( { \overline S , \overline \varepsilon , \overline R } \right ) = \bigcap { \left \{ { P \in { \mathbb { R } ^ + } : \left ( { \overline S , \overline R } \right ) \in { { \mathcal { R } } } _ { { \rm { S v B } } } ^ { { \rm { N } } } \left ( { W , P , K , \overline \varepsilon } \right ) } \right \} } . \end{align*}


\begin{align*} ( \nu S \textbf { \textit { u } } ) ^ i = \nu ^ j ( S \textbf { \textit { u } } ) ^ i _ j & = \nu ^ j ( \nabla ^ i u _ j + \nabla _ j u ^ i ) \end{align*}


\begin{align*} A \cdot B = 2 \Phi \frac { G } { r ^ { 2 } } + \nabla \cdot ( \Phi \nabla \theta \times A ) . \end{align*}


\begin{align*} { \rm V o l } ( \Sigma ) \geq C ( n ) \sum _ { i = 1 } ^ { k } { \rm V o l } ( \Sigma \cap B _ { p _ i } ( 3 r ) ) \geq \sum _ { i = 1 } ^ { k } \frac { C ( n ) } { r } { \rm V o l } ( B \cap B _ { p _ i } ( r ) ) \geq \frac { C ( n ) } { r } { \rm V o l } ( B ) . \end{align*}


\begin{align*} L \ge L _ G : = 4 \sqrt { d } \max _ { x \in [ - 2 b , 2 b ] ^ d } \| \nabla g ( x ) \| + 1 , \end{align*}


\begin{align*} \theta _ 1 & = \frac { 1 } { 2 } ( \arg \alpha _ 1 - \arg \alpha _ 2 + \arg \beta _ 1 - \arg \beta _ 2 ) , & \delta _ 1 & = u ( \theta _ 1 ) \\ \theta _ 2 & = \frac { 1 } { 2 } ( \arg \alpha _ 1 - \arg \alpha _ 2 - \arg \beta _ 1 + \arg \beta _ 2 ) , & \delta _ 2 & = u ( \theta _ 2 ) \end{align*}


\begin{align*} \forall \varepsilon > 0 , \lim _ { n \to \infty } \sum _ { m = 1 } ^ n \mathbb { E } _ { n , m } [ \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } ^ 2 : \| \chi _ { n , m } \| _ { L ^ 2 ( \mu ) } > \varepsilon ] = 0 \end{align*}


\begin{align*} \frac { \dd } { \dd t } \int _ { \mathbb { R } ^ { 3 } } \Phi _ + \frac { G } { r ^ { 2 } } \dd x = - \mu \int _ { \mathbb { R } ^ { 3 } } \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x , \end{align*}


\begin{align*} ( f _ { i , L } ) \subseteq \begin{cases} & \tilde { D } _ { N _ i , \leqslant L } , \ ; L = \max \{ \tilde { L } \in 2 ^ { \N _ 0 } : \tilde { L } \leqslant N _ 3 ^ { 5 - 2 \alpha + \varepsilon } \} , \\ & \tilde { D } _ { N _ i , L } , \ ; . \end{cases} \end{align*}


\begin{align*} \widetilde { \mu } ( \phi _ u ( A ) ) = \widetilde { \mu } ( A ) . \end{align*}


\begin{align*} F ( W ) = \frac { 1 } { 2 \pi { \bf i } } \int _ { \gamma } \frac { F ( Z ) } { Z - W } d Z + \frac { 1 } { 2 \pi { \bf i } } \int _ { \Gamma } \frac { \dfrac { \partial F } { \partial Z ^ * } } { Z - W } d Z \wedge d Z ^ * , \end{align*}


\begin{align*} \left | \bigcup _ { i \in I } F _ i \right | \le | I | e _ 0 - \underbrace { | J | | I \setminus J | } _ { \mathclap { = c _ 1 ( | I | - c _ 1 ) } } \theta _ d q ^ { d ^ 2 - d - 2 } \end{align*}


\begin{align*} \widehat { C } _ { \psi } = C _ { \widehat { \psi } } , \end{align*}


\begin{align*} \Sigma _ 2 ( { \frak y } ) = \left [ \begin{array} { c c } 0 _ { n + 1 } & 0 _ { n + 1 } \\ 0 _ { n + 1 } & D _ 2 ( { \frak y } ) \end{array} \right ] . \end{align*}


\begin{align*} d v v ^ { - 1 } = \frac { 1 } { 2 } a I + \frac { 1 } { \sqrt { \tau _ 0 } } \left ( \frac { \tau _ 0 } { 4 } d \overline { z } + Q _ 0 d z \right ) i J . \end{align*}


\begin{align*} H _ { \mathrm { C V } } ( s , i , p , q , l ) & \coloneqq ( s + i ) l w + \beta ( 1 - \theta l ) ^ 2 s i \left ( q - p \right ) + i \phi ( i ) \left ( \frac { w } { r } + \chi \right ) - ( \gamma + \phi ( i ) ) i q \\ & = \beta \theta ^ 2 s i ( q - p ) l ^ 2 + \left [ ( s + i ) w - 2 \beta \theta s i ( q - p ) \right ] l \\ & + \beta s i ( q - p ) + i \phi ( i ) \left ( \frac { w } { r } + \chi \right ) - ( \gamma + \phi ( i ) ) i q \ , , \end{align*}


\begin{align*} \| A \| _ \textrm { H S } = \sqrt { \frac { 1 } { n } \mathrm { t r } ( A ^ * A ) } \forall A \in \mathrm { U } ( n ) . \end{align*}


\begin{align*} \pi \circ \phi = \mbox { i d } _ Y , \end{align*}


\begin{align*} \tilde { \delta } ( \alpha , \theta ) = \limsup _ { n \to \infty } \frac { \ln q _ { n + 1 } + \sum _ { \theta ' : c _ { \lambda } ( \theta ' ) = 0 } \ln \| q _ n ( \theta - \theta ' ) \| } { q _ n } , \end{align*}


\begin{align*} \mathcal { L } = \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ 3 \Lambda _ { i j } \partial _ { x _ i x _ j } + x _ 1 x _ 3 \partial _ { x _ 1 } - x _ 2 x _ 3 \partial _ { x _ 2 } + ( x _ 2 ^ 2 - x _ 1 ^ 2 ) \partial _ { x _ 3 } - x _ 2 \sum _ { j = 1 } ^ 3 a _ j \partial _ { x _ j } . \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \nabla ^ \perp \varphi \cdot \nu | _ { \partial \Omega } = v _ n | _ { \partial \Omega } \ln \frac { 1 } { \varepsilon } . \end{cases} \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow \infty } \norm { \theta ^ k - \tilde { \theta } ^ k } _ { \mathcal { L } ^ 2 , G } ^ 2 = 0 . \end{align*}


\begin{align*} L ( E ) = \lim _ { \epsilon \rightarrow + \infty } L ( E , \epsilon ) = \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | . \end{align*}


\begin{align*} \| w _ n \| _ { L ^ { \infty } } = \left \| F ^ n e _ 1 \right \| _ { L ^ { \infty } } \leq \frac { 1 } { n ! } \left \| \widetilde { Q } ( u ) \right \| _ { L ^ 1 } ^ n . \end{align*}


\begin{align*} a ( x ) = \nabla \times ( \eta ( z , r ) \nabla \theta ) + \phi ( z , r ) \nabla \theta , \end{align*}


\begin{align*} \int _ { \mathfrak X ( n ) } | \omega ( n ) | = ( \mathfrak h ( n ) _ 0 ) _ ! \int _ { \mathfrak Y ( n ) } | \mathfrak h ( n ) _ { \eta } ^ * \omega ( n ) | . \end{align*}


\begin{align*} \theta _ { j , ( 1 ) } = \sum _ { k = 1 } ^ d - a _ { j , k } \theta _ { k , ( 2 ) } , \end{align*}


\begin{align*} ( p - L _ X h ) ^ \intercal & = - 2 \zeta A ' ( h ) \\ H ' ( p - L _ X h ) & = \zeta A \cdot A ' ( h ) . \end{align*}


\begin{align*} - \Delta p + \zeta p & = \overline { y } - g , \\ \langle p + \alpha \overline { u } , u - \overline { u } \rangle & \ge 0 u \in \mathcal { C } _ { a d } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m \mathcal { W } _ 2 ^ 2 ( P _ i , Q _ i ) \le \mathcal { W } _ 2 ^ 2 ( P , Q ) . \end{align*}


\begin{align*} m ( x ; z ) - \widetilde { m } ( x ; z ) = \int _ { - \infty } ^ x e ^ { 2 i z ( x - y ) } [ w ( y ) - \widetilde { w } ( y ) ] d y , \end{align*}


\begin{align*} d _ x T ( x , y ) = - \frac { d _ x \Phi ( x , y , T ( x , y ) ) } { \partial _ z \Phi ( x , y , T ( x , y ) ) } . \end{align*}


\begin{align*} \begin{cases} c _ { \alpha + n + 1 } - c _ { n + 1 } = b _ { \alpha + n } - b _ { \alpha + n + 1 } \\ d _ n = c _ { \alpha + 1 } + ( b _ { \alpha + 1 } - b _ \alpha ) = d \\ b _ { \alpha + n } = b _ \alpha + ( d _ 1 - c _ { \alpha + 1 } ) + \sum _ { j = 2 } ^ n ( c _ j - c _ { \alpha + j } ) \\ b = b _ \alpha + d - c _ { \alpha + 1 } + \sum _ { j = 2 } ^ \infty ( c _ j - c _ { \alpha + j } ) \end{cases} \end{align*}


\begin{align*} { R ^ * } = \mathop { \arg \max } \limits _ { { W _ m } \in \left [ { 0 , W } \right ] } \ ; R \left ( { p _ { b , m } ^ * , p _ { b , o } ^ * | { W _ m } , \overline S } \right ) . \end{align*}


\begin{align*} M s - A = B - M s \end{align*}


\begin{align*} & \phi _ x + i k ^ 2 \sigma _ 3 \phi = k P _ x \phi , \\ & \phi _ t + i \eta ^ 2 \sigma _ 3 \phi = H \phi , \end{align*}


\begin{align*} d | \tilde { x } _ t | ^ 2 = 2 \lambda _ A | \tilde { x _ t } | ^ 2 d t - 2 A \tilde { x } _ t \cdot \tilde { x } _ t d t + 2 e ^ { \lambda _ A t } \tilde { x } _ t \cdot \sigma d W _ t + C _ \sigma e ^ { 2 \lambda _ A t } d t . \end{align*}


\begin{align*} \check { Y } ^ { \epsilon } ( t ) & = \big ( 3 \delta ^ { 1 / 2 } + \sup _ { t \in [ 0 , 1 ] } | \varphi ( t ) - \varphi _ m ( t ) | ^ { 1 / 2 } \big ) + \int ^ { t } _ { 0 } \big ( b ( \hat { X } ^ { \epsilon } ( s ) ) - b ( X ^ { \epsilon } ( s ) ) \big ) d s \\ & \quad - \lambda ( \epsilon ) \epsilon \int ^ { t } _ { 0 } \big ( \sigma ( \hat { X } ^ { \epsilon } ( s - ) ) - \sigma ( X ^ { \epsilon } ( s - ) ) \big ) \epsilon \theta ^ { \epsilon } ( s - ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) , \end{align*}


\begin{align*} \zeta _ t = \left ( \psi _ t \right ) ^ { \otimes 2 } - \langle \langle \psi \rangle \rangle _ t , \end{align*}


\begin{align*} T = 2 n \exp ( \log 2 n / 1 0 3 . 9 5 \log \log 2 n ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { n } \sum _ { k = 0 } ^ { n - 1 } | \mu ( h ^ { - k } ( A ) \cap B ) - \mu ( A ) \mu ( B ) | = 0 ; \end{align*}


\begin{align*} \int _ { \{ z \in M : - \Psi ( z ) \in N \} } | \tilde { F } | ^ 2 _ h = 0 \end{align*}


\begin{align*} \Re \bigg \{ \left ( \frac { L ' } { L } \right ) ' ( s , \chi ) \bigg \} = \sum _ { \gamma _ \chi } g _ { a } ( t - \gamma _ \chi ) + \sum _ { \gamma } g _ { a } ( \gamma ) + O ( 1 ) , \end{align*}


\begin{align*} & \left | \sum _ { i = 1 } ^ n \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \frac { \rho } { \Delta _ n ^ { ( \mathfrak H + \epsilon ) } } \int _ { 2 - ( i \Delta _ n - s ) } ^ 2 x ^ { \mathfrak H + \epsilon } d x d s \right | \\ \leq & \rho \Delta _ n ^ { 1 - ( \mathfrak H + \epsilon ) } 2 ^ { \mathfrak H + \epsilon } . \end{align*}


\begin{align*} \limsup _ { s \rightarrow + \infty } \frac { J _ 2 ( s ) } { s } \leq - R e \left ( r e ^ { i \theta } \right ) = - \inf _ { \omega \in \Gamma ^ { \prime } } R e \left ( \omega e ^ { i \theta } \right ) . \end{align*}


\begin{align*} \mathbf { P } _ { v } = \mathbf { I } _ { n - 1 } - \mathbf { v } _ { 2 : n } ( \mathbf { v } _ { 2 : n } ) ^ T / \left \| \mathbf { v } _ { 2 : n } \right \| ^ 2 . \end{align*}


\begin{align*} X ^ { \mathfrak h } _ 1 , \ldots , X ^ { \mathfrak h } _ m , X ^ { \mathfrak n } _ { m + 1 } , \ldots , X ^ { \mathfrak n } _ k \mbox { i s a b a s i s f o r } \Delta . \end{align*}


\begin{align*} D ( j , k ) = \max ( D ( j , k - 1 ) , D ( j - 1 , k ) ) + e _ { k j } , k \geq 0 , j = 1 , \ldots , d . \end{align*}


\begin{align*} t ( v ' ) t ( w '' ) = t ( w ' ) t ( v '' ) \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ \S \int _ \S W _ r ( z - x ) W _ r ( y - x ) W _ r ( z - y ) \sin ( \Theta ( t , z ) + \Theta ( t , y ) - 2 \Theta ( t , x ) ) \ \d y \d z . \end{align*}


\begin{align*} R ^ \nabla _ { X , Y } Z = \left ( R ^ W _ { X , Y } \right ) ^ { J , + } Z & - \frac { 1 } { 2 } ( d J \theta ) ( X , Y ) J Z - \frac { 1 } { 2 } ( d \theta ) ( X , Y ) Z \\ & + \frac { 1 } { 4 } \left ( \left ( D ^ W _ X J \right ) \left ( D ^ W _ Y J \right ) - \left ( D ^ W _ Y J \right ) \left ( D ^ W _ X J \right ) \right ) Z , \end{align*}


\begin{align*} \bigcup _ { k = 1 } ^ { m } B \left ( y _ { k } , 2 r \right ) = \bigcup _ { k = 1 } ^ { m } E \left ( \mathbf { y } \right ) _ { k } \end{align*}


\begin{align*} \bigg | \frac { 1 } { k } \int _ { r } ^ { r + k } g ( t ) d t - g ( r ) \bigg | & \leq \frac { 1 } { k } \cdot \frac { \epsilon } { 2 } \cdot m ( [ r , r + k ] ) + \frac { 1 } { k } \cdot 2 M \cdot m ( [ r , r + k ] \setminus B _ r ) \\ & < \frac { 1 } { k } \cdot \frac { \epsilon } { 2 } \cdot k + \frac { 1 } { k } \cdot 2 M \cdot \frac { \epsilon k } { 4 M } \\ & = \epsilon \end{align*}


\begin{align*} \hat \Phi _ 2 ( ( a , 0 ) ^ \top , s ) & = \langle z _ 2 ^ * , \Phi ( a u _ \ell , p ( s ) ) \rangle \\ & = \langle z _ 2 ^ * , Q F ^ q ( a u _ \ell + \psi ( a u _ \ell , p ( s ) ) , p ( s ) ) \rangle \\ & = 0 , \end{align*}


\begin{align*} \lim _ { A \rightarrow + \infty } A ^ { - n p + n } \int _ { | x | > 1 } \frac { w ( A x ) } { { | x | } ^ { n p } } d x = 0 , \end{align*}


\begin{align*} & \psi \equiv 1 \ ; \ ; o n Q _ 0 a n d s u p p ( \psi ) \subset Q , \\ [ 2 m m ] & \phi \equiv 1 \ ; \ ; o n \ ; \ ; Q _ 1 a n d s u p p ( \phi ) \subset Q _ 0 \subset Q . \end{align*}


\begin{align*} M ^ k ( g h ) = ( \Lambda \ , \cdot \ , \Lambda ) ^ k ( g , h ) = \sum ^ { 2 k } _ { j = 0 } A _ { j , 2 k - j } ( g , h ) , \end{align*}


\begin{align*} \left \Vert d \varphi _ { r i } \left ( i \right ) \right \Vert = \left \langle l ^ { \prime } \left ( r \right ) i , \theta ^ { \prime } \left ( r \right ) i \right \rangle = l ^ { \prime } \left ( r \right ) \theta ^ { \prime } \left ( r \right ) \end{align*}


\begin{align*} f ( t ) = \displaystyle \sum _ { n = 0 } ^ { \infty } \frac { \left ( - \ ; \alpha ^ { 2 } t ^ { 2 } \right ) ^ { n } } { ( 2 n ) ! } = \cos \alpha t . \end{align*}


\begin{gather*} f \in L , \ , \ , f \le c \quad Q ( f ) = \sup _ { g \in L , \ , g \le c } Q ( g ) \quad Q \in M ; \end{gather*}


\begin{align*} { { \bf { F } } _ { { \mathop { \rm i n t } } } } = - G \psi \left ( { \bf { x } } \right ) \sum \limits _ { i = 1 } ^ { 1 8 } { \varpi \left ( { { { \left | { { { \bf { c } } _ i } } \right | } ^ 2 } } \right ) } \psi \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t } \right ) { { \bf { c } } _ i } , \end{align*}


\begin{align*} D ( p ) = \frac { \Gamma ( 3 - p ) } { \left [ \Gamma \left ( 2 - \frac { p } { 2 } \right ) \right ] ^ 2 \Gamma \left ( 3 - \frac { p } { 2 } \right ) } , \end{align*}


\begin{align*} \forall \eta _ { \Lambda _ n } \in \Omega _ { \Lambda _ n } : \nu ( \eta _ { \Lambda _ n } ) = 0 , \end{align*}


\begin{align*} S _ { \ell } = \left \{ \ k \in ( p ^ { \ell - 1 } , p ^ { \ell } ] : \ k \equiv 0 \bmod p ^ { \ell - \lfloor C _ 5 \log \ell \rfloor } \right \} \end{align*}


\begin{align*} & [ x , y ] = - \varepsilon ( x , y ) [ y , x ] , & & , \\ & \circlearrowleft _ { x , y , z } \varepsilon ( z , x ) [ \alpha ( x ) , [ y , z ] ] = 0 , & & \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow \infty } \norm { \tilde { \theta } ^ k - \bar { \theta } ^ k } _ { \mathcal { L } ^ 2 , G } = \lim \limits _ { k \rightarrow \infty } \norm { e _ { G } ( \tilde { \theta } ^ k , T , \mathcal { K } ) } _ { \mathcal { L } ^ 2 , G } \leq \lim \limits _ { k \rightarrow \infty } K \norm { \theta ^ k - \tilde { \theta } ^ k } _ { \mathcal { L } ^ 2 , G } = 0 . \end{align*}


\begin{align*} \nu _ x = \nu ^ G \Bigl ( \max _ { [ b ] \in B ( G ) _ x } [ b ] \Bigr ) = \nu ^ G \Bigl ( \max _ { [ b ] \in B ( \tilde G ) _ { x \gamma } } [ b \gamma ^ { - 1 } ] \Bigr ) . \end{align*}


\begin{align*} h ^ { q - 1 } = \frac { r + 1 } { r } = 1 + \frac { 1 } { r } . \end{align*}


\begin{align*} & \sum _ { y \neq 0 , 1 } \chi _ 4 ( y ) \chi _ 4 ( 1 - y ) \sum _ { x \neq 0 , 1 , y } \overline { \chi _ 4 } ( x ) \overline { \chi _ 4 } ( x - y ) \\ & = \sum _ { y \neq 0 , 1 } \chi _ 4 ( y ) \chi _ 4 ( 1 - y ) \left [ \varphi ( y ) \rho - \overline { \chi _ 4 } ( y - 1 ) \right ] \\ & = \rho \sum _ y \overline { \chi _ 4 } ( y ) \chi _ 4 ( 1 - y ) - \sum _ y \chi _ 4 ( y ) \\ & = - \rho + 1 . \end{align*}


\begin{align*} P _ N v \cdot \partial _ x P _ N ( P _ { \ll N } v \cdot P _ N u _ 1 ) = P _ N v \cdot \partial _ x P _ { \ll N } v \cdot P _ N u _ 1 + P _ N v \cdot P _ { \ll N } v \cdot \partial _ x P _ N u _ 1 \end{align*}


\begin{align*} \dot \theta _ k = \frac { - 1 } M \sum _ { j = 1 } ^ M b _ { k - j } \sin ( \theta _ j - \theta _ k ) + \frac { 1 } M \sum _ { j = 1 } ^ M b _ { 1 - j } \sin ( \theta _ j ) . \end{align*}


\begin{align*} \forall x \in E , \forall \lambda \in \mathbf { C } , \ \overline { x \otimes _ \gamma \lambda } : = x \otimes _ { _ { \overline { \gamma } } } \overline { \lambda } . \end{align*}


\begin{align*} d ( h _ z ) = \frac { 2 } { \mu } \mu _ z h _ z d z + \frac { \sqrt { \tau _ 0 } } { 2 } \left ( \frac { 1 } { 2 } d \overline { z } + ( \alpha - I \beta ) d z \right ) \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \eta = \dot { s } ( \varepsilon ( z - 1 ) - \varepsilon ( z ) ) \eta + \dot { s } \varepsilon ( z ) \varpi . \end{align*}


\begin{align*} h : = b \vert _ { E _ 1 } + \sum \limits _ { k = 2 } ^ \delta e ^ { 2 f _ k } b \vert _ { E _ k } + \sum \limits _ { k = 2 } ^ \delta \sum \limits _ { k ' = 2 } ^ \delta e ^ { 2 f _ { k , k ' } } b _ { k , k ' } + g . \end{align*}


\begin{align*} \partial _ { t } u = \nabla \cdot ( | \nabla u | ^ { p - 2 } \nabla u ) - c | u | ^ { p - 2 } u , \end{align*}


\begin{align*} a _ d ( n ) = [ z ^ n ] \frac { z } { M _ d ( z ) } . \end{align*}


\begin{align*} & \ell ^ \alpha = \ell \circ ( \alpha ^ 2 \otimes I d ) r ^ \alpha = r \circ ( \alpha ^ 2 \otimes I d ) . \end{align*}


\begin{align*} A _ w : = w ^ { - 1 } ( R ^ + ) \cap R ^ - . \end{align*}


\begin{align*} x \diamond y = x \cdot D ( y ) , f o r ~ a l l ~ x , y \in \mathcal { H } ( A ) . \end{align*}


\begin{align*} v _ 0 = c X _ { - \beta _ m } X _ { - \beta _ { m - 1 } } \cdots X _ { - \beta _ { m - k + 1 } } v _ k . \end{align*}


\begin{align*} E _ { \xi ^ { \bar { \theta } } _ 1 ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { n } Z ^ { { \bar { \theta } } [ u ] } _ i \biggr ) \biggr ] = & E _ { \xi ^ { \bar { \theta } } _ 1 ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z ^ { \theta ^ { [ n - 1 ] } } _ i \biggr ) \biggr ] \\ = & V \bigg ( \frac { \xi _ 0 f _ 1 ( u ) } { \xi _ 0 f _ 1 ( u ) + ( 1 - \xi _ 0 ) f _ 2 ( u ) } , n - 1 , x + u \bigg ) . \end{align*}


\begin{align*} f _ \xi ( z ) = \begin{cases} \frac { f ( z ) } { \xi - z } & , \\ 0 & . \end{cases} \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \limsup _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \big ( X ^ { \epsilon } \in B ( \varphi , \delta ) \big ) \leq - \mathcal { I } ( \varphi ) . \end{align*}


\begin{align*} w \mu = ( w s _ \alpha ) ( \mu - \langle \mu , \alpha \rangle \alpha ^ \vee ) = ( w s _ \alpha ) \mu + \langle \mu , \alpha \rangle w \alpha ^ \vee \geq ( w s _ \alpha ) \mu \underset { } \geq \mu . \end{align*}


\begin{align*} \begin{cases} \min \{ \frac { 1 } { 2 } + \frac { \alpha } { 2 \beta } ( r + \lambda - \kappa ) , 1 \} , & \mbox { i f } \alpha + \gamma = 1 , \\ \min \{ \frac { \alpha } { 2 \beta } \min \{ \kappa , r + \lambda \} + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \} , & \mbox { i f } \alpha + \gamma \neq 1 \end{cases} \end{align*}


\begin{align*} \eta _ t : = \exp \int _ t ^ T ( A - B ^ 2 R ^ { - 1 } P _ s ) d s . \end{align*}


\begin{align*} \sigma \cdot X : = \{ S ^ k \sigma ( x ) \mid x \in X , 0 \leq k < | \sigma ( x _ 0 ) | \} . \end{align*}


\begin{align*} d z _ t = ( \Pi _ { \mathrm { k e r } A } ( B ( y _ t , z _ t ) + B ( z _ t , y _ t ) + B ( y _ t , y _ t ) - A y _ t ) d t + \Pi _ { \mathrm { k e r } A } \sigma d W _ t . \end{align*}


\begin{align*} \begin{aligned} & \left ( \Psi ^ - - e ^ { - i c _ - ( x ) \sigma _ 3 } \right ) - \left ( \widetilde { \Psi } ^ - - e ^ { - i c _ - ( x ) \sigma _ 3 } ( \tilde { u } ) \right ) \\ & = ( I - F ) ^ { - 1 } ( T - \widetilde { T } ) + ( I - F ) ^ { - 1 } ( F - \tilde { F } ) ( I - \tilde { F } ) ^ { - 1 } \widetilde { T } , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} d _ { C } ( ( z , t ) , ( z ' , t ' ) ) & = \big \lvert \lvert z - z ' \rvert ^ { 4 } + \lvert t - t ' + 2 \Im ( z \overline { z ' } ) \rvert ^ { 2 } \big \rvert ^ { 1 / 4 } \\ & = \lvert 2 \langle \psi ( z , t , 0 ) , \psi ( z ' , t ' , 0 ) \rangle \rvert ^ { 1 / 2 } , \end{aligned} \end{align*}


\begin{align*} w _ k = \theta _ k u _ k . \end{align*}


\begin{align*} \int _ { B _ \rho ( x _ 0 ) } F ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) \ , d v = o ( \rho ^ \lambda ) \rho \rightarrow \infty , \end{align*}


\begin{align*} \tilde { \mu } ( \xi ) = \frac { 1 } { 2 } \sum _ { j = 1 } ^ N \mu ^ 2 _ j e _ j ^ 2 ( \xi ) . \end{align*}


\begin{align*} | F ^ { i j } ( \nabla _ \beta \varphi ) _ { i j } | \leq C \sqrt { b _ \alpha } \left ( \sum _ { i = 1 } ^ { \alpha } F ^ { i i } + \sum _ { i = \alpha + 1 } ^ { n - 1 } \frac { b _ i } { b _ \alpha } F ^ { i i } \right ) \mbox { i n } \omega _ \epsilon . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m \mathcal { W } _ 2 ^ 2 ( P _ { S _ i } , Q _ { S _ i } ) \le \mathcal { W } _ 2 ^ 2 ( P _ { S _ 1 \cup \cdots \cup S _ m } , Q _ { S _ 1 \cup \cdots \cup S _ m } ) \le \mathcal { W } _ 2 ^ 2 ( P , Q ) . \end{align*}


\begin{align*} { C ^ { ( p ) } _ { x , x } ( t ) } \ge T _ - , x = 0 , \ldots , n . \end{align*}


\begin{align*} \| \mathbf { c } _ X \| _ 2 \leq \| \mathbf { c } _ X \| _ 1 = \| \mathbf { x } \| _ 1 \leq \sqrt { N L } \| \mathbf { x } \| _ 2 , \| \mathbf { c } _ Y \| _ 2 \leq \| \mathbf { c } _ Y \| _ 1 = \| \mathbf { y } \| _ 1 \leq \sqrt { N L } \| \mathbf { y } \| _ 2 , \end{align*}


\begin{align*} Y _ { i j } = \bigcap _ { ( a , b ) \neq ( i , j ) } Z _ { a b } \end{align*}


\begin{align*} \partial _ t w = g ( c ) \left ( \frac { \partial _ { x x } w } { 1 + | \partial _ x w | ^ 2 } - \frac { 1 } { w } \right ) . \end{align*}


\begin{align*} \partial _ i \Gamma ^ j _ { k j } + \Gamma ^ j _ { k j } \Gamma ^ j _ { i j } - \Gamma ^ j _ { k j } \Gamma ^ k _ { i k } - \Gamma ^ j _ { i j } \Gamma ^ i _ { k i } + g _ { i k } = 0 . \end{align*}


\begin{align*} r ' ( X ) = \begin{cases} r ( X ) + 1 , ~ X ~ ~ n p - ~ ~ ~ \\ r ( X ) , \end{cases} \end{align*}


\begin{align*} W ( \xi _ 0 , k , x , \mathsf { M } ^ k ) = W ( 1 - \xi _ 0 , k , x , \mathsf { M } ^ k ) , ~ \forall \xi _ 0 \in [ 0 , 1 ] , \forall x \in \mathbb { R } . \end{align*}


\begin{align*} ( C ^ \bullet ) ^ { \Pi ' } : = ( S _ { H ' } ( C ^ \bullet ) ) ^ { \Pi ' / H ' } . \end{align*}


\begin{align*} \mathrm { c o e v } _ S : = \mathrm { c o e v } _ { P _ { S } } \colon & I \to P _ { S } \otimes P _ { - S } \\ \mathrm { e v } _ { S } : = \mathrm { e v } _ { P _ S } \colon & P _ { - S } \otimes P _ { S } \to I \end{align*}


\begin{align*} \max _ { | \eta - \lambda | \le \delta } \max _ { x \in G ( \lambda ) } \| F _ n ( \eta ) p _ n x - p _ n F ( \eta ) x \| = \max _ { x \in G ( \lambda ) } \left \| ( T _ n p _ n - p _ n T ) x \right \| \le \left \| T _ n p _ n - p _ n T \right \| . \end{align*}


\begin{align*} \begin{aligned} \root n \of { a _ 1 \cdot a _ n } & \le \frac { a _ 1 + \cdots + a _ n } { n } \ , , a _ 1 , \cdots a _ n > 0 \ , , \\ \ , ` ` & = " \ , \ , \ , \ , \ , \ , a _ 1 = \cdots = a _ n \ , . \end{aligned} \end{align*}


\begin{align*} \lim _ { m \to \infty } \mathbb { P } \left ( \sup _ { x \in \tilde { S } _ { n , m } ^ { \delta } } \Phi ^ { S _ { n } , \tilde { S } _ { n , m } } ( x ) > 2 \sqrt { 2 \gamma } \log ( 2 ^ { m } ) + c \sqrt { \log ( 2 ^ { m } ) } \right ) = 0 . \end{align*}


\begin{align*} \mathbb { E } e ^ { - \langle \tilde { Z } _ { \lambda } ^ { S _ { n } , M , m } , f _ { m , \delta } \rangle } \leq \mathbb { E } \left [ \exp \left ( - e ^ { - \varepsilon } \sum _ { j = 1 } ^ { 1 6 ^ { m } } \mathbb { E } [ \tilde { X } _ { j } 1 _ { \{ X _ { j } \leq \varepsilon \} } | \Phi ^ { S _ { n } , \tilde { S } _ { n , m } } ] \right ) \right ] . \end{align*}


\begin{align*} \partial ^ \square c ( t , p ) = \partial ^ \circ c ( t , p ) = \frac { \mathrm { d } } { \mathrm { d } t } c ( t ) = - \frac { m d } { \alpha _ d } R ( t ) ^ { - d - 1 } R ' ( t ) \end{align*}


\begin{align*} d ^ 1 _ { p , q } = \begin{cases} 0 , & p \ , \ , \\ i _ { * } , & p \ , \ , , \end{cases} \end{align*}


\begin{align*} T ^ { - 1 } ( \hat \Phi ( T ( \rho , \chi ) , s ) ) = \begin{pmatrix} \hat h ( \rho , s ) \\ 0 \end{pmatrix} . \end{align*}


\begin{align*} \langle A ( u ) , v \rangle = - \int _ { \mathbb { T } ^ d } \langle a ( u ( x ) ) \nabla u ( x ) + b ( u ( x ) ) , \nabla v ( x ) \rangle d x . \end{align*}


\begin{align*} x x ' + x ' y + y y ' = 2 n \ , . \end{align*}


\begin{align*} & M i n i m i z e _ { x \in \mathbb { R } ^ n } \ , T r ( A _ J , X ) + 2 b _ J ^ T x \\ & s . t . \ \ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \le 0 \ , k = 1 , . . . , m \\ & X = x x ^ T \end{align*}


\begin{align*} \int _ K | | ( \eta + g ^ { - 1 } ) ^ { \frac { 1 } { 2 } } u _ 0 | | _ { \omega } + \int _ K | | \tau _ 0 | | _ { \omega } \le \int _ M \langle ( B + \lambda I ) ^ { - 1 } v , v \rangle _ { \omega , h } d V _ { \omega } . \end{align*}


\begin{align*} V ( \xi _ 0 , n , x ) : = \sup _ { \theta \in \Theta _ n } W ( \xi _ 0 , n , x , \theta ) = \sup _ { \theta \in \Theta _ n } E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] . \end{align*}


\begin{align*} h ( x ) = x \cdot \frac { f ' ( x ) } { f ( x ) } - x \cdot \frac { f ' ( x ) } { f ( x ) } \cdot \log \left ( \frac { f ' ( x ) } { f ( x ) } \right ) - \log { f ( x ) } \end{align*}


\begin{align*} f ( r ) : = k \hat W _ { \frac { r } { k } } ( k ) \end{align*}


\begin{align*} { \mathcal { I } } _ h = \inf \left \{ \frac { 1 } { 2 } \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in L ^ { 2 } _ { \sigma } ( \Omega ) , \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = h \right \} . \end{align*}


\begin{align*} ( z f ) ( \sigma ) = z _ p \star f ( \sigma z ) . \end{align*}


\begin{align*} Y \times H \to ( Y \times H ) / H _ r = Y \times ^ { H _ r } H \to X ^ { r e p } , \end{align*}


\begin{align*} \mathbf { y ' } ^ H \mathbf { y } ' = & y _ c ^ * y _ c + y _ 1 ^ * y _ 1 + \cdots + y _ M ^ * y _ M = \left ( \sum _ { l = 1 } ^ M y _ l ^ * \right ) \left ( \sum _ { j = 1 } ^ M y _ j \right ) + \sum _ { i = 1 } ^ M y _ i ^ * y _ i . \end{align*}


\begin{align*} \begin{aligned} u ( x ) e ^ { - i ( 2 c _ - ( x ) + c ) } & = \frac { 1 } { \pi } \int _ { \mathbb { R } } z ^ { - 1 } \bar { r } _ { \delta , 1 } ( z ) e ^ { - 2 i z x } M _ { \delta , + , 1 1 } ( x ; z ) d z , \end{aligned} \end{align*}


\begin{align*} R ( X , Y ) Z = \dfrac { \tilde { c } } { 4 } ( X \wedge _ g ^ c Y ) Z , \end{align*}


\begin{align*} \begin{aligned} G _ { m , \epsilon } ( t _ 1 , \cdots , t _ k ) & = \left ( \int ^ { 1 } _ { 0 } K _ { m } ( t _ 1 , r ) \theta _ { \epsilon } ( r ) d r , \cdots , \int ^ { 1 } _ { 0 } K _ { m } ( t _ k , r ) \theta _ { \epsilon } ( r ) d r \right ) ^ { \tau } \\ & = \Big ( G _ { m , \epsilon } ( t _ 1 ) , \cdots , G _ { m , \epsilon } ( t _ k ) \Big ) . \end{aligned} \end{align*}


\begin{align*} u _ \eta ( p ) = u ( \eta \cdot p ) , f _ \eta ( p ) = f ( \eta \cdot p ) , p \in \Omega _ \eta . \end{align*}


\begin{align*} \kappa _ 1 = \underset { [ 0 , h ] } { \mathrm { m a x } } \norm { U ( \tau ) A _ d } , \kappa _ 2 = \underset { [ - h , h ] } { \mathrm { m a x } } \norm { A _ d ^ \top U ( \tau ) A _ d } . \end{align*}


\begin{align*} p _ \alpha = 1 / m _ \alpha \end{align*}


\begin{align*} m : = [ \log \| x - y \| ] \end{align*}


\begin{align*} \begin{cases} - \Delta v _ 0 = f & \Omega , \\ v _ 0 = \gamma ^ - _ { \partial \Omega } ( v ) & \partial \Omega , \end{cases} \end{align*}


\begin{align*} R _ { i a } = \sum _ { j , b } \mathcal { B } _ { b i } ^ j \mathcal { B } _ { a j } ^ b = \sum _ { j , b } \mathcal { B } _ { b i j } \mathcal { B } _ { a j b } ( - \varepsilon _ { j ' } ) \varepsilon _ b = \sum _ { j , b } \mathcal { B } _ { b i j } \mathcal { B } _ { j a b } \varepsilon _ { j ' } \varepsilon _ b \end{align*}


\begin{align*} \forall \Delta \Subset \Z ^ d \ \forall \xi _ { \Delta } : c ^ { B _ { N - 1 } ( x ( \Delta ) ) } _ { \Delta } ( \eta _ { \Lambda _ N } , \xi _ { \Delta } ) > 0 \forall n \in \N : \ c ^ { B _ { n - 1 } ( x ( \Delta ) ) } _ { \Delta } ( \eta _ { \Lambda _ n } , \xi _ { \Delta } ) = 0 . \end{align*}


\begin{align*} \rho { c _ v } \frac { { \partial T } } { { \partial t } } + \rho { c _ v } { \bf { u } } \cdot \nabla T = \nabla \cdot \left ( { \lambda \nabla T } \right ) - T { \left ( { \frac { { \partial { p _ { E O S } } } } { { \partial T } } } \right ) _ \rho } \nabla \cdot { \bf { u } } . \end{align*}


\begin{align*} \| A _ { | I | } ( \theta _ { \ell _ 1 } ) \| = \frac { 1 } { \prod _ { \ell \in I } | \cos ( \pi \theta _ { \ell } ) | } \| F _ { | I | } ( \theta _ { \ell _ 1 } ) \| \leq C ( \varepsilon ) e ^ { \varepsilon ( 2 q _ n - | I | ) } e ^ { | I | \ln 2 } e ^ { | I | ( \tilde { L } + \varepsilon ) } \leq e ^ { 3 \varepsilon q _ n } e ^ { L | I | } . \end{align*}


\begin{align*} j = j _ k = \Big \lfloor \frac { \omega _ k ( k + 1 ) } { 1 + \omega _ k } \Big \rfloor . \end{align*}


\begin{align*} a ( u _ n , v _ n ) = ( p _ n f , v _ n ) v _ n \in X _ n . \end{align*}


\begin{align*} { \frak M } : = \sup _ m \left \{ \sum _ { x = 0 } ^ n | M _ { x , 0 } ( m ) | ^ 2 \right \} ^ { 1 / 2 } < + \infty . \end{align*}


\begin{align*} M _ \delta ( x ; z ) = I + \mathcal { C } \left ( M _ { \delta , - } ( x ; \cdot ) \hat { R } _ \delta ( x ; \cdot ) \right ) ( z ) , z \in \mathbb { C } ^ { \pm } . \end{align*}


\begin{align*} \exp \left ( a _ { \alpha + 1 } + d _ { \alpha + 1 } - d _ \alpha \right ) = 1 - \exp \left ( c _ { n + 1 } + b _ { n + 1 } - b _ n \right ) \ . \end{align*}


\begin{align*} & \ \eta = \sum _ { x \in \bar { \mathbb { F } } _ q } a _ x \dot { s } \varepsilon ( u + x ) { \bf 1 } _ { \theta } + \sum _ { y , z \in \bar { \mathbb { F } } _ q } b ' _ { y , z } \varepsilon ( - ( u + y ) ^ { - 1 } ) \dot { s } \varepsilon ( ( u + y ) ^ 2 z - ( u + y ) ) { { \bf 1 } } _ { \theta } , \end{align*}


\begin{align*} \partial _ t \mu ^ \phi = - \partial _ x \Big \{ \Big [ b ( t , \cdot , \mu ^ \phi ) + \sigma \phi ( t , \cdot , \mu ^ \phi ) \Big ] \mu ^ \phi \Big \} . \end{align*}


\begin{align*} T _ { W _ \ell , \phi } ( \mu ) = \sum _ { i = 1 } ^ L c _ i \int _ { [ 0 , 1 ] ^ 2 } \bar a _ i ( u ) \bar b _ i ( v ) W _ \ell ( u , v ) d u d v , \end{align*}


\begin{align*} G ( x ) = \begin{cases} \frac { \Gamma ( 1 / s , ( \frac { \mu - x } { \sigma } ) ^ s ) } { 2 \Gamma ( 1 / s ) } , & x \leq \mu , \\ 1 - \frac { \Gamma ( 1 / s , ( \frac { x - \mu } { \sigma } ) ^ s ) } { 2 \Gamma ( 1 / s ) } , & x > \mu , \end{cases} \end{align*}


\begin{align*} \overline { \pi } _ 1 ( q ) = \eta ( q ) \overline { \pi } ( q ) \end{align*}


\begin{align*} V ( \xi _ 0 , n , x ) = \sup _ { \theta \in \Theta _ n } E _ { \xi _ 0 } \left [ h ^ \theta ( x , Z _ 1 ^ \theta ) \right ] , \end{align*}


\begin{align*} ( 1 . 4 4 ) ^ { 2 } \cdot \frac { 3 2 \pi ^ { 8 } } { 9 p ^ { 3 } } \sum _ { n = a p } ^ { b p } n \log ( n + 2 ) \sum _ { d = 1 } ^ { \infty } \psi \left ( d \sqrt { \frac { n } { p } } \right ) \leq \frac { 6 9 9 5 8 } { p ^ { 3 } } \cdot \left [ b p \log ( b p + 2 ) \right ] ( b - a ) p \sum _ { d = 1 } ^ { \infty } \psi ( d \sqrt { a } ) . \end{align*}