\begin{align*} \hat { c } _ { \Delta } ( \eta , \xi _ { \Delta } ) : = c _ { \Delta } ( \xi _ { \Delta } \eta _ { \Delta ^ c } , \eta _ { \Delta } ) \frac { \gamma _ { \Delta } ( \xi _ { \Delta } | \eta _ { \Delta ^ c } ) } { \gamma _ { \Delta } ( \eta _ { \Delta } | \eta _ { \Delta ^ c } ) } . \end{align*}


\begin{align*} g = W ( s _ 1 , \ldots , s _ d ) = \prod _ { k = 1 } ^ l s _ { \iota ( k ) } ; \end{align*}


\begin{align*} F _ k ( r , s ) = \cos { ( \frac { 2 \pi } { n - 1 } ( r - s + 1 ) k ) } . \end{align*}


\begin{align*} \Phi _ { n , e } ^ { ( i ) } ( x , y ) = \prod _ { \substack { m | n \\ ( m , e ) = 1 \\ \overline m m \equiv i \bmod e } } ( x ^ { n / m } - \zeta ^ { \overline m } _ { e } y ^ { n / m } ) ^ { \mu ( m ) } . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta u + V ( x ) u + \gamma \phi ( x ) u = b \ , | u | ^ { p - 2 } u & \mathbb { R } ^ 2 \\ \Delta \phi = u ^ 2 & \mathbb { R } ^ 2 , \end{array} \right . \end{align*}


\begin{align*} \Sigma \big ( X , q \big ) = \sum \limits _ { 1 \leq n \leq q \atop { n ^ 2 + n + 1 \equiv 0 \ , ( q ) } } \Omega ( X , q , n ) \ , . \end{align*}


\begin{align*} ( \partial _ v \Gamma _ { u v } ^ v - 2 \Gamma _ { u v } ^ v \Gamma _ { v u } ^ v ) \tau = \partial _ u \Gamma _ { v u } ^ u - 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v . \end{align*}


\begin{align*} { \mathcal { S } } _ 1 = \left \{ U \in L ^ { 2 } _ { \sigma } ( \Omega ) \ \middle | \ \nabla \times U = f _ 1 ^ { + } U , \ \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } U \cdot U \dd x = 1 \right \} . \end{align*}


\begin{align*} \norm { \varphi _ n ( r _ m \cdot ) - \varphi _ n ( r _ { m + 1 } \cdot ) } _ \infty \leq \sum _ { \abs { z } = m } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z \varphi _ n } _ \infty . \end{align*}


\begin{align*} ( R i c ^ g ) ^ { J , - } & = - \frac { 1 } { 2 } ( \theta \otimes \theta ) ^ { J , - } . \end{align*}


\begin{align*} F ( x ) = \operatorname { I } _ { \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } ( a , b ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int _ 0 ^ { \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } \omega ^ { a - 1 } ( 1 - \omega ) ^ { b - 1 } \mathrm { d } \omega , \end{align*}


\begin{align*} \left < K _ i , K _ j \right > = q ^ { - a _ { i j } / 2 } , \left < E _ i , K _ j \right > = \left < K _ i , F _ j \right > = 0 , \left < E _ i , F _ j \right > = \frac { \delta _ { i j } } { q _ j ^ { - 1 } - q _ j } \end{align*}


\begin{align*} \tilde H _ y ^ { ( n ) } = \frac { 1 } { 2 ( n + 1 ) } \sum _ { j = - n - 1 } ^ n \cos \left ( \frac { \pi j ( 2 y + 1 ) } { n + 1 } \right ) \Xi \left ( \frac { j } { n + 1 } \right ) . \end{align*}


\begin{align*} x \ , h _ s ( i ) ^ { \tilde \varepsilon x / 6 4 } \le x \exp \bigg ( - \frac { 1 2 8 \ , \log n } { \tilde \varepsilon ^ 2 ( 1 + \tilde \varepsilon ) ^ { - i } m _ s } \frac { \tilde \varepsilon x } { 6 4 } \bigg ) \le x \exp ( - 2 \log n ) = n ^ { - 1 } , \end{align*}


\begin{align*} m = \left \{ \begin{array} { l l l } x , & \hbox { i f $ i \in \{ 1 , 2 \} $ , } \\ 1 - x , & \hbox { } \end{array} \right . \end{align*}


\begin{align*} ( a ^ \alpha _ \beta ) : = \begin{pmatrix} a ^ 1 _ 1 & \dots & a ^ 1 _ { n - 1 } \\ \vdots & \ddots & \vdots \\ a ^ { n - 1 } _ 1 & \dots & a ^ { n - 1 } _ { n - 1 } \\ \end{pmatrix} , \end{align*}


\begin{align*} \begin{array} { l l l } \begin{pmatrix} a _ i & b _ i \\ c _ i & d _ i \end{pmatrix} , & a _ i d _ i - b _ i c _ i = 1 , & c _ i > 0 , \\ \begin{pmatrix} a ' _ i & b ' _ i \\ c ' _ i & d ' _ i \end{pmatrix} , & a ' _ i d ' _ i - b ' _ i c ' _ i = 1 , & c ' _ i > 0 , \end{array} \end{align*}


\begin{align*} A _ \phi = \begin{pmatrix} \cos ( 2 \pi \ell \phi ) & \sin ( 2 \pi \ell \phi ) \\ - \sin ( 2 \pi \ell \phi ) & \cos ( 2 \pi \ell \phi ) \end{pmatrix} \end{align*}


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


\begin{align*} v = v _ 1 , \dotsc , v _ k \in W \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \limsup _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { 0 \leq t - s \leq \delta , t , s \in [ 0 , 1 ] } \big | X ^ { \epsilon } ( t ) - X ^ { \epsilon } ( s ) \big | > \delta _ 0 \Big ) = - \infty . \end{align*}


\begin{align*} x _ i ( g ) = x ( f _ i ( g ) ) \forall g \in G . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb N } } \left ( \frac { \mu ( f ^ { k + n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \ \ \& \ \ \lim _ { n \rightarrow \infty } \inf _ { k \in - { \mathbb N } _ 0 } \left ( \frac { \mu ( f ^ { k - n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 3 $ } \end{align*}


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


\begin{align*} e ^ { \gamma x } f _ r \ast f _ G ( x ) = \int _ { - \infty } ^ { x / 2 } \big \{ e ^ { \gamma ( x - y ) } f _ r ( x - y ) e ^ { \gamma y } f _ G ( y ) + e ^ { \gamma ( x - y ) } f _ G ( x - y ) e ^ { \gamma y } f _ r ( y ) \big \} d y < \infty \end{align*}


\begin{align*} A ( x , x ) = \frac { 1 } { 2 } \int _ { x } ^ { \infty } q ( t ) d t , \end{align*}


\begin{align*} d _ A ( f , g ) = \begin{cases} \left | e ^ { 2 \pi i d ' _ A ( f , g ) } - 1 \right | & d ' _ A ( f , g ) < \frac { 1 } { 2 } \\ 2 & d ' _ A ( f , g ) \ge \frac { 1 } { 2 } . \end{cases} \end{align*}


\begin{align*} L ( x ) = 2 \sinh { \left ( \tfrac { x } { 2 } \right ) } . \end{align*}


\begin{align*} S _ { R } ^ { \alpha } = C _ { \beta , \delta } \int _ { 0 } ^ { 1 } ( 1 - t ) ^ { \beta - 1 } t ^ \delta S _ { R t } ^ { \delta } d t , \end{align*}


\begin{align*} \phi _ A ( t _ 1 , \hdots , t _ n ) = \left ( \prod _ { i = 1 } ^ n t _ i ^ { a _ { i 1 } } , \prod _ { i = 1 } ^ n t _ i ^ { a _ { i 2 } } , \hdots , \prod _ { i = 1 } ^ n t _ i ^ { a _ { i m } } \right ) . \end{align*}


\begin{align*} g ( \overline { t } ) = \mathop { \max } \limits _ { t > 0 } g ( t ) . \end{align*}


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


\begin{align*} \left \{ \begin{aligned} \Delta u & = f ( u ) & & \quad \mbox { i n } \Omega , \\ u & = \infty & & \quad \mbox { o n } \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { a _ { N } } { \log N } = 2 \lambda \sqrt { 2 \gamma } , \end{align*}


\begin{align*} \gamma _ t = \gamma _ 0 \delta ^ { n _ t } , \ \beta _ t = \frac { \gamma _ { t - 1 } ( 1 - \kappa _ t ) } { \gamma _ t ( 1 - \kappa _ { t - 1 } ) } \left ( 1 + \frac { 2 \mu \gamma _ { t - 1 } } { 1 - \kappa _ { t - 1 } } \right ) ^ { - 1 } , \ \alpha _ t = \frac { \kappa _ { t - 1 } \gamma _ { t } \beta _ t } { \gamma _ { t - 1 } } , \end{align*}


\begin{align*} \forall \ \sigma \geq 0 , \widehat { M ^ \sigma g } ( m , \eta ) = \rho _ \sigma \hat g ( m , \eta ) \end{align*}


\begin{align*} | F \cap M | \le \begin{cases} \Delta & U \le H , \\ \Delta + | H \cap W | q ^ { d ^ 2 - d } \theta _ { d - 1 } & . \end{cases} \end{align*}


\begin{align*} - L \phi = \mu ^ { 2 } ( \phi - \phi _ \infty ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 2 } _ { + } , \phi = 0 \textrm { o n } \ \partial \mathbb { R } ^ { 2 } _ { + } , \end{align*}


\begin{align*} \frac { 1 } { | x _ d | } \le \frac { 1 } { d } \Big | \frac { p ' ( c ) } { p ( c ) } \Big | = \frac { 1 } { d } \Big | \sum _ { j = 1 } ^ d \frac { 1 } { c - x _ j } \Big | \end{align*}


\begin{align*} \Pi _ V J _ { z / | z | } ^ \perp x = \lambda J _ { z / | z | } ^ \perp \Pi _ V x \forall x \in \C ^ n \end{align*}


\begin{align*} \partial _ x T - \mathcal { P } ^ + \left ( \partial _ x T \right ) R _ + - \mathcal { P } ^ - \left ( \partial _ x T \right ) R _ - = \widetilde { F } , \end{align*}


\begin{align*} [ N ( x ) , N ( y ) ] & = N ( x ) \cdot N ( y ) - N ( y ) \cdot N ( x ) \\ & = N \big ( N ( x ) \cdot y + x \cdot N ( y ) - N ( x \cdot y ) - N ( y ) \cdot x - y \cdot N ( x ) + N ( y \cdot x ) \big ) \\ & = N \big ( [ N ( x ) , y ] + [ x , N ( y ) ] - N [ x , y ] \big ) . \end{align*}


\begin{align*} \omega ( v _ \lambda , - ) = \lambda , \end{align*}


\begin{align*} \rho _ E ( \lambda , z ) = \frac { 1 } { 2 \pi i } \int _ { \gamma } \omega _ { \lambda , z } \end{align*}


\begin{align*} Q _ { i j } ( h ) = \partial ^ 2 _ { i j } h - \Gamma _ { j i } ^ i \partial _ i h - \Gamma _ { i j } ^ j \partial _ j h + g _ { i j } h = 0 , \forall i < j , \end{align*}


\begin{align*} \chi \cap \psi = \psi \setminus \{ x \} . \end{align*}


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


\begin{align*} f ( m ; z ) : = \frac { \eta ( 5 z ) } { \eta ( z ) } \eta ^ a ( 5 m z ) \eta ^ b ( m z ) , \end{align*}


\begin{align*} ( - 1 ) ^ n { k - 1 / 2 \choose n } = \frac { ( - 1 ) ^ k \Gamma ( k + 1 / 2 ) \Gamma ( n + 1 / 2 - k ) } { \pi \Gamma ( n + 1 ) } . \end{align*}


\begin{align*} R ^ { \nabla ^ t } = R ^ \nabla + t ( d ^ \nabla A ) + t ^ 2 [ A , A ] \ , , \end{align*}


\begin{align*} s _ { \min } ( A ) = s _ n ( A ) = \min \limits _ { x : \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 , s _ { \max } ( A ) = s _ 1 ( A ) = \max \limits _ { x : \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 . \end{align*}


\begin{align*} u _ n & = \sum _ { j = 0 } ^ \infty \Xi _ j ( \vec u _ { 0 , n } ) = \sum _ { j = 0 } ^ \infty \Xi _ j ( \vec u _ { 0 } + \vec \phi _ n ) , \end{align*}


\begin{align*} f = ( a _ 0 , b _ 0 , c _ 0 , d _ 0 ) _ q , \end{align*}


\begin{align*} F ( x ) = \max _ { i = 1 , \ldots , 2 ^ k } x _ i . \end{align*}


\begin{align*} \overline { \mathbf { p } } ^ { \oplus } = \{ \alpha _ { i _ k , j _ \ell } \mid 1 \leq k , \ell \leq n \} . \end{align*}


\begin{align*} \frac { 1 } { k } \int _ { 0 } ^ { 1 } \ln | \tilde { P } _ k ( \theta ) | \mathrm { d } \theta = \frac { 1 } { k } \int _ { 0 } ^ { 1 } \ln | \tilde { P } _ k ( 2 \theta ) | \mathrm { d } \theta \geq L - \ln 2 = \tilde { L } \end{align*}


\begin{align*} K _ q ( z , w ) = E _ q ( z \overline { w } ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { ( z \overline { w } ) ^ n } { \Gamma ( q n + 1 ) } , \end{align*}


\begin{align*} \liminf _ { n \to \infty } \frac { g _ + ( b ^ n ( x _ 0 + v ) ) } { n g _ + ( b ^ n x _ 0 ) } = \liminf _ { n \to \infty } \frac { \nu \big ( ( b ^ n ( x _ 0 + v ) , b ^ n ( x _ 0 + v ) + 1 ] \big ) } { n \nu \big ( ( b ^ n x _ 0 , b ^ n x _ 0 + 1 ] \big ) } \ge c _ 0 , \end{align*}


\begin{align*} \int _ 0 ^ \infty \left ( 1 - \frac { u ^ 2 } { 6 s } + \frac { u ^ 4 } { 7 2 s ^ 2 } \right ) e ^ { - u } u ^ { p / 2 - 1 } \dd u = \Gamma \left ( p / 2 \right ) - \frac { \Gamma ( p / 2 + 2 ) } { 6 s } + \frac { \Gamma ( p / 2 + 4 ) } { 7 2 s ^ 2 } \end{align*}


\begin{align*} f ( m ; z ) : = \frac { \eta ( 3 z ) } { \eta ( z ) } \eta ^ a ( 3 m z ) \eta ^ b ( m z ) , \end{align*}


\begin{align*} \begin{aligned} \mathbf { \Phi } _ { \mathrm { r } } & = \mathsf { d i a g } ( \phi _ { \mathrm { r } , 1 } , \ldots , \phi _ { \mathrm { r } , M } ) , ~ ~ \mathbf { \Phi } _ { \mathrm { t } } & = \mathsf { d i a g } ( \phi _ { \mathrm { t } , 1 } , \ldots , \phi _ { \mathrm { t } , M } ) , \end{aligned} \end{align*}


\begin{align*} \sum _ x \overline { \chi _ 4 } ( x ) \overline { \chi _ 4 } ( x - y ) & = \sum _ { x ' } \overline { \chi _ 4 } ( x ' y ) \overline { \chi _ 4 } ( x ' y - y ) \\ & = \varphi ( y ) \rho . \end{align*}


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


\begin{align*} \begin{aligned} d u _ m & = i 2 ^ m \left ( \overline { u _ { m + 1 } } u _ { m + 2 } - \frac { \delta } { 2 } \overline { u _ { m - 1 } } u _ { m + 1 } - \frac { \delta - 1 } { 4 } u _ { m - 2 } u _ { m - 1 } \right ) \\ & - \delta 2 ^ { 2 m } u _ m + q _ m d W _ t ^ { ( m ; R ) } + i p _ m d W _ t ^ { ( m ; I ) } , \end{aligned} \end{align*}


\begin{align*} \mathcal { P } ^ { \pm } ( h ) ( z ) : = \lim _ { \varepsilon \downarrow 0 } \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } \frac { h ( s ) } { s - ( z \pm i \varepsilon ) } d s z \in \mathbb { R } . \end{align*}


\begin{align*} 2 \pi i \cdot f ( z ) = \int _ { e ^ { - i \alpha } [ 0 , + \infty ) } \frac { f ( \zeta ) e ^ { p ( \zeta - z ) } } { \zeta - z } d \zeta - \int _ { e ^ { i \alpha } [ 0 , + \infty ) } \frac { f ( \zeta ) e ^ { p ( \zeta - z ) } } { \zeta - z } d \zeta , z \in \Delta . \end{align*}


\begin{align*} \Phi : = \sup \{ \zeta \in U S C ( \overline { \Omega } \times \mathbb R ^ m ) : \zeta | _ { \Omega \times \mathbb R ^ m } F \star \mathcal P \zeta | _ { \partial \Omega \times \mathbb R ^ m } \le \phi \} \end{align*}


\begin{align*} \partial _ u \Gamma _ { v u } ^ u = 2 \Gamma _ { u v } ^ u \Gamma _ { v u } ^ v = \partial _ v \Gamma _ { u v } ^ v . \end{align*}


\begin{align*} \mathbf { Q } ^ i = \begin{pmatrix} 1 & 0 \\ 0 & - \mathbf { I } _ { n _ i - 1 } \\ \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} \| ( \gamma - c ) \varphi \| _ { W ^ { \frac { 1 } { 3 } , \frac { 3 } { 2 } } ( \partial \Omega ) } + & \| c \nabla \varphi \cdot \nu \| _ { W ^ { \frac { 1 } { 3 } , \frac { 3 } { 2 } } ( \partial \Omega ) } \\ & \le C ( \gamma \| \varphi \| _ { W ^ { 2 , \frac { 3 } { 2 } } ( \Omega ) } + \| \nabla c \| _ { L ^ { 2 } ( \Omega ) } \| \varphi \| _ { W ^ { 1 , 6 } ( \Omega ) } ) . \end{aligned} \end{align*}


\begin{align*} \mathbf { x } { { \left ( \alpha \right ) } ^ { T } } \mathbf { s } \left ( \alpha \right ) = \left ( 1 - \left ( 1 - \nu \right ) \alpha \right ) { \mathbf { x } } ^ { T } \mathbf { s } . \end{align*}


\begin{align*} \begin{aligned} \| x ( t ) \| _ { \mathcal { A } } & \le \| x ( t ) - x _ a ( t ) \| + \| x _ a ( t ) \| _ \mathcal { A } \\ & \le \tilde { d } + \beta ( \| x _ 0 \| _ { \mathcal { A } } , \gamma t ) & & \forall \ , t \in [ 0 , \bar { t } / \gamma ] \end{aligned} \end{align*}


\begin{align*} F f ( x ; z ) = ( h _ { 1 } , h _ { 2 } ) ^ T , \end{align*}


\begin{align*} H _ { N } ( t ) : = H ( f _ { N , t } ; \nu _ { N , t } ) , \forall \ , t \in [ 0 , T ] . \end{align*}


\begin{align*} \mathcal { M } _ { ( i , j ) } [ m , n ] : = \mathcal { M } _ { ( i + m , j + n ) } \ , , \end{align*}


\begin{align*} \Tilde { L } ( x ) : = \gamma _ 0 f _ 0 ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge 0 \ \ x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} \Gamma _ 1 ( X ) & = X \sum \limits _ { 1 \leq d \leq z } \frac { \mu ( d ) \lambda ( d ^ 2 ) } { d ^ 2 } + \mathcal { O } \big ( z X ^ \varepsilon \big ) \\ & = \sigma X - X \sum \limits _ { d > z } \frac { \mu ( d ) \lambda ( d ^ 2 ) } { d ^ 2 } + \mathcal { O } \big ( z X ^ \varepsilon \big ) \ , , \end{align*}


\begin{align*} [ - T , T ] = \bigcup _ { j = - [ T / \tau ] - 1 } ^ { [ T / \tau ] } I _ j , I _ j = [ j \tau , ( j + 1 ) \tau ] \cap [ - T , T ] , \end{align*}


\begin{align*} a _ { i , d } ( \underline { b } ) + a ' _ { i , d } ( \underline { b } ) = x _ { i , d } i , d , \end{align*}


\begin{align*} \epsilon ( x , t ) : = \int _ 0 ^ t f ( x , \tau ) - f _ a ( x ) \ , d \tau , \end{align*}


\begin{align*} \int _ { \Omega } | \nabla v _ { - } | ^ { 2 } + \int _ { \Omega } u | v _ { - } | ^ { 2 } + \int _ { \partial \Omega } | v _ { - } | ^ { 2 } = - \eta \int _ { \partial \Omega } v _ { - } . \end{align*}


\begin{align*} g ^ { q p } u _ { , p q i } = - v _ i + u _ { , i q } \xi ^ q \end{align*}


\begin{align*} d ^ D _ t = \sum _ { u , v : t ( u , v ) = t } \sum _ { s = 0 } ^ { \omega _ { u v } - 1 } d _ { ( u , v , s ) } . \end{align*}


\begin{align*} \mu ^ { N , i } _ { \boldsymbol { \alpha } } : = \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } \alpha ^ j , \nu ^ { N , i } _ { \boldsymbol { x } } : = \frac { 1 } { N - 1 } \underset { j \neq i } { \sum } x ^ j . \end{align*}


\begin{align*} \alpha _ * ^ 2 \| u _ * \| ^ 2 + 2 \alpha _ * \langle u _ * , \lambda x _ v + ( 1 - \lambda ) x _ w \rangle - \lambda ( 1 - \lambda ) \| x _ v - x _ w \| ^ 2 = 0 \end{align*}


\begin{align*} b ( q ^ { \frac { 1 } { 3 } } ) = a ( q ) - c ( q ) , \end{align*}


\begin{align*} \frac { h _ { j t } + h _ { j } } { h _ { t } + h } & = w ^ { i k } \nabla _ { j } w _ { i k } + \nabla _ { j } \chi \\ & = w ^ { i i } ( h _ { j i i } + h _ { i } \delta _ { i j } ) + \nabla _ { j } \chi , \end{align*}


\begin{align*} U \cdot ( w ^ * \otimes w ) = ( U \cdot w ^ * ) \otimes w , \end{align*}


\begin{align*} \mathbf { U } _ { k l } \mathbf { h } _ { k l } = \left [ \mathbf { a } ( \overline { \theta } _ { 1 } ) , \mathbf { a } ( \overline { \theta } _ { 2 } ) , . . . , \mathbf { a } ( \overline { \theta } _ { N } ) \right ] ^ { \mathbf { H } } \mathbf { h } _ { k l } , \end{align*}


\begin{align*} a _ c \left [ t + 1 \right ] & = a _ c \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \varepsilon \right ] } { \partial a _ c } \\ a _ i \left [ t + 1 \right ] & = a _ i \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \varepsilon \right ] } { \partial a _ i } . \end{align*}


\begin{align*} \ell _ { v } = Z _ { 1 } \left ( v , v \right ) \rho _ { v } = Z _ { 1 } \left ( v , - v \right ) . \end{align*}


\begin{align*} X _ 1 : = ( \ln \vert \alpha \vert , \ln \vert \sigma ( \alpha ) \vert , \ln \vert \sigma ^ 2 ( \alpha ) \vert ) , & & X _ 2 : = ( \ln \vert \sigma ( \alpha ) \vert , \ln \vert \sigma ^ 2 ( \alpha ) \vert , \ln \vert \alpha \vert ) \end{align*}


\begin{align*} h _ { \alpha , \rho } ( t ) = \frac { t ^ { \alpha - 1 } } { \Gamma ( \alpha ) } \ , e ^ { - \rho t } , \ \ \alpha > 0 , \ \rho \in \R , \ t > 0 . \end{align*}


\begin{align*} W ^ 2 ( r ) + f ( r ) \frac { d W ( r ) } { d r } = W ^ { \prime 2 } ( r ) - f ( r ) \frac { d W ' ( r ) } { d r } + E _ 1 - E _ 0 , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ 2 d t = 0 . \end{align*}


\begin{align*} h ( t ) = \begin{cases} t + 2 ^ \alpha t ^ { 1 + \alpha } & \ 0 \le t < \frac { 1 } { 2 } \\ 2 t - 1 & \ \frac { 1 } { 2 } \le t \le 1 . \end{cases} \end{align*}


\begin{align*} \sum _ { i = 0 } ^ m b _ i ( x ^ { q ^ i } \ \ f ) = 0 . \end{align*}


\begin{align*} \sigma ( x _ i ) = x _ { \sigma ( i ) } , \ \sigma ( \epsilon ) = \epsilon , \sigma \in S _ n , \ 1 \leq i \leq n . \end{align*}


\begin{align*} D = D ( S , u ) = l _ S ( u ) + 2 \max \left \{ l _ S \big ( t ^ j \big ) \mid - r _ S \le j \le r _ S \right \} \in \mathbb { N } . \end{align*}


\begin{align*} ( X _ \ast ( T ) _ { \Gamma _ 0 } \otimes \mathbb Q ) ^ { \sigma w } = \{ \mu \in X _ \ast ( T ) _ { \Gamma _ 0 } \otimes \mathbb Q \mid \sigma ( \mu ) = \mu \langle \mu , \Phi \rangle = \{ 0 \} \} . \end{align*}


\begin{align*} d ^ * ( \lfloor x \rfloor ) & = q ( c ( \lfloor x \rfloor ) ) \\ & = q ( c _ b c _ b c _ a c _ a ) \\ & = b ^ 2 a ^ 2 \end{align*}


\begin{align*} R ( X , Y ) Z = \nabla _ X \nabla _ Y Z - \nabla _ Y \nabla _ X Z - \nabla _ { [ X , Y ] } Z , \end{align*}


\begin{align*} \nabla _ X \varphi = - \frac { 1 } { 2 } ( \sqrt { | c _ 1 | } \ X _ 1 \cdot \nu _ 1 + \sqrt { | c _ 2 | } \ X _ 2 \cdot \nu _ 2 ) \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} D ^ { k } B ^ C _ n ( j _ { V _ C } ) = D ^ { k } \check { B } ^ C _ n ( j _ { V _ C } ) \end{align*}


\begin{align*} \cos ^ 2 \theta & = \frac { 1 + \cos ( 2 \theta ) } { 2 } = \frac { \sqrt { m } - \delta _ { m , l } } { 2 \sqrt { m } - \delta _ { m , k } - \delta _ { m , l } } , \\ \sin ^ 2 \theta & = \frac { 1 - \cos ( 2 \theta ) } { 2 } = \frac { \sqrt { m } - \delta _ { m , k } } { 2 \sqrt { m } - \delta _ { m , k } - \delta _ { m , l } } , \end{align*}


\begin{align*} \liminf _ { i \to \infty } \frac { w _ { n _ i } } { n _ i ^ { r - \frac { 1 } { 2 } } } = 0 \limsup _ { i \to \infty } \frac { w _ { n _ i } } { n _ i ^ { r - \frac { 1 } { 2 } } } = \infty \end{align*}


\begin{align*} h ( M ) = \inf \frac { _ { n - 1 } ( \Sigma ) } { \min \{ _ n ( A ) , _ n ( B ) \} } , \end{align*}


\begin{align*} [ q ^ \alpha r ^ \beta ] C _ k ( q , r , t ) = C _ k ( t ) ^ { \beta + 1 } \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - ( k - 1 ) i } { i } t ^ i - \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - \beta - 1 - ( k - 1 ) i } { i } t ^ i . \end{align*}


\begin{align*} \begin{cases} - \Delta u _ 0 = f & \Omega , \\ u _ 0 = 0 & \partial \Omega . \end{cases} \end{align*}


\begin{align*} S = \frac { { W I } } { { K L } } \varepsilon \left ( { K , \gamma } \right ) . \end{align*}


\begin{align*} q _ n = x _ 1 ^ 2 + x _ 0 x _ 1 + x _ 0 x _ 2 + \dotsb + x _ 0 x _ n . \end{align*}


\begin{align*} 1 = N _ 0 \rhd N _ 1 \rhd \cdots \rhd N _ r = G , \end{align*}


\begin{align*} h ( \delta ) = C _ 3 \delta ^ 2 \ ; \ ; \ ; \ ; C _ 3 \stackrel { \rm d e f } { = } \frac { 1 } { 2 } \eta _ 1 \eta _ 2 \kappa _ { \rm f c d } \min \left \{ \frac { \eta _ 2 } { \kappa _ { \rm b h m } } , 1 \right \} . \end{align*}


\begin{align*} G _ { - } ( \rho ) = \frac { 1 } { 2 i \rho } \int _ { 0 } ^ { \infty } q ( s ) e ^ { - 2 i s \rho } d s + \frac { 1 } { \left ( 2 i \rho \right ) ^ { 2 } } \int _ { 0 } ^ { \infty } Q ^ { \prime } ( s ) e ^ { - 2 i s \rho } d s \end{align*}


\begin{align*} z ^ j \cdot \left ( \prod _ { i = 1 } ^ n x _ i ^ { \alpha _ i } . y _ i ^ { \beta _ i } \right ) ^ k , 0 \le j \le p - 1 , 1 \le k \le p - 1 . \end{align*}


\begin{align*} \rho _ { Z } = \begin{cases} n _ j z , & Z = V ( f _ j ) \\ \lambda _ j , & Z = V ( x _ j ) . \end{cases} \end{align*}


\begin{align*} \psi _ { 0 , H } ( t ) = \frac { ( n - 1 ) H } { ( n - 1 ) + t H } . \end{align*}


\begin{align*} \mathbf { H } _ { \mathrm { a l l } , k } = \left \{ \begin{array} { c c } \mathbf { H } _ { \mathrm { d } , k } + \overline { \mathbf { H } } _ { k , 1 } \mathbf { \Phi } _ { 1 , 1 } \overline { \mathbf { G } } _ { 1 } , & k \in \mathcal { K } _ { 1 } , \\ \mathbf { H } _ { \mathrm { d } , k } + \overline { \mathbf { H } } _ { k , 2 } \mathbf { \Phi } _ { 2 , 1 } \overline { \mathbf { G } } _ { 1 } , & k \in \mathcal { K } _ { 2 } . \end{array} \right . \end{align*}


\begin{align*} g ^ { q p } u _ { , p q i } = - v _ { i } + u _ { , i q } \xi ^ q \end{align*}


\begin{align*} b _ n = a _ { \alpha + 1 } + ( d _ { \alpha + 1 } - d _ \alpha ) = b \ , \end{align*}


\begin{align*} \mathcal { F } _ { \mathcal { R } } & = \{ f = \frac { \sum _ { h = 1 } ^ { H } c _ h \chi _ { Q _ h } } { \sum _ { i = 1 } ^ { H } c _ h \nu ( Q _ h ) } : \ ; c _ h \in \mathbb { R } ^ + , \ ; Q _ i \cap Q _ j = \emptyset \} , \end{align*}


\begin{align*} \begin{aligned} \gamma _ { k } = & \frac { | \tilde { \mathbf { h } } _ { k } ^ H \mathbf { w } _ { k } | ^ 2 } { \sum _ { p \in \mathcal { K } , p \ne k } | \tilde { \mathbf { h } } _ { k } ^ H \mathbf { w } _ { p } | ^ 2 + \sigma _ { k } ^ 2 } , \forall i \in \{ \mathrm { t , r } \} , \forall k \in \mathcal { K } _ i . \end{aligned} \end{align*}


\begin{align*} ( \mathbf { w } _ { x s } - \mu \mathbf { e } ) ^ T \mathbf { e } = 0 . \end{align*}


\begin{align*} \tilde U ^ { i j } D _ { i j } \tilde w = - \gamma ^ { - 1 } f , \end{align*}


\begin{align*} d = R \sqrt { 2 } \sqrt { \frac { n + 1 } { n } } \end{align*}


\begin{align*} \{ \tilde x \in \widetilde W \mid \exists x _ L \in \widetilde W _ L , x _ R \in \widetilde W _ R : ~ x = x _ L \tilde x x _ R \ell ( x ) = \ell ( x _ L ) + \ell ( \tilde x ) + \ell ( x _ R ) \} , \end{align*}


\begin{align*} Q _ { n + 1 } ( \lambda ) = \frac { 1 } { d _ n } [ ( \lambda - b _ n ) Q _ n ( x ) - c _ { n - 1 } Q _ { n - 1 } ( \lambda ) ] . \end{align*}


\begin{align*} x ^ G = \{ g ^ { - 1 } x g \mid g \in G \} , \end{align*}


\begin{align*} \begin{aligned} & I _ { h , W , \gamma } = \\ & \inf \left \{ \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } | b | ^ { 2 } \dd x \ \middle | \ b = \nabla \times ( \phi \nabla \theta ) + G \nabla \theta \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) , 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ { \infty } ) _ { + } \frac { G } { r ^ { 2 } } \dd x = h \ \right \} , \end{aligned} \end{align*}


\begin{align*} \dot H ^ r : = D \left ( \partial _ { x x } ^ { \frac r 2 } \right ) = \left \{ h \in H : \| h \| _ { \dot H ^ r } ^ 2 : = \sum _ { j = 1 } ^ { \infty } \lambda _ j ^ { r } \langle e _ j , h \rangle ^ 2 < \infty \right \} . \end{align*}


\begin{align*} \dfrac { \hat { k } _ 1 { ( G ) } } { \hat { k } _ 1 ( G \setminus \sigma ) } = \dfrac { D _ { \ell _ 1 } D _ { \ell _ 2 } } { D _ { \ell _ 1 - 1 } D _ { \ell _ 2 - 1 } } . \end{align*}


\begin{align*} p ^ { C _ 5 } = e . \end{align*}


\begin{align*} V _ { \mathrm { d e t } } = \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } g \ , d \gamma _ { y , T } - \frac { 1 } { n } H ( \gamma _ { y , T } \ , | \ , \gamma _ T ) \right ) = \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } g \ , d \gamma _ { y , T } - \frac { | y | ^ 2 } { 2 n T } \right ) , \end{align*}


\begin{align*} \dot { x } ( t ) & = - A x ( t ) + B u ( t ) \quad ( t > 0 ) , \\ x ( 0 ) & = x _ 0 \in X , \end{align*}


\begin{align*} H \left ( P ^ { ( n , n ) } [ t ] \mid \mu ^ { \otimes n } [ t ] \right ) = \mathbb { E } _ \mathbb { P } [ Z _ t ^ n \log Z _ t ^ n ] . \end{align*}


\begin{align*} b ( x ) = \nabla \times ( \phi ( z , r ) \nabla \theta ) + G ( z , r ) \nabla \theta . \end{align*}


\begin{align*} x _ { n + 1 } = \begin{cases} x _ n - 1 , & x _ n > 1 \\ n + 1 , & x _ n = 1 . \end{cases} \end{align*}


\begin{align*} d X _ t ^ i = \left ( b _ 0 ( t , X ^ i ) + \frac { 1 } { n - 1 } \sum _ { j = 1 , j \neq i } ^ n b \left ( t , X ^ i , X ^ j \right ) \right ) d t + d W ^ i _ t , i = 1 , \cdots , n , \end{align*}


\begin{align*} s ( \nu | \mu ) : = \lim _ { n \to \infty } \frac { 1 } { \abs { \Lambda _ n } } s _ n ( \nu | \mu ) . \end{align*}


\begin{align*} \mathbf { x } \circ \mathbf { s } = ( \mathbf { x } ) ( \mathbf { s } ) \mathbf { e } , \end{align*}


\begin{align*} H ( t , y , z , u , \mu , p , q ) : = \big < f ( t , y , z , u , \mu ) , p \big > + \big < g ( t , y , z , u , \mu ) , q \big > + h ( t , y , z , u , \mu ) . \end{align*}


\begin{align*} [ [ x , y ] , [ x , z ] ] = 0 , \forall \ y , z \in A . \end{align*}


\begin{align*} \Phi ^ + : \Phi \rightarrow \{ 0 , 1 \} , \alpha \mapsto \begin{cases} 1 , & \alpha \in \Phi ^ + , \\ 0 , & \alpha \in \Phi ^ - . \end{cases} \end{align*}


\begin{align*} \delta _ j ^ 1 & = \min \bigl \{ 1 , \frac { d ( a _ j , c _ j ) } { C _ { n + 1 } \alpha _ { j } } \bigr \} , \\ \delta _ j ^ 2 & = \min \bigl \{ 1 , \frac { d ( c _ j , d _ j ) } { C _ { n + 1 } ( \alpha _ { m + j } + \beta _ k ) } \bigr \} , \\ \delta _ j ^ 3 & = \min \bigl \{ 1 , \frac { d ( d _ j , b _ j ) } { C _ { n + 1 } \alpha _ { 2 m + j } } \bigr \} . \end{align*}


\begin{align*} p _ { _ { l , 2 } } \varphi _ l = - { \bf i } q _ { _ { l , 1 } } \vartheta _ l \ \textrm { o n } \ \C \ \textrm { f o r } \ l = 1 , 2 . \end{align*}


\begin{align*} & \o ^ \beta ( t , x ) = \o _ 0 ^ \beta ( X ^ \beta ( 0 ; t , x ) ) , \\ & \frac { d } { d s } X ^ \beta ( s ; t , x ) = u ^ \beta ( s , X ^ \beta ( s ; t , x ) ) , \ \ X ^ \beta ( s ; t , x ) | _ { s = t } = x . \end{align*}


\begin{align*} F ^ { q , \dagger } _ { v v } ( 0 , p ) [ u _ k , u _ k ] & = c _ 2 ( q , k , p ) u _ { 2 k } \end{align*}


\begin{align*} \mathbf { v } \cdot \overrightarrow { \zeta } = 0 , \end{align*}


\begin{align*} F ( x ) = \sum _ { { \bf k } \in \Gamma _ { r } \backslash \Lambda _ { 1 } } \begin{pmatrix} a _ { \bf k } & 0 \\ 0 & - a _ { \bf k } \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } + \sum _ { { \bf k } \in \Gamma _ { r } \backslash \Lambda _ { 2 } } \begin{pmatrix} 0 & b _ { \bf k } \\ c _ { \bf k } & 0 \end{pmatrix} { \rm e } ^ { { \rm i } \langle { \bf k } , x \rangle } . \end{align*}


\begin{align*} D _ { \xi _ 0 } ( k ) : = \left \lbrace z \in \mathbb { C } : \ | z - \xi _ 0 | < k \mu ( \xi _ 0 ) \right \rbrace \ . \end{align*}


\begin{align*} \mathcal { G } ^ i \triangleq \left \{ \theta ^ i \mathbf { G } ^ i : \theta ^ i > 0 , \mathbf { G } ^ i \in \mathbf { R } ^ { n _ i \times n _ i } , ( \mathbf { G } ^ i ) ^ T \mathbf { Q } ^ i \mathbf { G } ^ i = \mathbf { Q } ^ i \right \} , \end{align*}


\begin{align*} 0 = D ( i y ) D ( x ) - D ( x ) D ( i y ) = D ( v ) , \end{align*}


\begin{align*} \Pi _ \lambda f = \frac { 1 } { 2 \pi } \int _ { - \pi } ^ { \pi } e ^ { i \frac t 2 ( \lambda - \mathcal H ) } f d t , f \in \mathcal S ( \mathbb R ^ d ) . \end{align*}


\begin{align*} f ' ( t ) = \frac { ( t - a ) ^ { \alpha - 2 } } { \Gamma ( \alpha ) } \left [ \frac { \alpha - 1 } { \alpha - \beta } ( b - a ) - ( t - a ) \right ] . \end{align*}


\begin{align*} a d - b c = \frac { 1 } { 2 } [ ( a - b ) ( c + d ) - ( a + b ) ( c - d ) ] \end{align*}


\begin{align*} { \rm R e } \langle u _ { t t } , v \rangle + { \rm R e } \int _ { \mathbb { H } ^ n } \nabla _ { H } u \cdot \nabla _ { H } v d x + m { \rm R e } \int _ { \mathbb { H } ^ n } u v d x + b { \rm R e } \int _ { \mathbb { H } ^ n } u _ t v d x \ = { \rm R e } \int _ { \mathbb { H } ^ n } f ( u ) v d x , \end{align*}


\begin{align*} - y ^ { \prime \prime } + q ( x ) y + \frac { 1 } { 4 } y = 0 . \end{align*}


\begin{align*} \abs { F ( k p _ i ) - \left ( 1 - \frac { 1 } { p _ i } \right ) F ( k ) } \le \sum _ { d | k } \mu ^ 2 ( d ) \left ( 1 + \frac 1 p _ i \right ) \sqrt p ( \log p + 1 ) = ( 1 + 1 / p _ i ) W ( k ) \sqrt { p } ( \log p + 1 ) \end{align*}


\begin{align*} | u _ { m ( p ) - 1 } \dots u _ n | _ p \geq \prod ^ { m ( p ) - n } _ { r = 1 } ( | t _ r | _ p | a ' b ' | _ p ) . \end{align*}


\begin{align*} \begin{aligned} & \mathcal { K } _ { L } ^ { n } = \{ \mathbf { v } \in { { \mathbf { R } } ^ { n } } : \mathbf { v } \ge 0 \} , n \ge 1 , \\ & \mathcal { K } _ { S } ^ { n } = \{ \mathbf { v } \in { { \mathbf { R } } ^ { n } } : { { \mathbf { v } } _ { 1 } } \ge \left \| { { \mathbf { v } } _ { 2 : n } } \right \| \} , n \ge 2 , \end{aligned} \end{align*}


\begin{align*} \kappa \begin{pmatrix} 1 \\ \vdots \\ \theta ^ { n - 1 } \end{pmatrix} = \omega ^ { - 1 } \begin{pmatrix} a _ { 1 } \\ \vdots \\ a _ { n } \end{pmatrix} , \end{align*}


\begin{align*} \mu _ t ^ { * , \xi } = k ( t , \nu _ t ^ { * , \xi } , \varphi _ t ^ { * , \xi } ) . \end{align*}


\begin{align*} \langle z _ 1 ^ * , v \rangle & = 2 \int _ \S \sin ( 2 \pi \ell x ) v ( x ) \ \d x , \\ \langle z _ 2 ^ * , v \rangle & = - 2 \int _ \S \cos ( 2 \pi \ell x ) v ( x ) \ \d x , \end{align*}


\begin{align*} \frac { \partial \tilde x } { \partial x } = h ^ { \frac { 1 } { 3 } } + x \cdot \frac { 1 } { 3 } h ^ { - \frac { 2 } { 3 } } \cdot h _ x ' , \end{align*}


\begin{align*} \big | M ^ * _ v ( t ) - M ^ * _ v ( \delta n ) \big | & \le e ^ { - ( \lambda ^ * - \lambda ) t } \big | M _ v ( t ) - M _ v ( \delta n ) \big | + \big ( e ^ { - ( \lambda ^ * - \lambda ) \delta n } - e ^ { - ( \lambda ^ * - \lambda ) t } \big ) | M _ v ( \delta n ) | \\ & \le \delta e ^ { - ( \lambda ^ * - \lambda ) ( t - \delta n ) } + \delta \big ( 1 - e ^ { - ( \lambda ^ * - \lambda ) ( t - \delta n ) } \big ) = \delta , \end{align*}


\begin{align*} \tilde c ( t ) : = c ( t + \alpha ) . \end{align*}


\begin{align*} \frac { p ' _ { h + 1 } ( x ) } { p _ { h + 1 } ( x ) } = \frac { 1 } { 2 \sqrt x } \Big ( \frac { p _ h ( \sqrt x ) } { p _ h ( \sqrt { - x } ) } - \frac { p _ h ( \sqrt { - x } ) } { p ' _ h ( \sqrt x ) } \Big ) , ~ h = 0 , 1 , \dots , \end{align*}


\begin{align*} { \tt P s i 1 = p l g c i r m a p ( v e r s , h a l p h a ) } \end{align*}


\begin{align*} h _ d ( q ) = \log ( c _ { d , 2 } ( q ) ^ q ) - \log ( c _ { d , \infty } ( q ) ^ q ) . \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | e ^ { i k \phi } \cos ( n \theta ) \left | l _ 2 , m _ 2 \right \rangle = \end{align*}


\begin{align*} \sup _ { y \in \Z ^ d } \sum _ { x \neq y } \sum _ { \Delta \ni y } \sum _ { \xi _ \Delta } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ x c _ \Delta ( \cdot , \xi _ \Delta ) } _ \infty < \infty . \end{align*}


\begin{align*} \frac { 2 \epsilon _ f + 4 / a + r } { h ( \gamma \bar \Delta ) } T + \left [ \sum _ { k = 0 } ^ { T - 1 } \Theta _ k ( 1 - \Lambda _ k ' ) + ( 1 - \Theta _ k ) \Lambda _ k \right ] < \left ( \frac { 2 \epsilon _ f + 4 / a + r } { h ( \gamma \bar \Delta ) } + \frac { 1 } { 2 } \right ) T + \frac { 1 } { 2 } \left | \log _ \gamma \frac { \bar \Delta } { \delta _ 0 } \right | + \frac { 1 } { 2 } . \end{align*}


\begin{align*} G ^ { D _ { N } } ( [ x N ] , [ x N ] ) = \gamma \log N + s _ { D } ( x ) + o ( 1 ) \end{align*}


\begin{align*} \left [ \sum _ { m = 0 } ^ \infty \frac { ( - 1 ) ^ m } { ( 1 / s + m ) m ! } y ^ m \right ] ^ j = \sum _ { m = 0 } ^ \infty c _ { m , j } y ^ { m } , \end{align*}


\begin{align*} M ( x ; z ) : = \begin{cases} e ^ { i c _ + ( x ) \sigma _ 3 } \left ( \frac { \Psi _ 1 ^ - ( x ; z ) } { a ( z ) } , \Psi ^ + _ 2 ( x ; z ) \right ) , \ ; z \in \mathbb { C } ^ + , \\ [ 5 p t ] e ^ { i c _ + ( x ) \sigma _ 3 } \left ( \Psi _ 1 ^ + ( x ; z ) , \frac { \Psi ^ - _ 2 ( x ; z ) } { \bar { a } ( z ) } \right ) , \ ; z \in \mathbb { C } ^ - , \end{cases} \end{align*}


\begin{align*} \mathcal { Z } ^ * _ K ( \underline { \mathrm { p t } ^ { J } } , \{ \emptyset \} ) ( \underline { \mathrm { p t } ^ J } , \{ \emptyset \} ) & = \left \{ \left ( { \bigsqcup } _ { j _ i } \mathrm { p t } , \{ \emptyset \} \right ) \right \} _ { i \in \{ 1 , \ldots , m \} } . \end{align*}


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


\begin{align*} ( x , . . . , x ) \in X ^ { \sum _ i M _ i } = p _ X ^ { \otimes \sum _ i M _ i } [ \gamma _ { 1 , 1 } , . . . , \gamma _ { N , M _ N } ] , \end{align*}


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


\begin{align*} F ( k p _ i ) = \sum _ { d | k p _ i } \mu ( d ) N ( d ) = \sum _ { d | k } \mu ( d ) ( N ( d ) - N ( d p _ i ) ) = \sum _ { d | k } \frac { \mu ( d ) ( p + 1 ) } { 2 d } \left ( 1 - \frac 1 { p _ i } \right ) + \sum _ { d | k } \mu ( d ) ( \xi ( d ) - \xi ( d p _ i ) ) . \end{align*}


\begin{align*} \mathbf { a } ( \theta ) = \frac { 1 } { \sqrt { N } } { [ e ^ { - j 2 \pi \theta v } ] } _ { v \in \mathcal { V } } , \end{align*}


\begin{align*} ( a b ) ^ i { } _ { j k l } & = a ^ i { } _ { \alpha \beta \gamma } b ^ \alpha { } _ j b ^ \beta { } _ k b ^ \gamma { } _ l + a ^ i { } _ { \alpha \beta } b ^ \alpha { } _ { j l } b ^ \gamma { } _ k + a ^ i { } _ { \alpha \beta } b ^ \alpha { } _ j b ^ \gamma { } _ { k l } + a ^ i { } _ { \alpha \beta } b ^ \beta { } _ l b ^ \gamma { } _ { j k } + a ^ i { } _ \alpha b ^ \alpha { } _ { j k l } \end{align*}


\begin{align*} \widetilde h ( t , x , y ) = h ( \widetilde k ( t , x , y ) ) \quad \widetilde b ( t , x , y ) = b ( \widetilde k ( t , x , y ) ) . \end{align*}


\begin{align*} - \Delta w = w ^ p _ + , \ \ \ \mathbb { R } ^ 2 . \end{align*}


\begin{align*} \chi \cap \psi = \psi \setminus \{ x \} . \end{align*}


\begin{align*} R _ { 0 , j } ^ n = B _ { \rm R F } \log _ 2 \left ( 1 + \frac { p _ { 0 , j } ^ n \left | G _ { 0 , j } ^ m \right | ^ 2 } { N _ { \rm R F } B _ { \rm R F } } \right ) , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ 3 x _ k \psi _ k \rightarrow ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \end{align*}


\begin{align*} & R _ + ( x ; z ) = \begin{pmatrix} 0 & \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \\ 0 & 0 \end{pmatrix} , R _ - ( x ; z ) = \begin{pmatrix} 0 & 0 \\ r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } & 0 \end{pmatrix} , \\ & F = \left ( \mathcal { P } ^ { - } ( r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } ) e _ 2 , \mathcal { P } ^ + ( \bar { r } _ 1 ( z ) e ^ { - 2 i z x } ) e _ 1 \right ) , \end{align*}


\begin{align*} S ^ { \rm P R } ( B ) = \kappa ( \rho ) W \in \R ^ { N L \times N L } , \end{align*}


\begin{align*} W _ n ( t ) = Q _ n W ( t ) : = \sum _ { i = 1 } ^ n \langle W ( t ) , h _ i \rangle h _ i , \end{align*}


\begin{align*} \langle \overline { y } - g , S ' ( \overline { u } ; h ) \rangle = \langle \overline { y } - g , z \rangle = \langle \zeta z , p \rangle + \langle - \Delta z , p \rangle = \langle \zeta z , p \rangle - \langle F ' ( \overline { y } ; z ) , p \rangle + \langle h , p \rangle \ge - \alpha \langle \overline { u } , h \rangle . \end{align*}


\begin{align*} q ( x ) = x e ^ { - x ^ { 2 } } . \end{align*}


\begin{align*} Z ^ 2 ( v , s ) = \Psi ^ q + v + \frac 1 2 \psi _ { v v } ( 0 , p _ 0 ) [ v , v ] \end{align*}


\begin{align*} \psi _ { \tau } ( p ) = | | \tau | | _ p \le C _ 1 ( \beta , \gamma , L ) \ ( \beta - p ) ^ { - ( \gamma + 1 ) / \beta } \ L ^ { 1 / \beta } ( 1 / ( \beta - p ) ) , \end{align*}


\begin{align*} \mathcal { I } ( x , t ) = \bigcup _ { i = 1 } ^ p [ t _ { i , e n } ( x , t ) , t _ { i , e x } ( x , t ) ] \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \| u _ { \epsilon } \| _ { L ^ p ( ( 0 , T ) \times \Omega ) } = \| u _ 0 \| _ { L ^ p ( ( 0 , T ) \times \Omega \times Y ) } . \end{align*}


\begin{align*} A = B \circ g \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ { f } \ , d \gamma _ t - \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } f \ , d \gamma _ { y , t } - H ( \gamma _ { y , t } \ , | \ , \gamma _ t ) \right ) = \inf _ { y \in \R ^ n } H ( \gamma _ { y , t } \ , | \ , P ) . \end{align*}


\begin{align*} \int _ B \tilde { p } ( X _ { 1 : n } | \theta ) d \tilde { Q } _ k ( \theta ) = \sum _ { \sigma \in S _ k } \int _ B p ( X _ { 1 : n } | \theta ) d Q _ k ( \theta [ \sigma ] ) = \int _ { \bigcup _ { \sigma \in S _ k } B [ \sigma ] } p ( X _ { 1 : n } | \theta ) d Q _ k ( \theta ) \end{align*}


\begin{align*} T ^ { ( N ) } : M _ N ( E _ 1 ) \times \ldots \times M _ N ( E _ n ) \to M _ N ( E ) \end{align*}


\begin{align*} \lim \sup _ { t \rightarrow \infty } t ^ { - 1 / \eta } L ^ { \ast } \left ( t \right ) = \infty \end{align*}


\begin{align*} L = a _ n x ^ { q ^ n } + a _ { n - 1 } x ^ { q ^ { n - 1 } } + \cdots + a _ 0 x , \end{align*}


\begin{align*} a _ d ( n ) = \sum _ { k = 2 } ^ { n + 1 } - \mu _ d ( k ) a _ d ( n + 1 - k ) \end{align*}


\begin{align*} 0 \rightarrow E \rightarrow \bigoplus _ { j = 1 } ^ n E _ j \rightarrow \mathcal { T } _ E \rightarrow 0 , \end{align*}


\begin{align*} g _ 1 ' : = \frac { 1 } { 2 } ( 1 + i I ) v ' \end{align*}


\begin{align*} ( r ( t ) u ' ) ' + q ( t ) u = 0 , \end{align*}


\begin{align*} \frac { 1 - e ^ { a _ { \alpha + n + 1 } + a _ { \alpha + n + 2 } } } { \left ( e ^ { - a _ { \alpha + 1 } } - 1 \right ) \left ( 1 - e ^ { a _ { n + 1 } } \right ) \exp \left [ \sum _ { j = 2 } ^ n ( a _ j - a _ { \alpha + j } ) \right ] } \end{align*}


\begin{align*} M _ { I J } = \begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \cdots & G _ { I _ s } \\ G _ { J _ 1 } & G _ { J _ 2 } & \cdots & G _ { J _ s } \end{pmatrix} . \end{align*}


\begin{align*} C _ 2 ( \mathcal { H } _ { \ast } ) = \mathrm { I m } \ , \varphi _ 1 \oplus s _ { _ { 2 } } ( \mathrm { I m } \ , \partial ' _ 0 ) = s _ { _ { 2 } } ( \mathrm { I m } \ , \partial ' _ 0 ) . \end{align*}


\begin{align*} \left ( - \sqrt { f ( r ) } \frac { d } { d r } f ( r ) \frac { d } { d r } \sqrt { f ( r ) } + V ( r ) - E \right ) \psi ( r ) = 0 , \end{align*}


\begin{align*} \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | _ n ^ 2 = \sum _ { | I | = n } ( \det \ U _ I ) ^ 2 . \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( 0 , p _ 0 ) \psi ^ \dagger _ p ( 0 , p _ 0 ) + ( I - Q ^ \dagger ) F ^ { q , \dagger } _ p ( 0 , p _ 0 ) = 0 . \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } ( \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 + \sqrt { c _ 2 } \ X _ 2 \cdot \nu _ 2 ) \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} & P _ 1 = e ^ { \frac { i } { 2 } \int _ { - \infty } ^ x | u _ y ( y , t ) | ^ 2 d y \widehat { \sigma } _ { 3 } } ( k P _ x + \frac { i } { 2 } | u _ x | ^ 2 \sigma _ { 3 } ) . \end{align*}


\begin{align*} s _ i ( A ) = \min \limits _ { E : \ ; \dim ( E ) = n - i + 1 } \max \limits _ { x \in E , \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 , 1 \leq i \leq n , \end{align*}


\begin{align*} \mathcal { I } : G _ { \kappa } \rightarrow S O \left ( M _ { k } \right ) \mathcal { I } \left ( g \right ) = \left ( d g \right ) _ { o } \end{align*}


\begin{align*} a \succ b = \mathfrak { l } ( T ( a ) ) b \ \ \ \ a \prec b = \mathfrak { r } ( T ( b ) ) a . \end{align*}


\begin{align*} H _ { m i n } \left ( ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) \otimes F \right ) & = H _ { m i n } ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) H _ { m i n } ( F ) \\ & = H _ { m i n } ( E _ 1 [ t ] ) H _ { m i n } ( F ) \\ & = t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) . \end{align*}


\begin{align*} z _ 0 ^ { p ^ r } x _ 0 + \cdots + z _ n ^ { p ^ r } x _ n = 0 . \end{align*}


\begin{align*} \sum _ { d = 0 } ^ j x _ { i , d } = x _ i , \ , \sum _ { i = 1 } ^ r x _ i = t . \end{align*}


\begin{align*} \mathcal { L } ( u ) : = u _ t - A ( x , t ) u _ { x x } + B ( x , t ) u _ x + C ( x , t ) u , \end{align*}


\begin{align*} v ( p ) = \sup \{ u ( p ) : \} , p \in \Omega . \end{align*}


\begin{align*} N ( f ) & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( I - A ^ i M ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( A ^ { - i } ) \det ( I - A ^ i M ) \det ( A ^ i ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( I - M A ^ i ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( I - M ) \rvert = N ( \tilde f ) . \end{align*}


\begin{align*} \mu ( [ \gamma , u ] ) = n + C Z ( [ \gamma , u ] ) \end{align*}


\begin{align*} y _ n = \rho ( x ) - \rho ( x _ 0 ) \geq \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta + \kappa | y ' | ^ 2 , \end{align*}


\begin{align*} \sum _ { \Delta \Subset \Z ^ d } \sum _ { \xi _ { \Delta } } \nabla _ { \Lambda } \left ( c _ { \Delta } ( \cdot , \xi _ { \Delta } ) - \hat { c } ( \cdot , \xi _ { \Delta } ) \right ) ( \eta ) = 0 , \end{align*}


\begin{align*} \lambda _ { 2 } \left ( A _ { 3 } \right ) \leq \lambda _ { 2 } \left ( M _ { 3 } \right ) + \lambda _ { 3 } \left ( D _ { 3 } \right ) = \lambda _ { 2 } \left ( M _ { 3 } \right ) + a _ { 3 } . \end{align*}


\begin{align*} \epsilon ( \pi ^ * D , y ) = \epsilon ( D , \pi ( y ) ) . \end{align*}


\begin{align*} h _ { \Delta } ( \eta ) : = \sum _ { \zeta _ { \Delta } } \nabla _ { \Lambda } ( c _ { \Delta } ( \cdot , \zeta _ { \Delta } ) - \hat { c } _ { \Delta } ( \cdot , \zeta _ { \Delta } ) ) ( \eta ) , \eta \in \Omega . \end{align*}


\begin{align*} \mathbf { T } = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix} . \end{align*}


\begin{align*} \left ( Q _ - J + D _ j \right ) _ { i 1 } = \left ( M _ - V _ j J \right ) _ { i 1 } = \left ( M _ - R V _ j \right ) _ { i 1 } = \left ( M _ - R \right ) _ { i 1 } , i = 1 , 2 , \end{align*}


\begin{align*} \begin{aligned} C _ m ( U ) & = \sup \left \{ \int _ { U } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } : u \in Q S H _ m ( \Omega ) \cap C ( \Omega ) , - 1 \leq u < 0 \right \} \\ & = \sup \left \{ \int _ { U } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } : u \in Q S H _ m ( \Omega ) \cap C ^ { \infty } ( \Omega ) , - 1 \leq u < 0 \right \} \end{aligned} \end{align*}


\begin{align*} \mathbf { c } _ Y ^ T \mathbf { c } _ X = \| \mathbf { c } _ X \| _ 2 \| \mathbf { c } _ Y \| _ 2 \cos \phi , \end{align*}


\begin{align*} \langle Q ^ M _ x ( v ) , v \rangle _ M = \sum ^ m _ { i = 1 } \bigg ( 2 \langle B ( v , e _ i ) , B ( v , e _ i ) \rangle _ { \mathbb R ^ q } - \langle B ( v , v ) , B ( e _ i , e _ i ) \rangle _ { \mathbb R ^ q } \bigg ) \end{align*}


\begin{align*} \left \{ \begin{aligned} - d y _ t = & f ( t , y _ t , z _ t , u _ t , \mathcal L ( y _ t , z _ t , u _ t ) ) d t + g ( t , y _ t , z _ t , u _ t , \mathcal L ( y _ t , z _ t , u _ t ) ) d \overleftarrow B _ t \\ & - z _ t d W _ t , \ , \ , \ , \ , t \in [ 0 , T ] , \\ y _ T = & \xi . \end{aligned} \right . \end{align*}


\begin{align*} 1 + \sum _ { v ^ * } \ ! \Bigg ( \prod _ { s \ge 1 } \frac { q ^ { \langle v ^ s \ ! , \ , v ^ s \rangle } } { q ^ { \langle \beta ^ s \ ! , \ , \beta ^ s \rangle } } \frac { c ( v ^ s , q ) } { \mathit { g l } ( v ^ s , q ) } X ^ { s v ^ s } \Bigg ) = \ , \mathrm { E x p } \Bigg ( \sum _ { v \in \mathbb { N } ^ n \backslash \{ 0 \} } \ ! \frac { a ( v , q ) } { q - 1 } X ^ v \Bigg ) , \end{align*}


\begin{align*} \sum _ { \Delta \Subset \Z ^ d } \sum _ { \xi _ { \Delta } } \nabla _ { \Lambda } \left ( c _ { \Delta } ( \cdot , \xi _ { \Delta } ) - \hat { c } ( \cdot , \xi _ { \Delta } ) \right ) ( \eta ) = 0 . \end{align*}


\begin{align*} 2 m + n = \sum _ { x \in G } { \frac { | G | } { | x ^ G | } } = | G | \cdot \sum _ { x \in G } { \frac { 1 } { | x ^ G | } } = n \cdot k . \end{align*}


\begin{align*} _ { \rm r e f } [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x + y ) } \ \prod _ { i = 1 } ^ L X _ i \ \prod _ { 1 \leq i < j \leq L } X _ { i , j } \end{align*}


\begin{align*} \left | \dfrac { \partial a _ { 0 , \delta } ( x ) } { \partial x } \right | _ { x = x _ a } \le 2 \int _ 0 ^ 1 \left | \left . \dfrac { \partial h ( x ) } { \partial x } \right | _ { x = x _ a + \delta u ( t ) } \right | \ , d t \le 2 L _ r . \end{align*}


\begin{align*} \sum _ { k \geq N } \langle X _ n e _ k , e _ k \rangle = \sum _ { k \geq N } \| X _ n ^ { \frac 1 2 } e _ k \| ^ 2 \geq \sum _ { k , l \geq N } \langle X _ n ^ { \frac 1 2 } e _ k , e _ l \rangle ^ 2 = \sum _ { k , l \geq N } \langle X _ n ^ { \frac 1 2 } , e _ k \otimes e _ l \rangle _ { \mathcal H } ^ 2 . \end{align*}


\begin{align*} \sigma ( t ) = \max ( t , 0 ) , t \in \mathbb { R } . \end{align*}


\begin{align*} e _ c \left [ t + 1 \right ] = & \left \{ 1 - 4 \mu \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - 2 \mu f ^ { \left ( c \right ) } \vphantom { \sum _ { j = 1 } ^ { M } } \right \} e _ c \left [ t \right ] - 2 \mu \left ( 2 \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 + \mu f ^ { \left ( c \right ) } \right ) a _ c ^ { \left ( o \right ) } \\ & - 2 \mu \sum _ { l } ^ { M } \Re \left \{ \phi ^ { \left ( l , c \right ) } \right \} . \end{align*}


\begin{align*} [ V ^ W ] = [ V \otimes F ' ] \end{align*}


\begin{align*} g = \widetilde { g } \ , t ^ { - \sigma ( l ) } \widetilde { g } = \prod _ { k = 1 } ^ l \widetilde { s _ { \iota ( k ) } } ^ { \ , t ^ { \sigma ( k - 1 ) } } , \end{align*}


\begin{align*} [ T ] = \frac { 1 } { \mu } J ( \partial _ x h + I \partial _ y h ) = \frac { 2 } { \mu } h _ z J \end{align*}


\begin{align*} \gamma _ 1 + \gamma _ 2 = \beta _ 1 + \beta _ 2 \gamma _ 1 \leq \beta _ 1 , \beta _ 2 \leq \gamma _ 2 \end{align*}


\begin{align*} R _ { K } = \log _ { 2 } \bigg ( 1 + \frac { P _ { K } { \lvert \mathbf { d } _ { K } ^ { H } \mathbf { h } _ { K } \rvert } ^ { 2 } } { \sum _ { \pi _ { k ' , i ' } > \pi _ { K } } P _ { k ' , i ' } { \lvert \mathbf { d } _ { k ' , i ' } ^ { H } \mathbf { h } _ { k ' } \rvert } ^ { 2 } + \sigma _ { \textrm { u l } } ^ { 2 } } \bigg ) . \end{align*}


\begin{align*} \omega ^ { \prime } = p - \omega . \end{align*}


\begin{align*} \nabla _ X \psi _ 1 = \frac { 1 } { 2 } X _ 1 \cdot \psi _ 1 - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi _ 1 \end{align*}


\begin{align*} T _ { e u c l } ( w ) = \inf \left \{ \max \{ f ( p ) , f ( q ) \} : w \in [ p , q ] , p \in \Omega , q \in \Omega \right \} , w \in \R ^ 3 . \end{align*}


\begin{align*} v ^ { { \rm c o l } ( \mu ) } \prod _ { ( i , j ) \in \mu } X _ { a - j + 1 } = \prod _ { i = 1 } ^ { { \rm c o l } ( \mu ) } \bigg ( v \prod _ { j = 1 } ^ { \mu ' _ i } X _ { a - j + 1 } \bigg ) , \end{align*}


\begin{align*} \beta _ n = \sum \limits _ { l = 0 } ^ { n - 1 } \omega ^ l \wedge \omega ^ { n + l } , \beta _ n ^ n = \wedge ^ n \beta _ n = n ! ~ \Omega _ { 2 n } , \end{align*}


\begin{align*} ( ( \C ^ * ) ^ n , \omega _ { \mathrm { l o g } } ) , \omega _ { \mathrm { l o g } } = \i \sum \frac { d z ^ i } { z _ i } \wedge \frac { d \bar { z } ^ i } { \bar { z _ i } } , \end{align*}


\begin{align*} \begin{aligned} & P \Big ( \sup _ { t \in [ \iota _ 1 , \tau ] } \lambda ( \epsilon ) \epsilon \int ^ { t } _ { \iota _ 1 } \gamma ( s ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) > a \Big ) \\ & \leq \exp \left \{ - \lambda ^ { - 2 } ( \epsilon ) \Big ( \eta a - \frac { 1 } { 2 } \eta ^ 2 \rho ^ 2 ( \iota _ 2 - \iota _ 1 ) e ^ { \epsilon \lambda ^ { - 1 } ( \epsilon ) \eta \rho } \Big ) \right \} . \end{aligned} \end{align*}


\begin{align*} B \cdot n = 0 . \end{align*}


\begin{align*} \mathbb { E } \left [ y _ i ^ * y _ i \right ] = a _ c ^ 2 \lvert \mathbf { h } _ { i , * } \mathbf { p } _ c \rvert ^ 2 + \sum _ { l = 1 } ^ M a _ l ^ 2 \lvert \mathbf { h } _ { i , * } \mathbf { p } _ l \rvert ^ 2 + \sigma _ n ^ 2 . \end{align*}


\begin{align*} [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x + y ) } \ \prod _ { i = 1 } ^ L X _ i \ \prod _ { 1 \leq i < j \leq L } X _ { i , j } \ \Big \vert _ { y = 1 } \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } _ v ( v + \psi ^ \dagger ( v , p ) , p ) \psi _ p ( v , p ) + ( I - Q ^ \dagger ) F ^ { q , \dagger } _ p ( v + \psi ^ \dagger ( v , p ) , p ) = 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} \triangle u & = | x | ^ a v ^ p & & \quad \mbox { i n } \ B _ { R } , \\ \triangle v & = | x | ^ { b } v ^ q | \nabla u | ^ { s } & & \quad \mbox { i n } \ B _ { R } . \end{aligned} \right . \end{align*}


\begin{align*} h _ c ( G ) - h _ c ( G \backslash v ) & = f _ c ( b ) - f _ c ( 0 ) + \sum _ { i = 1 } ^ b \Delta _ c ( d _ i ) \\ & \le f _ c ( b ) - f _ c ( 0 ) + b \cdot \Delta _ c \left ( \frac { m + \binom { b } { 2 } } { b } \right ) \\ & : = L H S ( m , b , c ) . \end{align*}


\begin{align*} z m ( x ; z ) e _ 2 - ( I - F ) \left ( \widehat { \Psi } ^ - _ { 1 1 } e _ 1 + \widehat { \Psi } ^ - _ { 2 1 } e _ 2 \right ) = \widetilde { m } ( x ; z ) e _ 2 , \end{align*}


\begin{align*} | H \cap W ' | q ^ { d ^ 2 - d } \theta _ { d - 1 } \le \begin{cases} | W ' \cap \tau _ i | q ^ { d ^ 2 - d } \theta _ { d - 1 } & H \cap U = \tau _ i i \in \{ 1 , 2 \} , \\ ( q ^ { d - 1 } + \theta _ { d - 1 } ) q ^ { d ^ 2 - d } \theta _ { d - 1 } & \end{cases} \end{align*}


\begin{align*} Q ( n ) = \mathcal { K } m ^ { n } \bigl ( { 1 + o ( 1 ) } \bigr ) n \to \infty , \end{align*}


\begin{align*} \mathcal { W } _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , g , \psi ) : = \inf \bigg \{ \sum _ { i } c _ { i } ^ { - \alpha n _ { i } + S _ { 1 , n _ { i } } \psi ( x _ { i } , \epsilon ) } \bigg \} , \end{align*}


\begin{align*} C _ m ( K ) = \sup \left \{ \int _ { K } \Delta u _ 1 \wedge \cdots \wedge \Delta u _ m \wedge \beta _ n ^ { n - m } ; u _ j \in Q S H _ m ( \Omega ) \cap C ( \Omega ) , - 1 \leq u _ j < 0 \right \} . \end{align*}


\begin{align*} \beta _ i ( \gamma , x ) + \beta _ i ( \gamma , y ) = f _ i ( \gamma x , \gamma y ) - f _ i ( x , y ) ( \gamma \in \Gamma , \ x \ne y \in X , \ i = 1 , 2 ) . \end{align*}


\begin{align*} \psi _ { \tau } ( p ) \stackrel { d e f } { = } \left [ \ p \int _ 0 ^ { \infty } \ t ^ { p - 1 } \ T _ { \tau } ( t ) \ d t \ \right ] ^ { 1 / p } = | | \tau | | L _ p ( \Omega ) , \end{align*}


\begin{align*} & \int _ { X _ j } | F _ { \epsilon , j } - ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ h e ^ { v _ \epsilon ( \Psi ) - \delta \max \{ \psi + T , 2 \log | F | \} } c ( - v _ \epsilon ( \Psi ) ) \\ \le & \int _ { X _ j } v '' _ { \epsilon } ( \Psi ) | f F | ^ 2 _ h e ^ { - u ( - v _ { \epsilon } ( \Psi ) ) } < + \infty . \end{align*}


\begin{align*} \left ( \int \varphi d | \mu - \nu | \right ) ^ 2 \leq 4 C H \left ( \mu \mid \nu \right ) , C = \frac { 1 } { 6 } \int \varphi ^ 2 d \mu + \frac { 1 } { 3 } \int \varphi ^ 2 d \nu . \end{align*}


\begin{align*} \varprojlim _ { n \in \Delta } \mathcal F ( Y _ n ) = \varprojlim _ { n \in \Delta } \varprojlim _ { k \in \Delta } \mathcal F ( \tilde Y _ { n , k } ) = \varprojlim _ { k \in \Delta } \left ( \varprojlim _ { n \in \Delta } \mathcal F ( \tilde Y _ { n , k } ) \right ) . \end{align*}


\begin{align*} [ x , y ] = x \cdot y - y \cdot x , \end{align*}


\begin{align*} \overline { p ( z ) } = \sum _ { n = 0 } ^ k \overline { a } _ n \overline { z } ^ n = \sum _ { n = 0 } ^ k \overline { a } _ n z ^ { - n } = z ^ { - k } \sum _ { n = 0 } ^ \infty \overline { a } _ n z ^ { k - n } = z ^ { - k } q ( z ) . \end{align*}


\begin{align*} \int _ { B _ R } F ( x , D \tilde { u } ) d x = \lim _ { j _ 0 \to + \infty } \int _ { B _ R } F _ { j _ 0 } ( x , D \tilde { u } ) d x \leq \int _ { B _ R } F ( x , D u ) d x , \end{align*}


\begin{align*} K _ { V , I } = \sum _ { J \subset I } \kappa _ { V , J } . \end{align*}


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


\begin{align*} \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h } \int _ { \Omega } \frac { \partial } { \partial z _ { j , h } } ( - \delta ^ 2 ( K ( z _ j ) \nabla V _ { \delta , Z , j } ) ) \frac { \partial V _ { \delta , Z , i } } { \partial z _ { i , \hbar } } = \int _ { \Omega } u \frac { \partial V _ { \delta , Z , i } } { \partial z _ { i , \hbar } } , \ \ \forall i = 1 , \cdots , m , \ \hbar = 1 , 2 . \end{align*}


\begin{align*} { \bf h } | _ { \partial \bf N } & = \pm 2 \bar u v d t ^ 2 + h ^ \intercal \\ A ' | _ { \bar { \bf g } } ( \bf h ) & = \pm \big ( v \nu ( \bar u ) + \bar u ( \nu ( u ) ) ' \big ) d t ^ 2 + A ' | _ g ( h ) . \end{align*}


\begin{align*} \hat z = \Psi _ 2 ( \tilde z ) = \hat z _ 3 + \frac { ( \hat z _ 3 - \hat z _ 1 ) ( \hat z _ 2 - \hat z _ 3 ) ( \tilde z _ 2 - \tilde z _ 1 ) ( \tilde z - \tilde z _ 3 ) } { ( \hat z _ 2 - \hat z _ 1 ) ( \tilde z _ 2 - \tilde z _ 3 ) ( \tilde z - \tilde z _ 1 ) - ( \hat z _ 2 - \hat z _ 3 ) ( \tilde z _ 2 - \tilde z _ 1 ) ( \tilde z - \tilde z _ 3 ) } \ , , \end{align*}


\begin{align*} \widetilde { I } _ { t _ { 1 } , \cdots , t _ { k } } ( x _ { 1 } , \cdots , x _ { k } ) = \sup _ { \delta > 0 } \liminf _ { m \to \infty } \inf _ { | y - x | \le \delta } I _ { m , t _ { 1 } , \cdots , t _ { k } } ( y _ { 1 } , \cdots , y _ { k } ) , ~ ~ x = ( x _ { 1 } , \cdots , x _ { k } ) . \end{align*}


\begin{align*} f ( x ) = \log \Gamma ( x ) - \log \Gamma \left ( x + \frac 1 2 \right ) - \log \Gamma \left ( x + \frac 3 2 \right ) \end{align*}


\begin{align*} - \Gamma ^ k + \bar u ^ { - 1 } \bar g ^ { i k } \frac { \partial \bar u } { \partial x _ i } = 0 . \end{align*}


\begin{align*} & \circlearrowleft _ { x , y , z } \varepsilon ( z , x ) \Big ( \varepsilon ( h , x + y ) [ [ \alpha ( x ) , \alpha ( y ) ] , \alpha ( h \cdot z ) ] + \varepsilon ( x + y , z ) [ [ h \cdot z , \alpha ( x ) ] , \alpha ^ 2 ( y ) ] \\ & + \varepsilon ( x , y + z ) \big ( 2 [ \alpha ( h ) \cdot [ y , z ] , \alpha ^ 2 ( x ) ] - \varepsilon ( x , y + z ) [ [ h \cdot y , \alpha ( z ) ] , \alpha ^ 2 ( x ) ] \big ) \Big ) = 0 . \end{align*}


\begin{align*} W _ { 1 } = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle , W _ { 2 } = \left \langle \left [ \vartheta _ { 1 } \right ] , \left [ \vartheta _ { 2 } \right ] , \dots , \left [ \vartheta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) , \end{align*}


\begin{align*} [ M , \ , \ , \partial _ t + v \ , \cdot \ , \partial _ x ] = \partial _ { v _ 1 } ^ 2 , \end{align*}


\begin{align*} \sup _ { n \in \mathbb N } \mathbb P \left [ \sum _ { l \geq N _ k ^ { \epsilon } } \langle Y _ n , e _ l \rangle ^ 2 > \frac 1 { l _ k } \right ] \leq \epsilon \frac 1 { l _ k ^ 2 \sum _ { j = 1 } ^ { \infty } \frac 1 { l _ j ^ 2 } } . \end{align*}


\begin{align*} v ( u ) \leq d _ u ( \pi _ N ( x ) + L ) + ( i _ u - d _ u ) L = d _ u ( \pi _ N ( x ) ) + i _ u L < 2 i _ u \pi _ N ( x ) , L < \pi _ N ( x ) . \end{align*}


\begin{align*} \tilde { Y } _ t = e ^ { | z | J ^ \perp t } \tilde { Y } _ 0 + \int _ 0 ^ t e ^ { | z | J ^ \perp ( t - s ) } P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ s . \end{align*}


\begin{align*} x _ { \lambda } x _ { \mu } x _ { \nu } = ( x _ { \lambda } x _ { \mu } ) x _ { \nu } ( \lambda \mu ) \nu \end{align*}


\begin{align*} e ^ { \gamma x } f _ r \ast f _ G ( x ) = \int _ { - \infty } ^ { x / 2 } \big \{ e ^ { \gamma ( x - y ) } f _ r ( x - y ) e ^ { \gamma y } f _ G ( y ) + e ^ { \gamma ( x - y ) } f _ G ( x - y ) e ^ { \gamma y } f _ r ( y ) \big \} d y < \infty \end{align*}


\begin{align*} [ V _ f ] = \prod _ { i = 0 } ^ r e ^ { f ( x _ i ) \omega _ i } \frac { 1 } { D } . \end{align*}


\begin{align*} \gamma ( { \bf { v } } _ n , { \bf { u } } _ k , \psi _ n ) = 1 5 \left ( | c o s ( \rho _ { n , k } ) | + 2 | s i n ( \varphi _ { n , k } + \psi _ n ) | \right ) , \end{align*}


\begin{align*} J _ { i , k } ^ { ( s ) } = \int _ 0 ^ \infty z ^ i \ , \phi _ s ( z ) \ , \Phi _ s ( z ) ^ k \mathrm { d } z . \end{align*}


\begin{align*} q _ 1 = \begin{pmatrix} | \xi | I _ { n - 1 } + \frac { s _ { 1 } } { | \xi | } ( g ^ { \alpha \beta } \xi _ \alpha \xi _ \gamma ) & i s _ { 1 } ( g ^ { \alpha \beta } \xi _ \beta ) & 0 \\ [ 1 m m ] i s _ { 1 } ( \xi _ \beta ) & ( 1 - s _ { 1 } ) | \xi | & 0 \\ [ 1 m m ] 0 & 0 & | \xi | \end{pmatrix} , \end{align*}


\begin{align*} \mathfrak { R } \int _ { \{ \Psi < - t \} } \langle F _ t , \bar { q } \rangle _ h c ( - \Psi ) = 0 , \end{align*}


\begin{align*} q = \frac { b _ { 0 } ^ { \prime \prime } + b _ { 0 } ^ { \prime } } { b _ { 0 } + 1 } . \end{align*}


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


\begin{align*} \mathbf { R } _ y = \sum \limits _ { i = 1 } ^ { N } \lambda _ { i } \mathbf { e } _ { i } \mathbf { e } _ { i } ^ { H } = \mathbf { U _ S \Sigma _ S U _ S ^ { H } + U _ V \Sigma _ V U _ V ^ { H } } . \end{align*}


\begin{align*} \begin{cases} d X _ { t , 3 } = - X _ { t , 1 } X _ { t , 2 } d t \\ d X _ { t , j } = 0 & j \neq 3 \end{cases} \end{align*}


\begin{align*} \mathcal { P } [ \varrho ] = 0 . \end{align*}


\begin{align*} S _ q = B _ q \circ \mathcal F \circ B _ q ^ { - 1 } . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { n - 1 } } e ^ { - C | \xi | } \frac { \xi _ { \alpha } ^ 2 } { | \xi | ^ 2 } \ , d \xi = \frac { \Gamma ( n - 1 ) \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) } { ( n - 1 ) C ^ { n - 1 } } , C > 0 , \ n \geqslant 2 , \end{align*}


\begin{align*} \Lambda ( \mu , \nu ) = \varphi ( \mu ) + \varphi ( \nu ) + \Psi ( \mu , \nu ) , ( \mu , \nu ) \in ( 0 , \infty ) ^ { 2 } , \end{align*}


\begin{align*} w = f ( z ) = ( \Psi \circ \Phi ) ( z ) = ( \Psi _ 1 ^ { - 1 } \circ \Psi _ 2 ^ { - 1 } \circ \Phi ) ( z ) \end{align*}


\begin{align*} - L \phi = \mu ^ { 2 } ( \phi - \phi _ { \infty } ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 2 } _ { + } , \phi = 0 \textrm { o n } \ \partial \mathbb { R } ^ { 2 } _ { + } , \end{align*}


\begin{align*} ( \sum \limits _ { i = 1 } ^ { j } w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + ( j - 1 ) r < \chi _ j < ( \sum \limits _ { i = 1 } ^ { j } w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j - 1 } \chi _ i + j r , \end{align*}


\begin{align*} I _ { h , \gamma } & = \inf \left \{ E [ b ] \ \Bigg | \ b \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) , \ H [ b ] = h \right \} , \\ S _ { h , \gamma } & = \left \{ b \in L ^ { 2 } _ { \sigma , \textrm { a x i } } ( \mathbb { R } ^ { 3 } ) \ \middle | \ E [ b ] = I _ { h , \gamma } , \ H [ b ] = h \right \} . \end{align*}


\begin{align*} ( F f ) ( x ; z ) : = \int _ { - \infty } ^ x \left ( 1 , e ^ { 2 i z ( x - y ) } \right ) \widetilde { Q } f ( y ) \mathrm { d } y , \end{align*}


\begin{align*} V _ n ( s ) : = \begin{cases} \mathcal { V } ( y ) = 0 & \ \Omega _ s \\ y = Y _ { n } & \ \Gamma _ s \\ \end{cases} \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sqrt [ n ] { \binom { n + \lceil \alpha n \rceil } { \lceil \alpha n \rceil } } \le \frac { ( 1 + \alpha ) ^ { 1 + \alpha } } { \alpha ^ { \alpha } } . \end{align*}


\begin{align*} \frac { \mu ( x ) } { \nu ( x ) } \frac { \nu ( y ) } { \mu ( y ) } = 1 \end{align*}


\begin{align*} \angle \mathbf { H } ^ { } _ { m , n } = \angle e ^ { - j \theta _ { m , n } } , \forall m \in \left [ R _ { d } \right ] , n \in \left [ N _ { b } \right ] , \end{align*}


\begin{align*} \hat { A } ^ { \pm } = \mp \sqrt { f ( r ) } \frac { d } { d r } \sqrt { f ( r ) } + W ( r ) . \end{align*}


\begin{align*} | W | q ^ { d ^ 2 - 1 } \theta _ d = | M | & \le \delta ( \alpha + 4 ) q ^ { d ^ 2 + d - 2 } + \sum _ { l \in \Omega _ 1 } | l \cap W | q ^ { d ^ 2 - 1 } \\ & \hphantom { \le { } } \mathrel { + } \omega _ 2 | W | q ^ { d ^ 2 - d - 1 } + \sum _ { \mathclap { ( P , H ) \in \Omega _ 3 } } | W | q ^ { d ^ 2 - d } + | \widehat { \Omega } _ 4 | . \end{align*}


\begin{align*} X _ { n + 1 } = A X _ { n } + B u _ n + \xi _ { n + 1 } \end{align*}


\begin{align*} E _ { - m } = b _ 0 q _ { - m } + \frac { \partial q _ { - m } } { \partial x _ n } - i \sum _ \alpha \frac { \partial b _ 1 } { \partial \xi _ \alpha } \frac { \partial q _ { - m } } { \partial x _ \alpha } - \sum _ { \substack { - m \leqslant j , k \leqslant 1 \\ | J | = j + k + m } } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } q _ j \ , \partial _ { x ^ \prime } ^ { J } q _ k , m \geqslant 0 . \end{align*}


\begin{align*} \lambda _ k = \frac 1 8 \sum _ { \ell \in \Z } \Big [ & \hat W _ r ( - k + \ell - q ) \hat W _ r ( \ell + q ) + \hat W _ r ( - k + \ell + q ) \hat W _ r ( \ell - q ) \\ & + \hat W _ r ( \ell - q ) \hat W _ r ( k + q + \ell ) + \hat W _ r ( \ell + q ) \hat W _ r ( k - q + \ell ) \\ & - 4 \hat W _ r ( \ell - q ) \hat W _ r ( \ell + q ) \Big ] \end{align*}


\begin{align*} E X ^ { n } = \int _ { 0 } ^ { \infty } x ^ { n } f _ { g } ( x | \beta ) d x = \left ( \beta \right ) ^ { \left ( n \right ) } . \end{align*}


\begin{align*} | | u _ N | | _ { L ^ \infty ( B _ { L s _ { \delta _ N , i } } ( z _ { N , i } ) ) } = o ( 1 ) . \end{align*}


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


\begin{align*} d \mu _ { \beta , f , g } ( x ) = K _ { \beta , f } ( x ) g ( x ) \ , d x . \end{align*}


\begin{align*} \bigcup _ { y = 1 } ^ z I _ y = \{ 1 , \dots , n \} \end{align*}


\begin{align*} \sum _ { u = 1 } ^ t i _ u = 2 r , \end{align*}


\begin{align*} { f _ i } \left ( { { \bf { x } } + { { \bf { c } } _ i } \Delta t , t + \Delta t } \right ) = { f _ i } ^ * \left ( { { \bf { x } } , t } \right ) . \end{align*}


\begin{align*} \rho ( \xi _ { \Delta } \lvert \eta _ { \Lambda \setminus \Delta } ) = \prod _ { j = 1 } ^ { k - 1 } \rho ( \eta _ { i _ j } | \eta _ { [ i _ { j + 1 } , i _ k ] } \eta _ { \Lambda \setminus \Delta } ) . \end{align*}


\begin{align*} \varphi _ { 0 } \left ( \rho \right ) = G _ { - } \left ( \rho \right ) e ^ { i S ( \rho , x ) } \end{align*}


\begin{align*} \eta ( x ) = \sum _ { k = 1 } ^ \infty a _ k \sin ( 2 \pi k x ) + b _ k ( 1 - \cos ( 2 \pi k x ) ) , \end{align*}


\begin{align*} | | B _ q ( \psi ) | | _ { M L _ q ( \mathbb { C } ) } = | | \psi | | _ { L ^ 2 ( \mathbb { R } ) } . \end{align*}


\begin{align*} \frac { d } { d b } R L ( b , c ) = & \frac { 2 b + 1 } { 2 } \left ( \log \left ( \binom { b + 1 } { 2 } + c \right ) + 1 \right ) \\ & - \frac { 2 b - 1 } { 2 } \left ( \log \left ( \binom { b } { 2 } + c \right ) + 1 \right ) + f _ c ( 1 ) - f _ c ( 0 ) \\ \frac { d } { d b } L L ( b , c ) = & ( b + 1 ) \left ( \log \left ( b + c \right ) + 1 \right ) \\ & - b \left ( \log \left ( b + c - 1 \right ) + 1 \right ) + f _ c ( b ) - f _ c ( b - 1 ) \end{align*}


\begin{align*} \begin{aligned} & \omega \in [ 0 , 1 ] { { \xi } _ { o } } \le 0 \\ & \omega \in \left [ \frac { { { \xi } _ { o } } } { { { \xi } _ { o } } + \left ( \gamma - { { \gamma } _ { o } } \right ) \mu ( { { \mathbf { q } } _ { o } } ) } , 1 \right ] , { { \xi } _ { o } } > 0 . \\ \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \left \| \dfrac { \partial \epsilon ( s , t ) } { \partial s } \right \| \le \int _ { N T } ^ t \left \| \dfrac { \partial f ( s , \tau ) } { \partial s } \right \| + \left \| \dfrac { \partial f _ a ( s ) } { \partial s } \right \| \ , d \tau \le 2 L _ \mathcal { D } T , \end{aligned} \end{align*}


\begin{align*} 4 N _ 1 \leq \begin{cases} C _ 6 & \ \tau \leq t ^ { \beta } , \\ \displaystyle C _ 6 t ^ { - 2 \beta } \tau ^ 2 & \ t ^ { \beta } \leq \tau . \\ \end{cases} \end{align*}


\begin{align*} \int _ 0 ^ { 2 - i \Delta _ n + s } | x - y | ^ { 2 \mathfrak H - 1 } d y \leq \int _ 0 ^ 2 | x - y | ^ { 2 \mathfrak H - 1 } d y = & \int _ 0 ^ x ( x - y ) ^ { 2 \mathfrak H - 1 } d y + \int _ { x } ^ { 2 } ( y - x ) ^ { 2 \mathfrak H - 1 } d y \\ = & \frac { x ^ { 2 \mathfrak H } + ( 2 - x ) ^ { 2 \mathfrak H } } { 2 \mathfrak H } \\ \leq & \frac { 2 ^ { 2 \mathfrak H } } { \mathfrak H } , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k a _ i \neq 1 \ , \ , { \rm a n d } \ , \ , \sum _ { i \in \Omega _ t ^ o } a _ i - \sum _ { i \in \Omega _ t ^ e } a _ i \neq 1 . \end{align*}


\begin{align*} \lambda \Lambda _ 1 L _ { g _ 1 } ( z ) = \log \big ( 1 - ( 1 - e ^ { \lambda \Lambda _ 1 } ) L _ { f _ 1 } ( z ) \big ) . \end{align*}


\begin{align*} \frac { 2 \epsilon _ f + r } { h ( \gamma \bar \Delta ) } \left [ \sum _ { k = 0 } ^ { T - 1 } 1 \right ] - \left [ \sum _ { k = 0 } ^ { T - 1 } ( 1 - \Theta _ k ) ( 1 - \Lambda _ k ) + \Theta _ k \Lambda _ k ' \right ] < \left ( \frac { 2 \epsilon _ f + r } { h ( \gamma \bar \Delta ) } - \frac { 1 } { 2 } \right ) T + \frac { 1 } { 2 } \left | \log _ \gamma \frac { \bar \Delta } { \delta _ 0 } \right | + \frac { 1 } { 2 } . \end{align*}


\begin{align*} & \hphantom { { } = { } } D _ { ( o , o ) } ( \{ f , g \} . \{ f ' , g ' \} ) \\ * & = \begin{pmatrix} D f ( o ) D f ' ( o ) \\ D g ( o ) D f ' ( o ) + D f ( o ) D g ' ( o ) & D f ( o ) D f ' ( o ) \end{pmatrix} \\ * & = \begin{pmatrix} D f ( o ) \\ D g ( o ) & D f ( o ) \end{pmatrix} \begin{pmatrix} D f ' ( o ) \\ D g ' ( o ) & D f ' ( o ) \end{pmatrix} . \end{align*}


\begin{align*} \frac { i } { ( 2 \pi ) ^ { n } } \int _ { \mathbb { R } ^ { n - 1 } } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 1 } \ , d \tau \ , d \xi = a _ 0 ( x ) t ^ { 1 - n } , \end{align*}


\begin{align*} A : = \left [ \begin{array} { c c c c c } \alpha _ 1 & \alpha _ 1 ^ q & \alpha _ 1 ^ { q ^ 2 } & \cdots & \alpha _ 1 ^ { q ^ { n } } \\ \alpha _ 2 & \alpha _ 2 ^ q & \alpha _ 2 ^ { q ^ 2 } & \cdots & \alpha _ 2 ^ { q ^ { n } } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha _ n & \alpha _ n ^ q & \alpha _ n ^ { q ^ 2 } & \cdots & \alpha _ n ^ { q ^ { n } } \\ x & x ^ q & x ^ { q ^ 2 } & \cdots & x ^ { q ^ { n } } \\ \end{array} \right ] . \end{align*}


\begin{align*} g ^ { \prime } ( \rho , x ) = e ^ { - i \rho x } \left ( - \frac { z - 1 } { 2 ( z + 1 ) } + \frac { 1 } { 2 } \int _ { - \infty } ^ { x } q ( t ) d t + ( z + 1 ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } c _ { n } ( x ) \right ) \end{align*}


\begin{align*} \mathcal { L } u = \lambda | u | ^ 2 u , \lambda \in \R \end{align*}


\begin{align*} G ( x - 2 G ( x ) ) = - G ( x ) \end{align*}


\begin{align*} & \abs { \ell ( x , \alpha + \beta ) - \ell ( x , \alpha ) - \ell ( x , \beta ) } \\ = & \lvert \underbrace { \Phi ^ + ( \alpha + \beta ) - \Phi ^ + ( \alpha ) - \Phi ^ + ( \beta ) } _ { \in \{ - 1 , 0 \} } \underbrace { - \Phi ^ + ( w ( \alpha + \beta ) ) + \Phi ^ + ( w \alpha ) + \Phi ^ + ( w \beta ) } _ { \in \{ 0 , 1 \} } \rvert \leq 1 . \end{align*}


\begin{align*} \lim _ { s \rightarrow \infty } s L _ { q } [ f ( t ) ] = \lim _ { s \rightarrow \infty } \frac { f ( t ) } { 1 + ( 1 - q ) } . \end{align*}


\begin{align*} \lim \limits _ { r \downarrow 0 } \frac { \mu ( B _ r ( x ) \cap \mathfrak { J } _ k ) } { \mu ( B _ r ( x ) ) } = 1 . \end{align*}


\begin{align*} \langle P ( u _ t ( s ) ) , \phi ( s ) \rangle _ V \Bigr | ^ t _ 0 = \int ^ t _ 0 \Bigl [ \langle P ( u _ t ( s ) ) , \phi _ t ( s ) \rangle _ V - \langle A ( u ( s ) ) , \phi ( s ) \rangle _ X - \langle B ( s , x , u _ t ( s ) ) , \phi ( s ) \rangle _ W \Bigr ] d s \end{align*}


\begin{align*} U _ { r _ n } ( q _ n ) = \varphi ( q _ n ) + \delta _ n , \end{align*}


\begin{align*} \widehat { \Psi } ( \hat { A } ) ^ { \times } = \{ P \in \widehat { \Psi } ( \mathcal { O } _ { \hat { A } } ) \ ; | \ ; \pi ( P ) \in G _ { \hat { A } } \} \end{align*}


\begin{align*} S _ r = \min _ { k } \mathbb { E } \left [ \bar { R _ { c , k } } \right ] + \sum _ { k = 1 } ^ K \mathbb { E } \left [ \bar { R } _ k \right ] , \end{align*}


\begin{align*} \mathrm { H } ( X ) = & \ln \left \{ \sqrt { 2 \pi } \sigma \mathrm { B } ( \alpha , \beta ) \right \} + \frac { 1 } { 2 \sigma ^ 2 } \left [ \mathrm { E } ( X ^ 2 ) - 2 \mu \mathrm { E } ( X ) + \mu ^ 2 \right ] \\ & + \frac { 1 } { \mathrm { B } ( \alpha , \beta ) } \sum _ { n = 1 } ^ { \infty } \frac { 1 } { n } \left [ ( \alpha - 1 ) \mathrm { B } ( \alpha , n + \beta ) + ( \beta - 1 ) \mathrm { B } ( n + \alpha , \beta ) \right ] . \end{align*}


\begin{align*} \abs { \eta ( y ) - \eta ( x ) } ^ 2 & = \abs { \int _ x ^ y \partial \eta ( z ) \ \d z } ^ 2 \\ & \le \int _ \S \partial \eta ( z ) ^ 2 \ \d z \\ & = \norm { \partial \eta } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} & \ln \left \{ \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right \} = \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ n } { n } \ , \left [ 1 - \Phi \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ n \end{align*}


\begin{align*} \omega ^ * : = \R \setminus [ - R , 0 ] = ( - \infty , - R ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \mu \left ( \left \{ x \in X \mid \frac { S _ n f ( x ) } { \sqrt { n } } \le z \right \} \right ) = \frac { 1 } { \sigma _ f \sqrt { 2 \pi } } \int _ { - \infty } ^ z \exp \left ( - \frac { t ^ 2 } { 2 \sigma _ f ^ 2 } \right ) d t . \end{align*}


\begin{align*} \lambda _ 1 \Lambda _ 1 g _ 1 ( x ) \stackrel { a . e . } { = } - \sum _ { n = 1 } ^ \infty n ^ { - 1 } ( 1 - e ^ { \lambda \Lambda _ 1 } ) ^ n f _ 1 ^ { \ast n } ( x ) = : \lambda _ 1 \Lambda _ 1 \breve g _ 1 ( x ) , \end{align*}


\begin{align*} H _ r ( E [ t ] ) = t H _ r ( E ) . \end{align*}


\begin{align*} \mathcal { M } ^ { r } ( t ) - \mathcal { M } ^ { r } ( \delta ) = \int _ { \delta } ^ { t } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \mu } \omega ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \Lambda ( \mu , \nu ) \ d \nu d \mu d s , \end{align*}


\begin{align*} \vec { r } = ( r _ 1 , r _ 2 , \ldots ) . \end{align*}


\begin{align*} R _ { x } = I \left ( \cos \left ( \left \vert x \right \vert / 2 \right ) + \sin \left ( \left \vert x \right \vert / 2 \right ) x / \left \vert x \right \vert \right ) \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*} \begin{aligned} x _ n - y _ n = \ , & \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) + O ( | x ' - y ' | ^ 2 ) \\ \geq \ , & \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( y ' ) ( x _ \beta - y _ \beta ) - \kappa | x ' - y ' | ^ 2 \end{aligned} \end{align*}


\begin{align*} \partial _ t \varrho + m \cdot \nabla J ( \varrho ) = 0 , \varrho ( 0 , \cdot ) = \varrho ^ { \rm i n i } , \end{align*}


\begin{align*} S t _ { D N M } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( z ^ { 2 } , z ) - z ^ { 2 } | | _ { \infty } = 0 . \\ \end{align*}


\begin{align*} \mu ( n ) = ( 2 n + d ) ^ { - \alpha } e ^ { ( 2 n + d ) i t } . \end{align*}


\begin{align*} a ( Z ) = a \big ( ( Z Z ^ * ) ^ \frac { 1 } { 2 } \big ) \end{align*}


\begin{align*} ( r _ 0 , \ldots , r _ { \ell - 1 } ) = \big ( \ , \underbrace { s _ L , \ldots , s _ L } _ { m _ L } , \ldots , \underbrace { s _ 1 , \ldots , s _ 1 } _ { m _ 1 } \ , \big ) , 0 < s _ 1 < \cdots s _ L , m _ i > 0 , \ 1 \leq i \leq L . \end{align*}


\begin{align*} c ( \gamma , \gamma ^ - ) = - c ( \gamma , \gamma ^ + ) . \end{align*}


\begin{align*} \epsilon _ n = \max _ { | \eta - \lambda | \le \delta } \max _ { v \in G ( \lambda ) } \| F _ n ( \eta ) p _ n v - q _ n F ( \eta ) v \| _ { X _ n } \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { m \to \infty } \limsup _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { t \in [ 0 , \chi \wedge 1 ] } | \hat { X } ^ { \epsilon } ( t ) - X ^ { \epsilon } ( t ) | \geq \delta ^ { 1 / 4 } \Big ) = - \infty , \end{align*}


\begin{align*} \beta ^ { q ^ n } + a _ { n - k } \beta ^ { q ^ { n - k } } + \cdots + a _ 0 \beta = 0 \end{align*}


\begin{align*} \begin{aligned} \left \| M _ { + , 1 2 } ( x ; z ) \right \| _ { L _ { z } ^ { 2 } } \leq c \left \| \mathcal { P } ^ + \left ( \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \right ) \right \| _ { L _ { z } ^ 2 } , \end{aligned} \end{align*}


\begin{align*} \| \tilde { X } ( k ) \| ^ 2 \leq \frac { \sum \limits _ { i = 1 } ^ { p } \left \| \tilde { Y } _ i ( k ) \right \| ^ 2 + \sum \limits _ { j = 1 } ^ { q } \left \| \tilde { Z } _ j ( k ) \right \| ^ 2 } { p + q } . \end{align*}


\begin{align*} M P A P P ( m , n ) = & \biggl \{ A \in P A P P ( m , n ) : A \mbox { i s t h e m i n i m a l e l e m e n t i n t h e s e t } \\ & \{ A , \sigma ( A ) \} \mbox { u n d e r t h e l e x i c o g r a p h i c o r d e r } \biggr \} \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } t } _ { | t = 0 } \int _ { \Gamma _ t } ( c + t w ) \circ \theta _ t ^ { - 1 } \ , \mathrm { d } \mathcal { H } ^ d & = \int _ \Sigma \partial ^ \square \big ( ( c + t w ) \circ \theta _ t ^ { - 1 } \big ) - \big ( ( c + t w ) \circ \theta _ t ^ { - 1 } \big ) _ { | t = 0 } H V \ , \mathrm { d } \mathcal { H } ^ d \\ & = \int _ \Sigma w - c H V \ , \mathrm { d } \mathcal { H } ^ d = 0 \end{align*}


\begin{align*} \norm { g } _ { L ^ 2 } ^ 2 \le \prod _ { i = 1 } ^ n \norm { \eta _ i } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} w \varepsilon ^ \lambda = y \cdot w ' \varepsilon ^ { \lambda ' } . \end{align*}


\begin{align*} \mathcal { C } \big ( \mathcal { U } _ { ( i , j ) } \big ) \cong \begin{cases} C ^ \infty \big ( \mathcal { U } _ { ( 0 , 0 ) } \big ) & \\ \mathcal { S } \big ( \mathcal { U } _ { ( i , j ) } \big ) & \\ \mathcal { A } \big ( \mathcal { U } _ { ( i , j ) } \big ) & \end{cases} \ , , \end{align*}


\begin{align*} \eta ( q _ 1 ) \eta ( q _ 2 ) \overline { \pi } ( q _ 1 ) \overline { \pi } ( q _ 2 ) = \eta ( q _ 1 q _ 2 ) c _ { \chi _ 1 } ( q _ 1 , q _ 2 ) \overline { \pi } ( q _ 1 q _ 2 ) . \end{align*}


\begin{align*} \sum _ { \alpha , \beta = 1 } ^ d \xi _ \alpha \cdot \left ( \nabla G ( U , x , t ) ^ T B _ { \alpha \beta } ( U , x , t ) \ , \xi _ \beta \right ) = \sum _ { \alpha , \beta = 1 } ^ d \sum _ { i , j = 1 } ^ n \xi _ \alpha ^ i D _ { \alpha \beta } ^ { i j } \xi _ \beta ^ j > 0 \end{align*}


\begin{align*} F ( \theta _ { ( \ell + 1 ) q _ n + m _ n } ) F _ { q _ n - 1 } ( \theta _ { \ell q _ n + m _ n + 1 } ) = : \left ( \begin{matrix} b _ 1 & b _ 2 \\ b _ 3 & b _ 4 \end{matrix} \right ) , \end{align*}


\begin{align*} G : = { \rm G a l } ( M ' / K ' ) = { \rm G a l } ( M ' / M _ 1 ' ) \times { \rm G a l } ( M ' / M _ 2 ' ) \cong C _ 2 \times C _ 2 \end{align*}


\begin{align*} W _ r ( x ) : = \begin{cases} 1 & \min ( x , 1 - x ) \le r \\ 0 & \end{cases} . \end{align*}


\begin{align*} F _ 1 = \frac { 1 } { \sqrt { c _ 1 } } \langle \langle \nu _ 1 \cdot \varphi , \varphi \rangle \rangle . \end{align*}


\begin{align*} \varepsilon ^ { w ( \lambda - \lambda ' ) } = w \varepsilon ^ \lambda \varepsilon ^ { - \lambda ' } w ^ { - 1 } = y w ' \varepsilon ^ { \lambda ' } \varepsilon ^ { - \lambda ' } w ^ { - 1 } = y \underbrace { w ' w ^ { - 1 } } _ { \in W _ J } \in \widetilde W _ J . \end{align*}


\begin{align*} \chi _ i = \left \{ \begin{aligned} & \psi _ i & & ( i \neq j ) \\ & \sigma & & ( i = j ) . \end{aligned} \right . \end{align*}


\begin{align*} & \xi ( \xi + 1 ) = L ( L + 1 ) , \\ & - \kappa \xi ^ 2 + 2 \xi \zeta + \eta ^ 2 - \frac { \zeta ^ 2 } { \kappa } = - E _ 0 , \\ & 2 \xi \eta = - Q , \\ & 2 \eta \zeta = \kappa B _ 1 , \\ & \frac { \zeta } { \kappa } \left ( \frac { \zeta } { \kappa } - 1 \right ) = B _ 2 . \end{align*}


\begin{align*} & \limsup _ { m \to \infty } \limsup _ { \epsilon \to 0 } \epsilon ^ { 2 } \log \mathbb P \Big ( \epsilon \left | G _ { m , \epsilon } ( t _ 1 , \cdots , t _ k ) - G _ { \epsilon } ( t _ 1 , \cdots , t _ k ) \right | > \delta \Big ) = - \infty , \end{align*}


\begin{align*} \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \phi | ^ { 2 } \frac { 1 } { r } \dd z \dd r = 1 . \end{align*}


\begin{align*} c _ k = 0 \ \ c _ k = \left ( \frac { 1 } { p ^ { d / \tau } } \right ) ^ { p ^ { v _ p ( k ) } } \ \end{align*}


\begin{align*} \mathbf { x } ^ { ( k ) } = \mathbf { r } ^ { ( k ) } + \mathcal { B } ^ { ( k ) } \left [ \widetilde { \mathcal { H } } ^ { ( k ) } \left [ \mbox { s o f t } \left [ \mathcal { H } ^ { ( k ) } \left ( \mathcal { M } ^ { ( k ) } \left ( \mathbf { r } ^ { ( k ) } \right ) \right ) , \theta ^ { ( k ) } \right ] \right ] \right ] \end{align*}


\begin{align*} \P \left ( \frac { 1 } { m } \sum _ { i = 1 } ^ m U _ i \geq \left ( \frac { \pi } { 4 } + \frac { 1 9 9 \pi } { 3 2 0 0 k } \right ) \| \mathbf { x } \| \right ) \leq 2 \exp \left ( - \frac { c _ 8 m } { k } \right ) . \end{align*}


\begin{align*} \int _ { \Omega } | \nabla ( v - \tilde { v } ) | ^ { 2 } + \int _ { \partial \Omega } | v - \tilde { v } | ^ { 2 } + \int _ { \Omega } u | v - \tilde { v } | ^ { 2 } = 0 , \end{align*}


\begin{align*} \begin{aligned} | x ( t ) - x _ a ( t ) | & = | x ( t ) - z ( t ) - x _ a ( t ) + z ( t ) | \\ & \leq | x ( t ) - z ( t ) | + \gamma \bar \epsilon ( L _ r , M _ r , \delta ) . \end{aligned} \end{align*}


\begin{align*} W ( F ) ( \tau , \tau ' ) : = F _ { z = 0 } . \end{align*}


\begin{align*} \P \left ( \Omega _ 0 = \left \{ \omega \in \Omega : \int _ 0 ^ \eta | y _ t | ^ 2 d t \le \sqrt { \delta _ 1 } \eta K ^ { 2 r } , \sup _ { 0 \le t \le \eta } | z _ t - z _ 0 | \le R K ^ { \frac { r - 1 } { 3 } } \right \} \right ) \ge 1 - \frac { \beta } { 2 } . \end{align*}


\begin{align*} g _ k : = { \rm s g n } ( v ) \star \eta _ k , \end{align*}


\begin{align*} q ( A ) = \sum _ i q ( E _ i ) p ( A | E _ i ) , A \in \bf { A } . \end{align*}


\begin{align*} \begin{aligned} k ^ { - 1 } \psi ^ { \pm } _ { 2 1 } ( x ; k ) & = - \int _ { \pm \infty } ^ { x } e ^ { 2 i z ( x - y ) } \bar { u } _ y e ^ { - i c _ \pm } d y \\ & - \int _ { \pm \infty } ^ { x } e ^ { 2 i z ( x - y ) } \bar { u } _ y \left ( \psi ^ { \pm } _ { 1 1 } ( y ; k ) - e ^ { - i c _ \pm } \right ) d y . \end{aligned} \end{align*}


\begin{align*} \aligned \int ( I _ \alpha \ast f ) f d x = \int \left ( I _ { \frac { \alpha } { 2 } } \ast f \right ) ^ 2 d x \geq 0 \endaligned \end{align*}


\begin{align*} F ( \lambda ) x = 0 . \end{align*}


\begin{align*} S ( t ) : = \exp \left ( t \begin{pmatrix} 0 & x _ { 0 , n - 1 } \\ x _ { 0 , 2 } & 0 \end{pmatrix} \right ) . \end{align*}


\begin{align*} 0 = \frac { \dd } { \dd \tau } \int _ { \Omega } | U + \tau \tilde { U } + s ( \tau ) U _ 0 | ^ { 2 } \dd x \Bigg | _ { \tau = 0 } = 2 \int _ { \Omega } U \cdot ( \tilde { U } + \dot { s } ( 0 ) U _ 0 ) \dd x . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ g [ \alpha _ i , \beta _ i ] \prod _ { j = 1 } ^ n \gamma _ j = e . \end{align*}


\begin{align*} S t _ { D N P } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( f , z ) - f ( z ) | | _ { \infty } = 0 \end{align*}


\begin{align*} \mathcal S ( t - s ) e _ i - e _ i = \int _ s ^ t \mathcal S ( u - s ) \mathcal A e _ i d u , \end{align*}


\begin{align*} \partial _ { t } ^ { k + 1 } U ( x , t ) = \int _ { \R ^ { d } } \partial _ { t } ^ { k } M ( t / 2 , x , y ) \partial _ { t } u ( y , t / 2 ) d y . \end{align*}


\begin{align*} \delta _ n < \beta _ n = \frac { \ln q _ { n + 1 } } { q _ n } < \frac { \ln ( 4 q _ n ) } { q _ n } , \end{align*}


\begin{align*} \nabla _ X \varphi = \frac { 1 } { 2 } \sqrt { c _ 1 } \ X _ 1 \cdot \nu _ 1 \cdot \varphi - \frac { 1 } { 2 } B ( X ) \cdot \varphi \end{align*}


\begin{align*} p : w = w _ 1 \rightarrow w _ 2 \rightarrow \cdots \rightarrow w _ { \ell ( p ) + 1 } = w ' . \end{align*}


\begin{align*} \frac { \partial } { \partial s } f ( p , s ) = \log a - \frac { p / 2 + 2 } { s } \geq \log a - \frac { p _ k / 2 + 2 } { s _ k } \end{align*}


\begin{align*} \dot { { x } } _ a = - \gamma \int _ 0 ^ 1 y _ \delta ( { x } _ a , t ) u ( t ) d t \ , . \end{align*}


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x & \leq \theta \int _ { B _ { 8 \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x \rho \leq 1 / 8 \varepsilon \leq \varepsilon _ 0 , \end{align*}


\begin{align*} u ( t , x ) \approx u ^ h ( t , x ) = \sum _ i w _ i ( t ) \sigma _ i ( x ) . \end{align*}


\begin{align*} \mathbf { D } = ( \mathbf { \Theta } \mathbf { G } ) ^ { - 1 } . \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa B _ 1 \frac { r } { f } + \kappa B _ 2 \frac { 1 } { f ^ 2 } , B _ 2 > 0 , \end{align*}


\begin{align*} \mathbb { M } ( \theta ) _ J = \sum _ { w \in X _ J } \Bbbk { \bf U } \dot { w } \eta ( \theta ) _ J = \sum _ { w \in X _ J } \Bbbk { \bf U } _ { w _ J w ^ { - 1 } } \dot { w } \eta ( \theta ) _ J . \end{align*}


\begin{align*} \left \{ \begin{aligned} d x ^ i _ t = & ~ [ A x ^ i _ t + B \alpha ^ i _ t + f ( \nu ^ { N , i } _ { \boldsymbol { x } _ t } ) + b ( \mu ^ { N , i } _ { \boldsymbol { \alpha } _ t } ) ] d t + \sigma d W ^ i _ t + \sigma _ 0 d W ^ 0 _ t , \\ x ^ i _ 0 = & ~ \xi ^ i , \end{aligned} \right . \end{align*}


\begin{align*} \begin{aligned} p ( 5 n + 4 ) & \equiv 0 \ ( \mathrm { m o d } \ 5 ) , \\ p ( 7 n + 5 ) & \equiv 0 \ ( \mathrm { m o d } \ 7 ) , \\ p ( 1 1 n + 6 ) & \equiv 0 \ ( \mathrm { m o d } \ 1 1 ) . \end{aligned} \end{align*}


\begin{align*} d _ { 2 } ( \mathbf { x } , \mathbf { s } ) = \sqrt { 2 } \| \mathbf { w } _ { x s } - \mu \mathbf { e } \| = \min _ { \mathbf { D } \in \mathcal { G } } \| ( \mathbf { D } ^ { - T } \mathbf { x } ) \circ ( \mathbf { D } \mathbf { s } ) - \mu \mathbf { e } \| \end{align*}


\begin{align*} \begin{cases} \partial _ t \mathbf { w } + ( \mathbf { v } \cdot \nabla ) \mathbf { w } = ( \mathbf { w } \cdot \nabla ) \mathbf { v } , \ \ & D \times ( 0 , T ) , \\ \mathbf { w } ( \cdot , 0 ) = \nabla \times \mathbf { v } _ 0 ( \cdot ) , \ \ & D . \end{cases} \end{align*}


\begin{align*} \deg _ H ( x ) = \deg _ { H _ i } ( x ) + \deg _ F ( x ) \leq \binom { p - 1 } { k - 1 } + \frac { d } { 2 } \leq \frac { d } { 2 } + \frac { d } { 2 } = d . \end{align*}


\begin{align*} \varphi _ n ( \eta ) : = \sum _ { \Delta \cap \Delta _ n \neq \emptyset } \sum _ { \xi _ { \Delta } } \left ( c _ \Delta ( \eta , \xi _ \Delta ) - \hat { c } _ \Delta ( \eta , \xi _ \Delta ) \right ) , \end{align*}


\begin{align*} N _ n ( B ) = \left \{ \sum _ { i = 1 } ^ k \lambda _ i e _ i : \lambda _ i \in \{ - n , \dots , 0 , \dots , n \} \sum _ { i = 1 } ^ k | \lambda _ i | \leq n \right \} \ , . \end{align*}


\begin{align*} | T \cap M ' | & = | H \cap W ' | q ^ { d ^ 2 - d } \\ & \le \begin{cases} | W ' \cap \tau _ i | q ^ { d ^ 2 - d } & H \cap U = \tau _ i i \in \{ 1 , 2 \} , \\ ( q ^ { d - 1 } + \theta _ { d - 1 } ) q ^ { d ^ 2 - d } & \end{cases} \end{align*}


\begin{align*} & \phi = \frac { 2 \pi Z } { N } - A = 2 \pi \frac { d } { \lambda } { s i n } \theta _ 0 \\ & { s i n } \theta _ 0 = \frac { \lambda ( Z \sqrt { N } - 1 ) } { N \sqrt { N } d } . \end{align*}


\begin{align*} \lim _ { t \to 0 } f ( t ) = 0 f ( T ) \leq \frac { 1 } { 1 6 n } \cdot \end{align*}


\begin{align*} h _ { p } ^ { \left ( 0 \right ) } = \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } \varphi _ { 0 } ^ { - } \left ( \tau \right ) \frac { z ^ { p } ( \tau ) } { \left ( \frac { 1 } { 2 } - i \tau \right ) ^ { 2 } } d \tau , p = 1 , 2 , \ldots \end{align*}


\begin{align*} k ( w t ^ \mu , \alpha ) = \langle \mu , \alpha ^ \vee \rangle + \Phi ^ + ( ( \alpha ) ( w ^ { - 1 } ) ) - \Phi ^ + ( \alpha ) . \end{align*}


\begin{align*} \nabla _ X \psi _ 2 = - \frac { 1 } { 2 } X _ 1 \cdot \psi _ 2 - \frac { 1 } { 2 } S ( X ) \cdot N \cdot \psi _ 2 \end{align*}


\begin{align*} f ( x ) = ( 2 \pi ) ^ { - n / 2 } \int _ { - \infty } ^ \infty \int _ { S ^ { n - 1 } } \widehat { f } ( \lambda , \omega ) e _ { \lambda , \omega } ( x ) | \lambda | ^ { n - 1 } d \omega \ , d \lambda \end{align*}


\begin{align*} \mathcal { G } ^ { \mathrm { e n d } } = \left ( \begin{array} { c c } A & g ^ { - 1 } \\ h & A ^ * \end{array} \right ) \end{align*}


\begin{align*} A = \{ \omega \in \Omega \mid \exists ~ x \in C ( \omega ) ~ s . t . ~ \langle \mathcal { F } ( x ^ * ) ( \omega ) , x - x ^ * ( \omega ) \rangle < 0 \} . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } 0 = \Delta c - \tilde { n } c , & x \in \Omega , \\ \nabla c \cdot \nu = \gamma - c , & x \in \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} D _ j ( U ^ { i j } D _ i w ) = f \quad \ , \ , \ , \ , B _ 1 \end{align*}


\begin{align*} \begin{aligned} u ( \bar { x } ) \leq \ , & u ( x ^ * ) + C \epsilon _ 0 \delta ^ 2 b _ { \alpha } ^ 2 + C b _ { \alpha } \bar { x } _ n + C \delta ^ 3 b _ \alpha ^ 2 \\ \leq \ , & u ( x ^ * ) + C \left ( \epsilon _ 0 + \delta \right ) \delta ^ 2 b _ \alpha ^ 2 \leq \left ( 1 + \frac { c _ 0 } { 3 } \right ) u ( x ^ * ) \end{aligned} \end{align*}


\begin{align*} \mathcal { L } w = - \mathcal { L } c = \partial ^ \square c - \big ( \Delta _ \Gamma G ' ( c ) + c V H \big ) = 0 [ 0 , T ] \times M \end{align*}


\begin{align*} r ^ 2 + r q + q ^ 2 = 2 n \ , . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty b _ 3 \left ( \frac { m n - 1 } { 1 2 } \right ) q ^ n \in S _ { 3 m - 3 } ( \Gamma _ 0 ( 4 3 2 ) , \chi _ { 1 2 } ) _ m , \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ - _ 1 ( x ; z ) = e ^ { - i c _ - ( x ) } e _ 1 . \end{align*}


\begin{align*} F = \bigcup _ { n = 0 } ^ \infty K _ n , \end{align*}


\begin{align*} H _ { j , n } = \frac { \sqrt { n ! } } { \Gamma ( \beta ) \sqrt { \left ( \beta \right ) ^ { \left ( n \right ) } } } \int _ { 0 } ^ { \infty } x ^ { j } L _ { n } ^ { ( \beta ) } ( x ) x ^ { \beta - 1 } \exp ( - x ) d x . \end{align*}


\begin{align*} S ' u _ k + T v _ k = ( \frac { - 1 } { 8 \cos ^ { 2 } ( \frac { \pi k } { n - 1 } ) } ) S ' S v _ k + T v _ k . \end{align*}


\begin{align*} \Phi _ s ( z ) ^ \alpha = \{ 1 - [ 1 - \Phi _ s ( z ) ] \} ^ \alpha = \sum _ { j = 0 } ^ \infty ( - 1 ) ^ j \ , \binom { \alpha } { j } \ , [ 1 - \Phi _ s ( z ) ] ^ j . \end{align*}


\begin{align*} \frac { \partial h } { \partial t } \geq \frac { 1 } { f _ { \max } } h [ h ^ { - \varepsilon } \beta _ { 1 } - f _ { \max } ] . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \langle A _ 0 ( u _ n ) , u _ n - u \rangle = 0 . \end{align*}


\begin{align*} X ( t ) = X _ 0 + \int _ 0 ^ t A ( s , \bar { X } ( s ) ) d s + \int _ 0 ^ t B ( s , \bar { X } ( s ) ) d W ( s ) , \quad \forall \ t \in [ 0 , T ] , \end{align*}


\begin{align*} \begin{array} { l } \circlearrowleft _ { x , y , z } \varepsilon ( z , x ) \Big ( [ \alpha ( h ) , [ x , y ] , \alpha ^ 2 ( z ) ] \Big ) \\ + \circlearrowleft _ { x , y , z } \varepsilon ( z , x ) \Big ( \varepsilon ( h , x + y ) [ \alpha ( [ x , y ] ) , \alpha ( h \cdot z ) ] \Big ) = 0 . \end{array} \end{align*}


\begin{align*} \phi _ 2 ' = \phi _ 4 , \end{align*}


\begin{align*} \left \{ \begin{array} { r l } \div ( A ( x ) \nabla \varphi ) = 0 & B _ 1 \\ A ( x ) \nabla \varphi \cdot \nu = g & \partial B _ 1 , \end{array} \right . \end{align*}


\begin{align*} a ( z ) = 1 + k \int _ { \mathbb { R } } u _ y ( y ) \psi ^ - _ { 2 1 } ( x ; k ) \mathrm { d } y . \end{align*}


\begin{align*} \mathbb { P } ( h ^ { U } ( x ) \geq a ) & \leq \mathbb { P } ( h ^ { U } ( x ) \geq a , \varphi ^ { V , U } ( x ) \geq 0 ) + \mathbb { P } ( h ^ { U } ( x ) \geq a , \varphi ^ { V , U } ( x ) \leq 0 ) \\ & = 2 \mathbb { P } ( h ^ { U } ( x ) \geq a , \varphi ^ { V , U } ( x ) \geq 0 ) \leq 2 \mathbb { P } ( h ^ { V } ( x ) \geq a ) . \end{align*}


\begin{align*} \lim _ { j \to \infty } \lim _ { m \to \infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a = \lim _ { m \to \infty } \lim _ { j \to \infty } T ^ { c _ 1 ( j ) + c _ 2 ( m ) } a . \end{align*}


\begin{align*} \lim _ { n \to \infty } \mathbb E \left [ Z g ( U _ n ( Y ) ) \right ] = \tilde { \mathbb E } \left [ Z g ( U ( Y ) ) \right ] , \end{align*}


\begin{align*} \begin{aligned} & ~ & & \| g _ k - \nabla \phi ( x _ k ) \| = \| g _ k - L _ 1 x _ k \| \le \kappa _ { \rm e g } \delta _ k + \epsilon _ g \ ; \ ; & & \end{aligned} \end{align*}


\begin{align*} \frac { d g ( t ) } { d t } g ( t ) ^ { - 1 } = v ( t ) . \end{align*}


\begin{align*} ( \pi ( \prod _ { i = 2 } ^ g [ \alpha _ i . \beta _ i ] \prod _ { 1 = 1 } ^ n \gamma _ j ) ) ^ { - 1 } = \begin{pmatrix} a & b \\ c & d \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} & \| c \varphi \| _ { W ^ { 1 , 6 } ( \Omega ) } \\ & \le C ( \| \Delta ( c \varphi ) \| _ { L ^ { \frac { 3 } { 2 } } ( \Omega ) } + \| c \varphi \| _ { L ^ { \frac { 3 } { 2 } } ( \Omega ) } + \| ( \gamma - c ) \varphi \| _ { W ^ { \frac { 1 } { 3 } , \frac { 3 } { 2 } } ( \partial \Omega ) } + \| c \nabla \varphi \cdot \nu \| _ { W ^ { \frac { 1 } { 3 } , \frac { 3 } { 2 } } ( \partial \Omega ) } ) . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} d ( x , x ' ) & \leq d ( x , x H _ 2 \cap x ' H _ 1 ) + d ( x H _ 2 \cap x ' H _ 1 , x ' ) \leq 2 d ( x ' , x H _ 2 \cap x ' H _ 1 ) \\ & \leq 2 d ( x ' , x ' H _ 1 ) \leq 2 d ( 1 , \pi _ { H _ 1 } ( x ^ { - 1 } x ' ) ) . \\ \end{aligned} \end{align*}


\begin{align*} \chi _ i = \left \{ \begin{aligned} & \psi _ i & & ( i \neq j ) \\ & \phi _ j & & ( i = j ) . \end{aligned} \right . \end{align*}


\begin{align*} n H = \eta x _ { n + 1 } ^ m + \lambda . \end{align*}


\begin{align*} d ( D , E ) = \max \left \{ \sup _ { p \in D } d ( p , E ) , \ \sup _ { p \in E } d ( p , D ) \right \} . \end{align*}


\begin{align*} \mathbf { R } _ { x s } ^ i \triangleq \begin{pmatrix} & \dot { \mathbf { w } } _ 1 & \dot { \mathbf { w } } _ { 2 : n _ i } ^ T \\ & \dot { \mathbf { w } } _ { 2 : n _ i } & \tilde { \mathbf { R } } _ { x s } ^ i \end{pmatrix} \end{align*}


\begin{align*} ( F \circ G ) ^ 2 ( x ) & = ( f ^ i ( g ^ m ( x ) ) , f ^ i { } _ \alpha ( g ^ m ( x ) ) g ^ \alpha { } _ j ( x ) , \\ * & \hphantom { { } = { } } f ^ i { } _ { \alpha \beta } ( g ^ m ( x ) ) g ^ \alpha { } _ j ( x ) g ^ \beta { } _ k ( x ) + f ^ i { } _ \alpha ( g ^ m ( x ) ) g ^ \alpha { } _ { j k } ( x ) ) . \end{align*}


\begin{align*} M ( - t ) = \frac { 1 } { \sqrt 2 \pi } \sum _ { r , j = 0 } ^ \infty \ , \pi _ r \ , \ , c _ { r , j } \ , \int _ { - \infty } ^ { \infty } \ , x ^ { j } \ , \exp \left ( - t x - \frac { x ^ 2 } { 2 } \right ) \mathrm { d } x . \end{align*}


\begin{align*} \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } \operatorname { T r } \phi _ { - 1 } \ , d \tau = e ^ { - t \alpha | \xi | } + ( n - 2 ) e ^ { - t \mu | \xi | } + e ^ { - 2 t \mu | \xi | } + e ^ { - 2 t \mu s _ 1 | \xi | } . \end{align*}


\begin{align*} V ^ { - 1 } \rho \circ \varphi V = V ^ { - 1 } \varphi \circ \rho V = V ^ { - 1 } \rho V . \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ \S \int _ \S W _ r ( m _ 1 z + m _ 2 y + m _ 3 x ) ~ \sin ( n _ 1 \Theta ( t , z ) + n _ 2 \Theta ( t , y ) + n _ 3 \Theta ( t , x ) ) \ \d y \d z , \end{align*}


\begin{align*} \{ \lambda _ { t } ^ { N , j } \} _ { 1 \leq j \leq N } \subset \Big \{ \lambda \in \mathbb C \ , : \ , \left | \lambda - 1 \right | \leq \frac { 1 } { N - 1 } \sum _ { j \not = i } h ' ( \mu ^ { N , j } _ { \boldsymbol { \alpha } ^ * _ t } ) \Big \} . \end{align*}


\begin{align*} I _ k \cap ( H \setminus P ) = \bigcup _ { n = n _ k } ^ { \infty } ( I _ k \cap ( H _ n \setminus K ) \cap ( \bigcap _ { m = 1 } ^ { \infty } ( H _ n \setminus P _ m ) ) ) \end{align*}


\begin{align*} { { \left ( \mathbf { x } + \Delta \mathbf { x } \right ) } ^ { T } } \left ( \mathbf { s } + \Delta \mathbf { s } \right ) = \nu { { \mathbf { x } } ^ { T } } \mathbf { s } + \Delta \mathbf { x } ^ T \Delta \mathbf { s } . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \chi _ 3 < 1 \Big ) \leq \lim _ { K \to + \infty } \lim _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { t \in [ 0 , 1 ] } | X ^ { \epsilon } ( t ) | \geq K \Big ) = - \infty . \end{aligned} \end{align*}


\begin{align*} \left \langle l _ 1 , m _ 1 \right | e ^ { i k \phi } \sin ( n \theta ) \left | l _ 2 , m _ 2 \right \rangle = \end{align*}


\begin{align*} g ( \rho , x ) = e ^ { - i \rho x } + \int _ { - \infty } ^ { x } B ( x , t ) e ^ { - i \rho t } d t \end{align*}


\begin{align*} - F v = \norm { v } _ { V } F \left ( \frac { - v } { \norm { v } _ { V } } \right ) = \norm { v } _ { V } ( G \widetilde { w } + z ) = G ( \norm { v } _ { V } \widetilde { w } ) + \norm { v } _ { V } z , \end{align*}


\begin{align*} \begin{array} { r c l l } \xi ^ { \mathrm { i n } } & = & 0 & a . e . ~ \mbox { i n } ( R _ { 0 } , \infty ) , \\ \mathcal { I } _ { 1 } & = & 0 & a . e . ~ \mbox { i n } ( 0 , t _ { 0 } ) \times ( R _ { 0 } , \infty ) , \\ \mathcal { I } _ { 2 } & = & 0 & a . e . ~ \mbox { i n } ~ ( 0 , t _ { 0 } ) \times ( R _ { 0 } , \infty ) . \end{array} \end{align*}


\begin{align*} \int _ \Omega \left ( P ( t ) f ( \eta ) \right ) g ( \eta ) \mu ( d \eta ) = \int _ \Omega f ( \eta ) ( \hat { P } ( t ) g ( \eta ) ) \mu ( d \eta ) , t \geq 0 . \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll \max \left \{ \frac { p ^ { 3 + \epsilon } } { ( \min Q ^ { * } ) ^ { 2 } } , p ^ { 5 / 2 + \epsilon } \right \} . \end{align*}


\begin{align*} B _ 2 = L ( L + 1 ) , \zeta = \kappa ( L + 1 ) , \end{align*}


\begin{align*} { \rm G a l } ( L ' / K ' ) = { \rm G a l } ( L ' / N _ 1 ) \times { \rm G a l } ( L ' / N _ 2 ) . \end{align*}


\begin{align*} P _ \delta ( z ) = \frac { \pi \delta ^ 2 } { \ln \frac { R } { \varepsilon } } q ^ 2 \sqrt { d e t ( K _ H ) } ( z ) + N _ \delta ( z ) , \end{align*}


\begin{align*} \pi ( \lambda ) _ { r + 1 } - \pi ( \lambda ) _ r - 1 = ( \pi s _ r ) ( \lambda ) _ r - \pi ( \lambda ) _ r \end{align*}


\begin{align*} \hat { \Lambda } _ n = [ - 2 ^ n + n + L + 1 , 2 ^ n - n - L - 1 ] ^ d , L : = \max _ { \Delta \ni 0 : c _ { \Delta } > 0 } ( \Delta ) + 1 . \end{align*}


\begin{align*} \Delta u ( t ) = U ( t , 0 ) \Delta u _ 0 & + \int _ 0 ^ t U ( t , s ) [ 2 i ( D _ j \nabla \tilde { b } ^ j ( s ) + \nabla \tilde { b } ^ j ( s ) D _ j + \nabla \tilde { c } ( s ) ) \nabla u \\ & + i ( D _ j \Delta \tilde { b } ^ j ( s ) + \Delta \tilde { b } ^ j ( s ) D _ j + \Delta \tilde { c } ( s ) ) u + \Delta f ( s ) ] d s . \end{align*}


\begin{align*} C ^ { \ast } \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } ^ { \ast } g \right ) \left ( X , Y \right ) = - C \left ( X , Y , Z \right ) \end{align*}


\begin{align*} \max \bigg \lbrace \sum _ { i , j = 1 } ^ n & t _ i t _ j | U ^ \top U | _ { i j } : t \in \mathbb { R } ^ n , \ ; \| t \| _ 2 = 1 , U \in \mathbb { R } ^ { m \times n } , \ ; U U ^ \top = { \rm I } _ m \bigg \rbrace \\ & \le \frac { m } { n } \left ( 1 + \sqrt { \frac { ( n - 1 ) ( n - m ) } { m } } \right ) , \end{align*}


\begin{align*} { p _ { E O S } } = \frac { { \rho R T } } { { 1 - b \rho } } - \frac { { a \xi \left ( T \right ) { \rho ^ 2 } } } { { 1 + 2 b \rho - { b ^ 2 } { \rho ^ 2 } } } , \end{align*}


\begin{align*} \left ( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \cdot P \right ) ( X ) = \det \begin{pmatrix} a & b \\ c & d \end{pmatrix} ^ { \frac { w - n } { 2 } } ( a + c X ) ^ n P \left ( \frac { b + d X } { a + c X } \right ) . \end{align*}


\begin{align*} d _ \textrm { H S } ( A , B ) = d _ \textrm { H S } ( C A D , C B D ) \forall A , B , C , D \in \mathrm { U } ( n ) . \end{align*}


\begin{align*} \sigma ( \sqrt { a } ) = - \sqrt { a } , \ & \sigma ( \sqrt { d } ) = \sqrt { d } \\ \tau ( \sqrt { a } ) = \sqrt { a } , \ & \tau ( \sqrt { d } ) = - \sqrt { d } . \end{align*}


\begin{align*} { } ^ { \psi } \mathcal D \circ { } ^ { \psi } \mathcal T [ \mathfrak { f } ] = \mathfrak { f } , \ \ \forall \mathfrak { f } \in L _ 2 ( \Omega , \mathbb H ) \cup C ( \Omega , \mathbb H ) . \end{align*}


\begin{align*} g _ { x _ 1 x _ 2 \ldots x _ d } = ( - 1 ) ^ d \prod _ { j = 1 } ^ d \lambda _ j g . \end{align*}


\begin{align*} \varphi _ i : = \frac { \langle \alpha ^ f ( X _ i , X _ i ) , \alpha ^ f ( X _ i , X _ i ) \rangle } { \langle \beta ( X _ i , X _ i ) , \beta ( X _ i , X _ i ) \rangle } \neq 0 , \end{align*}


\begin{align*} d = 4 + 4 n = 8 + 8 l \equiv 0 \mod 8 \end{align*}


\begin{align*} ( 1 + \rho ) ^ { k } = \frac { 1 } { 2 ^ { k } } \sum _ { j = 0 } ^ { 2 k } \sum _ { m = 0 } ^ { \left \lfloor j / 2 \right \rfloor } ( 2 \rho ) ^ { j - 2 m } \binom { k } { j - 2 m } \binom { k - j + 2 m } { m } . \end{align*}


\begin{align*} I _ n : = \left [ n \dfrac { \bar { t } } { \gamma } , \ , ( n + 1 ) \dfrac { \bar { t } } { \gamma } \right ) n \in { \mathbb { N } } \end{align*}


\begin{align*} \varphi = \begin{bmatrix} v \\ b \\ \end{bmatrix} , F = \begin{bmatrix} ( d + B _ { \infty } ) \otimes b - ( w + u _ { \infty } ) \otimes v \\ ( d + B _ { \infty } ) \otimes v - ( w + u _ { \infty } ) \otimes b \\ \end{bmatrix} , { \mathcal { L } } = \begin{bmatrix} \nu & 0 \\ 0 & \mu \\ \end{bmatrix} \Delta . \end{align*}


\begin{align*} \int _ { X } ( \alpha + \sqrt { - 1 } \omega ) ^ { 2 } = \int _ { X } ( \alpha ^ { 2 } - \omega ^ { 2 } ) + 2 \sqrt { - 1 } \int _ { X } \alpha \wedge \omega = 3 2 + 2 6 \sqrt { - 1 } . \end{align*}


\begin{align*} | \tilde { \varphi } ( t , x ) - \tilde { \varphi } ( t ' , x ' ) | \leq \varepsilon , ( t ' , x ' ) \in O _ * : = O _ \nu ( t , x ) \cap ( [ \tau , \tau + \delta _ * ] \times \mathbb R ^ n ) . \end{align*}


\begin{align*} C ( H , K , M ) : = \bigoplus _ { n \geq 0 } C ^ n ( H , K , M ) \end{align*}


\begin{align*} T : = T ^ e _ A ( x ) . \end{align*}


\begin{align*} ( \mathcal { R } _ \lambda ( \sigma s _ i ) v _ T , v _ T ) = \pm \frac { 1 } { a _ { i + 1 } - a _ { i } } ( \mathcal { R } _ \lambda ( \sigma ) v _ T , v _ T ) . \end{align*}


\begin{align*} \lim _ { j \rightarrow \infty } M _ j ( C _ \varphi ) = \frac { 1 } { k } \sum _ { i = 0 } ^ { k - 1 } C _ { \rho \circ \varphi _ i } . \end{align*}


\begin{align*} N _ { k , l } = \frac 1 { \sqrt 2 } \sum _ { c , b = 1 } ^ d \int _ 0 ^ t \left ( \hat { \sigma } _ { k l , b c } ( s ) + \hat { \sigma } _ { l k , b c } ( s ) \right ) d B ^ { c b } _ s . \end{align*}