\begin{align*} \begin{array} { l } \displaystyle w = \frac { 1 } { \tilde { m } } \left [ \Lambda ^ m ( x ) - m \int _ 0 ^ \infty \sigma ( s ) \int _ 0 ^ s f ^ 6 ( \tau ) d \tau d s \right ] , \end{array} \end{align*}


\begin{align*} G _ 1 ( x ) = n ! G ( x ) \end{align*}


\begin{align*} \begin{aligned} p ( t , v , g ) : = \left ( \frac { 1 } { V _ F } \sum _ { k = - \infty } ^ { + \infty } c _ k \exp \left ( i k \frac { 2 \pi } { V _ F } ( v - \int _ 0 ^ t g _ { } ( s ) d s ) \right ) \right ) \frac { 1 } { \sqrt { 2 \pi a ( t ) } } \exp \left ( - \frac { ( g - g _ { } ( t ) ) ^ 2 } { 2 a ( t ) } \right ) , \end{aligned} \end{align*}


\begin{align*} \liminf _ n - \Delta _ 2 \varphi ( \gamma _ { 2 n } ( \omega ) ) = 0 \end{align*}


\begin{align*} I ( h ) : = \int _ { D _ h } q _ v ^ { \sum _ i m _ i t _ i } \ , d t . \end{align*}


\begin{align*} t _ 2 : = & \big ( \lambda _ h u _ { \mathrm { n c } } , ( I - \widehat { I } ) ( u - \widehat { u } _ { \mathrm { n c } } ) \big ) _ { 1 + \delta } \le 2 \Vert h _ { \mathcal { T } } ^ { m } \lambda _ h u _ { \mathrm { n c } } \Vert _ { L ^ 2 ( \mathcal { T } \setminus \widehat { \mathcal { T } } ) } \Vert h _ { \mathcal { T } } ^ { - { m } } ( I - \widehat { I } ) ( u - \widehat { u } _ { \mathrm { n c } } ) \Vert _ { L ^ 2 ( \Omega ) } . \end{align*}


\begin{align*} \sum _ { \ell = 0 } ^ { n } \beta _ { j , \ell } ^ k \ , [ \alpha ^ * ] ^ \ell k = 1 , \ldots , m , \ j = 1 , \ldots , L \end{align*}


\begin{align*} & \sum _ { j \in N _ { i ' } } \max \left \{ a _ { i ' j } , b - \sum _ { k \in M _ C - i ' } a _ { k t _ k } \right \} x _ { i ' j } \\ & + \sum _ { i \in M _ C - i ' } \left ( \sum _ { j = 1 } ^ { t _ i } a _ { i t _ i } \max \left \{ 1 , \frac { a _ { i j } } { b - \sum _ { k \in M _ C - { i , i ' } } a _ { k t _ k } - a _ { i ' n _ { i ' } } } \right \} x _ { i j } + \sum _ { j = t _ i + 1 } ^ { n _ i } a _ { i j } x _ { i j } \right ) \leq b \end{align*}


\begin{align*} A \ , = \ , \begin{pmatrix} \lambda _ 1 & 1 & 0 & 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & \lambda _ 2 & 1 & 0 & 0 & \dots & 0 & 0 \\ 0 & 0 & 0 & 0 & \lambda _ 3 & 1 & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \lambda _ g & 1 \\ \end{pmatrix} . \end{align*}


\begin{align*} h : \mathcal { N } _ { \mathrm { S p } } ( 2 m , \alpha , L ) & \longrightarrow \mathcal { A } \coloneqq \bigoplus _ { i = 1 } ^ { m } H ^ 0 ( X , K ^ { 2 i } ( D ^ { 2 i - 1 } ) ) \\ ( E _ * , \varphi , \Phi ) & \longmapsto ( s _ 2 , s _ 4 , \dots , s _ { 2 m } ) . \end{align*}


\begin{align*} \sup _ { ( x , y ) \in \bar B } \sum _ { i = 1 } ^ { n - 2 } ( \varphi _ i ^ 1 ( x , y ) - \Phi _ i ( x , y ) ) ^ 2 < \frac { \varepsilon } { 9 } . \end{align*}


\begin{align*} g ( w ^ * ) = ( z + a f _ k ) ( w ^ * ) = b + \sum _ { j = 1 } ^ m a c _ j = 0 \end{align*}


\begin{align*} ( D : _ K A ) = \{ x \in K \mid x A \subseteq D \} . \end{align*}


\begin{align*} \left \{ \begin{array} { c c c } a _ { x , \mathrm { s } } a _ { y , \mathrm { t } } + a _ { y , \mathrm { t } } a _ { x , \mathrm { s } } & = & 0 \\ a _ { x , \mathrm { s } } ^ { \ast } a _ { y , \mathrm { t } } + a _ { y , \mathrm { t } } a _ { x , \mathrm { s } } ^ { \ast } & = & \delta _ { x , y } \delta _ { \mathrm { s } , \mathrm { t } } \mathfrak { 1 } \end{array} \right . , x , y \in \Lambda , \mathrm { s } , \mathrm { t } \in \mathrm { S } , \end{align*}


\begin{align*} \bold { q } ( t ) = - \beta \theta _ x ( x , t ) - \int _ 0 ^ { \infty } g ( s ) \theta _ x ( x , t - s ) d s , \end{align*}


\begin{align*} p ^ { - 1 } R ^ { - p } & = p ^ { - 1 } \left ( \frac { \sqrt { \frac { \kappa } { C A } + \left ( \frac { c _ d ' | \nabla f ( a ) | } { 2 C A } \right ) ^ 2 } + \frac { c _ d ' | \nabla f ( a ) | } { 2 C A } } { \frac \kappa { C A } } \right ) ^ p \\ & = \frac { c _ { d , p } } { \kappa ^ p } \left ( \sqrt { \tfrac { C A } { c _ d '^ 2 } \kappa + \left ( \tfrac 1 2 | \nabla f ( a ) | \right ) ^ 2 } + \tfrac 1 2 | \nabla f ( a ) | \right ) ^ p . \end{align*}


\begin{align*} | \lambda ^ n | \to \infty \| U ^ n \| _ { \mathcal { H } } = \| ( u ^ n , v ^ n , y ^ n , z ^ n , w ^ n , \eta ^ n ) \| _ { \mathcal { H } } = 1 , \end{align*}


\begin{align*} \frac { d x } { d t } = \sum _ { \alpha = 1 } ^ { r } b _ \alpha ( t ) X _ \alpha , \end{align*}


\begin{align*} \norm { h } ^ 2 = u ( a ) . \end{align*}


\begin{align*} Y ^ { \infty , 1 } _ t = \kappa _ 2 \int _ 0 ^ t ( t - s ) ^ \alpha ( \sigma ' \sigma ) ( X _ s ) d B _ s + \int _ 0 ^ t ( t - s ) ^ \alpha \sigma ' ( X _ s ) Y ^ { \infty , 1 } _ s d W _ s , \end{align*}


\begin{align*} z = - \frac { 1 } { u ( 1 + u ^ 2 ) } . \end{align*}


\begin{align*} \widetilde { A } ^ n _ t = n ^ { \alpha + \frac 1 2 } \sigma ( X _ t ) \int ^ t _ 0 [ ( t - \eta _ n ( s ) ) ^ { \alpha } - ( t - s ) ^ { \alpha } ] d W _ s , \end{align*}


\begin{align*} D _ 4 F _ p \big ( u ( t ) , \zeta ( t ) , v ( t ) , c ( t ) \big ) \big [ \dot c ( t ) \big ] = \nabla F _ p \big ( u ( t ) , \zeta ( t ) , v ( t ) , c ( t ) \big ) \cdot \dot c ( t ) \ . \end{align*}


\begin{align*} \begin{array} [ c ] { c } ( \frac { \partial } { \partial t } - \Delta _ { B } ) \log | X | _ { \omega } ^ { 2 } = \frac { 1 } { | X | _ { \omega } ^ { 2 } } \left ( - g ^ { T i \overline { j } } g _ { k \overline { l } } ^ { T } ( \partial _ { i } ^ { k } X ) ( \overline { \partial _ { j } ^ { l } X } ) + \frac { \left \vert \bigtriangledown ^ { T } | X | _ { \omega } ^ { 2 } \right \vert _ { \omega } ^ { 2 } } { | X | _ { \omega } ^ { 2 } } \right ) \leq 0 . \end{array} \end{align*}


\begin{align*} a = a \textsf { 1 } _ A & = a q ( k ) \stackrel { ( L 2 ) } = q ( a k ) \\ & = q ( a ' k ' ) \stackrel { ( L 2 ) } = a ' q ( k ' ) = a ' \textsf { 1 } _ A = a ' . \end{align*}


\begin{align*} \boldsymbol { V } _ { B \rightarrow A } ^ { e x t } & = \left ( ( \boldsymbol { V } _ { B } ^ { p o s t } ) ^ { - 1 } - ( \boldsymbol { V } _ { B } ^ { p r i } ) ^ { - 1 } \right ) ^ { - 1 } , \\ \boldsymbol { x } _ { B \rightarrow A } ^ { e x t } & = \boldsymbol { V } _ { B \rightarrow A } ^ { e x t } \left ( ( \boldsymbol { V } _ { B } ^ { p o s t } ) ^ { - 1 } \boldsymbol { x } _ { B } ^ { p o s t } - ( \boldsymbol { V } _ { B } ^ { p r i } ) ^ { - 1 } \boldsymbol { x } _ { B } ^ { p r i } \right ) , \end{align*}


\begin{align*} F ( z + \omega _ 2 \ , | \ , \overline \omega _ 1 , \omega _ 2 ) = \exp \Bigg ( \int _ { C } \frac { e ^ { ( z + \omega _ 2 ) s } } { ( e ^ { \overline \omega _ 1 s } 1 - 1 ) ( e ^ { \omega _ 2 s } - 1 ) } \frac { d s } { s } \Bigg ) , 0 < ( z ) < ( \overline \omega _ 1 ) . \end{align*}


\begin{align*} f - l ( h _ \sigma ) = \sum _ { i = 1 } ^ { \deg _ X ( f ) } \partial _ i l ( h _ \sigma ) Q _ \sigma ^ i \end{align*}


\begin{align*} \sum _ { p \in \pi _ j \cap J } s _ { k _ j , \lambda } ( p , \pi _ j ) = \frac { 1 } { { \lambda + k _ j \choose k _ j } } \sum _ { p \in \pi _ j \cap J } \tilde S _ { k _ j } ( p , \pi _ j ( p ) , \alpha , \lambda ) = 1 . \end{align*}


\begin{align*} \dfrac { V _ r } { e _ 1 \cdots e _ { r - 1 } } = \sum _ { 1 \le j < r } \dfrac { m _ r } { m _ j } \ , \dfrac { h _ j } { e _ 1 \cdots e _ j } . \end{align*}


\begin{align*} F = \tilde { B } _ { k + 1 } \setminus \bigcup _ { i = 1 } ^ k \tilde { B _ i } . \end{align*}


\begin{align*} \langle ( y _ 1 , \ldots , y _ s ) , \gamma _ j \rangle = d _ j - y _ { s + 3 } \end{align*}


\begin{align*} Z _ t = \lim _ { n \to \infty } n ^ { - 1 } \sum _ { i = 1 } ^ n \delta _ { X _ i ( t ) } \end{align*}


\begin{align*} \tfrac { 1 } { 2 } ( \log \lambda ) ^ { \theta / 4 } = \tfrac { 1 } { 2 } s _ 0 ^ { \theta / 4 } \gg \log A . \end{align*}


\begin{align*} e ^ { x ( e _ { \lambda } ( t ) - 1 ) } = \sum _ { n = 0 } ^ { \infty } \phi _ { n , \lambda } ( x ) \frac { t ^ { n } } { n ! } , ( \mathrm { s e e } \ [ 1 4 , 1 5 ] ) . \end{align*}


\begin{align*} \rho _ { \lambda } = \rho ( \lambda ^ { 2 } t , \lambda x ) , u _ { \lambda } = \lambda u ( \lambda ^ { 2 } t , \lambda x ) , \end{align*}


\begin{align*} \frac { d } { d \left \| f _ j \right \| _ 2 ^ 2 } \left ( \prod _ { i = 1 } ^ d \left \| f _ i \right \| _ 2 ^ 2 - \prod _ { i = 1 } ^ d \left ( \left \| f _ i \right \| _ 2 ^ 2 - \frac { L ^ 2 } { 1 2 b ^ 2 } \right ) \right ) & = \prod _ { i \neq j } \left \| f _ i \right \| _ 2 ^ 2 - \prod _ { i \neq j } \left ( \left \| f _ i \right \| _ 2 ^ 2 - \frac { L ^ 2 } { 1 2 b ^ 2 } \right ) \\ & \ge 0 . \end{align*}


\begin{align*} \begin{pmatrix} 1 & 1 + i & 1 & 0 \\ 1 + i & 4 i & 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} & \sigma _ a s _ { ( k ) } = s _ { ( k ) } \sigma _ a = s _ { ( k ) } , & & \epsilon _ a s _ { ( k ) } = s _ { ( k ) } \epsilon _ a = 0 , & & a = 1 , \ldots , k - 1 . \end{align*}


\begin{align*} H _ { ( Z _ - , D _ { Z , - } ) } ( y ) = e ^ { \frac { t _ - } { 2 \pi i } } \sum _ { d \in \mathbb K _ { - } } y ^ d \left ( \frac { \prod _ { i = 1 } ^ 2 \Gamma ( 1 + \frac { v _ i } { z } + v _ i \cdot d ) } { \prod _ { i \in M _ 0 } \Gamma ( 1 + \frac { \bar D _ i } { z } + D _ i \cdot d ) } \right ) \textbf { 1 } _ { [ d ] } [ \textbf { 1 } ] _ { ( D _ i \cdot d ) _ { i \in I _ - } , v _ 1 \cdot d } . \end{align*}


\begin{align*} \nu = \frac { 1 8 } { \pi ^ { 4 } } \sum \nolimits _ { m \in \mathbb { Z } } \frac { 1 } { m ^ 2 } \left \{ \sum \nolimits _ { n \in \mathbb { N } } \frac { 1 } { 2 ^ { n } n ^ { 2 } } \sum \nolimits _ { j = 1 } ^ { 2 ^ { n } } \delta _ { m + j 2 ^ { - n } } \right \} \end{align*}


\begin{align*} & \rho ( I ) : = \sup \left \{ { n } / { r } \ \left . \right | \ , I ^ { ( n ) } \not \subset I ^ r \right \} , \\ & \ \ \ \ \ \widehat { \rho } ( I ) : = \sup \left \{ { n } / { r } \ \left . \right | \ , I ^ { ( n t ) } \not \subset I ^ { r t } t \gg 0 \right \} , \\ & \ \ \ \ \ \ \ \ \ \ \ \rho _ { i c } ( I ) : = \sup \left \{ { n } / { r } \ \left . \right | \ , I ^ { ( n ) } \not \subset \overline { I ^ { r } } \right \} , \ . \end{align*}


\begin{align*} \tilde { \Omega } = \lim \limits _ { t \rightarrow \infty } e ^ { i t H } J U _ 0 ( t ) \end{align*}


\begin{align*} \frac { k ' a } { 2 } \left [ \tanh ( k ' a / 2 ) - \coth ( k ' a / 2 ) \right ] = - 1 \end{align*}


\begin{align*} ( 1 - b ^ 2 ) ^ { \frac { 3 } { 2 } } r + \lambda \left ( \frac { r } { \sqrt { r ^ 2 + \dot { r } ^ 2 } } + b ( \left ( - \dot { r } \sin ( \theta - t ) + r \cos ( \theta - t ) ) \theta _ { r } + \sin ( \theta - t ) \right ) \right ) \\ - \lambda \frac { d } { d t } \left ( \frac { \dot { r } } { \sqrt { r ^ 2 + \dot { r } ^ 2 } } + b \cos ( \theta - t ) \right ) = 0 . \end{align*}


\begin{align*} S _ p : = \sum _ { j = 1 } ^ \infty \frac { j } { p ^ j } < \infty . \end{align*}


\begin{align*} \Theta _ \theta : = \left ( \begin{matrix} \cos \theta & \sin \theta & 0 \\ \\ - \sin \theta & \cos \theta & 0 \\ \\ 0 & 0 & I _ { n - 2 \times n - 2 } \end{matrix} \right ) . \end{align*}


\begin{align*} q ( a , t ) = \frac { \pi ^ 2 } { \abs { ( a + i b ) ( a + i b - 1 ) } } \end{align*}


\begin{align*} F _ m ( \varphi ) = F _ m ( \varphi _ 1 ) + F _ m ( \varphi _ 2 ) . \end{align*}


\begin{align*} \begin{pmatrix} \lambda _ { \delta } \\ \mu _ { \delta } \end{pmatrix} , \begin{pmatrix} \lambda _ { \theta } \\ \mu _ { \theta } \end{pmatrix} \end{align*}


\begin{align*} Y _ { x } : = \left \{ y \in \left ( \gg _ { x } \right ) _ { 1 } \mid \left [ y , y \right ] = 0 \right \} . \end{align*}


\begin{align*} \sum _ { m = D ^ 2 } ^ { \infty } [ \tilde p ^ \# _ m ( A , A ) - \tilde \pi ( A ) ] \lesssim \frac { \log ( D / \delta ) } { f ( n ) } . \end{align*}


\begin{align*} \int _ I D T ( p ( t ) ) u \ d t = \int _ { I _ 1 } D T ( p ( t ) ) u \ d t + \int _ { I _ 2 } D T ( p ( t ) ) u \ d t , \end{align*}


\begin{align*} 2 i \partial _ z F _ { m , Q } ( z , s ) = \sum _ { \substack { \gamma \in \Gamma _ \infty \backslash \Gamma \\ \gamma \neq \gamma _ 1 , \gamma _ 2 } } f _ { 2 , m } ( \gamma z , s ) \frac { d ( \gamma z ) } { d z } , \end{align*}


\begin{align*} \epsilon ( f ) = \max _ { 1 \leq b \leq \deg ( f ) } \left \{ \frac { \nu ( f ) - \nu ( \partial _ b f ) } { b } \right \} , \end{align*}


\begin{align*} f ^ { ( j ) } ( \lambda ) = \binom { j } { p } p ! g ^ { ( j - p ) } ( \lambda ) = ( j ) _ p g ^ { ( j - p ) } ( \lambda ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } } | y p _ { t , k } ( y ) | d y & = e ^ { - t } \int _ { \mathbb { R } } | y p _ { 0 , k } ( y ) | d y \\ & \leq e ^ { - t } \int _ { \mathbb { R } } \int _ { 0 } ^ { V _ F } | y | p _ { } ( v , y ) d v d y . \end{align*}


\begin{align*} \P \{ \theta _ \infty < \infty \} = 1 . \end{align*}


\begin{align*} \sum _ { t = \delta ^ 2 } ^ { D ^ 2 } \tilde p ^ \# _ t ( A , A ) \lesssim \sum _ { t = \delta ^ 2 } ^ { D ^ 2 } \frac { 1 } { t f ( n ) } \lesssim \frac { \log ( D / \delta ) } { f ( n ) } . \end{align*}


\begin{align*} \oplus _ { \mu \not = 0 } \gg _ { 1 } ^ { \mu } = \oplus _ { \alpha \in \Delta _ { 1 } \backslash { A } ^ { \perp } } \gg _ { \alpha } \end{align*}


\begin{align*} \Sigma ^ * R ^ { 1 / 2 } = R ^ { 1 / 2 } A . \end{align*}


\begin{align*} \frac { 1 } { 2 T } \mu ( \Lambda _ { S ^ * _ \Gamma M \backslash S ^ * \Gamma , \ , T } ) = 0 . \end{align*}


\begin{align*} J : = \{ 1 , \ldots , d \} \setminus I . \end{align*}


\begin{align*} \kappa _ \gamma ( a , b ) \lesssim _ N \sum _ { \ell = - 1 } ^ \iota 2 ^ { - \iota \gamma N + \iota Q / 2 - \ell d _ 2 / 2 } \norm { F ^ { ( \iota ) } } _ 2 \lesssim \norm { F ^ { ( \iota ) } } _ 2 . \end{align*}


\begin{align*} t _ { n , k } = \left \{ \begin{array} { l l } { \displaystyle \frac { k - [ m _ n ] } { [ m _ n ] - [ m _ { n - 1 } ] } } , & \ \mbox { i f } \ m _ { n - 1 } < | k | \leq m _ n , \\ & \\ { \displaystyle \frac { [ m _ { n + 1 } ] - k } { [ m _ { n + 1 } ] - [ m _ n ] } } , & \ \mbox { i f } \ m _ { n } < | k | \leq m _ { n + 1 } . \end{array} \right . \end{align*}


\begin{align*} \hat { H } \phi _ { 0 } ( x ) = E _ { 0 } \phi _ { 0 } = 0 \end{align*}


\begin{align*} \int _ 0 ^ \infty \ 1 ( c _ d ' r | \nabla f ( a ) | + C A r ^ 2 > \kappa ) \ , \frac { d r } { r ^ { p + 1 } } = p ^ { - 1 } R ^ { - p } \ , . \end{align*}


\begin{align*} T _ E E ' = \mathcal { H } \nabla _ { \nu E } \nu E ' + \nu \nabla _ { \nu E } \mathcal { H } E ' , \end{align*}


\begin{align*} s _ 1 : = \inf \{ s ' > 0 : Z \ge \zeta [ s ' , s ] \} . \end{align*}


\begin{align*} & K _ a v _ b = q ^ { \delta _ { a b } } v _ b , E _ a v _ b = \delta _ { a + 1 , b } v _ a , \ \ F _ a v _ b = \delta _ { a , b } v _ { a + 1 } , \\ & K _ a w _ b = q ^ { - \delta _ { a b } } w _ b , E _ a w _ b = \delta _ { a , b } w _ { a + 1 } , F _ a w _ b = \delta _ { a + 1 , b } w _ { a } . \end{align*}


\begin{align*} \max ( A ) = \{ s \in A : | s | = \ell _ A \} . \end{align*}


\begin{align*} \mathcal { S } ^ \infty : = \{ m ( \xi ) : m : \R ^ 3 \longrightarrow \R , \| \mathcal { F } ^ { - 1 } [ m ] ( x ) \| _ { L ^ 1 _ x } < + \infty \} . \end{align*}


\begin{align*} \mathrm { ( I I ) } ~ & \mathrm { M D T C o v } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { Y } ) = \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } \left [ \frac { \boldsymbol { \Omega } } { F _ { \mathbf { Z } } ( \boldsymbol { \xi _ { a } } , \boldsymbol { \xi _ { b } } ) } - \frac { \boldsymbol { \delta } \boldsymbol { \delta } ^ { T } } { F _ { \mathbf { Z } } ^ { 2 } ( \boldsymbol { \xi _ { a } } , \boldsymbol { \xi _ { b } } ) } \right ] \mathbf { \Sigma } ^ { \frac { 1 } { 2 } } , \end{align*}


\begin{align*} K ^ { ( \alpha - 1 ) } \subseteq \bigcap _ { n = 1 } ^ \infty \operatorname { K e r } ( u _ n + a _ n f _ { q ( n ) } ) . \end{align*}


\begin{align*} \| \phi _ w \| _ r ^ r = \int _ { G _ \infty \times G _ S } \phi _ w ^ r \ , d ( m _ \infty \times m _ p ) = \| w \| ^ r _ { L ^ r ( G _ \infty ) } < \infty . \end{align*}


\begin{align*} 1 = \sum _ { e \in \widehat E } e ; ~ ~ ~ ~ ~ \overline e = e , ~ \forall ~ e \in \widehat E ; ~ ~ ~ ~ ~ e e ' = \begin{cases} e , \ ! & e = e ' ; \\ 0 , \ ! & e \ne e ' ; \end{cases} ~ \forall ~ e , e ' \in \widehat E ; \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \in [ 0 , T ] } E [ | C ^ n _ t - \widetilde { C } ^ n _ t | ^ 2 ] = 0 \end{align*}


\begin{align*} \alpha = a / q + \theta , | \theta | \leq 1 / q Q \end{align*}


\begin{align*} P ( T _ 0 , x ) & = \mathcal { G } ( r ( T ) , x ) \prod _ { j = 1 } ^ { k } P ( T _ j , x ) ^ { \alpha _ j } . \end{align*}


\begin{gather*} \hat \phi ( \tau , z ) = \hat h ( \tau ) ^ { \rm t } \theta _ { m } ( \tau , z ) \end{gather*}


\begin{align*} \widetilde { G } ( x , y ) = \frac { G ( x , y ) } { m ( x ) \ , m ( y ) } , x , y \in \Omega , \end{align*}


\begin{align*} \chi ^ { \epsilon , L } _ { \infty } : = \epsilon ^ d \sum _ { x \in \Lambda _ { \epsilon , L } } \big < \varphi _ 0 \varphi _ x \big > ^ { \epsilon , L } _ { \lambda , \mu } \leq \bar \chi . \end{align*}


\begin{align*} [ Q , Y ( x ) + \widehat { \varrho ( x ) } ) ] = 0 . \end{align*}


\begin{align*} \mu _ 1 \sin ( \mu _ 1 \pi ) = - \beta _ 1 \qquad \mu _ 2 \sin ( \mu _ 2 \pi ) = - \beta _ 2 , \end{align*}


\begin{align*} w ( x ) = \Psi ^ \alpha ( x ) \psi ( x ) v ( x ) \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty p _ k ( n ) q ^ n = \prod _ { j = 1 } ^ k \frac 1 { 1 - q ^ j } . \end{align*}


\begin{align*} \chi _ 0 ( y , s ) : = \left \{ \begin{array} { l l } 1 & \mbox { i f } \ | y | \le s + R , \\ 0 & \mbox { o t h e r w i s e } . \end{array} \right . \end{align*}


\begin{align*} A x = \lambda x \end{align*}


\begin{align*} \lim _ { n \to \infty } \min _ { 1 \le i \le k _ { n } } \mu _ { n , i } ( ( 1 - \varepsilon , 1 + \varepsilon ) ) = 1 \end{align*}


\begin{align*} \cap _ { n = 0 } ^ { \infty } \mathrm { F i x } ( S _ { n } ) \subset \cap _ { n = 0 } ^ { j _ { f } } \mathrm { F i x } ( S _ { n } ) \subset \cap _ { i = 1 } ^ { m } C _ { i } \end{align*}


\begin{align*} \frac { \partial \psi } { \partial t } = - { \rm i } \widehat H _ 0 ( t ) \psi , \psi \in \mathcal { H } , \end{align*}


\begin{align*} \psi = \Omega ^ - \varphi \end{align*}


\begin{align*} H ( m ) = \theta \frac { H ( m ) } { m } + ( m - \theta ) \frac { H ( m ) } { m } \leq \theta \frac { H ( \theta ) } { \theta } + ( m - \theta ) \frac { H ( m - \theta ) } { m - \theta } . \end{align*}


\begin{align*} c _ { n - 1 , \xi _ { \boldsymbol { s } k } } ^ { \ast \ast } = \frac { \Gamma \left ( ( n - 1 ) / 2 \right ) } { ( 2 \pi ) ^ { ( n - 1 ) / 2 } } \left [ \int _ { 0 } ^ { \infty } x ^ { ( n - 3 ) / 2 } \overline { \mathcal { G } } _ { n } \left ( \frac { 1 } { 2 } \xi _ { \boldsymbol { s } k } ^ { 2 } + x \right ) \mathrm { d } x \right ] ^ { - 1 } \end{align*}


\begin{align*} I _ 2 \le & \frac { C ( \ln N ) ^ 2 } { N } \sum _ { k = N } ^ { N + \log _ 4 ( \ln 4 \cdot \ln N ) } \iint \phi ( \xi ) \phi ( \xi - \eta + 2 ^ { k } e _ 1 ) \phi ( \eta - 2 ^ { k } e _ 1 ) \dd \eta \dd \xi \le \frac { C ( \ln N ) ^ 3 } { N } . \end{align*}


\begin{align*} \tau ( M ) = \frac { 1 } { 1 2 \pi ^ 2 } \int _ M \left ( | W _ + | ^ 2 - | W _ - | ^ 2 \right ) d \mu _ g \end{align*}


\begin{align*} W ( x _ 1 , \ldots , x _ a ) = \sum _ { i = 1 } ^ a x _ i ^ { r _ i } , \end{align*}


\begin{align*} \Big \langle \sum \limits _ { i = 1 } ^ { m } \alpha _ i \tilde { f } ^ { | u _ i \rangle } \Big | \sum \limits _ { i = 1 } ^ { m } \beta _ j \tilde { f } ^ { | v _ j \rangle } \Big \rangle _ { { } _ { \mathfrak { J } , t } } = \sum \limits _ { i , j = 1 } ^ { m } \overline { \alpha _ i } \beta _ j \langle u _ i | v _ j \rangle _ t , \end{align*}


\begin{align*} & t \in ( 0 , \infty ) \mapsto \| E \| _ { L ^ 1 \cap L ^ \infty } ; \\ & \lim _ { t \to 0 } \| E \| _ { L ^ 1 \cap L ^ \infty } = 0 . \end{align*}


\begin{align*} \Gamma ( \lambda + a ) = ( \lambda + ( a - 1 ) ) \Gamma ( \lambda + ( a - 1 ) ) , \ ; \ ; \ ; a \in \mathbb { Z } , \lambda \in \mathfrak { a } _ \mathbb { C } ^ * \end{align*}


\begin{align*} u _ { A , q } ^ { \infty } ( \hat { x } , d ) = \frac { 1 } { 4 \pi } \int _ D e ^ { - i k \hat { x } \cdot y } Q _ { A , q } u _ { A , q } ( y , d ) d y . \end{align*}


\begin{align*} I _ { D _ { Z , - } , d } = \frac { \prod _ { 0 < a < v _ 1 \cdot d } ( v _ 1 + a z ) } { \prod _ { i \in I _ - , D _ i \cdot d > 0 } ( \bar D _ i + ( D _ i \cdot d ) z ) } [ \textbf { 1 } ] _ { ( - D _ i \cdot d ) _ { i \in I _ - } , v _ 1 \cdot d } . \end{align*}


\begin{align*} H ^ { * } ( \overline { f } ) = \left [ \ \overline { f } , H \overline { f } , \cdots , H ^ { n - 1 } \overline { f } \ \right ] _ { n \times n } \in \mathbb { R } ^ { n \times n } \end{align*}


\begin{align*} \left \{ \begin{array} { r l } | ( \lambda _ i ( A ) E _ n - A ) _ { p p } | ^ { r o w } & = \prod _ { k = 1 ; k \ne i } ^ n ( \lambda _ i ( A ) - \lambda _ k ( A ) ) | v _ { i p } | ^ 2 , \\ - | ( \lambda _ i ( A ) E _ n - A ) _ { q p } | ^ { r o w } & = \prod _ { k = 1 ; k \ne i } ^ n ( \lambda _ i ( A ) - \lambda _ k ( A ) ) v _ { i p } v _ { i q } , p \ne q . \end{array} \right . \end{align*}


\begin{align*} \mathbb { V } ( \overline { T } _ n ) = \frac { 1 } { n ^ 2 } \Big ( \sum _ { 1 \leq i , j \leq n } \mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) - \mathbb { E } ^ 2 ( \sum _ { 1 \leq i \leq n } X _ i X _ { i + 1 } ) \Big ) . \end{align*}


\begin{align*} 0 \leq \psi _ 0 \leq 1 , \psi _ 0 ( \xi ) = 1 \quad | \xi | \leq 1 \psi _ 0 ( \xi ) = 0 | \xi | \geq 2 \end{align*}


\begin{align*} S = S _ { n , s , m } = D _ { n - 1 } ( \varphi ) \hookrightarrow \mathbb P ^ s . \end{align*}


\begin{align*} \tilde { g } = 2 d \tau d u + ( | \nabla u | ^ 2 + a ^ 2 + b ^ 2 ) d u ^ 2 + 2 a d u d x _ 1 + 2 b d u d x _ 2 + d x _ 1 ^ 2 + d x _ 2 ^ 2 \end{align*}


\begin{align*} \limsup _ { \ell \to \infty } \sum _ { 0 \le i < k _ { \sigma ( \ell ) } } \mu _ { \sigma ( \ell ) } ( B ^ i _ { \sigma ( \ell ) } ) \le \sum _ { 0 \le i < k } \norm { \mu ^ i } + \limsup _ { \ell \to \infty } ( \ell \wedge k ) 2 ^ { - \ell } = \sum _ { 0 \le i < k } \norm { \mu ^ i } , \end{align*}


\begin{align*} \sup _ { k , \tau } \sum _ { \substack { | k _ 2 | \gtrsim | k | \\ k _ 2 ^ * \ll | k | \\ | \Phi | \gtrsim k ^ 2 } } \frac { \langle k \rangle ^ { 2 s + 2 \varepsilon - 2 + } } { \prod _ { i = 1 } ^ p \langle k _ i \rangle ^ { 2 s } } \lesssim \langle k \rangle ^ { 2 \varepsilon - 2 + ( p - 1 ) \max ( 1 - 2 s , 0 ) + } , \end{align*}


\begin{align*} d ( F _ 1 d x _ 1 + F _ 2 d x _ 2 ) = & \frac { \partial F _ 1 } { \partial x _ 2 } d x _ 2 \wedge d x _ 1 + \frac { \partial F _ 2 } { \partial x _ 1 } d x _ 1 \wedge d x _ 2 \\ = & ( | \nabla u | ^ { - 2 } A _ { 1 2 } - | \nabla u | ^ { - 2 } A _ { 1 2 } ) d x _ 2 \wedge d x _ 1 = 0 . \end{align*}


\begin{align*} _ J ( L ) = \int _ L e ^ { - i \theta } \Omega \geq | \int _ { L _ 1 } \Omega | + | \int _ { L _ 2 } \Omega | . \end{align*}


\begin{align*} Q _ j ^ k ( \alpha _ i ) = \sum _ { \ell = 0 } ^ { n } \beta _ { j , \ell } ^ k \ , \alpha _ i ^ \ell + \tilde \eta , k = 1 , \ldots , m , \ j = 1 , \ldots , L , \ i = 1 , \ldots , p , \end{align*}


\begin{align*} r ( K ) & = r | _ C ( K ) . \end{align*}


\begin{align*} \omega = p ^ { * } ( \omega _ { 0 } ) \wedge q ^ { * } ( \mu ) . \end{align*}


\begin{align*} \int _ 0 ^ L \abs { w _ x } ^ 2 d x = o ( 1 ) \int _ 0 ^ L \abs { w } ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} u ^ s _ { A , q } ( x , y ) = - \frac { k ^ 2 } { 4 \pi } \underset { \underset { ( \ell _ 2 , m _ 2 ) \in \Gamma } { ( \ell _ 1 , m _ 1 ) \in \Gamma } } { \sum } i ^ { \ell _ 1 - \ell _ 2 } \mu _ { ( \ell _ 1 , m _ 1 ; \ell _ 2 , m _ 2 ) } h ^ { ( 1 ) } _ { \ell _ 1 } ( k \vert x \vert ) h ^ { ( 1 ) } _ { \ell _ 2 } ( k \vert y \vert ) Y ^ { m _ 1 } _ { \ell _ 1 } \left ( \hat x \right ) Y ^ { m _ 2 } _ { \ell _ 2 } \left ( \hat y \right ) , \end{align*}


\begin{align*} & { \mathrm { H } } ( \boldsymbol { \beta } ) = \prod _ { v \in { { \mathfrak { M } } } _ K } \max \{ 1 , | \beta _ 0 | _ v , \ldots , | \beta _ m | _ v \} \enspace , \end{align*}


\begin{align*} \Tilde { f } ( u , \xi ) = \bigl ( 1 + u ^ { p ( \frac { 1 } { p ^ \star } - 1 ) } \bigr ) \abs { \xi } ^ p \end{align*}


\begin{align*} \langle \Lambda _ { \gamma _ 1 } f , g \rangle = \langle \Lambda _ { \gamma _ 2 } f , g \rangle \Leftrightarrow \int _ { \R ^ n } ( q _ 1 - q _ 2 ) u ^ { ( 1 ) } _ f u ^ { ( 2 ) } _ g \ , d x = 0 , \end{align*}


\begin{align*} d ^ k ( f ' ) ( x , y _ 1 , \ldots , y _ k ) \ ; = \ ; d ^ { k + 1 } f ( x , y _ 1 , \ldots , y _ k , \cdot ) \colon E \to F \end{align*}


\begin{align*} m _ k ^ { b _ 0 , \ldots b _ k } ( p _ k , \ldots p _ 1 ) = \sum m _ l ( b _ k , \ldots b _ k , p _ k , b _ { k - 1 } , \ldots , p _ { k - 1 } , \ldots , p _ 1 , b _ 0 , \ldots b _ 0 ) . \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' ( X _ s ) ) ^ 2 \Psi ^ { n , 1 } _ s d s \\ & = \kappa _ 3 \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s , \end{align*}


\begin{align*} K _ 1 ( s ) = ( s ^ \beta { + } A ) ^ { - 1 } . \end{align*}


\begin{align*} e ^ T B _ { S S } e \geq e ^ T \prod _ { t = 1 } ^ k A _ { S S } ( t ) e \geq \min _ { \substack { \forall t : X ( t ) \geq \gamma I , X ( t ) e \leq e \cr e ^ T \sum _ { t = 1 } ^ k X ( t ) e \geq k | S | - \Lambda } } e ^ T \prod _ { t = 1 } ^ k X ( t ) e \geq ? ? \end{align*}


\begin{align*} \frac { d \Gamma } { d t } = G _ 1 ( t ) + G _ 2 ( t ) \Gamma + \Gamma G _ 3 ( t ) + \Gamma G _ 4 ( t ) \Gamma , \end{align*}


\begin{align*} T _ q ( x ) : = D _ { q } u ( x ) + u ( x ) u ( q x ) + \tilde { A } ( x ) u ( q x ) + r ( x ) . \end{align*}


\begin{align*} & Q _ { \vec q } = \sum _ { i , j } \varepsilon _ i \varepsilon _ j q _ i q _ j ^ { - 1 } E _ i ^ j \otimes E _ { - i } ^ { - j } , & & ( Q _ { \vec q } ) ^ { i l } _ { j m } = \varepsilon _ i \varepsilon _ j q _ i q _ j ^ { - 1 } \delta ^ { i , - l } \delta _ { j , - m } , \end{align*}


\begin{align*} ( \mathbf { U } ^ 0 \cdot \mathbf { n } ) | _ { Y = 0 } = ( \mathbf { B } ^ 0 \cdot \mathbf { n } ) | _ { Y = 0 } = 0 . \end{align*}


\begin{align*} & x _ 1 \cdot \frac { \partial f } { \partial x _ 1 } , . . . , x _ n \cdot \frac { \partial f } { \partial x _ n } , \\ & w _ { - \alpha } ( f ) : = \sum \limits _ { m \in \Delta \cap M } h t _ { - \alpha } ( m ) \cdot a _ m \cdot x ^ { m - \alpha } , \alpha \in R ( N , \Sigma _ { C ( \Delta ) } ) . \end{align*}


\begin{align*} f _ { ( n ) } ( y ) : = \left \{ \begin{aligned} & n f \left ( \frac { 1 } { n } \right ) y & & \ ; y \leq \frac { 1 } { n } \\ & f ( y ) & & \ ; y \geq \frac { 1 } { n } \end{aligned} \right . . \end{align*}


\begin{align*} F l _ { J } \coloneqq X _ { w _ J } \cong \prod _ { k = 1 } ^ m F l _ { n _ k } , \end{align*}


\begin{align*} S _ n & = \big ( n + \frac { 1 } { 2 } \big ) - n + B ^ \prime - r _ n \\ n ! & = C n ^ { n + 1 / 2 } e ^ { - n } e ^ { r _ n } \end{align*}


\begin{align*} \mathop { \lim } \limits _ { \varepsilon _ 2 \to \varepsilon _ 1 } \mathop { \sup } \limits _ { ( \xi , j ) \in K \times S } P \left ( { \| { u ^ { \varepsilon _ 2 } \left ( { t , s , \xi , j } \right ) - u ^ { \varepsilon _ 1 } \left ( { t , s , \xi , j } \right ) } \| \ge \eta } \right ) = 0 . \end{align*}


\begin{align*} \mathrm { S O T } - \lim _ { n \to \infty } \frac { 1 } { n } \sum _ { i = 1 } ^ n U ( \gamma _ i ) = A _ 0 . \end{align*}


\begin{align*} E ( \mathbf { R } _ n ) _ 2 = \varepsilon n \left [ \frac { n - 1 } { 2 } a _ 2 + b _ 2 \right ] , \end{align*}


\begin{align*} h ( F _ \ast X , F _ \ast Y ) = \lambda ^ 2 ( p ) g ( X , Y ) ~ ~ X , Y \in \Gamma ( T _ p M ) . \end{align*}


\begin{align*} R _ { \beta } ( \varphi ) = \left \{ x \in [ 0 , 1 ] : | T _ { \beta } ^ n x - x | < \varphi ( n ) n \in \N \right \} , \end{align*}


\begin{align*} f _ { \mathfrak { m } } ^ { \sharp } \doteq \Delta _ { \mathfrak { a } } + f _ { \Phi } = \Delta _ { \mathfrak { a } } + e _ { \Phi } - \beta ^ { - 1 } s \ . \end{align*}


\begin{align*} \alpha _ j & = - \sum _ { i = 0 } ^ J { \sf A } _ { i j } ^ { - 1 } T ^ { \frac { \gamma } { 2 } + j } \int _ { - \log T } ^ \infty e ^ { ( \frac { \gamma } { 2 } + j ) \tau _ 1 } ( T - t ) ( G , e _ i ) _ \rho d \tau _ 1 \\ & \lesssim T ^ { \frac { \gamma } { 2 } + j } \int _ 0 ^ T ( T - t ) ^ { - 1 + J - j + { \sf c } _ 1 } d t \lesssim T ^ { \frac { \gamma } { 2 } + J + { \sf c } _ 1 } . \end{align*}


\begin{align*} \frac 1 { p _ 1 } + \frac 1 { p _ 2 } + \dotsb + \frac 1 { p _ n } = \frac 1 p , \end{align*}


\begin{align*} \phi ( x , v ) = \left ( 1 - \gamma m - \beta \gamma \psi ( m ) m \right ) e ^ { - \gamma M ( x ) } \end{align*}


\begin{align*} Z ^ { \nu } _ i : = B ^ { - 1 } ( D ^ { \nu t } _ i + F _ i ^ { \nu t } ) \end{align*}


\begin{align*} \left | \sum _ { | c | > h ^ \delta } \sum _ { k = 1 } ^ \infty k ^ { - 4 } S _ { k } ( c ; \xi ) I _ { k , \ell } ( { } ^ t g c ) \right | \ll _ { w , \ell , \theta } \| g \| _ E ^ \theta h ^ { 5 - ( \theta - 4 ) \delta } . \end{align*}


\begin{align*} \tilde B _ { i i } ( k _ 1 : k _ 0 ) & \geq \eta _ i 0 \leq k _ 0 \leq k _ 1 \leq \tilde k _ L , \cr \tilde B _ { j j } ( k _ 1 : k _ 0 ) & \geq \tilde \eta _ j 0 \leq k _ 0 \leq k _ 1 \leq \sigma , \cr \sum _ { k = 0 } ^ { \sigma - 1 } B _ { j i } ( k ) & \geq \tilde \delta , \end{align*}


\begin{align*} \mathbb { E } ( X _ i X _ { i + 1 } ) = ( 1 - \alpha ) M _ 1 + \alpha M _ 2 , \end{align*}


\begin{align*} & \eta _ { x , p ; \delta } ( y ) = e ^ { i p ( y - x ) } \eta _ \delta ( y - x ) \end{align*}


\begin{align*} \Delta _ { \hat { \beta } } ( t ) = \frac { 1 - t } { 1 - t ^ n } d e t ( { \mathcal B } _ { - t } ( \beta ) - I ) \end{align*}


\begin{align*} d \lambda _ i ( t ) = \sqrt { 2 } d B _ i ( t ) + { \beta } \sum _ { j ( \neq i ) } \frac { 1 } { \lambda _ i ( t ) - \lambda _ j ( t ) } d t , \ \ \beta > 0 , \ \ i = 1 , \dots , N . \end{align*}


\begin{align*} \{ | ( \xi ' , \xi _ n ) | _ x ^ 2 + V ( x ) - E ( h ) , \chi _ \alpha ( x _ n ) \xi _ n \} = 2 \chi _ \alpha ' ( x _ n ) \xi _ n ^ 2 - \chi _ \alpha ( x _ n ) \partial _ { x _ n } ( R + V ) , \end{align*}


\begin{align*} p _ { n + 1 } ( j ) = p _ n ( j ) + \gamma f ( j ) . \end{align*}


\begin{align*} v _ { e x } ^ i - u _ { e Y } ^ i = 0 , u _ { e x } ^ i + v _ { e Y } ^ i = 0 . \end{align*}


\begin{align*} 2 \mathrm { R e } ( \bar \partial _ { \Phi } ^ * \alpha , \alpha \llcorner ( \bar \partial \eta ) ^ { \sharp } ) _ { D _ j , \Phi } \geq & - \int _ { D _ j } g ^ { - 1 } | \bar \partial _ { \Phi } ^ * \alpha | ^ 2 e ^ { - \Phi } \\ & + \sum _ { j , k = 1 } ^ n \int _ { D _ j } - g ( \partial _ j \eta ) ( \bar \partial _ k \eta ) \alpha _ { \bar j } \bar \alpha _ { \bar k } e ^ { - \Phi } , \end{align*}


\begin{align*} \Phi ( H ) = \left [ \begin{array} { c c } H & 0 \\ 0 & H \end{array} \right ] \quad { \rm a n d \ h e n c e } \Phi ^ \dagger \left ( \left [ \begin{array} { c c } Y _ { 1 } & 0 \\ 0 & Y _ { 2 } \end{array} \right ] \right ) = Y _ { 1 } + Y _ { 2 } \ . \end{align*}


\begin{align*} \mathcal { S } _ { L _ 0 } ( L ) = \mathcal { S } _ { L _ 0 ' } ( L ) + \mathcal { S } _ { L _ 0 } ( L _ 0 ' ) . \end{align*}


\begin{align*} - \Delta u ^ s - k ^ 2 u ^ s = Q _ { A , q } ( u ^ s + v ) \mathrm { i n } \ ; \R ^ 3 , \end{align*}


\begin{align*} \mathcal { G } ( r ( T ) , x ) & = \dfrac { P ( T , x ) } { \displaystyle \prod _ { j = 1 } ^ { d } P ( T _ j , x ) } . \end{align*}


\begin{align*} \varphi ( z ) = 2 \sqrt y \sum _ { n \neq 0 } a _ \varphi ( n ) K _ { i r } ( 2 \pi | n | y ) e ( n x ) \end{align*}


\begin{align*} \pi _ i \coloneqq \pi _ { \{ i \} } = y _ 1 + y _ 2 + \cdots + y _ i ( 1 \le i \le n - 1 ) \end{align*}


\begin{align*} \sigma ( \Delta _ n ) = \bigg \{ \left . \frac { n ^ 2 } { \pi ^ 2 } \sum _ { j = 1 } ^ \alpha \sin ^ 2 \Big ( \frac { \pi k _ j } { n } \Big ) \right | \forall _ { j = 1 , \ldots , \alpha } : k _ j = 0 \ , , \cdots , n - 1 \bigg \} \end{align*}


\begin{align*} g : = g _ { 2 ^ { | I | } - 1 } \circ \cdots \circ g _ 1 . \end{align*}


\begin{align*} \gamma _ 0 = ( a \cos t , a \sin t ) \end{align*}


\begin{align*} 0 \longrightarrow \tilde { \mathbb { L } } : = ( \tilde { \beta } ) \longrightarrow \mathbb { Z } ^ { m + 2 } \stackrel { \tilde { \beta } } { \longrightarrow } N \oplus \mathbb Z \longrightarrow 0 , \end{align*}


\begin{align*} \dim \mathcal { M } _ { } ( 2 m , \alpha , L ) = m ( 2 m + 1 ) ( g - 1 ) + m ^ 2 n , \end{align*}


\begin{align*} k = b _ 1 > b _ 2 > \cdots > b _ { \ell } . \end{align*}


\begin{align*} \int _ { B _ { 1 / 2 } } \big ( | \nabla U _ k | ^ 2 - | \nabla V _ { k , r } | ^ 2 \big ) \ , d x & = \int _ { B _ { 1 / 2 } } \big ( \nabla ( U _ k - V _ { k , r } ) \cdot \nabla ( U _ k + V _ { k , r } ) \big ) \ , d x . \end{align*}


\begin{align*} 0 < 1 - \lambda _ h ( j ) \kappa _ { m } ^ 2 h _ { \max } ^ { 2 { m } } \le 1 - \kappa _ { m } ^ 2 h _ { \max } ^ { 2 m } \Vert w _ { \mathrm { p w } } \Vert _ { L ^ 2 ( \Omega ) } \le \Vert v _ { \mathrm { p w } } - \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 m } w _ { \mathrm { p w } } \Vert _ { L ^ 2 ( \Omega ) } = \Vert v _ { \mathrm { n c } } \Vert _ { L ^ 2 ( \Omega ) } . \end{align*}


\begin{align*} \mathbb { E } ( X _ i ) = \sum _ { \substack { 0 \leq l \leq i \\ \gcd ( l , i ) \ { \rm { i s } } \ k - { \rm { f r e e } } } } C _ { \alpha } ( i , l ) = f _ k ( i ) + O _ \alpha \Big ( \frac { \tau _ 3 ( i ) } { \sqrt { i } } \Big ) , \end{align*}


\begin{align*} & ( \tilde { u _ p ^ n } , \tilde { v _ p ^ n } , \tilde { h _ p ^ n } , \tilde { g _ p ^ n } ) | _ { y \rightarrow \infty } = ( 0 , 0 , 0 , 0 ) , \\ & ( \tilde { u _ p ^ n } , \tilde { v _ p ^ n } , \tilde { h _ p ^ n } , \tilde { g _ p ^ n } ) | _ { y = 0 } = ( - \overline u _ e ^ n ( x ) , 0 , - \overline h _ e ^ n ( x ) , 0 ) . \end{align*}


\begin{align*} \nabla ^ s u ( x , y ) = \sqrt { \frac { C _ { n , s } } { 2 } } \frac { u ( x ) - u ( y ) } { | x - y | ^ { n / 2 + s + 1 } } ( x - y ) , \end{align*}


\begin{align*} v _ { \epsilon } ( t ) : = \int _ { 0 } ^ { t } \left ( \int _ { - \infty } ^ { t _ { 1 } } \left ( \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ { 0 } - B + 2 \epsilon , - t _ { 0 } - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } \right ) ( s ) d s \right ) d t _ { 1 } , \end{align*}


\begin{align*} \sum _ { z \in \mathbb { Z } ^ { d } } \left \vert f \left ( z \right ) \right \vert = \left \vert f \left ( 0 \right ) \right \vert + \sum _ { n = 1 } ^ { d } \sum _ { \mathcal { I } \in 2 ^ { \left \{ 1 , \dots , d \right \} } : \left \vert \mathcal { I } \right \vert = n } \ \sum _ { z \in \mathbb { Z } ^ { n } \backslash \left \{ 0 \right \} } \left \vert f \circ \xi _ { \mathcal { I } } \left ( z \right ) \right \vert \ . \end{align*}


\begin{align*} E ( G / S ' ) = E ( G / H ) E ( H / S ) \neq 0 . \end{align*}


\begin{align*} \Big | \bar { u } _ R ( y ) - \frac { 1 } { 2 } y ^ T A y \Big | = | w _ R ( y ) | \leq \frac { 1 6 c _ 1 } { R ^ 2 } \Big | x + \frac { R } { 4 } y \Big | ^ { 2 - \varepsilon } \leq C ( A , \varepsilon , c _ 1 ) R ^ { - \varepsilon } , \end{align*}


\begin{align*} \operatorname { H e s s } _ { p } ( V , W ) : = g ( \nabla _ { V } \operatorname { g r a d } ( p ) , W ) \end{align*}


\begin{align*} ( d , 1 ^ k ) = ( d , \underbrace { 1 , 1 , \ldots , 1 } _ { k } , 0 , 0 , \cdots ) . \end{align*}


\begin{align*} V [ g ] = V [ g ] _ - \oplus V [ g ] _ 0 \oplus V [ g ] _ + , \end{align*}


\begin{align*} \hat { S } _ n ( f ) = \varphi \left ( X _ 1 , \ , f ( X _ 1 ) , \ , \hdots , \ , X _ n , \ , f ( X _ n ) \right ) , \end{align*}


\begin{align*} c _ 0 h + c _ 1 g _ 1 + \ldots + c _ n g _ n = 0 , \end{align*}


\begin{align*} \tau ^ \mathrm { a n / t o p } ( M / S , \rho \oplus \rho ' ) = \tau ^ \mathrm { a n / t o p } ( M / S , \rho ) + \tau ^ \mathrm { a n / t o p } ( M / S , \rho ' ) \end{align*}


\begin{align*} [ 0 , 1 ] = \bigcup _ { w \in \Sigma _ { \beta } ^ n } I _ { n , \beta } ( w ) \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty u _ m ( n ) \ , q ^ n = \sum _ { k = 0 } ^ \infty \frac { q ^ { m k } } { ( q ; q ) ^ 2 _ k } . \end{align*}


\begin{align*} u = \frac { \partial u } { \partial \nu } = 0 \Gamma _ 1 . \end{align*}


\begin{align*} g ( x , y ) : = x + \gamma \frac { \| y \| _ d ^ 2 } { \tilde b ( x ) } . \end{align*}


\begin{align*} C _ k ( a _ 1 , \ldots , a _ n ) = \begin{pmatrix} a _ 1 & a _ 2 & \cdots & a _ n \\ a _ { n - k + 1 } & a _ { n - k + 2 } & \cdots & a _ { n - k } \\ a _ { n - 2 k + 1 } & a _ { n - 2 k + 2 } & \cdots & a _ { n - 2 k } \\ \vdots & \vdots & \vdots & \vdots \\ a _ { k + 1 } & a _ { k + 2 } & \cdots & a _ k \end{pmatrix} . \end{align*}


\begin{align*} \sum _ { 1 \leq i \leq n } \mathbb { E } ( { X _ i } ^ 2 ) = \sum _ { 1 \leq i \leq n } \mathbb { E } ( { X _ i } ) = O ( n ) , \end{align*}


\begin{align*} t \ge \tau : = 1 . 1 4 0 3 \cdots , \end{align*}


\begin{align*} A _ { n } = \sum _ { \gamma \in \mathcal { M } _ { n } } w ( \gamma ) = \sum _ { \gamma \in \mathcal { A } _ { n } } w ( \gamma ) + \sum _ { \gamma \in \mathcal { B } _ { n } } w ( \gamma ) . \end{align*}


\begin{align*} \sup \limits _ { y } | V ( x , y ) | = \begin{cases} | x | ^ { - 2 } & | x | > 1 \\ 1 & | x | < 1 \end{cases} \end{align*}


\begin{align*} \nu _ \sigma \left ( f - l ( h _ \sigma ) \right ) = \beta _ b + b \gamma _ \sigma \end{align*}


\begin{align*} \begin{array} { l l } \frac { \partial \hat { \rho } ( t , 0 ) } { \partial t } = \frac { i } { \hbar } [ \hat { \rho } ( t , 0 ) , \hat { H } ( t ) ] = \left . \frac { i } { \hbar } \partial _ s ^ 2 { \rm L o g } \left ( e ^ { \int \hat { \rho } ( t , s ) d s } e ^ { \int \hat { H } ( t ) d s } \right ) \right | _ { s = 0 } , \end{array} \end{align*}


\begin{align*} Q = \{ s + i t : s \in [ A , B ] , | t - f _ { \nu } ( s _ { 0 } ) | \le \varepsilon \} \end{align*}


\begin{align*} \delta : = \frac { 1 } { 1 - \lambda _ h \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 { m } } } - 1 = \frac { \lambda _ h \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 { m } } } { 1 - \lambda _ h \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 { m } } } = \lambda _ h \kappa _ { m } ^ 2 h _ { \mathcal { T } } ^ { 2 { m } } ( 1 + \delta ) \in P _ 0 ( \mathcal { T } ) . \end{align*}


\begin{align*} \begin{array} { l l } \mbox { m i n i m i z e } & f ( x ) + g ( y ) + h ( x ) \\ \mbox { s u b j e c t t o } & A x = y , \end{array} \end{align*}


\begin{align*} W H W ^ T = d i a g ( m _ 1 S , \dots , m _ N S , 0 , \dots , 0 ) . \end{align*}


\begin{align*} \frac { \abs { f ' ( z ) } } { \abs { f ' ( z _ 0 ) } } = \frac { \abs { g ' ( \textbf { z } ) } } { \abs { g ' ( \textbf { z } _ 0 ) } } \frac { \abs { \textbf { z } + \lambda i } ^ 2 } { \abs { \textbf { z } _ 0 + \lambda i } ^ 2 } \geq \frac { \abs { g ' ( \textbf { z } ) } } { \abs { g ' ( \textbf { z } _ 0 ) } } \approx 1 , \end{align*}


\begin{align*} \Lambda ^ { ( 2 ) } _ n = \Lambda ^ { ( 3 ) } _ { n , \delta } + \Lambda ^ { ( 4 ) } _ { n , \delta } , \end{align*}


\begin{align*} P _ { n - 2 r , n } \left ( \frac { n } { 2 } \right ) = P _ { 2 r - 1 } \left ( n - 1 , \frac { n } { 2 } \right ) = 0 . \end{align*}


\begin{align*} J m = \sum \limits _ { j = 1 } ^ n \binom { n - 1 } { j - 1 } c _ j . \end{align*}


\begin{align*} f \circ ( \phi _ 1 * \phi _ 2 ) ( g ) = f _ + ' ( 0 ) \phi _ 1 * \phi _ 2 ( g ) + \int _ { 0 } ^ { \infty } f _ t \circ ( \phi _ 1 * \phi _ 2 ) ( g ) d \nu ( t ) \end{align*}


\begin{align*} \frac { \partial } { \partial x } ( 4 p _ t - 6 p p _ x - p _ { x x x } ) \ , = \ , 3 p _ { y y } , \end{align*}


\begin{align*} \label [ I e q ] { t r a n s i t i o n s e m i } T _ t \varphi ( x ) : = \int _ E \varphi ( y ) P _ t ( x , \dd y ) , \end{align*}


\begin{align*} u ( x , t ) \ge M _ n \left \{ \log \left ( \frac { 1 + t - x } { 1 + l _ n R } \right ) \right \} ^ { a _ n } \quad \mbox { i n } \ D _ { n } \end{align*}


\begin{align*} \pi _ { S } ^ T ( k + 1 ) A _ { S \bar { S } } ( k ) e + \pi _ { \bar { S } } ^ T ( k + 1 ) \left ( e - A _ { \bar { S } S } ( k ) e \right ) = \pi _ { \bar { S } } ^ T ( k ) e \end{align*}


\begin{align*} \dot \kappa ^ { \epsilon , L } _ t : = \frac { 1 } { t } - \frac { \chi ^ { \epsilon , L } _ t ( \lambda , \mu , m ^ 2 ) } { t ^ 2 } . \end{align*}


\begin{align*} ( L ( x ) , x ) _ V & = \sum \limits _ { v \in V } \delta _ V ( v ) [ \mathcal { Q } ( x ) ( v ) - n ( v ) x ( v ) ) ] x ( v ) \\ & = \sum \limits _ { v \in V } \delta _ V ( v ) [ ( a v g ^ * \circ a v g ) ( x ) ( v ) - n ( v ) x ( v ) ) ] x ( v ) \\ & = \sum \limits _ { v \in V } \sum \limits _ { e \in E _ v } \frac { \delta _ E ( e ) } { | e | } [ ( a v g ( x ) ) ( e ) - x ( v ) ) ] x ( v ) . \end{align*}


\begin{align*} \langle \mathcal { H } _ { q } ^ { n } \ , e _ { r } , e _ { 0 } \rangle = \sum _ { \gamma \in \mathcal { S } _ { 1 } ( n , r ) } w ( \gamma ) \end{align*}


\begin{align*} G _ V ( u ) = \int _ 1 ^ u \frac { \d s } { V ( s ) } = \int _ 1 ^ u \frac { \d s } { 1 + s ^ { \xi } } \sim 1 + u ^ { 1 - \xi } , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ d t _ i s _ i = n . \end{align*}


\begin{align*} | k ( t _ j ) | \le V a r ( \mathbb { K } _ h ( x - X _ 0 ) ) ^ \frac { 1 } { 2 } V a r ( \mathbb { K } _ h ( x - X _ { t _ j } ) ) ^ \frac { 1 } { 2 } \le \int _ { \mathbb { R } ^ d } ( \mathbb { K } _ h ( x - y ) ) ^ 2 \pi ( y ) d y \le \frac { c } { \prod _ { l = 1 } ^ d h _ l } . \end{align*}


\begin{align*} \phi _ h ( x _ 1 , x _ 2 ) = e ^ { \frac { i } h x _ 1 } , h ^ { - 1 } \in 2 \pi \mathbb { Z } . \end{align*}


\begin{align*} d _ i & = d _ 0 + U ( - \Delta _ d , \Delta _ d ) , \\ s _ i & = s _ 0 + U ( - \Delta _ s , \Delta _ s ) , \end{align*}


\begin{align*} \frac { d U } { d t } ( t ) U ^ \dagger ( t ) = P ( { \rm A d } _ { U ( t ) } { \rm i } \widehat H ( t ) ) . \end{align*}


\begin{align*} & \mathrm { M D T C o v } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { X } ) = \mathrm { E } \left [ ( \mathbf { X } - \mathrm { M D T E } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { X } ) ) ( \mathbf { X } - \mathrm { M D T E } _ { ( \boldsymbol { a } , \boldsymbol { b } ) } ( \mathbf { X } ) ) ^ { T } | \boldsymbol { a } < \mathbf { X } \leq \boldsymbol { b } \right ] . \end{align*}


\begin{align*} \nu _ \sigma ( f ) = \nu ( l ( h _ \sigma ) ) = \beta _ b + b \gamma _ \sigma . \end{align*}


\begin{align*} M _ 0 = \{ i \in \{ 1 , \ldots , m \} : D _ i \cdot e = 0 \} ; \end{align*}


\begin{align*} n ( z , x _ 0 - 2 x ) \cdot g = \begin{pmatrix} a '' & * & * \\ b ' & * & * \\ c & * & * \end{pmatrix} , a '' = a ' - \sqrt { - 3 } x c . \end{align*}


\begin{align*} \Delta ( F ^ { ( n ) } ) = \sum _ { a = 0 } ^ n q ^ { a ( n - a ) } F ^ { ( a ) } \otimes F ^ { ( n - a ) } K ^ { - a } . \end{align*}


\begin{align*} v ' _ { c } = 0 ; \end{align*}


\begin{align*} \int _ { \mathfrak { M } _ { i } } f _ { i } ( \alpha ) S _ { i } ^ { 2 } ( \alpha ) T _ { i } ^ { 2 } ( \alpha ) e ( - n \alpha ) d \alpha = \frac { 1 } { 3 ^ { 4 } } \mathfrak { S } ( n ) \mathfrak { J } _ { i } ( n ) + O ( N _ { i } ^ { \frac { 1 1 } { 9 } } L ^ { - 1 } ) \end{align*}


\begin{align*} C _ { \beta , \varphi } = \frac 1 { \pi } ( \cos \varphi ) ^ { \beta - 1 } \Gamma ( 1 - \beta ) . \end{align*}


\begin{align*} \widetilde { Y } ^ n _ t - Y ^ n _ t = R ^ n _ t + \int _ 0 ^ t ( t - s ) ^ { \alpha } \sigma ' ( X _ s ) ( \widetilde { Y } ^ n _ s - Y ^ n _ s ) \ , d W _ s , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { \tau } ( t - s ) ^ \alpha \sigma ' ( X _ s ) \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \ , \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) d s = 0 . \end{align*}


\begin{align*} \Theta ^ n _ s & = \sigma ( X _ s ) \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u + X ^ n _ { \eta _ n ( s ) } - X ^ n _ s \\ & = \sigma ( X _ s ) \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u + \int ^ s _ { \eta _ n ( s ) } \left ( s - \eta _ n ( u ) \right ) ^ { \alpha } \sigma ( X ^ n _ { \eta _ n ( u ) } ) \ , d W _ u \\ & + \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \ , \sigma ( X ^ n _ { \eta _ n ( u ) } ) \ , d W _ u . \end{align*}


\begin{align*} h ( \tau , p ( \tau ; t , x ) ) = h ( \tau , p ( \tau ; \sigma , p ( \sigma ; t , x ) ) ) . \end{align*}


\begin{align*} \lim _ { j \to \infty } - \Delta _ k \varphi ( \gamma _ { n _ j } ( \omega ) ) = 0 \end{align*}


\begin{align*} \partial _ t \vec { x } = \vec { H } . \end{align*}


\begin{align*} - \Delta _ p G _ n - \lambda G _ n ^ { p - 1 } = f _ n \Omega . \end{align*}


\begin{align*} p ( \boldsymbol { s } ) = p \left ( s _ { 1 } \right ) \prod _ { m = 1 } ^ { M - 1 } \mathop { p ( s _ { m + 1 } | s _ { m } ) } , \end{align*}


\begin{align*} \rho \ , H = \sqrt { \frac { 2 H ^ 2 A } { A + G } } \leq \sqrt { \frac { H ^ 2 A } { G } } = \sqrt { H G } \leq G , \end{align*}


\begin{align*} \sum \nolimits _ { S , m } a _ { S , m } \ , \zeta _ { S , m } = 0 , \end{align*}


\begin{align*} \frac { 1 } { p } = \frac { 1 } { p _ 1 } + \frac { 1 } { r } \end{align*}


\begin{align*} f \times _ \lambda g ( z ) = \int _ { \R ^ { 2 n } } f ( z - w ) g ( w ) e ^ { \frac i 2 \lambda \omega ( z , w ) } \ , d w , z \in \R ^ { 2 n } , \end{align*}


\begin{align*} \nu _ { n } = \mu _ { n , 1 } \boxplus \cdots \boxplus \mu _ { n , k _ { n } } , \end{align*}


\begin{align*} k _ \tau [ \mathfrak { M } ^ { e v } ( \mathcal { X } , \tau ) ] = m _ \tau k _ { * , r e d } [ \mathfrak { M } ^ { e v } ( \mathcal { X } , \tau ) _ { s , r e d } ] . \end{align*}


\begin{align*} \langle f , I ^ { \circ } ( w _ 1 , z _ 2 ) u _ { k - 1 + \frac { j _ 2 } { T } } w _ 2 \rangle = 0 . \end{align*}


\begin{align*} u _ 1 ( x ) = \cos \left ( \frac { \pi } { 2 } x \right ) u _ 2 ( x ) = \sin \left ( \frac { \pi } { 2 } x \right ) \end{align*}


\begin{align*} C _ { \beta , \varphi } = \frac 1 { \pi } ( \cos \varphi ) ^ { \beta - 1 } \Gamma ( 1 - \beta ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l } P _ 1 e = T Q _ 1 B f , \\ Q _ 0 B f = 0 . \end{array} \right . \end{align*}


\begin{align*} \norm { g _ { \le \iota } ^ { ( 1 ) } } _ p ^ p \lesssim _ \iota ( \iota + 2 ) ^ { p - 1 } \sum _ { \ell = - 1 } ^ \iota \sum _ { m = 1 } ^ { M _ { \ell } } \Vert \tilde g _ { m } ^ { ( \ell ) } \Vert _ p ^ p . \end{align*}


\begin{align*} \hat { \Gamma } _ { ( X _ + , D _ + ) } : = \bigoplus _ { \vec d \in ( \mathbb Z ) ^ { | I _ + | } , f \in \mathbb K _ { + } / \mathbb L } \prod _ { i \in M _ 0 } \Gamma ( 1 + \bar D _ i - \langle D _ i \cdot f \rangle ) \prod _ { i \in I _ + , d _ i < 0 } \frac { 1 } { \bar D _ i - d _ i } \textbf { 1 } _ { f } [ \textbf 1 ] _ { ( d _ i ) _ { i \in I _ + } } . \end{align*}


\begin{align*} f ( x _ 1 , \ldots , x _ r ) = ( x _ 1 v _ 1 + \ldots + x _ r v _ r ) ^ n \in A _ n \simeq k . \end{align*}


\begin{align*} & ( X , \delta _ X f \wedge \delta _ X ( D ^ 2 g ( \nu _ { X } D f , \cdot ) - D ^ 2 f ( \nu _ { X } D g , \cdot ) ) ) \\ & = D ^ 2 g ( \nu _ { X } D f , \nu _ { X } D h ) - D ^ 2 f ( \nu _ { X } D g , \nu _ { X } D h ) + \circlearrowright , \end{align*}


\begin{align*} W : = \Sigma \circ h : X ^ \vee \rightarrow \mathbb C . \end{align*}


\begin{align*} Z ^ n _ t = n ^ { \alpha + \frac 1 2 } \int ^ t _ 0 ( t - s ) ^ { \alpha } [ \sigma ( X ^ n _ { s } ) - \sigma ( X _ { s } ) ] \ , d W _ s . \end{align*}


\begin{align*} \widetilde { Z } ^ n _ t = n ^ { \alpha + \frac 1 2 } \int ^ t _ 0 ( t - s ) ^ { \alpha } \sigma ' ( X _ s ) \left ( X ^ n _ { s } - X _ { s } \right ) \ , d W _ s . \end{align*}


\begin{align*} S ^ { n , M , 2 } _ \tau = n ^ { \alpha + \frac 1 2 } \sum _ { i = 0 } ^ { M - 1 } \int _ { \tau _ i } ^ { \tau _ { i + 1 } } ( t - s ) ^ \alpha [ \sigma ' ( X _ s ) - \sigma ' ( X _ { \tau _ i } ) ] \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s \end{align*}


\begin{align*} \prod _ { k = 1 } ^ { \infty } \left ( \frac { z - \mu _ k ^ 2 } { z - \lambda _ k ^ 2 } \right ) & = 1 - \sum \limits _ { k = 1 } ^ { \infty } a _ { k } \frac { ( \lambda _ { k } ^ 2 - z ) + z } { \lambda _ { k } ^ 2 ( \lambda _ { k } ^ 2 - z ) } \\ & = 1 - \sum \limits _ { k \geq 1 } \frac { a _ k } { \lambda _ k ^ 2 } - \sum \limits _ { k \geq 1 } \frac { a _ k z } { \lambda _ k ^ 2 ( \lambda _ k ^ 2 - z ) } , \end{align*}


\begin{align*} W _ { n , m ; \mathrm { f a r } } = W _ { n , m ; \mathrm { o u t } } \sqcup W _ { n , m ; \mathrm { i n } } \end{align*}


\begin{align*} ( [ w , x ] , y , z ) = [ w , ( x , y , z ) ] + [ x , ( w , y , z ) ] \mbox { i f } \mathsf { c h a r } ( F ) = 2 ; \end{align*}


\begin{align*} \beta _ j : = \nu _ \sigma \left ( \partial _ j l ( h _ \sigma ) \right ) = \nu \left ( \partial _ j l ( h _ \sigma ) \right ) \mbox { f o r e v e r y } j , 1 \leq j \leq \deg _ X ( l ) \mbox { a n d } \rho < \sigma . \end{align*}


\begin{align*} \sum _ { n = 1 } ^ \infty a _ n r _ n ( x ) \end{align*}


\begin{align*} e ^ { \sqrt { c _ { 0 } t } \Lambda _ 1 } T _ { a } b = \sum _ { j \in \mathbb { Z } } W _ { j } \ , \ , \ , \mbox { w i t h } \ , \ , \ , W _ { j } \triangleq \mathcal { B } _ t ( \dot S _ { j - 1 } A , \dot \Delta _ j B ) . \end{align*}


\begin{align*} w ( \eta ) = \sup \limits _ { a } \left ( \eta a - J ( a ) \right ) \end{align*}


\begin{align*} d _ i \Delta ^ { n - i } e _ i & = \Delta ^ { n - i } \phi ^ { n - i } ( d _ i ) e _ i & ( \ ; d _ i \Delta = \Delta \phi ( d _ i ) ) \\ & = \Delta ^ { n - i } \phi ^ { n - i } ( d _ i ) \left ( \phi ^ { n - i } ( d _ i ) \backslash \Delta \right ) & ( \ ; e _ i = \phi ^ { n - i } ( d _ i ) \backslash \Delta ) \\ & = \Delta ^ { n - i + 1 } & ( \ ; \phi ^ { n - i } ( d _ i ) \left ( \phi ^ { n - i } ( d _ i ) \backslash \Delta \right ) = \Delta ) . \end{align*}


\begin{align*} \abs { f ' ( A ( z ) ) } = \abs { \frac { b } { 2 \lambda } } \frac { 1 } { \abs { g ( z ) } ^ 2 } \abs { g ' ( z ) } \abs { z + \lambda i } ^ 2 . \end{align*}


\begin{align*} \frac { 1 } { \gamma } \leq \int _ 0 ^ \infty \exp \Big [ - 2 \int _ 0 ^ t \dot \kappa ^ { \epsilon , L } _ s \ , d s \Big ] \ , d t , \dot \kappa ^ { \epsilon , L } _ t : = \frac { 1 } { t } - \frac { \chi _ t ^ { \epsilon , L } } { t ^ 2 } . \end{align*}


\begin{align*} x _ n = \prod _ { k \leq n } \left ( p _ k ^ 2 - \chi ( p _ k ) \right ) , y _ { n } = \prod _ { k \leq n } p _ k ^ 2 . \end{align*}


\begin{align*} E ( G / H ) = \dfrac { E ( G / G ^ 0 ) } { E ( H / H ^ 0 ) } \equiv 0 \mod p , \end{align*}


\begin{align*} \sum _ { \nu = 1 } ^ { \infty } \frac { \nu ^ t } { p ^ { \nu k } } \ll \frac 1 { p ^ k } . \end{align*}


\begin{align*} \mathrm { T C E } _ { p } ( X ) = \mu + \sigma \frac { c _ { 1 } \overline { G } _ { 1 } \left ( \frac { 1 } { 2 } \xi _ { p } ^ { 2 } \right ) } { \overline { F } _ { Y } ( \xi _ { p } ) } , \end{align*}


\begin{align*} W _ { j } ( z ) = \frac { 1 } { z ^ { j + 1 } } + O \left ( \frac { 1 } { z ^ { j + 2 } } \right ) . \end{align*}


\begin{align*} \sum _ { p \in \pi _ j \cap J } s _ { k _ j } ( p , e ( \pi _ j ) ) = 1 \end{align*}


\begin{align*} 2 ^ a \Phi ( - 1 , a , 2 b ) & = \sum _ { k = 0 } ^ { \infty } \frac { 1 } { \left ( k + b \right ) ^ a } - \sum _ { k = 0 } ^ { \infty } \frac { 1 } { \left ( k + b + \frac { 1 } { 2 } \right ) ^ a } = \zeta ( a , b ) - \zeta \left ( a , b + \frac { 1 } { 2 } \right ) . \end{align*}


\begin{align*} \mathcal { P } _ { [ n , 0 ] } = \bigcup _ { j = - 1 } ^ { p } \mathcal { U } _ { [ n , j ] } , n \geq 1 , \end{align*}


\begin{align*} g ( \nabla _ { \partial _ { i } } X , \partial _ { j } ) = - g ( \nabla _ { \partial _ { j } } X , \partial _ { i } ) \end{align*}


\begin{align*} \lim _ { \alpha \to 0 } \left < \frac { i } h [ - h ^ 2 \Delta + V ( x ) - E ( h ) , \chi _ \alpha ( x _ n ) h D _ n ] u _ h , \ , u _ h \right > _ { L ^ 2 ( \Omega _ \Gamma ) } = o ( 1 ) . \end{align*}


\begin{align*} \Lambda ^ m _ { x x } = - c ^ { - 1 } \left ( f ^ 5 + \delta f ^ 1 _ x \right ) , \end{align*}


\begin{align*} L ^ { n , 1 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) ( s - \eta _ n ( s ) ) ^ \alpha ( W _ s - W _ { \eta _ n ( s ) } ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) d s . \end{align*}


\begin{align*} \mu ( T _ 0 , \lambda ) \ge \sum _ { j = 1 } ^ { k } \mu ( T _ j , \lambda ) ( \forall \lambda \in \mathbb { R } ) . \end{align*}


\begin{align*} \operatorname { g r a d } ( f ^ { 2 } ) = F \operatorname { g r a d } ( \| X \| ^ { 2 } ) \end{align*}


\begin{align*} A d _ { \Theta _ n ( \phi ) ^ { - 1 } } ( d \Theta _ n ( e _ { \phi ( \lambda ) } ) ) \stackrel { ( \ref { e q - d i f f e r e n t i a l - o f - T h e t a - n - s e n d s - e - l a m b d a - t o } ) } { = } \Theta _ n ( \phi ) ^ { - 1 } e _ { \theta ( \phi ( \lambda ) ) } \Theta _ n ( \phi ) = \frac { t ( \lambda , \lambda ) } { 2 } e _ { \theta ( \lambda ) } ^ \vee . \end{align*}


\begin{align*} \mu _ P ( [ B ] ) = p _ { b _ 1 } p _ { b _ 2 } \cdots p _ { b _ k } . \end{align*}


\begin{align*} e _ { \Psi } \left ( \rho \right ) = \rho \left ( \mathfrak { e } _ { \Psi } \right ) \mathfrak { e } _ { \Psi } \doteq \sum \limits _ { \mathcal { Z } \in \mathcal { P } _ { \mathrm { f } } , \ ; \mathcal { Z } \ni 0 } \frac { \Psi _ { \mathcal { Z } } } { \left \vert \mathcal { Z } \right \vert } \in \mathcal { U } \end{align*}


\begin{align*} \mathcal { E } : = \mathcal { E } _ { u } \cup \mathcal { E } _ { \ell } \cup \mathcal { E } _ { d } , \end{align*}


\begin{align*} \phi ^ { \prime } _ { n , \lambda } ( x ) & = \frac { d } { d x } \phi _ { n , \lambda } ( x ) = \sum _ { k = 0 } ^ { n } \binom { n } { k } ( 1 ) _ { n - k , \lambda } \phi _ { k , \lambda } ( x ) - \phi _ { n , \lambda } ( x ) \\ & = \sum _ { k = 0 } ^ { n - 1 } \binom { n } { k } ( 1 ) _ { n - k , \lambda } \phi _ { k , \lambda } ( x ) , ( n \ge 1 ) . \end{align*}


\begin{align*} \label [ I e q ] { e q : L i n f t y d e f } \mathcal { L } _ \infty \Phi ( \nu ) = \frac { \dd } { \dd t } T _ { \infty , t } \Phi ( \nu ) \Big | _ { t = 0 } = \frac { \dd } { \dd t } \Phi \big ( \overline { S } _ t ( \nu ) \big ) \Big | _ { t = 0 } & = { \left \langle \dd \Phi [ \nu ] , \frac { \dd } { \dd t } \overline { S } _ t ( \nu ) \Big | _ { t = 0 } \right \rangle } \\ & = \langle \dd \Phi [ \nu ] , Q ( \nu ) \rangle . \end{align*}


\begin{align*} \delta : R ( G ) & \rightarrow H ^ { \mathrm { e v e n } \geqslant 2 } ( S ) \\ \rho & \mapsto \tau ^ \mathrm { a n } ( M / S , \rho ) - \tau ^ \mathrm { t o p } ( M / S , \rho ) \ ; . \end{align*}


\begin{align*} \overline { G } _ { ( 1 ) } ( u ) = ( 1 + \sqrt { 2 u } ) \exp ( - \sqrt { 2 u } ) \end{align*}


\begin{align*} r h = r h _ 0 + r h _ 1 + r h _ 2 + \cdots \end{align*}


\begin{align*} \exists m _ \ast \geq 0 , \begin{cases} H [ 0 , m _ \ast ] , \\ H ( m _ \ast , + \infty ) . \end{cases} \end{align*}


\begin{align*} \lim \limits _ { T \to \infty } \frac 1 T \ln \langle { x ^ \eta ( T ) } \rangle = \mathop \sum \limits _ n \frac { w ^ { ( n ) } } { { n ! } } \eta ^ n \end{align*}


\begin{align*} \Phi _ { \Lambda + x } = \alpha _ { x } \left ( \Phi _ { \Lambda } \right ) \ , x \in \mathbb { Z } ^ { d } , \ \Lambda \in \mathcal { P } _ { \mathrm { f } } , \end{align*}


\begin{align*} \begin{aligned} T _ 2 ( h _ i { } \leq { } 1 ) { } = { } h _ i + h _ { i - 1 } - { h _ i } - { h _ { i - 1 } } \left [ ( 1 + \kappa ) \tfrac { \alpha _ { i - 1 } } { B } - \kappa \right ] { } = { } h _ { i - 1 } ( 1 + \kappa ) \left ( 1 - \tfrac { \alpha _ { i - 1 } } { B } \right ) \end{aligned} \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } \int _ 0 ^ L \abs { \omega } ^ 2 d x + c ( 1 - m ) \int _ 0 ^ L \abs { \omega _ x } ^ 2 d x + \Re \left ( c m \int _ 0 ^ L \int _ 0 ^ { \infty } \sigma ( s ) \eta _ x ( s ) \overline { \omega } _ x d s d x \right ) - \delta \int _ 0 ^ L u _ t \overline { w } _ x d x = 0 . \end{align*}


\begin{align*} P _ 0 ( D ) = \sum \limits _ { j = 0 } ^ k P _ 0 ( D _ j ) \end{align*}


\begin{align*} ( B ( \Lambda ' _ i - 1 ) ) _ { k j } - ( B ( \Lambda ' _ j - 1 ) ) _ { k i } = \pi L _ { i j k } ( \Lambda ' ) . \end{align*}


\begin{align*} B ^ n = B ^ n ( a , r ) = B ^ { n _ 1 } ( a _ 1 , r _ 1 ) \times \cdots \times B ^ { n _ k } ( a _ k , r _ k ) \end{align*}


\begin{align*} \int _ 1 ^ T c _ p ( t ) d t \leq & \langle B ( t ) : \phi ( t ) \rangle _ t \vert _ { t = T } - \langle B ( t ) : \phi ( t ) \rangle _ t \vert _ { t = 1 } + \| g ( t ) \| _ { L ^ 1 _ t [ 1 , \infty ) } \\ \leq & \left ( \sup \limits _ { t \in \R } \| \psi ( t ) \| _ { \mathcal { H } _ x ^ a } \right ) ^ 2 + \| g ( t ) \| _ { L ^ 1 _ t [ 1 , \infty ) } . \end{align*}


\begin{align*} \kappa _ t ^ { \epsilon , L } \geq \log ( m ^ 2 t + 1 ) - c , c : = \int _ 0 ^ \infty \frac { \tilde p _ { d , t , m ^ 2 } ( \lambda ) } { t ^ 2 } . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \in [ 0 , T ] } E [ | \widetilde { Y } ^ n _ t - Y ^ n _ t | ^ 2 ] = 0 . \end{align*}


\begin{align*} \label [ I e q ] { e q : w e a k p d e m e a n f i e l d } \forall \varphi \in \mathcal { F } , \quad \frac { \dd } { \dd t } \langle f _ t , \varphi \rangle = \langle f _ t , L _ { f _ t } \varphi \rangle . \end{align*}


\begin{align*} \begin{aligned} & \int \limits _ { \mathbb R ^ 2 } \ ( \mathcal E - \mathcal L ( \Phi ) + \mathcal N ( \Phi ) \ ) \partial _ \rho W _ \rho \\ & = \mathfrak v _ \infty \mathfrak A \frac { k } { \rho ^ { \nu + 1 } } ( 1 + o ( 1 ) ) - \mathfrak B \sqrt { \frac k \rho } e ^ { - 2 \rho \frac \pi k } ( 1 + o ( 1 ) ) - \beta \mathfrak C \sqrt { \frac k \rho } e ^ { - 4 \rho \frac { \pi } { d k } } ( 1 + o ( 1 ) ) . \end{aligned} \end{align*}


\begin{align*} \mathbf j ( z _ 1 + z _ 2 \mathbf j ) = & \mathbf j ( x _ 0 + x _ 1 \mathbf i ) + \mathbf j ( x _ 2 + x _ 3 \mathbf i ) \mathbf j \\ = & - ( x _ 2 - x _ 3 \mathbf i ) + ( x _ 0 - x _ 1 \mathbf i ) \mathbf j \\ = & - \overline z _ 2 + \overline z _ 1 \mathbf j \end{align*}


\begin{align*} H _ { 2 \mathrm { - m o d e } } = \ ; & E _ 0 + E ^ w _ N + \mathcal { T } \big ( a ^ \dagger _ 1 a _ 2 + a ^ \dagger _ 2 a _ 1 \big ) - \mu \mathcal { N } _ \perp + \frac { \lambda U } { N - 1 } \left ( \mathcal { N } _ 1 - \mathcal { N } _ 2 \right ) ^ 2 \\ & + \frac { 2 \lambda } { N - 1 } w _ { 1 1 2 2 } \mathcal { N } ^ 2 _ - + \frac { \lambda } { 4 ( N - 1 ) } ( w _ { 1 1 1 1 } - 2 w _ { 1 1 2 2 } + w _ { 1 2 1 2 } ) \mathcal { N } _ \perp ^ 2 . \end{align*}


\begin{align*} \tilde B _ { m } ^ { ( \ell ) } : = B _ { 2 ^ { \gamma \iota + 1 } C R _ \ell } ( x _ { m } ^ { ( \ell ) } ) \times B _ { 3 C R ^ 2 } ( 0 ) , 1 \le m \le M _ { \ell } \end{align*}


\begin{align*} \| \phi _ n ( a b ) - \phi _ n ( a ) \phi _ n ( b ) \| & = \| P _ n a b P _ n - P _ n a P _ n b P _ n \| \\ & \leq \| P _ n a b - P _ n a P _ n b \| \\ & \leq \| P _ n a - P _ n a P _ n \| \| b \| \approx 0 , \end{align*}


\begin{align*} D ( n - 1 ) = \frac { \mathbb { G } ( n - 1 ) } { \mathbb { G } ( n - 1 ) - \mathbb { G } ( n ) } \end{align*}


\begin{align*} \P \{ \ell ^ { ( A ) } _ i > m - i \} \leq \sum \limits _ { k = 0 } ^ \infty \mu ( k ) \left ( 1 \wedge k \P \left \{ \exists t > 0 : t \leq \sum \limits ^ { i + S _ { t } } _ { z = i + 1 } \frac { 1 } { A ( z ) } S _ { t } \geq m - i \right \} \right ) . \end{align*}


\begin{align*} \langle m _ f ( u ) , v \rangle \vcentcolon = \lim _ { i \to \infty } \langle f , u _ i v _ i \rangle \mbox { f o r a l l } ( u , v ) \in H ^ r ( \R ^ n ) \times H ^ { - t } ( \R ^ n ) , \end{align*}


\begin{align*} \xi _ { ( w _ i ) _ i , \phi } ^ { \mathbf { G } } ( y _ { \sigma ( 1 ) } ^ 1 , \dots , y _ { \sigma ( d ) } ^ d ) = \widehat { \xi ^ \mathbf { G } } ( \sqrt { w _ 2 - w _ 1 } , \dots , \sqrt { w _ d - w _ { d - 1 } } ) . \end{align*}


\begin{align*} \phi ( x ) = e ^ { \omega \langle x \rangle } . \end{align*}


\begin{align*} R = R _ 0 \supseteq R _ 1 \supseteq R _ 2 \supseteq \cdots \end{align*}


\begin{align*} r ^ C _ A ( b ) = l ^ B _ A ( r ^ C _ A ( b ) ) r ^ B _ A ( r ^ C _ A ( b ) ) . \end{align*}


\begin{align*} \Theta _ { \gamma } \colon \R ^ { 2 n } \to \R ^ { n \times n } , \Theta _ { \gamma } ( x , y ) \vcentcolon = \gamma ^ { 1 / 2 } ( x ) \gamma ^ { 1 / 2 } ( y ) \mathbf { 1 } _ { n \times n } \end{align*}


\begin{align*} & \lim _ { n \rightarrow \infty } n ^ { 2 \alpha + 1 } \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' ( X _ s ) ) ^ 2 \Psi ^ { n , 5 } _ s d s \\ & = \kappa _ 5 \int _ 0 ^ { \tau \wedge t _ 1 \wedge t _ 2 } ( t _ 1 - s ) ^ { \alpha } ( t _ 2 - s ) ^ { \alpha } ( \sigma ' \sigma ) ^ 2 ( X _ s ) d s , \end{align*}


\begin{align*} \sum _ { 1 \leq i \leq n } \sum _ { i + 1 < j \leq n } \mathbb { E } ( X _ i X _ { i + 1 } X _ j X _ { j + 1 } ) = & V _ k ( n ) + O _ { \alpha } \Big ( \sum _ { 1 \leq i \leq n } \sum _ { i + 1 < j \leq n } \frac { \tau _ 3 ( i ) \tau _ 3 ( i + 1 ) \tau _ 3 ( j ) \tau _ 3 ( j + 1 ) } { \sqrt { i } } \Big ) \\ & + O _ { \alpha } \Big ( \sum _ { 1 \leq i \leq n } \sum _ { i + 1 < j \leq n } \frac { \tau _ 3 ( j ) \tau _ 3 ( j + 1 ) } { \sqrt { j - i - 1 } } \Big ) , \end{align*}


\begin{align*} q _ { l , n } ( \lambda _ 1 , \lambda _ 2 ) : = q _ { l _ 1 , n _ 1 } ( \lambda _ 1 ) \cdot q _ { l _ 2 , n _ 2 } ( \lambda _ 2 ) , \ ; \ ; \ ; \ ; ( \lambda _ 1 , \lambda _ 2 ) \in \mathbb { C } \times \mathbb { C } \cong \mathfrak { a } ^ * _ \mathbb { C } , \end{align*}


\begin{align*} \mathcal { F } | M = \{ F \in \mathcal { F } : F \in \mathcal { A D } ( M ) \} . \end{align*}


\begin{align*} T ^ { i , j } \circ S ^ { i - 1 , j + 1 } \circ T ^ { i , j } = T ^ { i , j } , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ \infty \sum _ { n = 1 } ^ { d _ i } f ( b _ { i } + 2 ^ i n ) = \sum _ { n = 1 } ^ \infty f ( a ( n ) ) = \infty , \end{align*}


\begin{align*} g ( t ^ m \otimes u ) = e ^ { - 2 \pi i m } ( t ^ m \otimes g u ) . \end{align*}


\begin{align*} \gamma _ i = ( \underbrace { 0 , \ldots , 0 } _ { | \mathfrak { A } | } , \underbrace { 2 , \ldots , 2 } _ { | N _ G ( \mathfrak { A } ) | } , { 1 , \ldots , 1 } ) \end{align*}


\begin{align*} y ^ * - w ^ * - \sum _ { j = 1 } ^ m \lambda _ j \widetilde { u } _ j \in U ^ \perp \subseteq \overline { K ^ { ( \alpha ) } } . \end{align*}


\begin{align*} \psi _ n ( y ) : = \psi _ n ( T _ i y ) \quad \textrm { o n } \ \ { \C } _ i , \ i = 2 , 3 , 4 . \end{align*}


\begin{align*} \int _ 0 ^ \infty \ 1 _ { \{ \overline f > \tau \} } ( r ) r ^ p g ( r ) \ , d r & = \int _ 0 ^ { b _ \tau } r ^ p g ( r ) \ , d r \leq b _ \tau \sup _ { \beta > 0 } \beta ^ { - 1 } \int _ 0 ^ \beta r ^ p g ( r ) \ , d r \\ & = \int _ 0 ^ \infty \ 1 _ { \{ \overline f > \tau \} } ( r ) \ , d r \ \underline \nu _ p ( g ) \ , . \end{align*}


\begin{align*} p _ n = \frac { 1 } { 2 } \frac { \mathbb { G } ( n - 1 ) + \mathbb { G } ( n + 1 ) - 2 \mathbb { G } ( n ) } { \mathbb { G } ( n - 1 ) - \mathbb { G } ( n + 1 ) } . \end{align*}


\begin{align*} \frac { 1 } { q } D _ { q ^ { - 1 } } D _ q y ( x ) - y ( x ) = 0 ( x \in \mathbb { R } ) \end{align*}


\begin{align*} u ' ( x ) + p ( x ) u ( x ) + r ( x ) = 0 . \end{align*}


\begin{align*} a _ m z ^ m = a _ { m _ 1 } z ^ { m _ 1 } \cdots a _ { m _ n } z ^ { m _ n } \end{align*}


\begin{align*} H _ { ( Y _ + , D _ { Y , + } ) } ( y ) = e ^ { \frac { t _ + } { 2 \pi i } } \sum _ { d \in \mathbb K _ { + } } y ^ d \left ( \frac { \prod _ { i = 1 } ^ k \Gamma ( 1 + \frac { v _ i } { z } + v _ i \cdot d ) } { \prod _ { i \in M _ 0 } \Gamma ( 1 + \frac { \bar D _ i } { z } + D _ i \cdot d ) } \right ) \textbf { 1 } _ { [ d ] } [ \textbf { 1 } ] _ { ( D _ i \cdot d ) _ { i \in I _ + } } . \end{align*}


\begin{align*} \mathcal { F } _ B = \{ F \in \mathcal { F } : B \sqsubset F \} . \end{align*}


\begin{align*} \left . \left ( \prod _ { x \in X / S } \left ( 1 + a _ x t + b _ x t + a _ x b _ x t \right ) \right ) \right \rvert _ { t = 1 } . \end{align*}


\begin{align*} \frac { \partial h } { \partial x ^ i } - \frac { d } { d t } \left ( \frac { \partial h } { \partial \dot { x } ^ i } \right ) = 0 , ~ i = 1 , 2 . \end{align*}


\begin{align*} u ( t ) = u ^ 0 + \int _ 0 ^ t k _ 1 ( t - \tau ) ( f ( \tau ) - A u ^ 0 ) d \tau , \end{align*}


\begin{align*} y _ i ( v ) = \begin{cases} 1 & v = u _ 0 , \\ - 1 & v = u _ i , \\ 0 & \end{cases} \end{align*}


\begin{align*} E ( G / T ) = E ( G / H ) E ( H / T ) = E ( N ) E ( H / T ) = \pm E ( H / T ) \neq 0 , \end{align*}


\begin{align*} \mbox { d e x p } ^ { - 1 } _ { \Omega } ( H ) = \sum _ { j = 0 } ^ { \infty } \frac { B _ j } { j ! } \operatorname { a d } _ { \Omega } ^ j ( H ) , \ , \end{align*}


\begin{align*} \overline { I ^ n } = ( \{ t ^ a \mid a / n \in { \rm N P } ( I ) \} ) = ( \{ t ^ a \mid \langle a , u _ i \rangle \geq n \mbox { f o r } i = 1 , \ldots , p \} ) \end{align*}


\begin{align*} \frac { 1 } { 2 } \{ g ( \nabla _ U \xi , V ) + g ( \nabla _ V \xi , U ) \} + R i c ( U , V ) + \mu g ( U , V ) = 0 . \end{align*}


\begin{align*} \binom { z } { k } : = \frac { z ( z - 1 ) \cdots ( z - k + 1 ) } { k ! } , \binom { - z } { k } = ( - 1 ) ^ k \binom { z + k - 1 } { k } . \end{align*}


\begin{align*} \int _ { \mathcal { M } _ 3 } \mathcal { I } = \int _ { \mathcal { M } _ 1 } \mathcal { I } + \int _ { \mathcal { M } _ 2 } \mathcal { I } . \end{align*}


\begin{align*} \mathcal { G } _ j = \dfrac { W _ { l ( T ) + 1 - j } } { W _ { l ( T ) - j } } \end{align*}


\begin{align*} = 2 \sum ( ) + \sum ( ) + \sum ( ) + n . \end{align*}


\begin{align*} \bigcup _ { k = 1 } ^ { \infty } \alpha ^ 1 _ k ( A _ { 1 , k } ) = \bigcup _ { m = 1 } ^ { \infty } \alpha ^ m _ 1 ( A _ { m , 1 } ) \end{align*}


\begin{align*} B _ r ( \nu ) : = \{ \mu \in \mathcal { M } _ 1 ( \mathbb { R } ^ k ) \colon \rho _ \mathrm { L P } ( \mu , \nu ) < r \} \end{align*}


\begin{align*} z \cdot W ( \sigma ) = R _ { \varphi ( \sigma ) } ( u ) \wedge v + u \wedge R _ { \varphi ( \sigma ) } ( v ) . \end{align*}


\begin{align*} N _ c ^ R : & = \big \{ y \in N : \exists ( y _ i ) _ { i = 1 , . . . , n } \in N , y _ 1 = y , y _ n = x , ( y _ i , y _ { i + 1 } ) \in E \backslash c \big \} , \\ E _ c ^ R : & = \big \{ ( y , z ) \in E : y , z \in N _ c ^ R \big \} , H _ c ^ R : = \big \{ h \cap N _ c ^ R : h \in H \big \} \backslash \{ \emptyset \} , \\ L _ c ^ R : & N _ c ^ R \to \{ 1 , . . . , d \} , L _ c ^ R = L | _ { N _ c ^ R } \end{align*}


\begin{align*} k ! ( x _ i - x _ { i + 1 } ) e _ k ( x _ 1 , \ldots , x _ i ) = 0 . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } u _ t = \Delta u - b | \nabla u | ^ { q } + | u | ^ { p - 1 } u , \ & x \in \Omega , t > 0 \\ u = 0 , \ & x \in \partial \Omega , t > 0 \\ u ( x , 0 ) = \phi ( x ) \geq 0 , \ & x \in \Omega , \end{array} \right . \end{align*}


\begin{align*} & \int _ 0 ^ r \left [ ( \phi '' + \frac { z } { 2 } \phi ' - \frac { z } { 2 } e _ \sigma \psi ' ) \cdot \phi '' + ( \psi '' + \frac { z } { 2 } \psi ' - \frac { z } { 2 } e _ \sigma \phi ' ) \cdot \psi '' \right ] { \rm { d } } z \\ & = \int _ 0 ^ r | ( \phi '' , \psi '' ) | ^ 2 + \frac { 1 } { 2 } \int _ 0 ^ r z ( \phi ' \phi '' + \psi ' \psi '' ) - \frac { 1 } { 2 } \int _ 0 ^ r z e _ \sigma ( \psi ' \phi '' + \phi ' \psi '' ) . \end{align*}


\begin{align*} D L _ { ( g , P ( g ) ) } ( 0 , h ) = 2 \Delta ^ { P ( g ) } _ { g } \left ( 2 \Delta ^ { P ( g ) } _ { g } h + h \right ) - ( 4 \pi ) ^ { - \frac { n } { 2 } } \int _ { M } h e ^ { - P ( g ) } \ , d v _ { g } , \quad g \in \mathcal { U } . \end{align*}


\begin{align*} P _ 1 \theta & = \Delta \theta - q { \sf U } _ \infty ^ { q - 1 } \theta \\ & = - f ( { \sf U } _ \infty ) - \sum _ { i = 1 } ^ N \tfrac { f ^ { ( i ) } ( { \sf U } _ \infty ) } { i ! } ( \theta - \theta _ L ) ^ i + \sum _ { i = 2 } ^ N \tfrac { f _ 2 ^ { ( i ) } ( { \sf U } _ \infty ) } { i ! } ( \theta - \theta _ L ) ^ i . \end{align*}


\begin{align*} I _ { \mathbb P ^ 2 } ( y , z ) = z e ^ { H \log y / z } \sum _ { d \geq 0 } y ^ { d } \left ( \frac { 1 } { \prod _ { a = 1 } ^ d ( H + a z ) ^ 3 } \right ) , \end{align*}


\begin{align*} J _ 1 & = \int _ 0 ^ r \left [ \frac { z } { 2 } ( \bar \phi + e _ \delta - \delta ) ( \bar \phi ' , \bar \psi ' ) \cdot z ^ { 2 } ( \phi , \psi ) \right ] { \rm { d } } z \\ & \leq \Vert z ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert z ( \bar \phi ' , \bar \psi ' ) \Vert _ { L ^ 2 } \cdot \frac { 1 } { 2 } \left [ \Vert z \bar \phi \Vert _ { L ^ \infty } + \Vert z ( e _ \delta - \delta ) \Vert _ { L ^ \infty } \right ] , \end{align*}


\begin{align*} X _ { \{ i \} } \cap \Omega _ { J } = \emptyset \end{align*}


\begin{align*} F ^ { ( \iota ) } ( \sqrt L ) f _ j = \frac { 1 } { 2 \pi } \int _ { 2 ^ { \iota - 1 } \le | \tau | \le 2 ^ { \iota + 1 } } \chi _ \iota ( \tau ) \hat F ( \tau ) \cos ( \tau \sqrt L ) f _ j \ , d \tau . \end{align*}


\begin{align*} \mathbf S | _ { \mathcal H ^ 0 _ { 1 2 3 } } = \mathbf S ^ 0 _ { 1 2 3 } \ , \end{align*}


\begin{align*} \| U ^ n \| _ { \mathcal { H } } = \| \left ( u ^ n , v ^ n , y ^ n , z ^ n , w ^ n , \eta ^ n \right ) ^ { \top } \| _ { \mathcal { H } } = 1 , \end{align*}


\begin{align*} s _ { n } = \sum _ { \gamma \in \mathcal { M } _ { n } } w ( \gamma ) . \end{align*}


\begin{align*} P ( u ) \cdot v = \begin{bmatrix} \nabla u \\ \nabla \nabla u \end{bmatrix} v = \begin{bmatrix} f \\ h \end{bmatrix} , v ( x ) \perp \mathrm { K e r } P ( u ) ( x ) . \end{align*}


\begin{align*} a = a _ 0 \cos \theta - b _ 0 \rho ^ { - 1 } \sin \theta , \ ; \ ; b = a _ 0 \sin \theta + b _ 0 \rho ^ { - 1 } \cos \theta \end{align*}


\begin{align*} s = - 1 - l , l \geq 0 s = - \frac { H } { z } - \frac { m } { 3 } , m \geq 1 . \end{align*}


\begin{align*} d V _ { B H } = \frac { \int \limits _ { 0 } ^ { \pi } d t } { \int \limits _ { 0 } ^ { \pi } ( b ' \cos t ) ^ 2 d t } \sqrt { \det A } d x = \frac { 2 } { b '^ 2 } \sqrt { \det A } d x . \end{align*}


\begin{align*} f ( y , v _ 1 , \ldots , v _ k ) - f ( x , v _ 1 , \ldots , v _ k ) = \int _ 0 ^ 1 d f ( x + t ( y - x ) , v _ 1 , \ldots , v _ k , y - x , 0 , \ldots , 0 ) \ , d t \end{align*}


\begin{align*} \mathcal { G } _ 1 ( r ( T ) , x ) & = x \displaystyle \prod _ { j = 1 } ^ { k } \mathcal { G } _ 1 ( r ( T _ j ) , x ) - \displaystyle \sum _ { j = 1 } ^ { k } \left ( \alpha _ j \ \mathcal { G } _ 2 ( r ( T _ j ) , x ) \displaystyle \prod _ { i = 1 , i \neq j } ^ { k } \mathcal { G } _ 1 ( r ( T _ i ) , x ) \right ) . \end{align*}


\begin{align*} Q = \left \{ \begin{array} { l l l } x ^ { 3 / 2 } , & i f \ k = 2 , \\ x ^ { 1 2 / 7 } , & i f \ k = 3 . \end{array} \right . \end{align*}


\begin{align*} R ^ { n , 2 } _ \tau = n ^ { 2 \alpha + 1 } \int _ 0 ^ \tau \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ s ) \left [ \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) ^ 2 - \int _ 0 ^ s \psi ^ 2 _ { n , 1 } ( u , s ) d u \right ] d s , \end{align*}


\begin{align*} \abs { w } _ { X \cup \mathcal { P } } = \abs { v } \geq \frac { \abs { \iota ( w ) } } { N } - ( N - 2 \mu ) \end{align*}


\begin{align*} j ( a * b ) ( \gamma ) = \sum _ { \eta \in G ^ { r ( \gamma ) } } j ( a ) ( \eta ) \ , j ( b ) ( \eta ^ { - 1 } \gamma ) \end{align*}


\begin{align*} V _ r ( a , b ) = \sup _ { f \in \mathcal { M } ^ + ( a , b ) } \frac { \bigg ( \int _ { a } ^ { b } f ^ r v \bigg ) ^ { \frac { 1 } { r } } } { \int _ { a } ^ { b } f } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k \int _ { B _ R ( \bar x _ n ^ i ) } ( v _ n ^ i ) ^ 2 \ , d x \to \int _ { B _ R ( \bar x ^ i ) } ( V ^ i ) ^ 2 \ , d x , \end{align*}


\begin{align*} f ( x ) = e ^ { - c x } ( 1 - x ) ^ c \ ; , \ , \end{align*}


\begin{align*} d _ k = g c _ i ^ { ( k ) } = h d _ { k - 1 } , \end{align*}


\begin{align*} A = \bigcup _ { j \ge 1 } \bigg ( S _ j \setminus \bigcup _ { 1 \le i < j } { \rm M } _ { I _ i } \bigg ) . \end{align*}


\begin{align*} & \dot { x } ( t , s ) = \lambda x ( t , s ) + \partial _ s ^ 2 x ( t , s ) , s \in [ 0 , 1 ] , \\ & \dot { \hat x } ( t , s ) = \lambda \hat x ( t , s ) + \partial _ s ^ 2 \hat x ( t , s ) + l ( s ) \left ( \partial _ s x ( t , 1 ) - \partial _ s \hat x ( t , 1 ) \right ) , \\ & \partial _ s x ( t , 0 ) = 0 , x ( t , 1 ) = 0 , \\ & \partial _ s \hat x ( t , 0 ) = 0 , \hat x ( t , 1 ) = 0 . \end{align*}


\begin{align*} P ^ n _ \tau & : = n ^ { \alpha + \frac 1 2 } \int ^ { \tau } _ 0 ( t - s ) ^ { \alpha } \sigma ' ( X _ s ) \ , \left ( \int ^ s _ { \eta _ n ( s ) } ( s - \eta _ n ( u ) ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) d s \\ & = n ^ { \alpha + \frac 1 2 } \int ^ { \tau } _ 0 ( t - s ) ^ { \alpha } \sigma ' ( X _ { s } ) \sigma ( X _ { \eta _ n ( s ) } ) \Xi ^ { n , 2 } _ s d s , \end{align*}


\begin{align*} r _ k = \max \left \{ r \left | \frac { j } { n } \right | ^ { \frac { 1 } { n - j } } , \ , j = 1 , 2 , \dots , k \right \} \end{align*}


\begin{align*} \begin{aligned} r _ { i j } ^ n w _ n d x ^ i \otimes d x ^ j : = & \{ 2 R _ i { } ^ k { } _ j { } ^ n \nabla _ k w _ n + R _ { i } { } ^ k { } _ j { } ^ m w _ n ( - \Gamma _ { k m } ^ n ) + \nabla ^ k ( { R _ { i k j } } ^ n ) w _ n + g ^ { k l } { R _ { i m j } } ^ n w _ n ( - \Gamma _ { l k } ^ m ) \\ & + R _ i { } ^ m w _ n ( - \Gamma _ { m j } ^ n ) \} d x ^ i \otimes d x ^ j . \end{aligned} \end{align*}


\begin{align*} B _ T = \begin{pmatrix} A _ 1 & \ 0 & \cdots & \ 0 \\ \ 0 & A _ 2 & \cdots & \ 0 \\ \vdots & \vdots & \ddots & \vdots \\ \ 0 & \ 0 & \cdots & A _ { \ell } \\ \end{pmatrix} \end{align*}


\begin{align*} \nu _ p ( l ! ) = \frac { l - \sigma _ p ( l ) } { p - 1 } \end{align*}


\begin{align*} 2 \sum _ { 1 \leq i < j \leq n } f _ k ( i ) f _ k ( j ) = \big ( \sum _ { 1 \leq i \leq n } f _ k ( i ) \big ) ^ 2 - \sum _ { 1 \leq i \leq n } f _ k ^ 2 ( i ) = \big ( \sum _ { 1 \leq i \leq n } f _ k ( i ) \big ) ^ 2 + O _ { \varepsilon } ( n ^ { 1 + \varepsilon } ) , \end{align*}


\begin{align*} E ( G / A ) = E ( G / H ) E ( H / A ) \neq 0 , \end{align*}


\begin{align*} W = x ^ r + y ^ s . \end{align*}


\begin{align*} t N _ n ( t , 1 ) = N _ n ( t , t ) . \end{align*}


\begin{align*} \textrm { r a n k } & \left ( \mathbf { V } \right ) = \lim _ { \varepsilon \rightarrow 0 } \frac { M \log \left ( \frac { 1 } { \varepsilon } \right ) + \log \left | \mathbf { V } + \varepsilon \mathbf { I } \right | } { \log \left ( 1 + \frac { 1 } { \varepsilon } \right ) } . \end{align*}


\begin{align*} \abs * { \sum _ { t = 1 } ^ { T } ( f _ t ( M ) - F ( M ) ) } \le C \sqrt { 2 T H d \log \frac { 3 T ^ 2 } { \delta } } \end{align*}


\begin{align*} & Q : C ^ { k , \alpha } ( M \times [ 0 , T ] ) \rightarrow C ^ { k + 2 , \alpha } ( M \times [ 0 , T ] ) , \\ & Q : C ^ { k + 2 , \alpha } ( M \times [ 0 , T ] ) \rightarrow s \ C ^ { k + 2 , \alpha } ( M \times [ 0 , T ] ) . \end{align*}


\begin{align*} K ^ { n , 4 } _ \tau & : = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ { ( \delta + \frac 1 n ) \wedge \tau } \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) \\ & \times \left [ \left ( \int _ { \eta _ n ( s ) - \delta } ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) d W _ u \right ) ^ 2 - \left ( \int _ 0 ^ { \eta _ n ( s ) } \psi _ { n , 2 } ( u , s ) d W _ u \right ) ^ 2 \right ] d s \end{align*}


\begin{align*} \int _ B e ^ { - i \varphi } q ( x ) u _ 1 u _ 2 d x = & \int e ^ { - i \varphi } ( A + \nabla \vartheta ) \cdot ( A _ 1 + A _ 2 ) u _ 1 u _ 2 d x \cr & + \int _ B i e ^ { - i \varphi } ( A + \nabla \vartheta ) \cdot ( u _ 1 \nabla u _ 2 - u _ 2 \nabla u _ 1 ) d x \cr & + \int _ B e ^ { - i \varphi } ( A + \nabla \vartheta ) \cdot \nabla \varphi u _ 1 u _ 2 d x + \mathcal { R } \cr : = & \mathcal { J } _ 1 + \mathcal { J } _ 2 + \mathcal { J } _ 3 + \mathcal { R } , \end{align*}


\begin{align*} u ( x , t ) = \frac 1 4 \int _ 0 ^ \infty \exp ( - \frac { k ^ 2 } { 8 } ) J _ 0 ( k | x | ) k \cos ( k t ) d k , x \in \Omega ^ - , \end{align*}


\begin{align*} \overline { S } _ { t + s } ( \nu ) = \int _ E P ^ { \overline { S } _ t ( \nu ) } _ { s } ( y , \cdot ) { \left ( \int _ E P ^ { \nu } _ { t } ( x , \dd y ) \nu ( \dd x ) \right ) } = \overline { S } _ { s } ( \overline { S } _ { t } ( \nu ) ) . \end{align*}


\begin{align*} \sum _ { \ell = - 1 } ^ \infty F _ \ell ( L , U ) f = F ( \sqrt L ) f . \end{align*}


\begin{align*} P _ c ( \Psi _ t ) v ' = P _ c ( \Psi _ t ) P ^ T ( \Psi _ t ) [ P ( \Psi _ t ) P ^ T ( \Psi _ t ) ] ^ { - 1 } \begin{bmatrix} 0 \\ h \end{bmatrix} = ( I _ { \frac { n ( n + 3 ) } { 2 } } - \frac { 1 } { n } \begin{bmatrix} 0 & 0 \\ 0 & J _ n \end{bmatrix} ) \begin{bmatrix} 0 \\ h \end{bmatrix} = \begin{bmatrix} 0 \\ h \end{bmatrix} , \end{align*}


\begin{align*} \lim _ { z _ 2 \to z _ 1 } \frac { K _ { \mathcal A _ 0 } ( \boldsymbol z , \boldsymbol z ) } { | z _ 1 - z _ 2 | ^ 2 } = \frac { 1 } { 2 } \frac { 1 } { ( 1 - | z _ 1 | ^ 2 ) ^ 4 } , ( z _ 1 , z _ 2 ) \in \mathbb D ^ 2 . \end{align*}


\begin{align*} v ^ 2 | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 = f ( u ) ^ 2 | \nabla u | ^ 2 _ g . \end{align*}


\begin{align*} \mathcal { K } _ { \mathcal { S } } \coloneqq \left \{ ( y ^ { n } _ { \mathcal { T } } , \tilde { y } ^ { n } _ { \mathcal { T } } ) \in \mathcal { Y } ^ { n } _ { \mathcal { T } } \times \mathcal { Y } ^ { n } _ { \mathcal { T } } \ : y ^ { n } _ { \mathcal { S } } = \tilde { y } ^ { n } _ { \mathcal { S } } \ \wedge \ y ^ { n } _ { i } \neq \tilde { y } ^ { n } _ { i } , \ \forall i \in \mathcal { S } ^ { c } \right \} . \end{align*}


\begin{align*} N ^ { n , 1 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) d W _ u \right ) ^ 2 \ , d s \end{align*}


\begin{align*} \frac { 1 } { x } \sum _ { \substack { n \le x \\ n \in { \rm L } _ { A \cap ( y , \infty ) } } } 1 \le \frac { 1 } { x } \sum _ { a \in A , a > y } \left \lfloor \frac { x } { a } \right \rfloor \le \sum _ { a \in A , a > y } \frac { 1 } { a } = o _ y ( 1 ) . \end{align*}


\begin{align*} T _ { \lambda ^ 2 } : = 6 \lambda ^ 2 \Big [ C \star \big ( S ^ 3 - C ^ 3 \big ) \star S + C \star \big ( { \bf 1 } ^ \epsilon _ 0 \| C ^ 3 \| _ { L ^ 1 } - { \bf 1 } ^ \epsilon _ 0 \| C _ \infty ^ 3 \| _ { L ^ 1 } \big ) \star S + C \star \psi \star S \Big ] , \end{align*}


\begin{align*} \beta ^ * ( g , \lambda ) : = ( \beta ( g ^ { - 1 } , \cdot ) | _ { E _ { \alpha ( g , x ) } } ^ { E _ x } ) ' ( \lambda ) \end{align*}


\begin{align*} P _ j ( \hat \phi _ j ) = 0 , 1 \leq j \leq n . \end{align*}


\begin{align*} \langle f , u _ { p + \frac { [ j _ 1 + j _ 2 ] } { T } } I ^ { \circ } ( w _ 1 , z ) w _ 2 \rangle = 0 \end{align*}


\begin{align*} \partial \bar { \partial } \log K ( z , z ) \geq \sup \Big \{ \tfrac { | f ^ \prime ( z ) | ^ 2 } { ( 1 - | f ( z ) | ^ 2 ) ^ 2 } : f \in { \rm R a t } ( \Omega ) , \| f \| _ { \Omega , \infty } \leq 1 \Big \} = \mathbb S _ \Omega ( z , z ) ^ 2 , \end{align*}


\begin{align*} \langle \alpha _ i ^ \vee , v \lambda \rangle = 1 + \langle \alpha _ i ^ \vee , \alpha _ { j _ { p + 1 } } \rangle + \dots + \langle \alpha _ i ^ \vee , \alpha _ { j _ r } \rangle \le 1 \end{align*}


\begin{align*} \begin{pmatrix} \lambda \\ \mu \end{pmatrix} \in \mathbb { C } ^ 2 \end{align*}


\begin{align*} ( - \Delta ) ^ { \sigma } _ { p } u ( x ) - \lim _ { i \rightarrow \infty } ( - \Delta ) ^ { \sigma } _ { p } u _ { i } ( x ) = \lim _ { R \rightarrow \infty } \lim _ { i \rightarrow \infty } \Psi _ { i } ( x , R ) . \end{align*}


\begin{align*} \omega ( a * b * c ) & = \omega ( b * c * \sigma _ i ^ R ( \nabla ^ { - 1 } a ) ) = \omega ( c * \sigma _ i ^ R ( \nabla ^ { - 1 } a ) * \sigma _ i ^ R ( \nabla ^ { - 1 } b ) ) \\ & = \omega ( \nabla \sigma _ { - i } ^ R ( \sigma _ i ^ R ( \nabla ^ { - 1 } a ) * \sigma _ i ^ R ( \nabla ^ { - 1 } b ) ) * c ) . \end{align*}


\begin{align*} W _ { c T } \stackrel { \mathrm { d } } { = } \widehat { W } _ T . \end{align*}


\begin{align*} R ( e _ i , e _ j ) e _ k = { R _ { i j k } } ^ { l } e _ l , \end{align*}


\begin{align*} \frac { n - 1 } { ( n + 1 ) \sigma ^ 2 - \| A \| _ F ^ 2 } \not = 0 , \end{align*}


\begin{align*} a \ell _ 1 + ( N / 8 - 2 - a ) \ell _ 2 = N / 1 6 \cdot ( k - 3 ) . \end{align*}


\begin{align*} \hat { \tau } = \hat { \tau } ^ * K _ 0 ( \hat { \tau } ^ * ) + ( \bar { \tau } - \hat { \tau } ^ * ) K _ { \bar { \tau } } ( \hat { \tau } ^ * ) , \end{align*}


\begin{align*} R _ { \ell , i , s } ( z ) = \sum _ { k = n } ^ { \infty } \dfrac { \psi _ { { i , s } } ( t ^ k P _ { \ell } ( t ) ) } { z ^ { k + 1 } } \ \ 0 \le \ell \le r m , \ 1 \le i \le m \ \ 0 \le s \le r - 1 \enspace , \end{align*}


\begin{align*} \mathsf { L } ( b ) = \mathsf { L } _ M ( b ) = \{ | z | \mid z \in \mathsf { Z } ( b ) \} . \end{align*}


\begin{align*} \forall t \leq 1 / ( 2 | \nu | + 1 ) , \dot \kappa _ t \geq \frac { 1 } { t } \Big ( 1 - \frac { 1 } { 1 + t \nu } \Big ) = \frac { \nu } { 1 + t \nu } . \end{align*}


\begin{align*} W ^ { s , p } ( E ) : = \left \lbrace u \in L ^ p ( E ) : [ u ] _ { W ^ { s , p } ( E ) } < \infty \right \rbrace \end{align*}


\begin{align*} \left \{ \begin{array} { c l } i \partial _ t u + \Delta u & = \pm | u | ^ 4 u , ( t , x ) \in \R \times \R , \\ u ( 0 ) & = u _ 0 \in M ^ s _ { 6 , 2 } ( \R ) + L ^ 2 ( \R ) \end{array} \right . \end{align*}


\begin{align*} F ( z ) : = ( ( W - z I ) ^ { - 1 } p , p ) = \int _ \R \frac { d \rho ( s ) } { s - z } \forall z \in \C \setminus \R . \end{align*}


\begin{align*} \lim _ { \gamma _ { - } \rightarrow 0 ^ { + } } \inf h _ { \gamma _ { - } } \left ( E _ { 1 } \right ) = \inf f _ { ( \Phi , \mathfrak { a } _ { \mathfrak { b } _ { + } } - \mathfrak { a } _ { \mathfrak { b } _ { - } } ) } ^ { \flat } \left ( E _ { 1 } \right ) = - \mathrm { P } _ { ( \Phi , \mathfrak { a } _ { \mathfrak { b } _ { + } } - \mathfrak { a } _ { \mathfrak { b } _ { - } } ) } ^ { \flat } \ . \end{align*}


\begin{align*} y ( z x ) = - ( - 1 ) ^ { \vert y \vert \vert z \vert } z ( y x ) . \end{align*}


\begin{align*} \left \langle [ \gamma _ 1 ] _ { \vec s ^ 1 } \bar { \psi } ^ { a _ 1 } , \ldots , [ \gamma _ m ] _ { \vec s ^ m } \bar { \psi } ^ { a _ m } \right \rangle _ { g , \{ \vec s ^ j \} _ { j = 1 } ^ m , \beta } ^ { X _ { D , \infty } } : = \left [ \left ( \prod _ { i = 1 } ^ n r _ i ^ { s _ { i , - } } \right ) \left \langle \gamma _ 1 \bar { \psi } ^ { a _ 1 } , \ldots , \gamma _ m \bar { \psi } ^ { a _ m } \right \rangle _ { g , \{ \vec s ^ j \} _ { j = 1 } ^ m , \beta } ^ { X _ { D , \vec r } } \right ] _ { \prod _ { i = 1 } ^ n r _ i ^ 0 } \end{align*}


\begin{align*} \mathcal { D } _ { [ n , 0 ] } = \bigcup _ { j = 0 } ^ { p } \mathcal { L } _ { [ n , j ] } \end{align*}


\begin{align*} ( X , \dot { \delta _ X } D f \wedge \delta _ X D g ) & = - ( X , \delta _ X D g \wedge \dot { \delta _ X } D f ) = - ( \rho _ X \delta _ X D g , \dot \delta _ X f ) \\ & = - ( \dot { d _ X } \rho _ X \delta _ X D g , D f ) = - ( [ \dot X , \rho _ X \delta _ X D g ] , D f ) \\ & = ( \dot X , - O _ { X , D g } D f ) . \end{align*}


\begin{align*} \phi \frac { \partial } { \partial x _ i } \phi ^ { - 1 } ( x _ j ) = \delta _ { i j } \end{align*}


\begin{align*} & \dot { \hat x } ( t , s ) = \lambda \hat x ( t , s ) + \partial _ s ^ 2 \hat x ( t , s ) \\ & + \int _ 0 ^ 1 l ( s ) w ( \theta ) \left ( \partial _ s x ( t , \theta ) - \partial _ s \hat x ( t , \theta ) \right ) d \theta . \end{align*}


\begin{align*} \left [ \left ( \prod _ i x _ i ^ { n _ i } \right ) \partial _ { x _ j } , \left ( \prod _ i x _ i ^ { n _ i ' } \right ) \partial _ { x _ k } \right ] = \left ( n _ j ' \prod _ { i = 1 } ^ n x _ i ^ { n _ i + n _ i ' - \delta _ { i j } } \right ) \partial _ { x _ k } - \left ( n _ k \prod _ { i = 1 } ^ n x _ i ^ { n _ i + n _ i ' - \delta _ { i k } } \right ) \partial _ { x _ j } . \end{align*}


\begin{align*} E = \{ e _ 0 \} \cup \{ e _ 1 , \cdots , e _ r \} \cup \{ e _ { r + 1 } , \overline e _ { r + 1 } , \ , \cdots , \ , e _ { r + s } , \overline e _ { r + s } \} , \end{align*}


\begin{align*} \theta _ { t + 1 } & = \theta _ t - \beta _ t \left ( \phi ( s _ t , a _ t ) ^ T \omega _ t \right ) \frac { d } { d \theta } \log \pi _ { \theta } ( a _ t | s _ t ) \\ \omega _ { t + 1 } & = P _ { \Omega } \left [ \omega _ t + \alpha _ t \left ( c _ t ' + \gamma \phi ( s _ t '' , a _ t '' ) ^ T \omega _ t - \phi ( s _ t ' , a _ t ' ) ^ T \omega _ t \right ) \phi ( s _ t ' , a _ t ' ) \right ] \end{align*}


\begin{align*} j ( g , z ) = A + B z . \end{align*}


\begin{align*} | \nabla H _ \lambda | = O ( | \nabla \Gamma | ) \hbox { i n } \Omega \end{align*}


\begin{align*} \mathcal { L } ( V , g ) = \oplus _ { r = 0 } ^ { r = T - 1 } t ^ { \frac { r } { T } } \mathbb { C } [ t , t ^ { - 1 } ] \otimes V ^ r . \end{align*}


\begin{align*} f _ { \sigma } ^ { ( j ) } ( x ) = \sum _ { k = j } ^ n ( k ) _ j \sigma ( c _ k ) x ^ { k - j } . \end{align*}


\begin{align*} \sum _ { j = j _ { \delta _ 1 } + 1 } ^ { j _ { \delta _ 2 } } \frac { 1 } { t _ j } \le \frac { c } { \Delta _ n } \log ( \frac { j _ { \delta _ 2 } } { j _ { \delta _ 1 } } ) = \frac { c } { \Delta _ n } \log ( \frac { \Delta _ n j _ { \delta _ 2 } } { \Delta _ n j _ { \delta _ 1 } } ) = \frac { c } { \Delta _ n } \log ( \frac { \delta _ 2 } { \delta _ 1 } ) . \end{align*}


\begin{align*} R _ { \kappa } ( x ) = N _ { \kappa } \sum _ { m = 0 } ^ n c _ m x ^ m \ \ , \end{align*}


\begin{align*} G ( x , y ) = \int _ { 0 } ^ { \infty } h _ t ( x , y ) \mathrm { d } t . \end{align*}


\begin{align*} | f ^ \dagger f | = | f | ^ 2 \end{align*}


\begin{align*} f ( V _ F ) p ( t , V _ F , g ) - f ( 0 ) p ( t , 0 , g ) = 0 , g \in \mathbb { R } , t > 0 , N ( t ) : = \int _ { 0 } ^ { \infty } g f ( V _ F ) p ( t , V _ F , g ) d g . \end{align*}


\begin{align*} K = \{ r e ^ { i \theta } : | r - r _ { 0 } | \le \varepsilon , | \theta - h _ { \nu } ( r _ { 0 } ) | \le \varepsilon \} \end{align*}


\begin{align*} = \left ( \frac { \alpha p - \beta } { k - 1 } \right ) ^ 2 - F \frac { ( \alpha p - \beta ) } { k - 1 } + \frac { 1 } { ( k - 1 ) ^ 2 ( t - \tau ) ^ 2 } . \end{align*}


\begin{align*} \lim _ { T \rightarrow \infty } \left ( \sum _ { t = 1 } ^ { T } \frac { \gamma ^ { t } } { t } \right ) & \geq \int _ { 1 } ^ { \infty } \frac { \gamma ^ { t } } { t } d t \\ & = \mathrm { E i } \left ( \infty \ln \gamma \right ) - \mathrm { E i } \left ( \ln \gamma \right ) = - \mathrm { E i } \left ( \ln \gamma \right ) \\ & = \mathrm { E _ { 1 } } \left ( \ln \frac { 1 } { \gamma } \right ) > 0 , \end{align*}


\begin{align*} \sum _ { Q ' \in f ^ { - 1 } ( Q ) } \mathrm { C o n t } ( Q ' ) | _ { b _ l = 0 } = ( l - 2 ) \mathrm { C o n t } ( Q ) . \end{align*}


\begin{align*} \frac { | \pi ( w ) | ^ 2 } { z - w } = \overline { \pi ( w ) } \frac { \pi ( w ) - \pi ( z ) } { z - w } + \frac { \pi ( z ) \overline { \pi ( w ) } } { z - w } . \end{align*}


\begin{align*} [ x \ast [ y , [ z , u ] ] , v ] = - [ x \ast [ y , [ u , z ] ] , v ] \overset { \eqref { c h a n - 3 5 } } { \equiv } [ x \ast [ y , [ v , z ] ] , u ] \equiv - [ x \ast [ y , [ z , v ] ] , u ] . \end{align*}


\begin{align*} \prod _ { j = 1 } ^ { | I | + 1 } \mu ( [ B _ j ^ { \bar r _ j } ] ) . \end{align*}


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


\begin{align*} G : \begin{cases} 0 \to 0 \\ 1 \to 0 1 \end{cases} \widetilde { G } : \begin{cases} 0 \to 0 \\ 1 \to 1 0 \end{cases} D : \begin{cases} 0 \to 1 0 \\ 1 \to 1 \end{cases} \widetilde { D } : \begin{cases} 0 \to 0 1 \\ 1 \to 1 \end{cases} . \end{align*}


\begin{align*} - \Delta _ p ( \mathit \Gamma + \mathcal { H } ) + \Delta _ p \mathit \Gamma = \mathcal G B _ 2 ( 0 ) \setminus \{ 0 \} \end{align*}


\begin{align*} \Delta \circ { { \tilde { \psi } } } _ { \alpha , s } \bigcirc _ { w = 1 } ^ { k _ s } ( \theta _ { X _ s } + \xi _ { s , w } ) ( V ( { \hat { P } } ) ) = \Delta \circ { { \tilde { \psi } } } _ { \alpha , s } \bigcirc _ { w = 1 } ^ { s + 1 } ( \theta _ { X _ s } + \gamma _ { r - s + w - 1 } ) \circ \theta _ { X _ s } ^ { k _ s - s - 1 } ( V ( { \hat { P } } ) ) \enspace , \end{align*}


\begin{align*} W _ { \beta } ( f , \varphi ) = \bigcap _ { N = 1 } ^ { \infty } \bigcup _ { n = N } ^ { \infty } W _ n . \end{align*}


\begin{align*} \nu _ p ( \{ ( a , r ) \in \omega \times \R _ + : \ c _ d ' r | \nabla f ( a ) | > \kappa \} ) & = \int _ { \omega } \int _ { \kappa / ( c _ d ' | \nabla f ( a ) | ) } ^ \infty \frac { d r } { r ^ { p + 1 } } \ , d a \\ & = p ^ { - 1 } \int _ \omega \left ( \frac { c _ d ' | \nabla f ( a ) | } { \kappa } \right ) ^ p d a \\ & = c _ { d , p } \kappa ^ { - p } \int _ \omega | \nabla f ( a ) | ^ p \ , d a \ , . \end{align*}


\begin{align*} \mathbf { h } ^ { c } = \sum _ { l = 1 } ^ { L } x _ { l } ^ { c } \boldsymbol { a } \left ( \theta _ { l } ^ { c } \right ) , \end{align*}


\begin{align*} \tilde { a } _ { i j } ( x ) D _ { i j } \widetilde { w } = F ( A + D ^ 2 \widetilde { w } ) - F ( A ) = g ( x ) - 1 \quad \ | x | > r _ 3 , \end{align*}


\begin{align*} \langle A _ { i j } ( t ) g [ \mathbf { A } ] \rangle = \sum _ { n = 0 } ^ { \infty } \frac { 1 } { n ! } \int _ { \mathbb { R } ^ n } \bigl \langle A _ { i j } ( t ) A _ { i _ 1 j _ 1 } ( t _ 1 ) \dots A _ { i _ n j _ n } ( t _ n ) \bigr \rangle _ c \ , \biggl \langle \frac { \delta ^ n \ , g [ \mathbf { A } ] } { \delta A _ { i _ 1 j _ 1 } ( t _ 1 ) \ , \dots \ , \delta A _ { i _ n j _ n } ( t _ n ) } \biggr \rangle \mathrm { d } t _ 1 \dots \mathrm { d } t _ n \end{align*}


\begin{align*} R ^ { n , M , 2 } _ \tau = n ^ { 2 \alpha + 1 } \sum _ { i = 0 } ^ { M - 1 } \int _ { \tau _ i } ^ { \tau _ { i + 1 } } \gamma _ s \left [ ( \sigma ' \sigma ) ^ 2 ( X _ s ) - ( \sigma ' \sigma ) ^ 2 ( X _ { \tau _ i } ) \right ] \Xi ^ { n , 1 } _ s d s \end{align*}


\begin{align*} p ^ * \sum _ { t = k _ 0 } ^ k e ^ T A _ { S \bar { S } } ( t ) e & \leq \left ( \pi _ { \bar { S } } ^ T ( k _ 0 ) - \pi _ { \bar { S } } ^ T ( k ) \right ) e + \sum _ { t = k _ 0 } ^ k e ^ T A _ { \bar { S } S } ( t ) e \cr & \leq 1 + \sum _ { t = k _ 0 } ^ k e ^ T A _ { \bar { S } S } ( t ) e \end{align*}


\begin{align*} ( \partial _ s + \Delta ) u = \mathcal { H } v + \frac { f ' ( u ) } { f ( u ) } ( m + v ^ 2 - 1 ) . \end{align*}


\begin{align*} \left ( \zeta _ { t \wedge t _ k } - \zeta _ { t \wedge s _ k } \right ) \ 1 { s _ k < \rho _ r } & = \left ( \zeta _ { t \wedge t _ k } - \zeta _ { s _ k } \right ) \ 1 { s _ k < \rho _ r \wedge t } \\ & = \left ( \zeta ^ { ( f , \theta ) } _ { t - s _ k } - \zeta ^ { ( f , \theta ) } _ 0 \right ) \ 1 { s _ k < \rho _ r \wedge t } , \end{align*}


\begin{align*} \lim _ { N \to \infty } Q _ { N } \Big [ \varepsilon _ 0 \Big ( \frac { \gamma } { \beta + \gamma } \Big ) ^ { 1 / D } N ^ { \frac { 1 } { D } \left ( d - \frac { 2 ( d - D ) } { D + 2 } \right ) } & \leq R _ { N , d , D } \\ & \leq K _ { 0 } \Big ( \frac { \beta + \gamma } { \beta } \Big ) ^ { 1 / 2 } N ^ { \frac { d } { 2 } + \frac { d - D } { D + 2 } } \Big ] = 1 . \end{align*}


\begin{align*} T _ { p ^ 2 } \psi = a _ \varphi ( p ) \psi . \end{align*}


\begin{align*} K _ { i j } = \frac 1 { \sin ^ 2 r } \left ( 1 - \alpha \cos ^ 2 r \right ) \geq 0 . \end{align*}


\begin{align*} \Lambda ( x , y ) = \Phi _ 0 ( [ x , y ] _ \mathfrak { g } ) - [ \Phi _ 0 ( x ) , \Phi _ 0 ( y ) ] . \end{align*}


\begin{align*} \begin{aligned} B ( n , q ; \chi _ { 1 } , \ldots , \chi _ { 5 } ) = \sum \limits _ { a = 1 \atop ( a , q ) = 1 } ^ { q } C _ { 1 } ( \chi _ { 1 } , a ) C _ { 3 } ( \chi _ { 2 } , a ) \ldots C _ { 3 } ( \chi _ { 5 } , a ) e \left ( - \frac { a n } { q } \right ) \end{aligned} \end{align*}


\begin{align*} f ^ \infty _ s = f _ o \circ \Psi _ { s - s _ 0 } . \end{align*}


\begin{align*} \mathbb { M } _ n ( \hat t _ n ) & = \sqrt { n } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \hat \theta _ n } ) + o _ { \mathbb { P } } ( 1 ) \\ & \leq \inf _ { \theta \in \Theta } \sqrt { n } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ \theta ) + o _ { \mathbb { P } } ( 1 ) \\ & = \inf _ { t \in \R ^ { d _ 0 } } \mathbb { M } _ n ( t ) + o _ { \mathbb { P } } ( 1 ) . \end{align*}


\begin{align*} \Delta _ h ^ { j _ \ell } \dots \Delta _ h ^ { j _ 2 } \Delta _ h ^ { j _ 1 } G ( x _ 0 , \dots , x _ { m - 1 } ) = \begin{cases} 1 & \{ j _ 1 , \dots , j _ \ell \} = A \\ 0 & . \end{cases} \end{align*}


\begin{align*} M = \mbox { s p a n } \{ u _ n a | u \in V , n \in \frac { 1 } { T } \mathbf { N } \} \end{align*}


\begin{align*} \operatorname { p a r d e g } ( E _ * ) \coloneqq \deg ( E ) + \sum \limits _ { p \in D } \sum \limits _ { i = 1 } ^ { r _ p } \alpha _ i ( p ) \cdot \dim ( E _ { p , i } / E _ { p , i + 1 } ) \end{align*}


\begin{align*} I _ { ( Y _ - , D _ { Y , - } ) } ( y , z ) = z e ^ { t _ - / z } \sum _ { d \in \mathbb K _ { - } } \tilde y ^ { d } \left ( \prod _ { i \in M _ 0 } \frac { \prod _ { a \leq 0 , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } { \prod _ { a \leq D _ i \cdot d , \langle a \rangle = \langle D _ i \cdot d \rangle } ( \bar { D } _ i + a z ) } \right ) \\ \left ( \prod _ { i = 1 } ^ k \prod _ { 0 < a \leq v _ i \cdot d } ( v _ i + a z ) \right ) \textbf { 1 } _ { [ - d ] } I _ { D _ { Y , - } , d } , \end{align*}


\begin{align*} P _ { o u t } ( i ) = \textrm { P r } \bigg [ \bar { \gamma _ i } < 1 \bigg ] . \end{align*}


\begin{align*} n = p D \ \ \gcd ( D , p ) = 1 \end{align*}


\begin{align*} F ^ { \nu } _ i + F ^ { \nu t } _ i = 0 . \end{align*}


\begin{align*} f ^ { \tilde { \phi } ^ { - n } } ( v _ m ) = \beta _ m \end{align*}


\begin{align*} u \times v \mapsto \langle u | v \rangle _ t = e ^ { t ( ( \tilde { f } ^ { | u \rangle } , \tilde { f } ^ { | v \rangle } ) _ { { } _ { \mathfrak { J } } } + 4 \pi ) } = e ^ { - t 4 \pi ( \lambda \textrm { c o t h } \lambda - 1 ) } \end{align*}


\begin{align*} { \mathcal G } _ { K ^ { - 1 } } ( z , w ) : = \Big ( \ ! \ ! \Big ( \ , K ( z , w ) \partial _ i \bar { \partial } _ j K ( z , w ) - \partial _ i K ( z , w ) \bar { \partial } _ j K ( z , w ) \Big ) \ ! \ ! \Big ) _ { i , j = 1 } ^ m , \ ; \ ; z , w \in \Omega , \end{align*}


\begin{align*} \tau ^ \mathrm { a n } ( M / S , \rho ) = \tau ^ \mathrm { t o p } ( M / S , \rho ) \ ; . \end{align*}


\begin{align*} \overline { H } ( \rho , t ) & \approx r ^ { - n } e ^ { - \frac { \rho ^ 2 } { 4 t } - \lambda _ 1 ( M ) t } = \max \left ( 1 , \frac { \rho } { t } \right ) ^ n e ^ { - \frac { \rho ^ 2 } { 4 t } - \lambda _ 1 ( M ) t } \\ & \approx \left ( \frac { \rho + t } { t } \right ) ^ n e ^ { - \frac { \rho ^ 2 } { 4 t } - \lambda _ 1 ( M ) t } \lesssim ( 1 + \rho ^ n ) e ^ { - \frac { \rho ^ 2 } { 4 t } - \frac { ( n - 1 ) ^ 2 a ^ 2 } { 4 } t } , \end{align*}


\begin{align*} s = \theta s _ 1 + ( 1 - \theta ) s _ 2 \quad \frac { 1 } { p } = \frac { \theta } { p _ 1 } + \frac { 1 - \theta } { p _ 2 } , \end{align*}


\begin{align*} [ Q , \Bar { \Psi } ^ { ( n + 1 ) } ] = \Bar \Psi ^ { ( n ) } \circ Q _ \mathfrak { g } ^ { ( 1 ) } - \Bar { Q } ^ { ( 1 ) } \circ \Bar \Psi ^ { ( n ) } . \end{align*}


\begin{align*} \sup _ { t > 0 } \frac { c \gamma _ t } { m _ t } \leq c ' ( \mu ) \lim _ { \mu \rightarrow \infty } c ' ( \mu ) = 0 . \end{align*}


\begin{align*} \frac { 1 } { 2 } \frac { d } { d t } c m \int _ 0 ^ { L } \int _ 0 ^ { \infty } \sigma ( s ) \abs { \eta _ x } ^ 2 d s d x - \frac { c m } { 2 } \int _ 0 ^ { L } \int _ 0 ^ { \infty } \sigma ' ( s ) \abs { \eta _ x } ^ 2 d s d x = \Re \left ( c m \int _ 0 ^ L \int _ 0 ^ { \infty } \sigma ( s ) \eta _ x \overline { w } _ x d s d x \right ) \end{align*}


\begin{align*} \pi _ { i _ 2 } \cdots \pi _ { i _ d } = \sum _ { J \subseteq [ n - 1 ] } c _ { J } \pi _ J \end{align*}


\begin{align*} \Phi ( S ) = \frac { Q ( S , S ^ c ) } { \pi ( S ) } , \end{align*}


\begin{align*} \nabla ( G _ \lambda - \Gamma ) \in L ^ { \bar q } ( \Omega ) , \bar q = \frac { N ( p - 1 ) } { N - 1 } , \end{align*}


\begin{align*} b _ h ( u , v , w ) : = \langle n l _ h ( u , v ) , w \rangle _ { H ^ { - 1 } \times H _ { \# } ^ 1 } , \end{align*}


\begin{align*} \left \{ \alpha \oplus \frac { \varepsilon _ 1 } { 2 ^ { k _ 1 } } \oplus \ldots \oplus \frac { \varepsilon _ l } { 2 ^ { k _ l } } : \ , \varepsilon _ j = 0 , 1 \right \} \subset E . \end{align*}


\begin{align*} B \overline { \overline { u } } \ , ' ( t ) + ( A + B Q ) \overline { \overline { u } } ( t ) = - H _ 1 ( { u } _ 2 ' ( t ) + Q { u } _ 2 ( t ) ) - H _ 2 ( { u } _ 1 ' ( t ) + Q { u } _ 1 ( t ) ) , \overline { \overline { u } } ( 0 ) = 0 . \end{align*}


\begin{align*} \mathrm { P } _ { ( \Phi , \hat { f } \left ( 0 \right ) \delta _ { \Psi } ) } = - \inf _ { \rho \in E _ { 1 } } \{ f _ { \Phi } \left ( \rho \right ) + \hat { f } \left ( 0 \right ) \Delta _ { \mathfrak { e } _ { \Psi } } \left ( \rho \right ) \} \ . \end{align*}


\begin{align*} d i v ( X ) = \sum _ { i = 1 } ^ { m } g ( \nabla _ { e _ i } X , e _ i ) , ~ \forall X \in \Gamma ( T M ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } { \| B ^ n \| } ^ { 1 / n } = \lim _ { n \to \infty } { \left | w \right | } ^ { - \frac { n + 1 } { 2 } } = 0 , \end{align*}


\begin{align*} s i g n ( \widehat { \alpha . \beta } ) = s i g n ( \hat { \alpha } ) + s i g n ( \hat { \beta } ) - M e y e r ( { \mathcal B } _ { - 1 } ( \alpha ) , { \mathcal B } _ { - 1 } ( \beta ) ) \end{align*}


\begin{align*} \phi ^ * ( D ) - t F \equiv \phi ^ \ast \left ( \frac { 3 } { 2 } C _ x \right ) - t F = \frac { 3 } { 2 } \left ( \hat { L } _ { x y } + \hat { R } _ 0 + \hat { R } _ 1 \right ) + \left ( \frac { 3 ( 2 n + 1 ) } { 2 ( 4 n + 1 ) } - t \right ) F , \end{align*}


\begin{align*} ( - \Delta ) ^ s u + \sum _ { | \alpha | \leq m } a _ \alpha ( D ^ \alpha u ) & = F \mbox { i n } \ ; \ ; \Omega , \\ u & = f \mbox { i n } \ ; \ ; \Omega _ e \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \ge M } \sup _ { \tau \in [ 0 , T ] } \| R ^ { n , M , 2 } _ \tau \| _ 2 = 0 . \end{align*}


\begin{align*} \frac { G ( z + \omega _ 1 \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) } { G ( z \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) } = & F ( z + \overline \omega _ 1 \ , | \ , \widetilde \omega _ 1 , \omega _ 2 ) ^ { - 1 } , \\ \frac { G ( z + \widetilde \omega _ 1 \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) } { G ( z \ , | \ , \omega _ 1 , \widetilde \omega _ 1 , \omega _ 2 ) } = & F ( z + \overline \omega _ 1 \ , | \ , \omega _ 1 , \omega _ 2 ) ^ { - 1 } . \end{align*}


\begin{align*} \lim \kappa ^ p \ , \tilde \nu _ \gamma ( \{ ( x , y ) \in \mathcal X : \ | f ( x ) - f ( y ) | / | x - y | ^ { 1 + \gamma / p } > \kappa \} ) = | \gamma | ^ { - 1 } \tilde c _ { d , p } \ , \| \nabla f \| _ { L ^ p ( \R ^ d ) } ^ p \ , , \end{align*}


\begin{align*} \mathcal { I } _ 2 = \underbrace { \iint _ { Q _ T } \nabla \cdot ( m \nabla p ) | \nabla p | ^ 2 } _ { \mathcal { I } _ { 2 , 1 } } + \iint _ { Q _ T } m F | \nabla p | ^ 2 , \end{align*}


\begin{align*} J ^ { e s s ; 0 } _ { k , m ' } ( t , x ) = \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } e ^ { i x \cdot \xi + i ( t - s ) \omega \cdot \xi } \frac { \varphi ^ { e s s } ( \omega , \xi ) } { i ( \omega + \hat { v } ) \cdot \xi } \hat { f } ( s , \xi , v ) \varphi _ k ( \xi ) \varphi _ j ( v ) \varphi _ { m ' ; - k } ( t - s ) \varphi _ { m ; - 1 0 M _ t } ( t - s ) d v d \omega d \xi \big | _ { s = 0 } ^ t \end{align*}