\begin{align*} \Omega _ { i j } ( \partial _ j , \partial _ r ) = \frac { - \Gamma _ { j i } ^ i \Gamma _ { r j } ^ j + \Gamma _ { r i } ^ i \Gamma _ { j r } ^ r - \Gamma _ { r i } ^ i \Gamma _ { j r } ^ r } { \varphi _ j } . \end{align*}


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


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


\begin{align*} \begin{aligned} < u \times n , \xi > _ { \partial \Omega } & = \int _ { \Omega } \nabla \times u \cdot \xi \dd x - \int _ { \Omega } u \cdot \nabla \times \xi \dd x , \\ < u \cdot n , \varphi > _ { \partial \Omega } & = \int _ { \Omega } u \cdot \nabla \varphi \dd x + \int _ { \Omega } \nabla \cdot u \varphi \dd x , \end{aligned} \end{align*}


\begin{align*} \int _ { M \setminus \Omega } \bigg \langle S ' ( h , v ) , \ , \kappa _ 0 ( \bar { g } , \bar { u } , X ) \bigg \rangle _ { \bar g } \ , d \mathrm { v o l } _ { \bar g } & = 0 \end{align*}


\begin{align*} \int ( I _ \alpha \ast f ) g d x = \int \left ( I _ { \frac { \alpha } { 2 } } \ast f \right ) \left ( I _ { \frac { \alpha } { 2 } } \ast g \right ) d x . \end{align*}


\begin{align*} l ( y _ p ) \leq \bigvee _ { 1 \leq i \leq n } l ( \Psi ( v _ i ) ) = \bigvee _ { 1 \leq i \leq n } \Phi ( l ( v _ i ) ) = \bigvee _ { 1 \leq i \leq n } \Phi ( p _ i ) = \Phi \left ( \bigvee _ { 1 \leq i \leq n } p _ i \right ) = \Phi ( p ) . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } s } \log \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) & = \frac { 1 } { s } + \frac { \psi ( 1 / s ) } { s ^ 2 } - \left | \frac { x - \mu } { \sigma } \right | ^ s \log \left ( \left | \frac { x - \mu } { \sigma } \right | \right ) . \end{align*}


\begin{align*} q _ X ( \alpha ) : = \frac { n } { 2 } \int _ X ( \sigma \overline \sigma ) ^ { n - 1 } \alpha ^ 2 + ( 1 - n ) \int _ X \sigma ^ n \overline \sigma ^ { n - 1 } \alpha \int _ X \sigma ^ { n - 1 } \overline \sigma ^ n \alpha \end{align*}


\begin{align*} \begin{cases} 1 < \alpha < \infty , & 1 \le d \le 4 , \\ 1 < \alpha < 1 + \frac { 2 } { d - 2 } , \ 2 \le \alpha < 1 + \frac { 4 } { d - 4 } , & 5 \le d \le 7 , \\ 1 < \alpha < 1 + \frac { 2 } { d - 2 } , & d \ge 8 . \end{cases} \end{align*}


\begin{align*} \dot \phi _ i = \omega _ i + \frac 1 M \sum _ { j = 1 } ^ M a _ { i j } \sin ( \phi _ j - \phi _ i ) , \end{align*}


\begin{align*} \mathbf { y } _ { \mathrm { r } } \left [ l \right ] & = \mathbf { H } _ { \mathrm { r } } \mathbf { x } \left [ l \right ] + \mathbf { m } \left [ l \right ] \\ & = \alpha e ^ { j 2 \pi \mathcal { F } _ { D } l T } { \mathbf { b } } \left ( \theta \right ) { \mathbf { a } } ^ { H } \left ( \theta \right ) \mathbf { x } \left [ l \right ] + \mathbf { m } \left [ l \right ] , \end{align*}


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


\begin{align*} \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { s } ^ { \left ( \right ) } \right ] = & \mathbb { E } \left [ s _ c ^ * s _ c \right ] + \mathbb { E } \left [ s _ 1 ^ * s _ 1 \right ] \cdots + \mathbb { E } \left [ s _ M ^ * s _ M \right ] , = M + 1 . \end{align*}


\begin{align*} \frac { d } { d t } ( \mathcal { Q } ^ { r } ( t ) ) = \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } ( r - 1 ) ( \mathcal { M } ^ { r } ( t ) ) ^ { \sigma + 1 } \ge \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } ( r - 1 ) ( \mathcal { Q } ^ { r } ( t ) ) ^ { \sigma + 1 } , t \in [ \delta , t _ { 0 } ) . \end{align*}


\begin{align*} \begin{cases} x _ 1 \mapsto \frac { 4 a ^ 2 } { s ^ 2 } ( x _ 1 - \tilde { \nu _ 1 } ' ( l s ) x _ 2 - \tilde { \nu _ 1 } ( l s ) x _ 3 + \tilde { \nu _ 1 } ' ( l s ) l s x _ 3 ) \\ x _ 2 \mapsto \frac { 2 a } { s } ( x _ 2 - l s x _ 3 ) \\ x _ 3 \mapsto x _ 3 \end{cases} \end{align*}


\begin{align*} M ( X ) = \sum _ { 0 \le i \le n - 1 } C _ i X ^ { 2 ^ i } + D . \end{align*}


\begin{align*} \forall ( x , y ) \in E \times E \ , \ \ b _ { \sigma } ( x , y ) = \sigma ( x ) ( y ) . \end{align*}


\begin{align*} \deg \varphi = [ S : \varphi ^ { * } ( S ) ] \end{align*}


\begin{align*} R L ( b , c ) - L L ( b , c ) & = \sum _ { i = 1 } ^ b \left [ \Delta _ c \left ( \binom { b } { 2 } + i \right ) - \Delta _ c ( i ) \right ] - b \left ( \Delta _ c ( b ) - \Delta _ c ( 1 ) \right ) \\ & > b \left [ \Delta _ c \left ( \binom { b } { 2 } + b \right ) - \Delta _ c ( b ) \right ] - b \left ( \Delta _ c ( b ) - \Delta _ c ( 1 ) \right ) \\ & = b \left [ \Delta _ c \left ( \binom { b + 1 } { 2 } \right ) + \Delta _ c ( 1 ) - 2 \Delta _ c ( b ) \right ] \end{align*}


\begin{align*} \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } q \ , \partial _ { x ^ \prime } ^ { J } q - \sum _ { J } \frac { ( - i ) ^ { | J | } } { J ! } \partial _ { \xi } ^ { J } b \ , \partial _ { x ^ \prime } ^ { J } q - \frac { \partial q } { \partial x _ n } + c = 0 , \end{align*}


\begin{align*} | | \nabla ( u - \tilde { U } ) | | _ { L _ 2 ( \Omega ) } ^ 2 = | | \Delta \tilde { U } + F | | ^ 2 _ { H ^ { - 1 } ( \Omega ) } \to \min \end{align*}


\begin{align*} ( f \star g ) ( z ) = k _ 1 k _ 2 ^ { - 1 } ( c z + d ) k _ 2 ( \det ( g ) ) f ( g z ) , \end{align*}


\begin{align*} \int _ { M \setminus \Omega } \bigg \langle S ( g , u ) , \ , \kappa _ 0 ( g , u , X ) \bigg \rangle _ g \ , d \mathrm { v o l } _ g & = 0 , \end{align*}


\begin{align*} \begin{aligned} \left | H [ b _ 0 ] - H [ b _ 1 ] - H [ b _ 2 ] \right | & \lesssim \left ( \max _ { 0 \leq i \leq 2 } | | b _ i | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } ^ { 5 / 3 } \right ) | | b _ 0 - b _ 1 - b _ 2 | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } , \end{aligned} \end{align*}


\begin{align*} \mathcal { L } _ H \varPhi = W = f _ \varepsilon \left ( \varPhi - \frac { \alpha } { 2 } | x ' | ^ 2 | \ln \varepsilon | \right ) \ \ \ B _ { R ^ * } ( 0 ) , \end{align*}


\begin{align*} S _ { R _ n } ^ { \beta + \delta } - \frac { C _ { \beta , \delta } } { C _ { \beta , \delta + 1 } } S _ { R _ n } ^ { \beta + \delta + 1 } & = C _ { \beta , \delta } R _ n ^ { - ( \beta + \delta ) } \int ^ { R _ n } _ 0 ( R _ n - t ) ^ { \beta - 1 } t ^ \delta \big ( S _ t ^ { \delta } - S _ t ^ { \delta + 1 } \big ) d t \\ & \ \ \ + C _ { \beta , \delta } R _ n ^ { - ( \beta + \delta ) } \int ^ { R _ n } _ 0 ( R _ n - t ) ^ { \beta - 1 } t ^ \delta ( 1 - R _ n ^ { - 1 } t ) S _ t ^ { \delta + 1 } d t \\ & = : I _ { R _ n } + I I _ { R _ n } . \end{align*}


\begin{align*} \Vert M _ { \widetilde { \phi } } ( a _ 1 , \ldots , a _ n ) \Vert _ { S _ p ^ N } \leq \Vert T _ { \phi } ^ { ( N ) } \Vert \Vert a _ 1 \Vert _ { S _ { p _ 1 } ^ N } \ldots \Vert a _ n \Vert _ { S _ { p _ n } ^ N } . \end{align*}


\begin{align*} \begin{aligned} & E _ { P } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \biggr ] \\ = & \xi _ 0 E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \vert H _ 1 \biggr ] + ( 1 - \xi _ 0 ) E _ P \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ \theta \biggr ) \vert H _ 2 \biggr ] , \end{aligned} \end{align*}


\begin{align*} | G _ { \gamma , \omega } ( s ) - 1 | = \dfrac { | \alpha \omega - \omega | } { | b | | c \omega + d | } \leq \dfrac { 3 ^ { - 2 } } { | b | } < \varepsilon , & & \mathrm { i f } \ | b | > \frac { 3 ^ { - 2 } } { \varepsilon } . \end{align*}


\begin{align*} h \cdot a = a , \end{align*}


\begin{align*} f ( x ) = e ^ { - \Lambda _ u } f _ r ( x ) + e ^ { - \Lambda _ u } \sum _ { n = 1 } ^ \infty ( \Lambda _ u ^ n / n ! ) g _ u ^ { \ast n } \ast f _ r ( x ) , \end{align*}


\begin{align*} \phi _ { \lambda } ( \partial a ) = ( \partial + \lambda ) \phi _ { \lambda } ( a ) , \forall \ a \in \mathcal { A } . \end{align*}


\begin{align*} \begin{aligned} \frac { d } { d t } H _ t ^ k \leq & \frac { k } { ( n - 1 ) ^ 2 } \mathbb { E } \left [ \left | \sum _ { j = 2 } ^ k \left ( K ( X _ t ^ 1 - X _ t ^ j ) - \langle \mu _ t , K ( X _ t ^ 1 - \cdot ) \right ) \right | ^ 2 \right ] \\ & + \frac { k ( n - k ) ^ 2 } { ( n - 1 ) ^ 2 } \mathbb { E } \left [ \left | \mathbb { E } [ K ( X _ t ^ 1 - X _ t ^ n ) \mid X _ t ^ 1 , \cdots , X _ t ^ k ] - \langle \mu _ t , K ( X _ t ^ 1 - \cdot ) \right | \right ] \end{aligned} . \end{align*}


\begin{align*} \frac { \partial } { \partial t } \zeta ( t ) = - \phi ' ( \zeta ( t ) ) , \end{align*}


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


\begin{align*} \langle u , v \rangle : = \ , \ , ^ { t } u \psi _ { 2 n } v \end{align*}


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


\begin{align*} \int _ { B _ { \rho } } r ^ { 2 - n } u _ r ^ 2 \d x & \leq C \int _ { B _ { 2 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { 4 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x \end{align*}


\begin{align*} \vert \mathcal { D } _ { n , h } ^ k \vert = [ t ^ n ] C _ k ( t ) ^ { h + 1 } = \frac { h + 1 } { k n + h + 1 } \binom { k n + h + 1 } { n } . \end{align*}


\begin{align*} \big [ \theta _ { \rho ^ \varepsilon } ( 0 , z _ l ) \big ] _ 1 = - \varepsilon \rho _ 0 < \varepsilon \rho _ 0 = \big [ \theta _ { \rho ^ \varepsilon } ( 0 , z _ r ) \big ] _ 1 \end{align*}


\begin{align*} \frac { 1 } { \tau } \int _ 0 ^ { \tau / 2 } | \Pi _ { \tilde { V } } e ^ { | z | J ^ \perp t } \tilde { Y } _ 0 | d t = \frac { 1 } { \tau } \int _ 0 ^ { \tau / 2 } | \Pi _ { \tilde { V } } \tilde { Y } _ 0 + t | z | | \Pi _ V \tilde { Y } _ 0 | d t \gtrsim R | z | \tau ^ { 3 / 2 } . \end{align*}


\begin{align*} \begin{aligned} d ( \phi ( y ) , \psi ( y ) ) & \leq d ( \phi ( y ) , \psi ( y ) ) + d ( \psi ( y ) , \eta ( y ) ) \\ & \leq \tilde L \min \{ d ( \eta ( \hat y ) , \eta ( y ) ) , d ( \psi ( \hat y ) , \phi ( y ) ) \} + M _ \phi + M _ \eta , \\ & = \tilde L \min \{ d ( \eta ( \hat y ) , \eta ( y ) ) , d ( \eta ( \hat y ) , \phi ( y ) ) \} + M _ \phi + M _ \eta , \\ \end{aligned} \end{align*}


\begin{align*} | h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | + | \nabla h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) | \le C , \ \ 1 \leq i , j \leq m , \end{align*}


\begin{align*} \sum _ { \sigma \subset \mu _ \ell } c _ { \ell } \frac { r _ \ell } { v ( \mu _ \ell ) } + c _ { \kappa } \frac { 1 } { v ( \kappa ) } = 0 , \sigma \subset \kappa , & \sum _ { \sigma \subset \mu _ \ell } c _ { \ell } \frac { r _ \ell } { v ( \mu _ \ell ) } = 0 , \sigma \not \subset \kappa , \end{align*}


\begin{align*} \mathbf { v } ^ { 0 } = [ \cdots , 1 , 0 , \cdots , 1 , \cdots , 1 , \cdots ] ^ { T } , m , n , k = 1 , \cdots , N . \end{align*}


\begin{align*} \kappa ( t ) = h _ \alpha ( t ) , \ \ k ( t ) = h _ { 1 - \alpha } ( t ) , \ 0 < \alpha < 1 , \end{align*}


\begin{align*} a _ { 2 c - i } = \sum _ { j = 0 } ^ { \lfloor i / 2 \rfloor } \frac { c ! \cdot 2 ^ j } { ( c + j - i ) ! ( i - 2 j ) ! j ! } \le c ^ i \sum _ { j = 0 } ^ { \lfloor i / 2 \rfloor } \underbrace { \frac { 2 ^ j } { c ^ j ( i - 2 j ) ! j ! } } _ { = : b _ { i j } } . \end{align*}


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


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


\begin{align*} \int _ 0 ^ t U ( t , s ) ( \partial _ t G ) ( y ( s ) ) d s = & ( \alpha - 1 ) \int _ 0 ^ t U ( t , s ) e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) d W ( s ) \\ & + ( \alpha - 1 ) ^ 2 \int _ 0 ^ t U ( t , s ) \mu e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) d s , \\ | \nabla G ( y ) | \le & | e ^ { ( \alpha - 1 ) W } | [ ( \alpha - 1 ) | \nabla W | | g ( y ) | + | \nabla g ( y ) | ] . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ M \sum \limits _ { \substack { j = 1 \\ j \neq l } } ^ M \sum _ { l = 1 } ^ M a _ l ^ 2 \phi ^ { \left ( i , l \right ) ^ * } \phi ^ { \left ( j , l \right ) } = 2 \sum _ { i = 1 } ^ { M - 1 } \sum _ { q = i + 1 } ^ { M } \sum _ { r = 1 } ^ M a _ r ^ 2 \Re \left [ \phi ^ { \left ( i , r \right ) ^ * } \phi ^ { \left ( q , r \right ) } \right ] . \end{align*}


\begin{align*} \int _ X f \ , d \mu = \int _ X f \circ \varphi _ \alpha \ , d \mu , \end{align*}


\begin{align*} r ' ( X ~ \cup ~ \{ z , \gamma \} ) = \begin{cases} r ( X ) + 1 , ~ e \in c l ( X ) ; \\ r ( X ) + 2 , ~ e \notin c l ( X ) ; ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \end{cases} \end{align*}


\begin{align*} \frac { \partial z } { \partial \alpha } & = \frac { \phi ( z ) [ 1 - \Phi ( z ) ] } { - \frac { \partial s ( z ) } { \partial z } } \mathrm { s i g n } \left ( \frac { \partial z } { \partial \alpha } \right ) = - \mathrm { s i g n } \left ( \frac { \partial s ( z ) } { \partial z } \right ) , \end{align*}


\begin{align*} & \lim _ { \epsilon \downarrow 0 } \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } \frac { \mathrm { e } ^ { i s ( \xi - 2 x ) } } { s - ( z + i \epsilon ) } d s = \lim _ { \epsilon \downarrow 0 } \begin{cases} \mathrm { e } ^ { i ( z + i \epsilon ) ( \xi - 2 x ) } , & \xi - 2 x > 0 \\ 0 , & \xi - 2 x < 0 \end{cases} \\ [ 6 p t ] & = \chi ( \xi - 2 x ) e ^ { i z ( \xi - 2 x ) } , \end{align*}


\begin{align*} \nu ( \eta _ { \Lambda _ n } ) = \nu ( \xi _ { \Delta } \eta _ { \Lambda _ n \setminus \Delta } ) = 0 . \end{align*}


\begin{align*} \operatorname { W S } = \frac { 1 } { 2 } \sqrt { \operatorname { t r } ( \Sigma _ i ^ { - 1 } ( \Sigma _ i + \eta ) - I ) + ( n + 1 ) ^ 2 \eta ^ 2 \sum \Sigma _ i ^ { - 1 } - \log \operatorname { d e t } ( ( \Sigma _ i + \eta ) \Sigma _ i ^ { - 1 } ) } \end{align*}


\begin{align*} \Gamma _ \rho ( t ) = \Bigg \{ \begin{bmatrix} x \\ w ( t , x ) \cos \varphi \\ w ( t , x ) \sin \varphi \end{bmatrix} \ , \Bigg | \ , x \in [ \tilde { x _ 0 } ( t ) , \tilde { x _ 5 } ( t ) ] , \varphi \in [ 0 , 2 \pi ] \Bigg \} \end{align*}


\begin{align*} \int _ \Omega \left ( { \mathcal D } _ { \psi } f ( w ) + ( \frac { \partial p _ 0 } { \partial x } + \frac { \partial q _ 0 } { \partial y } ) f + ( \frac { \partial p _ 1 } { \partial x } + \frac { \partial q _ 1 } { \partial y } ) { i } f \right ) d x d y = \int _ { \partial \Omega } f d \sigma _ { \psi } ( w ) , \end{align*}


\begin{align*} l _ S ( g ) = | \sigma ^ - | + ( \sigma ^ + - \sigma ^ - ) + | \sigma ^ + | + \sum _ { i = \sigma ^ - } ^ { \sigma ^ + } l _ { S _ 0 } \ ! \big ( w _ i ( a _ 1 , \ldots , a _ d ) \big ) . \end{align*}


\begin{align*} \mathbf { U } _ { v } = \begin{pmatrix} & 0 & 0 \\ & 0 & ( \mathbf { v } _ { 1 } - \beta _ { v } ) \mathbf { P } _ { v } \end{pmatrix} \end{align*}


\begin{align*} \frac { \partial s ( z ) } { \partial z } = - \Phi ( z ) \ , [ 1 - \Phi ( z ) ] - z \ , \phi ( z ) \ , \left [ \alpha - ( \alpha + \beta ) \ , \Phi ( z ) \right ] + ( 2 - \alpha - \beta ) \ , \phi ^ 2 ( z ) . \end{align*}


\begin{align*} X ^ v \circ X ^ w = q ^ { - \langle v , w \rangle } X ^ { v + w } \ , v , w \in \mathbb { N } ^ n . \end{align*}


\begin{align*} V _ 1 ( r ) = V ( r ) - E _ 0 , \end{align*}


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


\begin{align*} g _ n \big ( t , q , x \big ) : = \sup _ { z \in \mathbb { R } ^ d } \Big ( q \cdot z - f _ n ( t , z , x ) \Big ) . \end{align*}


\begin{align*} \lim _ { s \rightarrow \infty } s L _ { q } [ f ( t ) ] = \lim _ { t \rightarrow 0 } \frac { f ( t ) } { 1 + ( 1 - q ) } , \end{align*}


\begin{align*} \abs { \mathbf { I I } } \leq \sup _ { \Delta \ni 0 : \xi _ { \Delta } } \norm { c _ { \Delta } ( \cdot , \xi _ { \Delta } ) } _ { \infty } q ^ { \Delta } \log \left ( \frac { 1 } { \delta ( \mu ) } \right ) \abs { \Lambda _ n \setminus \hat { \Lambda } _ n } = o ( \abs { \Lambda _ n } ) , \end{align*}


\begin{align*} \mathcal { J } ^ i ( \alpha ^ i , \boldsymbol { \alpha } ^ { - i } ) : = \frac { 1 } { 2 } \mathbb { E } \Big \{ \int _ 0 ^ T \Big [ Q \left ( x ^ i _ t + l ( \nu ^ { N , i } _ { \boldsymbol { x } _ t } ) \right ) ^ 2 + R \left ( \alpha ^ i _ t + h ( \mu ^ { N , i } _ { \boldsymbol { \alpha } _ t } ) \right ) ^ 2 \Big ] d t + G \left ( x ^ i _ T + g ( \nu ^ { N , i } _ { \boldsymbol { x } _ T } ) \right ) ^ 2 \Big \} , \end{align*}


\begin{align*} x = \sum _ { n = 0 } ^ \infty \frac { \varepsilon _ n } { \beta ^ { n + 1 } } = \sum _ { n = 0 } ^ { s - 1 } \frac { \varepsilon _ n } { \beta ^ { n + 1 } } + \left ( \sum _ { n = s } ^ { s + t - 1 } \frac { \varepsilon _ n } { \beta ^ { n + 1 } } \right ) \frac { \beta ^ t } { \beta ^ t - 1 } . \end{align*}


\begin{align*} \begin{array} { l } { \rm S I N R } _ { k , j } ^ m = 2 y _ { k , j } ^ m \sqrt { { p _ { k , j } ^ m \left | G _ { k , j } ^ m \right | ^ 2 } } - y _ { k , j } ^ m \left ( { { \sum \limits _ { j ' \ne j } \sum \limits _ { k ' \ne { { k } } } { p _ { k ' , j ' } ^ { { m } } { { { \left | G _ { k ' , j } ^ m \right | ^ 2 } } } } } + N _ { \rm R F } B _ { \rm R F } } \right ) , \end{array} \end{align*}


\begin{align*} \widetilde { \varphi } ( g ) = \begin{cases} \varphi ( g ) & g \in H \\ 0 & g \notin H . \end{cases} \end{align*}


\begin{align*} H ( Q \ , | \ , P ) \le \inf \Big \{ H ( R \ , | \ , P ) : R \in \P ( \R ^ n ) , \ H ( R ) < \infty \Big \} = 0 , \end{align*}


\begin{align*} \mathcal { M } _ { \Gamma } = \left \lbrace p R \ , \vert \ , p _ { i } = \dfrac { 1 } { Z ( \theta ) } \prod \limits _ { F \in \mathrm { f a c e t } ( \Gamma ) } \theta ^ { F } _ { i _ { F } } \right \rbrace \end{align*}


\begin{align*} \partial _ r ( d _ { i s } ) + ( d _ { i s } - d _ { i r } ) \Gamma _ { r s } ^ s = \sum _ j \phi _ { i j } ( \partial _ r ) d _ { j s } \varphi _ j = \phi _ { i s } ( \partial _ r ) ( d _ { s s } - 1 ) \varphi _ s + \sum _ j \phi _ { i j } ( \partial _ r ) d _ j \varphi _ j = \phi _ { i s } ( \partial _ r ) , \end{align*}


\begin{align*} \underline { \delta } ^ \star ( s ) : = 2 \int _ { 0 } ^ { 1 / 2 } \alpha ( s \sin ( 2 \pi t ) ) \ , d t \end{align*}


\begin{align*} \mathcal { Y M } _ e ^ 0 ( \nabla ) = \int _ M \big ( \exp ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) - 1 \big ) \ , d v \ , , \end{align*}


\begin{align*} A ( 1 + B ) ^ k = \sum _ { j \in \mathbb Z _ 0 } \lambda _ j , \ \ \ \ B ( 1 + A ) ^ k = \sum _ { j \in \mathbb Z _ 0 } \lambda _ j . \end{align*}


\begin{align*} D ( A ^ { \circ } ) & = \{ x ^ { \circ } \in X ^ { \circ } \ | \ \exists y ^ { \circ } \in X ^ { \circ } \ \forall x \in D ( A ) : \langle - A x , x ^ { \circ } \rangle = \langle x , y ^ { \circ } \rangle \} , \\ - A ^ { \circ } x ^ { \circ } & = y ^ { \circ } \quad ( x ^ { \circ } \in D ( A ^ { \circ } ) ) , \end{align*}


\begin{align*} & S ( k ) = \sigma _ 2 \overline { S ( \bar { k } ) } \sigma _ 2 , \ \ \ S ( k ) = \sigma _ 3 S ( - k ) \sigma _ 3 \end{align*}


\begin{align*} J _ { l } \left ( \rho \right ) : = \int _ { \mathbb { R } } \frac { \varphi _ { 0 } \left ( \tau \right ) \left ( \frac { 1 } { 2 } - i \tau \right ) ^ { l - 1 } } { \left ( \tau - \rho \right ) ^ { l + 1 } } d \tau = \sum _ { j = 0 } ^ { l - 1 } C _ { l - 1 } ^ { j } \left ( \frac { 1 } { 2 } \right ) ^ { j } \left ( - i \right ) ^ { l - 1 - j } \int _ { \mathbb { R } } \frac { \varphi _ { 0 } \left ( \tau \right ) \tau ^ { l - 1 - j } } { \left ( \tau - \rho \right ) ^ { l + 1 } } d \tau . \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + S ( u ) \ , d t + \sum _ { n = 0 } ^ { + \infty } \xi _ { n } \cdot \nabla u \ , d B _ t ^ { n } , \end{align*}


\begin{align*} \mathcal { Y M } _ e ( \nabla ^ t ) = \int _ M \exp \ , ( \frac 1 2 | | R ^ \nabla | | ^ 2 + t \langle R ^ \nabla , d ^ \nabla A \rangle + \varepsilon ( t ^ 2 ) ) \ , d v \end{align*}


\begin{align*} \sum _ { \Theta \Subset \Z ^ d } h _ { \Theta } ( \omega ) = 0 . \end{align*}


\begin{align*} \mathbb { E } \Big [ \underset { t _ 0 \leq t \leq T } { \sup } | x _ t ^ i - x _ t ^ { * , i } | ^ 2 \Big ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } \mathbb { E } [ | \xi ^ i - \xi ^ j | ^ 2 ] + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } \mathbb { E } [ | \xi ^ { i } - \xi ^ { j } | ^ 2 ] \Big ) , \end{align*}


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


\begin{align*} \Vert M _ { \widetilde { H } \vert _ { \mathbb { Z } _ N \times \mathbb { Z } _ N } } : S _ { p _ 1 } ^ { 2 N + 1 } \times S _ { p _ 2 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert \leq A _ { p _ 1 , p _ 2 , 2 N + 1 } . \end{align*}


\begin{align*} \Omega _ 0 \cap \ ( - \mathbb { R } _ + ^ { m + 1 } ) = \emptyset . \end{align*}


\begin{align*} h = 0 , \nabla h = 0 \mbox { o n } \hat \Sigma \end{align*}


\begin{align*} a ( u , v ) = ( f , v ) v \in H _ 0 ^ 2 ( D ) . \end{align*}


\begin{align*} \Delta _ H w = | \nabla _ { H } \rho | ^ 2 \left \{ f '' ( \rho ) + \frac { Q - 1 } { \rho } f ' ( \rho ) \right \} , \ \ \ \ \ \ \ ; \mathbb { G } \setminus \{ e \} . \end{align*}


\begin{align*} \int _ M \langle d ^ \nabla \sigma , \rho \rangle d v _ g = \int _ M \langle \sigma , \delta ^ \nabla \rho \rangle d v _ g \ , , \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { g _ u ^ { \ast 2 } \ast f _ r ( x ) } { g _ u ( x ) } = 2 . \end{align*}


\begin{align*} C _ { \mathrm { o b s } } = \frac { C _ 1 } { T ^ { 1 / r } } \exp \left ( \frac { C _ 2 } { T ^ { \frac { \gamma _ 1 \gamma _ 3 } { \gamma _ 2 - \gamma _ 1 } } } + C _ 3 T \right ) , \end{align*}


\begin{align*} \gamma ^ { q ^ n } = \gamma ^ { q ^ { n - i } } . \end{align*}


\begin{align*} ( \hat h , \hat v ) : = \big ( h + L _ V \bar g , v + V ( \bar u ) \big ) \end{align*}


\begin{align*} \Gamma ( Y ) & = - \Gamma ( Z ^ { ( i ) } ) \mbox { i n } M \setminus \Omega \\ Y & = - Z ^ { ( i ) } \mbox { o n } \Sigma . \end{align*}


\begin{align*} P ( X _ a = c X _ b ) = \int \int & P ( X _ a = c t | \sum _ { i = 1 } ^ { a - 1 } X _ i = s X _ b = t ) \\ & \quad \times ( \sum _ { i = 1 } ^ { a - 1 } X _ i X _ b s t ) d s d t . \end{align*}


\begin{align*} i _ { \chi \theta } ( f ) ( F _ \bullet ) = \sum _ { F ' _ \bullet \in \partial _ { \chi \theta } ^ { - 1 } ( F _ \bullet ) } f ( F ' _ \bullet ) . \end{align*}


\begin{align*} d v ' = \left \{ ( \log \sqrt { \tau _ 0 } ) _ z \ d z + \frac { i } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) J \right \} v ' . \end{align*}


\begin{align*} \Phi ( B _ \phi v , p ) & = Q F ^ q ( B _ \phi v + \psi ( B _ \phi v , p ) , p ) \\ & = Q F ^ q ( B _ \phi v + B _ \phi \psi ( v , p ) , p ) \\ & = Q F ^ q ( B _ \phi [ v + \psi ( v , p ) ] , p ) \\ & = Q B _ \phi F ^ q ( v + \psi ( v , p ) , p ) \\ & = B _ \phi Q F ^ q ( v + \psi ( v , p ) , p ) \\ & = B _ \phi \Phi ( v , p ) , \end{align*}


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


\begin{align*} ( u , v ) = \langle u , v \rangle , \ \ \ \forall \ , u \in H , \ \ \forall \ , v \in V . \end{align*}


\begin{align*} A = \frac { N ^ { \frac 1 d } } { ( \log N ) ^ { \frac 1 { 1 6 d } } } , R = 1 , T = \frac 1 { N ( \log N ) ^ { \frac 1 { 1 6 } } } . \end{align*}


\begin{align*} \phi ( g H ) : = \pi _ N ( g ) \psi ( \pi _ N ( g ) ) , \forall g \in G \end{align*}


\begin{align*} F _ { \delta , Z } = \{ u \in L ^ p ( \Omega ) \mid \int _ { \Omega } \frac { \partial V _ { \delta , Z , j } } { \partial z _ { j , h } } u = 0 , \ \ \forall j = 1 , \cdots , m , \ h = 1 , 2 \} , \end{align*}


\begin{align*} F ( m ; z ) \ | \ T ( l ) = \sum _ { n = 0 } ^ \infty \left ( b _ 3 \left ( \frac { m l n - 1 } { 1 2 } \right ) + \left ( \frac { 3 } { l } \right ) l ^ { 3 m - 4 } b _ 3 \left ( \frac { m n - l } { 1 2 l } \right ) \right ) q ^ n \equiv 0 \ ( \mathrm { m o d } \ m ) . \end{align*}


\begin{align*} M _ + ( x ; z ) = M _ - ( x ; z ) ( I + R ( x ; z ) ) , z \in \mathbb { R } , \end{align*}


\begin{align*} \Phi ( t , y ) = H ( t / 4 , x , y ) . \end{align*}


\begin{align*} s _ \alpha \bullet _ 2 \lambda & = s _ \alpha ( \lambda + 2 \rho ) - 2 \rho \\ & = s _ \alpha ( \lambda ) + 2 \rho - 2 \alpha - 2 \rho \\ & = s _ \alpha ( \lambda ) - 2 \alpha \end{align*}


\begin{align*} \widetilde { \mu } ( n ) \ r _ n ( m ) = H _ { m } \ \widetilde { r } _ { n } ( m + 1 ) + I _ m \ \widetilde { r } _ { n } ( m ) + J _ m \ \widetilde { r } _ { n } ( m - 1 ) \ , , \end{align*}


\begin{align*} \nabla _ Y \mathcal { P } ( X ) = \mathcal { P } ( B ( Y , X ) ) - B ( Y , f ( X ) ) + B ^ * ( Y , h ( X ) ) . \end{align*}


\begin{align*} I ' ( u , v ) = \mu J ' ( u , v ) . \end{align*}


\begin{align*} \mathcal { I } & = \bigg \{ E \in \mathbb { C } : \log | \lambda | > \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | \bigg \} , \\ \mathcal { O } & = \bigg \{ E \in \mathbb { C } : \log | \lambda | < \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | \bigg \} . \end{align*}


\begin{align*} V _ { \alpha , \beta } = \mathbb { C } [ \partial ] v , L _ \lambda v = ( \partial + \alpha \lambda + \beta ) v . \end{align*}


\begin{align*} | | \mathbf { v } ( T , \mathbf { v } ^ 0 ) | | _ { \mathcal { E } } \leq C | | \mathbf { v } ^ 0 | | _ { \mathcal { E } } \leq C \gamma = \gamma _ 1 . \end{align*}


\begin{align*} \partial _ i \varphi _ i = - 2 \sum _ { j \neq i } \Gamma _ { i j } ^ j \varphi _ j . \end{align*}


\begin{align*} M ( A _ 1 , \dots , A _ m ) = \binom { n } { m } \det ( A _ 1 , \dots , A _ m , I , \dots , I ) . \end{align*}


\begin{align*} L _ { q } \left [ \exp \left ( - \alpha t ^ { 2 } \right ) \right ] = \displaystyle \int _ { 0 } ^ { \infty } d t \ ; \exp _ { q } \left ( - s t \right ) \exp \left ( - \alpha t ^ { 2 } \right ) . \end{align*}


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


\begin{align*} T ( x , y , w , z ) = ( x + \alpha , y + x , w + \alpha , z + w ) \end{align*}


\begin{align*} N = n + \frac { n ( n - 1 ) } { 2 } , \end{align*}


\begin{align*} w ' \varepsilon ^ { \mu ' } = w \varepsilon ^ \mu s _ { v \alpha } \varepsilon ^ { \Phi ^ + ( - v \alpha ) v \alpha ^ \vee } = w s _ { v \alpha } \varepsilon ^ { s _ { v \alpha } ( \mu ) + \Phi ^ + ( - v \alpha ) v \alpha ^ \vee } . \end{align*}


\begin{align*} \mathbf { x } _ { l } = \sum _ { i = 1 } ^ { K } \mathbf { D } _ { i l } \mathbf { w } _ { i l } q _ { i } , \end{align*}


\begin{align*} M = \left ( \begin{array} { c c } \varepsilon _ 1 \mu & \varepsilon _ 1 \nu \\ \varepsilon _ 3 \mu & \varepsilon _ { 3 } \nu \end{array} \right ) \end{align*}


\begin{align*} \epsilon ( x _ a , t ) : = \int _ 0 ^ t h ( x _ a + \delta u ( \tau ) ) u ( \tau ) - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \ , d \tau \end{align*}


\begin{align*} A _ t + B \times u + \nabla Q = - \mu \nabla \times B . \end{align*}


\begin{align*} a s _ { A } ( x , y , z ) : = \alpha ( x ) \cdot ( y \cdot z ) - ( x \cdot y ) \cdot \alpha ( z ) = 0 . & & \mbox { ( H o m - a s s o c i a t i v i t y ) } \end{align*}


\begin{align*} a \circ \widehat { \varphi } = f \circ \rho \end{align*}


\begin{align*} \begin{aligned} u _ t + u \cdot \nabla u + \nabla p & = B \cdot \nabla B + \nu \Delta u , \\ B _ t + u \cdot \nabla B & = B \cdot \nabla u + \mu \Delta B , \\ \nabla \cdot u = \nabla \cdot B & = 0 , \end{aligned} \end{align*}


\begin{align*} w ( \alpha + w ^ { - 1 } \gamma ) = \gamma + w \alpha \in \Phi . \end{align*}


\begin{align*} Q _ b ( \omega _ b ) = e ^ { - \big ( \Phi _ b ( \omega _ b ) + | \partial x | ^ { - 1 } \Phi _ { \{ x \} } ( \omega ( x ) ) + | \partial y | ^ { - 1 } \Phi _ { \{ y \} } ( \omega ( y ) ) \big ) } , \end{align*}


\begin{align*} M _ \beta : = \max _ { \eta > 0 } \bigg | \displaystyle \int _ 0 ^ \eta e ^ { - i u } u ^ { - \beta } d u \bigg | ^ 2 ; \end{align*}


\begin{align*} \left \{ \begin{array} { c l } \partial _ t u - D ^ { \alpha } _ x \partial _ { x } u - \partial _ { x } ^ { - 1 } \partial _ y ^ 2 u & = u \partial _ x u , ( t , x , y ) \in \R \times \R \times \R , \\ u ( 0 ) & = u _ 0 \in H ^ { s _ 1 , s _ 2 } ( \R ^ 2 ) , \end{array} \right . \end{align*}


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


\begin{align*} \varrho - \mathcal { P } ^ { - } ( \varrho J ) = I , \end{align*}


\begin{align*} \tilde u ( x , t ) : = [ G _ k ( \cdot , t ) * u _ 0 ( | \cdot | ) ] ( x ) + \int _ 0 ^ t [ G _ k ( \cdot , t - s ) * F ( | \cdot | , s ) ] ( x ) \ , { \rm d } s \end{align*}


\begin{align*} \begin{aligned} \frac { d } { d s } \mathcal { Y M } _ { e , g _ s } ^ 0 ( { \nabla } ) _ { \big { | } _ { s = 0 } } & = \frac 1 2 \int _ M \langle S _ { e , \mathcal { Y M } ^ 0 } , \delta g \rangle d v _ g \\ \end{aligned} \end{align*}


\begin{align*} f ( a , s ) = \int _ 0 ^ L \frac { s } { ( a - x ) ^ 2 + ( s ) ^ 2 } d x = \arctan ( \frac { L - a } { s } ) - \arctan ( \frac { - a } { s } ) \le \pi , \end{align*}


\begin{align*} w _ i ( \alpha , \beta ) = \frac { ( - 1 ) ^ i \ , \binom { \beta - 1 } { i } } { \mathrm { B } ( \alpha , \beta ) } w _ { i , j , r } ( \alpha , \beta ) = \frac { ( - 1 ) ^ { i + j + r } \ , \binom { \alpha + i - 1 } { j } \ , \binom { \beta - 1 } { i } \ , \binom { j } { r } } { \mathrm { B } ( \alpha , \beta ) } . \end{align*}


\begin{align*} H \left ( P ^ { ( n , k ) } [ t ] \mid \mu ^ { \otimes k } [ t ] \right ) \leq \frac { k } { n } M , k = 1 , \cdots , n , t \in [ 0 , T _ M ] . \end{align*}


\begin{align*} \big | T _ { W _ \ell , \phi } ( \mu ) - T _ { W , \phi } ( \mu ) \big | \le \sum _ { i = 1 } ^ L | c _ i | d _ \square ( W _ \ell , W ) . \end{align*}


\begin{align*} d ^ a _ 1 ( x , y ) & = x - \eta ^ a y \\ d ^ a _ 0 ( x , y ) & = \prod _ { i \neq a } ( x - \eta ^ i y ) \end{align*}


\begin{align*} \begin{aligned} \limsup _ { k \to \infty } \Bigg \{ & \left | | | b _ { n _ k } | | _ { L ^ { 2 } ( B ( z _ k e _ z , R _ 0 ) ) } ^ { 2 } - \alpha \right | + \left | | | b _ { n _ k } | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } \backslash B ( z _ k e _ z , R _ k ) ) } ^ { 2 } - ( l - \alpha ) \right | \\ & + | | b _ { n _ k } | | _ { L ^ { 2 } ( B ( z _ k e _ z , R _ k ) \backslash B ( z _ k e _ z , R _ 0 ) ) } ^ { 2 } \Bigg \} \leq \varepsilon . \end{aligned} \end{align*}


\begin{align*} \omega = \frac { 1 } { 2 } \left ( C \pi _ { 1 } ^ { \ast } \omega ^ { 1 } + \frac { 1 } { C } \ , \pi _ { 2 } ^ { \ast } \omega ^ { 2 } \right ) \end{align*}


\begin{align*} \left < K _ \ell , x _ { i j } \right > = q ^ { - \delta _ { \ell , i } } \delta _ { i , j } \left < E _ \ell , x _ { i j } \right > = \delta _ { \ell , i } \delta _ { j , i + 1 } \left < F _ \ell , x _ { i j } \right > = \delta _ { \ell , j } \delta _ { i , j + 1 } \end{align*}


\begin{align*} { \rm V o l } ( { \tilde \Sigma } ^ { 3 r } ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( B _ { p _ i } ( r ) ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) , \end{align*}


\begin{align*} \begin{aligned} | H ( p ( s , k ) ) | & = \exp ( k \eta ( k ) - 2 | s _ 1 | - 2 | s _ 2 | ) \\ & \geq \exp \ ! \left ( \frac 1 3 k \eta ( k ) \right ) \geq \exp \ ! \left ( \frac 1 5 | p ( s , k ) | \right ) . \end{aligned} \end{align*}


\begin{align*} ( \partial \alpha ) ( u , v , w ) = \sum _ { \mathfrak { S } } \langle \alpha _ u v , w \rangle u , v , w \in E , \end{align*}


\begin{align*} \tilde { P } _ { 2 s q _ { n - n _ 0 } - 1 } ( \theta ) = \sum _ { x _ 1 \in I _ 1 \cup I _ 2 } \tilde { P } _ { 2 s q _ { n - n _ 0 } - 1 } ( \theta _ { x _ 1 } ) \prod _ { \substack { j \in I _ 1 \cup I _ 2 \\ j \neq x _ 1 } } \frac { \sin \pi ( \theta - \theta _ j ) } { \sin \pi ( \theta _ { x _ 1 } - \theta _ j ) } . \end{align*}


\begin{align*} { } \hat { T } _ N : = \arg \min _ { T \in \mathcal { T } } M _ N ( T ) , M _ N ( T ) : = \frac { 1 } { 2 N } \sum _ { i = 1 } ^ N d ^ 2 _ { \mathcal { W } } ( T \# \mu _ i , \nu _ i ) . \end{align*}


\begin{align*} B ( 0 , R ) \subset \overline { B } ( 0 , R _ 1 ) = f ^ k \ ! \left ( Y _ 1 \right ) \subset f ^ k \ ! \left ( B ( x _ 1 , 2 r ) \right ) . \end{align*}


\begin{align*} \frac { \partial } { \partial x _ j } F ( u ( x ) ) & = g ' ( | u ( x ) | ) \frac { \partial | u ( x ) | } { \partial x _ j } \\ & = \frac { g ' ( | u ( x ) | ) } { 2 | u ( x ) | } \left ( u ( x ) \frac { \partial \overline { u } ( x ) } { \partial x _ j } + \frac { \partial u ( x ) } { \partial x _ j } \overline { u } ( x ) \right ) \\ & = { \rm R e } \left ( f ( u ( x ) ) \frac { \partial \overline { u } ( x ) } { \partial x _ j } \right ) . \end{align*}


\begin{align*} \begin{aligned} | \langle \boldsymbol { \nu } , G ( \lambda | \bar U , x , \tau ) \rangle | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; , \end{aligned} \end{align*}


\begin{align*} h ( x + \delta u ) = h ( x ) + R ( x , \delta u ) \end{align*}


\begin{align*} \lim _ { \delta \rightarrow 0 } \sup _ { n \in \mathbb { N } } \mathbb { E } | \langle Y _ n ^ M ( \zeta _ n + \delta ) - Y _ n ^ M ( \zeta _ n ) , e \rangle | ^ { \alpha } = 0 . \end{align*}


\begin{align*} \P _ x ( S ( 1 ) \in d y , \tau > 1 ) = \prod _ { i = 1 } ^ d \lambda _ i e ^ { - \sum _ { i = 1 } ^ d \lambda _ i ( y _ i - x _ i ) } \det ( q _ { 1 + i - j } ( y _ j - x _ i ) ) _ { i , j = 1 } ^ d \end{align*}


\begin{align*} \sum \limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \ ( x _ 1 \otimes \cdots \otimes x _ d ) & = \sum \limits _ { j = 1 } ^ d \| X _ { j } \| ^ 2 ( x _ 1 \otimes \cdots \otimes x _ d ) \\ & = \left \| \sum \limits _ { j = 1 } ^ d A _ { j } ^ * A _ { j } \right \| ( x _ 1 \otimes \cdots \otimes x _ d ) . \end{align*}


\begin{align*} \begin{cases} - \varepsilon ^ 2 ( K ( x ) \nabla u ) = ( u - q | \ln \varepsilon | ) ^ { p } _ + , \ \ & x \in \Omega , \\ u = 0 , \ \ & x \in \partial \Omega , \end{cases} \end{align*}


\begin{align*} \liminf _ { N \to \infty } \frac { 1 } { \# ( \Lambda _ N ^ { \vec { v } } ( b ) ) } \sum _ { ( m , n ) \in \Lambda _ N ^ { \vec { v } } ( b ) } \max _ { 1 \leq i < j \leq k } d ( T ^ { ( m , n ) } x _ i , T ^ { ( m , n ) } x _ j ) = 0 \end{align*}


\begin{align*} F ( \theta _ { ( \ell + 1 ) q _ n + m _ n } ) = \left ( \begin{matrix} \cos ( \pi \theta _ { ( \ell + 1 ) q _ n + m _ n } ) E - \lambda \sin ( \pi \theta _ { ( \ell + 1 ) q _ n + m _ n } ) \ \ & - \cos ( \pi \theta _ { ( \ell + 1 ) q _ n + m _ n } ) \\ \cos ( \pi \theta _ { ( \ell + 1 ) q _ n + m _ n } ) & 0 \end{matrix} \right ) \end{align*}


\begin{align*} \begin{aligned} & \mathbf { T } _ x \mathbf { X } ^ { - 1 } = \mathbf { X } ^ { - 1 } \mathbf { T } _ x = \left ( \mathbf { I } - \mathbf { U } _ { x ^ { ( i ) } } / \mathbf { x } ^ { ( i ) } _ 1 : i = 1 , \cdots , k \right ) , \\ & \mathbf { T } _ x \mathbf { X } ^ { - 1 } \mathbf { e } = \mathbf { e } ; \end{aligned} \end{align*}


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


\begin{align*} c _ \varepsilon \leq \inf \limits _ { w \in \mathcal { H } ( \Omega ) , w _ + \neq 0 } \max \limits _ { t \geq 0 } I _ \varepsilon ( t w ) = \inf \limits _ { w \in \mathcal { N } _ \varepsilon } I _ \varepsilon ( w ) . \end{align*}


\begin{align*} S _ k ^ { i j } H _ { i j } = - \sum _ i S _ k ^ { p q , r s } u _ { p q i } u _ { r s i } + B ( n - k + 1 ) S _ { k - 1 } + \Delta f . \end{align*}


\begin{align*} | | \zeta | | G \kappa \stackrel { d e f } { = } \sup _ { p \in U } \left \{ \ \frac { | \zeta | _ p } { \kappa ( p ) } \ \right \} . \end{align*}


\begin{align*} 0 = \partial ^ 2 _ { x _ i x _ j } \phi - \partial ^ 2 _ { x _ i x _ j } \psi - \partial ^ 2 _ { x _ i y } \psi \partial _ { x _ j } T - \partial ^ 2 _ { x _ j y } \psi \partial _ { x _ i } T - \partial ^ 2 _ y \psi \partial _ { x _ j } T \partial _ { x _ i } T - \partial _ y \psi \partial ^ 2 _ { x _ j x _ i } T \\ + { ( \partial _ y T ) ^ { - 2 } } ( \partial _ y T \partial ^ 3 _ { x _ j x _ i y } T - \partial ^ 2 _ { x _ i y } T \partial ^ 2 _ { x _ j y } T ) \end{align*}


\begin{align*} d f _ t = \left ( \partial _ x f _ t + \alpha ( \sigma _ s ) \right ) d s + \sigma _ s d W _ s , \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { \ln \frac { ( A _ \varepsilon , \partial \Omega ) } { ( A _ \varepsilon ) } } { \ln \frac { 1 } { \varepsilon } } = 1 . \end{align*}


\begin{align*} \begin{aligned} | \gamma ^ { \epsilon } ( s ) | ^ { 2 } & \leq M ^ 2 \left ( \rho ^ { 2 } + | Y ^ { \epsilon } ( s - ) | ^ { 2 } \right ) \\ & \leq \frac { M ^ 2 } { \big ( 1 - \epsilon \lambda ( \epsilon ) M \big ) ^ { 2 } } \cdot \left ( \rho ^ { 2 } + | \tilde { Y } ^ { \epsilon } ( s ) | ^ { 2 } \right ) \\ & \leq 2 M ^ { 2 } \left ( \rho ^ { 2 } + | \tilde { Y } ^ { \epsilon } ( s ) | ^ { 2 } \right ) , \end{aligned} \end{align*}


\begin{align*} T ( x ) = \begin{cases} 1 - | x | / 2 , & 1 \leq | x | \leq 2 , \\ 1 / 2 , & | x | < 1 . \end{cases} \end{align*}


\begin{align*} & \Sigma _ 4 = \sum \limits _ { 1 \leq | m | \leq M } \frac { \Theta _ m } { 2 \pi i m } \ , , \\ & \Theta _ m = \sum \limits _ { D \leq d < 2 D } e \left ( \frac { \sqrt { X } m } { d } \right ) \sum \limits _ { n \in \mathcal { N } ( d ) } e \left ( - \frac { n m } { d } \right ) \ , , \\ & \Sigma _ 5 = \sum \limits _ { D \leq d < 2 D } \sum \limits _ { n \in \mathcal { N } ( d ) } f _ { M } \left ( \frac { \sqrt { X } - n } { d } \right ) \ , . \end{align*}


\begin{align*} \lim \limits _ { q \to \infty } \dfrac { k _ 4 ( P ^ \ast ( q ) ) } { q ^ 4 } = \frac { 1 } { 1 5 3 6 } . \end{align*}


\begin{align*} v ^ { - 1 } \frac 1 N \sum _ { k = 1 } ^ N ( \sigma \circ w ) ^ k ( \mu ) \in X _ \ast ( T ) _ { \Gamma _ 0 } \otimes \mathbb Q \end{align*}


\begin{align*} A ( 0 ) \nabla ( L u ) \cdot \nabla u = - f ' ( u ) | \nabla u | _ { A ( 0 ) } ^ 2 . \end{align*}


\begin{align*} U _ n ( X ) _ t = \sup _ { s \in [ 0 , t ] } \| \Delta _ n ^ { 1 - \frac m 2 } \sum _ { i = 1 } ^ { \lfloor s / \Delta _ n \rfloor - k + 1 } \bigotimes _ { j = 1 } ^ k \tilde { \Delta } _ { i + j - 1 } ^ n X ^ { \otimes m _ j } - \int _ 0 ^ t \rho _ { \Sigma _ s } ^ { \otimes k } ( m _ 1 , . . . , m _ k ) d s \| _ { \mathcal H ^ m } . \end{align*}


\begin{align*} \eta ^ { z , i } _ x : = \begin{cases} \eta _ x , & x \neq z , \\ \ i , & x = z , \end{cases} \end{align*}


\begin{align*} \dot y = A y '' + M y , y ( 0 , x ) = y _ 0 ( x ) , \end{align*}


\begin{align*} \lim _ { s \to \infty } \Phi _ s ( z ) = \begin{cases} 0 , & z < - 1 , \\ \frac { 1 } { 2 } ( z + 1 ) , & | z | < 1 , \\ 1 , & z > 1 . \end{cases} \end{align*}


\begin{align*} \log \frac { 1 } { x } - \gamma + \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { n \cdot n ! } x ^ { n } \ \ ( \gamma : \rm { E u l e r ' s \ c o n s t a n t } ) ; \end{align*}


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


\begin{align*} { P } _ k ( \mathcal { T } ) : = \left \{ q _ h \in H ^ 1 ( \Omega ) : q _ h | _ T \in P _ k ( T ) T \in \mathcal { T } \right \} , \end{align*}


\begin{align*} B ( s , | s | ^ \beta ) \cap B ( s ' , | s ' | ^ \beta ) = \emptyset \quad \ s , s ' \in S . \end{align*}


\begin{align*} ( X \wedge _ g ^ c Y ) Z = ( X \wedge _ g Y ) Z + ( J X \wedge _ g J Y ) Z - 2 g ( J X , Y ) J Z , \end{align*}


\begin{align*} \Lambda : = \{ n \in \mathbb { N } : g _ n ( t , \omega ) > 0 \} . \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } d _ { q } s \ , F _ { q } ( s ) = \int _ { 0 } ^ { \infty } \frac { 1 } { t } \exp _ { q } ( - s t ) f ( t ) d t = L _ { q } \left [ \frac { f ( t ) } { t } \right ] . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + f ( \cdot , y ) & = u , \ ; \ ; \Omega , \\ y & = 0 , \ ; \ ; \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} W ( 0 ^ \infty | [ 0 ^ k 1 ] ) = \sum _ { n \in \N } ( a _ { n + k } - a ) \ . \end{align*}


\begin{align*} W ( r ) = \frac { \xi } { r } f + \eta + \zeta \frac { r } { f } + \frac { \sigma } { f ^ 2 } , \xi \le 0 , \sigma > 0 , \end{align*}


\begin{align*} r _ { b , k } ( 1 + \alpha _ { b , k } ) R _ b ( j _ b ) = 0 . \end{align*}


\begin{align*} \sum _ { n < x ^ 2 } \frac { \Lambda _ x ( n ) ^ 2 } { n ^ { 2 \sigma } } & < \sum _ { n = 1 } ^ \infty \frac { \Lambda ( n ) \log n } { n ^ { 2 \sigma } } = \frac { d } { d s } \left [ \frac { \zeta ' } { \zeta } ( s ) \right ] _ { s = 2 \sigma } . \end{align*}


\begin{align*} \Psi ( t , h ( x ) ) = h \left ( \Phi ( t , x ) \right ) \end{align*}


\begin{align*} ( - R _ { k \ell j i } + R _ { \ell i k j } ) V _ j V _ \ell = ( R _ { k \ell i j } + R _ { \ell i k j } ) V _ j V _ \ell = - R _ { i k \ell j } V _ j V _ \ell = 0 \end{align*}


\begin{align*} \phi ( y ) = \sum _ { z \in \partial I ( y ) } G _ { I ( y ) } ( z , y ) \phi ( z ' ) . \end{align*}


\begin{align*} { g _ i } \left ( { { { \bf { x } } _ b } , t } \right ) = g _ i ^ { \left ( { e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) + g _ i ^ { \left ( { n e q } \right ) } \left ( { { { \bf { x } } _ b } , t } \right ) , \end{align*}


\begin{align*} P _ { n , k } ( z ) = \sum _ { i = 0 } ^ { k + d _ n } \alpha _ { n , k , i } \binom { z } { i } \end{align*}


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


\begin{align*} \begin{aligned} C _ m ( K ) = \inf \left \{ \int _ { \Omega } ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } : u \in \mathcal { U } ^ * ( K , \Omega ) \right \} \end{aligned} \end{align*}


\begin{align*} \mathbb { E } \exp ( \lambda | X | ) = \mathbb { E } \sum _ { p = 0 } ^ \infty \frac { 1 } { p ! } ( \lambda | X | ) ^ p = 1 + \sum _ { p = 1 } ^ \infty \frac { 1 } { p ! } \lambda ^ p \mathbb { E } | X | ^ p \end{align*}


\begin{align*} \delta : = ( n - 1 ) B _ 0 - 2 d _ e \| R ^ \nabla \| ^ 2 _ \infty \sqrt { A } \coth \sqrt { A } \geq 0 \ , , \end{align*}


\begin{align*} \mathcal C '^ \kappa = \varprojlim _ n ( \mathcal C ^ n ) ^ \kappa = \varprojlim _ n \varinjlim _ i ( \mathcal C _ i ^ n ) ^ \kappa = \varinjlim _ i \varprojlim _ n ( \mathcal C _ i ^ n ) ^ \kappa = \varinjlim _ i \mathcal C ^ \kappa = \mathcal C ^ \kappa , \end{align*}


\begin{align*} \langle f , g \rangle = f _ 1 g _ 4 - \frac { 1 } { 3 } f _ 2 g _ 3 + \frac { 1 } { 3 } f _ 3 g _ 2 - f _ 4 g _ 1 . \end{align*}


\begin{align*} r b = \nabla \phi \times r \nabla \theta + G r \nabla \theta . \end{align*}


\begin{align*} \tilde { \nabla } \xi = d \xi + \omega \xi . \end{align*}


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


\begin{align*} \zeta ' & \ = \sum _ { z \in \mathbb { F } ^ * _ { q ^ n } } \overline { U _ { q ^ n } } \Lambda ( z ) { \bf 1 } _ { - } - ( q ^ n - 1 ) \overline { U _ { q ^ n } } { \bf 1 } _ { - } \\ & \ = \sum _ { x , y \in \mathbb { F } ^ * _ { q ^ n } , x y \ne 1 } \varepsilon ( x ) \Lambda ( y ) { \bf 1 } _ { - } - ( q ^ n - 1 ) \overline { U _ { q ^ n } } { \bf 1 } _ { - } \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty b _ 5 \left ( \frac { m n - 1 } { 6 } \right ) q ^ n \in S _ { 2 m - 2 } ( \Gamma _ 0 ( 1 8 0 ) , \chi _ { 5 } ) _ m . \end{align*}


\begin{align*} 0 < \frac { \sum \limits _ { i = 1 } ^ j \chi _ i - r ( j - 1 ) } { \chi } < \frac { \sum \limits _ { i = 1 } ^ j \chi _ j - r j } { \chi } < 1 . \end{align*}


\begin{align*} - \theta \Delta t \mathcal { L } ( \tilde { u } ^ { [ n ] } ) + & \tilde { u } ^ { [ n ] } ( x ) = \\ & \tilde { u } ^ { [ n - 1 ] } ( x ) + \theta \Delta t f ( t _ n , x ) + ( 1 - \theta ) \Delta t \left [ \mathcal { L } ( \tilde { u } ^ { [ n - 1 ] } ) + f ( t _ { n - 1 } , x ) \right ] . \end{align*}


\begin{align*} \phi ( \tau _ a , p ) : = \gamma ( \tau _ a , p ) \gamma ^ { - 1 } = ( \tau _ { \lambda \iota a } , \gamma _ N p \gamma _ N ^ { - 1 } ) . \end{align*}


\begin{align*} h ( g \xi , g \eta , g \omega ) - h ( \xi , \eta , \omega ) = 2 \bar \beta ( g , \xi ) . \end{align*}


\begin{align*} { \bf { \bar U } } ^ { \star } = { \left [ { { \bf { \tilde \Sigma } } { { \bf { \tilde B } } ^ H } } \right ] _ { 1 : { M _ t } , : } } , \end{align*}


\begin{align*} \left \{ \sum _ { i = 1 } ^ r d _ i p ^ { k _ i n _ i } \colon n _ i \in \N _ 0 \right \} , \end{align*}


\begin{align*} u = X \cdot \mathbf { e } , \nu ^ { n + 1 } = \mathbf { e } \cdot \nu , \end{align*}


\begin{align*} H _ { / f } ^ 1 ( G _ L , \sigma ) : = H ^ 1 ( G _ L , \sigma ) / H _ { f } ^ 1 ( G _ L , \sigma ) . \end{align*}


\begin{align*} f _ 2 ( x ) = ( e ^ { \lambda \Lambda _ 2 } - 1 ) ^ { - 1 } \sum _ { n = 1 } ^ \infty \big ( ( \lambda \Lambda _ 2 ) ^ n / n ! \big ) \ , g _ 2 ^ { \ast n } ( x ) , \end{align*}


\begin{align*} V _ { E _ k } ^ \# ( D ( p - 1 ) ) \subseteq V ( p _ k ) \cup \bigcup _ { i = 1 } ^ k V ( x _ i ) . \end{align*}


\begin{align*} \mathcal A _ { \mathcal X } ^ { \mathcal D } = \left \{ ( s , \alpha ) | s \in S , \alpha \in A _ { X _ s } ^ { D _ s } \right \} \end{align*}


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


\begin{align*} R ^ i _ { n _ 0 ; l _ 1 , \ldots , l _ { n _ 0 } } = \underset { n _ 0 \geq j \geq i } { \bigsqcup } Q ^ j _ { n _ 0 ; l _ 1 , \ldots , l _ { n _ 0 } } . \end{align*}


\begin{align*} \gamma _ { \Theta } ^ s ( E ) = \sup \{ | \langle \nu , 1 \rangle | \} , \end{align*}


\begin{align*} A ' _ n ( j ) = A _ n ( j _ V ) \end{align*}


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


\begin{align*} a _ I : = \sum _ { w \in W _ I } w \textrm { a n d } b _ I : = \sum _ { w \in W _ I } \epsilon ( w ) \ , w \ , , \end{align*}


\begin{align*} \dim K = | \sigma ^ 1 | - 1 > | \tau ^ 1 | - 1 = \dim L , \end{align*}


\begin{align*} F ( x ' , t ) : = \Phi _ t * f ( x ' ) \quad H ( x ' , t ) : = \widetilde { \Phi } _ t * h ( x ' ) , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} \left \{ \begin{array} { c l } \partial _ t u - D ^ { 4 } _ x \partial _ { x } u - \partial _ { x } ^ { - 1 } \partial _ y ^ 2 u & = u \partial _ x u , ( t , x , y ) \in \R \times \R \times \R , \\ u ( 0 ) & = u _ 0 \in H ^ { s _ 1 , s _ 2 } ( \R ^ 2 ) \end{array} \right . \end{align*}


\begin{align*} P ^ \infty & = \varprojlim P ^ r = \left \{ u = ( u ^ r ) _ { r \in \mathbb { N } } \in \prod _ { r \in \mathbb { N } } P ^ r \ ; \middle | \ ; \pi ^ r _ s u ^ r = u ^ s \ \right \} , \\ * \widetilde { P } ^ \infty & = \varprojlim \widetilde { P } ^ r , \\ * G ^ \infty & = P ^ \infty ( \mathbb { R } ^ n , o ) , \\ * \widetilde { G } ^ \infty & = \widetilde { P } ^ \infty ( \mathbb { R } ^ n , o ) , \end{align*}


\begin{align*} \min \{ c _ { d , 2 } ( q ) , c _ { d , \infty } ( q ) \} = \begin{cases} c _ { d , 2 } ( q ) , & - ( d - 1 ) < q \leq q _ d ^ * , \\ c _ { d , \infty } ( q ) , & q _ d ^ * \leq q \leq 2 , \end{cases} \end{align*}


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


\begin{align*} \int _ K | | \tau _ 0 | | _ { \omega _ { \epsilon _ 1 } } \le & \liminf _ { \epsilon \to 0 } \int _ K | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon _ 1 } } \\ \le & \liminf _ { \epsilon \to 0 } \int _ K | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon } } \\ \le & \liminf _ { \epsilon \to 0 } \int _ M | | \tau _ { \epsilon } | | _ { \omega _ { \epsilon } } . \end{align*}


\begin{align*} { \bf X } ' _ \ell ( t ) = e ^ { - A ( t + \ell \theta ) } { \bf X } ' + \int _ { - \ell \theta } ^ t e ^ { - A ( t - s ) } \Sigma \Big ( { \bf p } _ \ell ( s - ) \Big ) \dd M ( s ) , t \ge - \ell \theta . \end{align*}


\begin{align*} \int ^ \infty _ 0 x ^ { - 3 \alpha } \exp \left \{ - ( a \alpha + n ) \frac { \theta } { x ^ 2 } \right \} \mathrm { d } x = & \int ^ \infty _ 0 u ^ { \frac { 3 ( \alpha - 1 ) } { 2 } } \exp \left \{ - ( a \alpha + j ) u \right \} \mathrm { d } u \\ = & \left ( \frac { 1 } { a \alpha + n } \right ) ^ { \frac { 3 \alpha - 1 } { 2 } } \Gamma \left ( \frac { 3 \alpha - 1 } { 2 } \right ) . \end{align*}


\begin{align*} & < u _ t , \xi > + \int _ { \Omega } ( u \cdot \nabla u - B \cdot \nabla B ) \cdot \xi \dd x + \nu \int _ { \Omega } \nabla u : \nabla \xi \dd x = 0 , \\ & < B _ t , \zeta > + \int _ { \Omega } ( B \times u ) \cdot \nabla \times \zeta \dd x + \mu \int _ { \Omega } \nabla \times B \cdot \nabla \times \zeta \dd x = 0 , \end{align*}


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


\begin{align*} R _ c = \pi \sqrt { 1 6 . 6 7 / 0 . 6 5 } \approx 1 5 . 9 \ , . \end{align*}


\begin{align*} \mathbf { a } ( \theta _ { 0 } ) ^ { T } \mathbf { v } ^ { 0 } = e ^ { - j m \phi } + e ^ { - j n \phi } = e ^ { - j m \phi } ( 1 + e ^ { - j ( n - m ) \phi } ) \end{align*}


\begin{align*} F ^ { \pm } ( x ) = \sum _ { k = 1 } ^ { N } \alpha _ { k } ^ { \pm } e ^ { \mp \tau _ { k } x } + \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } s ^ { \pm } \left ( \rho \right ) e ^ { \pm i \rho x } d \rho . \end{align*}


\begin{align*} \ell _ \beta ( \langle \gamma \rangle ) = \beta ( \gamma , \gamma ^ + ) \end{align*}


\begin{align*} d _ 1 = \begin{pmatrix} 0 & i \mu ( g ^ { \alpha \beta } \xi _ \beta ) & 0 \\ i \lambda ( \xi _ \beta ) & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} , d _ 0 = \begin{pmatrix} 0 & 0 & 0 \\ \lambda ( \Gamma ^ \alpha _ { \alpha \beta } ) & \lambda \Gamma ^ \alpha _ { \alpha n } & - \beta \\ 0 & 0 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \overline { \mathbf { p } } ^ { \oplus } = ( \overline { T _ { \mathbf { p } } } ^ { \oplus } ) ^ { \tau } T _ { \mathbf { p } } = \bigcup _ { \alpha ^ { \tau } \in \mathbf { p } } \{ \gamma \in \alpha ^ { \tau } \} \subset R ^ - _ { { \tt A } _ { 2 n - 1 } } . \end{align*}


\begin{align*} \rho ( t ) = \begin{cases} 0 , & t = 0 , \\ \exp ( - 1 / t ) , & 0 < t < T , \end{cases} \end{align*}


\begin{align*} \pi _ N ( m \ell n ) = m C _ \ell ( n ) . \end{align*}


\begin{align*} \gamma ^ i ( x ) = \frac 1 N \sum _ { j = 1 } ^ N K ( x ^ i - x ^ j ) \end{align*}


\begin{align*} 0 = H _ i = ( \Delta u ) _ i + B x _ i \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ g \textbf { \textit { U } } = 0 & \ \Omega , \\ \textbf { \textit { U } } = \textbf { \textit { V } } & \ \partial \Omega , \end{cases} \end{align*}


\begin{align*} I ( \widetilde { u } _ { k } , \widetilde { v } _ { k } ) = I ( \overline { u } _ { k } , \overline { v } _ { k } ) + o _ { k } ( 1 ) \geq \widetilde { E } _ { 1 } + \widetilde { E } _ { \omega } + o _ { k } ( 1 ) . \end{align*}


\begin{align*} 0 = \int _ \Omega \varphi _ n ( \eta ) \mu ( d \eta ) = \sum _ { m = 0 } ^ \infty \int _ \Omega ( \varphi _ n ( r _ m \eta ) - \varphi _ n ( r _ { m + 1 } \eta ) ) \mu ( d \eta ) + \varphi _ n ( \mathbf { 1 } ) , \end{align*}


\begin{align*} \Sigma ( k ) = \theta \big ( \chi ( 2 k + 1 ) + 1 + \lceil \ln ( 2 L ( k + 1 ) ) \rceil \big ) + 1 . \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { y ' } ^ H \mathbf { s } ^ { \left ( \right ) } \right ] = a _ c \sum _ { j = 1 } ^ { M } \left ( \mathbf { h } _ { l , * } \mathbf { p } _ c \right ) ^ { * } + \sum _ { l = 1 } ^ M a _ l \left ( \mathbf { h } _ { l , * } \mathbf { p } _ l \right ) ^ { * } . \end{align*}


\begin{align*} j ( \tau , s ( \tau ) ) = h . \end{align*}


\begin{align*} \phi = \alpha + \beta - \gamma \end{align*}


\begin{align*} d g = \mathcal { L } _ { g } = _ { } \mathcal { I } \circ L _ { g } \circ \mathcal { I } ^ { - 1 } : S O \left ( M _ { \kappa } \right ) \rightarrow S O \left ( M _ { \kappa } \right ) \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \Vert X _ n ( t ) - X ( t ) \Vert _ H ^ { \kappa } d t = 0 , \forall \ \kappa \in [ 1 , \alpha ) . \end{align*}


\begin{align*} \lambda _ { \min } ( ( \mathbf { v } ) ) = \mathbf { v } _ 1 - \| \mathbf { v } _ { 2 : n } \| , \lambda _ { \max } ( ( \mathbf { v } ) ) = \mathbf { v } _ 1 + \| \mathbf { v } _ { 2 : n } \| . \end{align*}


\begin{align*} G _ 1 ( Q _ 1 , \dots , Q _ n ) \coloneqq \int _ { \R ^ n } f ( x _ 1 , \dots , x _ n ) \prod _ { i = 1 } ^ { n } Q _ i ( d x _ i ) , G _ 2 ( Q _ 1 , \dots , Q _ n ) \coloneqq H ( Q _ 1 \times \cdots \times Q _ n ) . \end{align*}


\begin{align*} I _ h = \inf \left \{ E [ b ] \ \middle | \ b \in \tilde { T } _ h \right \} . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } A = f ( B ) ; \\ B = f ( A ) . \end{array} \right . \end{align*}


\begin{align*} \mathcal { C } ( h ) ( z ) : = \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } \frac { h ( s ) } { s - z } d s , z \in \mathbb { C } \backslash \mathbb { R } . \end{align*}


\begin{align*} s = \max \{ q : j _ { r } = \ell _ { r } r \in [ q ] \} . \end{align*}


\begin{align*} S t _ { D N P } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( 1 , z ) - 1 | | _ { \infty } = 0 , \end{align*}


\begin{align*} B _ q \circ T ( f ) = f . \end{align*}


\begin{align*} T ( 3 , a , x ) = & G _ { 2 } ^ 3 \ , { } _ 3 ^ 0 \left ( x \left | \begin{matrix} 0 , 0 \\ - 1 , a - 1 , - 1 \end{matrix} \right . \right ) \\ = & \frac { x ^ { a - 1 } { } _ 2 \mathrm { F } _ 2 \left ( a , a ; 1 + a , 1 + a ; - x \right ) } { a ^ 2 } - \frac { \Gamma ( a ) \log ( x ) } { x } + \frac { \Gamma ( a ) \psi ( a ) } { x } , \end{align*}


\begin{align*} V ( G [ H ; U ] ) & = V ( G ) \times V ( H ) , \\ E ( G [ H ; U ] ) & = \left \{ ( u _ 1 , v _ 1 ) ( u _ 2 , v _ 2 ) \ \middle | \ \left . \begin{aligned} & u _ 1 = u _ 2 v _ 1 v _ 2 \in E ( H ) , \\ & \\ & u _ 1 u _ 2 \in E ( G ) v _ 1 , v _ 2 \in U \end{aligned} \right . \right \} . \end{align*}


\begin{align*} \Gamma \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } G \right ) \left ( X , Y \right ) = Z G \left ( X , Y \right ) - G \left ( \nabla _ { Z } X , Y \right ) - G \left ( X , \nabla _ { Z } Y \right ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| p _ n x \| _ { X _ n } = \| x \| _ X , \ , x \in X \lim _ { n \to \infty } \| q _ n y \| _ { Y _ n } = \| y \| _ Y , \ , y \in Y . \end{align*}


\begin{align*} \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} = - \frac { 1 } { k ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 } \begin{pmatrix} x _ 1 x _ 2 & - k ^ 2 - x _ 1 ^ 2 \\ k ^ 2 + x _ 2 ^ 2 & - x _ 1 x _ 2 \end{pmatrix} \begin{pmatrix} \partial _ { x _ 1 } \varphi \\ \partial _ { x _ 2 } \varphi \end{pmatrix} . \end{align*}


\begin{align*} \mu ^ { N , i } _ { \boldsymbol { \alpha } ^ * _ t } = \rho _ i ^ N ( \Delta _ { * , t } ^ { N , 1 } , \ldots , \Delta _ { * , t } ^ { N , N } ) , \end{align*}


\begin{align*} \mathcal { E } ( t ) = \mathcal { P } ^ * ( u _ t ( t ) ) + \mathcal { A } ( u ( t ) ) , \end{align*}


\begin{align*} L K ^ L _ { \cdot \beta } = 0 \ , \ , \ , \ , \R ^ n _ { + } \beta \in \{ 1 , \dots , N \} , \end{align*}


\begin{align*} \varphi ( \tau , w ( - 0 ) , w ( \cdot ) ) = \hat { \varphi } ( \tau , w ( \cdot ) ) , ( \tau , w ( \cdot ) ) \in [ 0 , \vartheta ] \times \mathrm { L i p } . \end{align*}


\begin{align*} \lim _ { \substack { n \rightarrow \infty \\ n \in \Lambda } } X _ n ( t , \omega ) = X ( t , \omega ) , \end{align*}


\begin{align*} \tilde { m } _ n = \begin{cases} m _ n - q _ { n - 1 } m _ n + q _ n - q _ { n - 1 } , \ell _ n = 2 \\ m _ n + q _ { n - 1 } m _ n - q _ n + q _ { n - 1 } , \ell _ n = - 2 \end{cases} . \end{align*}


\begin{align*} R _ { i j k l } = - ( \delta _ { i k } \delta _ { j l } - \delta _ { i l } \delta _ { j k } ) + h _ { i k } h _ { j l } - h _ { i l } h _ { j k } , \end{align*}


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


\begin{align*} f _ k ( d ) = \lim _ { n \rightarrow \infty } \frac { \vartheta ( n , d , k ) } { \log n } , \end{align*}


\begin{align*} \phi _ { f ' } ( x ) = \alpha ( v x + w ) ^ { q + 1 } ( \phi _ f \circ \mu ( x ) ) , \end{align*}


\begin{align*} \int _ { 0 } ^ { T } \sum _ { j = 1 } ^ { \infty } V _ { i , j } \psi _ j ( t ) d t < \infty , \end{align*}


\begin{align*} K ( p , \tilde { \pi } ' ) = & K ( p , \pi ) + \varepsilon Q ( g , R ) ( \pi , \bar { \pi } ) + \varepsilon Q ( g , R ) ( \pi , J \bar { \pi } ) + O ( \varepsilon ^ 2 ) \\ = & K ( p , \pi ) + \varepsilon Q ^ c ( g , R ) ( \pi ; \bar { \pi } ) + O ( \varepsilon ^ 2 ) . \end{align*}


\begin{align*} \phi : \mathbb { R } ^ { 3 } \cong \operatorname { I m } \mathbb { H } \rightarrow \mathbb { R } ^ { 3 } \rtimes S ^ { 3 } \phi \left ( r u \right ) = \left ( \ell \left ( r \right ) u , \exp \left ( r u / 2 \right ) \right ) \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { k } \dbinom { k } { j } ^ 2 a ^ { p _ k ( k - j ) / k } b ^ { p _ k j / k } \leq ( a + b ) ^ { p _ k } , \end{align*}


\begin{align*} | C \widehat p _ t + D \widehat q _ t | ^ 2 = 0 , t \in [ 0 , T ] . \end{align*}


\begin{align*} \boldsymbol { \lambda } _ k = \left ( \langle y _ k | A _ { 1 , k } y _ k \rangle , \dots , \langle y _ k | A _ { d , k } y _ k \rangle \right ) , \end{align*}


\begin{align*} \hat M ^ { ( k ) } = G _ k G _ { k - 1 } \cdots G _ 1 ( M + \tilde E ^ { ( k ) } ) , \end{align*}


\begin{align*} \Delta [ M _ v ] ( t ) = ( \Delta M _ v ( t ) ) ^ 2 = e ^ { - 2 \lambda t } ( v A ^ * _ k ) ^ 2 = e ^ { - 2 \lambda t } \lambda ^ 2 v _ { k } ^ 2 , \end{align*}


\begin{align*} \lambda _ s = \sum _ { a = m - s + 1 } ^ { m } i _ a + \frac { L } { 2 } \medspace ( 1 \leq s \leq m ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n { ( 1 \otimes a _ i \otimes a _ i + a _ i \otimes 1 \otimes a _ i + a _ i \otimes a _ i \otimes 1 ) } + 1 \otimes 1 \otimes y + 1 \otimes y \otimes 1 + y \otimes 1 \otimes 1 . \end{align*}


\begin{align*} \iota ( \upsilon ) = \frac { - 1 } { 2 \pi } \sin ( 2 \pi \upsilon ) + \frac { - \sin ( 2 \pi \upsilon ) + 4 \pi \upsilon - \sin ( 4 \pi \upsilon ) + \frac { 1 } { 3 } \sin ( 6 \pi \upsilon ) } { 8 \left ( - \sin ( 2 \pi \upsilon ) + \frac 1 3 \sin ( 6 \pi \upsilon ) - ( 2 \pi \upsilon + \frac 1 2 \sin ( 4 \pi \upsilon ) - 2 \sin ( 2 \pi \upsilon ) ) \right ) } \left ( - 4 \upsilon + \frac { 1 } { \pi } \sin ( 4 \pi \upsilon ) \right ) \end{align*}


\begin{align*} \partial _ t h _ { i j } = g \Delta h _ { i j } + \sum _ { k , l } g ^ { k l } \nabla _ i g \nabla _ j h _ { k l } + \end{align*}


\begin{align*} \frac { 1 } { 2 } \left \{ 1 - F _ \Gamma \left [ \left ( - \frac { x - \mu } { \sigma } \right ) ^ s \right ] \right \} = u . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { 1 6 ^ { m } } \mathbb { E } [ \tilde { X } _ { j } 1 _ { \{ X _ { j } > \varepsilon \} } | \Phi ^ { S _ { n } , \tilde { S } _ { n , m } } ] \to 0 , m \to \infty \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } \tilde { X } _ t = B ( \tilde { X } _ t , \tilde { X } _ t ) \\ \tilde { X } _ t | _ { t = 0 } = \Pi _ { \mathrm { k e r } A } x _ 0 . \end{cases} \end{align*}


\begin{align*} S _ { h _ C , W , 0 } = \left \{ U _ { C } ( \cdot + z e _ z ) - B _ { \infty } \ \big | \ z \in \mathbb { R } \ \right \} . \end{align*}


\begin{align*} \partial _ { i } [ \tau ] = \sum _ { \hat { \imath } = 1 } ^ { i + 1 } ( - 1 ) ^ { \hat { \imath } - 1 } [ \tau \setminus v _ { j _ { \hat { \imath } } } ] . \end{align*}


\begin{align*} \rho ^ { k _ 1 } = ( a / b ) ^ { x k _ 1 } ( \alpha / \beta ) ^ { y k _ 1 } = ( \alpha / \beta ) ^ { x l _ 1 } \zeta ^ x ( \alpha / \beta ) ^ { y k _ 1 } = ( \alpha / \beta ) \zeta ^ x . \end{align*}


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


\begin{align*} x = a _ 2 ^ 2 + \dots + a _ n ^ 2 + a _ { n + 1 } ^ 2 \end{align*}


\begin{align*} 0 \equiv \hat { \phi } ( \hat { h } ) = \hat { h } ( u ^ 2 , \hat { g } ( u ^ 2 ) ) , \end{align*}


\begin{align*} ( 1 - e ^ { - 2 } ) f ( x ) = ( 1 - e ^ { - 1 } ) e ^ { - 1 } ( \underline { f } ( x ) + \overline { f } ( x ) ) + ( 1 - e ^ { - 1 } ) ^ 2 \underline { f } \ast \overline { f } ( x ) \end{align*}


\begin{align*} - \Delta _ { H } u = V u \ ; \ ; , \end{align*}


\begin{align*} \psi ( p ) = \psi _ { p _ 0 } [ \eta ] ( p ) \stackrel { d e f } { = } \inf _ { r \in R ( p ) } g ( p , r , | \eta | _ r ) , \ p > p _ 0 ; \end{align*}


\begin{align*} Z _ { V , [ \alpha ] } ( t ) = \frac { 1 } { 2 } \int _ { V } ( t ^ { 2 } \omega ^ { 2 } - \alpha ^ { 2 } ) + \sqrt { - 1 } t \int _ { V } \alpha \wedge \omega . \end{align*}


\begin{align*} \mathcal { T } ^ * ( f ) ( x ) = \sup _ { t > 0 } | e ^ { - t L } f ( x ) | = \sup _ { t > 0 } \Big | \int _ { \mathbb { R } ^ n } k _ t ( x , y ) f ( y ) d y \Big | , \end{align*}


\begin{gather*} P ( A ) = P _ 0 ( A ) = 0 \quad A \in \mathcal { R } P ^ * ( A ) = 0 . \end{gather*}


\begin{align*} u v = T _ u v + T _ v u + R ( u , v ) , \end{align*}


\begin{align*} \prod _ { j = x _ 1 } ^ k | \cos ( \pi \theta _ j ) | \leq C ( \varepsilon ) e ^ { - ( \ln 2 - \varepsilon ) | x _ 1 - k | } , \prod _ { j = k } ^ { x _ 2 } | \cos ( \pi \theta _ j ) | \leq C ( \varepsilon ) e ^ { - ( \ln 2 - \varepsilon ) | x _ 2 - k | } c _ { n , \ell } c _ { n , \ell + 1 } . \end{align*}


\begin{align*} V ( t , p ) = \partial _ t \theta ( t , z ) \cdot \nu ( t , p ) = \frac { R ' ( t ) } { R _ 0 } z \cdot \frac { p } { R ( t ) } = \frac { R ' ( t ) } { R _ 0 } z \cdot \frac { z } { R _ 0 } = R ' ( t ) . \end{align*}


\begin{gather*} P ( f ) = P ( \tilde { f } ) = \int Q ( \tilde { f } ) \ , P _ \mathcal { U } ( d Q ) = \int Q ( f ) \ , P _ \mathcal { U } ( d Q ) = 0 . \end{gather*}


\begin{align*} 0 = \frac { ( \tilde { \kappa } _ 1 ) _ i } { \tilde { \kappa } _ 1 } - N \frac { ( \nu ^ { n + 1 } ) _ i } { \nu ^ { n + 1 } } , \end{align*}


\begin{align*} \| p _ N \mathcal S ( r ) \sigma _ { s } \| _ { L _ { } ( U , H ) } ^ 2 = \| \sigma _ { s } ^ * \mathcal S ( r ) ^ * p _ N \| _ { L _ { } ( H , U ) } ^ 2 = \sum _ { k = N } ^ { \infty } \| \sigma _ { s } ^ * \mathcal S ( r ) ^ * e _ k \| ^ 2 \to 0 , N \to \infty , \end{align*}


\begin{align*} \mathcal { Q } ^ { r } ( t ) : = \mathcal { M } ^ { r } ( \delta ) + ( r - 1 ) \theta _ { 1 } \varrho _ { 0 } ^ { 1 - \sigma } \int _ { \delta } ^ { t } ( \mathcal { M } ^ { r } ( s ) ) ^ { \sigma + 1 } \ d s . \end{align*}


\begin{align*} A ( ( v _ { i , j } ) _ { 1 \leq i < j \leq 2 d } ) = \begin{pmatrix} M _ 1 ( ( v _ { i , j } ) _ { 1 \leq i < j \leq 2 d } ) \\ M _ 2 ( ( v _ { i , j } ) _ { 1 \leq i < j \leq 2 d } ) \\ \vdots \\ M _ { 2 d } ( ( v _ { i , j } ) _ { 1 \leq i < j \leq 2 d } ) \\ \end{pmatrix} . \end{align*}


\begin{align*} q ( x ) = - \frac { c } { 2 } \operatorname * { s e c h } { } ^ { 2 } \left ( \frac { \sqrt { c } x } { 2 } \right ) . \end{align*}


\begin{align*} L \cong M / N \cong \bigoplus _ { i = 1 } ^ l M _ i / N _ i \end{align*}


\begin{align*} Z ^ k _ j ( n ) = \inf \{ t \geq 0 : M _ { d - j + 1 } ^ k ( t ) \geq n \} , n \geq 0 . \end{align*}


\begin{align*} S t ( e _ { 1 } ) = \left \{ \begin{pmatrix} 1 & c & ^ { t } u \\ 0 & 1 & 0 \\ 0 & x & B \end{pmatrix} \Bigg | \ , \ , u + \ , ^ { t } B \psi x = 0 ; \ , \ , 2 c + \langle x , x \rangle = 0 ; \ , \ , B \in O _ { n - 1 , n - 1 } . \right \} . \end{align*}


\begin{align*} [ \mathbf { h } ' _ 0 , \mathbf { h } _ 0 ] = 1 . \end{align*}


\begin{align*} u ^ K _ { n , h } ( p ) + H ( p , u ^ K _ { n , h } ( p ) , \nabla _ H u ^ K _ { n , h } ( p ) ) = u ^ K _ { n - 1 } ( p ) \end{align*}


\begin{align*} l r ( e p o c h s ) = \begin{cases} 1 0 ^ { - 3 } , \textit { 1 $ \leq $ e p o c h s $ < $ 2 , 0 0 0 } \\ 1 0 ^ { - 4 } , \textit { e p o c h s $ \geq $ 2 , 0 0 0 } . \end{cases} \end{align*}


\begin{align*} H \left ( P ^ { ( n , k ) } [ t ] \mid \mu ^ { \otimes k } [ t ] \right ) \leq \frac { k } { n } M , k = 1 , \cdots , n , t \in [ 0 , T _ M ] . \end{align*}


\begin{align*} \hat M ^ { ( k ) } _ { i , j } & = \begin{cases} 0 , & \mbox { i f } i \in [ k + 1 , n ] \mbox { a n d } j = k , \\ { \rm f l } \Big ( \hat M ^ { ( k - 1 ) } _ { i , j } - { \rm f l } \Big ( { \rm f l } \Big ( \frac { \hat M ^ { ( k - 1 ) } _ { i , k } } { \hat M ^ { ( k - 1 ) } _ { k , k } } \Big ) \ , \hat M ^ { ( k - 1 ) } _ { k , j } \Big ) \Big ) , & \mbox { i f } i , j \in [ k + 1 , n ] , \\ \hat M ^ { ( k - 1 ) } _ { i , j } , & \mbox { o t h e r w i s e } \end{cases} \end{align*}


\begin{align*} \mathbf { y } ' = \begin{bmatrix} y _ c \\ y _ 1 \\ \vdots \\ y _ M \end{bmatrix} = \begin{bmatrix} \sum _ { i = 1 } ^ { M } y _ i \\ y _ 1 \\ \vdots \\ y _ M \end{bmatrix} , \end{align*}


\begin{align*} & \lim _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \int _ { \{ z \in M : - \Psi ( z ) \in I _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h \\ \le & \frac { G ( T _ 1 ) } { \int _ { T _ 1 } ^ { + \infty } c ( s ) e ^ { - s } d s } \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { 1 } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } c ( s ) e ^ { - s } d s . \end{align*}


\begin{align*} \begin{aligned} & \int _ { - \infty } ^ { \infty } \bar { u } _ y e ^ { - 2 i z y } \mathrm { d } y = \bar { u } ( y ) e ^ { - 2 i z y } | _ { - \infty } ^ { + \infty } + 2 i z \int _ { - \infty } ^ { \infty } \bar { u } e ^ { - 2 i z y } \mathrm { d } y = i z \widehat { \bar { u } } ( z ) . \end{aligned} \end{align*}


\begin{align*} M ( S ) = \varinjlim _ i M ( S _ i ) . \end{align*}


\begin{align*} a _ { E } ( n ) = \prod _ { p \leq \infty } \beta _ { p } ( n ) \end{align*}


\begin{align*} \nabla _ X \Psi _ 1 = - \frac { 1 } { 2 } \left ( X - \langle X , T _ 1 \rangle ( T _ 1 - f _ 1 ) - \langle X , T _ 2 \rangle ( T _ 2 - f _ 2 ) \right ) \cdot \Psi _ 2 - \frac { 1 } { 2 } S ( X ) \cdot \Psi _ 1 \end{align*}


\begin{align*} A _ n ( \ell ) a _ { \ell } = Q _ { n , 1 } ( \ell ) \beta _ { n , 1 } ^ { \ell } + \cdots + Q _ { n , s ( n ) } ( \ell ) \beta _ { n , s ( n ) } ^ { \ell } . \end{align*}


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


\begin{align*} \mathcal { Q } _ { | _ { x \times C } } \cong \bigoplus _ { i = 1 } ^ { n } j _ { i * } ( E _ i ) \mathcal { R } _ { | _ { x \times C } } \cong \bigoplus _ { i = 1 } ^ { n - 1 } j _ { p _ i * } ( j _ { p _ i } ^ * ( j _ { i + 1 * } ( E _ { i + 1 } ) ) ) . \end{align*}


\begin{align*} & \ \dot { s } \varepsilon ( z ) \dot { s } ( e - \varepsilon ( 1 ) ) \lambda _ i = \sum _ { j , \mu } g ^ i _ { j , \mu } \dot { s } ( \varepsilon ( z ) - \varepsilon ( x ^ i _ { j , \mu } + z ) ) \lambda _ j \\ & \ = \sum _ { j , \mu } g ^ i _ { j , \mu } \sum _ { k , \nu } g ^ j _ { k , \nu } ( \varepsilon ( z ^ { - 1 } x ^ j _ { k , \nu } ) - \varepsilon ( ( x ^ i _ { j , \mu } + z ) ^ { - 1 } x ^ j _ { k , \nu } ) ) \lambda _ k \end{align*}


\begin{align*} \ell ( x , \alpha ) : = \langle \mu , \alpha \rangle + \Phi ^ + ( \alpha ) - \Phi ^ + ( w \alpha ) . \end{align*}


\begin{align*} \sup _ { x \in \mathbb { R } , t > 0 } | \mathcal { I } ( x , t ) | \leq \bar { \tau } = \max \left ( \sum _ { i = 1 } ^ p \frac { 2 R } { \lambda _ i } , \sum _ { i = p + 1 } ^ n \frac { 2 R } { \lambda _ i } \right ) . \end{align*}


\begin{align*} { \langle { q _ \delta } , } { q _ \delta \rangle } = \langle u _ \delta , u _ \delta \rangle + \langle v _ \delta , v _ \delta \rangle = n - 1 . \end{align*}


\begin{align*} p ^ \mathfrak { t } ( s ) = ( r ^ \mathfrak { t } ( s ) , \lambda ^ \mathfrak { t } ( s ) , \mu ^ \mathfrak { t } ( s ) ) ^ \top = ( r _ 0 , 4 s - 2 \mathfrak { t } + H ( q , r _ 0 ) / 4 , 2 s + 4 \mathfrak { t } ) ^ \top , \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \epsilon \int _ 0 ^ T \int _ { \Gamma _ { \epsilon } } u _ { \epsilon } ( t , x ) \phi \left ( t , x , \frac { x } { \epsilon } \right ) d \sigma d t = \int _ 0 ^ T \int _ { \Omega } \int _ { \Gamma } u _ 0 ( t , x , y ) \phi ( t , x , y ) d \sigma _ y d x d t . \end{align*}


\begin{align*} \left < a _ 1 b _ 1 c _ 1 , a _ 2 b _ 2 c _ 2 \right > = \left < b _ 1 , b _ 2 \right > \left < S ( c _ 1 ) , a _ 2 \right > \left < c _ 2 , S ( a _ 1 ) \right > \end{align*}


\begin{align*} \begin{aligned} | ( \langle \boldsymbol { \nu } , G \rangle - \bar G ) \cdot ( \langle \boldsymbol { \nu } , R \rangle - \bar R ) | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} \mathcal G _ t \mathcal E _ x = j _ { x - 1 , x } - j _ { x , x + 1 } , \end{align*}


\begin{align*} \tilde { B } ( \theta _ 0 , \varepsilon ) = \bigcup _ { \sigma \in S _ k } \big \{ \theta \in \Theta : d _ \Theta ( \theta , \theta _ 0 [ \sigma ] ) < \varepsilon \big \} . \end{align*}


\begin{align*} & \mathcal { P } ^ - ( f ( z ) e ^ { 2 i z x } ) = \frac { 1 } { 2 \pi i } \lim _ { \varepsilon \rightarrow 0 } \int _ { R } \frac { f ( s ) e ^ { 2 i s x } } { s - ( z - i \varepsilon ) } \mathrm { d } s . \end{align*}


\begin{align*} { N } ( \mathbf { v } ) = \mathbf { 0 } . \end{align*}


\begin{align*} ( \mu _ { 2 } ) _ \gamma ( d x ) = \frac { e ^ { \gamma x } } { \int _ { [ 0 , \infty ) } e ^ { \gamma x } \mu _ { 2 } ( d x ) } \mu _ { 2 } ( d x ) . \end{align*}


\begin{align*} \{ ( \xi , \rho ) \} = \mathcal { S } ( \beta ) \cap \mathcal { S } ( \beta ) ^ \perp . \end{align*}


\begin{align*} g ( E ) = \frac { 1 } { \left ( \hbar \omega \right ) ^ { D N } \left ( D N - 1 \right ) ! } E ^ { D N - 1 } , \end{align*}


\begin{align*} G = i \ \overline { v } \ \widehat { v } . \end{align*}


\begin{align*} p + \alpha \overline { u } + \mu = 0 , \\ u _ { a } \le \overline { u } \le u _ { b } , \ ; \ ; \mu _ { a } \ge 0 , \ ; \ ; \mu _ { b } \ge 0 , \\ \mu _ { a } ( u _ { a } - \overline { u } ) = \mu _ { b } ( \overline { u } - u _ { b } ) = 0 , \end{align*}


\begin{align*} \tilde X _ { n + 1 } ^ i = f ( \tilde X _ n ^ i , \tau ) + \gamma ^ i ( \tilde X _ n ) \tau + \sigma W _ \tau ^ i . \end{align*}


\begin{align*} L _ { q } \left [ \exp _ { q ^ { \prime } } \left ( - \alpha t ^ { 2 } \right ) \right ] = \displaystyle \int _ { 0 } ^ { \infty } d t \ ; \exp _ { q } \left ( - s t \right ) \exp _ { q ^ { \prime } } \left ( - \alpha t ^ { 2 } \right ) , \end{align*}


\begin{align*} & \mathcal { P } ^ + \left ( f ( z ) e ^ { 2 i z x } \right ) = - \int _ { - \infty } ^ { 2 x } \widehat { f } ( \xi ) e ^ { - i z ( \xi - 2 x ) } \mathrm { d } \xi , \\ & \mathcal { P } ^ - \left ( f ( z ) e ^ { - 2 i z x } \right ) = \int _ { - \infty } ^ { 2 x } \widehat { f } ( \xi ) e ^ { i z ( \xi - 2 x ) } \mathrm { d } \xi . \end{align*}


\begin{align*} \begin{aligned} ( \Delta \varphi _ { p , N } ^ { + } ) ^ m \wedge \beta _ n ^ { n - m } = 2 ^ { - m } ( \Delta u _ p + \Delta u _ { p + N } ) ^ m \wedge \beta _ n ^ m = 2 ^ { - m } \sum \limits _ { k = 0 } ^ m \binom m k ( \Delta u _ p ) ^ k \wedge ( \Delta u _ { p + N } ) ^ { m - k } \wedge \beta _ n ^ m . \end{aligned} \end{align*}


\begin{align*} \vec { h } = \mathcal { H } ( \vec { r } , h _ 1 ) . \end{align*}


\begin{align*} [ M ] & = \frac { [ \mathbb C _ x ] } { D } \\ [ \mathbb C _ x ] & = \prod _ { i = 0 } ^ N e ^ { x _ i \alpha _ i } \\ D & = \prod _ { n = 1 } ^ \infty ( 1 - e ^ { - n \delta } ) ^ { N } \prod _ { n = 1 } ^ \infty \prod _ { 1 \leq i < j \leq r } ( 1 - e ^ { \alpha _ j - \alpha _ i } e ^ { ( 1 - n ) \delta } ) ( 1 - e ^ { \alpha _ i - \alpha _ j } e ^ { - n \delta } ) . \end{align*}


\begin{align*} \left | \begin{array} { c c } x + y = y ' \ , \\ x ' + y ' = x \end{array} \right . . \end{align*}


\begin{align*} \delta ( \alpha , \theta ) = \limsup _ { n \to \infty } \max ( 0 , \delta _ n ) = \limsup _ { n \to \infty } \max ( 0 , \frac { \ln q _ { n + 1 } + \ln | c _ { n , 0 } | } { q _ n } ) . \end{align*}


\begin{align*} H = \displaystyle \sum _ { i = 1 } ^ { D N } \frac { p _ { i } ^ { 2 } } { 2 m } , \end{align*}


\begin{align*} f _ i = \sum _ { L _ i \geqslant N _ 3 ^ { ( 5 - 2 \alpha ) + \varepsilon } } f _ { i , L _ i } . \end{align*}


\begin{align*} \Sigma ^ 2 = \left ( \frac { 1 } { 2 L _ { x } } + \frac { \omega } { L _ { x y } } \right ) \frac { \sigma _ { F , r } ^ 2 } { n } + \left ( \frac { 1 } { 2 L _ { y } } + \frac { 1 } { L _ { x y } \omega } \right ) \frac { \sigma _ { G , r } ^ 2 } { n } . \end{align*}


\begin{align*} \gamma \times T ^ n = \gamma ( n \bmod \Z ) \ , T ^ n \end{align*}


\begin{align*} \Phi = \{ e _ j - e _ k \mid j , k = 1 , \dots , n , \ ; j \not = k \} . \end{align*}


\begin{align*} 2 \pi r \left \Vert d \phi _ { r i } \left ( z \right ) \right \Vert ^ { 1 / 2 } = \sqrt { 2 } ~ \pi \left ( \ell \left ( r \right ) \sin r \right ) ^ { 1 / 2 } \end{align*}


\begin{align*} \widehat \Phi ^ { \phi ^ \varepsilon } ( t ) \overset { d } { = } \Phi ^ { q _ \varepsilon } ( t ) \end{align*}


\begin{align*} \partial _ Y \left ( \Phi ( \mathcal { P } ( X ) ) \right ) = \Phi ( \nabla _ Y \mathcal { P } ( X ) ) - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle Y _ i , \mathcal { P } ( X ) \rangle N _ i \\ + \Phi ( B ( Y , \mathcal { P } ( X ) _ T ) - B ^ * ( Y , \mathcal { P } ( X ) _ N ) ) \end{align*}


\begin{align*} x _ 0 = \frac { 1 - x _ 0 } { 1 + \sum _ { l \in \mathbb Z _ 0 } { \lambda _ l \widetilde { z } _ l } } + x _ 0 \cdot c _ { 0 0 } , \ \ x _ j = \frac { ( 1 - x _ 0 ) \lambda _ { j } \widetilde { z } _ { j } } { 1 + \sum _ { l \in \mathbb Z _ 0 } { \lambda _ l \widetilde { z } _ l } } + x _ 0 \cdot c _ { 0 j } , \ \ j \in \mathbb Z _ 0 . \end{align*}


\begin{align*} V \left ( \mathcal { K } _ { m , n } \right ) = \left \{ \left ( i , j \right ) : 1 \leq i \leq m , 1 \leq j \leq n \right \} , \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) = \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } ( \delta _ { j , m } ) = \frac { z ^ m } { \sqrt { \Gamma ( q j + 1 ) } } : = e _ { m , q } ( z ) . \end{align*}


\begin{align*} F ( \eta ) : = T - \frac { 1 } { \eta } I , \eta \in \Omega , \end{align*}


\begin{align*} \mathcal { M } = \{ Z = ( z _ 1 , z _ 2 , \cdots , z _ m ) \in \mathbb { R } ^ { ( 2 m ) } \mid z _ i \in B _ { \bar { \rho } } ( x _ { 0 , i } ) , \ i = 1 , \cdots , m \} . \end{align*}


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


\begin{align*} ( \epsilon _ N \circ T _ \alpha G _ \alpha f ) ( s \otimes g ) & = \epsilon _ N ( s \otimes ( G _ \alpha f ) ( g ) ) \\ & = \epsilon _ N ( s \otimes ( f \circ g ) ) \\ & = f ( g ( s ) ) \\ & = f ( \epsilon _ N ( s \otimes g ) ) \\ & = ( f \circ \epsilon _ N ) ( s \otimes g ) \end{align*}


\begin{align*} \partial _ t M ^ k f + v \cdot \partial _ x M ^ k f + \mathcal L M ^ k f = \Gamma ( f , \ M ^ k f ) + M ^ k \Gamma ( f , f ) - \Gamma ( f , \ M ^ k f ) - \big [ M ^ k , \ \mathcal L \big ] f - k \partial _ { v _ 1 } ^ 2 M ^ { k - 1 } f . \end{align*}


\begin{align*} p _ 0 + \langle l , p \rangle > 0 , l \in L = F ^ \eta ( z , w ( - h ) ) + \partial ^ { c i } _ { \tau , w } g ( \tau , w ( \cdot ) ) . \end{align*}


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


\begin{align*} \boldsymbol { \lambda } = \sum \limits _ { k = 1 } ^ m s _ k \boldsymbol { \lambda } _ k . \end{align*}


\begin{align*} R _ c = \pi / 0 . 0 6 7 \approx 4 6 . 9 . \end{align*}


\begin{align*} & \int _ { - \infty } ^ { \infty } \bar { u } _ y ( y ) \left ( \psi ^ - _ { 1 1 } - 1 \right ) e ^ { - 2 i z y } \mathrm { d } y = - z \int _ { - \infty } ^ { \infty } \bar { u } _ y ( y ) e ^ { - 2 i z y } K \psi ^ - _ { 1 1 } \mathrm { d } y \\ & = - \frac { 1 } { 2 } z \widehat { \bar { u } _ x K \psi ^ - _ { 1 1 } } ( z ) . \end{align*}


\begin{align*} D ^ W _ X Y = D ^ g _ X Y - \frac { 1 } { 2 } \theta ( X ) Y - \frac { 1 } { 2 } \theta ( Y ) X + \frac { 1 } { 2 } g ( X , Y ) \theta ^ \sharp , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s | c _ { i } | \leq \sqrt { s } \sqrt { \sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } } . \end{align*}


\begin{align*} X ( t ) = X ( 0 ) + \int _ 0 ^ t F ( X ( s ) ) d s + \int _ 0 ^ t G ( X ( s ) ) d W ( s ) , \end{align*}


\begin{align*} \psi _ i = m \cdot \int _ { \kappa ^ { - 1 } ( K _ i ) } \psi \ , \mathrm { d } \mu _ \varphi ( \psi ) \end{align*}


\begin{align*} & \mathbb { E } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ 2 \\ = & \mathbb { E } \int _ { t \wedge \tau _ n ^ M } ^ { ( t + \delta ) \wedge \tau _ n ^ M } 2 \langle A ( r , Y _ n ( r ) ) , Y _ n ( r ) - Y _ n ( t \wedge \tau _ n ^ M ) \rangle d r \\ & + \mathbb { E } \int _ { t \wedge \tau _ n ^ M } ^ { ( t + \delta ) \wedge \tau _ n ^ M } \Vert P _ n B ( r , Y _ n ( r ) ) Q _ n \Vert _ { L _ 2 } ^ 2 d r . \end{align*}


\begin{align*} \mathbf { r } _ p ( \mathbf { x } , \mathbf { y } , \mathbf { s } , \kappa , \tau ) = \mathbf { A x } - \tau \mathbf { b } , \mathbf { r } _ d ( \mathbf { x } , \mathbf { y } , \mathbf { s } , \kappa , \tau ) = { { \mathbf { A } } ^ { T } } \mathbf { y } + \mathbf { s } - \tau \mathbf { c } . \end{align*}


\begin{align*} \sum _ { x \in E } \sum _ { y \neq x } \nu ( x ) \frac { \mu ( y ) } { \mu ( x ) } L _ { y x } = \sum _ { x \in E } \frac { \nu ( x ) } { \mu ( x ) } \sum _ { y \neq x } \mu ( y ) L _ { y x } = \sum _ { x \in E } \nu ( x ) L _ { x x } = \sum _ { x \in E } \nu ( x ) \sum _ { y \neq x } L _ { x y } , \end{align*}


\begin{align*} \int _ { \partial D } S _ { e , \mathcal { Y M } ^ 0 } ( X , \nu ) d s _ g = \int _ D \langle S _ { e , \mathcal { Y M } ^ 0 } , \nabla X ^ { \flat } \rangle d v _ g \ , , \end{align*}


\begin{align*} | m _ n + \ell _ n q _ n - \tilde { m } _ n | = q _ { n + 1 } = q _ n + q _ { n - 1 } , \end{align*}


\begin{align*} \phi ( j q _ n + m _ n + 1 ) + \phi ( j q _ n + m _ n - 1 ) = ( E - \lambda \tan ( \pi \theta _ { j q _ n + m _ n } ) ) \phi ( j q _ n + m _ n ) . \end{align*}


\begin{align*} T ( g , \rho ) = \sum _ { l \geq 1 } ( U ( l ) - U ( l - 1 ) ) ^ 2 \widehat { g } \left ( \frac { l } { \pi _ N ( x ) } \right ) , \end{align*}


\begin{align*} ^ F V = \bigoplus _ { p \in \frac { 1 } { 2 } \mathbb Z } F _ p V / F _ { p - \frac { 1 } { 2 } } V \end{align*}


\begin{align*} W ( v , x ) \stackrel { d } { = } e ^ { - \lambda ^ * t } W ( v , z ) . \end{align*}


\begin{align*} Z \colon Q \times \R \to \R ^ d , Z ( x _ 1 , \dots , x _ d ) = e ^ { x _ d } h ( x _ 1 , \dots , x _ { d - 1 } ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ 3 ( I _ N ^ i ( p ) - J _ N ^ i ( p ) ) = O ( p ^ { \mu - \nu } ) \quad \mbox { f o r l a r g e } p > 0 , \end{align*}


\begin{align*} \lim _ { m \to \infty } \lim _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \chi _ 4 < 1 \Big ) = - \infty . \end{align*}


\begin{align*} \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( | v | ^ { 2 } + | b | ^ { 2 } \right ) \dd x + \int _ { \delta } ^ { t } \int _ { \mathbb { R } ^ { 3 } } \left ( \nu | \nabla v | ^ { 2 } + \mu | \nabla b | ^ { 2 } \right ) \dd x \dd s = \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( | v | ^ { 2 } + | b | ^ { 2 } \right ) ( \delta ) \dd x . \end{align*}


\begin{align*} \Bigg | \dfrac { \partial \epsilon _ 1 ( \eta , t ) } { \partial \eta } \Bigg | _ { \eta = \eta _ a } \dot { \eta } _ a \Bigg | \le \gamma \delta L _ r ^ 2 3 \end{align*}


\begin{align*} X ( t , \xi ) = e ^ { W ( t , \xi ) } y ( t , \xi ) . \end{align*}


\begin{align*} L ( p , \pi , \bar { \pi } ) = \frac { R \cdot R ( v , w , w , v ; x , y ) } { Q ( g , R ) ( v , w , w , v ; x , y ) } . \end{align*}


\begin{align*} \P _ x ( \rho > n ) \sim C _ { \lambda } n ^ { - d ( d - 1 ) / 2 - d / 2 } e ^ { - \gamma n } e ^ { \sum _ { i = 1 } ^ d ( \lambda _ i - \bar { \lambda } ) x _ i } \Delta ( x ) , n \rightarrow \infty \end{align*}


\begin{align*} [ J _ + ^ \alpha , J _ - ^ \alpha ] = 2 \omega ^ { - 2 p + 1 } ( \alpha ) J _ 0 ^ \alpha . \end{align*}


\begin{align*} \pi _ 0 ( e ) = \pi _ 0 ( e ^ 2 ) = \pi _ 0 ( e ) ^ 2 + \sum _ { \omega \in W _ f ^ + } \pi _ { - \omega } ( e ) \pi _ { \omega } ( e ) . \end{align*}


\begin{align*} z = \prod _ { c = 0 } ^ { \ell - 1 } X _ c . \end{align*}


\begin{align*} \sigma ( \pi ) = \sum _ { i \ge 2 } ( i - 1 ) \pi _ i . \end{align*}


\begin{align*} \| q _ n ( \theta - \frac { 1 } { 2 } ) \| = e ^ { \delta _ n q _ n } \| q _ n \alpha \| . \end{align*}


\begin{align*} { S ^ { \rm { N } } } = \frac { { W I } } { { K L } } { \widetilde \varepsilon _ K } \left ( { { \gamma ^ { \rm { N } } } } \right ) , \end{align*}


\begin{align*} [ q ^ \alpha , q ^ \beta ] = \sum _ { \gamma \in S } { f ^ { \alpha \beta } } _ \gamma q ^ \gamma . \end{align*}


\begin{align*} z _ { i , x } = \lambda _ i \prod _ { y \in S ( x ) } { 1 \over 1 + \sum _ { j \in \mathbb { Z } _ 0 } z _ { j , y } } , \ \ i \in \mathbb { Z } _ 0 . \end{align*}


\begin{align*} \left \| \partial _ { z } \mathcal { H } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) \right \| _ { L ^ 2 } = \left \| \partial _ { z } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) \right \| _ { L ^ 2 } . \end{align*}


\begin{align*} \limsup _ { s \rightarrow + \infty } \frac { J _ 3 ( s ) } { s } \leq - R e \left ( q e ^ { i \theta } \right ) = - \inf _ { \omega \in \Gamma ^ { \prime } } R e \left ( \omega e ^ { i \theta } \right ) . \end{align*}


\begin{align*} ( S _ \bullet \to S ) = \varprojlim _ i ( T _ { i , \bullet } \to T _ i ) . \end{align*}


\begin{align*} \varphi \left ( \rho \right ) = T _ { + } \left ( \rho \right ) \varphi _ { 0 } \left ( \rho \right ) \end{align*}


\begin{align*} \mathrm { Q u a d } ^ i : = \{ z ^ i _ { j k } z ^ i _ { j ' k ' } - z ^ i _ { j k ' } z ^ i _ { j ' k } \mid j , j ' \in [ s _ i ] , k , k ' \in [ t _ i ] \} . \end{align*}


\begin{align*} \varepsilon ( z ) \Lambda ( z ^ { - 1 } ) = - \Lambda ( - z ^ { - 1 } ) , \dot { s } \Lambda ( z ) = - \Lambda ( - z ^ { - 1 } ) , \quad \\ \dot { s } \varepsilon ( x ) \Lambda ( y ) = \varepsilon ( - x ^ { - 1 } ) \Lambda ( x ( x y - 1 ) ) + \Lambda ( x ) - \Lambda ( y ^ { - 1 } ( x y - 1 ) ) , \end{align*}


\begin{align*} P _ s ( x , t ) = t ^ { \frac { - N } { 2 s } } \phi \left ( | x | t ^ { - \frac 1 { 2 s } } \right ) \end{align*}


\begin{align*} \begin{aligned} & ( I - F ) \left [ z \left ( \Psi ^ { - } _ 1 - e ^ { - i c _ - } e _ 1 \right ) - \left ( \widehat { \Psi } ^ - _ { 1 1 } e _ 1 + \widehat { \Psi } ^ - _ { 2 1 } e _ { 2 } \right ) \right ] \\ = & z d e _ { 2 } - ( I - F ) \left ( \widehat { \Psi } ^ - _ { 1 1 } e _ 1 + \widehat { \Psi } ^ - _ { 2 1 } e _ 2 \right ) , \end{aligned} \end{align*}


\begin{align*} h _ { n + 1 } = r _ n h _ n + \frac { r _ n ( r _ n - 1 ) } { 2 } . \end{align*}


\begin{align*} \bar \Delta = \min _ { 0 \le k \le T - 1 } \beta ( X _ k ) . \end{align*}


\begin{align*} \gamma \varphi ( h ) G ( \nabla h ) \sigma _ { k } ( x ) = f ( x ) \ { \rm f o r } \ \gamma = 1 . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \dot { P } _ t + 2 A P _ t - B ^ 2 R ^ { - 1 } P ^ 2 _ t + Q = 0 , \\ & P _ T = G , \end{aligned} \right . \end{align*}


\begin{align*} \langle V , z _ \tau \rangle = v _ \tau + \sum _ { q = 1 } ^ { d + 1 } ( - 1 ) ^ q v _ { \tau _ q } . \end{align*}


\begin{align*} \pi _ 1 ^ * ( Z ) = m _ Z E _ 1 + \tilde { Z } \end{align*}


\begin{align*} \begin{pmatrix} \bar { x } _ { t , n } \\ \bar { x } _ { t , 1 } \\ \end{pmatrix} = \int _ 0 ^ t S ( t - s ) \begin{pmatrix} F ( s ) \\ G ( s ) \\ \end{pmatrix} d s - \int _ 0 ^ t S ( t - s ) \begin{pmatrix} 0 \\ \sigma _ 1 d W _ s ^ { ( 1 ) } \\ \end{pmatrix} . \end{align*}


\begin{align*} [ D \widetilde { H } ] ( x _ 0 , Y _ 0 ) ( x , Y ) = \big < D \widetilde { H } ( x _ 0 , Y _ 0 ) , ( x , Y ) \big > _ { \mathcal H } \ , . \end{align*}


\begin{align*} \mathcal { I } & = \{ ( i , j ) : i \leq 0 \mbox { o r } j \leq 0 \mbox { o r } i \leq a , j \leq b \} \subseteq \Z ^ 2 \\ m _ { i , j } & = e _ \alpha ^ { a - i } \overline { e } _ \alpha ^ { b - j } \cdot m \mbox { i f } i \leq a , j \leq b \\ m _ { i , j } & = 0 \mbox { o t h e r w i s e . } \end{align*}


\begin{align*} \begin{array} { l } R _ { \max } = \mathop { \max } \limits _ { { \bf p } , { \bf y } } { R _ T } \\ { \rm { s } } { \rm { . t } } { \rm { . } } \\ C 2 , \ , C 3 , \ , C 8 , \ , { \rm a n d } \ , C 9 . \end{array} \end{align*}


\begin{align*} \alpha ^ \xi _ t : = - R ^ { - 1 } B ( P _ t x ^ \xi _ t + \varphi _ t ^ \xi ) - h ( \mu _ t ) \end{align*}


\begin{align*} \alpha _ j = \alpha _ j ' \dfrac { d ( a _ j , c _ j ) } { d ( a _ j , b _ j ) } , \alpha _ { m + j } = \alpha _ j ' \dfrac { d ( c _ j , d _ j ) } { d ( a _ j , b _ j ) } , \alpha _ { 2 m + j } = \alpha _ j ' \dfrac { d ( d _ j , b _ j ) } { d ( a _ j , b _ j ) } . \end{align*}


\begin{align*} K _ i = \langle \{ a _ i , b _ i \} \rangle , \ L _ i = \langle \{ b _ i \} \rangle . \end{align*}


\begin{align*} \gamma _ i + \bar { \gamma } _ 4 = 1 \ , \ , { \rm a n d } \ , \ , \gamma _ s + \bar { \gamma } _ t = 1 . \end{align*}


\begin{align*} \left [ g , h ^ n \right ] = \left [ g , h ^ { n - 1 } \right ] \left [ g , h \right ] ^ { h ^ { n - 1 } } = \left [ g , h ^ { n - 1 } \right ] t ^ { h ^ { n - 1 } } = \left [ g , h ^ { n - 1 } \right ] t \end{align*}


\begin{align*} \aligned e _ 1 & \ , : = \ , ( 1 - y ) \ , \partial _ x + x \ , \partial _ u , \\ e _ 2 & \ , : = \ , ( 1 - y ) \ , \partial _ y + u \ , \partial _ u , \\ e _ 3 & \ , : = \ , x \ , \partial _ x + 2 \ , u \ , \partial _ u , \\ e _ 4 & \ , : = \ , - \ , u \ , \partial _ x + x \ , \partial _ y , \endaligned \end{align*}


\begin{align*} \frac { \tilde { P } _ k ( \theta ) } { ( \cos { \pi \theta } ) ^ k } = : g _ k ( \tan \pi \theta ) . \end{align*}


\begin{align*} \tilde { w } _ i ( t , x , y ) : = w _ i ( t , x , S _ 0 ^ { - 1 } ( t , x , y ) ) \mbox { f o r } ( t , x , y ) \in ( 0 , T ) \times \Omega \times Y ^ { \ast } ( t , x ) , \end{align*}


\begin{align*} { \rm V o l } ( B ) \geq \min \{ { \rm V o l } ( A ) , { \rm V o l } ( B ) \} = \mathfrak h ^ { - 1 } { \rm V o l } ( \Sigma ) . \end{align*}


\begin{align*} \phi ^ { ( k , \sigma ) } _ i = \left \{ \begin{aligned} & \phi _ i & & ( i \neq k ) \\ & \sigma & & ( i = k ) . \end{aligned} \right . \end{align*}


\begin{align*} ( \varphi _ 1 \cdot \varphi _ 2 ) ( Z _ i ) = F _ i \big ( \varphi _ 1 ( Z _ 1 ) , \ldots , \varphi _ 1 ( Z _ d ) , \varphi _ 2 ( Z _ 1 ) , \ldots , \varphi _ 2 ( Z _ d ) \big ) , \end{align*}


\begin{align*} & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ + \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \right ) \right \| _ { L _ z ^ \infty } \leq \frac { 1 } { \sqrt { 2 } } \left \| z ^ { - i } r _ 1 ( z ) \right \| _ { H ^ 1 } , i = 0 , 1 , \\ & \sup _ { x \in \mathbb { R } } \left \| \mathcal { P } ^ - \left ( r _ 2 ( z ) \mathrm { e } ^ { 2 i z x } \right ) \right \| _ { L _ { z } ^ { \infty } } \leq \frac { 1 } { \sqrt { 2 } } \left \| r _ 2 ( z ) \right \| _ { H ^ { 1 } } . \end{align*}


\begin{align*} A D ( x ) + D ( x ) B = D ( v ) . \end{align*}


\begin{align*} Q : = \left \{ x \in \R ^ { d - 1 } \colon \| x \| _ \infty \leq 1 \right \} = [ - 1 , 1 ] ^ { d - 1 } \end{align*}


\begin{align*} f \vee g = \frac { 1 } { 2 } \big ( f + g + | f - g | \big ) , \end{align*}


\begin{align*} \lambda _ { - , 1 } = - 2 \sqrt { \delta ( 1 - \delta ) } , E _ { - , 1 } = \left \{ \begin{pmatrix} \bar { a } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ - \bar { b } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ 0 \\ 1 \end{pmatrix} , \begin{pmatrix} - \bar { b } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ - \bar { a } _ 1 \sqrt { \frac { \delta } { \delta - 1 } } \\ 1 \\ 0 \end{pmatrix} \right \} . \end{align*}


\begin{align*} \gamma _ { i - 1 } \alpha _ i ^ 2 = ( 1 - \alpha _ i ) \alpha _ { i - 1 } ^ 2 \gamma _ i + \mu \alpha _ i \gamma _ i \gamma _ { i - 1 } \geq ( 1 - \alpha _ i ) \mu \gamma _ { i - 1 } \gamma _ i + \mu \alpha _ i \gamma _ i \gamma _ { i - 1 } = \mu \gamma _ i \gamma _ { i - 1 } , \end{align*}


\begin{align*} \gamma \cdot f ( x , y ) = \frac { f ( ( x , y ) \gamma ) } { | \det \gamma | } , \end{align*}


\begin{align*} \mathcal U ( s , i ) = [ 0 , \bar L ] , ( s , i ) \in C . \end{align*}


\begin{align*} \begin{aligned} \langle \partial _ t ( \bar { J } _ 0 u _ 0 ) , & \phi \rangle _ { H ^ 1 ( \Omega ) ' , H ^ 1 ( \Omega ) } - \int _ { \Omega } q u _ 0 \phi d x + \int _ { \Omega } D _ 0 ^ { \ast } \nabla u _ 0 \cdot \nabla \phi d x \\ & = \int _ { \Omega } \int _ { Y ^ { \ast } } J _ 0 f _ 0 d y \phi d x - \int _ { \Omega } \partial _ t R _ 0 ( u _ 0 - \rho ) \phi \vert \Gamma ( t , x ) \vert d x . \end{aligned} \end{align*}


\begin{align*} \sum _ { \ell + 2 p = j , \ \ell + 2 q = 2 k - j } c ^ { k , j } _ { \ell , p , q } = { { 2 k } \choose j } . \end{align*}


\begin{align*} V ' = \{ v \in V \mid ( v , e _ 1 ) \in E ( v , f _ 1 ) \in E \} . \end{align*}


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


\begin{align*} I _ { h , W , \gamma } = \left ( \frac { W } { 2 } \right ) ^ { - 2 } I _ { \tilde { h } , 2 , \tilde { \gamma } } . \end{align*}


\begin{align*} g ^ \iota = \begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix} ( g ^ { - 1 } ) ^ t \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} . \end{align*}


\begin{align*} M ( S ) = \varinjlim _ i M ( S _ i ) . \end{align*}


\begin{align*} \mathcal D _ { E / F } = \prod _ { \mathfrak p \in \mathcal R _ { E / F } } \mathfrak p ^ { e _ { \mathfrak p } - 1 } . \end{align*}


\begin{align*} R \cdot R ( X , J X , J X , X ; U , J U ) = f \ , \ , Q ^ c ( g , R ) ( X , J X , J X , X ; U , J U ) \end{align*}


\begin{align*} [ \alpha ( x ) , y \cdot z ] = \varepsilon ( x , y ) \alpha ( y ) \cdot [ x , z ] + \varepsilon ( x + y , z ) \alpha ( z ) \cdot [ x , y ] . \end{align*}


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


\begin{align*} \sigma ( \alpha + \beta ) = \sigma ( \alpha ) + \sigma ( \beta ) \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { l } ( 1 - Z _ i ) \ge \exp \big ( - 2 \sum _ { i = 1 } ^ { l } Z _ i \big ) = & \exp \bigg ( - 2 \sum _ { s = 1 } ^ { \lceil \log _ 2 ( l / 2 ) \rceil } 2 ^ s \log \big ( \tau ^ { - 1 / 2 ^ s } l / 2 ^ s \big ) \frac { 3 } { l } \bigg ) . \end{align*}


\begin{align*} B _ 1 + \cdots + B _ k = \{ b _ 1 + \cdots + b _ k : b _ 1 \in B _ 1 , \dots , b _ k \in B _ k \} \end{align*}


\begin{align*} \mathbb E \left [ \sup _ { s \leq t } \| \int _ 0 ^ t \sigma _ s d W _ s \| ^ m \right ] & \leq C _ m \mathbb E \left [ \left ( \int _ 0 ^ t ( \Sigma _ s ) d s \right ) ^ { \frac m 2 } \right ] \\ & = C _ m \mathbb E \left [ \left ( \int _ 0 ^ t \| \sigma _ s \| _ { L _ { } ( U , H ) } ^ 2 d s \right ) ^ { \frac m 2 } \right ] , \end{align*}


\begin{align*} ( d / d t ) v + A v = 0 , v \in \R ^ D , \end{align*}


\begin{align*} a ( x ) = \sum _ { i = 0 } ^ r a _ i x ^ i \end{align*}


\begin{align*} \mathbf { d } _ { h x } = 0 , \mathbf { d } _ { h s } = 0 . \end{align*}


\begin{align*} \int _ \Omega \Delta u _ 1 \wedge \dots \wedge \Delta u _ m \wedge \beta _ n ^ { n - m } = \int _ \Omega \Delta v _ 1 \wedge \dots \wedge \Delta v _ m \wedge \beta _ n ^ { n - m } . \end{align*}


\begin{align*} \begin{aligned} \| \Phi ( v _ { 1 } , v _ { 2 } ) \| _ { { { \bf X } } _ T } & \le \tilde { C } ( T ^ { - 2 \tilde { \nu } - \frac 1 2 } + ( \varepsilon _ { 1 } + \varepsilon _ { 2 } ) T ^ { - \tilde { \nu } - \frac 1 4 } + \varepsilon _ { 1 } \varepsilon _ { 2 } T ^ { - \frac { \delta } 2 } \\ & \ \ \ \ \quad + ( \varepsilon _ { 1 } + \varepsilon _ { 2 } ) T ^ { \tilde { \nu } - \frac 1 2 } ( \log T ) ^ 2 + \varepsilon _ { 1 } ^ 2 ) \end{aligned} \end{align*}


\begin{align*} ( d ^ \nabla \sigma ) ( X _ 1 , X _ 2 ) = ( \nabla _ { X _ 1 } \sigma ) ( X _ 2 ) - ( \nabla _ { X _ 2 } \sigma ) ( X _ 1 ) \ , . \end{align*}


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


\begin{align*} t _ 1 = \max \bigl \{ t \colon d ( \gamma _ { a _ 1 , b _ 1 } ( t ) , a _ 1 ) = R \bigr \} \end{align*}


\begin{align*} & \lim _ { | z | \rightarrow 0 } \Psi ^ { \pm } ( x ; z ) = \begin{pmatrix} 1 & - u _ x ( x ) \\ - \bar { u } _ x ( x ) & 1 \end{pmatrix} , \\ [ 4 p t ] & \lim _ { | z | \rightarrow \infty } \Psi ^ { \pm } ( x ; z ) = e ^ { - i c _ \pm \sigma _ 3 } . \end{align*}


\begin{align*} \begin{aligned} \frac { 1 } { 2 } \frac { d } { d t } \| \Lambda ^ s & \theta ( t ) \| ^ 2 + \mu \| \Lambda ^ { s + 1 } \theta ( t ) \| ^ 2 \\ & = - \int _ { \R ^ 3 } \Lambda ^ s ( ( u \cdot \nabla ) \theta ) \Lambda ^ s \theta \ , d x - \int _ { \R ^ 3 } \Lambda ^ s ( \mathrm { d i v } \ , v ) \Lambda ^ s \theta \ , d x . \end{aligned} \end{align*}


\begin{align*} \mathcal { P } _ s = \{ V _ 1 \subset V _ 2 \subset \ldots V _ { l - 1 } \subset V _ { l + 1 } \subset \ldots \subset V _ { n } = \mathbb { C } ^ n ; \dim _ { \mathbb { C } } ( V _ i ) = i \} . \end{align*}


\begin{align*} u ( t ) = u _ 0 + \int _ 0 ^ t \Delta _ p u \ , \ , d s + \int _ 0 ^ t S ( u ) \ , \ , d s + \sum _ { n = 0 } ^ { + \infty } \int _ 0 ^ t \xi _ { n } ( x ) \cdot \nabla u ( s ) \ , d B _ s ^ { n } \end{align*}


\begin{align*} \begin{gathered} \min _ { \mathbf { a } } \mathbb { E } \left [ \lVert \mathbf { s } ^ { \left ( \right ) } - \mathbf { y } ' \rVert ^ 2 \right ] \\ ~ ~ \left ( \mathbf { P } ^ { \left ( \right ) } \left ( \mathbf { a } ^ { \left ( \right ) } \odot \mathbf { a } ^ { \left ( \right ) } \right ) \mathbf { P } ^ { \left ( \right ) ^ { H } } \right ) = E _ { t r } , \end{gathered} \end{align*}


\begin{align*} \varphi _ n ( \eta ) = \varphi _ n ( \mathbf { 1 } ) + \sum _ { m = 0 } ^ \infty ( \varphi _ n ( r _ m \eta ) - \varphi _ n ( r _ { m + 1 } \eta ) ) , \end{align*}


\begin{align*} \beta _ { p , d } = \frac { \sqrt { \pi } \Gamma \left ( \frac { d - p } { 2 } \right ) } { \Gamma \left ( \frac { 1 - p } { 2 } \right ) \Gamma \left ( \frac { d } { 2 } \right ) } . \end{align*}


\begin{align*} \mu _ k ^ { \star } = { \left [ { \frac { 1 } { \nu } - \frac { 1 } { { { \left \| { { { \bf { \bar e } } _ k } } \right \| } ^ 2 } } } \right ] ^ + } , \ \forall k , \end{align*}


\begin{align*} \| q _ { k - 1 } \alpha \| = \min _ { 1 \leq n < q _ k } \| n \alpha \| , \end{align*}


\begin{align*} y ( \tau ) = \left ( z ( \tau ) + 1 \right ) \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } ( \tau ) a _ { n } ( x ) = \frac { 1 } { \frac { 1 } { 2 } - i \tau } \sum _ { n = 0 } ^ { \infty } \left ( - 1 \right ) ^ { n } z ^ { n } ( \tau ) a _ { n } ( x ) . \end{align*}


\begin{align*} \begin{cases} \partial _ t u ( t ) = - \Delta ^ 2 u + \Delta \varphi ( u ) , \\ \nabla u \cdot \nu = \nabla ( \Delta u ) \cdot \nu = 0 \ \partial \mathcal { O } , \\ u ( 0 ) = u _ 0 , \end{cases} \end{align*}


\begin{align*} A _ q ( z , x ) = A _ q ^ z ( x ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { \overline { z } ^ n } { \sqrt { \Gamma ( q n + 1 ) } } \psi _ n ( x ) , \end{align*}


\begin{align*} [ \mathbb C _ x ] = \prod _ { i = 0 } ^ r e ^ { x _ i \omega _ i } \end{align*}


\begin{align*} a _ { r } = \sum _ { j = 1 } ^ { n } \delta _ { r j } a _ { j } , \end{align*}


\begin{align*} K _ { \lambda } = \chi \int _ { W ^ d } e ^ { - \sum _ { j = 1 } ^ d ( \lambda _ j - \bar { \lambda } ) z _ j } \hat { h } ^ { ( \bar { \lambda } , \ldots , \bar { \lambda } ) } ( z ) d z _ 1 \ldots d z _ d . \end{align*}


\begin{align*} \mathbb E \left [ \| \Delta _ n ^ { - \frac 1 2 } ( 3 ) _ t \| _ { \mathcal H } ^ 2 \right ] = & 2 \Delta _ n \sum _ { i = 1 } ^ n ( i - 1 ) \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \mathbb E \left [ \| \sigma _ s ^ { \mathcal S _ n } \| _ { L _ { } ( U , H ) } ^ 2 \right ] d s \leq 2 \sup _ { r \in [ 0 , T ] } \| \mathcal S ( r ) \| _ { } ^ 2 . \end{align*}


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


\begin{align*} \partial _ X \langle \langle \varphi , \varphi ' \rangle \rangle = \langle \langle \nabla _ X \varphi , \varphi ' \rangle \rangle + \langle \langle \varphi , \nabla _ X \varphi ' \rangle \rangle \end{align*}


\begin{align*} ( T _ 1 + f _ 1 ) \cdot ( T _ 2 - f _ 2 ) \cdot \Psi _ 2 = i \Psi _ 2 \end{align*}


\begin{align*} \tau ( K ) : = \bigcap _ { K \subset U } \ker ( r : S H _ { M } ( M ; \Lambda ) \rightarrow S H _ { M } ( M - U ; \Lambda ) ) \end{align*}