\begin{align*} \frac { \partial q _ { \theta } } { \partial \theta _ i } = u ^ T ( I - \gamma P _ { \theta } ) ^ { - 1 } \left [ - \gamma \frac { \partial P _ { \theta } } { \partial \theta _ i } \right ] ( I - \gamma P _ { \theta } ) ^ { - 1 } v \end{align*}


\begin{align*} \tilde { T } _ { \vec s , k } = [ T _ { I _ { \vec s } , k } ] _ { \vec s } . \end{align*}


\begin{align*} D = a W + b L _ { x y } + c R + \Omega , \end{align*}


\begin{align*} \int _ 0 ^ T \langle V ' ( \vec p ) \widehat { \vec p } _ 1 ( { u } _ 2 ^ \prime ( t ) + Q { u } _ 2 ( t ) ) , w ( t ) \big \rangle _ X \d t & = - \int _ 0 ^ T \langle u _ 2 , V ' ( \vec p ) \widehat { \vec p } _ 1 ( w ^ \prime ( t ) + Q w ( t ) ) \big \rangle _ X \d t \\ & = \int _ 0 ^ T \langle V ' ( \vec p ) \widehat { \vec p } _ 2 ( { u } ^ \prime ( t ) + Q { u } ( t ) ) , z ( t ) \big \rangle _ X \d t . \end{align*}


\begin{align*} \min \limits _ { 0 \leq i \leq N } \{ \| \nabla f ( x _ i ) \| ^ 2 \} & \leq \frac { f ( x _ 0 ) - f _ * } { q + \sum \limits _ { i = 0 } ^ { N - 1 } p _ i } \end{align*}


\begin{align*} \int _ a ^ b \left | \alpha - f ( x ) \right | d x & = \int _ a ^ b \alpha - f ( x ) d x \\ & = \int _ a ^ b \alpha - \max \left ( f ( a ) , f ( b ) \right ) + \max ( f ( a ) , f ( b ) ) - f ( x ) d x \\ & = \int _ a ^ b \left | \alpha - \max ( f ( a ) , f ( b ) ) \right | + \left | \max ( f ( a ) , f ( b ) ) - f ( x ) \right | d x \\ & > \int _ a ^ b \left | \max ( f ( a ) , f ( b ) ) - f ( x ) \right | d x . \end{align*}


\begin{align*} H ( \omega ) = H _ 0 + V _ \omega \end{align*}


\begin{align*} \begin{aligned} & v ^ { \Delta t } _ { h } , \ , u ^ { \Delta t } _ { h } \to u \textrm { s t r o n g l y i n } L ^ { 2 } ( 0 , T ; L ^ { 2 } _ { \# } ) \quad \textrm { a s } ( \Delta t , h ) \to ( 0 , 0 ) . \end{aligned} \end{align*}


\begin{align*} \Vert J _ { 2 } \Vert _ { p } = \mathcal { O } \left ( \left ( v ( | z _ { 1 } | ) + v ( | z _ { 2 } | ) \right ) \left ( \frac { \omega ( h _ { 1 } ) } { v ( h _ { 1 } ) } + \frac { \omega ( h _ { 2 } ) } { v ( h _ { 2 } ) } \right ) \right ) . \end{align*}


\begin{align*} A \mapsto A _ t : = \bigcup \big \{ J _ v \mid J \in F ( D ) J \subseteq A \big \} . \end{align*}


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


\begin{align*} | \det D T u | = e ^ { [ \tau ( 1 - t ) + \epsilon ] ( u - u _ 0 ) } [ t f + ( 1 - t ) f ^ * \circ T u _ 0 | \det ( D T u _ 0 ) | ] / f ^ * \circ T u , { \rm i n } \ \Omega , \\ \end{align*}


\begin{align*} J _ W : = \left ( \exp \left ( \frac { 2 \pi i } { 3 } \right ) , \exp \left ( \frac { 2 \pi i } { 3 } \right ) , \exp \left ( \frac { 2 \pi i } { 3 } \right ) , \exp \left ( \frac { 2 \pi i } { 3 } \right ) \right ) \in ( \mathbb C ^ \times ) ^ 4 . \end{align*}


\begin{align*} P = c \big ( u _ 1 \otimes u _ 1 + u _ 2 \otimes u _ 2 \big ) \end{align*}


\begin{align*} 6 A + B q = 1 \end{align*}


\begin{align*} S ( F ( c ( x ) ) ) = F ( S ( c ( x ) ) ) = F ( c ( x ) ) , \end{align*}


\begin{align*} 2 g ( Y _ t ) - 2 - K _ { X _ t } \cdot h _ t ( Y _ t ) = \deg N _ { h _ t / X _ t } . \end{align*}


\begin{align*} E = \lim _ { r \rightarrow \infty } \frac { 1 } { 1 6 \pi } \int _ { S _ { r } } \sum _ i \left ( g _ { i j , i } - g _ { i i , j } \right ) \upsilon ^ j d A , \quad P _ i = \lim _ { r \rightarrow \infty } \frac { 1 } { 8 \pi } \int _ { S _ { r } } \left ( k _ { i j } - ( \mathrm { T r } _ g k ) g _ { i j } \right ) \upsilon ^ j d A \end{align*}


\begin{align*} \frac { 1 } { \sigma _ { k - 1 } } \lambda _ { i } ( \sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \left [ \lambda _ { i } ( \sigma _ { k - 1 } Z _ { k - 1 } - X ( x _ k ) ) \right ] _ { + } \to 0 , \ i = 1 , \dots , m , \end{align*}


\begin{align*} D = \bigcup \limits _ { j = 0 } ^ k D _ j . \end{align*}


\begin{align*} | \nabla u | _ g ^ 2 = \frac { f ( u ) ^ 2 } { f ( u ) ^ 2 - | \widetilde { \nabla } u | _ { \widetilde { g } } ^ 2 } - 1 = v ^ 2 - 1 . \end{align*}


\begin{align*} \big \langle \mu _ { \mathbf { x } ^ N } ^ { \otimes k } , \varphi _ k \big \rangle & = \frac { 1 } { N ^ k } \sum _ { i _ 1 , \ldots , i _ k } \varphi _ k { \left ( x ^ { i _ 1 } , \ldots , x ^ { i _ k } \right ) } \\ & = \frac { 1 } { N ^ k } \sum _ { \substack { i _ 1 , \ldots , i _ k \\ } } \varphi _ k { \left ( x ^ { i _ 1 } , \ldots , x ^ { i _ k } \right ) } + R _ { k , N } , \end{align*}


\begin{align*} \int _ { c _ 1 + \delta } ^ { c _ 2 - \delta } \abs { y _ x } ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} \frac { 2 0 n + 4 } { 2 0 n + 5 } = \frac { 1 } { \lambda } < b + \mu \leq \frac { 1 6 n + 3 } { 1 6 n + 4 } + \epsilon . \end{align*}


\begin{align*} \omega _ j ( t ) = 0 t \leq t _ j . \end{align*}


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


\begin{align*} \theta ( k ) : = \sum _ { k = 1 } ^ 3 \theta ( k _ i ) , \theta ( k _ i ) : = \frac { 4 } { \epsilon ^ 2 } \sin ^ 2 \Big ( \frac { k _ i \epsilon } { 2 } \Big ) , k \in \Lambda ^ * . \end{align*}


\begin{align*} \bar { u } _ { j } ( x ) : = \varepsilon _ { 0 } ^ { s } u _ { j } ( \varepsilon _ { 0 } ( x + 4 e _ { 1 } ) ) , \quad \mathrm { a n d } \ ; \ , \bar { K } _ { j } ( x ) : = \varepsilon _ { 0 } ^ { \sigma p - s ( p - 1 ) ( q - 1 ) } K _ { j } ( \varepsilon _ { 0 } ( x + 4 e _ { 1 } ) ) . \end{align*}


\begin{align*} M = \mbox { s p a n } \{ u ( n ) a | u ( n ) \in V [ g ] \} \end{align*}


\begin{align*} \partial ^ { 2 } _ { x _ { k } x _ { l } } K _ { j } ( 0 ) = 0 , \quad \end{align*}


\begin{align*} \sum _ { { z = 1 } } ^ \infty \frac { 1 } { A ( z ) } < \infty \end{align*}


\begin{align*} [ G , G ] = N _ 1 S \end{align*}


\begin{align*} S _ { 2 , \lambda } ( n + 1 , k ) = S _ { 2 , \lambda } ( n , k - 1 ) + ( k - n \lambda ) S _ { 2 , \lambda } ( n , k ) , \end{align*}


\begin{align*} 1 + \frac { \sum _ { i = 1 } ^ { n } \alpha _ { i } } { \sum _ { i = 1 } ^ { n } \frac { 1 + \beta _ { i } } { 1 - \beta _ { i } } \alpha _ { i } } = \phi . \end{align*}


\begin{align*} \Gamma ^ { k } _ { i j } = \frac { 1 } { 2 } \left ( \sum _ { a = 1 } ^ { 2 } g ^ { k a } \left ( \partial _ { i } g _ { j a } + \partial _ { j } g _ { i a } - \partial _ { a } g _ { i j } \right ) \right ) . \end{align*}


\begin{align*} \left ( r \tilde { \rho } \right ) '' - \dfrac { f ^ { 2 } } { 2 r } \tilde { \rho } = - \dfrac { 1 } { 4 } r \left ( A - B \right ) ^ { 2 } \tilde { \rho } + \dfrac { \lambda } { 2 } r \tilde { \rho } \left ( \tilde { \rho } ^ { 2 } - \dfrac { 2 \mu ^ { 2 } } { \lambda } \right ) \end{align*}


\begin{align*} t _ { k + 1 } & = \min \{ t > t _ { k } : \mathsf { G } ( u ( t _ { k } ) , u ( t ) ) = \lambda \} , \end{align*}


\begin{align*} { \mathfrak m } { \mathfrak c } _ { u , v } : = - \int _ { M } g ( [ u , v ] , [ u , v ] ) \mu - \int _ { M } g ( u , [ [ u , v ] , v ] ) \mu , \end{align*}


\begin{align*} \partial _ t \frac { 1 } { 2 } \| u \| _ { L ^ 2 ( \mathbb { T } ) } ^ 2 & = \frac { 1 } { 2 } \int _ \mathbb { T } \overline { u } \left ( i u _ x x - i | u | ^ { p - 1 } u - \gamma u + i f \right ) + u \left ( - i \overline { u } _ x x + i | u | ^ { p - 1 } \overline { u } - \gamma \overline { u } - i \overline { f } \right ) \ , d x \\ & = - \gamma \| u \| _ { L ^ 2 _ x ( \mathbb { T } ) } ^ 2 + \int _ \mathbb { T } \Im ( u \overline { f } ) \ , d x \end{align*}


\begin{align*} \left < \tau ( \beta ^ S _ h ) \phi , \phi \right > = \frac { 1 } { m _ { G _ S } ( B ^ S _ h ) } \int _ { U } m _ S \big ( B ^ S _ h \cap x p ^ { - 1 } ( U ) ^ { - 1 } \big ) \ , d \nu ( x L _ S ) . \end{align*}


\begin{align*} H _ I = \bigcap _ { j \in I } H _ j . \end{align*}


\begin{align*} \binom { k } { j _ 1 , j _ 2 , \dots , j _ m } 1 j _ 1 + 2 j _ 2 + \dots + m j _ m = n . \end{align*}


\begin{align*} \lim _ { y \to \infty } \sup _ { \substack { A \subset [ y , \infty ) \\ \textnormal { L - p r i m i t i v e } A \not \ni q } } f ( A _ q ) \ = \ \sup _ { \textnormal { L - p r i m i t i v e } A \not \ni q } f ( A _ q ) \ = \ e ^ \gamma { \rm d } ( { \rm L } _ q ) . \end{align*}


\begin{align*} \theta _ { a , b } ( z , \tau ) = \sum _ { m \in \Z ^ g } \exp \left ( i \pi ( m + \tfrac a 2 ) ^ t \tau ( m + \tfrac a 2 ) + 2 i \pi ( m + \tfrac a 2 ) ^ t ( z + \tfrac b 2 ) \right ) . \end{align*}


\begin{align*} \Tilde { \Psi } _ t ^ * g _ { \mathrm { c a n } } = g + O ( t ^ l ) \end{align*}


\begin{align*} \langle \Theta \nabla ^ s u , \nabla ^ s \phi \rangle _ { L ^ 2 ( \R ^ { 2 n } ) } = \langle ( - \Delta ) ^ { s / 2 } ( \gamma ^ { 1 / 2 } u ) , ( - \Delta ) ^ { s / 2 } ( \gamma ^ { 1 / 2 } \phi ) \rangle _ { L ^ 2 ( \R ^ n ) } + \langle q \gamma ^ { 1 / 2 } u , \gamma ^ { 1 / 2 } \phi \rangle _ { L ^ 2 ( \R ^ n ) } \end{align*}


\begin{align*} m ^ { ( i ) } ( x ) = \prod _ { j \in C _ i } ( x - \alpha ^ j ) , \end{align*}


\begin{align*} [ T ^ 0 _ g , J x _ m J ] = & \sum _ n E _ n T ^ { i ( n ) } _ g [ E _ n , J x _ m J ] + \sum _ n [ E _ n , J x _ m J ] T ^ { i ( n ) } _ g E _ n \\ & + \sum _ n E _ n [ T ^ { i ( n ) } _ g , J x _ m J ] E _ n . \end{align*}


\begin{align*} f = \sum _ { D \in F } \left ( \sum _ { \gamma \in I _ D } ( f _ D ) | _ { D _ { \gamma , i _ { \gamma } } } \right ) = \sum _ { D \in F } \left ( \sum _ { i = 1 } ^ k \sum _ { \gamma \in I _ D } ( f _ D ) | _ { D _ { \gamma , i } } \right ) = \sum _ { i = 1 } ^ k \left ( \sum _ { D \in F } \sum _ { \gamma \in I _ D } ( f _ D ) | _ { D _ { \gamma , i } } \right ) . \end{align*}


\begin{align*} \mathbb { V } ( \overline { T } _ n ) = O _ { k , \alpha } ( n ^ { - 1 / 2 + \varepsilon } ) \end{align*}


\begin{align*} g ( x ) : = g ( x , n , s ) = ( - 1 ) ^ n L _ n ^ { ( - n - s - 1 ) } = \displaystyle \sum _ { j = 0 } ^ { n } \binom { n + s - j } { n - j } \frac { x ^ j } { j ! } = \displaystyle \sum _ { j = 0 } ^ { n } b _ j \frac { x ^ j } { j ! } \end{align*}


\begin{align*} 1 + p ' u = \frac { - \frac { 1 } { 2 } \min u } { u - \frac { 1 } { 2 } \min u } < 0 . \end{align*}


\begin{align*} \max _ { I \in \mathcal { B } ( \bigcap _ { i = 1 } ^ m \mathcal { M } _ i ) } | I | \leqslant \min \sum _ { i = 1 } ^ m r _ i ( X _ i ) \end{align*}


\begin{align*} \begin{aligned} \tfrac { 2 L \sigma _ N } { B } = 1 + h _ { N - 1 } \tfrac { 1 } { 2 - ( 1 + \kappa ) h _ { N - 1 } } \ , \ , , \tfrac { \alpha _ { N - 1 } } { B } = \tfrac { 1 - \kappa h _ { N - 1 } } { 2 - ( 1 + \kappa ) h _ { N - 1 } } \end{aligned} \end{align*}


\begin{align*} \left \vert \frac { k _ { \phi } ( z ) } { k ' _ { \phi } ( z ) } \right \vert \leq \frac { r } { m ( r ) } , \vert z \vert = r . \end{align*}


\begin{align*} c _ { d , s } = \begin{cases} \frac { 1 } { Z _ N } e ^ { - d ^ 2 / 4 \sigma _ N ^ 2 } & \\ 0 & \end{cases} \end{align*}


\begin{align*} \label [ I e q ] { e q : c l o s u r e f 2 } f ^ { 2 , N } _ t = f ^ { 1 , N } _ t \otimes f ^ { 1 , N } _ t , \end{align*}


\begin{align*} \tilde { \lambda _ i } = r \sigma ( \lambda _ i ) + s \end{align*}


\begin{align*} \| f \| _ { L ^ p } : = \Big ( \epsilon ^ d \sum _ { x \in \Lambda _ { \epsilon , L } } | f ( x ) | ^ p \Big ) ^ { 1 / p } , \| f \| _ { L ^ \infty } : = \max _ { x \in \Lambda _ { \epsilon , L } } | f ( x ) | , f \in \R ^ { \Lambda _ { \epsilon , L } } . \end{align*}


\begin{align*} f ( z ) = \int _ { 0 } ^ { z } \frac { 1 } { y } \int _ { 0 } ^ { y } G ' ( t ) \psi ( \omega ( t ) ) d t d y . \end{align*}


\begin{align*} \eta \delta _ n ( \beta ) = \sum _ { w \in W _ { \rho } ^ { m _ 0 } } \dfrac { f ' ( v _ { n + 1 } + w ) } { f ( v _ { n + 1 } + w ) } = \operatorname { T } _ { n + 1 , n } ( \delta _ { n + 1 } ( \beta ) ) . \end{align*}


\begin{align*} h ( y ) = - \frac { d ^ 2 } { d x ^ 2 } + V ( x , y ) \end{align*}


\begin{align*} \Psi ^ { n , 6 } _ s = 2 \left ( \int ^ s _ { \eta _ n ( s ) } \left ( s - \eta _ n ( u ) \right ) ^ { \alpha } \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \ , \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) . \end{align*}


\begin{align*} \xi ( A ) = X ^ 2 - s _ 1 X + s _ 2 . \end{align*}


\begin{align*} p ( t , g ) = ( p _ { t , 0 } * G _ t ) ( g ) . \end{align*}


\begin{align*} D = D ^ h + \sum _ { s \geq 0 } D ^ { v _ s } \end{align*}


\begin{align*} P _ { n , s , p } ( \Omega ) = P _ { k , s , p } ( \omega ) . \end{align*}


\begin{align*} \delta ^ { ( s ) } w = \beta w ^ { ( p ^ s ) } , \end{align*}


\begin{align*} j _ n ^ * ( s _ l ) = \begin{cases} s _ l ' , & \\ 0 , & \end{cases} \end{align*}


\begin{align*} & \angle _ { f ( \gamma _ j \gamma ) } ( \xi , f ( e ) ) = \angle _ { f ( \gamma _ j \gamma ) } ( \xi , q _ j ) + \angle _ { f ( \gamma _ j \gamma ) } ( q _ j , f ( e ) ) , \\ & \angle _ { f ( \gamma _ j \gamma ) } ( \xi , q _ j ) = \angle _ { q _ j } ( \xi , f ( \gamma _ j \gamma ) ) = \frac { \pi } { 2 } . \end{align*}


\begin{align*} \delta ( x y ) = x \delta ( y ) + \delta ( x ) y \ \ \ \ \ x , y \in D ( \delta ) . \end{align*}


\begin{align*} \P \{ \theta _ \infty < \infty \} = 0 , \end{align*}


\begin{align*} K _ { \tilde X } = f _ - ^ * K _ { X _ - } + \left ( 1 + \sum _ { i \in M _ - } D _ i \cdot e \right ) E = f _ + ^ * K _ { X _ + } + \left ( 1 - \sum _ { i \in M _ + } D _ i \cdot e \right ) E . \end{align*}


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


\begin{align*} \nabla _ { \vec s , k } L ( t , z ) \alpha = 0 , \end{align*}


\begin{align*} p _ t ( x ) = t ^ { - d / \alpha } p _ 1 ( t ^ { - { 1 / \alpha } } x ) \ , . \end{align*}


\begin{align*} \ell ^ { ( A ) } _ x = \max \Big \{ k \in \Z _ + : \exists t > 0 , j \in \overline { 1 , \eta ( x ) } t \leq \sum \limits ^ { x + S _ { t } ^ { ( x , j ) } } _ { z = x + 1 } \frac { 1 } { A ( z ) } S _ { t } ^ { ( x , j ) } \geq k \Big \} \vee 0 . \end{align*}


\begin{align*} \nabla _ 3 k _ { \alpha 3 } - \nabla _ \alpha k _ { 3 3 } = - \nabla _ 3 \frac { \nabla ^ 2 _ { \alpha 3 } u } { | \nabla u | } + \nabla _ \alpha \frac { \nabla ^ 2 _ { 3 3 } u } { | \nabla u | } = R _ { \alpha 3 3 3 } = 0 \end{align*}


\begin{align*} \frac { \beta \sigma _ k ^ { 1 1 } } { k \sigma _ k } \theta _ 1 ^ 2 \le \frac { \beta \sigma _ k ^ { 1 1 } } { k \sigma _ k } a _ { 1 1 } \le \beta . \end{align*}


\begin{align*} E B ^ 2 ( t , s , x , \omega , v ) = \big ( \sum _ { i = 1 , 2 , 3 } - ( \hat { v } + \omega ) _ 2 ( \omega \times \mathbf { e } _ 3 ) \cdot \mathbf { e } _ i ( B \cdot ( \omega \times \mathbf { e } _ i ) ) - ( \hat { v } + \omega ) _ 3 B _ 2 , \sum _ { i = 1 , 2 , 3 } ( \hat { v } + \omega ) _ 1 ( \omega \times \mathbf { e } _ 3 ) \cdot \mathbf { e } _ i ( B \cdot ( \omega \times \mathbf { e } _ i ) ) \end{align*}


\begin{align*} R ^ 2 - R _ 1 ^ 2 = p p ^ * . \end{align*}


\begin{align*} \Psi ( E _ j ( x , a ) ) & = 4 \sqrt { a } \sum _ { h = 1 } ^ { \lfloor \frac { j } { 2 } \rfloor } \frac { \zeta ^ h + \zeta ^ { - h } } { 2 } \\ & = 2 \sqrt { a } \sum _ { h = 1 } ^ { \lfloor \frac { j } { 2 } \rfloor } \left ( \zeta ^ h + \zeta ^ { - h } \right ) \\ & = 2 \sqrt { a } \left ( \left ( \sum _ { h = - \lfloor \frac { j } { 2 } \rfloor } ^ { \lfloor \frac { j } { 2 } \rfloor } \zeta ^ h \right ) - 1 \right ) . \end{align*}


\begin{align*} p _ k ( t , g ) = e ^ { i k ( \frac { 2 \pi } { V _ F } ) g } ( p _ { t , k } \ast G _ { t , k } ) ( g ) . \end{align*}


\begin{align*} \nu = \mathcal { D } \Big ( \Big ( \frac { 1 } { \beta } \Big ) ^ { 1 / p } \sum _ { j = 1 } ^ \infty A _ { \bullet , j } Z _ j + \sigma _ { p , \beta } ( I _ k - A A ^ T ) ^ { 1 / 2 } N _ k \Big ) \end{align*}


\begin{align*} \begin{aligned} & S ( p ) ^ d \ ( \prod _ { j = 1 } ^ d { \lambda + k _ j \choose k _ j } \ ) \abs { W ( ( \pi _ j ) _ j , \alpha ) - W ( ( \pi _ j ) _ j , \alpha ^ \prime ) } \\ & = \abs { \prod _ { j = 1 } ^ d \ ( \tilde S ( p , \pi _ j , \alpha ^ \prime , \lambda ) - ( \tilde S ( p , \pi _ j , \alpha ^ \prime , \lambda ) - \tilde S ( p , \pi _ j , \alpha , \lambda ) ) \ ) - \prod _ { j = 1 } ^ d \tilde S ( p , \pi _ j , \alpha ^ \prime , \lambda ) } . \end{aligned} \end{align*}


\begin{align*} ( - 1 ) ^ { ( P _ 2 z - 1 ) / 2 + ( x - 1 ) / 2 + ( y - 1 ) / 2 } \left ( \frac { x y } { P _ 2 z } \right ) \left ( \frac { P _ 2 z x } { y } \right ) \left ( \frac { P _ 2 z y } { x } \right ) = 1 . \end{align*}


\begin{gather*} \frac { | C _ { \varphi } ( D ) | ^ 2 } { \langle \varphi , \varphi \rangle } = \frac { \sqrt { | D | } } { 2 \pi } \frac { L ( f \otimes D , 1 ) } { \langle f , f \rangle } \end{gather*}


\begin{align*} D = \{ x + i y : x \in ( \alpha - \varepsilon , \alpha + \varepsilon ) | y - f ( x ) | < \delta \} , \end{align*}


\begin{align*} \pi \wedge \rho : = \{ V \cap W \mid V \in \pi , \ , W \in \rho , \ , V \cap W \neq \emptyset \} . \end{align*}


\begin{align*} g ( x ) \ge 0 , \int _ 0 ^ 1 g ( x ) d x = 1 , \end{align*}


\begin{align*} \widehat { T } [ W _ t u _ 0 , v , \cdots , v ] _ k = \sum _ { \ell \ , \ , o d d } \widehat { T _ \ell } [ v , \cdots , v , \underset { \ell ' t h } { \underbrace { W _ t u _ 0 } } , v , \cdots , v ] _ k = \frac { p + 1 } { 2 } \widehat { T _ 1 } [ W _ t u _ 0 , v , \cdots , v ] _ k , \end{align*}


\begin{align*} \begin{aligned} \big [ 2 - h ( 1 + \kappa ) \big ] \big ( f _ { 0 } - f _ { 1 } \big ) { } \geq { } \frac { h } { 2 L } \big [ \kappa h ^ 2 - 2 h ( 1 + \kappa ) + 3 \big ] \big \| g _ { 0 } \big \| ^ 2 + \frac { h } { 2 L } \| g _ 1 \| ^ 2 \end{aligned} \end{align*}


\begin{align*} \int _ G f ( x ) \ , d \mu _ G ( x ) = \Delta _ G ( y ) \int _ G f ( x y ) \ , d \mu _ G ( x ) , \end{align*}


\begin{align*} \eta ^ 2 ( \mathcal { T } , \mathcal { M } ) : = \big ( \eta ( \mathcal { T } , \mathcal { M } ) \big ) ^ 2 : = \sum _ { T \in \mathcal { M } } \eta ^ 2 ( \mathcal { T } , T ) \quad \eta ^ 2 ( \mathcal { T } ) : = \eta ( \mathcal { T } , \mathcal { T } ) . \end{align*}


\begin{align*} w _ 2 ( \pi ) & = \left ( \sum _ { 1 \leq i < j \leq n } N _ i N _ j c _ i c _ j + \sum _ { i = 1 } ^ { n } N _ i \binom { c _ i } { 2 } + \sum _ { i = 1 } ^ { n } c _ i ^ 2 \binom { N _ i } { 2 } \right ) \left ( \sum _ { t = 1 } ^ { n } v _ t ^ 2 \right ) \\ & + \left ( \sum _ { i = 1 } ^ { n } c _ i N _ i '' + 2 \sum \limits _ { 1 \leq i < j \leq n } N _ i N _ j c _ i c _ j \right ) \cdot \left ( \sum _ { k < l } v _ k v _ l \right ) . \end{align*}


\begin{align*} \lim \limits _ { t \rightarrow \infty } \| \chi _ { S _ { v t } ^ c } e ^ { - i t H } \psi \| = 0 \end{align*}


\begin{align*} \widetilde { ( B + C ) } T ( t , A ) x = ( \tilde { B } + \tilde { C } ) T ( t , A ) x \forall x \in \mathcal { H } , \end{align*}


\begin{align*} \boldsymbol { \theta } ^ { * } & = \underset { \boldsymbol { \theta } } { } \ln p ( \boldsymbol { y } | \boldsymbol { \theta } ) . \end{align*}


\begin{align*} \psi _ { k } ( x ) : = \tilde { \psi } ( | x | / 2 ^ k ) - \tilde { \psi } ( | x | / 2 ^ { k - 1 } ) , \varphi _ 0 ( x ) : = \tilde { \psi } ( | x | ) , \quad \forall j \in ( 0 , \infty ) \cap \Z , \varphi _ j ( x ) : = \psi _ j ( x ) . \end{align*}


\begin{align*} \mathcal { W } = \{ x _ n \} _ { n \in \mathbb { N } } \cup \{ u _ n \} _ { n \in \mathbb { N } } \end{align*}


\begin{align*} \int _ L \Omega = \int _ { \mathcal { M } } \tilde { \Omega } _ { L } , \int _ { L _ i } \Omega = \int _ { \mathcal { M } } \tilde { \Omega } _ { L _ i } , i = 1 , 2 . \end{align*}


\begin{align*} 0 \in \begin{bmatrix} 0 & A ^ T \\ - A & 0 \end{bmatrix} \begin{bmatrix} x \\ z \end{bmatrix} + \begin{bmatrix} \partial f ( x ) + \nabla h ( x ) \\ \partial g ^ \ast ( z ) \end{bmatrix} . \end{align*}


\begin{align*} e ^ { \max } = \max _ { j = 1 , \dots , N } \| P _ h u ( t _ j ) - U _ j \| _ { L ^ 2 ( \Omega ) } , \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { 1 } { x } \int _ 1 ^ \infty [ f ( e ^ { - y } x ) - f ( x ) + ( e ^ { - y } - 1 ) x f ' ( x ) ] \Pi ( d y ) = 0 . \end{align*}


\begin{align*} \alpha _ { x , y } : = \psi _ { x , y } ^ { - 1 } \circ \psi _ { y , x } \circ \iota : \mathcal O _ { x , y } \to \mathcal O _ { x , y } \ , \end{align*}


\begin{align*} g ( f ( x _ 1 , \ldots , x _ k ) , f ( x _ 2 , \ldots , x _ { k + 1 } ) , \ldots ) = x _ 1 . \end{align*}


\begin{align*} V = \{ a _ 0 + a _ 1 z + a _ 2 z ^ 2 B ( z ) + a _ 3 z ^ 3 B ( z ) + \cdots \ : : \ : \sum \abs { a _ j } ^ 2 < \infty \} \end{align*}


\begin{align*} ( - \Delta ) ^ s u ( x ) = - \frac { C _ { n , s } } { 2 } \int _ { \R ^ n } \frac { \delta u ( x , y ) } { | y | ^ { n + 2 s } } \ , d y \quad x \in \R ^ n . \end{align*}


\begin{align*} g ^ { - 1 } = ( g ^ { i j } ) = \begin{pmatrix} 1 & 0 \\ 0 & c _ { 1 } ^ { - 2 } \end{pmatrix} . \end{align*}


\begin{align*} s ( A ) = \{ f \in A : g \in A \mbox { f o r a l l h o l o m o r p h i c } g \mbox { w i t h } | \hat { g } ( k ) | \leq | \hat { f } ( k ) | \mbox { f o r a l l } k \} \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { \lambda _ N ^ 2 } \prod \limits _ { k = 1 } ^ { \infty } \Bigg ( \frac { \lambda _ N ^ 2 + \mu _ { k } ^ { 2 } } { \lambda _ N ^ 2 + \lambda _ { k } ^ { 2 } } \Bigg ) = \infty \end{align*}


\begin{align*} \label [ I e q ] { e q : F P g e n e r a t o r } \forall \varphi \in C ^ 2 _ b ( \R ^ d ) , L _ \mu \varphi ( x ) : = b ( x , \mu ) \cdot \nabla \varphi + \frac { 1 } { 2 } \sum _ { i , j = 1 } ^ d a _ { i j } ( x , \mu ) \partial _ { x _ i } \partial _ { x _ j } \varphi , \end{align*}


\begin{align*} G ( v , x ) = \frac { F } { \bigtriangleup + i \gamma } . \end{align*}


\begin{align*} \begin{aligned} \mathsf { T } _ { 2 } ( t _ { k } ) & = \frac { \phi \lambda - \sum _ { i = 1 } ^ { n } \beta _ { i } u _ { i } ( t _ { k } ) } { \sum _ { i = 1 } ^ { n } \alpha _ { i } } , \\ u _ i ( t _ { k + 1 } ) & = \beta _ { i } u _ i ( t _ { k } ) + \alpha _ { i } \mathsf { T } _ { 2 } ( t _ { k } ) . \end{aligned} \end{align*}


\begin{align*} \mathcal { P } : & \underset { \left \{ \boldsymbol { v } _ { 2 , p } \right \} _ { p = 1 } ^ { P _ { 2 } } } { \max } \lambda \\ & \textrm { t r } \left ( \boldsymbol { v } _ { 2 , p } \boldsymbol { v } _ { 2 , p } ^ { H } \right ) \leq P _ { t } , \\ & \mathbf { J } _ { \textrm { e f f } } \left ( \left \{ \boldsymbol { v } _ { 2 , p } \right \} _ { p = 1 } ^ { P _ { 2 } } \right ) \succeq \lambda \mathbf { I } , \end{align*}


\begin{align*} w _ 2 = \sum _ { j = 0 } ^ J a _ j ( \tau ) e _ j ( z ) + w _ 2 ^ \bot a _ j ( \tau ) = ( w _ 2 ( \cdot , \tau ) , e _ j ) _ \rho . \end{align*}


\begin{align*} ( C _ R ( J ) ) ^ h = C _ { R [ y ] } ( J ^ h ) . \end{align*}


\begin{align*} { \bar \Omega } _ { A B } - k _ A k _ B \theta _ A \wedge \theta _ B = \Omega _ { A B } \end{align*}


\begin{align*} \label [ I e q ] { e q : O m e g a k } \Omega _ k \left ( f ^ N , f \right ) : = W _ p { \left ( f ^ { k , N } , f ^ { \otimes k } \right ) } \underset { N \to + \infty } { \longrightarrow } 0 . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { x \in M , \| x \| \leq 1 } | \langle x \xi u _ n , \eta \rangle | = 0 . \end{align*}


\begin{align*} e ^ { \max } _ { } = \max _ { j = 1 , \dots , N } E ^ 2 _ { } ( t _ j ) , \end{align*}


\begin{align*} \mathcal { G } _ { l ( T ) } & = x , \\ \mathcal { G } _ j & = x - \dfrac { c _ j } { \mathcal { G } _ { j + 1 } } ( \forall j \in \overline { 1 , l ( T ) - 1 } ) , \end{align*}


\begin{align*} \abs { m } = \abs { m _ 1 } + \cdots + \abs { m _ n } , \end{align*}


\begin{align*} \nabla _ \beta k _ { \alpha \beta } - \nabla _ \alpha k _ { \beta \beta } + \nabla _ 3 k _ { \alpha 3 } - \nabla _ \alpha k _ { 3 3 } = 0 . \end{align*}


\begin{align*} y ^ d \mapsto \prod _ { i = 1 } ^ { \mathrm { r } } \tilde y _ i ^ { p _ i ^ - \cdot d } . \end{align*}


\begin{align*} \forall \varphi \in \R ^ \Lambda , V _ t ( \varphi ) = - \log { \bf E } _ { C _ t } \Big [ e ^ { - V _ 0 ( \varphi + \cdot ) } \Big ] . \end{align*}


\begin{align*} \emph { c o l s u m } _ 0 ( M ) = \mu _ { 1 , 0 } + \mu _ { 2 , 0 } = n . \end{align*}


\begin{align*} ( ( - \Delta ) ^ s + q ) v & = F \quad \quad \Omega , \\ v & = f \quad \quad \Omega _ e , \end{align*}


\begin{align*} d \sigma _ { \mu } ^ { + } ( s ) = p c \ , s ^ { - ( a + 1 ) } \ , d \lambda ( s ) , \quad 1 \leq s < + \infty , \end{align*}


\begin{align*} \begin{aligned} h _ * = \arg \max _ { 1 { } \leq { } h < 2 } 2 h + \frac { \kappa h ^ 3 } { 2 - h \left ( 1 + \kappa \right ) } \end{aligned} , \end{align*}


\begin{align*} Y ^ { n , 1 } _ t = n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { t } ( t - s ) ^ \alpha \xi ^ n _ s d W _ s , \end{align*}


\begin{align*} \begin{aligned} \frac { d B ( t ) } { d t } & = g _ 0 + g _ 1 N ( B , C ) - B ( t ) + g _ 1 \epsilon ( t ) , \\ \frac { d C ( t ) } { d t } & = 2 a _ 0 + 2 a _ 1 N ( B , C ) - 2 C ( t ) + 2 a _ 1 \epsilon ( t ) . \end{aligned} \end{align*}


\begin{align*} \frac { d F } { d t } = [ F , H ] + \frac { \partial F } { \partial t } \ , \end{align*}


\begin{align*} \label [ I e q ] { e q : L N h a t } \widehat { \mathcal { L } } _ N \Phi ( \mu _ { \mathbf { x } ^ N } ) = \mathcal { L } _ N [ \Phi \circ \boldsymbol { \mu } _ N ] ( \mathbf { x } ^ N ) . \end{align*}


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


\begin{align*} \label [ I e q ] { e q : r e c u r s i v e j u m p i n g t i m e s } t _ { k + 1 } = \max _ { \ell , p , n } \big \{ T ^ { i _ \ell , p } _ n \ , | \ , T ^ { i _ \ell , p } _ n < t _ k , \ , \ell \leq k \big \} . \end{align*}


\begin{align*} v ' = v - ( u \cdot \omega ) \omega , \mbox { a n d } v _ * ' = v _ * - ( u \cdot \omega ) \omega . \end{align*}


\begin{align*} L _ { Z } \cdot C = \int _ { C } R _ { h _ { Z } } \geq 0 . \end{align*}


\begin{align*} \mathcal { D } _ { [ n , j , k ] } : = \{ \gamma \in \mathcal { D } _ { [ n , j ] } : \kappa _ { 0 } ( \gamma ) = k \} , 0 \leq k \leq n - j . \end{align*}


\begin{align*} \lvert v _ { i , j } \rvert ^ 2 \prod _ { k = 1 ; k \ne i } ^ { n - 1 } ( { \lambda _ i } ( A ) - { \lambda _ k } ( A ) ) = \prod _ { k = 1 } ^ { n - 1 } ( { \lambda _ i } ( A ) - { \lambda _ k } ( M _ j ) ) . \end{align*}


\begin{align*} \norm { f _ m } ^ 2 = \norm { g _ 0 } ^ 2 + \norm { g _ 1 } ^ 2 + \cdots + \norm { g _ m } ^ 2 \leq \norm { r h } ^ 2 \end{align*}


\begin{align*} p _ i : = x _ { a _ i b _ i } \prod _ { j = 1 } ^ { \ell _ i - 1 } \left ( - \sum _ { u \in S ( w _ { i j } ) } x _ { \phi ( u ) } + \sum _ { u \in S ( v _ i ) } x _ { \phi ( u ) } \right ) . \end{align*}


\begin{align*} | \nabla H _ { \lambda } | \leq \frac { C ' } { | x | ^ \frac { N - 1 } { p - 1 } } = C | \nabla \Gamma | \hbox { o n } \partial B _ R ( 0 ) \end{align*}


\begin{align*} \sqrt { 1 + 1 6 i m \beta \lambda / 3 } = \pm \left ( \sqrt { \frac { \sqrt { 1 ^ { 2 } + ( 1 6 m \beta \lambda / 3 ) ^ { 2 } } + 1 } { 2 } } + i \sqrt { \frac { \sqrt { 1 ^ { 2 } + ( 1 6 m \beta \lambda / 3 ) ^ { 2 } } - 1 } { 2 } } \right ) \end{align*}


\begin{align*} C ( v ) = \frac { \sigma _ Z ^ 2 v ^ 2 } { 2 } . \end{align*}


\begin{align*} d \nu ^ i _ t : = d \nu ^ i _ { x _ i , s _ i \ , | \ , t } = v _ { i , t } \ , d g _ { i , t } , \end{align*}


\begin{align*} d ^ \gamma ( { \mathcal K } , Y _ n ) _ X : = \inf _ { \Phi _ n } \sup _ { f \in { \mathcal K } } \inf _ { y \in B _ { Y _ n } } \| f - \Phi _ n ( y ) \| _ X , \end{align*}


\begin{align*} \langle \nabla u _ { q , r } , \nabla \nu \rangle = \frac { 1 + 2 \delta } { ( 1 + \delta ) ^ 2 } \xi ^ 2 _ { \tilde K } \langle \nabla w , \nabla w \rangle + ( 1 + u _ { q , r } ) ^ { 2 \delta + 1 } \langle \nabla u _ { q , r } , \nabla ( \xi _ { \tilde K } ) ^ 2 \rangle , \end{align*}


\begin{align*} \times \varphi _ { \tilde { j } , \tilde { l } } ( u , \tilde { \omega } ) d \omega d \tilde { \omega } d u d s d \tau \lesssim \sum _ { \kappa \in [ - 1 0 M _ t , \epsilon M _ t ] \cap \Z } H _ { \kappa } ( t , x ) , H _ { \kappa } ( t , x ) : = \int _ 0 ^ t \int _ { \tau } ^ { t } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \int _ { \mathbb { S } ^ 2 } f ( \tau , x + ( t - s ) \omega + ( s - \tau ) \tilde { \omega } , u ) \end{align*}


\begin{align*} \lim _ { t \to \infty } L _ t = L _ 1 + \ldots L _ N . \end{align*}


\begin{align*} \sigma _ j = \sigma ^ * - \frac { \sigma ^ * j } { 2 k } , j = 1 , \ldots , k . \end{align*}


\begin{align*} \ i _ * = \ j _ * . \end{align*}


\begin{align*} ( X , W ) = ( \C ^ n , W = \sum _ i x _ i ^ { r _ i } ) . \end{align*}


\begin{align*} ( L y ) ( v ) & = - \sum _ { e \in E _ V } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } \frac { 1 } { | e | ^ 2 } \sum _ { u \in e \cap w } | V _ 2 | \\ & = - k ( v ) | V _ 2 | \end{align*}


\begin{align*} C _ 1 & : = \{ ( x , y , z ) ^ \top \in \mathbb { R } ^ 3 \colon 0 \leq x , 0 \leq y , 0 \leq z \leq x + y \} , \\ C _ 2 & : = \{ ( x , y , z ) ^ \top \in \mathbb { R } ^ 3 \colon 0 \leq x , 0 \geq y , y \leq z \leq x \} , \\ C _ 3 & : = \{ ( 0 , 0 , z ) ^ \top \in \mathbb { R } ^ 3 \colon 0 \leq z \} \end{align*}


\begin{align*} N ^ { n , 4 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ^ 2 ( u , s ) d u \ , d s . \end{align*}


\begin{align*} \theta _ 1 ( t ) = \sum _ { n = \ell + 1 } ^ \infty \psi _ { 1 } ^ { [ * n ] } ( t ) , \end{align*}


\begin{align*} - \Delta _ p z _ n = a _ n | \hat { u } _ n | ^ { p - 2 } \hat { u } _ n + \hat { f } _ n \hbox { i n } \Omega . \end{align*}


\begin{align*} I S [ x ] ^ { [ q ] } : I S [ x ] = ( I ^ { [ q ] } : I ) S [ x ] . \end{align*}


\begin{align*} h ( \R \times \{ - \infty \} ) & = W _ 1 , h ( \{ + \infty \} \times \R ) = W _ 2 , \\ h ( \{ - \infty \} \times \R ) & = W _ 1 ' , h ( \R \times \{ + \infty \} ) = W _ 2 ' . \end{align*}


\begin{align*} K _ { \emph { Z } } ^ { o r b } = \varphi ^ { \ast } ( K _ { Z } + [ \Delta ] ) . \end{align*}


\begin{align*} \sigma _ { m , k ; n , \ell } : = & \left ( 1 + \frac { m } { k } \right ) \left ( 1 + \frac { n } { \ell } \right ) \sigma _ { m + k - 1 , n + \ell - 1 } - \left ( 1 + \frac { m } { k } \right ) \frac { n } { \ell } \sigma _ { m + k - 1 , n - 1 } \\ & - \frac { m } { k } \left ( 1 + \frac { n } { \ell } \right ) \sigma _ { m - 1 , n + \ell - 1 } + \frac { m n } { k \ell } \sigma _ { m - 1 , n - 1 } , \end{align*}


\begin{align*} \int f ( x ) h ( x / q ) S _ q ( x ) y ( x ) d _ q x = f ( x / q ) h ( x / q ) \left ( y ( x / q ) u ( x / q ) - D _ { q ^ { - 1 } } y ( x ) \right ) , \end{align*}


\begin{align*} \overline { \widetilde { N } } _ f & = \overline { M _ f S _ \zeta } = \overline { M } _ f S _ \zeta = M _ f S _ \zeta + \sum _ { g \prec f } r _ { g f } M _ g S _ \zeta = \widetilde { N } _ f + \sum _ { g \prec f ; g \in \Z ^ { m | n } _ { \zeta - } } r ' _ { g f } \widetilde { N } _ g . \end{align*}


\begin{align*} L u = \textrm { d i v } ( A \nabla u ) , A = ( a _ { i j } ( x ) ) _ { i , j = 1 } ^ n , a _ { i j } = a _ { j i } \in L ^ \infty ( \Omega ) , \end{align*}


\begin{align*} \psi _ n ( x ) = e ^ { - \gamma R } \lambda _ n ^ { \frac 1 { p - 2 } } e ^ { - \gamma \ , \lambda _ n ^ { \frac { 1 } { 2 } } | x - P ^ i _ n | } , \end{align*}


\begin{align*} \int _ \mathbb { R } f \circ ( \phi _ 1 ^ * * \phi _ 2 ^ * ) ( x ) d x - \int _ \mathbb { R } f \circ ( \phi _ 1 * \phi _ 2 ) ( x ) d x = 4 ( \lambda f ( y _ 1 ) + ( 1 - \lambda ) f ( y _ 2 ) - f ( \lambda y _ 1 + ( 1 - \lambda ) y _ 2 ) ) . \end{align*}


\begin{align*} F ( k , s , t ) & = \min \{ N \mid \exists \ , \mathit { O A } ( N , k , s , t ) \} , \\ F ^ * ( k , s , t ) & = \min \{ N \mid \exists \mathit { O A } ( N , k , s , t ) \} . \\ \end{align*}


\begin{align*} & \left ( 4 \sum _ { e \in E } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum \limits _ { \{ u , v \} \subset e \cap V _ 1 } y ( u ) y ( v ) + \alpha \right ) \\ & = \left ( 2 \sum _ { e \in E } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum \limits _ { \{ u , v \} \subset e \cap V _ 1 } y ( u ) y ( v ) + \sum _ { e \in \partial V _ 1 } \frac { \delta _ E ( e ) } { | e | ^ 2 } \sum \limits _ { u \in e \cap V _ 2 ; v \in e \cap V _ 1 } z _ 2 ( u ) z _ 2 ( v ) \right ) \ge 0 . \end{align*}


\begin{align*} & v _ a = \frac { 1 + \sigma _ a } 2 - \frac { \epsilon _ a } { \omega } , & & a = 1 , \ldots , k - 1 , \end{align*}


\begin{align*} \lambda _ * H ^ { ( 1 ) } ( \mu _ N ) - \lambda _ * \Psi ( \zeta ^ { \binom { N } { 2 } } ) \cong \Psi _ * \binom { N } { 2 } , \end{align*}


\begin{align*} \nu = a + 2 k \end{align*}


\begin{align*} J ^ { e s s ; 1 } _ { k , m ' } ( t , x ) : = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \mathbb { S } ^ 2 } \big ( E ( s , x - y + ( t - s ) \omega ) + \hat { v } \times B ( s , x - y + ( t - s ) \omega ) \big ) \cdot \nabla _ v \mathcal { K } _ { k , j } ^ { e s s } ( y , \omega , v ) \end{align*}


\begin{align*} E ( t ) = \frac { 1 } { 2 } \int _ 0 ^ L \left ( \abs { u _ t } ^ 2 + \abs { \nabla u } ^ 2 + \abs { y _ t } ^ 2 + \abs { \nabla y } ^ 2 \right ) d x . \end{align*}


\begin{align*} { \bar \Omega } _ { A B } = d \theta _ { A B } - \sum \theta _ { A C } \wedge \theta _ { C B } \end{align*}


\begin{align*} ( \widehat { u } , \widehat { v } , \widehat { h } , \widehat { g } , \widehat { p } ) = ( u _ 1 - u _ 2 , v _ 1 - v _ 2 , h _ 1 - h _ 2 , g _ 1 - g _ 2 , p _ 1 - p _ 2 ) , \end{align*}


\begin{align*} \widetilde { K } ^ { m ; p , q ; \tilde { m } } _ { k , j ; n , l ; \tilde { k } ; \tilde { j } , \tilde { l } } ( t , x , \zeta ) : = \int _ 0 ^ t \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ 0 ^ { 2 \pi } \int _ 0 ^ { \pi } 2 ^ { m + 3 k + 2 n - j + \max \{ p , l \} + 3 \epsilon M _ t } f ( s , x - y + ( t - s ) \omega , v ) \big | B ^ { \tilde { m } } _ { \tilde { k } ; \tilde { j } , \tilde { l } } ( s , x - y + ( t - s ) \omega ) \big | \end{align*}


\begin{align*} \widetilde { H } ( \widetilde { \Theta } _ n ( \phi _ 1 ) ) = \det ( \phi _ 1 ) ^ { n + 1 } ( B _ { - \delta / 2 } \circ \tilde { \iota } _ { \phi _ 1 } \circ B _ { \delta / 2 } ) . \end{align*}


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


\begin{align*} \lim _ { n \rightarrow \infty } \bigg | \frac { H _ n ( 1 - s ) } { H _ n ( s ) } \bigg | = 1 . \end{align*}


\begin{align*} U = \frac { 2 L \Delta } { 1 + \sum \limits _ { i = 0 } ^ { N - 1 } \left [ 2 h _ i - h _ i ^ 2 \frac { - \kappa } { 2 \min \left ( 1 , \frac { 1 } { h _ i } \right ) - ( 1 + \kappa ) } \right ] } \end{align*}


\begin{align*} \pi _ { n } ( z ) = \frac { Q _ { n } ( z ) } { P _ { n } ( z ) } \end{align*}


\begin{align*} N ( \ell ) \lesssim n ^ \ell ( d + 1 ) ^ { k ^ 2 \delta _ { } } = n ^ { \ell + o ( 1 ) } . \end{align*}


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


\begin{align*} b \widetilde { z } + \sum _ { j = 1 } ^ m Q _ { c _ j } ( \alpha _ j , A _ j , x ^ k _ j + a _ j f _ { p ( j ) } ) \subseteq Q _ b ( \alpha , A , z + a f _ k ) . \end{align*}


\begin{align*} \Psi _ n = \{ Q \in K [ x ] \mid Q \mbox { i s a k e y p o l y n o m i a l f o r } \nu \mbox { a n d } \deg ( Q ) = n \} . \end{align*}


\begin{align*} M _ f S _ 0 & = M _ f \sum _ { \sigma \in W _ f } q ^ { \ell ( w _ 0 ^ f ) - \ell ( \sigma ) } H _ \sigma \sum _ { \tau \in W ^ f } q ^ { \ell ( ^ f w _ 0 ) - \ell ( \tau ) } H _ \tau \\ & = M _ f \sum _ { \sigma \in W _ f } q ^ { \ell ( w _ 0 ^ f ) - 2 \ell ( \sigma ) } \sum _ { \tau \in W ^ f } q ^ { \ell ( ^ f w _ 0 ) - \ell ( \tau ) } H _ \tau \\ & = [ | W _ f | ] M _ f \sum _ { \tau \in W ^ f } q ^ { \ell ( ^ f w _ 0 ) - \ell ( \tau ) } H _ \tau \\ & = [ | W _ f | ] \sum _ { \tau \in W ^ f } q ^ { \ell ( ^ f w _ 0 ) - \ell ( \tau ) } M _ { f \cdot \tau } . \end{align*}


\begin{align*} \begin{pmatrix} - \frac { | R _ 1 | - 2 } { 3 } & 0 & 0 & \cdots & 0 \end{pmatrix} . \end{align*}


\begin{align*} \tilde { M } _ { n } ( t ) = \sup _ { s \in [ 0 , t ] } M _ { n } ( s ) + ( 1 + t ) ^ { n ^ 3 } \lesssim \big ( \tilde { M } _ { n } ( t ) \big ) ^ { 1 - \epsilon } + ( 1 + t ) ^ { n ^ 3 } , \Longrightarrow \tilde { M } _ { n } ( t ) \lesssim ( 1 + t ) ^ { n ^ 3 } . \end{align*}


\begin{align*} z _ { n _ j } \in B ^ { n _ j } , \ , \ , j = 1 , \ldots , k \end{align*}


\begin{align*} \overline { U } ( t , r ) = c e ^ { - k t } w ( \dfrac { r } { \overline { s } _ { 1 } ( t ) } ) , \ 0 \leq r < \overline { s } _ { 1 } ( t ) , t \geq 0 \ \ \overline { V } ( t , r ) = d e ^ { - h t } w ( \dfrac { r } { \overline { s } _ { 2 } ( t ) } ) , \ 0 \leq r < \overline { s } _ { 2 } ( t ) , t \geq 0 , \end{align*}


\begin{align*} L _ { \alpha , \beta } = L \left ( H ^ { * } ( \overline { \alpha } \otimes \overline { \beta } ) \right ) . \end{align*}


\begin{align*} \mathrm { D T E } _ { ( p , q ) } ( X ) = \mu + \sigma \frac { \phi ( \xi _ { p } ) - \phi ( \xi _ { q } ) } { F _ { Y } ( \xi _ { p } , \xi _ { q } ) } , \end{align*}


\begin{align*} \begin{array} { l l l l } d = d _ - & \tilde d = d _ \mathrm { p c v } & \tilde \phi = \phi _ \mathrm { p c v } & \mbox { f o r B r e g m a n p r i m a l C o n d a t - - V \ ~ u ~ \eqref { e - b c v } , } \\ d = d _ + & \tilde d = d _ \mathrm { d c v } & \tilde \phi = \phi _ \mathrm { d c v } & \mbox { f o r B r e g m a n d u a l C o n d a t - - V \ ~ u ~ \eqref { e - b c v - d u a l } . } \end{array} \end{align*}


\begin{align*} \Delta _ h \psi ( x ) = \sum _ { i = 1 } ^ { N } \frac { \psi ( x + h e _ i ) + \psi ( x - h e _ i ) - 2 \psi ( x ) } { h ^ 2 } . \end{align*}


\begin{align*} \pi _ J = 0 . \end{align*}


\begin{align*} | A | = q ^ { n - 1 - \lfloor \frac { n - 1 } { k } \rfloor } , \end{align*}


\begin{align*} 0 = \frac { \partial h ( \tau , p ( \tau ; \cdot ) ) } { \partial x } \left [ \frac { \partial p ( \tau ; \cdot ) } { \partial t } + \frac { \partial p ( \tau ; \cdot ) } { \partial x } \frac { \partial p ( t ; \cdot ) } { \partial \tau } \right ] \ , , \end{align*}


\begin{align*} X _ 1 = \frac { 1 } { \sqrt { 1 + \epsilon } } \begin{pmatrix} 1 & 1 \\ \epsilon ^ { \frac { 1 } { 2 } } & - \epsilon ^ { \frac { 1 } { 2 } } \\ 0 & 0 \end{pmatrix} , X _ 2 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} . \end{align*}


\begin{align*} G ( y ) : = \sum _ { j = 1 } ^ m \max \{ F ( x ) : Z ( y ) = h ^ 1 , h ^ 2 , \ldots , h ^ m \land x \leq h ^ j \land x \in P ( \mathcal { F } ) \} . \end{align*}


\begin{align*} \textstyle a b = a _ 0 b _ 0 + \sum _ { i = 1 } ^ r a _ i b _ i + \sum _ { j = 1 } ^ s a _ { r + j } b _ { r + j } . \end{align*}


\begin{align*} [ t ^ k ] \circ \theta ^ { n } _ t ( t ^ m ) = m ^ n t ^ { m + k } \enspace . \end{align*}


\begin{align*} \lim _ { n \to \infty } \sup _ { x \in M , \| x \| \leq 1 } | \langle x \xi y _ n , \xi \rangle | = 0 , \end{align*}


\begin{align*} \omega _ { t + 1 } = P _ { \Omega } \left ( ( I + \alpha _ t A _ t ) \omega _ t - \alpha _ t b _ t ) \right ) . \end{align*}


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


\begin{align*} \int \limits _ 0 ^ T \int \limits _ { V } \eta p ( \rho _ m ) ( \psi \rho _ m - \psi \alpha _ m ) \ d x \ d t = J _ 1 + J _ 2 , \end{align*}


\begin{align*} ( x , y ) \cdot ( x ' , y ' ) = ( x + x ' , y + y ' + ( x x '^ T - x ' x ^ T ) ) \end{align*}


\begin{align*} \Theta _ { \sf a } & = \Theta _ J - w , \\ P _ 1 \theta & = \Delta _ x \theta - q { \sf U } _ \infty ^ { q - 1 } \theta . \end{align*}


\begin{align*} \mu \int _ { \Omega } \varepsilon y _ \delta : \varepsilon v + \nu \int _ { \Omega } \mathsf { m } _ \delta ( \varepsilon y _ \delta ) : \varepsilon v = \int _ { \Omega } u \cdot v , \forall v \in Y . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \log p _ n ( X _ n , X _ 0 ) } { \log \mathbb { G } ( X _ n , X _ 0 ) } = 1 \end{align*}


\begin{align*} \sum _ { i \in M _ C } \sum _ { j = 1 } ^ { r _ i - 1 } a _ { i r _ i } x _ { i j } + \sum _ { i \in M _ C } \sum _ { j = r _ i } ^ { n _ i } \max \left \{ a _ { i j } , b - \sum _ { k \in M _ C - i } a _ { k r _ k } \right \} x _ { i j } \leq b \end{align*}


\begin{align*} \mathbb { E } _ q ( x ) = \prod _ { k \geq 0 } ( 1 - x q ^ k ) , \end{align*}


\begin{align*} | \psi | ( x , u , p ) = \frac { f ( x ) } { f ^ * \circ Y ( x , u , p ) } , \end{align*}


\begin{align*} S = \left \{ s \in \mathbb { R } : ( - \Im G _ { \mu \boxplus \nu } ) ^ { * } ( s ) = + \infty \right \} \end{align*}


\begin{align*} | G | \leqslant \sum _ { m = N } ^ { \infty } C _ 2 ^ m n ^ { - 2 ( m - N ) } \sum _ { j = 0 } ^ m \frac { ( x ^ 2 + y ^ 2 ) ^ j } { ( x ^ 2 + y ^ 2 + z ^ 2 ) ^ { j } } \leqslant C , \end{align*}


\begin{align*} P _ { i + 1 } = P _ i + \begin{cases} ( 1 , 0 ) & , \\ ( 0 , 1 ) & , \end{cases} \end{align*}


\begin{align*} K ( \Omega ) & = \inf \{ K > 0 : K f \in \mathcal { F } \} , \\ K _ ( \Omega ) & = \inf \{ K > 0 : K f \in \mathcal { F } _ \} . \end{align*}


\begin{align*} A _ E E ' = \mathcal { H } \nabla _ { \mathcal { H } E } \nu E ' + \nu \nabla _ { \mathcal { H } E } \mathcal { H } E ' , \end{align*}


\begin{align*} B _ { \pi } : = \{ b _ { \ell } , \ldots , b _ 1 , b _ 0 \} \subseteq \{ 1 , \ldots , k + 1 \} . \end{align*}


\begin{align*} \sum _ { e \in E ( H ) \colon i \in e } t ' _ { e } = \big | V ^ * _ i \big | t ' _ { \hat e } \ge z \ge \frac { n } { 5 m } \end{align*}


\begin{align*} \begin{pmatrix} - \gamma ^ - \partial _ t u \\ 0 \end{pmatrix} + \begin{pmatrix} 0 & \frac 1 2 I \\ - \frac 1 2 I & 0 \end{pmatrix} \begin{pmatrix} \varphi \\ \psi \end{pmatrix} + B ( \partial _ t ) \begin{pmatrix} \varphi \\ \psi \end{pmatrix} = \begin{pmatrix} - \partial _ t \beta _ 0 \\ 0 \end{pmatrix} \end{align*}


\begin{align*} \mathcal { E } ^ \mathrm { H } [ u ] : = \ ; & \int _ { \mathbb { R } ^ d } | \nabla u ( x ) | ^ 2 d x + \int _ { \mathbb { R } ^ d } V _ \mathrm { D W } ( x ) | u ( x ) | ^ 2 d x \\ & + \frac { \lambda } { 2 } \iint | u ( x ) | ^ 2 w ( x - y ) | u ( y ) | ^ 2 d x d y , \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { S } _ { 2 } ( n + 1 , r ) } w ( \gamma ) & = \sum _ { \gamma \in \mathcal { S } _ { 2 } ( n , r - 1 ) } w ( \gamma ) + \sum _ { k = 0 } ^ { p } a _ { - k - q - r } ^ { ( k ) } \Big ( \sum _ { \gamma \in \mathcal { S } _ { 2 } ( n , r + k ) } w ( \gamma ) \Big ) \\ & = \langle \mathcal { B } _ { q } ^ { n } e _ { r - 1 } , e _ { 0 } \rangle + \sum _ { k = 0 } ^ { p } a _ { - k - q - r } ^ { ( k ) } \langle \mathcal { B } _ { q } ^ { n } e _ { r + k } , e _ { 0 } \rangle \\ & = \langle \mathcal { B } _ { q } ^ { n + 1 } e _ { r } , e _ { 0 } \rangle \end{align*}


\begin{align*} | \det ( \sigma ^ 2 I _ n - A ^ H A ) | \left ( \frac { n - 1 } { ( n + 1 ) \sigma ^ 2 - \| A \| _ F ^ 2 } \right ) ^ { n - 1 } = 0 . \end{align*}


\begin{align*} \sum _ { n _ 1 \cdots n _ k \le x } ( n _ 1 , \ldots , n _ k ) = x \ , Q _ { k - 1 } ( \log x ) + O ( x ^ { \theta _ k + \varepsilon } ) , \end{align*}


\begin{align*} \liminf _ { N \to \infty } \frac { C _ N } { D _ N } \le \frac 1 { \nu ( [ 1 ] ) } \sum _ { n = 0 } ^ \infty \lambda ( [ 1 , * ^ n , 1 ] ) \cdot \nu ( [ 1 , 0 ^ n , 1 ] ) = \sum _ { n = 0 } ^ \infty \lambda ( [ 1 , * ^ n , 1 ] ) \cdot c _ n , \end{align*}


\begin{align*} \chi _ { j _ 1 j _ 2 j _ 3 } \left ( a _ 1 , a _ 2 , a _ 3 \right ) : = e ^ { \left ( \pi a _ 1 j _ 1 \right ) \sqrt { - 1 } } e ^ { \left ( \pi a _ 2 j _ 2 \right ) \sqrt { - 1 } } e ^ { \left ( \pi a _ 3 j _ 3 \right ) \sqrt { - 1 } } \end{align*}


\begin{align*} g _ { r , s , j } ( \kappa _ i ) = r ^ { j + \sum _ { i = 1 } ^ { m } \mu _ { i , j } } g _ j \left ( r \kappa _ i + s \right ) = r ^ { j + \sum _ { i = 1 } ^ { m } \mu _ { i , j } } g _ j \left ( \lambda _ i \right ) \neq 0 \end{align*}


\begin{align*} \langle M ^ { ( n ) } , W \rangle _ \tau = n ^ { \alpha + \frac 1 2 } \int _ 0 ^ { \tau } \sum _ { i = 1 } ^ Q \rho _ i \mathbf { 1 } _ { [ 0 , t _ i ] } ( \tau ) ( t _ i - s ) ^ { \alpha } \sigma ' ( X _ s ) \Theta ^ n _ s d s \end{align*}


\begin{align*} \tilde { H } _ { \lambda } ( \Z ; q ) = \sum _ { \tau } \tilde { K } _ { \tau , \lambda } ( q ) s _ { \tau } ( \Z ) . \end{align*}


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


\begin{align*} \mathbb { S } \doteq \left \{ \Phi \in \mathcal { W } _ { 1 } : \left \Vert \Phi \right \Vert _ { \mathcal { U } } = 1 \right \} \end{align*}


\begin{align*} f ( x ) = x ^ 4 - a x = x \left ( x ^ 3 - a \right ) \end{align*}


\begin{align*} \sum _ { n \le x } \tau _ k ( n ) = \frac 1 { ( k - 1 ) ! } x ( \log x ) ^ { k - 1 } + O ( x ( \log x ) ^ { k - 2 } ) , \end{align*}


\begin{align*} \sum _ { \gamma \in \mathcal { S } _ { 2 } ( n + 1 , 0 ) } w ( \gamma ) = \sum _ { k = 0 } ^ { p } a _ { - k - q } ^ { ( k ) } \Big ( \sum _ { \gamma \in \mathcal { S } _ { 2 } ( n , k ) } w ( \gamma ) \Big ) = \sum _ { k = 0 } ^ { p } a _ { - k - q } ^ { ( k ) } \langle \mathcal { B } _ { q } ^ { n } e _ { k } , e _ { 0 } \rangle = \langle \mathcal { B } _ { q } ^ { n + 1 } e _ { 0 } , e _ { 0 } \rangle \end{align*}


\begin{align*} c _ { 1 } = \frac { \Gamma \left ( ( m + 1 ) / 2 \right ) } { \Gamma ( m / 2 ) ( m \pi ) ^ { \frac { 1 } { 2 } } } , \end{align*}


\begin{align*} \overline \varepsilon & = \frac { 1 } { 4 } \min _ { i \in \mathcal I } \left \{ | a _ i | : a _ i \neq 0 \right \} = \frac { 1 } { 4 } \min _ { i \in \mathcal I } | a _ i | , \end{align*}


\begin{align*} S _ \mathcal { E } = \left \{ x \in H : x = \sum _ { k = 0 } ^ \infty x _ k E _ k , \{ x _ k \} _ { k = 0 } ^ \infty \textrm { i s R D } \right \} . \end{align*}


\begin{align*} u ( t , x ) = ( 1 + t ^ \beta + H ( t - r ) t ^ \beta ) \sin x \end{align*}


\begin{align*} - \Delta _ p u _ i - a u _ i ^ { p - 1 } = f _ i \Omega \end{align*}


\begin{align*} 0 \geq J _ { 1 } \geq - \int _ { B _ { 1 } ( x ) ^ { c } } \frac { d y } { | x - y | ^ { n + \sigma p } } = - \gamma , \end{align*}


\begin{align*} \sum _ { i _ 1 + \cdot + i _ r = k } \sum _ { \sigma \in \mathfrak { S } ( i _ 1 , \ldots , i _ r ) } \epsilon ( \sigma ) \frac { 1 } { r ! } \prod _ { j = 1 } ^ r \Phi _ { i _ j - 1 } ( x _ { \sigma ( i _ { 1 } + \cdots + i _ { j - 1 } + 1 ) } , \ldots , x _ { \sigma ( i _ { 1 } + \cdots + i _ j ) } ) . \end{align*}


\begin{align*} \hbox { \rm G r a p h } f = \{ ( x , f ( x ) ) \ , | \ , x \in M \} \end{align*}


\begin{align*} q _ A ( x ) = \sup \{ | f ( x ) | : \ ; f \in A \cap B _ { X ^ * } \} \end{align*}


\begin{align*} \vartheta _ p \cdot \vartheta _ q = \sum _ { r \in B ( \mathbb { Z } ) } \sum _ { \textbf { A } \in N E ( X ) } \alpha _ { p , q , \textbf { A } } ^ r t ^ \textbf { A } \vartheta _ r . \end{align*}


\begin{align*} \beta ( s ) = \left ( \frac { \pi } { 2 } \right ) ^ { s - 1 } \cos \left ( \frac { \pi s } { 2 } \right ) \Gamma ( 1 - s ) \beta ( 1 - s ) . \end{align*}


\begin{align*} \min \limits _ { 0 \leq i \leq N } \{ \| \nabla f ( x _ i ) \| ^ 2 \} & \leq \frac { f ( x _ 0 ) - f ( x _ { N } ) } { \sum \limits _ { i = 0 } ^ { N - 1 } p _ i } \end{align*}


\begin{align*} ( - \Delta ) ^ s u _ f ^ * + \sum _ { | \alpha | \leq m } ( - 1 ) ^ { | \alpha | } D ^ \alpha ( a _ \alpha u _ f ^ * ) & = 0 \mbox { i n } \quad \Omega , u _ f ^ * = f \mbox { i n } \quad \Omega _ e , \end{align*}


\begin{align*} \mathfrak { g } _ { A _ { I , \mathrm { s y m } } , k } ^ { r , s } : = \widetilde \psi _ I ^ { - 1 } ( \mathfrak { g } ^ { r , s } _ { A _ I , k } ) . \end{align*}


\begin{align*} | \tau ^ * ( n ) | \leq \tau _ 2 ( n ) = O _ { \varepsilon } ( n ^ { \varepsilon } ) . \end{align*}


\begin{align*} \min \{ \lambda _ h ( { k } ) , \lambda _ k \} \kappa ^ 2 _ m h _ { \max } ^ { 2 { m } } \le 1 \quad \quad \lambda _ h ( { k } ) \le \lambda _ { k } \quad k = 1 , \dots , M . \end{align*}


\begin{align*} \big ( F '' ( V ( \vec p ) ) [ V ' ( \vec p ) \widehat { \vec p } _ 1 , V ' ( \vec p ) \ , \cdot ] ^ * \vec g \big ) \widehat { \vec p } _ 2 & = \int _ 0 ^ T \langle V ' ( \vec p ) \widehat { \vec p } _ 1 ( { u } _ 2 ^ \prime ( t ) + Q { u } _ 2 ( t ) ) , w ( t ) \big \rangle _ X \d t \\ & \qquad \qquad + \int _ 0 ^ T \langle V ' ( \vec p ) \widehat { \vec p } _ 2 ( { u } _ 1 ^ \prime ( t ) + Q { u } _ 1 ( t ) ) , w ( t ) \big \rangle _ X \d t . \end{align*}


\begin{align*} ( \alpha , \beta ) _ { L , n } = T _ { L | K } ( \lambda _ { \rho } ( \alpha ) \Psi _ { L , n } ( \beta ) ) \cdot _ { \rho } v _ n \end{align*}


\begin{align*} - \Delta U + U = U ^ 3 \ \hbox { i n } \ \mathbb R ^ n . \end{align*}


\begin{align*} \Xi ^ { n , 1 } _ s = \left [ \left ( \int ^ { s } _ 0 \psi _ { n , 1 } ( u , s ) d W _ u \right ) ^ 2 - \int ^ { s } _ 0 \psi _ { n , 1 } ^ 2 ( u , s ) d u \ , \right ] , \end{align*}


\begin{align*} \langle \mathcal { B } _ { q } ^ { n } \ , e _ { r } , e _ { 0 } \rangle = \sum _ { \gamma \in \mathcal { S } _ { 2 } ( n , r ) } w ( \gamma ) , \end{align*}


\begin{align*} r _ X ( k + h ^ \vee _ { X ^ + } ) ( \ell + h ^ \vee _ { Y ^ - } ) = 1 \end{align*}


\begin{align*} d \varphi _ t = ( \Delta ^ \epsilon \varphi _ t - \lambda \varphi _ t ^ 3 - ( \mu + a ^ \epsilon ( \lambda ) ) \varphi _ t ) \ , d t + \sqrt { 2 } d W _ t ^ { \epsilon , L } \end{align*}


\begin{align*} a b + c + 1 & = \frac { \gamma ^ 2 + \gamma + 1 } { \gamma + 1 } \cdot \frac { \gamma ^ 2 + 1 } { \gamma ^ 3 + \gamma ^ 2 + \gamma } + \frac { \gamma + 1 } { \gamma ^ 3 + \gamma ^ 2 + \gamma } + 1 = \frac { ( \gamma ^ 3 + 1 ) + ( \gamma + 1 ) + ( \gamma ^ 3 + \gamma ^ 2 + \gamma ) } { \gamma ^ 3 + \gamma ^ 2 + \gamma } \\ & = \frac { \gamma ^ 2 } { \gamma ^ 3 + \gamma ^ 2 + \gamma } \ne 0 \end{align*}


\begin{align*} \mathcal { Q } ^ { \tau } _ { \lambda } ( q ) = \sum _ { \nu } \chi ^ { \nu } _ { \lambda } \tilde { K } _ { \nu \tau } ( q ) . \end{align*}


\begin{align*} v ( t ) : = \inf \{ r > 0 : \int _ r ^ \infty \psi _ \Lambda ( u ) ^ { - 1 } d u > t \} , \end{align*}


\begin{align*} \| T ( f ) \| \le \underset { x \in \R / \Z } { \textrm { s u p } } | f ( x ) | = \| f \| _ \infty \ , . \end{align*}


\begin{align*} d _ 0 ( { \mathcal K } ) _ X : = \sup _ { f \in { \mathcal K } } \| f \| _ X , d _ n ( { \mathcal K } ) _ X : = \inf _ { \dim ( X _ n ) = n } \sup _ { f \in { \mathcal K } } { \rm d i s t } ( f , X _ n ) _ X , n \geq 1 , \end{align*}


\begin{align*} f ( A ) = \sum _ q f ( A _ q ) < \sum _ q \max \{ f ( q ) , e ^ \gamma { \rm d } ( { \rm L } _ q ) \} = f ( \mathcal Q ) + e ^ \gamma \big ( 1 - { \rm d } ( { \rm L } _ { \mathcal Q } ) ) . \end{align*}


\begin{align*} \tau ( F ) = ( n - 2 ) \frac { \lambda ^ 2 } { 2 } F _ \ast \left ( \nabla _ { \mathcal { H } } \frac { 1 } { \lambda ^ 2 } \right ) - ( m - n ) F _ \ast ( H ) . \end{align*}


\begin{align*} \det ( \lambda - \Phi ) = \lambda ^ { 2 m } + s _ 2 \lambda ^ { 2 m - 2 } + \cdots + s _ { 2 m } , \end{align*}


\begin{align*} ( \phi , \psi ) = T ( \bar \phi , \bar \psi ) \in B _ { R ( \delta , \sigma ) } \subset H _ w ^ 2 , ~ { \rm { f o r } } ~ { \rm { a n y } } ~ ( \bar \phi , \bar \psi ) \in B _ { R ( \delta , \sigma ) } \subset H _ w ^ 2 , \end{align*}


\begin{align*} \Theta _ n ( \phi ) ^ { - 1 } e _ { ( \widetilde { H } ( \Theta _ n ( \phi ) ) ) ( \theta ( \lambda ) ) } \Theta _ n ( \phi ) = \frac { t ' ( \lambda , \lambda ) } { 2 } e _ { \theta ( \lambda ) } ^ \vee . \end{align*}


\begin{align*} g _ { 1 } ( u ) = \exp ( - \sqrt { 2 u } ) , \end{align*}


\begin{align*} S ^ m ( f _ 1 , \dots , f _ m ) ( x ) & \geq \int _ { B ^ { k } ( 0 , 1 ) } \prod _ { i = 1 } ^ k | f _ i ( x - x \sqrt { k } y _ i ) | d y _ 1 \dots d y _ k \\ & \gtrsim | x | ^ { - k } \end{align*}


\begin{align*} \bar { \mathcal { S } } ( L ) = \left ( \sum _ 1 ^ N ( \sup _ { L _ i } f _ { L _ i } ) e ^ { - i \hat { \theta } } \int _ { L _ i } \Omega \right ) - ( \sup _ { L _ 0 } f _ { L _ 0 } ) ( e ^ { - i \hat { \theta } } \int _ { L _ 0 } \Omega ) . \end{align*}


\begin{align*} \begin{aligned} T _ 4 ( h _ i { } \leq { } 1 ) = 0 \end{aligned} \end{align*}


\begin{align*} \mathcal { W } = \bigcup _ { i \in \mathbb { N } } \{ u _ i \} \cup \{ x ^ t _ s : \ : s \in \mathbb { N } , \ ; t \in A _ i \cup \{ q ( i ) \} \} , \end{align*}


\begin{align*} s _ { \lambda } ( \Z ) = \sum _ { \tau } K _ { \lambda , \tau } ( q ) P _ { \tau } ( \Z ; q ) , \end{align*}


\begin{align*} = \int _ { \R ^ 3 } \int _ { | y - x | \leq t } \frac { \varphi _ { m ; - 1 0 M _ t } ( | y - x | ) } { | y - x | } ( E ( t - | y - x | , y ) + \hat { v } \times B ( t - | y - x | , y ) ) \cdot \nabla _ v \big ( \frac { ( \hat { v } _ 2 \omega _ 3 - \hat { v } _ 3 \omega _ 2 ) \varphi _ { j , l } ( v , \omega ) } { 1 + \hat { v } \cdot \omega } \big ) \end{align*}


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


\begin{align*} d _ V ^ { i } A ^ { i } \alpha = A ^ { i + 1 } D ^ { i } \alpha . \end{align*}


\begin{align*} A = \min \left ( P _ { \ , } , P _ { \ , } \right ) \ ; . \ , \end{align*}


\begin{align*} ( - \Delta ) ^ s m - \frac { ( - \Delta ) ^ s m _ 1 } { \gamma _ 1 ^ { 1 / 2 } } m & = 0 \quad \Omega , \\ m & = m _ 0 \quad \Omega _ e . \end{align*}


\begin{align*} a \# b \rightharpoonup m \otimes x = \sum _ { ( b ) } a ( b _ { - 1 } \rightharpoonup m ) \otimes b _ 0 x = \sum _ { ( b ) } ( a \# b _ { - 1 } ) m \otimes b _ 0 x \end{align*}


\begin{align*} T ( t , A + B ) x - T ( t , A ) x = \int _ { 0 } ^ { t } T ( t - s , A + B ) \tilde { B } T ( s , A ) x \ , \textnormal { d } s . \end{align*}


\begin{align*} \begin{aligned} x _ 1 ^ { a _ 1 } \cdots x _ { i - 1 } ^ { a _ { i - 1 } } x _ i ^ { r _ i - 1 } x _ { i + 1 } ^ { a _ { i + 1 } } \cdots x _ n ^ { a _ n } \Omega & = 0 \\ x _ i ^ r x _ 1 ^ { a _ 1 } \cdots x _ n ^ { a _ n } \Omega & = - \hbar \tfrac { a _ i + 1 } { r _ i } x _ 1 ^ { a _ 1 } \cdots x _ n ^ { a _ n } \Omega . \end{aligned} \end{align*}


\begin{align*} L : = F ^ { i j } ( M [ u _ 0 ] ) D _ { i j } , \end{align*}


\begin{align*} p ( t , g ) = \int _ { \mathbb { R } } \frac { 1 } { \sqrt { 2 \pi C ( t ) } } \exp \left ( - \frac { ( g - B ( t ) - y ) ^ 2 } { 2 C ( t ) } \right ) p _ { t , 0 } ( y ) d y . \end{align*}


\begin{align*} \nabla _ U V = T _ U V + \nu \nabla _ U V , \end{align*}


\begin{align*} h ( z ) = z + \sum _ { n = 2 } ^ \infty a _ n z ^ n . \end{align*}


\begin{align*} e _ i : = \frac { 2 \sum _ j a _ { j , i } + ( - 1 + \sum _ { T : i \in T } k _ T ) ( r _ i - 2 ) } { r _ i } \in \Z , ~ ~ i = 1 , \ldots , a , \end{align*}


\begin{align*} \tau _ l ( n ) = O _ { l , \varepsilon } ( n ^ \varepsilon ) \end{align*}


\begin{align*} \Big \{ \tilde { \phi } ^ { T , \Upsilon , Y } [ \tilde { h } _ j ^ Y ] : \tilde { h } _ j ^ Y \in Z ^ a [ Y _ 1 , . . . , Y _ n ] , \tilde { h } _ j ^ \Upsilon \in Z ^ a [ \Upsilon _ 1 , . . . , \Upsilon _ n ] \Big \} = Z ^ a [ T _ 1 , . . . , T _ n ] . \end{align*}


\begin{align*} \widehat { \mathcal { R } ^ 1 } [ u , \cdots , u ] _ k : = \sum _ { \substack { \ell = 1 \\ o d d } } ^ { p } \sum _ { \substack { k = k _ 1 - k _ 2 + \cdots - k _ { p - 1 } + k _ p \\ k _ \ell = k } } \prod _ { \substack { i = 1 \\ o d d } } ^ { p } \widehat { u } _ { k _ i } \prod _ { \substack { i = 2 \\ e v e n } } ^ p \overline { \widehat { u } _ { k _ i } } = \frac { p + 1 } { 4 \pi } \widehat { u } _ k \int _ \mathbb { T } | u | ^ { p - 1 } \ , d x . \end{align*}


\begin{align*} \mu ( \alpha ) = \sum _ { w \in W _ { \rho } ^ n } \min \{ \mu ( \xi ) , \mu ( w ) \} . \end{align*}


\begin{align*} \mbox { d e g } ( t ^ m \otimes u ) = \mbox { w t } u - m - 1 \end{align*}


\begin{align*} \Theta = { \rm { d e t } } { \begin{pmatrix} \vec { q } _ { 0 } \ \cdots \ \vec { q } _ { r m - 1 } \end{pmatrix} } \enspace . \end{align*}


\begin{align*} F ( z ) = a + b z + \int _ { \mathbb { R } } \frac { 1 + s z } { s - z } \ , d \rho ( s ) , z \in \mathbb { C } ^ { + } , \end{align*}


\begin{align*} f ^ c _ { k - 2 } ( z , \overline \omega _ 1 ) = \int _ { c \cdot C } \frac { e ^ { z s } \ , s ^ { k - 2 } } { e ^ { \overline \omega _ 1 s } - 1 } \ d s , \end{align*}


\begin{align*} \norm { A _ j } _ { W ^ { 2 , \infty } } \leq M , \norm { q _ j } _ { L ^ \infty } \leq M , j = 1 , 2 , \end{align*}


\begin{align*} \lim _ { t \to 0 } \tau _ x \left ( [ P ^ E _ t ] \right ) = \tau _ x \left ( [ P ^ E _ 0 ] \right ) . \end{align*}


\begin{align*} & I \coloneqq ( e _ k ( y _ 1 , y _ 2 , \ldots , y _ n ) \mid 1 \le k \le n ) , \\ & I ' \coloneqq ( ( y _ i - y _ { i + 1 } ) e _ k ( y _ 1 , \ldots , y _ i ) \mid 1 \le i \le n - 1 , \ 1 \le k \le \min \{ i , n - i \} ) . \end{align*}


\begin{align*} \int _ \Omega f = \int _ { \Omega ^ * } f ^ * . \end{align*}


\begin{align*} c _ { ( k ) } ^ { \ast } = \frac { 1 } { \sqrt { 2 } } \left [ \int _ { 0 } ^ { \infty } s ^ { - 1 / 2 } \overline { G } _ { ( k ) } ( s ) \mathrm { d } s \right ] ^ { - 1 } . \end{align*}


\begin{align*} \mathcal Z _ K = \bigsqcup _ { i = 1 } ^ s \widetilde U _ { \tau _ i } . \end{align*}


\begin{align*} \overline C = \{ x \ , \omega _ 1 + y \ , \omega _ 2 : x , y \ge 0 \} \end{align*}


\begin{align*} t _ n = \inf \left \{ m > t _ { n - 1 } : Z _ m < a _ n Z _ { t _ { n - 1 } } Z _ m < ( 1 - n ^ { z ^ \prime - 1 } ) Z _ m ^ * \right \} \end{align*}


\begin{align*} p _ { s u c c } = \sum _ { i = 1 } ^ { m i n ( m , ( 2 | E | - \gamma _ { R S U } ) k ) } \mbox { P r } ( C _ { i } ) \sum _ { j = 0 } ^ { k } \binom { m - i } { j } ( p _ { m a t c h , j } ) ^ { 2 } , \end{align*}


\begin{align*} \begin{cases} - \phi '' - \frac { z } { 2 } \phi ' + \frac { z } { 2 } e _ \sigma \psi ' = f _ 1 , \\ - \psi '' - \frac { z } { 2 } \psi ' + \frac { z } { 2 } e _ \sigma \phi ' = f _ 2 , \\ \phi ( 0 ) = \phi ( r ) = 0 , \psi ( 0 ) = \psi ( r ) = 0 . \end{cases} \end{align*}


\begin{align*} ( \alpha , \beta ) _ { \rho , L , n } = \Phi _ L ( \beta ) ( \xi ) - \xi ; ~ ~ \rho _ { \eta ^ n } ( \xi ) = \alpha . \end{align*}


\begin{align*} \lim _ { x \rightarrow 0 } \rho ( x ) = \infty . \end{align*}


\begin{align*} \int _ { \R } e ^ { - i t s } \int _ { \sf T _ \varphi \cal Q _ { k } } e ^ { i \psi ( u ) } \big ( 1 - \kappa ( \psi ) \big ) \sum _ { n \in \N } \big ( \cal L _ { \varphi + i \psi } ^ n p _ F \big ) ( x ) \cal F v _ G ( \psi ) d \psi _ 0 d s = O ( t ^ { - N } ) \end{align*}


\begin{align*} { \rm R C } ( I ^ \vee ) = \left ( \bigcap _ { i = 1 } ^ { s + 1 } H ^ + _ { e _ i } \right ) \bigcap \left ( \bigcap _ { i = 1 } ^ q H ^ + _ { ( \delta _ i , - f _ i ) } \right ) \bigcap \left ( \bigcap _ { i = q + 1 } ^ { p _ 1 } H ^ + _ { ( \delta _ i , - f _ i ) } \right ) , \end{align*}


\begin{align*} s = \sum \limits _ { i = n + 1 } ^ { m } R i c ( U _ i , V _ i ) + \sum \limits _ { j = 1 } ^ { n } R i c ( X _ j , X _ j ) , \end{align*}


\begin{align*} P = \sum _ { | \alpha | \leq m } a _ \alpha D ^ \alpha , \end{align*}


\begin{align*} I ^ \alpha = \{ f \in S ^ \alpha \ , \vert \ , \Lambda ( f g ) = 0 \ \mbox { f o r a l l } \ g \in S ^ { N - \alpha } \} ; \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } V o l ( h ) = - \sum _ { l \ne i _ j } \langle \chi , \Lambda ( \rho _ { l } ) \rangle \partial _ l \partial _ { i _ 1 } ^ { k _ 1 - 1 } \ldots \partial _ { i _ s } ^ { k _ s } V o l ( h ) \end{align*}


\begin{align*} & { { \tilde { \psi } } } _ { \alpha , s , \boldsymbol { \xi } _ s , \ell _ s } = { { \tilde { \psi } } } _ { \alpha , s } \bigcirc _ { w = 1 } ^ { \ell _ s } ( \theta _ { X _ s } + \xi _ { s , w } ) \ \ 0 \le s \le r - 1 \enspace , \\ & \Psi _ { \alpha , \boldsymbol { \Xi } _ { \boldsymbol { \ell } } } = \bigcirc _ { s = 0 } ^ { r - 1 } { { \tilde { \psi } } } _ { \alpha , s , \boldsymbol { \xi } _ s , \ell _ s } \enspace , \end{align*}


\begin{align*} & \int \limits _ 0 ^ \infty \| \chi _ { B _ { r _ 0 } } e ^ { - i t H _ 0 ^ \perp } P _ \delta ^ \perp ( W ^ \perp _ { n ; \textrm { o u t } } ) \| _ \textrm { o p } \ , d t \leq \int \limits _ 0 ^ \frac { 1 } { n } 1 \ , d t + C \int \limits _ { \frac { 1 } { n } } ^ \infty t ^ { - \frac { 1 } { 2 } } ( n + t m ) ^ { - \ell } \ , d t \\ & \leq \frac { 1 } { n } + C \sqrt { n } \int \limits _ { \frac { 1 } { n } } ^ \infty ( n + t m ) ^ { - \ell } \ , d t = \frac { 1 } { n } + C \frac { \sqrt { n } } { m } ( n + \frac { m } { n } ) ^ { - \ell + 1 } \end{align*}


\begin{align*} [ D / \mathcal { W } \vec { y } ] _ { v _ i } = \sum _ { j = 1 , j \neq i } ^ { k } d _ G ( w _ i , w _ j ) | W _ j | \vec { x } _ { w _ j } + c _ i ( | W _ i | - 1 ) \vec { x } _ { w _ i } , \end{align*}


\begin{align*} A _ n = \{ x \in A : S ^ n ( x ) = x \} = \{ x \in A : x _ i = x _ { i + n } , i \in \Z \} . \end{align*}


\begin{align*} t _ + = \sum _ { a = 1 } ^ \mathrm { r } \bar p _ a ^ + \log y _ a , y ^ d = y _ 1 ^ { p _ 1 ^ + \cdot d } \cdots y _ \mathrm { r } ^ { p _ \mathrm { r } ^ + \cdot d } \end{align*}


\begin{align*} \Psi ( E _ j ( x , a ) ) & = \sum _ { h = 1 } ^ { j } \left | 2 \sqrt { a } \cos \left ( \frac { h } { j + 1 } \pi \right ) \right | = 2 \sqrt { a } \sum _ { h = 1 } ^ { j } \left | \cos \left ( \frac { h } { j + 1 } \pi \right ) \right | . \end{align*}


\begin{align*} P ( T , x ) & = \prod _ { j = 1 } ^ { l ( T ) } \mathcal { G } _ j ^ { n ( T , j ) } . \end{align*}


\begin{align*} \mathbb { E } ( \tau ) = \sum _ { n = 1 } ^ { \infty } i b _ i = \sum _ { j = 1 } ^ { \infty } \left \lceil { \frac { j } { k } } \right \rceil a _ j ' \leq \frac { 1 } { k } \left ( k + \sum _ { j = 1 } ^ { \infty } j a _ j ' \right ) = 1 + \sum _ { n = 1 } ^ { \infty } n a _ n , \end{align*}


\begin{align*} \label [ I e q ] { e q : a b s t r a c t t h e o r e m s p l i t t i n g } \big \langle f ^ { k , N } _ t - f _ t ^ { \otimes k } , \varphi _ k \big \rangle = \big \langle f ^ { k , N } _ t - F ^ { k , N } _ t , \varphi _ k \big \rangle & + \big \langle F ^ { k , N } _ t - \overline { F } { } ^ { k , N } _ t , \varphi _ k \big \rangle + \big \langle \overline { F } { } ^ { k , N } _ t - f _ t ^ { \otimes k } , \varphi _ k \big \rangle . \end{align*}


\begin{align*} | \pi | _ 1 + 2 \ , | \pi | _ 2 = k , \mbox { t h e t o t a l n u m b e r o f p o i n t s p a r t i t i o n e d b y $ \pi $ } . \end{align*}


\begin{align*} \sup \left \{ s \in [ 0 , 1 ] : \frac { x - 1 } { \log x } \le s \sqrt { x } + ( 1 - s ) \left ( \frac { x + 1 } { 2 } \right ) , ( x > 0 ) \right \} = \frac { 2 } { 3 } . \end{align*}


\begin{align*} \operatorname { d i v } _ { \mu _ { H } } ( Y ) = \operatorname { d i v } _ { \mu } ( H Y ) . \end{align*}


\begin{align*} p _ { \lambda } ( \Z ) = & \sum _ { \tau \in \mathcal { P } } \chi ^ { \tau } _ { \lambda } s _ { \tau } ( \Z ) , \\ s _ { \lambda } ( \Z ) = & \sum _ { \tau } z _ { \tau } ^ { - 1 } \chi ^ { \lambda } _ { \tau } p _ { \tau } ( \Z ) , \end{align*}


\begin{align*} \bar q ( z ) = \left ( z + | \hat \lambda _ { 1 } | \right ) \left ( z + | \hat \lambda _ { 2 } | \right ) \cdots \left ( z + | \hat \lambda _ { r } | \right ) = z ^ r + \bar \sigma _ 1 z ^ { r - 1 } + \bar \sigma _ 2 z ^ { r - 2 } + \cdots + \bar \sigma _ r . \end{align*}


\begin{align*} & \int _ { \{ \Psi < - t \} \cap D _ 0 } | \tilde F | ^ 2 e ^ { - k \varphi - a \Psi } \\ \le & 2 \int _ { \{ \psi < - t \} \cap D _ 0 } | f | ^ 2 e ^ { - a \Psi } + 2 \int _ { \{ \Psi < - t \} \cap D _ 0 } | \tilde F - f F ^ { 2 k } | ^ 2 e ^ { - k \varphi - a \Psi } \\ < & + \infty . \end{align*}


\begin{align*} w ( \gamma ) = \prod _ { e \subset \gamma } w ( e ) , \end{align*}


\begin{align*} \int _ 0 ^ 1 e ^ { 2 \pi i m ( x + u ) } & \prod _ { k = 1 } ^ s \cos ( 2 \pi n _ { k _ j } u ) d u \\ & \qquad = \left \{ \begin{array} { l l } 2 ^ { - s } e ^ { 2 \pi i m x } , & \hbox { i f } m = \varepsilon _ 1 n _ { k _ 1 } + \varepsilon _ { 2 } n _ { k _ 2 } + \ldots + \varepsilon _ 1 n _ { k _ s } , \\ 0 , & \hbox { o t h e r w i s e } , \end{array} \right . \end{align*}


\begin{align*} \int _ { \mathcal { G } } | u | ^ 2 \ , d x = \mu > 0 , \end{align*}


\begin{align*} \frac { \partial C } { \partial z ^ i _ { \epsilon } } v ^ i = 0 , \end{align*}


\begin{align*} ( A _ G x ) ( v ) = \sum \limits _ { e \in E _ v } \frac { \delta _ E ( e ) } { \delta _ V ( v ) } \frac { 1 } { | e | ^ 2 } \sum \limits _ { u \in e ; u \neq v } x ( u ) . \end{align*}


\begin{align*} \phi ( x , v ) = H ( x , v ) + \alpha x \cdot v + \beta | x | ^ 2 , \end{align*}


\begin{align*} \mathbf { D } _ { L _ I } \big ( \prod _ { j = 1 } ^ { 2 k } g ^ { - 1 } t _ j \sigma ( t _ j ) g \big ) = \mathbf { D } _ { L _ I } \big ( \prod _ { j = 1 } ^ { 2 k } ( w _ j s _ j w _ j ^ { - 1 } ) \sigma ( w _ j s _ j w _ j ^ { - 1 } ) \big ) , \end{align*}


\begin{align*} \psi ^ * ( d x _ 1 \wedge \cdots \wedge d x _ n ) = d x _ 1 \wedge \cdots \wedge d x _ n . \end{align*}


\begin{align*} \Gamma ^ { k ( s ) } _ { i j } ( 1 ) = 0 \ \ \textup { f o r a l l } \ \ i , j , k \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} \mu ( T _ 0 , \lambda ) \ge \sum _ { j = 1 } ^ { k } ( \alpha _ j - 1 ) \mu ( T _ j , \lambda ) \end{align*}


\begin{align*} - \lim _ { \epsilon \rightarrow 0 ^ + } \frac { \Im \varphi _ \mu ( x + i \epsilon ) } { \epsilon } = \int _ \mathbb { R } \frac { 1 + s ^ 2 } { ( x - s ) ^ 2 } \ , d \sigma _ { \mu } ( s ) > 1 , x \in \mathbb { R } , \end{align*}


\begin{align*} \phi ( z ) = \phi _ * ( z ) - e _ \delta ( z ) , \psi ( z ) = \psi _ * ( z ) - e _ \sigma ( z ) . \end{align*}


\begin{align*} & \partial _ t u + \nabla _ { u } u = - \operatorname { g r a d } p t \in [ 0 , t _ { 0 } ] , \\ & \operatorname { d i v } u = 0 , \\ & u | _ { t = 0 } = \dot { \eta } ( 0 ) , \end{align*}


\begin{align*} \mathcal { V } _ { [ n , \ell , k ] } : = \{ \gamma \in \mathcal { V } _ { [ n , \ell ] } : \eta ( \gamma ) = k \} , \ell \leq k \leq n - 1 . \end{align*}


\begin{align*} j ( a ^ { * } ) ( \gamma ) = j ( a ) ( \gamma ^ { - 1 } ) ^ { * } \end{align*}


\begin{align*} \tau _ { i , j } = \delta _ { i , j } \sum _ { k = 1 } ^ { n } B _ { i , k } - B _ { i , j } , i , j = 1 , \cdots , n . \end{align*}


\begin{align*} v o l ( D ^ 2 _ x ( 1 ) ) : = \frac { v o l ( B ^ 2 ( 1 ) ) b ^ 2 A ^ { \gamma \tau } \left [ \cos ^ 2 \theta z ^ { 1 } _ { \gamma } z ^ { 1 } _ { \tau } + \sin ^ 2 \theta z ^ { 2 } _ { \gamma } z ^ { 2 } _ { \tau } + 2 \sin \theta \cos \theta z ^ { 1 } _ { \gamma } z ^ { 2 } _ { \tau } \right ] } { 2 \sqrt { \det A } } . \end{align*}


\begin{align*} E = b ^ 2 f ^ 2 ( x ^ 1 ) \left [ f '^ 2 ( x ^ 1 ) + \sin ^ { 2 } \left ( x ^ 2 \right ) \right ] . \end{align*}


\begin{align*} C ( v ) = - \ln \left ( 1 - \frac { v ^ 2 } { q ^ 2 } \right ) . \end{align*}


\begin{align*} | f _ k ( n ) | \leq \tau _ 3 ( n ) = O _ { \varepsilon } ( n ^ { \varepsilon } ) . \end{align*}


\begin{align*} g _ i = x _ i ( h _ 1 , \ldots , h _ s ) . \end{align*}


\begin{align*} m _ 0 ^ b = m _ 0 + m _ 1 ( b ) + m _ 2 ( b , b ) + \ldots = 0 \in C F _ { s e l f } ^ 2 ( L , L ) . \end{align*}


\begin{align*} \tau ( x , y , t ) \ , \ , = \ , \ , \sum _ { I \in \binom { [ n ] } { k } } p _ I \prod _ { \substack { i , j \in I \\ i < j } } ( \kappa _ i - \kappa _ j ) \cdot \exp \biggl [ x \cdot \sum _ { i \in I } \kappa _ i + y \cdot \sum _ { i \in I } \kappa _ i ^ 2 + t \cdot \sum _ { i \in I } \kappa _ i ^ 3 \biggr ] , \end{align*}


\begin{align*} C = \frac { 1 } { 1 - \beta \kappa ^ 2 e ^ { - \tau \| \sigma \| _ \infty } } , \mbox { a n d } \lambda = - \frac { \log ( 1 - \beta \kappa ^ 2 e ^ { - \tau \| \sigma \| _ \infty } ) } { \tau } . \end{align*}


\begin{align*} h _ \mu ( \P ) = \lim _ { n \to \infty } \frac 1 n H _ \mu \ ! \left ( \bigvee _ { i = 0 } ^ { n - 1 } T ^ { - i } \P \right ) , \end{align*}


\begin{align*} d V _ { \max } = ( 1 + \| \beta \| _ { \alpha } ) ^ { n + 1 } d V _ { \alpha } \end{align*}


\begin{align*} D _ + = \sum _ { i \in M _ - } \bar { D } _ i \subset X _ + D _ - = \sum _ { i \in ( M _ - \cup \{ j \} ) } \bar { D } _ i \subset X _ - . \end{align*}


\begin{align*} \underline { \gamma } & = n \chi ( \underline { \epsilon } _ n ) \\ \overline { \gamma } & = n \chi ( \overline { \epsilon } _ n ) . \end{align*}


\begin{align*} \nabla _ X Y = A _ X Y + \mathcal { H } \nabla _ X Y , \end{align*}


\begin{align*} \sum _ { p = 1 } ^ { n - 1 } \epsilon _ p & = \sum _ { p = 1 } ^ { \infty } \epsilon _ p - \sum _ { p = n } ^ { \infty } \epsilon _ p \\ & = B - r _ n \end{align*}


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


\begin{align*} A _ m ( \alpha ) { \tilde { \bf u } } _ m ( \alpha ) = { \bf f } _ m ( \alpha ) + \boldsymbol { \eta } _ m , \end{align*}


\begin{align*} \frac { d } { d t } \rho _ t = \mathcal { L } ( \rho _ t ) , \mathcal { L } ( \rho ) = - \frac 1 2 \sum _ { j = 1 } ^ { N } [ C _ j , [ C _ j , \rho ] ] , \end{align*}


\begin{align*} \frac { 1 } { \mathbb { E } _ { \alpha } [ T _ { A } ] } = \lambda = \frac { \langle ( I - P ) f _ { 1 } , f _ { 1 } \rangle _ { \pi } } { \| f _ 1 \| _ 2 ^ 2 } = \mathrm { V a r } _ { \pi } f _ 1 \frac { \langle ( I - P ) f _ { 1 } , f _ { 1 } \rangle _ { \pi } } { \mathrm { V a r } _ { \pi } f _ 1 } . \end{align*}


\begin{align*} \begin{aligned} \mathcal { Q } _ { j , } ^ k \sim \mathcal { G P } ( q _ { j , } ^ k , z _ { j , } ^ k ) & k = 1 , \ldots , m , \ j = 1 , \ldots , L , \end{aligned} \end{align*}


\begin{align*} \mathcal { C } = \mathcal { C } _ X \cup \bigcup _ { x \in D } \mathcal { C } _ x . \end{align*}


\begin{align*} Q _ { \kappa } ( x ) = \frac { d \ , R _ { \kappa } ( x ) } { d \ , x } + N _ { \kappa } \ , x ^ { n - 1 } \ \ . \end{align*}


\begin{align*} B ^ C & = \{ ( P _ f ) _ { f \in G } \in \wp ( G ) ^ G \ ; | \ ; { \rm t h e r e \ ; e x i s t s } \ ; g \in P _ f \ ; { \rm a n d } \ ; h \in P _ { g f } \ ; { \rm s u c h \ ; t h a t } \ ; h g \notin P _ { f } \ ; { \rm f o r \ ; s o m e \ ; } f \in G \} \\ & = \bigcup _ { f , g , h \in G } ( W _ { ( f , g ) } \cap W _ { ( g f , h ) } \cap W _ { ( f , h g ) } ^ C ) . \end{align*}


\begin{align*} \sum _ { \substack { i \in M _ P , \\ i \neq i ^ * } } \sum _ { j \in N _ i } a _ { i j } \tilde x _ { i j } + \sum _ { \substack { i j \in P , \\ i \notin M _ 0 , i \neq i ^ * } } ( b - s ) \tilde x _ { i j } & \leq s + ( | M _ P - M _ 0 | - 1 ) ( b - s ) \\ & = b + ( | M _ P - M _ 0 | - 2 ) ( b - s ) . \end{align*}


\begin{align*} J _ 5 & = \int _ 0 ^ r z ^ { 2 } ( \bar \phi + e _ \delta - \delta ) ( 1 - \delta + \bar \phi + e _ \delta ) \cdot ( e '' _ \delta \phi + e '' _ \sigma \psi ) { \rm { d } } z \\ & \leq \Vert z ( \phi , \psi ) \Vert _ { L ^ 2 } \cdot \Vert z ( e '' _ \delta , e '' _ \sigma ) \Vert _ { L ^ 2 } \cdot \Vert ( \bar \phi + e _ \delta - \delta ) ( 1 - \delta + \bar \phi + e _ \delta ) \Vert _ { L ^ \infty } , \end{align*}


\begin{align*} | \Phi _ { n } ( r \zeta ) | \begin{cases} < 1 , & r < R _ { n } ( \zeta ) , \\ = 1 , & r = R _ { n } ( \zeta ) , \\ > 1 , & r > R _ { n } ( \zeta ) . \end{cases} \end{align*}


\begin{align*} V _ s = - X _ s + \Phi _ { s * } X = - X _ s + \widehat \psi _ { s * } X \end{align*}


\begin{align*} D _ { q } u ( x ) + p ( x ) u ( q x ) + r ( x ) = 0 , \end{align*}


\begin{align*} \mathrm { T C E } _ { q } ( X ) = E ( X | X > x _ { q } ) \end{align*}


\begin{align*} \Delta \lambda _ i = - \frac { \Delta f ( \lambda _ i ) + 2 \nabla f _ { \lambda } ( \lambda _ i ) \cdot \nabla \lambda _ i + f _ { \lambda \lambda } ( \lambda _ i ) \nabla \lambda _ i \cdot \nabla \lambda _ i } { f _ \lambda ( \lambda _ i ) } , \end{align*}


\begin{align*} \partial _ { x _ i } \partial _ { x _ j } \bigl ( a ^ { i j } \varrho \bigr ) - \partial _ { x _ i } \bigl ( b ^ i \varrho \bigr ) = 0 \end{align*}


\begin{align*} X _ c \cdot ( f , a ) = f ( c ) + X _ { c + a } . \end{align*}


\begin{align*} \mathcal { F } ( x , z ) = \frac { 2 C ^ 3 } { E } . \end{align*}


\begin{align*} S ( \delta _ { } ) = o ( 1 ) . \end{align*}


\begin{align*} \left ( I + ( \mu - \lambda ) R ( \lambda , A _ 2 ) \right ) & \left ( R ( \mu , A _ 2 ) - R ( \mu , A _ 1 ) \right ) \\ & = \left ( R ( \lambda , A _ 2 ) - R ( \lambda , A _ 1 ) \right ) \left ( 1 + ( \lambda - \mu ) R ( \mu , A _ 1 ) \right ) , \end{align*}


\begin{align*} l ( s ) = - \sqrt { \lambda } \frac { I _ 1 \left ( \sqrt { \lambda ( 1 - s ^ 2 ) } \right ) } { \sqrt { 1 - s ^ 2 } } . \end{align*}


\begin{align*} \sum _ { k = - m } ^ { m } c _ k e ^ { i n _ k x } , m = 1 , 2 , \ldots , ( n _ { - k } = n _ k ) \end{align*}


\begin{align*} \forall x \in \Lambda _ { \epsilon , L } , C _ { ( m ) } ( x ) : = C _ { ( m ) } ( 0 , x ) = \frac { 1 } { L ^ d } \sum _ { k \in \Lambda ^ * } e ^ { i k \cdot x } \frac { 1 } { m ^ 2 + \theta ( k ) } . \end{align*}


\begin{align*} \int _ { \R ^ d } \frac { \phi _ { \sigma } ^ { p } ( x - y ) } { \phi ^ { p - 1 } _ { \sigma } ( y ) } d y = \exp \left ( \frac { p ( p - 1 ) | x | ^ 2 } { 2 \sigma ^ 2 } \right ) , \end{align*}


\begin{align*} D _ { q ^ { - 1 } } \widetilde { h } _ n ( x ; q ) = q ^ { 1 - n } [ n ] _ q \widetilde { h } _ { n - 1 } ( x ; q ) , \end{align*}


\begin{align*} \mathbb { E } [ \ell ] = \dfrac { \ell } { 2 ^ { \ell + 1 } } \left ( 2 ^ \ell + \dbinom { \ell } { \ell / 2 } \right ) . \end{align*}


\begin{align*} A _ { \bar S S } ( \tau : t _ 0 ) & = \sum _ { k = t _ 0 } ^ { \tau - 2 } A _ { \bar S } ( \tau - 2 ) \cdots A _ { \bar S } ( k + 1 ) A _ { \bar S S } ( k ) A _ S ( k : t _ 0 ) \cr & { = } \sum _ { k = t _ 0 } ^ { \tau - 2 } \tilde A ( \tau - 1 : k + 1 ) A _ { \bar S S } ( k ) A _ S ( k : t _ 0 ) \cr & \stackrel { ( a ) } { \geq } e ^ { - M ( \gamma ) \lambda } \sum _ { k = t _ 0 } ^ { \tau - 2 } \tilde A ( \tau - 1 : k + 1 ) A _ { \bar S S } ( k ) , \end{align*}


\begin{align*} f ( \lambda ) = \lambda ^ N + \sum _ { k = 1 } ^ { N } ( - 1 ) ^ k \lambda ^ { N - k } \sum _ { 1 \le j _ 1 < \cdots < j _ k \le N } \underset { 1 \le \ell , m \le k } { { \rm d e t } } ( H _ { j _ \ell j _ m } ) , \end{align*}


\begin{align*} B _ { i , j } : = b _ { i , j } \rho _ i \rho _ j \end{align*}


\begin{align*} g ^ { - 1 } b _ 2 \sigma ( g ) = b _ 1 . \end{align*}


\begin{align*} f ( X ) D [ X ] = ( A P ) _ t = ( A _ t P ) _ t = ( ( A _ f ) _ t P ) _ t = ( A _ f P ) _ t , \end{align*}


\begin{align*} \sum _ { k = N } ^ M c _ k b _ k = \sum _ { k = N + 1 } ^ M ( b _ k - b _ { k - 1 } ) \sum _ { i = k } ^ M c _ i + \bigg ( \sum _ { k = N } ^ M c _ k \bigg ) b _ N . \end{align*}


\begin{align*} \label [ I e q ] { e q : g i b b s m e a s u r e } \mu ^ N \big ( \dd \mathbf { x } ^ N \big ) = \frac { 1 } { Z _ N } \exp { \left [ N G ( \mu _ { \mathbf { x } ^ N } ) \right ] } \mu _ 0 ^ { \otimes N } \big ( \dd \mathbf { x } ^ N \big ) , \end{align*}


\begin{align*} \| N ^ { n , 5 } _ \tau \| _ 2 \le \alpha ^ 2 \sup _ { 0 \le s \le \tau } \| ( \sigma ' \sigma ) ^ 2 ( X _ s ) \| _ 2 \frac 1 n \sum _ { j = 0 } ^ { \lfloor n \tau \rfloor } \int _ { 0 } ^ { 1 \wedge ( n \tau - j ) } \gamma _ { \frac { x + j } n } d x \sum _ { k = j + 1 } ^ { \infty } k ^ { 2 \alpha - 2 } . \end{align*}


\begin{align*} - A < A F - B E \leq A ( B - 1 ) - B A = - A , \end{align*}


\begin{align*} \widehat { \varphi } _ 1 ( f ) = \begin{pmatrix} \varphi _ 1 ( f ) & 0 \\ 0 & 1 \end{pmatrix} \widehat { \varphi } _ 2 ( f ) = \begin{pmatrix} 1 & 0 \\ 0 & \varphi _ 2 ( f ) \end{pmatrix} . \end{align*}


\begin{align*} k ( t ) = \frac 1 { 2 \pi i } \int _ \Gamma e ^ { s t } K ( s ) d s \end{align*}


\begin{align*} s _ { i , j } = \begin{cases} s _ i s _ { i + 1 } \cdots s _ { j } & \hbox { i f } i \leq j , \\ s _ i s _ { i - 1 } \cdots s _ { j } & \hbox { i f } i > j . \end{cases} \end{align*}


\begin{align*} h ^ 0 \left ( L _ { \chi } + K _ Y \right ) = 0 \end{align*}


\begin{align*} M _ { \mathbb { Z } } ^ { * } = \left \{ H ^ { * } ( \overline { f } ) \mid \overline { f } \in \mathbb { Z } ^ { n } \right \} . \end{align*}


\begin{align*} f _ m ( z , s ) = \frac { \Gamma ( s ) } { 2 \pi \sqrt { | m | } \Gamma ( 2 s ) } M _ { 0 , s - \frac 1 2 } ( 4 \pi | m | y ) e ( m x ) . \end{align*}


\begin{align*} R ( K ) \to K ^ * ( C ^ * _ r G ) \colon [ E ] \to [ P ^ E _ t ] - \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \end{align*}


\begin{align*} u _ { j } ( x ) = & \begin{cases} j + j ^ { q } \phi ( x ) , & \quad \mathrm { i n } \ ; B _ { R } , \\ ( 1 - \varphi ( x ) ) ( j + j ^ { q } ) + \varphi ( x ) | x | ^ { - s } , & \quad \mathrm { i n } \ ; B _ { R } ^ { c } , \end{cases} \end{align*}


\begin{align*} \textstyle C ^ { \le \delta } = \big \{ c \ ; \big | \ ; c \in C , ~ \frac { { \rm w } ( c ) } { n } \le \delta \big \} . \end{align*}


\begin{align*} \dim H ^ 1 ( G ^ { \mathcal C } , V ) = \dim H ^ 1 ( G ^ { \mathcal C } , V ^ \vee ) \textrm { a n d } H ^ 2 ( G ^ { \mathcal C } , V ) ^ { \mathcal C , \tau } = H ^ 2 ( G ^ { \mathcal C } , V ^ \vee ) ^ { \mathcal C , \tau } = 0 \end{align*}


\begin{align*} \int _ 0 ^ 1 \frac { 1 } { u ( 1 \vee \log \Phi ^ { - 1 } ( u ) ) } \ \mathrm { d } u = \infty \limsup _ { n \to \infty } \frac { \mathbb { G } ( X _ n , X _ m ) } { \Phi ( n ) } < \infty \end{align*}


\begin{align*} \label [ I e q ] { e q : s i m u l t a n e o u s j u m p s o p e r a t o r N } \widetilde { L } ^ N _ \mu \varphi ( x ) : = N \iint _ { E \times E } \lambda ( z , \mu ) \{ \varphi ( y ) - \varphi ( x ) \} \widetilde { P } _ \mu ^ N ( x , z , \dd y ) \mu ( \dd z ) . \end{align*}


\begin{align*} \mathit { M } _ { \mathfrak { m } } \doteq \left \{ \omega \in E _ { 1 } : f _ { \mathfrak { m } } ^ { \sharp } \left ( \omega \right ) = \inf \ , f _ { \mathfrak { m } } ^ { \sharp } ( E _ { 1 } ) = - \mathrm { P } _ { \mathfrak { m } } ^ { \sharp } \right \} \ . \end{align*}


\begin{align*} \frac { 1 } { t } \int _ { \R ^ d } \varphi d ( \mu _ t - \nu _ t ) = \frac { 1 } { t } \iint _ { \R ^ d \times \R ^ d } \{ \varphi ( x ) - \varphi ( y ) \} d \pi _ t ( x , y ) . \end{align*}


\begin{align*} p _ j = P ( d ) \le P ( c ) < P ( a ^ * ) ^ { 1 / \sqrt { v } } . \end{align*}


\begin{align*} \bigg | \frac { \Omega ( 1 - s ) } { \Omega ( s ) } \bigg | = 1 . \end{align*}


\begin{align*} \begin{cases} - \Delta _ p G _ { \lambda , R } - \lambda R ^ p G _ { \lambda , R } ^ { p - 1 } = \delta _ 0 & \Omega _ R \\ G _ { \lambda , R } \geq 0 & \Omega _ R \\ G _ { \lambda , R } = 0 & \partial \Omega _ R . \end{cases} \end{align*}


\begin{align*} 4 k d + ( 8 k - 1 ) e + \ell \leq 4 k d + 8 k e = 4 k N < 4 k N + j , \end{align*}


\begin{align*} Q ^ { n , 7 } _ \tau = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 [ \gamma _ s - \gamma _ { \eta _ n ( s ) } ] ( s - \eta _ n ( s ) ) ^ { 2 \alpha + 1 } ( \sigma ' \sigma ) ^ 2 ( X _ { \eta _ n ( s ) } ) d s , \end{align*}


\begin{align*} \bigcup _ { k = 1 } ^ N \overline { \Omega _ k ( t ) } = \R ^ 2 , \end{align*}


\begin{align*} u ( \cdot , 0 ) = u _ 0 ( \cdot ) , \ u _ t ( \cdot , 0 ) = u _ 1 ( \cdot ) , \ y ( \cdot , 0 ) = y _ 0 ( \cdot ) \ \ y _ t ( \cdot , 0 ) = y _ 1 ( \cdot ) \ \ \Omega , \end{align*}


\begin{align*} u ( t _ { k + 1 } ) = \left \{ \begin{aligned} & A _ { 1 } u ( t _ { k } ) + B _ { 1 } & & \beta ^ { \top } u ( t _ { k } ) \geq ( 2 - \phi ) \lambda \\ & A _ { 2 } u ( t _ { k } ) + B _ { 2 } & & \beta ^ { \top } u ( t _ { k } ) < ( 2 - \phi ) \lambda \end{aligned} \right . \end{align*}


\begin{align*} \alpha = s _ { 0 } - \varphi _ { \mu } \left ( F _ { \nu } ^ { * } ( \alpha ) \right ) = s _ { 0 } - \gamma _ { \mu } + \int _ { \mathbb { R } } \frac { 1 + s F _ { \nu } ^ { * } ( \alpha ) } { s - F _ { \nu } ^ { * } ( \alpha ) } \ , d \sigma _ { \mu } ( s ) , \end{align*}


\begin{align*} \varphi ( e , f ) = \langle z ( e a \oplus b e ) , ( f c \oplus d f ) \rangle . \end{align*}


\begin{align*} K = \left \{ s + i t : s \in [ A , B ] , \delta \le t \le \frac { 1 } { \delta } \right \} \end{align*}


\begin{align*} \lim _ { | y | \to \infty } \| B R ( x + i y , A ) \| _ q = 0 \forall x < - \omega _ 0 ( A ) . \end{align*}


\begin{align*} \eta ( x ) = \eta ^ \parallel ( x ^ \parallel ) \eta ^ \perp ( x ^ \perp ) \end{align*}


\begin{align*} & \int _ { 0 } ^ { T } \left ( \partial _ t v ^ { \Delta t } _ { h } , P _ h ( u ^ { \Delta t } _ { h } \phi ) \right ) \ , d t = \int _ { 0 } ^ { T } \left ( \partial _ t v ^ { \Delta t } _ { h } , u ^ { \Delta t } _ { h } \phi \right ) \ , d t + \int _ { 0 } ^ { T } \left ( \partial _ t v ^ { \Delta t } _ { h } , P _ h ( u ^ { \Delta t } _ { h } \phi ) - u ^ { \Delta t } _ { h } \phi \right ) \ , d t = : I _ 1 + I _ 2 . \end{align*}


\begin{align*} g _ s = 2 \pi i / \overline \delta , Q = \exp ( - 2 \pi i \overline \mu ) , \end{align*}


\begin{align*} a _ n = - \frac { N ! } { \pi n } \sum _ { k = 0 } ^ { N - 1 } \frac { ( - 1 ) ^ k F _ k ( n ) } { ( N - k ) ! } \cos \left ( \frac { \pi } { 2 } k \right ) , \end{align*}


\begin{align*} W \left [ { \eta \left ( t \right ) } \right ] = \mathop \sum \limits _ n \frac { 1 } { { n ! } } \mathop \int { W ^ { \left ( n \right ) } } \left ( { { t _ 1 } - { t _ 2 } , \ldots , { t _ 1 } - { t _ n } } \right ) \eta \left ( { { t _ 1 } } \right ) . . . \eta \left ( { { t _ n } } \right ) d { t _ 1 } . . . d { t _ n } \end{align*}


\begin{align*} \nu \left ( a _ { \rho 0 } ( f ) \right ) = \nu _ \rho ( f ) = \nu ( l ( h _ \rho ) ) . \end{align*}


\begin{align*} \Pi ^ t ( s ) = ( \pi ^ t _ 1 ( s ) , \pi ^ t _ 2 ( s ) , \dots , \pi ^ t _ { N ^ t _ s } ( s ) ) , \end{align*}


\begin{align*} \tilde { \mathcal { B } } : = \{ ( \phi _ 1 , \phi _ 2 , \cdots , \phi _ N ) \mid \phi _ 1 , \phi _ 2 , \cdots , \phi _ N \colon G \to \mathbb { R } _ { \geq 0 } , \ ; \| \phi _ 1 \| _ { p _ 1 } = \| \phi _ 2 \| _ { p _ 2 } = \cdots = \| \phi _ N \| _ { p _ N } = 1 \} \end{align*}


\begin{align*} F _ { i , j } = F _ { j , i } , \end{align*}


\begin{align*} \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } S ( h ) = n ! \cdot \partial _ { i _ 1 } ^ { k _ 1 } \ldots \partial _ { i _ s } ^ { k _ s } V o l ( h ) , \end{align*}


\begin{align*} \rho = \sum _ { n \ge 1 } a _ n \delta _ { \lambda _ n ^ 2 } , a _ n > 0 . \end{align*}


\begin{align*} b _ { i j } = h ^ { i k } g _ { j k } = h _ { i j } = D _ { i j } ^ 2 u + u \delta _ { i j } . \end{align*}


\begin{align*} \varpi _ { [ a , i ] } \cdot \varpi _ { [ i + 1 , b ] } = \frac { 1 } { ( i - a + 1 ) ! } \cdot \frac { 1 } { ( b - i ) ! } \cdot \varpi _ { a } \varpi _ { a + 1 } \cdots \varpi _ { b } = \binom { b - a + 1 } { i - a + 1 } \varpi _ { [ a , b ] } \end{align*}


\begin{align*} \lambda _ { 1 } = + \lambda _ { + } , \lambda _ { 2 } = + \lambda _ { - } , \lambda _ { 3 } = - \lambda _ { + } , \lambda _ { 4 } = - \lambda _ { - } , \end{align*}


\begin{align*} \| \nabla ( \xi _ { \tilde K } w ) \| ^ 2 _ { L ^ 2 ( M ) } = \| \xi _ { \tilde K } \nabla w + w \nabla \xi _ { \tilde K } \| ^ 2 _ { L ^ 2 ( M ) } . \end{align*}


\begin{align*} W _ { n } ( t ) : = \frac { 1 } { \sqrt { n } } \bigg [ \sum _ { j = 0 } ^ { [ n t ] - 1 } v \circ T ^ j + ( n t - [ n t ] ) v \circ T ^ { [ n t ] } \bigg ] , t \in [ 0 , 1 ] , \end{align*}


\begin{align*} k = 0 , 1 , 2 , \ldots , m , i _ k = \tilde { \omega } _ k = j _ k . \end{align*}


\begin{align*} \int _ { 1 } ^ { \infty } \dfrac { \widehat { \varPhi } _ { x } ^ { - 1 } ( \tau ) } { \tau ^ { \frac { N + s } { N } } } d \tau = \infty ~ ~ x \in \overline { \Omega } . \end{align*}


\begin{align*} 1 _ A ^ { - 1 } ( \mathbb { R } _ { > t } ) = \left \{ \begin{aligned} & A & & \ ; t < 1 \\ & \emptyset & & \ ; t \geq 1 \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { 1 } { n } \log \frac { 1 } { \epsilon _ n } = \lim _ { n \to \infty } \frac { 1 } { n } \log \frac { 1 } { 1 - \epsilon _ n } = 0 , \end{align*}


\begin{align*} \alpha _ { k } = ( v _ { \mathrm { n c } } , \phi _ { k } ) _ { 1 + \delta } = ( z _ { \mathrm { n c } } , \phi _ { k } ) _ { 1 + \delta } - t ( u _ { \mathrm { n c } } , \phi _ { k } ) _ { 1 + \delta } = 0 . \end{align*}


\begin{align*} & \lim \limits _ { n \rightarrow \infty } \| \varphi _ { n ; \mathrm { i n } } \| = 0 \\ \end{align*}


\begin{align*} \phi _ { 0 } ( x ) = \frac { 1 } { \sqrt { a } } , \end{align*}


\begin{align*} I _ m f ( x , \xi ) = \int \limits _ { - \infty } ^ \infty \L f ( x + t \xi ) , \xi ^ m \R \ , d t \end{align*}


\begin{align*} A _ t = ( A , B C ) _ t = ( A , A C , B C ) _ t = ( A , ( A , B ) C ) _ t = ( A , ( A , B ) _ t C ) _ t = ( A , C ) _ t , \end{align*}


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


\begin{align*} \lim _ { t \to 0 } \kappa _ t ^ s ( k \exp ( t v ) ) = \kappa _ 0 ^ s ( k , v ) . \end{align*}


\begin{align*} \psi : = \sum _ { j = - 2 } ^ 2 \chi _ j \end{align*}


\begin{align*} d \beta ( x , y ) = \sum _ { \nu = 1 } ^ k \beta ( x _ 1 , \ldots , x _ { \nu - 1 } , y _ \nu , x _ { \nu + 1 } , \ldots , x _ k ) , \end{align*}


\begin{align*} h _ \mu ( T ) = \sup _ \P h _ \mu ( \P ) , \end{align*}


\begin{align*} \bigcap _ { j \in M _ - } \{ z _ j = 0 \} \subset X _ + , \bigcap _ { j \in M _ + } \{ z _ j = 0 \} \subset X _ - . \end{align*}


\begin{align*} \begin{aligned} b ^ * & \geq \frac { g _ 0 } { 1 - g _ 1 / V _ F } > 0 , \\ c ^ * & \geq a _ 0 + \frac { a _ 1 } { V _ F } \frac { g _ 0 } { 1 - g _ 1 / V _ F } > 0 . \end{aligned} \end{align*}


\begin{align*} \chi _ j ( \lambda ) = \chi ( \lambda / 2 ^ j ) j \in \Z . \end{align*}


\begin{align*} \psi _ I : A _ { I , \mathrm { s y m } } \rightarrow & A _ I \\ t _ { \alpha , \beta , d } \mapsto & \sum _ { i \in I : ( a _ i , b _ i ) = ( \alpha , \beta ) } u _ { i , d } . \end{align*}


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


\begin{align*} P _ j = \sum _ { \abs { \alpha } \leq m } ( a _ { j , \alpha } + b _ { j , \alpha } ) D ^ \alpha , a _ { j , \alpha } \in M _ 0 ( H ^ { s - | \alpha | } \rightarrow H ^ { - s } ) , P _ { j , s } \vcentcolon = \sum _ { \abs { \alpha } \leq m } { b _ { j , \alpha } } D ^ \alpha \in \mathcal { P } _ { m , s } ( \Omega ) \end{align*}


\begin{align*} y = \lim _ { n \rightarrow \infty } y _ { n } \geq \lim _ { n \rightarrow \infty } f ( x _ { n } ) = f ( x ) , \end{align*}


\begin{align*} & \deg ( x _ i ) = \deg ( y _ i ) = 1 + ( n - 1 ) \delta _ { i , 0 } , \end{align*}


\begin{align*} \omega _ i = \frac { \lambda A [ ( 2 i + 1 ) / k - 1 ] } { ( \sigma _ N ^ 2 + \sigma _ Z ^ 2 ) \lambda ^ 2 + \rho } , \end{align*}


\begin{align*} Q ^ { n , 3 } _ \tau & = n ^ { 2 \alpha + 1 } \int ^ { \tau } _ 0 \gamma _ s ( s - \eta _ n ( s ) ) ^ { 2 \alpha } \left ( ( \sigma ' ( X _ s ) ) ^ 2 - ( \sigma ' ( X _ { \eta _ n ( s ) } ) ) ^ 2 \right ) \sigma ^ 2 ( X _ { \eta _ n ( s ) } ) \ , \\ & \times \left [ \left ( W _ s - W _ { \eta _ n ( s ) } \right ) ^ 2 - ( s - \eta _ n ( s ) ) \right ] \ , d s \end{align*}


\begin{align*} | I _ 2 | \le c \frac { \delta _ 2 ^ { \frac { 1 } { 2 } } } { T _ n } \frac { 1 } { \prod _ { l \ge 2 } h _ l } . \end{align*}


\begin{align*} P ( S _ i ) = \frac { { m _ x \choose i } \sum _ { j = 0 } ^ { i } ( - 1 ) ^ { j } { { i } \choose { j } } ( i - j ) ^ { \gamma _ x k _ x } } { m _ x ^ { \gamma _ x k _ x } } . \end{align*}


\begin{align*} f _ + ^ * ( D _ + ) = \sum _ { i = 1 } ^ m a _ { i , + } \tilde { D } _ i + \left ( \sum _ { i = 1 } ^ m a _ { i , + } D _ i \cdot e \right ) E , \end{align*}


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


\begin{align*} \Psi ^ { n , 5 } _ s = 2 \sigma ( X _ s ) \left ( \int _ 0 ^ s \psi _ { n , 1 } ( u , s ) d W _ u \right ) \left ( \int ^ { \eta _ n ( s ) } _ 0 \psi _ { n , 2 } ( u , s ) \ , \sigma ( X _ { \eta _ n ( u ) } ) \ , d W _ u \right ) \end{align*}


\begin{align*} S ^ { n , 1 } _ \tau = S ^ { n , M , 1 } _ \tau + S ^ { n , M , 2 } _ \tau , \end{align*}


\begin{align*} \varphi ( A ) = N ( A ) \prod _ { P | A } ( 1 - \frac { 1 } { N ( P ) } ) \end{align*}


\begin{align*} \sum _ { t = 1 } ^ n y _ t \leq n \left ( \rho P _ { \max } + \lambda + \delta \right ) \ , , \ , \end{align*}


\begin{align*} r ( m m ' ) = & r ( l ( m ) \cdot a \cdot r ( m ) \cdot r ( m ' ) ) \\ \stackrel { ( R 2 ) } = & r ( l ( m ) a ) \cdot r ( m ) \cdot r ( m ' ) \\ = & \textsf { 1 } _ B \cdot r ( m ) \cdot r ( m ' ) = r ( m ) \cdot r ( m ' ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ N \det \left ( ( \lambda _ i ( t ) I _ N - H _ { \alpha } ( t ) ( \lambda _ j ( t ) I _ N - H _ { \alpha } ( t ) ) _ { k | k } \right ) & = \sum _ { k = 1 } ^ N \prod _ { p ( \neq k ) } ( \lambda _ i ( t ) - \lambda _ p ( t ) ) ( \lambda _ j ( t ) - \lambda _ p ( t ) ) \\ & = \begin{cases} f _ { \lambda } ( \lambda _ i ( t ) ) ^ 2 & i = j \\ 0 & i \neq j \end{cases} . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { \tau \in [ 0 , T ] } \| Q ^ { n , 7 } _ \tau \| _ 2 = 0 . \end{align*}


\begin{align*} \inf _ { \theta \in \Theta } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \theta } ) = \inf _ { \theta \in N } { \sf W } _ p ^ { ( \sigma ) } ( \hat \mu _ n , \nu _ { \theta } ) \end{align*}


\begin{align*} S = S _ 1 + S _ 2 , \end{align*}


\begin{align*} \frac { d \xi ( t ) } { d t } = \xi ( t ) - \mu , \end{align*}


\begin{align*} \max \{ | S | : S \in \bigcap _ { i = 1 } ^ { k + 1 } \mathcal { F } _ i \} & = \max \{ | S | : S \in ( \bigcap _ { i = 1 } ^ { k } \mathcal { F } _ i ) \bigcap \mathcal { F } _ { k + 1 } \} \\ & = \min \{ \Bar { r } _ k ( X ) + r _ { k + 1 } ( E \backslash X ) : X \subseteq E \} \end{align*}


\begin{align*} h ( M ) = \sum _ { 0 \leq 2 j \leq M - 1 } ( j + 1 ) \binom { M - j - 1 } j . \end{align*}


\begin{align*} R i c ( e _ 1 , e _ 2 ) = 0 . \end{align*}


\begin{align*} \left ( 1 + 2 f '^ 2 ( x ^ 1 ) \right ) \left \lbrace f '^ 2 ( x ^ 1 ) + 3 f ( x ^ 1 ) f '' ( x ^ 1 ) \right \rbrace - 1 = 0 . \end{align*}


\begin{align*} B ^ m _ { j , l } ( t , x ) = \int _ { \R ^ 3 } \int _ { | y - x | \leq t } \varphi _ { m ; - 1 0 M _ t } ( | y - x | ) \frac { \varphi _ { j , l } ( v , \omega ) } { | y - x | } \hat { v } \times \nabla _ x f ( t - | y - x | , y , v ) d y d v , \omega : = \frac { y - x } { | y - x | } . \end{align*}


\begin{align*} \delta = \begin{cases} 0 , & , \\ 1 , & . \end{cases} \end{align*}


\begin{align*} \frac { d } { d x } H ( x + i f ( x ) ) = a - b f ' ( x ) = a ( 1 + f ' ( x ) ^ { 2 } ) . \end{align*}


\begin{align*} \overline { G } _ { ( 1 ) } ( u ) = \int _ { u } ^ { \infty } g _ { 1 } ( s ) \mathrm { d } s \end{align*}


\begin{align*} \int _ { x - \Delta x / 2 } ^ { x + \Delta x / 2 } | \phi ( x ) | ^ { 2 } d x = \frac { \Delta x } { a } , \end{align*}


\begin{align*} Y = \frac { \partial u } { \partial s } d s , \eta = \omega ( \cdot , \frac { \partial u } { \partial s } ) d s . \end{align*}


\begin{align*} t _ x ( S ) = \begin{cases} S \cup \{ x \} , & x \not \in S S \cup \{ x \} \in J ( P ) \\ S \setminus \{ x \} , & x \in S S \setminus \{ x \} \in J ( P ) \\ S & \end{cases} \end{align*}


\begin{align*} Y _ O ( P , G ) \leq C ( P ) < 1 = \tilde { Y } _ O ( P , G ) \end{align*}


\begin{align*} \mathfrak { g } _ { A _ { I , \mathrm { s y m } } , ( k _ 1 , k _ 2 ) } ^ { r , s } : = \widetilde \psi _ I ^ { - 1 } ( \mathfrak { g } ^ { r , s } _ { A _ I , ( k _ 1 , k _ 2 ) } ) \end{align*}


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


\begin{align*} B _ h ^ S : = \{ g \in G _ S : \ , \hbox { H } _ S ( g ) \le h \} . \end{align*}


\begin{align*} H : = L ^ 2 ( \Omega ) \times L ^ 2 ( \partial \Omega , \beta ^ { - 1 } d S ) \end{align*}


\begin{align*} F ( \mathbf { B } _ U ) = ( \kappa - \nu ) ( \kappa - \nu - 1 ) ( \kappa - 2 ) \ , . \end{align*}


\begin{align*} \big ( f _ 1 ( \tau , \theta ) \cdot y _ { \gamma _ { m 1 } } \big ) \widehat { \ast } \big ( f _ 2 ( \tau , \theta ) \cdot y _ { \gamma _ { m 2 } } \big ) = f _ 1 ( \tau , \theta - \langle \gamma _ { m 2 } , - \rangle \tau / 2 ) f _ 2 ( \tau , \theta + \langle \gamma _ { m 1 } , - \rangle \tau / 2 ) \cdot y _ { \gamma _ { m 1 } + \gamma _ { m 2 } } , \end{align*}


\begin{align*} m _ q : = \inf _ { \substack { p \ge q \\ p \in \mathcal P } } \mu _ p , \quad M _ x : = \sup _ { \substack { y \ge x \\ y \in \R } } \mu _ y . \end{align*}


\begin{align*} \left [ y \frac { \partial } { \partial x } , x \frac { \partial } { \partial u } \right ] = y \frac { \partial } { \partial u } = \delta \left ( \frac { \partial } { \partial z } \right ) \end{align*}


\begin{gather*} f _ i ( v ) = \begin{cases} s _ i v & s _ i v > v s _ i v \in W ^ J \\ 0 & \end{cases} e _ i ( v ) = \begin{cases} s _ i v & s _ i v < v \\ 0 & \end{cases} \end{gather*}


\begin{align*} \big \langle \Phi ' ( \vec p ) ^ * \vec g , \widehat { \vec p } \big \rangle _ { ( L ^ \infty ( D ) ^ 5 ) ' \times L ^ \infty ( D ) ^ 5 } & = \big \langle F ' ( V ( \vec p ) ) ^ * \vec g , V ' ( \vec p ) \widehat { \vec p } \big \rangle _ { \mathcal { L } ( X ) ' \times \mathcal { L } ( X ) } \\ [ 1 m m ] & = \int _ 0 ^ T \ ! \big \langle V ' ( \vec p ) \widehat { \vec p } \big ( u ' ( t ) + Q u ( t ) \big ) , w ( t ) \rangle _ X \ , \d t \end{align*}


\begin{align*} P _ { n m _ 0 } = Q _ { n m _ 0 } Q _ { n m _ 0 - 1 } \cdots Q _ 1 \end{align*}


\begin{align*} E r r M ^ 2 _ { i , i _ 1 } ( t _ 1 , t _ 2 ) = \int _ { t _ 1 } ^ { t _ 2 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } \int _ { \R ^ 3 } e ^ { i X ( s ) \cdot \xi - i s \hat { v } \cdot \eta + i s \mu _ 1 | \xi - \eta | } \mathcal { F } \big [ ( \hat { v } - \hat { V } ( s ) ) \times B f \big ] \cdot \nabla _ v \big [ ( \hat { V } ( s ) \cdot \xi - \hat { v } \cdot \eta + \mu _ 1 | \xi - \eta | ) ^ { - 1 } \end{align*}


\begin{align*} f _ i ( x ) = f \circ S ^ { 1 - i - w } ( x ) = f ( x _ { i + w } , \dotsc , x _ { i + w + k - 1 } ) i \in \Z . \end{align*}


\begin{align*} \eta _ { x , p ; \delta } ( y ) = \eta ^ \parallel _ { x ^ \parallel , p ^ \parallel ; \delta } ( y ^ \parallel ) \eta _ { x ^ \perp , p ^ \perp ; \delta } ^ \perp ( y ^ \perp ) \end{align*}


\begin{align*} W _ { i / n } - \widetilde W ^ { ( n ) } _ { i / n } - ( W _ { 1 } - \widetilde W ^ { ( n ) } _ { 1 } ) i / n = \sum _ { k = 1 } ^ i \eta _ { n , k } . \end{align*}


\begin{align*} \widetilde { \Delta } _ { r a d } ( f ) : = \frac { 1 } { 2 } f '' + \frac { \alpha } { 2 } f ' , \end{align*}


\begin{align*} U c _ 1 \int \limits _ { t _ 0 } ^ { t _ 1 } \cos t d t + U c _ 2 \int \limits _ { t _ 0 } ^ { t _ 1 } \sin t d t + U \mu \int \limits _ { t _ 0 } ^ { t _ 1 } \frac { a ^ 2 } { \lambda } d t = 0 . \end{align*}


\begin{align*} c x ^ { \sum _ { j = 1 } ^ h k _ { I _ j } } \hbar ^ { - h } \prod _ { j = 1 } ^ h \langle \prod _ { i \in I _ j } \tau ^ { a _ i } \sigma ^ { k _ { I _ j } + 1 } \rangle ^ { \tfrac 1 r , o } \end{align*}


\begin{align*} \mathrm { P } _ { \Phi } = - \inf f _ { \Phi } ( E _ { 1 } ) < \infty , \end{align*}


\begin{align*} \delta _ n = c \rho ^ 2 \epsilon _ n = c \rho ^ 2 a n ^ { \frac { 1 } { 2 } ( b - 1 ) } \ ; , \ , \end{align*}


\begin{align*} \phi _ { n } ^ { \pm } ( x ) = \frac { e ^ { \pm i \left ( 2 n \pi x / a - \theta \right ) } } { \sqrt { a } } ; E _ { n } ^ { \pm } = \frac { \hslash ^ { 2 } } { 2 m } \left ( \frac { 2 n \pi } { a } \right ) ^ { 2 } + \frac { \beta \hslash ^ { 4 } } { 3 m } \left ( \frac { 2 n \pi } { a } \right ) ^ { 4 } , \end{align*}


\begin{align*} \lVert \frac { 1 } { N } \sum _ { n = 1 } ^ N u _ n ^ * T u _ n \rVert _ { \infty , 2 } ^ 2 & \leq \frac { 1 } { N ^ 2 } \sum _ { n , m = 1 } ^ N \lVert u _ m ^ * T u _ m u _ n ^ * T u _ n \rVert _ { \infty , 1 } \\ & \leq \frac { 2 } { N ^ 2 } \sum _ { 1 \leq m < n \leq N } 2 ^ { - n } + \frac { 1 } { N ^ 2 } \sum _ { n = 1 } ^ N \lVert u _ n ^ * T ^ 2 u _ n \rVert _ { \infty , 1 } \\ & \leq \frac { 2 } { N } + \frac { \| T \| ^ 2 } { N } \to 0 . \end{align*}


\begin{align*} [ X , Y ] _ { \pi _ 1 ( M ) } = \sum _ { p \in \alpha \cap \beta } \epsilon ( p ; \alpha , \beta ) \{ \alpha _ p \beta _ p \} \end{align*}


\begin{align*} g \left ( x \right ) = \sum _ { z \in \mathbb { Z } ^ { d } } u _ { z } \mathrm { e } ^ { 2 \pi \gamma ^ { - 1 } i z \cdot x } , x \in \mathbb { R } ^ { d } \ , \end{align*}


\begin{align*} \liminf _ { n \to \infty } | T _ { \beta } ^ n x - x _ 0 | = 0 \end{align*}


\begin{align*} \mathcal R _ 1 ( f ) = \frac { 1 } { \sqrt { \alpha \beta ( \alpha + \beta ) } } \begin{pmatrix} ( \beta \partial _ { 1 } f - \alpha \partial _ { m + 1 } f ) _ { | \Delta } \\ \vdots \\ ( \beta \partial _ { m } f - \alpha \partial _ { 2 m } f ) _ { | \Delta } \end{pmatrix} \end{align*}


\begin{align*} \textstyle \lim \limits _ { i \to \infty } \frac { \log _ q n _ i } { \mu _ q ( n _ i ) } = 0 ; ~ ~ ~ \mbox { a n d } ~ ~ ~ - 1 \in \langle q \rangle _ { n _ i } , \forall ~ i = 1 , 2 , \cdots ; \end{align*}


\begin{align*} \psi = s _ \psi \varphi + \psi _ 0 \end{align*}


\begin{align*} \begin{aligned} \nabla _ x \mathcal { L } ( x ) & = \tfrac { 1 } { N } A ^ T \big ( \sigma ( A _ i x ) - y \big ) \\ \nabla ^ 2 _ x \mathcal { L } ( x ) & = \tfrac { 1 } { N } A ^ T B A \end{aligned} \end{align*}


\begin{align*} Q _ { N , d , D , \beta , \gamma } ( A ) = \frac { 1 } { Z _ { N , d , D , \beta , \gamma } } E \big [ \mathcal { E } _ { N , d , D , \gamma } \mathbf { 1 } _ A \big ] . \end{align*}


\begin{align*} \frac { 1 } { k ! } \big ( e _ { \lambda } ( t ) - 1 \big ) ^ { k } = \sum _ { n = k } ^ { \infty } S _ { 2 , \lambda } ( n , k ) \frac { t ^ { n } } { n ! } , ( k \ge 0 ) , ( \mathrm { s e e } \ [ 8 ] ) . \end{align*}


\begin{align*} a _ { i j } = a _ { i j } ( t ) : = \frac { \eta } { N ( ( \lambda _ i ( t ) - \lambda _ j ( t ) ) ^ 2 + \eta ^ 2 ) } , \end{align*}


\begin{align*} \begin{array} { l l } A _ n ( t ) = \psi ^ { - 1 } \partial _ t ^ { n } \psi = \Pi _ { k = 1 } ^ n \left [ ( \partial _ t ^ { k - 1 } \psi ) ^ { - 1 } \partial _ t ^ { k } \psi \right ] . \end{array} \end{align*}


\begin{align*} n p _ t ( o , x ) \leq \frac { c _ 1 n } { t f ( n ) } e ^ { - c _ 2 / a } = \frac { c _ 1 } { a } e ^ { - c _ 2 / a } . \end{align*}


\begin{align*} T _ { \lambda ^ 0 } : = | \mu - m ^ 2 | \ , C \star S = | \mu - m ^ 2 | \big ( C \star C + C \star E \big ) . \end{align*}


\begin{align*} \Psi _ \kappa ^ { - 1 } x _ 1 & = x _ 1 , \\ \Psi _ \kappa ^ { - 1 } x _ \ell & = x _ \ell - \sum _ { j = 1 } ^ { \ell - 1 } \Big ( \# W ^ u ( x _ \ell ; P ) \cap W ^ s ( x _ j ; \tilde P ) \ \ 2 \Big ) \cdot \Phi _ \kappa ^ { - 1 } x _ j , \forall \ell \geq 2 . \end{align*}