\begin{align*} F _ { 1 ^ s } ( \theta _ 1 , \theta _ 2 , \ldots , \theta _ n ) = \theta _ 1 F _ { 1 ^ { s - 1 } } ( \theta _ 2 , \ldots , \theta _ n ) + F _ { 1 ^ s } ( \theta _ 2 , \theta _ 3 , \ldots , \theta _ n ) . \end{align*}


\begin{align*} I ( h ) : = \sum _ { j = 1 } ^ l h _ j \R [ x ] \ , . \end{align*}


\begin{align*} d \sigma - ( \log \sqrt { \tau _ 0 } ) _ z d z \sigma = e ^ { 2 i \theta } \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { \sigma } I \end{align*}


\begin{align*} \Psi ( s ) = \left \{ \begin{array} { l } 1 , \ , \ , \ , \ , - \infty < s \leq 0 , \\ 0 , \ , \ , \ , \ , s \geq 1 . \end{array} \right . \end{align*}


\begin{align*} \lim _ { n \to \infty } \| T _ n p _ n f - p _ n T f \| = 0 f \in X . \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { 1 } { \varepsilon ^ 2 \ln \frac { 1 } { \varepsilon } } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ p _ + v ^ \tau _ \varepsilon d x = q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p } _ + d y . \end{align*}


\begin{align*} v = D '' u _ 0 + P _ { \omega , h } ( \sqrt { \lambda } \tau _ 0 ) . \end{align*}


\begin{align*} q _ \pi ( G _ 0 ) = q _ \pi ( V \times K ) = q _ \pi ( V ) q _ \pi ( K ) . \end{align*}


\begin{align*} T ( \delta _ \xi ) = \frac { \delta _ \xi } { \sqrt { 1 + \gamma ( \xi ) ^ 2 } } . \end{align*}


\begin{align*} v = 0 \partial A ( r _ { 1 } , r _ { 2 } ) . \end{align*}


\begin{align*} \eta : = - \tau ( \mathbf d + \mathrm i \mathbf d ^ \perp ) . \end{align*}


\begin{align*} \Psi = \rho ( x ' ) - x _ n - \theta | x ' | ^ 2 + K x _ n ^ 2 . \end{align*}


\begin{align*} \operatorname { I } ^ { - 1 } _ { u } ( a , b ) = \sum _ { i = 1 } ^ { \infty } q _ i \ , [ a \operatorname { B } ( a , b ) u ] ^ { i / a } , \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n { \binom n k \frac { F _ { j k + m } } { L _ j ^ { k } } B _ { n - k } } = \begin{cases} F _ m B _ n \Big ( \frac { \alpha ^ j } { L _ j } \Big ) , & \\ [ 6 p t ] \frac { { L _ m } } { { \sqrt 5 } } B _ n \Big ( \frac { \alpha ^ j } { L _ j } \Big ) , & \end{cases} \end{align*}


\begin{align*} \lambda _ x | _ { E _ { 2 , m } } = \lambda _ y , \lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } = \lambda ^ j \lambda _ x | _ { E _ { 2 , [ m - e _ j , m + e _ i ] } } = \phi ^ j \end{align*}


\begin{align*} \bigg | \frac { \partial h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \partial z _ { i , \hbar } } \bigg | \leq C , \ \ 1 \leq i , j \leq m , \hbar = 1 , 2 , \end{align*}


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


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


\begin{align*} \begin{aligned} \frac { ( - 1 ) ^ r } { r ! } \frac { \mathrm { \partial } ^ r } { \mathrm { \partial } \alpha ^ r } Z ^ { \star } _ { I } ( \{ k _ i \} _ { i = 1 } ^ { n } ; ( \alpha , \beta ) ) = \sum _ { \begin{subarray} { c } r _ 1 + \cdots + r _ { n - 1 } = r \\ r _ i \in \mathbb { Z } _ { \ge 0 } \end{subarray} } Z ^ { \star } _ { I } ( \{ k _ i , \{ 1 \} ^ { r _ i } \} _ { i = 1 } ^ { n - 1 } , k _ n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


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


\begin{align*} \langle v , w \rangle : = \sum _ { i \in Q _ 0 } v _ i w _ i - \sum _ { h \in Q _ 1 } v _ { h ' } w _ { h '' } \ , \ , v , w \in \mathbb { Z } ^ n . \end{align*}


\begin{align*} ( 2 \pi ) ^ { - n } | \lambda | ^ n \ , \int _ { \C ^ n } \widehat { f } ( a , w ) \varphi _ { k , \lambda } ^ { n - 1 } ( z - w ) e ^ { \frac { i } { 2 } \lambda \Im ( z \cdot \bar { w } ) } d w = f ^ { - \lambda } \ast _ { - \lambda } \varphi _ { k , \lambda } ^ { n - 1 } ( z ) = \widehat { f } ( a , z ) . \end{align*}


\begin{align*} \int _ { B ( 0 , R ) } \partial _ t a _ { j } \cdot \xi \dd x + \int _ { B ( 0 , R ) } B _ j \times u _ j \cdot \mathbb { P } \xi \dd x = - \mu _ j \int _ { B ( 0 , R ) } b _ j \cdot \nabla \times \xi \dd x . \end{align*}


\begin{align*} [ v _ a , v _ b ] = \alpha _ c \varepsilon _ c v _ c , \forall \quad \mbox { c y c l i c } \quad ( a , b , c ) \in \mathfrak { S } _ 3 , H = h \mathrm { v o l } _ g , \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \varphi | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


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


\begin{align*} \tilde { h } _ d ' ( x ) = \log d + \left ( \psi ( x ) - \psi \left ( x + \frac 1 2 \right ) \right ) - \psi \left ( x + \frac { d - 1 } { 2 } \right ) \end{align*}


\begin{align*} \tau _ y = \tau ' _ y v _ y , v _ y = \sum _ { \substack { 1 \leq r \leq j \\ C _ r \in O _ y } } \lambda _ r ( e _ { i _ { r , 1 } } + \dots + e _ { i _ { r , d _ r } } ) \end{align*}


\begin{align*} A ^ { 1 , \dots , k } : = A ^ { 1 } A ^ { 2 } \dots A ^ { k } . \end{align*}


\begin{align*} \mu _ { i + 1 } = ( I - \gamma g _ { \mu _ i } ) \# \mu _ i . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h ( x ) } { 1 / x } = & 2 \theta ( b - 1 ) \lim _ { x \to \infty } \left [ \frac { 1 / x ^ 2 } { 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } \right ] - 2 . \end{align*}


\begin{align*} u z _ 1 \cdots z _ r = t _ 1 \cdots t _ s = z _ 1 ^ { l _ { 1 1 } + \dots + l _ { s 1 } } \cdots z _ r ^ { l _ { 1 r } + \dots + l _ { s r } } . \end{align*}


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


\begin{align*} h ( x ) = \frac { 2 \theta } { \operatorname { B } ( a , b ) \ , x ^ 3 } \frac { \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) } { I _ { 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) } ( b , a ) } . \end{align*}


\begin{align*} 0 = D f | _ { \Gamma _ 0 } ( \Delta ) = \phi ( C \Delta + \Delta ^ T C ^ T + \Delta ^ T D \Gamma _ 0 + \Gamma _ 0 ^ T D \Delta ) . \end{align*}


\begin{align*} \langle \langle \varphi , \varphi ' \rangle \rangle = \tau \langle \langle \varphi ' , \varphi \rangle \rangle \end{align*}


\begin{align*} \mathcal { B } ( u , v , w ) = \langle [ u , v ] _ H , w \rangle , u , v , w \in E . \end{align*}


\begin{align*} \begin{aligned} \Xi = \{ \gamma | \ , \Phi _ 1 ( \gamma ) \cap \Phi _ 2 ( \gamma ) \neq \emptyset , - 3 < \gamma \leq 1 \} \ , , \end{aligned} \end{align*}


\begin{align*} \sigma _ { w _ { \rm d } } ^ 2 = \mathbb { E } \left \{ \mu ^ 2 _ { \rm a } ( x ) \right \} - \mathbb { E } ^ 2 \left \{ \mu _ { \rm a } ( x ) \right \} - \alpha ^ 2 . \end{align*}


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


\begin{align*} \Gamma ( x , y ) : = - \frac { 1 } { e ^ { - { \psi } } } \left ( d \phi ( x ) \int _ { - \infty } ^ y e ^ { - { \psi ( x , v ) } } d v - \int _ { - \infty } ^ y e ^ { - { \psi ( x , v ) } } d _ x { \psi } ( x , v ) d v \right ) . \end{align*}


\begin{align*} \operatorname { d e t } \left ( M _ { n , 1 } \right ) = ( - 1 ) ^ { \frac { n - 1 } { 2 } } \left ( \frac { n - 1 } { 2 } ! \right ) ^ { 2 } . \end{align*}


\begin{align*} \int _ 0 ^ \infty y ^ { - r / 2 } \exp \left \{ - ( a + n ) y \right \} \mathrm { d } y = \frac { \Gamma \left ( 1 - \frac { r } { 2 } \right ) } { ( a + n ) ^ { 1 - r / 2 } } , r < 2 . \end{align*}


\begin{align*} T & = \left ( [ x ] \times [ y ] \right ) \backslash \left ( [ x - r + 1 . . x ] \times y \right ) \backslash \left ( x \times [ y - s + 1 . . y - 1 ] \right ) \mbox { a n d } \\ T ' & = \left ( [ x - w ] \times [ y ] \right ) \backslash \left ( ( x - w ) \times [ y - r ' + 1 . . y ] \right ) , \end{align*}


\begin{align*} \Delta _ { B } \textbf { \textit { u } } = ( \nabla ^ j \nabla _ j u ^ i ) \frac { \partial } { \partial x _ i } , \textbf { \textit { u } } = u ^ i \frac { \partial } { \partial x _ i } \in \mathfrak { X } ( \Omega ) . \end{align*}


\begin{align*} \begin{aligned} y \left [ i \right ] & = \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { d } } \left [ { i - { n _ l } } \right ] } + z \left [ i \right ] \\ & = \sum \limits _ { t = 0 } ^ { { n _ { { \rm { s p a n } } } } } { { { \bf { g } } ^ H } \left [ t \right ] { \bf { d } } \left [ { i - t - { n _ { \min } } } \right ] } + z \left [ i \right ] , \end{aligned} \end{align*}


\begin{align*} B S ( m , n ) = \langle a , b \mid a b ^ m = b ^ n a \rangle \end{align*}


\begin{align*} \vert \vert \psi ( \mathbf x ) \vert \vert _ { H ^ { 1 , 8 } } = \mathcal O ( \tau ^ { - \frac { 2 } { 3 } } ) , \end{align*}


\begin{align*} | \mathfrak { I } _ { q _ 1 , \ldots , q _ 4 , \kappa } ( \ell ) | \leq \frac { C } { \ell ^ 3 \sqrt { \ell } } \log \ell , \end{align*}


\begin{align*} L _ X g = h . \end{align*}


\begin{align*} \begin{cases} 2 + \gamma + \frac { 2 \theta } { \alpha } < p , & \alpha \leq 2 , \\ 4 + \gamma + \theta - \alpha < p , & \alpha > 2 . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta v + u v = f , & x \in \Omega , \\ \nabla v \cdot \nu + v = \eta , & x \in \partial \Omega \end{array} \right . \end{align*}


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


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


\begin{align*} \int _ { \mathbb { R } ^ { 2 } _ { + } } \nabla \phi \cdot \nabla \tilde { \phi } \frac { 1 } { r } \dd z \dd r = \mu ^ { 2 } \int _ { \mathbb { R } ^ { 2 } _ { + } } ( \phi - \phi _ { \infty } ) _ { + } \tilde { \phi } \frac { 1 } { r } \dd z \dd r , \end{align*}


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


\begin{align*} \int _ { \mathfrak U } | \omega | & = { \pi ' _ s } _ ! \int _ { \mathfrak U ' } | \omega | = { \pi ' _ s } _ ! \Big ( { i ' _ s } ^ * \int _ { \mathfrak X ' } | \omega | \Big ) \\ & = ( \mathfrak i _ 0 ^ * \circ { \pi _ s } _ ! ) \int _ { \mathfrak X ' } | \omega | \\ & = \mathfrak i _ 0 ^ * \int _ { \mathfrak X } | \omega | , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' + B = 0 & \textnormal { i n } \ I \\ \varphi ( r _ 1 ) = 1 \\ \varphi ' ( r _ 1 ) = b < 0 \end{array} \right . \end{align*}


\begin{align*} \mbox { $ ( ( X _ { 2 , 6 } \setminus N ( c _ 2 ) ) \cup \{ c _ 6 \} ) \cap { \cal N } = \emptyset $ . } \end{align*}


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


\begin{align*} ( \delta _ n ^ k ) _ ! = \sum _ { J = \{ i _ 0 < \hdots < \widehat { i _ k } < \hdots < i _ n \} \subset K = \{ i _ 0 < \hdots < i _ n \} } ( \nu _ { K } ^ J ) _ ! \end{align*}


\begin{align*} f ( x ) = c \ , \int _ { - \infty } ^ \infty \int _ { K / M } \widetilde { f } ( \lambda , b ) e ^ { ( i \lambda + \rho ) A ( x , b ) } | c ( \lambda ) | ^ { - 2 } d b \ , d \lambda \end{align*}


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


\begin{align*} n = \frac { \beta ^ { 3 } - 3 } { 6 } \end{align*}


\begin{align*} a \in \sigma _ l ( A ) , \ , v = ( 1 , x ) ^ T \in V _ l ( a ) , \forall x \in S _ L ( b , 0 ) \cap S _ L ( a - d , 0 ) ; \end{align*}


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


\begin{align*} w ( x ' , t ) = W ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x ' ) ) , \ \ \varphi ( x ' , t ) = \varPhi ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x ' ) ) , \end{align*}


\begin{align*} \begin{array} { l } { \rm S I N R } _ { v , j } ^ q = 2 y _ { v , j } ^ q \sqrt { p _ { v , j } ^ { q } { \left ( { { R _ { \rm { P D } } } { G _ { v , j } ^ { q } } } \right ) } ^ 2 } - y _ { v , j } ^ q \left ( { \sum \limits _ { j ' \ne j } { \sum \limits _ { v ' \ne v } p _ { v ' , j ' } ^ { q } { \left ( { { R _ { \rm { P D } } } { G _ { v ' , j } ^ { q } } } \right ) } ^ 2 } + N _ { \rm V L C } B _ { \rm V L C } } \right ) . \end{array} \end{align*}


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


\begin{align*} & L _ 4 \colon \mathbf { Z } = 2 \eta \mathbf { X } - \iota \mathbf { W } = 0 , & L ' _ 4 \colon \mathbf { W } - \rho \mathbf { Z } = 2 \eta ' \mathbf { X } - \iota ' \mathbf { W } = 0 , \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \Delta _ \Psi \omega = f ( u _ 2 ) - f ( u _ 1 ) & \textnormal { i n i n t } M \\ \omega = 0 & \textnormal { o n } \ \partial M , \end{array} \right . \end{align*}


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


\begin{align*} \operatorname { E } ( X ) = \frac { \sqrt { \pi \theta } } { \operatorname { B } ( a , b ) } \sum _ { n = 0 } ^ b \binom { b - 1 } { n } \frac { 1 } { \sqrt { a + n } } . \end{align*}


\begin{align*} A = \begin{pmatrix} 1 & - 1 \\ 0 & 1 \\ 0 & - 1 \end{pmatrix} B = \begin{pmatrix} 1 & 1 & - 1 \end{pmatrix} . \end{align*}


\begin{align*} \mathcal { C } _ { i j } & \simeq \Omega _ { X } | _ { D _ { i j } } & \mathcal { C } _ { i j } ^ l & \simeq \omega _ l | _ { D _ { i j } } , l = i , j \\ \omega _ X | _ { D _ i } & \simeq \mathcal { C } _ i \ \otimes \omega _ i , & \det ( \mathcal { C } _ { i j } ) & \simeq \mathcal { C } _ { i j } ^ l \otimes \mathcal { C } _ l | _ { D _ { i j } } l = i , j \end{align*}


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


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


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + , \end{align*}


\begin{align*} \gamma ^ i ( x ) = \frac 1 { p - 1 } \sum _ { j \neq i , j \in \C } K ( x ^ i - x ^ j ) , ~ ~ ~ ~ \forall x \in \mathbb R ^ { N d } , \end{align*}


\begin{align*} \gamma \alpha \omega - \gamma \omega = \dfrac { ( \alpha \omega - \omega ) \det ( \gamma ) } { ( c \omega + d ) ( c \alpha \omega + d ) } . \end{align*}


\begin{align*} \kappa : = D _ { a , n } = \begin{pmatrix} a \\ & a ^ { - 1 } \\ & & \ddots \\ & & & a \\ & & & & a ^ { - 1 } \end{pmatrix} . \end{align*}


\begin{align*} M _ { I J } ^ \prime = \begin{pmatrix} G _ { I _ 1 } ^ \prime & G _ { I _ 2 } ^ \prime & \cdots & G _ { I _ { 2 k } } ^ \prime \\ G _ { J _ 1 } ^ \prime & G _ { J _ 2 } ^ \prime & \cdots & G _ { J _ { 2 k } } ^ \prime \end{pmatrix} \end{align*}


\begin{align*} \eta ( \mathbf 0 ) v ( \mathbf 0 ) \int _ { \Gamma _ h } e ^ { \rho \cdot \mathbf x } \mathrm d \sigma = \sum _ { i = 1 } ^ 6 I _ i + J _ 1 + J _ 2 , \end{align*}


\begin{align*} H ^ 2 ( \mathbb { D } ) & = \{ \sum _ { n = 0 } ^ { \infty } a _ n z ^ n \mid ( a _ n ) \in \ell ^ 2 \} \\ Y & = \{ \sum _ { n = 0 } ^ { \infty } a _ n f _ n \mid ( a _ n ) \in \ell ^ 2 \} \end{align*}


\begin{align*} \mathcal { D } _ { G , \kappa } ^ { \mathrm { c h } } \simeq \bigoplus _ { \lambda \in P ^ + } \mathbb { V } ^ { \kappa } _ \lambda \otimes \mathbb { V } ^ { \kappa ^ * } _ { \lambda ^ * } , \frac { 1 } { \kappa + h ^ \vee } + \frac { 1 } { \kappa ^ * + h ^ \vee } = 0 . \end{align*}


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


\begin{align*} \left \{ \begin{array} { l l } - L u = f ( u ) & B _ 1 \\ u = 0 & \partial B _ 1 \end{array} \right . \end{align*}


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


\begin{align*} h ^ \theta ( x , u ) \le \sup _ { \rho \in \Theta _ { n - 1 } } E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \biggr ) \biggr ] = V \left ( \xi _ 1 ^ \theta ( u ) , n - 1 , x + u \right ) . \end{align*}


\begin{align*} \frac { \left ( 6 n - \beta ^ { 3 } + 3 \right ) \left ( n ^ { 2 } + n + 1 \right ) } { 3 } = 0 \end{align*}


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


\begin{align*} \partial _ t u + ( u \cdot \nabla ) \ , u & = - \nabla p , \\ \nabla \cdot u & = 0 \end{align*}


\begin{align*} \mathcal { F } = \left \{ f \in \mathcal { H } ( \mathbb { C } ) : \frac { 1 } { \pi } \iint _ { \mathbb C } | f ( z ) | ^ 2 e ^ { - | z | ^ 2 } d A ( z ) < \infty \right \} , \end{align*}


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


\begin{align*} d \hat { \sigma } _ i ( T , X _ j ) = - T ( r ^ j _ i ) + X _ j ( \lambda _ i ( T ) ) - \lambda _ i ( \nabla ^ v _ { X _ j } T ) + \lambda _ j ( T ) r ^ j _ i = 0 , \quad \forall j \neq i . \end{align*}


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


\begin{align*} \mbox { $ ( ( X _ { 2 , 7 } \setminus N ( c _ 2 ) ) \cup ( X _ { 3 , 7 } \setminus N ( c _ 3 ) ) \cup \{ c _ 7 \} ) \cap { \cal N } = \emptyset $ . } \end{align*}


\begin{align*} \lambda _ \mathbb { I } - \rho _ { \mathbb { I } } = d ^ { \mathbb { I } } h + h d ^ { \mathbb { I I } } , \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) = \displaystyle \frac { z ^ m } { \sqrt { \Gamma ( q m + 1 ) } } , \end{align*}


\begin{align*} \lambda _ c = 2 \sum _ { j = 1 } ^ { M } \lvert \phi ^ { \left ( j , c \right ) } \rvert ^ 2 - f ^ { \left ( c \right ) } \end{align*}


\begin{align*} \nabla \Phi ^ g ( x , y ) = \begin{bmatrix} e ^ { y _ 1 } \nabla g _ 1 ( x ) & \dots & e ^ { y _ m } \nabla g _ m ( x ) \\ e ^ { y _ 1 } g _ 1 ( x ) & \dots & 0 \\ . & \dots & . \\ 0 & \dots & e ^ { y _ m } g _ m ( x ) \end{bmatrix} = \varphi ^ g ( x ) \times \begin{bmatrix} e ^ { y _ 1 } & \dots & 0 \\ . & \dots & . \\ 0 & \dots & e ^ { y _ m } \end{bmatrix} \ , . \end{align*}


\begin{align*} v _ { t _ 0 } ( t ) : = \lim _ { j \to + \infty } v _ { t _ 0 , B _ j } ( t ) = \left \{ \begin{aligned} & - t _ 0 & & x < - t _ 0 , \\ & \ t & & x \ge t _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} \Vert f \Vert _ { H ^ s _ \gamma ( G ) } : = \left ( \int _ { G ^ \wedge } \left ( 1 + \gamma ( \xi ) ^ 2 \right ) ^ s \vert \widehat { f } ( \xi ) \vert ^ 2 d \nu ( \xi ) \right ) ^ { 1 / 2 } . \end{align*}


\begin{align*} \sum _ { 1 \leq i , j \leq n } \Psi ( v _ i v _ i ^ * v ' _ j ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi \left ( \sum _ { i = 1 } ^ n v _ i v _ i ^ * v ' _ j \right ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi \left ( p v ' _ j \right ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi ( v ' _ j ) { v ' _ j } ^ * . \end{align*}


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


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


\begin{align*} \dot { x } ( t ) & = - A ^ { \circ } x ( t ) \quad ( t > 0 ) , \\ y ( t ) & = B ^ { \circ } x ( t ) \quad ( t \geq 0 ) , \\ x ( 0 ) & = x _ 0 \in X ^ \circ . \end{align*}


\begin{align*} & \Tilde { x } = \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \in \mathcal { S } ^ n \\ & \| \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \end{align*}


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


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X _ n ( t , \omega ) - X ( t , \omega ) \Vert _ H = 0 , \ ( t , \omega ) . \end{align*}


\begin{align*} \phi ( x , y ) = \sum _ { i = 1 } ^ L c _ i a _ i ( x ) b _ i ( y ) , \end{align*}


\begin{align*} \phi ( z ) = \sum _ { q \geq 2 } \frac { | b _ q | } { q ! } H _ { q } ( z ) , \end{align*}


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


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


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


\begin{align*} \Phi ^ \dagger ( v , p ) = 0 , \end{align*}


\begin{align*} { \rm D } ^ 2 \widetilde { W } ( { \rm D } \phi ) . ( \xi \otimes \eta , \xi \otimes \eta ) > 0 \forall \ \eta , \xi \in \mathbb { R } ^ 3 , \ \ \lVert \eta \rVert = \lVert \xi \rVert = 1 , \end{align*}


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


\begin{align*} z _ 0 ^ { p ^ r } x _ 0 + z _ 1 ^ { p ^ r } x _ 1 \cdots + z _ n ^ { p ^ r } x _ n = 0 . \end{align*}


\begin{align*} \Phi ^ g ( x , y ) = ( g _ 1 ( x ) e ^ { y _ 1 } , \dots , g _ m ( x ) e ^ { y _ m } ) \ , . \end{align*}


\begin{align*} \gamma _ 2 \dot s ( 0 ) = - \dot { \nu } ( 0 ) , \end{align*}


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


\begin{align*} d ^ * d u + d d ^ * u = \Delta u = d ^ * ( - i a ) \end{align*}


\begin{align*} \nabla _ i u : = \nabla _ { e _ i } u = e _ i ^ j \partial _ j u = e _ i ^ j u _ j \end{align*}


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


\begin{align*} ( \sqrt { - 1 } & \Theta _ { T M } + \varpi \otimes I d _ { T M } ) ( \kappa _ 1 \otimes \kappa _ 2 , \kappa _ 1 \otimes \kappa _ 2 ) \ge 0 \\ & \forall \kappa _ 1 , \kappa _ 2 \in T M w i t h \langle \kappa _ 1 , \kappa _ 2 \rangle = 0 \end{align*}


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


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { c _ \varepsilon } { \ln \frac { 1 } { \varepsilon } } = \pi \min \limits _ { \overline { \Omega } } q ^ 2 \sqrt { d e t ( K _ H ) } . \end{align*}


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


\begin{align*} \begin{aligned} & \chi ( x , r , s ) = \chi ( y , r , s ) \quad \mbox { i f } | x | = | y | \quad \mbox { a n d } \\ & \chi ( x , r , s ) + | \partial _ { r } \chi ( x , r , s ) | \le \chi _ { 0 } ( s ) \quad \mbox { w i t h s o m e } \ , \ , \chi _ { 0 } \in \mathcal { C } ( \R _ { + } ) . \end{aligned} \end{align*}


\begin{align*} \operatorname { d i v } \textbf { \textit { u } } = \nabla _ i u ^ i = \frac { \partial u ^ i } { \partial x _ i } + \Gamma ^ j _ { i j } u ^ i , \textbf { \textit { u } } = u ^ i \frac { \partial } { \partial x _ i } \in \mathfrak { X } ( \Omega ) , \end{align*}


\begin{align*} | \langle B ^ { \circ } S _ { T - t } ^ { \circ } x ^ { \circ } , u \rangle | = | \langle x ^ { \circ } , S _ { T - t } B u \rangle | \leq C p _ { \gamma _ U } ( u ) \end{align*}


\begin{align*} \kappa _ { i [ j k ] } = \mathbf { A } _ { [ j k ] , i } = \frac { 1 } { 2 } \left ( A _ { j k , i } - A _ { k j , i } \right ) = \frac { 1 } { 2 } \left ( - \epsilon _ { j k l } \vartheta _ { l , i } + \epsilon _ { k j l } \vartheta _ { l , i } \right ) = \frac { 1 } { 2 } \left ( \epsilon _ { k j l } + \epsilon _ { k j l } \right ) \vartheta _ { l , i } = \epsilon _ { k j l } \ , \vartheta _ { l , i } . \end{align*}


\begin{align*} u = 0 , ( \nabla \times B ) \times n = 0 , B \cdot n = 0 , \end{align*}


\begin{align*} \widetilde { F } _ { 2 } ( s ) = \prod _ { p } ^ { } \left ( 1 + \frac { \lambda ^ { 2 } _ { s y m ^ { j } f } ( p ) v ( p ) } { p ^ { s } } + \cdots + \frac { \lambda ^ { 2 } _ { s y m ^ { j } f } ( p ^ { m } ) v ( p ^ { m } ) } { p ^ { m s } } + \cdots \right ) . \end{align*}


\begin{align*} \sum _ { k , \ell = 1 } ^ n A _ { k \ell } ( y , \omega ) \ , \xi _ k \ , \xi _ \ell \geqslant a _ 0 { \vert \xi \vert } ^ 2 . \end{align*}


\begin{align*} A ^ { \tilde \Psi + \eta _ n } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ { n - 1 } ] - A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ n ] = g _ 1 ( x ) + g _ 2 ( x ) . \end{align*}


\begin{align*} s _ t = \frac { \varphi ( m ) - 1 } { m - p _ t } ( r _ { p _ t } - r _ m ) . \end{align*}


\begin{align*} \inf \left \{ | | b _ 0 - \tilde { b } | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } \ | \ \tilde { b } \in S _ { h _ C , W , 0 } \ \right \} & = \inf \left \{ | | b _ 0 - \tilde { b } | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } \ \middle | \ \tilde { b } = U _ C ( \cdot + z e _ z ) - B _ { \infty } , \ z \in \mathbb { R } \right \} \\ & = \inf \left \{ | | b _ 0 + B _ { \infty } - U _ C ( \cdot + z e _ z ) | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } \ \middle | \ z \in \mathbb { R } \right \} . \end{align*}


\begin{align*} W _ { \rm { m p } } ( \overline { \mathbf { U } } ) = 0 \Rightarrow \boldsymbol { \alpha } = 0 ( \boldsymbol { \mathfrak { K } } = 0 ) . \end{align*}


\begin{align*} \pi ( u , v ) = \pi ( v , u ) \in [ 0 , \infty ) . \end{align*}


\begin{align*} D ^ k \phi ( Z ) = \sum _ { q \geq 2 \vee k } \frac { | b _ { q } | } { ( q - k ) ! } H _ { q - k } ( Z ) D ^ k L \phi ( Z ) = - \sum _ { q \geq 2 \vee k } q \frac { | b _ { q } | } { ( q - k ) ! } H _ { q - k } ( Z ) \end{align*}


\begin{align*} N _ - ( x ; k ) = I + \mathcal { P } ^ { - } \left ( N _ { - } J \right ) ( z ) , z \in \mathbb { R } , \end{align*}


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


\begin{align*} \phi ( \sigma , \tau ) = \epsilon ^ { \sum _ { r = 1 } ^ j \lambda _ r ( d _ r - 1 ) } . \end{align*}


\begin{align*} \partial _ t \left [ t ^ 2 \left [ \left ( \frac { \alpha } { t ^ 4 } + \beta + \gamma \frac { u ^ 2 } { t ^ 4 } \right ) ^ 2 + \delta ^ 2 \frac { u ^ 2 } { t ^ 4 } \right ] \right ] = 0 . \end{align*}


\begin{align*} a _ { n , k } = \lambda ^ m a _ { n - 1 , k - 1 } + ( \frac { \lambda ^ m - 1 } { \lambda - 1 } ) ^ n a _ { n - 1 , k } . \end{align*}


\begin{align*} \langle \nabla f _ { \delta } ( x ^ g _ k ) - \nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \rangle & \leq f ( x _ f ^ k ) - f _ { \delta } ( x _ g ^ k ) - \langle \nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \rangle \\ & \leq f ( x _ f ^ k ) - f ( x _ g ^ k ) - \langle \nabla f ( x ^ * ) , x _ f ^ k - x _ g ^ k \rangle + \delta _ x \\ & = B _ f ( x _ f ^ k , x ^ * ) - B _ f ( x _ g ^ k ) + \delta _ x . \end{align*}


\begin{align*} ( I - \bar t _ n \Gamma ) d _ n = \mu _ n ^ { 1 / 2 } \beta . \end{align*}


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


\begin{align*} f ( z ) - f ( 0 ) & = \int _ { 0 } ^ 1 \frac { d f ( t z ) } { d t } d t \\ & = \int _ { 0 } ^ 1 \left ( \frac { \partial f } { \partial x } ( t z ) x + \frac { \partial f } { \partial y } ( t z ) y \right ) d t . \end{align*}


\begin{align*} \partial \Omega _ \varepsilon = \{ x + \varepsilon \rho ( x ) \mathbf { n } _ x \mid x \in \partial \Omega \} , \end{align*}


\begin{align*} H ( v , w , t ^ { - 1 } ) = t ^ { ( _ { \bullet } v - w ) \cdot ( _ { \bullet } v - w ) - _ { \bullet } v \cdot _ { \bullet } v } \frac { \mathit { g l } ( _ { \bullet } v , t ) } { \mathit { g l } ( _ { \bullet } v - w , t ) } . \end{align*}


\begin{align*} \sigma _ { n - 1 } ( \kappa ) = \sigma , \partial \Sigma = \Gamma , \end{align*}


\begin{align*} u _ { 0 } ( \mathbf x ) = ( 1 + \psi ( \mathbf x ) ) e ^ { \rho \cdot \mathbf x } , \ \mathbf x \in \mathbb R ^ n \end{align*}


\begin{align*} \frac { d } { d t } \theta _ i = \omega _ i + \frac { K } { N } \sum _ { j = 1 } ^ N \sin ( \theta _ j - \theta _ i ) , i = 1 , \ldots , N . \end{align*}


\begin{align*} \begin{array} { l l l l l } \partial \phi ( [ \mathbf { a } , \mathbf { a ' } ] ) & = & & \partial \left ( [ \mathbf { a } ] \otimes ( s ^ 0 ) ^ { k + 1 } [ \mathbf { a ' } ] \right ) \\ & = & & \partial [ \mathbf { a } ] \otimes ( s ^ 0 ) ^ { k + 1 } [ \mathbf { a ' } ] + ( - 1 ) ^ { | \mathbf a | + 1 } [ \mathbf { a } ] \otimes \partial ( s ^ 0 ) ^ { k + 1 } [ \mathbf { a ' } ] \\ & = & & \phi ( \partial ( [ \mathbf { a } , \mathbf { a ' } ] ) ) \\ \end{array} \end{align*}


\begin{align*} { \bf { x } } \left [ n \right ] = \sum \limits _ { l = 1 } ^ L { { { \bf { f } } _ l } s \left [ { n - { \kappa _ l } } \right ] } , \end{align*}


\begin{align*} p _ { \rm r e v } ( x ) : = x ^ d p \Big ( \frac { 1 } { x } \Big ) = \sum _ { i = 0 } ^ d p _ i x ^ { d - i } , ~ p _ { \rm r e v } ( x ) = p _ 0 \prod _ { j = 1 } ^ d \Big ( x - \frac { 1 } { x _ j } \Big ) ~ { \rm i f } ~ p _ 0 \neq 0 , \end{align*}


\begin{align*} J ( x ) \ge J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \end{align*}


\begin{align*} \mu _ t = j _ t - \frac { L } { 2 } \medspace ( 1 \leq t \leq n ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} d x _ t ^ { * , i } = & ~ \Big [ A x _ t ^ { * , i } - B ^ 2 R ^ { - 1 } y _ t ^ { * , i } - B h \big ( \rho _ i ^ N ( \Delta ^ { N , 1 } _ { * , t } , \ldots , \Delta ^ { N , N } _ { * , t } ) \big ) \Big ] d t + \sigma d W _ t ^ i + \sigma _ 0 d W ^ 0 _ t , \\ x ^ { * , i } _ 0 = & ~ \xi ^ i . \end{aligned} \right . \end{align*}


\begin{align*} W _ r ^ M ( x ) = a _ i i \in \left [ \frac { i - 1 } M , \frac i M \right ) , \end{align*}


\begin{align*} \dot { x } = - \gamma \ , y _ \delta ( x , t ) \ , u ( t ) x ( 0 ) = x _ 0 \end{align*}


\begin{align*} m ^ a \ell ^ { t _ { \widetilde K } } = m ^ a ( \ell ' v ) ^ { t _ { \widetilde K } } = m ^ a { \ell ' } ^ { t _ { \widetilde K } } \cdot ( v + \ell ' ( v ) + \dots + \ell '^ { t _ { \widetilde K } - 1 } ( v ) ) . \end{align*}


\begin{align*} \dim { ( C / E ) / ( A / E ) } = \dim { ( D / E ) } + \dim B '' . \end{align*}


\begin{align*} M = \left ( \begin{aligned} & - 1 2 . 4 6 \\ & 0 . 5 2 - 1 \end{aligned} \right ) , A = \left ( \begin{aligned} & 1 . 1 0 \\ & \ , \ , 0 5 0 \end{aligned} \right ) . \end{align*}


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


\begin{align*} = \int F ( M ) \sum _ { k \tau \in [ 1 / K , K ] } \int \ 1 ( S < t \le W ) Q _ { \tau } ( R , d M ) G _ { \tau } ( R , S ; W ) \ , d W P _ { ( k - 1 ) \tau } ( \mu , s ; S , d R ) \ , d S . \end{align*}


\begin{align*} \left \| \psi \left ( \frac { \mathbf y } { \tau } \right ) \right \| _ { L ^ 8 ( \mathcal C ) } = \tau ^ { \frac { 3 } { 8 } } \| \psi ( \mathbf x ) \| _ { L ^ 8 ( \mathcal C ) } \leq \tau ^ { \frac { 3 } { 8 } } \| \psi ( \mathbf x ) \| _ { H ^ { 1 , 8 } ( \mathcal C ) } = \mathcal O ( \tau ^ { - \frac { 1 } { 4 0 } } ) , \end{align*}


\begin{align*} T _ 1 = \sum _ { X \in S ( 0 ) } \Theta \begin{bmatrix} m _ 1 ' & X \\ 0 & m _ 2 ' \end{bmatrix} \overline { \psi _ A ( { m _ 1 ' } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 ( q - 1 ) ^ 3 . \end{align*}


\begin{align*} | W _ i \cap C ^ { ( i ) } | = \prod _ { l = 1 } ^ { i } \frac { \beta _ { j + l - 1 } } { c _ { l } } . \end{align*}


\begin{align*} u \cdot n = 0 \end{align*}


\begin{align*} \Phi = y _ i \xi ^ i + c , \end{align*}


\begin{align*} \begin{aligned} | \epsilon _ 2 ( \eta _ a , t ) | \le \int _ N ^ t \left | h ( x _ a + \delta u ( \tau ) ) - h ( \bar { x } _ 1 ) \right | \ , d \tau \le L _ r 2 \delta . \end{aligned} \end{align*}


\begin{align*} B _ 2 = \frac { ( L + 1 ) B _ 1 } { Q } \left ( \frac { ( L + 1 ) B _ 1 } { Q } - 1 \right ) , \end{align*}


\begin{align*} \varpi _ 1 ( \mu , \nu ) = \nu \varpi ' ( \mu ) - \varpi ( \nu ) , \end{align*}


\begin{align*} g ( u \times v , w ) = \mathrm { v o l } _ g ( u , v , w ) , \end{align*}


\begin{align*} \phi ( \sigma , \tau ) = [ \widetilde \sigma , \widetilde \tau ] = [ \widetilde \tau , \widetilde \sigma ] ^ { - 1 } = [ \widetilde \tau , \widetilde \sigma ] = \phi ( \tau , \sigma ) . \end{align*}


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


\begin{align*} \partial _ { t } ^ n \sqrt { P _ { \pm , \pm } \left ( t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) } \Bigg | _ { t = t _ 0 } \asymp _ n \frac { 1 } { t _ 0 ^ { n - 1 } } , \end{align*}


\begin{align*} \| u \| _ { 2 , h _ n } = \sum _ { K \in \mathcal { T } _ n } ( u , u ) _ { H ^ 2 ( K ) } , \end{align*}


\begin{align*} \begin{cases} \Delta _ { \Psi } u = f ( u ) & \O \\ u = \phi _ { j } & ( \partial \O ) _ { j } \end{cases} \end{align*}


\begin{align*} W ( 1 ^ \infty | 1 ^ \infty ) = c - d + \sum _ { n = 2 } ^ { \infty } ( c - c _ n ) \end{align*}


\begin{align*} \gamma ( \| \cdot \| _ { \infty } , \tau _ { \operatorname { c o } } ) = \beta _ { 0 } = \mu ( \mathrm { C } _ { \operatorname { b } } ( \Omega ) , \mathrm { M } _ { \operatorname { t } } ( \Omega ) ) \end{align*}


\begin{align*} s _ { \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor } ( F ) \geq g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) , j = 1 , 2 , \dots , i , \end{align*}


\begin{align*} ( r + \nu ) V ( s , i ) = H \left ( s , i , p , q \right ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \dot { X } & = \frac { 1 } { \tau } ( Z - X ) , \\ \dot { Z } & = - \nabla f ( X ) \end{aligned} \right . \end{align*}


\begin{align*} C _ j ( S _ j ^ { \top } R _ j ) ^ { \top } ( S _ j ^ { \top } R _ j ) C _ j ^ { \top } v = \lambda C _ j C _ j ^ { \top } v , \end{align*}


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


\begin{align*} \tau = \tau ' v , v = \sum _ { r = 1 } ^ j \lambda _ r ( e _ { i _ { r , 1 } } + \dots + e _ { i _ { r , d _ r } } ) \end{align*}


\begin{align*} \begin{aligned} \abs { c _ 2 ( S ^ 2 \setminus S _ i ) } \le \abs { c _ 1 ( S ^ 1 \setminus S _ i ) } + \abs { c _ 2 ( P ^ 2 ) } & \le \abs { S } - \abs { S \cap S _ i } + q - q _ i + s _ 2 \\ & = ( 2 k - t ' - p ) - \abs { S \cap S _ i } + q - q _ i + ( p - q - s _ 1 ) \\ & = 2 k - t ' - s _ 1 - \abs { S \cap S _ i } - q _ i . \end{aligned} \end{align*}


\begin{align*} & \lim _ { j \rightarrow \infty } \sup _ { z \in V ^ { - 1 } \mathbb { B } _ n } \Big { | } ( ( V ^ { - 1 } \varphi _ { k j } V ) ^ 1 , . . . , ( V ^ { - 1 } \varphi _ { k j } V ) ^ s ) ( z ) \Big { | } \\ & \qquad = \lim _ { j \rightarrow \infty } \| V ^ { - 1 } ( \varphi _ { k j } - \rho ) \| _ \infty \\ & \leq C \lim _ { j \rightarrow \infty } \| \varphi _ { k j } - \rho \| _ \infty = 0 . \end{align*}


\begin{align*} [ a _ 1 , a _ 2 , \cdots , a _ m ] : = \cfrac { 1 } { a _ 1 - \cfrac { 1 } { a _ 2 - \cfrac { 1 } { \ddots - \cfrac { 1 } { a _ { m } } } } } \end{align*}


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


\begin{align*} \frac { 1 } { 1 - y } = \sum _ { n = 0 } ^ \infty y ^ n . \end{align*}


\begin{align*} \begin{cases} | \mathcal { K } ( c ' , x ) | \leqslant ( c ' + 1 ) C , & | x | < 1 , \\ \mathcal { K } ( c ' , x ) \leqslant ( - \alpha + \frac { \beta } { | x | } + \frac { C } { | x | } + c ' C ) , & | x | \geqslant 1 , \end{cases} \end{align*}


\begin{align*} \Phi ( x ' ) : = \partial _ n K ^ L ( x ' , 1 ) , x ' \in \R ^ { n - 1 } . \end{align*}


\begin{align*} \begin{aligned} | \varepsilon _ \delta ( z _ 1 , \bar { y } & , t ) - \varepsilon _ \delta ( x _ a , \bar { y } _ a , t ) | \\ \le & L _ r \gamma | \epsilon _ 1 ( \eta _ a , t ) | + | \bar { y } ( t ) - \bar { y } _ a ( t ) | \\ \le & \gamma \delta L _ r \left ( 2 + L _ r \ ( 3 + L _ r ) \right ) + | \bar { y } - z _ 2 | . \end{aligned} \end{align*}


\begin{align*} \tilde H - ( k + 2 ) B & = K _ { S ^ { [ 2 ] } } - K _ { S ^ { [ 2 ] } } + \tilde H - ( k + 2 ) B \\ & = \widetilde { K _ S } + \widetilde { H - K _ S } - ( k + 2 ) B . \end{align*}


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


\begin{align*} \chi _ j = \chi - \sum _ { \substack { i = 1 , \\ i \neq j } } ^ { n } \chi _ i + ( n - 1 ) r . \end{align*}


\begin{align*} u = \int ^ x \frac { \dd x } { \sqrt { f _ 5 ( x ) } } + \int ^ { x ' } \frac { \dd x ' } { \sqrt { f _ 5 ( x ' ) } } , v = \int ^ x \frac { x \dd x } { \sqrt { f _ 5 ( x ) } } + \int ^ { x ' } \frac { x ' \dd x ' } { \sqrt { f _ 5 ( x ' ) } } , \end{align*}


\begin{align*} n = \frac { - 3 \left ( \alpha ^ { 2 } + 8 \right ) \pm \sqrt { - 3 \left ( \alpha ^ { 4 } + 8 \left ( \alpha ^ { 3 } - 6 \alpha ^ { 2 } + 1 3 \alpha + 2 \right ) \right ) } } { 6 \left ( \alpha + 2 \right ) } \end{align*}


\begin{align*} Q ( h , v ) = \Big ( v A + \bar u A ' ( h ) - ( \nu ( u ) ) ' \bar g ^ \intercal , \ , 2 v \Big ) \mbox { o n } \hat \Sigma . \end{align*}


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


\begin{align*} \sum _ { \deg ( v ) \mid \ell } \deg ( v ) \mathrm { t r } \left ( \rho ( \mathrm { F r o b } _ { v } ) ^ { \frac { \ell } { \deg ( v ) } } \right ) = - \sum _ { \lambda } \lambda ^ { \ell } . \end{align*}


\begin{align*} \mathcal { N } _ \varepsilon = & \{ u \in \mathcal { H } ( \Omega ) \setminus \{ 0 \} \mid \langle I ' _ \varepsilon ( u ) , u \rangle = 0 \} \\ = & \left \{ u \in \mathcal { H } ( \Omega ) \setminus \{ 0 \} \mid \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x = \frac { 1 } { \varepsilon ^ 2 } \int _ { \Omega } \left ( u - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } _ + u d x \right \} . \end{align*}


\begin{align*} L ( E , \epsilon ) = \max \{ \log | \lambda | + | \epsilon | , L ( E , 0 ) \} . \end{align*}


\begin{align*} ( \Pi _ V \tilde { Y } _ { t _ * } ) = \int _ 0 ^ { t _ * } \left \| \Pi _ V e ^ { | z | J ^ \perp ( t _ * - s ) } P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma \right \| _ F ^ 2 d s , \end{align*}


\begin{align*} \rho { u _ y } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i y } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ y } } , \end{align*}


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


\begin{align*} \lim _ { x \to - \infty } \frac { h _ { } ( x ) } { \phi ( \frac { x - \mu } { \sigma } ) } = \frac { 1 } { \sigma } \lim _ { x \to - \infty } \frac { \phi ( \frac { x - \mu } { \sigma } ) } { 1 - \Phi ( \frac { x - \mu } { \sigma } ) } \frac { 1 } { \phi ( \frac { x - \mu } { \sigma } ) } = \frac { 1 } { \sigma } \lim _ { x \to - \infty } \frac { 1 } { 1 - \Phi ( \frac { x - \mu } { \sigma } ) } = \frac { 1 } { \sigma } . \end{align*}


\begin{align*} X _ { i ; j k } = - R _ { \ell k j i } X ^ \ell + \tfrac { 1 } { 2 } ( h _ { i j ; k } + h _ { i k ; j } - h _ { j k ; i } ) . \end{align*}


\begin{align*} q ^ \perp = 1 \oplus 0 \oplus 1 \oplus \begin{pmatrix} b ^ 2 & - a b \\ - a b & a ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} a _ 0 b _ 0 = 1 , \ \sum _ { k = 0 } ^ n \Gamma ( k + 1 - \alpha ) \Gamma ( \alpha + n - k ) a _ { n - k } b _ k = 0 , \ n \ge 1 . \end{align*}


\begin{align*} \begin{aligned} F ^ { i j } ( b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \pm ( u - \varphi ) ) _ { i j } \geq \ , & 0 \mbox { i n } \omega \\ b _ { n - 1 } ( A _ 1 v - A _ 2 w ) \pm ( u - \varphi ) \leq \ , & 0 \mbox { o n } \partial \omega \end{aligned} \end{align*}


\begin{align*} { { \bf { W } } ^ { \star } } = \left [ { { { \bf { W } } ^ { \star } } , { { \bf { 0 } } _ { \left ( { \sum { { { \bar r } _ { l ' } } } } \right ) \times \left ( { { M _ t } - K } \right ) } } } \right ] \left [ \begin{array} { l } { { \bf { I } } _ K } \\ { { \bf { 0 } } _ { \left ( { { M _ t } - K } \right ) \times K } } \end{array} \right ] . \end{align*}


\begin{align*} \xi ' = - L - 2 , \eta ' = \frac { Q } { 2 ( L + 2 ) } - \sqrt { \kappa B _ 4 } , \zeta ' = \kappa \left ( \frac { B _ 3 } { 2 \sqrt { \kappa B _ 4 } } + 3 \right ) , \sigma ' = \sqrt { \kappa B _ 4 } , \end{align*}


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


\begin{align*} \sum _ { k = 1 } ^ { \infty } e ^ { - t \tau _ k } = \int _ { \partial \Omega } \bigg [ \frac { 1 } { ( 2 \pi ) ^ { n - 1 } } \int _ { \mathbb { R } ^ { n - 1 } } \bigg ( \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } \sum _ { j \leqslant - 1 } \operatorname { T r } \phi _ { j } \ , d \tau \bigg ) \ , d \xi \bigg ] \ , d S . \end{align*}


\begin{align*} d \sigma - ( \log \sqrt { \tau _ 0 } ) _ z d z \sigma = e ^ { 2 i \theta } \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { \sigma } I \end{align*}


\begin{align*} d = k ^ 2 + 2 n k = 4 m k + 2 k - k ^ 2 \equiv 2 k - k ^ 2 \mod 4 k \end{align*}


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


\begin{align*} l ( y _ p ) \sim r ( y _ p ) = p = p _ 1 + \cdots + p _ n = r ( y _ { p _ 1 } ) + \cdots + r ( y _ { p _ n } ) \succ l ( y _ { p _ 1 } ) \vee \cdots \vee l ( y _ { p _ n } ) . \end{align*}


\begin{align*} | f _ \alpha ( U | \bar U ; x , t ) | & \le | f _ \alpha - \bar f _ \alpha - \nabla \bar f _ \alpha ( U - \bar U ) ) | \\ & + \left | \nabla \bar f _ \alpha \nabla \bar A ^ { - 1 } \big ( A ( U , x , t ) - \bar A - \nabla \bar A ( U - \bar U ) \big ) \right | \\ & \le k _ 4 | A ( U , x , t ) - \bar A | ^ 2 \ , , \end{align*}


\begin{align*} \| \tilde { F } \| _ { h } \leq \| F \| _ { h } \| P \| ^ { 2 } \leq { \rm e } ^ { 2 | \log \varepsilon | ^ { \frac { 2 } { 2 + \eta } } } \varepsilon = : \tilde { \varepsilon } . \end{align*}


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


\begin{align*} \frac { d } { d t } \bar { y } _ t = - Z _ { t , 3 } \bar { y } _ t - \bar { z } _ { t , 3 } y _ t , \end{align*}


\begin{align*} \int _ { \Omega } ( K _ H ( x ) \nabla v ^ \tau _ \varepsilon | \nabla v ^ \tau _ \varepsilon ) d x = & 2 \pi q ^ 2 \sqrt { ( K _ { H } ) _ { 1 1 } ( K _ { H } ) _ { 2 2 } } ( \bar { x } ) \ln \frac { 1 } { \varepsilon } + O ( 1 ) + C | \ln \tau | \\ = & 2 \pi q ^ 2 \sqrt { d e t ( K _ { H } ) } ( \bar { x } ) \ln \frac { 1 } { \varepsilon } + O ( 1 ) + C | \ln \tau | , \end{align*}


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


\begin{align*} \mu _ T ( e _ i , \mu _ T ( e _ j , e _ k ) ) = 0 \forall i , j , k \in \{ 1 , \ldots , n \} \end{align*}


\begin{align*} \int _ { \R ^ n } f \ , d Q - H ( Q ) = - H ( Q \ , | \ , P ) + \log Z . \end{align*}


\begin{align*} \overline { W ^ { k , i } } = W ^ { - k , i } , \quad \big [ W ^ { k , i } , W ^ { l , j } \big ] _ { t } = 2 t \delta _ { k , - l } \delta _ { i , j } . \end{align*}


\begin{align*} \phi ( s , t ) = \gamma ( s ) + \sum _ { i = 1 } ^ { n - 1 } t _ i w _ i , t = ( t _ 1 , \ldots , t _ { n - 1 } ) , \end{align*}


\begin{align*} E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \biggr ) \biggr ] = E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \biggr ) \biggr ] . \end{align*}


\begin{align*} ( Q ( \xi ) ) _ { i j } = Q _ { i j } ( \xi ) = \partial ^ 2 _ { i j } \xi + a _ { i j } ^ j \partial _ j \xi + a _ { j i } ^ i \partial _ i \xi + b _ { i j } \xi = 0 \quad \forall \ , 0 \leq i < j \leq p , \end{align*}


\begin{align*} V _ { \delta , Z } ( x ) - q ( x ) = V _ { \delta , z _ i , \hat { q } _ { \delta , i } , z _ i } ( x ) - \hat { q } _ { \delta , i } + O \left ( \frac { \ln | \ln \varepsilon | } { | \ln \varepsilon | ^ 2 } \right ) . \end{align*}


\begin{align*} \mathbf { c } ^ { ( 1 ) } = ( \underbrace { 1 , 1 , \cdots , 1 } _ { 2 k } , \ast , \ast , \cdots , \ast ) . \end{align*}


\begin{align*} h ^ \theta ( x , u ) \le \sup _ { \rho \in \Theta _ { n - 1 } } E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \biggr ) \biggr ] = V \left ( \xi _ 1 ^ \theta ( u ) , n - 1 , x + u \right ) . \end{align*}


\begin{align*} \hat \Phi _ s ( 0 , 0 ) = 0 . \end{align*}


\begin{align*} \hat { A } ^ * ( y | x ) = A ( y _ 2 y _ 1 x ) + \left [ A ( y _ 1 x ' ) - A ( y _ 2 x ' ) \right ] . \end{align*}


\begin{align*} T _ \ell ( x ) = \int _ { \mathbb { S } ^ d } f _ { x ; \ell } ( y ) W ( d y ) , x \in \mathbb { S } ^ d , \end{align*}


\begin{align*} G \frac { d } { d t } w ( t ) = L w ( t ) + f ( t ) \end{align*}


\begin{align*} A ( u ) = - \Delta ^ 2 u + \Delta \varphi ( u ) . \end{align*}


\begin{align*} \frac { g ( t ) - g ( r ) } { t - r } = \lambda \frac { g ( t ) - g ( s ) } { t - s } + ( 1 - \lambda ) \frac { g ( s ) - g ( r ) } { s - r } . \end{align*}


\begin{align*} \mathbf G ^ { [ 2 ] } ( \kappa h ; \alpha ) = \int _ { - \pi } ^ { \pi } \frac { \sin ^ 2 ( \theta ) \ , | \widehat { \alpha } ( \theta ) | ^ 2 } { \sinh ^ 2 ( 2 \kappa h ) + \sin ^ 2 ( \theta ) } \ , \frac { d \theta } { 2 \pi } . \end{align*}


\begin{align*} \alpha _ n ( 0 ) = h [ P _ { \leq N } \psi _ 0 ] ( h n ) + ( - 1 ) ^ n h [ P _ { \leq N } \phi _ 0 ] ( h n ) . \end{align*}


\begin{align*} \lambda _ { f \times i d _ { X } } [ ( f \times i d _ { X } ) ( r , x ) , \alpha ] ( 0 ) = ( f ( r ) , x ) = ( f \times i d _ { X } ) ( r , x ) \end{align*}


\begin{align*} { [ k + 1 + \ell ] \choose [ q ] } _ + \to \bigvee \limits _ { r = 0 } ^ q \left ( { [ k ] \choose [ r ] } \times { [ \ell ] \choose [ q - r ] } \right ) _ + \end{align*}


\begin{align*} \| f | \sigma _ 1 \| _ { B V ( \sigma _ 1 ) } = \lim _ { n \to \infty } \| p _ n | \sigma _ 1 \| _ { B V ( \sigma _ 1 ) } \leq \lim _ { n \to \infty } \| p _ n \| _ { B V ( \sigma ) } = \| f \| _ { B V ( \sigma ) } . \end{align*}


\begin{align*} \mathbf z \cdot \omega _ 1 = 0 a n d z \cdot \omega _ j < - 1 , j = 2 , \cdots , n . \end{align*}


\begin{align*} { \bf P } ( \lim _ { n \to \infty } \kappa _ n = \kappa ) = 1 . \end{align*}


\begin{align*} ( \mathbf { x } , - \mathbf { y } ) \underbrace { \begin{pmatrix} G _ { 1 } & G _ { 2 } & \cdots & G _ { 2 k - 1 } \\ G _ { 2 } & G _ { 3 } & \cdots & G _ { 2 k } \end{pmatrix} } _ M = 0 . \end{align*}


\begin{align*} \Sigma _ { t + a } = r ^ { - 1 } ( t ) = \{ t + a \} \times N , \end{align*}


\begin{align*} L ( \lambda \alpha ) = \lambda L ( \alpha ) = 0 \end{align*}


\begin{align*} \int _ \Omega u ( t , x ) \cdot \nabla h ( x ) \ , d x = 0 \forall h \in H ^ 1 _ { \rm l o c } ( \Omega ) \end{align*}


\begin{align*} \pi ( u ) : = \sum _ { v \in V } \pi ( u , v ) > 0 ; A : = \sup _ { u \in V } \pi ( u ) < \infty . \end{align*}


\begin{align*} \overline { \mathfrak { D } _ { q ^ n } } \ \overline { U _ { q ^ n } } \zeta = ( A + B ) \overline { U _ { q ^ n } } \dot { s } \overline { U ^ * _ { q ^ n } } { \bf 1 } _ { - } + ( q ^ n - 1 ) C \overline { U _ { q ^ n } } { \bf 1 } _ { - } \in \Bbbk { \bf G } \zeta . \end{align*}


\begin{align*} \left . \begin{aligned} & \boldsymbol { y } _ t ^ { * } : = ( y _ t ^ { * , i } ) _ { 1 \leq i \leq N } : = ( P _ t x _ t ^ { * , i } + \varphi _ t ^ { * , i } ) _ { 1 \leq i \leq N } , \\ & \boldsymbol { z } _ t ^ { * } : = ( z _ t ^ { * , i , j } ) _ { 1 \leq i , j \leq N } : = ( P _ t \sigma + \Lambda _ t ^ { * , i , j } ) _ { 1 \leq i , j \leq N } , \boldsymbol { z } _ t ^ { 0 , * } : = ( z _ t ^ { 0 , * , i } ) _ { 1 \leq i \leq N } : = ( P _ t \sigma _ 0 + \Lambda _ t ^ { 0 , * , i } ) _ { 1 \leq i \leq N } . \end{aligned} \right . \end{align*}


\begin{align*} \mu \Big ( \bigcap _ { j = 0 } ^ { t } T ^ { j h ( C ) } J \Big ) \geq ( 1 - \epsilon ) \mu ( J ) . \end{align*}


\begin{align*} w ( x ) : = x _ n - y _ n - L ( x ) + \kappa | x ' - y ' | ^ 2 \geq 0 \mbox { o n } \partial \omega \cap \partial \Omega . \end{align*}


\begin{align*} h _ { \hat { \mu } } ( \hat { \sigma } ) = h _ { \hat { \mu } } ( \hat { \sigma } , \hat { \P } ) \ \ h _ \mu ( \sigma ) = h _ \mu ( \sigma , \P ) \ . \end{align*}


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


\begin{align*} A ( \tau _ y ( x ) ) + \sum _ { n = 1 } ^ N \left [ \hat { A } \circ \hat { \sigma } ^ { - ( n + 1 ) } ( y | x ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( \sigma ( y ) | x ' ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ) + \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] & = \\ A ( \tau _ { y , N + 1 } ( x ) ) + \sum _ { n = 1 } ^ N \left [ \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( \sigma ( y ) | x ' ) \right ] \ . \end{align*}


\begin{align*} p ^ { \ell } = p ^ { \ell - 1 } \ast \frac { A ^ { i } d } { A ^ { i } p ^ { \ell - 1 } } \end{align*}


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


\begin{align*} h ( x , u ) = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \Bigr ) \right ] = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \Bigr ) \right ] . \end{align*}


\begin{align*} H = \Theta : L ^ 2 ( \R ^ { n - 1 } , \mathbb { C } ^ N ) \to L ^ 2 ( \R ^ n _ + , d x ' d t / t ) ^ N . \end{align*}


\begin{align*} \mathcal { F } _ { \geq p } t e l ( \mathcal { C } ) : = \lbrace x \in t e l ( \mathcal { C } ) \mid \mathcal { A } ( x ) \geq p \rbrace . \end{align*}


\begin{align*} | \Gamma _ { \ell , m , n } | = ( q - 1 ) ^ 3 q ^ { 3 \ell + m - n + 6 } \end{align*}


\begin{align*} ( \tilde { \kappa } _ 1 ) _ i = \tilde { h } _ { 1 1 i } = h _ { 1 1 i } , \end{align*}


\begin{align*} { \sigma } _ { j i } & = \mu ^ * ( u _ { i , j } + u _ { j , i } ) + \varkappa \ , ( u _ { i , j } - \epsilon _ { j i s } \vartheta _ s ) + \lambda \ , u _ { k , k } \delta _ { j i } , \\ * m _ { j i } & = \beta \ , \vartheta _ { j , i } + \gamma \ , \vartheta _ { i , j } + \alpha \ , \vartheta _ { k , k } \delta _ { j i } \ , \end{align*}


\begin{align*} \begin{aligned} & \dot y _ { b e n } = a y '' _ { b e n } + M y _ { b e n } , x \in ( 0 , R ) \\ & \dot y _ { n b } = b y '' _ { n b } + ( M - \mu \ 1 ) y _ { n b } , x \in ( R , R + r ) , \end{aligned} \end{align*}


\begin{align*} \# R ^ i _ { n _ 0 ; l _ 1 , \cdots , l _ { n _ 0 } } = \underset { \tilde { l } = ( \tilde { l } _ { i } , \tilde { l } _ { i + 1 } , \cdots , \tilde { l } _ { n _ 0 } ) \in L } { \sum } c _ { \tilde { l } } \# P _ { l _ 1 , l _ 2 , \cdots , l _ { i - 1 } , \tilde { l } _ i , \tilde { l } _ { i + 1 } , \cdots , \tilde { l } _ { n _ 0 } } \end{align*}


\begin{align*} \tau ( \widetilde K _ n ) \ge [ \Lambda ( g , 2 A , M ) ] ^ { - 1 } ( 1 + \varepsilon ) & \geq [ \Lambda ( g , A , M ) ] ^ { - 2 } ( 1 + \varepsilon ) \\ & > \left [ \frac { 1 + \varepsilon } { 1 + \frac { \varepsilon } { 2 } } \right ] ^ { - 1 } ( 1 + \varepsilon ) \\ & = 1 + \frac { \varepsilon } { 2 } , \end{align*}


\begin{align*} \phi _ t + u \cdot \nabla \phi = \mu \Delta \phi . \end{align*}


\begin{align*} H ^ n ( G , M ) = H ^ n \mathcal C ^ \bullet , . \end{align*}


\begin{align*} \left ( 2 n - \beta \left ( 3 \beta ^ { 2 } + 4 \right ) + 1 \right ) \left ( n ^ { 2 } + \left ( 2 \beta + 1 \right ) n + \beta ^ { 2 } + \beta + 1 \right ) = 0 \end{align*}


\begin{align*} u _ n = P _ 1 ( n ) \alpha _ 1 ^ n + \cdots + P _ s ( n ) \alpha _ s ^ n \end{align*}


\begin{align*} d g _ 1 '' = ( \log \sqrt { \tau _ 0 } ) _ z d z \ g '' _ 1 + e ^ { 2 i \theta } \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { g _ 1 '' } I ; \end{align*}


\begin{align*} \rho \ , u _ { , t t } & = \ , \boldsymbol { \sigma } \ , , \\ \rho \eta \ , \tau _ { \rm c } ^ 2 \ , \mathbf { A } _ { , t t } & = - \ , \ , \boldsymbol { m } + \ , \boldsymbol { \sigma } \ , , \end{align*}


\begin{align*} \begin{aligned} \eta ( U | \bar U ; x , t ) & \doteq \eta ( U , x , t ) - \eta ( \bar U , x , t ) - G ( \bar U , x , t ) \cdot ( A ( U , x , t ) - A ( \bar U , x , t ) ) \\ & = \tilde { \eta } ( V , x , t ) - \tilde { \eta } ( \bar V , x , t ) - ( \nabla _ V \tilde { \eta } ) ( \bar V , x , t ) \ ; ( A ( U , x , t ) - A ( \bar U , x , t ) ) \ , \\ & \doteq \tilde { \eta } \big ( V | \bar V ; x , t \big ) \ , , \end{aligned} \end{align*}


\begin{align*} \Tilde { x } = \sum \limits _ { i = 1 } ^ p \alpha _ i u _ i + \lambda x _ v + ( 1 - \lambda ) x _ w \in \mathcal { S } ^ n , \end{align*}


\begin{align*} n = \frac { \beta \left ( 3 \beta ^ { 2 } + 4 \right ) + 1 } { 2 } \end{align*}


\begin{align*} \widehat { \partial _ j f ( 0 , \xi ) } = 2 \pi i \ ; y _ j ( \xi ) \widehat { f ( 0 , \xi ) } . \end{align*}


\begin{align*} { { \bf { \bar X } } ^ { \star } } = \left [ { { { \bf { W } } ^ { \star } } , { { \bf { 0 } } _ { \left ( { \sum { { { \bar r } _ { l ' } } } } \right ) \times \left ( { { M _ t } - K } \right ) } } } \right ] , \end{align*}


\begin{align*} [ T ] + \nu [ N ] = \frac { 2 } { \mu } h _ z J + \nu I , \end{align*}


\begin{align*} Z _ j \leq \frac { D ' } { Q _ 1 } + 1 0 \sqrt { \frac { D ' } { Q _ 1 ^ 2 Q _ 2 ^ 2 } } \sqrt { \log ( k ) } = \frac { 1 } { Q _ 1 } \left ( D ' + 1 0 \frac { \sqrt { D ' \log k } } { Q _ 2 } \right ) . \end{align*}


\begin{align*} \begin{dcases} \dfrac { d } { d t } \Upsilon ( y , t ) = v ( \Upsilon ( y , t ) ) , t \in \mathbb { R } , \\ \Upsilon ( y , 0 ) = y . \end{dcases} \end{align*}


\begin{align*} \int _ X | | \xi ( x ) | | G \ \mu ( d x ) = \int _ X \beta ( x ) \ \mu ( d x ) , \end{align*}


\begin{align*} E _ { \delta , Z } = \{ u \in W ^ { 2 , p } \cap H ^ 1 _ 0 ( \Omega ) \mid \int _ { \Omega } ( K ( x ) \nabla \frac { \partial V _ { \delta , Z , j } } { \partial z _ { j , h } } ) u = 0 , \ \ \forall j = 1 , \cdots , m , \ h = 1 , 2 \} . \end{align*}


\begin{align*} \mathcal { E } : = u _ 1 ^ t \Delta u _ 2 ^ t \wedge \dots \wedge \Delta u _ { k } ^ t \wedge \beta _ n ^ { n - m } - u _ 1 \Delta u _ 2 \wedge \dots \wedge \Delta u _ { k } \wedge \beta _ n ^ { n - m } \longrightarrow 0 , t \to \infty , \end{align*}


\begin{align*} \iota ( \upsilon ) = - \frac { 1 } { 4 } f ( \upsilon ) + \frac { h ( \upsilon ) } { 4 g ( \upsilon ) } \left ( - 4 \upsilon + \frac 1 2 f ( 2 \upsilon ) \right ) . \end{align*}


\begin{align*} Z ( \theta ) = \sum \limits _ { i \in R } \prod \limits _ { F \in \mathrm { f a c e t } ( \Gamma ) } \theta ^ { F } _ { i _ { F } } . \end{align*}


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


\begin{align*} p = 1 \oplus 0 \oplus 0 \oplus \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} , q = 0 \oplus 1 \oplus 0 \oplus \begin{pmatrix} a ^ 2 & a b \\ a b & b ^ 2 \end{pmatrix} , \end{align*}


\begin{align*} J _ 1 + J _ 3 = 0 . \end{align*}


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


\begin{align*} \frac { 1 } { \ln \frac { R } { s _ { \delta , j } } } = \frac { 1 } { \ln \frac { R } { \varepsilon } } + O \left ( \frac { \ln | \ln \varepsilon | } { | \ln \varepsilon | ^ 2 } \right ) . \end{align*}


\begin{align*} \phi = \frac { 1 } { 2 } \left ( u _ { , p } \xi ^ p + v _ q x ^ q \right ) . \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\ X _ 0 = x _ 0 \end{cases} \end{align*}


\begin{align*} \lambda _ 1 ^ { - \mathcal { L } } ( D ) \leq Q ( v ) = Q _ { 1 } ( v ) . \end{align*}


\begin{align*} \int _ 0 ^ T \left | \frac { \zeta ' } { \zeta } ( \sigma + i t ) \right | ^ 2 d t = O \left ( \frac { T } { ( \sigma - 1 / 2 ) ^ 2 } \right ) . \end{align*}


\begin{align*} \widehat { \iota } ( D ) = \frac { \beta - e } { d } \end{align*}


\begin{align*} \tilde { h } _ d ' ( x ) = \log d + \left ( \psi ( x ) - \psi \left ( x + \frac 1 2 \right ) \right ) - \psi \left ( x + \frac { d - 1 } { 2 } \right ) \end{align*}


\begin{align*} \hat \Phi ^ \dagger _ s ( 0 , 0 ) = 0 , \qquad \hat \Phi ^ \dagger _ { a s } ( 0 , 0 ) = \gamma _ 2 , \end{align*}


\begin{align*} F ( x ) = \sum _ { n = 0 } ^ \infty c _ n ( a , b ) \ , G ( x ; ( a + n ) \theta ) . \end{align*}


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


\begin{align*} A _ { k j } = \begin{cases} \sum _ { l = 1 } ^ { n _ 1 } \Psi _ { k l } \bar \Psi _ { j l } , & \lambda _ k = \lambda _ j , \\ 0 , & . \end{cases} \end{align*}


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


\begin{align*} \varphi ( y ) = \frac { 1 } { 8 \pi ^ { 2 } } \int _ { \{ \varphi > W / 2 \} } \frac { \mu ^ { 2 } } { | y - w | ^ { 3 } } \left ( \varphi - \frac { W } { 2 } \right ) _ { + } \dd w . \end{align*}


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


\begin{align*} ( b ^ { [ j ] } \wedge a ) ( \alpha ) ( \xi ) = \begin{cases} b ( \alpha ) ( \xi ) & \alpha < \beta , \\ x _ j ( \beta ) ( \xi ) & \alpha = \beta \xi \le \zeta , \\ b ( \beta ) ( \xi ) & \alpha = \beta \xi > \zeta , \\ a ( \alpha ) ( \xi ) & \end{cases} \end{align*}


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


\begin{align*} \begin{aligned} & \mathcal { g } _ { D } ( v , w ) = 2 ^ { m } v _ { n - 1 } \int _ { M } g ( v , w ) ~ v o l _ { g } , \\ & \mathcal { G } _ { D } ( v , w ) = 2 ^ { m } \frac { v _ { n - 1 } } { 6 } \int _ { M } G ( v , w ) ~ v o l _ { g } , \end{aligned} \end{align*}


\begin{align*} l ( y _ p ) \sim r ( y _ p ) = p = p _ 1 + \cdots + p _ n = r ( y _ { p _ 1 } ) + \cdots + r ( y _ { p _ n } ) \succ l ( y _ { p _ 1 } ) \vee \cdots \vee l ( y _ { p _ n } ) . \end{align*}


\begin{align*} H _ { j , n } = ( - 1 ) ^ { n } \binom { j } { n } ( \beta ) ^ { ( j ) } \sqrt { \frac { n ! } { \left ( \beta \right ) ^ { \left ( n \right ) } } } . \end{align*}


\begin{align*} x ( t ) = \frac { 1 } { \sqrt { 2 t + 1 } } \end{align*}


\begin{align*} \tilde { Q } = \ln \tilde { \kappa } _ 1 - N \ln \nu ^ { n + 1 } . \end{align*}


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


\begin{align*} \sigma ( \gamma _ 1 ) & = \gamma _ 3 , & \sigma ( \gamma _ 2 ) & = \gamma _ 2 , & \sigma ( \gamma _ 3 ) & = \gamma _ 4 , & \sigma ( \gamma _ 4 ) & = \gamma _ 1 . \end{align*}


\begin{align*} - \int _ { \R } \left ( T _ i ' ( x _ i ) - 1 \right ) Q _ i ( x _ i ) d x _ i = \int _ \R \left ( T _ i ( x _ i ) - x _ i \right ) Q _ i ' ( x _ i ) \ , d x _ i . \end{align*}


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


\begin{align*} \left \{ \begin{aligned} & \Delta w + k ^ 2 q w = 0 & \mathrm { i n } \ \mathcal K , \\ & \Delta v + k ^ 2 v = 0 & \mathrm { i n } \ \mathcal K , \\ & w = v , \ \partial _ { \nu } w = \partial _ { \nu } v & \mathrm { i n } \ \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \chi _ j = \chi - \sum \limits _ { i \neq j } \chi _ i - ( n - 1 ) r \end{align*}


\begin{align*} \hat X ( p ) = X ( p ) \mbox { a n d } \omega ( p ) = ( \nabla X ) ( p ) . \end{align*}


\begin{align*} p \ast q : = \sum _ { n \geq 1 } p _ n q _ n T ^ n . \end{align*}


\begin{align*} M _ i ( t ) = Q ^ * _ i \circ ( ( 1 - t ) \mathrm { I d } + t T _ i ) ^ { - 1 } , \end{align*}


\begin{align*} e ^ { i c _ + ( x ) } \partial _ { x } \left ( \bar { u } _ x ( x ) e ^ { i c _ + ( x ) } \right ) & = - \frac { 1 } { \pi } \int _ { \mathbb { R } } r _ 2 ( z ) e ^ { 2 i s x } \mathrm { d } z - \frac { 1 } { \pi } \int _ { \mathbb { R } } r _ 2 ( z ) e ^ { 2 i z x } \left [ M _ { + , 2 2 } - 1 \right ] d z \\ & : = I _ 3 ( x ) + I _ 4 ( x ) . \end{align*}


\begin{align*} \begin{aligned} E _ { \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { k + 1 } Z _ i ^ { \mathsf { L } ^ { k + 1 } [ u ] } \biggr ) \biggr ] & = E _ { \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { k } Z _ i ^ { \mathsf { M } ^ { k } } \biggr ) \biggr ] \\ & = W ( \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) , k , x + u , \mathsf { M } ^ { k } ) . \end{aligned} \end{align*}


\begin{align*} f ( n , \alpha , t ) \leq \left ( \sum _ { i = 1 } ^ \alpha \binom { n } { i } \right ) ^ t \leq \left ( 2 \binom { n } { \alpha } \right ) ^ { t } . \end{align*}


\begin{align*} \Gamma \left ( X , Y , Z \right ) = C \left ( X , J Y , Z \right ) + g \left ( X , \left ( \nabla _ { Z } J \right ) Y \right ) . \end{align*}


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


\begin{align*} \inf _ { x \in E , p _ n x = x _ n } \| x \| _ E \le b \| x _ n \| _ { E _ n } ( n \in \mathbb N , \forall x _ n \in E _ n ) \end{align*}


\begin{align*} I _ f ( p , q ) & = \sum _ { i \in A _ { 1 + \kappa , \infty } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) + \sum _ { i \in A _ { 1 , 1 + \kappa } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) \\ & + \sum _ { i \in A _ { 1 / ( 1 + \kappa ) , 1 } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) + \sum _ { i \in A _ { 0 , 1 / ( 1 + \kappa ) } } q _ i f \left ( \frac { p _ i } { q _ i } \right ) , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H = 0 , \ \mathbb { P } . \end{align*}


\begin{align*} - \Delta _ { y } \varphi = \mu ^ { 2 } \left ( \varphi - \frac { W } { 2 } \right ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 5 } . \end{align*}


\begin{align*} & L _ 4 \colon \mathbf { Z } = 2 \eta \mathbf { X } - \iota \mathbf { W } = 0 , & L ' _ 4 \colon \mathbf { W } - \rho \mathbf { Z } = 2 \eta ' \mathbf { X } - \iota ' \mathbf { W } = 0 , \end{align*}


\begin{align*} \int _ 0 ^ T \int _ { \mathbb { R } ^ { N } } & | u | ^ { p } ( I ^ \gamma _ { T - } \varphi ) d x d t + \int _ 0 ^ T \int _ { \mathbb { R } ^ { N } } \omega \varphi d x d t + \int _ { \mathbb { R } ^ { N } } u _ 0 ( \varphi ( 0 , x ) - k \Delta \varphi ( 0 , x ) ) d x \\ & = - \int _ 0 ^ T \int _ { \mathbb { R } ^ { N } } u \varphi _ t d x d t + k \int _ 0 ^ T \int _ { \mathbb { R } ^ { N } } u \Delta \varphi _ t d x d t - \int _ 0 ^ T \int _ { \mathbb { R } ^ { N } } u \Delta \varphi d x d t , \end{align*}


\begin{align*} \Lambda ^ { ( s ) } ( h _ k ) = G _ k ( s ) = - \frac { \zeta ( s ) } { s } ( k ^ { - s } - k ^ { - 1 } ) \end{align*}


\begin{align*} \sum _ { X _ k } \sum _ { X _ k ' } \int \widehat { h _ { X _ k } } \overline { \widehat { h _ { X _ k ' } } * \widehat { w _ U } } & = \sum _ { X _ k } \sum _ { X _ k ' \sim X _ k } \int h _ { X _ k } \overline { h _ { X _ k ' } } w _ U \\ & \le \sum _ { X _ k } \sum _ { X _ k ' \sim X _ k } \int ( | h _ { X _ k } | ^ 2 + | h _ { X _ k ' } | ^ 2 ) w _ U \\ & \le \sum _ { X _ k } \sum _ { X _ k ' \sim X _ k } \int ( | h _ { X _ k } | ^ 2 + | h _ { X _ k ' } | ^ 2 ) w _ U \le 2 L \sum _ { X _ k } \int | h _ { X _ k } | ^ 2 w _ U . \end{align*}


\begin{align*} { { \bf { \bar x } } _ { \rm t } } \left [ m , n \right ] { \rm = } \frac { 1 } { \sqrt K } \sum \limits _ { k = 0 } ^ { K - 1 } { { { \bf { u } } _ k } s \left [ m , k \right ] { e ^ { j \frac { { 2 \pi } } { K } k n } } } , \ n { \rm = } - { { N } _ { \rm C P } } . \cdots , K - 1 . \end{align*}


\begin{align*} u _ { \varepsilon } ( x ' , t ) : = u ( x ' , t + \varepsilon ) \quad f _ { \varepsilon } ( x ' ) : = u ( x ' , \varepsilon ) , x ' \in \R ^ n , \ , t > 0 . \end{align*}


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


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


\begin{align*} D \phi ( F ) = \sum _ { r = 1 } ^ k \frac { \partial \phi } { \partial x _ r } ( F ) D F _ r . \end{align*}


\begin{align*} \begin{aligned} u ( x _ 0 ) & = \varphi ( x _ 0 ) = \frac { 1 } { 2 } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 + \frac { 1 } { 6 } \sum _ { \xi , \beta , \gamma \geq \alpha + 1 } \varphi _ { \xi \beta \gamma } ( 0 ) x _ \xi x _ \beta x _ \gamma + O ( | \tilde { x } | ^ 4 ) \\ & \geq \frac { 1 } { 2 } \delta ^ 2 b _ \alpha ^ 2 - C \delta ^ 3 b _ \alpha ^ 2 \geq \frac { 7 } { 1 6 } \delta ^ 2 b _ \alpha ^ 2 \end{aligned} \end{align*}


\begin{align*} \left < p , q \right > = \overline { p ( \partial _ { \theta _ 1 } , \partial _ { \theta _ 2 } , \ldots , \partial _ { \theta _ n } ) } q ( \theta _ 1 , \theta _ 2 , \ldots , \theta _ n ) | _ { \theta _ 1 = \theta _ 2 = \cdots = \theta _ n = 0 } ~ . \end{align*}


\begin{align*} A ^ { - 1 } \mathcal { L } _ g = \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) \end{align*}


\begin{align*} k _ \theta \cdot f _ - & = \frac { 1 } { \sqrt { 2 } } \left ( s ( \theta ) , c ( \theta ) , s ( \theta ) , c ( \theta ) \right ) \\ k _ \theta \cdot f _ + & = \frac { 1 } { ( 1 0 8 ) ^ { \frac { 1 } { 4 } } } \left ( s ( 3 \theta ) , 3 c ( 3 \theta ) , - 3 s ( 3 \theta ) , - c ( 3 \theta ) \right ) . \end{align*}


\begin{align*} \begin{aligned} \mathsf { M } ^ n = & \{ \mathsf { m } _ 1 , \cdots , \mathsf { m } _ n \} \\ = & \left \{ g \bigl ( \xi _ 0 \bigr ) , g \bigl ( \xi _ 1 ^ { \mathsf { M } } \bigr ) , \cdots , g \bigl ( \xi _ { n - 1 } ^ { \mathsf { M } } \bigr ) \right \} . \end{aligned} \end{align*}


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


\begin{align*} \mathbb { E } \left [ y _ i ^ * y _ i | \hat { \mathbf { H } } \right ] = & a _ c ^ 2 \left ( \lvert \hat { \phi } ^ { \left ( i , c \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ c \rVert ^ 2 \right ) + \sum _ { j = 1 } ^ { K } a _ j ^ 2 \left ( \lvert \hat { \phi } ^ { \left ( i , j \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 \right ) + \sigma _ n ^ 2 . \end{align*}


\begin{align*} \overline { \mathfrak { D } _ { q ^ n } } \ \overline { U _ { q ^ n } } \dot { s } \varepsilon ( v ) \zeta = ( A + B + C ) \overline { U _ { q ^ n } } \dot { s } \overline { U ^ * _ { q ^ n } } { \bf 1 } _ { - } \in \Bbbk { \bf G } \zeta . \end{align*}


\begin{align*} w = 1 + 2 \gamma _ 1 a _ 1 + 2 \gamma _ 2 a _ 2 > 0 \end{align*}


\begin{align*} L ( s , \pi \otimes \chi _ { 8 d } ) = \prod _ p \prod ^ M _ { j = 1 } ( 1 - \chi _ { 8 d } ( p ) \alpha _ { \pi } ( p , j ) p ^ { - s } ) ^ { - 1 } . \end{align*}


\begin{align*} \delta = \frac { P _ { \rm m a x } - P _ { \rm m i n } } { \kappa _ { \rm t } - \kappa _ { \rm b } } , \ , \ , { \rm a n d } \ , \ , P _ { \rm b i a s } = \frac { P _ { \rm m i n } \kappa _ { \rm t } - P _ { \rm m a x } \kappa _ { \rm b } } { \kappa _ { \rm t } - \kappa _ { \rm b } } . \end{align*}


\begin{align*} \rho ^ { \sigma } ( g ) : = \rho ( g ^ { \sigma } ) , \ \ g \in G . \end{align*}


\begin{align*} \rho c _ v \left [ g _ i + \Delta t D _ i g _ i + \frac { \Delta t ^ 2 } { 2 } D _ i ^ 2 g _ i \right ] - g _ i + ( 1 - \rho c _ v ) \left [ g _ i + \Delta t d _ i g _ i + \frac { \Delta t ^ 2 } { 2 } d _ i ^ 2 g _ i \right ] = - { \left ( { { { \bf { M } } ^ { - 1 } } \Lambda { \bf { M } } } \right ) _ { i j } } \left [ { g _ i - g _ i ^ { ( e q ) } } \right ] + \Delta t { \bar F _ i } + \vartheta \Delta t { S _ i } , \end{align*}


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


\begin{align*} f ( n , \phi , \tilde { p } , \tilde { q } ) : = \P _ { ( x _ 1 , \dots , x _ n ) \sim \tilde { p } ^ { \otimes n } } ( \phi ( y _ 1 , \dots , y _ n ) \ne p ) + \P _ { ( x _ 1 , \dots , x _ n ) \sim \tilde { q } ^ { \otimes n } } ( \phi ( y _ 1 , \dots , y _ n ) \ne q ) . \end{align*}


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


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


\begin{align*} \bar { N } ( v ) = \{ u \in V \mid ( u , v ) \in E \} \cup \{ v \} . \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \varphi | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


\begin{align*} \vert I \vert \leq \sum _ { m = 1 } ^ 9 \vert I _ m \vert + \vert J _ 1 \vert + \vert J _ 2 \vert . \end{align*}


\begin{align*} \prod ^ { n e } _ { \substack { j = 1 \\ j \equiv i \bmod e } } ( x - \zeta ^ j _ { n e } y ) = x ^ n - \zeta ^ i _ e y ^ n \end{align*}


\begin{align*} | \nabla f | = \alpha ( f ) \Delta _ { \Psi } f = \beta ( f ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = | x | ^ { a } v ^ { p } & & \quad \mbox { i n } \Omega , \\ \Delta v & = | x | ^ { b } v ^ { q } f ( | \nabla u | ) & & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \big ( h ( t _ j ) , v ( t _ j ) \big ) & = \big ( \hat h ( t _ j ) , \hat v ( t _ j ) \big ) - \big ( L _ V g _ { t _ j } , V ( u _ { t _ j } ) \big ) \\ ( p , z ) & = ( \hat p , \hat z ) - \Big ( L _ V L _ X g _ a , V \big ( X ( u _ a ) \big ) \Big ) \end{align*}


\begin{align*} I _ 2 = ( y - 1 ) \cdot f ( y ) + y \left ( f \left ( \left ( a + 1 + \frac { b } { y } \right ) ( y - 1 ) \right ) - f \left ( \left ( a + \frac { b } { y } \right ) ( y - 1 ) \right ) \right ) . \\ \end{align*}


\begin{align*} N K B L ( m , n ) & = \sum _ { A \in M N A P P ( m , n ) } N ( A ) \cdot ( m - 1 ) ! \\ & \ge \sum _ { i = 1 } ^ { 6 } N ( A _ i ) \cdot ( m - 1 ) ! \ge ( 6 \cdot 2 ^ m ) \cdot ( m - 1 ) ! . \end{align*}


\begin{align*} \ker ( D _ { \varphi } ) = \bigg \{ \sum _ j \varphi _ j e _ j \bigg \} . \end{align*}


\begin{align*} \| \psi _ { n + m } - \psi _ n \| _ 1 \leq \sum _ { i = 1 } ^ m \frac { 1 } { 2 \pi } \int _ 0 ^ { 2 \pi } \left | \arg \frac { \hat { b } _ { n + i } \left ( e ^ { i \theta } \right ) } { e ^ { i \theta } } \right | \ , d \theta . \end{align*}


\begin{align*} \| \boldsymbol { \kappa } \| ^ 2 = \| { \rm D } \mathbf { A } \| ^ 2 = 2 \ , \| { \rm D } { \rm a x l } ( \mathbf { A } ) \| ^ 2 = 2 \ , \| \boldsymbol { \mathfrak { K } } \| ^ 2 & = 2 \ , \| { \rm D } \vartheta \| ^ 2 \end{align*}


\begin{align*} \int | h _ X | ^ 2 w _ U & = \int h _ X \overline { h _ X w _ U } = \int \widehat { h _ X } \overline { \widehat { h _ X } * \widehat { w _ U } } \end{align*}


\begin{align*} \varphi _ a ( z ) = \dfrac { a - P _ a ( z ) - s _ a Q _ a ( z ) } { 1 - \langle z , a \rangle } , z \in \mathbb { B } _ n , \end{align*}


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


\begin{align*} X _ { j } ( u ) ( x _ { 0 } ) = 0 , \forall j = 1 , \cdots , k , \ \ \ \forall x _ 0 \in \bar \O . \end{align*}


\begin{align*} 0 & = \nabla _ { X _ { k + 1 } } ( \Omega _ { k - 1 } ( X _ 0 , \ldots , X _ k ) \varphi ) = \nabla _ { X _ { k + 1 } } \Omega _ { k - 1 } ( X _ 0 , \ldots , X _ k ) \varphi - \Omega _ { k - 1 } ( X _ 0 , \ldots , X _ k ) \circ \omega ( X _ { k + 1 } ) \varphi \\ & = \Omega _ k ( X _ 0 , \ldots , X _ { k + 1 } ) \varphi , \quad \forall X _ i \in \Gamma ( T L ) , \end{align*}


\begin{align*} { \mathbb { E } } \left \{ x [ n ] w _ { \rm d } [ n ] \right \} = 0 , \end{align*}


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


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


\begin{align*} | A | ^ { m - 1 } \left | P _ { [ u _ 1 , \dots , u _ m ] ^ \bot } A \right | _ { n - m } \le \prod _ { i = 1 } ^ m \left | P _ { u _ i ^ \bot } A \right | _ { n - 1 } . \end{align*}


\begin{align*} \Phi _ p ( x ) = \begin{cases} \phi _ p ( x ) , & x \geq 1 , \\ 2 \phi _ p ( 1 ) - \phi _ p ( 2 - x ) , & 0 \leq x \leq 1 . \end{cases} \end{align*}


\begin{align*} \frac { d } { d t } E \big [ [ M _ v ] ( t ) \big ] = \sum _ { k = 1 } ^ N e ^ { - 2 \lambda t } \lambda ^ 2 v _ { k } ^ 2 E | Z _ k ( t ) | \le 4 | v | ^ 2 e ^ { - 2 \lambda t } E \bigg [ \sum _ { k = 1 } ^ N | Z _ k ( t ) | \bigg ] . \end{align*}


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


\begin{align*} e ^ { - \phi ( x ) } = \int _ { \mathbb R } e ^ { - { \psi } ( x , y ) } d y . \end{align*}


\begin{align*} H = \sum _ \alpha \kappa _ \alpha = - \Gamma ^ { \alpha } _ { n \alpha } = - \frac { 1 } { 2 } \frac { \partial g _ { \alpha \alpha } } { \partial x _ n } , \end{align*}


\begin{align*} \begin{aligned} Z ^ { \star } _ { I } ( 1 , \{ \{ 1 \} ^ { m - 1 } , 2 \} ^ { n } ; ( \alpha , \beta ) ) = Z ( m n | n + 1 ; ( \alpha , \beta ) ) + m Z ( m n + 1 | n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \widetilde { f } ( \lambda , b ) = \int _ { G / K } f ( x ) e ^ { ( - i \lambda + \rho ) A ( x , b ) } d x \end{align*}


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


\begin{align*} a _ c \left [ t + 1 \right ] & = a _ c \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \lvert \varepsilon \rvert ^ 2 | \hat { \mathbf { H } } ^ { } \right ] } { \partial a _ c } , \\ a _ i \left [ t + 1 \right ] & = a _ i \left [ t \right ] - \mu \frac { \partial \mathbb { E } \left [ \lvert \varepsilon \rvert ^ 2 | \hat { \mathbf { H } } ^ { } \right ] } { \partial a _ i } , \end{align*}


\begin{align*} F _ { 1 ^ r } ( \theta _ 1 , \theta _ 2 , \ldots , \theta _ n ) : = \sum _ { \substack { A \subseteq [ n ] \\ | A | = r } } \theta _ A ~ . \end{align*}


\begin{align*} \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { + } ^ { 2 } ( \partial _ t \varphi + ( v + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s = - \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } ^ { 2 } \varphi _ 0 \dd x . \end{align*}


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


\begin{align*} f ( x ) = \left \{ \begin{array} { l l l } x & & x \geq 0 \\ a ( e ^ { x } - 1 ) & & x < 0 \end{array} \right . a > 0 . \end{align*}


\begin{align*} p ' _ { 1 } \circ ( p _ { 1 2 } \circ f \circ \iota _ { 1 2 } ) = p _ { 1 } \circ f \circ \iota _ { 1 2 } = p _ { 1 } \circ f \circ l _ { 1 } \circ \iota _ { 1 2 } = p _ { 1 } \circ f \circ \iota _ { 1 2 } \circ l ' _ { 1 } \\ = p ' _ { 1 } \circ ( p _ { 1 2 } \circ f \circ \iota _ { 1 2 } ) \circ l ' _ { 1 } , \end{align*}


\begin{align*} x _ { n - 1 } x _ { n + 1 } = \begin{cases} 1 + x _ n ^ b & \\ 1 + x _ n ^ c & . \end{cases} \end{align*}


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


\begin{align*} \nabla h ( \Omega , x ) = h _ { i } e _ { i } + h x , F _ { i } ( x ) = ( h _ { i j } + h \delta _ { i j } ) e _ { j } . \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 G ( - t ) H ( t ) - \varphi ( t ) ^ 2 G ( t ) H ( t ) = 2 t \varphi ( t ^ 4 ) ^ 2 H ( t ^ 4 ) H ( t ) \end{align*}


\begin{align*} \begin{aligned} & \dot y _ { b e n } = A y '' _ { b e n } + M _ { b e n } y _ { b e n } \\ & \dot y _ { n b } = a A y '' _ { n b } + M _ { n b } y _ { n b } \end{aligned} \end{align*}


\begin{align*} & \sigma _ 3 = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} , \ \ \ \ P = \begin{pmatrix} 0 & u \\ - \bar { u } & 0 \end{pmatrix} , \\ & \eta = \sqrt { \alpha } \left ( k - \frac { \beta } { 2 k } \right ) , H = \alpha k U _ x + \frac { i \alpha \beta ^ 2 } { 2 } \sigma _ 3 \left ( \frac { 1 } { k } P - P ^ 2 \right ) . \end{align*}


\begin{align*} \psi _ \epsilon ( x , y ) = A _ { \epsilon } \| y \| ^ 2 + \tilde { v } ( x ) \| y \| \ge R + 1 \end{align*}


\begin{align*} G ( U , x , t ) = ( \nabla _ V \tilde { \eta } ) ( A ( U , x , t ) , x , t ) \end{align*}


\begin{align*} T : L ^ 2 ( D ) \rightarrow L ^ 2 ( D ) T f = u . \end{align*}


\begin{align*} K _ w ( z ) = \frac { \overline { E ( w ) } E ( z ) - \overline { E ^ * ( w ) } E ^ * ( z ) } { 2 \pi i ( \overline w - z ) } = \frac { \overline { A ( w ) } B ( z ) - \overline { B ( w ) } A ( z ) } { \pi ( z - \overline w ) } , \end{align*}


\begin{align*} \begin{aligned} \Big [ | \langle \boldsymbol { \nu } , A \rangle - \bar A | + | \langle \boldsymbol { \nu } , f _ \alpha \rangle - \bar f _ \alpha | \Big ] & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} \int _ { B ( 0 , 2 ) \backslash B ( 0 , 1 ) } | \nabla \varphi _ n | ^ { 2 } \dd y = \int _ { D ( 0 , 2 ) \backslash D ( 0 , 1 ) } | \nabla \phi _ n | ^ { 2 } \frac { 2 \pi ^ { 2 } } { r } \dd z \dd r , \end{align*}


\begin{align*} x ' _ 1 = \varphi _ { \sigma , s } ^ L ( x _ 1 ) , x ' _ 2 = \varphi _ { \sigma , s } ^ L ( x _ 2 ) , y ' _ 1 = \varphi _ { \sigma , p } ^ R ( y _ 1 ) y ' _ 2 = \varphi _ { \sigma , p } ^ R ( y _ 2 ) . \end{align*}


\begin{align*} A _ k \ & = \ \sum _ { j = 0 } ^ { k } \ , j ! \ , F _ { j + 1 } \begin{Bmatrix} k \\ j \end{Bmatrix} , \\ B _ k \ & = \ \sum _ { j = 0 } ^ { k } \ , j ! \ , F _ { j + 2 } \begin{Bmatrix} k \\ j \end{Bmatrix} . \end{align*}


\begin{align*} g = D ^ 2 u , u \in C ^ \infty ( N , \R ) . \end{align*}


\begin{align*} ( \hat h , \hat v ) = \left \{ \begin{array} { l l } ( h , v ) & \mbox { i n } M \setminus \Omega \\ ( 0 , 0 ) & \mbox { i n } \overline U \end{array} \right . . \end{align*}


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


\begin{align*} M _ I = \bigoplus _ { J \subset I } R _ J . \end{align*}


\begin{align*} \nabla ^ \perp _ T \eta _ i = 0 , \quad \forall T \in \Gamma . \end{align*}


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


\begin{align*} C _ 1 ( t ) = \sum _ { u : \ , i _ u \geq 3 \atop { 0 \leq d _ u < i _ u } } i _ u = \sum _ { i \geq 3 } \sum _ { d = 0 } ^ { i - 1 } i x _ { i , d } , \end{align*}


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


\begin{align*} & \nabla _ V \left ( \nabla _ Y \hat X - \omega ( Y ) \right ) \\ & = \nabla _ Y \nabla _ V \hat X + R ( V , Y ) \hat X - ( \nabla _ V \omega ) ( Y ) - \omega ( \nabla _ V Y ) \\ & = \nabla _ Y ( \omega ( V ) ) + R ( V , Y ) \hat X + R ( \hat X , V ) Y - T _ h ( V , Y ) - \omega ( \nabla _ V Y ) \mbox { ( b y \eqref { e q : O D E } ) } \\ & = ( \nabla _ Y \omega ) ( V ) - R ( Y , \hat X ) V - T _ h ( V , Y ) \mbox { ( b y B i a n c h i i d e n t i t y ) } \end{align*}


\begin{align*} \begin{aligned} \Phi _ 1 ( \gamma ) : = \{ b _ 0 | \ , b _ 0 \in [ 0 , 1 - \gamma ] \textrm { i f } - 1 < \gamma \leq 1 , \textrm { a n d } b _ 0 \in ( - \gamma - 1 , 2 ) \textrm { i f } - 3 < \gamma \leq - 1 \} \ , . \end{aligned} \end{align*}


\begin{align*} [ a _ \lambda b ] = \sum _ { j \in \mathbb { Z _ { + } } } ( a _ { ( j ) } b ) \frac { \lambda ^ { j } } { j ! } . \end{align*}


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


\begin{gather*} \mathcal { U } = \bigl \{ Q \in \mathbb { P } : Q \ll P ^ * ( \pi _ 1 , \ldots , \pi _ n ) Q \bigr \} . \end{gather*}


\begin{align*} Q _ { i - 1 } ( x , w x ) = \frac { Q _ i ( x , w x ) - Q _ i ( x , s _ { \alpha _ { i - 1 } } w x ) } { \alpha _ i ( w x ) } . \end{align*}


\begin{align*} A _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 1 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = \theta ( a ) q ^ 8 . \end{align*}


\begin{align*} a \in \sigma _ l ( A ) , \ , v = ( 1 , x ) ^ T \in V _ l ( a ) , \forall x \in S _ L ( a - d , c ) . \end{align*}


\begin{align*} \begin{aligned} \partial _ x \left ( e ^ { 2 i ( c _ - ( x ) + c ) } u ( x ) \right ) & = I _ 1 ' ( x ) + I _ 2 ' ( x ) . \end{aligned} \end{align*}


\begin{align*} S ( t ) = \begin{bmatrix} e ^ { t \nu \Delta } & 0 \\ 0 & e ^ { t \mu \Delta } \\ \end{bmatrix} . \end{align*}


\begin{align*} \Delta x _ k + \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ k = 0 \mbox { f o r a l l } k = 1 , \dots , n . \end{align*}


\begin{align*} D _ 1 = \displaystyle \sum _ { \substack { X \in S ( 0 ) \\ t = 2 } } \Theta _ { \theta } \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - q ^ 8 ( 1 - q ) . \end{align*}


\begin{align*} \frac { t _ n - s _ l } { t _ n - s _ m } \bigg ( \frac { \overline { A ( s _ l ) } } { \bar s _ l - \bar t _ n } - \overline { B ( s _ l ) } \bigg ) = d _ { l , m } \bigg ( \frac { \overline { A ( s _ m ) } } { \bar s _ m - \bar t _ n } - \overline { B ( s _ m ) } \bigg ) . \end{align*}


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


\begin{align*} { \left ( { { { { D \mathord { \left / { \vphantom { D D } } \right . \kern - \nulldelimiterspace } D } } _ 0 } } \right ) ^ 2 } = 1 - K t , \end{align*}


\begin{align*} \begin{aligned} N _ 3 ^ { ( m / 2 ) } [ \hat { f } ] < \infty \end{aligned} \end{align*}


\begin{align*} x \cdot ( x \wedge y ) = x \cdot y \in I . \end{align*}


\begin{align*} \deg _ H ( S ) = \deg _ F ( S ) \leq 1 , \end{align*}


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


\begin{align*} \lim _ { n \rightarrow + \infty } \int _ { \{ \Psi < - l \} } | f | _ h ^ 2 c _ t ^ n ( - \Psi ) = \int _ { \{ - t \leq \Psi < - l \} } | f | _ h ^ 2 . \end{align*}


\begin{align*} \widetilde k ( t , x , y ) : = \rho ( - R ^ { - 1 } B P _ t \eta _ t ^ { - 1 } x - R ^ { - 1 } B \eta _ t y ) \quad \end{align*}


\begin{align*} c h ( \mathfrak { s } ^ + _ { 3 / 2 } \otimes L ) - c h ( \mathfrak { s } ^ - _ { 3 / 2 } \otimes L ) = ( c h ( \mathfrak { s } ^ { + } _ { 1 / 2 } \otimes L ) - c h ( \mathfrak { s } ^ { - } _ { 1 / 2 } \otimes L ) ) ( c h ( T X ) + 1 ) . \end{align*}


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


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


\begin{align*} \frac { d } { d t } F ( \gamma ( t ) ) = \left \langle v , \dot { \gamma } ( t ) \right \rangle \forall v \in D _ F ( \gamma ( t ) ) , t \in [ 0 , 1 ] . \end{align*}


\begin{align*} T = \{ k _ 1 , k _ 2 , \ldots , k _ m \} \mbox { \ \ \ s u c h t h a t } k _ i < k _ { i + 1 } , \mbox { f o r } 1 \le i < m . \end{align*}


\begin{align*} \left | \sum _ { k = 1 } ^ m \mu ( k ) \left ( \left \lfloor \frac { n } { k } \right \rfloor - \left \lfloor \frac { n ' } { k } \right \rfloor \right ) \right | & = \left | \sum _ { t = 1 } ^ q \left ( M ( n / t ) - M ( n ' / t ) \right ) \right | \\ \leq \sum _ { t = 1 } ^ q \left ( \left \lfloor \frac { | d - d ' | } { t } \right \rfloor + 1 \right ) & \leq q + | d - d ' | \sum _ { t = 1 } ^ r \frac { 1 } { t } . \end{align*}


\begin{align*} w _ n = \sum _ { i = 0 } ^ { r + \tilde { r } - 1 } c _ i w _ { n + i } \end{align*}


\begin{align*} [ r ^ \beta ] C _ k ( q , r , t ) = \frac { C _ k ( t ) ^ { \beta + 1 } - q ^ { \beta + 1 } } { 1 - q + q ^ k t } = \left ( C _ k ( t ) ^ { \beta + 1 } - q ^ { \beta + 1 } \right ) \left ( 1 + ( q - q ^ k t ) + ( q - q ^ k t ) ^ 2 + \hdots \right ) . \end{align*}


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


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


\begin{align*} M _ { \delta , \pm } ( x ; z ) : = M _ { \pm } ( x ; z ) \delta _ { \pm } ( z ) ^ { - \delta _ 3 } , \end{align*}


\begin{align*} & P \otimes _ { G } Q \rightarrow P \otimes _ { G } Q \\ & p \otimes q \mapsto p \kappa ^ { - 1 } \otimes \kappa q = p \otimes q \end{align*}


\begin{align*} \varphi ^ L _ { \sigma , s } ( x ) = a = \varphi ^ L _ { \sigma , s } ( y ) \end{align*}


\begin{align*} \rho { u _ z } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i z } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ z } } . \end{align*}


\begin{align*} \tau ( x ) = \dfrac { \gamma } { 2 } \int _ { - \infty } ^ 0 e ^ { \gamma \tau } a _ { 0 , \delta } ( \hat { x } _ a ( \tau , x ) ) \ , d \tau . \end{align*}


\begin{gather*} \Omega = \Omega _ 1 \times \ldots \times \Omega _ n \quad \quad \mathcal { A } = \mathcal { A } _ 1 \otimes \ldots \otimes \mathcal { A } _ n \end{gather*}


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


\begin{align*} \frac { 1 } { 2 } L ( z ) e ^ { - \frac { L _ j } { 2 } z } = { \coth \Big ( \frac { \sqrt 5 F _ j } { 2 } z \Big ) } \sinh \Big ( \frac { \sqrt 5 F _ j } { 2 } z \Big ) . \end{align*}


\begin{align*} \begin{aligned} \langle \L f , f \rangle \geq \lambda \| f \| ^ 2 _ { L ^ 2 ( \nu ) } \end{aligned} \end{align*}


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


\begin{align*} x y - ( - 1 ) ^ { p ( x ) p ( y ) } y x = 0 \end{align*}


\begin{align*} [ L _ \lambda L ] = ( \partial + 2 \lambda ) L , [ L _ \lambda H ] = ( \partial + \lambda ) H , [ H _ \lambda L ] = \lambda H , [ H _ \lambda H ] = 0 . \end{align*}


\begin{align*} \begin{aligned} \| x ( 2 \bar { t } / \gamma ) \| _ { \mathcal { A } } & \le \beta ( \| x ( \bar { t } / \gamma ) \| _ { \mathcal { A } } , \bar { t } ) + \tilde { d } \\ & \le \beta ( \underline { \rho } , \bar { t } ) + \tilde { d } \\ & \le \beta ( \rho , \bar { t } ) + \tilde { d } \\ & \le \underline { \rho } \end{aligned} \end{align*}


\begin{align*} c h ( \mathfrak { s } ^ { \pm } _ { 1 / 2 } \otimes L ) \ , c h ( T X ) = c h ( \mathfrak { s } ^ { \pm } \otimes L \otimes T X ) = c h ( \mathfrak { s } ^ { \pm } _ { 3 / 2 } \otimes L ) + c h ( \mathfrak { s } ^ { \mp } _ { 1 / 2 } \otimes L ) . \end{align*}


\begin{align*} \varphi ( - t ) ^ 2 G ( t ) H ( - t ) + \varphi ( t ) ^ 2 G ( t ) H ( t ) = 2 \varphi ( t ^ 4 ) ^ 2 G ( t ^ 4 ) G ( t ) . \end{align*}


\begin{align*} \hat { p } ^ i _ { j k } = \frac { \hat { p } ^ i _ j ( B ) \hat { p } ^ i _ k ( C ) } { \hat { p } ^ i ( D ) } . \end{align*}


\begin{align*} G _ { 0 ^ n } = F _ { 1 ^ 0 } ( \theta _ 1 , \theta _ 2 , \ldots , \theta _ n ) = 1 = G _ { 0 ^ { n - 1 } } ( \theta _ 2 , \theta _ 3 , \ldots , \theta _ n ) ~ . \end{align*}


\begin{align*} C ( G , x ) : = 1 + \sum _ { \emptyset \neq U \subseteq V ( G ) : G [ U ] ~ i s ~ a ~ c l i q u e } x ^ { \vert U \vert } , \end{align*}


\begin{align*} R : ( x , y ) \mapsto R ( x , y ) = ( f ( x , y ) , g ( x , y ) ) \end{align*}


\begin{align*} D _ 1 = \displaystyle \sum _ { \substack { X \in S ( 0 ) \\ t = 2 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = - \theta ( a ) q ^ 8 ( 1 - q ) . \end{align*}


\begin{align*} \begin{cases} d Y _ t = ( \Pi _ { V _ j ^ \perp } L _ { \Pi _ { V _ j } x _ 0 } \Pi _ { V _ j ^ \perp } ) Y _ t d t + \Pi _ { V _ j ^ \perp } \sigma d W _ t \\ Y _ 0 = \Pi _ { V _ j ^ \perp } x _ 0 . \end{cases} \end{align*}


\begin{align*} | | \mathcal { B } _ q ( \psi _ m ) | | _ { Q M L _ q ( \mathbb { H } ) } = 1 = | | \psi _ m | | _ { L ^ 2 ( \mathbb { R , , \mathbb { H } } ) } . \end{align*}


\begin{align*} \begin{aligned} | x ( t ) | _ { \mathcal { A } _ \delta } & \le | x _ 1 ( t ) | _ { \mathcal { A } _ \delta } + | x ( t ) - x _ 1 ( t ) | \\ & \le \kappa \bar c _ 0 e ^ { - \gamma \lambda ( t - \bar { t } / \gamma ) } + \bar c \\ & \le \kappa \bar c _ 0 + \bar c \leq d & & \forall \ , t \in I _ 1 \end{aligned} \end{align*}


\begin{align*} S ' | _ { g _ { t _ j } } \big ( h ( t _ j ) , v ( t _ j ) \big ) = 0 \mbox { a n d } S ' | _ { g _ { t _ j } } \big ( \psi _ { t _ j } ^ * ( h , v ) \big ) = 0 \mbox { i n } M \setminus \Omega . \end{align*}


\begin{align*} \nabla \mathcal { N } _ { \epsilon } ( x ) = W _ { L } \cdot \sigma _ { \epsilon } ' ( N _ { \epsilon } ^ { ( L - 1 ) } ( x ) ) \cdot W _ { L - 1 } \cdot \ldots \cdot \sigma _ { \epsilon } ' ( N _ { \epsilon } ^ { ( 1 ) } ( x ) ) \cdot W _ { 1 } . \end{align*}


\begin{align*} s ( n , k , \ell ) = \max \{ \vartheta ( S ) : \ : S \} \end{align*}


\begin{align*} \begin{aligned} E _ { \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { k + 1 } Z _ i ^ { \mathsf { L } ^ { k + 1 } [ u ] } \biggr ) \biggr ] & = E _ { \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { k } Z _ i ^ { \mathsf { M } ^ { k } } \biggr ) \biggr ] \\ & = W ( \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) , k , x + u , \mathsf { M } ^ { k } ) . \end{aligned} \end{align*}


\begin{align*} \mathbf d \cdot \theta \leq - \frac { 2 } { 3 } \sum _ { j = 2 } ^ n \alpha _ j \frac { 1 } { \sqrt { 1 + s ^ 2 } } \leq - \frac { 1 } { 2 } , \end{align*}


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa B _ 1 \frac { r } { f } + \kappa B _ 2 \frac { 1 } { f ^ 2 } + \kappa B _ 3 \frac { r } { f ^ 3 } + \kappa B _ 4 \frac { 1 } { f ^ 4 } + \kappa B _ 5 \frac { r } { f ^ 5 } + \kappa B _ 6 \frac { 1 } { f ^ 6 } . \end{align*}


\begin{align*} \bigl ( T \xRightarrow { Q _ 1 ^ 0 } Q _ 1 \circ A \xRightarrow { \beta * 1 _ A } Q _ 2 \circ A \bigr ) = \bigl ( T \xRightarrow { Q _ 2 ^ 0 } Q _ 2 \circ A \bigr ) \end{align*}


\begin{align*} \begin{aligned} \partial _ t F _ \kappa + v \cdot \nabla _ x F _ \kappa = \tfrac { 1 } { \kappa } \mathcal { B } ( F _ \kappa , F _ \kappa ) \ , , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} F _ a ( \eta _ a ) & : = \left ( \begin{array} { c } - \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \\ - \bar { y } _ a + \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } \end{array} \right ) . \end{aligned} \end{align*}


\begin{align*} Z _ n = \delta ( u _ n ) + r _ n N _ n , \end{align*}


\begin{align*} A _ t : = t A _ 1 + ( 1 - t ) A _ 0 t \in [ 0 , 1 ] \end{align*}


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


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


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + . \end{align*}


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


\begin{align*} f ( x ) = c \ , \int _ { - \infty } ^ \infty \int _ { K / M } \widetilde { f } ( \lambda , b ) e ^ { ( i \lambda + \rho ) A ( x , b ) } | c ( \lambda ) | ^ { - 2 } d b \ , d \lambda \end{align*}


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


\begin{align*} \Delta ^ 2 u = \lambda u & D , \\ [ 1 m m ] u = \frac { \partial u } { \partial \nu } = 0 & \partial D . \end{align*}


\begin{align*} \phi ( u _ 0 e _ 1 u _ 1 ) = - \tau ( u _ 0 , e _ 1 ) \phi ( e _ 1 ) . \end{align*}


\begin{align*} \begin{aligned} \vert J _ 2 \vert & \leq \| \partial _ { \nu } u _ 0 \| _ { L ^ { 2 } ( \Lambda _ { h } ) } \| w - v \| _ { L ^ { 2 } ( \Lambda _ { h } ) } \leq C \| \partial _ { \nu } u _ 0 \| _ { L ^ { 2 } ( \Lambda _ { h } ) } \| w - v \| _ { H ^ { 1 } ( \mathcal C ^ h ) } \\ & \lesssim ( 1 + \tau ) ( 1 + \tau ^ { - \frac { 2 } { 5 } } ) e ^ { - \zeta h \tau } , \end{aligned} \end{align*}


\begin{align*} u ( x ) \geq l ( y ' ) + \frac { 1 } { 4 } \sum _ { \beta = 1 } ^ { n - 1 } b _ \beta y _ \beta ^ 2 - C b _ \alpha | \hat { y } | ^ 2 \end{align*}


\begin{align*} \phi ( \sigma , \tau ) = [ \widetilde \sigma , \widetilde \tau ] = \widetilde \sigma \widetilde \tau { \widetilde \sigma } ^ { - 1 } { \widetilde \tau } ^ { - 1 } \in \langle \epsilon \rangle \cong \mathbb Z / 2 \mathbb Z . \end{align*}


\begin{align*} \mathcal { A } = & \int _ { 0 } ^ { T } \int _ \Omega \bigg ( \frac { 1 } { 2 } \ , \rho \ , \| \dot { u } \| ^ 2 + \ , \rho \ , j \ , \| \dot { \vartheta } \| ^ 2 - W ) \ , d v \ , d t , \end{align*}


\begin{align*} \frac { \partial ^ 2 v } { \partial n _ e ^ 2 } = n _ e \cdot ( \triangle v ) n _ e \end{align*}


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


\begin{align*} \alpha _ n ( 0 ) = h \psi _ 0 ( h n ) + ( - 1 ) ^ n h \phi _ 0 ( h n ) , \end{align*}


\begin{align*} \int _ { x } ^ { 2 x } \left | \theta ( y + \delta y ) - \theta ( y ) - \delta y \right | ^ 2 d y & \geq B \log ^ 4 x \int _ I d y = B g x \log ^ 4 x . \end{align*}


\begin{align*} g ^ * f _ ! = f ' _ ! g '^ * \end{align*}


\begin{align*} I _ 2 ( x ) = - \int _ { \mathbb { R } } r _ 2 ( z ) M _ { + , 1 2 } ( x ; z ) \mathrm { e } ^ { 2 i s x } \mathcal { P } ^ + \left ( z ^ { - 1 } \bar { r } _ 1 ( z ) \mathrm { e } ^ { - 2 i z x } \right ) \mathrm { d } z . \end{align*}


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


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


\begin{align*} \int _ { 0 } ^ { 1 } h ( s ) \sin ( 2 \pi t ) \ , d t = 0 \end{align*}


\begin{align*} \mu ^ { \rm N } & = \mu ^ * + \frac { \varkappa } { 2 } , \varkappa ^ { \rm N } = \frac { \varkappa } { 2 } , \qquad \ , \lambda ^ { \rm N } = \lambda , \\ \gamma ^ { \rm N } & = \frac { \beta + \gamma } { 2 } , \qquad \ \ \beta ^ { \rm N } = \frac { \gamma - \beta } { 2 } , \ \ \ \ \ \ \ \alpha ^ { \rm N } = \alpha \ , . \end{align*}


\begin{align*} \zeta _ \mathrm { V L C } = \frac { A _ { \rm d } } { L ^ 2 } R _ o ( \upsilon _ { \rm t } ) G ( \upsilon _ { \rm r } ) \mathrm { c o s } ( \upsilon _ { \rm r } ) , \end{align*}