\begin{align*} 0 = J ^ { ' } ( \widehat { u } ) + \lambda _ + \varphi _ { w ^ + } ^ * + \lambda _ - \varphi _ { w ^ - } ^ * \ \ \ ( W ^ { \alpha , G } , ( \mathbb { R } ^ d ) ) ^ * , \ \ \varphi ^ * _ { \widehat { w } ^ + } \in \partial \varphi _ + ( \widehat { w } ) \ \ \varphi ^ * _ { \widehat { w } ^ - } \in \partial \varphi _ - ( \widehat { w } ) . \end{align*}


\begin{align*} { R } _ { M } f & = \sum \limits _ { k = - \infty } ^ \infty \sum \limits _ { \{ j \in \Bbb Z : \ , | k - j | > { M } \} } { D } _ k { D } _ j = \sum \limits _ { \{ \ell \in \Bbb Z : \ , | \ell | > M \} } \sum \limits _ { k = - \infty } ^ \infty D _ k D _ { k + \ell } f \\ & = \sum \limits _ { k = - \infty } ^ \infty D _ k ( I - S _ { k + M } ) f + \sum \limits _ { k = - \infty } ^ \infty D _ k S _ { k - M - 1 } f . \end{align*}


\begin{align*} 2 m _ 0 & \leq J ( t _ { \widehat { w } ^ + } \widehat { w } ^ + ) + J ( s _ { \widehat { w } ^ - } \widehat { w } ^ - ) \\ & < J ( t _ { \widehat { w } ^ + } \widehat { w } ^ + + s _ { \widehat { w } ^ - } \widehat { w } ^ - ) \\ & \leq J ( \widehat { w } ) \\ & = \inf \limits _ { \mathcal { M } } J = m _ 1 . \end{align*}


\begin{align*} \frac { 2 ( g - 1 ) } { | G | } = 2 h + u - \delta _ u - \frac { 1 } { p ^ { e ( \alpha ) } } + \sum \limits _ { i = 1 } ^ { r } \Big ( 1 - \frac { 1 } { p ^ { d _ { i } } } \Big ) + \sum \limits _ { t \in \Omega } \Big ( 1 - \frac { 1 } { p ^ { c _ t } } \Big ) \end{align*}


\begin{align*} a _ { i \tau ( i , t - 1 ) } \leq m ^ t _ { i j } \leq a _ { i \max \{ \tau ( i , t - 1 ) , \min \{ j , \tau ( i , t ) \} \} } m ^ { t } _ { i \tau ( i , t ) } = a _ { \tau ( i , t ) } , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { n } k ^ { \ell - 1 } S ( k ) = \frac { 1 } { \ell } B _ \ell ( n + 1 ) t - \frac { 1 } { \ell } \sum _ { a \in S } B _ \ell ( a ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ d } \chi ^ k ( x , \omega ) \partial _ i \eta ( x ) d x = \int _ { \mathbb { R } ^ d } U _ i ^ k ( \tau _ y \omega ) \int _ 0 ^ 1 \sum _ { j \neq i } \partial _ j \left ( \frac { y _ j } { t ^ d } \eta \left ( \frac { y } { t } \right ) \right ) + \frac { y _ i } { t ^ { d + 1 } } \partial _ i \eta \left ( \frac { y } { t } \right ) d t d y . \end{align*}


\begin{align*} c _ n : = \sum _ { s _ 1 , \dots , s _ k \in { \mathbb N } s _ 1 + \cdots + s _ k = n } { n \choose s _ 1 , \dots , s _ k } \ ; \prod _ { ( v _ i , v _ j ) \in E ( D ) } 2 ^ { s _ i s _ j } . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ { n + 1 } S _ { 2 , \lambda } ( n + 1 , k ) k ! x ^ { k } = - n \lambda F _ { n , \lambda } ( x ) + x F _ { n , \lambda } ( x ) + ( x + x ^ { 2 } ) F _ { n , \lambda } ^ { \prime } ( x ) . \end{align*}


\begin{align*} \tilde \Lambda _ { n , 2 } ^ { ( 1 , 3 ) } = 1 + o ( 1 ) , \end{align*}


\begin{align*} x \cdot J ( y \cdot x ) = J y \end{align*}


\begin{align*} \langle \operatorname { t r } _ e , \operatorname { I n d } _ { C ^ * _ r ( \Gamma ) } ( \widetilde { D } ) \rangle = \operatorname { i n d } ( D ) . \end{align*}


\begin{align*} \max \{ u ( y _ i , t ) , u ( z _ i , t ) \} < u ( x _ i , t ) + \varepsilon < h + \varepsilon \ i = 1 , 2 . \end{align*}


\begin{align*} \int _ { \R ^ n } \frac { v ( y ) + v ( - y ) } { \vert y \vert ^ { n + 2 s } } \dd y & = 0 . \end{align*}


\begin{align*} \langle J ^ { ' } ( u ) , v \rangle & = \int _ { \mathbb { R } ^ { d } } \int _ { \mathbb { R } ^ { d } } g \left ( \frac { u ( x ) - u ( y ) } { \vert x - y \vert ^ { \alpha } } \right ) \frac { v ( x ) - v ( y ) } { \vert x - y \vert ^ { \alpha + d } } d x d y + \int _ { \mathbb { R } ^ { d } } g ( u ) v \ d x - \int _ { \mathbb { R } ^ { d } } K ( x ) f ( x , u ) v \ d x , \end{align*}


\begin{align*} \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land \gamma ^ { - 1 } ( x - _ X p ) \in A p \rightarrow p = _ X J ^ A _ { \gamma } x \right ) . \end{align*}


\begin{align*} \phi _ { \infty } '' = \left [ \begin{array} { c c } i _ n & 0 \\ 0 & - i _ n \end{array} \right ] , \psi _ { \infty } '' = \left [ \begin{array} { c c } 0 & - i _ n \\ - i _ n & 0 \end{array} \right ] . \end{align*}


\begin{align*} \varepsilon _ 1 ( x , T ) = 2 \frac { ( \log x ) ^ 2 } { T } . \end{align*}


\begin{align*} \norm { \sum _ { k = N _ { 2 n } + 1 } ^ { N _ { 2 n + 1 } } e _ { \sigma ( k ) } } _ \mathcal { B } & = \norm { \sum _ { \gamma \in F _ { 2 n + 1 } } e _ \gamma - \sum _ { \gamma \in F _ { 2 n } } e _ \gamma } _ \mathcal { B } \\ & \geq \norm { f - \sum _ { \gamma \in F _ { 2 n } } e _ \gamma } _ \mathcal { B } - \norm { f - \sum _ { \gamma \in F _ { 2 n + 1 } } e _ \gamma } _ \mathcal { B } > \varepsilon - \frac { \varepsilon } { 2 } = \frac { \varepsilon } { 2 } . \end{align*}


\begin{align*} f ( \bar { x } ) = g ( \bar { x } ) \leq f ( x ) . \end{align*}


\begin{align*} h _ { i j } ( \tilde { x } _ 0 ) M ^ i _ k ( \tilde { x } ^ 0 ) M ^ j _ l ( \tilde { x } ^ 0 ) = \delta _ { k l } \ , , \end{align*}


\begin{align*} \bar { X } _ W = \{ V \in X ( k ) : V \subseteq W \} . \end{align*}


\begin{align*} \tilde { f } _ \gamma = \int _ { 0 } ^ { + \infty } P _ t ^ \gamma ( f ) d t = ( - \mathcal { L } ^ \gamma ) ^ { - 1 } ( f ) . \end{align*}


\begin{align*} \widehat { K ' } _ { \mathrm { a - c p t - d } } ^ \mathbb { R } = \Re \{ \mathbf { s } ^ H \mathbf { y } \} / ( N _ 2 \sqrt { \rho } ) \end{align*}


\begin{align*} F ( v ) ( x ) = f ( v ( x ) ) = v ^ 3 ( x ) - v ( x ) , x \in D , v \in L ^ 6 ( D ; \mathbb { R } ) . \end{align*}


\begin{align*} \| \delta _ \alpha ( f ) \| _ { C _ b ( \Omega _ X ) } & = \max ( | f | _ { C ^ \alpha ( X ) } , \| f \| _ { C ( X ) } ) , f , f _ 1 , f _ 2 \in C ^ \alpha ( X ) . \end{align*}


\begin{align*} y \coloneqq \left ( \prod _ { i = 1 } ^ { d _ c - 1 } \operatorname { s g n } ( \psi _ i ) \right ) \sum _ { i = 1 } ^ { d _ c - 1 } ( - \log | \psi _ i | ) . \end{align*}


\begin{align*} E ( s _ 1 , s _ 3 ; n , M ) = O \left ( \frac { 1 } { n ^ { \sigma _ 1 + \sigma _ 3 } } \right ) \end{align*}


\begin{align*} n = m ^ 2 , \theta = m - 3 , \hat \theta = \frac { | m - 3 | } { m ( m - 1 ) } \sqrt { \frac { m + 1 } { 2 } } . \end{align*}


\begin{align*} \frac { \mathrm { d } } { \mathrm { d } t } \left ( \| \phi \| ^ 2 \right ) = 2 \mathrm { R e } \langle \phi , \dot { \phi } \rangle = 0 \quad \mbox { a n d } \frac { \mathrm { d } ^ 2 } { \mathrm { d } t ^ 2 } \left ( \| \phi \| ^ 2 \right ) = 2 \left ( \mathrm { R e } \ , \langle \phi , \ddot { \phi } \rangle + \| \dot { \phi } \| ^ 2 \right ) = 0 , \forall t > 0 . \end{align*}


\begin{align*} g ( p ) : = \inf \big \{ r : \P ( w \ge r ) \le p \big \} = \inf \big \{ r : \Lambda ( e ^ r - 1 ) \le p \big \} . \end{align*}


\begin{align*} \begin{cases} L _ { \alpha _ j } < k _ 1 & \alpha = 1 \pi _ { 1 } ^ { \circ } = 1 \\ L _ { \alpha _ j } \geq k _ { m - 1 } & \alpha = \mu - 1 \pi _ { \mu } ^ { \circ } = 1 \\ k _ { r _ { \alpha } } \leq L _ { \alpha _ j } < k _ { r _ { \alpha } + 1 } & \end{cases} \end{align*}


\begin{align*} f ^ * ( g _ { \textnormal { e u c } } ) = e ^ { 2 u } \ , g _ { \textnormal { p o i n } } , \end{align*}


\begin{align*} \| ( \mathbf { x } _ { k + 1 } , \mathbf { y } _ { k + 1 } ) - \mathcal { P } ( \mathbf { x } _ { k } , \mathbf { y } _ { k } ) \| \leq \varepsilon _ k \min \{ 1 , \| ( \mathbf { x } _ { k + 1 } , \mathbf { y } _ { k + 1 } ) - ( \mathbf { x } _ { k } , \mathbf { y } _ { k } ) \| \} , \hbox { w h e r e } \sum _ { k = 0 } ^ { + \infty } \varepsilon _ k < \infty . \end{align*}


\begin{align*} \varphi _ S = \varphi \circ S , \varphi _ { S ^ { - 1 } } = \varphi \circ S ^ { - 1 } . \end{align*}


\begin{align*} \frac { 2 \log \log t _ 0 + 1 } { \log ^ { 5 / 3 } t _ 0 ( \log \log t _ 0 ) ^ { 4 / 3 } } = \frac { 3 c } { \log x } \end{align*}


\begin{align*} \partial \Phi ^ n _ { U } ( f ) [ \phi ] & = \int _ U f ( u , v ) ( \partial _ u \phi ) ^ n ( u ) \ , \mathrm { d } u \mathrm { d } v \\ & = \int _ { \mathcal { I } } \left ( \int _ { \mathbb { R } } f ( u , v ) \ , \mathrm { d } v \right ) ( \partial _ u \phi ) ^ n ( u ) \ , \mathrm { d } u \\ & = \Psi ^ n ( \eta f ) [ \partial _ { \Sigma } \phi ] \end{align*}


\begin{align*} \P \Big [ \ , \bigcap _ { j = 1 } ^ n F _ { z _ j , r _ j } \Big ] \le c ' ( \xi ) ^ { K } + \P [ | \mathcal S | < K ] \end{align*}


\begin{align*} S = \begin{pmatrix} a & b \\ c & d \end{pmatrix} S ^ { - 1 } = \begin{pmatrix} d & - b \\ - c & a \end{pmatrix} . \end{align*}


\begin{align*} \gamma = & c \mathcal E ( \rho _ 0 , u _ 0 | r ( 0 , \cdot ) , U ( 0 , \cdot ) ) + c ( D , T ) ( \varepsilon ^ \alpha + R ^ { - 1 } + ( \varepsilon R ) ^ { - 1 } + \nu + \varepsilon ^ 2 \nu ^ 2 ( 1 + R ^ { - 2 } ) + \varepsilon \nu ^ { - 1 } ) , \\ \delta ( \tau ) = & c ( D , T ) \Bigl ( \frac { 1 } { | \Omega _ R | } \int _ { \Omega _ R } p ( \rho ) + \varepsilon ^ \frac { 4 } { 3 } \nu ^ { - 1 } ( 1 + R ^ { - 4 } ) + ( 1 + \varepsilon ^ 2 ) ( 1 + R ^ { - 2 } ) + \varepsilon ^ 2 \nu + 1 \Bigr ) . \end{align*}


\begin{align*} \Delta _ { g _ M } \xi + \sum _ { j = 1 } ^ d a _ j e ^ { ( \iota ^ \ast u ^ j \xi ) } \iota ^ \ast u ^ j = w . \end{align*}


\begin{align*} [ D _ { G , K } \hat { \otimes } 1 , 1 \hat { \otimes } D _ { Z } ] = 0 , D ^ 2 = D _ { G , K } ^ 2 + D _ { Z } ^ 2 . \end{align*}


\begin{align*} f _ { \rho , L } ( x , y ) = 1 + \rho \sin ( 2 ^ { L + 1 } \pi x ) \sin ( 2 ^ { L + 1 } \pi x ) \end{align*}


\begin{align*} \begin{gathered} A _ 0 ( a _ i ) ^ 2 \subseteq A _ 0 ( a _ i ) , A _ { \eta } ( a _ i ) ^ 2 \subseteq A _ 0 ( a _ i ) + A _ 1 ( a _ i ) , \\ ( A _ 0 ( a _ i ) + A _ 1 ( a _ i ) ) A _ { \eta } ( a _ i ) \subseteq A _ { \eta } ( a _ i ) , A _ 0 ( a _ i ) A _ 1 ( a _ i ) = ( 0 ) . \end{gathered} \end{align*}


\begin{align*} d \theta ^ 1 & \equiv \theta ^ 1 \wedge \omega _ 1 { } ^ 1 \mod \theta \wedge \theta ^ { \bar 1 } , \\ \omega _ { 1 \bar 1 } + \omega _ { \bar 1 1 } & = d h _ { 1 \bar 1 } , \end{align*}


\begin{align*} \widehat M = \left \{ f : \Omega \rightarrow \R , \ ; \ ; \left \| \frac { \nabla ^ r f } { g } \right \| ^ { p _ 1 } _ { L _ { p _ 1 } ( \Omega ) } + \left \| \frac { f } { g _ 0 } \right \| ^ { p _ 1 } _ { L _ { p _ 1 } ( \Omega ) } \le 1 , \ ; \ ; \| w f \| _ { L _ { p _ 0 } ( \Omega ) } \le 1 \right \} , \end{align*}


\begin{align*} \varphi ( b _ 0 a _ 1 b _ 1 a _ 2 b _ 2 \cdots a _ n b _ n ) = \varphi ( a _ 1 a _ 2 \cdots a _ n ) \varphi ( b _ 0 ) \varphi ( b _ 1 ) \cdots \varphi ( b _ { n - 1 } ) \varphi ( b _ n ) . \end{align*}


\begin{align*} f _ j ( z _ k ) = ( \bar A z _ k - b ) ^ \intercal S _ j ( S _ j ^ \intercal \bar A B ^ { - 1 } \bar { A } ^ \intercal S _ { i _ k } ) ^ \dagger S _ { i _ k } ^ \intercal ( \bar A z _ k - b ) , ~ j = 1 , \ldots , \epsilon . \end{align*}


\begin{align*} \widehat { D } ^ { ( \ell ) } = \mathrm { D i a g } \bigg ( \underbrace { a _ + \xi _ i ^ { ( \ell ) } + a _ - ( 1 - \xi _ { i } ^ { ( \ell ) } ) } _ { = : d _ { i } ^ { ( \ell ) } } , \ , i = 1 , \ldots , n _ \ell \bigg ) , \xi _ i ^ { ( \ell ) } \sim \mathrm { B e r n o u l l i } ( 1 / 2 ) \ , \ , i i d . \end{align*}


\begin{align*} 0 & = \varepsilon \circ \psi ( u v u v ^ { - 1 } [ a _ 1 , a _ 2 ] \cdots [ a _ { 2 m - 1 } , a _ { 2 m } ] ) = \varepsilon \circ \psi ( u v u v ^ { - 1 } ) \\ & = \varepsilon \bigl ( w _ u \ldotp g ^ { 2 k + 1 } w _ v g ^ { - 2 k - 1 } \ldotp g ^ { 2 k + 1 + b } w _ u g ^ { - 2 k - 1 - b } \ldotp \Delta _ { 4 k + 2 } ^ 2 \ldotp w _ v ^ { - 1 } \bigr ) = 2 \varepsilon ( w _ u ) + \varepsilon ( \Delta _ { 4 k + 2 } ^ 2 ) . \end{align*}


\begin{align*} H _ { N } ( z q , q ^ 2 ) = \Psi _ { N } ( z q , z q , q / z , q / z ) . \end{align*}


\begin{align*} p _ { \varepsilon } ( y ) = \frac { 1 } { ( 2 \pi \varepsilon ) ^ { \frac { d } { 2 } } } \exp \left ( - \dfrac { \| y \| ^ 2 } { 2 \varepsilon } \right ) . \end{align*}


\begin{align*} d _ { q - 1 } \tilde p = ( I - \varPhi _ q \ , d _ q ) \Big ( \varPsi _ { \mu , q } f + \varPsi _ { \mu , q , 0 } u _ 0 - \varPsi _ \mu { \mathcal N } _ q u \Big ) . \end{align*}


\begin{align*} \Lambda ^ { ( 2 : 2 ) } _ { n , 2 , \lambda } & = \frac { 1 6 \cdot \left ( 1 + o ( 1 ) \right ) } { 9 0 ^ 2 n ^ 4 \delta ^ 4 _ n ( k ) } \cdot \sum \limits _ { \mathbf { P } _ { k } \in \mathcal { W } _ { k } ( \lambda ) } \sum \limits _ { \mathbf { i } \in \mathcal { I } _ 2 \times \mathcal I _ 2 } \varphi _ { k - 1 } \left ( \mathbf { P } _ { k } , \mathbf { i } \right ) . \end{align*}


\begin{align*} | I _ 3 ( x ) | ~ \leq ~ \O ( 1 ) , \big | I ' _ 3 ( x ) \big | ~ = ~ \left | \int _ { 3 x / 2 } ^ { 2 } { g _ b ' ( y ) \over y - x } ~ d y + { 3 \over 2 } \cdot g ' _ b \left ( { 3 x \over 2 } \right ) \cdot \ln \left ( { x \over 2 } \right ) \right | ~ \leq ~ \O ( 1 ) \cdot \ln ^ 2 x , \end{align*}


\begin{align*} & \frac { 1 } { ( 2 m - 1 ) ! ! } E \left [ \left ( \dfrac { S _ n } { \sqrt { n / ( 1 - 2 \alpha ) } } \right ) ^ { 2 m } \right ] - 1 \sim \frac { m ( m - 1 ) } { 2 } \cdot c ( \alpha ) \cdot n ^ { - 1 } , \end{align*}


\begin{align*} F ( z ) : = f ( x + z ) - f ( x ) - z f ' ( x ) . \end{align*}


\begin{align*} g = \kappa ( g ) \mu ( g ) e ^ { H ( g ) } n \in K M A N = G . \end{align*}


\begin{align*} \mathcal { L } ^ { \psi } ( g _ i ) ( x ) = \Delta ( g _ i ) ( x ) + \left \langle \frac { \nabla ( \psi ) ( x ) } { \psi ( x ) } ; \nabla ( g _ i ) ( x ) \right \rangle . \end{align*}


\begin{align*} u _ t + \big \{ m ( K \cap \{ u ( \cdot , t ) < u ( x , t ) \} ) - 1 \big \} | \nabla u | = 0 \end{align*}


\begin{align*} \big \{ ( x , x ^ \prime , y , y ^ \prime ) \ \ y = f ( x ) \big \} \approx \Gamma _ f \times \R ^ { n ^ \prime } \times \R ^ { m ^ \prime } \end{align*}


\begin{align*} ( I \ ! + \ ! \varPhi _ q \ , d _ q \ , \varPsi _ { \mu , q } { \mathcal N } _ q ) u = \varPhi _ q \ , d _ q \ , ( \varPsi _ { \mu , q } f + \varPsi _ { \mu , q , 0 } u _ 0 ) \end{align*}


\begin{gather*} \int _ 0 ^ 1 x ^ { 2 N + 1 } \Lambda ( x ) \ , d x \geq \int _ { 1 - 1 / N } ^ 1 x ^ { 2 N + 1 } \Lambda ( x ) \ , d x \geq \alpha \sum _ { n = N } ^ \infty | I _ n | c _ n . \end{gather*}


\begin{align*} \lim _ { l \to \infty } \gamma _ F ( P \cap \tau _ { v _ l } P ) = ( \gamma _ F ( P ) ) ^ 2 . \end{align*}


\begin{align*} \lim _ { n \to \infty } \left ( \langle X g _ n , P g _ n \rangle - \langle P g _ n , X g _ n \rangle \right ) = \langle X g , P g \rangle - \langle P g , X g \rangle = 0 . \end{align*}


\begin{align*} \Gamma ( \pi ) \ge \frac { 1 } { N } \sum _ { \ell = 1 } ^ N \left ( \frac { \rho _ \ell T _ { \ell , \pi } ^ { a v } } { 2 } + \rho _ \ell \gamma _ \ell + \frac { c _ \ell } { T _ { \ell , \pi } ^ { a v } } \right ) . \end{align*}


\begin{align*} y _ t = x + \int _ 0 ^ t \tilde { f } ( s , y _ s ) d s - \tilde { \sigma } B _ t ^ H , 0 \le t \le 1 , H \in ( 0 , \frac { 1 } { 2 } ) , \end{align*}


\begin{align*} \lambda _ 1 + \lambda _ { k + 1 } + \lambda _ { 2 k + 1 } + \cdots + \lambda _ { ( t - 1 ) k + 1 } = \sum \limits _ { j = 1 } ^ { t } \sum \limits _ { i = 1 } ^ k a _ { i , j } , \end{align*}


\begin{align*} B ( G ) _ { \rm b a s i c } \simeq \pi _ 1 ( G ) _ \Gamma = ( \pi _ 1 ( G ) _ I ) _ \phi = \Omega _ \phi , \end{align*}


\begin{align*} \Gamma ( f , g ) = \frac 1 2 \left \lbrace \mathcal { L } ^ { ( 1 ) } ( f g ) - f \mathcal { L } ^ { ( 1 ) } g - g \mathcal { L } ^ { ( 1 ) } f \right \rbrace \end{align*}


\begin{align*} | g ' ( y ) | = O \bigg ( \frac { 1 } { | y | } \bigg ) \ ; \ ; | y | \rightarrow + \infty . \end{align*}


\begin{align*} \begin{aligned} A ^ { 0 } U _ { t } + A ^ { i } \partial _ { i } U - \epsilon B ^ { i j } \partial _ { i } \partial _ { j } U + D U & = F , \\ \left . V \right \rvert _ { t = 0 } & = U _ { 0 } , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} e ^ { - t D _ { { \rm l e f t } } } u ( x ) & = u ( x - t ) \\ e ^ { - t D _ { { \rm r i g h t } } } v ( x ) & = v ( x + t ) \end{aligned} \end{align*}


\begin{align*} | p - q | \leq 1 , p + q = 1 + s \geq 2 , 2 p + r = 2 + s = p + q + 1 , r = q - p + 1 , 0 \leq r \leq 2 . \end{align*}


\begin{align*} \mathcal { L } ( q , \lambda ( q , t ) ) = \frac { \partial } { \partial t } ( e ^ { \lambda ( q , t ) } - 1 ) - 1 . \end{align*}


\begin{align*} \lim _ { n \to + \infty } \tilde { \lambda } _ { n } = \tilde { \lambda } _ { 0 } . \end{align*}


\begin{align*} t _ n < t _ n ' , \ ; \forall n \in \mathbb { N } , \delta ( t _ n ) = \varepsilon _ 1 , \delta ( t ) < \varepsilon _ 1 , \ ; \forall t \in [ t _ n , t _ n ' ) . \end{align*}


\begin{align*} \| \Pi ^ \perp \mathcal { S } _ L ( t ) \| _ { H ^ 2 _ x L ^ 2 ( \mu ^ { - ( 1 / 2 + \varepsilon ) } ) \rightarrow H ^ 2 _ x L ^ 2 ( \mu ^ { - 1 / 2 } ) } \leq C e ^ { - \lambda t ^ b } , \Pi ^ \perp = I - \Pi \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { n } S _ { 1 , \lambda } ( n , k ) \mathcal { E } _ { k , \lambda } ( x ) = n ! \sum _ { k = 0 } ^ { n } \binom { x } { k } \bigg ( - \frac { 1 } { 2 } \bigg ) ^ { n - k } . \end{align*}


\begin{align*} \begin{bmatrix} 0 & - 1 \\ - 1 & 0 \end{bmatrix} ^ n \cdot \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} \cdot \cal { M } ^ { v _ 1 } _ { v _ 2 } ( \psi _ { j , i , \Delta } ) . \end{align*}


\begin{align*} b _ { i j } ( \delta ) = m _ { i j 1 } + \sum _ { t = 1 } ^ { l _ i - 1 } ( m _ { i j t + 1 } - m _ { i j t } ) \delta _ { i t } i \in \{ 1 , \ldots , d \} \ ; j \in \{ 0 , \ldots , n \} . \end{align*}


\begin{align*} \frac { f ( x ) - g ( x ) } { h ( x ) + 1 } \ = \ r . \end{align*}


\begin{align*} \min _ { ( \rho , m ) \in C _ F ( \rho ^ a , \rho ^ b ) } \mathcal A ( \rho , m ) & = \sup _ { S } \Big \{ \ < S ( 1 ) , \rho ^ b \ > - \ < S ( 0 ) , \rho ^ a \ > : \sup _ { \rho } \{ \ < \dot S , \rho \ > + \frac 1 4 \sum _ { i j } v _ { i j } ^ 2 \theta _ { i j } ( \rho ) \\ & \quad + \frac 1 2 \sum _ { i j } ( \Sigma _ i - \Sigma _ j ) ( S _ { i } - S _ { j } ) \theta _ { i j } ( \rho ) d W ^ { \delta } ( t ) \} = 0 \Big \} . \end{align*}


\begin{align*} - \phi _ { t , t t } + \phi _ { t , x x } = ( | \phi _ t | ^ 2 - | \phi _ x | ^ 2 ) \phi _ t + 2 ( \phi _ t \cdot \phi _ { t t } - \phi _ x \cdot \phi _ { t x } ) \phi + a \phi _ { t t } . \end{align*}


\begin{align*} \varphi = \frac { 1 } { \langle g , \widetilde { g } \rangle } \iint _ { \R ^ { 2 d } } V _ { \widetilde { g } } \varphi ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) \end{align*}


\begin{align*} \tau \leqslant \inf _ { B _ { 1 / 2 } ^ + } \frac { \tilde u ( x ) } { x _ 1 } = \rho \inf _ { B _ { \rho / 2 } ^ + } \frac { u ( x ) } { x _ 1 } . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { \lfloor a / 2 \rfloor } \binom { a + b - 1 } { k , b + k - 1 , a - 2 k } + \sum _ { k = 0 } ^ { \lfloor ( a - 1 ) / 2 \rfloor } \binom { a + b - 1 } { k , b + k , a - ( 2 k + 1 ) } , \end{align*}


\begin{align*} W _ { \mathbb { M } ^ 2 } ( u , v , u ' , v ' ) = \lim _ { \epsilon \searrow 0 } \frac { - 1 } { 4 \pi } \ln \left ( \frac { - ( u - u ' ) ( v - v ' ) + i \epsilon t } { \Lambda ^ 2 } \right ) . \end{align*}


\begin{align*} u ( x , y ) = \tanh \left [ 2 \left ( x ^ 3 - y ^ 4 \right ) \right ] . \end{align*}


\begin{align*} \alpha _ i + \alpha _ { n + 1 - i } = 0 . \end{align*}


\begin{align*} J \cap K & = ( ( A \setminus \{ \bar { a } _ n \} ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G ) ) \cap ( C _ 1 \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( H ) ) \\ & \subseteq ( ( A \setminus \{ \bar { a } _ n \} ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( G ) ) \cap ( ( C \setminus \{ \bar { a } _ n \} ) \cup [ \bar { b } _ n \mapsto \bar { a } _ n ] ( H ) ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { m } x _ i f ( z _ i ) = \int _ { \gamma } f ( z ) w ( z ) d z \end{align*}


\begin{align*} \begin{aligned} \lfloor \frac { j - 1 } { 2 } \rfloor - \lceil \frac { s - 1 } { 2 } \rceil + \frac { l } { 2 } + q + 1 & \geq \lceil \frac { j } { 2 } \rceil + \frac { l } { 2 } + \lfloor \frac { q } { 2 } \rfloor \\ & \geq \lceil \frac { j } { 2 } \rceil + \frac { l } { 2 } + \lfloor \frac { j - l } { 2 } \rfloor \\ & \geq j - \frac { 1 } { 2 } . \end{aligned} \end{align*}


\begin{align*} M = \mathcal { O } ( \log T ) \end{align*}


\begin{align*} \left ( 1 + \frac { \tau } { 2 } \eta ^ n + \frac { \tau ^ 2 } { 2 } \vartheta ^ n \right ) \tilde { \phi } ^ { n + 1 } _ { j k } - \frac { \tau ^ 2 } { 4 } \left ( \Delta _ h \tilde { \phi } ^ { n + 1 } \right ) _ { j k } = \mathcal { H } ^ n _ { j k } , j , k = 0 , 1 , \ldots , M - 1 , \end{align*}


\begin{align*} s ( c , d ) + s ( d , c ) = { c ^ 2 + d ^ 2 - 3 \vert c d \vert + 1 \over 1 2 c d } , \ \ \ \ \ ( c , d \in { \mathbb Z } \setminus \{ 0 \} , \ \gcd ( c , d ) = 1 ) . \end{align*}


\begin{align*} E _ g : = E ( \phi _ g ) = \inf _ { \phi \in S } E ( \phi ) , \end{align*}


\begin{align*} z = \sum _ { i = 0 } ^ { 3 n - 6 j - 4 } \gamma _ { i } e _ { i } , \quad \gamma _ { 3 n - 6 j - 4 } \neq 0 . \end{align*}


\begin{align*} H _ i = \left \lbrace x \in \mathbb { R } ^ n \middle | \left \langle \tilde a _ i , x \right \rangle = b _ i \right \rbrace . \end{align*}


\begin{align*} F _ { K , Q } ^ { \rm C } ( n ) : = F _ n ^ C F _ { K , Q } ^ { \rm D } ( n ) : = \frac { n - 1 } n F _ n ^ D . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \int _ 0 ^ T \int _ { \mathbb { R } } \rho _ { 3 } ( s , u ) \partial _ s G _ { k } ( s , u ) d u d s = - \int _ 0 ^ T \int _ { \mathbb { R } } [ \rho _ { 3 } ( s , u ) ] ^ 2 \rho _ { 4 } ( s , u ) d u d s . \end{align*}


\begin{align*} \langle \hat \varepsilon _ - , \phi _ { j , \infty } \rangle _ { L ^ 2 _ \rho } = 0 , \forall j \le \ell . \end{align*}


\begin{align*} \frac { \mathbf { P } ( X _ n / \sqrt { \langle X \rangle _ n } > x ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) . \end{align*}


\begin{align*} f = \mathop { \sum \sum } _ { k , l \in \Z ^ d } c _ { k , l } M _ { \beta l } T _ { \alpha k } g , \end{align*}


\begin{align*} H o m _ { S h } ( \Z _ { V , X } ^ \# , \mathcal I ) = H o m _ { P S h } ( \Z _ { V , X } , \mathcal I ) = \mathcal I ( V ) \end{align*}


\begin{align*} Y _ t = H _ t X _ t + V _ t , \ \ t = 1 , \ldots , n , \ \ \frac { 1 } { n } { \bf E } \Big \{ \sum _ { t = 1 } ^ { n } | | X _ t | | _ { { \mathbb R } ^ { n _ x } } ^ 2 \Big \} \leq \kappa \end{align*}


\begin{align*} \Sigma _ 1 = \sum _ { \substack { 0 \leq m , n < \infty \\ m + h _ p = n + k _ p } } \Big ( D _ { 1 , m , n } + D _ { 2 , m , n } + D _ { 3 , m , n } \Big ) \frac { 1 } { p ^ { \frac { m } { 2 } + \frac { n } { 2 } } } , \end{align*}


\begin{align*} \bigcup _ { t \in \R } Y _ t = \Sigma , \bigcap _ { t \in \R } Y _ t = \emptyset Y _ s \subset \mathring { Y } _ t s < t . \end{align*}


\begin{align*} x ^ \prime : y ^ \prime = ( a _ 1 x + b _ 1 y ) : ( a _ 2 x + b _ 2 y ) \end{align*}


\begin{align*} \gamma a = \frac { \log \left ( n + \sqrt { n ^ 2 + 4 } \right ) - \log ( 2 ) } { n } . \end{align*}


\begin{align*} \lim _ { x \rightarrow 0 ^ - } f ( x ) = \lim _ { x \rightarrow 0 ^ + } f ( x ) = f ( 0 ) = 0 , \lim _ { x \rightarrow 0 ^ - } f ' ( x ) = \lim _ { x \rightarrow 0 ^ + } f ' ( x ) = f ' ( 0 ) = 1 . \end{align*}


\begin{align*} h _ A ( t ) = t \ , f _ A ' ( t ) h _ B ( t ) = t \ , f _ B ' ( t ) . \end{align*}


\begin{align*} \bar x = \arg \max \left \{ \left \langle c , x \right \rangle \middle | A x \leqslant b , x \in \mathbb { R } ^ n \right \} , \end{align*}


\begin{align*} T _ m M = E _ 0 ( m ) \oplus E _ 1 ( m ) \oplus . . . \oplus E _ r ( m ) . \end{align*}


\begin{align*} ( k - 0 . 1 ) n c < \sum _ { u \sim z } \sum _ { \substack { w \sim u \\ w \in N _ 2 ( z ) } } \mathbf { \mathrm { v } } _ w = \sum _ { u \sim z } \sum _ { \substack { w \sim u \\ w \in M _ 2 } } \mathbf { \mathrm { v } } _ w + \sum _ { u \sim z } \sum _ { \substack { w \sim u \\ w \in S _ 2 \setminus M _ 2 } } \mathbf { \mathrm { v } } _ w < 0 . 9 n c + e ( N _ 1 , S _ 2 \setminus M _ 2 ) \frac { \alpha } { 3 } , \end{align*}


\begin{align*} [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( K ) \cup [ \bar { a } _ n \mapsto \bar { b } _ n ] ( L ) & = [ \bar { a } _ n \mapsto \bar { b } _ n ] ( J \cup K \cup L ) \\ & = [ \bar { a } _ n \mapsto \bar { b } _ n ] ( \mathbb { N } ) = \{ \bar { b } _ n \} \end{align*}


\begin{align*} \Omega _ + = \{ ( e ^ { t H _ p } ( y , \eta ) , y , \eta ) \ , : \ , p ( y , \eta ) = 0 \} \subset T ^ \ast ( X \times X ) \simeq T ^ \ast X \times T ^ \ast X , \end{align*}


\begin{align*} F _ \pi ( v ) = \sum _ { E \in { \cal T } _ h } \int _ E \left ( \Pi _ { E , q - 1 } f \right ) v \ , , \end{align*}


\begin{align*} \mathfrak { D } ^ d f = \mathfrak { D D } ^ { d - 1 } f , \ , \ , \overline { \mathfrak { D } } ^ d f = \overline { \mathfrak { D } } \overline { \mathfrak { D } } ^ { d - 1 } f , \ , \ , \mathfrak { C } ^ d f = \mathfrak { C C } ^ { d - 1 } f , \ , \ , \mathfrak { D } ^ 0 f = \overline { \mathfrak { D } } ^ 0 f = \mathfrak { C } f = f . \end{align*}


\begin{align*} B f ( z ) = e ^ { q ( z ) } . \end{align*}


\begin{align*} \Omega _ 0 ( X + \xi , Y + \eta ) : = \frac { 1 } { 2 } ( \xi ( Y ) - \eta ( X ) ) . \end{align*}


\begin{align*} \begin{aligned} & \rho _ 1 ^ { i n } ( x ) = 2 - x ^ 2 , \rho _ 2 ^ { i n } ( x ) = 2 + s i n ( \pi x ) , \\ & \phi ( 0 , t ) = - 1 , \ \phi ( 1 , t ) = 1 . \end{aligned} \end{align*}


\begin{align*} B _ n ( x ) = \sum _ { i = 0 } ^ { n } \binom n i B _ i x ^ { n - i } , \end{align*}


\begin{align*} N _ 1 = x _ 1 \sqrt { \frac { z B } { F _ { k , 1 } ( B ) } } . \end{align*}


\begin{align*} & \sum _ { i = 1 } ^ 2 [ \mathfrak { a } _ i ^ + ( t ) ] ^ 2 \leq c _ 0 \ , \sup _ { \tau \geq t } \bigg ( \frac { 1 } { [ \mathfrak { q } _ 1 ( \tau ) - \mathfrak { q } _ 2 ( \tau ) ] ^ 4 } + \sum _ { i = 1 } ^ 2 [ \mathfrak { a } _ i ^ - ( \tau ) ] ^ 2 \bigg ) , \\ & \| \epsilon ( t ) \| _ { H ^ 1 } ^ 2 \leq C _ 0 \ , \sup _ { \tau \geq t } \bigg ( \frac { 1 } { [ \mathfrak { q } _ 1 ( \tau ) - \mathfrak { q } _ 2 ( \tau ) ] ^ 4 } + \sum _ { i = 1 } ^ 2 [ \mathfrak { a } _ i ^ - ( \tau ) ] ^ 2 \bigg ) . \end{align*}


\begin{align*} \tilde { \Theta } _ { a , k , \zeta , \mu , \sigma } ( x ) & = \Psi _ { a , \zeta , \mu , \sigma } ( x _ 1 , \dots , x _ { k - 1 } , x _ { k + 1 } , \dots , x _ d ) , \\ \tilde { W } _ { a , k , \zeta , \mu , \sigma } ( x ) & = \tilde { \Psi } _ { a , \zeta , \mu , \sigma } ( x _ 1 , \dots , x _ { k - 1 } , x _ { k + 1 } , \dots , x _ d ) e _ k . \end{align*}


\begin{align*} ( m \otimes m ) X = ( m \otimes m ) ( Y ) , \end{align*}


\begin{align*} \lim _ { h \to 0 } \norm { u ( \ , \cdot \ , + h ) - u } { L ^ 2 ( J _ h ; U ) } = 0 . \end{align*}


\begin{align*} ( x , \omega , \tau ) \odot ( x ' , \omega ' , \tau ' ) = ( x + x ' , \omega + \omega ' , \tau + \tau ' + x ' \cdot \omega ) . \end{align*}


\begin{align*} \begin{aligned} \min \ & f ( x ) \\ \enspace & h ( x ) = 0 , \end{aligned} \end{align*}


\begin{align*} C ( { \sf S N R } ) = \max _ { \rho \in \{ 1 , 2 , \ldots , N \} } \rho \log _ 2 \left ( 1 + \frac { { \sf S N R } } { \rho } \frac { N ^ 2 } { \rho } \right ) . \end{align*}


\begin{align*} \begin{cases} \forall r ^ 1 , x ^ X , y ^ X \Big ( r > _ \mathbb { R } 0 _ \mathbb { R } \rightarrow \norm { J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y } _ X \\ \qquad \qquad \qquad \leq _ \mathbb { R } \norm { r ( x - _ X y ) + _ X ( 1 - r ) ( J ^ A _ { 1 } x - _ X J ^ A _ { 1 } y ) } _ X \Big ) , \\ \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \land \gamma ^ { - 1 } ( x - _ X p ) \in A p \rightarrow p = _ X J ^ A _ { \gamma } x \right ) , \end{cases} \end{align*}


\begin{align*} ( I - P ) v = | D | ^ { - 1 } \int _ D v \ , \dd x . \end{align*}


\begin{align*} \begin{pmatrix} 1 \\ b _ 1 \\ \vdots \\ b _ r \\ 0 \end{pmatrix} = A ^ { - 1 } \begin{pmatrix} 1 \\ Z _ 1 ( z _ 1 , \ldots , z _ { r + 1 } ) \\ \vdots \\ Z _ { r + 1 } ( z _ 1 , \ldots , z _ { r + 1 } ) \end{pmatrix} . \end{align*}


\begin{align*} & | \dot { x } _ i ( t ) - \mu _ i ( t ) | \lesssim \frac { 1 } { t ^ { 5 / 4 - \delta _ 0 } } , \\ & \bigg | \dot { \mu } _ i ( t ) + \frac { \alpha _ i } { t ^ { \frac { 3 } { 2 } } } - \sum ^ n _ { j = 1 } \frac { m _ { i j } x _ j ( t ) } { t ^ 2 } + \sum ^ n _ { \substack { k , j = 1 , \\ j \not = i } } \frac { b _ { i j k } \alpha _ k } { ( \alpha _ i - \alpha _ j ) ^ 3 t ^ 2 } \bigg | \lesssim \frac { 1 } { t ^ { 9 / 4 - \delta _ 0 } } , \end{align*}


\begin{align*} \varphi ( x + k , \omega + l ) = \varphi ( x , \omega ) + k \cdot \omega + \kappa ( k , l ) . \end{align*}


\begin{align*} \Gamma \mapsto \big \{ ( s , \xi _ s ) & \in \dot { T } ^ { * } \Sigma _ 0 \ , | \ , \\ & \exists ( u , v ; \xi _ u , \xi _ v ) \in \Gamma , ( s , u , v ; \xi _ s , \xi _ u , \xi _ v , ) \in \mathrm { W F } ( K ) \big \} \end{align*}


\begin{align*} L : = H ( W _ 1 ) = H ( W _ 2 ) = \cdots = H ( W _ K ) , \end{align*}


\begin{align*} & S _ { 1 , \lambda } ( k + 1 , m + 1 ) \\ & = \sum _ { l = m } ^ { k } S _ { 1 , \lambda } ( k , l ) \bigg \{ \binom { l } { m } \langle 1 \rangle _ { l - m , \lambda } ( - 1 ) ^ { l - m } + l \lambda \binom { l - 1 } { m } ( - 1 ) ^ { l - m - 1 } \langle 1 \rangle _ { l - m - 1 , \lambda } \bigg \} . \end{align*}


\begin{align*} \sum _ { i \in \{ 1 , \dots , \ell \} } \alpha _ i = \sum _ { j \in \{ \ell + 1 , \dots , k \} } \beta _ j . \end{align*}


\begin{align*} \mathcal { U } _ { k } ( \lambda ) & = \big \{ \mathbf { P } _ k \in \mathcal { U } _ k : c _ { k - 1 } ( \mathbf P _ k ) = 2 k - 2 - \lambda \big \} , \\ \mathcal { W } _ { k } ( \lambda ) & = \big \{ \mathbf { P } _ k \in \mathcal { W } _ k : c _ { k - 1 } ( \mathbf P _ k ) = 2 k - 2 - \lambda \big \} , \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } \omega ( Q ) \psi _ { Q } ( x , x _ { Q } ) q _ { Q } h ( x _ { Q } ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} u ( x ) + \frac { 1 } { 2 } | u ^ \prime ( x ) | ^ 2 - f ( x ) & \leq 0 \ , ( - 1 , 1 ) , \\ u ( x ) + \frac { 1 } { 2 } | u ^ \prime ( x ) | ^ 2 - f ( x ) & \geq 0 [ - 1 , 1 ] . \end{aligned} \right . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \norm { \mathcal { R } ^ { n } Z } _ { y } = 0 . \end{align*}


\begin{align*} F ( t , \rho , G , g ) : = & \langle \rho _ t , G _ t \rangle - \langle g , G _ 0 \rangle - \int _ 0 ^ t \langle \rho _ s , \partial _ s G _ s \rangle d s - \int _ 0 ^ t \langle \rho ^ 2 _ s , [ - ( - \Delta ) ^ { \gamma / 2 } G _ s ] \rangle d s ; \end{align*}


\begin{align*} \Delta _ B \xi + \sum _ { j = 1 } ^ d a _ j e ^ { ( \iota ^ \ast u ^ j , \xi ) } \iota ^ \ast u ^ j = w . \end{align*}


\begin{align*} T \log \left ( \frac { a ^ 2 + T ^ 2 } { 4 } \right ) = T \log \left ( \frac { T ^ 2 } { 4 } \left ( 1 + \frac { a ^ 2 } { T ^ 2 } \right ) \right ) \end{align*}


\begin{align*} \mathcal { K } _ { S } ( x , k ) = \chi _ + ( x ) \mathcal { K } _ + ( x , k ) + \chi _ - ( x ) \mathcal { K } _ - ( x , k ) \end{align*}


\begin{align*} \mathbb { E } \Big [ \sup _ { 1 \leq m \leq M } \Big \| \sum _ { i = 1 } ^ m E _ { k , N } ^ { m - i + 1 } P _ N \Delta W _ i \Big \| _ \mu ^ p \Big ] \leq C ( \| A ^ { \frac { \gamma - 2 } 2 } P Q ^ { \frac 1 2 } \| ^ p _ { \mathcal { L } _ 2 ( H ) } + \| ( I - P ) Q ^ { \frac 1 2 } \| ^ p _ { \mathcal { L } _ 2 ( H ) } ) < \infty . \end{align*}


\begin{align*} \int \left ( \partial _ { x _ j } Q \right ) f _ * = \int ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p ) Q ^ { p - 1 } f _ * = 0 , \forall 1 \le j \le N . \end{align*}


\begin{align*} J ^ { \mathcal { \psi } } ( \xi ) = \left ( \begin{array} { c c } 1 - r _ \eta r ^ * _ \eta & - r ^ * _ { \eta } \\ r _ { \eta } & 1 \end{array} \right ) . \end{align*}


\begin{align*} 2 p + r = n + 1 = p + q + 1 , r = q - p + 1 , \end{align*}


\begin{align*} \langle \varepsilon , \partial _ \tau \phi _ i \rangle _ { L ^ 2 _ \rho } = \sum _ { j = 0 } ^ { M } \varepsilon _ j \langle \phi _ j , \partial _ \tau \phi _ i \rangle _ { L ^ 2 _ \rho } + \langle \varepsilon _ { - } , \partial _ \tau \phi _ i \rangle _ { L ^ 2 _ \rho } \end{align*}


\begin{align*} v _ - : = \sup \big \{ v \in \mathbb { R } \colon \liminf _ { H \to \infty } \tilde { p } _ H ( v ) = 0 \big \} . \end{align*}


\begin{align*} \partial _ { ( X _ t , U _ t ) } H _ t ( X _ t , U _ t , y _ { t + 1 } ) = \partial _ { X _ t } H _ t ( X _ t , U _ t , y _ { t + 1 } ) \times \partial _ { U _ t } H _ t ( X _ t , U _ t , y _ { t + 1 } ) , \end{align*}


\begin{align*} - \partial _ r ^ 2 Q - \frac { N - 1 } { r } \partial _ r Q + Q - \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ { p - 1 } = 0 . \end{align*}


\begin{align*} \norm { M _ \omega T _ x f } _ { M ^ { p , q } } = \norm { T _ { ( x , \omega ) } V _ { g _ 0 } f } _ { L ^ { p , q } } \leq C \norm { V _ { g _ 0 } f } _ { L ^ { p , q } } = C \norm { f } _ { M ^ { p , q } } . \end{align*}


\begin{align*} \varphi _ R ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) = ( w _ k , \emptyset ) . \end{align*}


\begin{align*} \delta = \Im \omega _ 2 ( i \varepsilon ) = \frac { | \lambda | ^ 2 } { s ( | \lambda | , \varepsilon ) - \varepsilon } . \end{align*}


\begin{align*} Z _ { b ^ { - 1 } } \phi ( x , \omega ) & = \sqrt { \tfrac { \pi } { 2 } } \ , b \ , \vartheta _ 1 ' ( 0 , i b ) \ , e ^ { - \pi b x ^ 2 } \frac { \vartheta _ 3 ( \omega - i b x , i b ) } { \vartheta _ 4 ( \omega , i b ) \vartheta _ 4 ( x , i / b ) } \\ & = \sqrt { \tfrac { \pi } { 2 ^ { 3 / 2 } } } \ , b ^ { 3 / 4 } \ , \vartheta _ 1 ' ( 0 , i b ) \ , \frac { Z _ { b ^ { - 1 / 2 } } \ , \varphi ( x , \omega ) } { \vartheta _ 4 ( \omega , i b ) \vartheta _ 4 ( x , i / b ) } , \end{align*}


\begin{align*} \gamma _ i ( g ) = g - \frac { \left \langle \tilde a _ i , g \right \rangle - b _ i } { \left \langle \tilde a _ i , c \right \rangle } c . \end{align*}


\begin{align*} \frac { \lambda _ p + \lambda _ q } { 2 } + i \frac { \mu _ p + \mu _ q } { 2 } = \frac { \lambda _ p + i \mu _ p } { 2 } + \frac { \lambda _ q + i \mu _ q } { 2 } = 1 . \end{align*}


\begin{align*} W _ { t } ( w ) = w _ { t } \end{align*}


\begin{align*} K ( s , u , v ) = - \partial _ u \left ( \delta ( u + s ) \delta _ { \epsilon } ( \tfrac { u + v } { 2 } ) \right ) , \end{align*}


\begin{align*} \langle f , D ( c _ \gamma ) \rangle _ \mathcal { H } = \langle f , \sum _ { \gamma \in \Gamma } c _ \gamma e _ \gamma \rangle _ \mathcal { H } = \sum _ { \gamma \in \Gamma } \overline { c _ \gamma } \langle f , e _ \gamma \rangle _ \mathcal { H } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { | X | } | \mathcal { X } ( L ^ { ( i ) } ) | \end{align*}


\begin{align*} \left ( \frac { \partial } { \partial t } - \mathcal L \right ) \ , u _ { \mathbf { 0 } } ( t , x ) + \ , q _ { \mathbf { 0 } } ( x ) \ , u _ { \mathbf { 0 } } ( t , x ) = f _ { \mathbf { 0 } } ( t , x ) , u _ { \mathbf { 0 } } ( 0 , x ) = g _ { \mathbf { 0 } } ( x ) . \end{align*}


\begin{align*} \varphi _ R ( A , b , \lbrace x _ j : j \leq k \rbrace , \lbrace W _ j : j < k \rbrace , \lbrace \zeta _ j : j < k \rbrace ) = ( E _ { \pi ( k ) } , \emptyset ) . \end{align*}


\begin{align*} \lambda _ { \phi } ( t _ n ) = \ ! \int _ { \mathbb { R } ^ d } \ ! \left ( \frac 1 2 | \nabla \phi ( \cdot , t _ n ) | ^ 2 + V | \phi ( \cdot , t _ n ) | ^ 2 + \beta | \phi ( \cdot , t _ n ) | ^ 4 - \Omega \overline { \phi } ( \cdot , t _ n ) L _ z \phi ( \cdot , t _ n ) \ ! \right ) \ ! \mathrm { d } \mathbf { x } - \| \dot { \phi } ( \cdot , t _ n ) \| ^ 2 . \end{align*}


\begin{align*} S _ j = 2 \beta b H ^ { - 1 } \left [ \left ( \frac { \alpha } { 2 } - i + \tilde \lambda + \frac { 1 } { 2 } \Lambda \right ) \left ( S _ j + c _ { i , j } ( 2 \beta ) ^ { j + 1 } T _ { j + 1 } \right ) \right ] . \end{align*}


\begin{align*} \gamma _ * = \theta , \ ; \ ; s _ * = \frac r d , \ ; \ ; \mu _ * = \beta + \lambda ‐ r ‐ \frac d q + \frac { d } { p _ 1 } , \ ; \ ; \alpha _ * = \sigma ‐ \lambda + \frac d q ‐ \frac { d } { p _ 0 } . \end{align*}


\begin{align*} x _ t & = \frac { x ^ { - \tilde { \theta } } } { \tilde { \theta } } + \int _ 0 ^ t f ( s , \frac { y _ s ^ { - \tilde { \theta } } } { \tilde { \theta } } ) d s + \int _ 0 ^ t \tilde { \sigma } y _ s ^ { - ( \tilde { \theta } + 1 ) } d B _ s ^ H \\ & = \frac { x ^ { - \tilde { \theta } } } { \tilde { \theta } } + \int _ 0 ^ t f ( s , \frac { y _ s ^ { - \tilde { \theta } } } { \tilde { \theta } } ) d s + \int _ 0 ^ t \tilde { \sigma } \tilde { \theta } ^ { \theta } ( x _ s ) ^ { \theta } d B _ s ^ H , \end{align*}


\begin{align*} P _ n ( x ) = 2 ^ n E _ n \left ( { 1 + x ^ 2 \over 2 } , x \right ) . \end{align*}


\begin{align*} v ^ \alpha w ^ \beta x ^ n = v ^ \alpha w ^ { \beta - 1 } x ^ n + v ^ \alpha w ^ { \beta - 1 } ( v - 1 ) x ^ { n + 1 } = v ^ \alpha w ^ { \beta - 1 } x ^ n + v ^ { \alpha + 1 } w ^ { \beta - 1 } x ^ { n + 1 } - v ^ \alpha w ^ { \beta - 1 } x ^ { n + 1 } . \end{align*}


\begin{align*} \xi ( t ; r , x ) & = S _ { t - r } x + \int _ r ^ t S _ { t - u } \alpha ( u , \xi ( u ; r , x ) ) d u \\ & = S _ { t - s } S _ { s - r } x + S _ { t - s } \int _ r ^ s S _ { s - u } \alpha ( u , \xi ( u ; r , x ) ) d u + \int _ s ^ t S _ { t - u } \alpha ( u , \xi ( u ; r , x ) ) d u \\ & = S _ { t - s } \xi ( s ; r , x ) + \int _ s ^ t S _ { t - u } \alpha ( u , \xi ( u ; r , x ) ) d u , t \in [ s , \infty ) . \end{align*}


\begin{align*} \eta ( x + 1 ) - \overrightarrow { \eta } ^ { \varepsilon n } ( x + 1 ) = \displaystyle \frac { 1 } { \varepsilon n } \sum _ { r = 1 } ^ { \varepsilon n } [ \eta ( x + 1 ) - \eta ( x + 1 + r ) ] , \end{align*}


\begin{align*} M _ { d _ 0 } ( p , H ) = \frac { \pi ^ 2 } { 6 } \left \{ \prod _ { q \mid d _ 0 } \left ( 1 - \frac { 1 } { q ^ 2 } \right ) \right \} \left ( 1 + \frac { N _ { d _ 0 } ( p , H ) } { p } \right ) , \end{align*}


\begin{align*} \partial _ t \rho + \nabla \rho \cdot u = 0 . \end{align*}


\begin{align*} Q e _ j = \lambda _ j e _ j \end{align*}


\begin{align*} \begin{aligned} & | H _ k ( x , y ) - H _ k ( x ' , y ) | \\ & = \bigg | \int _ { \Bbb R ^ N } \big [ D _ k ( x , z ) - D _ k ( x ' , z ) \big ] S _ { k - M - 1 } ( z , y ) d \omega ( z ) \bigg | \\ & \leqslant C \int _ { \{ \| x - z \| \leqslant r ^ { 2 - k } \ \ \| x ' - z \| \leqslant r ^ { 2 - k } \} } \frac { r ^ k \| x - x ' \| } { V _ k ( x ) + V _ k ( z ) } ( V _ { k - M - 1 } ( y ) ) ^ { - 1 } d \omega ( z ) \\ & \leqslant C r ^ k \| x - x ' \| ( V _ { k - M - 1 } ( y ) ) ^ { - 1 } . \end{aligned} \end{align*}


\begin{align*} ( x , \omega , e ^ { 2 \pi i \tau } ) \circ ( x ' , \omega ' , e ^ { 2 \pi i \tau ' } ) = ( x + x ' , \omega + \omega ' , e ^ { 2 \pi i ( \tau + \tau ' ) } e ^ { \pi i ( x ' \cdot \omega - x \cdot \omega ' ) } ) . \end{align*}


\begin{align*} h _ { i j } \dot { M } ^ i _ k M ^ j _ l = h _ { i j } M ^ i _ k \dot { M } ^ j _ l \ , , \end{align*}


\begin{align*} \Big | \frac { 1 } { 2 m \ell n } \sum _ { x } \Phi _ { n } ( s , \tfrac { x } { n } ) \sum _ { j = 1 } ^ { m - 1 } \int _ { \Omega _ 2 ^ j ( x ) } \sum _ { z = x - \ell } ^ { x - 1 } [ \eta ( z ) - \eta ( z - j \ell ) ] \overrightarrow { \eta } ^ { \varepsilon n } ( x + 1 ) [ f ( \eta _ { 2 , x , j , z } ) - f ( \eta _ { 3 , x , j , z } ) ] d \nu _ { b } \Big | , \end{align*}


\begin{align*} \Psi _ { n } ( { \bf x } ) = \begin{bmatrix} \Psi ^ { 1 } _ { n } ( { \bf x } ) \\ \Psi ^ { 2 } _ { n } ( { \bf x } ) \\ \vdots \\ \Psi ^ { 2 d } _ { n } ( { \bf x } ) \end{bmatrix} \in \mathbb { C } ^ { 2 d } . \end{align*}


\begin{align*} u ( x ) & = C \int _ { \R ^ n \setminus B _ 1 } \bigg ( \frac { 1 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - 1 } \bigg ) ^ s \bigg ( \frac 1 { \vert x - y \vert ^ n } - \frac 1 { \vert x _ \ast - y \vert ^ n } \bigg ) u ( y ) \dd y \\ & \geqslant C x _ 1 \int _ { \R ^ n \setminus B _ 1 } \frac { y _ 1 u ( y ) } { \big ( \vert y \vert ^ 2 - 1 \big ) ^ s \vert y \vert ^ { n + 2 } } \dd y \end{align*}


\begin{align*} \begin{pmatrix} X _ 1 & 0 \\ 0 & - X _ 2 \end{pmatrix} \le \begin{pmatrix} Z & - Z \\ - Z & Z \end{pmatrix} + \alpha \begin{pmatrix} I & I \\ I & I \end{pmatrix} \end{align*}


\begin{align*} \mathcal H f ( x ) = \frac { 1 } { 2 \pi } P . V . \int _ { - \pi } ^ { \pi } f ( y ) \cot \left ( \frac { x - y } { 2 } \right ) \ , d y \end{align*}


\begin{align*} & f _ { k , j } ( x ) \equiv 0 , k + 1 < j , \\ & f _ { k , k + 1 } ( x ) \equiv 1 , k = \overline { 1 , n - 1 } , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \Phi _ x + i k \sigma _ 3 \Phi = Q \Phi , \\ & \Phi _ t + 4 i k ^ 3 \sigma _ 3 \Phi = V \Phi , \end{aligned} \right . \end{align*}


\begin{align*} \hat u _ i = \sum _ { ( i , j ) \in E } \hat l _ { i j } ( \hat z _ j - \hat z _ i ) , \end{align*}


\begin{align*} f ( x ) = x ^ n + a _ 1 x ^ { n - 1 } + a _ 2 x ^ { n - 2 } + \cdots + a _ n , \end{align*}


\begin{align*} ( - 1 ) ^ \ell \prod _ { a = 1 } ^ \ell i _ a \alpha ^ { i _ a } _ { r - ( j - i _ 1 - . . . - i _ a ) d - u _ a } \end{align*}


\begin{align*} | \mathcal { S } _ k | & = \binom { d } { 2 k } \cdot \binom { 2 k - 1 } { k - 1 } . \end{align*}


\begin{align*} I ( s , x ) : = \int _ 0 ^ \infty \mathcal { Q } ( x , k ) \phi ( k ) e ^ { i k ^ { 2 } s } h ( k ) \ , d k & = \int _ 0 ^ { \infty } \mathcal { Q } ( x , \sqrt { \lambda } ) \phi ( \sqrt { \lambda } ) e ^ { i s \lambda } h ( \sqrt { \lambda } ) \frac { 1 } { 2 \sqrt { \lambda } } \ , d \lambda . \end{align*}


\begin{align*} - \Delta _ g u ( x ) - \mu \frac { u ( x ) } { d ^ 2 _ g ( x _ 0 , x ) } + u ( x ) = \lambda \alpha ( x ) f ( u ( x ) ) , \ \ x \in M , \end{align*}


\begin{align*} \sum _ { r \in \mathbb { Z } } c ^ F _ { - r } \big ( \overline { \pi _ E ( w _ n ) } W , \sigma _ F ^ { ( l ) } ( w _ { n - 1 } ) W ' \big ) q _ F ^ { - \frac { r } { 2 } } X ^ { - l r } = r _ l \big ( \gamma \big ( X , \pi _ E , \widehat { \sigma _ E } , \psi _ E \big ) \big ) \sum _ { r \in \mathbb { Z } } c ^ F _ r \big ( W , W ' \big ) q _ F ^ { \frac { r } { 2 } } X ^ { l r } , \end{align*}


\begin{align*} m _ n ( t ) = \sum _ { k = 1 } ^ n \sum _ { i _ 1 + \dotsb + i _ k = n } ( i _ 1 + 1 ) \dotsm ( i _ 1 + \dotsb + i _ { k - 1 } + 1 ) h _ { i _ 1 } \dotsm h _ { i _ k } \frac { t ^ k } { k ! } . \end{align*}


\begin{align*} \sum _ { i \in \mathcal { I } _ { n } ^ { \sigma } } \underbrace { e ^ { \varphi _ { n } ^ { \sigma } ( i ) } \prod _ { \ell = 1 } ^ n \big ( \lambda _ i ^ { ( \sigma _ { \ell } ) } \big ) ^ { T _ { \ell } ^ { \sigma } - T _ { \ell - 1 } ^ \sigma } } _ { = : \ , p _ { \sigma _ n } ^ { \ast } ( i ) } = 1 , \end{align*}


\begin{align*} \omega ^ \perp = \check x ^ \perp = x ^ \perp - \check { \mathcal T } _ { t , x _ 1 , v _ 1 } v ^ \perp , \end{align*}


\begin{align*} \mathrm { i n d e x } ( f ) ^ 2 \cdot \mathrm { D i s c } ( K ) = \mathrm { d i s c } ( f ) > H ^ { n ^ 2 - n - 2 } . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d - 1 } } \psi \tilde \psi \ , d x & = 1 \\ \int _ { \mathbb { R } ^ { d - 1 } } \beta ( | a | ^ \frac { 1 } { p } \psi ) \tilde \psi \ , d x & = \sigma | a | ^ \frac { 1 } { p } . \end{align*}


\begin{align*} \phi _ { n , \lambda } ( x ) & = \frac { 1 } { e ^ { x } } \sum _ { k = 0 } ^ { \infty } \frac { ( k ) _ { n , \lambda } } { k ! } x ^ { k } = \frac { 1 } { e ^ { x } } \bigg ( x \frac { d } { d x } \bigg ) _ { n , \lambda } e ^ { x } , ( n \ge 1 ) . \end{align*}


\begin{align*} \begin{aligned} & \frac { | \tilde { x } - \tilde { y } | ^ 4 } { \varepsilon ^ 4 } + \alpha ( | \tilde { x } | ^ 2 + | \tilde { y } | ^ 2 ) \\ & \leq u ( \tilde { x } , \tilde { t } ) - v ( \tilde { y } , \tilde { t } ) - u ( x _ 1 , t _ 1 ) + v ( x _ 1 , t _ 1 ) + 2 \alpha | x _ 1 | ^ 2 + { \lambda \over T - t _ 1 } - { \lambda \over T - \tilde { t } } . \end{aligned} \end{align*}


\begin{align*} \Sigma = \bigcup _ { g \in \mathbb { Z } / d \mathbb { Z } } g ( \tilde { p } ) = \bigcup _ { g \in ( \mathbb { Z } / l \mathbb { Z } ) } g ( \tilde { p } ) , \end{align*}


\begin{align*} b \cdot \beta ' ( x ) = b \cdot \beta ' ( u . x ) = \beta ( u . x ) = \beta ( x ) \end{align*}


\begin{align*} L ( s , \mathbf { f } , \chi ) = \prod _ { p \nmid \mathfrak { n } } L _ p ( s , \mathbf { f } , \chi ) , \end{align*}


\begin{align*} \check H ^ q _ { c t } ( \Z ^ n ; A ) = \begin{cases} A \oplus A & n = 1 , q = 0 \\ A & n \not = 1 , q = 0 \vee q = n - 1 \\ 0 & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} & \forall n ^ 0 , x ^ X , y ^ X \exists z ^ X \preceq _ X L 1 _ X \exists w ^ X \preceq _ X L 2 ^ { \alpha _ n + 1 } 1 _ X \\ & \qquad \qquad \qquad \left ( y \in A ( \tilde { x } ^ L ) \rightarrow ( w \in A z \land \tilde { x } ^ L = _ X z + _ X \gamma _ n w ) \right ) \end{align*}


\begin{align*} \int _ M | \nabla \dot u | ^ 2 \ , d V _ g & = \int _ M V \dot u \ , r \ , d V _ g + \int _ M V \dot u ^ 2 \ , d V _ g . \end{align*}


\begin{align*} ( D _ A ) _ { i i } = \frac { \widehat { \Theta } _ { i i } ^ { - 1 } - \Theta _ { i i } ^ { - 1 } } { 1 + \delta ^ 2 ( { \Theta } _ { i i } ^ { - 1 } + \rho ) ( \widehat { \Theta } _ { i i } ^ { - 1 } + \rho ) + \delta ( { \Theta } _ { i i } ^ { - 1 } + \rho ) + \delta ( \widehat { \Theta } _ { i i } ^ { - 1 } + \rho ) } . \end{align*}


\begin{align*} \lim _ { n \to \infty } \Lambda ^ { ( 2 : 2 ) } _ { n , 2 } = \frac 1 { 2 } , \end{align*}


\begin{align*} u ( r , t ) = \frac { 1 } { \mu ( t ) } w ( y , \tau ) , y = \frac { r } { \sqrt { \mu ( t ) } } \frac { d \tau } { d t } = \frac { 1 } { \mu ( t ) } , \tau | _ { t = 0 } = \tau _ 0 , \end{align*}


\begin{align*} \lim _ { K \rightarrow + \infty } \mathbb { P } \Big ( \sup _ { i \neq j } \frac { | \beta ^ K _ i ( 0 ) - \beta ^ K _ j ( 0 ) | } { \rho ( i \ , \delta _ K , j \ , \delta _ K ) } > A \Big ) = 0 . \end{align*}


\begin{align*} g _ 0 g _ 1 = \phi ( h ^ + ) ^ { - 1 } \cdot g _ 0 g _ 2 \cdot h ^ + . \end{align*}


\begin{align*} ( \hat { n } ) _ { k + 1 , \lambda } & = ( \hat { n } - \lambda ) _ { k , \lambda } \hat { n } = a ^ { + } \big ( ( \hat { n } + 1 - \lambda ) \cdots ( \hat { n } + 1 - k \lambda ) \big ) a \\ & = a ^ { + } ( \hat { n } + 1 - \lambda ) _ { k , \lambda } a . \end{align*}


\begin{align*} \pi ^ { M p } ( \widehat { S } _ 1 \widehat { S } _ 2 ) = \pi ^ { M p } ( \widehat { S } _ 1 ) \pi ^ { M p } ( \widehat { S } _ 2 ) . \end{align*}


\begin{align*} B _ t + ( B \cdot \nabla ) J - ( J \cdot \nabla ) B + \mu \Lambda ^ \alpha B = 0 , \ \ \ \Lambda = ( - \Delta ) ^ { \frac 1 2 } \end{align*}


\begin{align*} \tilde { u } _ { t } - \tilde { u } _ { x x t } - \tilde { u } _ { x x } + \gamma \tilde { u } _ { x x x } & = 0 , \ \ x \in \R , \ t > 0 , \\ \tilde { u } ( x , 0 ) & = u _ { 0 } ( x ) , \ \ x \in \R . \end{align*}


\begin{align*} \check { S } ( \omega ) = \omega \circ S ^ { - 1 } . \end{align*}


\begin{align*} M _ { \lambda , \beta } ( n ) : = \int _ { \mathbb { R } } x ^ { n } \ , \mathrm { d } \pi _ { \lambda , \beta } ( x ) = \sum _ { m = 0 } ^ { n } \frac { m ! } { \Gamma ( m \beta + 1 ) } S ( n , m ) \lambda ^ { m } , \end{align*}


\begin{align*} \Phi & : = \eta _ { \ell } \ast \varphi , \\ \tilde { \Phi } & : = \eta _ { \ell } \ast \tilde { \varphi } . \end{align*}


\begin{align*} \eta ^ { \prime \prime \prime } ( x ) + \lambda \xi ^ \prime ( x ) - \lambda \eta ^ \prime ( x ) = 0 , \ \ \forall x \in ( 0 , 1 ) . \end{align*}


\begin{align*} \sum \limits _ { \ell = 0 } ^ { 2 } \frac { d } { d t } \langle \nabla ^ { \ell } u ^ { \epsilon } , \epsilon \nabla ^ { \ell + 1 } \phi ^ { \epsilon } \rangle + \frac { 3 } { 4 } \| \nabla \phi ^ { \epsilon } \| _ { H ^ 2 _ { \frac { P ' ( \rho ^ { \epsilon } ) } { \rho ^ { \epsilon } } } } ^ 2 \lesssim D ( u ^ { \epsilon } , \eta ^ { \epsilon } , \tau ^ { \epsilon } ) + \| \mathrm { d i v } u ^ { \epsilon } \| _ { H ^ 2 } ^ 2 . \end{align*}


\begin{align*} \frac { 1 } { ( a + X ) ( b + X ) } = \frac { 1 } { b - a } \Bigl ( \frac { 1 } { a + X } - \frac { 1 } { b + X } \Bigr ) , \end{align*}


\begin{align*} v _ t + a _ n ( t , x ) \cdot v _ x ~ = ~ 0 , v ( \tau _ 0 , \cdot ) ~ = ~ \bar { v } ~ \in ~ H ^ 2 ( \R \backslash I ^ { \tau } _ { t } ) , \end{align*}


\begin{align*} | \mathcal { F } ( a + b , - b \ , ; \ , \{ U \} , \{ U \} ) | = \sum _ { i = 0 } ^ { a } \binom { a + b - 2 } { \lfloor i / 2 \rfloor } \binom { a + b - 1 - \lfloor i / 2 \rfloor } { a - i } . \end{align*}


\begin{align*} \gamma _ 1 ( x ^ 2 ) - \gamma _ 2 ( x ) = \omega ( x ) ^ 2 \ ; \ ; \forall x \in A \end{align*}


\begin{align*} F _ r * _ { \mathbf { H } _ r } G _ r = ( F \natural G ) _ r . \end{align*}


\begin{align*} x = \sum _ { i = 0 } ^ { 3 j + 2 } \alpha _ { i } e _ { i } , \quad \alpha _ { 3 j + 2 } \neq 0 \quad ~ \langle x \rangle _ { T ^ { * 3 } } = j + 1 . \end{align*}


\begin{align*} \left ( \frac { R } { b _ { i + 1 } } \right ) ^ { \frac { 3 } { 2 } } \exp \left ( \frac { 2 } { \sqrt { R } } \sqrt { b _ { i + 1 } } \right ) \left ( \frac { b _ { i } } { R } \right ) ^ { \frac { 3 } { 2 } } \exp \left ( \frac { - 2 } { \sqrt { R } } \sqrt { b _ { i } } \right ) = \left ( \frac { b _ i } { b _ { i + 1 } } \right ) ^ { 3 / 2 } \exp \left ( \frac { 2 } { \sqrt { R } } ( \sqrt { b _ { i + 1 } } - \sqrt { b _ { i } } ) \right ) \end{align*}


\begin{align*} \sum _ { \substack { 0 \leq b < p \\ \gcd ( m , p - 1 ) = 1 } } e ^ { \frac { i 2 \pi b ( \tau ^ { e m } - v ) } { p } } = 0 \end{align*}


\begin{align*} \mathbf { f } | [ K _ 1 ( \mathfrak { n } ) \xi K _ 1 ( \mathfrak { n } ) ] = \lambda _ { \mathbf { f } } ( \xi ) \mathbf { f } \xi \in \mathfrak { X } . \end{align*}


\begin{align*} T = ( 1 - \alpha ) I d + \alpha N \end{align*}


\begin{align*} \nu _ { \alpha , 1 } ( d u ) = c _ { \alpha } \frac { d u } { | u | ^ { \alpha + 1 } } , \end{align*}


\begin{align*} \alpha ( \alpha \beta ) \gamma \delta \varepsilon ( \zeta \zeta \eta ) = \alpha ( \beta \alpha ) \gamma \delta \varepsilon ( \zeta \eta \zeta ) \neq \alpha ( \beta \alpha ) \gamma \varepsilon \delta ( \zeta \eta \zeta ) , \alpha , \ldots , \eta \in [ d ] \end{align*}


\begin{align*} N _ 6 ( f , H ) = - f + S ( H , f ) - 2 S ( H _ 2 , 2 f ) - \frac { 3 } { 2 } S ( H _ 3 , 3 f ) + 3 S ( H _ 6 , 6 f ) = \begin{cases} - 2 a - 1 & \hbox { i f $ a \equiv 0 \pmod 6 $ , } \\ - 3 & \hbox { i f $ a \equiv 2 , 3 \pmod 6 $ , } \\ 2 a + 1 & \hbox { i f $ a \equiv 5 \pmod 6 $ . } \end{cases} \end{align*}


\begin{align*} \int _ M V ^ \nu _ { 2 , p } \ d \nu = \left \| G _ \lambda \nu \right \| _ { L ^ q ( M ) } ^ q . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ M | \varepsilon _ j | + \| \varepsilon _ - \| _ { L ^ 2 _ \rho } \le 1 . \end{align*}


\begin{align*} p ^ 2 _ k - q ^ 2 _ k = a _ k \Longrightarrow 4 p _ k ^ 4 - 4 a _ k p _ k ^ 2 - b ^ 2 _ k = 0 , \end{align*}


\begin{align*} \begin{aligned} & \int _ { t } ^ { T _ { x , \xi } } C _ p q \left | \dot { \xi } ( s ) \right | ^ q d s = - C _ p ( q - 1 ) p ^ q \left ( f ( \xi ( t ) ) - u ( \xi ( t ) ) \right ) + f ( \xi ( t ) ) \\ \Longrightarrow & \int _ { t } ^ { T _ { x , \xi } } p \left ( f ( \xi ( s ) ) - u ( \xi ( s ) ) \right ) d s = u ( \xi ( t ) ) . \end{aligned} \end{align*}


\begin{align*} \Gamma _ k = \left \{ \rho \colon \frac { \pi ( k - 1 ) } { n } < \arg \rho < \frac { \pi k } { n } \right \} , k = \overline { 1 , 2 n } . \end{align*}


\begin{align*} c _ { F _ k } = 2 m n + 1 1 n - 3 . \end{align*}


\begin{align*} \frac { c \left ( 1 + o ( 1 ) \right ) } { n ^ { h + h ' } \delta ^ 4 _ n ( k ) } \cdot \sum _ { \lambda = 0 } ^ { k - 1 } | \mathcal { U } _ k ( \lambda ) | \cdot | \mathcal { I } _ { h } \times \mathcal { I } _ { h ' } | . \end{align*}


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


\begin{align*} \langle x ^ X , y ^ X \rangle _ X : = _ 1 \frac { 1 } { 4 } \left ( \norm { x + _ X y } ^ 2 _ X - \norm { x - _ X y } ^ 2 _ X \right ) . \end{align*}


\begin{align*} \lim _ { \alpha \rightarrow 0 ^ + } \delta ( \alpha ) = 0 . \end{align*}


\begin{align*} & g _ 1 ( t ) ^ { ( - 1 ) } ( - 1 ) ^ { s _ r } ( t - \theta ) ^ { w - s _ r } \Omega ^ w \alpha _ r \mathcal { L i } _ { K , ( s _ 1 , \ldots , s _ { r - 1 } ) } ( \alpha _ 1 , \ldots , \alpha _ { r - 1 } ) + g _ 1 ( t ) ^ { ( - 1 ) } ( t - \theta ) ^ w \Omega ^ w \mathcal { L i } _ { K , \mathfrak { s } } ( { \boldsymbol \alpha } ) \\ & + g _ 2 ( t ) ^ { ( - 1 ) } = 0 . \end{align*}


\begin{align*} \eta \cdot f : = \sum _ { \alpha \in \mathbb { N } ^ n } \eta _ { \alpha } \frac { f ^ { ( \alpha ) } ( z _ 0 ) } { \alpha ! } . \end{align*}


\begin{align*} & \mathcal { K } _ { S , \# } '' ( x , k ) : = \chi _ 0 ( x ) T ( 0 ) \partial _ x m _ { + } ( x , 0 ) e ^ { i k x } + \partial _ x \chi _ + ( x ) \mathcal { K } ' _ { + , \# } ( x , k ) + \partial _ x \chi _ - ( x ) \mathcal { K } ' _ { - , \# } ( x , k ) . \end{align*}


\begin{align*} ( a , b , c , d ) = \left ( \frac { p r + 3 q s } { 8 } , \frac { 9 r s - p q } { 8 } , \frac { p q - r s } { 8 } , \frac { 3 p r + q s } { 8 } \right ) \end{align*}


\begin{align*} \mathcal { M } \mathcal { E } [ u ] = 1 . \end{align*}


\begin{align*} \sum _ i a _ i c \otimes b _ i d = 0 , \forall u \in I , \forall c \in A _ u ^ s , \forall d \in B ^ u _ w . \end{align*}


\begin{align*} \left ( \begin{array} { c } u _ 1 ( - t + \bar { T } ) \\ u _ 2 ( - t + \bar { T } ) \\ u _ 3 ( - t + \bar { T } ) \end{array} \right ) = M ( 2 p - 1 ) \left ( \begin{array} { c } u _ 1 ( t + \bar { T } ) \\ u _ 2 ( t + \bar { T } ) \\ u _ 3 ( t + \bar { T } ) \end{array} \right ) . \end{align*}


\begin{align*} d \theta ^ c = \omega + \theta \wedge \theta ^ c . \end{align*}


\begin{align*} J _ t = & J - \frac \partial { \partial x } \otimes d ^ c \left ( \frac 1 2 t ^ 2 \left ( \frac { 1 - \left | w \right | ^ 2 } { 1 + \left | w \right | ^ 2 } \right ) ^ 2 \right ) - \frac \partial { \partial y } \otimes d ^ c \left ( \frac 1 2 t ^ 2 \left ( \frac { 1 - \left | w \right | ^ 2 } { 1 + \left | w \right | ^ 2 } \right ) ^ 2 \right ) \circ J . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \frac { a _ { n + 1 } } { n ! } X ^ n & = \alpha ' = \exp ( X ) \exp ( \exp ( X ) - 1 ) \\ & = \Bigl ( \sum _ { k = 0 } ^ \infty \frac { 1 } { k ! } X ^ k \Bigr ) \Bigl ( \sum _ { k = 0 } ^ \infty \frac { a _ k } { k ! } X ^ k \Bigr ) = \sum _ { n = 0 } ^ \infty \sum _ { k = 0 } ^ n \frac { a _ k } { k ! ( n - k ) ! } X ^ n . \end{align*}


\begin{align*} B ( \varepsilon { x _ k } , { \delta _ 0 } { a _ \varepsilon } ) \subset T _ { \varepsilon , k } = \varepsilon { x _ k } + a _ \varepsilon T _ { 0 } \subset \subset B ( \varepsilon { x _ k } , { \delta _ 1 } { a _ \varepsilon } ) \subset \subset B ( \varepsilon { x _ k } , { \delta _ 2 } { a _ \varepsilon } ) \subset B ( \varepsilon { x _ k } , \delta _ 3 \varepsilon ) \subset \varepsilon { Q _ k } , \end{align*}


\begin{align*} r _ l \big ( \epsilon ( X , \widetilde { \chi } _ 0 \pi _ E , \psi _ E ) \big ) = \epsilon \big ( X , \chi _ 0 r _ l ( \pi _ F ) , \overline { \psi } _ F \big ) ^ l = \epsilon ( X ^ l , \chi r _ l ( \pi _ F ) ^ { ( l ) } , \overline { \psi } _ F ^ l \big ) \end{align*}


\begin{align*} h ^ { - \frac { 1 } { 2 } + \alpha } k ^ { - \frac { 1 } { 2 } + \beta - \alpha ^ * - \beta ^ * } \mathcal { G } ( 2 - \alpha - \beta + \alpha ^ * + \beta ^ * , \alpha , \beta - \alpha ^ * - \beta ^ * ; A , B _ { - \alpha ^ * - \beta ^ * } , h , k ) \\ = \mathcal { K } ( 0 , 0 , 2 - \alpha - \beta + \alpha ^ * + \beta ^ * ; A , B _ { - \alpha ^ * - \beta ^ * } , \alpha , \beta - \alpha ^ * - \beta ^ * , h , k ) . \end{align*}


\begin{align*} \int _ { t - d } ^ { t + d } \int _ { \lambda > 1 } e ^ { i s \lambda } \hat { g } ( \lambda ) d \lambda d s & = \int _ { \lambda > 1 } \hat { g } ( \lambda ) \int _ { t - d } ^ { t + d } e ^ { i s \lambda } d s d \lambda \\ & = - i \int _ { \lambda > 1 } \hat { g } ( \lambda ) \left ( \frac { e ^ { i ( t + d ) \lambda } } { \lambda } - \frac { e ^ { i ( t - d ) \lambda } } { \lambda } \right ) d \lambda . \end{align*}


\begin{align*} \lim _ { T \to \infty } \ , \sup _ { \tau \in [ 0 , 1 ] } S \left ( \omega _ { V ( 0 ) } \circ \alpha _ T ^ { 0 \to \tau } \bigg | \omega _ { V ( \tau ) } \right ) = 0 . \end{align*}


\begin{align*} \mu ^ { \# } ( k , \ell , m , n ) : = \int \overline { \mathcal { K } ^ { \# } ( x , k ) } \mathcal { K } ^ { \# } ( x , \ell ) \overline { \mathcal { K } ^ { \# } ( x , m ) } \mathcal { K } ^ { \# } ( x , n ) \ , d x . \end{align*}


\begin{align*} L \psi _ { \eta } = \psi _ { \eta } ( 0 ) = 0 . \end{align*}


\begin{align*} \tfrac { \partial } { \partial \eta } A ( \tfrac { \eta } { 3 } , \tfrac { 1 } { \eta } ) \Bigr | _ { \substack { \eta = 3 . 9 } } > 0 \tfrac { \partial } { \partial \eta } B ( \tfrac { \eta } { 3 } , \tfrac { 1 } { \eta } ) \Bigr | _ { \substack { \eta = 3 . 9 } } > 0 , \end{align*}


\begin{align*} \lim _ { \varepsilon \rightarrow 0 ^ { + } } | \langle \pi _ s , \overleftarrow { i _ { \varepsilon } ^ { u } } \rangle - \rho ( s , u ) | = \lim _ { \varepsilon \rightarrow 0 ^ { + } } | \langle \pi _ s , \overrightarrow { i _ { \varepsilon } ^ { u } } \rangle - \rho ( s , u ) | = 0 , \end{align*}


\begin{align*} \Lambda _ m = \{ n \in \Gamma _ { 2 , y , ( r , s ) } ^ { \delta } : A _ n ^ \delta \subseteq A _ m ^ \delta \} \end{align*}


\begin{align*} \norm { A \oplus B } _ { p } = ( \norm { A } _ { p } ^ { p } + \norm { B } _ { p } ^ { p } ) ^ { \frac { 1 } { p } } , \end{align*}


\begin{align*} x ^ \prime = z ( j ^ 2 z , j z ) \ , , y ^ \prime = y \ , , z ^ \prime = x ( j x , j ^ 2 x ) \ , , w ^ \prime = w \ , . \end{align*}


\begin{align*} n a + ( n - 2 ) b = 0 \end{align*}


\begin{align*} V _ s = V _ s ( { r _ v - 1 } ) \oplus V _ s ( { r _ v } ) . \end{align*}


\begin{align*} A ^ { \pm h } _ { a , b } = c ^ { - \frac { 1 } { \rho } } \left [ ( a \vee 0 ) ^ \frac { 1 } { \rho } \mp h , \ , ( b \vee 0 ) ^ \frac { 1 } { \rho } \pm h \right ] , \end{align*}


\begin{align*} \partial _ y ( \mathcal { L } \mathfrak { B } _ { i j k } ) = G _ { i j k } - b _ { i j k } \sigma _ i \Lambda Q . \end{align*}


\begin{align*} Q ( y ) = \operatorname { s g n } ( y ) \min \left ( \lfloor | y | / 2 ^ r \rfloor + 1 , 2 ^ { w - 1 } \right ) . \end{align*}


\begin{align*} ( g - 1 ) \left ( \frac { c _ x } { \pi _ 1 } - \frac { a ^ 2 } { 2 } \right ) = - \kappa ( x ) ( g ) . \end{align*}


\begin{align*} \beta _ v ( \rho ( t ) ) = a _ { v , \xi } t + b _ { v , x , \xi } \end{align*}


\begin{align*} F _ h ( v ) = \sum _ { E \in { \cal T } _ h } \sum _ { \iota \in I ^ E } [ f v ] ( \xi ^ E _ \iota ) \ , \omega ^ E _ \iota \ , . \end{align*}


\begin{align*} \left | \frac { 1 } { s - \rho } - \frac { 1 } { 2 + i t - \rho } \right | = ( 2 - \sigma ) \left | \frac { 1 } { ( s - \rho ) ( 2 + i t - \rho ) } \right | \leq \frac { 3 } { | \gamma - t | ^ 2 } \end{align*}


\begin{align*} X ( \cdot ) = \lim _ { n \rightarrow \infty } X ^ { n } ( \cdot ) \ ; \ ; C _ W ( [ 0 , T ] ; H ^ { \frac d 3 } ) , d = 1 , 2 , 3 . \end{align*}


\begin{align*} J _ \psi : = { \rm i n d } _ { N _ 2 ( K ) } ^ { P _ 2 ( K ) } ( \psi ) , \end{align*}


\begin{align*} t _ 0 = t _ 0 ( x ) = \exp \Big ( \sqrt { \frac { \log x } R } \Big ) , \ T = t _ 0 ^ c , \ \ \sigma _ 2 = 1 - \frac { 2 } { R \log t _ 0 } . \end{align*}


\begin{align*} \mathcal { R } ^ { 2 } ( \eta , \xi ) = \Lambda ( ( \eta ^ { r _ 0 } \circ \xi ) ^ { r _ 1 } \circ \eta , \eta ^ { r _ 0 } \circ \xi ) \end{align*}


\begin{align*} \check H _ { c t } ^ 0 ( X ; A _ X ) = A _ X ( X ) = \begin{cases} A ^ { e ( X ) } & e ( X ) < \infty \\ \bigoplus _ { \N } A & e ( X ) = \infty . \end{cases} \end{align*}


\begin{align*} d _ { p + q } ( \phi \vee \psi ) = d _ p \phi \vee \psi + ( - 1 ) ^ p \phi \vee d _ q \psi \end{align*}


\begin{align*} \Phi ^ { \pm \infty } ( x , t , k ) = \Delta _ { R , L } ( k ) e ^ { - i ( X _ { R , L } ( k ) x + \Omega _ { R , L } ( k ) t ) \sigma _ 3 } , \Phi ^ { \pm \infty } : \mathbb { R } \times [ 0 , + \infty ) \times ( \mathbb { C } \backslash [ - C , C ] ) \rightarrow \mathbb { C } , \end{align*}


\begin{align*} \sum _ { y \leq q , \ , g ( y ) = p } \mu _ { Q } ( y , q ) = \begin{cases} \mu _ { Q } ( y , q ) , & y \in Q \ a n d \ p = g ( y ) , \\ 0 , & o t h e r w i s e . \end{cases} \end{align*}


\begin{align*} \ll Q \sum _ { 1 \leq c \leq C } \frac { 1 } { c } \sum _ { \substack { 1 \leq m , n \ll X \\ m h = n k } } \frac { ( m n ) ^ \varepsilon } { \sqrt { m n } } . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l l } D ^ + \psi = \Phi & \Omega \\ P ^ \pm \psi _ { | \Sigma } = P ^ \pm \Psi & \Sigma \end{array} \right . \end{align*}


\begin{align*} F _ { \mu \boxplus \nu } ( z ) = F _ \mu ( \omega _ 1 ( z ) ) = F _ \nu ( \omega _ 2 ( z ) ) . \end{align*}


\begin{align*} X h = g _ 1 \cdot \ldots \cdot g _ { n + 1 } \ , . \end{align*}


\begin{align*} \sqrt { f \left ( \eta ^ { x , x + r } \right ) } - \sqrt { f \left ( \eta \right ) } = \sum _ { k = 1 } ^ { 6 } [ \sqrt { f \left ( \eta _ { k } \right ) } - \sqrt { f \left ( \eta _ { k - 1 } \right ) } ] . \end{align*}


\begin{align*} R _ 2 ^ { \infty } = \mathbb { E } \left [ \log _ 2 \left ( 1 + \frac { \alpha _ 2 } { \alpha _ 1 } + \frac { P _ { t h } | h _ { 1 2 } | ^ 2 } { \sigma ^ 2 } \right ) \right ] , \end{align*}


\begin{align*} \int _ { \R ^ n _ + } \frac { \dd z } { \vert e _ 1 + z \vert ^ { n + 2 s } } & = \int _ 0 ^ \infty \int _ { \R ^ { n - 1 } } \frac { \dd z _ 1 \dd z ' } { \big ( ( z _ 1 + 1 ) ^ 2 + \vert z ' \vert ^ 2 \big ) ^ { \frac { n + 2 s } 2 } } \end{align*}


\begin{align*} s ( | \lambda | , t ) = W ( t ) = \Im \omega _ 1 ( i t ) . \end{align*}


\begin{align*} Q ( y ) = \frac { \kappa _ 0 } { y ^ 2 } + O \bigg ( \frac { 1 } { y ^ 4 } \bigg ) , \ ; \ ; y \rightarrow + \infty . \end{align*}


\begin{align*} h _ n ^ { ( 2 m ) } \sim - \frac { m ( m - 1 ) } { 3 } \cdot ( 2 \alpha ^ 2 + 1 ) \cdot n ^ { - 2 } ( n \to \infty ) . \end{align*}


\begin{align*} \lim \limits _ { x \rightarrow \infty } \psi ' ( x ) = 1 - 1 + 0 = 0 . \end{align*}


\begin{align*} C : = \int _ Z \frac { | f ( y ) - f ( x _ 0 ) | ^ p } { \nu ( B ( y , d ( x _ 0 , y ) ) ) d ( x _ 0 , y ) ^ { \theta p } } d \nu ( y ) < \infty . \end{align*}


\begin{align*} J _ { n , m } = J \times J _ { n , m } ^ { 2 } \cdots \times J _ { n , m } ^ d , \end{align*}


\begin{align*} \Gamma ( \pi ) & = \lim _ { t \to \infty } \frac { 1 } { N } \sum _ { \ell = 1 } ^ { N } \frac { \sum _ { i = 1 } ^ { R _ \ell ^ \pi ( t ) } \rho _ \ell ( \frac { ( T _ { \ell i } ^ \pi ) ^ 2 } { 2 } + T _ { \ell i } ^ \pi Z _ { \ell i } ^ \pi ) + c _ \ell \tilde { R } _ \ell ^ \pi ( t ) } { t } . \end{align*}


\begin{align*} 2 e ( S _ 1 ) \alpha + 2 e ( L _ 1 ) \leq 2 e ( N _ 1 ) \alpha + 2 \binom { | L | } { 2 } \leq ( 2 k - 1 ) n \alpha + | L | ( | L | - 1 ) < 2 k n \alpha , \end{align*}


\begin{align*} - \Delta w = \tilde { \lambda } _ 0 \tilde { m } _ 0 w \R ^ N \ , . \end{align*}


\begin{align*} ( \overline { R } ( e ) ) ( x ^ * ) = R ( e \otimes x ^ * \vert _ { Q ( X ) } ) \ , \ , e \in E x ^ * \in X ^ * . \end{align*}


\begin{align*} \begin{array} { l l } \displaystyle \sum _ { i = 1 } ^ { m } \overline { z _ i } a _ { i j } = \eta , \ ; \ ; \ ; j = 1 , 2 , . . . , n \\ \displaystyle \sum _ { i = 1 } ^ { m } \overline { z _ i } = 1 \\ \lvert a r g \overline { z _ i } \rvert \leq \alpha , \ ; \ ; \ ; i = 1 , 2 , . . . , m \end{array} \end{align*}


\begin{align*} x : = n \epsilon . \end{align*}


\begin{align*} \Psi ^ n _ { ( \Sigma , U ) } ( f ) [ \psi ] : = \int _ { \mathcal { I } } f ( s ) \psi ^ n ( s ) \gamma ' ( s ) ^ { \tfrac { n } { 2 } } \mathrm { d } s \end{align*}


\begin{align*} \int _ \sigma ^ t ( t - s ) ^ { \alpha - 1 } ( s - \sigma ) ^ { - \alpha } \ , \dd s = \frac { \pi } { \sin \pi \alpha } , \ ; 0 \leq \sigma \leq t , \ ; 0 < \alpha < 1 . \end{align*}


\begin{align*} \partial _ t X _ t \cdot \nu ( X _ t ) & = \frac { - \dot { x } ( t ) \cdot \nabla u ( t , x ( t ) ) } { \sqrt { 1 + | \nabla u ( t , x ( t ) ) | ^ 2 } } + \frac { \dot { x } ( t ) \cdot \nabla u ( t , x ( t ) ) } { \sqrt { 1 + | \nabla u ( t , x ( t ) ) | ^ 2 } } + \frac { \partial _ t u ( t , x ( t ) ) } { \sqrt { 1 + | \nabla u ( t , x ( t ) ) | ^ 2 } } \\ & = \frac { \partial _ t u ( t , x ( t ) ) } { \sqrt { 1 + | \nabla u ( t , x ( t ) ) | ^ 2 } } . \end{align*}


\begin{align*} G _ k = \Bigl \{ \rho \in \mathbb C \colon \arg \rho \in \left ( ( - 1 ) ^ { n - k } - 1 ) \tfrac { \pi } { 2 n } , ( ( - 1 ) ^ { n - k } + 3 ) \tfrac { \pi } { 2 n } \right ) \Bigr \} , \end{align*}


\begin{align*} u ( x ) & = C \int _ { \R ^ n \setminus B _ 1 } \bigg ( \frac { 1 - \vert x \vert ^ 2 } { \vert y \vert ^ 2 - 1 } \bigg ) ^ s \bigg ( \frac 1 { \vert x - y \vert ^ n } - \frac 1 { \vert x _ \ast - y \vert ^ n } \bigg ) u ( y ) \dd y \\ & \leqslant C x _ 1 \int _ { \R ^ n \setminus B _ 1 } \frac { y _ 1 u ( y ) } { \big ( \vert y \vert ^ 2 - 1 \big ) ^ s \vert y \vert ^ { n + 2 } } \dd y . \end{align*}


\begin{align*} b ( s ) = \begin{cases} 0 & s \leq \overline \rho - \alpha _ 1 \\ - \log ( \overline \rho - s ) & \overline \rho - \alpha _ 2 \leq s < \overline { \rho } \end{cases} \end{align*}


\begin{align*} \delta ( t ) \simeq \| h ( t ) \| _ { H ^ 1 } \simeq | \alpha ( t ) | = \Big | \int _ t ^ \infty \dot { \alpha } ( s ) d s \Big | \lesssim \int _ t ^ \infty \delta ( s ) d s \le C e ^ { - c t } , \ ; \forall t \ge 0 , \end{align*}


\begin{align*} \langle a , b \rangle = \varphi ( a ^ * b ) . \end{align*}


\begin{align*} Y ^ { ( \ell + 1 ) } [ \partial _ \mu z , \partial _ \mu z ; f , g ] = Y ^ { = ( \ell + 1 ) } [ \partial _ \mu z , \partial _ \mu z ; f , g ] + Y ^ { < ( \ell + 1 ) } [ \partial _ \mu z , \partial _ \mu z ; f , g ] , \end{align*}


\begin{align*} D _ { k n } \le { \log 2 \over \pi ( K + 1 ) } + { 1 \over \pi | \bar { u } _ { k n } | } \sum _ { k = 1 } ^ K { 1 \over k } \min \left ( | \bar { u } _ { k n } | , { 1 \over \{ k \rho _ * \} } \right ) , \end{align*}


\begin{align*} \begin{aligned} d X _ t & = \mu ( X _ t ) \ , d t + \sigma ( X _ t ) \ , d W _ t , t \in [ 0 , \infty ) , \\ X _ 0 & = x _ 0 , \end{aligned} \end{align*}


\begin{align*} f ( x ) = \int _ 0 ^ \infty \psi _ { t } \ast q _ { t } \ast f ( x ) \frac { d t } { t } , \end{align*}


\begin{align*} \partial _ { \alpha \beta \gamma } g = \nabla _ \lambda g S ^ \lambda _ { \alpha \beta \gamma } + \nabla _ { \mu \nu } g S ^ { \mu \nu } _ { \alpha \beta \gamma } + \nabla _ { \lambda \mu \nu } g S ^ { \lambda \mu \nu } _ { \alpha \beta \gamma } \end{align*}


\begin{align*} \begin{cases} \ \\ [ 5 p t ] \ \end{cases} \in \ \R _ { \ell } . \end{align*}


\begin{align*} \lim _ { s \to \overline \rho _ - } \frac { p ( s ) } { ( b ( s ) ) ^ \gamma } = \lim _ { s \to \overline \rho _ - } \frac { P ( s ) } { ( b ( s ) ) ^ \gamma } = \lim _ { s \to \overline \rho _ - } \frac { p ( s ) } { ( b ' ( s ) ) ^ \beta } = \lim _ { s \to \overline \rho _ - } \frac { P ( s ) } { ( b ( s ) ) ^ { \beta - 1 } } = + \infty . \end{align*}


\begin{align*} \mu _ { R , 1 } ^ { ( 1 ) } ( k , \ell , m , n ) & : = \int \overline { \mathcal { K } } _ { R } ( x , k ) \mathcal { K } _ { M _ { 1 } } ( x , \ell ) \overline { \mathcal { K } _ { M _ { 2 } } ( x , n ) } \mathcal { K } _ { M _ { 3 } } ( x , m ) \ , d x , \\ \mu _ { R , 1 } ^ { ( 2 ) } ( k , \ell , m , n ) & : = \int \overline { \mathcal { K } } _ { S } ( x , k ) \mathcal { K } _ { M _ { 1 } } ( x , \ell ) \overline { \mathcal { K } _ { M _ { 2 } } ( x , n ) } \mathcal { K } _ { R } ( x , m ) \ , d x , M _ { i } \in \{ S , R \} . \end{align*}


\begin{align*} \sigma _ 1 ^ 2 \norm { f } ^ 2 \leq \norm { C _ \mathbf { F } f } ^ 2 = \langle S _ \mathbf { F } f , f \rangle = \sum _ { k = 1 } ^ K | \langle f , v _ k \rangle | ^ 2 \leq \sigma _ d ^ 2 \norm { f } ^ 2 . \end{align*}


\begin{align*} \int ^ { T } _ 0 \int _ { \Omega } ( \rho \partial _ t \psi + \rho u \cdot \nabla _ x \psi ) \ , d x d t + \int _ { \Omega } \rho _ 0 \psi ( 0 , x ) \ , d x = 0 ; \end{align*}


\begin{align*} I _ 1 & = ( - \Delta ) ^ s \bar u ( x ) - \frac 1 { 2 s } c _ { n , s } n \vert B _ 1 \vert r ^ { - 2 s } u ( x ) - c _ { n , s } \int _ B \frac { u ( x ) - u ( y ) } { \vert x _ \ast - y \vert ^ { n + 2 s } } \dd y \end{align*}


\begin{align*} e ^ { - t } C _ p q \left | \dot { \xi } ( t ) \right | ^ { q - 2 } \dot { \xi } ( t ) = \int _ 0 ^ t e ^ { - s } D f \left ( \xi ( s ) \right ) d s + \tilde { c } \end{align*}


\begin{align*} M _ { \alpha } : = \sum _ { 0 < i _ 2 < \cdots < i _ n } x _ 0 ^ { \alpha _ 1 } x _ { i _ 2 } ^ { \alpha _ 2 } \cdots x _ { i _ k } ^ { \alpha _ k } F _ { \alpha } : = \sum _ { { 0 \leq i _ 1 \leq i _ 2 \leq \cdots \leq i _ n } \atop { j \in D ( \alpha ) \Rightarrow i _ j < i _ { j + 1 } } } x _ { i _ 1 } x _ { i _ 2 } \cdots x _ { i _ n } , \end{align*}


\begin{align*} \mathcal { D } _ p : = \left \lbrace f \in \mathcal { D } ( \mathcal { L } ) \cap L ^ p ( M ) : \mathcal { L } f \in L ^ p ( M ) \right \rbrace . \end{align*}


\begin{align*} & \alpha _ i = \sum _ { j = i } ^ k \theta _ j , \forall i = 1 , \ldots , k , \\ & \alpha _ { n + 1 - i } = - \alpha _ i , \forall i = 1 , \ldots , k . \end{align*}


\begin{align*} S ( S ( a ) ^ * ) ^ * = a , \forall a \in A , \end{align*}


\begin{align*} B [ \mathfrak { h } _ \omega ] & = p [ \mathfrak { c } _ \omega ] \in H C _ \lambda ^ { p } ( \mathfrak { A } ) \\ S [ \mathfrak { c } _ \omega ] & = 0 \in H C _ \lambda ^ { p + 2 } ( \mathfrak { A } ) . \end{align*}


\begin{align*} Z ( M _ \eta T _ \xi f ) ( x , \omega ) = e ^ { 2 \pi i \eta \cdot x } Z f ( x - \xi , \omega - \eta ) , \end{align*}


\begin{align*} ( r / t ) ^ \perp _ p ( G ) = \sup \left \{ \left . \| r ^ \perp _ p ( g ) \| / d ( p , g ( p ) ) \ , \right | \ , g \neq e \in G , \ ; \| r _ p ^ \perp ( g ) \| \right \} , \end{align*}


\begin{align*} ( P ^ { \nu _ \alpha } _ t ) ^ * ( g ) = ( ( M _ \alpha ) ^ { - 1 } \circ T ^ { \alpha } _ t \circ M _ \alpha ) ( g ) . \end{align*}


\begin{align*} \mathbb { B } = \Q \oplus \Q \zeta \oplus \Q \xi \oplus \Q \zeta \xi , \end{align*}


\begin{align*} J ^ E ( x , t , k ) = \left \{ \begin{aligned} & \Delta _ \eta J ^ { ( 3 ) } ( x , t , k ) { \Delta _ \eta } ^ { - 1 } , & k \in \Gamma ^ { ( 3 ) } \backslash \left ( U _ { \delta } ( \eta ) \cup U _ { \delta } ( - \eta ) \right ) , \\ & m _ { \pm \eta } ( x , t , k ) \Delta _ \eta ^ { - 1 } , & k \in \partial U _ { \delta } ( \pm \eta ) , \end{aligned} \right . \end{align*}


\begin{align*} H ( Y ^ n ) = \frac { 1 } { 2 } \sum _ { t = 1 } ^ n \ln \big ( ( 2 \pi e ) ^ { n _ y } \det \big ( { \bf C } _ t \Pi _ t { \bf C } _ t ^ T + { \bf D } _ t K _ { \overline { W } _ t } { \bf D } _ t ^ T \big ) \big ) \end{align*}


\begin{align*} \tilde { c } _ { x , y } ( \eta ) : = \eta ( x - 1 ) + \eta ( x + 1 ) + \eta ( y - 1 ) + \eta ( y + 1 ) \end{align*}


\begin{align*} | \alpha _ - ( t ) | \lesssim e ^ { e _ 0 t } \sum _ { n = [ t ] } ^ \infty \Bigg | \int _ { n } ^ { n + 1 } e ^ { - e _ 0 s } B ( R , \mathcal { Y } _ + ) ( s ) d s \Bigg | \lesssim e ^ { e _ 0 t } \sum _ { n = [ t ] } ^ \infty e ^ { - ( e _ 0 + c _ 1 ) n } \lesssim e ^ { - c _ 1 t } . \end{align*}


\begin{align*} i s ( | \lambda | , t ) + \omega _ 2 ( i t ) = i t + \frac { 1 } { - i h ( s ( | \lambda | , t ) ) } . \end{align*}


\begin{align*} \begin{aligned} \| \hat { V } ^ { T } \hat { V } \| _ { 2 } & \leq 1 + \xi , \\ \| \hat { V } \| ^ 2 _ F & \leq k + \sqrt { k } \xi , \\ \| \hat { V } \| _ { 2 } & \leq \sqrt { 1 + \xi } . \end{aligned} \end{align*}


\begin{align*} A _ k : = Z _ k D _ { A _ k } W _ k ^ T , B _ k : = Z _ k D _ { B _ k } U _ k ^ T , G _ k : = V _ k D _ { G _ k } W _ k ^ T , \end{align*}


\begin{align*} \| \delta _ \alpha ( f ) \| _ { C _ b ( \Omega _ X ) } & = \max \left ( \sup _ { x \in X } \sup _ { y \in \tilde { X } \setminus \{ x \} } | \delta _ \alpha ( f ) ( x , y ) | , \sup _ { y \in X } | \delta _ \alpha ( f ) ( \infty , y ) | \right ) \\ & = \max \left ( \sup _ { x , y \in X , \ , x \neq y } | \delta _ \alpha ( f ) ( x , y ) | , \sup _ { y \in X } | f ( y ) | \right ) = \max ( | f | _ { C ^ \alpha ( X ) } , \| f \| _ { C ( X ) } ) . \end{align*}


\begin{align*} \begin{aligned} u ( x + h ) + u ( x - h ) - 2 u ( x ) & \leq C \left ( 1 + \frac { 1 } { \mathrm { d i s t } \left ( x , \partial \Omega \right ) } \right ) | h | ^ 2 \\ & \leq \frac { C } { \mathrm { d i s t } \left ( x , \partial \Omega \right ) } | h | ^ 2 , \end{aligned} \end{align*}


\begin{align*} \forall \gamma ^ 1 , p ^ X , x ^ X \left ( \gamma > _ \mathbb { R } 0 \rightarrow ( p = _ X J ^ A _ { \gamma } x \leftrightarrow \gamma ^ { - 1 } ( x - _ X p ) \in A p ) \right ) \end{align*}


\begin{align*} \eta _ { n + 1 } = h \cdot \phi ( \eta _ n ) \cdot h ^ { - 1 } . \end{align*}


\begin{align*} \| \phi \| _ { \mathcal { W } } : = \| ( \phi _ { x } , \phi _ { t } ) \| _ { L ^ 2 _ t \times L ^ { \infty } _ x \cap L ^ { \infty } _ t \times L ^ { 2 } _ x ( [ 0 , T ] \times \S ) } . \end{align*}


\begin{align*} \rho ( \l ) = M _ { \omega / 2 } T _ x M _ { \omega / 2 } = T _ { x / 2 } M _ \omega T _ { x / 2 } = e ^ { - \pi i \omega \cdot x } \pi ( \l ) . \end{align*}


\begin{align*} \int \left ( | \cdot | ^ { - ( N - \gamma ) } * Q ^ p \right ) Q ^ p d x = \frac { 2 p } { \gamma + 2 p - N ( p - 1 ) } \| Q \| _ 2 ^ 2 = \frac { 2 p } { N ( p - 1 ) - \gamma } \| \nabla Q \| _ 2 ^ 2 . \end{align*}


\begin{align*} \gamma _ F ( P ) = ( 2 \pi ) ^ { - \frac { m k } { 2 } } ( \det \Lambda ) ^ { - \frac { 1 } { 2 } } \int _ B e ^ { - \frac { 1 } { 2 } \langle \Lambda ^ { - 1 } t , t \rangle } \ d t , \end{align*}


\begin{align*} \bar { G } ( x ) = M _ Y ( x ) & = \mathbb { E } [ e ^ { Y x } ] \\ & = \mathbb { E } [ e ^ { - Z x } ] \\ & = \int _ { 0 } ^ { \infty } e ^ { - z x } f _ Z ( z ) d z \\ & = \mathcal { L } [ f _ Z ] ( x ) \end{align*}


\begin{align*} \mathcal { F } ^ \# [ f ] ( k ) = f ^ \# ( k ) = \int \overline { \mathcal { K } ^ \# ( x , k ) } f ( x ) \ , d x . \end{align*}


\begin{align*} \alpha ^ 2 ( \norm { x } _ X ^ 2 - \norm { ( 1 - \alpha ^ { - 1 } ) x + _ X \alpha ^ { - 1 } y } _ X ^ 2 ) = \alpha ( \norm { x } _ X ^ 2 - \alpha ^ { - 1 } ( 1 - \alpha ) \norm { x - _ X y } _ X ^ 2 - \norm { y } _ X ^ 2 ) . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ n { n \choose k } L _ k ( x , y ) L _ { n - k } ( x , y ) = \sum _ { k = 0 } ^ n { n \choose k } M _ { k } ( x , y ) M _ { n - k } ( x , y ) . \end{align*}


\begin{align*} \| x - P _ A ( x ) \| _ \omega & \ = \ \left \| x - P _ A ( x ) - P _ B ( x ) + \sum _ { n \in B } e _ n ^ * ( x ) e _ n \right \| _ \omega \\ & \ \leqslant \ \left \| P _ { ( A \cup B ) ^ c } ( x ) + t 1 _ { B } \right \| _ \omega \\ & \ \leqslant \ \left \| P _ { ( A \cup B ) ^ c } ( x ) + t 1 _ F \right \| _ \omega \\ & \ \leqslant \ \left \| P _ { ( A \cup B ) ^ c } ( x ) + P _ F ( x ) \right \| _ \omega \ = \ \left \| x - S _ k ( x ) \right \| _ \omega , \end{align*}


\begin{align*} f ( x ) = \sum \limits _ { l = - \infty } ^ \infty \sum \limits _ { Q \in B _ l } \omega ( Q ) D _ k ( x , x _ Q ) { \widetilde D } _ k ( f ) ( x _ Q ) , \end{align*}


\begin{align*} \eta ^ T \cdot k _ w ( t ) \cdot \eta = \frac { t ^ 2 } { 2 } \eta ^ T \cdot ( k _ w '' ( 0 ) + \sqrt { t } R ) \cdot \eta \geq \frac { t ^ 2 } { 2 } \cdot k _ 1 . \end{align*}


\begin{align*} \lim _ { k \rightarrow \infty } \int _ 0 ^ T \int _ { \mathbb { R } } \rho _ 5 ( s , u ) [ - ( - \Delta ) ^ { \gamma / 2 } G _ k ] ( s , u ) d u d s = - \frac { c _ { \gamma } } { 4 } \iint _ { \mathbb { R } ^ 2 } \frac { [ \int _ 0 ^ T \rho _ 5 ( r , u ) d r - \int _ 0 ^ T \rho _ 5 ( s , v ) d s ] ^ 2 } { | u - v | ^ { 1 + \gamma } } d u d v . \end{align*}


\begin{align*} \forall x ^ X , p ^ X \left ( p = _ X J ^ A _ { 1 } x \rightarrow ( x - _ X p ) \in A p \right ) . \end{align*}


\begin{align*} M _ n ( x ) = \sum _ { k = 0 } ^ { \lfloor ( n - 1 ) / 2 \rfloor } M ( n , k ) x ^ { k } . \end{align*}


\begin{align*} z _ k ^ { l + k } = \Phi ^ { k } _ { Z } ( \zeta _ k ^ { l + k } ) , \xi _ k = \Phi ^ { k } _ { Z } ( c _ k ) . \end{align*}


\begin{align*} \left \{ T _ \Sigma ( h ) , \Psi _ \Sigma ( f ) \right \} _ \ell ^ \Sigma = - \Psi _ \Sigma ( h f ' ) = \frac { d } { d t } \left ( \mathfrak { P } _ { \ell } \rho ^ { ( t ) } \Psi _ \Sigma ( f ) \right ) \big | _ { t = 0 } . \end{align*}


\begin{align*} \frac { ( c - 1 ) \phi ^ n } { \tau ^ 2 } + \eta ^ n \frac { ( c - 1 ) \phi ^ n } { 2 \tau } = - G ( \phi ^ n ) + \mu ^ n \phi ^ n . \end{align*}


\begin{align*} \int _ \R Q _ \alpha \ , d \rho & : = \int _ \R Q _ \alpha ( x ) \ , d \rho ( x ) \\ \int _ { - 1 } ^ 1 K _ \alpha & : = \int _ { - 1 } ^ 1 K _ \alpha ( x ) \ , d x \\ ( K _ \alpha * \rho ) ( x ) & : = \int _ \R K _ \alpha ( x - y ) \ , d \rho ( y ) , \end{align*}


\begin{align*} { i } & \longmapsto \begin{cases} L & i \le \alpha ( 0 ) , \\ j & \exists j \colon i \in [ \alpha ( j - 1 ) + 1 , \alpha ( j ) ] , \\ R & i > \alpha ( p ) . \end{cases} \end{align*}


\begin{align*} \frac { \mathcal { H } _ { d - 1 } ( W \cap \partial O ) } { | O | ^ \frac { d - \zeta } { d } } \geq \frac { c _ H } { | O | ^ \frac { d - \zeta } { d } } \geq \frac { c _ H } { R ^ { \theta \cdot \frac { d - \zeta } { d } } } = \frac { c _ H } { R ^ { 1 - \zeta } } . \end{align*}


\begin{align*} \eta : = \frac { \delta } { 2 \sqrt { K _ 1 e ^ { K _ 2 T } } } \wedge \frac { \delta } { 2 } , \end{align*}


\begin{align*} c _ k ( s ) = \alpha ( s ) \pi ^ { \frac { n ( n - 1 ) } { 2 } } \frac { \Gamma ( s + k - 2 n + 3 ) \Gamma ( s + k - 2 n + 5 ) \ldots \Gamma ( s + k - 1 ) } { \Gamma ( s + k - n + 2 ) \Gamma ( s + k - n + 3 ) \ldots \Gamma ( s + k ) } , \end{align*}


\begin{align*} \Lambda _ y w = 2 w + y \partial _ y w . \end{align*}


\begin{align*} P _ { \theta } ^ { \mathrm { o u t } } ( M ) : = P _ { - \theta } ^ { \mathrm { i n } } ( M ) . \end{align*}


\begin{align*} \begin{aligned} \max \left \{ u ( y _ j , t ) , u ( z _ j , t ) \right \} & \leq \left ( { 1 \over \min \{ \lambda , 1 - \lambda \} } \right ) ^ { 1 \over q } ( u _ { q , \lambda } ( x , t ) + 1 ) \\ & \leq \left ( { 1 \over \min \{ \lambda , 1 - \lambda \} } \right ) ^ { 1 \over q } ( u _ { \star , \lambda } ( x , t ) + 1 ) \end{aligned} \end{align*}


\begin{align*} \lim _ { t \mapsto 0 } \frac { D ^ G ( x \cdot \exp ( t X ) ) } { D ^ { G _ x } ( t X ) } = D ^ G ( x ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r c l } d X _ r ( t ) & = & \big ( A X _ r ( t ) + \alpha ( X _ r ( t ) ) \big ) d t + \sum _ { j = 1 } ^ r \sigma ^ j ( X _ r ( t ) ) d B ^ j ( t ) \medskip \\ X _ r ( 0 ) & = & x , \end{array} \right . \end{align*}


\begin{align*} & \langle V ( T _ { \alpha , N } ) , \xi \rangle = \langle T _ { \alpha , N } , V ^ * ( \xi ) \rangle = \lim _ { \varepsilon \to 0 } \int _ { \vert f \vert \geq \varepsilon } \vert f \vert ^ { 2 \alpha } f ^ { - N } V ^ * ( \xi ) \\ & = \lim _ { \varepsilon \to 0 } \int _ { \vert f \vert \geq \varepsilon } V ( \vert f \vert ^ { 2 \alpha } f ^ { - N } ) \xi \end{align*}


\begin{align*} m ( \pi ) + m ( \overline { \pi } ) = \begin{cases} n + 1 , & d _ { m ( \pi ) } = m ( \pi ) - 1 \\ n , & \end{cases} \end{align*}


\begin{align*} \sum _ { n \geq 0 } A _ n ( x ) { t ^ n \over n ! } = { 1 - x \over 1 - x e ^ { ( 1 - x ) t } } . \end{align*}


\begin{align*} F \left ( { \bf w } , u \right ) = I _ { 2 d } - u \widehat { M } _ A ( { \bf w } ) \ \ \ a n d \ \ \ \widehat { M } _ A ( { \bf w } ) = \sum _ { j = 1 } ^ { d } \Big ( e ^ { i w _ j } P _ { 2 j - 1 } A + e ^ { - i w _ j } P _ { 2 j } A \Big ) , \end{align*}


\begin{align*} \widehat { \psi _ { p ^ { 2 k } } } ( \mathbf { u } ) = \frac { 1 } { p ^ { 2 n k } } \sum _ { \mathbf { c } \in ( \mathbb { Z } / p ^ { 2 k } \mathbb { Z } ) ^ n } \psi _ { p ^ { 2 k } } ( f _ \mathbf { c } ) \exp \left ( \frac { 2 \pi i c _ 1 u _ 1 } { p ^ { 2 k } } \right ) . \end{align*}


\begin{align*} k ( s , t ) = \bigg ( 1 - \frac { t } { s } \bigg ) s \bigg ( \frac { 1 } { h ( s ) } - s \bigg ) + ( s - t ) t = k ( s , 0 ) \bigg ( 1 - \frac { t } { s } \bigg ) + ( s - t ) t . \end{align*}


\begin{align*} \partial \varphi ( \bar x ; u ) : = \left \{ \eta \in \mathbb X \ , \middle | \ , \begin{aligned} & \exists \{ u _ k \} _ { k \in \N } \subset \mathbb X , \ , \exists \{ t _ k \} _ { k \in \N } \subset \R _ + , \ , \exists \{ \eta _ k \} _ { k \in \N } \subset \mathbb X \colon \\ & u _ k \to u , \ , t _ k \searrow 0 , \ , \eta _ k \to \eta , \ , \eta _ k \in \widehat \partial \varphi ( \bar x + t _ k u _ k ) \ , \forall k \in \N \end{aligned} \right \} \end{align*}


\begin{align*} T ( V ) : = \bigoplus _ { n \ge 0 } V ^ { \otimes n } \end{align*}


\begin{align*} \frac { \mathbf { P } ( U _ n > x ) } { 1 - \Phi \left ( x \right ) } = 1 + o ( 1 ) . \end{align*}


\begin{align*} \forall V \in X ( i ) : \pi _ i ( V ) = \frac { 1 } { { d \choose i } _ q } \sum \limits _ { W \in X ( d ) : W \supset V } \pi _ d ( W ) , \end{align*}


\begin{align*} \bigg | { \rm I } - C _ 0 \int _ { \{ | y | < 2 | y - x | \} } \frac { 2 x F ( x ) } { y - x } \ , \dd x \bigg | \lesssim \frac { 1 } { y ^ 2 } \int _ { \mathbb { R } } ( 1 + | x | ^ 2 ) F ( x ) \ , \dd x = O \bigg ( \frac { 1 } { y ^ 2 } \bigg ) . \end{align*}


\begin{align*} & \int _ { \mathbb { R } } x ^ k \ d \mu _ { A _ N + U _ N B _ N U _ N ^ * } ( x ) \\ & = \tau _ { A _ N , U _ N B _ N U _ N ^ * } ( ( X + Y ) ^ k ) = \tau ( ( X + Y ) ^ k ) + o ( 1 ) = \int _ { \mathbb { R } } x ^ k \ d \mu _ 1 \boxplus \mu _ 2 ( x ) \end{align*}


\begin{align*} \nu ( B ( x , r ) ) \approx \ , \frac { \mu ( B ( x , r ) ) } { r ^ \Theta } \ , \theta = \frac { p - \Theta } { p } , \end{align*}


\begin{align*} \sum _ { k = 1 } ^ \infty k \tau _ k Y ^ k & = Y \tau ' = \tau \sum _ { k = 1 } ^ n \frac { X _ k Y } { 1 - X _ k Y } = \tau \sum _ { k = 1 } ^ n \sum _ { i = 1 } ^ \infty ( X _ k Y ) ^ i \\ & = \tau \sum _ { i = 1 } ^ \infty \rho _ i Y ^ i = \sum _ { k = 1 } ^ \infty \Bigl ( \sum _ { i = 1 } ^ k \rho _ i \tau _ { k - i } \Bigr ) Y ^ k . \end{align*}


\begin{align*} ( c , d ) = \begin{cases} 1 , & c = d , \\ 0 , & | c d | = 2 , \\ \frac { \eta } { 2 } , & | c d | = 3 . \end{cases} \end{align*}


\begin{align*} ( M ) \subseteq \bigcup _ { i = 1 } ^ k \left [ \lambda _ i - c _ i , \lambda _ i + c _ i \right ] . \end{align*}


\begin{align*} \left | \int _ { \R } \int _ { \S } ( | \phi _ t | ^ 2 + | \phi _ x | ^ 2 ) ( t , x ) \psi ( t ) d x d t - E ( 0 ) \right | = \left | \int _ { \R } \psi ( t ) \big ( E ( t ) - E ( 0 ) \big ) d t \right | \lesssim \delta E ( 0 ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } \partial _ t W = W _ 1 + W _ 2 + W _ 3 + W _ 4 , \end{align*}


\begin{align*} \lim _ { t \rightarrow 0 ^ + } \omega _ 2 ( i t ) = \begin{cases} \infty , & 0 < | \lambda | \leq \lambda _ 1 ( \mu ) ; \\ i \frac { | \lambda | ^ 2 } { s ( | \lambda | , 0 ) } , & | \lambda | \in ( \lambda _ 1 ( \mu ) , \lambda _ 2 ( \mu ) ) ; \\ 0 , & , | \lambda | \geq \lambda _ 2 ( \mu ) . \end{cases} \end{align*}


\begin{align*} \tau _ 1 : = \frac { \beta } { } \bigg ( \sum _ { i = 1 } ^ K \mathbf { C } _ i ( \psi ) - L \int _ D \psi \ ; \mathrm { d } q \ , \mathbb { I } \bigg ) , \end{align*}


\begin{align*} x _ 1 ^ \prime = \alpha _ 1 y _ 1 \ , , x _ 2 ^ \prime = \alpha _ 2 y _ 2 \ , , y _ 1 ^ \prime = \beta _ 1 x _ 1 \ , , y _ 2 ^ \prime = \beta _ 2 x _ 2 \ , . \end{align*}


\begin{align*} \sum _ { i , j = 1 } ^ d a _ { i j } ( x ) \xi _ i \xi _ j \geq \eta ( x ) | \xi | ^ 2 x , \xi \in \R ^ d , \end{align*}


\begin{align*} s ^ { ( 2 m ) } _ n = \left ( \frac { n } { 1 - 2 \alpha } \right ) ^ m ( 2 m - 1 ) ! ! , \end{align*}


\begin{align*} I ( \alpha , \beta , Z ) + I ( \beta , \gamma , Z ' ) = I ( \alpha , \gamma , Z + Z ' ) . \end{align*}


\begin{align*} A _ { k n } = G _ { k n } \circ B _ { k n } , \end{align*}


\begin{align*} \| a ( h ) - a ( g ) \| & \leq L \| h - g \| , \\ \| \sigma ^ j ( h ) - \sigma ^ j ( g ) \| & \leq L \| h - g \| , j = 1 , \ldots , r , \end{align*}


\begin{align*} \Delta _ { \mathbf { L } ( \mathbf { t } ) } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { t } ) } ( \mathcal { I } ) = \Delta _ { \mathbf { L } ( \mathbf { 1 } ) } ( \mathcal { I } ) \cdot \Delta _ { \mathbf { R } ( \mathbf { 1 } ) } ( \mathcal { I } ) \cdot \mathbf { t } ^ { \mathbf { \Psi } ( \mathcal { I } ) } , \mathcal { I } \in \mathfrak { G } ( \mathbf { L } ( \mathbf { 1 } ) ) . \end{align*}


\begin{align*} d \rho _ i = \sum _ { j \in N ( i ) } \omega _ { i j } ( S _ i - S _ j ) \theta _ { i j } ( \rho ) d t ; & \\ d S _ i + ( \sum _ { j \in N ( i ) } \frac { \omega _ { i j } } { 2 } ( S _ i - S _ j ) ^ 2 \frac { \partial \theta _ { i j } } { \partial \rho _ i } + \frac 1 8 \frac { \partial } { \partial \rho _ i } I ( \rho ) + \mathbb V _ i & + \sum _ { j \in N ( i ) } \mathbb W _ { i j } \rho _ j - \log ( \rho _ i ) ) d t + \sigma _ i d W _ t = 0 . \end{align*}


\begin{align*} \Lambda Q = 2 Q + y \partial _ y Q = - \frac { \epsilon ' _ x ( x ) } { e ^ { 2 x } } < 0 \forall x \in ( - \infty , \infty ) . \end{align*}


\begin{align*} \iint _ { \R ^ { 2 d } } W f ( x , \omega ) \ , d ( x , \omega ) = 1 . \end{align*}


\begin{align*} \Psi _ { ( \Sigma , U ) } ( \eta _ { ( \Sigma , U ) } h ) [ \partial _ { \Sigma , \epsilon } \phi ] = \Psi _ { ( \widetilde { \Sigma } , U ) } ( \eta _ { ( \widetilde { \Sigma } , U ) } h ) [ \partial _ { \widetilde { \Sigma } , \widetilde { \epsilon } } \phi ] . \end{align*}


\begin{align*} ( \sigma , v ) = \sum _ { k \geq K _ 0 } a _ k e ^ { \lambda ^ p _ k ( T - t ) } ( \xi _ { \lambda ^ p _ k } , \eta _ { \lambda ^ p _ k } ) + \sum _ { | k | \geq K _ 0 } b _ k e ^ { \lambda ^ h _ k ( T - t ) } ( \xi _ { \lambda ^ h _ k } , \eta _ { \lambda ^ h _ k } ) , \end{align*}


\begin{align*} S ( U ) = \left ( \begin{array} { c c c c } \frac { p _ { \rho } } { \rho } & & & \\ & \mathbb { I } _ { 3 \times 3 } & & \\ & & \frac { 1 } { \theta } & \\ & & & \frac { 1 } { \kappa \theta } \mathbb { I } _ { 3 \times 3 } \\ \end{array} \right ) . \end{align*}


\begin{align*} f = \frac { 1 } { \langle g , \widetilde { g } \rangle } \iint _ { \R ^ { 2 d } } V _ { \widetilde { g } } f ( x , \omega ) M _ \omega T _ x g \ , d ( x , \omega ) \end{align*}


\begin{align*} \mathfrak { F } _ { ( \alpha , \beta ) } ( \cosh ( \pi t ) ^ { - 1 } ) = \{ \alpha \Z \times \beta \Z \mid \alpha \beta < 1 \} \end{align*}


\begin{align*} D _ n : \ell ^ 2 ( \Gamma ) \to \mathcal { H } , D _ n ( c _ \gamma ) _ { \gamma \in \Gamma } = \sum _ { | \gamma | \leq n } c _ \gamma e _ \gamma . \end{align*}


\begin{align*} \| v ^ { n } ( t ) - v ^ { n } ( s ) \| _ { L ^ p ( \Omega ; H ^ \beta ) } \leq & \| ( E ( t - s ) - I ) v ^ { n } ( s ) \| _ { L ^ p ( \Omega ; H ^ \beta ) } + \Big \| \int _ s ^ t E ( t - r ) P _ n A P F ( X ^ { n } ( r ) ) \ , \dd r \Big \| _ { L ^ p ( \Omega ; H ^ \beta ) } \\ \leq & C ( t - s ) ^ { \frac { \gamma - \beta } 4 } \| v ^ { n } ( s ) \| _ { L ^ p ( \Omega ; H ^ \gamma ) } + C ( t - s ) ^ { \frac { \gamma - \beta } 4 } . \\ \leq & C ( T , \gamma , X _ 0 , p ) ( t - s ) ^ { \frac { \gamma - \beta } 4 } . \end{align*}


\begin{align*} 0 & = \langle \pi ( F ) \pi ( \xi , \eta ) f , \pi ( \xi , \eta ) g \rangle \\ & = \iint _ { \R ^ { 2 d } } F ( x , \omega ) \langle \pi ( \xi , \eta ) ^ { - 1 } \pi ( x , \omega ) \pi ( \xi , \eta ) f , g \rangle \ , d ( x , \omega ) . \end{align*}


\begin{align*} \quad \ ; [ u , v , w ] _ { T _ t } & = \displaystyle \sum _ { k = 0 } ^ { + \infty } \sum _ { i + j = k } \Big ( D ( T _ i ( u ) , T _ j ( v ) ) w + \theta ( T _ i ( v ) , T _ j ( w ) ) u - \theta ( T _ i ( u ) , T _ j ( w ) ) v \Big ) t ^ { k } , \end{align*}


\begin{align*} \sigma ( y ) & = \sup \{ 0 < \varepsilon ' < c \ ; | \ ; \exists \delta > 0 \ \ ( \mathcal B _ d ( x , \delta ) \times \mathcal B _ { n - d } ( y , \varepsilon ' ) ) \cap X \\ & \} . \end{align*}


\begin{align*} H ^ { - \alpha } = \langle X , \nu \rangle , \end{align*}


\begin{align*} i \neq j \Rightarrow \langle ( \mathcal H ^ A _ \emph { C K } ) _ { i } , ( \mathcal H ^ A _ \emph { G L } ) ^ { j } \rangle = 0 , \langle x , ( \mathcal H ^ A _ \emph { G L } ) ^ { | x | } \rangle = 0 \Rightarrow x = 0 , \langle ( \mathcal H ^ A _ \emph { C K } ) _ { | y | } , y \rangle = 0 \Rightarrow y = 0 \end{align*}


\begin{align*} D _ { 1 , m , n } = \tau _ { A \smallsetminus \{ \alpha \} \cup \{ - \beta \} } ( p ^ m ) \tau _ { B \smallsetminus \{ \beta \} } ( p ^ n ) . \end{align*}


\begin{align*} \langle \xi ; \nabla ( \varphi _ \mu ) ( \xi ) \rangle = - \langle \xi ; \Sigma ( \xi ) \rangle \varphi _ \mu ( \xi ) . \end{align*}


\begin{align*} \inf _ { \gamma ( x ) \ge R } u _ 0 ( x ) \ge \sigma ( R ) \ge c _ 0 \ ; \ ; R \in [ 0 , \infty ) . \end{align*}


\begin{align*} E ( u , v , u ' , v ' ) = - \frac { 1 } { 4 } \left ( \mathrm { s g n } ( u - u ' ) + \mathrm { s g n } ( v - v ' ) \right ) . \end{align*}


\begin{align*} \beta _ { H ' - H } F = \sum _ { n = 0 } ^ { \infty } \frac { \hbar ^ n } { 2 ^ n n ! } \left \langle ( H ' _ { \Sigma } - H _ { \Sigma } ) ^ { \otimes n } , F ^ { ( 2 n ) } \right \rangle \end{align*}


\begin{align*} [ X , Y ] _ \star : = \mathcal { L } ^ \mathcal { F } _ X Y = [ \overline { \mathcal { F } } _ 1 \rhd X , \overline { \mathcal { F } } _ 2 \rhd Y ] = X \star Y - ( \overline { \mathcal { R } } _ 1 \rhd Y ) \star ( \overline { \mathcal { R } } _ 2 \rhd X ) , \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Phi ^ H ( q , t ) - 1 = \Phi ^ T ( q , t ) , \end{align*}


\begin{align*} J ^ { G P } = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} , k \in ( - C _ L , C _ L ) \end{align*}


\begin{align*} \int _ { 0 } ^ { T } z \dd ( y - u ) = \int _ { 0 } ^ { t } z \dd ( y - u ) + \int _ { t } ^ { t + \varepsilon } z \dd ( y - u ) + \int _ { t + \varepsilon } ^ { T } z \dd ( y - u ) > 0 . \end{align*}


\begin{align*} ( \mathbf { f } | [ K _ 1 ( \mathfrak { n } ) \xi K _ 1 ( \mathfrak { n } ) ] ) ( g ) = \sum _ { y \in Y } \mathbf { f } ( g y ^ { - 1 } ) , \end{align*}


\begin{align*} \begin{aligned} f _ t - \frac { 1 } { 2 \sqrt { t } } f '' + \frac { 1 } { 2 \sqrt { t } } & \geq 0 , \\ f - f ^ * & \geq 0 , \\ \left ( f _ t - \frac { 1 } { 2 \sqrt { t } } f '' + \frac { 1 } { 2 \sqrt { t } } = 0 \right ) \vee ( f - f ^ * & = 0 ) , \end{aligned} \end{align*}


\begin{align*} \mathbb { E } \left \| x _ \ast - \frac { 1 } { M } \sum _ { k = 1 } ^ { M } x _ { k } \right \| \leq \frac { 1 + \| A \| _ { F } \left \| A ^ { - 1 } \right \| } { \sqrt { M } } \left \| x _ \ast - x _ { 0 } \right \| . \end{align*}


\begin{align*} \begin{aligned} U ^ T \ ! A X & = \Gamma = ( \gamma _ 1 , \dots , \gamma _ n ) , & \gamma _ i \in [ 0 , 1 ] , \\ V ^ T \ ! G X & = \Sigma = ( \sigma _ 1 , \dots , \sigma _ n ) , & \sigma _ i \in [ 0 , 1 ] , \end{aligned} \end{align*}


\begin{align*} h _ B ( x ) = h _ B ( y ) \Longleftrightarrow x = y . \end{align*}


\begin{align*} \frac { \partial ^ 2 } { \partial \eta ^ 2 } \left [ \log \left ( \tfrac { 1 } { n } A ( \tfrac { \eta } { n } , \tfrac { 1 } { \eta } ) \right ) \right ] = \frac { \tanh '' ( \tfrac { \eta } { 2 } ) \tanh ( \tfrac { \eta } { 2 } ) - \tanh ' ( \tfrac { \eta } { 2 } ) ^ 2 } { 4 \tanh ( \tfrac { \eta } { 2 } ) ^ 2 } + \frac { \psi '' ( \eta ) \psi ( \eta ) - \psi ' ( \eta ) ^ 2 } { \psi ( \eta ) ^ 2 } . \end{align*}


\begin{align*} & \Phi _ { R , L } ( x , t , k ) \sim \Phi ^ { \pm \infty } ( x , t , k ) , & x \rightarrow \pm \infty , \\ & \Phi _ { R , L } ( x , k ) e ^ { i X _ { R , L } ( k ) x \sigma _ 3 } = I + \mathcal { O } ( k ^ { - 1 } ) , & k \rightarrow \infty . \end{align*}


\begin{align*} n ^ { 2 - p } \norm { T } _ { p } ^ { p } \leq \overset { n } { \underset { i , j = 1 } { \sum } } \norm { T _ { i j } } _ { p } ^ { p } \leq \norm { T } _ { p } ^ { p } , \end{align*}


\begin{align*} I _ 1 ( u ) = \int _ { \R } u ( x , t ) d x , \end{align*}


\begin{align*} \exp \Big ( 2 \sum ^ j _ { i = 0 } { \mathcal M } _ { i , j } ( p ) \Big ) \ll \prod ^ j _ { i = 1 } \Big | E _ { e ^ 2 \alpha ^ { - 3 / 4 } _ i } ( { \mathcal M } _ { i , j } ( p ) ) \Big | ^ 2 , \end{align*}


\begin{align*} \langle B ( \epsilon ) , \phi _ 0 \rangle _ { L ^ 2 _ \rho } \lesssim ( \sum _ { j = 0 } ^ M \varepsilon _ j ^ 2 + \epsilon _ - ^ 2 ) . \end{align*}


\begin{align*} \operatorname { h r m } ( P ) = \sum _ { s \in S _ p } \frac { l ( s ) ^ 2 } { A ( P ) } \end{align*}


\begin{align*} \frac { 2 } { q } + \frac { N } { r } = \frac { N } { 2 } - s , 2 \le q , r \le \infty , ( q , r , N ) \not = ( 2 , \infty , 2 ) . \end{align*}


\begin{align*} C _ { n } ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \coloneqq \bigg ( \ ! 1 - \ ! \sum _ { m = n + 1 } ^ { d } C _ m ^ { ( d ) , \sigma } ( \mathbf { P } _ { \ ! \sigma } ) \cdot \chi _ n ^ { \sigma } ( \mathbf { p } _ { \sigma _ m } ) \ ! \bigg ) \frac { 1 } { \chi _ n ^ { \sigma } ( \mathbf { p } _ { \sigma _ n } ) } . \end{align*}


\begin{align*} \int \nabla _ x f _ 0 ( x - t v ) g ( v ) \dd v & = \frac { 1 } { t } \int f _ 0 ( x - t v ) \nabla g ( v ) \dd v . \end{align*}


\begin{align*} d _ 0 \tilde \varphi ( x , x + n ) & = \sum _ { i = 0 } ^ { x + n - 1 } \varphi ( i , i + 1 ) - \sum _ { i = 0 } ^ { x - 1 } \varphi ( i , i + 1 ) \\ & = \sum _ { i = x } ^ { x + n - 1 } \varphi ( i , i + 1 ) \\ & = \begin{cases} \varphi ( x , x + n ) - \varphi _ n ( x ) & x = x _ 1 , \ldots , x _ { r _ n } \\ \varphi ( x , x + n ) & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} \sigma _ { } ( \mathcal { P } ) = [ 0 , + \infty ) . \end{align*}


\begin{align*} X \cdot e _ j = X , Y \cdot e _ j = x _ { 1 j } X + Y \quad \mbox { a n d } Z \cdot e _ j = ( x _ { 1 j } ^ 2 + x _ { 2 j } ) X + 2 x _ { 1 j } Y + Z . \end{align*}


\begin{align*} u ( r , t ) = \lambda _ { \ell } ^ { - 1 } ( t ) Q \left ( \frac { r } { \sqrt { \lambda _ { \ell } ( t ) } } \right ) + \tilde u ( r , t ) , \end{align*}


\begin{align*} & \bigg | D _ k T _ M ( f ) ( x ) \bigg | = \bigg | D _ k \bigg ( - \ln r \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } w ( Q ) \psi _ { Q } ( \cdot , x _ { Q } ) q _ { Q } f ( x _ { Q } ) \bigg ) ( x ) \bigg | \\ & \lesssim \sum \limits _ { j = - \infty } ^ \infty \sum \limits _ { Q \in Q ^ j } w ( Q ) | D _ k \psi _ { Q } ( x , x _ { Q } ) | | q _ { Q } f ( x _ { Q } ) | . \end{align*}


\begin{align*} 2 ( \xi _ i \eta _ j - \xi _ j \eta _ i ) = ( \xi _ i - \eta _ i ) ( \xi _ j + \eta _ j ) - ( \xi _ j - \eta _ j ) ( \xi _ i + \eta _ i ) \end{align*}


\begin{align*} \mathcal { H } = \{ z \in \C _ n ^ n : U ( z ) \in \Omega \} , \ , \ , \ , \ , U ( z ) : = \left [ \begin{array} { c } z \\ u _ 0 \end{array} \right ] , \end{align*}


\begin{align*} \cos { \rho } _ n & = \cos \left ( n + \frac { 1 } { 2 } \right ) \pi \cos \xi _ n - \sin \left ( n + \frac { 1 } { 2 } \right ) \pi \sin \xi _ n = - \left ( - 1 \right ) ^ n \sin \xi _ n , \\ [ 0 . 2 e m ] \sin { \rho } _ n & = \sin \left ( n + \frac { 1 } { 2 } \right ) \pi \cos \xi _ n + \cos \left ( n + \frac { 1 } { 2 } \right ) \pi \sin \xi _ n = \left ( - 1 \right ) ^ n \cos \xi _ n . \end{align*}


\begin{align*} \int _ { 0 } ^ { 1 } u _ x \bar { \sigma } - \int _ { 0 } ^ { 1 } \rho \bar { \sigma } _ x + \rho \bar { \sigma } | _ { 0 } ^ { 1 } + \lambda \int _ { 0 } ^ { 1 } \rho \bar { \sigma } = \int _ { 0 } ^ { 1 } f \bar { \sigma } . \end{align*}


\begin{align*} ( u \rhd _ s v ) \rhd _ s w = ( u \rhd _ s w ) \rhd _ s ( v \rhd _ s w ) . \end{align*}


\begin{align*} \alpha _ { n - j } ( F _ i ) = \alpha _ { n - j } ( T _ i ) + \alpha _ { n - j } ( F _ i / T _ i ) . \end{align*}


\begin{align*} \mathfrak { A } _ { \omega } = \bigcup _ e \{ g ^ { - 1 } f : f \in M _ { \tau _ e } , 0 \neq g \in M _ e \} , \end{align*}


\begin{align*} A _ n ( q ) = \begin{cases} { \displaystyle \sum _ { k = 1 } ^ { n } A ( n , k ) q ^ k , } & n \geq 1 , \\ 1 , & n = 0 . \end{cases} \end{align*}


\begin{align*} \dot \Phi _ t = \rho \ast \Phi _ t = \Phi _ t * \rho . \end{align*}


\begin{align*} a ( m b ) = ( a m ) b , \forall a , b \in A . \end{align*}


\begin{align*} \Lambda ( \phi \succ \gamma ) = \sum _ { \substack { u _ 1 , \dotsc , u _ k \in \{ \ 1 \} \cup \N ^ * \\ { \substack { u \in \N ^ * \\ v = i _ 1 \dotsm i _ k \in \N ^ * } } } } f _ v g _ u g _ { u _ 1 } \dotsm g _ { u _ k } x _ { u } x _ { i _ 1 } x _ { u _ 1 } \dotsm x _ { i _ k } x _ { u _ k } = ( g ( x ) - 1 ) f ( x g ( x ) ) . \end{align*}


\begin{align*} \begin{aligned} \phi ( f _ 1 , f _ 2 ) & = \phi ( 1 , 1 ) + \bigl ( \phi ( f _ 1 , 1 ) - \phi ( 1 , 1 ) \bigr ) + \bigl ( \phi ( f _ 1 , f _ 2 ) - \phi ( f _ 1 , 1 ) \bigr ) \\ & \leq \phi ( f _ 1 , 1 ) + \bigl ( \phi ( 2 , f _ 2 ) - \phi ( 2 , 1 ) \bigr ) . \end{aligned} \end{align*}


\begin{align*} D _ n ( 2 u , v ) = 2 ^ n E _ n ( u , v ) . \end{align*}


\begin{align*} u ( x ) = - \int _ x ^ b ( f ( t ) - u ( t ) ) ^ \frac { 1 } { p } d t < 0 \end{align*}


\begin{align*} \norm { g } _ 2 ^ 2 = \iint _ { \mathcal { Q } \times \mathcal { Q } } | Z g ( x , \omega ) | ^ 2 \ , d ( x , \omega ) = 1 . \end{align*}


\begin{align*} \mathbb { D } _ i ^ { ( - 1 ) } = \begin{cases} ( \theta - t ) ^ { q ^ { i - 1 } } \mathbb { D } _ { i - 1 } \ & , \\ 1 \ & . \end{cases} \end{align*}


\begin{align*} d _ { q + 1 } \circ d _ q = 0 , d ^ * _ q d _ q + d _ { q - 1 } d ^ * _ { q - 1 } = - E _ { m ( q ) } \varDelta , \ , 0 \leq q \leq n , \end{align*}


\begin{align*} V _ { \widetilde { g } } f = \frac { 1 } { \langle \widetilde { g } , g \rangle } V _ { \widetilde { g } } V ^ * _ { \widetilde { g } } ( V _ g f ) . \end{align*}


\begin{align*} \alpha _ { H ' - H } \partial _ { \Sigma , \epsilon } ^ { * } = \partial _ { \Sigma , \epsilon } ^ { * } \beta _ { H ' - H } , \end{align*}


\begin{align*} \Xi ^ { - 1 } _ 0 ( x \wedge y ) = \tfrac 1 2 x y , \end{align*}


\begin{align*} D _ { n } ( u , v ) = M _ n ( x , y ) , \end{align*}


\begin{align*} L B ( \chi ) = L \chi \quad \mbox { f o r a l l } \quad \chi \in C ^ 1 . \end{align*}


\begin{align*} \int _ { G ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \mathbf { E } ( g \times h , s ) \mathbf { f } ( h ) \mathbf { d } h = \int _ { G ( \Q ) \backslash G ( \mathbb { A } ) / K _ 1 ( \mathfrak { n } ) K _ { \infty } } \mathbf { E } _ m ( g \times h , s ) \mathbf { f } ( h ) \mathbf { d } h . \end{align*}


\begin{align*} \begin{pmatrix} \varphi ^ { 2 } \\ \psi ^ { 2 } \end{pmatrix} = \rho ^ { 2 \lambda _ { p } - 1 } X _ { 1 } ( \theta ) + \rho ^ { 2 \lambda _ { p } - 2 m + 1 } X _ { 2 } ( \theta ) , \end{align*}


\begin{align*} \mathbf { f } ( g \times h ) = \sum _ { j = 1 } ^ e \mathbf { g } _ j ( g ) \overline { \mathbf { h } _ j ( h ) } . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { d } } \left ( h ( U ^ { 1 } ) - h ( U ^ { 0 } ) \right ) \partial _ { i } \phi d x & = \int _ { 0 } ^ { 1 } \left ( \int _ { \mathbb { R } ^ { d } } D h ( U _ { r } ) ( U ^ { 1 } - U ^ { 0 } ) \partial _ { i } \phi d x \right ) d r , \\ & = - \int _ { \mathbb { R } ^ { d } } \int _ { 0 } ^ { 1 } \partial _ { i } \left ( D h ( U _ { r } ) ( U ^ { 1 } - U ^ { 0 } ) \right ) d r \phi d x \end{align*}


\begin{align*} \lim _ { h \to 0 } \frac { ( - \Delta ) ^ s v ( h e _ 1 ) } h = ( - \Delta ) ^ s \partial _ 1 v ( 0 ) . \end{align*}


\begin{align*} Y _ { 1 } = x + \eta b ( x ) + Z _ { \eta } , \end{align*}


\begin{align*} \xi ( t ; s , x ) = S _ { t - s } x + \int _ s ^ t S _ { t - u } \alpha ( u , \xi ( u ; s , x ) ) d u , t \in [ s , \infty ) . \end{align*}