\begin{align*} \int _ { - \delta } ^ { \delta } f ( v | x - e ^ { \xi } y | ) \ , d \xi - \int _ { - \delta } ^ { \delta } f ( v | x - y | ) \ , d \xi & = \int _ { - \delta } ^ { \delta } \int _ 1 ^ { e ^ { \xi } } \frac { d } { d t } f ( v | x - t y | ) \ , d t \ , d \xi \\ & \ll | v y | \int _ { - \delta } ^ { \delta } | \xi | \ , d \xi \ll | v y | \delta ^ 2 . \end{align*}


\begin{align*} \widehat { N } _ k ^ j = D ^ { k + j } _ { k } U ^ { k + j } _ { k } , \end{align*}


\begin{align*} \Omega _ \varepsilon : = \Omega \backslash \bigcup \limits _ { k \in K _ \varepsilon } { T _ { \varepsilon , k } } , ~ ~ ~ ~ { K _ \varepsilon } : = \{ k \in \mathbb { Z } ^ 3 : \varepsilon \overline { Q _ k } \subset \Omega \} . \end{align*}


\begin{align*} _ { s + 1 } F _ s ( 1 , \ldots , 1 ; 2 , \ldots , 2 ) ( z ) ^ q = L i _ { K , s } ( z ) . \end{align*}


\begin{align*} \operatorname { P } \left ( \left | \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \psi _ { \tau } ( y _ { i } ) z _ { i } - \mathbb { E } \psi _ { \tau } ( y _ { i } ) z _ { i } \right | \leq C \sqrt { \frac { \tau ^ { 1 - \delta } M _ { \delta } ^ { 1 / k } t } { n } } + C \frac { \tau t } { n } \right ) \geq 1 - 2 e ^ { - t } . \end{align*}


\begin{align*} \begin{aligned} & \sum _ { j = 1 } ^ { n - N } ( \frac j n ) ^ { c ' k n } = n \sum _ { j = 1 } ^ { n - N } \frac 1 n ( \frac j n ) ^ { c ' k n } \le n \int _ 0 ^ { 1 - \frac { N - 1 } n } x ^ { c ' k n } d x \le \\ & \frac n { c ' k n + 1 } ( 1 - \frac { N - 1 } n ) ^ { c ' k n + 1 } \le \frac n { c ' k n + 1 } e ^ { - \frac { N - 1 } n ( c ' k n + 1 ) } . \end{aligned} \end{align*}


\begin{align*} \sum _ { j = 1 } ^ \infty | e _ j ( z ) | ^ 2 \le \Big ( \frac { C } { \delta ^ 2 } \log \frac { 1 } { \delta } \Big ) ^ 2 , \end{align*}


\begin{align*} \norm { Q _ \Omega P _ T } _ { H . S . } ^ 2 = \iint _ { \R ^ { 2 d } } | k ( x , t ) | ^ 2 \ , d ( x , t ) . \end{align*}


\begin{align*} \left ( \frac { \partial } { \partial t } - { \mathcal L } \right ) U ( t , x , \omega ) + { Q ( x , \omega ) } \lozenge U ( t , x , \omega ) & = F ( t , x , \omega ) , t \in ( 0 , T ] , \ , x \in \mathbb R ^ d , \ , \omega \in \Omega \\ U ( 0 , x , \omega ) & = G ( x , \omega ) x \in \mathbb R ^ d , \ , \omega \in \Omega . \end{align*}


\begin{align*} \begin{aligned} \min \ & f ( x ) \\ \enspace & A x = b \\ & g ( x ) \le 0 , \end{aligned} \end{align*}


\begin{align*} \begin{aligned} & F ( u ( y _ q , t _ q ) , \nabla \varphi ( x _ q , t _ q ) , Y _ q ' , W _ \star [ x _ 0 , t _ 0 ] ) \\ & \geq F ( u ( y _ q , t _ q ) , \nabla \varphi ( x _ q , t _ q ) , Y _ q ' , W _ \star [ x _ 0 , t _ 0 ] \cap U [ y _ q , t _ q ] ) , \end{aligned} \end{align*}


\begin{align*} \nabla \psi _ Q & \neq 0 \Rightarrow \chi _ Q = 0 \\ \nabla \chi _ Q & \neq 0 \Rightarrow \psi _ Q = 1 . \end{align*}


\begin{align*} { \mathcal M } _ { i , j } ( p ) = & \sum _ { \substack { X ^ { \alpha _ { i - 1 } } < q \leq X ^ { \alpha _ { i } } \\ q } } \frac { \chi _ { 8 p } ( q ) } { q ^ { 1 / 2 + 1 / ( \log X ^ { \alpha _ { j } } ) } } \frac { \log ( X ^ { \alpha _ { j } } / q ) } { \log X ^ { \alpha _ { j } } } , 1 \leq i \leq j \leq \mathcal { J } . \end{align*}


\begin{align*} \liminf _ { j \to \infty } \tilde { d } _ j / d _ j = 0 . \end{align*}


\begin{align*} | E _ i ( x ) | = \left | \alpha + \int _ 0 ^ x ( n _ i - b _ i ) d y \right | \le \alpha + \int _ 0 ^ 1 ( n _ i + b _ i ) d y \le \alpha + 2 \bar { b } _ i , \quad \forall x \in [ 0 , 1 ] , i = 1 , 2 . \end{align*}


\begin{align*} \abs { \xi _ \ell - C _ 1 \ell ^ { - \psi } } \leq C _ 1 ' \ell ^ { - 1 - \psi } , \abs { \zeta _ \ell - C _ 2 \ell ^ { - 1 } } \leq C _ 2 ' \ell ^ { - 2 } . \end{align*}


\begin{align*} \sum _ { i < j } ( 1 - \lambda _ i ^ 2 \lambda _ j ^ 2 ) \ , \sec _ R ( v _ i \wedge v _ j ) = 0 . \end{align*}


\begin{align*} g \mapsto \left \| g \right \| _ \varphi : = \int _ M \sqrt { \Gamma ( g ) } ( x ) \varphi ( x ) \mu ( d x ) \end{align*}


\begin{align*} \check { \Delta } ( \omega ) ( \chi b \otimes \theta c ) = ( \check { \Delta } ( \omega ) ( \chi \otimes \theta ) ) ( b \otimes c ) , ( b \chi \otimes c \theta ) \check { \Delta } ( \omega ) = ( b \otimes c ) ( ( \chi \otimes \theta ) \check { \Delta } ( \omega ) . \end{align*}


\begin{align*} A ^ { ( q ) } _ n : = \varphi _ n ( q , W _ { 1 : K } ) , \end{align*}


\begin{align*} \Delta a _ { t + 1 } & \in \partial _ { x } H _ t ( x _ { t } , m _ t + a _ t ) , \\ \Delta x _ t & \in \partial _ { y } [ - H _ t ] ( x _ { t } , m _ t + a _ t ) , \end{align*}


\begin{align*} & \frac { d \rho } { d t } = \frac { \partial } { \partial S } \mathcal H _ 0 ( \rho , S ) + \frac { \partial } { \partial S } \mathcal H _ 1 ( \rho , S ) d W _ { \delta } ( t ) , \\ & \frac { d S } { d t } = - \frac { \partial } { \partial \rho } \mathcal H _ 0 ( \rho , S ) - \frac { \partial } { \partial \rho } \mathcal H _ 1 ( \rho , S ) d W _ { \delta } ( t ) . \end{align*}


\begin{align*} \mathcal { S } _ k = \left \{ z = x + i y \in \mathbb { C } \ : \ 1 - \frac { \pi } { 2 } \leq x \leq 1 + \frac { \pi } { 2 } , \ \ 2 k \pi - \frac { \pi } { 2 } \leq y \leq 2 k \pi + \frac { \pi } { 2 } \right \} . \end{align*}


\begin{align*} \mathbb { P } \left ( V _ j = i \right ) = \frac { \mathbb { P } \left ( \left \{ \varsigma _ { Q _ j , m } = 1 , \varsigma _ { Q _ j , i } = 1 \right \} \bigcap _ { r \in [ i - 1 ] } \left \{ \varsigma _ { Q _ j , r } = 0 \right \} \right ) } { \mathbb { P } \left ( \varsigma _ { Q _ j , m } = 1 \right ) } . \end{align*}


\begin{align*} \zeta _ f ( z ) = \zeta _ f ^ { ( 1 ) } ( z ) - \zeta _ f ^ { ( 2 ) } ( z ) . \end{align*}


\begin{align*} f ( t ) = \sum _ { k = 1 } ^ n c _ k M _ { \omega _ k } T _ { x _ k } g _ 0 . \end{align*}


\begin{align*} ( F \circ \psi ) ^ * ( g _ { \textnormal { e u c } } ) = \psi ^ * ( e ^ { 2 u } \ , g _ { \textnormal { p o i n } } ) = e ^ { 2 u \circ \psi + 2 v } \ , g _ { \textnormal { e u c } } \textnormal { o n } \ , \ , B ^ 2 _ 1 ( 0 ) . \end{align*}


\begin{align*} \mathcal R _ c = : \int _ 0 ^ t \int _ { \R ^ 3 } \mathfrak r _ c ( s , x , v ) \dd v \dd s . \end{align*}


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


\begin{align*} 0 \leq \int _ 0 ^ T ( h - \eta ) \dd \bar p = \int _ 0 ^ T h \dd \bar p - \int _ 0 ^ T \eta \dd \bar p , \end{align*}


\begin{align*} \frac { \partial f ( t ) } { \partial t } = e _ { \lambda } ^ { 1 - \lambda } ( t ) | z | ^ { 2 } f ( t ) \ \Longleftrightarrow \ \frac { f ^ { \prime } ( t ) } { f ( t ) } = e _ { \lambda } ^ { 1 - \lambda } ( t ) | z | ^ { 2 } , \bigg ( f ^ { \prime } ( t ) = \frac { d } { d t } f ( t ) \bigg ) . \end{align*}


\begin{align*} \mu = \sum _ { i = 1 } ^ m p _ i \mu \circ f _ i ^ { - 1 } . \end{align*}


\begin{align*} B _ { 4 , 1 } : = B _ { 4 } \cap \{ ( s , v ) \in [ 0 , t ] \times \R ^ 3 : | v | \geq \frac { \langle x ^ \perp \rangle } { 2 \langle \check { \mathcal T } _ { t , x _ 1 , v _ 1 } \rangle } \} . \end{align*}


\begin{align*} L _ { n + 1 } ( x ) = x \sum _ { k = 0 } ^ n { n \choose k } L _ { k } ( x ) M _ { n - k } ( x ) . \end{align*}


\begin{align*} g _ n ' ( 0 ) = f _ n ( 0 ) = \sqrt { n } , \end{align*}


\begin{align*} ( s - t ) ^ 2 - \frac { s - t } { h ( s ) } + | \lambda | ^ 2 = 0 . \end{align*}


\begin{align*} \inf _ { f \in \mathcal { K } } \Phi _ 1 ( f ) = \inf _ { f \in \mathcal { K } } \frac { 1 } { 2 } \int \left ( L _ + f \right ) f d x , \end{align*}


\begin{align*} s _ { K , L } = \sum _ { \substack { k \in \Z ^ d \\ | k _ j | \leq K } } \sum _ { \substack { l \in \Z ^ d \\ | l _ j | \leq L } } c _ { k , l } M _ { \beta l } T _ { \alpha k } g \end{align*}


\begin{align*} p _ { \mathcal { S } } ( \psi _ i , \Phi ) = p _ Y ( _ { \mathcal { S } / Y } ( \psi _ i ) , \Phi ' ) = p _ Y ( _ { \mathcal { S } / Y } ( \phi _ i ) , \Phi ' ) = p _ { \mathcal { S } } ( \phi _ i , \Phi ) , \end{align*}


\begin{align*} R ( \lambda ) - R ( \mu ) = ( \mu - \lambda ) R ( \lambda ) R ( \mu ) ( \Re \lambda , \ , \Re \mu > 0 ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \frac { ( q + 1 ) ^ 2 | S _ { \ell } | } { | G | } = \frac { q + 1 } { q | T _ { \ell ' } | } & \mbox { i f } a = 1 , \mbox { a n d } \\ \frac { ( q + 1 ) | S _ { \ell } | } { | G | } \left ( \alpha ( a ) + \alpha ( a ^ { - 1 } ) \right ) = \frac { 1 } { q | T _ { \ell ' } | } \left ( \alpha ( a ) + \alpha ( a ^ { - 1 } ) \right ) & \mbox { i f } 1 \neq a \in ( \mu _ { q - 1 } ) _ { \ell ' } . \end{array} \right . \end{align*}


\begin{align*} \int _ 0 ^ \infty \norm { 1 _ { U _ t } } _ { L ^ \frac { d } { d - 1 } ( W _ R ) } d t \geq \int _ 0 ^ \infty \norm { g 1 _ { U _ t } } _ { L ^ 1 ( W _ R ) } \ ; d t = \int _ { W _ R } g ( x ) \int _ 0 ^ \infty 1 _ { U _ t } ( x ) \ ; d t d x = \norm { g u } _ { L ^ 1 ( W _ R ) } . \end{align*}


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


\begin{align*} \frac { 4 ^ m } { 2 m + 1 } \leq \binom { 2 m } { m } , \end{align*}


\begin{align*} H = G _ m \cap G _ { s ( K _ j ) } = G _ m \cap G _ { s ( K _ 1 ) } \cap \dotsb \cap G _ { s ( K _ t ) } . \end{align*}


\begin{align*} \mathcal { S } ( h , k ) = \mathcal { I } _ 0 ( h , k ) + \mathcal { I } _ 1 ( h , k ) + \mathcal { E } ( h , k ) , \end{align*}


\begin{align*} U _ i f ( y ) = & ~ \frac { 1 } { R ( i + 1 , i ) } \sum \limits _ { x \lessdot y } f ( x ) , \\ D _ { i + 1 } f ( x ) = & ~ \frac { 1 } { \pi _ { i + 1 } ( X _ { x } ) } \sum \limits _ { y \gtrdot x } \pi _ { i + 1 } ( y ) f ( y ) , \end{align*}


\begin{align*} \langle W f , W ( M _ { - \omega } T _ x g _ 0 ) \rangle = \iint _ { \R ^ { 2 d } } W f ( \xi , \eta ) W ( M _ { - \omega } T _ x g _ 0 ) ( \xi , \eta ) \ , d ( \xi , \eta ) > 0 , \forall ( x , \omega ) \in \R ^ { 2 d } . \end{align*}


\begin{align*} f = \sum \limits _ { k = - \infty } ^ { \infty } \sum \limits _ { Q } \mu ( Q ) { \widetilde { \widetilde { D } } } _ { k } ( x , x _ Q ) D _ { k } ( f ) ( x _ Q ) = \sum \limits _ { k = - \infty } ^ { \infty } \sum \limits _ { Q } \mu ( Q ) D _ { k } ( x , x _ Q ) { \bar { \bar { D } } } _ { k } ( f ) ( x _ Q ) , \end{align*}


\begin{align*} \frac { \partial } { \partial t } ( e ^ { \lambda ( q , t ) } - 1 ) - 1 & { } \stackrel { \phantom { \eqref { e q _ l a m b d a _ p r i m o } } } { = } e ^ { \lambda ( q , t ) } \frac { \partial } { \partial t } ( \lambda ( q , t ) ) - 1 = \\ & { } \stackrel { \eqref { e q _ l a m b d a _ p r i m o } } { = } \frac { e ^ { \lambda ( q , t ) } ( 1 - q ) } { e ^ { \lambda ( q , t ) \cdot q } - q e ^ { \lambda ( q , t ) } } - 1 = \frac { e ^ { \lambda ( q , t ) } - e ^ { \lambda ( q , t ) \cdot q } } { e ^ { \lambda ( q , t ) \cdot q } - q e ^ { \lambda ( q , t ) } } . \end{align*}


\begin{align*} W ( f , g ) ( x , \omega ) = 2 ^ d e ^ { 4 \pi i x \cdot \omega } V _ { g ^ \vee } f ( 2 x , 2 \omega ) = 2 ^ d A ( f , g ^ \vee ) ( 2 x , 2 \omega ) . \end{align*}


\begin{align*} \sigma = e ^ { - z _ 6 \cdot } . \end{align*}


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


\begin{align*} y _ n ( x ) : = \sum _ { k = 0 } ^ n \frac { ( n + k ) ! } { ( n - k ) ! k ! } \left ( \frac { x } { 2 } \right ) ^ k . \end{align*}


\begin{align*} \theta ( \tau ) = \sqrt { \tfrac { i } { \tau } } \ \theta ( - \tfrac { 1 } { \tau } ) \end{align*}


\begin{align*} f ( x _ 1 , \ldots , x _ n ) = \sum _ { i = 1 } ^ n c _ i \prod _ { j = 1 } ^ n x _ j ^ { E _ { i j } } \end{align*}


\begin{align*} \partial _ \pm = \{ x = ( x ^ 1 , x ^ 2 ) \in \R ^ 2 : x ^ 1 = \pm R \} . \end{align*}


\begin{align*} B _ r ( \theta _ I ) = \sqrt { \frac { 1 } { N _ s } } \textbf { a } _ 1 ^ T ( \theta _ I ) \textbf { a } _ 1 ^ * ( \theta _ I ) = \sqrt { N _ s } . \end{align*}


\begin{align*} H _ { 0 , \lambda } = 0 , H _ { n , \lambda } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \lambda } \binom { \lambda } { k } ( - 1 ) ^ { k - 1 } , ( n \in \mathbb { N } ) , ( \mathrm { s e e } \ [ 9 ] ) . \end{align*}


\begin{align*} \kappa _ { \alpha \alpha } ^ { ( \ell ) } = K _ { \alpha \alpha } ^ { ( \ell ) } + O ( n ^ { - 1 } ) . \end{align*}


\begin{align*} & T ( k ) = \alpha k + \mathcal { O } ( k ^ 2 ) , \alpha \neq 0 , \ k \rightarrow 0 , \\ & 1 + R _ { \pm } ( k ) = \alpha _ { \pm } k + \mathcal { O } ( k ^ 2 ) , k \rightarrow 0 . \end{align*}


\begin{align*} d x ( t ) = ( \alpha _ { - 1 } x ( t ) ^ { - 1 } - \alpha _ { 0 } + \alpha _ { 1 } x ( t ) - \alpha _ { 2 } x ( t ) ^ { 2 } ) d t + \sigma x ( t ) ^ { \theta } d B _ t \end{align*}


\begin{align*} h _ A ( \tfrac { \eta } { n } ) = h _ A ( \tfrac { 1 } { \eta } ) { \Longleftrightarrow } \frac { \eta } { n } = \frac { 1 } { \eta } \ ; \vee \ ; \frac { \eta } { n } = \eta \Longleftrightarrow \eta = \sqrt { n } \ ; \vee \ ; n = 1 . \end{align*}


\begin{align*} P _ n = \sum _ { i = 1 } ^ n X _ i ^ 2 - n . \end{align*}


\begin{align*} \frac { T } { 4 } \int _ { \frac { 1 } { 2 } - \epsilon } ^ { \frac { 1 } { 2 } + \epsilon } \log \left ( \frac { \sigma ^ 2 + T ^ 2 } { 4 } \right ) \ d \sigma & = \epsilon T \log \left ( \frac { T } { 2 } \right ) - \epsilon T + \epsilon T + \epsilon \ \mathcal { O } \left ( \frac { 1 } { T } \right ) \end{align*}


\begin{align*} \begin{aligned} \sup _ { Q _ s \in \mathfrak { Q } _ s , s \in \lbrace 1 , \ldots , k \rbrace } \min _ { G \in \mathcal { G } ( Q _ 0 , \ldots , Q _ k ) } \det ( G ^ \intercal G ) \geq \sup _ { Q _ s \in \mathfrak { Q } _ s ' , s \in \lbrace 1 , \ldots , k \rbrace } \min _ { G \in \mathcal { G } ( Q _ 0 , \ldots , Q _ k ) } \det ( G ^ \intercal G ) , \end{aligned} \end{align*}


\begin{align*} \rho ( x ) : = 2 \int _ { x ' = 0 } ^ x \frac { \mathrm { d } x ' } { t _ + ( x ' ) } . \end{align*}


\begin{align*} T : = \frac { 1 } { \max _ { t \ge H } \frac { x ^ { - \nu _ 2 ( t ) } } { t } } = \min _ { t \ge H } t x ^ { \nu _ 2 ( t ) } . \end{align*}


\begin{align*} T : = \left \{ \begin{pmatrix} \mathrm { e } ^ { \mathbf { i } \beta _ { 1 } } & & \\ & \ddots & \\ & & \mathrm { e } ^ { \mathbf { i } \beta _ { p + q } } \end{pmatrix} \ ; , \ ; \beta _ { 1 } , \ldots , \beta _ { p + q } \ , \in \mathbb { R } \right \} \ ; , \end{align*}


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


\begin{align*} \sum _ { a \in S } a = \frac { 1 } { 2 } n ^ { 3 / 2 } + O ( n ^ { 1 1 1 / 8 0 + \varepsilon } ) . \end{align*}


\begin{align*} \widetilde { g } = S _ { g , \L } ^ { - 1 } g \end{align*}


\begin{align*} ( \eta _ { ( \Sigma , U ) } h ) ( s ) : = \int _ { \mathbb { R } } h ( - s , v ) \mathrm { d } v , \end{align*}


\begin{align*} \tilde { g } _ i ( z _ 1 \dots , z _ n ) = \prod _ { ( \sigma ( z _ 1 ) , \dots , \sigma ( z _ n ) ) \in S _ n } \ ; g _ i ( \sigma ( z _ 1 ) , \dots , \sigma ( z _ n ) ) \ , , \end{align*}


\begin{align*} \| C \hat { V } \| _ F ^ 2 = { \rm T r } \left ( \hat { V } ^ { T } S ^ { T } S \hat { V } \right ) + { \rm T r } \left [ \hat { V } ^ { T } ( C ^ { T } C - S ^ { T } S ) \hat { V } \right ] \geq \| S \hat { V } \| ^ 2 _ F - \theta ( k + \sqrt { k } \xi ) \| C \| ^ 2 _ F . \end{align*}


\begin{align*} \theta _ 0 : = \frac { 1 } { 4 } ( \lambda _ 1 + \gamma ) ^ 2 , \theta ^ * : = \frac { ( \lambda _ 1 + \gamma ) ^ 2 } { 2 ( \lambda _ 2 + \gamma ) } \end{align*}


\begin{align*} b _ { G } : = \inf \Big \{ \forall s \in [ 0 , T ] , \tilde { b } \geq 0 : \sup _ { | u | \geq \tilde { b } } | G ( s , u ) | = 0 \Big \} . \end{align*}


\begin{gather*} \varphi = \exp \left ( \log \varphi \right ) = \exp f , \\ w = \exp \left ( \log w \right ) = \exp g , \end{gather*}


\begin{align*} \lambda + \mu X = \lambda ^ { 2 ^ h } + \mu ^ { 2 ^ h } X ^ { 2 ^ h } \end{align*}


\begin{align*} \lambda ( t ) = C ( u _ 0 ) ( T - t ) ^ { \frac { 2 } { \alpha } } ( 1 + O ( \left | \ln ( T - t ) \right | ^ { - 1 } ) ) t \to T . \end{align*}


\begin{align*} P _ { \theta } ^ { \mathrm { i n } } ( M ) : = P _ { \theta } ^ { \mathrm { i n } } ( M - a ) \cup { ( a ) } \end{align*}


\begin{align*} \pi _ { 3 , \ell / r } \circ ( \chi _ 2 \circ \chi _ 1 ) = : ( \chi _ 2 \circ \chi _ 1 ) _ { \ell / r } \circ \pi _ { 1 , \ell / r } . = ( \chi _ { 2 , \ell / r } \circ \chi _ { 1 , \ell / r } ) \circ \pi _ { 1 , \ell / r } \end{align*}


\begin{align*} \mathbf { E } ( \mathfrak { g } , s ) : = E ^ n _ k ( \mathfrak { g } \ , \sigma ^ { - 1 } , s ; \chi ) , \ , \ , \ , \ , \mathfrak { g } \in G _ N ( \mathbb { A } ) , \end{align*}


\begin{align*} ( a b ) b = \frac { 1 } 2 ( \alpha b + a b ) , ( a b ) a = \frac { 1 } 2 ( \alpha a + a b ) , ( a b ) ( a b ) = \frac { \alpha } 4 ( a + b + 2 a b ) . \end{align*}


\begin{align*} \sigma ^ { \varphi } ( \delta _ { \varphi } ) = \delta _ { \varphi } \nu ^ { - 1 } . \end{align*}


\begin{align*} W _ { \mathbf { e } , \mathbf { f } } ( X , Y ) = \prod _ { i = 1 } ^ r \left ( \frac { 1 } { 1 - X ^ { f _ i } } \right ) \sum _ { I \subseteq [ r ] } ( - 1 ) ^ { | I | } \frac { X ^ { \sum _ { i \in I } f _ i } } { 1 - X ^ { 4 n + \sum _ { i \in I } e _ i f _ i } Y ^ { n + 2 } } , \end{align*}


\begin{align*} M _ g = \left ( g ( x _ j - y _ k ) \right ) _ { j , k = 1 } ^ N \end{align*}


\begin{align*} \rho _ n \ge 0 \ ; \R ^ N , \int _ { \R ^ N } \rho _ n = \lambda \end{align*}


\begin{align*} \lim _ { s \rightarrow \infty } s h ( s ) = \lim _ { s \rightarrow \infty } \int _ 0 ^ \infty \frac { s ^ 2 } { s ^ 2 + u ^ 2 } d \mu ( u ) = 1 . \end{align*}


\begin{align*} \lim _ { M \to \infty } \frac { | \Lambda ( M ) | } { | \Lambda ( M , \Gamma ) | } = \lim _ { M \to \infty } \frac { | \Lambda ( M ) | } { M ^ { 2 } } \frac { M ^ { 2 } } { | \Lambda ( M , \Gamma ) | } = | H _ { 1 } ( Y ) | . \end{align*}


\begin{align*} \chi _ A ( x ) : = \left \{ \begin{array} { l l } 1 & x \in A \\ 0 & \end{array} \right . \end{align*}


\begin{align*} \sum \limits _ { Y \in \mathcal Y } \alpha _ Y \lambda _ { Y , \delta , \ell } & = \frac { 1 } { q ^ { ( k - t ) ^ 2 } { k \choose t } _ q } \sum \limits _ { i = 0 } ^ { k - t } ( - 1 ) ^ { k - t - i } q ^ { { k - t - i \choose 2 } } { k - t \choose i } _ q { k + i - \ell \choose i } _ q \\ & = \frac { { k - j \choose \ell } _ q } { { k \choose \ell } _ q } . \end{align*}


\begin{align*} T B ( f ) = 1 3 T B _ { 6 } ( f ) . \end{align*}


\begin{align*} 0 < \Im \int x \cdot \nabla v ( x , t ) \bar { v } ( x , t ) d x = - \Im \int x \cdot \nabla u ( x , - t ) \bar { u } ( x , - t ) d x , \ ; \forall t > 0 , \end{align*}


\begin{align*} \norm { u } _ { W ( \sigma _ { j + 1 } R ) , 2 \alpha _ { j + 1 } p ^ * } \leq \prod _ { k = 1 } ^ j \left ( C \frac { ( \rho / p ^ * ) ^ { 2 k } } { ( \sigma - \sigma ' ) ^ 2 } \norm { \lambda ^ { - 1 } } _ { W ( R ) , q } \norm { \Lambda } _ { W ( R ) , p } \right ) ^ { 1 / ( 2 \alpha _ k ) } \norm { u } _ { W ( \sigma R ) , \rho } ^ { \prod _ { k = 1 } ^ j \gamma _ k } . \end{align*}


\begin{align*} \beta = ( 1 - \phi ) \gamma \end{align*}


\begin{align*} \widetilde { m } x : = \sum _ i ( m y _ i ) z _ i , x \in A \otimes B , E x = \sum _ i y _ i z _ i \textrm { f o r } y _ i \in A \ , { } ^ I \ ! \otimes ^ I B , z _ i \in A \otimes B , \end{align*}


\begin{align*} h _ i ( \boldsymbol { f } _ O ^ { d _ i - 1 } \left ( \boldsymbol { f } ( \mathbf { x } _ k , \tilde { \mathbf { u } } _ k ) \right ) ) = 0 , \forall i \in \mathbb { Z } _ { [ 1 , m ] } . \end{align*}


\begin{align*} | E _ 1 | ~ & = ~ | \{ x \in K : | B f _ 1 ( x ) | > \lambda / 2 \} | \\ & = ~ \bigg | \bigg ( \bigcup _ i W _ i \bigg ) \bigcap \{ x \in K : | B f _ 1 ( x ) | > \lambda / 2 \} \bigg | ~ + ~ \bigg | \bigg ( \bigcup _ i W _ i \bigg ) ^ \complement \bigcap \{ x \in K : | B f _ 1 ( x ) | > \lambda / 2 \} \bigg | . \end{align*}


\begin{align*} M = \left \{ x \in \mathbb { R } ^ n \middle | A x \leqslant b \right \} . \end{align*}


\begin{align*} \gamma _ i ( x ) = x - \sigma _ i ( x ) c , \end{align*}


\begin{align*} \begin{aligned} D _ { { \rm l e f t } } u ( x ) = \lim _ { t \rightarrow 0 ^ + } \frac { u ( x ) - u ( x - t ) } { t } \\ D _ { { \rm r i g h t } } u ( x ) = \lim _ { t \rightarrow 0 ^ + } \frac { u ( x ) - u ( x + t ) } { t } , \end{aligned} \end{align*}


\begin{align*} ( - \Delta ) ^ s u ( x ) & = c _ { n , s } \lim _ { \delta \to 0 ^ + } \int _ { \R ^ n _ + \setminus B _ \delta ( x ) } \left ( \frac 1 { \vert x - y \vert ^ { n + 2 s } } - \frac 1 { \vert x _ \ast - y \vert ^ { n + 2 s } } \right ) \big ( u ( x ) - u ( y ) \big ) \dd y \\ & + 2 c _ { n , s } u ( x ) \int _ { \R ^ n _ + } \frac { \dd y } { \vert x _ \ast - y \vert ^ { n + 2 s } } . \end{align*}


\begin{align*} C _ { G \rtimes S } ( \underline { \lambda } , \sigma ) = \langle { \rm K e r } \ , A _ \sigma \cap G , ( \underline { \lambda } , \sigma ) \rangle \ , . \end{align*}


\begin{align*} { L } f ( \xi ) = \frac { 1 } { \xi ^ { d + 1 } \Lambda Q ( \xi ) } \int _ 0 ^ \xi f ( \xi ' ) \Lambda Q ( \xi ' ) ( \xi ' ) ^ { d + 1 } d \xi ' . \end{align*}


\begin{align*} \begin{dcases} \rho _ t + \rho _ x + b u _ x = 0 & ( 0 , + \infty ) \times ( 0 , \delta L ) , \\ u _ t - u _ { x x } + u _ x + b \rho _ x = 0 \ & ( 0 , + \infty ) \times ( 0 , \delta L ) , \end{dcases} \end{align*}


\begin{align*} h _ \ell = \sqrt { \beta _ { s s } \beta _ { s \ell } } \mathbf { h } _ { s s } ^ H \mathbf { \Theta } \mathbf { h } _ { s \ell } , \end{align*}


\begin{align*} ( K _ H ^ * y ) ( s ) = K _ H ( T , s ) y ( s ) + \int _ s ^ T ( y ( t ) - y ( s ) ) \frac { \partial } { \partial t } K _ H ( t , s ) d t \end{align*}


\begin{align*} \partial _ t \rho = & \nabla \cdot \left ( \nabla \rho + \rho \nabla \phi \right ) , \\ - \Delta \phi = & \rho + f ( x , y ) , \end{align*}


\begin{align*} \widehat { \mathbf { 1 } _ { P _ { \textbf { c } } } } ( \pmb { \xi } ) = \begin{cases} \exp ( 2 \pi i \frac { \pmb { \xi } \cdot \textbf { c } } { p ^ { 2 k } } ) \cdot \frac { p ^ { w - k } } { p ^ { k n } } , & \pmb { \xi } \equiv \alpha \textbf { D } _ { \textbf { c } } ( p ^ k ) \alpha \in \mathbb { Z } / p ^ { k } \mathbb { Z } \\ 0 & . \end{cases} \end{align*}


\begin{align*} S ^ { - 1 } = \begin{pmatrix} D ^ T & - B ^ T \\ - C ^ T & A ^ T \end{pmatrix} . \end{align*}


\begin{align*} M ( p , \{ 1 \} ) = { \pi ^ 2 \over 6 } \left ( 1 - { 1 \over p } \right ) \left ( 1 - { 2 \over p } \right ) \leq \frac { \pi ^ 2 } { 6 } \ \ \ \ \ \hbox { ( $ p \geq 3 $ ) } . \end{align*}


\begin{align*} \underline { \alpha _ 2 } = \min \ ! \left ( \alpha _ { 0 2 } , \dfrac { 3 | 2 c _ 2 - 1 | \varepsilon _ 2 } { M _ g } \right ) . \end{align*}


\begin{align*} \mathcal { E } _ \alpha ( f , g ) = \int _ { 0 } ^ { + \infty } x f ' ( x ) g ' ( x ) \gamma _ { \alpha , 1 } ( d x ) . \end{align*}


\begin{align*} F ( x , \omega ) = \left ( V _ g f \ , \overline { V _ f g } \right ) ( x , \omega ) . \end{align*}


\begin{align*} \begin{array} { c l } \langle \bar R ( X , Y ) Z , V \rangle = & \langle \nabla ^ { \perp } _ X \sigma ( Y , Z ) , V \rangle - \langle \sigma ( \nabla _ X Y , Z ) , V \rangle - \langle \sigma ( Y , \nabla _ X Z ) , V \rangle \\ \\ & - \langle \nabla ^ { \perp } _ Y \sigma ( X , Z ) , V \rangle + \langle \sigma ( \nabla _ Y X , Z ) , V \rangle + \langle \sigma ( X , \nabla _ Y Z ) , V \rangle . \end{array} \end{align*}


\begin{align*} 1 - F _ n ( \mu _ n + s _ n y ) \sim 1 - \Phi ( y ) , y = o ( N _ n ^ { 1 / 6 } ) . \end{align*}


\begin{align*} [ x , y , z ] _ N = & [ N x , N y , z ] + [ x , N y , N z ] + [ N x , y , N z ] \\ - & N \Big ( [ N x , y , z ] + [ x , N y , z ] + [ x , y , N z ] - N [ x , y , z ] \Big ) , \end{align*}


\begin{align*} M _ { q _ 1 , q _ 2 } ( p ) : = \frac { 2 } { \phi ( p ) } \sum _ { \chi \in X _ p ^ - } \chi ( q _ 1 ) \overline { \chi } ( q _ 2 ) \vert L ( 1 , \chi ) \vert ^ 2 . \end{align*}


\begin{align*} f ( t ) & = \langle z | e _ { \lambda } ^ { a ^ { \dagger } a } ( t ) | z \rangle = \sum _ { k = 0 } ^ { \infty } \frac { t ^ { k } } { k ! } \langle z | ( a ^ { \dagger } a ) _ { k , \lambda } | z \rangle \\ & = \sum _ { k = 0 } ^ { \infty } \frac { t ^ { k } } { k ! } \sum _ { l = 0 } ^ { k } S _ { 2 , \lambda } ( k , l ) | z | ^ { 2 l } = \sum _ { k = 0 } ^ { \infty } \phi _ { k , \lambda } ( | z | ^ { 2 } ) \frac { t ^ { k } } { k ! } . \end{align*}


\begin{align*} A ( f , g ) ( \l ) = \langle \pi ( - \l / 2 ) f , \pi ( \l / 2 ) g \rangle = \langle f , \rho ( \l ) g \rangle . \end{align*}


\begin{align*} E _ { U } = U \cap E _ { \mu } ^ { [ \alpha , \beta ] , \varepsilon _ 2 } \cap \limsup _ { n \to + \infty } B _ n . \end{align*}


\begin{align*} Y _ { \alpha \beta } ^ { i j } \cdot ( Y _ { \alpha \omega } ^ { m i } Y _ { \beta \omega } ^ { m j } ) = - ( Y _ { \alpha \omega } ^ { m j } Y _ { \beta \omega } ^ { m i } ) . \end{align*}


\begin{align*} \pi _ { a , b } ( x ) = \left ( \int _ { a } ^ { b } m ( x , \theta ) \mathrm { d } x \right ) ^ { - 1 } = \frac { e ^ { - \frac { 2 } { \sigma ^ { 2 } } \int _ { a } ^ { x } f ( y , \theta ) \mathrm { d } y } } { \int _ { a } ^ { b } e ^ { - \frac { 2 } { \sigma ^ { 2 } } \int _ { a } ^ { x } f ( y , \theta ) \mathrm { d } y } \mathrm { d } x } . \end{align*}


\begin{align*} ( - \triangle _ { g } ) ^ { \alpha } u ( x ) & = \int _ { \mathbb { R } ^ { d } } g \bigg { ( } \frac { | u ( x ) - u ( y ) | } { | x - y | ^ { \alpha } } \bigg { ) } \frac { u ( x ) - u ( y ) } { | u ( x ) - u ( y ) | } \frac { d y } { | x - y | ^ { d + \alpha } } , \end{align*}


\begin{align*} f ( 0 , x _ 2 , \ldots , x _ m ) \ = \ g ( x _ 2 , \ldots , x _ m ) \end{align*}


\begin{align*} \vartheta _ 3 ( z , \tau ) = \sum _ { k \in \Z } e ^ { \pi i \tau k ^ 2 } e ^ { 2 \pi i k z } , \ , z \in \C , \tau \in \mathbb { H } . \end{align*}


\begin{align*} X ( t ) = E ( t ) X _ 0 - \int _ 0 ^ t E ( t - s ) A P F ( X ( s ) ) \ , \mathrm { d } s + \int _ 0 ^ t E ( t - s ) \mathrm { d } W ( s ) . \end{align*}


\begin{align*} \frac { \int _ { \tilde { \Omega } _ { \varepsilon } } | \nabla P _ { \tilde { \Omega } _ { \varepsilon } } | ^ 2 } { \int _ { \tilde { \Omega } _ { \varepsilon } } \tilde { m } _ 0 \ , | P _ { \tilde { \Omega } _ { \varepsilon } } | ^ 2 } = \tilde { \lambda } _ 0 + \Phi e ^ { - \beta _ { \varepsilon } \tilde { \Psi } _ { \varepsilon } ( \mathbf { x } _ { \varepsilon } ) } + o ( e ^ { - \beta _ { \varepsilon } \tilde { \Psi } _ { \varepsilon } ( \mathbf { x } _ { \varepsilon } ) } ) . \end{align*}


\begin{align*} \lambda _ { Z , n } = \lambda _ { Z } \cdot \lambda _ { \mathcal { R } Z } \cdot \ldots \cdot \lambda _ { \mathcal { R } ^ { n - 1 } Z } . \end{align*}


\begin{align*} \begin{cases} A & = ( L R \times \overline { L } R ) \cdot ( L ^ { 2 } R \times \overline { L } R ) , \\ B & = ( L R \times \overline { L } R ) \cdot ( L ^ { 2 } R \times L R ) , \\ C & = ( L R \times \overline { L } R ) \cdot ( L R \times \overline { L } R ) . \end{cases} \end{align*}


\begin{align*} 0 = E _ { 0 } \subset E _ { 1 } \subset . . . . \subset E _ { n - 1 } \subset E _ n = E , \end{align*}


\begin{align*} \begin{aligned} & b \Phi ( a _ 0 \otimes \dots \otimes a _ { k + 1 } ) \\ = & \sum _ { i = 0 } ^ k ( - 1 ) ^ i \Phi ( a _ 0 \otimes \dots \otimes a _ i a _ { i + 1 } \otimes \dots \otimes a _ { k + 1 } ) + ( - 1 ) ^ { k + 1 } \Phi ( a _ { k + 1 } a _ 0 \otimes a _ 1 \otimes \dots \otimes a _ { k } ) . \end{aligned} \end{align*}


\begin{align*} q ( x , t ) = 2 i D ^ { - 2 } _ { \infty } ( \xi ) \left [ E _ { 1 } ^ { ( 1 2 ) } + \lim _ { k \rightarrow \infty } k \Delta _ { \eta } ^ { ( 1 2 ) } ( k ) \right ] . \end{align*}


\begin{align*} \bigg | \bigg ( \bigcup _ i W _ i \bigg ) \bigcap \{ x \in K : | B f _ 1 ( x ) | > \lambda / 2 \} \bigg | & ~ = ~ \bigg | \bigcup _ i W _ i \cap \{ x \in K : | B f _ 1 ( x ) | > \lambda / 2 \} \bigg | \\ & ~ \leq ~ \bigg | \bigcup _ i W _ i \bigg | \\ & ~ \leq ~ \sum \limits _ i | W _ i | \\ & ~ \leq ~ \| f \| _ { L ^ 1 ( K ) } ~ \lambda ^ { - 1 } . \end{align*}


\begin{align*} \Omega = \{ ( z _ 1 , \dots , z _ n ) \in \mathbb D ^ n \ , : \ , f _ i ( z _ 1 , \dots , z _ n ) = 0 \ , , \ , i = 1 , \dots , n \} . \end{align*}


\begin{align*} u ( t ) = U ( t ) \phi - \int _ 0 ^ t U ( t - \tau ) \vartheta ( \tau ) d \tau , \end{align*}


\begin{align*} U ^ 1 ( x ) = \lambda _ 2 Q ( \mu _ 2 x + x _ 2 ) , \end{align*}


\begin{align*} x ^ n \triangleleft x ^ m = ( n + 1 ) x ^ { n + m } , \end{align*}


\begin{align*} U ( w ) T ( w ) = V ( w ) S ( w ) . \end{align*}


\begin{align*} g ( k ) = \alpha \ \Leftrightarrow \ \exists t ( \forall s \geq t ) ( g _ s ( k ) = \alpha ) . \end{align*}


\begin{align*} H = ( H _ 1 , \dots , H _ m ) : G \to \mathfrak { a } . \end{align*}


\begin{align*} S ( H _ 3 , 3 f ) = \frac { 5 f + c _ a '' } { 1 8 } \hbox { w h e r e } c _ a '' : = \begin{cases} - 8 a - 5 & \hbox { i f $ a \equiv 0 \pmod 3 $ } , \\ 8 a + 3 & \hbox { i f $ a \equiv 2 \pmod 3 $ } , \end{cases} \end{align*}


\begin{align*} \frac { g - 1 } { | G | } = b - 1 \ge \Big [ \frac { { \rm r k } ( G ) + 1 } { 2 } \Big ] - 1 . \end{align*}


\begin{align*} p ( \rho ) - p ( r ) - p ' ( r ) ( \rho - r ) \leq \begin{cases} c ( \rho - r ) ^ 2 , & \rho \in ( \alpha _ 1 , \bar \rho - \alpha _ 1 ) , \\ 1 + p ' ( r ) r - p ( r ) , & \rho \in [ 0 , \alpha _ 1 ] , \\ 2 p ( \rho ) , & \rho \in [ \bar \rho - \alpha _ 1 , \bar \rho ) . \end{cases} \end{align*}


\begin{align*} \phi ( u , v ) = \phi _ \ell ( u ) + \phi _ r ( v ) + \frac { p } { 2 \pi } ( u + v ) , \end{align*}


\begin{align*} \mu _ h : = \dd c \otimes \P _ h \end{align*}


\begin{align*} \widetilde { \mathfrak { L } } _ \alpha f & : = \Big ( \sqrt { x } \Big ( \frac { d } { d x } - \Big ( \frac { \alpha - 1 / 2 } { x } - 1 \Big ) \Big ) \Big ( \sqrt { x } \frac { d f } { d x } \Big ) \\ & = x \frac { d ^ 2 f } { d x ^ 2 } + ( - \alpha + x + 1 ) \frac { d f } { d x } . \end{align*}


\begin{align*} \gamma _ 1 \colon & = - \limsup _ { n \to \infty } \frac { \max _ { Q \in \mathcal { Q } _ n } \log _ { b ^ { - 1 } } P _ n ( Q ) } { n } , \\ \gamma _ 2 \colon & = - \limsup _ { n \to \infty } \frac { \min _ { Q \in \mathcal { Q } _ n } \log _ { b ^ { - 1 } } P _ n ( Q ) } { n } . \end{align*}


\begin{align*} j ( g _ { \infty } , z _ 0 ) ^ { - k } \delta ( z ) ^ { \frac { s - k } { 2 } } \int _ { \mathfrak { Z } } \delta ( w , z ) ^ { - k } | \delta ( w , z ) | ^ { k - s } \delta ( w ) ^ { \frac { k + s } { 2 } } f ( w ) d w = \end{align*}


\begin{align*} & \int ^ { T } _ 0 \int _ { \Omega } ( \rho _ { N } \partial _ t \psi + \rho _ { N } u _ { N } \cdot \nabla \psi - \varepsilon \nabla \rho _ { N } \cdot \nabla \psi ) \ , d x d t + \int _ { \Omega } \rho _ 0 \psi ( 0 , x ) \ , d x \\ & \qquad \qquad \qquad = \int _ 0 ^ T \int _ { \Gamma _ { \rm { i n } } } \rho _ B u _ B \cdot \nu ( x ) \psi \ , d \sigma ( x ) d t ; \end{align*}


\begin{align*} & \sum _ { i , j \geq 1 } \frac { \omega _ { i , j } ^ { [ 0 ] } } { \lambda ^ { i + 1 } \mu ^ { j + 1 } } = \frac { B ( \lambda ) B ( \mu ) ( ( \lambda - v ) ( \mu - v ) - 4 w ) - 1 } { 2 \ , ( \lambda - \mu ) ^ 2 } , \\ & \frac 1 \lambda + \sum \limits _ { i \geq 1 } \frac { \varphi _ i ^ { [ 0 ] } } { \lambda ^ { i + 1 } } = B ( \lambda ) , \end{align*}


\begin{align*} f - \frac { \partial f } { \partial \kappa _ { i } } \kappa _ { i } = H _ { k } ^ { \frac { 1 } { k } } \left ( 1 - \frac { \sigma _ { k - 1 } ( \kappa | i ) \kappa _ { i } } { k \sigma _ { k } ( \kappa ) } \right ) \geq \frac { 1 } { k } H _ { k } ^ { \frac { 1 } { k } } \left ( 1 - \frac { \sigma _ { k - 1 } ( \kappa | i ) \kappa _ { i } } { \sigma _ { k } ( \kappa ) } \right ) . \end{align*}


\begin{align*} [ x ^ e , g ] _ { \circ } = [ x , g ] _ { \circ } ^ e = 0 . \end{align*}


\begin{align*} H _ j ( X ^ { m + 2 } ) = \frac { 1 } { A _ j ^ n } D ^ j \left [ h ( X ^ { m + 2 } ) \right ] \ , . \end{align*}


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


\begin{align*} \mathbf { g } _ t ^ \mathcal { F } : = ( \mathrm { p r } _ { t \star } \otimes _ { \mathcal { X } _ \star } \mathrm { p r } _ { t \star } ) ( \mathbf { g } ) = ( \mathrm { p r } _ { t } \otimes _ { \mathcal { X } } \mathrm { p r } _ { t } ) ( \mathbf { g } ) = \mathbf { g } _ t \end{align*}


\begin{align*} \begin{gathered} \phi _ k : = \begin{bmatrix} - y _ { k - 1 } & - y _ { k - 2 } & u _ { k - 1 } & u _ { k - 2 } \end{bmatrix} ^ \top , \\ \theta ^ * _ 1 : = \begin{bmatrix} 0 . 5 & - 0 . 1 & 1 & - 0 . 4 \end{bmatrix} ^ \top , \\ \theta ^ * _ 2 : = \begin{bmatrix} - 1 . 4 & 0 . 3 & 1 & - 1 . 3 \end{bmatrix} ^ \top , \end{gathered} \end{align*}


\begin{align*} \tilde { \lambda } _ 0 = \frac { 1 } { \int _ { \R ^ N } m u ^ 2 } \ ; . \end{align*}


\begin{align*} \mathcal { L } ^ \alpha ( f ) ( x ) = - \langle x ; \nabla ( f ) ( x ) \rangle + \sum _ { j = 1 } ^ d \partial _ j D ^ { \alpha - 1 } _ j ( f ) ( x ) , \end{align*}


\begin{align*} \lim _ { h \to 0 } \frac 1 h \bigg ( \frac 1 { \vert h e _ 1 - y \vert ^ n } - \frac 1 { \vert h e _ 1 + y \vert ^ n } \bigg ) & = \frac { 2 n y _ 1 } { \vert y \vert ^ { n + 2 } } . \end{align*}


\begin{align*} H & = \nabla ^ 2 f ( x ) - \sum _ { i = 1 } ^ { m } \lambda _ i ( x ) \nabla ^ 2 h _ i ( x ) . \end{align*}


\begin{align*} T _ \Omega f ( x ) = \lim _ { \epsilon \to 0 } \int _ { | y | > \epsilon } \dfrac { \Omega ( y / | y | ) } { | y | ^ n } f ( x - y ) d y . \end{align*}


\begin{align*} \partial _ \xi ^ k \Theta _ i ( \xi ) = O \left ( \xi ^ { - \gamma + 2 i - k - 2 } \ln \xi \right ) \xi \to + \infty . \end{align*}


\begin{align*} \big \{ ( x , y ) \in \ \R ^ 2 \ y ^ 2 = x , y \geq 0 \big \} , \end{align*}


\begin{align*} \mathbb { P } \left ( V _ j = i \right ) = \frac { \epsilon _ { i - 1 } - \epsilon _ i } { 1 - \epsilon _ m } . \end{align*}


\begin{align*} | \dot { y } _ R ( t ) | = \Big | 2 R \Im \int \nabla \varphi \left ( \frac { x } { R } \right ) \cdot \nabla u \bar { u } d x \Big | \le R \| \nabla \varphi \| _ \infty \| \nabla u \| _ 2 \| u \| _ 2 \le C R t > 0 . \end{align*}


\begin{align*} \sigma ^ { \varphi _ S } ( a ) = \delta _ { \varphi } \sigma ^ { \varphi } ( a ) \delta _ { \varphi } ^ { - 1 } \end{align*}


\begin{align*} S = \{ t \in T \mid d ( t _ 0 , t ) \in \N _ 0 \} . \end{align*}


\begin{align*} \begin{cases} \partial _ t \rho ( t , u ) = [ - ( - \Delta ) ^ { \gamma / 2 } \rho ^ 2 ] ( t , u ) , & ( t , u ) \in [ 0 , T ] \times \mathbb { R } , \\ \rho ( 0 , u ) = g ( u ) , & u \in \mathbb { R } , \end{cases} \end{align*}


\begin{align*} \Lambda ( \rho ) = \int _ 0 ^ \infty \exp \left ( - \rho y ^ { - 1 } \right ) \ , q ( y ) \ , d y \sim b \ , \rho ^ { - ( \alpha - 1 ) } , \textrm { a s } \rho \to \infty , \end{align*}


\begin{align*} M ( p , H ) = \frac { \pi ^ 2 } { 6 } + o ( 1 ) \end{align*}


\begin{align*} H ( \rho _ 0 , u _ 0 , p ) = ( \rho , u ) . \end{align*}


\begin{align*} \langle u , v \rangle _ { U _ 0 } : = \langle Q ^ { - 1 / 2 } u , Q ^ { - 1 / 2 } v \rangle _ U \end{align*}


\begin{align*} \nabla _ v f ( x ) & = \left \langle \nabla f ( x ) , v \right \rangle = \lim _ { \varepsilon \to 0 } \frac { f ( x + \varepsilon v ) - f ( x ) } { \varepsilon } , \\ \nabla _ { v _ 2 } \nabla _ { v _ 1 } f ( x ) & = \left \langle \nabla ^ 2 f ( x ) , v _ 1 v ^ \top _ 2 \right \rangle _ { \mathrm { H S } } = \lim _ { \varepsilon \to 0 } \frac { \nabla _ { v _ 1 } f ( x + \varepsilon v _ 2 ) - \nabla _ { v _ 1 } f ( x ) } { \varepsilon } , \end{align*}


\begin{align*} \Phi ( S ) = \underset { v \sim S } { \mathbb { E } } \left [ M ( v , X ( k ) \setminus S ) \right ] , \end{align*}


\begin{align*} d = q ( u , c ) \end{align*}


\begin{align*} \begin{cases} - k _ { \varepsilon } \Delta \Psi _ { \varepsilon } + | \nabla \Psi _ { \varepsilon } | ^ 2 - \tilde { \lambda } _ 0 \underline { m } = 0 & \Omega \\ \Psi _ { \varepsilon } = - k _ { \varepsilon } \log ( w ) ( \frac { \mathbf { x } - \mathbf { q } } { k _ { \varepsilon } } ) & \partial \Omega \end{cases} \end{align*}


\begin{align*} g _ 0 ( x ) = { \rm d i s t } ^ { r ‐ \beta } ( x , \ , \Gamma ) \end{align*}


\begin{align*} & \sigma \tau | _ E \sigma ^ { - 1 } ( a , b , c ) \\ & = \left ( a + ( 1 - e ) b , e b , - 3 \frac { ( a + b ) ^ 2 } { e b } + 3 ( a + b ) - e b + e ^ { - 2 } \left ( 3 \frac { a ^ 2 } { b } + 3 a + b + c \right ) \right ) . \end{align*}


\begin{align*} M : = \biggl \{ ( \lambda , w ) \biggm | \lambda \in [ 0 , 1 ] ^ d , \ w _ I = \prod _ { i \in I } \lambda _ i , \ I \subseteq \{ 1 , \ldots , d \} , \ \lvert I \rvert \geq 2 \biggr \} . \end{align*}


\begin{align*} \kappa ^ n ( a ) \otimes \rho ^ { n } ( b ) = \sum _ { i = 1 } ^ n \Delta ( p _ i ) ( 1 \otimes \rho ^ n ( q _ i ) ) \end{align*}


\begin{align*} \eta _ { \alpha } ^ { \beta } [ \tau ] ( t ) : = \begin{cases} \mu ( \frac { t - \beta } { \tau } ) , \ ; \ ; \forall t \in ( \beta , + \infty ) , \\ 1 , \ ; \ ; \forall t \in [ \alpha , \beta ] , \\ \mu ( \frac { t - \alpha } { \tau } ) , \ ; \ ; \forall t \in ( - \infty , \alpha ) , \end{cases} \end{align*}


\begin{align*} \L = r S \Z ^ { 2 d } , S \in S p ( \R , 2 d ) \end{align*}


\begin{align*} \sum _ { \substack { F \in \mathcal { F } \\ { G \in \mathcal { G } } } } \varphi ( F , G ) \leq | \mathcal { G } | = f _ k ( P ) , \end{align*}


\begin{align*} D _ { u } ( z , \chi ) = L ( z , \chi ) L ( 2 z , \chi ^ { 2 } ) G ( z , \chi ) F _ { u } ( z , \chi ) . \end{align*}


\begin{align*} D _ { s } \phi ( X _ { t } ^ { x ; l ^ \epsilon } ) = \nabla \phi ( X _ { t } ^ { x ; l ^ \epsilon } ) D _ { s } X _ { t } ^ { x ; l ^ \epsilon } = \nabla \phi ( X _ { t } ^ { x ; l ^ \epsilon } ) \nabla _ { x } X _ { t } ^ { x ; l ^ \epsilon } ( \nabla _ { x } X _ { \gamma ^ { \epsilon } _ { s } } ^ { x ; l ^ \epsilon } ) ^ { - 1 } . \end{align*}


\begin{align*} \frac { d } { d t } \int x ^ 2 u ( x , t ) d x = \int x u ^ 2 ( x , t ) d x , \end{align*}


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


\begin{align*} \partial _ \sigma ( f ) ( x ) : = \sqrt { x } f ' ( x ) , \end{align*}


\begin{align*} \langle f , g \rangle = \int _ \R f ( t ) \ , \overline { g ( t ) } \ , d t \norm { f } _ 2 ^ 2 = \langle f , f \rangle . \end{align*}


\begin{align*} w _ k = s _ k \cdot s _ { k - 1 } \cdots s _ 2 \cdot s _ 1 \cdot s _ { 1 ^ { \prime } } \cdot s _ 2 \cdots s _ { k - 1 } \cdot s _ k . \end{align*}


\begin{align*} E ( \rho , \phi ) = \int _ { \Omega } \bigg ( \sum _ { i = 1 } ^ s \rho _ i \log \rho _ i + \frac { 1 } { 2 } \epsilon ( x ) | \nabla \phi | ^ 2 \bigg ) d x + \frac { 1 } { 2 \beta _ R } \int _ { \Gamma _ R } \phi ^ 2 d s \end{align*}


\begin{align*} \liminf _ { t - s \to \infty } \inf _ { x \in \Omega } u ( t , x ; s , u _ n , a , b ) \le m _ n : = \frac { 1 } { n } . \end{align*}


\begin{align*} \int _ X \left ( \frac { g } { \rho } \right ) ^ 2 \ , d \mu _ \omega = \int _ X g ^ 2 \ , d \mu _ X , \end{align*}


\begin{align*} V ( \mathbf { x } ) = ( 1 - \theta ) \frac { x ^ 2 + y ^ 2 } { 2 } + \frac { \kappa \left ( x ^ { 2 } + y ^ { 2 } \right ) ^ { 2 } } { 4 } \end{align*}


\begin{align*} \hat \Sigma : = { \rm d i a g } \left ( \lambda _ 1 + \lambda _ { 1 , 1 } \epsilon , \cdots , \lambda _ 1 + \lambda _ { 1 , k _ 1 } \epsilon , \lambda _ 2 + \lambda _ { 2 , 1 } \epsilon , \cdots , \lambda _ r + \lambda _ { r , k _ r } \epsilon \right ) , \end{align*}


\begin{align*} R ( \lambda ) g = \sigma \lim _ { \mathclap { n \to \infty } } R ( \lambda ) T ( t _ n ) f = \sigma \lim _ { \mathclap { n \to \infty } } T ( t _ n ) R ( \lambda ) f = T ( t ) R ( \lambda ) f = R ( \lambda ) T ( t ) f ; \end{align*}


\begin{align*} f ( x _ 1 , \ldots , x _ k , j ) \ = \ 0 \end{align*}


\begin{align*} i \partial _ t v + \sqrt { H + 1 } v \pm \big ( \frac { v - \bar { v } } { - 2 i \sqrt { H + 1 } } \big ) ^ 3 = 0 . \end{align*}


\begin{align*} x ^ 2 ( x y ) = 0 , \end{align*}


\begin{align*} \mbox { f o r $ k \geq 0 $ , } \sqrt { 2 \pi } \mathcal { K } ( x , k ) & = T ( k ) m _ + ( x , k ) e ^ { i x k } , \\ \mbox { f o r $ k < 0 $ , } \sqrt { 2 \pi } \mathcal { K } ( x , k ) & = T ( - k ) m _ - ( x , - k ) e ^ { i x k } . \end{align*}


\begin{align*} T B _ { 1 2 } ( f ) = 4 T B _ { 6 } ( f ) , \end{align*}


\begin{align*} l & \leq \sum _ { p + f \leq s } \binom { | T | } { p } \binom { | T | - p } { f } + \sum _ { q + g \leq s } \binom { | T | } { q } \binom { | T | - q } { g } \\ & = 2 \sum _ { p + f \leq s } \binom { | T | } { p } \binom { | T | - p } { f } \\ & \leq 2 \sum _ { p + f \leq s } | T | ^ s \leq 2 s ^ 2 | T | ^ s . \end{align*}


\begin{align*} \gamma _ F ( A ) = \limsup _ { l \to \infty } \gamma _ F ( A \cap \tau _ { v _ l } A ) \leq \limsup _ { l \to \infty } \gamma _ F ( P \cap \tau _ { v _ l } P ) + 2 \epsilon \leq ( \gamma _ F ( A ) + \epsilon ) ^ 2 + 2 \epsilon . \end{align*}


\begin{align*} A ( x , \sigma ) : = \left [ s _ 1 '' ( x , \sigma ) + s _ 2 '' ( x , \sigma ) + s _ 3 '' ( x , \sigma ) \right ] \left [ \exp \left ( \frac { B _ 2 ( 5 - 8 \sigma ) } { 3 } r \right ) ( B _ 2 r ) ^ { 5 - 2 \sigma } \right ] ^ { - 1 } \end{align*}


\begin{align*} Q _ 3 ( z , y ) = D b ( z + y ) - D b ( z ) , z , y \in \R ^ 2 , \end{align*}