\begin{align*} A = A 1 = A 1 ( t ) = ( A 1 _ { i j } ) _ { i , j = 0 } ^ { \infty } = \begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \dots \\ 0 _ { r 0 } & 0 & 0 & 0 & 0 & 0 & \dots \\ a _ 0 \cdots a _ r & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & a _ 1 \cdots a _ { r + 1 } & 0 & 0 & 0 & 0 & \dots \\ \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \dots \end{pmatrix} . \end{align*}


\begin{align*} L ( q ^ \beta ) \ \geq \ \beta L ( q ) - ( \beta - 1 ) \ = \ \beta \ell - ( \beta - 1 ) \ = \ \beta ( \ell - 1 ) + 1 . \end{align*}


\begin{align*} g \left ( { A } _ { \xi } X , Y \right ) = g \left ( h ( X , Y ) , \xi \right ) . \end{align*}


\begin{align*} \left | \sum \limits _ { j = 0 } ^ { n - 1 } g _ { \lambda _ i } ^ { - \frac { 1 } { 2 } } ( \alpha \theta _ n ^ { j } ) \theta _ n ^ { - j k } \right | & = \left | \lambda _ i ^ { - 1 / 2 } \sum \limits _ { j = 0 } ^ { n - 1 } \theta _ n ^ { - j k } + \sum \limits _ { j = 0 } ^ { n - 1 } ( g _ { \lambda _ i } ^ { - \frac { 1 } { 2 } } ( \alpha \theta _ n ^ { j } ) - \lambda _ i ^ { - 1 / 2 } ) \theta _ n ^ { - j k } \right | . \end{align*}


\begin{align*} ( q _ n , \lambda _ n ) \neq ( q _ n ' , \lambda _ n ' ) E ( q _ n , \lambda _ n ) = E ( q _ n ' , \lambda _ n ' ) , \qquad H ( \lambda _ n ) , \ , H ( \lambda _ n ' ) \to 0 . \end{align*}


\begin{align*} L p ( y _ j ) = \sum _ { x _ i \in X _ j ^ \mathrm { n d f } } w _ { j i } p ( x _ i ) , \forall p \in P _ j ^ \mathrm { n d f } . \end{align*}


\begin{align*} \gamma _ R ( x ) : = \gamma \left ( \frac { | x | } { R } \right ) \textrm { f o r a l l } \ ; \ ; x \in \R ^ N \ , ; \end{align*}


\begin{align*} \left . \begin{array} { l l } \dfrac { d } { d \tau } \mathcal { S } \left ( \xi ^ { \tau } , - \eta ^ { \tau } \right ) \Big | _ { \tau = 1 } = 0 . \end{array} \right . \end{align*}


\begin{align*} \omega ( X , Y ) = - \langle X ( t ) , Y ' ( t ) \rangle + \langle X ' ( t ) , Y ( t ) \rangle \end{align*}


\begin{align*} f ( | x | \bar { \otimes } | y | ) = \int _ { [ 0 , \infty ) } f ( \lambda ) \ , e _ { | x | \bar { \otimes } | y | } ( d \lambda ) = \int _ { [ 0 , \infty ) ^ 2 } f ( \lambda _ 1 \lambda _ 2 ) \ , e _ { | x | , | y | } ( d \lambda _ 1 , d \lambda _ 2 ) \end{align*}


\begin{align*} p _ \Psi ( x _ 0 , \xi , \tau ) = \{ p _ \Psi , \Psi \} ( x _ 0 , \xi ) = 0 \implies \tau = 0 . \end{align*}


\begin{align*} | S _ { \omega , \mathbf { c } } ( U _ n ( t _ n ) ) - \mu _ { \omega , \mathbf { c } } | = | S _ { \omega , \mathbf { c } } ( U _ { n , 0 } ) - S _ { \omega , \mathbf { c } } ( \Phi _ { \omega , \mathbf { c } , n } ) | \ \rightarrow \ 0 \end{align*}


\begin{align*} P _ \gamma ^ { ( \alpha , \beta ) } ( 1 ) = \frac { \Gamma ( \alpha + \gamma + 1 ) } { \Gamma ( \alpha + 1 ) \Gamma ( \gamma + 1 ) } , \end{align*}


\begin{align*} R ^ * ( \sigma \curvearrowright \tau ) = \sigma \curvearrowright R ^ * \tau , \end{align*}


\begin{align*} h ( s ) = M \left [ 1 + \boldsymbol { d } _ { \mathcal { S } _ 1 } ( s _ 1 ^ \star , s _ 1 ) ^ { p _ 1 ' } + \boldsymbol { d } _ { \mathcal { S } _ 2 } ( s _ 2 ^ \star , s _ 2 ) ^ { p _ 2 ' } \right ] , \end{align*}


\begin{align*} z _ j = z _ 0 + \frac { \zeta _ j } { \sqrt { n } } , j = 1 , 2 , \dotsc , m , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ 3 \int _ { \R ^ d } e ^ { p | x | } \left ( | \varphi _ j ( x ) | ^ 2 + | \nabla \varphi _ j ( x ) | ^ 2 \right ) d x < \infty \end{align*}


\begin{align*} d _ { \Phi _ { T } } ( \alpha ) = & \mu \left \{ ( x , \chi ) \in G \times \widehat { G } : | \Phi _ T ( x , \chi ) | > \alpha \right \} \leq \left ( \frac { \| T \| _ { \mathcal { B } _ 2 ( L ^ 2 ( G ) ) } } { \alpha } \right ) ^ 2 , \\ & \mu \left \{ ( x , \chi ) \in G \times \widehat { G } : | \Phi _ T ( x , \chi ) | > \alpha \right \} \lesssim \frac { M _ \psi \| T \| _ { \mathcal { B } _ 1 ( L ^ 1 ( G ) ) } } { \alpha } . \end{align*}


\begin{align*} f - f _ { a d d } = \sum _ { k = 1 } ^ N c _ k x _ 1 ^ { E _ { k 1 } } \cdots x _ N ^ { E _ { k N } } . \end{align*}


\begin{align*} u ( z ) = \zeta + \ln ' \big [ p ( z ) \big ] \end{align*}


\begin{align*} F _ { | g _ i | ^ 2 } ( x ) = \gamma ( M , M x ) / \Gamma ( M ) = 1 - \sum _ { i = 0 } ^ M \frac { ( M x ) ^ i } { i ! } e ^ { - M x } , \end{align*}


\begin{align*} \Delta ^ M = \frac { \sum _ { n m \in \mathcal { L } _ S } \delta ^ M _ { n m } } { | \mathcal { L } _ S | } \end{align*}


\begin{align*} \dot { x } = f ( x ) , \dot { y } = f ( y ) , \dot { z } = \frac { f ( x ) - f ( y ) } { \epsilon } . \end{align*}


\begin{align*} \frac { 1 } { n + 1 } \sum _ { l = 0 } ^ { n } \kappa _ l ( x ) \leq C _ w \frac { 1 } { 3 n + 1 } \sum _ { l = 0 } ^ { 3 n } \kappa _ 1 ^ l ( x ) . \end{align*}


\begin{align*} H _ N = \sum _ { j = 1 } ^ N ( - \Delta _ j ) + \frac { 1 } { N - 1 } \sum _ { 1 \leq j < k \leq N } w _ N ( x _ j - x _ k ) \ , , \end{align*}


\begin{align*} \forall n \geq 0 , F _ n ( O [ g _ 1 ] ) = \sum _ { k = 0 } ^ n ( F _ k O ) \cdot g _ 1 ^ { n - k } , \mathrm { g r } _ n ( O [ g _ 1 ] ) \simeq \mathbb C \cdot X ^ n \oplus Y ^ 2 \cdot \mathbb C [ X , Y ] _ { n - 2 } , \end{align*}


\begin{align*} \lim _ { h \to + \infty } F _ h ( v _ h , \gamma _ h , B _ s ) = : \alpha ( s ) \end{align*}


\begin{align*} \aligned \left \{ \begin{array} { l l l } - \Delta u + \lambda _ 1 u = | u | ^ { 2 r _ 1 - 2 } u + \nu p | v | ^ q | u | ^ { p - 2 } u \ & \mathbb { R } ^ N , \\ - \Delta v + \lambda _ 2 v = | v | ^ { 2 r _ 2 - 2 } v + \nu q | u | ^ p | v | ^ { q - 2 } v \ & \mathbb { R } ^ N , \\ \int _ { \mathbb { R } ^ N } u ^ 2 = a ^ 2 , \ \int _ { \mathbb { R } ^ N } v ^ 2 = b ^ 2 . \end{array} \right . \endaligned \end{align*}


\begin{align*} x ^ 4 + a x y + y ^ 3 = 0 \end{align*}


\begin{align*} \vartheta _ { m } ^ { i + 1 } = - \frac { \gamma } { ( \gamma - 1 ) \Delta x } \big ( \widetilde { m } _ { i + 1 } ^ { \gamma - 1 } - \widetilde { m } _ { i } ^ { \gamma - 1 } \big ) + \frac 1 2 \chi \left ( \frac { ( c _ x ) ^ i } { 1 + \widetilde { m } _ { i } } + \frac { ( c _ x ) ^ { i + 1 } } { 1 + \widetilde { m } _ { i + 1 } } \right ) , \end{align*}


\begin{align*} \nu _ { y ' } ^ n ( x ) = \langle y ' \sigma _ n ( x ) \Omega _ \rho , \Omega _ \rho \rangle _ { \rho } . \end{align*}


\begin{align*} \int _ { \Omega } \nabla v \cdot \nabla \varphi \ : x = \int _ { \Omega } u \varphi \ : x \end{align*}


\begin{align*} \| h _ j \| _ { p ^ * , q , \mu } \geq \| u _ j \| _ { p ^ * , q , \mu } - \| u _ j - h _ j \| _ { p ^ * , q , \mu } \geq \lambda - r = \varepsilon _ 0 . \end{align*}


\begin{align*} ( \pounds _ { \xi _ i } \ , d \eta ^ j ) ( X , Y ) = \xi _ i ( d \eta ^ j ( X , Y ) ) - d \eta ^ j ( [ \xi _ i , X ] , Y ) - d \eta ^ j ( X , [ \xi _ i , Y ] ) , \end{align*}


\begin{align*} \widetilde { m } ( x , \xi ) = ( 1 - \chi _ m ( x , \xi ) ) m ( x , \xi ) \log | \xi | \end{align*}


\begin{align*} v _ n ( t ) = v _ 0 ( t ) + \int _ 0 ^ t S ( t - s ) F _ n ( s ) v _ n ( s ) \ , d s + \int _ 0 ^ t S ( t - s ) \sigma ' ( u ( s ) ) v _ n ( s ) B \ , d W ( s ) , \end{align*}


\begin{align*} \widehat \psi _ r ( \beta _ 2 ) = & \inf _ { x \in [ 0 , 1 ] } J _ r ( x , \beta _ 2 ) \\ \geq \ , & \inf _ { x \in [ 0 , 1 ] } J _ r ( x , \beta _ 1 ) + \theta ( x , \beta _ 1 ) ( \beta _ 2 - \beta _ 1 ) \\ \geq \ , & \widehat \psi _ r ( \beta _ 1 ) + \inf _ { x \in [ 0 , 1 ] } \theta ( x , \beta _ 1 ) ( \beta _ 2 - \beta _ 1 ) > \widehat \psi _ r ( \beta _ 1 ) . \end{align*}


\begin{align*} H : = \bigcup _ { Q ' \in \mathcal { F } } Q ' \ , . \end{align*}


\begin{align*} \frac { d } { d s } \left [ \begin{array} { c c } U \\ V \end{array} \right ] = \left [ \begin{array} { c c } V \\ \alpha V + H _ { \lambda } \left ( \phi _ { 2 } ( \lambda ) - U \right ) \end{array} \right ] \end{align*}


\begin{align*} r _ 1 = \begin{cases} \prod \limits _ { \varpi ^ k \in P _ 1 ( r ) } \varpi ^ k & c _ 2 \neq 0 \\ \prod \limits _ { \varpi ^ k \in P _ 2 ( r ) } \varpi ^ k & \end{cases} \end{align*}


\begin{align*} \mathcal { R } _ \varepsilon c ( x ) = \int _ { \mathbb { R } ^ n _ - } J _ \varepsilon ( | x - y | ) ( c ( x ) - \tilde { c } ( y ) ) y = \int _ { \mathbb { R } ^ n _ + } J _ \varepsilon ( | x - \hat { y } | ) ( c ( x ) - c ( \hat { y } ) ) y \end{align*}


\begin{align*} v ( t , x ) = e ^ { h ^ { - 1 } \int _ 0 ^ t q _ 1 ( \varphi ^ s ( x ) ) d s } w ( t , \varphi ^ t ( x ) ) , \end{align*}


\begin{align*} \frac { \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \geq x \right ) } { 1 - \Phi ( x ) } = 1 + o ( 1 ) \quad \frac { \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \leq - x \right ) } { \Phi ( - x ) } = 1 + o ( 1 ) \end{align*}


\begin{align*} \omega _ f ( r ) : = \sup _ { \substack { z , \zeta \in H \\ d ( z , \zeta ) < r } } | f ( z ) - f ( \zeta ) | . \end{align*}


\begin{align*} \Pr ( X _ t \le k \ , \vert \ , X _ 0 = r ) = \frac { ( n - r ) \cdots ( n - k ) } { n ^ t } \frac { 1 } { ( k - r ) ! } \Delta ^ { k - r } \left [ \frac { x ^ t } { n - x } \right ] _ { x = r } ^ { } , \end{align*}


\begin{align*} ( 2 ^ * - 1 ) \alpha _ n \sum _ { i = 1 } ^ k \mu _ i ^ { \frac { n - 2 } { 2 } } \int _ \Omega U _ { \mu _ i , \xi _ i } ^ { 2 ^ * - 2 } H ( x , \xi _ i ) \psi ^ 0 _ { \mu _ j , \xi _ j } d x = o \Big ( \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big ) . \end{align*}


\begin{align*} \Psi _ + \left ( \Xi _ { \mathfrak { l } } \prod _ { i } \mathcal { I } _ { \mathbf { m } _ i } ( \tau _ i ) \right ) & = \Psi _ + \left ( \prod _ { i } \mathcal { I } _ { \mathbf { m } _ i } ( \tau _ i ) \curvearrowright \Xi _ { \mathfrak { l } } \right ) \\ & = \rho \left ( \Psi \left ( \prod _ { i } \mathcal { I } _ { \mathbf { m } _ i } ( \tau _ i ) \right ) \right ) \Psi \left ( \Xi _ { \mathfrak { l } } . \right ) \end{align*}


\begin{align*} \phi _ { \ast } \left ( \bar { m } _ { K } \times \eta \right ) & = \int _ { G / Q } \int _ { K \cap Q } \tau ( y ) k . \lambda d m _ { K \cap Q } ( k ) d \bar { m } _ { K } ( y ) \\ & = \left ( \int _ { G / Q } \tau ( y ) . m _ { K \cap Q } d \bar { m } _ { K } ( y ) \right ) \ast \lambda = m _ { K } \ast \lambda = \nu . \end{align*}


\begin{align*} \overline { E } = \{ \ , ( x , y ) \in V ^ 2 : \ , \} . \end{align*}


\begin{align*} \mathbb { V } ( \overline { S } _ n ) = \mathbb { E } ( \overline { S } _ n ^ 2 ) - \mathbb { E } ( \overline { S } _ n ) ^ 2 \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } p q + p c = p b ^ 2 + a ^ 2 b + a b c \\ p ( c - a ) = ( b - d ) \{ p ( b + d ) - q ( a + c ) \} \\ p ( d - b ) = a ^ 2 - c ^ 2 \\ q ^ 2 + p d = a ^ 2 + q b ^ 2 + a b ^ 2 + a b d \\ q ( d - b ) = a b - c d . \\ \end{array} \right . \end{align*}


\begin{align*} f ( Z ) = f ( 0 ) P _ 0 + \sum _ { j = 1 } ^ m f ( z _ j ^ { \uparrow } ) P _ j , \end{align*}


\begin{align*} ( \Psi \circ \Phi ) ( A , B ) = \Psi ( \Phi ( A , B ) ) = \Psi ( A ^ r \otimes B ^ { 1 - r } ) = A ^ r \circ B ^ { 1 - r } . \end{align*}


\begin{align*} \lambda _ k B _ 1 ^ { ( \alpha ) } - 2 B _ 2 ^ { ( \alpha ) } = D _ { \alpha } ^ { - 1 } \mathbb { F } ^ { * } \Lambda _ { \alpha } \mathbb { F } D _ { \alpha } , \end{align*}


\begin{align*} R _ { N _ m } ^ 2 ( L _ m , \alpha , \Delta ) = L _ m + o ( 1 ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \| \nabla u _ n \| _ { p , q , \mu } = \| \nabla u _ \mu ^ \bigstar \| _ { p , q , \mu } . \end{align*}


\begin{align*} \varphi _ \lambda ( x , y ) = - ( - 1 ) ^ { | x | | y | } \varphi _ { - \partial - \lambda } ( y , x ) , \end{align*}


\begin{align*} W = B A = B _ { I } A _ { I } + B _ { I ^ { c } } A _ { I ^ { c } } . \end{align*}


\begin{align*} \ < \Psi _ { \mathrm { S l a t e r } } , a _ p ^ * a _ q \Psi _ { \mathrm { S l a t e r } } \ > = \delta ( p - q ) \chi ( p ) \ , \forall p , q \in \Lambda ^ * \ . \end{align*}


\begin{align*} \lambda : = - \frac { L _ { \omega , \mathbf { c } } ( U ) } { 3 N ( U ) } , \end{align*}


\begin{align*} ( P _ \omega ( E ) ) = \widetilde \Theta _ { \omega , E } , \mu _ \omega E . \end{align*}


\begin{align*} \langle \ddot { s } , \nabla ^ 2 f ( x ) \ddot { s } \rangle = \langle T _ { y , s } ^ { - 1 } \ddot { s } , \nabla ^ 2 \hat { f } ( s ) T _ { y , s } ^ { - 1 } \ddot { s } \rangle - \langle \nabla f ( x ) , c _ { y , s , \dot { s } } '' ( 0 ) \rangle = - \langle g , T _ { y , s } ^ { - 1 } c _ { y , s , \dot { s } } '' ( 0 ) \rangle . \end{align*}


\begin{align*} \langle h , A g \rangle _ { \mu _ p } = 0 \end{align*}


\begin{align*} w : = \begin{cases} u & \mbox { i n } B _ \rho \setminus \overline { B _ { s \rho } } \\ w _ s & \mbox { i n } B _ { s \rho } , \end{cases} \end{align*}


\begin{align*} R _ { C F } = \log _ { 2 } \left ( \det \left [ \textbf { R } + \textbf { I } _ K \right ] \right ) \end{align*}


\begin{align*} \mathcal { P } _ { r } \left ( \mathbb { R } ^ { n } \right ) = \left \{ \mathbb { P } _ { \xi } \left \vert \xi \in L ^ { r } \left ( \Omega ^ { 1 } , \mathcal { G } , \mathbb { P } ^ { 1 } \right ) \right . \right \} , \end{align*}


\begin{align*} \Lambda ^ C _ { h , t } = \{ \exp _ x ( h \phi _ t ( x ) \nu ( x ) ) \in M \ ; | \ ; x \in B _ { t } \setminus B _ { t ^ 2 } \} , \end{align*}


\begin{align*} 0 \leq u _ { n \bar n } ( 0 ) = u _ { x _ n x _ n } + u _ { y _ n y _ n } ( 0 ) \leq C . \end{align*}


\begin{align*} ( g _ 0 ) _ j = [ u _ { j + 1 } , \ldots u _ { r + 1 } , a _ j , \ldots , a _ r , g _ { r + 1 } ] \in Q _ { \langle z _ j \rangle } \end{align*}


\begin{align*} = \exp \left \{ \frac { 1 } { 2 } \log \left ( \frac { 1 } { 1 - z } \right ) + \frac { 1 } { 2 } \frac { z } { 1 - z } \right \} , \end{align*}


\begin{align*} S ^ { - 1 } r S = r ^ { - 1 } . \end{align*}


\begin{align*} \begin{aligned} T _ 1 \geq C s ^ 3 \lambda ^ 4 \iint _ Q \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d t - C s ^ 3 \lambda ^ 3 \int _ 0 ^ T \int _ { \omega } \xi ^ 3 | u | ^ 2 d x d t . \end{aligned} \end{align*}


\begin{align*} k \cdot r _ { b } ( n ) \ = \ n , \end{align*}


\begin{align*} \iota \beta ^ { - 1 } r \beta \iota = \beta ^ { - 1 } r ^ { - 1 } \beta . \end{align*}


\begin{align*} | \nabla u _ \nu | ^ 2 _ 2 + | \nabla v _ \nu | ^ 2 _ 2 = ( 1 + o _ \nu ( 1 ) ) [ ( \gamma _ p + \gamma _ q ) \nu D _ 0 ] ^ { \frac { 2 } { 2 - \gamma _ p - \gamma _ q } } . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ^ { \star } ( \lambda ) = \sup _ { \pi \in \mathcal { G } _ { \mathrm { D } , \lambda } } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda d \pi = \sup _ { \pi \in \bar { \Gamma } } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ { \lambda } d \pi . \end{align*}


\begin{align*} \| ( a _ j ) _ { j = 1 } ^ n \| _ p \coloneqq \sum _ { j = 1 } ^ { n } | a _ j | ^ p , \forall ( a _ j ) _ { j = 1 } ^ n \in \mathbb { K } ^ n , \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D M R } } ( \delta ) = \inf _ { \lambda \in \mathbb { R } _ { + } } \left \{ \lambda \delta + \int _ { \mathcal { X } } \sup _ { x ' \in \mathcal { X } } [ f ( x ' ) - \lambda c ( x , x ' ) ] \ , d \mu ( x ) \right \} , \end{align*}


\begin{align*} \nabla = \partial _ z + Z - \sum ^ r _ { i = 1 } \ln ' \Big [ q ^ i _ + ( z ) \Big ] \check { \alpha } _ i + \sum ^ { r } _ { i = 1 } \Lambda _ i ( z ) e _ i . \end{align*}


\begin{align*} \mathcal { E } ( u , \overline { J _ u } , \Omega ) = \mathcal { G } ( u , \Omega ) \leq \mathcal { G } ( v , \Omega ) \leq \mathcal { E } ( v , \Gamma , \Omega ) . \end{align*}


\begin{align*} X _ T ^ { ( \ast , n ) } : = \sup _ { s \in [ 0 , T ] } X _ s ^ { ( n ) } . \end{align*}


\begin{align*} & \tau _ { \Lambda _ { t - \cdot } } = \inf \{ s \in [ 0 , t + 1 ] \ ! : ( R _ s - \Lambda _ { t - s } ) ( R _ 0 - \Lambda _ t ) \leq 0 \} , \end{align*}


\begin{align*} \mathrm { d } Z _ t = A Z _ t \mathrm { d } t + F ( Z _ t ) \mathrm { d } t + G ( Z _ t ) \circ \mathrm { d } \mathbf { X } _ t , \ \ \ \ Z _ { 0 } = \xi \in \mathcal { B } _ { \alpha } . \end{align*}


\begin{align*} ( \rho ( \cdot , t ) , u ( \cdot , t ) , \theta ( \cdot , t ) , q ( \cdot , t ) , S ( \cdot , t ) = ( 1 , 0 , 1 , 0 , 0 ) : = ( \bar \rho , \bar u , \bar \theta , \bar q , \bar S ) \end{align*}


\begin{align*} { _ { 2 } F _ { 1 } } \left [ \begin{array} { c c c c c c c c } a , b \\ c \end{array} ; x \right ] = ( 1 - x ) ^ { c - a - b } { _ { 2 } F _ { 1 } } \left [ \begin{array} { c c c c c c c c } c - a , c - b \\ c \end{array} ; x \right ] , \end{align*}


\begin{align*} D ^ { 3 } \bar { w } = D ^ { 3 } w \end{align*}


\begin{align*} E ( \widetilde { \Delta } _ { 2 } ) = E ( _ { n , m } ^ { 2 } ) - \Delta _ { 0 } - E ( \Delta _ { 1 } ) = 0 , \end{align*}


\begin{align*} A ( x _ 1 t , \ldots , x _ m t ) \tilde { A } ( x _ 1 , \ldots , x _ m , t ) + B ( x _ 1 t , \ldots , x _ m t ) \tilde { B } ( x _ 1 , \ldots , x _ m , t ) = G . \end{align*}


\begin{align*} B ( M ) = B ( N ) \cup \left \{ \{ j , j ^ * \} \ , | \ , j \in A ( M ) \right \} . \end{align*}


\begin{align*} \varphi ( M _ I ) = \begin{cases} a ^ n & \\ 0 & \end{cases} \end{align*}


\begin{align*} S _ \alpha ( X ) = \lim _ { \epsilon \to 0 } \frac { 1 } { 4 \pi i } \int _ { \Gamma } f _ \alpha ( 1 + \epsilon , \lambda ) \frac { d \log D _ X ( \lambda ) } { d \lambda } d \lambda , \end{align*}


\begin{align*} L i _ 2 ( z ) = \sum _ { r = 1 } ^ { \infty } \frac { z ^ r } { r ^ 2 } , \ ; \ ; | z | \leq 1 . \end{align*}


\begin{align*} \iota ( \alpha ) \ : = \ \begin{cases} 0 , & , \\ \alpha , & . \end{cases} \end{align*}


\begin{align*} T _ { \alpha , \beta } ( x ) = \begin{cases} x ( 1 + 2 ^ { \alpha } x ^ { \alpha } ) x \in [ 0 , \tfrac 1 2 ) , \\ 2 ^ { \beta } ( x - \tfrac 1 2 ) ^ { \beta } , x \in [ \tfrac 1 2 , 1 ] . \end{cases} \end{align*}


\begin{align*} \mathcal { E C S } _ { { \rm { ( r a n k \ , 2 ) } } } = \{ S ( p , q , a , b , c , d ) \in \mathcal { E C } : { \rm { r a n k } } \ , S ( p , q , a , b , c , d ) = 2 \} . \end{align*}


\begin{align*} T ( r , M ) & = T \left ( r , \frac { P ' } { P } - \frac { Q ' } { Q } + P _ 0 \right ) \\ & = m \left ( r , \frac { P ' } { P } - \frac { Q ' } { Q } + P _ 0 \right ) + N \left ( r , \frac { P ' } { P } - \frac { Q ' } { Q } + P _ 0 \right ) \\ & \leq S ( r , P ) + S ( r , Q ) + O ( \log r ) + N \left ( r , \frac { P ' } { P } \right ) + N \left ( r , \frac { Q ' } { Q } \right ) + N ( r , P _ 0 ) \\ & = N ( r , \frac { 1 } { P } ) + N ( r , \frac { 1 } { Q } ) + S ( r , W ) \\ & \leq T ( r , P ) + T ( r , Q ) + S ( r , W ) , \end{align*}


\begin{align*} b _ t : = d i a g ( e ^ { t / m } , \ldots , e ^ { t / m } , e ^ { - t / n } , \ldots , e ^ { - t / n } ) , \ t > 0 \end{align*}


\begin{align*} F _ X ( x ) & = M \int _ { - \infty } ^ x \mathrm { e } ^ { \beta t } ( - t ) ^ \nu K _ { \nu } ( - \alpha t ) \ , \mathrm { d } t = M \sum _ { k = 0 } ^ \infty \frac { \beta ^ k } { k ! } \int _ { - \infty } ^ x ( - 1 ) ^ k ( - t ) ^ { \nu + k } K _ \nu ( - \alpha t ) \ , \mathrm { d } t \\ & = M \sum _ { k = 0 } ^ \infty \frac { ( - \beta ) ^ k } { k ! } \int _ { - x } ^ \infty y ^ { \nu + k } K _ \nu ( \alpha y ) \ , \mathrm { d } y , \end{align*}


\begin{align*} & \overline { R } ^ { R ^ x _ { t } } _ s = R ^ x _ { t - s } , \ ; \ ; s \in [ 0 , t ] , \\ & R ^ { \overline { R } ^ x _ { t } } _ s = \overline { R } ^ x _ { t - s } , \ ; \ ; s \in [ 0 , t ] \quad \overline { R } ^ x _ { t } > 0 . \end{align*}


\begin{align*} \lim _ { \substack { \lambda \to 1 ^ + \\ \kappa \to 1 ^ + } } \limsup _ { m , n \to \infty } \max _ { \substack { P _ m \leq P _ i \leq \lambda P _ { m } \\ Q _ n \leq Q _ j \leq \kappa Q _ { n } } } \left | u _ { i j } - u _ { m n } \right | = 0 ; \end{align*}


\begin{align*} \varphi _ { \lambda , \ell } ( x _ 1 , x _ 2 ) = ( \lambda _ 1 + \lambda _ 2 ) h _ \ell ( x _ 1 ) , \forall \ell = 1 , 2 . \end{align*}


\begin{align*} - \partial ^ { \rm s t i f f ( s o f t ) } _ \nu \bigl ( \mathcal { A } _ { 0 , \chi } ^ { \rm s t i f f ( s o f t ) } \bigr ) ^ { - 1 } = : \Xi _ \chi ^ { \rm s t i f f ( s o f t ) } : \mathcal { H } ^ { \rm s t i f f ( s o f t ) } \to { \mathcal E } . \end{align*}


\begin{align*} \int _ M | T ^ \varphi | ^ 2 | \nabla w | ^ 2 = 0 \ , . \end{align*}


\begin{align*} \begin{aligned} \mathbb E \big [ ( X _ { N , M } ^ { ( l ) } ) ^ 4 \big ] \le \frac { \theta _ N ^ 2 } { N ^ 4 } \ , \bigg ( \ , \Xi ( 1 ) + \Xi ( 2 ) + \sum _ { r = 3 } ^ \infty \Xi ( r ) \ , \bigg ) \ , , \end{aligned} \end{align*}


\begin{align*} \| f ^ { = 1 } \| _ 2 = \sigma , \end{align*}


\begin{align*} \alpha _ n = \frac { 1 - a q ^ { 2 n } } { 1 - a } \sum _ { j \leq n } \frac { ( a ) _ { n + j } } { ( q ) _ { n - j } } ( - 1 ) ^ { n - j } q ^ { \left ( { n - j } \atop { 2 } \right ) } \beta _ j \ ; \ ; \ ; \ ; \forall \ , n \in \mathbb { Z } . \end{align*}


\begin{align*} ( \pi _ { \varphi _ 1 } ( a ) \otimes \pi _ { \varphi _ 2 } ( b ) ) ( \lambda ^ { \varphi _ 1 } ( t _ 1 ) \otimes \lambda ^ { \varphi _ 2 } ( t _ 2 ) ) = \pi _ \sigma ( a _ 1 \otimes a _ 2 ) \lambda ^ \sigma ( t _ 1 , t _ 2 ) , a \in M _ 1 , b \in M _ 2 , ( t _ 1 , t _ 2 ) \in G \end{align*}


\begin{align*} x _ i v _ j = - v _ { 3 i + 5 j \bmod 7 } , i , j \in \{ 1 , 2 , \dots , 7 \} . \end{align*}


\begin{align*} I ' = I _ { < i _ 0 } : = \{ i \in I \mid i < i _ 0 \} . \end{align*}


\begin{align*} \prod _ { j = 1 } ^ { m } A _ { p _ { j } } & \subsetneq A _ { \vec { P } } . \end{align*}


\begin{align*} \begin{aligned} - \operatorname { d i v } \left ( | \nabla u | ^ { p - 2 } \nabla u \right ) & = \lambda | u | ^ { p - 2 } u & & \Omega , \\ u & = 0 & & \partial \Omega . \end{aligned} \end{align*}


\begin{align*} \begin{cases} i \partial _ t u + \frac { 1 } { 2 m } \Delta u = \overline { u } v , \\ i \partial _ t v + \frac { 1 } { 2 M } \Delta v = u ^ 2 , \end{cases} ( t , x ) \in \R \times \R ^ d , \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( U ) = \frac { 1 } { 2 } L _ { \omega , \mathbf { c } } ( U ) + N ( U ) = \frac { 1 } { 3 } K _ { \omega , \mathbf { c } } ( U ) + \frac { 1 } { 6 } L _ { \omega , \mathbf { c } } ( U ) \end{align*}


\begin{align*} w _ b = \sum _ { \ell } c _ \ell a _ \ell + \Psi _ b . \end{align*}


\begin{align*} ( \widetilde { \varphi } \otimes \widetilde { \alpha } ) \circ \widetilde { \Delta } ( x , m ) = ( \varphi _ L \otimes \alpha ) \Delta ( x ) + ( \varphi _ M \otimes \alpha ) ( \rho _ { \beta } ( m ) ) - ( \varphi _ M \otimes \alpha ) ( \tau \circ ( \rho _ { \beta } ( m ) ) ) . \end{align*}


\begin{align*} D = \begin{bmatrix} - \frac { \sqrt { n } } { 2 } I _ n & \left ( \frac { - \sqrt { n } } { 2 } + 1 \right ) I _ n \\ \left ( - \frac { \sqrt { n } } { 2 } + 1 \right ) I _ n & - \frac { \sqrt { n } } { 2 } I _ n \end{bmatrix} = \begin{bmatrix} - \frac { \sqrt { n } } { 2 } & \left ( - \frac { \sqrt { n } } { 2 } + 1 \right ) \\ \left ( - \frac { \sqrt { n } } { 2 } + 1 \right ) & - \frac { \sqrt { n } } { 2 } \end{bmatrix} \otimes I _ n , \end{align*}


\begin{align*} D _ { r , z } u ( t , x ) = u ( r , z ) v ^ { ( r , z ) } ( t , x ) , \end{align*}


\begin{align*} \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } = I + z \mathcal { A } _ 0 ^ { - 1 } \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } = I + z ( \mathcal { A } _ 0 - z I ) ^ { - 1 } , \end{align*}


\begin{align*} \langle v ^ { \otimes d } , ( v ^ * ) ^ { \otimes d } \rangle = \langle v , v ^ * \rangle ^ d . \end{align*}


\begin{align*} \Pi _ \chi ^ { \rm s o f t } = { \rm e } ^ { - { \rm i } \chi y } \Pi _ 0 ^ { \rm s o f t } { \rm e } ^ { { \rm i } \chi y } , \Lambda _ \chi ^ { \rm s o f t } = { \rm e } ^ { - { \rm i } \chi y } \Lambda _ 0 ^ { \rm s o f t } { \rm e } ^ { { \rm i } \chi y } , \Gamma _ { \chi , 1 } ^ { \rm s o f t } = { \rm e } ^ { - { \rm i } \chi y } \Gamma _ { 0 , 1 } ^ { \rm s o f t } { \rm e } ^ { { \rm i } \chi y } . \end{align*}


\begin{align*} f _ { \gamma , 1 } ( x ) = x ( 1 + 2 ^ { \alpha } x ^ { \alpha } ) , f _ { \gamma , 2 } ( x ) = 2 ^ { \beta } ( x - \tfrac 1 2 ) ^ { \beta } . \end{align*}


\begin{align*} \frac { \kappa _ { \ell + 1 } - \kappa _ \ell } { \kappa _ \ell - \kappa _ { \ell - 1 } } \ , = \ , \frac { \mu ( 1 ) + \nu _ 1 ( \ell ) } { \mu ( 1 ) + \nu _ 3 ( \ell ) } \ , \ , \ , \textrm { f o r $ \ell \geq a $ } \ , . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta u ^ * & = \lambda ^ * u ^ * \mbox { i n } \ \ \Omega ; \\ u ^ * & = 0 \quad \ \ \ \mbox { o n } \ \ \partial \Omega ; \\ \| u ^ * \| _ { L ^ 2 ( \Omega ) } & = 1 . \end{aligned} \right . \end{align*}


\begin{align*} ( \partial _ t - \partial _ x ^ 2 ) u = f ( u ) + g ( u ) \partial _ x u + h ( u ) ( \partial _ x u ) ^ 2 + \sigma ( u ) \xi . \end{align*}


\begin{align*} \tilde { v } ( t ) : = \psi \left ( t \right ) - \phi . \end{align*}


\begin{align*} F ( m ' ) - F ( m ) = \int _ { 0 } ^ { 1 } \int _ { \mathcal { M } } \frac { \delta F } { \delta m } ( m + \lambda ( m ' - m ) , x ) \left ( m ' - m \right ) ( \mathrm { d } x ) \ , \mathrm { d } \lambda . \end{align*}


\begin{align*} \theta _ { 3 / 2 } ( \tfrac { a z + b } { c z + d } , \epsilon ) = \lambda ^ { \pm } ( \gamma , \epsilon ) ( \sqrt { \det \gamma ( c z + d ) } ) ^ 3 \sum _ { n \in \Z } n e ^ { i ( \det \gamma ) \epsilon \pi n ^ 2 z } , \end{align*}


\begin{align*} \mathcal { W } _ n ^ { S A W } ( x _ 0 , x _ { 0 } ) = \emptyset = \mathcal { W } _ 0 ^ { S A W } ( x _ 0 , x ) \forall x _ { 0 } \neq x , n \geq 1 . \end{align*}


\begin{align*} \bar { F } _ { P _ s } ( \theta , x ) \overset { \Delta } { = } \Pr \left ( P _ s ( \theta ) > x \right ) , x \in [ 0 , 1 ] , \end{align*}


\begin{align*} 3 L < \frac { 2 a ^ 2 } { 2 5 } = \frac { t } { 2 5 \pi } . \end{align*}


\begin{align*} ( - \Delta ) ^ { - 1 } : \{ c \in ( H ^ 1 ( \Omega ) ) ^ \prime : c _ \Omega = 0 \} \rightarrow \{ c \in H ^ 1 ( \Omega ) : c _ \Omega = 0 \} . \end{align*}


\begin{align*} F ( u ) : = \prod _ { \deg P \in S ^ c } ( 1 - u ^ { \deg P } ) ^ { - 1 } = \exp \left ( \sum _ { \deg P \in S ^ c } \sum _ { k \ge 1 } \frac { u ^ { k \deg P } } { k } \right ) = : \sum _ { i = 0 } ^ { \infty } \widetilde { A } _ i u ^ i . \end{align*}


\begin{align*} T _ 0 : = L _ 0 + C \hbox { a n d } \widetilde { T } _ 0 : = \widetilde { L } _ 0 + C ^ * \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ^ \star ( \lambda ) = \sup _ { \gamma \in \mathcal { P } _ { \mathrm { D } } } I _ { \mathrm { D } , \lambda } [ \gamma ] = \sup _ { \pi \in \Gamma ( \Pi , \varphi _ \lambda ) } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda \ , d \pi = \sup _ { \pi \in \Pi ( \mu _ 1 , \ldots , \mu _ L ) } \int _ { \mathcal { V } } \ , g _ \lambda d \pi . \end{align*}


\begin{align*} \mathcal { H } = - \mathcal { L } + \frac { 1 } { | x | ^ { 2 } } . \end{align*}


\begin{align*} \| \varphi _ 1 \| _ { W ^ { 2 , p _ 0 } } = \| ( - \alpha \Delta + 2 \omega + i \mathbf { c } \cdot \nabla ) ^ { - 1 } \{ ( \nabla \cdot \varphi _ 3 ) \varphi _ 2 \} \| _ { W ^ { 2 , p _ 0 } } \lesssim \| ( \nabla \cdot \varphi _ 3 ) \varphi _ 2 \| _ { L ^ { p _ 0 } } < \infty \end{align*}


\begin{align*} S _ N ( \ell ) = \sum _ { j = 1 } ^ N \chi _ \ell ( x _ j ) . \end{align*}


\begin{align*} [ g , \epsilon ] [ k ( t ) , 1 ] & = [ g k ( t ) , \overline { \overline { C } } _ { X ^ { \ast } } ( g , k ( t ) ) \epsilon ] \\ & = [ g k ( t ) , \epsilon ] \\ & = [ p _ g , \epsilon ] [ k _ g , 1 ] [ k ( t ) , 1 ] . \end{align*}


\begin{align*} \mathcal { A } \left ( C , u \right ) & : = \arg \min \sum \limits _ { i = 1 } ^ { m } \left ( u _ { i } \right ) ^ { p } d _ { S _ { i } } ^ { p } \left ( x \right ) + I _ { C } \left ( x \right ) , \bigskip \\ \mathcal { A } \left ( l \right ) & : = \arg \min \sum \limits _ { i = 1 } ^ { m } \left ( l _ { i } \right ) ^ { p } d _ { S _ { i } } ^ { p } \left ( x \right ) . \end{align*}


\begin{align*} ( x , y , z ) \star ( x ' , y ' , z ' ) = \bigg ( x + x ' , y + y ' , z + z ' + \frac 1 2 ( x y ' - x ' y ) \bigg ) , \qquad \forall \ , ( x , y , z ) , ( x ' , y ' , z ' ) \in \R ^ 3 , \end{align*}


\begin{align*} f \circ _ { } g \coloneqq \sum _ { i = 0 } ^ { \infty } \left ( \sum _ { j + k + \ell = i } \frac { i ! } { j ! k ! \ell ! } \alpha ^ { j } \beta ^ k \gamma ^ \ell a ( j + k ) b ( \ell + k ) \right ) t ^ { i } \in k [ [ t ] ] \ , , \end{align*}


\begin{align*} = \exp \left \{ \frac { 1 } { 4 } L i _ 2 ( z ) + \frac { 1 } { 3 } \log \left ( \frac { 1 } { 1 - z } \right ) + \frac { 1 } { 4 } \frac { z } { 1 - z } \right \} , \end{align*}


\begin{align*} \| e ^ { - t H ^ \beta } g \| _ { L ^ q } \le \begin{cases} C e ^ { - t d ^ \beta } \| g \| _ { L ^ p } & t \geq 1 \\ C t ^ { - \sigma _ \beta } \| g \| _ { L ^ p } & 0 < t \leq 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} \boldsymbol { W } _ { p } ( \nu ( t _ 0 ) , \nu ( t _ 1 ) ) & = \boldsymbol { W } _ { p } \left ( \nu ( t _ 0 ) , \gamma + \Delta \pi ^ \prime \right ) \\ & \leq ( 1 - \Delta ) \boldsymbol { W } _ { p } \left ( \nu ( t _ 0 ) , ( 1 - \Delta ) ^ { - 1 } \gamma \right ) + \underbrace { \Delta \boldsymbol { W } _ { p } \left ( \nu ( t _ 0 ) , \pi ^ \prime \right ) } _ { = O ( \Delta ) } . \\ \end{aligned} \end{align*}


\begin{align*} \sigma ( S ) = \{ \lambda \in \mathbb C \colon \ , S - \lambda I \} . \end{align*}


\begin{align*} \tau _ a ( \alpha ) & = \frac { 1 } { q } \sum _ { u \in \mathbb { F } _ { q } } \psi _ 1 \left ( _ { \mathbb { F } _ { q ^ m } / \mathbb { F } _ { q } } ( u \alpha ) - u a \right ) \\ & = \frac { 1 } { q } \sum _ { u \in \mathbb { F } _ { q } } \hat { \psi _ 1 } ( u \alpha ) \psi _ 1 ( - u a ) , \end{align*}


\begin{align*} 1 + e v _ y ^ 2 v _ z ^ 2 - e u _ z ^ 2 = 0 , \end{align*}


\begin{align*} \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) = \langle \lambda , \delta \rangle + \sup _ { x ^ { \prime } \in \mathbb { R } ^ d } \left [ \sum _ { 1 \leq \ell \leq 2 } \varphi _ \ell ( x ^ { \prime } , \lambda _ \ell ) \right ] \mathrm { R } _ { \mathrm { D } } ( \lambda , \delta ) = \langle \lambda , \delta \rangle + \sum _ { 1 \leq \ell \leq 2 } \sup _ { x ^ { \prime } \in \mathbb { R } ^ d } \varphi _ \ell ( x ^ { \prime } , \lambda _ \ell ) . \end{align*}


\begin{align*} | - 2 z + \lambda _ i z ^ 2 | & \leq 2 c _ 1 + \lambda _ i c _ 1 ^ 2 \\ & \leq 2 \left ( \sqrt { \frac { 1 } { 2 } + \frac { 1 } { \lambda _ i ^ 2 } } - \frac { 1 } { \lambda _ i } \right ) + \lambda _ i \left ( \sqrt { \frac { 1 } { 2 } + \frac { 1 } { \lambda _ i ^ 2 } } - \frac { 1 } { \lambda _ i } \right ) ^ 2 = \frac { \lambda _ i } { 2 } . \end{align*}


\begin{align*} ( \underline { X } , \underline { A } ) ^ { K , c } = \mbox { c o l i m } _ { \sigma \subseteq K } ( \underline { X } , \underline { A } ) ^ { \sigma , c } . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( L _ a - c ) w = 0 & \textnormal { i n } & \Omega \\ w = 0 & \textnormal { o n } & \partial \Omega \\ w \in C ^ { 2 } ( \Omega ) \cap C ^ 0 ( \bar \Omega ) . & & \\ \end{array} \right . \end{align*}


\begin{align*} \sum _ { 1 \leq i , j \leq k } | \alpha _ i \alpha _ j | \ , 2 d _ { \square } ( f _ n , f ) , \end{align*}


\begin{align*} \delta = \frac { ( n - 1 ) ( p - 1 - \alpha ) } { p - 1 } > 0 . \end{align*}


\begin{align*} j - \frac { 3 } { p } = \left ( m - \frac { 3 } { r } \right ) \alpha + ( 1 - \alpha ) \left ( - \frac { 3 } { q } \right ) , \frac { j } { m } \leq \alpha \leq 1 . \end{align*}


\begin{align*} \varphi ' ( t ) - N c = - N ^ 2 \ , , \end{align*}


\begin{align*} u _ n ( x , t ) = \begin{cases} u _ { \mathcal { K } _ n } ^ { \{ L _ \mathcal { P } \} _ { \mathcal { P } \in \mathcal { N } _ { \mathcal { K } _ n } } } ( x , t ) & t \in [ 0 , 5 0 ] , \\ - u _ { \mathcal { K } _ n } ^ { \{ L _ \mathcal { P } \} _ { \mathcal { P } \in \mathcal { N } _ { \mathcal { K } _ n } } } ( x , 1 0 0 - t ) & t \in [ 5 0 , 1 0 0 ] . \end{cases} \end{align*}


\begin{align*} ( \rho \eta ) _ t + \left ( \rho u \eta + \frac { q } { \theta } \right ) _ x = \frac { q ^ 2 } { \kappa ( \theta ) \theta ^ 2 } + \frac { S ^ 2 } { \mu \theta } . \end{align*}


\begin{align*} \det ( \cos t I - \sin t S ) = ( - \sin t ) ^ n \det ( S - \cot t I ) = 0 . \end{align*}


\begin{align*} ( x , y , z , t ) = ( u v , w r , u w , v r ) \ , \ , u , v , w , r . \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( \frac { 1 } { 1 - z ^ n } \right ) ^ { \frac { l m ^ 2 } { n ^ 4 } } = \exp \left \{ \sum _ { n = 1 } ^ { \infty } \left ( \frac { n } { 2 } + \frac { n ^ 2 } { 2 } \right ) \left ( \frac { n } { 6 } + \frac { n ^ 2 } { 2 } + \frac { n ^ 3 } { 3 } \right ) \frac { z ^ n } { n ^ 4 } \right \} \end{align*}


\begin{gather*} R ( t ) = \begin{bmatrix} R _ 1 ( t ) _ { k \times k } & R _ 2 ( t ) _ { k \times ( n - k ) } \\ R _ 3 ( t ) _ { ( n - k ) \times k } & R _ 4 ( t ) _ { ( n - k ) \times ( n - k ) } \end{bmatrix} , \\ G ( t ) = \begin{bmatrix} G _ 1 ( t ) _ { k \times k } & G _ 2 ( t ) _ { k \times ( n - k ) } \\ G _ 3 ( t ) _ { ( n - k ) \times k } & G _ 4 ( t ) _ { ( n - k ) \times ( n - k ) } \end{bmatrix} , \end{gather*}


\begin{align*} \mathrm { d i m } _ { \mathbb { C } } \ , \mathbb { S } _ k ( \Gamma , \mathbb { C } ) = 2 \ , \mathrm { d i m } _ { \mathbb { C } } \ , { S } _ k ( \Gamma ) . \end{align*}


\begin{align*} 1 + \cos \alpha _ { 1 0 2 } + \cos \alpha _ { 1 0 3 } + \cos \alpha _ { 1 0 4 } = 0 . \end{align*}


\begin{align*} ( 1 + \overline { c } _ 1 + \dots + \overline { c } _ n ) \cdot ( 1 + c _ 1 + \dots + c _ { k + n } ) = 1 \ . \end{align*}


\begin{align*} M _ \psi : = \| \psi \| _ { L ^ { 1 , \infty } ( G \times \widehat { G } ) } = \sup _ { s > 0 } \ s \underset { \underset { | \psi ( x , \chi ) | > s } { ( x , \chi ) \in G \times \widehat { G } } } { \int } d x d \chi . \end{align*}


\begin{align*} \aligned \left \{ \begin{array} { l l l } - \Delta u + \lambda _ 1 u = { ( K ( x ) \ast | u | ^ { r _ 1 } ) | u | ^ { r _ 1 - 2 } u } + \nu p ( K ( x ) \ast | v | ^ q ) | u | ^ { p - 2 } u \ & \mathbb { R } ^ N , \\ - \Delta v + \lambda _ 2 v = ( K ( x ) \ast | v | ^ { r _ 2 } ) | v | ^ { r _ 2 - 2 } v + \nu q ( K ( x ) \ast | u | ^ p ) | v | ^ { q - 2 } v \ & \mathbb { R } ^ N , \end{array} \right . \endaligned \end{align*}


\begin{align*} ( u | v ) _ { L ^ 2 ( \mathrm { M } _ 0 ) } = ( u | ( \Delta + \lambda - q ) w ) _ { L ^ 2 ( \mathrm { M } _ 0 ) } = ( ( \Delta + \lambda - q ) u | w ) _ { L ^ 2 ( \mathrm { M } _ 0 ) } = 0 . \end{align*}


\begin{align*} \pounds _ { \xi _ i } = \iota _ { { \xi _ i } } \ , d + d \ , \iota _ { { \xi _ i } } , \end{align*}


\begin{align*} O _ { B } ^ { H } ( M ) & = \min \left \{ \sum _ { i = 1 } ^ k 2 ^ i x _ i \mid \sum _ { i = 1 } ^ k ( 2 ^ i - 1 ) x _ i = M , \ x _ i \in \mathbb { N } , 1 \leq i \leq k \right \} \\ & = \min \left \{ M + \sum _ { i = 1 } ^ k x _ i \mid \sum _ { i = 1 } ^ k ( 2 ^ i - 1 ) x _ i = M , \ x _ i \in \mathbb { N } , 1 \leq i \leq k \right \} . \end{align*}


\begin{align*} F _ j : = F ( f , B _ { 4 \rho _ j } ( x _ 0 ) ) . \end{align*}


\begin{align*} \bigcup \limits _ { \mathcal { H } \times J \in N H } \mathcal { H } \times J = \bigcup _ { \mathcal { K } \times I \in P ^ { 1 } N D _ { \lambda } } \mathcal { K } \times I = \bigcup _ { \mathcal { K } \times I \in P ^ { 2 } N D _ { \lambda } } \mathcal { K } \times I . \end{align*}


\begin{align*} \mathcal { I } ^ \star ( \lambda ) = \sup _ { \pi \in \Gamma \left ( \Pi \left ( \mu _ { 1 3 } , \mu _ { 2 3 } \right ) , \phi _ \lambda \right ) } \int _ { \mathcal { V } \times \mathcal { V } } \phi _ \lambda \ , d \pi . \end{align*}


\begin{align*} M _ a = \begin{pmatrix} 1 & 0 \\ a & 0 \end{pmatrix} \end{align*}


\begin{align*} & E [ X _ { \ell , \Delta } | { \cal F } _ { ( \ell + 1 ) \Delta } ] = \nu \int _ { \ell \Delta } ^ { ( \ell + 1 ) \Delta } Z _ 0 ( s ) d s = \Delta \nu Z _ 0 ( \ell \Delta ) ( 1 + O ( \Delta ) ) \end{align*}


\begin{align*} b _ { i , j } > 0 \iff \varepsilon ( i ) = + \varepsilon ( j ) = - . \end{align*}


\begin{align*} E = \frac { 1 } { 2 } \rho u ^ 2 + \frac { \tau _ 2 } { 2 \mu } \rho S ^ 2 + \rho e ( \theta , q ) , \end{align*}


\begin{align*} R \circ \alpha = \alpha \circ R , \end{align*}


\begin{align*} D _ a = a - d , D _ b = b - d , D _ c = c - d \end{align*}


\begin{align*} h _ { \mu } ( Z , \lambda ) & = h _ { \mu } ( Y , \nu ) + \lim _ { n \to \infty } \int _ { X } \left ( H \left ( \xi _ { n } ^ { x } \right ) - H \left ( \xi _ { n - 1 } ^ { x } \right ) \right ) d \eta ( x ) \\ & = h _ { \mu } ( Y , \nu ) + \lim _ { n \to \infty } \frac { 1 } { n } H \left ( \xi _ { n } | X , \eta \right ) . \end{align*}


\begin{align*} \mathcal { S } ^ * ( 1 / 2 ) = \bigg \{ f \in \mathcal { A } : { \rm R e } \left ( \frac { z f ^ { \prime } ( z ) } { f ( z ) } \right ) > 1 / 2 , \ ; z \in \mathbb { D } \bigg \} \end{align*}


\begin{align*} | \nabla \chi _ m ^ i ( x ) | \leq \frac { 2 } { \sqrt { \mu _ m ^ i \mu _ m ^ { i - 1 } } } { \rm a n d } | \nabla ^ 2 \chi _ m ^ i ( x ) | \leq \frac { 4 } { \mu _ m ^ i \mu _ m ^ { i - 1 } } , \quad \mbox { f o r \ a n y } \ i = 1 , \cdots , k . \end{align*}


\begin{align*} B _ \mathbf { m } ( s ) = \int _ { \mathbb { R } ^ 3 } \ ! & \mathcal { D } _ { \widetilde { P } _ 1 ' ( \bar { t } ) s } ^ { ( 1 ) } f _ { 1 , m _ 1 } \left ( x + \widetilde { P } _ 1 ( t ) , y \right ) \mathcal { D } _ { \widetilde { P } _ 2 ' ( \bar { t } ) s } ^ { ( 2 ) } f _ { 2 , m _ 2 } \left ( x , y + \widetilde { P } _ 2 ( t ) \right ) \\ & \widetilde { \zeta } _ \mathbf { m } ( x , y , t , s ) \ , \mathrm { d } x \mathrm { d } y \mathrm { d } t . \end{align*}


\begin{align*} \begin{gathered} f ^ { 2 K } ( \cdot , t ) \xrightarrow { K \to \infty } f _ 0 \circ \big ( \tilde { y } ^ \mathrm { e v e n } _ t \big ) ^ { - 1 } , \\ f ^ { 2 K + 1 } ( \cdot , t ) \xrightarrow { K \to \infty } f _ 0 \circ \big ( \tilde { y } ^ \mathrm { o d d } _ t \big ) ^ { - 1 } , \end{gathered} \end{align*}


\begin{align*} \nabla = \partial _ z + u ( z ) \check { \alpha } . \end{align*}


\begin{align*} \frac { a } { q } = [ 0 ; c _ 1 , \dots , c _ s ] = \cfrac { 1 } { c _ 1 + \cfrac { 1 } { c _ 2 + \cfrac { 1 } { c _ 3 + \cdots + \cfrac { 1 } { c _ s } } } } \ , , c _ j \le M \ , , \forall j \in [ s ] \ , . \end{align*}


\begin{align*} \mathsf { S } \mathbb { P } & = \{ S \subseteq \mathbb { P } : S \} \\ & = \{ S \subseteq \mathbb { P } : S \} \\ & = \{ S \subseteq \mathbb { P } : S \} . \end{align*}


\begin{align*} \widetilde { \Delta } ( x , m ) & = \Delta ( x ) + \rho _ { \beta } ( m ) - \tau \circ ( \rho _ { \beta } ( m ) ) , \\ \widetilde { \varphi } ( x , m ) & = ( \varphi _ L ( x ) , \varphi _ M ( m ) ) , \\ \widetilde { \alpha } ( x , m ) & = ( \alpha ( x ) , \beta ( m ) ) , \end{align*}


\begin{align*} T ( f \cdot g ) = f T ( g ) + T ( f ) g + 2 B ( A ( f ) , A ( g ) ) \end{align*}


\begin{align*} f _ \lambda ( v ) : = \sup _ { s ^ \prime \in \mathcal { S } } \phi _ \lambda ( v , s ^ { \prime } ) . \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( \frac { 1 } { 1 - x ^ l y ^ m z ^ n } \right ) ^ { \frac { l m ^ 2 } { n ^ 4 } } \end{align*}


\begin{align*} ( \ast ) & = w _ { i j } R ^ \varphi _ { i k } R ^ \varphi _ { k j } + w _ j R ^ \varphi _ { i k } R ^ \varphi _ { k j , i } + w _ j R ^ \varphi _ { i k , i } R ^ \varphi _ { k j } \\ & \phantom { = \ ; } + w _ { t i } R _ { i j k t } R ^ \varphi _ { j k } + w _ t R _ { i j k t , i } R ^ \varphi _ { j k } + w _ t R _ { i j k t } R ^ \varphi _ { j k , i } \ , . \end{align*}


\begin{align*} 1 2 \cdot \mu ^ { p \delta } _ { D R } = - [ ( \mu _ { D D , \delta } ) _ c ] . \end{align*}


\begin{align*} \gamma ( a ) = p , \gamma ( b ) = q , \end{align*}


\begin{align*} \Lambda ( z ) = q _ { + } ^ 2 ( z ) p ^ 2 ( z ) L ( z ) \end{align*}


\begin{align*} J ( \sigma ) = \int _ a ^ b F ( \sigma , \dot \sigma ) \ , d t \end{align*}


\begin{align*} \acute { \rho } _ \sigma ( \hat { s } ) = & \sum _ { 0 \leq a < b \leq 3 } \acute { \rho } _ \sigma ^ { a b } ( \hat { s } ) [ \det \acute { J } _ { a b } ^ \sigma ] ( \hat { s } ) , \\ | \acute { \rho } _ \sigma | ( \hat { s } ) = & \left | \sum _ { 0 \leq a < b \leq 3 } \acute { \rho } _ \sigma ^ { a b } ( \hat { s } ) [ \det \acute { J } _ { a b } ^ \sigma ] ( \hat { s } ) \right | . \end{align*}


\begin{align*} \langle G ^ { \prime } ( u ) - G ^ { \prime } ( v ) , u - v \rangle = 0 . \end{align*}


\begin{align*} \frac { \delta F } { \delta \nu } ( \nu _ t , \mu _ t , x ) + \frac { \sigma ^ 2 } { 2 } \log \frac { \Psi ( \nu _ t , \mu _ t ) ( x ) } { \pi ( x ) } = C _ t , \end{align*}


\begin{align*} ( b b ' ) \ast - = - \langle b \ast - , b ' \ast - \rangle , \forall b , b ' \in B , \end{align*}


\begin{align*} d ( S , ( \delta , 1 ) ) \leq d ( ( \delta , f ( \delta ) ) , ( \delta , 1 ) ) = 1 - f ( \delta ) \end{align*}


\begin{align*} \lim _ { r \to 0 ^ + } \frac { A ( r ) ^ 2 } { A ' ( r ) } \cdot \left [ \mathcal W ( r ) ^ { p - 1 - \alpha } \right ] ' & = \lim _ { r \to 0 ^ + } r ^ \delta u _ 0 ' ( r ) ^ { p - 1 - \alpha } \cdot \frac { g _ 1 ( v ( r ) ) } { h _ 1 ( v _ 0 ( r ) ) } = 0 \end{align*}


\begin{align*} { \rm e l l } _ h ( A ) = \{ ( x _ 0 , \xi _ 0 ) \in \overline { T } ^ * \mathcal { M } \ , : \ , ( \langle \xi \rangle ^ { - k } \sigma _ h ( A ) ) ( x _ 0 , \xi _ 0 ) \neq 0 \} . \end{align*}


\begin{align*} { \bf E } \left ( { \bf A } , { \mathbb V } \right ) = \left \{ { \bf A } _ { \theta } : \theta \left ( x , y \right ) = \sum _ { i = 1 } ^ { s } \theta _ { i } \left ( x , y \right ) e _ { i } \ \ \ \ \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in { \bf T } _ { s } ( { \bf A } ) \right \} . \end{align*}


\begin{align*} E : = \bigcap _ { m = 1 } ^ \infty \bigcup _ { k = m } ^ \infty E _ k \end{align*}


\begin{align*} \left | \langle T ( f ) _ b , \varphi \rangle - \langle T ( f ) ( \cdot ) , \varphi \rangle \right | & = \left | \lim _ { r \nearrow 1 } \int _ 0 ^ { 2 \pi } T ( f ) ( r e ^ { i \theta } ) \varphi ( \theta ) \ , d \theta - \int _ 0 ^ { 2 \pi } T ( f ) ( e ^ { i \theta } ) \varphi ( \theta ) \ , d \theta \right | \\ & \leq C _ \varphi \lim _ { r \nearrow 1 } \int _ 0 ^ { 2 \pi } | T ( f ) ( r e ^ { i \theta } ) - T ( f ) ( e ^ { i \theta } ) | \ , d \theta = 0 , \end{align*}


\begin{align*} \varphi _ \lambda ( v , v ^ \prime ) = g \left ( s _ 1 ^ \prime , s _ 2 ^ \prime \right ) - \lambda _ 1 c _ 1 \left ( s _ 1 , s _ 1 ^ \prime \right ) - \lambda _ 2 c _ 2 \left ( s _ 2 , s _ 2 ^ \prime \right ) , \end{align*}


\begin{align*} S _ n \left ( \frac { 1 - q t | } { | 1 - q } A \right ) = ( 1 - q t ) \sum _ { k = 0 } ^ n ( - q t ) ^ k R _ { 1 ^ k , n - k } \left ( \frac { A } { 1 - q } \right ) \end{align*}


\begin{align*} \begin{array} { l l } \underset { x \in X } { } & \sum \limits _ { i = 1 } ^ { m } f _ { i } \left ( x \right ) , \end{array} \end{align*}


\begin{align*} \tilde { \mathbf { m } } _ \lambda = \lambda \mathbf { b } _ \lambda \mathbf { e } _ \lambda . \end{align*}


\begin{align*} H ^ s ( \Sigma ) = \{ \varphi = h _ { | \Sigma } ; \ ; h \in H ^ { 1 / 2 } ( \partial \mathrm { M } ) \} , \end{align*}


\begin{align*} \frac \nu 2 \| \bar u ( t ) \| ^ 2 = \max _ { v \in \R ^ k } \frac \nu 2 \| v \| ^ 2 . \end{align*}


\begin{align*} \int _ { \R ^ 2 } \sup _ { y \in \R } \Big ( \int _ { \R ^ 2 } I _ k d x d z \Big ) ^ 2 | \hbar | ^ { 2 H - 2 } | h | ^ { 2 H - 2 } d h d \hbar < C k = 2 , 3 , 4 . \end{align*}


\begin{align*} u _ { t } + \mathcal { L } ^ { \Phi } _ { A } u = ( - \Delta ) ^ { \frac { s } { 2 } } f + g \quad \Omega _ { T } \equiv \Omega \times ( 0 , T ) , \end{align*}


\begin{align*} P _ r ( \theta ) = \frac { 1 - r ^ 2 } { 1 - 2 r \cos ( \theta ) + r ^ 2 } , \end{align*}


\begin{align*} l o n g t a i l ( \lambda ) = \max _ { \Sigma \in \mathbb { S } } \sharp \Sigma \cap C _ { \lambda } \end{align*}


\begin{align*} \bigg \| \lambda _ s ( b ) \bigg ( \sum _ { t \in G } b _ t \delta _ t \bigg ) \bigg \| ^ 2 & = \bigg \| \sum _ { t \in G } b b _ t \delta _ { s t } \bigg \| ^ 2 = \bigg \| \sum _ { t \in G } ( b b _ t ) ^ * b b _ t \bigg \| = \bigg \| \sum _ { t \in G } b _ t ^ * ( b ^ * b ) b _ t \bigg \| \\ & \leq \bigg \| \sum _ { t \in G } \| b \| ^ 2 b _ t ^ * b _ t \bigg \| = \| b \| ^ 2 \bigg \| \sum _ { t \in G } b _ t ^ * b _ t \bigg \| = \| b \| ^ 2 \bigg \| \sum _ { t \in G } b _ t \delta _ t \bigg \| ^ 2 \end{align*}


\begin{align*} g ^ { ( k ) } + A _ 1 g = 0 \end{align*}


\begin{align*} \mathcal { M } _ { \alpha } ( \vec { f } ) ( x ) & \le \prod _ { i = 1 } ^ { m } M _ { \alpha _ { i } } f _ { i } ( x ) \ \ \ x \in \mathbb { R } ^ { n } . \end{align*}


\begin{align*} \tilde { t } _ { \nu , \nu } ( x ) = \mathbf { L } _ \nu ( x ) . \end{align*}


\begin{align*} T ( f ^ { 2 } ) = 2 f T ( f ) + 2 B ( A ( f ) , A ( f ) ) \left ( f \in U \subset P \right ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { k } \lambda _ i ^ { \gamma } ( C + D ) + \sum _ { i = 1 } ^ { k } \lambda _ i ^ { \gamma } ( C - D ) \geq 2 \sum _ { i = 1 } ^ k \lambda _ i ^ { \gamma } ( C ) . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } M _ j ( t ) & = \frac { \nu q Y } { \lambda } \int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } ( 1 - y ) \sum _ { k = j } ^ { \infty } y ^ { k - 1 } d y \\ & = \frac { \nu q Y } { \lambda } \int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y . \end{align*}


\begin{align*} ( \partial _ T + P \cdot \nabla _ X ) F = 4 \pi & \int \d P _ 2 \d P _ 3 \d P _ 4 \delta ( P + P _ 2 - P _ 3 - P _ 4 ) \delta ( P ^ 2 + P _ 2 ^ 2 - P _ 3 ^ 2 - P _ 4 ^ 2 ) \\ & \times | \hat V ( P - P _ 3 ) - \hat V ( P - P _ 4 ) | ^ 2 ( F F _ 2 \widetilde F _ 3 \widetilde F _ 4 - F _ 4 F _ 3 \widetilde F _ 2 \widetilde F ) \ . \end{align*}


\begin{align*} | { \Sigma _ t } | = | { \Sigma _ 0 } | - \frac { 1 } { 2 } | \lambda | t ^ 2 + O ( t ^ 3 ) . \end{align*}


\begin{align*} P _ \mu = \frac { P _ \mu } { P _ { \mu - 1 } } P _ { \mu - 1 } \leq \left ( 1 + \frac { \delta } { 4 } \right ) P _ m \left ( 1 + \frac { \delta } { 2 } \right ) \leq P _ m ( 1 + \delta ) = \lambda P _ m , \end{align*}


\begin{align*} \lim _ { t \to \infty } t ^ { \frac { d + 2 \beta } { \alpha } \frac { q - 1 } { q } } \| P _ t ^ { \Gamma } f - A \Psi _ t \| _ { q , M _ { \Gamma } } = 0 . \end{align*}


\begin{align*} B ^ { k + 2 } = B ^ { k + 1 } \cup \{ s \} \setminus \{ p \} = B ^ { k } \cup \{ s \} \setminus \{ q \} . \end{align*}


\begin{align*} ( \partial _ t - \partial _ x ^ 2 ) ( u - u ( x ) ) = u ( x ) \xi + ( u - u ( x ) ) \xi . \end{align*}


\begin{align*} ( x , y ) \sim _ { X \boxtimes Y } ( u , v ) \Leftrightarrow \begin{cases} x = u \mbox { a n d } y \sim _ Y v \\ y = v \mbox { a n d } x \sim _ X u \\ x \sim _ X u \mbox { a n d } y \sim _ Y v . \end{cases} \end{align*}


\begin{align*} ( d - d ^ { \sigma } ) g = D ^ { k - 1 } ( h ^ { \sigma } - h ) \end{align*}


\begin{align*} & \sum _ { k = 0 } ^ { \infty } \frac { \binom { 2 k } { k } \binom { 3 k } { k } } { x ^ k } ( 3 H _ { 3 k } - H _ k ) = \bigg ( \log \frac { x } { x - 2 7 } \bigg ) \sum _ { k = 0 } ^ { \infty } \frac { \binom { 2 k } { k } \binom { 3 k } { k } } { x ^ k } , \end{align*}


\begin{align*} \frac { S _ k ^ l ( t ) } { S _ { l - 1 } ^ l ( t ) } = \tilde S _ k ^ l ( t ) , l = 1 , \dots , p ; k \in \mathbb { Z } _ + . \end{align*}


\begin{align*} { 1 \over 2 } \sum _ j E _ j \R ^ \bullet \pi _ * ( d _ j ) = \Theta ^ 2 ( V ) ^ { - 1 } \end{align*}


\begin{align*} \int _ \Omega \Big | V ^ p \Big ( \ln \ln ( e + V ) \Big ) ^ 2 \psi ^ h _ { \mu _ j , \xi _ j } \Big | d x = O ( ( \ln | \ln \mu _ i | ) ^ 2 ) = O \bigg ( \Big ( \ln \Big | \ln \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big | \Big ) ^ 2 \bigg ) . \end{align*}


\begin{align*} \left | \lambda _ { \min } ( T _ n ) - \lambda _ { \min } ( T ) \right | = \left | \inf _ { \| f \| _ 2 = 1 } \langle f , T _ n f \rangle - \inf _ { \| f \| _ 2 = 1 } \langle f , T f \rangle \right | \leq \sup _ { \| f \| _ 2 = 1 } | \langle f , T _ n f \rangle - \langle f , T f \rangle | \to 0 \end{align*}


\begin{align*} e _ { i j } = \phi _ i - \phi _ j = \sum _ { m = 1 } ^ { \infty } C _ { q _ m } h ^ { q _ m } \big [ r ^ { ( i - j ) q _ m } - 1 \big ] r ^ { ( j - 1 ) q _ m } \end{align*}


\begin{align*} \bar { F } _ { P _ s } ( \theta , x ) \overset { \Delta } { = } \Pr \left ( P _ s ( \theta ) > x \right ) , x \in [ 0 , 1 ] , \end{align*}


\begin{align*} \alpha _ n = ( - 1 ) ^ { n + m } q ^ { \left ( { n + m \atop 2 } \right ) } \frac { 1 - a q ^ { 2 n } } { 1 - a } \frac { ( a ) _ { n - m } } { ( q ) _ { n + m } } \mbox { a n d } \quad \beta _ n = \delta _ { n , - m } , \end{align*}


\begin{align*} \mathcal { O } _ i = \left \{ ( A , B ) : \ \mbox { $ A , B \in \binom { [ n ] } { k } $ s u c h t h a t } | A \cap B | = k - i \right \} , \end{align*}


\begin{align*} \mathcal { F } ' _ { X ^ { \ast } X } : & L ^ 2 ( X ^ { \ast } ) \longrightarrow L ^ 2 ( X ) ; \\ & g \longmapsto \mathcal { F } ' _ { X ^ { \ast } X } ( g ) ( x ) = \int _ { X ^ { \ast } } \psi ( \langle x ^ { \ast } , x \rangle ) g ( x ^ { \ast } ) d x ^ { \ast } , \end{align*}


\begin{align*} { \rm I I } = \int _ { X } \int _ { G ^ { \mathbb { N } } } \log \frac { d \left ( \omega _ { 1 } \lambda \right ) _ { x } } { d \lambda _ { x } } \left ( \psi _ { x } ( \omega ) \right ) d \mathbb { P } _ { \mu } ^ { \pi ( x ) } ( \omega ) d \eta ( x ) , \end{align*}


\begin{align*} F ( p ) : = \begin{cases} \min _ { v } & \sum _ { i = 1 } ^ n c p _ i \max ( - x _ i - y _ i , - d _ i ) + \sum _ { i \neq j } \alpha _ { i j } v _ { i j } \\ s . t . \quad & \begin{cases} - v \leq 0 \\ \sum _ { j \neq i } v _ { i j } - x _ { 0 i } \leq 0 , \forall i \in I \end{cases} \end{cases} \end{align*}


\begin{align*} \| \langle X , t \rangle \| _ { L _ k } \leq L k ^ { 1 / \alpha } \| \langle X , t \rangle \| _ { L _ 2 } = L k ^ { 1 / \alpha } \| t \| _ 2 . \end{align*}


\begin{align*} F ( x ) : = \frac { x ^ { N } } { x + m ^ 2 } . \end{align*}


\begin{align*} | f ( x ) - f ( y ) | \leqslant C _ n \cdot \frac { ( \Vert Q \Vert _ 1 ) ^ { \frac { 1 } { q } } } { \log ^ { \frac { 1 } { n } } \left ( 1 + \frac { r _ 0 } { 2 | x - y | } \right ) } \ , , \ , \ , r _ 0 = d ( K , \partial D ) , \end{align*}


\begin{align*} E _ S ( E _ { T ^ c } L _ T [ f ] ) _ { S \rightarrow x } = E _ S E _ { T ^ c } L _ { T ' } ( L _ S [ f ] ) _ { S \rightarrow x } = E _ { T '^ c } L _ { T ' } [ D _ { S , x } [ f ] ] = ( D _ { S , x } f ) ^ { = T \setminus S } . \end{align*}


\begin{align*} \begin{cases} \tau _ 0 ( A ) & = \sigma ( A ) \\ \tau _ i ( A ) & = \mu ( A , i ) \left ( \tau _ { i - 1 } ( A ) \right ) , \mbox { f o r } 1 \leq i \leq k , \end{cases} \end{align*}


\begin{align*} f ( \alpha , s ) & = \rho ( s ^ 2 t \cos ( \gamma + \beta ) + s \sqrt { 1 - s ^ 2 } \cos ( \alpha + \beta ) ) . \end{align*}


\begin{align*} r = \Omega \left ( \frac { b ^ { 5 + 6 \kappa } } { \chi \beta \Delta e ^ { \beta \Delta } \cdot ( \Delta \log n ) ^ { \kappa } ( \log \log n ) ^ { 1 + \kappa } \cdot e ^ { 1 3 \kappa } } \right ) . \end{align*}


\begin{align*} N _ { d r } = \left ( \sum _ { i = 1 } ^ k x _ i \right ) a + r ( a + d ) = \left ( \sum _ { i = 1 } ^ k 2 ^ i x _ i \right ) a + r d . \end{align*}


\begin{align*} J _ 3 = & \int _ \Omega \Big [ f _ 0 ( V ) - f _ \epsilon ( V ) \Big ] \psi ^ h _ { \mu _ j , \xi _ j } d x \\ = & \epsilon \int _ \Omega V ^ p \ln \ln ( e + V ) \psi ^ h _ { \mu _ j , \xi _ j } d x - \epsilon ^ 2 \int _ \Omega V ^ p \Big ( \ln \ln ( e + V ) \Big ) ^ 2 \psi ^ h _ { \mu _ j , \xi _ j } d x . \end{align*}


\begin{align*} r _ 1 ( u , \bar v ) & = \sum _ { j = 1 } ^ n R ( e _ j , \bar e _ j , u , \bar v ) , & r _ 2 ( u , \bar v ) & = \sum _ { j = 1 } ^ n R ( e _ j , u , \bar v , \bar e _ j ) , \\ r _ 3 ( u , \bar v ) & = \sum _ { j = 1 } ^ n R ( u , \bar v , e _ j , \bar e _ j ) , & r _ 4 ( u , \bar v ) & = \sum _ { j = 1 } ^ n R ( u , e _ j , \bar e _ j , \bar v ) , \end{align*}


\begin{align*} \binom i s \alpha ^ s ( 1 - \alpha ) ^ { i - s } \binom { j - i } r \alpha ^ r ( 1 - \alpha ) ^ { j - i - r } , \end{align*}


\begin{align*} \left \{ \begin{aligned} d \xi _ t ^ u & = e ^ { - \frac { \beta } { 2 } t } h ( x _ t ^ u , u _ t ) d t + d \tilde { W } _ t , \ t \in [ 0 , \infty ) , \\ \xi _ 0 & = 0 . \end{aligned} \right . \end{align*}


\begin{align*} K _ j = K _ j ^ 2 = \left ( \prod _ { i : v _ i \in I _ j } P _ i \right ) ^ 2 = \left ( \prod _ { i : v _ i \in I _ j } P _ i \prod _ { i : v _ i \notin I _ j } I \right ) ^ 2 = \prod _ { i = 1 } ^ { n } \hat { P } _ i \prod _ { i = 0 } ^ { n - 1 } \hat { P } _ { n - i } . \end{align*}


\begin{align*} [ ( d , D ) , ( d ' , D ' ) ] = ( d \circ d ' - d ' \circ d , D \circ d ' - d ' \circ D ) . \end{align*}


\begin{align*} A f ( x ) = f ( \neg x ) , \end{align*}


\begin{align*} L ( t ) = \begin{bmatrix} - S _ 1 ( t ) & - S _ 2 ( t ) \\ 0 & I \\ I & 0 \\ S _ 3 ( t ) & S _ 4 ( t ) \\ \end{bmatrix} , \end{align*}


\begin{align*} f ^ * ( z _ 1 , \ldots , z _ { m } ) = \sum _ { n = m } ^ { \infty } P _ n ( z _ 1 , \ldots , z _ { m } ) = \sum _ { i _ 1 \geq 1 , \ldots , i _ m \geq 1 } c _ { i _ 1 , \ldots , i _ m } z _ 1 ^ { i _ 1 } \cdots z _ m ^ { i _ m } , \end{align*}


\begin{align*} [ D \widetilde { \psi } : D \tau _ M ] _ t = [ D \psi : D \varphi ] _ t \ , \lambda ^ { \varphi } ( t ) \end{align*}


\begin{align*} \rho _ i ( u ) = \log ( \theta _ i ) + B ^ * _ i ( u ) + \frac { 1 } { 2 } u + \log \left ( \frac { T _ i ^ { \infty } - T _ i ( u ) } { T _ i ^ { \infty } } \right ) . \end{align*}


\begin{align*} b \ast ( b ' \ast x ) = 0 , \forall b , b ' \in B , \ ; \forall x \in X . \end{align*}


\begin{align*} & \lim _ { t \to \infty } e ^ { - \lambda t } S _ { j , + } ( t ) = \nu Y \int _ 0 ^ \infty e ^ { - \lambda s } \left ( \sum _ { k = 1 } ^ \infty p _ { j , + } ^ k ( s ) \right ) d s , \\ & \lim _ { t \to \infty } e ^ { - \lambda t } S _ { j , - } ( t ) = \nu Y \int _ 0 ^ \infty e ^ { - \lambda s } \left ( \sum _ { k = 1 } ^ \infty p _ { j , - } ^ k ( s ) \right ) d s , \end{align*}


\begin{align*} ( - \Delta ) ^ s u + \langle b , \ , \nabla u \rangle + \rho \ , c u = 0 \ , \ , \ , \R ^ N , \\ \end{align*}


\begin{align*} I _ { i + j + \alpha - 1 } ( 2 \sqrt { x } ) = \sqrt { x } \frac { d } { d x } I _ { i + j + \alpha } ( 2 \sqrt { x } ) + \frac { \alpha + i + j } { 2 \sqrt { x } } I _ { i + j + \alpha } ( 2 \sqrt { x } ) , \end{align*}


\begin{align*} \bar { \mu } : = \norm { \cdot } _ 1 - \lim _ { a \to \infty } M _ a ( \mu ) . \end{align*}


\begin{align*} \lim _ { t \to \infty } e ^ { - \lambda t } M _ - ( t ) = \nu Y \int _ 0 ^ { \infty } e ^ { - \lambda s } p _ 0 ( s ) d s , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | u _ n | ^ { p } ) | v _ n | ^ { q } = \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | u | ^ { p } ) | v | ^ { q } + o _ n ( 1 ) . \end{align*}


\begin{align*} u _ \mathrm { l i n } ( t , r ) = \begin{cases*} 2 & i f $ 0 < r < t $ , \\ \frac { 2 t } { r } & i f $ t \leq r < 1 - t $ , \\ \frac { 1 - r + t } { r } & i f $ 1 - t \leq r < 1 + t $ , \\ 0 & o t h e r w i s e . \end{cases*} \end{align*}


\begin{align*} [ D \widetilde { \psi } : D \tau _ M ] _ t = s ( \psi ) [ D \widetilde { \chi } : D \tau _ M ] _ t & = s ( \psi ) [ D \widetilde { \chi } : D \widetilde { \varphi } ] _ t \ , [ D \widetilde { \varphi } : D \tau _ M ] _ t \\ & = [ D \widetilde { \psi } : D \widetilde { \varphi } ] _ t \ , \lambda ^ { \varphi } ( t ) \end{align*}


\begin{align*} \Big \| \sum _ { n = 1 } ^ N f _ n \Big \| _ X \leq \sum _ { n = 1 } ^ N K _ X ^ n \| f _ n \| _ X . \end{align*}


\begin{align*} t ^ * = \begin{cases} \displaystyle \frac { 2 \mathbb S ^ \circ } { | \omega | } , & \omega \neq 0 , \\ \infty , & \omega = 0 . \end{cases} \end{align*}


\begin{align*} \begin{gathered} f ^ { \mathrm { e v e n } } ( \cdot , 1 ) = f _ 0 , \\ f ^ { \mathrm { o d d } } ( \cdot , 1 ) = f _ 0 \circ y _ { - 2 \tau _ 1 } ^ { ( i _ 1 ; L _ 1 ) } . \end{gathered} \end{align*}


\begin{align*} \Sigma _ { \mathrm { M } } ( \delta _ 0 ) : = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \boldsymbol { K } ( \mu _ { ( Y _ 1 , \dotsc , Y _ L ) | X = x } , \gamma _ { ( Y _ 1 , \dotsc , Y _ L ) | X = x } ) \le \delta _ 0 x , \ ; \mu _ { X } = \gamma _ X \right \} , \end{align*}


\begin{align*} \| v \| _ { V } = \min \{ \| u \| _ E \mid f ( u ) = v \} . \end{align*}


\begin{align*} \sigma \left ( b \right ) : = \left \{ g _ { i } ( x ) \leq b _ { i } , \ ; i = 1 , \ldots , m \right \} , \end{align*}


\begin{align*} \Gamma _ 0 : = \Gamma \setminus \{ 0 \} , \varrho ( z ) : = 1 \vee \mathbf { d } \left ( z , \Gamma _ 0 ^ c \right ) ^ { - 1 } \end{align*}


\begin{align*} \partial _ { i i } V = \bigg ( \frac { 1 } { 1 + \lambda _ i ^ 2 } - 2 e _ i ^ 2 \bigg ) V + e _ i ^ 2 V \\ \partial _ { i j } V = V e _ i e _ j . \end{align*}


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } ^ { 2 } u ( t , x ) + a ( t ) \mathcal { H } _ { V } u ( t , x ) + q ( t ) u ( t , x ) = f ( t , x ) , \quad ( t , x ) \in ( 0 , T ] \times \mathbb { R } ^ { n } , \\ u ( 0 , x ) = u _ { 0 } ( x ) , x \in \mathbb { R } ^ { n } , \\ \partial _ { t } u ( 0 , x ) = u _ { 1 } ( x ) , x \in \mathbb { R } ^ { n } , \end{array} \right . \end{align*}


\begin{align*} \tau _ { n + 1 , \omega } ^ { I } ( \chi ) \coloneqq \sup \big \lbrace \tau : \ \tau ^ { I } _ { n , \delta _ 1 , \omega } ( \chi ) \leq \tau \leq b \ \ \ \ W _ { \mathbf { X } ( \omega ) , \gamma , \delta _ 1 } ^ { { \gamma - \delta _ 1 } } ( \tau _ { n , \omega } ^ { I } ( \chi ) , \tau ) \leq \chi \big \rbrace . \end{align*}


\begin{align*} \mathcal { S } ^ * ( \alpha ) = \bigg \{ f \in \mathcal { A } : { \rm R e } \left ( \frac { z f ^ { \prime } ( z ) } { f ( z ) } \right ) > \alpha , \ ; z \in \mathbb { D } \bigg \} \end{align*}


\begin{align*} W _ M ^ + ( h ) : = \{ m \in M \ : X ^ M _ h ( m ) \in V _ + ( m ) \} \end{align*}


\begin{align*} e _ k ( t _ 1 , \ldots , t _ n ) : = \sum _ { 1 \le j _ 1 < j _ 2 < \cdots < j _ k \le n } t _ { j _ 1 } t _ { j _ 2 } \cdots t _ { j _ k } \end{align*}


\begin{align*} H \left ( \xi _ { n } | X , \eta \right ) = \int _ { X } H \left ( \xi _ { n } ^ { x } \right ) d \eta ( x ) . \end{align*}


\begin{align*} \P _ x ( \tau _ { \Gamma } > t ) = \P _ { t ^ { - 1 / \alpha } x } ( \tau _ { \Gamma } > 1 ) , x \in \Gamma , \ ; t > 0 . \end{align*}


\begin{align*} S _ { 1 , \frac { \mathbf { c } } { \sqrt { \omega } } } ( \Theta ^ { \omega } ) = \omega ^ { \frac { d } { 2 } - 2 } S _ { \omega , \mathbf { c } } ( \Theta ) . \end{align*}


\begin{align*} \| x \| - \| V \theta _ g x \| = \| \theta _ f x \| - \| V \theta _ g x \| \leq \| \theta _ f x - V \theta _ g x \| \leq ( \varepsilon + \delta ) \| x \| , \forall x \in \mathcal { X } . \end{align*}


\begin{align*} W ^ - = \begin{bmatrix} u _ n + p / u _ n \\ u _ { \tau } \end{bmatrix} , \Lambda ^ - = \begin{bmatrix} u _ n & 0 \\ 0 & u _ n \end{bmatrix} , W ^ + = p , \Lambda ^ + = - 1 / u _ n \end{align*}


\begin{align*} F ^ { \prime } ( u ) = \mu G ^ { \prime } ( u ) \mathcal { G } , \end{align*}


\begin{align*} \mathcal { B } ( X ) = \mathcal { B } ( X \setminus \{ x \} ) \cup \left | \operatorname { s t } ( \mathcal { N } ( x ) ) \right | . \\ \end{align*}


\begin{align*} \mathbb { P } ( E _ i ) = \mathbb { P } ( E _ { i , 1 } ) \mathbb { P } ( E _ { i , 2 } ) . \end{align*}


\begin{align*} w \rho _ w ( \theta , w ) = e ^ { c ( \theta , w ) + i d ( \theta , w ) } \end{align*}


\begin{align*} h ( v ) = \chi _ { [ 0 , 1 ] } ( v ) \frac { g ( v ) } { \sqrt { 1 + v } } . \end{align*}


\begin{align*} \frac { 2 \rho \beta b ^ { 2 } } { d } = \frac { 2 \rho \beta } { 3 / 2 + \left ( 1 + \beta \right ) \left ( T / 2 - \varepsilon \right ) ^ { 2 } / \left ( \varepsilon \left ( T - \varepsilon \right ) \right ) } . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ \gamma u + q u = 0 & \Omega , \\ u = f & \Omega ^ c , \end{cases} \end{align*}


\begin{align*} \mathrm { o s c } ( v , T ^ N ( I _ \alpha ^ { ( k ) } ) ) = \mathrm { o s c } ( v , I _ \alpha ^ { ( k ) } ) + O ( e ^ { ( - \lambda _ 1 ( 1 - a ) + \tau ) k } ) ( c _ 0 ( D \varphi ) + p _ a ( D \varphi ) ) . \end{align*}


\begin{align*} f _ n ( \pmb { t } _ n , \pmb { x } _ n , t , x ) = & f _ n ( t _ 1 , \dots , t _ n , x _ 1 , \dots , x _ n , t , x ) \\ \coloneqq & G _ { t - t _ n } ( x - x _ n ) \times \dots \times G _ { t _ 2 - t _ 1 } ( x _ 2 - x _ 1 ) \mathbf { 1 } _ { T _ n ( t ) } ( \pmb { t } _ n ) \end{align*}


\begin{align*} \log \left ( 1 + \frac { P _ k } { 1 + ( M - 1 ) P _ { k + 1 } } \right ) = R , 1 \leq k \leq K - 1 , \end{align*}


\begin{align*} P ( W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 , W ^ k _ { \ell _ 2 \Delta , t } ( j ) = 1 ) - \nu \Delta p ^ k _ { j , + } ( t - \Delta \ell _ 2 ) E [ Z _ 0 ( \Delta \ell _ 2 ) ; W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 ] . \end{align*}


\begin{align*} \left ( \mathcal { A } _ \varepsilon - z I \right ) ^ { - 1 } = \mathcal { G } _ \varepsilon ^ { - 1 } \left ( \int _ { Y ' } ^ \oplus \left ( \frac { 1 } { \varepsilon ^ 2 } \mathcal { A } _ { \chi , \varepsilon } - z I \right ) ^ { - 1 } d \chi \right ) \mathcal { G } _ \varepsilon , z \in \rho ( \mathcal { A } _ \varepsilon ) , \end{align*}


\begin{align*} M ( z ) = \Lambda + z \Pi ^ * \left ( I - z \mathcal { A } _ 0 ^ { - 1 } \right ) ^ { - 1 } \Pi , z \in \rho ( \mathcal { A } _ 0 ) . \end{align*}


\begin{align*} 2 \lambda + 1 = \left \{ 2 \lambda ^ { ( i ) } + 1 : i \in \mathcal { I } ( \lambda ) \cup \left \{ t + 1 \right \} \right \} . \end{align*}


\begin{align*} ( b , x ) \cdot ( b ' , x ' ) = ( b b ' , x x ' + b \ast x ' + x \ast b ' ) \end{align*}


\begin{align*} f [ k ] = f ( \underbrace { 1 , \dots , 1 } _ { k } ) . \end{align*}


\begin{align*} ( \underline { D } ^ { 2 } , \underline { S } ^ { 1 } ) ^ { \{ 1 \} , c } = \begin{cases} D ^ 2 & k = 1 ; \\ P ^ 2 ( k ) & k > 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} \int _ 0 ^ 1 \big | X _ N ( L , \alpha ) \big | ^ 2 \ , d \alpha & \ll \frac { L } { N ^ 3 } \sum _ { \substack { 1 \leq r \leq N ^ 2 \\ 1 \leq s \leq N ^ 2 } } W _ N ( w _ r , \mathcal { A } ) W _ N ( w _ s , \mathcal { A } ) \frac { \gcd ( w _ r , w _ s ) } { \sqrt { | w _ r w _ s | } } \\ & \ll \frac { L } { N ^ 3 } \sum _ { 1 \leq r \leq N ^ 2 } W _ N ( w _ r , \mathcal { A } ) ^ 2 \exp \left ( \frac { 1 0 \log r } { \log \log ( r + 1 ) } \right ) . \end{aligned} \end{align*}


\begin{align*} \varphi ( x ) = \varphi ( t x _ 0 + ( 1 - t ) x _ 1 ) \geq t \varphi ( x _ 0 ) + ( 1 - t ) \varphi ( x _ 1 ) = \infty . \end{align*}


\begin{align*} T _ n ( \phi ) = T _ n ( \phi _ 0 ) T _ n ( \phi _ 1 ) \cdots T _ n ( \phi _ R ) + P _ n K _ 1 P _ n + W _ n K _ 2 W _ n + C _ n \end{align*}


\begin{align*} \Phi ( X , Y ) = g ( X , { f } Y ) , X , Y \in \mathfrak { X } _ M . \end{align*}


\begin{align*} \mathcal { P } ( X _ \eta ) = \mathcal { P } ( X _ \varphi ) = \{ ( M _ H ) _ \ast ( m _ H \vee \kappa ) : \kappa \in \mathcal { P } ( \{ 0 , 1 \} ^ \Z ) \} . \end{align*}


\begin{align*} h ( B , \nu _ { B } ) = h \left ( Y , \nu \right ) + \int _ { Y } { \rm I } ( \mathbb { P } _ { \mu , 1 } ^ { y } , \mathcal { T } _ { y } ) d \nu ( y ) = h \left ( Y , \nu \right ) + \int _ { Y } \inf _ { n \in \mathbb { N } } { \rm I } ( \mathbb { P } _ { \mu , 1 } ^ { y } , \mathbb { P } _ { \mu , n } ^ { y } ) d \nu ( y ) . \end{align*}


\begin{align*} \Delta ( f ) ( t ) \coloneqq \sum _ { i = 0 } ^ { + \infty } a ( i , \ldots , i ) t ^ i \in k [ [ t ] ] \ , . \end{align*}


\begin{align*} u ( r ) = \max \Big \{ 0 , ( n - 1 ) \inf _ { r \geq r _ 0 } \log \frac { f ( r ) } { f ( r _ 0 ) } \Big \} . \end{align*}


\begin{align*} \mathcal { T } ^ \ast \mathcal { T } = \mu _ j \psi _ j , j \ge 1 . \end{align*}


\begin{align*} Q u ( k ) : = \frac { 1 } { V ( k ) + 1 } u ( k ) , u \in \ell ^ { 2 } ( \hbar \mathbb { Z } ^ { n } ) . \end{align*}


\begin{align*} \left [ \Sigma \phantom { ^ { } } \right ] _ { i , j } ^ { } = \begin{cases} 1 & i \le j ; \\ 0 & ; \end{cases} \end{align*}


\begin{align*} \bar u ( t ) = \hat \lambda _ t ^ * , \qquad t \in [ 0 , 1 ] . \end{align*}


\begin{align*} \overline { u } _ h : = \begin{cases} \zeta ( v + m _ h ) + ( 1 - \zeta ) z _ h & \mbox { o n } B _ { \sigma ' } \\ \varphi z _ h + ( 1 - \varphi ) u _ h & \mbox { o n } B _ r \setminus B _ { \sigma ' } . \end{cases} \end{align*}


\begin{align*} T _ { 5 1 } = & s \lambda ^ 3 \iint _ Q \xi u A \nabla u \cdot \nabla \eta \left ( A \nabla \eta \cdot \nabla \eta \right ) d x d t \\ \geq & - C s ^ 2 \lambda ^ 4 \iint _ Q \xi \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 | u | ^ 2 d x d t - C \lambda ^ 2 \iint _ Q \xi | A \nabla u \cdot \nabla \eta | ^ 2 d x d t , \end{align*}


\begin{align*} g = g ( a _ i ) + g ' ( a _ i ) ( x - a _ i ) + \ldots + ( x - a _ i ) ^ d . \end{align*}


\begin{align*} m ( r , k Y ) = m \left ( r , - \frac { R _ { k - 2 } } { M Y ^ { k - 2 } } + \frac { h } { M Y ^ { k - 2 } } \right ) \\ m ( r , Y ) \leq m \left ( r , - \frac { R _ { k - 2 } } { M Y ^ { k - 2 } } \right ) + m \left ( r , \frac { h } { M Y ^ { k - 2 } } \right ) + \log 2 . \\ & \end{align*}


\begin{align*} \lim _ { | x | \to \infty } u _ \eta = \eta . \end{align*}


\begin{align*} - \Delta ( \zeta \overline v ) = { \rm c u r l } ( \zeta \omega _ \theta e _ \theta ) - \nabla ( \overline v \cdot \nabla \zeta ) + { \rm c u r l } ( \nabla \zeta \times \overline v ) . \end{align*}


\begin{align*} & E \left [ \left ( S ^ k _ { j , + } ( t ) - \bar { S } ^ k _ { j , + , \Delta } ( t ) \right ) ^ 2 \right ] \\ & = \nu \Delta \sum _ { \ell = 0 } ^ { \lfloor t / \Delta \rfloor } p ^ k _ { j , + } ( t - \ell \Delta ) E [ Z _ 0 ( \ell \Delta ) ] \\ & + 2 \sum _ { \ell _ 1 < \ell _ 2 } E \left [ W ^ k _ { \ell _ 1 \Delta , t } ( j ) \left ( W ^ k _ { \ell _ 2 \Delta , t } ( j ) - \nu \Delta Z _ 0 ( \Delta \ell _ 2 ) p ^ k _ { j , + } ( t - \Delta \ell _ 2 ) \right ) \right ] . \end{align*}


\begin{align*} \limsup _ { r \to 0 ^ + } \Phi ( r ) = + \infty , \qquad { \mbox { w h e r e } } \ ; \Phi ( r ) : = \frac { \displaystyle \inf _ { B _ { r / 2 } ( x _ r ) } | u | } { r ^ { 2 s } } . \end{align*}


\begin{align*} \begin{cases} - \Delta v _ i = f ( v _ i ) - \mu _ i v _ i + \lambda v _ i \ \ \ \ \mathbb R ^ N , \\ \lim _ { | x | \to \infty } v _ i ( x ) = 0 , i = 1 , 2 , \cdots , \ell , \\ \sum _ { i = 1 } ^ \ell | v _ i | _ 2 ^ 2 = \alpha . \end{cases} \end{align*}


\begin{align*} \tau _ j ( f ) _ i ( 0 ) = \begin{cases} \sigma ( \tau _ j ) , & \tau _ i = \tau _ j , \\ 0 , & , \end{cases} \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ s u = \lambda u & , \\ u = 0 & , \end{cases} \end{align*}


\begin{align*} \bar { M } _ + ( t ) : = \nu \int _ 0 ^ t Z _ 0 ( s ) d s , \end{align*}


\begin{align*} \theta _ { Z , p , q } ( k ) \leq 2 \sup _ { ( a _ 1 , \dots , a _ p ) \in \Gamma _ { p , q } } \prod _ { i = 1 } ^ p 2 ^ { a _ i } \theta _ { X , p , q } ( k ) \ , , \end{align*}


\begin{align*} \exp ( 2 z ) \exp ( 2 x _ 1 ) \exp ( 2 z ) = \exp ( 2 x _ 1 ) , \end{align*}


\begin{align*} \mathfrak { a } _ { I } = \bigcap _ { \alpha \in I } \ker \alpha . \end{align*}


\begin{align*} \begin{aligned} \theta ( \bar { u } ) + I _ K ( \bar { \zeta } ) = \lim \limits _ { k \rightarrow + \infty } \theta ( u ^ k ) + I _ K ( \zeta ^ k ) \leq \theta ( \hat { u } ) + I _ K ( \hat { \zeta } ) + \beta \sum \limits _ { i = n } ^ { + \infty } \| r ^ { i } \| ^ 2 . \end{aligned} \end{align*}


\begin{align*} e ^ { \begin{bmatrix} 0 & \tfrac { t } { 2 } \\ \tfrac { t } { 2 } & 0 \end{bmatrix} } & = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} 0 & \tfrac { t } { 2 } \\ \tfrac { t } { 2 } & 0 \end{bmatrix} + o ( t ^ 2 ) , \\ e ^ { \begin{bmatrix} t / 2 & 0 \\ 0 & - t / 2 \end{bmatrix} } & = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \begin{bmatrix} t / 2 & 0 \\ 0 & - t / 2 \end{bmatrix} + o ( t ^ 2 ) ; \end{align*}


\begin{align*} 1 = \chi + \sum \limits _ j \chi _ j \end{align*}


\begin{align*} \lim _ { m \to \infty } \frac { P _ { \lambda _ m } } { P _ m } = \varphi ( \lambda ) \ , \ , \ , \ , \ , \end{align*}


\begin{align*} \mathcal { D } _ x \ : ( 1 + x ) _ r = ( 1 + x ) _ r H _ r ( x ) . \end{align*}


\begin{align*} \delta ( E ( q , \lambda ) ) = \sqrt { 2 H ( \lambda ) } , \qquad \forall \ , ( q , \lambda ) \in D _ \epsilon . \end{align*}


\begin{align*} g \lbrace \alpha , \beta \rbrace = \lbrace g ( \alpha ) , g ( \beta ) \rbrace , \end{align*}


\begin{align*} \lim _ { k \to \infty } f _ k = \liminf _ { k \to \infty } g _ k = h \end{align*}


\begin{align*} \frac { d \widetilde \beta _ i ^ { ( u ) } } { d \beta _ i } = \frac { 1 } { \Big ( 1 - \frac { \beta _ i } { \beta _ i ^ { ( u ) } } \Big ) ^ 2 } = \frac { 1 } { \phi _ i ( u ) ^ 2 } . \end{align*}


\begin{align*} \mathcal { H } = P _ { \rm s o f t } \mathcal { H } \oplus P _ { \rm s t i f f } \mathcal { H } = \mathcal { H } ^ { \rm s o f t } \oplus \mathcal { H } ^ { \rm s t i f f } . \end{align*}


\begin{align*} \begin{aligned} \mathcal { I } _ { \mathrm { D } } ( \eta _ 0 , \delta ) - \mathcal { I } _ { \mathrm { D } } ( \eta , 0 ) & = \mathcal { I } _ { \mathrm { D } } ( \eta _ 0 , \delta ) - \mathcal { I } _ { \mathrm { D } } ( \eta , \delta ) + \mathcal { I } _ { \mathrm { D } } ( \eta , \delta ) - \mathcal { I } _ { \mathrm { D } } ( \eta , 0 ) \\ & \leq \Psi ( \eta _ 0 - \eta , 0 ) + \Psi ( 0 , \delta ) = \Psi ( | \eta _ 0 - \eta | , 0 ) + \Psi ( 0 , \delta ) . \end{aligned} \end{align*}


\begin{align*} S _ n ( x ) = \frac { 1 } { n + 1 } \sum _ { 0 } ^ { n } \sigma _ k ( x ) , ~ n \in \N . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta _ 1 , 0 ) = \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \inf _ { \lambda _ 1 \in \mathbb { R } _ { + } } \left [ \lambda _ 1 \delta _ 1 + \int g _ { \lambda , 1 } \ , d \pi \right ] = \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \inf _ { \lambda \in \mathbb { R } ^ 2 _ { + } } \left [ \langle \lambda , \left ( \delta _ 1 , 0 \right ) \rangle + \int _ { \mathcal { V } } g _ { \lambda } \ , d \pi \right ] . \end{align*}


\begin{align*} \left ( \mathord { \operatorname { p r o j } } _ { K _ j \cap K _ { j + 1 } } \circ { \operatorname { p r o j } _ { K _ j } } ^ { - 1 } \right ) \# \mu _ j = \mathcal { N } ( 0 , 1 ) , \forall j \in [ 3 ] . \end{align*}


\begin{align*} \begin{cases} \partial _ t \nu _ t ( x ) = - a ( \nu _ t , \mu _ t , x ) \nu _ t ( x ) , \\ \partial _ t \mu _ t ( y ) = b ( \nu _ t , \mu _ t , y ) \mu _ t ( y ) , \end{cases} \end{align*}


\begin{align*} ( p , q ) & = p ^ < \cap q ^ > = \{ r \in \mathbb { P } : p < r < q \} , \\ [ p , q ] & = p ^ \leq \cap q ^ \geq = \{ r \in \mathbb { P } : p \leq r \leq q \} . \end{align*}


\begin{align*} \lim _ { n \to \infty } D _ \Theta X _ \cdot ^ { ( n ) } = D _ \Theta X _ \cdot \textrm { ~ ~ w e a k l y ~ ~ i n ~ ~ } L ^ p ( \Omega \times [ 0 , T ] ) . \end{align*}


\begin{align*} \tilde { a } _ 1 = \frac { \frac { p + \gamma - 1 } { 2 } - \frac { p + \gamma - 1 } { 4 q } } { \frac { p + \gamma - 1 } { 2 } + \frac { 1 } { 2 } } . \end{align*}


\begin{align*} x ' _ 1 - x ' _ 3 = ( n ' + m ' ) \delta + 2 \pi z ' _ 1 \quad \textrm { a n d } x ' _ 2 - x ' _ 3 = - m ' \delta + 2 \pi z ' _ 2 \textrm { f o r } \delta \in \mathbb { R } , \end{align*}


\begin{align*} T ( f ) ( z ) = - \frac { 1 } { \pi } \iint _ D \frac { f ( \zeta ) } { \zeta - z } \ , d \xi \ , d \eta , \end{align*}


\begin{align*} A = \left ( \begin{array} { c c } 1 & 2 \\ 0 & 1 \end{array} \right ) , \ B = \left ( \begin{array} { c c } 1 & 0 \\ 2 & 1 \end{array} \right ) . \end{align*}


\begin{align*} \delta ^ M _ { n m } = 1 0 0 \frac { M _ { n m } + M _ { m n } } { 2 M ^ 0 _ { n m } } \end{align*}


\begin{align*} 0 = \sum _ { i = 1 } ^ n c \beta _ i ( e _ { i \bullet } - e _ { \bullet i } ) + \nabla \left ( \sum _ { i \neq j } \alpha _ { i j } v _ { i j } - \langle \lambda , v \rangle + \sum _ { i \in I } \gamma _ i ( \sum _ { j \neq i } v _ { i j } - x _ { 0 i } ) \right ) . \end{align*}


\begin{align*} R _ k ( t , u , v , w ) : = \int _ { k = \ell - m + j } e ^ { i t \Delta \omega _ { k \ell m j } } u _ \ell \bar v _ m w _ j . \end{align*}


\begin{align*} \alpha = \alpha \left ( \left \{ \left . \min \limits _ { \ell \in I _ 0 ( g , i ) } \alpha _ \ell \ \right | \ Q _ i \in I _ { n _ 0 } \right \} \right ) . \end{align*}


\begin{align*} \partial _ { t } F + \partial _ { x } ( A ( F ) ) = \mathbf { S } [ F ] ( t , x ) \end{align*}


\begin{align*} \sum _ { y < p \le \sqrt { x } } \frac { x } { p \log ( x / p ) } & \ge \frac { x } { \log y } \left ( \frac { \log ( u - 1 ) } { u } - \frac { 2 c _ 1 } { u \log ^ 2 y } \right ) \\ & = \frac { x } { \log y } \left ( \omega ( u ) - \frac { 2 c _ 1 } { u \log ^ 2 y } \right ) - \frac { x } { \log x } . \end{align*}


\begin{align*} x ^ 2 y = z ^ 2 + 1 \end{align*}


\begin{align*} \mathbb { H } _ { \infty - } : = \bigcap _ { p \ge 1 } \mathbb { H } _ p , \mathbb { V } _ { \infty - } : = \bigcap _ { p \ge 1 } \mathbb { V } _ p . \end{align*}


\begin{align*} \Theta ^ * ( J _ u , x ) : = \limsup _ { r \to 0 } \frac { \mathcal { H } ^ 1 \left ( J _ u \cap B _ r ( x ) \right ) } { 2 \pi r } \geq \theta _ \delta > 0 . \end{align*}


\begin{align*} \partial f ^ { \lambda } : = \frac { I - \left ( I + \lambda \partial f \right ) ^ { - 1 } } { \lambda } , \lambda \in ( 0 , 1 ) , \end{align*}


\begin{align*} \xi _ { \theta } ( y ) = \xi _ { \theta _ 0 } ( y ) + ( \theta - \theta _ 0 ) ^ \prime \dot { \xi } _ { \theta _ 0 } ( y ) + r _ { \theta } ( y ) , \end{align*}


\begin{align*} V ( t _ 1 , \dots , t _ n ) = \frac { 1 } { n + k } \sum _ { i = 1 } ^ n t _ i ^ { n + k } = V _ { n + k } ( t _ 1 , \dots , t _ n ) \ . \end{align*}


\begin{align*} \tau _ h ^ G : = ( \kappa _ h ^ G ) ^ 2 \end{align*}


\begin{align*} & - \int _ \Omega \langle \nabla u , \nabla \varphi \rangle \ , d x - \frac { C _ { N , s } } { 2 } \iint _ { \R ^ { 2 N } } \frac { ( u ( x ) - u ( y ) ) ( \hat { \varphi } ( x ) - \hat { \varphi } ( y ) ) } { | x - y | ^ { N + 2 s } } \ , d x \ , d y - \int _ \Omega V ( x ) u \varphi \ , d x \\ & \geq \ , [ ] \ , \int _ \Omega f ( x ) \varphi \ , d x . \end{align*}


\begin{align*} \pi _ 1 ( f _ 1 ) \cdots \pi _ k ( f _ k ) \psi = \theta _ W ( f _ k \otimes \ldots \otimes f _ 1 \otimes \rho ) * \psi \end{align*}


\begin{align*} \eta ( B ^ { \prime } ) = \eta ( B ) + 1 \ge T _ 0 + \max _ { 1 \le i _ 1 < \ldots < i _ { q _ 0 } \le s } \sum _ { j = 1 } ^ { q _ 0 } a _ { i _ j } . \end{align*}


\begin{align*} \xi \cdot b ( s ) = \xi ( s ) b \xi \in X , \ b \in B _ e , \end{align*}


\begin{align*} \rho _ 2 ( \varpi ) = | \varpi | ^ { n - 1 } + O \left ( | \varpi | ^ { ( n + 1 ) / 2 } \right ) . \end{align*}


\begin{align*} H _ b ( X , Y ) & = \int _ a ^ b \langle X ' , Y ' \rangle + \langle R ( \dot { \gamma } , X ) \dot { \gamma } , Y \rangle d s \\ & = \int _ a ^ b \langle - X '' + R ( \dot { \gamma } , X ) \dot { \gamma } , Y \rangle d s . \end{align*}


\begin{align*} r z ^ { 2 } - ( p - s ) z - q = 0 . \end{align*}


\begin{align*} g ( X , Z ) & = - \Phi ( { f } X , Z ) - g ( X , \widetilde { Q } Z ) + \sum \nolimits _ { i } \big ( \eta ^ i ( X ) \ , \eta ^ i ( Z ) + \eta ^ i ( X ) \ , \eta ^ i ( \widetilde { Q } Z ) \big ) \\ & = - \Phi ( { f } X , Z ) + \sum \nolimits _ { i } \eta ^ i ( X ) \ , \eta ^ i ( Z ) - g ( X , \widetilde { Q } Z ) . \end{align*}


\begin{align*} ( \widetilde { \Delta } _ { l } ) = O ( N ^ { - 1 } p ^ { - l } ) . \end{align*}


\begin{align*} \Delta ^ L = \frac { \sum _ { n m \in \mathcal { L } } \delta ^ L _ { n m } } { | \mathcal { L } | } \end{align*}


\begin{align*} T ( u ) = \int _ 0 ^ u \exp ( 2 \rho ( v ) ) d v . \end{align*}


\begin{align*} b _ { i , \tau ( i ) } = 0 . \end{align*}


\begin{align*} \tilde { \mathbf { m } } _ \lambda = \lambda \mathbf { b } _ \lambda \mathbf { e } _ \lambda . \end{align*}


\begin{align*} Q = Q _ 1 \oplus ^ { \mathrm { f i n e } } _ P Q _ 2 \end{align*}


\begin{align*} \left \{ \begin{aligned} & \left [ r ^ \delta u ' ( r ) ^ { p - 1 - \alpha } \right ] ' = \frac { \delta } { n - 1 } \ , r ^ \delta f _ 1 ( r ) g _ 1 ( v ( r ) ) & & \quad , \\ & \left [ r ^ \delta u _ 0 ' ( r ) ^ { p - 1 - \alpha } \right ] ' = \frac { \delta } { n - 1 } \ , r ^ \delta f _ 1 ( r ) h _ 1 ( v _ 0 ( r ) ) & & \quad \end{aligned} \right . \end{align*}


\begin{align*} R _ F ( H ) & = R ( H ) [ F ( \left . \rho \right | _ H ) ^ { - 1 } ] \ . \end{align*}


\begin{align*} \Sigma _ n ^ { - 1 } U _ n ^ { } = \left [ \begin{array} { c | c } U _ { n - 1 } & \mathbf { 0 } \\ \hline \mathbf { 0 ' } & 1 \end{array} \right ] \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } S _ j ( \tau _ N ) = \nu \int _ 0 ^ \infty e ^ { - \lambda s } p _ j ( s ) d s , j \geq 1 , \end{align*}


\begin{align*} ( f _ { \mathcal { S } } ) _ { \lambda } ( s _ 1 , s _ 2 ) & = \sup _ { x ^ \prime \in \mathcal { X } } \left \{ - y _ 2 d ( x ' ) - y _ 1 ( 1 - d ( x ' ) ) - \sum _ { 1 \leq \ell \leq 2 } \lambda _ \ell \| x _ \ell - x ' \| \right \} \\ & = - \min \{ y _ 2 + \varphi _ { \lambda , 1 } ( x _ 1 , x _ 2 ) , y _ 1 + \varphi _ { \lambda , 0 } ( x _ 1 , x _ 2 ) \} , \end{align*}


\begin{align*} \mathcal { H } _ { \hbar , V } u ( k ) : = \left ( - \hbar ^ { - 2 } \mathcal { L } _ { \hbar } + V \right ) u ( k ) , k \in \hbar \mathbb { Z } ^ { n } , \end{align*}


\begin{align*} \boldsymbol { d } _ { \mathcal { S } _ \ell } ( ( y _ \ell , x ) , ( y ^ { \prime } _ \ell , x ^ \prime ) ) = \boldsymbol { d } _ { \mathcal { Y } _ \ell } ( y _ \ell , y ^ { \prime } _ \ell ) + \boldsymbol { d } _ { \mathcal { X } } ( x , x ^ { \prime } ) , \end{align*}


\begin{align*} \nu = B _ 1 ^ { - 1 } \theta ^ { \beta } \end{align*}


\begin{align*} u _ { t _ \alpha t _ \beta } ( 0 ) = - u _ { x _ n } ( 0 ) \rho _ { t _ \alpha t _ \beta } ( 0 ) , ~ ~ \alpha , \beta < 2 n . \end{align*}


\begin{align*} Z = \{ z = x + i y \in \mathbb { C } : x \in I , 0 < y < \gamma \} \end{align*}


\begin{align*} \left | \frac { \partial ( u , v , w ) } { \partial ( t _ 1 , t _ 2 , t _ 3 ) } \right | ( \mathbf { t } ) = \left | \widetilde { P } _ 1 ' ( t _ 2 ) \widetilde { P } _ 2 ' ( t _ 3 ) - \widetilde { P } _ 1 ' ( t _ 3 ) \widetilde { P } _ 2 ' ( t _ 2 ) \right | \asymp 2 ^ { - \mathfrak { c } | l | } | t _ 2 - t _ 3 | , \end{align*}


\begin{align*} g ( 0 ) = 2 L ( U ) > 0 , \ \ g ( 1 ) = I _ { \omega , \mathbf { c } } ( U ) < 0 \end{align*}


\begin{align*} \Psi \left ( - \frac 1 2 , x ' , 0 \right ) = 0 | x ' | < \frac 1 4 . \end{align*}


\begin{align*} ( \alpha \otimes \Delta _ c ) \circ \Delta _ c ( l ) & = \alpha ( l _ 1 ) \otimes T ( l _ { 1 2 } ) \otimes T ( \alpha ( l _ 2 ) ) - \alpha ( l _ 1 ) \otimes T ( \alpha ( l _ 2 ) ) \otimes T ( l _ { 1 2 } ) \\ & - T ( \alpha ( l _ 2 ) ) \otimes \alpha ( l _ 1 ) \otimes T ( l _ { 1 2 } ) + T ( \alpha ( l _ 2 ) ) \otimes T ( l _ { 1 2 } ) \otimes \alpha ( l _ 1 ) . \end{align*}


\begin{align*} \frac { 1 } { 2 \pi i } \ , \intop _ { \mu - i \infty } ^ { \mu + i \infty } \Gamma ( z ) \ , \Gamma ( s - z ) \ , \zeta _ { p } ( 2 z ) \ , x ^ { 2 z } d z = \frac { 1 } { 2 \pi i } \ , \intop _ { \frac { 1 } { 2 } - \mu - i \infty } ^ { \frac { 1 } { 2 } - \mu + i \infty } \Gamma ( z ) \ , \Gamma ( s - z ) \ , \zeta _ { p } ( 2 z ) \ , x ^ { 2 z } d z - \frac { 1 } { 2 } \ , \frac { \Gamma ( s ) } { 1 + \frac { 1 } { \pi p } } + \frac { \sqrt { \pi } \ , x } { 2 } \Gamma \left ( s - \frac { 1 } { 2 } \right ) . \end{align*}


\begin{align*} A _ t = \{ \exists \ : z _ 0 = z _ 0 ( \omega ) \in [ - k t , k t ] ^ d \ : \ : \ : \ : Z _ t ( B ( z _ 0 , r ( t ) ) ) \geq e ^ { \delta t } \} , \end{align*}


\begin{align*} | ( \gamma , \mathbf { n } ) | & : = | \gamma | + \eta - | \mathbf { n } | , \\ | i | & : = | e _ i | ; \end{align*}


\begin{align*} & \mathbf { b } _ \mu = \sqrt { 2 \cosh \left ( \sqrt { \mu } / 2 \right ) } , \mu > 0 , \\ & \mathbf { m } _ \lambda = \max \left ( \lambda ^ 2 , \mathbf { b } _ \lambda \right ) \lambda ^ 5 \mathbf { e } _ \lambda ^ 2 . \end{align*}


\begin{align*} & ~ ~ ~ ~ ( \beta M _ 1 ^ H + \alpha M _ 2 ^ H ) ^ H ( \beta M _ 1 ^ H - \alpha M _ 2 ^ H ) + ( \beta M _ 1 ^ H - \alpha M _ 2 ^ H ) ^ H ( \beta M _ 1 ^ H + \alpha M _ 2 ^ H ) \\ & = 2 \vert \beta \vert ^ 2 ( M _ 1 M _ 1 ^ H - \tfrac { \vert \alpha \vert ^ 2 } { \vert \beta \vert ^ 2 } M _ 2 M _ 2 ^ H ) \\ & \geq 2 \vert \beta \vert ^ 2 ( 1 - \tfrac { \vert \alpha \vert ^ 2 } { \vert \beta \vert ^ 2 } ) M _ 2 M _ 2 ^ H \\ & \geq 0 . \end{align*}


\begin{align*} { \rm W F } ' _ h ( B ) = \{ ( x , \xi , y , \eta ) \ , : \ , ( x , \xi , y , - \eta ) \in { \rm W F } _ h ( \mathcal { K } _ B ) \} \end{align*}


\begin{align*} y ^ m = P ( x ) = a _ n x ^ n + \dots + a _ 1 x + a _ 0 , \end{align*}


\begin{align*} S _ { \chi , \varepsilon } ( z ) & = S _ { \chi } ^ { \rm s o f t } ( z ) + S _ { \chi } ^ { \rm s t i f f } ( \varepsilon ^ 2 z ) , \\ [ 0 . 3 e m ] M _ { \chi , \varepsilon } ( z ) & = M _ { \chi } ^ { \rm s o f t } ( z ) + \varepsilon ^ { - 2 } M _ { \chi } ^ { \rm s t i f f } ( \varepsilon ^ 2 z ) , z \in \rho ( \mathcal { A } _ { \chi , \varepsilon , 0 } ) . \end{align*}


\begin{align*} L _ { \beta } ( u , \zeta ; \lambda ) = \theta ( u ) + I _ K ( \zeta ) + \langle \lambda , S u - \zeta \rangle + \frac { \beta } { 2 } \| S u - \zeta \| ^ 2 , \end{align*}


\begin{align*} P v ( x H ) = \begin{cases} 1 - \frac { | \mathcal { S } | } { | \Omega | } & \mbox { i f } x H \in \mathcal { S } \\ - \frac { | \mathcal { S } | } { | \Omega | } & \mbox { o t h e r w i s e } \end{cases} . \end{align*}


\begin{align*} \phi _ l = \Tilde { \phi } _ { e } + \sum _ { m = 1 } ^ { k } C _ { p _ m } ( r ^ { l - 1 } h ) ^ { p _ m } , \end{align*}


\begin{align*} a \tau ^ { p } - b \sigma ^ { p } = c r ^ { 2 } . \end{align*}


\begin{align*} \begin{aligned} & \mbox { ( i ) } B = B _ 1 \cup B _ 2 , \widetilde B = E _ 1 \cup E _ 2 \mbox { a n d } \omega ( E _ i ) \ge \frac 1 2 \omega ( \widetilde B ) , i = 1 , 2 ; \\ & \mbox { ( i i ) } b ( x ) - b ( y ) \mbox { d o e s n o t c h a n g e s i g n f o r a l l } ( x , y ) \in B _ i \times E _ i , i = 1 , 2 ; \\ & \mbox { ( i i i ) } | b ( x ) - m _ b ( \tilde B ) | \le | b ( x ) - b ( y ) | \mbox { f o r a l l } ( x , y ) \in B _ i \times E _ i , i = 1 , 2 . \end{aligned} \end{align*}


\begin{align*} \theta _ { L , Y ^ { \ast } } ( f ' ) ( w ) = \sum _ { l \in L / L \cap Y ^ { \ast } } f ' ( w + l ) \psi ( \tfrac { \langle l , w \rangle } { 2 } + \tfrac { \langle y _ { l } , y ^ { \ast } _ l \rangle } { 2 } ) , \end{align*}


\begin{align*} \left ( A ^ { T } \right ) ^ { \odot ( d - 1 ) } \ , { \rm d i a g } \ , A = 0 . \end{align*}


\begin{align*} Y _ 1 = 0 , \ldots , Y _ { n - r } = 0 . \end{align*}


\begin{align*} \mathrm { R W } _ 0 ( d ) : = \inf _ { \gamma \in \Sigma _ 0 ( \delta _ 0 ) } \mathbb { E } _ { \gamma } [ Y _ 1 ( 1 - d ( X ) ) + Y _ 2 d ( X ) ] , \end{align*}


\begin{align*} \tilde { u } _ m ^ i ( y ) = ( \mu _ m ^ i ) ^ { \frac { n - 2 } { 2 } } u _ m ( \mu _ m ^ i y + \xi _ m ) \chi _ m ^ i ( \mu _ m ^ i y + \xi _ m ) . \end{align*}


\begin{align*} \mu _ { T ' } ( \overline { T ' \cdot x } ) = \{ e _ i + e _ j \ : : \ : \{ i , j \} \in { \mathcal B } ' ( x ) \} . \end{align*}


\begin{align*} \lambda ( t ) = J _ K \lambda _ { S ( t ) } = e ^ { i t } J _ K \lambda _ { S ( 0 ) } \implies \lambda _ { S ( t ) } = e ^ { i t } \lambda _ { S ( 0 ) } . \end{align*}


\begin{align*} & G _ 2 ^ { \mathrm { r e g } } ( \xi , \sigma _ 1 , \sigma _ 2 , \alpha _ 1 , \alpha _ 2 , \beta _ 1 , \beta _ 2 ) = \phi _ 2 ( \sigma _ 1 , \sigma _ 2 , \alpha _ 1 , \alpha _ 2 , \beta _ 1 , \beta _ 2 ) \\ & + H _ 2 ( \xi , \sigma _ 1 , \sigma _ 2 , \alpha _ 1 - \tau \beta _ 1 , \alpha _ 2 - \tau \beta _ 2 , \beta _ 1 + \beta _ 2 , \beta _ 2 ) + \phi _ 1 ( \sigma _ 1 , \alpha _ 1 , \beta _ 1 ) H _ 1 ( \xi , \sigma _ 2 , \alpha _ 2 - \tau \beta _ 2 , \beta _ 2 ) \ , . \end{align*}


\begin{align*} d _ { p } \left ( \overline { b } , \Theta _ { c } \right ) ^ { p } \geq \inf _ { x \in \mathbb { R } ^ { n } } \sum \limits _ { i = 1 } ^ { m } \left ( l _ { i } \right ) ^ { p } d _ { S _ { i } } ^ { p } \left ( x \right ) . \end{align*}


\begin{align*} \psi ( z _ 1 , \ldots , z _ m ) = \sum _ { ( k _ 1 , \ldots , k _ m ) \in \Z ^ m _ { \geq 0 } } \Re ( c _ { k _ 1 , \ldots , k _ m } ) z _ 1 ^ { k _ 1 } \cdots z _ m ^ { k _ m } \in \Q [ [ z _ 1 , \ldots , z _ m ] ] \end{align*}


\begin{align*} u _ { q , \lambda } ( \varphi ) = \Phi - R _ q ( \lambda ) ( ( \Delta + \lambda - q ) \Phi ) . \end{align*}


\begin{align*} \displaystyle \lim _ { m \rightarrow \infty } \dfrac { \left ( C _ { D ^ { k - 1 } ( f ) } ( p ^ { 2 m + 1 } ) - C _ { D ^ { k - 1 } ( f ' ) } ( p ^ { 2 m + 1 } ) \right ) } { \beta ^ { 2 m } } = 0 . \end{align*}


\begin{align*} \{ Z _ 0 ( t ) = j \} & = \bigcup _ { k \geq 1 } \{ \tau _ j ^ + ( k ) \leq t , \tau _ j ^ - ( k ) > t \} \\ & = \bigcup _ { k \geq 1 } \{ \tau _ j ^ + ( k ) \leq t \} \backslash \{ \tau _ j ^ - ( k ) \leq t \} . \end{align*}


\begin{align*} \begin{array} { c c c } K ( T ) & = & \bigcup \{ L : L T \} \\ & = & \ , \ , \ , \ , \ , \bigcup \{ R : R T \} . \end{array} \end{align*}


\begin{align*} f ( k , t , \mathbf { p } _ { n + 1 } ) \overset { } { = } \Pr ( X _ t = k \ , \vert \ , X _ 0 = 0 , \mathbf { p } _ { n + 1 } ) , k \in S , \end{align*}


\begin{align*} \Psi ( z ) : = \sum _ { k = 1 } ^ { n - 1 } \left ( T ^ k ( \Phi _ k ) ( z ) \right ) + T ^ n ( f ) ( z ) , \end{align*}


\begin{align*} \begin{cases} - \Delta v _ i = f ( v _ i ) + \lambda v _ i \ \ \ \ \mathbb R ^ N , \\ v _ i ( x ) > 0 , \ \lim _ { | x | \to \infty } v _ i ( x ) = 0 , i = 1 , 2 , \cdots , \ell , \\ \sum _ { i = 1 } ^ { \ell } | v _ i | _ 2 ^ 2 = \alpha . \end{cases} \end{align*}


\begin{align*} \partial _ s \Big ( \frac { 1 } { z } \Big ) ( s ) & = - \frac { z ' ( s ) } { z ( s ) ^ 2 } \\ & = - \frac { h ^ 2 \beta _ h ' ( s ) + \frac { h ^ 2 } { 2 } \overline { \beta _ h ' ( s ) } + \frac { \sigma ^ 2 } { 2 } \overline { \beta _ \sigma ' ( s ) } } { \big ( 2 \sigma ^ 2 + \mathcal { O } ( h ^ 2 + s \sigma ^ 4 ) \big ) ^ 2 } \\ & \sim \frac { i \sigma ^ 4 } { 4 \sigma ^ 4 } \end{align*}


\begin{align*} \begin{aligned} | G _ { \omega , E , \Lambda _ L ( x ) } ( a , b ) | & \leq | G _ { \omega , E _ 0 , \Lambda _ L ( x ) } ( a , b ) | + | E - E _ 0 | \cdot | G _ { \omega , E , \Lambda _ L ( x ) } ( a , b ) | \cdot | G _ { \omega , E _ 0 , \Lambda _ L ( x ) } ( a , b ) | \\ & \leq e ^ { - m _ 0 | a - b | } + e ^ { - m ' L } e ^ { L ^ { 1 - \eta } + L ^ { 1 - \eta } } \\ & \leq e ^ { - m | a - b | } \end{aligned} \end{align*}


\begin{align*} v \ast a = v \cdot \alpha _ { ( e , \sigma ) } ( a ) \ . \end{align*}


\begin{align*} g ( s _ - ) = \frac { 1 } { 4 } ( - \sqrt { C - 4 } + \sqrt { C } ) \end{align*}


\begin{align*} r e s _ p ( \nabla ) ( F _ i ( E _ p ) ) \subseteq F _ i ( E _ p ) \ , , \ , \ , \textnormal { a n d } \ , \ , r e s _ p ( \nabla ) | _ { F _ i ( E _ p ) / F _ { i + 1 } ( E _ p ) } = \frac { k _ { i , p } } { r } \cdot \ , . \end{align*}


\begin{align*} A _ 0 & = \{ \varphi \in C _ c ^ \infty ( M ) ^ K : \rho _ i \cdot \varphi \in A _ 0 ^ { ( i ) } i = 1 , \ldots , N \} . \end{align*}


\begin{align*} \frac { \partial V } { \partial \bar { c } _ { n - ( j + 1 ) } } = ( - 1 ) ^ { n - j } c _ { k + j + 1 } \ . \end{align*}


\begin{align*} \mathrm { E n d } _ { \mathbb { C } } ( A _ g ) \otimes _ { \mathbb { Z } } \mathbb { Z } _ \ell = \mathcal { O } _ { K , \ell } . \end{align*}


\begin{align*} \widetilde { m } ( x , \xi ) = ( 1 - \chi _ m ( x , \xi ) ) m ( x , \xi ) \log | \xi | \end{align*}


\begin{align*} C _ r ^ n & = \psi ^ { \alpha _ 1 , n } ( g _ 1 ) \cdots \psi ^ { \alpha _ s , n } ( g _ s ) \psi ^ { \alpha _ { s + 1 } , n } ( f _ 1 ) \cdots \psi ^ { \alpha _ { r } , n } ( f _ { r - s } ) , \\ D _ s ^ n & = \psi ^ { \beta _ { 1 } , n } ( f _ { r - s + 1 } ) \cdots \psi ^ { \beta _ { s } , n } ( f _ r ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta _ p u & = c _ 1 | x | ^ { m _ 1 } \cdot g _ 1 ( v ) \cdot | \nabla u | ^ { \alpha } & \quad \mbox { i n } \Omega , \\ \Delta _ p v & = c _ 2 | x | ^ { m _ 2 } \cdot g _ 2 ( v ) \cdot g _ 3 ( | \nabla u | ) & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} h : \xi _ g \mapsto h ^ { * } ( \xi _ g ) : = \prod _ { k \in I _ g ^ c } h ^ { - 1 } _ k \ , \cdot \xi _ g \end{align*}


\begin{align*} \Delta q & = \partial _ j \bigl ( ( \delta _ { j k } - a _ { j i } a _ { k i } ) \partial _ k q \bigr ) + \partial _ j ( a _ { j i } a _ { k i } \partial _ k q ) = \partial _ j \bigl ( ( \delta _ { j k } - a _ { j i } a _ { k i } ) \partial _ k q \bigr ) + \partial _ t a _ { j i } \partial _ j v _ i , \end{align*}


\begin{align*} h _ { \mu } \left ( G / P , \nu _ { P } \right ) = \sum _ { 1 \le i < j \le d } \lambda _ { i } - \lambda _ { j } . \end{align*}


\begin{align*} F _ j ( E ) = F _ j ( V ) \oplus F _ j ( W ) . \end{align*}


\begin{align*} \tilde { c } _ { i j } = ( k \mapsto c _ { k i } ) - ( k \mapsto c _ { k j } ) = ( k \mapsto c _ { k i } - c _ { k j } ) = ( k \mapsto c _ { k j } + c _ { j i } - c _ { k j } ) = ( k \mapsto c _ { j i } ) \end{align*}


\begin{align*} \mathcal W _ \infty ^ { p - 2 - \alpha } \ , \lim _ { r \to \infty } \frac { u '' ( r ) } { u _ 0 '' ( r ) } = \lim _ { r \to \infty } \frac { g _ 1 ( v ( r ) ) } { h _ 1 ( v _ 0 ( r ) ) } = \lim _ { r \to \infty } Q _ 1 ( \mathcal V ( r ) , v _ 0 ( r ) ) \end{align*}


\begin{align*} D ^ l ( \varphi _ 1 ^ { s _ 1 } \varphi _ 2 ^ { r _ 2 } \dotsm \varphi _ k ^ { r _ k } ) = \sum _ { \substack { j _ 1 + j _ 2 + \cdots + j _ k = l \\ j _ 1 , j _ 2 , \ldots , j _ k \geq 0 } } \ , \frac { l ! } { j _ 1 ! j _ 2 ! \cdots j _ k ! } \ ; D ^ { j _ 1 } ( \varphi _ 1 ^ { s _ 1 } ) D ^ { j _ 2 } ( \varphi _ 2 ^ { r _ 2 } ) \cdots D ^ { j _ k } ( \varphi _ k ^ { r _ k } ) . \end{align*}


\begin{align*} \varphi ( d ) = 0 & \Rightarrow \varphi ( d ^ * ) \varphi ( d ) = 0 \Rightarrow f ( \varphi ( d ) ^ * \varphi ( d ) ) = 0 \Rightarrow f \varphi ( d ^ * d ) = 0 \\ & \Rightarrow g ( d ^ * d ) = 0 \Rightarrow d = 0 . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) & = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \sup _ { \pi \in \Pi ( \mu _ 1 , \mu _ 2 ) } \int g _ { \lambda } ( s _ 1 , s _ 2 ) \ , d \pi ( s _ 1 , s _ 2 ) \right ] . \end{align*}


\begin{align*} \begin{array} { l l } \underset { x \in X } { } & \sum \limits _ { i = 1 } ^ { m } [ g _ { i } \left ( x \right ) - b _ { i } ] _ { + } ^ { p } , \end{array} \end{align*}


\begin{align*} \mathrm { I } \left ( \xi _ { 1 } , \xi _ { n } | X \right ) = H \left ( \xi _ { n } | X \right ) - H \left ( \xi _ { n - 1 } | X \right ) . \end{align*}


\begin{align*} h _ i d \left ( \sum \limits _ { j = 1 } ^ { s _ \ell } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } \right ) \equiv h _ i g ' ( \eta ) \mod \mathcal J _ 1 . \end{align*}


\begin{align*} \Phi ( r ) = \left \lbrace \begin{array} { l l l } \dfrac { 1 } { 2 } r ^ 2 , & r \in [ 0 , + \infty ) , & K = 0 , \\ 1 - \cos r , & r \in [ 0 , \dfrac { \pi } { 2 } ) , & K = 1 , \\ \cosh r - 1 , & r \in [ 0 , + \infty ) , & K = - 1 . \end{array} \right . \end{align*}


\begin{align*} A ( \alpha ) = \langle x _ 1 , x _ 2 , x _ 3 \rangle / ( \alpha x _ 3 x _ 1 + x _ 1 x _ 3 , \alpha x _ 3 x _ 2 - x _ 2 x _ 3 , x _ 1 ^ 2 - x _ 2 ^ 2 ) . \end{align*}


\begin{align*} f ( k , t ; p , n ) = \overline { F } ( k - 1 , t ; p , n ) - \overline { F } ( k , t ; p , n ) . \end{align*}


\begin{align*} S ( \mathfrak M ( \lambda , L 1 ) ) ( t ) = \begin{pmatrix} S ^ { 1 , 1 } _ k \\ \vdots \\ S ^ { r , 1 } _ k \end{pmatrix} _ { k = 0 } ^ \infty ; S _ k ^ { m , 1 } = ( L 1 ^ k ) _ { m - 1 , \ , 0 } ; \end{align*}


\begin{align*} S ^ m _ j & : = \left \{ ( A _ 0 \subsetneq \dots \subsetneq A _ m ) \in ( K ^ n _ k ) _ m \mid \begin{array} { l } A _ m = [ n ] \\ k \not \in A _ { j - 1 } \\ A _ { j } = A _ { j - 1 } \cup \{ k \} \end{array} \right \} . \end{align*}


\begin{align*} \beta = ( 1 + \frac { c } { \sqrt { 2 } \alpha } ) ( \alpha - \frac { c } { \sqrt { 2 } } ) > \alpha - \frac { c } { \sqrt { 2 } } > \alpha - c = \eta \rho > 0 \end{align*}


\begin{align*} \overline { F } ( \underline { k } , t , \mathbf { p } _ { n + 1 } ) = h _ { t - k - 1 } ( q _ 0 , q _ 1 , . . . , q _ k , 1 ) \prod _ { i = 0 } ^ k p _ i , \end{align*}


\begin{align*} J _ { \Lambda } = - 2 \pi \int _ { 0 } ^ { \infty } \left [ \exp \left ( - \frac { 1 } { \Lambda } \left [ \left ( \frac { \sigma } { r } \right ) ^ { n } - \left ( \frac { \sigma } { r } \right ) ^ { m } \right ] \right ) \right ] \ , r ^ { 2 } d r . \end{align*}


\begin{align*} f _ { \lambda , A } ( v ) = \sup _ { s ' \in \mathcal { S } } \left \{ f ( s ' ) - \sum _ { \ell = 1 } ^ { L } \lambda _ \ell \tilde { c } _ { \ell } ( s _ { \ell } , s _ { \ell } ' ) \right \} , \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } p ' y ^ 2 + ( a ' + c ' ) x y = z \\ x ^ 2 + q ' y ^ 2 + ( b ' + d ' ) x y = w \\ p ' w ^ 2 + ( a ' + c ' ) z w = p x + q z \\ z ^ 2 + q ' w ^ 2 + ( b ' + d ' ) z w = p y + q w \\ p ' y w + a ' x w + c ' y z = a x + b z \\ x z + q ' y w + b ' x w + d ' y z = a y + b w \\ p ' y w + a ' y z + c ' x w = c x + d z \\ x z + q ' y w + b ' y z + d ' x w = c y + d w \end{array} \right . \end{align*}


\begin{align*} \xi _ { u ( t ) } ( q ) : = \xi ( q , u ( t ) ) = \sum _ { i = 1 } ^ k u _ i ( t ) X _ i ( q ) , \qquad \forall \ , q \in M , t \in [ 0 , 1 ] , \end{align*}


\begin{align*} \int _ E \phi ( s ) ^ q \dd s \leq \sum _ { j \in \mathcal J } \int _ { a _ j } ^ { b _ j } \phi ( s ) ^ q \dd s & \leq \sum _ { j \in \mathcal J } \int _ { \{ x \in \Sigma : u ^ * _ \mu ( b _ j ) < | u ( x ) | \leq u ^ * _ \mu ( a _ j ) \} } | \nabla u ( x ) | ^ q \dd \mu ( x ) \\ & = \sum _ { j \in \mathcal J } \int _ { \{ x \in \Sigma : u ^ * _ \mu ( b _ j ) < | u ( x ) | < u ^ * _ \mu ( a _ j ) \} } | \nabla u ( x ) | ^ q \dd \mu ( x ) . \end{align*}


\begin{align*} \partial _ { x _ j } \big | _ { x = 0 } f ( \xi + B x ) = \sum _ { k = 1 } ^ n B _ { k j } \partial _ { \xi _ k } f ( \xi ) = : \partial _ { B , \xi _ j } f ( \xi ) , j = 1 , \ldots , n . \end{align*}


\begin{align*} b _ m ( E ) = \| E \| . \end{align*}


\begin{align*} I _ { \omega , \mathbf { c } } ( \Phi ) : = 2 L ( \Phi ) + \left ( \frac { d } { 2 } + 1 \right ) N ( \Phi ) + \mathbf { c } \cdot \mathbf { P } ( \Phi ) . \end{align*}


\begin{align*} \dim \widehat { S _ { k } } ^ { \# , 0 } ( \Gamma _ { 0 } ( N ) ) = \dim S _ { k } ( \Gamma _ { 0 } ( N ) ) . \end{align*}


\begin{align*} \hat \psi _ r ( \beta ) = J _ r ( x _ * , \beta ) \geq \frac { 1 } { 2 v _ r ( x _ * ) } ( \beta - C _ r ) ^ 2 + O ( ( \beta - C _ r ) ) ^ 3 ) , \beta \downarrow C _ r . \end{align*}


\begin{align*} A _ p = \{ a \in A : a p = p a = 0 \} . \end{align*}


\begin{align*} \frac { 1 } { \prod _ { i = 0 } ^ n ( 1 - t q ^ { r i } ) } = \sum _ { k = \ell } ^ n \frac { ( t q ^ { r \cdot k } ) ^ { k - \ell } } { \prod _ { i = 0 } ^ k ( 1 - t q ^ { r i } ) } { n - \ell \brack k - \ell } _ { q ^ r } \end{align*}


\begin{align*} \lambda _ { 2 n - 1 } \left ( \begin{bmatrix} A & X \\ X ^ * & B \end{bmatrix} \right ) & \le \lambda _ { n } \left ( \begin{bmatrix} A & 0 \\ 0 & 0 \end{bmatrix} \right ) + \lambda _ { n } \left ( \begin{bmatrix} 0 & 0 \\ 0 & B \end{bmatrix} \right ) \\ & = \lambda _ { n } ( A ) + \lambda _ n ( B ) \\ & \le \lambda _ n ( A + B ) . \end{align*}


\begin{align*} U _ n = \left ( ( - \infty , 0 ) \times \{ 0 \} \right ) \cup \left ( [ 0 , \infty ) \times \{ n \} \right ) & & \chi _ n = \mathrm { p r o j } _ 1 | _ { U _ n } & & n \in \Z . \end{align*}


\begin{align*} \frac { \partial \phi } { \partial t } + a _ x \frac { \partial \phi } { \partial x } + a _ y \frac { \partial \phi } { \partial y } = 0 , \end{align*}


\begin{align*} \lambda _ f ( r ) \lambda _ f ( s ) = \sum _ { d | ( r , s ) } \lambda _ f \left ( \frac { r s } { d } \right ) , \end{align*}


\begin{align*} \sum _ { \substack { m , n \in \mathcal { M } ' \\ m \not = n } } r ( m ) r ( n ) \Phi \left ( T \log { \frac { m } { n } } \right ) \ll \sum _ { j \in \mathcal { J } } r ( m _ j ) ^ { 2 } . \end{align*}


\begin{align*} \int _ { a + P _ { r , s , A } } | f | ^ 2 e ^ { - \varphi } = | P _ { r , s , A } | e ^ { - \varphi ( a ) } . \end{align*}


\begin{align*} \mathrm { H } _ { \mathcal { H } _ { \hbar , V } } ^ { s } : = \left \{ u \in \mathrm { H } ^ { - \infty } _ { \mathcal { H } _ { \hbar , V } } : \left ( I + \mathcal { H } _ { \hbar , V } \right ) ^ { s / 2 } u \in \ell ^ { 2 } \left ( \hbar \mathbb { Z } ^ { n } \right ) \right \} , \end{align*}


\begin{align*} J = 2 \sum \limits _ { i = 1 } ^ N { { G _ { { \alpha _ 1 } , { \beta _ 1 } } } \left ( { { e _ i } } \right ) } + \sum \limits _ { i = 1 } ^ N { \sum \limits _ { j = 1 } ^ N { { G _ { { \alpha _ 2 } , { \beta _ 2 } } } \left ( { { e _ i } - { e _ j } } \right ) } } . \end{align*}


\begin{align*} S ^ \sqsubset = \mathop { \sqsupset } [ S ] = \{ b \in B : \exists s \in S \ ( s \sqsubset b ) \} . \\ \intertext { L i k e w i s e , t h e \emph { i m a g e } o f a n y $ T \subseteq B $ i s t h e p r e i m a g e o f t h e o p p o s i t e r e l a t i o n $ \sqsupset $ , i . e . } T ^ \sqsupset = \mathop { \sqsubset } [ T ] = \{ a \in A : \exists t \in T \ ( a \sqsubset t ) \} . \end{align*}


\begin{align*} \mathrm { R W } ( d ) = \sup _ { \lambda \ge 1 } \left [ \inf _ { \pi \in \Pi ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ { \mathcal { V } } \min \{ y _ 2 + \varphi _ { \lambda , 1 } ( x _ 1 , x _ 2 ) , y _ 1 + \varphi _ { \lambda , 0 } ( x _ 1 , x _ 2 ) \} d \pi ( v ) - \langle \lambda , \delta \rangle \right ] , \end{align*}


\begin{align*} \widetilde { \rho } _ t ( t , w ) = A \beta _ 1 t ^ { \beta _ 1 - 1 } + B ( 1 + \beta _ 1 ) t ^ { \beta _ 1 } + ( 1 - \beta _ 1 ) C t ^ { - \beta _ 1 } | w | ^ 2 + \end{align*}


\begin{align*} B ( A ( f ) , A ( 1 ) ) = B ( A ( f ) , 0 ) = 0 \end{align*}


\begin{align*} e ^ { r X ^ B } b = t ( e ^ { r X } b ) & & r \in \R , b \in B . \end{align*}


\begin{align*} M _ { \le r } = \bigcap _ { 0 \le j \le r } M _ { U _ j } . \end{align*}


\begin{align*} \omega _ \Pi = \inf _ { \{ \Gamma _ t \} \in \Pi } \sup _ { t \in I ^ n } | \Sigma _ t | , \end{align*}


\begin{align*} \rho _ \ell ( y _ \ell , y _ \ell ^ \prime ) : = \inf _ { x _ \ell , x _ \ell ^ \prime \in \mathcal { X } } \boldsymbol { d } _ { \mathcal { S } _ \ell } \left ( ( y _ \ell , x _ \ell ) , ( y _ \ell ^ { \prime } , x _ \ell ^ { \prime } ) \right ) ^ { p _ \ell } . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta _ 1 , 0 ) = \inf _ { \lambda _ 1 \in \mathbb { R } _ { + } } \left [ \lambda _ 1 \delta _ 1 + \sup _ { \varpi \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \int _ { \mathcal { V } } g _ { \lambda , 1 } ( v ) \ , d \varpi ( v ) \right ] . \end{align*}


\begin{align*} \begin{aligned} - \Delta _ p u + ( - \Delta _ p ) ^ s u = f \ ; \ ; \Omega , \end{aligned} \end{align*}


\begin{align*} T _ { \C } ( f ) ( z ) = - \frac { 1 } { \pi } \iint _ { \C } \frac { f ( \zeta ) } { \zeta - z } \ , d \xi \ , d \eta , \end{align*}