\begin{align*} - \frac { 2 \binom { n - k } { k } \binom { n - 1 } { k - 1 } } { 2 \binom { n } { k } - 2 \binom { n - 1 } { k - 1 } } . \end{align*}


\begin{align*} g \colon \Z / p \Z \to \Z / p \Z , g ( x ) = \alpha x . \end{align*}


\begin{align*} ( 1 _ { L ^ { \otimes ^ 3 } } - \tau ^ { 1 2 } ) ( ( \Delta _ 0 \otimes \alpha ) \circ \Delta _ 0 - ( \alpha \otimes \Delta _ 0 ) \circ \Delta _ 0 ) = 0 \end{align*}


\begin{align*} \begin{aligned} ( f g ) x = ( f x ) g + f ( g x ) + \alpha _ 1 \big ( f ( x g ) + f ( g x ) \big ) + \alpha _ 2 \big ( g ( x f ) + g ( f x ) \big ) \\ x ( f g ) = ( x f ) g - ( x g ) f + \beta _ 1 \big ( f ( x g ) + f ( g x ) \big ) + \beta _ 2 \big ( g ( x f ) + g ( f x ) \big ) \end{aligned} \end{align*}


\begin{align*} \varphi ( y , d _ 0 ) : = \begin{cases} \max _ { p , v } & \sum _ { i = 1 } ^ n ( 1 - c ) p _ i \cdot \min ( x _ i + y _ i , d _ i ) \\ s . t . \quad & \begin{cases} p _ i \in [ p _ { i , \min } , p _ { i , \max } ] , \forall i \in I \\ v \mbox { s o l v e s } F ( p ) . \end{cases} \end{cases} \end{align*}


\begin{align*} & F ( A ) = 2 a ^ 2 - \left ( 3 - d + 2 \left \lfloor \frac { a - 1 } { 3 } \right \rfloor \right ) a - d . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ r a _ j ^ p + \sum _ { j = r + 1 } ^ k b _ j ^ p \leq k . \end{align*}


\begin{align*} b _ p + k x _ p + l y _ p = a _ p + n x _ p \leq c _ p + m y _ p \end{align*}


\begin{align*} \epsilon _ 1 ( n , A , \rho ) : = \frac { 1 } { 2 } ( 1 0 n ) ^ { - 3 } \epsilon _ 0 ( n , A , \rho ) , C _ 1 ( n , A , \rho ) : = n \epsilon _ 1 ( n , A , \rho ) ^ { - 1 } . \end{align*}


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


\begin{align*} = \left ( 1 - y z \right ) ^ { \frac { y } { 1 - y } } \exp \left \{ \frac { y } { ( 1 - y ) ^ 2 } \left ( L i _ 2 ( z ) - L i _ 2 ( y z ) \right ) \right \} ; \end{align*}


\begin{align*} \mathrm { R W } ( d ) = \sup _ { \gamma \in \Sigma ( \delta ) } \mathbb { E } _ { \gamma } \left [ \sum _ { \ell = 1 } ^ L Y _ \ell I ( d ( X ) = \ell ) \right ] . \end{align*}


\begin{align*} \chi _ t = - \sqrt { \chi _ 1 ^ 2 + 2 G \left ( \frac 1 { \chi } - \frac 1 { \chi _ 0 } \right ) } . \end{align*}


\begin{align*} \xi _ { i _ 1 , \ldots , i _ n } ( f _ { i _ n , \ldots , i _ 1 , i _ 1 , \ldots , i _ n } ) & = \xi _ { i _ 1 , \ldots , i _ n } ( \psi ^ n _ { i _ n , \ldots , i _ 1 , \ldots , i _ n } ( f _ { i _ n , \ldots , i _ 1 , \ldots , i _ n } ) ) = \\ & = \xi _ { i _ 2 , \ldots , i _ n } ( \delta ^ n _ { i _ n , \ldots , i _ 1 , \ldots , i _ n } ( f _ { i _ n , \ldots , i _ 1 , \ldots , i _ n } ) ) = \xi _ { i _ 2 , \ldots , i _ n } ( f _ { i _ n , \ldots , i _ 2 , i _ 2 , \ldots , i _ n } ) = \\ & = \epsilon ( f ) . \end{align*}


\begin{align*} \rho _ a ( x ) = \underbrace { x \cdot \ldots \cdot x } _ { a } . \end{align*}


\begin{align*} \R \ni s \mapsto \gamma ( s ) : = G ( \phi , \omega , r ; s ) . \end{align*}


\begin{align*} \theta _ t = ( \theta ^ { ( 1 ) } _ t \bar { \otimes } \mathrm { i d } ) \ ! \upharpoonright _ { \widetilde { M _ 1 \bar { \otimes } M _ 2 } } = ( \mathrm { i d } \bar { \otimes } \theta _ t ^ { ( 2 ) } ) \ ! \upharpoonright _ { \widetilde { M _ 1 \bar { \otimes } M _ 2 } } , t \in \mathbb { R } . \end{align*}


\begin{align*} C _ F = \sum _ { G \le F } X _ G . \end{align*}


\begin{align*} \widetilde { k } \left ( x \right ) = 2 \left ( \nabla u _ { 0 , 1 } \left ( x \right ) \right ) ^ { - 2 } v \left ( x , T / 2 \right ) + F \left ( x \right ) , \end{align*}


\begin{align*} \begin{cases} c _ t - \Delta c + c = g , \ \ ( x , t ) \in \Omega \times ( 0 , T ) , \\ \displaystyle { \frac { \partial c } { \partial \nu } } = 0 , \ \ ( x , t ) \in \partial \Omega \times ( 0 , T ) , \\ c ( x , 0 ) = c _ 0 ( x ) , \ \ ( x , t ) \in \Omega . \end{cases} \end{align*}


\begin{align*} \int _ { T _ n ( t ) } \prod _ { j = 1 } ^ n ( t _ { j + 1 } - t _ j ) ^ { \alpha _ j } d \pmb { t } _ n = \frac { \prod _ { j = 1 } ^ { n } \Gamma ( \alpha _ j + 1 ) } { \Gamma ( | \alpha | + n + 1 ) } t ^ { | \alpha | + n } , \end{align*}


\begin{align*} R _ K ( X , \varepsilon , t , a , b ) = \mathcal { O } \left ( \frac { b + 1 } { \sqrt { \log g } } \right ) . \end{align*}


\begin{align*} H ^ \star = \nu P ( \Lambda ) P ^ T = \nu ( P _ + P _ + ^ T - P _ - P _ - ^ T ) . \end{align*}


\begin{align*} { \sqsupset _ \psi ^ * * \sqsupset _ \phi ^ * } = { \sqsupset _ \psi * \sqsupset _ \phi } = { \sqsupset _ { \psi \circ \phi } ^ * } . \end{align*}


\begin{align*} \widetilde { R } ( h , x ) = \begin{cases} ( R h , x ) , & \underline { \varphi } ( h ) ( 0 ) = \varphi ( h ) ( 0 ) , \\ ( R h , \sigma x ) , & 0 = \underline { \varphi } ( h ) ( 0 ) < \varphi ( h ) ( 0 ) = 1 . \end{cases} \end{align*}


\begin{align*} \Big | \widehat { J _ \varepsilon } ( 0 ) - \widehat { J _ \varepsilon } ( \xi ) - | \xi | ^ 2 \Big | & = \Big | \int _ { \mathbb { R } ^ n } J _ \varepsilon ( | x | ) \big ( f _ \xi ( x ) - f _ \xi ( 0 ) - \nabla f _ \xi ( 0 ) \cdot x - \frac { 1 } { 2 } x ^ T D ^ 2 f _ \xi ( 0 ) x \big ) \ : x \Big | \\ & \leq C ( 1 + | \xi | ^ 2 ) \int _ { \mathbb { R } ^ n } \rho _ \varepsilon ( | x | ) \ : x \leq C ( 1 + | \xi | ^ 2 ) , \end{align*}


\begin{align*} \alpha _ { { \bf Y } , 4 } ( 0 ) = 1 \alpha _ { { \bf Y } , 4 } ( n ) = \max _ { 1 \leq l \leq 4 } \ \sup _ { n \leq i _ 1 \leq \cdots \leq i _ l } \alpha ( { \mathcal F } _ 0 , ( Y _ { i _ 1 } , \dots , Y _ { i _ l } ) ) \ , . \end{align*}


\begin{align*} \beta _ n = v ( p ) - p \nu ( x - a _ n ) . \end{align*}


\begin{align*} \Delta w = - \frac { S ^ \varphi } { m - 1 } w \end{align*}


\begin{align*} M : = \sup _ { t \in [ 0 , T _ { \max } ) } \| U ( t ) \| _ { \mathcal { H } ^ 1 } < \infty . \end{align*}


\begin{align*} \mathfrak { f } _ { \bar t } ( \xi _ { [ ( \sigma , k , l ) ] } ) = 0 \mathfrak { o } ( \bar t ) < \mathfrak { o } ( \sigma , k ) . \end{align*}


\begin{align*} \frac { 1 } { 2 } M \leq ( 1 - \epsilon ) M = k - k _ \epsilon \leq | k _ \epsilon | + | k | \leq 2 M \end{align*}


\begin{align*} W = W ( \beta ) = \bigcup _ { s < t } [ \beta ( s ) , \beta ( t ) ] . \end{align*}


\begin{align*} v ( n ) & \ = \ v ( p ^ { \alpha _ 1 } _ 1 \cdots p ^ { \alpha _ k } _ k ) \ = \ v ( p ^ { \alpha _ 1 } _ 1 ) + \cdots + v ( p ^ { \alpha _ k } _ k ) \\ & \ \le \ p ^ { \alpha _ 1 } _ 1 + \cdots + p ^ { \alpha _ k } _ k \ \le \ p ^ { \alpha _ 1 } _ 1 \cdots p ^ { \alpha _ k } _ k \ = \ n . \end{align*}


\begin{align*} \limsup _ { N \to \infty } \sup _ { \lambda \in ( 1 , \infty ) } \sum _ j \norm { P _ { \geq N } g _ j ^ \lambda } _ { L ^ 4 _ { \lambda _ k } } ^ 2 = 0 . \end{align*}


\begin{align*} R ( Y ) : = \bigcup _ { x \in X } Y ( x ) \subset \bar { Y } . \end{align*}


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


\begin{align*} \int _ { - \infty } ^ { \infty } | R ( t ) | ^ { 2 } \Phi \left ( \frac { t } { T } \right ) d t = \sqrt { 2 \pi } T \sum _ { m , n \in \mathcal { M } ' } r ( m ) r ( n ) \Phi \left ( T \log { \frac { m } { n } } \right ) . \end{align*}


\begin{align*} g _ { w ( l ) } ( y ) = \prod _ { i = 1 } ^ r c _ y ( i ) \ast h _ l ( z _ y ( i ) ) ) \ast c _ y ( r + 1 ) \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } e ^ 2 = a _ 1 e + b _ 1 f \\ f ^ 2 = a _ 2 e + b _ 2 f \\ e f = a _ 3 e + b _ 3 f \\ f e = a _ 4 e + b _ 4 f \\ \end{array} \right . \end{align*}


\begin{align*} T ( u , v ) : = \inf _ { p } T ( p ) , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { q ^ n } | \mu _ j | ^ 4 = \sum _ { a , a ' } \left | \sum _ { b } M ( a , b ) M ( a ' , b ) \right | ^ 2 : = \sigma \ , , \end{align*}


\begin{align*} M = { T ^ { \vec { v } } _ \varepsilon } ^ T , \end{align*}


\begin{align*} I _ \mu ^ { \prime } ( u ) = \lambda _ a \Psi ^ { \prime } ( u ) , \ , \ , \ , \ , X ^ { \prime } \end{align*}


\begin{align*} \widetilde { T } _ l ( f _ 1 , f _ 2 ) = \sum _ { \mathbf { m } \in ( \mathbb { Z } ^ 2 ) ^ 2 } \widetilde { T } _ l \left ( f _ { 1 , m _ 1 } , f _ { 2 , m _ 2 } \right ) \textrm { w i t h $ \mathbf { m } = ( m _ 1 , m _ 2 ) $ } . \end{align*}


\begin{align*} \lim _ { k \to \infty } e _ k ( \mathrm { i n t } ( X ) ) = - \infty . \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } \left | \left | u _ n \right | ^ { p - 2 } u _ n - | u | ^ { p - 2 } u \right | ^ { p ' } d x & \leq C \int _ { \Omega } \left | u _ n - u \right | ^ { p ' } \left ( \left | u _ n \right | + | u | \right ) ^ { p ' ( p - 2 ) } d x \\ & \leq C \left \| u _ n - u \right \| _ { p } ^ { p ' } \left ( \left \| u _ n \right \| _ { p } + \| u \| _ { p } \right ) ^ { p ' ( p - 2 ) } . \end{aligned} \end{align*}


\begin{align*} \phi ( x ; r ) = O \big ( \varepsilon ^ { - 2 } _ 2 ( r + \varepsilon _ 2 - x ) ^ { 1 / 2 } ( x + \varepsilon _ 2 - r ) ^ { 1 / 2 } \big ) . \end{align*}


\begin{align*} \sum _ { i \le N } \left \langle \tilde { g } _ { i j } B _ { i } , x \right \rangle ^ { 2 } \le \sum _ { i \le N } ( \max _ { j \le n } \tilde { g } _ { i j } ^ { 2 } ) \Vert B _ { i } \Vert _ { 2 } ^ { 2 } = : \Xi ^ { 2 } . \end{align*}


\begin{align*} \left . u _ 1 \right | _ { \Omega ^ c } = f _ 1 = f _ 2 = \left . u _ 2 \right | _ { \Omega ^ c } \Omega ^ c . \end{align*}


\begin{align*} \underline { \mathcal { I } } _ { n , k } ( A ) : = \left \{ g : [ n - k ] \to [ n ] \setminus \operatorname { I m } ( A ) : \ g \mbox { i s i n j e c t i v e } \right \} . \end{align*}


\begin{align*} \zeta _ { R } ( \lambda ) = \prod _ { \gamma ^ \# } ( 1 - e ^ { - \lambda T _ { \gamma ^ \# } } ) , \Re \lambda \gg 1 . \end{align*}


\begin{align*} & E [ W ^ k _ { \ell _ 2 \Delta , t } ( j ) W ^ k _ { \ell _ 1 \Delta , t } ( j ) ] = P ( W ^ k _ { \ell _ 2 \Delta , t } ( j ) = 1 , W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 ) + O \left ( e ^ { \lambda \Delta \ell _ 1 } e ^ { 2 \lambda \Delta \ell _ 2 } \Delta ^ 3 \right ) , \\ & E [ Z _ 0 ( \ell _ 2 \Delta ) W ^ k _ { \ell _ 1 \Delta , t } ( j ) ] = E [ Z _ 0 ( \ell _ 2 \Delta ) ; W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 ] + O \left ( e ^ { \lambda \Delta \ell _ 1 } e ^ { 2 \lambda \Delta \ell _ 2 } \Delta ^ 2 \right ) . \end{align*}


\begin{align*} \beta _ n = \sum _ { j = 0 } ^ n \frac { \alpha _ j } { ( q ) _ { n - j } ( a q ) _ { n + j } } \ ; \ ; \ ; \ ; \forall \ , n \in \mathbb { N } . \end{align*}


\begin{align*} \begin{cases} \Delta v + \Lambda ( x ) v \ge 0 & \ , \Omega \\ v \equiv 0 & \ , \partial \Omega . \end{cases} \end{align*}


\begin{align*} \textsf { P } \{ \Xi ^ { 2 } > C z \log n \sum _ { i \le N } \Vert B _ { i } \Vert _ { 2 } ^ { 2 } \} & = \textsf { P } \{ \Xi ^ { 2 } > C \log n \sum _ { i \le N } \Vert B _ { i } \Vert _ { 2 } ^ { 2 } \} \\ & \le \exp ( - C _ { 1 } z \log n \sum _ { i \le N } \Vert B _ { i } \Vert _ { 2 } ^ { 2 } ) . \end{align*}


\begin{align*} \begin{aligned} \left ( \operatorname { p r o j } _ { K _ N \cap Q } \circ { \operatorname { p r o j } _ { K _ N } } ^ { - 1 } \right ) \# \mu _ N & = \left ( \operatorname { p r o j } _ { K _ N \cap Q } \circ { \operatorname { p r o j } _ { K _ \ell } } ^ { - 1 } \right ) \# \mu _ \ell . \end{aligned} \end{align*}


\begin{align*} 0 = \left \langle X , \pounds _ X g \right \rangle = \nabla _ X X + \frac 1 2 \nabla \left ( | X | ^ 2 \right ) = \nabla _ X X , \end{align*}


\begin{align*} ( \theta , p _ \theta , z , s , t ) \mapsto ( \vartheta , r , p _ \vartheta , p _ r , w ) = ( \theta , e ^ s , e ^ s p _ \theta , z , t + e ^ s z ) . \end{align*}


\begin{align*} \mathcal { M } ^ { 0 } _ { ( N ) , A } f \ ; = \sup _ { \substack { \phi \in \mathcal { S } , \| \phi \| _ { ( N ) } \leq 1 } } M _ { \phi , A } ^ { 0 } f , \end{align*}


\begin{align*} \begin{aligned} & G _ \alpha ' ( r ) = p r ( r ^ 2 + \alpha ) ^ { \frac p 2 - 1 } \\ & G _ \alpha '' ( r ) = p ( r ^ 2 + \alpha ) ^ { \frac p 2 - 2 } [ \alpha + r ^ 2 ( p - 1 ) ] \geq 0 \textrm { f o r a l l } \ ; \ ; r \in \R \ , . \end{aligned} \end{align*}


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


\begin{align*} W _ { K _ 1 K _ 2 K _ 3 K } ( s ) = \frac { \pi } { z ( s ) } \frac { 1 } { ( 2 \pi ) ^ 8 } | \beta _ h ( s ) | ^ 2 \beta _ h ( s ) \overline { \beta _ { \sigma } ( s ) } e ^ { \gamma ( s ) + \frac { \zeta ( s ) \cdot \zeta ( s ) } { 4 z ( s ) } } \end{align*}


\begin{align*} D \ ! F ( a , b , c , g ) = x ^ 2 ( x - 1 ) ^ 2 \partial ^ 3 + D \ ! F _ 1 \partial ^ 2 + D \ ! F _ 2 \partial + D \ ! F _ 3 , \end{align*}


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


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


\begin{align*} \exp \left [ - 2 \lambda \left ( \alpha \varepsilon \left ( T - \varepsilon \right ) - b ^ { 2 } \right ) \right ] = \exp \left ( - 2 \lambda \beta b ^ { 2 } \right ) . \end{align*}


\begin{align*} \left | \log \kappa _ { n } \right | = o \left ( \log n \right ) , n \rightarrow \infty . \end{align*}


\begin{align*} X ( K ) \cap \Gamma = \left ( X ( K ) \cap \tilde { \Gamma } \right ) \cap \Gamma , \end{align*}


\begin{align*} J _ { 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 y 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 y d t - C \lambda ^ 2 \iint _ Q \xi | A \nabla u \cdot \nabla \eta | ^ 2 d x d y d t , \end{align*}


\begin{align*} e ^ { - \lambda t } \hat { S } _ { j , + } ^ k ( t ) & = \nu e ^ { - \lambda t } \int _ 0 ^ t Y e ^ { \lambda s } p _ { j , + } ^ k ( t - s ) d s \\ & = \nu Y \int _ 0 ^ t e ^ { - \lambda s } p _ { j , + } ^ k ( s ) d s \\ & \leq \frac { \nu Y } { \lambda } \theta ^ k . \end{align*}


\begin{align*} \frac 1 { k ! } \sum _ { \ell ( I ) = k } \varphi ^ I \end{align*}


\begin{align*} u _ { x _ n } ( 0 ) \leq w _ { x _ n } ( 0 ) = : - \delta _ 0 < 0 . \end{align*}


\begin{align*} p : = P ( \Omega _ \infty ^ c ) = P ( Z _ 0 ( t ) = 0 \ ; \} , \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \sup _ { \varpi \in \Pi ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ { \mathcal { V } } f _ \lambda \ , d \varpi \right ] \end{align*}


\begin{align*} \langle x _ 1 , x _ 2 , \dots , x _ 7 : x _ i x _ j x _ i ^ { - 1 } = x _ { 3 i + 5 j \bmod 7 } \rangle . \end{align*}


\begin{align*} m ( t ; \lambda ) = \frac { 4 \pi } { 3 } \cdot \begin{cases*} ( \frac { t + 1 } { \lambda + 1 } ) ^ 3 - ( \frac { t - 1 } { 1 - \lambda } ) ^ 3 & i f $ 0 < \lambda < \frac { 1 } { t } $ , \\ 0 & o t h e r w i s e . \end{cases*} \end{align*}


\begin{align*} \| A \| = \sup \bigl \{ \| A ( x _ 1 , \ldots , x _ k ) \| \colon ( x _ 1 , \ldots , x _ k ) \in S _ { X _ 1 } \times \cdots \times S _ { X _ k } \bigr \} . \end{align*}


\begin{align*} \texttt { F p [ m ] } = 0 , M \le m \le 3 L - 3 , \end{align*}


\begin{align*} & \forall \ , x : n \forall \ , y : n ( h _ n ( x + _ n y ) = h _ n ( x ) + _ { n + 1 } h _ n ( y ) ) \\ & \forall \ , x : n \forall \ , a : 0 ( h _ n ( a \cdot _ n x ) = a \cdot _ n h _ n ( x ) ) \\ & h _ n ( \top _ n ) = \top _ { n + 1 } \end{align*}


\begin{align*} \mathcal { H } ^ 1 \left ( J _ u \cap B _ { 2 r } \setminus J \right ) = 0 , \end{align*}


\begin{align*} \phi ( b ) = \langle \theta ( b ) \xi , \xi \rangle , b \in B . \end{align*}


\begin{align*} ( R ) & \xrightarrow { { \sim } } \Omega _ { c l } \\ I & \mapsto ( I = ( 0 ) ) \end{align*}


\begin{align*} D _ \Theta X _ T ^ \ast \le \left ( D _ \Theta X \right ) _ T ^ \ast : = \sup _ { s \in [ 0 , T ] } D _ \Theta X _ s , \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c ) = 1 \\ a , b , c \geq 1 } } \left ( \frac { 1 } { 1 - x ^ a y ^ b z ^ c } \right ) ^ { \frac { c } { a b } } = \exp \left \{ L i _ 1 ( x ) L i _ 1 ( y ) L i _ { - 1 } ( z ) \right \} = \left ( 1 - x \right ) ^ { \frac { z \log ( 1 - y ) } { ( 1 - z ) ^ 2 } } , \end{align*}


\begin{align*} v t ^ { - 1 } = \bigcup _ { \substack { w \in V / L \\ w t = v } } w . \end{align*}


\begin{align*} H = \begin{pmatrix} h | _ { L _ 1 } & & & \\ & h | _ { L _ 2 } & & \\ & & \ddots & \\ & & & h | _ { L _ n } \end{pmatrix} \end{align*}


\begin{align*} \mathcal { R } ( A \circ B ) ^ { - 1 } & \leq \mathcal { R } ^ { - 1 } ( A \circ B ) & & \\ & \leq ( \mathcal { R } A \circ \mathcal { R } B ) ^ { - 1 } & & \\ & \leq \mathcal { R } ^ { - 1 } A \circ \mathcal { R } ^ { - 1 } B & & \\ & \leq \sec ^ 2 \theta \mathcal { R } A ^ { - 1 } \circ \sec ^ 2 \theta \mathcal { R } B ^ { - 1 } & & \\ & = \sec ^ 4 \theta ( \mathcal { R } A ^ { - 1 } \circ \mathcal { R } B ^ { - 1 } ) . & & \end{align*}


\begin{align*} S ( t ) = \left ( \begin{array} { c c } a ( t ) & 0 \\ 0 & 1 \end{array} \right ) , \end{align*}


\begin{align*} { \sqsupset \circ \vartriangleright } \ \subseteq \ { \sqsupset * \geq } \ = \ { \sqsupset ^ * \circ \geq } \ = \ { \sqsupset ^ * } . \end{align*}


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


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


\begin{align*} \begin{cases} \i \partial _ t \varphi ( t , x ) = \left ( - \Delta _ x + b | \varphi ( t , x ) | ^ 2 - \mu ( t ) \right ) \varphi ( t , x ) , \\ \varphi ( 0 , x ) = \varphi _ 0 ( x ) \ , , \end{cases} \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( \Psi ) = \partial _ { \lambda } S _ { \omega , \mathbf { c } } ( \lambda \Psi ) \big | _ { \lambda = 1 } = \sum _ { j = 1 } ^ 3 \langle D _ j S _ { \omega , \mathbf { c } } ( \Psi ) , \psi _ j \rangle = 0 . \end{align*}


\begin{align*} e _ { i j } ( a , b ) = \begin{cases} 1 \quad & a = i b = j , \\ 0 & \end{cases} \end{align*}


\begin{align*} \prod _ { k = 1 } ^ R \det T _ n ( \phi _ k ) & \sim n ^ \Omega E _ 4 \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 \Big ( \ln \ln ( e + V ) \Big ) \psi ^ h _ { \mu _ j , \xi _ j } d x + O \bigg ( \epsilon ^ 2 \Big ( \ln \Big | \ln \frac { \epsilon } { | \ln \epsilon | ^ 2 } \Big | \Big ) ^ 2 \bigg ) . \end{align*}


\begin{align*} c _ d = \begin{cases} 1 2 \pi & d = 3 , \\ 4 \pi & d = 2 , \\ 2 & d = 1 . \end{cases} \end{align*}


\begin{align*} \tau = \inf \{ s \geq 0 : \ , | W _ s | \geq \varepsilon _ { 2 } \} , \sigma ^ z = \inf \{ s \geq 0 : \ , | z + W _ s | \geq \widetilde { \Lambda } _ { s } \} , R _ s ^ z = | z + W _ s | . \end{align*}


\begin{align*} & \partial _ t v _ i + a _ { k i } \partial _ k q = 0 , i = 1 , 2 , 3 \\ & a _ { i k } \partial _ i v _ k = 0 , \end{align*}


\begin{align*} [ A x ] _ i = a _ { i i } x _ i + \sum _ { j \neq i } a _ { i j } x _ j = - \frac { \sum _ { j \neq i } a _ { i j } x _ j } { x _ i } x _ i + \sum _ { j \neq i } a _ { i j } x _ j = 0 . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c w ) ^ n = ( - \eta _ 1 w ) ^ n g \omega ^ n & \textnormal { o n } & \Omega \\ w = 0 & \textnormal { i n } & \partial \Omega \\ w < 0 . & & \end{array} \right . \end{align*}


\begin{align*} \sum _ { z : C } \exists _ { y : A } ( f ( y ) = x ) \wedge g ( y ) = z \end{align*}


\begin{align*} m _ 1 ( \tau \ge n ) = 2 m ( [ \tfrac 1 2 , \hat { b } _ { n - 1 } ] ) \le 2 ( \hat { b } _ n - \tfrac 1 2 ) \le 2 ^ { \frac { 1 } { \beta } ( \frac { 1 } { \alpha ^ 2 } + \frac { 2 } { \alpha } ) } n ^ { - \frac { 1 } { \alpha \beta } } \le C ( B ) n ^ { - \frac { 1 } { \alpha _ u \beta _ u } } . \end{align*}


\begin{align*} r _ { \theta } ( t ) = \begin{cases} 0 b = 0 , \\ b ^ { p \theta } t ^ { p - 1 } + \sum _ { m = 1 } ^ { p - 1 } \binom { p - 1 } { m } \left ( a ^ { p \theta - m } b ^ m t ^ { p \theta + p - 1 - m } + a ^ { - m } b ^ { p \theta + m } t ^ { p - 1 - m } \right ) . \end{cases} \end{align*}


\begin{align*} d _ { \overline { h } _ N } ( x ) = N \int _ { [ \frac { i - 1 } { N } , \frac { i } { N } ) } \d x \ , h ( x ) \leq \| d _ h \| _ { \infty } . \end{align*}


\begin{align*} \int _ \Omega f _ \epsilon ^ { ' } ( V _ m ) u _ m ^ 2 d x = o ( 1 ) . \end{align*}


\begin{align*} \cosh ( \alpha x _ { j - 1 } ) \sinh ( \alpha x _ { j } ) - \cosh ( \alpha x _ { j } ) \sinh ( \alpha x _ { j - 1 } ) = \sinh ( \alpha h _ j ) . \end{align*}


\begin{align*} \bar { \rho } = \Bigl ( \frac { 2 - \gamma _ p - \gamma _ q } { 2 2 ^ * _ \mu - \gamma _ p - \gamma _ q } S ^ { 2 ^ * _ \mu } _ { H , L } \Bigr ) ^ { \frac 1 { 2 2 ^ * _ \mu - 2 } } , \end{align*}


\begin{align*} L i _ 3 ( 0 ) = 0 , \end{align*}


\begin{align*} \alpha _ i = \nu ( Q _ i ' ) - \nu ( Q _ i ) , \ \beta _ i = \nu ( g ' ) - \nu _ i ( g ) \mbox { a n d } \tilde { \beta } _ i = \nu _ i ( g ' ) - \nu _ i ( g ) . \end{align*}


\begin{align*} \mathcal { L } ^ \vee \otimes \iota ^ * \Big ( \bigoplus _ { i \in Q _ 0 } \mathcal { U } _ i ^ \vee \boxtimes \mathcal { U } _ i \Big ) = \bigoplus _ { i \in Q _ 0 } ( ( \iota ^ * \circ p _ 1 ^ * ) ( \mathcal { U } _ i ) \otimes \mathcal { L } ) ^ \vee \otimes ( \iota ^ * \circ p _ 2 ^ * ) ( \mathcal { U } _ i ) \end{align*}


\begin{align*} ( J _ \varepsilon * 1 ) ( x ) = \mathcal { F } ( J _ \varepsilon ) ( 0 ) \end{align*}


\begin{align*} \begin{cases} \displaystyle \frac { d } { d t } x ( X , t ) = u ( x ( X , t ) , t ) , t > 0 , \smallskip \\ x ( X , 0 ) = X , \end{cases} \end{align*}


\begin{align*} K = K ( \omega ) = \bigcup _ { x _ i \in \ : ( \Pi ) } \bar { B } ( x _ i , a ) \end{align*}


\begin{align*} \begin{gathered} \left \| \phi _ \epsilon \circ \big ( \tilde { y } _ { T _ m } ^ K \big ) ^ { - 1 } - f _ 0 \circ \big ( \tilde { y } _ { T _ m } ^ K \big ) ^ { - 1 } \right \| _ { L ^ 1 ( \mathbb { T } ^ 2 ) } \le \epsilon , \\ \left \| \phi _ \epsilon \circ z _ m - f _ 0 \circ z _ m \right \| _ { L ^ 1 ( \mathbb { T } ^ 2 ) } \le \epsilon . \end{gathered} \end{align*}


\begin{align*} \bar { c } _ k ( t _ 1 , \dots , t _ n ) = \sum _ { 1 \leq i _ 1 < \dots < i _ k \leq n } t _ { i _ 1 } t _ { i _ 2 } \cdots t _ { i _ k } \end{align*}


\begin{align*} \epsilon ( \Gamma \mid \Delta ) = | \{ \gamma \in \Gamma \mid 0 \leq \gamma < \Delta _ { > 0 } \} | . \end{align*}


\begin{align*} \mathcal { S } _ { D } \left ( b \right ) = \left \{ x \in \mathbb { R } ^ { n } : a _ { t } ^ { \prime } x \leq b _ { t } , \ , t \in T \backslash D ; \ ; a _ { t } ^ { \prime } x = b _ { t } , \ , t \in D \right \} . \end{align*}


\begin{align*} d ( \tau , j ) _ i ^ { \sigma } = \begin{cases} p & j = i \mbox { a n d } \tau = \sigma \\ 1 & \mbox { e l s e } \end{cases} \end{align*}


\begin{align*} f ( k , t , n ) = \frac { n } { n - k } \Big ( F ( k , t , n ) - F ( k , t + 1 , n ) \Big ) . \end{align*}


\begin{align*} H = \mathcal { G } [ \ell ] \bigcap \mathcal { G } [ \ell , \sigma ] \bigcap \mathrm { K e r } ( \mu ) , \end{align*}


\begin{align*} F ( k , t + 1 , \mathbf { p } _ { n + 1 } ) = \begin{dcases} p _ { k } ^ { } F ( k - 1 , t , \mathbf { p } _ { n + 1 } ) + ( 1 - p _ { k } ^ { } ) F ( k , t , \mathbf { p } _ { n + 1 } ) & \textnormal { i f } \ ; k < t ; \\ 1 & \textnormal { i f } \ ; k \ge t ; \end{dcases} \end{align*}


\begin{align*} E : D \to M , E ( q , \lambda ) = \exp _ q ( \lambda ) , \end{align*}


\begin{align*} \log \det ( u _ { i \bar { j } } ) = \log \psi ( \cdot , v ) = g , \end{align*}


\begin{align*} \mathcal { P } _ { \alpha } ^ { - 1 } \mathcal { Y T } { \bf u } = \mathcal { P } _ { \alpha } ^ { - 1 } \mathcal { Y } { \bf f } . \end{align*}


\begin{align*} Q _ { 2 , \vec { v } _ { s , t , f , 0 } ^ { ( 2 ) } } = \left ( \left ( \begin{array} { r r r } t & 1 & 0 \\ s & 0 & 1 \\ 1 & 0 & 0 \end{array} \right ) , \left ( \begin{array} { r r r } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) , \left ( \begin{array} { r r r } 0 & 0 & 1 \\ 1 & 0 & - t \\ 0 & 1 & - s \end{array} \right ) \right ) , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } ( d d ^ c v _ \lambda ) ^ n = ( \Vert u _ { \lambda } \Vert _ { C ^ 0 ( \bar \Omega ) } ^ { - 1 } - \lambda v _ \lambda ) ^ n f ^ n \omega ^ n & \textnormal { i n } & \Omega \\ v _ \lambda = 0 & \textnormal { i n } & \partial \Omega \\ \Vert v _ \lambda \Vert _ { C ^ 0 ( \bar \Omega ) } = 1 . \end{array} \right . \end{align*}


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


\begin{align*} \phi & : C \rightarrow E , \\ \phi & ( x , y , r ) = ( 5 y / x , - 1 5 0 r / x ) , \end{align*}


\begin{align*} \textsf { P } \{ \sum _ { i \le N } ( \max _ { j \le n } \tilde { g } _ { i j } ^ { 2 } ) \Vert B _ { i } \Vert _ { 2 } ^ { 2 } > t \} \le e ^ { - \lambda t } \prod _ { i = 1 } ^ { N } \textsf { E } \exp ( \lambda ( \max _ { j \le n } \tilde { g } _ { i j } ^ { 2 } ) \Vert B _ { i } \Vert _ { 2 } ^ { 2 } ) . \end{align*}


\begin{align*} \nu ( \alpha , g ) = ( \det a _ 1 a _ 2 , y ) _ F \gamma ( y , \psi ^ { \tfrac { 1 } { 2 } } ) ^ { - | S | } , \end{align*}


\begin{align*} k x _ p + l y _ p = m y _ p < n x _ p . \end{align*}


\begin{align*} \| T _ { \rho } f \| _ q ^ q = \| T _ { \rho ' \frac { 1 } { \sqrt { q } } } f ' \| _ q ^ q \le \sum _ { S \subseteq [ n ] } \left ( \frac { \beta } { \sqrt { q } } \right ) ^ { q | S | } \mathbb { E } _ { x \sim \Omega ^ S } \| D _ { S , x } [ f ' ] \| _ 2 ^ q . \end{align*}


\begin{align*} { { { \boldsymbol { \hat x } } } _ { k ; t + 1 } } { } { \boldsymbol { f } } \left ( { { { { \boldsymbol { \hat x } } } _ { k ; t } } } \right ) = { \left ( { { \boldsymbol { W } } _ k ^ T { { \boldsymbol { \Omega } } _ { k ; t } } { { \boldsymbol { W } } _ k } } \right ) ^ { - 1 } } { \boldsymbol { W } } _ k ^ { } { { \boldsymbol { \Omega } } _ { k ; t } } { { \boldsymbol { d } } _ k } , \end{align*}


\begin{align*} \psi ( x ) = \frac { d } { d x } \big \{ \log \Gamma ( x ) \big \} , \end{align*}


\begin{align*} u & = u _ 0 + p _ \mathrm { k } ( u ) + p _ \mathrm { \tilde k } ( u ) \\ u ' & = u _ 0 ' + p _ \mathrm { k } ( u ' ) + p _ \mathrm { \tilde k } ( u ' ) \ ; , \end{align*}


\begin{align*} \alpha _ i x + \beta _ i z = 0 , \mbox { f o r } i = 1 , \ldots , k , \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta _ p u _ 0 & = f _ 1 ( | x | ) \cdot h _ 1 ( v _ 0 ) \cdot | \nabla u _ 0 | ^ { \alpha } & \quad \mbox { i n } \Omega , \\ \Delta _ p v _ 0 & = f _ 2 ( | x | ) \cdot h _ 2 ( v _ 0 ) \cdot h _ 3 ( | \nabla u _ 0 | ) & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \inf _ { \lambda \in \mathbb { R } _ { + } ^ 2 } \left [ \langle \lambda , \delta \rangle + \sup _ { \varpi \in \Pi ( \mu _ { 1 3 } , \mu _ { 2 3 } ) } \int _ { \mathcal { V } } f _ \lambda \ , d \varpi \right ] \end{align*}


\begin{align*} \begin{cases} \mathbf { v } \cdot \nabla \mathbf { v } = - \nabla P & \ \ , D , \\ \nabla \cdot \mathbf { v } = 0 \ , \ \ , \ \ \ \ \ \ \ \ \ , & \ \ , D , \end{cases} \end{align*}


\begin{align*} \omega ^ + = a ^ + + m , \ , \ , \omega ^ - = a ^ - + m , \ , \ , \omega ^ { ( 0 ) } = n - \omega ^ + - \omega ^ - \end{align*}


\begin{align*} P _ n ^ { ( \alpha , \beta ) } ( \cos \theta ) : = \frac { \Gamma ( \alpha + 1 + n ) } { \Gamma ( n + 1 ) \ , \Gamma ( \alpha + \beta + 1 + n ) } \sum _ { k = 0 } ^ n ( - 1 ) ^ k \binom { n } { k } \frac { \Gamma ( \alpha + \beta + 1 + n + k ) } { \Gamma ( \alpha + 1 + k ) } \sin ^ { 2 k } ( \tfrac 1 2 \theta ) , \end{align*}


\begin{align*} \| \nabla ^ j u \| _ \infty \leq C \| u \| _ { W ^ { m , \mathcal { P } } ( V ) } , \quad \forall j = 0 , \cdots , m . \end{align*}


\begin{align*} ( f ^ * h ) \circ \phi : M ^ { \otimes r } \overunderset { \phi } { } { \longrightarrow } f ^ * L \overunderset { f ^ * h } { } { \longrightarrow } f ^ * ( \sigma ^ * L ) = ( \sigma \circ f ) ^ * L , \end{align*}


\begin{align*} | A _ \gamma ( \hat { x } , \hat { x } ) ( \hat { x } - \bar { y } ) | ^ 2 = | A ^ \prime ( \hat { x } ) ( \hat { x } ^ \prime - y ^ \prime ) | ^ 2 + | \hat { x } _ n + y _ n | ^ 2 , \end{align*}


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


\begin{align*} G _ k ( x ) = \frac { 1 } { 2 \pi i } \int _ { \mathrm { R e } ( t ) = 3 \slash 4 } \frac { \Gamma ( k + t ) ^ 2 } { ( 2 \pi ) ^ { 2 t } x ^ t } \frac { d t } { t } . \end{align*}


\begin{align*} i \partial _ t u = - \Delta u + \varepsilon | u | ^ 2 u \end{align*}


\begin{align*} f = \sum _ { i = 1 } ^ s a _ i { \tilde { \textbf { Q } } } ^ { \lambda _ i } \end{align*}


\begin{align*} a = { d i a g } ( e ^ { w _ 1 } , \ldots , e ^ { w _ m } , e ^ { - 1 } , \ldots , e ^ { - 1 } ) . \end{align*}


\begin{align*} t _ 0 : = \inf \{ t \in [ 0 , \zeta \wedge \widetilde \zeta ) : \ , \Lambda _ t \neq \widetilde { \Lambda } _ t \} . \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ 0 < u < 1 , & \ \ D , \\ u = 1 , \ \ | \nabla u | = c _ 1 > 0 , & \ \ , \partial \Omega _ 1 , \\ u = 0 , \ \ | \nabla u | = 0 , & \ \ , \partial \Omega _ 2 . \end{cases} \end{align*}


\begin{align*} \mathcal T ^ { 1 , 1 } _ { \rm l o c } ( \Omega ) = \left \{ u \ , \hbox { i s m e a s u r a b l e i n $ \Omega $ } : \hbox { $ T _ { t } ( u ) \in W ^ { 1 , 1 } _ { \rm l o c } ( \Omega ) $ f o r e v e r y $ t > 0 $ } \right \} . \end{align*}


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


\begin{align*} \nu \left ( \frac { a _ i h } { Q ' _ i } \right ) = \nu ( a _ i ) + \nu ( h ) - \nu ( Q _ i ' ) \geq 0 . \end{align*}


\begin{align*} \Delta _ g \frac { | u | ^ p } { p } = ( p - 1 ) | u | ^ { p - 2 } | \nabla _ g u | ^ 2 + | u | ^ { p - 2 } u \Delta _ g u , \end{align*}


\begin{align*} S ( t ) = \dfrac { \sin t I + \cos t S ( 0 ) } { \cos t I - \sin t S ( 0 ) } , \end{align*}


\begin{align*} ( E _ 0 = M \oplus K _ X ^ { n - 1 } \oplus K _ X ^ { n - 2 } \oplus \cdots \oplus K _ X ^ { 2 - n } \oplus K _ X ^ { 1 - n } \oplus M ^ { - 1 } , \quad \theta _ 0 = \begin{pmatrix} 0 & & & & & \\ \mu & 0 & & & & \\ & 1 & 0 & & & \\ & & \ddots & \ddots & \\ & & & 1 & 0 & \\ & & & & \mu & 0 \end{pmatrix} ) . \end{align*}


\begin{align*} \lim _ { R \to \infty } \frac { \sigma _ R ^ 2 ( t ) } { R } = \lim _ { R \to \infty } 4 \pi \sum _ { n \geq 2 } C _ H ^ n \int _ { T _ n ( t ) } \big ( \ell _ R * g _ { \pmb { t } _ n } ^ { ( n ) } \big ) ( 0 ) d \pmb { t } _ n = 4 \pi \sum _ { n \geq 2 } C _ H ^ n \int _ { T _ n ( t ) } g _ { \pmb { t } _ n } ^ { ( n ) } ( 0 ) d \pmb { t } _ n . \end{align*}


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


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


\begin{align*} | V | = \left | \bigcup _ { x H \in \Omega \setminus \mathcal { S } } \{ ( x H , y H ) : \ y H \in \mathcal { S } \} \right | = A ( X _ i / \pi ) _ { 1 2 } | \Omega \setminus \mathcal { S } | = A ( X _ i / \pi ) _ { 2 1 } \left ( | \Omega | - | \mathcal { S } | \right ) . \end{align*}


\begin{align*} \Bigg ( \sum _ { i = 1 } ^ k X _ i \Bigg ) \Bigg ( \sum _ { i = 1 } ^ k X _ { i + k } \Bigg ) = \sum _ { i = 1 } ^ { 2 k } X _ i + \sum _ { p \geq 2 } \sum _ { ( i _ 1 , \dots , i _ p ) \in \mathbb { I } _ { 2 k } ^ p } \alpha _ { ( i _ 1 , \dots , i _ p ) } [ X _ { i _ 1 } , \dots , X _ { i _ p } ] . \end{align*}


\begin{align*} \rightarrow \bigoplus _ { \substack { \sigma \in W ^ { a f f } , \\ l ( \sigma ) = r } } P _ { \sigma ( \rho ) - \rho } \rightarrow \bigoplus _ { \substack { \sigma \in W ^ { a f f } , \\ l ( \sigma ) = r - 1 } } P _ { \sigma ( \rho ) - \rho } \rightarrow \dots \bigoplus _ { \substack { \sigma \in W ^ { a f f } , \\ l ( \sigma ) = 1 } } P _ { \sigma ( \rho ) - \rho } \rightarrow P _ 0 \rightarrow V _ 0 . \end{align*}


\begin{align*} 1 \geq | \langle B \eta , \eta \rangle | \geq \langle \Re ( B ) \eta , \eta \rangle = 1 . \end{align*}


\begin{align*} \Omega : = W G W ^ * = [ \Omega _ { s , t } ] \end{align*}


\begin{align*} \prod _ { j = 0 } ^ { n - 1 } ( 1 - q ^ \alpha \zeta ^ \beta \zeta ^ { j / n } ) = 1 - q ^ { n \alpha } \zeta ^ { n \beta } . \end{align*}


\begin{align*} \Psi _ t ( x ) = t ^ { - ( d + \beta ) / \alpha } \Psi _ 1 \big ( t ^ { - 1 / \alpha } x \big ) , \end{align*}


\begin{align*} D ^ { 2 } \bar { w } = D ^ { 2 } w - \left ( D ^ { 2 } w \right ) _ { 1 } \end{align*}


\begin{align*} \overline { \overline { \Pi } } _ { \psi } [ u ( b ) ] f ( [ \epsilon , x ] ) = e ^ { \pi i \epsilon x ^ 2 b } f ( [ \epsilon , x ] ) ; \end{align*}


\begin{align*} \frac { \alpha _ { 1 } } { \alpha _ { 2 } } = \left ( \frac { \left \Vert x - P _ { 2 } ( x ) \right \Vert } { \left \Vert x - P _ { 1 } ( x ) \right \Vert } \right ) ^ { p - 1 } , \end{align*}


\begin{align*} b _ { i , j } ' = \begin{cases} - b _ { i , j } & i = k j = k , \\ \displaystyle b _ { i , j } + \frac { | b _ { i , k } | b _ { k , j } + b _ { i , k } | b _ { k , j } | } { 2 } & . \end{cases} \end{align*}


\begin{align*} ( 1 - \rho ) \phi ( z ) + \rho \phi ( z ) \zeta _ \infty ( z ) . \end{align*}


\begin{align*} d u _ \lambda + A u _ \lambda \ , d t = f _ \lambda ( u _ \lambda ) \ , d t + \sigma ( u _ \lambda ) B \ , d W , u _ \lambda ( 0 ) = u _ 0 \in C ( \overline { G } ) \end{align*}


\begin{align*} \begin{cases} & \i \partial _ t \Psi _ N ( t ) = H _ N \Psi _ N ( t ) , \\ & \Psi _ N ( 0 ) = \Psi _ { N , 0 } \ , , \end{cases} \end{align*}


\begin{align*} z ( s ) & = \frac { \sigma ^ 2 } { 2 } + \mathcal { O } ( h ^ 2 + s \sigma ^ 4 ) , \\ \zeta ( s ) & = i ( K _ 1 + K _ 3 - K _ 2 - K ) + \mathcal { O } ( s \sigma ^ 2 ) \end{align*}


\begin{align*} p _ { } ( \mathcal { D } _ a , C _ { } ) = 0 \end{align*}


\begin{align*} \acute { \rho } _ \sigma ^ { a b } : = & \frac { \det \acute { J } _ { a b } ^ \sigma } { \sqrt { \det \left [ \acute { J } _ { a b } ^ { \sigma , T } \acute { J } _ { a b } ^ \sigma + \acute { J } _ { c d } ^ { \sigma , T } \acute { J } _ { c d } ^ \sigma \right ] } } , \end{align*}


\begin{align*} \gamma _ p : = \frac { N ( p - 2 ) + \mu } { 2 } . \end{align*}


\begin{align*} \partial _ { t } \mu _ { B } = A _ { B } ^ { * - 1 } \Big [ G _ { D , 2 } - D _ { D } [ \partial _ { t } \mu _ { S } ] \Big ] \end{align*}


\begin{align*} \big \| B _ l ^ { ( j + 1 ) } \big \| _ l \leq \sum _ { q = 2 } ^ k \sum _ { ( p _ 1 , \dots , p _ q ) \in \mathbb { I } _ { 2 k } ^ q , \ , \deg ( S _ { p _ 1 } ) + \cdots + \deg ( S _ { p _ q } ) = l } | \alpha _ { p _ 1 , \dots , p _ q } | 2 ^ { p _ 1 \wedge \cdots \wedge p _ q } \prod _ { i = 1 } ^ q \tilde { Q } _ { p _ i } \big ( \nu _ 1 , \dots , \sqrt [ k ] { \nu _ k } \big ) , \end{align*}


\begin{align*} h ( \sigma ( x ) ) = h ( x ) \\ h ( x _ 1 + \dots + x _ r ) \leq h ( x _ 1 ) + \dots + h ( x _ r ) + \log r \\ h ( x _ 1 \dots x _ r ) \leq h ( x _ 1 ) + \dots + h ( x _ r ) . \end{align*}


\begin{align*} \begin{gathered} \sum _ { M = 0 } ^ \infty M ^ { \# S - 1 - k } e ^ { - M } \sum _ { N = 0 } ^ { \lfloor w \rfloor + M } N ^ k \leq \sum _ { M = 0 } ^ \infty M ^ { \# S - 1 - k } e ^ { - M } ( w + M ) ^ { k + 1 } \\ \lesssim \sum _ { M = 0 } ^ \infty M ^ { \# S } e ^ { - M } \end{gathered} \end{align*}


\begin{align*} & \{ \mu \in M ' / K \mid \pi _ { K _ 1 } ( p _ K ( \mu ) ) = \lambda / n \} \\ & \simeq \{ ( \gamma , \delta ) \in L ' _ 1 \times M ' / L \mid n | \gamma , \ \gamma | _ { K _ 1 } = \lambda , \ \pi ( \delta ) = \gamma / n \} , \end{align*}


\begin{align*} \eta _ 1 ^ n = \frac { E ( w ) } { I _ g ( w ) } \cdot \end{align*}


\begin{align*} \int _ { \Omega ' } \beta ^ { i j , k l } u ^ { h _ m } _ { i j } \eta _ { k l } d x = 0 \forall \eta \in C _ 0 ^ \infty ( \Omega ' ) \end{align*}


\begin{align*} \max M ( c , x , 1 ) = 1 0 2 4 \end{align*}


\begin{align*} \widetilde { D } _ { \alpha , z } ( \psi | | \varphi ) : = \frac { 1 } { \alpha - 1 } \log \frac { \widetilde { Q } _ { \alpha , z } ( \psi | | \varphi ) } { \psi ( 1 ) } \end{align*}


\begin{align*} P _ { \mu , x } \left ( L _ { x } g , A \right ) = \int _ { G } { \bf 1 } _ { A } \left ( L _ { x } g s \right ) \frac { \varphi _ { g s } ( \pi ( x ) ) } { \varphi _ { g } ( \pi ( x ) ) } d \mu ( s ) , \end{align*}


\begin{align*} M _ b ( \theta ) = \mathbb { E } _ { \Phi | \Phi ( \mathcal { A } ) > 0 } \{ P _ s ^ b ( \theta ) \} \Pr ( \Phi ( \mathcal { A } ) > 0 ) . \end{align*}


\begin{align*} \int _ { \mathbb { S } ^ n } u \mathrm { d } A = - \dfrac { n } { 2 } \dfrac { \phi ' ( \rho ) } { \phi ( \rho ) } \rho \int _ { \mathbb { S } ^ n } u ^ 2 \mathrm { d } A + O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 , \end{align*}


\begin{align*} \psi ( X ) = \mathbf { K } \psi ( X ) ^ { \square } \end{align*}


\begin{align*} \mathrm { K e r } ( \lambda ^ * ) \cap \overline { R ( S ) } = Y \subset \mathrm { K e r } ( \tilde { \lambda } ) \cap \overline { R ( S ) } . \end{align*}


\begin{align*} < \mathcal { R } A , B > = ( A , B ) _ { H _ 0 ^ 1 ( \mathcal { D } _ a ; K ) } , \forall A , B \in H _ 0 ^ 1 ( \mathcal { D } _ a ; K ) . \end{align*}


\begin{align*} \nu _ t ( x ) = e ^ { - \alpha t } \nu _ 0 ( x ) + \int _ 0 ^ t \alpha e ^ { - \alpha ( t - s ) } \Psi ( \nu _ s , \mu _ s ) ( x ) \mathrm { d } s , \end{align*}


\begin{align*} \widehat { \mathcal { I } } _ { \mathrm { D } } ( \delta _ 1 , \delta _ 2 ) = \sup _ { \gamma \in \widehat { \Sigma } _ { \mathrm { D } } ( \delta _ 1 , \delta _ 2 ) } \int _ { \mathcal { V } } g ( s _ 1 , s _ 2 ) \ , d \gamma ( s _ 1 , s _ 2 ) . \end{align*}


\begin{align*} \frac { n - 1 } { \sqrt { 1 + | D _ \tau \xi | ^ 2 } } - { \rm d i v } _ \tau \bigg ( \frac { D _ \tau \xi } { \sqrt { 1 + | D _ \tau \xi | ^ 2 } } \bigg ) - \mu e ^ { \xi } - R e ^ { \xi } = 0 . \end{align*}


\begin{align*} \sigma \left ( ( S , \underline { S } ) ^ * \right ) : = \begin{cases} ( \sigma ( S ) , \sigma ( \underline { S } ) ) ^ { \mathsf { s g n } ( \sigma ( \hat { S } ) , \sigma ( \underline { \hat { S } } ) ) } & \mbox { i f } * = + , \\ ( \sigma ( S ) , \sigma ( \underline { S } ) ) ^ { - \mathsf { s g n } ( \sigma ( \hat { S } ) , \sigma ( \underline { \hat { S } } ) ) } & \mbox { i f } * = - . \end{cases} \end{align*}


\begin{align*} \phi \left ( \bigotimes \limits _ { i = 1 } ^ { m - 1 } X _ i E _ { j _ i j _ i } X _ i ^ * \otimes B \right ) = \left ( Z _ { X } \left ( \bigotimes \limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \right ) Z _ { X } ^ * \right ) \otimes \varphi _ { j _ 1 , \ldots , j _ { m - 1 } , X } ( B ) \end{align*}


\begin{align*} d ( \tau , j ) ^ { \delta } _ { j } = \prod _ { \tau \subseteq \sigma } ( c ( \sigma , j ) ^ { \delta } _ { j } ) ^ { ( - 1 ) ^ { | \sigma \setminus \tau | } } = \prod _ { \tau \subseteq \sigma \subseteq \delta } ( c ( \sigma , j ) ^ { \delta } _ { j } ) ^ { ( - 1 ) ^ { | \sigma \setminus \tau | } } . \end{align*}


\begin{align*} Z ^ H | _ { W _ i } = \begin{pmatrix} \zeta _ i & 0 \\ 0 & - \zeta _ i - \sum _ { j \neq i } { a _ { j i } } \zeta _ j \end{pmatrix} . \end{align*}


\begin{align*} \prod _ { \substack { m , n \geq 1 \\ m \leq n ; \ , ( m , n ) = 1 } } \left ( \frac { 1 } { 1 - y ^ m z ^ n } \right ) ^ { \frac { m ^ 3 } { n ^ 4 } } = \left ( 1 - y z \right ) ^ { \frac { y } { 1 - y } } \end{align*}


\begin{align*} D _ \epsilon : = \{ ( s , t ) \in [ 0 , 1 ] ^ 2 \ , : \ , | t - s | < \epsilon \} , \end{align*}


\begin{align*} \Delta _ g & = \frac { 1 } { \sqrt { g } } \partial _ i ( \sqrt { g } g ^ { i j } \partial _ j ) \\ & = g ^ { i j } \partial _ { i j } + \frac { 1 } { \sqrt { g } } \partial _ i ( \sqrt { g } g ^ { i j } ) \partial _ j \\ & = g ^ { i j } \partial _ { i j } - g ^ { j p } \Theta _ q u _ { p q } \partial _ j . \end{align*}


\begin{align*} S ( k ) \varphi = S ( k ) ( \widetilde { v } _ 0 \circ T - \widetilde { v } _ 0 ) + Q ( k ) \chi . \end{align*}


\begin{align*} \widetilde { \rho } ( - i ^ - , w ) = - \widetilde { \rho } ( - i ^ + , w ) \ \ { \rm a n d } \ \ \widetilde { \rho } _ w ( - i ^ - , w ) = - \widetilde { \rho } _ w ( - i ^ + , w ) , \end{align*}


\begin{align*} - \Sigma _ 1 = \Sigma _ 1 , \quad - \Sigma _ 2 = \Sigma _ 3 , \quad \Sigma _ 1 \cup \Sigma _ 2 \cup \Sigma _ 3 = S ^ 2 , \end{align*}


\begin{align*} \left ( Q _ { \ell , \vec { v } } \right ) _ i M _ { \bullet , \ell } = \left ( \mathcal { Q } _ { \ell , \vec { v } } \left ( M _ { \bullet , \ell } \right ) \right ) _ { \bullet , i } . \end{align*}


\begin{align*} E ( t ) : = \frac 1 2 \chi _ t ^ 2 - \frac { G ( r ) } { \chi } = E ( 0 ) \end{align*}


\begin{align*} \left ( \bigcup _ { \ell < m } K _ { \ell } \right ) \cap K _ { m } = \bigcup _ { \ell < m } ( K _ { \ell } \cap K _ { m } ) = ( K _ { 1 } \cap K _ { m } ) \in 2 ^ { K _ { 1 } } \subset \bigcup _ { \ell < m } 2 ^ { K _ { \ell } } . \end{align*}


\begin{align*} \big \| \mathcal { R } _ \varepsilon c \big \| _ { L ^ 2 ( \Omega ^ \prime ) } ^ 2 & \leq K \int _ { \Omega ^ \prime } \left | \int _ { \Omega ^ c } J _ \varepsilon ( | x - y | ) c ( x ) \ : y \right | ^ 2 \ : x + K \int _ { \Omega ^ \prime } \left | \int _ { \Omega ^ c } J _ \varepsilon ( | x - y | ) \tilde { c } ( y ) \ : y \right | ^ 2 \ : x \\ & = : K \big ( I _ \varepsilon ^ 1 + I _ \varepsilon ^ 2 \big ) . \end{align*}


\begin{align*} \psi \circ \varphi _ { L _ 1 } = \varphi _ { L _ 2 } \circ \psi . \end{align*}


\begin{align*} \frac { d } { d \lambda _ n } g _ n ' ( 0 , 0 ) = \rho - | \alpha | \frac { d } { d \lambda _ n } h _ n ' ( 0 ) . \end{align*}


\begin{align*} \vartheta _ L ( f ) ( g ) = \sum _ { h \in A } \vartheta _ L ( f ; h ) ( g ) h ^ \vee \end{align*}


\begin{align*} Q _ { 2 , \vec { v } _ { s , t , f , 0 } } = \end{align*}


\begin{align*} \overline { \nabla } _ { X } Y = \nabla _ { X } Y + h ( X , Y ) \end{align*}


\begin{align*} v ( x , t ) : = \psi \left ( x - \gamma ( x , t ) - \frac { 1 } { N } \sigma ( t ) , t \right ) - \phi ( x ) , \end{align*}


\begin{align*} - \Delta \tilde { u } ^ i = f _ 0 ^ { ' } ( U _ { 1 , \sigma _ i } ) \tilde { u } ^ i \quad \mbox { i n } \ \R ^ n , \end{align*}


\begin{align*} _ { n , m } ^ { 2 } = \Delta _ { 0 } + \sum ^ { l - 1 } _ { s = 1 } \Delta _ { s } + \widetilde { \Delta } _ { l } , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle 2 \left ( \dfrac { a _ 2 } { 2 m _ 1 } \int _ { \mathbb { R } ^ N } | \nabla \xi | ^ 2 d x + \dfrac { a _ 1 } { 4 m _ 2 } \int _ { \mathbb { R } ^ N } | \nabla \eta | ^ 2 d x \right ) - \dfrac { N } { 2 } a _ 1 a _ 2 \int _ { \mathbb { R } ^ N } \eta \xi ^ 2 d x = 0 , \end{array} \right . \end{align*}


\begin{align*} M _ n ( f ) : = \frac { 1 } { n ^ d } \sum _ { ( j _ 1 , \ldots , j _ d ) \in \Z ^ d _ + \cap Q _ n } \int _ { Q _ { j _ 1 , \ldots , j _ d } } T _ t ( f ) d t , \end{align*}


\begin{align*} \mathrm { d i v } _ T \Big ( \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla v ) \big ) \Big ) \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla v ) \cdot \nu & - \nabla _ T \big [ \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla v ) \big ] \ , \big [ \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla v ) \big ] _ T \cdot \nu \\ & = H ^ 2 ( \nabla v ) \ , H ( \nu ) \ , \mathrm { t r } \ , \mathcal { B } ^ H \ , . \end{align*}


\begin{align*} \partial _ t { w } ^ { ( i ) } _ { \alpha - 1 } ( x _ i , t ) \ , + \ , \big ( v _ i ^ { ( i ) } ( x _ i ) \ , w ^ { ( i ) } _ { \alpha - 1 } ( x _ i , t ) \big ) ^ \prime = - \widehat { \varphi } ^ { ( i ) } ( x _ i , t ) , ( x _ i , t ) \in I _ \varepsilon ^ { ( i ) } \times ( 0 , T ) , \end{align*}


\begin{align*} l _ z ( x ) - l _ w ( x ) = \begin{cases} - 1 & z < x < w \ , \\ 1 & w < x < z \ , \\ 0 & \ . \end{cases} \end{align*}


\begin{align*} \mathcal { Q } ( s ) = \sum _ { e ^ { \ell } \leq \log p \leq e ^ { n _ { \mathcal { L } } } } \left ( \frac { a ( p ) } { p ^ s } + \frac { b ( p ) } { p ^ { 2 s } } \right ) , \end{align*}


\begin{align*} [ P ] _ { i , j } ^ { } = \frac { 1 } { 2 } \times \begin{dcases} 1 & j = i \neq n ; \\ 2 & j = i = n ; \\ 1 & j = i + 1 ; \\ 0 & . \end{dcases} \end{align*}


\begin{align*} t _ n \le 2 ^ r \ , t _ { n - 1 } , \mbox { f o r } \ ; 2 \le n \le N . \\ \end{align*}


\begin{align*} M \vec { v } = \varepsilon \vec { v } , \end{align*}


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


\begin{align*} G ( 1 + z ) = ( 2 \pi ) ^ { z / 2 } \exp \left ( - \frac { z + z ^ 2 ( 1 + \gamma ) } { 2 } \right ) \ , \prod _ { k = 1 } ^ \infty \left \{ \left ( 1 + \frac { z } { k } \right ) ^ k \exp \left ( \frac { z ^ 2 } { 2 k } - z \right ) \right \} . \end{align*}


\begin{gather*} \phi _ 2 ( X _ 1 \otimes X _ 2 ) = [ X _ 1 , X _ 2 ] , \\ \phi _ i ( X _ 1 \otimes \cdots \otimes X _ i ) = [ X _ 1 , \phi _ { i - 1 } ( X _ 2 \otimes \cdots \otimes X _ i ) ] , i \geq 3 . \end{gather*}


\begin{align*} H _ \beta = 2 \beta K _ { 1 / 2 \beta } = 2 \beta \widetilde { K } ^ { ( u ) } K ^ { ( u ) } = 2 \beta \widetilde { T } ( u ) \widetilde { H } ^ { ( u ) } K ^ { ( u ) } , \end{align*}


\begin{align*} | K _ { \alpha } ( z , \vec { y } ) - K _ { \alpha } ( x , \vec { y } ) | & \le \dfrac { A } { \Big ( \sum \limits _ { j = 1 } ^ { m } | z - y _ { j } | \Big ) ^ { m n - \alpha } } \omega \bigg ( \frac { | z - x | } { \sum \limits _ { j = 1 } ^ { m } | z - y _ { j } | } \bigg ) \le \dfrac { C \omega ( 2 ^ { - k } ) } { | 2 ^ { k } \sqrt { n } Q | ^ { m - \alpha / n } } . \end{align*}


\begin{align*} \begin{array} { c c c } x ^ { + } & = & f ( x , u ) \\ \zeta & = & g ( x , u ) \end{array} \end{align*}


\begin{align*} \top & = ( a \lor \neg a ) ^ k \to ( a \to b ) = ( a ^ k \lor ( \neg a ) ^ k ) \to ( a \to b ) \\ & = ( a ^ k \to ( a \to b ) ) \land ( ( \neg a ) ^ k \to ( a \to b ) ) = a ^ { k + 1 } \to b , \end{align*}


\begin{align*} w _ b = \Phi _ { 0 , b } + \Psi _ { b } , \end{align*}


\begin{align*} \begin{aligned} & \left \| { S } _ { N } ( t ) v \right \| _ { L _ N ^ 2 } \leq C ( 1 + t ) ^ { - \frac { 3 } { 4 } } \| v \| _ { L _ N ^ 1 \cap L _ N ^ 2 } , & & v \in L _ N ^ 2 , \end{aligned} \end{align*}


\begin{align*} \nu : = \frac { x - y } { \abs { x - y } } \end{align*}


\begin{align*} \varphi ( b ) = \varphi ( b ' ) = 1 _ { X } \end{align*}


\begin{align*} \sum _ { i \ge 1 } i \widetilde { A } _ i u ^ { i } = F ( u ) G ( u ) , \end{align*}


\begin{align*} C _ n = \int _ { \mathbb { S } ^ { n - 1 } } | e _ 1 \cdot \sigma | ^ 2 \ : \mathcal { H } ^ { n - 1 } ( \sigma ) = \int _ { \mathbb { S } ^ { n - 1 } } | e _ j \cdot \sigma | ^ 2 \ : \mathcal { H } ^ { n - 1 } ( \sigma ) = \frac { 1 } { n } \int _ { \mathbb { S } ^ { n - 1 } } | \sigma | ^ 2 \ : \mathcal { H } ^ { n - 1 } ( \sigma ) = \frac { \omega _ n } { n } . \end{align*}


\begin{align*} r ( t ) = \Pi _ 0 ^ t \exp f ( s ) \ , d s . \end{align*}


\begin{align*} c r ^ d ( 3 / 8 ) ^ { k } \mu \le \sum _ { \| h \| \le r } \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e p _ { e - h } = \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e \cdot \mathbb E [ | \gamma \cap \Lambda _ e | ] \le C r \sum _ { e \in E ( \mathbb Z ^ d ) } q _ e , \end{align*}


\begin{align*} \vert S _ { t } x \vert _ { \beta + { \sigma } _ 1 } & \lesssim t ^ { - { \sigma } _ 1 } \vert x \vert _ { { \beta } } , \\ \vert ( I - S _ { t } ) x \vert _ { { \beta } } & \lesssim t ^ { { \sigma } _ 1 } \vert x \vert _ { \beta + { \sigma } _ 1 } . \end{align*}


\begin{align*} | \xi | ^ { p } \geq | \eta | ^ { p } + p | \eta | ^ { p - 2 } ( \xi - \eta ) \eta = p | \eta | ^ { p - 2 } \xi \eta + ( 1 - p ) | \eta | ^ { p } , \qquad \forall \xi , \eta , \end{align*}


\begin{align*} \norm { u } _ { W ^ { 1 , p ( \cdot ) } ( \Omega ) } : = \norm { u } _ { L ^ { p ( \cdot ) } ( \Omega ) } + \norm { \nabla u } _ { L ^ { p ( \cdot ) } ( \Omega ) } , \end{align*}


\begin{align*} \mathfrak { I } : H ^ { - 1 / 2 } _ \Gamma ( \partial \mathrm { M } ) \rightarrow H ^ { 1 / 2 } _ \Gamma ( \partial \mathrm { M } ) : \langle u , v \rangle = ( \mathfrak { I } u | v ) _ { H ^ { 1 / 2 } _ \Gamma ( \mathrm { M } ) } , \end{align*}


\begin{align*} S a = - a S \nabla \eta a - r a - a . \end{align*}


\begin{align*} ( \det b ) ^ \frac { 1 } { n } = \frac { 1 } { n } \inf \{ t r ( a \cdot b ) : ~ a \in \mathcal { H } _ n \} , \end{align*}


\begin{align*} \Omega _ + = \{ ( e ^ { t H _ p } ( y , \eta ) , y , \eta ) \ , : \ , p ( y , \eta ) = 0 , y \in \mathcal { U } , \varphi ^ t ( y ) \in \mathcal { U } , t \geq 0 \} . \end{align*}


\begin{align*} o p t _ B ( M ) : = \min \left \{ \sum _ { i = 1 } ^ k x _ i \mid \sum _ { i = 1 } ^ k b _ i x _ i = M , \ \ M , x _ i \in \mathbb { N } , 1 \leq i \leq k \right \} . \end{align*}


\begin{align*} c \colon V _ p \otimes V _ p \to V _ p \otimes V _ p , c ( v _ i \otimes v _ j ) = \lambda v _ { 2 i - j \bmod p } \otimes v _ i , \end{align*}


\begin{align*} \forall r \geq l > 0 \forall i _ 0 , \dots , i _ { r - l } . H ^ l ( U _ { i _ 0 , \dots , i _ { r - l } } , \mathcal F ) = 0 \rlap { . } \end{align*}


\begin{align*} D \left ( \beta _ { x , 1 } \parallel \beta _ { x , 2 } \right ) = \sup _ { n } H _ { \beta _ { x , 1 } \parallel \beta _ { x , 2 } } ( \mathcal { P } _ { x , n } ) , \quad \textrm { w h e r e } H _ { \beta _ { x , 1 } \parallel \beta _ { x , 2 } } ( \mathcal { P } _ { x , n } ) = \sum _ { A \in \mathcal { P } _ { x , n } } \beta _ { x , 1 } ( A ) \log \frac { \beta _ { x , 1 } ( A ) } { \beta _ { x , 2 } ( A ) } . \end{align*}


\begin{align*} \begin{cases} - \Delta w = f ( w ) , \ \ 0 < w < 1 & \ \ \Omega _ { a , b } , \\ w ( x ) \equiv 1 , \ \ & \ \ | x | \le a , \\ w \in H ^ 1 _ 0 ( B _ b ) . \end{cases} \end{align*}


\begin{align*} \Delta \left ( \frac { m } { a } \right ) = \begin{cases} 1 - \tfrac { | m | } { a } , & | m | \leq a \\ 0 , & | m | > a . \end{cases} \end{align*}


\begin{align*} w _ s \vert _ { \partial \Omega } = v _ s \vert _ { \partial \Omega } = v \vert _ { \partial \Omega } . \end{align*}


\begin{align*} q _ { i j } ^ * T _ { \varrho _ { i j } } = q _ { i j } ^ * T _ { \eta _ { i j } } \leq T _ { \psi ^ * \delta _ { i j } } . \end{align*}


\begin{align*} \hat { q } ( \eta ) = \int _ { \mathrm { M } _ 0 } q u _ 1 u _ 2 d x - \int _ { \mathrm { M } _ 0 } q \varrho d x , \end{align*}


\begin{align*} \lim _ { \kappa \to 1 ^ + } \limsup _ { m , n \to \infty } \max _ { n \leq j \leq n ^ \kappa } | u _ { m j } - u _ { m n } | = 0 , \end{align*}


\begin{align*} & \Big \| \Big \{ \sum _ { k \in \mathbb Z } { \mathfrak R } ^ { - k \beta } | D _ k ( a ) | \Big \} \Big \| _ { L ^ { p ' } _ { \omega } } \\ & \le \Big \| \Big \{ \sum _ { k = k _ 0 + 1 } ^ \infty { \mathfrak R } ^ { - k \beta } | D _ k ( a ) | \Big \} \Big \| _ { L ^ { p ' } _ { \omega } } + \Big \| \Big \{ \sum _ { k = - \infty } ^ { k _ 0 } { \mathfrak R } ^ { - k \beta } | D _ k ( a ) | \Big \} \Big \| _ { L ^ { p ' } _ { \omega } } \\ & = : I _ 1 + I _ 2 . \end{align*}


\begin{align*} \mu ( f ) = \min \{ \mu ( a q ) , \mu ( b ) \} . \end{align*}


\begin{align*} | d X | ^ 2 = 4 c ^ 2 \left ( \lambda + \frac { c ^ 2 } { m } \right ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \bar { \Lambda } ( [ \bar { \zeta } _ 0 ] ) > 0 , \\ & \langle \lambda ^ * , \bar { \zeta } _ 0 \rangle > 0 . \end{aligned} \right . \end{align*}


\begin{align*} a \prod _ { i = 1 } ^ n x _ i ^ { \alpha _ i } + b \prod _ { i = 1 } ^ n x _ i ^ { \beta _ i } = c \prod _ { i = 1 } ^ n x _ i ^ { \gamma _ i } , \end{align*}


\begin{align*} ( W ^ { \natural } ; \mathcal { L } : = \mathcal { L } ' \cup \mathcal { L } '' ) , \end{align*}


\begin{align*} \bigg | h ( x ) - \sum ^ { l } _ { s = 0 } \frac { h ^ { ( s ) } ( 0 ) x ^ s } { s ! } \bigg | \leq C | x | ^ { l + 1 } \end{align*}


\begin{align*} a w _ { 2 } ^ { p } - b w _ { 1 } ^ { 2 p } = c \end{align*}


\begin{align*} \min \ , \alpha = \alpha _ \ell = \tilde { \beta } _ i . \end{align*}


\begin{align*} \partial _ t c & = m \Delta \mu \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \ ; \Omega _ T , \\ \mu & = \mathcal { L } _ \varepsilon c + f ^ \prime ( c ) \ ; \ ; \ ; \ ; \Omega _ T , \end{align*}


\begin{align*} y _ { i _ 1 , j _ 1 } \cdots y _ { i _ k , j _ k } = \left ( \prod ^ k _ { \ell = 1 } \frac { c _ { i _ { \ell } } ^ { \tau } } { c _ { i _ { \ell } } ^ { \{ i _ { \ell } \} } } \right ) y _ { \tau , \mathfrak { u } } . \end{align*}


\begin{align*} & \bullet \ , \ , L ^ p _ f ( \R ^ N ) = \Big \{ u : \R ^ N \to \R : \ , \int _ { \R ^ N } | u | ^ p f \ , d x < \infty \Big \} \quad ; \\ [ 0 . 1 c m ] & \bullet \ , \ , L ^ \infty _ f ( \R ^ N ) = \big \{ u : \R ^ N \to \R : \ , \mathrm { e s s \ , s u p } _ { \R ^ N } ( | u | f ) < \infty \} . \end{align*}


\begin{align*} N ( P ) = c \widehat { P } ^ { n - 5 } + O \left ( \widehat { P } ^ { n - 5 - \delta } \right ) , \end{align*}


\begin{align*} \sum _ { n = 1 } ^ { \infty } \sigma _ { - 1 } ( n ) \ , e ^ { - 2 n \alpha } - \sum _ { n = 1 } ^ { \infty } \sigma _ { - 1 } ( n ) \ , e ^ { - 2 n \beta } = \frac { \beta - \alpha } { 1 2 } + \frac { 1 } { 4 } \log \left ( \frac { \alpha } { \beta } \right ) . \end{align*}


\begin{align*} \psi ^ 2 ( \R ^ \bullet f _ * ( V ) ) = \R ^ \bullet f _ * ( \Theta ^ 2 ( \Omega _ f ) ^ { - 1 } \otimes \psi ^ 2 ( V ) ) \end{align*}


\begin{align*} \| u \mid W ^ 1 _ p ( D ) \| = \| u \mid L _ p ( D ) \| + \| \nabla u \mid L _ p ( D ) \| , \end{align*}


\begin{align*} y ' ( t ) = - 5 y ( t ) , y ( 0 ) = 1 , \end{align*}


\begin{align*} B [ \phi ] = \left ( \begin{array} { c c } 3 \phi _ { r } ^ 2 + \phi _ { i } ^ 2 & 2 \phi _ { r } \phi _ { i } \\ 2 \phi _ { r } \phi _ { i } & \phi _ { r } ^ 2 + 3 \phi _ { i } ^ 2 \end{array} \right ) , \end{align*}


\begin{align*} \dfrac { 1 } { 2 } \dfrac { d } { d t } \| \vec U \| _ { P \otimes P _ { \Omega } } ^ 2 + \vec U ^ T ( I _ n \otimes E ^ T P _ { \partial \Omega } N _ { i } E ) { \bf A _ i } \vec U + ( \vec U , { \vec L _ D } ) = 0 . \end{align*}


\begin{align*} \mathcal I _ 2 = \left ( \left . h _ i \sum \limits _ { j = 1 } ^ { s _ { i _ { } } } b _ { i _ { } i j } { \textbf { X } } ^ { \lambda _ j } \ \right | \ i \in I \setminus \{ i _ { } \} \right ) . \end{align*}


\begin{align*} x _ i = \left ( a \prod _ { j = 1 } ^ n u _ j ^ { \alpha _ j } + b \prod _ { j = 1 } ^ n u _ j ^ { \beta _ j } \right ) ^ { z _ i } \left ( c \prod _ { j = 1 } ^ n u _ j ^ { \gamma _ j } \right ) ^ { t _ i } w ^ { - z _ i - t _ i } \cdot u _ i , i = 1 , \dots , n , \end{align*}


\begin{align*} 1 = \| x \| ^ 2 = \phi ( x ^ * x ) = \psi ( x ^ * x ) \end{align*}


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


\begin{align*} \norm { r ' \alpha _ 2 z } = \norm { \theta ' ( z _ 1 t + z _ 2 ) + c z _ 2 / t } . \end{align*}


\begin{align*} \| \phi ( A \otimes B ) \| _ { ( p , k ) } = \| A \otimes B \| _ { ( p , k ) } \hbox { f o r a l l $ A \in M _ m $ a n d $ B \in M _ n $ , } \end{align*}


\begin{align*} 1 \leq e \sum _ { d = 1 } ^ m \left ( \frac { p } { 1 - p } \ , C \ , r \ , \sqrt { q \cdot A } \right ) ^ d . \end{align*}


\begin{align*} \nabla _ { \xi _ i } \ , \xi _ j + \nabla _ { \xi _ j } \ , \xi _ i = 0 , \end{align*}


\begin{align*} \dim { \mathcal G } _ { M , { \mathcal H } } = \dim \mathcal G _ M + \dim T _ M ' - \dim T _ { M } \end{align*}


\begin{align*} \| u \| _ { L ^ { p ( \cdot ) } ( \Omega ) } : = \inf \{ \lambda > 0 : \varrho _ { p ( \cdot ) } ( u / \lambda ) \leq 1 \} . \end{align*}


\begin{align*} \delta _ 1 = \sqrt { \frac { \alpha } { 2 \omega } } , \ \ \delta _ 2 = \sqrt { \frac { \beta } { \omega } } , \ \ \delta _ 3 = \sqrt { \frac { \gamma } { \omega } } . \end{align*}


\begin{align*} 2 { \rm R e } \left ( \widetilde { \rho } _ w ( \theta , \kappa ( \theta ) ) \frac { \partial \kappa } { \partial \theta } \right ) = - \widetilde { \rho } _ { \theta } ( \theta , \kappa ( \theta ) ) , \end{align*}


\begin{align*} \mathcal { B } _ p [ \mathcal { H } ; X ] : = & \left [ \mathcal { B } _ 1 [ \mathcal { H } ; X ] , \mathcal { B } _ \infty [ \mathcal { H } ; X ] \right ] _ { 1 / p } \mbox { ( i n t h e s e n s e o f c o m p l e x i n t e r p o l a t i o n ) } \\ \mathcal { B } _ { p , r } [ \mathcal { H } ; X ] : = & \left [ \mathcal { B } _ 1 [ \mathcal { H } ; X ] , \mathcal { B } _ \infty [ \mathcal { H } ; X ] \right ] _ { 1 / p , r } \mbox { ( i n t h e s e n s e o f r e a l i n t e r p o l a t i o n ) } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ k \lambda _ { i } ^ { \gamma } ( C + D ) = \sum _ { i = 1 } ^ k \lambda _ { i } \big ( ( C + D ) ^ { \gamma } \big ) \geq \mathrm { t r } ( \hat { U } ^ * ( C + D ) ^ { \gamma } \hat { U } ) \end{align*}


\begin{align*} S _ 2 = - \frac { 2 \beta } { \alpha } \Gamma ( \nu + 1 ) { } _ 2 F _ 1 \bigg ( 1 , \nu + 1 ; \frac { 3 } { 2 } ; \frac { \beta ^ 2 } { \alpha ^ 2 } \bigg ) . \end{align*}


\begin{align*} ( t , w ) \longmapsto \left \{ \begin{array} { r l } \widetilde { \rho } ( t , w ) = \frac { 1 } { t ^ s } \rho ( t , t ^ s w ) ; & t \ge 0 , w \in \C \\ { \rm I m } ( e ^ { i \pi ( 1 - \beta _ 1 ) } w ) ; & t \le 0 , w \in \C \end{array} \right . \end{align*}


\begin{align*} F X = \phi X + \omega X \end{align*}


\begin{align*} \phi _ t ( p , z ) = \exp _ z ( ( t - 1 ) d _ z \mathfrak f _ p ) , \forall \ , z \in \mathcal O _ q . \end{align*}


\begin{align*} g ( s ) \ge \sum _ { \ell = 1 } ^ { L } a _ \ell ( s _ \ell ) , \forall s = ( s _ 1 , \ldots , s _ L ) \in \prod _ { \ell \in [ L ] } \mathcal { S } _ \ell . \end{align*}


\begin{align*} E _ j = P _ 1 + P _ { 2 } + \cdots P _ { j } , j = 1 , 2 \cdots \end{align*}


\begin{align*} \Phi ( \sigma ) = \tilde { \delta } ^ { \prime } \mu _ m ^ { \prime } \tilde { \alpha } _ m ^ { \prime } \mu _ { m - 1 } ^ { \prime } \tilde { \alpha } _ { m - 1 } ^ { \prime } \cdots \mu _ 1 ^ { \prime } \tilde { \alpha } _ 1 ^ { \prime } \mu _ 0 ^ { \prime } \end{align*}


\begin{align*} S _ i \left ( f ' \right ) = \{ l - 1 \mid l \in S _ i ( f ) \setminus \{ 0 \} \mbox { a n d } p \nmid l \} . \end{align*}


\begin{align*} \begin{cases} \left ( \sigma _ y ( t ) , \partial D \right ) \to 0 \ \ , \ \ , u ( \sigma _ y ( t ) ) \to 0 \ \ , \ \ , t \to t _ y ^ - , \ \ & \\ | \sigma _ y ( t ) | \to 0 \ \ , \ \ , u ( \sigma _ y ( t ) ) \to u ( 0 ) \ \ , \ \ , t \to t _ y ^ + . & \end{cases} \end{align*}


\begin{align*} H ( \sigma ) = - \sum _ { \{ v , u \} \in E } K ( \sigma _ v , \sigma _ u ) - \sum _ { \{ v , u \} \in E : u \in V , v \in ( \partial V ) _ { \tau } } K ( \sigma _ v , \tau _ v ) - \sum _ { v \in V } U ( \sigma _ v ) , \end{align*}


\begin{align*} \psi ( b _ k d b _ j ^ * ) = \langle \pi ( d ) \pi ( b _ j ) ^ * \xi , \pi ( b _ k ) ^ * \xi \rangle = \psi ( b _ { k } ) \psi ( d ) \psi ( b _ j ) ^ * = \omega ( b _ { k } ) \psi ( d ) \omega ( b _ j ) ^ * \end{align*}


\begin{align*} \mathcal { L } ^ { \Phi } _ { A } u = ( - \Delta ) ^ { \frac { s } { 2 } } f + g \end{align*}


\begin{align*} f ( \alpha _ 0 ) = f ( \beta _ 0 ) = 0 \qquad f > 0 \ , \ \ \ , ( \alpha _ 0 , \beta _ 0 ) . \end{align*}


\begin{align*} \beta _ j - \tilde { \beta } _ j = \nu ( g ' ) - \nu _ j ( g ' ) < \epsilon \mbox { f o r e v e r y } j \in I ' \mbox { w i t h } j \geq j _ 0 . \end{align*}


\begin{align*} X ^ M ( m ) = \tfrac { d } { d r } ( e ^ { r X } \tau ( m ) ) \cdot m \Big | _ { r = 0 } & & m \in M . \end{align*}


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


\begin{align*} D _ { g _ 0 ^ n } & H ^ n \left ( x ^ 0 , \vec { a } , t \right ) \\ & = i m _ n \left [ c _ r k _ { 1 2 } ^ n \left ( \{ x ^ 0 \} _ { \tau = 1 } ^ { r + s } , \vec { a } \right ) \cos ( t c _ r m _ n ) \mp c _ s k _ { 2 1 } ^ n \left ( \{ x ^ 0 \} _ { \tau = 1 } ^ { r + s } , \vec { a } \right ) \cos ( t c _ s m _ n ) \right ] . \end{align*}


\begin{align*} \Pi _ { \psi } [ ( x , 0 ) + ( x ^ { \ast } , 0 ) + ( 0 , k ) ] f ( [ \epsilon , y ] ) = \psi ( \epsilon k + \epsilon \langle x + y , x ^ { \ast } \rangle ) f ( [ \epsilon , x + y ] ) , \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ u \in ( 0 , 1 ) , & \ \ D , \\ u = 1 , \ \ | \nabla u | = c _ 1 , & \ \ \partial \Omega _ 1 , \\ u = 0 , \ \ | \nabla u | = c _ 2 , & \ \ \partial \Omega _ 2 , \\ \end{cases} \end{align*}


\begin{align*} \alpha _ { i ' } = \min _ { \ell \in I _ 0 ( f , i ) } \alpha _ \ell . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \frac { \mathbb { E } ( \overline { S } _ n ) - \mathbb { E } ( \overline { S } _ { n - n ' } ) } { \mathbb { E } ( \overline { S } _ n ) } = 0 \ \ \ \ n ' \ll n \log ^ { - B } n , \end{align*}


\begin{align*} \Theta ( \delta ) = \left [ \min _ { \gamma \in \Sigma ( \delta ) } \int _ { \mathcal { S } } f \ , d \gamma , \max _ { \gamma \in \Sigma ( \delta ) } \int _ { \mathcal { S } } f \ , d \gamma \right ] , \end{align*}


\begin{align*} \vec { \delta } _ { 1 0 2 } = ( 1 + \cos \alpha _ { 1 0 2 } , \sin \alpha _ { 1 0 2 } , 0 ) \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f ( [ \epsilon , w ] ) = e ^ { \tfrac { \pi i ( 1 - \epsilon ) } { 4 } } ( c , \epsilon ) _ { \R } \beta ( d , \epsilon c ) ^ { - 1 } _ { \R } f ( [ \epsilon , w g ^ { s ( \epsilon ) } ] ) . \end{align*}


\begin{align*} x ^ { i , + } = f ^ { i } ( x , u ) \ , , i = 1 , \dots , n \end{align*}


\begin{align*} \dot \gamma ( t ) = \sum _ { i = 1 } ^ k \bar u _ i ( t ) X _ i ( \gamma ( t ) ) , \qquad \norm { \dot \gamma ( t ) } _ { \gamma ( t ) } = \norm { \bar u ( t ) } , \qquad t \in [ 0 , 1 ] . \end{align*}


\begin{align*} | E \Delta F | & \le | { \rm c o v } ( E ) \setminus E | + | { \rm c o v } ( E ) \setminus F | = 2 | { \rm c o v } ( E ) \setminus E | \\ & \le 2 ( \sqrt \pi ) ^ { - 1 } | { \rm c o v } ( E ) | ^ { \frac 1 2 } ( P ( E ) - P ( F ) ) \leq 2 \sqrt { 2 } ( P ( E ) - P ( F ) ) . \end{align*}


\begin{align*} \nu ( a _ p ^ * a _ q ) = \delta ( p - q ) \Big ( \delta ( p - h _ 1 ) + \ldots + \delta ( p - h _ n ) + \delta ( p - p _ 1 ) + \ldots + \delta ( p - p _ n ) \Big ) \ . \end{align*}


\begin{align*} \aligned m _ \nu ( a , b ) = J _ \nu ( u _ \nu , v _ \nu ) \geq \bar { f } \Bigl ( \int _ { \mathbb { R } ^ N } ( I _ \mu * | u _ \nu | ^ { p } ) | v _ \nu | ^ { q } \Bigr ) . \endaligned \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } S _ { \omega , \mathbf { c } } ( U _ n ) = \mu _ { \omega , \mathbf { c } } , \ \ \lim _ { n \rightarrow \infty } K _ { \omega , \mathbf { c } } ( U _ n ) = 0 . \end{align*}


\begin{align*} \| \mathbf { v } \| _ { \mathbb { V } _ p } : = \| \nabla _ z \mathbf { v } \| _ { \mathbb { L } _ p ^ 1 } + \| \mathbf { v } \varrho \| _ { \mathbb { L } _ p ^ 1 } < \infty , \end{align*}


\begin{align*} m = \frac { s } { v _ i ^ d } \frac { ( q ^ * ) ^ d } { q } ( U ' ) ^ d . \end{align*}


\begin{align*} \Omega _ u : = \bigcup _ { \delta > 0 } \Omega _ u ^ { ( \delta ) } \end{align*}


\begin{align*} 2 ( - 1 ) ^ j \binom { n - k - j } { k - j } , \end{align*}


\begin{align*} b _ { \varepsilon ' , \hat { T } } ( ( \hat { \ell } ) _ { \sf p } ) + b _ { \varepsilon '' , \hat { T } } ( ( \hat { \ell } ) _ { \sf p } ) = b _ { \varepsilon ' , \hat { T } } ( ( \hat { \ell } ) _ { \sf q } ) + b _ { \varepsilon '' , \hat { T } } ( ( \hat { \ell } ) _ { \sf q } ) \end{align*}


\begin{align*} \begin{aligned} \rho ( u ) = \log ( \theta ) + B ( u ) - \frac { 1 } { 2 } u - \int _ 0 ^ u \eta e ^ { \rho ( v ) } d v \end{aligned} \end{align*}


\begin{align*} \frac { 1 } { 2 } \Delta ( 1 + w ) ^ 2 = - \frac { 1 } { m - 1 } S ^ \varphi w ( 1 + w ) + | \nabla w | ^ 2 \ , , \end{align*}


\begin{align*} \langle u _ k , \psi \rangle = \int _ { \tilde { \gamma } _ k ^ 3 } \hat { \psi } ( \lambda ) \frac { d } { d \lambda } \log \zeta _ 1 ( \lambda ) d \lambda , \end{align*}


\begin{align*} a _ 0 \ \equiv \ b _ 0 + 2 \ \equiv \ \begin{cases} 3 , & , \\ 8 , & . \end{cases} \end{align*}


\begin{align*} \Pr ( X _ t \le k ) = ( 1 - \frac { 1 } { n } ) ^ t \sum _ { j = 0 } ^ k \Pr ( \widetilde { X } _ t = k \ , \vert \ , \widetilde { X } _ 0 = j ) , \end{align*}


\begin{align*} \hat \omega = & \ , \sqrt { \frac { \alpha ^ 2 } { 2 m n } + \frac { \lambda } { 2 n } } \hat { J } , \\ \hat { R } _ { i j } = & \ , \left ( \frac { \alpha ^ 2 } { m n } + \left ( 1 + \frac { 1 } { n } \right ) \lambda \right ) \hat { g } _ { i j } \end{align*}


\begin{align*} ( T ^ T \tilde F ) ^ T \Lambda ^ { - 1 } ( T ^ T \tilde F ) = \begin{bmatrix} \tilde F _ n \\ \tilde F _ { \tau } \\ - \tilde F _ n / u _ n \end{bmatrix} ^ T \begin{bmatrix} - 1 / u _ n & 0 & 0 \\ 0 & - 1 / u _ n & 0 \\ 0 & 0 & u _ n \end{bmatrix} \begin{bmatrix} \tilde F _ n \\ \tilde F _ { \tau } \\ - \tilde F _ n / u _ n \end{bmatrix} = - \tilde F _ { \tau } ^ 2 / u _ n \end{align*}


\begin{align*} \phi _ { \ell } ( z _ \ell ' , x ' ; x _ \ell ) & = a _ \ell z _ \ell ' - \lambda _ \ell \ \begin{bmatrix} z _ \ell ' \\ x ' - x _ \ell \end{bmatrix} ^ { \top } Q _ { \ell } \begin{bmatrix} z _ \ell ' \\ x ' - x _ \ell \end{bmatrix} . \end{align*}


\begin{align*} Z _ 0 ( \tau _ N ) & = Y e ^ { \lambda \tau _ N } + \omega _ 0 \left ( \tau _ N \right ) \\ & = N e ^ { \lambda ( \tau _ N - t _ N ) } + \omega _ 0 \left ( \tau _ N \right ) , \end{align*}


\begin{align*} \lim _ { N \to \infty } \frac { 1 } { N + 1 } \sum _ { n = 0 } ^ { N } s ^ { n } . \left ( f \circ \xi \right ) \left ( u , v , x _ { 0 } \right ) = \mathbb { E } _ { \lambda } \left ( \tilde { f } \left ( u , \cdot \right ) | \mathcal { F } ^ { s } \right ) ( x _ { 0 } ) , \end{align*}


\begin{align*} \psi \equiv \lim _ { n \to \infty } \psi _ n = \sum _ { k = 1 } ^ \infty \frac { ( 2 r ) ^ { k - 1 } } { ( 2 r + 1 ) ^ k } \phi _ k , \end{align*}


\begin{align*} a \prod _ { i = 1 } ^ n x _ i ^ { \alpha _ i } = b \prod _ { i = 1 } ^ n x _ i ^ { \gamma _ i } , \end{align*}


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


\begin{align*} g ( \alpha ) = \frac { a \alpha + b } { c \alpha + d } , \ , \ \ , \ g = \small { \left ( \begin{array} { c c c } a & b \\ c & d \end{array} \right ) } . \end{align*}


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


\begin{align*} \dfrac { 1 } { q ( x ) } = \dfrac { 1 } { q _ { 1 } ( x ) } + \cdots + \dfrac { 1 } { q _ { m } ( x ) } \mbox { f o r a . e . } \ x \in \mathbb { R } ^ { n } . \end{align*}


\begin{align*} K _ 1 & = T ( \phi ) - T ( \phi _ 0 ) T ( \phi _ 1 ) \cdots T ( \phi _ R ) \\ K _ 2 & = T ( \tilde { \phi } ) - T ( \tilde { \phi } _ 0 ) T ( \tilde { \phi } _ 1 ) \cdots T ( \tilde { \phi } _ R ) \end{align*}


\begin{align*} 0 = G ^ \prime ( \widetilde { y } , \widetilde { z } ) \widetilde { g } ( \widetilde { y } , \widetilde { z } ) = F ^ \prime ( \widetilde { y } ) \widetilde { g } _ Y ( \widetilde { y } , \widetilde { z } ) - \widetilde { g } _ Z ( \widetilde { y } , \widetilde { z } ) , \end{align*}


\begin{align*} b _ i = w _ { \lambda ^ i } : = a + \sum _ { j = 1 } ^ { k } w _ { \lambda _ j ^ i } , \end{align*}


\begin{align*} \beta _ K = \prod _ { j = 1 } ^ n \frac { k _ j ^ 2 } { \mu _ j ^ 2 } , K = ( k _ 1 , \ldots , k _ n ) \in \mathbb { N } ^ n . \end{align*}


\begin{align*} M _ a ( f ) : = \frac { 1 } { a ^ d } \int _ { Q _ a } T _ t ( f ) d t . \end{align*}


\begin{align*} \partial _ x ^ \alpha \big | _ { x = 0 } f ( \xi + B x ) = \partial _ { B , \xi } ^ \alpha f ( \xi ) . \end{align*}


\begin{align*} \boldsymbol { \Phi } = { \rm d i a g } ( e ^ { j \phi _ { 1 , n _ t } } , \cdots , e ^ { j \phi _ { l , n _ t } } , \cdots , e ^ { j \phi _ { L , n _ t } } ) , \end{align*}


\begin{align*} X ^ { \star } = \{ f \in H ^ * : f ( X ) \subseteq R \} . \end{align*}


\begin{align*} \div v = ( \delta _ { j i } - a _ { j i } ) \partial _ j v _ i . \end{align*}


\begin{align*} I - S ^ T S - ( R ^ T S ) ^ T \left ( \sum ^ { \infty } _ { k = 0 } ( R ^ T R ) ^ k \right ) ( R ^ T S ) \geq 0 . \end{align*}


\begin{align*} \int _ { \mathbb H ^ n } \frac { | y | _ h ^ { - Q / p } } { \max ( 1 , | y | _ h ^ Q ) } d y = \frac { \omega _ Q Q } { ( Q - Q / p ) Q / p } , \end{align*}


\begin{align*} \begin{aligned} \overline { \mathrm { m d i m } } _ M ( Y ^ \mathbb { N } \times X , F , D ) = \overline { \mathrm { m d i m } } _ { I _ \psi ( F ) } ( F , D ) & \le \overline { \mathrm { m d i m } } _ { Y ^ \mathbb { N } \times I _ \psi ( F ) } ( F , D ) \\ & \le \overline { \mathrm { u m d i m } } _ { Y ^ \mathbb { N } \times I _ \psi ( F ) } ( F , D ) . \end{aligned} \end{align*}


\begin{align*} \Pi _ { \psi } [ - 1 , 0 ] f ( \epsilon , x + x ^ { \ast } ) = f ( - \epsilon , x - x ^ { \ast } ) , \end{align*}


\begin{align*} x v = \lambda v , \end{align*}


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


\begin{align*} \mathcal { P } ( X _ \eta ) = \mathcal { P } ( X _ { \eta ' } ) = \mathcal { P } ( X _ { \eta ^ * } ) . \end{align*}


\begin{align*} M = \{ u \in \mathcal { U } : \ G ( u ) \in \mathcal { K } \} . \end{align*}


\begin{align*} I - \mathfrak { z } _ 1 T _ { 1 , 0 } ^ * - \mathfrak { z } _ 2 T _ { 2 , 0 } ^ * = \begin{pmatrix} 1 & \frac { 1 } { 2 } ( \mathfrak { z } _ 1 + \mathfrak { z } _ 2 ) & \frac { 1 } { 2 } ( \mathfrak { z } _ 1 + \mathfrak { z } _ 2 ) \\ 0 & 1 + \frac { 1 } { 2 } \mathfrak { z } _ 2 & - \frac { 1 } { 2 } \mathfrak { z } _ 2 \\ 0 & - \frac { 1 } { 2 } \mathfrak { z } _ 1 & 1 + \frac { 1 } { 2 } \mathfrak { z } _ 1 \end{pmatrix} . \end{align*}


\begin{align*} f _ { \textbf { Q } } : = f \left ( \textbf { Q } \right ) = \sum _ { j = 1 } ^ s b _ j \textbf { Q } ^ { \lambda _ j } \mbox { a n d } f _ { \tilde { \textbf { Q } } } : = f \left ( \tilde { \textbf { Q } } \right ) = \sum _ { j = 1 } ^ s b _ j \tilde { \textbf { Q } } ^ { \lambda _ j } . \end{align*}


\begin{align*} \mathbf { 1 } _ H ^ G = \sum _ { i = 1 } ^ t m _ i \phi _ i , \end{align*}


\begin{align*} \begin{cases} \partial _ t m = \nabla \cdot \left ( D ( m ) \nabla m - \chi f ( m ) \nabla c \right ) + M ( m ) , \\ \partial _ t c = \Delta c - c + d + m , \\ \partial _ t d = h ( m ) \left ( 1 - d \right ) , \end{cases} x \in \Omega , \ , t > 0 , \end{align*}


\begin{align*} s ' ( \nu ) = - \frac { \frac { \partial F } { \partial \mu } } { \frac { \partial F } { \partial s } } \Bigl | _ { \mu = \nu } = - \frac { ( \gamma _ p + \gamma _ q ) B } { \nu ( \gamma _ p + \gamma _ q ) ( \gamma _ p + \gamma _ q - 2 ) B + ( 2 2 ^ * _ \mu - 2 ) C } , \end{align*}


\begin{align*} C ( p , p ' , t , \tau , \rho ) = \frac { \bar C ( p , p ' , \rho ) } { \tau - t } . \end{align*}


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


\begin{align*} \mathcal { G } ^ M _ { r } ( K , \gamma ) \cap B ^ + _ { \rho ( r ) } ( q ) = C N ^ + _ { r } ( \xi ) \cap B ^ + _ { \rho ( r ) } ( q ) , \end{align*}


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


\begin{align*} \Psi & : Z \to M : = G / Q \times _ { \beta } \left ( \Omega _ { 0 } \times _ { \sigma } M _ { F } \right ) \\ & \left ( \left ( y , r , H \right ) , A \right ) \mapsto \left ( y , r , \left ( H , \bar { \psi } _ { H } \left ( r ^ { - 1 } \bar { p } _ { k } \left ( \tau ( y ) ^ { - 1 } A \right ) \right ) \right ) \right ) , \end{align*}


\begin{align*} \Pr [ \hat { M } = M ] \leq \int _ { \mathbb { R } ^ { n \times n } } \max _ { m } \frac { s ! } { ( n + s ) ! } f ( g - \mu m ) d g \end{align*}


\begin{align*} \langle X _ N ( L , \alpha ) ^ 2 \rangle = \int _ 0 ^ 1 \big | X _ N ( L , \alpha ) \big | ^ 2 \ , d \alpha \ll _ \varepsilon L N ^ { - 3 + \varepsilon } E _ N ( \mathcal { A } ) \end{align*}


\begin{align*} \gamma ( t ) G ( t , \sigma ( t ) ) = \gamma ( 0 ) G ( 0 , \sigma ( 0 ) ) 0 \leq t \leq t _ 0 , \end{align*}


\begin{align*} h ( H \times \{ 0 , 1 \} ^ \Z , \widetilde { R } ) = m _ H ( W \setminus \overline { W } ) = \overline { d } - d ^ * = h ( X _ \eta ) \end{align*}


\begin{align*} \eta = \int _ { G / H } \boldsymbol { \beta } _ { \eta } ( w ) d \nu _ { H } ( w ) = \int _ { G / H } g . \lambda d \nu _ { H } ( g H ) . \end{align*}


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


\begin{align*} z ^ { \vec x } _ j : = \begin{cases} ( M \vec x ) _ { i } - \max _ k ( M \vec x ) _ k & j \in J _ i , \\ 0 & j \notin \cup _ { i \in [ n ] } J _ i . \end{cases} \end{align*}


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } U _ { \varepsilon } ( t , \xi ) = i \langle \xi \rangle A _ { \varepsilon } ( t ) U _ { \varepsilon } ( t , \xi ) + i \langle \xi \rangle ^ { - 1 } Q _ { \varepsilon } ( t ) U _ { \varepsilon } ( t , \xi ) + F _ { \varepsilon } ( t , \xi ) , \xi \in \mathcal { I } _ { \hbar } , \\ U _ { \varepsilon } ( 0 , \xi ) = U _ { 0 } ( \xi ) , \xi \in \mathcal { I } _ { \hbar } , \end{array} \right . \end{align*}


\begin{align*} - \partial _ { x } ^ 2 \phi + \left ( \omega - \frac { c ^ 2 } { 4 } \right ) \phi + \frac { c } { 2 } \phi ^ 3 - \frac { 3 } { 1 6 } \phi ^ 5 = 0 . \end{align*}


\begin{align*} & \alpha ( f ( u ) ) = f ( \beta ( u ) ) , \\ & f ( \rho ( x , f ( u ) ) v ) = [ x , f ( u ) , f ( v ) ] , \end{align*}


\begin{align*} h ( X _ \eta ) = 0 \iff \mathcal { P } ( X _ \eta ) = \{ \nu _ { \eta } \} \iff X _ \eta \end{align*}


\begin{align*} \lim _ { t \uparrow T _ { \max } } \left \| \psi ( t ) \right \| _ { H ^ 2 _ N } = \infty . \end{align*}


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


\begin{align*} \langle S _ { \xi \chi } f , g \rangle & = \langle \int _ { \Theta } v ( w ) s ( w ) \pi _ { G ( w ) } \xi _ { w } ^ { \ast } \chi _ { w } \pi _ { F ( w ) } f d \mu ( w ) , g \rangle \\ & = \int _ { \Theta } v ( w ) s ( w ) \langle \chi _ { w } \pi _ { F ( w ) } f , \xi _ { w } \pi _ { G ( w ) } g \rangle d \mu ( w ) . \end{align*}


\begin{align*} E _ { \varepsilon } ( t , \xi ) : = ( S _ { \varepsilon } ( t ) U _ { \varepsilon } ( t , \xi ) , U _ { \varepsilon } ( t , \xi ) ) , \end{align*}


\begin{align*} \| T _ { \eta } f \| _ { L ^ p ( x ^ \gamma d x ) } ^ p & \lesssim \int _ 0 ^ \infty x ^ { - 1 - \tau + \gamma } \int _ x ^ \infty \abs { f ( z ) } ^ p z ^ { \tau } \ , d z \ , d x = \int _ 0 ^ \infty \abs { f ( z ) } ^ p z ^ { \tau } \int _ 0 ^ z x ^ { - \tau - 1 + \gamma } \ , d x \ , d z \\ & \simeq \| f \| _ { L ^ p ( x ^ \gamma d x ) } ^ p , \end{align*}


\begin{align*} u v ' + v u ' = u ' v + v ' u = - ( u , v ) \in \mathbb { F } \subseteq \mathcal { C } \mathrm { \ f o r \ e v e r y \ } u \mathrm { \ a n d \ } v \mathrm { \ i n \ } V . \end{align*}


\begin{align*} y _ { B 1 } = \sqrt { 1 - \alpha } h _ { A B } x + \sqrt { \alpha } h _ { C B } z _ d + n _ { B 1 } , \end{align*}


\begin{align*} a _ h = \mathcal { L } ^ 2 \left ( \left \{ f _ h \neq g _ h \right \} \cap B _ 1 \right ) \end{align*}


\begin{align*} L u _ n + C _ n u _ n = f = L u _ f + K f \ , , \end{align*}


\begin{align*} & \mathrm { i n t } ( F _ 0 ( \mathcal G ) ) = \mathrm { i n t } ( O _ S [ g _ 1 ] ) = \mathrm { i n t } ( \partial ( O _ S [ g _ 1 ] ) + \sum _ { s \in S } T _ { \tilde s } ( \mathrm { S p a n } _ { \mathbb C } \{ g _ n \ , | \ , n \geq 0 \} ) ) \\ & = \mathrm { i n t } ( \partial ( O _ S [ g _ 1 ] ) ) + \sum _ { s \in S , n \geq 0 } \mathbb C \cdot \mathrm { i n t } ( T _ { \tilde s } ( g _ n ) ) \subset F _ 0 ( \mathcal G ) + \sum _ { s \in S , n \geq 0 } F _ 1 ( \mathcal G ) = F _ 1 ( \mathcal G ) \end{align*}


\begin{align*} \partial ^ 2 _ v \psi _ { \mu _ N } = \frac 1 { N + 1 } \bigg ( - 1 + \sum _ { i = 2 } ^ { N + 1 } \partial ^ 2 _ v \psi _ { \delta _ { x _ i } } ( x _ 1 ) \bigg ) \geq \frac { - 1 + ( \alpha - 2 ) K _ N } { N + 1 } \ , . \end{align*}


\begin{align*} Y _ { t } = \xi + \int _ { t } ^ { T } f \left ( s , Y _ { s } , Z _ { s } \right ) \mathrm { d } s - \int _ { t } ^ { T } Z _ { s } \mathrm { d } W _ { s } \end{align*}


\begin{align*} \mathcal { R } _ \varepsilon c ( x ) : = \int _ { \Omega ^ c } J _ \varepsilon ( | x - y | ) \big ( c ( x ) - \tilde { c } ( y ) \big ) \ : y \end{align*}


\begin{align*} \theta ( I ) = \int _ I Q ( E ) d \rho ( E ) . \end{align*}


\begin{align*} y \left ( x \right ) & = \widetilde { y } \left ( \frac { x - 2 } { 3 } \right ) = 9 \left ( \frac { x - 2 } { 3 } \right ) ^ { 2 } + 1 2 \left ( \frac { x - 2 } { 3 } \right ) + 4 \\ & = x ^ { 2 } \end{align*}


\begin{align*} \tilde { y } _ { T _ { ( k , m , i , n ) } } ^ { \mathcal { K } ' } \circ \big ( \tilde { y } _ { t } ^ { \mathcal { K } ' } \big ) ^ { - 1 } = \tilde { y } _ { T _ { ( k , m , i , n ) } } ^ { \mathcal { K } } \circ \big ( \tilde { y } _ t ^ { \mathcal { K } } \big ) ^ { - 1 } . \end{align*}


\begin{gather*} \mathrm { e r f } ( x ) = \int _ { - \infty } ^ x e ^ { - y ^ 2 } \ ; d y , \\ C _ 0 = \mathrm { e r f } \left ( 0 \right ) , \\ a = 1 - 2 ^ { - \lfloor k / 2 \rfloor } . \end{gather*}


\begin{align*} & \upsilon _ 0 = \upsilon \wedge \frac { ( ( \Lambda _ 0 ) ^ 3 - \widehat { r } _ 0 ^ 3 ) ^ 2 } { 3 6 C _ \varepsilon ^ 2 \Lambda _ 0 ^ 4 } . \end{align*}


\begin{align*} u = \sum _ { k \ge 1 } \frac { ( f | \phi _ q ^ k ) } { \lambda _ q ^ k - \lambda } \phi _ q ^ k , \end{align*}


\begin{align*} L _ { \omega , \mathbf { c } } ( U _ n ) = 6 S _ { \omega , \mathbf { c } } ( U _ n ) - 2 K _ { \omega , \mathbf { c } } ( U _ n ) \ \rightarrow \ 6 \mu _ { \omega , \mathbf { c } } . \end{align*}


\begin{align*} x ^ n + y ^ m = z ^ k \end{align*}


\begin{align*} \vec U ^ T ( I _ 2 \otimes P _ x ) \vec U _ t + \vec U ^ T ( I _ 2 \otimes Q _ x ) { \bf A } \vec U ) + \vec U ^ T ( I _ 2 \otimes P _ x ) { \bf A } ^ T { \bf D _ x } ( \vec U ) = 0 . \end{align*}


\begin{align*} h ( r e ^ { i \theta } ) = \frac { 1 } { 2 \pi } \langle h _ b , P _ r ( \theta - \cdot ) \rangle . \end{align*}


\begin{align*} K _ { \omega , \mathbf { c } } ( V ) = 2 S _ { \omega , \mathbf { c } } ( V ) + N ( V ) < 0 . \end{align*}


\begin{align*} f ' ( \lambda ) = 2 \lambda L ( U ) + \left ( \frac { d } { 2 } + 1 \right ) \lambda ^ { \frac { d } { 2 } } N ( U ) + \mathbf { c } \cdot \mathbf { P } ( U ) . \end{align*}


\begin{align*} \| 1 + d Z \| _ { q } ^ { q } \ge 1 + \binom { \lfloor q \rfloor } { 2 } d ^ 2 \ge 1 + \frac { q ^ 2 } { 6 } d ^ 2 . \end{align*}


\begin{align*} \| w _ { \varepsilon } ( t , \cdot ) \| ^ { 2 } _ { \mathrm { H } _ { \mathcal { H } _ { \hbar , V } } ^ { 1 + s } } + \| \partial _ { t } w _ { \varepsilon } ( t , \cdot ) \| ^ { 2 } _ { \mathrm { H } _ { \mathcal { H } _ { \hbar , V } } ^ { s } } \lesssim \varepsilon ^ { - 2 L _ { 1 } - L _ { 2 } - 1 } \varepsilon ^ { 2 L _ { 1 } + L _ { 2 } + 1 + q } = \varepsilon ^ { q } , q \in \mathbb { N } _ { 0 } , \end{align*}


\begin{align*} p _ { a } ( D ^ { n } \varphi ) : = \max _ { \alpha \in \mathcal { A } } \sup _ { x \in ( l _ \alpha , r _ \alpha ) } \max \{ | D ^ { n + 1 } \varphi ( x ) ( x - l _ \alpha ) ^ { 1 + a } | , | D ^ { n + 1 } \varphi ( x ) ( r _ \alpha - x ) ^ { 1 + a } | \} \end{align*}


\begin{align*} \mu _ { \omega , \sqrt { \omega } \ \mathbf { c } _ 0 } = \omega \mu _ { 1 , \mathbf { c } _ 0 } = \omega S _ { 1 , \mathbf { c } _ 0 } ( \Phi ) = \omega ( Q ( \Phi ) - E ( \Phi ) ) . \end{align*}


\begin{align*} \Psi \left [ \sigma \curvearrowright ^ { \mathbf { n } } \tau \right ] = \tfrac { 1 } { \mathbf { n } ! } \Psi \left [ \sigma \right ] D ^ { ( \mathbf { n } ) } \Psi \left [ \tau \right ] . \end{align*}


\begin{align*} X _ 1 = \frac { \partial } { \partial x _ 1 } , \enspace \ldots , \enspace X _ { N - 1 } = \frac { \partial } { \partial x _ { N - 1 } } , \enspace X _ { N } = x _ { 1 } ^ \beta \frac { \partial } { \partial x _ { N } } \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) , \ \ 0 \le u \le 1 & \ \ B _ R ( 0 ) \backslash Q _ r ( 0 ) , \\ u \equiv 1 , \ \ & \ \ Q _ r ( 0 ) . \end{cases} \end{align*}


\begin{align*} \langle B _ l ' ( y ' ) x \Omega _ { \rho } , \Omega _ { \rho } \rangle _ { \rho } = \langle y ' B _ l ( x ) \Omega _ { \rho } , \Omega _ { \rho } \rangle _ { \rho } , x \in M , y ' \in M ' . \end{align*}


\begin{align*} \mathcal { N } _ { \Lambda _ { L , L ' } } = \bigcap _ { E \in \sigma ( H _ { \omega , \Lambda _ { L ' } } ) } \mathcal { N } _ { \Lambda _ { L , L ' } } ^ { ( E ) } \in \mathcal { F } _ { \Lambda _ L } \end{align*}


\begin{align*} \widetilde { \gamma } _ { \xi } = \frac { 1 - a \xi } { \xi - a } \ , \gamma _ { \xi } . \end{align*}


\begin{align*} a ^ * a - a a ^ * = 2 i ( b c - c b ) \end{align*}


\begin{align*} y = \mathcal { L } u , \end{align*}


\begin{align*} \tilde { \mathcal { L } } ( V ) = \tau ^ { - 1 } \langle c , A \psi - \phi \rangle + \tau ^ { - 1 } \sum _ { i = 1 } ^ m \langle c , B _ i \psi \rangle u _ i < 0 . \end{align*}


\begin{align*} \varphi ^ + ( F _ I ) = \tilde F _ I ( - X ) = ( - 1 ) ^ { | I | } \tilde F _ { \bar I ^ \sim } , \varphi ^ - ( F _ I ) = V \varphi _ + ( \tilde F _ I ) . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \Delta u + V ( \epsilon x ) u = \lambda u + u \log u ^ 2 , \hbox { i n } \mathbb { R } ^ N , \\ & \int _ { \mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } . \end{aligned} \right . \end{align*}


\begin{align*} S ^ { - 1 } \left ( \begin{bmatrix} \sqrt { | u _ n | } & 0 \\ 0 & \sqrt { | u _ n | } \end{bmatrix} \begin{bmatrix} u _ n + p / u _ n \\ u _ { \tau } \end{bmatrix} - \begin{bmatrix} R _ 1 \\ R _ 2 \end{bmatrix} p / \sqrt { | u _ n | } \right ) = G \end{align*}


\begin{align*} t ^ n = \sum _ { k = 0 } ^ { n } S _ B [ n , k ] ( t ) _ { k , q } ^ B , \end{align*}


\begin{align*} Q _ { m } u ( k ) : = \left \{ \begin{array} { c c } Q u ( k ) , & | k | \leq m , \\ 0 , & \mathrm { o t h e r w i s e } . \end{array} \right . \end{align*}


\begin{align*} \mathcal { Y } \mathcal { T } \mathbf { u } = \mathcal { Y } \mathbf { f } , \end{align*}


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


\begin{align*} 2 \nabla f \cdot \nabla \Delta f = \Delta | \nabla f | ^ 2 - 2 | D ^ 2 f | ^ 2 , \end{align*}


\begin{align*} \eta ( \sigma ) ^ * ( y _ { \tau , \mathfrak { u } } ) = \left ( \prod ^ k _ { \ell = 1 } \frac { c _ { i _ { \ell } } ^ { \sigma } } { c _ { i _ { \ell } } ^ { \tau } } \right ) x _ { \tau , \mathfrak { u } } \in H ^ * ( ( \underline { X } , \underline { \ast } ) ^ { \sigma } ; R ) . \end{align*}


\begin{align*} \mathbf { v } ( t , z ) = 0 , \forall z \in U . \end{align*}


\begin{align*} W ( q _ - , q _ + ) ( z ) + 2 \zeta q _ - ( z ) q _ + ( z ) = \Lambda ( z ) , \end{align*}


\begin{align*} L ( \Phi ^ { \lambda } ) = \lambda ^ 2 L ( \Phi ) , \ \ N ( \Phi ^ { \lambda } ) = \lambda ^ { \frac { d } { 2 } + 1 } N ( \Phi ) , \ \ M ( \Phi ^ { \lambda } ) = M ( \Phi ) , \ \ \mathbf { P } ( \Phi ^ { \lambda } ) = \lambda \mathbf { P } ( \Phi ) , \end{align*}


\begin{align*} B _ n ( t ) = \frac { 1 } { \sqrt { n } } \Big ( \sum _ { k = 1 } ^ { [ n t ] } X _ k + ( n t - [ n t ] ) X _ { [ n t ] } \Big ) \ , . \end{align*}


\begin{align*} \lim _ { p \rightarrow 0 ^ { + } , p ^ { \prime } \rightarrow \infty } \tilde { \zeta } _ { p , p ^ { \prime } } ( s , c ) = \sum _ { m \neq 0 , \ , n \in \mathbb { Z } } \frac { ( - 1 ) ^ { m } } { \left ( m ^ { 2 } + c \ , n ^ { 2 } \right ) ^ { s } } , \end{align*}


\begin{align*} g ( z ) : = | A | z ^ { n + m } + | B | \overline { z } ^ m - | C | , z \in \mathbb { C } , \end{align*}


\begin{align*} I ( A , B ) : = H ( A ) - H ( A | B ) = H ( B ) - H ( B | A ) = H ( A ) + H ( B ) - H ( A , B ) . \end{align*}


\begin{align*} \abs { v _ h } = t _ h \mbox { a . e . i n } B _ 1 , \end{align*}


\begin{align*} \Bigl \langle \sum _ { \gamma \in D _ L } a _ { \gamma } \mathfrak { e } _ \gamma , \sum _ { \delta \in D _ L } b _ \delta \mathfrak { e } _ \delta \Bigr \rangle _ L = \sum _ { \gamma \in D _ L } a _ \gamma \overline { b _ \gamma } . \end{align*}


\begin{align*} \boldsymbol { K } _ c ( \mu , \nu ) = \inf _ { \pi \in \Pi ( \mu , \nu ) } \int _ { \mathcal { X } \times \mathcal { X } } c \ , d \pi . \end{align*}


\begin{align*} \widetilde { \rho } _ j ( \xi , w ) = { \rm I m } \left ( e ^ { i \pi ( 1 - \beta _ j ) } \frac { ( \xi - 1 ) ^ { \delta _ 1 } ( \xi + 1 ) ^ { \delta _ { - 1 } } } { T _ j ( \xi ) ^ { \delta _ j } } w \right ) . \end{align*}


\begin{align*} \frac { P ( Z _ 0 ( \Delta \ell _ 1 ) = n ) } { P ( W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 , Z _ 0 ( \Delta \ell _ 1 ) = n ) } & = \frac 1 { P ( W ^ k _ { \ell _ 1 \Delta , t } ( j ) = 1 | Z _ 0 ( \Delta \ell _ 1 ) = n ) } \\ & = \frac 1 { n \nu \Delta p ^ k _ { j , + } ( t - \Delta \ell _ 1 ) } , \end{align*}


\begin{align*} R _ { \theta _ 0 } ( z _ 1 , z _ 2 , u ) & = ( z _ 1 \cos \theta _ 0 - z _ 2 \sin \theta _ 0 , z _ 1 \sin \theta _ 0 + z _ 2 \cos \theta _ 0 , u ) ; \\ \eta ( z _ 1 , z _ 2 , u ) & = ( \bar z _ 1 , \bar z _ 2 , u ) . \end{align*}


\begin{align*} \overline { \Pi } _ { \psi } ( p ) A ( [ \epsilon , x ] ) = a ^ { \tfrac { 1 } { 2 } } e ^ { - ( \det p ) \epsilon \pi x ^ 2 a ^ 2 } \cdot e ^ { \pi i \epsilon ( \det p ) x ^ 2 a b } = a ^ { \tfrac { 1 } { 2 } } e ^ { i \epsilon \pi x ^ 2 z _ p } . \end{align*}


\begin{align*} \left ( \lfloor y ^ 2 + f y + r \rfloor = r \right ) \wedge \left ( \lfloor y \rfloor \right ) = 0 . \end{align*}


\begin{align*} \limsup _ { h \to + \infty } { ( _ h ) } = c \left ( \alpha ( \rho ) - \alpha ( \rho ' ) \right ) , \end{align*}


\begin{align*} \int _ { B _ { 1 } } { b } ^ { i j , k l } f _ { i j } [ \tau ^ { 4 } f ] _ { k l } d x = 0 , \end{align*}


\begin{align*} \delta _ { 0 } = \delta _ { 0 } \left ( N , \varepsilon , \Omega , T , c , \rho \right ) \in \left ( 0 , 1 \right ) , \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle Q ( \xi _ n , - \eta _ n ) = Q \left ( ( u ^ * _ n ) _ { \beta _ n } , - ( v ^ * _ n ) _ { \beta _ n } \right ) = 0 . \end{array} \right . \end{align*}


\begin{align*} g _ { i j } = \rho ^ 2 u _ i u _ j + \phi ^ 2 s _ { i j } . \end{align*}


\begin{align*} \Psi ( \nu , \mu ) ( x ) = \frac { 1 } { Z ( \nu , \mu ) } \exp \left ( { - \frac { 2 } { \sigma ^ 2 } \frac { \delta F } { \delta \nu } ( \nu , \mu , x ) - U ^ { \pi } ( x ) } \right ) , \end{align*}


\begin{align*} M _ { r , c ( i ) } \in \begin{cases} - \infty & \emph { i f } \ , \ , r \neq i , \\ \R & \emph { i f } \ , \ , r = i . \end{cases} \end{align*}


\begin{align*} 0 = \int _ M \psi ( q _ 2 - q _ 1 ) u _ 2 u _ 1 d x + \int _ M [ \Delta , \psi ] u u _ 1 d x . \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = \mathfrak { S } ( f _ k | _ { \tilde { G } } ) \geq \tilde { k } + \tilde { r } - 1 - \ell ' + \ell _ + , \end{align*}


\begin{align*} \min \limits _ { u \in H _ 0 ^ 1 ( \Omega ) } \int _ \Omega \nabla u \cdot \nabla u \ d x \mbox { s . t . } \ \ \| u \| _ { L ^ 2 ( \Omega ) } ^ 2 = 1 . \end{align*}


\begin{align*} ( a ) K & = K \left ( x , t \right ) \\ ( b ) K & = K \left ( y \left ( t \right ) , t \right ) \\ ( c ) K & = K \left ( y ^ { \prime } \left ( t \right ) , t \right ) \\ ( d ) K & = K _ { 1 } \left ( x \right ) K _ { 2 } \left ( y \left ( t \right ) , t \right ) . \end{align*}


\begin{align*} \begin{cases} - \Delta u = f ( u ) + \lambda u , \\ \int _ { \R ^ N } u ^ 2 = \alpha . \end{cases} \end{align*}


\begin{align*} ( \rho \cdot \varphi ) ( x ) = \rho ( \tau ( x ) ) \varphi ( x ) & & \rho \in C ^ \infty ( B ) , \varphi \in C _ c ^ \infty ( M ) , x \in M . \end{align*}


\begin{align*} x _ 1 = \sqrt { \frac { 2 } { 3 } } \approx 0 . 8 1 6 4 9 7 . \end{align*}


\begin{align*} \partial _ t c = - \mathcal { L } _ \varepsilon c - f ^ \prime ( c ) \ ; \ ; \ ; \Omega _ T \end{align*}


\begin{align*} U ^ T ( n _ i A _ i ) U - \epsilon U ^ T ( n _ i D _ i ) = U ^ T \tilde A U - \epsilon U ^ T \tilde F - \epsilon \tilde F ^ T U = \begin{bmatrix} U \\ \epsilon \tilde F \end{bmatrix} ^ T \begin{bmatrix} \tilde A & - I \\ - I & 0 \end{bmatrix} \begin{bmatrix} U \\ \epsilon \tilde F \end{bmatrix} \end{align*}


\begin{align*} | q _ i | \leq 2 \epsilon \ \ \ i = 1 , \cdots , n - 1 . \end{align*}


\begin{align*} - c _ 2 F _ 1 ( x _ 1 , \dots , x _ n ) = g ( 0 , x _ 1 , \dots , x _ n ) h ( 0 , x _ 1 , \dots , x _ n ) . \end{align*}


\begin{align*} F _ \varepsilon ^ n = ( t + \varepsilon ) { \rm I d } . \end{align*}


\begin{align*} \mathfrak { h } _ { 0 } & \Big ( \varphi - \sum _ { 1 \leq i < i _ a } d ^ + _ { i , 0 } ( D ^ 2 \varphi ) h _ { i , 2 } - \sum _ { 1 \leq i \leq g } d ^ + _ { i , 1 } ( D \varphi ) h _ { i , 1 } - \sum _ { 1 \leq s < \gamma } d ^ 0 _ { s , 1 } ( D \varphi ) c _ { s , 1 } \Big ) \\ & = \sum _ { 1 \leq i \leq g } d ^ + _ { i , 2 } ( \varphi ) h _ i + \sum _ { 1 \leq s < \gamma } d ^ 0 _ { s , 2 } ( \varphi ) c _ s + \sum _ { 1 \leq j \leq g } d ^ - _ { - j , 2 } ( \varphi ) h _ { - j } . \end{align*}


\begin{align*} \left | 1 + \rho d X \right | ^ { q } \le 1 + q \rho d X + \binom { q } { 2 } \rho ^ 2 d ^ 2 X ^ 2 + \binom { q } { 3 } \rho ^ 3 | d X | ^ 3 ( 1 + | \rho d X | ) ^ { q - 3 } . \end{align*}


\begin{align*} y = \sqrt { P _ s } \sum \nolimits _ { l = 1 } ^ L h _ l e ^ { j \phi _ { l , n _ t } } g _ { l , n _ t } + n _ 0 , \end{align*}


\begin{align*} ( \underline { X } , \underline { A } ) ^ { \sigma } = \{ ( x _ 1 , \ldots , x _ m ) \in \prod ^ m _ { i = 1 } X _ i | x _ i \in A _ i i \notin \sigma \} . \end{align*}


\begin{align*} | \rho ( x ) | \leq 2 \epsilon , \rho ( x ) \neq - \epsilon , X \rho ( x ) = 0 \Longrightarrow X ^ 2 \rho ( x ) < 0 . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D } } ( \delta ) = \sup _ { \gamma \in \Sigma _ { \mathrm { D } } ( \delta ) } \int _ { \mathcal { Y } _ 1 \times \mathcal { Y } _ 2 } f ( y _ 1 , y _ 2 ) \ , d \gamma ( y _ 1 , y _ 2 ) \mathcal { I } ( \delta ) = \sup _ { \gamma \in \Sigma ( \delta ) } \int _ { \mathcal { S } } f ( y _ 1 , y _ 2 ) \ , d \gamma ( y _ 1 , y _ 2 , x ) . \end{align*}


\begin{align*} 5 \cdot 1 0 ^ m - 3 \ = \ 4 \underbrace { 9 \cdots 9 } _ { m - 1 } 7 \end{align*}


\begin{align*} c _ m \Delta u - S ^ \varphi u + \tilde S ^ { \tilde \varphi } u ^ { \frac { m + 2 } { m - 2 } } = 0 \ , \Omega , \end{align*}


\begin{align*} \frac 1 { \omega ( B ) } \| ( b - b _ B ) f _ 1 \| ^ t _ { L ^ t _ \omega } & = \frac 1 { \omega ( B ) } \int _ { 5 B } | b ( y ) - b _ B | ^ t | f | ^ t d \omega ( y ) \\ & \le { \Big ( \frac 1 { \omega ( B ) } \int _ { 5 B } | b ( y ) - b _ B | ^ { t s ' } d \omega ( y ) \Big ) ^ { \frac 1 { s ' } } } \Big ( \frac 1 { \omega ( B ) } \int _ { 5 B } | f | ^ { t s } d \omega ( y ) \Big ) ^ { \frac 1 s } . \end{align*}


\begin{align*} L _ { \omega , \mathbf { c } } ( \Psi ) = - 3 N ( \Psi ) \lesssim \| \nabla \cdot \psi _ 3 \| _ { L ^ 2 } \| \psi _ 1 \| _ { L ^ 4 } \| \psi _ 2 \| _ { L ^ 4 } \lesssim \| \Psi \| _ { \mathcal { H } ^ 1 } ^ 3 . \end{align*}


\begin{align*} \bigoplus _ { j = 1 } ^ n V _ j ^ { \oplus q _ j ( \xi _ I - \sigma \cdot \xi _ I ) } \to \bigoplus _ { j = 1 } ^ n V _ j ^ { \oplus q _ j ( \rho \cdot \xi _ I - \rho \cdot \sigma \cdot \xi _ I ) } \end{align*}


\begin{align*} A ^ { \ast } ( k x ) = k ^ { n } A ^ { \ast } ( x ) \left ( x \in G \right ) . \end{align*}


\begin{align*} \widetilde { \Psi } ( z ) : = \widetilde { T } ( \Phi _ 1 + \widetilde { T } ( \Phi _ 2 + \widetilde { T } ( \cdots \Phi _ { n - 1 } + \widetilde { T } ( f ) ) \cdots ) ) , \end{align*}


\begin{align*} \tau ( T ^ * _ { g } ( x ) ) = \tau ( x ) x \in M g \in G . \end{align*}


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


\begin{align*} \{ E _ { T ^ c } L _ T \} _ { T \subseteq [ n ] } = \bigotimes _ { i \in T } L _ i \otimes \bigotimes _ { i \in T ^ c } E _ i . \end{align*}


\begin{align*} e ^ h \mathrm { d i v } \Big ( e ^ { - h } V \ , \mathcal { L } _ h V \Big ) & = \big ( \mathcal { L } _ h V \big ) ^ 2 + V \cdot \nabla \mathcal { L } _ h V = \\ & = \big ( \mathcal { L } _ h V \big ) ^ 2 + V ^ i \ , \partial _ i \partial _ j V ^ j - \partial _ j h \ , V ^ i \ , \partial _ i V ^ j - \partial _ { i j } h \ , V ^ i \ , V ^ j . \end{align*}


\begin{align*} \eta = \int _ { B } \boldsymbol { \beta } _ { \eta } ( b ) d \nu _ { B } ( b ) , \end{align*}


\begin{align*} \left | \frac { A _ 1 } { A _ 2 } \right | = \left | \frac { B _ 1 } { B _ 2 } \right | = \left | \frac { C _ 1 } { C _ 2 } \right | \end{align*}


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


\begin{align*} ( \vec U , D _ { x _ i } \vec V ) = \vec U ^ T P _ { \Omega } ( D _ { x _ i } \vec V ) = - ( D _ { x _ i } \vec U , \vec V ) + ( E \vec U ) ^ T P _ { \partial \Omega } N _ { i } ( E \vec V ) . \end{align*}


\begin{align*} \Delta _ { x , g } ( n ) : = 2 \left ( \frac { C _ { x , g } } { \left ( 1 - \varepsilon _ { x } ( e , n ) \right ) \left ( 1 - \varepsilon _ { x } ( g , n ) \right ) } \right ) ^ { 1 / 2 } \varepsilon _ { x } ( g , n ) + \varepsilon _ { x } ( e , n ) . \end{align*}


\begin{align*} J _ K ( q ^ i , p ^ i ) = \begin{cases} ( - p ^ i , q ^ i ) , & \mbox { i f } i \in K ; \\ ( q ^ i , p ^ i ) , & \mbox { i f } i \notin K , \end{cases} \end{align*}


\begin{align*} \Sigma _ 0 ( \delta _ 0 ) : = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \ , \boldsymbol { K } _ c ( \mu , \gamma ) \le \delta _ 0 \right \} , \end{align*}


\begin{align*} G ( \theta ) = & \mathbb { E } _ { \Phi | \Phi ( \mathcal { A } ) > 0 } \left \{ \left ( \sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } e ^ { - m \eta \theta r _ 1 ^ { \alpha } / \rho } \right . \right . \\ & \left . \left . \prod \limits _ { x _ { i } \in \Phi \backslash \{ x _ 1 \} \cap { \mathcal { A } } } \frac { 1 } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M r _ i ^ { \alpha } } \right ) ^ M } \right ) ^ b \right \} , \end{align*}


\begin{align*} \hat { S } = S Q , \end{align*}


\begin{align*} a ' ( \nu ) = - \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | u _ \nu | ^ p ) | v _ \nu | ^ q . \end{align*}


\begin{align*} F ( k , t ; p , n ) = ( n - k ) \binom { n } { k } \sum _ { j = 0 } ^ k ( - 1 ) ^ { k - j } \binom { k } { j } \dfrac { q _ { j } ^ t } { ( n - j ) } \end{align*}


\begin{align*} \prod _ { j = 1 } ^ { n } | \eta _ j - \eta _ { j - 1 } | ^ { 1 - 2 H } \leq \sum _ { \pmb { \alpha } _ n \in D _ n } \prod _ { j = 1 } ^ n | \eta _ j | ^ { \alpha _ j } , \end{align*}


\begin{align*} \mathcal F \left ( e ^ { - z | \cdot | ^ 2 } \right ) = \frac { \pi } { z } e ^ { - \frac { | \xi | ^ 2 } { 4 z } } . \end{align*}


\begin{align*} \eta = \frac { 2 ( \mu _ { \omega , \mathbf { c } } - S _ { \omega , \mathbf { c } } ( \Phi ^ { \lambda _ 0 } _ 0 ) ) } { C ( 1 + M ^ 2 + \mu _ { \omega , \mathbf { c } } ) } . \end{align*}


\begin{align*} \nu _ 2 : = \frac { 1 } { 2 } \epsilon _ 1 ( n , A , \rho ) ^ { - 1 } \epsilon < \frac { 1 } { 2 } . \end{align*}


\begin{align*} \epsilon _ j = \begin{cases} 1 & j \notin Y \\ - 1 & j \in Y \end{cases} \end{align*}


\begin{align*} \frac { 1 - M ^ { - 1 } } { \log M } \sum _ { A \in \mathcal { P } _ { + } } \beta ( A ) \frac { \alpha ( A ) } { \beta ( A ) } \log \frac { \alpha ( A ) } { \beta ( A ) } \le \sum _ { A \in \mathcal { P } _ { + } } \beta ( A ) \left ( \frac { \alpha ( A ) } { \beta ( A ) } - 1 \right ) = d _ { { \rm T V } } ( \alpha , \beta ) . \end{align*}


\begin{align*} \operatorname { G r } ( B ) = \operatorname { G r } ( A _ n ) \oplus \mathbb { C } ( \phi _ + ^ n - \phi _ - ^ n , i \phi _ + ^ n + i \phi _ - ^ n ) = : \operatorname { G r } ( A _ n ) \oplus \mathbb { C } v _ n , \end{align*}


\begin{align*} S ( z ) = \Pi + z ( \mathcal { A } _ 0 - z I ) ^ { - 1 } \Pi . \end{align*}


\begin{align*} D _ \xi ( \det ) ( x ) = D ( \det ) ( x ) \cdot \xi = t r ( \tilde { x } ^ \tau \xi ) , \end{align*}


\begin{align*} \phi ^ { + } ( t ) : = \int ^ { \infty } _ { t } H ( s ) [ H ( s ) ] ^ T d s . \end{align*}


\begin{align*} v _ { A , \ell } & = \begin{cases} v _ { \ell } & \ell \in A , \\ 0 & \ell \notin A . \end{cases} \end{align*}


\begin{align*} \boldsymbol Y _ n = | A _ n | / | Q _ n | \end{align*}


\begin{align*} \Delta \left ( | X | ^ 2 \right ) = 2 | \nabla X | ^ 2 + 2 \left \langle X , \Delta X \right \rangle . \end{align*}


\begin{align*} \begin{aligned} M _ w ^ B ( Z , \lambda , N , \frac { \varepsilon } { 4 } , G , d ) & \ge \inf _ { \mathcal { G } _ w ( \mathcal { U } ) } \left \{ \sum _ { ( w _ { \mathbf { U } } , \mathbf { U } ) \in \mathcal { G } _ w ( \mathcal { U } ) } e ^ { - \lambda \mathfrak { l } ( \mathbf { U } ) } \right \} \\ & \ge M _ w ( Z , \lambda , N , \varepsilon , G , d ) . \end{aligned} \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*} J ' _ \nu ( u , v ) + \lambda _ 1 ( u , 0 ) + \lambda _ 2 ( 0 , v ) = 0 , \ u \geq 0 , \ v \geq 0 , \end{align*}


\begin{align*} \phi = \phi _ { } + \tau ^ m \frac H r , \end{align*}