\begin{align*} d \phi _ { i j } ( \partial _ j , \partial _ r ) = \partial _ j ( \phi _ { i j } ( \partial _ r ) ) - \partial _ r ( \phi _ { i j } ( \partial _ j ) ) = - \partial _ r \Big ( \frac { \Gamma _ { j i } ^ i } { \varphi _ j } \Big ) = \frac { - \partial _ r \Gamma _ { j i } ^ i + 2 \Gamma _ { r j } ^ j \Gamma _ { j i } ^ i } { \varphi _ j } , \end{align*}


\begin{align*} \Delta = \sum _ { a = 1 , 2 } \delta _ { a } ^ { 2 } . \end{align*}


\begin{align*} \alpha _ * ^ 2 \| u _ * \| ^ 2 + 2 \alpha _ * \langle u _ * , \lambda x _ v + ( 1 - \lambda ) x _ w \rangle - \lambda ( 1 - \lambda ) \| x _ v - x _ w \| ^ 2 = 0 \end{align*}


\begin{align*} \int _ 0 ^ 1 ( 1 + f ( x ) ) x ^ { s - 1 } d x = \frac { 1 } { s } \left ( 1 - \zeta ( s ) \sum _ { k = 2 } ^ N c _ k k ^ { - s } \right ) . \end{align*}


\begin{align*} \begin{aligned} \partial _ t u + \mathcal { L } u = & \ F , ( 0 , T ] \times \Omega \\ u ( 0 , \cdot ) = & \ u _ 0 , \ , \Omega \\ u = & \ g _ D ^ { } , ( 0 , T ] \times \Gamma _ D ^ { } , \\ \mathbf { n } \cdot \nabla u = & \ g _ N ^ { } , ( 0 , T ] \times \Gamma _ N ^ { } , \end{aligned} \end{align*}


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


\begin{align*} \rho = - \tau ( \mathbf { d } + \mathrm { i } \mathbf { d } ^ { \perp } ) , \end{align*}


\begin{align*} \frac { 1 } { 2 } B ( X ) : = \frac { 1 } { 2 } \sum _ { j = 1 } ^ p e _ j \cdot B ( X , e _ j ) \ \in C l ( T M \oplus E \oplus \mathcal { E } _ 2 ) \end{align*}


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


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


\begin{align*} U _ { i j } \subseteq U _ { i } \subseteq U U \backslash U _ { i } = B _ { i } \backslash U _ { i } U _ { j } \backslash U _ { i j } = B _ { i } \backslash U _ { i j } . \end{align*}


\begin{align*} t _ x : = \sup \{ t > 0 \ , | \ , X _ s ^ x \in \Omega \} \end{align*}


\begin{align*} C \log ( 2 N + 1 ) \leq \Vert M _ { H _ 0 } : S _ { 1 } ^ { 2 N + 1 } \rightarrow S _ { 1 } ^ { 2 N + 1 } \Vert . \end{align*}


\begin{align*} \int _ { t _ 0 } ^ { t _ 1 } \int _ { \Omega } \psi ( x ) ^ T u ( t , x ) \ , \omega ( t , x ) d x d t = \int _ { \Omega } \phi ( x ) \omega ( t _ 1 , x ) d x - \int _ { \Omega } \phi ( x ) \omega ( t _ 0 , x ) d x \end{align*}


\begin{align*} \Lambda ( s , \chi ) = \Lambda ( 1 - s , \chi ) . \end{align*}


\begin{align*} g _ \xi ( z ) = \begin{cases} 0 & z \in U ^ c , \\ \frac { g ( z ) } { \xi - z } & z \in U \end{cases} \end{align*}


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


\begin{align*} r _ k ( n ) = k \left \{ \frac { n } { k } \right \} , \end{align*}


\begin{align*} \Delta _ { \Psi } v = X ( \Delta _ { \Psi } u ) = X ( f ( u ) ) = f ' ( u ) X ( u ) = f ' ( u ) v . \end{align*}


\begin{align*} H _ { m i n } \left ( ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) \otimes F \right ) & = H _ { m i n } ( E _ 1 [ t ] \times E _ 1 [ t ] ^ { \vee } ) H _ { m i n } ( F ) \\ & = H _ { m i n } ( E _ 1 [ t ] ) H _ { m i n } ( F ) \\ & = t H _ { m i n } ( E _ 1 ) H _ { m i n } ( F ) . \end{align*}


\begin{align*} S t _ { D N P } \lim _ { n \rightarrow \infty } | | \mathcal { Y } _ { n } ( z , z ) - z | | _ { \infty } = 0 , \end{align*}


\begin{align*} \nu = [ h , \eta ] = h \eta h ^ { - 1 } \eta ^ { - 1 } = \rho \eta ^ { g } \rho ^ { - 1 } \eta ^ { - 1 } \in \Gamma ^ G . \end{align*}


\begin{align*} l ( \rho \circ \Psi \circ \pi ^ { - 1 } ( 1 \otimes e _ 0 ) ) = l ( \rho \circ \Psi ( p r ) ) = \rho ( l ( \Psi ( p r ) ) ) = \rho ( \Phi ( p r ) ) . \end{align*}


\begin{align*} L ^ { i } = \{ p \in \Delta _ { m - 1 } \vert \ , A ^ { i } p = A ^ { i } d \} \end{align*}


\begin{align*} g ( x ^ { \mu } ) : = \sum _ { i _ 1 = 0 } ^ { \infty } \cdots \sum _ { i _ n = 0 } ^ { \infty } \frac { 1 } { i _ 1 ! \cdots i _ n ! } \partial _ 1 ^ { i _ 1 } \cdots \partial _ n ^ { i _ n } g ( \epsilon ( x ^ { \mu } ) ) ( x ^ 1 - \epsilon ( x ^ 1 ) ) ^ { i _ 1 } \cdots ( x ^ n - \epsilon ( x ^ n ) ) ^ { i _ n } . \end{align*}


\begin{align*} \left \{ \begin{aligned} F _ 2 ^ { X , D } ( \alpha ) _ x & = H ^ 0 \left ( P ^ D _ X ( \alpha ) _ x , T _ { P ^ D _ X ( \alpha ) _ x } ( - [ s ( x ) ] ) \right ) \\ F _ 1 ^ { X , D } ( \alpha ) _ x & = H ^ 0 \left ( P ^ D _ X ( \alpha ) _ x , T _ { P ^ D _ X ( \alpha ) _ x } ( - 2 [ s ( x ) ] ) \right ) \end{aligned} \right . \end{align*}


\begin{align*} \sigma ( ( u ^ i , u ^ i { } _ j ) . a ) & = \sigma ( u ^ i , u ^ i { } _ \alpha a ^ \alpha { } _ j ) \\ * & = ( u ^ i , u ^ i { } _ \alpha a ^ \alpha { } _ j , - \Gamma ^ i { } _ { \alpha \beta } u ^ \alpha { } _ \gamma u ^ \beta { } _ \delta a ^ \gamma { } _ j a ^ \delta { } _ k ) \\ * & = \sigma ( u ^ i , u ^ i { } _ j ) . a \end{align*}


\begin{align*} ( g _ { j k } ) = 2 \left ( \begin{array} { c c c } \omega ^ 2 & 0 & 0 \\ 0 & 0 & 2 \omega ^ { - 2 p + 2 } \\ 0 & 2 \omega ^ { - 2 p + 2 } & 0 \end{array} \right ) . \end{align*}


\begin{align*} W ( 0 ^ \infty | [ 0 ^ k 1 ] ) = \sum _ { n \in \N } ( a _ { n + k } - a ) \ . \end{align*}


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


\begin{align*} \overline { X } ( t ) : = x + \int _ 0 ^ t \widetilde { \mathcal { A } } ( s ) d s + \int _ 0 ^ t \widetilde { \mathcal { B } } ( s ) d \widetilde { W } ( s ) . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \liminf _ { n \rightarrow \infty } ( I + I I + I I I ) = 0 . \end{align*}


\begin{align*} H ( p , s ) = G ( p , s ) - F ( p , s ) . \end{align*}


\begin{align*} \bigg ( \sum _ { j = 0 } ^ { k } \dbinom { k } { j } ( 1 - q ) ^ { k - j } q ^ j \dbinom { k } { j } \Big ( \frac { 1 } { 2 } \Big ) ^ k \bigg ) ^ { 1 / p _ k } \leq \Big ( \frac { 1 - q } { 2 } \Big ) ^ { k / p _ k } + \Big ( \frac { q } { 2 } \Big ) ^ { k / p _ k } \end{align*}


\begin{align*} t ( y ) = \frac { 6 } { y ^ 2 } + 4 ( b - 1 ) \frac { \exp ( y ) } { \left [ 1 - \exp ( y ) \right ] ^ 2 } . \end{align*}


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


\begin{align*} p ' _ { j } \circ p _ { 1 \ldots ( n - 1 ) } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } \circ u ' _ { j } = p _ { j } \circ f \circ u _ { j } \circ \iota _ { 1 \ldots ( n - 1 ) } = p _ { j } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } \\ = p ' _ { j } \circ p _ { 1 \ldots ( n - 1 ) } \circ f \circ \iota _ { 1 \ldots ( n - 1 ) } ~ ~ ~ ~ \forall j = 1 , \ldots , n - 1 . \end{align*}


\begin{align*} \alpha _ 2 = 0 \ , \Leftrightarrow \ , a _ 2 = 0 \ , \Leftrightarrow \ , \ell _ t = 2 \ , \ell _ b \ , \Leftrightarrow \ , \beta = \gamma 2 \ , \alpha _ 1 + 3 \ , \alpha _ 3 = 0 \ , \Leftrightarrow \ , a _ 3 = 0 \ , \Leftrightarrow \ , \Psi = \frac { 3 } { 2 } , \end{align*}


\begin{align*} P _ { \mu , f _ { 1 , \infty } } ( x , \psi ) : = \lim _ { \epsilon \to 0 } \liminf _ { n \to \infty } \dfrac { - \log \mu ( B _ { n } ( x , \epsilon ) ) + S _ { 1 , n } \psi ( x ) } { n } \end{align*}


\begin{align*} X _ { M _ { \mathbf i ' } } ( x _ 1 , \dots , x _ { i _ 1 } ' , \dots , x _ n ) = x _ { \mathbf i ' } ( x _ 1 , \dots , x _ { i _ 1 } ' , \dots x _ n ) = x _ \mathbf i ( x _ 1 , \dots , x _ { i _ 1 } , \dots , x _ n ) \in \mathcal A ( B ) . \end{align*}


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


\begin{align*} \left | \frac { \partial ^ 2 \varphi } { \partial x _ \beta \partial x _ \mu } ( x ' , \rho ( x ' ) ) \right | \leq C \delta \sqrt { \tilde { b } _ \beta \tilde { b } _ \mu } , \ \beta , \mu = 1 , \ldots , n - 1 ; \beta \neq \mu \end{align*}


\begin{align*} Z ^ { ( i ) } = \partial _ i \mbox { a n d } Z ^ { ( i , j ) } = x _ i \partial _ j - x _ j \partial _ i \mbox { f o r } i , j = 1 , \dots , n . \end{align*}


\begin{align*} 0 = V _ 0 \subset V _ 1 \subset V _ 2 \subset \cdots \subset V _ n \subseteq V , \end{align*}


\begin{align*} | A | ^ m | A ' + B _ 1 + \ldots + B _ m | \le ( 1 + \varepsilon ) | A ' | \prod _ { i = 1 } ^ m | A + B _ i | . \end{align*}


\begin{align*} \left \{ \begin{array} { l } \large \displaystyle u _ t - k \Delta u _ t - \Delta u = | u | ^ { p } + \omega ( x ) , \ , \ ( t , x ) \in ( 0 , \infty ) \times \mathbb { R } ^ N , \\ u ( 0 , x ) = u _ 0 ( x ) , \ , x \in \mathbb { R } ^ N , \end{array} \right . \end{align*}


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


\begin{align*} \begin{aligned} | \zeta _ \shortparallel | ^ 2 \geq \tfrac { 1 } { 2 } | \zeta _ \shortparallel - \tfrac { 1 } { 2 } V _ \shortparallel | ^ 2 - \tfrac { 1 } { 4 } | V _ \shortparallel | ^ 2 \ , . \end{aligned} \end{align*}


\begin{align*} G = \begin{pmatrix} 1 & 0 & \cdots & 0 & a _ 1 & 0 & \cdots & 0 & 1 \\ 0 & 1 & \cdots & 0 & a _ 2 & 0 & \cdots & 1 & 0 \\ \vdots & \vdots & & \vdots & \vdots & \vdots & & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & a _ k & 1 & \cdots & 0 & 0 \\ \end{pmatrix} _ { k \times ( 2 k + 1 ) } , \end{align*}


\begin{align*} { \tilde \Sigma } ^ t = \{ x \in M \mid { \rm d i s t } ( x , \tilde \Sigma ) \leq t \} . \end{align*}


\begin{align*} & M i n i m i z e _ { x \in \mathbb { R } ^ n } \ , T r ( A _ J , X ) + 2 b _ J ^ T x \\ & s . t . \ \ T r ( A _ k , X ) + 2 b _ k ^ T x + c _ k \le 0 \ , k = 1 , . . . , m \\ & \begin{pmatrix} 1 & x ^ T \\ x & X \end{pmatrix} \succeq 0 \end{align*}


\begin{align*} \frac { \sum _ { i = 1 } ^ p \log R _ i - m _ n } { s _ n } \overset { d } { \longrightarrow } S _ \alpha , n \to \infty \ , , \end{align*}


\begin{align*} \hat { f } ( \pi ) = \int _ { G } f ( g ) \pi ( g ) d g \end{align*}


\begin{align*} X _ { r } ^ i = \xi _ r \quad { X } ^ { i } _ t = \xi _ 0 + \int _ { 0 } ^ { t } \mu ( { \delta } _ i ( s ) , \mathcal { X } ^ { i } ) \ , d s + \int _ { 0 } ^ { t } \sigma ( { \delta } _ i ( s ) , \mathcal { X } ^ { i } ) \ , d W _ s . \end{align*}


\begin{align*} \begin{cases} \dim H ^ 1 ( S , T _ S ) = 4 0 ; \\ \dim H ^ 2 ( S , T _ S ) = 0 , \end{cases} \end{align*}


\begin{align*} G ( \Psi ) ( x ) = \int _ \S W _ r ( x - y ) \sin ( \Psi ( y ) - \Psi ( x ) ) \ \d y - \int _ \S W _ r ( y ) \sin ( \Psi ( y ) ) \ \d y . \end{align*}


\begin{align*} \Phi : = { \sup } ^ * \left \{ \zeta : \begin{array} { l } \zeta \overline { \Omega } \times \mathbb R ^ m , \\ F \star \mathcal P \Omega \times \mathbb R ^ m \Phi | _ { \partial \Omega \times \mathbb R ^ m } \le \phi \end{array} \right \} . \end{align*}


\begin{align*} \underline { u } = \frac { 1 } { 2 } \sum _ { 1 \leq j < k \leq N } \langle u ( e _ j ) , e _ k \rangle \ e _ j \cdot e _ k . \end{align*}


\begin{align*} h ( X ) = - \int f ( x ) \log f ( x ) d x . \end{align*}


\begin{align*} f '' ( x ) = a + b ' x + c ' x ^ q , \end{align*}


\begin{align*} \left | v _ { i , j } \right | ^ { 2 } \prod _ { k = 1 ; k \neq i } ^ { n } \left ( \lambda _ { i } ( A ) - \lambda _ { k } ( A ) \right ) = \prod _ { k = 1 } ^ { n - 1 } \left ( \lambda _ { i } ( A ) - \lambda _ { k } \left ( M _ { j } \right ) \right ) . \end{align*}


\begin{align*} a ( u , v ) = \lambda ( u , v ) v \in H _ 0 ^ 2 ( D ) . \end{align*}


\begin{align*} L _ \delta \omega _ \delta = l _ { 1 , \delta } + l _ { 2 , \delta } + R _ \delta ( \omega _ \delta ) , \end{align*}


\begin{align*} f ( z , t ) = \int _ \Omega \int _ { \C ^ n } F ( a , w ) e _ a ( ( - w , 0 ) ( z , t ) ) d w \ , d \nu ( a ) . \end{align*}


\begin{align*} N ( f ) & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( I - A ^ i M ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( A ^ { - i } ) \det ( I - A ^ i M ) \det ( A ^ i ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( I - M A ^ i ) \rvert \\ & = \frac { 1 } { d } \sum _ { i = 0 } ^ { d - 1 } \lvert \det ( I - M ) \rvert = N ( \tilde f ) . \end{align*}


\begin{align*} - L \phi = \mu ^ { 2 } ( \phi - \phi _ { \infty } ) _ { + } \textrm { i n } \ \mathbb { R } ^ { 2 } _ { + } , \phi = 0 \textrm { o n } \ \partial \mathbb { R } ^ { 2 } _ { + } , \end{align*}


\begin{align*} \mathcal { B } _ q ( \phi ) ( p ) : = \displaystyle \int _ { \mathbb { R } } A _ q ( p , x ) \phi ( x ) d x . \end{align*}


\begin{align*} c _ s ^ 2 \partial _ { y 1 } T = - \frac { 1 } { \Delta t } { \varsigma } _ 2 { m } _ 2 ^ { ( 1 ) } , \end{align*}


\begin{align*} T _ \ell ( x ) = \sqrt { \frac { \mu _ d } { n _ { \ell ; d } } } \sum _ { m = 1 } ^ { n _ { \ell ; d } } a _ { \ell , m } Y _ { \ell , m ; d } ( x ) , x \in \mathbb { S } ^ d , \end{align*}


\begin{align*} \mathcal { A } _ q ( p , x ) = \mathcal { A } _ q ^ p ( x ) : = \displaystyle \sum _ { n = 0 } ^ \infty \frac { \overline { p } ^ n } { \sqrt { \Gamma ( q n + 1 ) } } \psi _ n ( x ) , p \in \mathbb { H } , x \in \mathbb { R } \end{align*}


\begin{align*} \kappa ( t ) = h _ \alpha ( t ) , \ \ k ( t ) = h _ { 1 - \alpha } ( t ) , \ 0 < \alpha < 1 , \end{align*}


\begin{align*} F ^ { q , \dagger } \colon X ^ \dagger \times \mathcal P \to Z ^ \dagger , F ^ { q , \dagger } ( v , p ) : = F ^ q ( v , p ) , \end{align*}


\begin{align*} \Delta _ { 1 } ( x , \xi _ 0 ) = E _ { \xi _ 0 } \left [ \varphi \left ( x + X _ { 1 } \right ) \right ] - E _ { \xi _ { 0 } } \left [ \varphi \left ( x + Y _ { 1 } \right ) \right ] \end{align*}


\begin{align*} E ^ L _ Y ( u ) = \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \end{align*}


\begin{align*} 0 = \lim _ { m \to \infty } \int _ { \mathbb { R } ^ { 3 } } | b - b _ m | ^ { 2 } \dd x = \lim _ { m \to \infty } \int _ { \mathbb { R } ^ { 2 } _ { + } } \left ( | \nabla ( \phi - \phi _ m ) | ^ { 2 } + | G - G _ m | ^ { 2 } \right ) \frac { 2 \pi } { r } \dd z \dd r . \end{align*}


\begin{align*} C ^ { \ast } \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } ^ { \ast } g \right ) \left ( X , Y \right ) = - C \left ( X , Y , Z \right ) \end{align*}


\begin{align*} E ( \theta ) _ J = \sum _ { w \in Z _ J } \Bbbk { \bf U } _ { w _ J w ^ { - 1 } } \dot { w } C ( \theta ) _ J . \end{align*}


\begin{align*} y = \mathbf { X } \beta + \varepsilon , \varepsilon \sim \gamma _ { \sigma ^ 2 } , \end{align*}


\begin{align*} | [ - u _ 1 , u _ 1 ] \oplus _ 2 \cdots \oplus _ 2 [ - u _ m , u _ m ] | _ n = | B _ 2 ^ n | \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | _ n . \end{align*}


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


\begin{align*} \lambda : = \frac { \beta } { \log C _ 0 + \beta / ( 2 \alpha ) + 1 } , \end{align*}


\begin{align*} \sum _ { ( d , 2 ) = 1 } \left ( \frac { d } { p } \right ) W \left ( \frac { d \alpha ^ 2 } { X } \right ) = \sum _ { d } \left ( \frac { d } { p } \right ) W \left ( \frac { d \alpha ^ 2 } { X } \right ) - \left ( \frac { 2 } { p } \right ) \sum _ { d } \left ( \frac { d } { p } \right ) W \left ( \frac { 2 d \alpha ^ 2 } { X } \right ) . \end{align*}


\begin{align*} \hat M = ( M \setminus \Omega ) \cup \overline U \end{align*}


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


\begin{align*} 0 = N ^ { ( 0 ) } \subset N ^ { ( 1 ) } \subset \cdots \subset N ^ { ( k ) } \subset M \end{align*}


\begin{align*} \sigma _ { n } ^ M : = & T \wedge \inf \Big \{ t \geq 0 : \Vert X ( t , x _ n ) \Vert _ H > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \int _ 0 ^ t \Vert X ( s , x _ n ) \Vert _ { V } ^ { \alpha } d s > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \Vert X ( t , x ) \Vert _ H > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \int _ 0 ^ t \Vert X ( s , x ) \Vert _ { V } ^ { \alpha } d s > M \Big \} . \end{align*}


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


\begin{align*} N = n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } = \left ( n + a \right ) ^ { 3 } - \left ( n + a - \beta \right ) ^ { 3 } \end{align*}


\begin{align*} \vert \vert \psi ( \mathbf x ) \vert \vert _ { H ^ { 1 , p } } = \mathcal O \left ( \tau ^ { n ( \frac { 1 } { \tilde p } - \frac { 1 } { p } ) - 2 } \right ) , \end{align*}


\begin{align*} \begin{cases} d u = \nabla \cdot [ a ( u ) \nabla u + b ( u ) ] d t , ( 0 , T ] \times \mathbb { T } ^ d , \\ u ( 0 ) = u _ 0 , \end{cases} \end{align*}


\begin{align*} - \Delta p + \zeta p & = \overline { y } - g , \\ \langle p + \alpha \overline { u } , u - \overline { u } \rangle & \ge 0 u \in \mathcal { C } _ { a d } . \end{align*}


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


\begin{align*} & \Delta x _ j - \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ j = 0 \\ & x _ j ( p ) = 0 \ ; \mbox { a n d } \ ; \frac { \partial x _ j } { \partial y _ i } ( p ) = \delta _ { i j } . \end{align*}


\begin{align*} \Delta _ n ( x , \xi _ 0 ) : = & W ( \xi _ 0 , n , x , \mathsf { L } ^ n ) - W ( \xi _ 0 , n , x , \mathsf { R } ^ n ) \\ = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ { \mathsf { L } ^ n } \biggr ) \biggr ] - E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ { n } Z _ i ^ { \mathsf { R } ^ n } \biggr ) \biggr ] . \end{align*}


\begin{align*} \mu _ \varphi ( \{ \chi \in \widehat { H } \ : : \ : \chi ^ g = \chi \} ) > 0 . \end{align*}


\begin{align*} | \rho ^ \prime | = \left | \frac { 1 } { 1 + h ^ \prime } \right | \leq \frac { 1 } { \varepsilon _ 0 } . \end{align*}


\begin{align*} & \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { E } \int _ 0 ^ { T - \delta } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ { \alpha } d s \\ \leq & C \bigg ( \lim _ { \delta \rightarrow 0 + } \sup _ { n \in \mathbb { N } } \mathbb { E } \int _ 0 ^ { T - \delta } \Vert Y _ n ^ M ( t + \delta ) - Y _ n ^ M ( t ) \Vert _ { H } ^ { 2 } d s \bigg ) ^ { \frac { \alpha } { 2 } } = 0 . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ 2 d t = 0 . \end{align*}


\begin{align*} \begin{cases} d \tilde { Y } _ t = | z | J ^ \perp \tilde { Y } _ t d t + P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma d W _ s \\ \tilde { Y } _ 0 = P ^ { - 1 } Y _ 0 \in \C ^ n . \end{cases} \end{align*}


\begin{align*} A _ { 1 } = \sum _ { X \in S ( 0 ) } \Theta _ { \theta } \begin{bmatrix} m _ { 1 } & X \\ 0 & m _ { 2 } \end{bmatrix} \overline { \psi _ A ( { m _ { 1 } } ^ { - 1 } X } ) = q ^ { 8 } ( q - 1 ) ^ { 3 } . \end{align*}


\begin{align*} & H _ { \gamma } [ b _ n ] = H _ { \gamma } [ b _ { 0 , n } ] = h _ n , \\ & I _ { h _ n , \gamma } \leq { \mathcal { E } } [ v _ n , b _ n ] \leq { \mathcal { E } } [ v _ { 0 , n } , b _ { 0 , n } ] . \end{align*}


\begin{align*} \langle \nabla y _ \ell , \partial _ k \Q ^ t \partial _ k \Q \ , x \rangle = \left \langle \nabla y _ \ell , \sum _ { i = 1 } ^ d ( \partial _ k f _ i ) ^ 2 w ^ i \right \rangle , \end{align*}


\begin{align*} \gamma _ { i - 1 } \alpha _ i ^ 2 = ( 1 - \alpha _ i ) \alpha _ { i - 1 } ^ 2 \gamma _ i + \mu \alpha _ i \gamma _ i \gamma _ { i - 1 } \geq ( 1 - \alpha _ i ) \mu \gamma _ { i - 1 } \gamma _ i + \mu \alpha _ i \gamma _ i \gamma _ { i - 1 } = \mu \gamma _ i \gamma _ { i - 1 } , \end{align*}


\begin{align*} \begin{aligned} & ( C _ { a j } - C _ j - C _ a ) \ , [ C _ { i k } , C _ { k \ell } ] + ( C _ { a i } - C _ i - C _ a ) [ C _ { k \ell } , C _ { j k } ] + ( C _ { a \ell } - C _ \ell - C _ a ) [ C _ { i j } , C _ { j k } ] \\ & \qquad + ( C _ { a k } - C _ k - C _ a ) [ C _ { i \ell } , C _ { j \ell } ] = 0 \ , . \end{aligned} \end{align*}


\begin{align*} \mathrm { H o m } ( V , W ) = \bigoplus _ { \alpha \in K ( \mathcal { I } ) } \mathrm { H o m } ( V , W ) _ { \alpha } , \mathrm { H o m } ( V , W ) _ { \alpha } = \{ f \in \mathrm { H o m } ( V , W ) | f ( V _ { i } ) \subset W _ { j } , [ ( j , i ) ] = \alpha \} . \end{align*}


\begin{align*} \begin{cases} d _ n = \log ( 1 - e ^ { a _ { n + 1 } } ) \\ b _ n = \log ( 1 - e ^ { c _ { n + 1 } } ) \end{cases} \ \forall \ n \in \N \ , \end{align*}


\begin{align*} \begin{cases} \mathcal { L } _ H \varphi ( x ) = - \cdot ( K _ H ( x ) \nabla \varphi ( x ) ) = \frac { 1 } { \varepsilon ^ 2 } f ( \varphi ( x ) - \mu ) , \ & x \in \Omega , \\ \varphi ( x ) = 0 , \ & x \in \partial \Omega . \end{cases} \end{align*}


\begin{align*} \eta \{ F \} \stackrel { d e f } { = } { \bf E } \eta / F . \end{align*}


\begin{align*} \begin{aligned} B & = \nabla \times ( \Phi \nabla \theta ) + G \nabla \theta , \\ A & = \nabla \times ( \eta \nabla \theta ) + \Phi \nabla \theta , \\ \Phi & = \phi - \phi _ { \infty } . \end{aligned} \end{align*}


\begin{align*} f _ { * } \nabla _ X A D _ i Y & = \widetilde { \nabla } _ X f _ { * } A D _ i Y - \langle A X , A D _ i Y \rangle \rho = - \widetilde { \nabla } _ X h _ { * } \pi _ { * } D _ i Y - \langle h _ { * } \pi _ { * } X , h _ { * } \pi _ { * } D _ i Y \rangle h \circ \pi \\ & = - h _ { * } \nabla ' _ { \hat { X } } \hat { D } _ i \hat { Y } - \alpha ^ h ( \hat { X } , \hat { D } _ i \hat { Y } ) . \end{align*}


\begin{align*} \frac { \partial \tilde x } { \partial x } = h ^ { \frac { 1 } { 3 } } + x \cdot \frac { 1 } { 3 } h ^ { - \frac { 2 } { 3 } } \cdot h _ x ' , \end{align*}


\begin{align*} \begin{aligned} 3 \int _ { \Omega } n \log n & ( S ( x , n , c ) \cdot \nabla c ) \cdot \nabla \varphi \varphi ^ { 2 } \\ & \le \frac { 1 } { 4 } C _ { 3 } \| \nabla c \| _ { L ^ { 2 } ( \Omega \cap B _ { \delta } ) } \int _ { \Omega } \frac { | \nabla n | ^ { 2 } } { n } \varphi ^ { 3 } + \frac { 1 } { 5 } \int _ { \Omega } \frac { | \nabla n | ^ { 2 } } { n } \varphi ^ { 3 } + M . \end{aligned} \end{align*}


\begin{align*} A : = \left [ \begin{array} { c c c c c } \alpha _ 1 & \alpha _ 1 ^ q & \alpha _ 1 ^ { q ^ 2 } & \cdots & \alpha _ 1 ^ { q ^ { n } } \\ \alpha _ 2 & \alpha _ 2 ^ q & \alpha _ 2 ^ { q ^ 2 } & \cdots & \alpha _ 2 ^ { q ^ { n } } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha _ n & \alpha _ n ^ q & \alpha _ n ^ { q ^ 2 } & \cdots & \alpha _ n ^ { q ^ { n } } \\ x & x ^ q & x ^ { q ^ 2 } & \cdots & x ^ { q ^ { n } } \\ \end{array} \right ] . \end{align*}


\begin{align*} \partial _ { i i } f ( x ) & = V '' ( x _ i ) + \sum _ { j \neq i } J _ { i j } K '' ( x _ i - x _ j ) , \partial _ { i j } f ( x ) = - J _ { i j } K '' ( x _ i - x _ j ) . \end{align*}


\begin{align*} \chi ( x ) = \begin{cases} 1 & : \abs { x } \leq 1 \\ 0 & : \abs { x } \geq 2 \end{cases} \end{align*}


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


\begin{align*} \mathcal { J } ( \mu , \nu ; \alpha ) : = \frac { 1 } { 2 } \mathbb { E } \Big \{ \int _ 0 ^ T \Big [ Q \left ( x _ t ^ { \xi , \alpha } + l ( \nu _ t ) \right ) ^ 2 + R \left ( \alpha _ t + h ( \mu _ t ) \right ) ^ 2 \Big ] d t + G \left ( x _ T ^ { \xi , \alpha } + g ( \nu _ T ) \right ) ^ 2 \Big \} , \end{align*}


\begin{align*} C _ k ( t ) ^ { 2 \beta - m + 1 } \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - m - ( k - 1 ) i } { i } t ^ i - C _ k ( t ) ^ { \beta - m } \sum _ { i \geq 0 } ( - 1 ) ^ i \binom { \alpha - m - 1 - ( k - 1 ) i } { i } t ^ i . \end{align*}


\begin{align*} g ^ * f _ * \mathcal M = \varprojlim _ k g ^ * f _ * \tau _ { \le k } \mathcal M = \varprojlim _ k f ' _ * g '^ * \tau _ { \le k } \mathcal M = f ' _ * \varprojlim _ k g '^ * \tau _ { \le k } \mathcal M = f ' _ * g '^ * \mathcal M , \end{align*}


\begin{align*} G ^ { ( 1 ) } \vee . . . \vee G ^ { ( k ) } = \vee _ { \gamma \in \Gamma } G ^ { ( \gamma ) } . \end{align*}


\begin{align*} ( a \wedge b ) \cdot c & = ( a \cdot b ) \ : \cdot \ : ( a \cdot c ) \\ a \cdot ( b \wedge c ) & = ( a \cdot b ) \wedge ( a \cdot c ) . \end{align*}


\begin{align*} \epsilon _ 1 b _ { n - 1 } u _ n ( x ) \leq u ( y ) - u ( x ) + \sum _ { \alpha = 1 } ^ { n - 1 } ( y _ \alpha - x _ \alpha ) u _ \alpha ( x ) \leq C b _ { n - 1 } ^ 2 \end{align*}


\begin{align*} F _ j \subseteq \bigcup _ { n = 0 } ^ \infty [ x _ { n , j } , y _ { n , j } ] \end{align*}


\begin{align*} \langle B ^ { \circ } x ^ { \circ } , u \rangle = \langle x ^ { \circ } , B u \rangle = \lim _ { n \to \infty } \langle x ^ { \circ } , B u _ { n } \rangle , \end{align*}


\begin{align*} U _ C = \nabla \times ( \Phi _ C \nabla \theta ) + G _ C \nabla \theta = b ( \cdot - z _ 0 e _ z ) + B _ { \infty } . \end{align*}


\begin{align*} \Tilde { L } ( x ) : = \gamma _ 0 f _ 0 ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) \ge 0 \ \ x \in \mathbb { R } ^ n . \end{align*}


\begin{align*} N _ { P ^ * } ( c _ 4 ) = \emptyset . \end{align*}


\begin{align*} \frac { \partial } { \partial s } f ( p , s ) = \log a - \frac { p / 2 + 2 } { s } \geq \log a - \frac { p _ k / 2 + 2 } { s _ k } \end{align*}


\begin{align*} 0 = \frac { \dd } { \dd \tau } \int _ { \Omega } | U + \tau \tilde { U } + s ( \tau ) U _ 0 | ^ { 2 } \dd x \Bigg | _ { \tau = 0 } = 2 \int _ { \Omega } U \cdot ( \tilde { U } + \dot { s } ( 0 ) U _ 0 ) \dd x . \end{align*}


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


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } | \tilde { b } | ^ { 2 } \dd x & = \left ( \frac { W } { 2 } \right ) ^ { - 2 } \int _ { \mathbb { R } ^ { 3 } } | b | ^ { 2 } \dd x , \\ \int _ { \mathbb { R } ^ { 3 } } \left ( \tilde { \phi } - r ^ { 2 } - \tilde { \gamma } \right ) _ + \frac { \tilde { G } } { r ^ { 2 } } \dd x & = \left ( \frac { W } { 2 } \right ) ^ { - 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( \phi - \frac { W } { 2 } r ^ { 2 } - \gamma \right ) _ + \frac { G } { r ^ { 2 } } \dd x . \end{align*}


\begin{align*} x _ { n + 1 } & = ( 1 - \beta _ n ) U y _ n + \beta _ n y _ n , \\ y _ { n } & = ( 1 - \alpha _ n ) T x _ n + \alpha _ n u , \end{align*}


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


\begin{align*} \deg _ H ( S ) \leq \binom { p - \ell } { k - \ell } \leq d ^ { \frac { k - \ell } { k - 1 } } . \end{align*}


\begin{align*} H _ { \Psi } = H - g ( \nabla \Psi , \vec \nu ) \end{align*}


\begin{align*} W ( r ) : = M _ { 0 } r ^ { d } . \end{align*}


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


\begin{align*} H _ r ( E [ t ] ) = t H _ r ( E ) . \end{align*}


\begin{align*} \norm { g } _ { L ^ 2 } ^ 2 \le \prod _ { i = 1 } ^ n \norm { \eta _ i } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} x _ n = \rho ( x ' ) = \frac { 1 } { 2 } \sum _ { \alpha , \beta < n } B _ { \alpha \beta } x _ \alpha x _ \beta + O ( | x ' | ^ 3 ) , \end{align*}


\begin{align*} \{ n \in \Z : T ^ n a \in E \} = A . \end{align*}


\begin{align*} \begin{gathered} ( x _ { 1 0 } , x _ { 1 1 } ) \in \omega ( ( x _ { 0 0 } , x _ { 0 1 } ) , T \times T ) \quad \bigl ( x _ { 0 1 } , x _ { 1 1 } \bigr ) \in \omega \bigl ( ( x _ { 0 0 } , x _ { 1 0 } ) , T \times T \bigr ) . \end{gathered} \end{align*}


\begin{align*} c x ^ 2 + ( d - a ) x = 0 . \end{align*}


\begin{align*} \tilde U ^ { i j } D _ { i j } \tilde w = - \gamma ^ { - 1 } f , \end{align*}


\begin{align*} \tau ' _ y ( i ) = \begin{cases} \tau ' ( i ) & i \in I _ y \\ i & i \notin I _ y . \end{cases} \end{align*}


\begin{align*} R _ { c , k } = \log _ 2 \left ( 1 + \gamma _ { c , k } \right ) . \end{align*}


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


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) = \int _ { \S ^ d } W _ r \left ( \sum _ { i = 1 } ^ d m _ i y _ i + m _ { d + 1 } x \right ) ~ \sin \left ( \sum _ { i = 1 } ^ d n _ i \Theta ( t , y _ i ) + n _ { d + 1 } \Theta ( t , x ) \right ) \ \d y \end{align*}


\begin{align*} E Q C _ n : = R _ n / I _ n ~ . \end{align*}


\begin{align*} 0 = \oint q _ 1 ( z ) q _ 2 ( z ) d z = \int _ { \mathbb { R } } \left [ Q _ { j , - } \mathcal { P } ^ - ( D _ j ) + Q _ { j , - } J \right ] Q _ { j , - } ^ H \mathrm { d } z , \end{align*}


\begin{align*} \sigma ( F ^ q _ v ( 0 , p ) ) = \operatorname { c l } ( \{ \xi _ k , k \in \N \} ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ L z _ { s , k } \frac { \partial } { \partial z _ { s , k } } \Lambda _ a = \left \{ \begin{array} { l l } 0 & \mbox { f o r $ 1 \leq s \leq m - a $ , } \\ 1 & \mbox { f o r $ m - a < s \leq m $ . } \end{array} \right . \end{align*}


\begin{align*} \Psi _ L ( x ) = \begin{cases} \left ( A _ L J _ 1 ( ( \lambda - 1 ) ^ { 1 / 2 } r ) + \frac { W \lambda } { \lambda - 1 } r \right ) \sin \theta , & r \le a , \\ \frac { W a } { K _ 1 ( a ) } K _ 1 ( r ) \sin \theta , & r > a , \\ \end{cases} \end{align*}


\begin{align*} T _ { \xi } ( t ) = \exp \left ( \ - 0 . 5 \ \gamma ^ { - 1 } \ ( \ln ^ 2 t ) \ \right ) , \ t \ge e , \end{align*}


\begin{align*} \begin{aligned} ( \Delta ( u + v ) ) ^ m \wedge \beta _ n ^ { n - m } & = ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } + ( \Delta v ) ^ m \wedge \beta _ n ^ { n - m } + \sum \limits _ { p = 1 } ^ { m - 1 } \binom { m } { p } ( \Delta u ) ^ p \wedge ( \Delta v ) ^ { m - p } \wedge \beta _ n ^ { n - m } \\ & \geq ( \Delta u ) ^ m \wedge \beta _ n ^ { n - m } + ( \Delta v ) ^ m \wedge \beta _ n ^ { n - m } . \\ \end{aligned} \end{align*}


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


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


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


\begin{align*} M = \left ( \begin{array} { c c } - \varepsilon _ 1 \mu & \varepsilon _ 1 \nu \\ \varepsilon _ 3 \mu & - \varepsilon _ { 3 } \nu \end{array} \right ) \end{align*}


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


\begin{align*} v = \frac { 1 } { \sqrt { b _ \alpha } } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta y _ \beta ^ 2 + \sqrt { b _ \alpha } \bar { y } _ n . \end{align*}


\begin{align*} \Phi ( x ) : = \eta ( r ) s ( x ) \end{align*}


\begin{align*} \prod _ { i = 1 } ^ { \pi ( a ) - 1 } q _ i \neq \prod _ { i = 1 } ^ { \pi ( a ) - 1 } p _ i ^ { \alpha _ i } . \end{align*}


\begin{align*} & \| \alpha _ * u _ * + \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 = \\ & = \alpha _ * ^ 2 \| u _ * \| ^ 2 + \| \lambda x _ v + ( 1 - \lambda ) x _ w \| ^ 2 + 2 \alpha _ * \langle u _ * , \lambda x _ v + ( 1 - \lambda ) x _ w \rangle = \\ & = \lambda \| x _ v \| ^ 2 + ( 1 - \lambda ) \| x _ w \| ^ 2 \end{align*}


\begin{align*} B _ n \Big ( \frac { 2 \alpha ^ j } { L _ j } \Big ) - B _ n \Big ( \frac { F _ j \sqrt 5 } { L _ j } \Big ) = n \Bigl ( { \frac { { F _ j \sqrt 5 } } { { L _ j } } } \Bigr ) ^ { n - 1 } . \end{align*}


\begin{align*} \frac { 1 } { \varepsilon ^ 2 } \int _ { A _ \varepsilon } \left ( u _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ { p } q d x = \frac { 2 c _ \varepsilon } { \ln \frac { 1 } { \varepsilon } } + O \left ( \frac { 1 } { \ln \frac { 1 } { \varepsilon } } \right ) \leq C . \end{align*}


\begin{align*} Z ( A ) = V ( I ( A ) ) \ , . \end{align*}


\begin{align*} P _ { \mu , f _ { 1 , \infty } } ( x , \psi ) : = \lim _ { \epsilon \to 0 } \liminf _ { n \to \infty } \dfrac { - \log \mu ( B _ { n } ( x , \epsilon ) ) + S _ { 1 , n } \psi ( x ) } { n } \end{align*}


\begin{align*} c = \oint _ l \mathbf { v } \cdot \mathbf { t } d l = \iint _ { \sigma } \mathbf { w } \cdot \mathbf { n } d \sigma , \end{align*}


\begin{align*} z ^ \top \nabla ^ 2 f ( x ) z & = \sum _ { i , j = 1 } ^ n z _ i z _ j \partial _ { i j } f ( x ) = \sum _ { i = 1 } ^ n z _ i ^ 2 V '' ( x _ i ) + \sum _ { i , j = 1 } ^ n \big ( z _ i ^ 2 - z _ i z _ j \big ) J _ { i j } K '' ( x _ i - x _ j ) . \end{align*}


\begin{align*} \forall x \in E , \forall \lambda \in K _ v , \ \sigma _ v ( x \otimes \lambda ) = \sigma ( x ) \otimes \lambda \end{align*}


\begin{align*} \omega ( E , \epsilon ) = - 1 , L ( E , \epsilon ) = - \epsilon + \log | \lambda | . \end{align*}


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


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


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


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


\begin{align*} \lambda _ c ^ { \left ( \right ) } = 2 \sum _ { j = 1 } ^ { M } \lvert \hat { \phi } ^ { \left ( j , c \right ) } \rvert ^ 2 + \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 - f ^ { \left ( c \right ) } \end{align*}


\begin{align*} { \bf { h } } _ l ^ H { { \bf { f } } _ { l ' } } = 0 , \ \forall l \ne l ' . \end{align*}


\begin{align*} z ^ q D _ * ^ q ( f ) ( z ) = \sum _ { n = 1 } ^ { \infty } a _ n \frac { \Gamma ( q n + 1 ) } { \Gamma ( q ( n - 1 ) + 1 } z ^ { q m } . \end{align*}


\begin{align*} \sum _ { p \leq x } \sum _ { 1 \leq i \leq j \leq M } \alpha _ { \pi } ( p , i ) \alpha _ { \pi } ( p , j ) \log p = & \frac { 1 - \delta ( \pi ) } { 2 } x + O ( x ^ { 1 / 2 } \log ^ { 2 } x ) , \mbox { a n d } \\ \sum _ { p \leq x } \sum _ { 1 \leq i < j \leq M } \alpha _ { \pi } ( p , i ) \alpha _ { \pi } ( p , j ) \log p = & \frac { \delta ( \pi ) + 1 } { 2 } x + O ( x ^ { 1 / 2 } \log ^ { 2 } x ) , \end{align*}


\begin{align*} \int _ 0 ^ \infty \left | \frac { G ( e ^ { \tau } , \delta ) } { e ^ { \sigma \tau } } \right | ^ 2 d \tau = \int _ 1 ^ \infty \left | \frac { G ( y , \delta ) } { y ^ { \sigma + \frac { 1 } { 2 } } } \right | ^ 2 d y & = \int _ 1 ^ \infty \left | \frac { G ( y , \delta ) } { y ^ { 1 + \frac { \alpha } { \log ( 1 / \delta ) } } } \right | ^ 2 d y > \frac { 1 } { e ^ { \alpha b } } \int _ 1 ^ { 1 / \delta ^ b } \left | \frac { G ( y , \delta ) } { y } \right | ^ 2 d y \end{align*}


\begin{align*} \int _ T \gamma = \sum _ { \widetilde { K } \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde { K } } - 1 ) \langle [ \widetilde { K } ] , \alpha \rangle . \end{align*}


\begin{align*} \phi _ m ( x ) = \begin{cases} 1 & x \in A _ m ^ { ( 1 ) } \\ 0 & x \notin A _ m ^ { ( 2 ) } . \end{cases} \end{align*}


\begin{align*} \hat { R } _ { \delta } ( x ; z ) : = \begin{pmatrix} 0 & \bar { r } _ { \delta , 1 } ( z ) e ^ { - 2 i x z } \\ r _ { \delta , 2 } ( z ) e ^ { 2 i x z } & \bar { r } _ { \delta , 1 } ( z ) r _ { \delta , 2 } ( z ) \end{pmatrix} , \end{align*}


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


\begin{align*} \begin{aligned} | h ( x ( t ) + \delta u ( t ) ) - h ( z ( t ) + \delta u ( t ) ) | \le & \ , L _ r | x ( t ) - z ( t ) | \\ | h ( z ( t ) + \delta u ( t ) ) - h ( x _ a ( t ) + \delta u ( t ) ) | \le & \ , \gamma L _ r | \epsilon ( x _ a ( t ) , t ) | . \end{aligned} \end{align*}


\begin{align*} \epsilon _ 1 ( \eta _ a , t ) = \int _ { N } ^ { t } \varepsilon _ \delta ( { x } _ a , \bar { y } _ a , \tau ) u ( \tau ) - b _ { 1 , \delta } ( x _ a ) / 2 \ , d \tau \end{align*}


\begin{align*} \Psi ^ q + v + \psi ( v , p ( s ) ) = \Psi ^ q + v + \frac 1 2 \psi _ { v v } ( 0 , p _ 0 ) [ v , v ] + O ( \norm { ( v , s ) } ^ 3 ) \end{align*}


\begin{align*} \phi _ { f ' } ( x ) = \alpha \left ( a ( t x + u ) ^ { q + 1 } + b ( t x + u ) ^ q ( v x + w ) + c ( t x + u ) ( v x + w ) ^ q + d ( v x + w ) ^ { q + 1 } \right ) , \end{align*}


\begin{align*} \sum _ { q = r + 1 - \sum _ { d = j } ^ { i + 1 } \ell _ { d } } ^ { r - \sum _ { d = j + 1 } ^ { i + 1 } \ell _ { d } } a _ q ^ 2 \leq \frac { \beta ^ 2 } { g \big ( \lfloor ( 1 - ( 1 + \tilde \varepsilon ) ^ { - j } ) r \rfloor \big ) ^ 2 } \leq \frac { h ^ 2 } { 1 6 ^ 2 } \cdot 1 6 ^ { j - i } . \end{align*}


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


\begin{align*} 0 \in ( F _ k + B ) ( x ) = ( F + B ) ( x ) + \frac { 1 } { \rho _ k } ( x - z ^ k ) . \end{align*}


\begin{align*} \Theta f ( x ' , t ) : = \int _ { \R ^ { n - 1 } } \theta ( x ' , t ; y ' ) f ( y ' ) \ , d y ' , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} V ( \xi _ 0 , n , x ) = \sup _ { \theta \in \Theta _ n } E _ { \xi _ 0 } \left [ h ^ \theta ( x , Z _ 1 ^ \theta ) \right ] , \end{align*}


\begin{align*} d \in \sigma _ l ( A ) \mbox { w i t h e i g e n v e c t o r s } \left \{ \begin{aligned} v = ( x , 1 ) ^ T , \ , \forall x \in S _ L ( d - a , b ) \mbox { p r o v i d e d } S _ L ( d - a , b ) \neq \emptyset ; \\ v = ( 1 , x ) ^ T , \ , \forall x \in S _ L ( b , d - a ) \mbox { p r o v i d e d } S _ L ( b , d - a ) \neq \emptyset . \end{aligned} \right . \end{align*}


\begin{align*} \left \{ \begin{array} { l } \lambda ^ { N _ { i - 1 } } + \sum _ { \ell = 1 } ^ { r } c _ \ell u _ { i - 1 } ^ { \gamma _ \ell } \lambda ^ { ( i - 1 ) \gamma _ \ell + N _ { i - 1 } - \ell } = 0 \\ \lambda = u _ { i - 1 } v _ { i } \end{array} \right . \end{align*}


\begin{align*} f ( v _ { \ell , m , n } ) = \sqrt { q } ^ { 3 \ell + m - n } & \sum _ { \substack { ( i , j ) \in \{ 1 , 3 \} ^ 2 \\ i \neq j } } \{ F _ { i i j } z _ i ^ { \ell + m } z _ j ^ n + F _ { i j i } z _ i ^ { \ell + n } z _ j ^ m + F _ { i j j } z _ i ^ \ell z _ j ^ { m + n } \} , \\ \end{align*}


\begin{align*} X _ { n + 1 } = L ( X _ n ) + F ( X _ n ) + G ( X _ n ) \xi _ { n + 1 } , \end{align*}


\begin{align*} \nabla _ Y \mathcal { P } ( X ) = - \mathcal { P } ( B ^ * ( Y , X ) ) - B ( Y , s ( X ) ) + B ^ * ( Y , t ( X ) ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m U _ i < \sum _ { i = 1 } ^ m V _ i , \end{align*}


\begin{align*} \begin{cases} \partial _ t w + \nabla ^ \perp \varphi \cdot \nabla w = 0 , \ \ & \ B _ { R ^ * } ( 0 ) , \\ w = \mathcal { L } _ H \varphi , \ \ & \ B _ { R ^ * } ( 0 ) , \\ \varphi = 0 , \ \ & \ \partial B _ { R ^ * } ( 0 ) . \end{cases} \end{align*}


\begin{align*} \omega ( E , \epsilon ) = 0 , L ( E , \epsilon ) = \log \Big | \frac { E } { 2 } + \frac { \sqrt { E ^ { 2 } - 4 } } { 2 } \Big | . \end{align*}


\begin{align*} 0 = E ^ 0 \subset E ^ 1 \subset \cdots \subset E ^ { k - 1 } \subset E ^ { k } = E , \end{align*}


\begin{align*} D + E ^ + - E ^ - = f ^ * ( f _ * D ) . \end{align*}


\begin{align*} H _ k : = \begin{pmatrix} A _ k & b _ k \\ b _ k ^ T & c _ k \end{pmatrix} \ \ k = 1 , . . . , m H _ J : = \begin{pmatrix} A _ J & b _ k \\ b _ J ^ T & c _ J \end{pmatrix} \end{align*}


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


\begin{align*} \int _ 0 ^ \infty \left ( 1 - \frac { u ^ 2 } { 6 s } + \frac { u ^ 4 } { 7 2 s ^ 2 } \right ) e ^ { - u } u ^ { p / 2 - 1 } \dd u = \Gamma \left ( p / 2 \right ) - \frac { \Gamma ( p / 2 + 2 ) } { 6 s } + \frac { \Gamma ( p / 2 + 4 ) } { 7 2 s ^ 2 } \end{align*}


\begin{align*} W ( r ) = W ( r ; l , Q ) = - \frac { L + 1 } { r } f ( r ) + \frac { Q } { 2 L + 2 } \end{align*}


\begin{align*} \dfrac { \varepsilon } { 2 } \leq \beta ( \varphi _ { 2 k } ( a _ 1 ) , \rho ( a _ 1 ) ) = \beta ( \varphi _ { 2 k } ( a _ 1 ) , \rho \circ \varphi _ { 2 k } ( a _ 1 ) ) \leq \beta ( \varphi _ i ( a _ 1 ) , \rho \circ \varphi _ i ( a _ 1 ) ) . \end{align*}


\begin{align*} \aligned e _ 1 & \ , : = \ , \big ( x - y - \tfrac { 1 0 } { 9 } \ , u + 1 \big ) \ , \partial _ x + \big ( \tfrac { 1 0 } { 9 } \ , x - y - \tfrac { 1 0 } { 9 } \ , u \big ) \ , \partial _ y + \big ( x + 2 \ , u \big ) \ , \partial _ u , \\ e _ 2 & \ , : = \ , \big ( u - 2 \ , x \big ) \ , \partial _ x + \big ( \tfrac { 4 } { 3 } \ , x - y + \tfrac { 8 } { 9 } \ , u + 1 \big ) \ , \partial _ y - 3 \ , u \ , \partial _ u , \endaligned \end{align*}


\begin{align*} \epsilon ( s , \pi \otimes \chi _ { 8 d } ) = Q ^ { - s + 1 / 2 } _ { \pi \otimes \chi _ { 8 d } } , \end{align*}


\begin{align*} \epsilon ( \varphi ^ * ( f ) ) = \epsilon ( f ) \circ \tilde { \varphi } . \end{align*}


\begin{align*} \left \{ \begin{aligned} - \Delta y + \tilde { f } ( y ) & = u _ { 1 } - u _ { 2 } , \ ; \ ; \Omega , \\ y & = 0 , \ ; \quad \quad \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \left [ \Delta , \Delta ' \right ] _ \hbar : = \frac { 1 } { \hbar } [ \Delta , \Delta ' ] = \sum _ { n \geq 0 } \hbar ^ n \sum _ { \substack { i + j = n + 1 , \\ i , j \geq 0 } } [ \Delta _ { i } , \Delta _ { j } ] , \end{align*}


\begin{align*} & b _ k ^ T y = 0 \ \ k \in \{ 0 , . . . , m \} \\ & \begin{bmatrix} b ^ 0 _ 1 & \cdots & b ^ 0 _ n \\ \vdots & \ddots & \vdots \\ b ^ m _ 1 & \cdots & b ^ m _ n \end{bmatrix} \begin{bmatrix} y _ 1 \\ \vdots \\ y _ n \end{bmatrix} = \begin{bmatrix} 0 \\ \vdots \\ 0 \end{bmatrix} \end{align*}


\begin{align*} S \cap [ q ] = ( U \cap [ q ] ) \cup ( T \cap [ q ] ) = ( T \cap F _ { \ell + 1 } ) \cup T = T . \end{align*}


\begin{align*} j ( \tau , s ) = H [ b + \tau \tilde { b } + s b _ 0 ] , \tau , s \in \mathbb { R } . \end{align*}


\begin{align*} N ( u '' , V ' ) = N ( u , V ' ) . \end{align*}


\begin{align*} \lim _ { \lambda \to + 0 } \frac { 1 } { m ( B ( \mathbf 0 , \lambda ) \cap \Omega ) } \int _ { B ( \mathbf 0 , \lambda ) \cap \Omega } \vert v ( \mathbf x ) \vert \mathrm d \mathbf x = 0 , \end{align*}


\begin{align*} \mathbb { E } \left [ \mathbf { s } ^ { \left ( \right ) ^ H } \mathbf { y } ' | \hat { \mathbf { H } } \right ] = a _ c \sum _ { i = 1 } ^ M \hat { \phi } ^ { \left ( i , c \right ) } + \sum _ { j = 1 } ^ { M } a _ j \hat { \phi } ^ { \left ( j , j \right ) } . \end{align*}


\begin{align*} h _ s ( i ) : = & \exp \Big ( - \max \Big \{ \frac { 1 2 8 \alpha \ , \log n } { \tilde \varepsilon ^ 2 ( 1 + \tilde \varepsilon ) ^ { - i } m _ s } , \ , C _ h \Big \} \Big ) , \end{align*}


\begin{align*} \int ^ \infty _ 0 f ( x ) ^ \alpha \mathrm { d } x = \left [ \frac { 2 \theta } { \operatorname { B } ( a , b ) } \right ] ^ \alpha \frac { \Gamma \left ( \frac { 3 \alpha - 1 } { 2 } \right ) } { 2 \theta ^ { \frac { 3 ( \alpha - 1 ) } { 2 } + 1 } } \sum ^ \infty _ { n = 0 } \frac { ( - 1 ) ^ n } { ( a \alpha + n ) ^ { \frac { 3 \alpha - 1 } { 2 } } } \frac { \Gamma ( \alpha ( b - 1 ) + 1 ) } { \Gamma ( \alpha ( b - 1 ) + 1 - n ) n ! } . \end{align*}


\begin{align*} \left ( \frac { \left | \sum _ { i = 1 } ^ n [ 0 , z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , z _ i ] \right | } \right ) ^ 2 \ge \left ( \frac { \left | \sum _ { i = 1 } ^ n [ 0 , P _ E z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , P _ E z _ i ] \right | } \right ) ^ 2 + \left ( \frac { \left | \sum _ { i = 1 } ^ n [ 0 , P _ { E ^ \bot } z _ i ] \right | } { \left | \sum _ { i = 1 } ^ { n - 1 } [ 0 , P _ { E ^ \bot } z _ i ] \right | } \right ) ^ 2 . \end{align*}


\begin{align*} \begin{aligned} \left | H [ b _ 0 ] - H [ b _ 1 ] - H [ b _ 2 ] \right | & \lesssim \left ( \max _ { 0 \leq i \leq 2 } | | b _ i | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } ^ { 5 / 3 } \right ) | | b _ 0 - b _ 1 - b _ 2 | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } , \end{aligned} \end{align*}


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


\begin{align*} \mathcal { W } _ { f _ { 1 , \infty } } ( n , \alpha , \epsilon , g , \psi ) : = \inf \bigg \{ \sum _ { i } c _ { i } ^ { - \alpha n _ { i } + S _ { 1 , n _ { i } } \psi ( x _ { i } , \epsilon ) } \bigg \} , \end{align*}


\begin{align*} H _ 2 = \displaystyle \sum _ { \beta \in F ^ { \times } } \sum _ { \substack { X \in S ( \beta ) \\ t = 3 } } \Theta \begin{bmatrix} m _ 1 & X \\ 0 & m _ 2 \end{bmatrix} \overline { \psi _ A ( { m _ 1 } ^ { - 1 } X } ) = q ^ 8 ( 1 - q ) ( 1 - q ^ 2 ) . \end{align*}


\begin{align*} \kappa ( A ) : = \lVert A \rVert \ , \lVert A ^ { - 1 } \rVert = \frac { s _ { \max } ( A ) } { s _ { \min } ( A ) } . \end{align*}


\begin{align*} ( \alpha + \beta ) ^ { q ^ i } = \alpha ^ { q ^ i } + \beta ^ { q ^ i } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ T g _ n ^ { + } ( t ) d t = 0 . \end{align*}


\begin{align*} \Gamma \left ( X , Y , Z \right ) \equiv \left ( \nabla _ { Z } G \right ) \left ( X , Y \right ) = Z G \left ( X , Y \right ) - G \left ( \nabla _ { Z } X , Y \right ) - G \left ( X , \nabla _ { Z } Y \right ) . \end{align*}


\begin{align*} L _ X g ( V , \cdot ) = h ( V , \cdot ) \mbox { i n t h e c o l l a r n e i g h b o r h o o d o f } \Sigma . \end{align*}


\begin{align*} \gamma ( u ^ q + u ) + ( \gamma + 1 ) ( v ^ q + v ) & = c , \\ \gamma u ^ { q + 1 } + ( \gamma + 1 ) v ^ { q + 1 } & = d , \end{align*}


\begin{align*} \d Y _ t ^ i = b ^ i ( Y _ t ) \d t + \sigma \d W _ t ^ i , ~ ~ \tilde Y _ { n + 1 } ^ i = \tilde Y _ n ^ i + b ^ i ( \tilde Y _ n ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , ~ ~ i = 1 , \cdots , N . \end{align*}


\begin{align*} s ( q _ 1 ) s ( q _ 2 ) = c ( q _ 1 , q _ 2 ) s ( q _ 1 q _ 2 ) \forall q _ 1 , q _ 2 \in Q . \end{align*}


\begin{align*} N = n ^ { 3 } + \left ( n + 1 \right ) ^ { 3 } = \left ( n + a \right ) ^ { 3 } + \left ( n + b \right ) ^ { 3 } \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ f \ , d \gamma _ t \le \sup _ { y \in \R ^ n } \left ( \int _ { \R ^ n } f \ , d \gamma _ { y , t } - H ( \gamma _ { y , t } \ , | \ , \gamma _ t ) \right ) + \frac { t ^ 2 } { 2 } \sum _ { i , j = 1 } ^ n \int _ { \R ^ n } | \partial _ { i j } f | ^ 2 \ , d \gamma _ { y ^ * , t } . \end{align*}


\begin{align*} F ( \lambda ) x _ j + \frac { 1 } { 1 ! } F ^ { ( 1 ) } ( \lambda ) x _ { j - 1 } + \ldots + \frac { 1 } { j ! } F ^ { ( j ) } ( \lambda ) x _ 0 = 0 , j = 0 , 1 , \ldots , k , \end{align*}


\begin{align*} ( \mathbf { I } - \mathbf { P } ) ( v _ i v _ j \sqrt { \mu } ) = \Big \{ v _ i v _ j - \frac { | v | ^ 2 } { 3 } \delta _ { i j } \Big \} \sqrt { \mu } . \end{align*}


\begin{align*} A ^ { 1 } A ^ { 2 } : = \begin{bmatrix} A ^ { 1 } \\ \hline A ^ { 2 } \end{bmatrix} . \end{align*}


\begin{align*} \langle \langle Z \cdot \varphi , \varphi ' \rangle \rangle = \langle \langle \varphi , Z \cdot \varphi ' \rangle \rangle . \end{align*}


\begin{align*} \ell \binom { k } { 2 } m ^ { k - 2 } < s \binom { k } { 2 } m ^ { k - 2 } \leq \frac { d m ^ { k - 1 } } { 4 } \leq \frac { m ^ k } { 4 } \end{align*}


\begin{align*} \psi _ i = m \cdot \int _ { \kappa ^ { - 1 } ( K _ i ) } \psi \ , \mathrm { d } \mu _ \varphi ( \psi ) \end{align*}


\begin{align*} F _ j \subseteq \bigcup _ { n = 0 } ^ \infty [ x _ { n , j } , y _ { n , j } ] \end{align*}


\begin{align*} ( \Theta ^ { r ( p _ 1 , \ldots , p _ { r - 1 } ) } ) ^ i { } _ { j _ 1 , \ldots , j _ { r - 1 } } & = - v ^ i { } _ \alpha u ^ \alpha { } _ { j _ 1 , \ldots , \underset { \stackrel { \frown } { p _ { r - 1 } } } { l } , \ldots , j _ { r - 1 } , \beta } v ^ \beta { } _ \gamma d u ^ \gamma \wedge v ^ l { } _ \delta d u ^ \delta \\ * & \hphantom { { } = { } } + ( \textit { t e r m s w h i c h d o n o t i n v o l v e $ u ^ i { } _ { j _ 1 , \ldots , j _ r } $ } ) . \end{align*}


\begin{align*} \psi ( B _ \phi u , p ) = B _ \phi \psi ( u , p ) \end{align*}


\begin{align*} \lim _ { n \rightarrow 0 } g _ { n } ^ { + } ( t , \omega ) = 0 , ( t , \omega ) . \end{align*}


\begin{align*} [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x + y ) } \ \prod _ { i = 1 } ^ L X _ i \ \prod _ { 1 \leq i < j \leq L } X _ { i , j } \ \Big \vert _ { y = 1 } \end{align*}


\begin{align*} \Phi ( p _ 0 ) = R { s ( p _ 0 ) } \leq \Phi ( p ) \leq C ( \delta , m _ \delta ) \end{align*}


\begin{align*} { \rm V o l } ( { \tilde \Sigma } ^ { 3 r } ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( B _ { p _ i } ( r ) ) \leq C ( n ) \sum _ { i = 1 } ^ s { \rm V o l } ( A \cap B _ { p _ i } ( r ) ) , \end{align*}


\begin{align*} \nabla _ i \frac { A _ { j k } } { \mathcal { H } ^ 2 } & = \frac { 1 } { \mathcal { H } ^ 2 } \nabla _ i A _ { j k } - \frac { 1 } { \left ( \mathcal { H } ^ 2 \right ) ^ 2 } A _ { j k } \nabla _ i \mathcal { H } ^ 2 \\ & = \frac { 1 } { \left ( \mathcal { H } ^ 2 \right ) ^ 2 } ( \mathcal { H } ^ 2 \nabla _ i A _ { j k } - A _ { j k } \nabla _ i \mathcal { H } ^ 2 ) , \end{align*}


\begin{align*} A = \left ( \begin{array} { c c } a & b \\ c & d \\ \end{array} \right ) = a E _ 2 + \left ( \begin{array} { c c } 0 & b \\ c & d - a \\ \end{array} \right ) = : a E _ 2 + B . \end{align*}


\begin{align*} \sigma \geq v + \sum _ { k = 1 } ^ l a _ k \hat C _ { q _ k , N } . \end{align*}


\begin{align*} \frac { q _ { h + 1 } ( x ) } { p _ { h + 1 } ( x ) } = \frac { 1 } { 2 \sqrt x } \Big ( \frac { q _ h ( \sqrt x ) } { p _ h ( \sqrt { - x } ) } - \frac { q _ h ( \sqrt { - x } ) } { p _ h ( \sqrt x ) } \Big ) , ~ h = 0 , 1 , \dots \end{align*}


\begin{align*} x _ i ^ 2 = \epsilon ^ 2 = 1 , 1 \leq i \leq n , \end{align*}


\begin{align*} \mathsf { f } = \mathsf { f } ( y , \mathsf { m } ) = 2 \mathsf { m } \pi \frac { \mathsf { H } ( \| y \| ) } { \mathsf { H } ( b ) } , \mathsf { H } ( r ) = \int _ a ^ r \frac { d z } { z ^ { n + 1 } H ( z , z ^ 2 ) } . \end{align*}


\begin{align*} \lim _ { h \to 0 } \mathcal R \bigl [ \eqref { a h d u h } \bigr ] = \eqref { d u h a m e l 1 } \end{align*}


\begin{align*} \Phi \left ( \bigvee _ { l \geq j + 1 } p _ l \right ) ^ { \perp } \check { y } h _ n & = \Phi \left ( \bigvee _ { l \geq j + 1 } p _ l \right ) ^ { \perp } \check { y } ( h _ { n , 1 } + \cdots + h _ { n , j } ) \\ & = \Phi \left ( \bigvee _ { l \geq j + 1 } p _ l \right ) ^ { \perp } \check { y } ( p _ 1 + \cdots + p _ j ) h _ n . \end{align*}


\begin{align*} ( \sigma _ { S } \times \sigma _ { X } ) \circ \tau _ { * } \Psi \circ ( \sigma _ { S } \times \sigma _ { X } ) ^ { - 1 } \circ \Psi = \mathrm { i d } _ { S _ { \mathbb { C } } \times _ { \mathbb { C } } X _ { \mathbb { C } } } \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb N } } \left ( \frac { \mu ( f ^ { k + n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \ \ \& \ \ \lim _ { n \rightarrow \infty } \inf _ { k \in - { \mathbb N } _ 0 } \left ( \frac { \mu ( f ^ { k - n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 3 $ } \end{align*}


\begin{align*} V ( K \cup L ) = V ( K ) + V ( L ) - V ( K \cap L ) \end{align*}


\begin{align*} M _ { I J } = \begin{pmatrix} G _ { I _ 1 } & G _ { I _ 2 } & \cdots & G _ { I _ s } \\ G _ { J _ 1 } & G _ { J _ 2 } & \cdots & G _ { J _ s } \end{pmatrix} . \end{align*}


\begin{align*} b x ^ 2 + ( a - d ) x = 0 . \end{align*}


\begin{align*} \mathcal { A } _ { H _ { t } } ( [ \gamma , u ] ) = \int _ { \gamma } H _ { t } + \int u ^ { * } \omega . \end{align*}


\begin{align*} \gamma ^ { q ^ n } \beta ^ { q ^ n } + \gamma ^ { q ^ { n - k } } a _ { n - k } \beta ^ { q ^ { n - k } } + \cdots + \gamma a _ 0 \beta = 0 . \end{align*}


\begin{align*} - \Delta p _ { \epsilon } + F _ { \epsilon } ' ( y _ { \epsilon } ) p _ { \epsilon } & = y _ { \epsilon } - g , \\ \langle p _ { \epsilon } + \alpha u _ { \epsilon } , u - u _ { \epsilon } \rangle + \langle | u _ { \epsilon } - \overline { u } | ^ { p - 2 } ( u _ { \epsilon } - \overline { u } ) , u - u _ { \epsilon } \rangle & \ge 0 u \in \mathcal { C } _ { a d } . \end{align*}


\begin{align*} O _ { n , n } ( R ) \subseteq O _ { n + 1 , n + 1 } ( R ) : A \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & A \end{pmatrix} . \end{align*}


\begin{align*} \mathcal { M } _ { f _ { 1 , \infty } } ( N , \alpha + \delta , 6 \epsilon , \bigcup _ { n = N } ^ { \infty } Z _ { n , n ^ { - 2 } t } , \psi ) \leq \sum _ { n = N } ^ { \infty } \mathcal { M } _ { f _ { 1 , \infty } } ( N , \alpha + \delta , 6 \epsilon , Z _ { n , n ^ { - 2 } t } , \psi ) . \end{align*}


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


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


\begin{align*} M ( S ) = \varinjlim _ i M ( S _ i ) . \end{align*}


\begin{align*} I _ f ( p ' , q ' ) \geq \sum _ { j = 0 } ^ { D - 1 } q '' _ j f \left ( \frac { p '' _ j } { q '' _ j } \right ) . \end{align*}


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


\begin{align*} \beta _ y = \min \left \{ \frac { 1 } { 2 L _ x } , \frac { 1 } { 2 \eta _ y L _ { x y } ^ 2 } \right \} . \end{align*}


\begin{align*} \frac { 1 } { | x _ d | } \le \frac { 1 } { d } \Big | \frac { p ' ( c ) } { p ( c ) } \Big | = \frac { 1 } { d } \Big | \sum _ { j = 1 } ^ d \frac { 1 } { c - x _ j } \Big | \end{align*}


\begin{align*} A _ { \gamma _ j , \gamma _ { j + 1 } } = \{ i : p _ i / q _ i \in [ \gamma _ j , \gamma _ { j + 1 } ) \} = \{ i : \delta _ i \in [ \nu _ j , \nu _ { j + 1 } ) \} . \end{align*}


\begin{align*} w - \widetilde { w } = \left ( \bar { u } _ { x x } + \frac { 1 } { 2 i } | \tilde { u } _ x | ^ 2 \bar { u } _ x \right ) e ^ { i c _ - ( x ) } - \left ( \bar { \tilde { u } } _ { x x } + \frac { 1 } { 2 i } | \tilde { u } _ x | ^ 2 \bar { \tilde { u } } _ x \right ) e ^ { i c _ - ( x ) } ( \tilde { u } ) . \end{align*}


\begin{align*} \frac { 1 } { \rho _ K } \| ( x ^ { K + 1 } , \lambda ^ { K + 1 } ) - ( x ^ K , \lambda ^ K ) \| \leq \frac { r _ 0 + \theta } { \rho _ 0 \zeta ^ K } \overset { \eqref { K } } { \leq } \frac { \varepsilon } { 2 } , \eta _ K = \eta _ 0 \sigma ^ K \overset { \eqref { K } } { \leq } \frac { \varepsilon } { 2 } , \end{align*}


\begin{align*} s _ { h , q } ( p , c , \rho ) : = \frac { \rho } { q } \sum _ { g = 0 } ^ { q - 1 } \zeta ^ { ( h + 1 ) g } ~ \frac { p ' ( c + \rho \zeta ^ g ) } { p ( c + \rho \zeta ^ g ) } ~ { \rm f o r } ~ \zeta ~ { \rm o f ~ ( \ref { e q z t } ) ~ a n d } ~ h = 0 , 1 , \dots , q - 1 , \end{align*}


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


\begin{align*} \begin{cases} \partial _ t \mathbf { w } + ( \mathbf { v } \cdot \nabla ) \mathbf { w } = ( \mathbf { w } \cdot \nabla ) \mathbf { v } , \ \ & D \times ( 0 , T ) , \\ \mathbf { w } ( \cdot , 0 ) = \nabla \times \mathbf { v } _ 0 ( \cdot ) , \ \ & D . \end{cases} \end{align*}


\begin{align*} y _ n ^ 2 = \left ( \bar { y } _ n + \sum _ { \beta = 1 } ^ { n - 1 } \rho _ \beta ( x ' _ 0 ) y _ \beta \right ) ^ 2 \leq 2 \bar { y } _ n ^ 2 + C \epsilon ^ 2 \delta ^ 2 b _ \alpha | y ' | ^ 2 . \end{align*}


\begin{align*} \left | \sum _ { n = 1 } ^ { \infty } S _ N ( n ) \int _ { \frac { 1 } { n + 1 } } ^ { \frac { 1 } { n } } x ^ { s - 1 } d x \right | \leq \frac { 1 } { \Re s } \sum _ { n = 1 } ^ { \infty } \left | S _ N ( n ) \right | \left | \frac { 1 } { n ^ { \Re ( s ) } } - \frac { 1 } { ( n + 1 ) ^ { \Re ( s ) } } \right | . \end{align*}


\begin{align*} \abs { \eta ( y ) - \eta ( x ) } ^ 2 & = \abs { \int _ x ^ y \partial \eta ( z ) \ \d z } ^ 2 \\ & \le \int _ \S \partial \eta ( z ) ^ 2 \ \d z \\ & = \norm { \partial \eta } _ { L ^ 2 } ^ 2 . \end{align*}


\begin{align*} B _ j ( z ) = A ( z ) \sum _ n \frac { \overline { t _ n } ^ j \mu _ n } { z - t _ n } , j = 0 , \dots , N - 1 . \end{align*}


\begin{align*} \phi _ A ( t _ 1 , \hdots , t _ n ) = \left ( \prod _ { i = 1 } ^ n t _ i ^ { a _ { i 1 } } , \prod _ { i = 1 } ^ n t _ i ^ { a _ { i 2 } } , \hdots , \prod _ { i = 1 } ^ n t _ i ^ { a _ { i m } } \right ) . \end{align*}


\begin{align*} M ( n + 3 ) \coloneqq \begin{cases} F ^ { ( 0 ) } \circ F ^ { ( 1 ) } \circ \cdots \circ F ^ { ( n - 1 ) } ( E _ 1 ) & \\ F ^ { ( 0 ) } \circ F ^ { ( 1 ) } \circ \cdots \circ F ^ { ( n - 1 ) } ( E _ 2 ) & . \end{cases} \end{align*}


\begin{gather*} f = \bigl ( \pi _ { i + 1 } - \pi _ i \bigr ) \ , 1 ( \pi _ 1 \in A _ 1 ) \ , \ldots \ , 1 ( \pi _ i \in A _ i ) , \end{gather*}


\begin{align*} \begin{aligned} { \mathfrak a } _ { 2 } = & g ^ { a b } \xi _ { a } \xi _ { b } , { \mathfrak a } _ { 1 } = & \frac { - i } { \sqrt { ( g ) } } \partial _ { a } \bigl ( \sqrt { ( g ) } g ^ { a b } \bigr ) \xi _ { b } , { \mathfrak a } _ { 0 } = 0 . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \| \nabla u _ { 0 } ( \mathbf x ) \| _ { L ^ { 2 } ( \Lambda _ h ) } & \leq \sqrt { 2 } \tau \| u _ { 0 } \| _ { L ^ { 2 } ( \Lambda _ h ) } + \| e ^ { \rho \cdot \mathbf x } \| _ { L ^ { 4 } ( \Lambda _ h ) } \| \nabla \psi ( \mathbf x ) \| _ { L ^ { 4 } ( \Lambda _ h ) } \\ & \lesssim ( 1 + \tau ) ( 1 + \tau ^ { - \frac { 2 } { 3 } } ) e ^ { - \zeta h \tau } , \mbox { a s $ \tau \rightarrow \infty $ . } \end{aligned} \end{align*}


\begin{align*} \int _ K | | ( \eta + g ^ { - 1 } ) ^ { \frac { 1 } { 2 } } u _ 0 | | _ { \omega } + \int _ K | | \tau _ 0 | | _ { \omega } \le \int _ M \langle ( B + \lambda I ) ^ { - 1 } v , v \rangle _ { \omega , h } d V _ { \omega } . \end{align*}


\begin{align*} \beta _ i + \bar { \beta _ j } = 1 \ , \ , { \rm a n d } \ , \ , \beta _ u + \bar { \beta } _ 4 = 1 . \end{align*}


\begin{align*} W ( \xi _ 0 , n , x , \widetilde \theta ) = & E _ { \xi _ 0 } \biggl [ \varphi \biggl ( x + \sum _ { i = 1 } ^ n Z ^ { \widetilde \theta } \biggr ) \biggr ] \\ = & E _ { \xi _ 0 } \biggl [ V \biggl ( \frac { \xi _ 0 f _ 2 ( Y _ 1 ) } { \xi _ 0 f _ 2 ( Y _ 1 ) + ( 1 - \xi _ 0 ) f _ 1 ( Y _ 1 ) } , n - 1 , x + Y _ 1 \biggr ) \biggr ] . \end{align*}


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


\begin{align*} \sigma _ { X , \mathrm { t r } } ^ * \circ f ^ \sigma = f \circ \sigma _ { Y , \mathrm { t r } } ^ * . \end{align*}


\begin{align*} \Delta [ M _ v ] ( t ) = ( \Delta M _ v ( t ) ) ^ 2 = e ^ { - 2 \lambda t } ( v A ^ * _ k ) ^ 2 = e ^ { - 2 \lambda t } \lambda ^ 2 v _ { k } ^ 2 , \end{align*}


\begin{align*} \mathcal { W } ' ( X ) = \sum _ { m = 0 } ^ n V _ m ' ( X ) \end{align*}


\begin{align*} T _ \alpha = \partial _ { \alpha } + \sum _ { \beta < n } B _ { \alpha \beta } ( x _ { \beta } \partial _ { n } - x _ { n } \partial _ { \beta } ) , \ ; \ ; \mbox { f o r } \alpha < n , \end{align*}


\begin{align*} \{ A ( \xi ) \} \cap \mathrm { K e r } ( B ) = \{ 0 \} , \forall \xi \in \R ^ * . \end{align*}


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


\begin{align*} H _ r ( E ^ { \vee } ) = H _ r ( E ) ^ { - 1 } \end{align*}


\begin{align*} \forall \theta > 0 \ \Rightarrow \sup _ { p \ge 1 } \left [ \ \frac { L ( p ^ { \theta } ) } { L ( p ) } \ \right ] = C ( \theta ) < \infty . \end{align*}


\begin{align*} \begin{aligned} f _ { 1 } ^ { + } & = \inf \left \{ \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in L ^ { 2 } _ { \sigma } ( \Omega ) , \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = 1 \right \} , \\ - f _ { 1 } ^ { - } & = \inf \left \{ \int _ { \Omega } | B | ^ { 2 } \dd x \ \middle | \ B \in L ^ { 2 } _ { \sigma } ( \Omega ) , \ \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = - 1 \right \} . \end{aligned} \end{align*}


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


\begin{align*} \phi _ p ( x ) = ( 1 + x ) ^ { - p / 2 } \end{align*}


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


\begin{align*} \gamma _ { 1 , k , m } \le \epsilon = 1 / 2 ^ b ~ { \rm f o r } ~ k \ge ( b + 2 ) \log _ { \Delta _ { 1 , m } } ( 2 ) ~ { \rm i f } ~ 2 \Delta _ { 1 , m } ^ k \le 1 . \end{align*}


\begin{align*} { \partial _ t } = \sum \limits _ { n = 1 } ^ { + \infty } { { \varepsilon ^ n } \partial { t _ n } } , \ ; \ ; \ ; \nabla = \varepsilon { \nabla _ 1 } , \ ; \ ; \ ; { g _ i } = \sum \limits _ { n = 1 } ^ { + \infty } { { \varepsilon ^ n } g _ i ^ { \left ( n \right ) } } , \ ; \ ; \ ; \bar F = \varepsilon { \bar F ^ { \left ( 1 \right ) } } , \end{align*}


\begin{align*} \delta _ H ( k ) = 2 \phi _ H ( k + 1 ) - ( \phi _ H ( 2 k ) + \phi _ H ( 2 ) ) \end{align*}


\begin{align*} \Gamma _ { i j } ^ k = \frac { 1 } { 2 } g ^ { l k } u _ { , i j l } \end{align*}


\begin{align*} \widetilde { \sigma } _ { j i } & = ( 2 \ , \mu ^ * + \varkappa ) ( u _ { i , j } + u _ { j , i } ) + \varkappa \ , \epsilon _ { j i m } \Big ( \underbrace { \frac { 1 } { 2 } \epsilon _ { m k s } u _ { s , k } } _ { r _ m } - \vartheta _ m \Big ) + \lambda \ , u _ { k , k } \delta _ { j i } , \\ * m _ { j i } & = \beta \ , \vartheta _ { j , i } + \gamma \ , \vartheta _ { i , j } + \alpha \ , \vartheta _ { k , k } \delta _ { j i } \ , . \end{align*}


\begin{align*} f ( x ) = \log \Gamma ( x ) - \log \Gamma \left ( x + \frac 1 2 \right ) - \log \Gamma \left ( x + \frac 3 2 \right ) \end{align*}


\begin{align*} E ( g _ 1 , w ) = \int _ { - \infty } ^ \infty \int _ 0 ^ 1 \int _ 0 ^ \infty \int _ 0 ^ \infty e ^ { i \Phi \left ( \lambda , \theta , t , u + \left ( \frac { t _ 1 } { t } \right ) ^ 2 u _ 1 \right ) } W ( \lambda , t , u ) d \lambda d \theta d t d u . \end{align*}


\begin{align*} \partial _ t \mu ^ \phi = - \partial _ x \Big \{ \Big [ b ( t , \cdot ) + \sigma \phi ( t , \cdot ) \Big ] \mu ^ \phi \Big \} . \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Theta ( t , x ) & = \int _ \S W _ r ( x - y ) \sin ( \Theta ( t , y ) - \Theta ( t , x ) ) \ \d y \\ & + \lambda \int _ \S \int _ \S W _ r ( z + y - 2 x ) \sin ( \Theta ( t , z ) + \Theta ( t , y ) - 2 \Theta ( t , x ) ) \ \d y \d z \\ & + \mu \int _ \S \int _ \S \int _ \S W _ r ( z - y + w - x ) \sin ( \Theta ( t , z ) - \Theta ( t , y ) + \Theta ( t , w ) - \Theta ( t , x ) ) \ \d w \d y \d z , \end{align*}


\begin{align*} v _ { \alpha } ( x ) = \begin{cases} x _ n ^ { \alpha } ( | x ' | ^ 2 - C _ \alpha ) \quad C _ { \alpha } = \frac { 1 + 2 D ^ 2 } { \alpha ( 1 - \alpha ) } & \alpha \in [ \frac { 2 } { n } , 1 ) , \\ ( 1 + 2 D ^ 2 ) x _ n \log ( x _ n / D ) + x _ n ( | x ' | ^ 2 - D ^ 2 ) & \alpha = 1 . \end{cases} \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) : = \sum _ { K \in { \mathcal T } _ n } \int _ K D ^ 2 u _ n : D ^ 2 v _ n d K , u _ n , v _ n \in X _ n . \end{align*}


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


\begin{align*} \begin{aligned} v _ 1 ' = v _ 1 - [ ( v _ 1 - v ) \cdot \omega ] \omega \ , , \ v ' = v + [ ( v _ 1 - v ) \cdot \omega ] \omega \ , , \end{aligned} \end{align*}


\begin{align*} \psi _ \alpha ( f ) : & = \nu ( \{ x \in X : | f ( x ) | > \alpha \} ) \end{align*}


\begin{align*} \sum _ { k = c _ 1 } ^ { c _ 2 } { F _ { i w _ k + m } v _ k z ^ { w _ k } } = \frac { 1 } { \sqrt 5 } \big ( \alpha ^ m h ( \alpha ^ i z ) + \beta ^ m h ( \beta ^ i z ) \big ) \ , , \end{align*}


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


\begin{align*} ( \Theta ^ { r ( p _ 1 , \ldots , p _ { r - 1 } ) } ) ^ i { } _ { j _ 1 , \ldots , j _ { r - 1 } } & = - v ^ i { } _ \alpha u ^ \alpha { } _ { j _ 1 , \ldots , \underset { \stackrel { \frown } { p _ { r - 1 } } } { l } , \ldots , j _ { r - 1 } , \beta } v ^ \beta { } _ \gamma d u ^ \gamma \wedge v ^ l { } _ \delta d u ^ \delta \\ * & \hphantom { { } = { } } + ( \textit { t e r m s w h i c h d o n o t i n v o l v e $ u ^ i { } _ { j _ 1 , \ldots , j _ r } $ } ) . \end{align*}


\begin{align*} q , r , \tilde q , \tilde r \geq 2 \quad \tfrac { 2 } { q } + \tfrac { 1 } { r } = \tfrac { 1 } { 2 } = \tfrac { 2 } { \tilde q } + \tfrac { 1 } { \tilde r } . \end{align*}


\begin{align*} \left \{ \begin{aligned} \sigma _ k \big ( \lambda ( D ^ { 2 } u ) \big ) & = f \ ; \ ; \mbox { i n } ~ \Omega , \\ u & = \varphi \ ; \ ; \mbox { o n } ~ \partial \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \varphi _ 2 ( x ) = u ( x ) + t v _ 2 ( x ) \end{align*}


\begin{align*} O : = \{ f \in X : f ( x ) = - f ( - x ) \} . \end{align*}


\begin{align*} \mathbf { w } = \frac { w } { k } \overrightarrow { \zeta } , \end{align*}


\begin{align*} \frac { \partial } { \partial t } \Psi ( t , x ) = F ( \Psi , p ) ( x ) . \end{align*}


\begin{align*} \int _ { \mathcal C ^ h } k ^ 2 ( q - 1 ) v u _ 0 \mathrm d \mathbf x = \int _ { \Lambda _ h } ( w - v ) \partial _ \nu u _ 0 - u _ 0 \partial ( w - v ) \mathrm d \sigma - \int _ { \Gamma _ h } \eta u _ 0 v \mathrm d \sigma \end{align*}


\begin{align*} g _ 2 = - \frac { 1 } { \nu } I \widehat { g _ 1 } I + \frac { 2 i } { \sqrt { \tau _ 0 } } h _ z J g _ 1 . \end{align*}


\begin{align*} \widetilde { W } _ { B _ { r _ 1 } } ( x ) = \sum _ { k = 0 } ^ \infty \frac { 1 } { ( 2 ^ k ) ^ { 1 0 } } \int H ( x - y ) \chi _ { A _ k } ( y ) d y . \end{align*}


\begin{align*} \alpha _ i + \bar { \alpha } _ j = 1 . \end{align*}


\begin{align*} ( { } _ a D _ g ^ { \alpha } f ) ( x ) = & \left ( \frac { 1 } { g ' ( y ) } \frac { d } { d y } \right ) ( { } _ { a } I _ g ^ { 1 - \alpha } f ) ( y ) \\ = & \frac { \left ( \displaystyle \frac { 1 } { g ' ( y ) } \displaystyle \frac { d } { d y } \right ) } { \Gamma ( 1 - \alpha ) } \int _ a ^ x \frac { f ( y ) g ' ( y ) } { ( g ( x ) - g ( y ) ) ^ { \alpha } } d y . \end{align*}


\begin{align*} \hat { q } _ i = q ( z _ i ) + \frac { \hat { q } _ i } { \ln \frac { R } { \varepsilon } } g _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ i ) - \Sigma _ { j \neq i } \frac { \hat { q } _ j } { \ln \frac { R } { \varepsilon } } \bar { G } _ { z _ j } ( T _ { z _ j } z _ i , T _ { z _ j } z _ j ) , \end{align*}


\begin{align*} H _ { \mathrm { C V } } ( s , i , p , q , l ) & \coloneqq ( s + i ) l w + \beta ( 1 - \theta l ) ^ 2 s i \left ( q - p \right ) + i \phi ( i ) \left ( \frac { w } { r } + \chi \right ) - ( \gamma + \phi ( i ) ) i q \\ & = \beta \theta ^ 2 s i ( q - p ) l ^ 2 + \left [ ( s + i ) w - 2 \beta \theta s i ( q - p ) \right ] l \\ & + \beta s i ( q - p ) + i \phi ( i ) \left ( \frac { w } { r } + \chi \right ) - ( \gamma + \phi ( i ) ) i q \ , , \end{align*}


\begin{align*} \xi _ z = \frac { 1 } { r ! } \sum _ { j = 1 } ^ { r ! } T ^ j ( \xi _ { i _ z , z } ) \end{align*}


\begin{align*} d F ( X ) = \langle \langle X \cdot \varphi , \varphi \rangle \rangle , \end{align*}


\begin{align*} \begin{aligned} & T ^ { \star } ( k _ 1 , \ldots , k _ n ; ( \alpha , \beta ) ) - T ^ { \star } ( k _ n , k _ 1 , \ldots , k _ { n - 1 } ; ( \alpha , \beta ) ) \\ = & ( k _ n - 1 ) Z ( n | k - n + 1 ; ( \alpha , \beta ) ) + Z ( n + 1 | k - n ; ( \alpha , \beta ) ) \\ & - \sum _ { j = 0 } ^ { k _ n - 2 } Z ^ { \star } _ { I } ( j + 1 , k _ 1 , \ldots , k _ { n - 1 } , k _ n - j ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} x ^ G = \{ g ^ { - 1 } x g \mid g \in G \} , \end{align*}


\begin{align*} q ' _ j & = \sum _ { i \in A _ j } q _ i = \sum _ { i \in [ m ] : \tilde { \delta } _ i \in [ \nu _ j , \nu _ { j + 1 } ) } q _ i = \sum _ { i \in [ m ] : \tilde { \delta } _ i \in [ \nu _ j , \nu _ { j + 1 } ) } 0 . 5 \tilde { q } _ i = 0 . 5 \P \{ \tilde { X } \in [ \nu _ j , \nu _ { j + 1 } ) \} . \end{align*}


\begin{align*} \tilde { V } ( x ) = \frac { 1 } { T } \int _ 0 ^ T \mathcal { P } _ t V ( x ) d t \end{align*}


\begin{align*} M = \left ( \begin{array} { c c } \varepsilon _ 1 \mu & \varepsilon _ 1 \nu \\ \varepsilon _ 3 \mu & \varepsilon _ { 3 } \nu \end{array} \right ) \end{align*}


\begin{align*} \begin{cases} \min \{ \frac { 1 } { 2 } + \frac { \alpha } { 2 \beta } ( r + \lambda - \kappa ) , 1 \} , & \mbox { i f } \alpha + \gamma = 1 , \\ \min \{ \frac { \alpha } { 2 \beta } \min \{ \kappa , r + \lambda \} + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \} , & \mbox { i f } \alpha + \gamma \neq 1 \end{cases} \end{align*}


\begin{align*} g _ 1 ^ { \prime + } ( 0 ) & = \sum _ { i = 1 } ^ n \int _ { \R ^ n } \partial _ i f ( x ) \big ( T _ i ( x _ i ) - x _ i \big ) Q ( x ) d x , \\ g _ 2 ^ { \prime + } ( 0 ) & = - \sum _ { i = 1 } ^ n \int _ { \R } \left ( T _ i ' ( x _ i ) - 1 \right ) Q _ i ( x _ i ) d x _ i . \end{align*}


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


\begin{align*} X : = \{ \sum _ { n = 0 } ^ \infty a _ n z ^ n \mid ( a _ n n ^ { - \alpha } ) _ { n \in \mathbb { N ^ * } } \in \ell ^ 2 \} \end{align*}


\begin{align*} \sigma ( x _ i ) = x _ { \sigma ( i ) } , \ \sigma ( \epsilon ) = \epsilon , \sigma \in S _ n , \ 1 \leq i \leq n . \end{align*}


\begin{align*} - \Delta ( v \psi _ { m } ) ( x ) + F ' ( \overline { y } ; v \psi _ { m } ) ( x ) = 0 , x \in \mathcal { A } , \end{align*}


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \nabla ^ \perp \varphi \cdot \nu | _ { \partial \Omega } = v _ n | _ { \partial \Omega } \ln \frac { 1 } { \varepsilon } . \end{cases} \end{align*}


\begin{align*} & \lim _ { \epsilon \downarrow 0 } \frac { 1 } { 2 \pi i } \int _ { \mathbb { R } } \frac { \mathrm { e } ^ { i s ( \xi - 2 x ) } } { s - ( z + i \epsilon ) } d s = \lim _ { \epsilon \downarrow 0 } \begin{cases} \mathrm { e } ^ { i ( z + i \epsilon ) ( \xi - 2 x ) } , & \xi - 2 x > 0 \\ 0 , & \xi - 2 x < 0 \end{cases} \\ [ 6 p t ] & = \chi ( \xi - 2 x ) e ^ { i z ( \xi - 2 x ) } , \end{align*}


\begin{align*} f ( x ) = ( 2 \pi ) ^ { - n / 2 } \int _ { - \infty } ^ \infty \int _ { S ^ { n - 1 } } \widehat { f } ( \lambda , \omega ) e _ { \lambda , \omega } ( x ) | \lambda | ^ { n - 1 } d \omega \ , d \lambda \end{align*}


\begin{gather*} \Phi ( r ) = \displaystyle \int _ { B _ r } | D u | ^ { p + 2 \beta } d x , \end{gather*}


\begin{align*} F ' _ m ( n ) = F ' _ m ( n - 1 ) . \end{align*}


\begin{align*} \begin{aligned} & \begin{pmatrix} \delta _ - ( z ) & 0 \\ 0 & \delta _ { - } ^ { - 1 } ( z ) \end{pmatrix} ( I + R ( x ; z ) ) \begin{pmatrix} \delta _ + ^ { - 1 } ( z ) & 0 \\ 0 & \delta _ + ( z ) \end{pmatrix} \\ & = \begin{pmatrix} 1 & \delta _ - ( z ) \delta _ + ( z ) \bar { r } _ 1 ( z ) e ^ { - 2 i z x } \\ \bar { \delta } _ + ( z ) \bar { \delta } _ - ( z ) r _ 2 ( z ) e ^ { 2 i z x } & 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) \end{pmatrix} , \end{aligned} \end{align*}


\begin{align*} \Theta ^ q ( x ) = 2 \pi q x + \alpha . \end{align*}


\begin{align*} \begin{aligned} e ^ { - a } \int _ { | v | \leq 1 } | v | ^ k \d v \leq \int _ { | v | \leq 1 } | v | ^ k e ^ { - a | v | ^ 2 } \d v \leq \int _ { | v | \leq 1 } | v | ^ k \d v \ , . \end{aligned} \end{align*}


\begin{align*} \left | \det \begin{pmatrix} x _ j & x _ k \\ y _ j & y _ k \end{pmatrix} \right | \le | x _ j y _ k | + | x _ k y _ j | . \end{align*}


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


\begin{align*} \displaystyle \int _ { 0 } ^ { 1 } y _ \delta ( x _ a , \tau ) \ , d \tau & = \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } \\ \displaystyle \int _ { 0 } ^ { 1 } \left ( y _ \delta ( { x _ a } , \tau ) - \bar { y } _ a \right ) u ( \tau ) \ , d \tau & = \dfrac { b _ { 1 , \delta } ( x _ a ) } { 2 } \end{align*}


\begin{align*} D ( M _ { q } ) = \{ f \in \ell ^ { \infty } \ ; | \ ; q f \in \ell ^ { \infty } \} \end{align*}


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


\begin{align*} f _ { 1 } ^ { + } = 2 \mathcal { I } _ 1 = \int _ { \Omega } | U | ^ { 2 } \dd x = \int _ { \Omega } U \cdot \nabla \times ( \textrm { c u r l } ^ { - 1 } U ) \dd x = f \int _ { \Omega } U \cdot \textrm { c u r l } ^ { - 1 } U \dd x = f . \end{align*}


\begin{align*} \widehat { \mathbf { Q } } = \begin{footnotesize} \begin{pmatrix} \widehat { \mathbf { Q } } _ 1 & 0 \\ 0 & \widehat { \mathbf { Q } } _ 2 \end{pmatrix} \end{footnotesize} \end{align*}


\begin{align*} \begin{cases} \frac { \partial } { \partial t } u _ j ( x , t ) = c _ j ( t ) \frac { \partial } { \partial x } u _ j ( x , t ) , & x \in \left ( 0 , 1 \right ) , \ t \geq s , \\ u _ j ( x , s ) = f _ j ( x ) , & x \in \left ( 0 , 1 \right ) , \\ \phi ^ - _ { i j } c _ j ( t ) u _ j ( 1 , t ) = w _ { i j } \sum _ { k = 1 } ^ m { \phi ^ + _ { i k } c _ k ( t ) u _ k ( 0 , t ) } , & t \geq s , \end{cases} \end{align*}


\begin{align*} b _ t + \nabla \times ( B \times u ) + \mu \nabla \times ( \nabla \times b ) = 0 \textrm { o n } \ ( L ^ { 2 } _ { \sigma } \cap H ^ { 1 } ) ( \mathbb { R } ^ { 3 } ) ^ { * } , \end{align*}


\begin{align*} \mathbb { R } [ X ] / ( X ^ { 2 } + 1 ) \to \mathbb { R } [ X ] / ( X ^ { 2 } + 1 ) \otimes _ { \mathbb { R } } ( \mathbb { R } \times \mathbb { R } ) = \mathbb { R } [ X ] / ( X ^ { 2 } + 1 ) \times \mathbb { R } [ X ] / ( X ^ { 2 } + 1 ) , \ ; i \mapsto ( i , - i ) . \end{align*}


\begin{align*} \partial _ Y \left ( \Phi ( \mathcal { P } ( X ) ) \right ) = \Phi ( \nabla _ Y \mathcal { P } ( X ) ) - \sum _ { i = 1 , 2 } \sqrt { c _ i } \ \langle Y _ i , \mathcal { P } ( X ) \rangle N _ i \\ + \Phi ( B ( Y , \mathcal { P } ( X ) _ T ) - B ^ * ( Y , \mathcal { P } ( X ) _ N ) ) \end{align*}


\begin{align*} & \int _ { X _ j } | u _ { m , m ' , \epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { v _ \epsilon ( \Psi _ m ) - \delta M _ { \eta _ m } } c ( - v _ \epsilon ( \Psi _ m ) ) + \int _ { X _ j } | h _ { m , m ' , \epsilon , j } | ^ 2 _ { h _ { m ' } } e ^ { - \phi - \delta M _ { \eta _ m } } \\ \le & \int _ { X _ j } v '' _ { \epsilon } ( \Psi _ m ) | f F ^ { 1 + \delta } | ^ 2 _ { h _ { m ' } } e ^ { - \phi - \delta M _ { \eta _ m } } < + \infty . \end{align*}


\begin{align*} \pi _ 1 ( \theta _ 2 F _ { 1 } ( \theta _ 1 , \theta _ 2 , \theta _ 3 , \theta _ 4 ) ) = - \theta _ 1 \theta _ 2 + \theta _ 1 \theta _ 3 + \theta _ 1 \theta _ 4 . \end{align*}


\begin{align*} d = 2 n + 1 = 4 l + 3 \equiv 3 \mod 4 \end{align*}


\begin{align*} \frac { \partial j } { \partial s } ( 0 , 0 ) = 2 \int _ { \Omega } \textrm { c u r l } ^ { - 1 } U \cdot U _ 0 \dd x \neq 0 . \end{align*}


\begin{align*} | u _ { m ( p ) - 1 } \dots u _ n | _ p \geq \prod ^ { m ( p ) - n } _ { r = 1 } ( | t _ r | _ p | a ' b ' | _ p ) . \end{align*}


\begin{align*} f = ( a _ 0 , b _ 0 , c _ 0 , d _ 0 ) _ q , \end{align*}


\begin{align*} F _ { \delta } ( x ; z ) : = \left ( \mathcal { P } ^ + \left ( r _ { \delta , 2 } ( z ) e ^ { 2 i z x } \right ) e _ 2 , \mathcal { P } ^ - \left ( \bar { r } _ { \delta , 1 } ( z ) e ^ { - 2 i z x } \right ) e _ 1 \right ) . \end{align*}


\begin{align*} \beta _ n = \sum \limits _ { l = 0 } ^ { n - 1 } \omega ^ l \wedge \omega ^ { n + l } , \beta _ n ^ n = \wedge ^ n \beta _ n = n ! ~ \Omega _ { 2 n } , \end{align*}


\begin{align*} \gamma ^ { q ^ n } = \gamma ^ { q ^ { n - i } } . \end{align*}


\begin{align*} \tilde X _ { n + 1 } ^ i = f ( \tilde X _ n ^ i , \tau ) + \gamma ^ i ( \tilde X _ n ) \tau + \sigma W _ \tau ^ i . \end{align*}


\begin{align*} \left \{ \begin{aligned} a + b x = \lambda , \\ d x = \lambda x . \end{aligned} \right . \end{align*}


\begin{align*} \mathcal { U } _ 3 = \int _ { \kappa _ { \rm b } } ^ { \kappa _ { \rm t } } e ^ { - \frac { 1 } { 2 } \left ( x + \frac { 2 \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) ^ 2 } \ , \mathrm { d } x = \sqrt { { 2 \pi } } \left [ Q \left ( \kappa _ { \rm b } + \frac { 2 \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) - Q \left ( \kappa _ { \rm t } + \frac { 2 \psi _ { 1 } \tau _ { \rm d } } { N _ { \rm a } } \right ) \right ] . \end{align*}


\begin{align*} x y = y x = x \wedge y \end{align*}


\begin{align*} \begin{aligned} y \left [ n \right ] = & \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { { \bf { f } } _ l } s \left [ { n - { \kappa _ l } - { n _ l } } \right ] } + \\ & \sum \limits _ { l = 1 } ^ L { \sum \limits _ { l ' \ne l } ^ L { { \bf { h } } _ l ^ H { { \bf { f } } _ { l ' } } s \left [ { n - { \kappa _ { l ' } } - { n _ l } } \right ] } } + z \left [ n \right ] . \end{aligned} \end{align*}


\begin{align*} \theta ( x ) = \begin{cases} 1 & x \in q ( x ) ( [ 0 , \infty ) ) \\ - 1 & \end{cases} \end{align*}


\begin{align*} \langle \nabla ^ E _ X [ e _ i ] , [ e _ j ] \rangle = \sum _ k \varphi _ k \phi _ { i k } ( X ) d _ { k j } = \phi _ { i j } ( X ) + \sum _ k \varphi _ k \phi _ { i k } ( X ) = \phi _ { i j } ( X ) , \end{align*}


\begin{align*} P _ { n , k } ( z ) = z ^ k A _ n ( z ) . \end{align*}


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


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


\begin{align*} \deg ( F ) = \max \{ i \ : \ A _ i \ne 0 \} \end{align*}


\begin{align*} \Gamma _ k = \{ \lambda \in \mathbb { R } ^ n : \sigma _ j ( \lambda ) > 0 , 1 \leq j \leq k \} . \end{align*}


\begin{align*} v _ j = \int _ { \mathbb S ^ 2 } e ^ { \mathrm i k \xi \cdot \mathbf x } g _ j ( \xi ) \mathrm d \xi , \xi \in \mathbb S ^ 2 , j = 1 , 2 , \cdots , \end{align*}


\begin{align*} - \frac { 1 } { \sqrt { 1 + { f ' } ^ 2 } } = \frac { \eta } { m + 1 } f ^ { m + 1 } + \lambda f + c , \end{align*}


\begin{align*} Q _ { j , + } ( x ; k ) = ( I + J ) Q _ { j , - } ( x ; k ) + D _ j , k \in \mathbb { R } \cup i \mathbb { R } , \end{align*}


\begin{align*} \beta ( a + 1 ) = \begin{cases} 1 , & a + 1 \in \mathbb { P } , \\ 0 , & a + 1 \not \in \mathbb { P } . \end{cases} \end{align*}


\begin{align*} \kappa ( s ) = \eta x _ { n + 1 } ^ m ( s ) + \lambda . \end{align*}


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


\begin{align*} \begin{aligned} \tfrac { \chi ( \sqrt { | V _ \shortparallel | ^ 2 + | V _ \perp ' - \zeta _ \perp | ^ 2 } ) } { ( | V _ \shortparallel | ^ 2 + | V _ \perp ' - \zeta _ \perp | ^ 2 ) ^ { \frac { 1 - \gamma } { 2 } } } \leq \tfrac { 2 ^ { 1 - \gamma } } { ( 1 + | V _ \shortparallel | ^ 2 + | V _ \perp ' - \zeta _ \perp | ^ 2 ) ^ { \frac { 1 - \gamma } { 2 } } } \ , . \end{aligned} \end{align*}


\begin{align*} d _ 1 = \begin{pmatrix} 0 & i \mu ( g ^ { \alpha \beta } \xi _ \beta ) & 0 \\ i \lambda ( \xi _ \beta ) & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} , d _ 0 = \begin{pmatrix} 0 & 0 & 0 \\ \lambda ( \Gamma ^ \alpha _ { \alpha \beta } ) & \lambda \Gamma ^ \alpha _ { \alpha n } & - \beta \\ 0 & 0 & 0 \end{pmatrix} . \end{align*}


\begin{align*} \eta _ t : = \exp \int _ t ^ T ( A - B ^ 2 R ^ { - 1 } P _ s ) d s . \end{align*}


\begin{align*} { \bf T } _ { s } ( { \bf A } ) = \left \{ W = \left \langle \left [ \theta _ { 1 } \right ] , \left [ \theta _ { 2 } \right ] , \dots , \left [ \theta _ { s } \right ] \right \rangle \in G _ { s } \left ( { \rm H ^ { 2 } } \left ( { \bf A } , \mathbb C \right ) \right ) : \bigcap \limits _ { i = 1 } ^ { s } \operatorname { A n n } ( \theta _ { i } ) \cap \operatorname { A n n } ( { \bf A } ) = 0 \right \} , \end{align*}


\begin{align*} \Psi ' = ( \tau ^ { * } \alpha _ { \underline { S } \times \underline { X } } \circ \tau ^ { * } \Phi \circ \alpha _ { \underline { S } \times \underline { X } } ) \circ \Psi \circ \Phi ^ { - 1 } \end{align*}


\begin{align*} a _ n ( u _ n , v _ n ) : = \sum _ { K \in { \mathcal T } _ n } \int _ K D ^ 2 u _ n : D ^ 2 v _ n d K , u _ n , v _ n \in X _ n . \end{align*}


\begin{align*} D _ { e _ i } e _ j = 0 , h _ { i j } = \delta _ { i j } \kappa _ i , \kappa _ 1 \geq \cdots \geq \kappa _ n . \end{align*}


\begin{align*} W _ r ( x ) = \frac 1 2 \hat W _ r ( 0 ) + \sum _ { k = 1 } ^ \infty \hat W _ r ( k ) \cos ( 2 \pi k x ) , \end{align*}


\begin{align*} t ^ { \alpha } - s ^ { \alpha } = t ^ { \alpha - \varrho } t ^ { \varrho } - s ^ { \alpha - \varrho } s ^ { \varrho } \leq s ^ { \alpha - \varrho } t ^ { \varrho } - s ^ { \alpha - \varrho } s ^ { \varrho } \leq s ^ { \alpha - \varrho } ( t - s ) ^ { \varrho } , \end{align*}


\begin{align*} \varphi '' + \Big ( ( m - 1 ) \frac { \sigma ' } { \sigma } - \Phi ' \Big ) \varphi ' - f ' ( u ) \leq 0 \textnormal { i n } \ I = ( r _ 1 , r _ 2 ) . \end{align*}


\begin{align*} \gamma ( X _ i , X _ i ) = \langle A X _ i , X _ i \rangle [ e _ i ] \quad \forall i , \end{align*}


\begin{align*} T = \Big ( \Z + \frac { 1 } { 2 } \Big ) \cup i \Big ( \Z + \frac { 1 } { 2 } \Big ) , A ( z ) = \frac { 1 } { \pi } \cos ( \pi z ) \cos ( \pi i z ) \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \dfrac { \sup _ { | u | \leq M } \sum _ { i = 1 } ^ d a _ i ( t , x , u , z ) z _ i } { | z | + | z | ^ { \alpha - 1 } } = \infty . \end{align*}


\begin{align*} F ( s , x ) = \left | \frac { \zeta ' } { \zeta } ( s ) + \sum _ { n < x ^ 2 } \frac { \Lambda _ x ( n ) } { n ^ s } \right | \leq | Z _ 1 | + | Z _ 2 | + | Z _ 3 | \end{align*}


\begin{gather*} E ( X - Y ) ^ { 2 k } \allowbreak = \allowbreak E ( X + \rho Y - ( 1 + \rho ) Y ) ^ { 2 k } \allowbreak = \\ \allowbreak \sum _ { j = 0 } ^ { 2 k } \binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \rho ) ^ { 2 k - j } E ( X + \rho Y ) ^ { j } Y ^ { 2 k - j } \allowbreak = \\ \allowbreak \sum _ { j = 0 } ^ { 2 k } \binom { 2 k } { j } ( - 1 ) ^ { j } ( 1 + \rho ) ^ { 2 k - j } E ( E ( X + \rho Y ) ^ { j } | Y ) Y ^ { 2 k - j } \allowbreak . \end{gather*}


\begin{align*} C _ { B _ n } ( \sigma ) = \{ \tau \in B _ n : \sigma \tau = \tau \sigma \} \end{align*}


\begin{align*} \sum _ { \gamma } f _ { \frac { 1 } { 2 } } ( t - \gamma ) \leq \frac { \widehat { h } ( 0 ) } { 2 \pi } \log t + \frac { 1 } { \pi } \sum _ { n = 2 } ^ \infty \frac { \Lambda ( n ) } { \sqrt { n } } \left | \widehat { h } \left ( \frac { \log n } { 2 \pi } \right ) \right | . \end{align*}


\begin{align*} c _ { q ; d } = \begin{cases} \Big ( 2 ^ { \frac d 2 - 1 } \Big ( \frac d 2 - 1 \Big ) ! \Big ) ^ q \int _ 0 ^ \infty J _ { \frac d 2 - 1 } ( u ) ^ q u ^ { - q ( \frac d 2 - 1 ) + d - 1 } d u & \mbox { i f } q \geq 3 , \cr \frac { ( d - 1 ) ! \mu _ d } { 4 \mu _ { d - 1 } } & \mbox { i f } q = 2 , \\ \end{cases} \end{align*}


\begin{align*} \sum _ { j = 1 } ^ d y _ { n , j } < \min _ { k = 1 , 2 , \ldots , d } \left ( x _ { n + 1 , k } + \sum _ { j \neq k } x _ { 0 , j } \right ) \forall n \geq N _ 0 . \end{align*}


\begin{align*} a ( u , v ) = \lambda ( u , v ) ~ v \in H ^ 1 _ 0 ( D ) , \end{align*}


\begin{align*} E [ t ] ^ { \vee } = E ^ { \vee } [ t ^ { - 1 } ] t > 0 . \end{align*}


\begin{align*} Y _ \infty = \left ( \frac { m } { \textrm { G e V } } \right ) ^ { - 3 } \left [ 1 + \frac { 3 \ln ( m / \textrm { G e V } ) } { 1 5 } + \frac { \ln ( c _ 2 / 5 ) } { 1 5 } \right ] \end{align*}


\begin{align*} C \le M _ n ^ { \psi } ( Q ^ n ) / n = T _ { W _ { n J } , \psi } ( \mu _ n ( Q ^ n ) ) - I ( \mu _ n ( Q ^ n ) ) \le - I ( \mu _ n ( Q ^ n ) ) . \end{align*}


\begin{align*} i x , x ' \in E _ i , ~ \frac { q ( x ) } { p ( x ) } = \frac { q ( x ' ) } { p ( x ' ) } , \end{align*}


\begin{align*} \mathcal { U } ( E , \Omega ) : = \{ u \in Q S H _ m ( \Omega ) , u \vert _ { \Omega } \leq 0 , u \vert _ { E } \leq - 1 \} , \end{align*}


\begin{align*} G = \begin{pmatrix} \mathbf { 1 } _ { n - 1 } & 1 \\ G _ { n - 1 } & \mathbf { 0 } _ { n - 1 } ^ T \end{pmatrix} , \end{align*}


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


\begin{align*} \int _ { \R ^ { n - 1 } } P ^ L ( x ' ) \ , d x ' = I _ { N \times N } , \end{align*}


\begin{align*} A _ { l , u } & = \left \{ i \in [ k ] : \frac { p _ i } { q _ i } \in [ l , u ) \right \} \ , \ , \ , \ , \\ A _ { l , \infty } & = \left \{ i \in [ k ] : \frac { p _ i } { q _ i } \in [ l , \infty ] \right \} . \end{align*}


\begin{align*} \int _ { \hat { B } _ { \delta } ( 0 ) \setminus \hat { B } _ { \varepsilon } ( 0 ) } q ^ 2 ( K _ { H } ) _ { 2 2 } ( x + \bar { x } ) \partial _ 2 \hat { U } \left ( \frac { x } { \varepsilon } \right ) \partial _ 2 \hat { U } \left ( \frac { x } { \varepsilon } \right ) d x = \frac { l _ 1 } { l _ 2 } \pi q ^ 2 ( K _ { H } ) _ { 2 2 } ( \bar { x } ) \ln \frac { \delta } { \varepsilon } + O ( 1 ) . \end{align*}


\begin{align*} F ' _ m ( n ) = 1 - G ' ( n , m ) + m \left \lfloor \frac { n } { m } \right \rfloor \sum _ { \begin{smallmatrix} k = 1 \\ k | m \end{smallmatrix} } ^ m \frac { \mu ( k ) } { k } . \end{align*}


\begin{align*} \textup { S t o C } : = \min \Big \{ \alpha + \gamma - \frac { 1 } { 2 } - \frac { \alpha } { 2 \beta } ( \kappa - r ) , 1 - \varepsilon \Big \} = \min \Big \{ \frac { \alpha r } { 2 \beta } + ( \gamma - \frac { 1 } { 2 } ) ^ { + } , 1 - \varepsilon \Big \} . \end{align*}


\begin{align*} \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( X _ n \notin \Upsilon \right ) \leq & \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( X _ n \notin K _ M \right ) + \sum _ { k = 1 } ^ { \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( X _ n \notin \Gamma _ k \right ) \\ \leq & \frac { \varepsilon } { 2 } + \sum _ { k = 1 } ^ { \infty } \frac { \varepsilon } { 2 ^ { k + 1 } } \leq \varepsilon . \end{align*}


\begin{align*} \left \| \partial _ { z } \mathcal { H } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) \right \| _ { L ^ 2 } = \left \| \partial _ { z } \log \left ( 1 + \bar { r } _ 1 r _ 2 \right ) \right \| _ { L ^ 2 } . \end{align*}


\begin{align*} \mu ( u ) = \frac { \pi ( u ) } { \sum _ { v \in V } \pi ( v ) } , u \in V . \end{align*}


\begin{align*} b _ { \sigma \otimes \tau } = b _ { \sigma } \otimes b _ { \tau } . \end{align*}


\begin{align*} \beta _ { q } ( n ) = \lim _ { v \to \infty } \frac { \# \{ \vec { x } \in ( \Z / p ^ { v } \Z ) ^ { 4 } | Q ( \vec { x } ) \equiv n \pmod { p ^ { v } } \} } { p ^ { 3 v } } , \end{align*}


\begin{align*} \sum _ { \alpha , \beta = 1 } ^ d \xi _ \alpha \cdot \left ( \nabla G ( U , x , t ) ^ T B _ { \alpha \beta } ( U , x , t ) \ , \xi _ \beta \right ) = \sum _ { \alpha , \beta = 1 } ^ d \sum _ { i , j = 1 } ^ n \xi _ \alpha ^ i D _ { \alpha \beta } ^ { i j } \xi _ \beta ^ j > 0 \end{align*}


\begin{align*} \xi ( F ) = u _ \S ( F ) + 2 . \end{align*}


\begin{align*} \Phi _ 2 ( \mu ^ t ) = \int _ 0 ^ 1 H \big ( \mu ^ 0 _ u \circ ( ( 1 - t ) \mathrm { I d } + t T _ u ) ^ { - 1 } \big ) \ , d u \end{align*}


\begin{align*} \nabla u ( x _ { 0 } ) \in \mathrm { s p a n } { \nabla d ( x _ { 0 } ) } = ( T _ { x } \Sigma _ { d ( x _ { 0 } ) } ) ^ { \perp } . \end{align*}


\begin{align*} \omega : = \nabla \times u = \partial _ { x _ 1 } u _ { 2 } - \partial _ { x _ 2 } u _ { 1 } \end{align*}


\begin{align*} x \cdot B ( x , x ) = 0 . \end{align*}


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


\begin{align*} h = 0 , \nabla h = 0 , \nabla ^ { k + 2 } h = 0 \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} s _ { \min } ( A ) = s _ n ( A ) = \min \limits _ { x : \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 , s _ { \max } ( A ) = s _ 1 ( A ) = \max \limits _ { x : \ , \lVert x \rVert _ 2 = 1 } \lVert A x \rVert _ 2 . \end{align*}


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


\begin{align*} \Delta _ { h } = \sum _ { a = 1 } ^ { 4 } \chi ^ { - 1 } \cdot \delta _ { a } \cdot \chi \cdot \delta _ { a } \cdot \chi ^ { - 1 } , \end{align*}


\begin{align*} \mbox { $ ( ( X _ { 2 , 6 } \cap N ( c _ 2 ) ) \cup ( X _ { 2 , 7 } \cap N ( c _ 2 ) ) \cup \{ c _ 2 \} ) \cap { \cal N } = \emptyset $ . } \end{align*}


\begin{align*} y \left [ n \right ] = \left ( { \sum \limits _ { l = 1 } ^ L { { \bf { h } } _ l ^ H { \bf { H } } _ l ^ \bot { { \bf { b } } _ l } } } \right ) s \left [ { n - { n _ { \max } } } \right ] + z \left [ n \right ] . \end{align*}


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


\begin{align*} & \hphantom { { } = { } } D _ { ( o , o ) } ( \{ f , g \} . \{ f ' , g ' \} ) \\ * & = \begin{pmatrix} D f ( o ) D f ' ( o ) \\ D g ( o ) D f ' ( o ) + D f ( o ) D g ' ( o ) & D f ( o ) D f ' ( o ) \end{pmatrix} \\ * & = \begin{pmatrix} D f ( o ) \\ D g ( o ) & D f ( o ) \end{pmatrix} \begin{pmatrix} D f ' ( o ) \\ D g ' ( o ) & D f ' ( o ) \end{pmatrix} . \end{align*}


\begin{align*} N ( d ) \simeq N ( \ell ) \times _ { \psi } N ( d ' ) : = \{ ( n , n ' ) \in N ( \ell ) \times N ( d ' ) : \psi ( n ) = \psi ' ( n ' ) \} . \end{align*}


\begin{align*} B _ { \sigma , \lambda } = \dfrac { ( 2 \sigma - 1 ) \left ( 1 - \exp \left ( - \frac { 2 \lambda ( 1 - \sigma ) } { 2 \sigma - 1 } \right ) \right ) } { 2 \lambda ( 1 - \sigma ) ^ 2 } . \end{align*}


\begin{align*} ( \Delta \omega ^ * ( q , K , \Omega ) ) ^ m \wedge \beta _ n ^ { n - m } = 0 \Omega \setminus K . \end{align*}


\begin{align*} \widetilde { \Phi } ( x ' ) : = \Phi ^ { \top } ( - x ' ) , \widetilde { \Psi } _ { \varepsilon , \varepsilon ^ { - 1 } } ( x ' ) : = \Psi ^ { \top } _ { \varepsilon , \varepsilon ^ { - 1 } } ( - x ' ) , \widetilde { P } ^ L ( x ' ) : = ( P ^ L ) ^ { \top } ( - x ' ) , x ' \in \R ^ { n - 1 } , \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 \Delta w = ( w - a ) ^ { p } _ + , \ \ & \ B _ R ( 0 ) , \\ w = 0 , \ \ & \ \partial B _ R ( 0 ) , \end{cases} \end{align*}


\begin{align*} F ( x ) = \frac { 1 } { \operatorname { B } ( a , b ) } \int ^ { \infty } _ { \frac { \theta } { x ^ 2 } } \exp ( - a u ) \left \{ 1 - \exp ( - u ) \right \} ^ { b - 1 } \mathrm { d } u . \end{align*}


\begin{align*} \lambda _ t = \prod _ { i = 1 } ^ { t } ( 1 - \alpha _ i ) \leq \prod _ { i = 1 } ^ { t } \left ( 1 - \sqrt { \mu \gamma _ i } \right ) . \end{align*}


\begin{align*} H ( Q _ k \ , | \ , P ) = - \log P ( B _ k ) \to 0 . \end{align*}


\begin{align*} \left \{ \begin{aligned} d x _ t ^ { t _ 0 , x _ 0 , \xi , \alpha } = & ~ [ A x _ t ^ { t _ 0 , x _ 0 , \xi , \alpha } + B \alpha _ t + f ( \nu _ t ^ { * , t _ 0 , \xi } ) + b ( \mu _ t ^ { * , t _ 0 , \xi } ) ] d t + \sigma d W _ t + \sigma _ 0 d W ^ 0 _ t , \\ x _ { t _ 0 } ^ { t _ 0 , x _ 0 , \xi , \alpha } = & ~ x _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} - \sum \limits _ { j = 3 } ^ { n - 1 } + ( n - 2 ) r \geq - ( n - 3 ) r + ( n - 2 ) r = r \end{align*}


\begin{align*} \epsilon _ n = \max _ { | \eta - \lambda | \le \delta } \max _ { v \in G ( \lambda ) } \| F _ n ( \eta ) p _ n v - q _ n F ( \eta ) v \| _ { X _ n } \end{align*}


\begin{align*} \int _ { B _ R } F ( x , D \tilde { u } ) d x = \lim _ { j _ 0 \to + \infty } \int _ { B _ R } F _ { j _ 0 } ( x , D \tilde { u } ) d x \leq \int _ { B _ R } F ( x , D u ) d x , \end{align*}


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


\begin{align*} \lambda | _ { E _ { k , [ p , q ] } } ^ * ( a ) = \lambda ( p + a ) \end{align*}


\begin{align*} F ^ { q , \dagger } _ v ( 0 , p _ 0 ) \psi ^ \dagger _ { v v } ( 0 , p _ 0 ) [ u _ \ell , u _ \ell ] = - c _ 2 ( q , \ell , p _ 0 ) u _ { 2 \ell } . \end{align*}


\begin{align*} Y [ k ] = \alpha X [ k ] + W _ { \rm d } [ k ] + W _ { \rm s } [ k ] , \end{align*}


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


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


\begin{align*} D ( p ) = \frac { \Gamma ( 2 x ) } { \Gamma ( x + \frac 1 2 ) ^ 2 \Gamma ( x + \frac 3 2 ) } = \frac { 2 ^ { 2 x - 1 } \Gamma ( x ) } { \sqrt { \pi } \Gamma ( x + \frac { 1 } { 2 } ) \Gamma ( x + \frac { 3 } { 2 } ) } . \end{align*}


\begin{align*} \begin{pmatrix} A _ { \mathrm { c } \mathrm { c } } - A _ { \mathrm { R } \mathrm { c } } ^ { \top } R + R ^ { \top } A _ { \mathrm { R } \mathrm { c } } & B _ 0 ^ { \top } \\ B _ 0 & 0 \\ \end{pmatrix} \begin{pmatrix} U _ \mathrm { c } \\ P _ 0 \end{pmatrix} = \begin{pmatrix} F _ \mathrm { c } - R ^ \top F _ \mathrm { R } \\ 0 \end{pmatrix} , \end{align*}


\begin{align*} g ( \nabla \Psi , \nabla X ( u ) ) = D _ { X } g ( \nabla \Psi , \nabla u ) . \end{align*}


\begin{align*} \lambda _ 1 = n - 1 , \lambda _ 2 = n - 3 , \cdots , \lambda _ { i } = n + 1 - 2 i , \cdots , \lambda _ n = 1 - n . \end{align*}


\begin{align*} b - b _ 1 = \sum _ { i = 2 } ^ r b _ i = l ( p - q ) + ( p - q r ) + 1 . \end{align*}


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


\begin{align*} x ( [ \mathbf { a ' } , \mathbf { a } ] \otimes [ 0 , \mathbf { b } ] ) = 0 \end{align*}


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


\begin{align*} \left ( - \sqrt { f ( r ) } \frac { d } { d r } f ( r ) \frac { d } { d r } \sqrt { f ( r ) } + V ( r ) - E \right ) \psi ( r ) = 0 , \end{align*}


\begin{align*} \overline { \pi } _ 1 ( q ) = \eta ( q ) \overline { \pi } ( q ) \end{align*}


\begin{align*} \hat M ^ { ( k ) } _ { i , j } & = \begin{cases} 0 , & \mbox { i f } i \in [ k + 1 , n ] \mbox { a n d } j = k , \\ { \rm f l } \Big ( \hat M ^ { ( k - 1 ) } _ { i , j } - { \rm f l } \Big ( { \rm f l } \Big ( \frac { \hat M ^ { ( k - 1 ) } _ { i , k } } { \hat M ^ { ( k - 1 ) } _ { k , k } } \Big ) \ , \hat M ^ { ( k - 1 ) } _ { k , j } \Big ) \Big ) , & \mbox { i f } i , j \in [ k + 1 , n ] , \\ \hat M ^ { ( k - 1 ) } _ { i , j } , & \mbox { o t h e r w i s e } \end{cases} \end{align*}


\begin{align*} - \varepsilon ^ 2 ( K ( x ) \nabla u ) + V ( x ) u = u ^ { p } , \ \ \ \ x \in \mathbb { R } ^ n , \end{align*}


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


\begin{align*} \begin{aligned} \dot { x } ( t ) & - \dot { z } ( t ) = \\ & - \gamma ( h ( x ( t ) + \delta u ( t ) ) - h ( z ( t ) + \delta u ( t ) ) ) u ( t ) \\ & - \gamma ( h ( z ( t ) + \delta u ( t ) ) - h ( x _ a ( t ) + \delta u ( t ) ) ) u ( t ) \\ & - \left . \gamma ^ 2 \dfrac { \partial \epsilon ( x , t ) } { \partial x } \right | _ { x = x _ a ( t ) } \dfrac { b _ { 1 , \delta } ( x _ a ( t ) ) } { 2 } . \end{aligned} \end{align*}


\begin{align*} \begin{cases} - \delta ^ 2 ( K _ H ( x ) \nabla v ) = \left ( v - \left ( \frac { \alpha | x | ^ 2 } { 2 } + \beta \right ) \right ) ^ { p } _ + , \ \ & x \in B _ { R ^ * } ( 0 ) , \\ v = 0 , \ \ & x \in \partial B _ { R ^ * } ( 0 ) . \end{cases} \end{align*}


\begin{align*} x _ { n + 1 } = \begin{cases} x _ n - 1 , & x _ n > 1 \\ n + 1 , & x _ n = 1 . \end{cases} \end{align*}


\begin{align*} \begin{aligned} a _ { k , \delta } ( x _ a ) & : = 2 \int _ { 0 } ^ { 1 } y _ \delta ( x _ a , t ) \cos ( k 2 \pi t ) \ , d t \\ b _ { k , \delta } ( x _ a ) & : = 2 \int _ { 0 } ^ { 1 } y _ \delta ( x _ a , t ) \sin ( k 2 \pi t ) \ , d t \ , . \end{aligned} \end{align*}


\begin{align*} \varprojlim _ { n \in \Delta } \mathcal F ( Y _ n ) = \varprojlim _ { k \in \Delta } \mathcal C ^ k . \end{align*}


\begin{align*} V ( A , - C ) - \sqrt { | A | \ , | C | } = | A | . \end{align*}


\begin{align*} \partial \omega = \partial _ 1 \omega \cup \partial _ 2 \omega \cup \partial _ 3 \omega , \end{align*}


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


\begin{align*} ( d / d t ) v _ k + i \lambda _ k v _ k = 0 , \ ; \ ; 1 \leqslant k \leqslant n . \end{align*}


\begin{align*} E ^ L _ Y ( u ) \supset E ^ L _ Y ( s \sigma ( v ' ) w ) = \varphi _ { \sigma , s } ^ L ( E ^ L _ X ( v ' ) ) = \varphi _ { \sigma , s } ^ L ( E ^ L _ X ( v ) ) . \end{align*}