\begin{align*} & \mathcal A z + \frac { \partial f } { \partial y } ( x , y _ u ) z = v , \\ & \mathcal A z + \frac { \partial f } { \partial y } ( x , y _ u ) z = - \frac { \partial ^ 2 f } { \partial y ^ 2 } ( x , y _ u ) z _ { u , v _ 1 } z _ { u , v _ 2 } , \end{align*}


\begin{align*} c _ \Delta : = \sup _ { \xi _ \Delta } \norm { c _ \Delta ( \cdot , \xi _ \Delta ) } _ \infty > 0 . \end{align*}


\begin{align*} ( \ell ^ { \infty } , \beta _ { 0 } ) ' = ( \ell ^ { \infty } , \mu ( \ell ^ { \infty } , \ell ^ { 1 } ) ) ' = \ell ^ { 1 } . \end{align*}


\begin{align*} \sum _ { x y \ne 1 } f _ { x , y } = A + B , \sum _ { z \in \bar { \mathbb { F } } ^ * _ q } g _ { z } = C , \sum _ { u \in \bar { \mathbb { F } } _ q } h _ { u } = 0 . \end{align*}


\begin{align*} u ( p ) + H ( p , u ( p ) , \nabla _ H u ( p ) ) = f ( p ) , \end{align*}


\begin{align*} \begin{cases} d X _ { t , n } = X _ { t , 1 } X _ { t , n - 1 } d t + \sigma _ { n } d W _ t ^ { ( n ) } \\ d X _ { t , n - 1 } = \sigma _ { n - 1 } d W _ t ^ { ( n - 1 ) } \\ d X _ { t , j } = 0 & j \not \in \{ n , n - 1 \} \end{cases} \end{align*}


\begin{align*} a \in F \textup { a n d } \pi ( b ) = \pi ( a ) \implies b \in F \end{align*}


\begin{align*} \nabla h ( \Omega , x ) = h _ { i } e _ { i } + h x , F _ { i } ( x ) = ( h _ { i j } + h \delta _ { i j } ) e _ { j } . \end{align*}


\begin{align*} T _ 1 ( u , J u , J u , u , v ) = T _ 2 ( u , J u , J u , u , v ) , \end{align*}


\begin{align*} x \diamond y = x \cdot D ( y ) , \forall x , y \in \mathcal { H } ( A ) . \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*} W ' ( r ) = \frac { \xi ' } { r } f + \eta ' + \zeta ' \frac { r } { f } , \xi ' \le 0 , \zeta ' > 0 . \end{align*}


\begin{align*} | ( k + 1 ) ^ { 1 - s } - k ^ { 1 - s } | = | 1 - s | \left \lvert \int _ k ^ { k + 1 } y ^ { - s } \dd y \right \rvert \asymp | 1 - s | k ^ { - \Re ( s ) } \end{align*}


\begin{align*} x ' = 2 \ , . \end{align*}


\begin{align*} ( 1 + h ) \cdot p ( h ) = q \Big ( \frac { h } { 1 + h } \Big ) \in A _ * \P ^ n . \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*} W = \bigoplus _ { \alpha = 1 } ^ r \Omega ^ { a ( \alpha ) } \end{align*}


\begin{align*} \widetilde { q } _ { 2 d } ( 2 k ) = e ^ { \frac { i \pi ( 2 k ) ^ 2 } { 2 d } } = q _ d ( k ) . \end{align*}


\begin{align*} | | B _ q ( \psi _ m ) | | _ { M L _ q ( \mathbb { C } ) } = 1 = | | \psi _ m | | _ { L ^ 2 ( \mathbb { R } ) } . \end{align*}


\begin{align*} \prod ^ e _ { \substack { i = 1 \\ ( i , e ) = 1 } } \Phi _ { n , e } ^ { ( i ) } ( x , y ) = \Phi _ { n e } ( x , y ) . \end{align*}


\begin{align*} f _ { \alpha } : = D _ { \theta _ { ( 2 ) } } \left ( \frac { \theta _ { \alpha , ( 2 ) } ( f ) } { \alpha ! } \right ) , \end{align*}


\begin{align*} x = \sum _ { m = 0 } ^ \infty \frac { \psi ( \eta _ m ) } { \gamma ^ { m + 1 } } . \end{align*}


\begin{align*} | \phi | & = \sum _ { i \in \overline { \phi } } | \phi _ i | + \sum _ { i \notin \overline { \phi } } | \phi _ i | \\ & = | \overline { \phi } | ( \dim K + 1 ) + ( m - | \overline { \phi } | ) ( \dim L + 1 ) \\ & = ( \dim M + 1 ) ( \dim K + 1 ) + ( m - \dim M - 1 ) ( \dim L + 1 ) \end{align*}


\begin{align*} \inf \{ t \geq 0 : \sigma ( x , t ) = 0 \} & = T ( x ) , \\ \mu ( x , T ( x ) ) & = h ( x ) . \end{align*}


\begin{align*} | \phi ( y ) | \leq & \frac { | \tilde { P } _ { x _ 2 - y } ( \theta + ( y + 1 ) \alpha ) | } { | \tilde { P } _ { 2 q _ n - 1 } ( \theta + x _ 1 \alpha ) | } \prod _ { j = x _ 1 } ^ { y } | \cos ( \pi ( \theta + j \alpha ) ) | \cdot | \phi ( x _ 1 - 1 ) | \\ & + \frac { | \tilde { P } _ { y - x _ 1 } ( \theta + x _ 1 \alpha ) | } { | \tilde { P } _ { 2 q _ n - 1 } ( \theta + x _ 1 \alpha ) | } \prod _ { j = y } ^ { x _ 2 } | \cos ( \pi ( \theta + j \alpha ) ) | \cdot | \phi ( x _ 2 + 1 ) | , \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*} \partial _ i g ( x ) = \frac { 1 } { n } D _ m G \bigg ( \frac { 1 } { n } \sum _ { k = 1 } ^ n \delta _ { x _ k } , x _ i \bigg ) , \partial _ { i j } g ( x ) = \frac { 1 } { n ^ 2 } D _ m ^ 2 G \bigg ( \frac { 1 } { n } \sum _ { k = 1 } ^ n \delta _ { x _ k } , x _ i , x _ j \bigg ) , \ \ i \neq j . \end{align*}


\begin{align*} 0 = \ell ( 1 , \alpha ) = \ell ( x x ^ { - 1 } , \alpha ) = \ell ( x , w ^ { - 1 } \alpha ) + \ell ( x ^ { - 1 } , \alpha ) . \end{align*}


\begin{align*} a _ { j k } : = \int _ { T } \mathrm { d i v } ( { \boldsymbol \psi } ^ { \mathrm { R T } _ 2 } _ { j } ) \chi _ k \ , d { \boldsymbol x } = \frac { 4 \delta _ { j k } - 3 } { 1 8 0 } j , k = 1 , 2 , 3 , \end{align*}


\begin{align*} u ( \upsilon ) = \frac { 4 \upsilon + \frac 1 2 f ( 2 \upsilon ) - 2 f ( \upsilon ) } { f ( \upsilon ) } . \end{align*}


\begin{align*} I _ 2 = - \frac { 1 } { 2 i z } ( 1 - e ^ { 2 i z \delta } ) \nu ( x ; z ) . \end{align*}


\begin{align*} \int _ { R _ { 0 } } ^ { \infty } \nu \xi ( \nu , s ) \ d \nu = 0 , s \in ( 0 , t _ { 0 } ) . \end{align*}


\begin{align*} ( 1 - x ) ^ { - a } - 1 = a x { } _ { 2 } F _ { 1 } \left [ \left . \begin{matrix} 1 , a + 1 \\ 2 \end{matrix} \right | x \right ] , \end{align*}


\begin{align*} \forall x \in \Z ^ d \ \forall \Delta \Subset \Z ^ d \ \forall \eta \in \Omega : c _ { \Delta + x } ( \tau _ x \eta , \cdot ) = c _ \Delta ( \eta , \cdot ) , \end{align*}


\begin{align*} \lim _ { s \rightarrow 0 } s L _ { q } [ f ( t ) ] = \lim _ { t \rightarrow \infty } \frac { f ( t ) } { 1 + ( 1 - q ) } . \end{align*}


\begin{align*} \left \| \frac { 1 } { s ^ { \alpha } } \right \| _ { L ^ { q } ( \mathbb { R } ) } = C \frac { ( r r ' ) ^ { \alpha } } { | r - r ' | ^ { 2 \alpha - 1 / q } } , \alpha > \frac { 1 } { 2 q } , \end{align*}


\begin{align*} \begin{cases} \nabla W \cdot \nabla ^ \perp \left ( \varPhi - \frac { \alpha } { 2 } | x ' | ^ 2 | \ln \varepsilon | \right ) = 0 , \\ W = \mathcal { L } _ H \varPhi , \\ \varPhi | _ { \partial B _ { R ^ * } ( 0 ) } = 0 . \end{cases} \end{align*}


\begin{align*} N ( \eta ) : = \frac { \operatorname { v o l } ( B ^ { n - 1 } ) \operatorname { v o l } ( \partial \Omega ) } { ( 2 \pi ) ^ { n - 1 } } \eta ^ { n - 1 } + o ( \eta ^ { n - 1 } ) \ \eta \to + \infty , \end{align*}


\begin{align*} \alpha ( x \cdot y ) = \alpha ( x ) \cdot \alpha ( y ) , \forall x , y \in A , \end{align*}


\begin{align*} \mathcal { P } ' \left ( \partial _ Y \Phi ( X ) \right ) = \partial _ Y \left ( \Phi \left ( \mathcal { P } ( X ) \right ) \right ) . \end{align*}


\begin{align*} 0 = \int _ { B _ 1 ( 0 ) } \phi ^ { p - 1 } _ + \frac { \partial \phi } { \partial x _ h } \hat { u } _ i . \end{align*}


\begin{align*} \ell ( x _ 0 ) = \ell ( x _ 1 ) = \ell ( x _ L x _ 0 x _ R ) = \ell ( x _ 0 x _ R ) - \ell ( x _ L ) . \end{align*}


\begin{align*} \int _ { \Gamma ( t ) } c ( t ) \ , \mathrm { d } \mathcal { A } = m \end{align*}


\begin{align*} A _ k = \sum _ { j = 0 } ^ { k - 1 } \binom { k } { j } B _ j . \end{align*}


\begin{align*} t / v = \mu ( \infty ) \textrm { i s a z e r o o f $ f $ } \iff \infty \textrm { i s a z e r o o f } f ' \end{align*}


\begin{align*} \beta = \alpha - \alpha _ j j \geq s . \end{align*}


\begin{align*} d \log \varphi _ { \lambda , u } = \sum _ { j = 1 } ^ { l } u _ j \frac { d f _ j } { f _ j } + \sum _ { i = 0 } ^ { n } \lambda _ i \frac { d x _ i } { x _ i } \end{align*}


\begin{align*} F ( u , v ) & = \frac { p } { p \alpha - 1 } \Big ( \alpha a ( u , v ) + ( \alpha - 1 ) b ( u , v ) + ( 3 \alpha - 2 ) c ( u , v ) \Big ) \\ & \geq \frac { p } { p \alpha - 1 } ( \alpha - 1 ) \| ( u , v ) \| _ { E } ^ { 2 } . \end{align*}


\begin{align*} P _ { w , v } = \sum _ { i = 1 } ^ k w _ i F _ { v _ i } \end{align*}


\begin{align*} y \left [ i \right ] = \sum \limits _ { l ' = 1 } ^ { L ' } { \sum \limits _ { l \in { { \bar { \cal L } } _ { l ' } } } { { \bf { h } } _ l ^ H { \bf { \bar H } } _ { l ' } ^ \bot { \bf { \bar X } } _ { l ' } { \bf { d } } \left [ { i - { n _ l } - { \kappa _ { l ' } } } \right ] } } + z \left [ i \right ] , \end{align*}


\begin{align*} x = ( r , 0 , 0 , \dots , 0 ) , y = ( \rho , h , 0 , \dots , 0 ) . \end{align*}


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


\begin{align*} F \widehat { \otimes } _ K F ' : = \varinjlim _ { n , m } F _ n \widehat { \otimes } _ K F _ m ' . \end{align*}


\begin{align*} \delta _ { \lambda } ( g ) = e x p \circ \Delta _ { \lambda } \circ e x p ^ { - 1 } g , \end{align*}


\begin{gather*} 0 = P _ \alpha ( f ) = P _ \alpha \bigl \{ f \ , 1 ( \abs { f } \le k ) \bigr \} + P _ \alpha \bigl \{ f \ , 1 ( \abs { f } > k ) \bigr \} \\ \le P _ \alpha \bigl \{ f \ , 1 ( \abs { f } \le k ) \bigr \} + \sup _ { Q \in M } Q \bigl \{ \abs { f } \ , 1 ( \abs { f } > k ) \bigr \} \quad \quad k . \end{gather*}


\begin{align*} H \left ( \rho _ n ^ { ( k , \epsilon ) } ( t , \cdot ) \mid \bar { \rho } ^ { \epsilon \otimes k } ( \cdot ) \right ) = \sup _ { g \in L ^ \infty ( \mathbb { T } ^ { k d } ) } \mathbb { E } _ { \rho _ n ^ { ( k , \epsilon ) } } ( g ) - \mathbb { E } _ { \bar { \rho } ^ { \epsilon \otimes k } } ( e ^ g ) + 1 . \end{align*}


\begin{align*} D _ { q , j } = \sum _ { i = 1 } ^ { j } x _ { q , i } . \end{align*}


\begin{align*} \sup _ { | y | \le \alpha ( x ) } | \beta ( x + y ) - \beta ( x ) | = o ( \beta ( x ) ) \ x \to \infty , \ | y | \le \alpha ( x ) . \end{align*}


\begin{align*} \begin{aligned} f ( ) & \ge \ , ( f ) \ , , \\ \big ( \operatorname { r e s p . } f ( ) & \le \ , ( f ) \ , , \\ f ( ) & = \ , ( f ) \ , \big ) . \end{aligned} \end{align*}


\begin{align*} \dim H _ f ^ 1 ( G _ { F _ v } , \mathrm { a d } ^ 0 \rho ) = \dim H ^ 0 ( G _ { F _ v } , \mathrm { a d } ^ 0 \rho ) + \dim D _ { d R } ( \mathrm { a d } ^ 0 \rho ) / D _ { d R } ^ + ( \mathrm { a d } ^ 0 \rho ) . \end{align*}


\begin{align*} \overline { \mathcal F } ( X ) = \varinjlim _ { Y \in ( \mathcal C _ 0 ) _ { X / } } F ( Y ) , \end{align*}


\begin{align*} x ^ { ( 4 ) } = f ( x ) : = f _ \uparrow ( x ) + f _ \downarrow ( x ) , t \in ( 0 , 1 ) , \end{align*}


\begin{align*} Q ^ c ( g , R ) ( \pi ^ h ; \bar { \pi } ) = 2 Q ( g , R ) ( \pi ^ h ; \bar { \pi } ) , \end{align*}


\begin{align*} u z _ 1 \cdots z _ r = t _ 1 \cdots t _ s = z _ 1 ^ { l _ { 1 1 } + \dots + l _ { s 1 } } \cdots z _ r ^ { l _ { 1 r } + \dots + l _ { s r } } . \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*} f ( n , \alpha , t ) \leq \left ( \sum _ { i = 1 } ^ \alpha \binom { n } { i } \right ) ^ t \leq \left ( 2 \binom { n } { \alpha } \right ) ^ { t } . \end{align*}


\begin{align*} \overline { t } = \overline { \xi } , \overline { x } = \overline { y } . \end{align*}


\begin{align*} B _ q ( \psi _ m ) ( z ) & = \int \limits _ { \mathbb R } \overline { A _ q ( z , x ) } \psi _ m ( x ) d x \\ & = \int \limits _ { \mathbb R } \left ( \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } \psi _ j ( x ) \right ) \psi _ m ( x ) d x , \\ & = \sum _ { j = 0 } ^ { \infty } \frac { z ^ j } { \sqrt { \Gamma ( q j + 1 ) } } \left ( \int \limits _ { \mathbb R } \psi _ j ( x ) \psi _ m ( x ) d x \right ) . \\ \end{align*}


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


\begin{align*} \{ \widehat S _ i ( n + d - \pi ( i ) ) \in d z _ { \pi ( i ) } , i = 1 , \ldots , d , \widehat \tau < \infty \} \end{align*}


\begin{align*} \kappa ( \lambda ) : = \Phi _ \lambda ( u ) | _ { E ^ + _ \lambda \cap \partial B _ \rho ( 0 ) } \geq \left ( \frac { 1 } { 8 } - \frac { 1 } { 2 ^ { 2 p + 1 } } \right ) \left ( \frac { \beta _ \lambda ^ p } { 2 C ( N , \alpha , p ) } \right ) ^ { \frac { 1 } { p - 1 } } > 0 . \end{align*}


\begin{align*} L _ { q } [ f ( t ) \ast g ( t ) ] = F _ { q } ( s ) \ast G _ { q } ( s ) , \end{align*}


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


\begin{align*} \Phi _ s ( z ) ^ \alpha = \sum _ { r = 0 } ^ \infty \sum _ { m = r } ^ \infty ( - 1 ) ^ { m + r } \binom { \alpha } { m } \binom { m } { r } \Phi _ s ( z ) ^ r . \end{align*}


\begin{align*} \frac { f ( x + y ) } { f ( x ) } & = \frac { \inf _ { t \in [ x _ 0 , x + y ] } f ( t ) } { \sup _ { s \ge x } f ( s ) } \frac { f ( x + y ) } { \inf _ { t \in [ x _ 0 , x + y ] } f ( t ) } \frac { \sup _ { s \ge x } f ( s ) } { f ( x ) } \\ & \le \frac { F ( x + y - c + \Delta ) } { F ( x + \Delta ) } \frac { f ( x + y ) } { \inf _ { t \in [ x _ 0 , x + y ] } f ( t ) } \frac { \sup _ { s \ge x } f ( s ) } { f ( x ) } . \end{align*}


\begin{align*} \tilde X _ { n + 1 } ^ i = \tilde X _ n ^ i + \bigg ( b ( \tilde X _ n ^ i ) + \frac 1 { N - 1 } \sum _ { j \neq i } K ( \tilde X _ n ^ i - \tilde X _ n ^ j ) \bigg ) \tau + \sigma ( W _ { t _ { n + 1 } } ^ i - W _ { t _ n } ^ i ) , \end{align*}


\begin{align*} { \rm R e } \langle u _ { t t } , u \rangle & = { \rm R e } \int _ { \mathbb { H } ^ n } f ( u ) \overline { u } d x - \int _ { \mathbb { H } ^ n } | \nabla _ { H } u | ^ 2 d x - m \int _ { \mathbb { H } ^ n } | u | ^ 2 d x - b { \rm R e } \int _ { \mathbb { H } ^ n } \overline { u } u _ t d x \\ & = - I ( u ) - b { \rm R e } \int _ { \mathbb { H } ^ n } \overline { u } u _ t d x . \end{align*}


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


\begin{align*} D _ t \Delta u = & ( - \Delta + D _ j \widetilde { b } ^ j + \widetilde { b } ^ j D _ j + \widetilde { c } ) \Delta u + 2 ( D _ j \nabla \widetilde { b } ^ j + \nabla \widetilde { b } ^ j D _ j + \nabla \widetilde { c } ) \nabla u \\ & + ( D _ j \Delta \widetilde { b } ^ j + \Delta \widetilde { b } ^ j D _ j + \Delta \widetilde { c } ) u - i \Delta f . \end{align*}


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


\begin{align*} | \phi ^ { N , i } ( 0 , \boldsymbol { \xi } ) - \Psi ^ { N , i } ( 0 , \boldsymbol { \xi } ) | \leq \frac { C } { N } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } | \xi ^ { i } - \xi ^ { j } | ^ 2 + \frac { 1 } { N ^ 2 } \underset { i , j = 1 } { \overset { N } { \sum } } | \xi ^ { i } - \xi ^ { j } | ^ 2 \Big ) ^ { \frac { 1 } { 2 } } , \end{align*}


\begin{align*} k = \# \mathcal { N } ( d ) = \# \mathcal { N } ' ( d ) \ , , \end{align*}


\begin{align*} \mu ^ i + h ( \mu ^ i ) = - R ^ { - 1 } B \Big [ P _ t \nu ^ { N , i } _ { \boldsymbol { x } } + \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \Big ] . \end{align*}


\begin{align*} x _ { i , j } + x _ { i + 1 , j } + x _ { i + 1 , j } + x _ { i + 1 , j + 1 } = S , \mbox { \ \ f o r a l l } 1 \leq i \leq m \mbox { a n d } 1 \leq j \leq n _ 1 . \end{align*}


\begin{align*} w ^ { - 1 } s _ \beta ( \gamma ) = w ^ { - 1 } \gamma - \langle \beta ^ \vee , \gamma \rangle w ^ { - 1 } \beta > w ^ { - 1 } \gamma . \end{align*}


\begin{align*} \int _ { \Omega } \left ( \sum _ { \Theta \Subset \Z ^ d } h _ { \Theta } ( \omega ) \right ) ^ 2 \mu ( d \omega ) = 0 . \end{align*}


\begin{align*} F _ n \colon R _ { q } ^ \flat ( n ) & \to \mathcal { P } \ ! \left ( B _ S ( n + D ' ) \right ) , \\ g & \mapsto \big \{ \dot g ( J ) \mid J \subseteq [ \sigma _ g ^ - , \sigma _ g ^ + ] _ \mathbb { Z } | J | = 2 \big \} \end{align*}


\begin{align*} m ( s , t ) = \frac { s } { s - t } , s \in \mathbb { R } _ { > 0 } , t \in \mathbb { R } _ { < 0 } . \end{align*}


\begin{align*} \lim _ { k \to \infty } \deg _ p ( N _ k ' ) = \lim _ { k \to \infty } \deg _ p ( N _ k '' ) . \end{align*}


\begin{align*} \left \| \sum _ { k = 1 } ^ n a _ k \xi _ k \right \| _ q \geq c _ d ( q ) \left \| \sum _ { k = 1 } ^ n a _ k \xi _ k \right \| _ 2 . \end{align*}


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


\begin{align*} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } = \int _ 0 ^ 1 [ h ( x + \delta \sin ( 2 \pi t ) ) - h ( x ) ] \sin ( 2 \pi t ) \ , d t . \end{align*}


\begin{align*} & T _ { 3 , r } ( \underline { p , q } ) = 2 ^ r \sum _ { n _ 1 , n _ 2 , \dots , n _ r \atop { 1 \leq n _ i \leq \pi _ N ( x ) } } G ( n _ 1 ) G ( n _ 2 ) \dots G ( n _ r ) \\ & \sum _ { l _ 1 , l _ 2 , \dots , l _ r \atop { 0 \leq l _ i \leq L } } U ( l _ 1 ) U ( l _ 2 ) \dots U ( l _ r ) \sum _ { l _ 1 ' , l _ 2 ' , \dots , l _ r ' \atop { 0 \leq l _ i ' \leq L } } U ( l _ 1 ' ) U ( l _ 2 ' ) \dots U ( l _ r ' ) \prod _ { i = 1 } ^ r I ( p _ i , q _ i , n _ i , l _ i , l _ i ' ) , \end{align*}


\begin{align*} \lambda _ { i , j } = \begin{cases} 0 ~ ~ { \rm i f } ~ ~ i < a , \\ \\ \frac { \lambda _ { i } \lambda _ { j } } { \lambda _ a } ~ ~ { \rm i f } ~ ~ i \geq a . \\ \end{cases} \end{align*}


\begin{align*} | \prod _ { \alpha \in \Phi ^ + } \alpha ( X ) | = | D ( X ) | ^ { 1 / 2 } . \end{align*}


\begin{align*} p _ \delta = p + \delta _ 1 q _ 1 + \ldots + \delta _ k q _ k \end{align*}


\begin{align*} f ( z ) = \frac { 1 } { 2 \pi i } \int _ { \Gamma } g ( t ) e ^ { - z t } d t , \end{align*}


\begin{align*} [ u , v ] = L ( u \times v ) , \forall u , v \in \mathfrak { g } , \end{align*}


\begin{align*} \int _ { \Gamma _ t } c _ t \ , \mathrm { d } \mathcal { H } ^ d & = \int _ { \Gamma _ t } ( c + t w ) \circ \theta _ t ^ { - 1 } + m _ t \ , \mathrm { d } \mathcal { H } ^ d \\ & = \int _ { \Gamma _ t } ( c + t w ) \circ \theta _ t ^ { - 1 } \ , \mathrm { d } \mathcal { H } ^ d + \left ( m - \int _ { \Gamma _ t } ( c + t w ) \circ \theta _ t ^ { - 1 } \mathrm { d } \mathcal { H } ^ d \right ) = m \end{align*}


\begin{align*} { { \left ( \mathbf { x } + \Delta \mathbf { x } \right ) } ^ { T } } \left ( \mathbf { s } + \Delta \mathbf { s } \right ) + \Delta { { \mathbf { y } } ^ { T } } \mathbf { A x } - \Delta { { \mathbf { x } } ^ { T } } \left ( { { \mathbf { A } } ^ { T } } \mathbf { y } + \mathbf { C } \mathbf { x } + \mathbf { s } \right ) = \nu { \mathbf { x } } ^ { T } \mathbf { s } + \Delta \mathbf { x } ^ T \Delta \mathbf { s } . \end{align*}


\begin{align*} H ( \xi ) : = \begin{cases} 1 / b ' ( \xi ) & \ \xi \notin ( b ) \\ 0 & \ \xi \in ( b ) \end{cases} \end{align*}


\begin{align*} \rho c _ v \left [ g _ i + \Delta t D _ i g _ i + \frac { \Delta t ^ 2 } { 2 } D _ i ^ 2 g _ i \right ] - g _ i + ( 1 - \rho c _ v ) \left [ g _ i + \Delta t d _ i g _ i + \frac { \Delta t ^ 2 } { 2 } d _ i ^ 2 g _ i \right ] = - { \left ( { { { \bf { M } } ^ { - 1 } } \Lambda { \bf { M } } } \right ) _ { i j } } \left [ { g _ i - g _ i ^ { ( e q ) } } \right ] + \Delta t { \bar F _ i } + \vartheta \Delta t { S _ i } , \end{align*}


\begin{align*} \sup _ { n } \mathbb { E } _ { \mathbb { P } } \left [ \exp \left \{ \kappa \sum _ { i = 1 } ^ { n } \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \triangle K _ { t } ^ { i , n } \cdot d W _ { t } ^ { i } - \frac { \kappa } { 2 } \sum _ { i = 1 } ^ n \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \left | \triangle K _ { t } ^ { i , n } \right | ^ { 2 } d t \right \} \right ] \end{align*}


\begin{align*} \sup _ { z \in \mathbb { B } _ n } \beta ( \varphi _ { k j } ( z ) , \rho ( z ) ) & \leq \sup _ { z \in \mathbb { B } _ n } k _ { V ^ { - 1 } \mathbb { B } _ n } ( V ^ { - 1 } \varphi _ { k j } ( z ) , V ^ { - 1 } \rho ( z ) ) \\ & = \sup _ { z \in V ^ { - 1 } \mathbb { B } _ n } k _ { V ^ { - 1 } \mathbb { B } _ n } ( V ^ { - 1 } \varphi _ { k j } V ( z ) , V ^ { - 1 } \rho V ( z ) ) . \end{align*}


\begin{gather*} \inf _ { Q \in M } Q ( c ) = \sup _ { f _ 1 , \ldots , f _ n } \ , \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \end{gather*}


\begin{align*} \begin{aligned} & \min _ { \mathbf { x } } \mathbf { c } ^ T \mathbf { x } , \\ & \\ & \mathbf { A } \mathbf { x } = \mathbf { b } , \\ & \mathbf { x } \in \mathcal { K } , \\ & \mathcal { K } = \mathcal { K } _ { L } ^ { l } \times \mathcal { K } _ { S } ^ { { { n } _ { l + 1 } } } \times \mathcal { K } _ { S } ^ { { { n } _ { l + 2 } } } \cdots \times \mathcal { K } _ { S } ^ { { { n } _ { l + m } } } , \end{aligned} \end{align*}


\begin{align*} \operatorname { s u p p } \widetilde \mu _ { ( x , t ) } = \operatorname { s u p p } \mu _ { x } \times \{ t \} \end{align*}


\begin{align*} F ( r , \varphi ) = \int ^ { \varphi } _ 0 \frac { d \theta } { \sqrt { 1 - r ^ 2 \sin ^ 2 \theta } } = \int ^ z _ 0 \frac { d t } { \sqrt { ( 1 - t ^ 2 ) ( 1 - r ^ 2 t ^ 2 ) } } ( 0 \le r < 1 , \ z = \sin \varphi ) . \end{align*}


\begin{align*} p _ { \rm r e v } ( x ) : = x ^ d p \Big ( \frac { 1 } { x } \Big ) = \sum _ { i = 0 } ^ d p _ i x ^ { d - i } , ~ p _ { \rm r e v } ( x ) = p _ 0 \prod _ { j = 1 } ^ d \Big ( x - \frac { 1 } { x _ j } \Big ) ~ { \rm i f } ~ p _ 0 \neq 0 , \end{align*}


\begin{align*} \delta _ { + } ( z ) \delta _ { - } ^ { - 1 } ( z ) = \mathrm { e } ^ { \log \left ( 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) \right ) } = 1 + \bar { r } _ 1 ( z ) r _ 2 ( z ) , z \in \mathbb { R } . \end{align*}


\begin{align*} F _ 0 ( u ) : = \begin{cases} u - u \log ( u ) - 1 , & , \\ \ - 1 , & , \end{cases} \end{align*}


\begin{align*} \sup _ { n } \mathbb { E } _ { \mathbb { P } } \left [ \exp \left \{ \kappa \sum _ { i = 1 } ^ { n } \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \Delta K _ { t } ^ { i , n } \cdot d W _ { t } ^ { i } - \frac { \kappa } { 2 } \sum _ { i = 1 } ^ n \int _ { T _ { 0 } } ^ { T _ { 0 } + \delta } \left | \Delta K _ { t } ^ { i , n } \right | ^ { 2 } d t \right \} \right ] \end{align*}


\begin{align*} Y ^ { \ast } ( t , x ) : = Y \setminus \overline { B _ { R _ 0 ( t , x ) } ( x ) } , \end{align*}


\begin{align*} \norm { u _ j } _ { L ^ 2 } ^ 2 & = \int _ \S \left ( \int _ { x - r } ^ { x + r } \prod _ { \stackrel { i = 1 } { i \neq j } } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) ( - \partial \eta _ j ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y \right ) ^ 2 \ \d x \\ & \le \int _ \S \int _ \S \prod _ { \stackrel { i = 1 } { i \neq j } } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 ( \partial \eta _ j ( x ) ) ^ 2 \ \d y \ \d x \\ & = \prod _ { i = 1 } ^ n \norm { \partial \eta _ i } _ { L ^ 2 } ^ 2 \end{align*}


\begin{align*} & \int _ { A _ { n , j } \cap A _ n } \int _ 0 ^ { \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P \\ & = \int _ { A _ { n , j } \cap A _ n } \int _ 0 ^ { \eta _ j ( K ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P + \int _ { A _ { n , j } \cap A _ n } \int _ { \eta _ j ( K ) } ^ { \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P . \end{align*}


\begin{align*} \lambda _ \mathbb { I } - \rho _ { \mathbb { I } } = d ^ { \mathbb { I } } h + h d ^ { \mathbb { I I } } , \end{align*}


\begin{align*} \left \{ e _ { m , q } ( z ) = \frac { z ^ m } { \sqrt { \Gamma ( m q + 1 ) } } \ ; ; \ ; m \geq 0 \right \} \end{align*}


\begin{align*} h ( H _ { \rho } ( x ) ) = h ( x ) \end{align*}


\begin{align*} \begin{aligned} \abs { c _ 3 ( S ^ 3 \setminus S _ i ) } & \le \abs { c _ 2 ( S ^ 2 \setminus S _ i ) } + \abs { c _ 3 ( P ^ 3 \setminus S _ i ) } \\ & \le ( 2 k - t ' - \abs { S \cap S _ i } - q _ i - s _ 1 ) + ( s _ 1 - t _ i ) = 2 k - t ' - \abs { S \cap S _ i } - p _ i . \end{aligned} \end{align*}


\begin{align*} | \tilde H ^ { ( n ) } ( y ) | \le \frac { C } { \chi ^ 2 ( y ) } , y = 0 , \ldots , n , \ , n \ge 1 . \end{align*}


\begin{align*} \mathbf { x } ^ { ( k ) } = \underset { \mathbf { x } } { \arg \min } \frac { 1 } { 2 } \left \| \mathbf { x } - \mathbf { r } ^ { ( k ) } \right \| ^ { 2 } + \lambda \| \mathcal { F } ( \mathbf { x } ) \| _ { 1 } , \end{align*}


\begin{align*} s _ { - w ^ { - 1 } \gamma } ( \beta ) = \beta + \langle - w ^ { - 1 } \gamma ^ \vee , \beta \rangle w ^ { - 1 } \gamma . \end{align*}


\begin{align*} z \in \Lambda _ { n , m } : = [ - 2 ^ n + n + m + 1 , 2 ^ n - n - m - 1 ] ^ d , \end{align*}


\begin{align*} \eta ^ { D } ( \mathrm { d } x \mathrm { d } h ) = Z _ { \lambda } ^ { D } ( \mathrm { d } x ) \otimes e ^ { - \pi \lambda h } \mathrm { d } h \end{align*}


\begin{align*} D \widetilde H ( x , Y , Z , U ) = \Big ( \partial _ x H ( x , \mu ) , \partial _ { \mu _ y } H ( x , \mu ) , \partial _ { \mu _ z } H ( x , \mu ) , \partial _ { \mu _ u } H ( x , \mu ) \Big ) ( Y , Z , U ) . \end{align*}


\begin{align*} \begin{aligned} | | S ( t ) \varphi _ 0 | | _ { T } & \leq C | | \varphi _ 0 | | _ { L ^ { 2 } ( \mathbb { R } ^ { 3 } ) } , \\ \left \| \int _ { 0 } ^ { t } S ( t - s ) \mathbb { P } \nabla \cdot F _ 0 ( s ) \dd s \right \| _ T & \leq C ' | | F _ 0 | | _ { L ^ { 2 } ( 0 , T ; L ^ { 2 } ( \mathbb { R } ^ { 3 } ) ) } . \end{aligned} \end{align*}


\begin{align*} w ( t , x ) = w _ 0 ( x ) + \int _ 0 ^ t \partial _ t w ( s , x ) \mathrm { d } s \begin{cases} \leq w _ 0 ( x ) + \left ( - g \big ( c _ 0 ( x ) \big ) \frac { 1 } { w _ 0 ( x ) } + \varepsilon \right ) \cdot t , \\ \geq w _ 0 ( x ) + \left ( - g \big ( c _ 0 ( x ) \big ) \frac { 1 } { w _ 0 ( x ) } - \varepsilon \right ) \cdot t \end{cases} \end{align*}


\begin{align*} D \sigma _ T ( x ) = \frac { 1 } { \pi } \Big ( \log \Big ( \frac { \sin ( \pi x _ 1 ) } { \sin ( \pi ( x _ 1 + x _ 2 ) ) } , \log \Big ( \frac { \sin ( \pi x _ 2 ) } { \sin ( \pi ( x _ 1 + x _ 2 ) ) } \Big ) \Big ) . \end{align*}


\begin{align*} K ^ { ( 1 ) } ( S _ t , I _ t ) & = \frac { w } { 2 \theta ( \partial _ i V ( S _ t , I _ t ) - \partial _ s V ( S _ t , I _ t ) ) } , \\ K ^ { ( 2 ) } ( S _ t , I _ t ) & = \frac { w } { 2 \theta ( 1 - \theta \bar L ) ( \partial _ i V ( S _ t , I _ t ) - \partial _ s V ( S _ t , I _ t ) ) } . \end{align*}


\begin{align*} & \sup _ { n \in \mathbb N } \mathbb P \left [ \sup _ { t \leq T } \Delta _ n ^ { 1 - \frac m 2 } \| P _ N ^ m S A M P V _ t ^ n ( m _ 1 , . . . , m _ k ) \| _ { \mathcal H ^ m } > \epsilon \right ] \to 0 N \to \infty . \end{align*}


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


\begin{align*} a ( b ( x ) ) & \in g ( b ( x ) ) \ast g ' ( b ( x ) ) \subseteq [ g ( f ( x ) \ast f ' ( x ) ) ] \ast [ g ' ( f ( x ) \ast f ' ( x ) ) ] \\ & \subseteq [ g ( f ( x ) ) \ast g ( f ' ( x ) ) ] \ast [ g ' ( f ( x ) ) \ast g ' ( f ' ( x ) ) ] \\ & = ( g \circ f ) ( x ) \ast ( g \circ f ' ) ( x ) \ast ( g ' \circ f ) ( x ) \ast ( g ' \circ f ' ) ( x ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \mathbb { E } \Big [ \sup _ { t \leq T } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H ^ p \Big ] = 0 , \end{align*}


\begin{align*} y _ { b e n } ( R ) = y _ { n b } ( R ) , a y ' _ { b e n } ( R ) = b y ' _ { n b } ( R ) . \end{align*}


\begin{align*} \langle y ^ * - \tilde { y } ^ k , \tilde { \lambda } ^ k - \beta ( \tilde { y } ^ k - y ^ k ) - \beta ( y ^ k - \tilde { y } ^ k ) \rangle _ { \mathcal { L } ^ 2 } = \langle y ^ * - \tilde { y } ^ k , \tilde { \lambda } ^ k \rangle _ { \mathcal { L } ^ 2 } \geq 0 . \end{align*}


\begin{align*} \sum _ { \eta _ { \Lambda _ n } } \gamma _ { \Lambda _ n } ( \eta _ { \Lambda _ n } | r _ n \eta _ { \Lambda _ n ^ c } ) \varphi _ n ( \eta _ { \Lambda _ n } ) = 0 . \end{align*}


\begin{align*} \displaystyle \sum _ { i = 1 } ^ { n } a _ i ^ 2 \displaystyle \sum _ { i = 1 } ^ { n } b _ i ^ 2 - { \Big ( \displaystyle \sum _ { i = 1 } ^ { n } a _ i b _ i \Big ) } ^ 2 \leq \frac { n ^ 2 } { 4 } { ( M _ 1 M _ 2 - m _ 1 m _ 2 ) } ^ 2 . \end{align*}


\begin{align*} \lim _ { k \rightarrow { 0 } } \frac { G ( r + k ) - G ( r ) } { k } = g ( r ) \end{align*}


\begin{align*} A _ { m n } ( x ) : = ( - 1 ) ^ { m + n } \left ( \sum _ { k = 1 } ^ { N } \alpha _ { k } ^ { + } e ^ { - 2 \tau _ { k } x } \frac { \left ( \frac { 1 } { 2 } - \tau _ { k } \right ) ^ { m + n } } { \left ( \frac { 1 } { 2 } + \tau _ { k } \right ) ^ { m + n + 2 } } + \frac { 1 } { 2 \pi } \int _ { - \infty } ^ { \infty } s ^ { + } \left ( \rho \right ) e ^ { 2 i \rho x } \frac { \left ( \frac { 1 } { 2 } + i \rho \right ) ^ { m + n } } { \left ( \frac { 1 } { 2 } - i \rho \right ) ^ { m + n + 2 } } d \rho \right ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} & \partial _ t \chi + \gamma \nabla \cdot ( \chi \nabla \Phi ) , & \Omega \times ( 0 , T ] , \\ & \partial _ t \Phi + \frac { \gamma } { 2 } \lvert \nabla \Phi \rvert ^ 2 + \gamma \chi = 0 , & \Omega \times ( 0 , T ] , \end{aligned} \right . \end{align*}


\begin{align*} \partial v = N \partial _ { z , \alpha } N ^ { - 1 } v & = \gamma _ { \alpha } \partial _ z ( \gamma _ { - \alpha } v ) = \gamma _ { \alpha } ( \gamma _ { - \alpha } ' v + \gamma _ { - \alpha } \partial _ z v ) \\ & = \gamma _ { \alpha } ( - \alpha \gamma _ { - \alpha } v + \gamma _ { - \alpha } \partial _ z v ) \\ & = ( \partial _ z - \alpha ) v . \end{align*}


\begin{align*} \begin{aligned} & \mu ( { { \mathbf { q } } _ w } ) \le { { c } _ { \mu } } \mu ( { { \mathbf { q } } _ c } ) , \\ & \left ( 1 - \omega \right ) \left ( \mathbf { e } ^ T ( { { \mathbf { x } } _ { o } } + { { \mathbf { s } } _ { o } } ) / k + 1 \right ) = { { c } _ { x s } } , \\ & { { c } _ { \mu } } + { { c } _ { x s } } \le 1 . \\ \end{aligned} \end{align*}


\begin{align*} s _ h : = \sum _ { j = 1 } ^ w x _ j ^ { h } = \frac { 1 } { 2 \pi \sqrt { - 1 } } \int _ { \mathcal C } \frac { p ' ( x ) } { p ( x ) } ~ x ^ h ~ d x . \end{align*}


\begin{align*} \frac { 1 } { t } \int _ 0 ^ t \vert \langle \psi _ { 1 / 2 , f _ \delta } , e ^ { - i s H _ N } \psi _ { 1 / 2 , f _ \delta } \rangle \vert ^ 2 \ , d s = O ( \log ( t ) / t ) ) \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*} c _ \pm ( x ) = \frac { 1 } { 2 } \int _ { \pm \infty } ^ x | u _ y ( y , t ) | ^ 2 d y . \end{align*}


\begin{align*} S ^ { p q , r s } _ k = \frac { \partial ^ 2 S _ k [ D ^ 2 u ] } { \partial r _ { p q } \partial r _ { r s } } . \end{align*}


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


\begin{align*} J _ 1 ( b , c ) = & \frac { 2 b - 1 } { 2 } \left ( \log \left ( \binom { b + 1 } { 2 } + c \right ) - \log \left ( \binom { b } { 2 } + c \right ) - 2 \log \left ( b + c \right ) + 2 \log \left ( b + c - 1 \right ) \right ) , \\ J _ 2 ( b , c ) = & c \left ( \log ( c + 1 ) - \log ( c ) - \log ( b + c ) + \log ( b + c - 1 ) \right ) , \\ J _ 3 ( b , c ) = & \log \left ( \binom { b + 1 } { 2 } + c \right ) + \log ( c + 1 ) - 2 \log ( b + c ) . \end{align*}


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


\begin{align*} \prod _ { \ell \in I } | \cos ( \pi \theta _ { \ell } ) | = \frac { \prod _ { \ell \in \tilde { I } } | \cos ( \pi \theta _ { \ell } ) | } { \prod _ { \ell \in I ^ c } | \cos ( \pi \theta _ { \ell } ) | } \geq e ^ { - \varepsilon ( 2 q _ n - | I | ) } e ^ { ( - \ln 2 ) | I | } . \end{align*}


\begin{align*} \liminf _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \big ( X ^ { \epsilon } \in B ( \varphi , \gamma ) \big ) & \geq \lim _ { \delta \to 0 } \liminf _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \hat { X } ^ { \epsilon } \in B ( \varphi , \delta ) \Big ) \\ & \geq - \mathcal { I } ^ { \varphi } ( \varphi ) = - \mathcal { I } ( \varphi ) , \end{align*}


\begin{align*} \left \| \tilde { \mathbf { U } } \right \| \le ( 1 - \omega ) \rho , \ \rho = \max _ { i } \left ( 1 - \frac { 2 \omega ( x _ o ^ { ( i ) } ) _ 1 + 1 - \omega } { \beta _ { x _ w ^ { ( i ) } } + \omega \beta _ { x _ o ^ { ( i ) } } } \right ) . \end{align*}


\begin{align*} G ^ { \alpha \beta } _ 0 = \frac { \partial G } { \partial r _ { \alpha \beta } } ( \nabla _ { \alpha \beta } u ( x _ 0 ) ) , 1 \leq \alpha , \beta \leq n - 1 . \end{align*}


\begin{align*} x _ n - y _ n \geq ( 2 - \epsilon ) b _ { n - 1 } \mbox { o n } \partial \omega \cap \{ x _ n = 2 b _ { n - 1 } \} . \end{align*}


\begin{align*} & w ( x ; z ) = \sum _ { n = 0 } ^ \infty w _ n ( x ; z ) = \sum _ { n = 0 } ^ \infty F ^ n e _ 1 . \end{align*}


\begin{align*} \begin{aligned} & y '' _ { b e n } + N _ { b e n } ( E ) y _ { b e n } = 0 \\ & y '' _ { n b } + N _ { n b } ( E ) y _ { n b } = 0 . \end{aligned} \end{align*}


\begin{align*} \tau _ { u } ^ M : = & T \wedge \inf \Big \{ t \geq 0 : \Vert u ( t ) \Vert _ H ^ 2 > M \Big \} \\ & \wedge \inf \Big \{ t \geq 0 : \int _ 0 ^ t \Vert u ( s ) \Vert _ { V } ^ { \alpha } d s > M \Big \} . \end{align*}


\begin{align*} \begin{cases} \alpha a + ( \alpha - 1 ) b + ( 3 \alpha - 2 ) c - \frac { p \alpha - 1 } { p } d = 0 ; \\ ( 1 - 2 \alpha \mu ) a + ( 1 - 2 \mu ( \alpha - 1 ) ) b + 3 \big ( 1 - \mu ( 6 \alpha - 4 ) \big ) c - \big ( 1 - \mu ( 2 p \alpha - 2 ) \big ) d = 0 ; \\ \big ( 2 \mu ( \alpha - 1 ) - 1 \big ) b + 2 \big ( \mu ( 6 \alpha - 4 ) - 1 \big ) c - \frac { \mu ( 2 p \alpha - 2 ) - 1 } { p } d = 0 . \end{cases} \end{align*}


\begin{align*} \| x _ { k + 1 } - x ^ \star \| ^ 2 \leq \tfrac { 1 } { 2 } \| x _ k - x ^ \star \| ^ 2 , \textnormal { f o r a l l } k = 0 , 1 , 2 , \ldots , T . \end{align*}


\begin{align*} & \varepsilon ( h , x + y ) [ [ \alpha ( x ) , \alpha ( y ) ] , \alpha ( h \cdot z ) ] + \varepsilon ( x + y , z ) [ [ h \cdot z , \alpha ( x ) ] , \alpha ^ 2 ( y ) ] \\ & + \varepsilon ( x , y + z ) \big ( 2 [ \alpha ( h ) \cdot [ y , z ] , \alpha ^ 2 ( x ) ] - \varepsilon ( x , y + z ) [ [ h \cdot y , \alpha ( z ) ] , \alpha ^ 2 ( x ) ] \big ) = 0 . \end{align*}


\begin{align*} \dot { v } ( s ) = \begin{cases} \frac { 1 } { | G / G _ 0 | } \sum _ { t G _ 0 \in G / G _ 0 } u ( t ^ { - 1 } s t ) & s \in G _ 0 \\ 0 & \end{cases} \end{align*}


\begin{align*} \theta _ { \rho ^ \varepsilon } ( 0 , z _ 1 ) = F ( z _ 1 ) + \rho _ 0 ^ \varepsilon \nu _ \Sigma ( z _ 1 ) \neq F ( z _ 2 ) + \rho _ 0 ^ \varepsilon \nu _ \Sigma ( z _ 2 ) = \theta _ { \rho ^ \varepsilon } ( 0 , z _ 2 ) \end{align*}


\begin{align*} \phi _ i & = \phi \cap V ( K _ i ) \ ( i \in [ m ] ) , \\ \overline { \phi } & = \{ i \in [ m ] \ | \ \phi _ i \notin L _ i \} . \end{align*}


\begin{align*} \psi _ { K , H } ( t ) = ( n - 1 ) \sqrt K \cdot \frac { ( n - 1 ) \sqrt K \sinh \sqrt K t + H \cosh \sqrt K t } { ( n - 1 ) \sqrt K \cosh \sqrt K t + H \sinh \sqrt K t } . \end{align*}


\begin{align*} \norm { f } _ { H ^ 1 _ 0 } = \sqrt { f \cdot f } = \sqrt { \norm { f } _ { L ^ 2 } ^ 2 + \norm { D f } _ { L ^ 2 } ^ 2 } . \end{align*}


\begin{align*} \left ( \begin{matrix} \phi ( k ) \\ \phi ( k - 1 ) \end{matrix} \right ) = A _ k ( \theta , E ) \left ( \begin{matrix} \phi ( 0 ) \\ \phi ( - 1 ) \end{matrix} \right ) , \end{align*}


\begin{align*} \langle f \rangle ( a ) = \lim _ { { T ' } \to \infty } \frac { 1 } { { T ' } } \int _ 0 ^ { T ' } f ( \Phi _ { - t \Lambda } a ) d t , a \in \C ^ n . \end{align*}


\begin{align*} \begin{array} { l l l } \mu \circ ( ( R \circ \eta ) \otimes \eta ) & = & \mu \circ ( R \otimes i d _ M ) \circ ( \eta \otimes \eta ) \\ & = & \mu \circ ( i d _ M \otimes L ) \circ ( \eta \otimes \eta ) \\ & = & \mu \circ ( \eta \otimes i d _ M ) \circ ( i d _ I \otimes ( L \circ \eta ) ) \\ & = & \lambda _ M \circ ( i d _ I \otimes ( L \circ \eta ) ) \\ & = & L \circ \eta \circ \lambda _ I \end{array} \end{align*}


\begin{align*} m _ { \mathrm A D M } ( g ) = \frac { 1 } { 2 ( n - 1 ) \omega _ { n - 1 } } \lim _ { r \to \infty } \int _ { | x | = r } \sum _ { i , j = 1 } ^ n \left ( \frac { \partial g _ { i j } } { \partial x _ i } - \frac { \partial g _ { i j } } { \partial x _ j } \right ) \frac { x _ j } { | x | } \ , d \sigma . \end{align*}


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


\begin{align*} \phi ( \sigma , \tau ) = \epsilon ^ { \sum _ { r = 1 } ^ j \lambda _ r ( d _ r - 1 ) } . \end{align*}


\begin{align*} u ^ k = \arg \min _ { w \in H ^ 1 ( \Omega ) } \bigg ( \frac { | \Omega | } { 2 \mathrm { N } _ I } \sum _ { n = 1 } ^ { \mathrm { N } _ I } ( w - u ^ { k - 1 } ) ^ 2 ( x _ n ^ I ) + \tau _ k \tilde { N } ( w ; w ) \bigg ) . \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*} q _ { \pi ( a ) } = p _ { \pi ( a ) } . \end{align*}


\begin{align*} C _ j ( S _ j ^ { \top } R _ j ) ^ { \top } ( S _ j ^ { \top } R _ j ) C _ j ^ { \top } v = \lambda C _ j C _ j ^ { \top } v , \end{align*}


\begin{align*} \lim _ { \varepsilon \to 0 } \frac { 1 } { \varepsilon ^ 2 \ln \frac { 1 } { \varepsilon } } \int _ \Omega \left ( v ^ \tau _ \varepsilon - q \ln \frac { 1 } { \varepsilon } \right ) ^ p _ + v ^ \tau _ \varepsilon d x = q ^ { p + 1 } ( \bar { x } ) \int _ { \hat { B } _ \tau ( 0 ) } ( \hat { U } ( y ) + \ln \tau ) ^ { p } _ + d y . \end{align*}


\begin{align*} \pi ( s ) = { \frac { w ( s ) } { f _ { q } ( s ) } } \pi \bigl ( { f _ { q } ( s ) } \bigr ) . \end{align*}


\begin{align*} T _ { \textnormal { i n n e r } } = \left \lceil \left ( \tfrac { 2 ^ { p + 2 } ( 5 p + 1 ) } { p ! } \tfrac { L D ^ { p - 1 } } { \mu _ { \textnormal { M i n t y } } } \right ) ^ { \frac { 2 } { p } } + \left ( \tfrac { 2 ^ { p + 5 } } { p ! } \tfrac { L D ^ { p - 1 } } { \mu _ { \textnormal { M i n t y } } } \right ) ^ { \frac { 2 } { p + 1 } } \right \rceil , \end{align*}


\begin{align*} [ X _ 1 ] = ( \frac { \mu } { 2 } d \overline { z } - \frac { 2 } { \mu } h _ z ^ 2 d z + \frac { \tau _ 0 } { 2 \mu } d \overline { z } ) J - \frac { 1 } { \mu } d h \sqrt { \tau _ 0 } I . \end{align*}


\begin{align*} \lim \limits _ { i \to \infty } \left | \ , ^ { k _ i } \ ! \lambda / \left ( \ , ^ { k _ i } \ ! \lambda _ 1 \right ) \right | = \infty . \end{align*}


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


\begin{align*} \norm { \Theta _ { \norm { \cdot } } ( \nu ) } _ { ( C ( [ 0 , T ] ; ( U , \norm { \cdot } _ { U } ) ) , \norm { \cdot } _ \infty ) ' } = \norm { \nu } _ { v a r } ( \nu \in M ( [ 0 , T ] ; U ' ) ) \end{align*}


\begin{align*} \phi _ 2 ( t _ 1 , t _ 2 ) : = \left ( \frac { \alpha } { \sigma ( \alpha ) } \right ) ^ { \ln ( t _ 1 / \alpha ) } \left ( \frac { \sigma ( \alpha ) } { \sigma ^ 2 ( \alpha ) } \right ) ^ { \ln ( t _ 2 / \sigma ( \alpha ) ) } \\ \phi _ 3 ( t _ 1 , t _ 2 ) : = \left ( \frac { \alpha } { \sigma ^ 2 ( \alpha ) } \right ) ^ { \ln ( t _ 1 / \alpha ) } \left ( \frac { \sigma ( \alpha ) } { \alpha } \right ) ^ { \ln ( t _ 2 / \sigma ( \alpha ) ) } . \end{align*}


\begin{align*} f ( t ) = ( \kappa \ , * \ , \phi ) ( t ) \ , = \ , \int _ 0 ^ t \kappa ( t - \tau ) \phi ( \tau ) \ , d \tau , \ t > 0 \end{align*}


\begin{align*} \int _ M \sum _ i \lambda _ i ^ + ( x ) \ , d x = h _ \mu ( f ) , \end{align*}


\begin{align*} U ^ { \dagger } = A + H + \sum _ { k \in \nabla } \Delta _ k . \end{align*}


\begin{align*} \Delta _ { H } u = \sum _ { i = 1 } ^ m X _ { i } ^ 2 u . \end{align*}


\begin{align*} \small [ P ( u _ t ( t ) ) ] _ t + A u ( t ) + B ( t , x , u _ t ( t ) ) = 0 , t \in J = ( 0 , \infty ) , \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*} C _ m \left ( \bigcup _ { N = 1 } ^ \infty U _ { p , N } ( \sigma ) \right ) = C _ m ( U _ p ( \sigma ) ) = \lim \limits _ { N \to \infty } C _ m ( U _ { p , N } ( \sigma ) ) \end{align*}


\begin{align*} \langle \rho _ { \Sigma } ^ { \otimes k } ( m _ 1 , . . . , m _ k ) , \bigotimes _ { l = 1 } ^ { m _ 1 } h _ { 1 , l } \otimes . . . \otimes \bigotimes _ { l = 1 } ^ { m _ k } h _ { k , l } \rangle _ { \mathcal H ^ m } = & \prod _ { j = 1 } ^ { k } \sum _ { p \in \mathcal P ( m _ l ) } \prod _ { ( x , y ) \in p } \langle \Sigma _ s h _ { x , j } , h _ { y , j } \rangle . \end{align*}


\begin{align*} P _ N ( v u _ 1 ) & = P _ N ( P _ { \ll N } v \cdot u _ 1 ) + P _ N ( P _ { \gtrsim N } v \cdot u _ 1 ) \\ & \sim P _ N ( P _ { \ll N } v \cdot P _ N u _ 1 ) + P _ N ( P _ { \gtrsim N } v \cdot P _ { \gtrsim N } u _ 1 ) . \end{align*}


\begin{align*} \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ j \cdot B _ j \dd x + 2 \mu _ j \int _ { 0 } ^ { t } \int _ { \Omega } \nabla \times B _ j \cdot B _ j \dd x \dd s = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ { 0 } \cdot B _ 0 \dd x , \end{align*}


\begin{align*} \mathbb P \left [ \frac 1 n \sum _ { i = 1 } ^ n ( X _ i - \mathbb E [ X _ i ] ) < - \epsilon \right ] \le \exp \left ( - \frac { n \epsilon ^ 2 } { 2 \sigma ^ 2 } \right ) \end{align*}


\begin{align*} \langle \omega _ a , \beta \rangle = \begin{cases} 1 , & a = ( \beta , 0 ) , \\ 0 , & a \neq ( \beta , 0 ) . \end{cases} \end{align*}


\begin{align*} x _ t = \frac { x _ { y y } } { e ^ { 2 \phi } + x _ y ^ 2 } - \phi ' ( x ) \left ( 1 + \frac { x _ y ^ 2 } { e ^ { 2 \phi } + x _ y ^ 2 } \right ) = \frac { \partial } { \partial y } \left ( e ^ { - \phi } \tan ^ { - 1 } ( x _ { y } e ^ { - \phi } ) \right ) + \phi ' ( x ) \left ( x _ y e ^ { - \phi } \tan ^ { - 1 } ( x _ y ) e ^ { - \phi } - 1 \right ) , \end{align*}


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


\begin{align*} R _ q ^ f ( n ) = R _ { \mathcal { W } , q } ^ f ( n ) = \big \{ g \in R _ q ( n ) \mid g _ { | i } \in B _ { H , S _ 0 } ( f ( n ) ) i \in \mathbb { Z } \big \} . \end{align*}


\begin{align*} \bar \pi \bar p _ T = \bar \pi . \end{align*}


\begin{align*} \begin{aligned} \min & \ \frac { 1 } { 2 } x ^ T A x + b ^ T x \\ \mbox { s . t . } & \ \frac { 1 } { 2 } x ^ T B _ i x + c _ i ^ T x + d _ i \leq 0 , i = 1 , \dots , m , \\ & \ - 1 \leq x _ i \leq 1 , i = 1 , \dots , n , \end{aligned} \end{align*}


\begin{align*} \limsup _ { N \to \infty } \frac { 1 } { N } \sum _ { n = 0 } ^ { N - 1 } \min _ { 1 \leq i < j \leq k } { d } ( T ^ { ( n , [ n \beta + t ] ) } x _ i , T ^ { ( n , [ n \beta + t ] ) } x _ j ) \geq \eta > 0 . \end{align*}


\begin{align*} I _ { l } ^ { \left ( 1 \right ) } ( s , \rho ) = \frac { 1 } { 2 i } \int _ { \mathbb { R } } \frac { e ^ { i S ( \tau , x - s ) } \tau ^ { l - 2 } } { \left ( \tau - \rho \right ) ^ { l + 1 } } d \tau \end{align*}


\begin{align*} u _ { t } - 6 u u _ { x } + u _ { x x x } = 0 . \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*} \begin{aligned} \dfrac { | b _ { 1 , \delta } ( x _ a ) | } { 2 } = & \ , \left | \int _ 0 ^ 1 h ( x _ a + \delta u ( t ) ) u ( t ) \ , d t \right | \\ = & \ , \left | \int _ 0 ^ 1 ( h ( x _ a + \delta u ( t ) ) - h ( x _ a ) ) u ( t ) \ , d t \right | \\ \le & \ , \int _ 0 ^ 1 \left | h ( x _ a + \delta u ( t ) ) - h ( x _ a ) \right | \ , d t \le L _ r \delta . \end{aligned} \end{align*}


\begin{align*} C _ 0 ( \mathcal { H } _ { \ast } ) = \mathrm { I m } \ , \partial _ 0 \oplus s _ { _ 0 } ( \mathrm { I m } \ , \varphi _ 0 ) = \mathrm { I m } \ , \partial _ 0 . \end{align*}


\begin{align*} z & = \sqrt { \frac { a } { a + b } } \left ( \frac { \sqrt { 1 - b } \ , \cos ( t ) } { \sqrt { 1 + a } } + \frac { i \sqrt { 1 + b } \ , \sin ( t ) } { \sqrt { 1 - a } } \right ) , \\ w & = \pm \sqrt { \frac { b } { a + b } } \left ( \frac { \sqrt { 1 - a } \ , \sin ( t ) } { \sqrt { 1 + b } } - \frac { i \sqrt { 1 + a } \ , \cos ( t ) } { \sqrt { 1 - b } } \right ) , t \in [ 0 , 2 \pi ) , \end{align*}


\begin{align*} \tau = \sum _ { j = 1 } ^ \infty \frac { c _ j } { 2 ^ j } \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } | b | ^ { 2 } \dd x = 2 \pi \int _ { \mathbb { R } ^ { 2 } _ { + } } \left ( | \nabla \phi | ^ { 2 } + | G | ^ { 2 } \right ) \frac { 1 } { r } \dd z \dd r . \end{align*}


\begin{align*} h ( x , y ) \ge \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | + 2 \max \{ d i s t ( x , \partial \Omega ) , d i s t ( y , \partial \Omega ) \} } . \end{align*}


\begin{align*} \langle \mu , v \gamma _ 1 \rangle \geq - 1 , \langle \mu , v \gamma _ 2 \rangle \geq - 1 , \langle \mu , v \gamma _ 1 + v \gamma _ 2 \rangle = - 1 . \end{align*}


\begin{align*} \omega = \sum _ { i = 1 } ^ n d x ^ i \wedge d \theta _ i . \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*} \langle \alpha ^ \vee , 2 \rho \rangle = \sum _ { \beta \in I } \langle \alpha ^ \vee , \beta \rangle . \end{align*}


\begin{align*} n ! F _ { n + 1 } \ & = \ \sum _ { k = 1 } ^ { n } \begin{bmatrix} n \\ k \end{bmatrix} A _ k , \\ n ! F _ { n + 2 } \ & = \ \sum _ { k = 1 } ^ { n } \begin{bmatrix} n \\ k \end{bmatrix} B _ k . \\ \end{align*}


\begin{align*} \partial ^ \square c = - \frac { m d } { \alpha _ d } R ^ { - d - 1 } R ' = 0 + \frac { m } { \alpha _ d } R ^ { - d } \cdot \frac { - d } { R } \cdot R ' = \Delta _ \Gamma \big ( G ' ( c ) \big ) + c H V \end{align*}


\begin{align*} \Delta { { \mathbf { x } } ^ { T } } \Delta \mathbf { s } = 0 . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } ( \sigma _ { n } ^ M < T ) = 0 . \end{align*}


\begin{align*} { \bf { \hat \Lambda } } = d i a g \left [ { 1 , 1 , 1 , 1 , { { \hat s } _ e } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ \upsilon } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ q } , { { \hat s } _ \pi } , { { \hat s } _ \pi } , { { \hat s } _ \pi } } \right ] , \end{align*}


\begin{align*} J ( z ) = \int _ 0 ^ z x f ( x ) \mathrm { d } x = \frac { \sqrt \theta } { B ( a , b ) } \int _ { \theta / z ^ 2 } ^ \infty y ^ { - 1 / 2 } \exp ( - a y ) \left [ 1 - \exp ( - y ) \right ] ^ { b - 1 } \mathrm { d } y . \end{align*}


\begin{align*} \begin{aligned} \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \geq - A + \underline { \delta } ^ \star ( \delta ) & \forall \ , x \geq x ^ \star + \delta \\ \dfrac { b _ { 1 , \delta } ( x ) } { 2 } & \leq A - \underline { \delta } ^ \star ( \delta ) & \forall \ , x \leq x ^ \star - \delta \ , , \end{aligned} \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*} ( \sum \limits _ { i = 1 } ^ { j + 1 } w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j } \chi _ i + r j < \chi _ { j + 1 } < ( \sum \limits _ { i = 1 } ^ { j + 1 } w _ i ) \chi - \sum \limits _ { i = 1 } ^ { j } \chi _ i + r ( j + 1 ) . \end{align*}


\begin{align*} v _ \epsilon { '' } ( t ) = \frac { 1 } { B - 4 \epsilon } \mathbb { I } _ { ( - t _ 0 - B + 2 \epsilon , - t _ 0 - 2 \epsilon ) } * \rho _ { \frac { 1 } { 4 } \epsilon } ( t ) , \end{align*}


\begin{align*} \alpha ( H ) \leq \alpha \left ( \bigcup _ { i = 1 } ^ k H _ i \right ) \leq ( k - 1 ) \frac { n } { p } \end{align*}


\begin{align*} E ( X - Y ) ^ { n } = \left \{ \begin{array} { c c c } 0 & & n \\ \frac { n ! } { ( n / 2 ) ! } ( 1 - \rho ) ^ { n / 2 } & & n \end{array} \right . . \end{align*}


\begin{align*} B _ 2 = \frac { ( L + 1 ) B _ 1 } { Q } \left ( \frac { ( L + 1 ) B _ 1 } { Q } - 1 \right ) , \end{align*}


\begin{align*} S u p p ( F ) = \bigcup \limits _ { i = 0 } ^ l S u p p ( s _ { \lambda ^ i } ) . \end{align*}


\begin{align*} \langle x \rangle \vee \langle y \rangle = \langle x \wedge y \rangle . \end{align*}


\begin{align*} c ( q ^ { 3 } ) = \frac { a ( q ) - b ( q ) } { 3 } , \end{align*}


\begin{align*} \dot { x } = - \gamma h ( x + \delta u ) u = - \gamma ( h ( x ) + R ( x , \delta u ) ) u \end{align*}


\begin{align*} K _ i = \langle \{ a _ i , b _ i \} , \{ b _ i , c _ i \} , \{ d _ i \} \rangle , \ L _ i = \langle \{ c _ i \} , \{ d _ i \} \rangle . \end{align*}


\begin{align*} \tilde { \varphi } ( t , x ) = \varphi ( t , x , \kappa _ t ( \cdot ) ) , \kappa ( \cdot ) \in \Lambda _ 0 ( \tau , z , w ( \cdot ) ) , \end{align*}


\begin{align*} \gamma _ { \Lambda } ( \gamma _ { \Delta } ( \eta _ { \Delta } | \cdot ) | \eta _ { \Lambda ^ c } ) = \gamma _ \Lambda ( \eta _ \Delta | \eta _ { \Lambda ^ c } ) . \end{align*}


\begin{align*} P _ i ^ j = P _ 0 ^ 0 + j ( q ^ 2 + 1 ) + i . \end{align*}


\begin{align*} \pi _ A : \mathbb { R } ^ { | R ^ + | } \longrightarrow \mathbb { R } ^ { | A | } , \ ; e _ \alpha \mapsto \begin{cases} e _ \alpha & - \alpha \in A , \\ 0 & \end{cases} \end{align*}


\begin{align*} \P & = \left \{ x \in \mathcal { B } : x \mbox { i s n o n n e g a t i v e , n o n d e c r e a s i n g , a n d } \right . \\ & \phantom { = = } \left . ( y - T _ 1 ) ^ k x ( w ) \le ( w - T _ 1 ) ^ k x ( y ) \mbox { f o r a l l } y , w \in [ T _ 1 , T _ 2 ] \mbox { w i t h } y \le w \right \} . \end{align*}


\begin{align*} J ( t , j ) = \int _ { - \infty } ^ { \infty } \ , x ^ { j } \ , \exp \left ( - t x - \frac { x ^ 2 } { 2 } \right ) \mathrm { d } x = ( - 1 ) ^ j \ , \sqrt { 2 \pi } \ , \ , \ , \frac { \partial ^ j } { \partial t ^ j } \left \{ \exp \left ( \frac { t ^ 2 } { 2 } \right ) \right \} \end{align*}


\begin{align*} & ~ \sum _ { n \in A } \Big | \int _ { \R \times \R ^ 2 } ( f _ 1 \ast \tilde { f } _ 2 ) \cdot f _ 3 ~ d \xi d \tau d \eta \Big | \\ & \lesssim N N _ 1 ^ { 5 - 2 \alpha + \varepsilon } \sup _ { n \in A } \sum _ { N _ 1 ^ { 5 - 2 \alpha + \varepsilon } \leqslant L _ i \leqslant N _ 1 ^ \alpha N } \Big | \int _ { \R \times \R ^ 2 } ( f _ { 1 , L _ 1 } \ast f _ { 2 , L _ 2 } ) \cdot f _ { 3 , L _ 3 } d \xi d \eta d \tau \Big | , \end{align*}


\begin{align*} \frac { d } { d t } F ( \gamma ( t ) ) = \left \langle v , \dot { \gamma } ( t ) \right \rangle \forall v \in D _ F ( \gamma ( t ) ) , t \in [ 0 , 1 ] . \end{align*}


\begin{align*} \left ( \int \varphi d | \mu - \nu | \right ) ^ 2 \leq 4 C H \left ( \mu \mid \nu \right ) , C = \frac { 1 } { 6 } \int \varphi ^ 2 d \mu + \frac { 1 } { 3 } \int \varphi ^ 2 d \nu , \end{align*}


\begin{align*} \mbox { $ N _ { P _ 2 ^ * } ( c _ 6 ) = \emptyset $ , a n d $ T $ i s n o t a j u m p a c r o s s $ c _ 1 c _ 7 $ . } \end{align*}


\begin{align*} \prescript { } { l } { b } ( \lambda ) = \prescript { } { l } { \beta } ( \lambda ) \prescript { } { l } { \tilde { p } } ( \lambda ) \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac 1 n \sum _ x \varphi \left ( \frac x n \right ) \int _ 0 ^ \theta \left ( \bar { p ^ 2 _ x } ( t ) - T \left ( \frac x n \right ) \right ) ^ 2 d t = 0 . \end{align*}


\begin{align*} | \{ p \leq X : B ( p ) = B \} | = c ( B ) \int _ 2 ^ X \frac { d t } { \log t } + O _ K \left ( X e ^ { - c _ K \sqrt { \log X } } \right ) , \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*} \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*} \mathbf { H } _ { \mathrm { a l l } , k } = \mathbf { H } _ { \mathrm { d } , k } + \overline { \mathbf { H } } _ { k } \mathbf { \Phi } \overline { \mathbf { G } } , \forall k \in \mathcal { K } , \end{align*}


\begin{align*} { \rm e x } ( n , H _ k ^ r ) \ : \geq \ : \frac { 1 } { n } \sum _ { d = 0 } ^ { \left \lfloor \ ! \frac { ( k - 1 ) n } { r } \ ! \right \rfloor } N ( n , r , k - 1 , d ) \ , , \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*} \begin{aligned} & \left \| \mathbf { R } _ { x s } - \mathbf { W _ { x s } } \right \| \le 2 \gamma \dot { \nu } , \\ & \left \| \mathbf { W } _ { x s } - \dot { \nu } \mathbf { I } \right \| \le \gamma \dot { \nu } , \\ & \left \| \mathbf { R } _ { x s } - \dot { \nu } \mathbf { I } \right \| \le 3 \gamma \dot { \nu } , \end{aligned} \end{align*}


\begin{align*} P f ( x ) = \sum \limits _ { j = 1 } ^ { n + 1 } f \left ( x ^ { ( j ) } \right ) \lambda _ j ( x ) . \end{align*}


\begin{align*} | \sigma | ^ 2 _ g = \frac { \i ^ { n ^ 2 } \sigma \wedge \bar { \sigma } } { d V _ g } = c _ n \frac { ( d y _ 1 \wedge \dots \wedge d y _ n ) \wedge ( d \theta _ 1 \wedge \dots \wedge d \theta _ n ) } { ( d x ^ 1 \wedge \dots \wedge d x ^ n ) \wedge ( d \theta _ 1 \wedge \dots \wedge d \theta _ n ) } = c _ n \frac { d y _ 1 \wedge \dots \wedge d y _ n } { d x ^ 1 \wedge \dots \wedge d x ^ n } , \end{align*}


\begin{align*} S u p p ( F ) = \bigcup \limits _ { \mu } S u p p ( s _ \mu ) . \end{align*}


\begin{align*} \partial ^ { c i } _ { \tau , w } g ( \tau , w ( \cdot ) ) = \partial ^ { c i } _ { \tau , w } g _ * ( \tau , z , w ( \cdot ) ) . \end{align*}


\begin{align*} \bar \mu _ t ^ { * , t _ 0 , \xi } : = \rho \big ( - R ^ { - 1 } B \mathbb E \big [ \bar U ( t , \bar x _ t ^ { * , t _ 0 , \xi } , \nu _ t ^ { * , t _ 0 , \xi } ) | \mathcal { F } _ t ^ { W ^ 0 } \big ] \big ) \quad \bar \nu _ t ^ { * , t _ 0 , \xi } : = \mathbb E \big [ \bar x _ t ^ { * , t _ 0 , \xi } | \mathcal { F } _ t ^ { W ^ 0 } \big ] . \end{align*}


\begin{align*} F _ { q } ( s - s _ { 0 } ) = \int _ { 0 } ^ { \infty } \exp _ { q } ( - s t ) \exp _ { q } \bigg ( \frac { s _ { 0 } t } { 1 - ( 1 - q ) s t } \bigg ) f ( t ) d t . \end{align*}


\begin{align*} \operatorname { d i v } \textbf { \textit { u } } = \nabla _ i u ^ i = \frac { \partial u ^ i } { \partial x _ i } + \Gamma ^ j _ { i j } u ^ i , \textbf { \textit { u } } = u ^ i \frac { \partial } { \partial x _ i } \in \mathfrak { X } ( \Omega ) , \end{align*}


\begin{align*} \Phi ( a u _ \ell + b w _ \ell , p ) = c u _ \ell + d w _ \ell \end{align*}


\begin{align*} \min _ { \mathbf { \Phi } _ { g } } ~ & \tilde { f } _ g ( \mathbf { \Phi } _ { g } ) = \mathsf { T r } ( \mathbf { \Phi } _ { g } \mathbf { Y } _ { g , g } \mathbf { \Phi } _ { g } ^ H \mathbf { Z } _ { g } - 2 \Re \{ \mathbf { \Phi } _ { g } \tilde { \mathbf { X } } _ { g } \} ) \\ \mathrm { s . t . } ~ & \mathbf { \Phi } _ { g } ^ H \mathbf { \Phi } _ { g } = \mathbf { I } _ { \bar { M } } , \forall g \in \mathcal { G } . \end{align*}


\begin{align*} C _ 3 ( t ) = \sum _ { u : \ , i _ u \geq 3 \atop { 2 \leq d _ u \leq i _ u } } d _ u = \sum _ { i \geq 3 } \sum _ { d = 2 } ^ { i } d x _ { i , d } , \\ \end{align*}


\begin{align*} \int e ^ { W ( y | x ) } d \nu _ { A } ( y ) = \varphi _ A ( x ) . \end{align*}


\begin{align*} - \sum _ { K = 0 } ^ { p } \binom { p } { K } \ , B _ K \ , n ^ { p - K } \end{align*}


\begin{align*} \tilde V _ x ( m ) = \sum _ { x ' = 1 } ^ n M _ { x , x ' } ( m ) \tilde V _ { x ' } ( m ) + \tilde v _ x ( m ) . \end{align*}


\begin{align*} \mathcal { L } _ { } ( \Theta ) = \mathcal { L } _ { } + \gamma \cdot \mathcal { L } _ { } , \end{align*}


\begin{align*} \varphi _ 1 ( x ) = u ( x ) + t v _ 1 ( x ) , \end{align*}


\begin{align*} P ( x _ 1 , \ldots , x _ n ) = \lambda _ 0 x _ 1 ^ s + \lambda _ 1 x _ 1 ^ { s - 1 } x _ 2 + \cdots + \lambda _ s x _ 2 ^ s , \end{align*}


\begin{align*} k ( t ) = h _ { 1 - \alpha , \rho } ( t ) \ , + \ , \rho \ , \int _ 0 ^ t h _ { 1 - \alpha , \rho } ( \tau ) \ , d \tau , \end{align*}


\begin{align*} x \diamond y = x \cdot D ( y ) . \end{align*}


\begin{align*} \sum _ { j \in \mathbb Z _ 0 } z _ j = \frac { \sum _ { j \in \mathbb Z _ 0 } \lambda _ j } { ( 1 + A ) ^ k } , \end{align*}


\begin{align*} \begin{aligned} { \mathcal { G } } ( z , r , z ' , r ' ) & = \frac { \sqrt { r r ' } } { 2 \pi } F ( s ) , s = \frac { | z - z ' | ^ { 2 } + | r - r ' | ^ { 2 } } { r r ' } , \\ F ( s ) & = \int _ { 0 } ^ { \pi } \frac { \cos \theta } { \sqrt { 2 ( 1 - \cos \theta ) + s } } \dd \theta . \end{aligned} \end{align*}


\begin{align*} \{ w ' \in \Omega : O ( \chi _ { n , m } ' ( w ' , M , \cdot ) ) = O ( \chi _ { n , m } ' ( w , M , \cdot ) ) \} \end{align*}


\begin{align*} C _ 0 ( C _ { \ast } ) = \ell _ 0 ( H _ 0 ( C _ { \ast } ) ) . \end{align*}


\begin{align*} N _ { c o m p } ^ i & = N ( r _ { i , f } + r _ { i , g } ) \\ & = 2 N + \mathcal { O } \left ( \frac { \max \{ \omega , \omega ^ { - 1 } \} } { n L _ { x y } ( 1 - \theta ) ^ 2 \epsilon } \left ( \left ( \frac { 1 } { L _ { x } } + \frac { \omega } { L _ { x y } } \right ) \sigma _ { f , i } ^ 2 + \left ( \frac { 1 } { L _ { y } } + \frac { 1 } { L _ { x y } \omega } \right ) \sigma _ { g , i } ^ 2 \right ) \log \left ( \frac { \Psi ^ 0 \nu } { \epsilon } \right ) \right ) . \end{align*}


\begin{align*} h _ { k , S , ( n , g ) } ( \tau , z ; \tau _ 0 , z _ 0 ) : = \det ( \tau - \bar { \tau } _ 0 ) ^ { - k } e \left ( - ( \tau - \bar { \tau } _ 0 ) ^ { - 1 } S [ z - \overline { z } _ 0 ] \right ) . \end{align*}


\begin{align*} Q _ { \delta _ N } L _ { \delta _ N } u _ N = L _ { \delta _ N } u _ N - \sum _ { j = 1 } ^ m \sum _ { h = 1 } ^ 2 C _ { j , h , N } \frac { \partial } { \partial z _ { j , h } } ( - \delta _ N ^ 2 ( K ( z _ { N , j } ) \nabla V _ { \delta _ N , Z _ N , j } ) ) . \end{align*}


\begin{align*} \begin{array} { l } ( x + a ) \cdot ( y + b ) = \big ( x \cdot _ A y + l _ { B } ( a ) y + r _ { B } ( b ) x \big ) + \big ( a \cdot _ B b + l _ { A } ( x ) b + r _ { A } ( y ) a \big ) , \\ ( \alpha \oplus \beta ) ( x + a ) = \alpha ( x ) + \beta ( a ) . \end{array} \end{align*}


\begin{align*} | A | \ , | A + B + C | = & \sum _ { k + j + m = n } { n \choose k , j , m } | A | V ( A [ k ] , B [ j ] , C [ m ] ) \\ = & \sum _ { 0 \le j + m \le n } { n \choose j , m , n - j - m } | A | V ( A [ n - j - m ] , B [ j ] , C [ m ] ) , \end{align*}


\begin{align*} l ( y ' ) : = u ( x _ 0 ) + \sum _ { \beta = 1 } ^ { n - 1 } \nabla _ \beta u ( x _ 0 ) y _ \beta . \end{align*}


\begin{align*} x _ { \lambda } x _ { \mu } x _ { \nu } = x _ { \lambda } ( x _ { \mu } x _ { \nu } ) \lambda ( \mu \nu ) \end{align*}


\begin{align*} \begin{aligned} A _ { t } + B \times u + \nabla Q & = - \mu \nabla \times B . \end{aligned} \end{align*}


\begin{align*} & ( i ) \ \ \ \nabla ( \gamma _ 0 J + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ) ( x ^ * ) = 0 \\ & ( i i ) \ \ \gamma _ k f _ k ( x ^ * ) = 0 \ \ k \in \{ 1 , . . . , m \} \\ & ( i i i ) \ \gamma _ { 0 } A _ J + \sum \limits _ { k = 1 } ^ m \gamma _ k A _ k \succeq 0 \end{align*}


\begin{align*} \begin{aligned} \frac { ( - 1 ) ^ r } { r ! } \frac { \mathrm { \partial } ^ r } { \mathrm { \partial } \alpha ^ r } Z ^ { \star } _ { I } ( \{ k _ i \} _ { i = 1 } ^ { n } ; ( \alpha , \beta ) ) = \sum _ { \begin{subarray} { c } r _ 1 + \cdots + r _ { n - 1 } = r \\ r _ i \in \mathbb { Z } _ { \ge 0 } \end{subarray} } Z ^ { \star } _ { I } ( \{ k _ i , \{ 1 \} ^ { r _ i } \} _ { i = 1 } ^ { n - 1 } , k _ n ; ( \alpha , \beta ) ) \end{aligned} \end{align*}


\begin{align*} \Lambda = \Lambda ( i _ 1 , \cdots , i _ m ) = \prod _ { a = 1 } ^ { m } \Lambda _ a ^ { i _ a } , \medspace R = R ( j _ 1 , \cdots , j _ n ) = \prod _ { b = 1 } ^ { n } R _ b ( j _ b ) \cdot C . \end{align*}


\begin{align*} g ( \Gamma ) = h ^ 1 ( \Gamma ) + \sum _ { v \in V } g ( v ) , h ^ 1 ( \Gamma ) = | E | - | V | + 1 . \end{align*}


\begin{align*} e _ 2 ( C ) = e ( C ) , \end{align*}


\begin{align*} \mathcal { L } _ g : = \begin{pmatrix} \mu \Delta _ { B } + ( \lambda + \mu ) \operatorname { g r a d } \operatorname { d i v } + \mu \operatorname { R i c } + \rho \omega ^ 2 & - \beta \operatorname { g r a d } \\ i \omega \theta _ { 0 } \beta \operatorname { d i v } & \alpha \Delta _ { g } + i \omega \gamma \end{pmatrix} , \end{align*}


\begin{align*} f ^ { - 1 } ( \overline { V } _ \bullet ) = \{ V _ \bullet ; V _ j = \overline { V } _ j j \in [ n ] \setminus \{ l \} , V _ { l - 1 } + X V _ i \subset V _ l \subset V _ { l + 1 } \} \end{align*}


\begin{align*} \bar \pi \bar p _ t = \bar \pi , ~ ~ ~ ~ \forall t > 0 . \end{align*}


\begin{align*} \frac { 1 } { \nu ( \xi _ { \Delta } \eta _ { \Lambda _ n \setminus \Delta } ) } \int \mathbf { 1 } _ { \eta _ { \Lambda _ n } } ( \omega ) \frac { \gamma _ { \Delta } ( \xi _ { \Delta } | \omega _ { \Delta ^ c } ) } { \gamma _ { \Delta } ( \eta _ { \Delta } | \omega _ { \Delta ^ c } ) } \nu ( d \omega ) = 1 , \end{align*}


\begin{align*} F ^ { + } ( x + y ) + A ( x , y ) + \int _ { x } ^ { \infty } A ( x , t ) F ^ { + } ( t + y ) d t = 0 , y > x \end{align*}


\begin{align*} \sum \limits _ { i = 1 } ^ j \chi _ i \leq \chi ( \sum \limits _ { i = 1 } ^ j w _ i ) + j r , \end{align*}


\begin{align*} \left ( 2 \binom { n } { ( k - 1 ) n / p } \right ) ^ t \leq ( c p ) ^ { ( k - 1 ) n t / p } \end{align*}


\begin{align*} \begin{aligned} & \omega _ T = \frac { d ( x + \sqrt { \epsilon } y ) } { x + \sqrt { \epsilon } y } \wedge \frac { d ( x - \sqrt { \epsilon } y ) } { x - \sqrt { \epsilon } y } \\ & = \frac { ( d x + \sqrt { \epsilon } d y ) \wedge ( d x - \sqrt { \epsilon } d y ) } { x ^ 2 - \epsilon y ^ 2 } = \frac { - 2 \sqrt { \epsilon } } { N _ { E / F } ( x + \sqrt { \epsilon } y ) } d x \wedge d y , \end{aligned} \end{align*}


\begin{align*} \int _ { \mathfrak X } | \omega | = { \mathfrak h _ 0 } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} T _ { \vec { v } } ^ { - n } \phi _ u \widetilde { B } = & \phi _ u ( T ^ { ( 0 , 1 ) } \times i d ) T _ { \vec { v } } ^ { - n } ( \widetilde { B } \cap ( X \times A _ 1 ^ n ) ) \\ & \cup \phi _ u T _ { \vec { v } } ^ { - n } ( \widetilde { B } \cap ( X \times A _ 2 ^ n ) ) \cup \phi _ u ( T ^ { - ( 0 , 1 ) } \times i d ) T _ { \vec { v } } ^ { - n } ( \widetilde { B } \cap ( X \times A _ 3 ^ n ) ) . \end{align*}


\begin{align*} \left \{ \begin{array} { r l } a _ n = ( 1 - \frac 1 n ) a _ { n - 1 } + \frac 1 n b _ { n - 1 } \\ b _ n = ( 1 - \frac 1 n ) a _ { n - 1 } + \frac { e ^ t } { n } b _ { n - 1 } \end{array} \right . a _ 0 = b _ 0 = 1 . \end{align*}


\begin{align*} \left \| ( I - F ) ^ { - 1 } \right \| _ { L _ x ^ { \infty } L _ z ^ 2 \rightarrow L _ x ^ { \infty } L _ z ^ 2 } \leq \sum _ { n = 0 } ^ { \infty } \frac { 1 } { n ! } \left \| \widetilde { Q } ( u ) \right \| _ { L ^ 1 } ^ n = \mathrm { e } ^ { \left \| \widetilde { Q } ( u ) \right \| _ { L ^ 1 } } . \end{align*}


\begin{align*} \nu _ { \alpha } ^ { \prime } \left ( d x \right ) = \boldsymbol { 1 } _ { \left \vert x \right \vert > 1 } \nu _ { \alpha } \left ( d x \right ) \end{align*}


\begin{align*} \big \langle \mathrm { E } ( \Sigma , c ) , ( V , w ) \big \rangle = \big \langle ( V _ g , w _ g ) , ( V , w ) \big \rangle = \int _ \Sigma \big ( V _ g - u _ g c H \big ) V \ , \mathrm { d } \mathcal { H } ^ d + \int _ \Sigma u _ g w \ , \mathrm { d } \mathcal { H } ^ d \end{align*}


\begin{align*} g ^ * f _ ! = f ' _ ! g '^ * \end{align*}


\begin{align*} U _ p : = \prod _ { v \mid p } U ^ { e _ v } _ v , \ u _ p = \prod _ { v \mid p } u ^ { e _ v } _ v , \ S _ p = \prod S ^ { e _ v } _ v . \end{align*}


\begin{align*} J u _ 2 ( t ) = U ( t - t _ 0 ) J u _ 2 ( t _ 0 ) - i \lambda _ 6 \int ^ t _ { t _ 0 } U ( t - \tau ) J \mathcal { N } _ 2 ( u _ 1 , u _ 2 ) ( \tau ) \ d \tau . \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { w _ F } \sum _ { i = 1 } ^ h \phi ( a _ i x ) . \end{align*}


\begin{align*} ( I - Q ^ \dagger ) F ^ { q , \dagger } ( v + \psi ^ \dagger ( v , p ) , p ) = 0 \end{align*}


\begin{align*} h ^ { V } \overset { \mathrm { l a w } } { = } h ^ { U } + \varphi ^ { V , U } , \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*} C _ { 0 } ( K ; \mathrm { A d } _ { \rho } ) = C _ { 0 } ( \widetilde { K } ; \mathbb { Z } ) \otimes \mathfrak { g } / \sim \ ; \cong \mathfrak { g } . \end{align*}


\begin{align*} \delta ^ 2 \int _ { \Omega } \left ( K ( x ) \nabla u | \nabla \varphi \right ) = \int _ { \Omega } \hat { h } \varphi , \ \ \ \ \forall \varphi \in H ^ 1 _ 0 ( \Omega ) . \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 \\ u _ { i - 1 } = \lambda u _ { i } \end{array} \right . \end{align*}


\begin{align*} T \circ \pi _ 1 ( \exp ( X ) ) = \pi _ 2 ( \exp ( X ) ) \circ T , \end{align*}


\begin{align*} \begin{aligned} & { { \mathbf { x } } _ { w } } = \omega { { \mathbf { x } } _ { o } } + \left ( 1 - \omega \right ) \mathbf { e } \\ & { { \mathbf { y } } _ { w } } = \omega { { \mathbf { y } } _ { o } } \\ & { { \mathbf { s } } _ { w } } = \omega { { \mathbf { s } } _ { o } } + \left ( 1 - \omega \right ) \mathbf { e } \\ & { { \kappa } _ { w } } = { { \left ( { { \mathbf { x } } _ { w } } \right ) } ^ { T } } { { \mathbf { s } } _ { w } } / k \\ & { { \tau } _ { w } } = 1 , \end{aligned} \end{align*}


\begin{align*} T = \frac { 1 } { d } \int _ 0 ^ { R _ 0 } \frac { z } { g \left ( \frac { m } { \alpha _ d } z ^ { - d } \right ) } \ , \mathrm { d } z = C ( m , d , s ) \int _ 0 ^ { R _ 0 } z ^ { s d + 1 } \ , \mathrm { d } z = \frac { C ( m , d , s ) } { s d + 2 } \ , z ^ { s d + 2 } \Big | _ 0 ^ { R _ 0 } \end{align*}


\begin{align*} \begin{aligned} | \bar { y } ( t ) & - z _ 2 ( t ) | \le | \bar { y } ( 0 ) - z _ 2 ( 0 ) | + t \gamma ^ 2 \bar { k } _ 2 ( L _ r , \delta ) \\ & + \gamma \int _ 0 ^ t L _ r | x - z _ 1 | + | \bar { y } ( \tau ) - z _ 2 ( \tau ) | \ , d \tau \end{aligned} \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*} \lim _ { x \to \infty } \frac { f ( x ) } { g ( x ) } = \lim _ { x \to \infty } \frac { f ( x ) } { f _ 1 ( x ) } \frac { f _ 1 ( x ) } { g _ 1 ( x ) } \frac { g _ 1 ( x ) } { g ( x ) } = \lambda / ( 1 - e ^ { - \lambda } ) . \end{align*}


\begin{align*} \lim _ { x \to \infty } \frac { h _ { } ( x ) } { x } & = \lim _ { x \to \infty } \frac { \phi ( \frac { x - \mu } { \sigma } ) \frac { 1 } { x } } { \sigma [ 1 - \Phi ( \frac { x - \mu } { \sigma } ) ] } . \end{align*}


\begin{align*} \ell ( x \varepsilon ^ \lambda ) = \ell ( x ) + \ell ( \varepsilon ^ \lambda ) , \ell ( x ' \varepsilon ^ \lambda ) = \ell ( x ' ) + \ell ( \varepsilon ^ \lambda ) . \end{align*}


\begin{align*} I : = \{ \beta \in \Phi ^ + \mid s _ \alpha ( \beta ) \in \Phi ^ - \} . \end{align*}


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


\begin{align*} \Gamma _ { \lambda } ( t ) : = \int _ { 0 } ^ { \lambda } \mu \xi ( \mu , t ) \ d \mu \mbox { a n d } \Theta _ { \lambda } ( t ) : = \int _ { \lambda } ^ { \infty } \mu \xi ( \mu , t ) \ d \mu , \end{align*}


\begin{align*} \sum _ { r _ { Q } ( n ) = 0 } n \ll p ^ { 1 + \epsilon } m p ^ { - 1 / 4 + \epsilon } \ll p ^ { 5 / 2 + \epsilon } . \end{align*}


\begin{align*} q = \frac { a _ { 0 } ^ { \prime \prime } - a _ { 0 } ^ { \prime } } { a _ { 0 } + 1 } \end{align*}


\begin{align*} \delta ^ 2 ( 1 - \epsilon ^ 2 ) b _ { \alpha } u _ n ( x ) \leq u ( y ) - u ( x ) \leq C b _ { \alpha } ^ 2 , \mbox { w i t h } y = ( x ' , \delta ^ 2 b _ \alpha ) \in \partial _ 2 \omega \end{align*}


\begin{align*} S _ 2 \wr S _ n = ( \mathbb Z / 2 \mathbb Z ) \wr S _ n = ( \mathbb Z / 2 \mathbb Z ) ^ n \rtimes S _ n = B _ n . \end{align*}


\begin{align*} \int _ { \Omega } \tilde { A } \cdot B \dd x = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x , \end{align*}


\begin{align*} a _ { \ell } = R _ 1 ( \ell ) \beta _ 1 ^ { \ell } + \cdots + R _ s ( \ell ) \beta _ s ^ { \ell } \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*} ( U _ { \alpha } ( t ) \phi ) ^ { \wedge } ( \xi , \eta ) = e ^ { i t \omega _ \alpha ( \xi , \eta ) } \hat { \phi } ( \xi , \eta ) = e ^ { i t ( | \xi | ^ { \alpha } \xi + \frac { \eta ^ 2 } { \xi } ) } \hat { \phi } ( \xi , \eta ) . \end{align*}


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


\begin{align*} v o l ( \mathcal { H } ( A ) ) = \tfrac { 1 } { 2 } \mathcal { A } . \end{align*}


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


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


\begin{align*} c ( \gamma , \gamma ^ + ) + c ( \gamma , \gamma ^ - ) = C ( \gamma \gamma ^ + , \gamma \gamma ^ { - } ) - C ( \gamma ^ + , \gamma ^ - ) = 0 \end{align*}


\begin{align*} f ( z ) = \int _ { \mathbb { C } } e ^ { - \omega z } d \mu ( \omega ) , \end{align*}


\begin{align*} & I _ { m , t _ { 1 } , \cdots , t _ { k } } ( x _ { 1 } , \cdots , x _ { k } ) \\ & = \inf \left \{ J ( \varphi ) ; ~ \varphi ( t ) = \int ^ { t } _ { 0 } \phi ( r ) d r \in \mathcal { A C } , \int ^ { 1 } _ { 0 } K _ { m } ( t _ { i } , r ) \phi ( r ) d r = x _ { i } , 1 \leq i \leq k , t \in [ 0 , 1 ] \right \} , \end{align*}


\begin{align*} \varepsilon & \leq \| \varphi _ { k m _ { j _ 0 } } - \rho \| _ \infty \\ & = \| \varphi _ { k j } \circ \varphi _ { k ( m _ { j _ 0 } - j ) } - \rho \circ \varphi _ { k ( m _ { j _ 0 } - j ) } \| _ \infty \\ & = \sup _ { z \in \mathbb { B } _ n } | ( \varphi _ { k j } - \rho ) ( \varphi _ { k ( m _ { j _ 0 } - j ) } ( z ) ) | \\ & \leq \| \varphi _ { k j } - \rho \| _ \infty . \end{align*}


\begin{align*} \dot { s } \varepsilon ( z ) \dot { s } \varepsilon ( 1 ) \xi _ 2 = \varepsilon ( - z ^ { - 1 } ) \dot { s } h ( - z ) \varepsilon ( - z ^ { - 1 } ) \varepsilon ( 1 ) \xi _ 2 = \varepsilon ( - z ^ { - 1 } ) \dot { s } \varepsilon ( z ^ 2 - z ) \xi _ 2 . \end{align*}


\begin{align*} \mathbf { 1 } _ { N L } ^ T \mathbf { y } \mathbf { x } ^ T \mathbf { 1 } _ { N L } = \mathbf { 1 } _ { N } ^ T Y X ^ T \mathbf { 1 } _ { N } = \mathbf { c } _ Y ^ T \mathbf { c } _ X . \end{align*}


\begin{align*} y _ { n b } = A \exp \{ \sqrt { \frac { \mu + E } { b } } x \} + B \exp \{ - \sqrt { \frac { \mu + E } { b } } x \} , x \in ( R , R + r ) . \end{align*}


\begin{align*} q ( t , x ) & : = \int _ { Y ^ { \ast } } \nabla _ y \cdot V _ 0 ( t , x , y ) d y , \\ \bar { J } _ 0 ( t , x ) & : = \int _ { Y ^ { \ast } } J _ 0 ( t , x , y ) d y . \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { T - 1 } \Theta _ k ( 1 - \Lambda _ k ' ) - ( 1 - \Theta _ k ) ( 1 - \Lambda _ k ) + ( 1 - \Theta _ k ) \Lambda _ k - \Theta _ k \Lambda _ k ' \le \left | \log _ \gamma \frac { \bar \Delta ' } { \delta _ 0 } \right | < \left | \log _ \gamma \frac { \bar \Delta } { \delta _ 0 } \right | + 1 . \end{align*}


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


\begin{align*} \mathrm { e } ^ { \lambda \sum _ { i = 1 } ^ d ( y _ i - x _ i ) - d n \ln ( 1 + \lambda ) } \le \mathrm { e } ^ { \lambda d ( y _ 1 - x _ 1 ) + d ^ 2 - n d \lambda + n d \frac { \lambda ^ 2 } { 2 } } \le C _ d \mathrm { e } ^ { d \frac { ( y _ 1 - x _ 1 - n ) } { \sqrt n } } . \end{align*}


\begin{align*} - \left ( \partial _ { \rho } ^ { 2 } + \frac { 4 } { \rho } \partial _ { \rho } \right ) \tilde { \varphi } = \mu ^ { 2 } \tilde { \varphi } _ { + } , \rho > 0 . \end{align*}


\begin{align*} \theta _ { j , ( 2 ) } ( z _ { k } ) = \delta _ { j , k } . \end{align*}


\begin{align*} \mu = m _ 1 e _ 1 + \dots + m _ n e _ n , \end{align*}


\begin{align*} \frac { 1 } { h ^ + } = \begin{cases} \frac { 1 } { h } , & h > 0 , \\ \infty , & h \le 0 . \end{cases} \end{align*}


\begin{align*} Q = ] a , b [ \times B ( x _ 0 , r ) , \mbox { w i t h } 0 < a < b < + \infty , \ ; \ ; x _ 0 \in \R \ ; \ ; \mbox { a n d } \ ; \ ; 0 < r < + \infty , \end{align*}


\begin{align*} F \ast G ( x ) = \int _ { - \infty } ^ \infty F ( x - y ) G ( d y ) \end{align*}


\begin{align*} \Upsilon ( \cdot ) = \Phi ( \cdot ) + \Psi ( \cdot ) \end{align*}


\begin{align*} & W _ 0 = \{ ( p , q ) \in \hat A \times \hat B \mid d ( q , w ) \geq d ( p , w ) \} ; \\ & W _ 1 = \{ ( p , q ) \in \hat A \times \hat B \mid d ( p , \tilde { w } ) \geq d ( q , \tilde { w } ) \} . \end{align*}


\begin{align*} \begin{aligned} & ( i ) & \left | w ^ \varepsilon ( x , t ) \right | & \leq K , \medskip \\ & ( i i ) & \left | \partial _ x w ^ \varepsilon ( x , t ) \right | & \leq K , \medskip \\ & ( i i i ) & \left | w ^ \varepsilon ( x , t ) - w ^ \varepsilon ( x , s ) \right | & \leq K \left ( \left | t - s \right | ^ { 1 / 2 } + | t - s | \right ) . \end{aligned} \end{align*}


\begin{align*} q ( A | E _ i ) = p ( A | E _ i ) , i q ( E _ i ) > 0 A \in { \bf A } , \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*} f _ n = \sum _ { j = 0 } ^ n x _ 0 \dotsb \hat { x } _ j \dotsb x _ n \in \C [ x _ 0 , \dotsc , x _ n ] \end{align*}


\begin{align*} \mathbf { y } = \mathbf { H } \mathbf { P } ^ { \left ( \right ) } \left ( \mathbf { a } ^ { \left ( \right ) } \right ) \mathbf { s } ^ { \left ( \right ) } + \mathbf { n } , \end{align*}


\begin{align*} D ( x ) A + B D ( x ) = - D ( v ) , \end{align*}


\begin{align*} [ v ] _ { B ^ { 1 + \gamma } _ { p , \infty } ( B _ \rho ) } = \displaystyle \sup _ { h \in \mathbb { R } ^ { n } } \biggl ( \displaystyle \int _ { B _ \rho } \dfrac { | \tau ^ 2 _ h v ( x ) | ^ { p } } { | h | ^ { ( 1 + \gamma ) p } } d x \biggr ) ^ { \frac { 1 } { p } } . \end{align*}


\begin{align*} p _ 0 = 1 - 2 \exp \left ( a ( \epsilon _ f - r / 2 ) \right ) , \end{align*}


\begin{align*} R < \min \left \{ \begin{aligned} & 3 C , C + 2 C _ 0 + F _ 2 , \\ & 2 C + 4 C _ 0 + F _ 3 , F _ 5 , 3 C + F _ 7 . \end{aligned} \right \} \end{align*}


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


\begin{align*} \alpha _ t ( x ) : = - R ^ { - 1 } B ( P _ t x + \varphi _ t ^ \xi ) - h ( \mu _ t ) . \end{align*}


\begin{align*} I = ( I \cap X ) \vee ( I \cap Y ) = ( I \cap X ) \times ( I \cap Y ) . \end{align*}


\begin{align*} \begin{aligned} \| \Lambda ^ { s - 1 } \nabla v \| _ 3 & \leq C \| \nabla v \| ^ { 1 - \kappa } \| \Lambda ^ { s + 1 } v \| ^ \kappa \end{aligned} \end{align*}


\begin{align*} d _ \Theta ( \theta , \theta ' ) = \left \{ \begin{array} { l l } \min \{ \| \theta - \theta ' \| , 1 \} & k ( \theta ) = k ( \theta ' ) , \\ 1 & \end{array} \right . \end{align*}


\begin{align*} \frac { 1 + b ( z ) } { 1 - b ( z ) } = \int _ { \partial \mathbb { D } } \frac { t + z } { t - z } d \mu ( t ) \ , z \in \mathbb { D } \end{align*}


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


\begin{align*} \begin{aligned} \sigma ( a , r ) = \int _ { B ( a , r ) } \Delta u \wedge \beta _ n ^ { n - 1 } . \end{aligned} \end{align*}


\begin{align*} \int _ { \partial \Omega } \nabla c \cdot \nu c \psi ^ { 2 } = 0 . \end{align*}


\begin{align*} e ( \rho , x ) = e ^ { i \rho x } \left ( 1 + o ( 1 ) \right ) , e ^ { \prime } ( \rho , x ) = i \rho e ^ { i \rho x } \left ( 1 + o ( 1 ) \right ) , x \rightarrow \infty , \end{align*}


\begin{align*} Z \psi = 0 . \end{align*}


\begin{align*} T _ { \varphi _ A ^ * ( f ) } = A ^ t \circ T _ f \circ \varphi _ A \end{align*}


\begin{align*} \begin{aligned} f ( t ) \le & \ , f ( 0 ) + t \gamma ^ 2 \left ( \bar { k } _ 3 ( L _ r , \delta ) + \bar { k } _ 2 ( L _ r , \delta ) \right ) \\ & + \gamma A \int _ 0 ^ t f ( \tau ) \ , d \tau , \end{aligned} \end{align*}


\begin{align*} C ( G , x ) : = 1 + \sum _ { \emptyset \neq U \subseteq V ( G ) : G [ U ] ~ i s ~ a ~ c l i q u e } x ^ { \vert U \vert } , \end{align*}


\begin{align*} W ( y | x ) = \sum _ { n \in \N } \left [ \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ) - \hat { A } \circ \hat { \sigma } ^ { - n } ( y | x ' ) \right ] \end{align*}


\begin{align*} \widetilde { \mu } ( k - 1 ) : = \frac { k \mu ( k ) } { \sum _ { k ' = 1 } ^ \infty k ' \mu ( k ' ) } , k = 1 , 2 , \ldots . \end{align*}


\begin{align*} U \cdot ( A , B , x ) = ( U A , B , x ) , ( A , B , x ) \cdot V = ( A , B V , x ) . \end{align*}


\begin{align*} \Phi \left ( \lambda \right ) = \int _ { \left ( 0 , \infty \right ) } \left ( 1 - e ^ { - \lambda x } \right ) \nu _ { \alpha } ^ { \prime } \left ( d x \right ) \end{align*}


\begin{align*} U : = \left \{ x \in \R ^ { d } \colon \| x \| _ 2 = 1 , x _ d \geq 0 \right \} . \end{align*}


\begin{align*} \xi \left ( ( a ) \right ) = \left ( \frac { a } { | a | } \right ) ^ { k - 1 } \end{align*}


\begin{align*} \begin{aligned} \int _ { \Omega } f ^ { p + 1 } \le C \frac { ( p + 1 ) ^ { 2 } } { 2 \log s } \int _ { \Omega } ( f \log f + e ^ { - 1 } ) \int _ { \Omega } f ^ { p - 2 } | \nabla f | ^ { 2 } + 2 C \| w \| _ { L ^ { 1 } ( \Omega ) } ^ { 2 } + 3 s ^ { p + 1 } | \Omega | , \end{aligned} \end{align*}


\begin{align*} \Delta _ g & = \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + \Gamma ^ \alpha _ { n \alpha } \frac { \partial } { \partial x _ n } + g ^ { \alpha \beta } \frac { \partial ^ 2 } { \partial x _ \alpha \partial x _ \beta } + \biggl ( g ^ { \alpha \beta } \Gamma ^ \gamma _ { \gamma \alpha } + \frac { \partial g ^ { \alpha \beta } } { \partial x _ \alpha } \biggr ) \frac { \partial } { \partial x _ \beta } . \end{align*}


\begin{align*} \frac { d _ 1 } { u ^ { q + 1 } } = \frac { h ^ q } { h } \end{align*}


\begin{align*} A _ n : = B _ n + C _ n \longrightarrow B + C = : A . \end{align*}


\begin{align*} \lim _ { n \to \infty } \mu _ { n } ^ { D , \beta } ( A ) = \mu _ { \infty } ^ { D , \beta } ( A ) , \quad \mathrm { a . s . } \end{align*}


\begin{align*} \lim _ { \gamma \rightarrow 0 } \| g _ { b , \gamma } ( f ) - g _ b ( f ) \| _ { L ^ p ( w ) \rightarrow L ^ p ( w ) } = 0 . \end{align*}


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


\begin{align*} M _ { - , 1 } ( x ; z ) - e _ 1 = \mathcal { P } ^ - \left ( r _ 2 ( z ) e ^ { 2 i z x } M _ { + , 2 } ( x ; z ) \right ) ( z ) , z \in \mathbb { R } . \end{align*}


\begin{align*} \theta \left ( S ( r + s ) \right ) & = \int ^ { T _ 2 } _ { T _ 1 } G \left ( \frac { 3 T _ 1 + T _ 2 } { 4 } , \tau \right ) f _ \downarrow ( r ( \tau ) + s ( \tau ) ) \ ; d \tau \\ & \le f _ \downarrow ( 0 ) \int ^ { T _ 2 } _ { T _ 1 } G \left ( \frac { 3 T _ 1 + T _ 2 } { 4 } , \tau \right ) d \tau \\ & < c . \end{align*}


\begin{align*} x ^ { t + 1 } - y ^ t = \ & ( 1 - \alpha _ t ) x ^ t + \alpha _ t z ^ { t + 1 } - y ^ t = ( 1 - \alpha _ t ) ( x ^ t - y ^ t ) + \alpha _ t z ^ { t + 1 } - \alpha _ t y ^ t \\ \overset { \eqref { l 2 - e 2 } } { = } \ & \alpha _ t ( 1 - \beta _ t ) ( y ^ t - z ^ t ) + \alpha _ t z ^ { t + 1 } - \alpha _ t y ^ t = \alpha _ t \left ( z ^ { t + 1 } - \beta _ t y ^ t - ( 1 - \beta _ t ) z ^ t \right ) . \end{align*}


\begin{align*} \sigma ^ L ( G ^ L ( X ) ) = G ( \{ \varphi ^ L _ { \sigma , s } ( E ^ L _ X ( v ) ) \mid ( s , v ) \in D \} ) . \end{align*}


\begin{align*} \epsilon = \sum _ { i \in I } \bar { \nu } _ i ^ ! & \ ; & ( \delta ^ n _ k ) ^ ! = \sum _ { J = \{ i _ 0 < \hdots < \widehat { i _ k } < \hdots < i _ n \} \subset K = \{ i _ 0 < \hdots < i _ n \} } ( \nu _ { K } ^ J ) ^ ! \end{align*}


\begin{align*} { \mathbb { E } } \left [ { { { \left \| { { \bf { x } } \left [ n \right ] } \right \| } ^ 2 } } \right ] = \sum \limits _ { l = 1 } ^ L { { \mathbb { E } } \left [ { { { \left \| { { { \bf { f } } _ l } s \left [ { n - { \kappa _ l } } \right ] } \right \| } ^ 2 } } \right ] } { \rm = } \sum \limits _ { l = 1 } ^ L { { { \left \| { { { \bf { f } } _ l } } \right \| } ^ 2 } } \le P , \end{align*}


\begin{align*} x = \sum _ { n = 0 } ^ \infty \frac { \varepsilon _ n } { \gamma ^ { n + 1 } } . \end{align*}


\begin{align*} \varrho : = | z | ^ 2 + | w | ^ 2 + \Re ( a z ^ 2 + b w ^ 2 ) - 1 . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta ( c ( t ) - c ( s ) ) + n ( t ) ( c ( t ) - c ( s ) ) = - c ( s ) ( n ( t ) - n ( s ) ) , & x \in \Omega , \\ \nabla ( c ( t ) - c ( s ) ) \cdot \nu + ( c ( t ) - c ( s ) ) = 0 , & x \in \partial \Omega , \end{array} \right . \end{align*}


\begin{align*} \min _ { \substack { \omega _ { n , k } , \ , { { \bf { u } } } _ k } } & \mathcal { L } ^ * \\ & \omega _ { n , k } \in \{ 0 , 1 \} , \forall k , \forall n , \end{align*}


\begin{align*} L = \frac { 1 } { \delta } \det A . \end{align*}


\begin{align*} \langle T ( f , g ) , h \rangle = \langle T ^ { * 1 } ( h , g ) , f \rangle = \langle T ^ { * 2 } ( f , h ) , g \rangle . \end{align*}


\begin{align*} \frac { 1 } { n } \sum _ { i = 1 } ^ n \delta _ { X _ i } \to \int _ 0 ^ 1 \mu ^ * _ u \ , d u , . \end{align*}


\begin{align*} \begin{array} { l } \mu ^ { N , 1 } _ { \boldsymbol { \alpha } ^ * _ t } + h _ 1 ^ N ( \mu ^ { N , 1 } _ { \boldsymbol { \alpha } ^ * _ t } , \ldots , \mu ^ { N , N } _ { \boldsymbol { \alpha } ^ * _ t } ) = \Delta ^ { N , 1 } _ { * , t } , \\ \quad \vdots \qquad \qquad ~ ~ \vdots \qquad ~ ~ ~ ~ \vdots \\ \mu ^ { N , N } _ { \boldsymbol { \alpha } ^ * _ t } + h _ N ^ N ( \mu ^ { N , 1 } _ { \boldsymbol { \alpha } ^ * _ t } , \ldots , \mu ^ { N , N } _ { \boldsymbol { \alpha } ^ * _ t } ) = \Delta ^ { N , N } _ { * , t } , \end{array} \end{align*}


\begin{align*} r _ g ( \mathbf { x } , \mathbf { y } , \mathbf { s } , \kappa , \tau ) = { { \mathbf { b } } ^ { T } } \mathbf { y } - { { \mathbf { c } } ^ { T } } \mathbf { x } - \kappa . \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { 3 m } \alpha _ j + \sum _ { k = 1 } ^ l \beta _ k < \sum _ { i = 1 } ^ { n + 1 } \lambda _ i + \varepsilon . \end{align*}


\begin{align*} \rho ^ { \circ } ( A ) = \inf \{ t > 0 : A \mbox { i s c o v e r e d b y f i n i t e l y m a n y s u b s e t s o f s i z e } \leqslant t \} , \end{align*}


\begin{align*} \bar { \mathbf { X } } = ( \bar { \mathbf { x } } ) , \bar { \mathbf { S } } = ( \bar { \mathbf { s } } ) , \bar { \mathbf { x } } = \mathbf { D } ^ { - T } \mathbf { x } , \bar { \mathbf { s } } = \mathbf { D } \mathbf { s } , \mathbf { D } \in \mathcal { G } . \end{align*}


\begin{align*} \mathcal { P } [ \varrho ] = \frac { 1 } { 2 } a \left ( \alpha ^ 2 + \gamma ^ 2 \right ) ^ 3 P ( a , \tau ) \end{align*}


\begin{align*} H ^ * ( Y _ { K } ^ 1 , \mathcal { D } _ { \Omega } ) = H ^ * ( Y ^ 1 _ { K } , \mathcal { D } _ { \Omega } ) _ { \leq h } \oplus H ^ * ( Y ^ 1 _ { K } , \mathcal { D } _ { \Omega } ) _ { > h } \end{align*}


\begin{align*} d ( u \Rightarrow w v ) = d ( u \Rightarrow w \rho _ x ( u ) ) + d ( w \rho _ x ( u ) \Rightarrow w v ) . \end{align*}


\begin{align*} \{ l _ 1 , l _ 2 , \dots , l _ r , l _ 1 ' , l _ 2 ' , \dots , l _ r ' \} = \cup _ { u = 1 } ^ t \mathcal I ( s _ u ) , \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*} \mathbb E \left [ [ ( \mathcal S ( \Delta _ n ) - I ) Y _ { ( i - 1 ) } ] ^ { \otimes 2 } \Big | \sigma \right ] = \int _ 0 ^ { ( i - 1 ) \Delta _ n } ( \mathcal S ( \Delta _ n ) - I ) \Sigma _ s ^ { \mathcal S _ n } ( \mathcal S ( \Delta _ n ) - I ) ^ * d s . \end{align*}


\begin{align*} \begin{aligned} | x ( t ) - z & ( t ) | \le | x ( 0 ) - z ( 0 ) | + \int _ 0 ^ t \left | \dot { x } ( \tau ) - \dot { z } ( \tau ) \right | \ , d \tau \\ \le & \ , | x ( 0 ) - z ( 0 ) | + \gamma L _ r \int _ 0 ^ t | x ( \tau ) - z ( \tau ) | \ , d \tau \\ & + \gamma ^ 2 \bar k ( L _ r , M _ r , \delta ) t \end{aligned} \end{align*}


\begin{align*} \tilde { \nu _ 1 } ( 0 ) = 0 \qquad \frac { d } { d \tilde { \nu _ 2 } } \tilde { \nu _ 1 } ( 0 ) = 0 . \end{align*}


\begin{align*} \sum _ { j = 0 } ^ { n } ( - 1 ) ^ { j } \binom { n } { j } \frac { \left ( \beta \right ) ^ { \left ( j + m \right ) } } { \left ( \beta \right ) ^ { \left ( j \right ) } } = ( - 1 ) ^ { n } n ! \binom { m } { n } \frac { \left ( \beta \right ) ^ { \left ( m \right ) } } { \left ( \beta \right ) ^ { \left ( n \right ) } } . \end{align*}


\begin{align*} \begin{aligned} \| u _ 2 ( t ) \| _ { H ^ 1 } + \| J u _ 2 ( t ) \| _ { L ^ 2 } & \le ( 1 + C \varepsilon ^ 2 _ 1 \langle T _ 0 \rangle ^ { 1 + 2 C _ 1 \varepsilon _ 1 ^ 2 } e ^ { C \varepsilon ^ 2 _ 1 \langle T _ 0 \rangle ^ { 1 + 2 C _ 1 \varepsilon _ 1 ^ 2 } } ) \varepsilon _ 2 \\ & < 2 \varepsilon _ 2 \end{aligned} \end{align*}


\begin{align*} H : = \frac { 1 } { n - 1 } \begin{bmatrix} 0 & \ 0 ' & \ 0 ' \\ \ 0 & T & O \\ \ 0 & O & O \end{bmatrix} , \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*} r = 2 r _ 0 \ , , q = 2 q _ 0 \ , . \end{align*}


\begin{align*} \exists \ C = C [ \psi ] \ \in ( 0 , \infty ) \ \Rightarrow \ | | \zeta | | G \psi \le C ( \psi ) . \end{align*}


\begin{align*} V _ { p , \alpha _ z } = \bigoplus _ { v \mid p } V _ { v , \alpha _ { z , v } } , \end{align*}


\begin{align*} { \mathcal { E } } = \frac { 1 } { 2 } \int _ { \mathbb { R } ^ { 3 } } \left ( | v | ^ { 2 } + | b | ^ { 2 } \right ) \dd x , { \mathcal { H } } = 2 \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ { \infty } ) \frac { G } { r ^ { 2 } } \dd x . \end{align*}


\begin{align*} a ( u _ n , v _ n ) = \lambda _ n ( u _ n , v _ n ) v _ n \in X _ n . \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \sup _ { n \in \mathbb { N } } \mathbb { P } \left ( \int _ 0 ^ T \Vert X _ n ( t ) \Vert _ V ^ p d t > M \right ) = 0 , \end{align*}


\begin{align*} A ( u ) = - \Delta ^ 2 u + \Delta \varphi ( u ) . \end{align*}


\begin{align*} 0 \le \chi ( z ) & \le 1 z \in \R , \\ \chi ( z ) & = 1 \mbox { i f } | z | \le \frac { \delta _ 0 } { 2 } , \\ \chi ( z ) & = 0 \mbox { i f } | z | \geq \delta _ 0 , \\ z \chi ' ( z ) & < 0 \mbox { i f } | z | \in \left ( \frac { \delta _ 0 } { 2 } , \delta _ 0 \right ) . \end{align*}


\begin{align*} m : = b _ { n - 1 } \cdots b _ { \alpha + 1 } b _ \alpha ^ { k + \alpha - n } . \end{align*}


\begin{align*} t _ n = \frac { \alpha ^ n - \beta ^ n } { \alpha - \beta } \end{align*}


\begin{align*} | Z _ 3 | & = \left | \frac { 1 } { \log x } \sum _ { \rho } \frac { x ^ { \rho - s } - x ^ { 2 ( \rho - s ) } } { ( s - \rho ) ^ 2 } \right | \\ & \leq \frac { x ^ { \frac { 1 } { 2 } - \sigma } + x ^ { 1 - 2 \sigma } } { \log x } \sum _ { \gamma } \frac { 1 } { ( \sigma - \beta ) ^ 2 + ( t - \gamma ) ^ 2 } . \end{align*}


\begin{align*} \left | e ^ { - \omega ^ { \prime } \zeta } \right | = 1 . \end{align*}


\begin{align*} I _ 1 = & \mathbb { E } \int _ 0 ^ { T \wedge \tau _ n ^ M } \big [ 2 \langle A ( r , Y _ n ( r ) ) , Y _ n ( r ) \rangle + \Vert P _ n B ( r , Y _ n ( r ) ) Q _ n \Vert _ { L _ 2 } ^ 2 \big ] d r \int _ { 0 \vee ( r - \delta ) } ^ { r } \mathbf { 1 } _ { \{ \tau _ n ^ M > t \} } d t \\ \leq & \delta \ , \mathbb { E } \int _ 0 ^ { T \wedge \tau _ n ^ M } f ( s ) ( 1 + \Vert Y _ n ( s ) \Vert _ H ^ 2 ) d s \\ \leq & C _ M \delta . \end{align*}


\begin{align*} y ( t + 0 ) = - r y ( t - 0 ) \end{align*}


\begin{align*} P _ A ( \lambda ) + P _ A ( \mu ) = P _ A ( \lambda + \mu ) S _ A ( \lambda ) + S _ A ( \mu ) = S _ A ( \lambda + \mu ) . \end{align*}


\begin{align*} \dot { { x } } _ a = \ , - \gamma c _ 1 \delta \left . \dfrac { \partial h } { \partial x } \right | _ { { x } _ a } \ , . \end{align*}


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


\begin{align*} r ^ { 2 } < \phi ( z , r ) \leq \phi ( z ' , r ) - \phi ( z ' , 0 ) = \int _ { 0 } ^ { r } \partial _ { r ' } \phi ( z ' , r ' ) \dd r . \end{align*}


\begin{align*} d X _ { t } = \left ( \langle \mu _ t , b ( t , X _ t , \cdot ) \rangle \right ) d t + d W _ t , \mu _ t = \operatorname { L a w } ( X _ t ) , \end{align*}


\begin{align*} \lim \limits _ { q \to \infty } \dfrac { k _ 4 ( P ^ \ast ( q ) ) } { q ^ 4 } = \frac { 1 } { 1 5 3 6 } + 3 \times \lim \limits _ { q \to \infty } { _ { 3 } } F _ { 2 } \left ( \begin{array} { c c c } \chi _ 4 , & \chi _ 4 , & \chi _ 4 ^ 3 \\ & \varepsilon , & \varepsilon \end{array} | 1 \right ) . \end{align*}


\begin{align*} \tilde { \theta } ^ { k } = \Pi _ { \mathcal { K } , G } \big \{ \theta ^ { k } - G ^ { - 1 } [ T ( \tilde { \theta } ^ { k } ) - \beta \Lambda ( y ^ { k } - \tilde { y } ^ { k } ) + \zeta ^ k ] \big \} . \end{align*}


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


\begin{align*} I _ \alpha \ast I _ \beta = I _ { \alpha + \beta } , \forall \alpha , \beta > 0 ~ ~ \alpha + \beta < N . \end{align*}


\begin{align*} \exp \bigg [ \epsilon ^ { - 2 } ( t _ { k } - t _ { k - 1 } ) ( \Lambda ( \bar \alpha _ { k , k } ) - \delta ) \bigg ] \le \mathbb E \left ( e ^ { \Delta _ k } \Big | \mathcal { F } _ { t _ { k - 1 } } \right ) \le \exp \bigg [ \epsilon ^ { - 2 } ( t _ { k } - t _ { k - 1 } ) ( \Lambda ( \bar \alpha _ { k , k } ) + \delta ) \bigg ] . \end{align*}


\begin{align*} \begin{aligned} B \times u \in L ^ { 4 / 3 } ( 0 , T ; L ^ { 2 } ( \mathbb { R } ^ { 3 } ) ) , \\ \nabla \times B \in L ^ { 2 } ( 0 , T ; L ^ { 2 } ( \mathbb { R } ^ { 3 } ) ) . \end{aligned} \end{align*}


\begin{align*} \operatorname { E } ( X ^ r ) = & \frac { 2 \theta } { \operatorname { B } ( a , b ) } \int _ 0 ^ \infty x ^ { r - 3 } \exp \left ( - \frac { a \theta } { x ^ 2 } \right ) \left [ 1 - \exp \left ( - \frac { \theta } { x ^ 2 } \right ) \right ] ^ { b - 1 } \mathrm { d } x . \end{align*}


\begin{align*} \begin{aligned} | x & ( t ) - z _ 1 ( t ) | \le \ , | x ( 0 ) - z _ 1 ( 0 ) | + t \gamma ^ 2 \bar { k } _ 3 ( L _ r , \delta ) \\ & + \gamma \int _ 0 ^ t L _ r | x ( \tau ) - z _ 1 ( \tau ) | + | \bar { y } ( \tau ) - z _ 2 ( \tau ) | d \tau \end{aligned} \end{align*}


\begin{align*} \widehat { \Phi _ t } ( \xi ) = \frac { 1 } { t } e ^ { - \frac { \pi | \xi | ^ 2 } { t ^ 2 } } , \xi \in \mathbb { R } . \end{align*}


\begin{align*} \int _ { \Omega } - \delta _ N ^ 2 \left ( \frac { \partial K ( z _ { N , i } ) } { \partial z _ { i , h } } \nabla V _ { \delta _ N , Z _ N , i } \right ) u _ N = O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}


\begin{align*} G ( c ) - G ' ( c _ 0 ) c \geq G ( c _ 0 ) - G ' ( c _ 0 ) c _ 0 = g ( c _ 0 ) > 0 \phantom { x } \Leftrightarrow \phantom { x } \frac { G ( c ) } { c } \geq G ' ( c _ 0 ) \end{align*}


\begin{align*} \lambda _ i \neq 0 , i = 1 , \dots , n . \end{align*}


\begin{align*} c _ n ( a , b ) = \frac { ( - 1 ) ^ n \Gamma ( a + b ) } { ( a + n ) \Gamma ( a ) \Gamma ( b - n ) n ! } , \end{align*}


\begin{align*} \root n \of { a _ 1 \cdot a _ n } = \overset { ` ` f ( a v e r a g e ) " \swarrow } { \exp ( n ^ { - 1 } \sum _ { j = 1 } ^ n A _ j ) } \overset { ( \ , ) } \le \overset { ` ` a v e r a g e ( f ) " \swarrow } { n ^ { - 1 } \sum _ { j = 1 } ^ n \exp A _ j } = n ^ { - 1 } \sum _ { j = 1 } ^ n a _ j \ , . \end{align*}


\begin{align*} ( f _ { I _ 1 } ( \mathbf { x } ) , f _ { I _ 2 } ( \mathbf { x } ) , \cdots , f _ { I _ s } ( \mathbf { x } ) ) = ( f _ { J _ 1 } ( \mathbf { y } ) , f _ { J _ 2 } ( \mathbf { y } ) , \cdots , f _ { J _ s } ( \mathbf { y } ) ) . \end{align*}


\begin{align*} \gamma _ { \textrm { p } , k } = \frac { { \lvert \mathbf { h } _ { k } ^ { H } \mathbf { p } _ { \textrm { p } , k } \rvert } ^ { 2 } } { \sum _ { i = 1 , i \neq k } ^ { K } { \lvert \mathbf { h } _ { k } ^ { H } \mathbf { p } _ { \textrm { p } , i } \rvert } ^ { 2 } + \sigma _ { z } ^ { 2 } } . \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*} 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*} x _ { 2 i , n - j } + x _ { 2 i , n - j + 1 } = a _ j , \mbox { a n d } \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*} \int \rho ^ { - l } \psi Z ( v ^ 2 ) = - ( Q - l ) \int \rho ^ { - l } \psi v ^ 2 . \end{align*}


\begin{align*} \frac { 1 } { q } = \left ( \frac { 1 - \kappa } { p _ 1 } + \frac { \kappa } { p _ 2 } \right ) - \frac { \kappa s - r } { n } , \ , \ , \mbox { a n d } \ , \ , r \le \kappa s . \end{align*}


\begin{align*} g _ d ( \ell ) = \begin{cases} \ell , & d = 1 , \\ \log \ell , & d = 2 , \\ 1 , & d \geq 3 . \end{cases} \end{align*}


\begin{align*} s _ \ast \beta '' ( \gamma , \xi ) = \beta '' ( s \gamma s ^ { - 1 } , s \xi ) ( \gamma \in \Gamma ' ) . \end{align*}


\begin{align*} \Delta _ { g } f = g ^ { i j } \Bigl ( \frac { \partial ^ 2 f } { \partial x _ i \partial x _ j } - \Gamma ^ k _ { i j } \frac { \partial f } { \partial x _ k } \Bigr ) , f \in C ^ { \infty } ( \Omega ) . \end{align*}


\begin{align*} \begin{aligned} L ^ { p } ( \Omega ) & = L ^ { p } _ { \sigma } ( \Omega ) \oplus G ^ { p } ( \Omega ) , 1 < p < \infty , \\ L ^ { p } _ { \sigma } ( \Omega ) & = \{ u \in L ^ { p } ( \Omega ) \ | \ \nabla \cdot u = 0 \ \textrm { i n } \ \Omega , \ u \cdot n = 0 \ \textrm { o n } \ \partial \Omega \} , \\ G ^ { p } ( \Omega ) & = \{ \nabla q \in L ^ { p } ( \Omega ) \ | \ q \in L ^ { 1 } _ { \textrm { l o c } } ( \Omega ) \} , \end{aligned} \end{align*}


\begin{align*} \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B \cdot B \dd x = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ 0 \cdot B _ 0 \dd x , \end{align*}


\begin{align*} C _ { { 3 i + 2 } } ( \mathcal { H } _ { \ast } ) = \mathrm { I m } \ , \partial ' _ i \oplus s _ { _ { 3 i + 2 } } ( \mathrm { I m } \ , \partial _ i ) = s _ { _ { 3 i + 2 } } ( \mathrm { I m } \ , \partial _ i ) . \end{align*}


\begin{align*} \phi ( z , r ) \longmapsto \varphi ( y ) = \frac { \phi ( z , r ) } { r ^ { 2 } } , y = { } ^ { t } ( y _ 1 , y ' ) , \ y _ 1 = z , \ | y ' | = r , \end{align*}


\begin{align*} g _ i : = \left | G _ i \cap \bigcup _ { j = 1 } ^ { i - 1 } G _ j \right | . \end{align*}


\begin{align*} h = 2 \mu \int _ { \mathbb { R } ^ { 3 } } ( \phi - \phi _ \infty ) _ { + } ^ { 2 } \frac { 1 } { r ^ { 2 } } \dd x . \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*} \partial _ t \gamma = \frac { c } { 4 \pi } | \ln \varepsilon | ( \partial _ s \gamma \times \partial _ { s s } \gamma ) = \frac { c \bar { K } } { 4 \pi } | \ln \varepsilon | \mathbf { b } _ { \gamma ( t ) } , \end{align*}


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


\begin{align*} \mathcal { I } - d ( t _ 0 , H \setminus D _ { t _ 0 } ) = 0 , \ \mbox { s i n c e } \ \mathcal { I } - d ( t _ 0 , D _ { t _ 0 } ) = 1 \end{align*}


\begin{align*} d ( x , y ) = d ( y , x ) = d ( x , x ) = d ( y , y ) \implies x = y . \end{align*}


\begin{align*} u \ \mbox { i s b o u n d e d i n } \ B _ { R } \ \mbox { a n d } \ \lim \limits _ { | x | \nearrow R } v ( x ) = \infty \end{align*}


\begin{align*} T \circ \beta & = \alpha \circ T , \\ T ( a ) \cdot T ( b ) & = T \big ( \ell ( T ( a ) ) b + r ( T ( b ) ) a \big ) , \quad \forall \ a , b \in V . \end{align*}


\begin{align*} K _ Y = ( K _ X + A _ X ) \vert _ Y = ( 2 - n ) A _ X \vert _ Y \end{align*}


\begin{align*} \alpha = \lambda ^ 0 < \dots < \lambda ^ l = \beta , \end{align*}


\begin{align*} I _ k \cap ( H \setminus P ) = \bigcup _ { n = 0 } ^ { \infty } ( I _ k \cap ( H _ n \setminus K ) \cap ( \bigcap _ { m = 1 } ^ { \infty } ( H _ n \setminus P _ m ) ) ) \end{align*}


\begin{align*} \nabla _ X \psi = - \frac { 1 } { 2 } S ( X ) \cdot \nu _ 1 \cdot \psi \end{align*}


\begin{align*} | s _ { h , q } - s _ h | \le \frac { d \theta ^ { h } } { \theta ^ { q } - 1 } ~ { \rm f o r } ~ h = 0 , 1 , \dots , q - 1 , \end{align*}


\begin{align*} d X _ { t } ^ { i } = \left ( \frac { 1 } { n - 1 } \sum _ { j \neq i } b \left ( t , X _ { t } ^ { i } , X _ { t } ^ { j } \right ) \right ) d t + d W _ { t } ^ { i } , i = 1 \cdots n , \end{align*}


\begin{align*} & \int _ { B _ { R / 2 } } \vert \nabla u \vert ^ 2 d x \lesssim \left ( \int _ { \mathcal { C } ( R / 2 , R ) } \vert \nabla u \vert ^ { \frac { p } { 2 } } d x + \int _ { \mathcal { C } ( R / 2 , R ) } \vert u \vert ^ { p } d x \right ) ^ { \frac { 2 } { p } } R ^ { 2 - \frac { 9 } { p } } \left ( \int _ { \mathcal { C } ( R / 2 , R ) } \vert u \vert ^ { p } d x \right ) ^ { \frac { 1 } { p } } \end{align*}


\begin{align*} Z ^ { i , j } _ \infty = \cap _ r \ , Z _ r ^ { i , j } , \end{align*}