\begin{align*} \frac { W \left \{ f _ { \lambda } , f _ { \nu } \right \} } { \lambda ^ { 2 } - \nu ^ { 2 } } \mathbf { = - } \int _ { x } ^ { \infty } f _ { \lambda } \left ( s \right ) f _ { \nu } \left ( s \right ) \mathrm { d } s . \end{align*}


\begin{align*} ( L ) ^ \sigma = \mathrm { i m } ( V _ p ) = \ker ( F _ p ) . \end{align*}


\begin{align*} d F ( Z _ \alpha ) = ( Z _ \alpha w ^ \lambda ) Z ' _ \lambda \ ; , Z _ \alpha w ^ { \bar \mu } = 0 \ ; , Z _ \alpha s = ( Z _ \alpha w ^ \lambda ) K ' _ \lambda \circ F \ ; . \end{align*}


\begin{align*} \delta ( k , \xi ) = \frac { \left ( k + k _ 0 \right ) ^ { i \nu ( - k _ 0 ) } } { \left ( k - k _ 0 \right ) ^ { i \overline { \nu ( - k _ 0 ) } } } e ^ { \hat { \chi } ( \xi , k ) } , \end{align*}


\begin{align*} \mathbf { P r } \left ( \bigg | \frac { d } { n \epsilon } \sum _ { i = 1 } ^ { n } \big \langle J ( u _ 0 + \epsilon v _ { 0 , i } ) v _ { 0 , i } v _ 0 ' \big \rangle - \big \langle \nabla \hat { J } ( u _ 0 ) , v _ 0 ' \big \rangle \bigg | \geq t \right ) \leq 2 \exp \left ( - \frac { c n t ^ 2 } { K ^ 2 } \right ) , \end{align*}


\begin{gather*} d ^ 1 _ { Q } ( \tau , \sigma ) = \big ( d ^ 1 \tau , d _ { r } ^ 2 \sigma - \frac { 1 } { 2 } B \left ( \tau \wedge d ^ 1 \tau \right ) \big ) . \end{gather*}


\begin{align*} \| g _ n \| _ { L ^ q ( Q _ { R , \tau } ) } = \frac { r _ n ^ { 2 - \frac { N + 2 } { q } } } { M _ n } \| f _ n \| _ { L ^ q ( \widetilde { Q } _ { n , R , \tau } ) } , \widetilde { Q } _ { n , R , \tau } = B _ { R r _ n } ( \bar x _ n ) \times ( \bar t _ n , \bar t _ n + r _ n ^ 2 \tau ) . \end{align*}


\begin{align*} \{ g , h \} = \pi ( d g , d h ) . \end{align*}


\begin{align*} \Delta ( - k _ 0 ) : = \int _ { - \infty } ^ { - k _ 0 } d \arg \left ( 1 + r _ 1 ( s ) r _ 2 ( s ) \right ) . \end{align*}


\begin{align*} \left ( a ( y ) D ^ { 2 } h ( y ) \right ) = \frac { 1 } { \| y \| } \left ( \ , a ( y ) - \sum \limits _ { i , j = 1 } ^ { m } a _ { i j } ( y ) \frac { y _ { i } y _ { j } } { \| y \| ^ { 2 } } \right ) \leq \frac { m \overline { \Lambda } } { \| y \| } . \end{align*}


\begin{align*} J _ 2 \ge \frac 1 2 \frac d { d t } \sum \limits _ { | \alpha | = 0 } ^ { 2 } \left \| \sqrt { A } \partial _ \tau ^ \alpha \partial _ y \tilde { h } \right \| _ { L ^ 2 } ^ 2 - C D ( t ) ^ { \frac 1 2 } E ( t ) . \end{align*}


\begin{align*} u _ t ( x ) \cdot \partial _ x u _ t ( x ) = 0 a . e . \ , \ , ( t , x ) \in [ 0 , T ] \times D \ , . \end{align*}


\begin{align*} \Phi _ { \pmb { \theta } } ( \sigma ) = \Psi _ { \theta _ 1 } ( \sigma ) \oplus \cdots \oplus \Psi _ { \theta _ m } ( \sigma ) , \end{align*}


\begin{align*} W ( z ) = 9 \sqrt { \pi } \ \frac { \Gamma ( z + \frac 1 2 ) } { \Gamma ( z + 1 ) } \ \frac { z - 1 } { z ( 2 z - 1 ) ( 2 z - 3 ) } . \end{align*}


\begin{align*} \big ( b K _ { k , t } + K _ { k - 1 , t } b \big ) c \ , & = \int _ t ^ 1 b H _ { k , \frac s 2 } \big ( \partial _ s \Psi _ { k , s } c \big ) + H _ { k - 1 , \frac s 2 } b \big ( \partial _ s \Psi _ { k , s } c \big ) d s = \\ & = \int _ t ^ 1 \partial _ s \Psi _ { k , s } c - \Psi _ { k , \frac s 2 } ( \partial _ s \Psi _ { k , s } c ) d s = \Psi _ { k , 1 } c - \Psi _ { k , t } c \ : , \end{align*}


\begin{align*} \alpha ( G ) : = \min _ { W \neq 0 } \frac { \dim G - \dim G _ W } { \dim W } , \end{align*}


\begin{align*} \partial _ { x _ k } h ( x ) = \int \limits _ { \mathbb { S } ^ { n - 1 } } \partial _ k P ( x , t ) \big ( h ( t ) - h ( e _ n ) \big ) d \sigma ( t ) \ , . \end{align*}


\begin{align*} K _ 4 & = - \sum \limits _ { | \alpha | = 0 } ^ { 3 } \left \langle B ( { \bf { v } } ) \partial _ y ^ 2 \partial _ \tau ^ \alpha { \bf { v } } , \partial _ \tau ^ \alpha { \bf { v } } \right \rangle - \sum \limits _ { | \alpha | = 0 } ^ { 3 } \sum \limits _ { | \beta | = 1 } ^ { | \alpha | } \left \langle \partial _ \tau ^ \beta B ( { \bf { v } } ) \partial _ \tau ^ { \alpha - \beta } \partial _ y ^ 2 { \bf { v } } , \partial _ \tau ^ \alpha { \bf { v } } \right \rangle \\ & = : K _ 4 ^ 1 + K _ 4 ^ 2 . \end{align*}


\begin{align*} I _ F ( b ^ * ) - \frac { I _ F ' ( b ^ * ) \varphi _ F ( b ^ * ) } { \varphi _ F ' ( b ^ * ) } = \frac { \psi ( b ^ * ) } { \psi ' ( b ^ * ) } - \frac { \varphi _ F ( b ^ * ) } { \varphi _ F ' ( b ^ * ) } . \end{align*}


\begin{align*} G = \bigcup _ { j = 1 } ^ n ( t _ j + P _ U ( x ) ) \end{align*}


\begin{align*} \lambda _ k = \min \limits _ { g \in V , < g , g _ j > = 0 \ f o r \ j = k + 1 , \cdots , n } \frac { < A g , g > } { < g , g > } = \max \limits _ { g \in V , < g , g _ l > = 0 \ f o r \ l = 1 , \cdots , k - 1 } \frac { < A g , g > } { < g , g > } \end{align*}


\begin{align*} \left | \smash { \prod _ { j = 1 } ^ n } \vphantom { \prod } ( \partial ^ { \gamma _ j } | \bullet | ^ 2 ) ( \tau ) \right | \leq 2 ^ n \cdot \prod _ { j = 1 } ^ n | \tau | ^ { 2 - | \gamma _ j | } \leq 2 ^ { | \alpha | } \cdot | \tau | ^ { 2 n - | \alpha | } . \end{align*}


\begin{align*} ( x ' , z ) , ( z ' , y ) \in U ' \ , , x = \alpha ( t , x ' ) \mbox { a n d } z = \alpha ( s , z ' ) \ , . \end{align*}


\begin{align*} a _ i x _ i = b _ { i + 1 } x _ { i + 1 } . \end{align*}


\begin{align*} \left \{ \ ; \begin{aligned} & \Phi '' ( z ) - \eta \Phi ' ( z ) - \gamma \Phi ( z ) = 0 , & \ ; z \in [ R , n ] \\ & \Phi ' ( R ) = 0 \Phi ( n ) = 1 & \end{aligned} \right . \end{align*}


\begin{align*} V = \{ x _ 4 ^ { n - 2 } Q + R = 0 \} \end{align*}


\begin{align*} \sum _ { i + j = \ell } e _ { i } ( \sigma ) e _ { j } ( \hat { \sigma } ) = \hat { g } _ { \ell } ^ { 2 d } ( \sigma , m , q _ { 2 d } ) , \ell = 1 , \ldots , n \ / . \end{align*}


\begin{align*} \begin{array} { c c } \left ( B ^ 2 u ^ l \right ) _ j ( x , t ) = \displaystyle \sum _ { k \not = j } \sum _ { i \not = k } \int _ { x _ j } ^ x \int _ \eta ^ x d _ { j k i } ( \xi , \eta , x , t ) b _ { j k } ( \xi , \omega _ j ( \xi ) ) \\ \displaystyle \times u _ i ^ l ( \eta , \omega _ k ( \eta , \xi , \omega _ j ( \xi ) ) ) d \xi d \eta \end{array} \end{align*}


\begin{align*} B ^ S _ t = 0 , C ^ S _ t = \int _ 0 ^ t \bigg ( \sum _ { | J | \leq n } \ell ^ J _ W \langle \epsilon _ { J } , \mathbb { X } _ { s ^ - } \rangle \bigg ) ^ 2 d s , \nu ^ S ( d t \times d x ) = K _ t ( d x ) \times d t , \end{align*}


\begin{align*} | \bar { \mathcal { C } } ( f , g , h ) | \le \tilde { \mathcal { C } } ( f , g , h ) = 2 \iint _ { 0 < \omega _ 3 < \omega _ 4 , 0 < \omega _ 2 } W [ ( f _ 1 + f _ 2 ) g _ 3 h _ 4 + ( g _ 3 + g _ 4 ) h _ 1 f _ 2 ] \ , \dd \omega _ 3 \ , \dd \omega _ 4 . \end{align*}


\begin{align*} r _ { q \alpha } = \sum _ { a , b , c , d , e , f , g , x , y } t _ { x a \alpha } t _ { x b y } t _ { y c q } M _ { x b d } M _ { f e } M _ { f g } v _ g \delta _ { d a c e } \ ; , \end{align*}


\begin{align*} \chi ( \P \operatorname { E x t } ^ 1 ( M , N ) ) X _ M X _ N = \sum _ E ( \chi ( \P \operatorname { E x t } ^ 1 ( M , N ) _ E ) + \chi ( \P \operatorname { E x t } ^ 1 ( N , M ) _ E ) ) X _ E . \end{align*}


\begin{align*} x _ i - x _ j = 0 , \ f o r \ a n y \ \{ i , j \} \in e \in E ( T ) , Y _ { p + 1 } = 0 , \cdots , Y _ { n - r } = 0 \end{align*}


\begin{align*} [ e ( z ) , f ( w ) ] = \frac { 1 } { \beta _ 1 } \delta \left ( \frac { z } { w } \right ) \left ( \psi ^ + ( z ) - \psi ^ - ( z ) \right ) \ , , \end{align*}


\begin{align*} m ( t ) = \frac { 1 } { 1 - \alpha } \sup _ { s \leq t } \left ( w ( s ) - \frac { \beta } { 1 - \beta } \sup _ { u \leq s } \left ( - w ( u ) - \alpha m ( u ) \right ) \right ) = : \phi ^ + ( w , m ) ( t ) . \end{align*}


\begin{align*} \partial _ t u ( t , x ) - \div _ x A ( t , x ) \nabla _ x u ( t , x ) = f ( t , x ) , u ( 0 ) = 0 . \end{align*}


\begin{align*} G _ 0 \sigma ^ { e } ( G _ 0 ^ { T } ) = \begin{pmatrix*} O _ { l \times l } & H _ { l \times r } \\ O _ { r \times l } & P _ { r \times r } \\ \end{pmatrix*} , \end{align*}


\begin{align*} \partial _ t U - \Delta U = - f \hbox { i n ~ } Q , U ( T ) = u _ 0 \hbox { i n ~ } \Omega , \end{align*}


\begin{align*} 0 \geq g _ H ( N , \dot \gamma ) & = g _ H ( X , N ) g _ H ( X , \dot \gamma ) + g _ H ( X ^ \perp , N ) g _ H ( X ^ \perp , \dot \gamma ) \\ & = g _ H ( X , N ) \sin ( \alpha ) + g _ H ( X ^ \perp , N ) \cos ( \alpha ) . \end{align*}


\begin{align*} X ^ \epsilon ( t ) = \varphi ( \zeta ^ \epsilon ( t ) , X ^ \epsilon ( 0 ) ) \end{align*}


\begin{align*} T _ a Q _ 1 = e ^ { - a ( x + \vartheta u ) } Q _ 1 ( x + \vartheta u ) , \quad T _ b Q _ 2 = e ^ { - b ( x + \vartheta v ) } Q _ 2 ( x + \vartheta v ) . \end{align*}


\begin{align*} \rho = { \rm i n t } [ \overline { \mathbf { P } } ^ { x } ] . \end{align*}


\begin{align*} x \to y : = \min \{ z * y \colon [ x ) \cap [ y ) \subseteq [ z ) \subseteq [ y ) \} . \end{align*}


\begin{align*} J ( x , f ) : = \det f ^ { \ , \prime } ( x ) , l \left ( f ^ { \ , \prime } ( x ) \right ) \ , = \ , \ , \ , \min \limits _ { h \in { \Bbb R } ^ n \setminus \{ 0 \} } \frac { | f ^ { \ , \prime } ( x ) h | } { | h | } \ , \ , , \end{align*}


\begin{align*} \widehat { Q } ^ 1 _ n ( F _ 1 \vee \cdots \vee F _ n ) = ( Q ' ) ^ 1 _ n \circ ( F _ 1 \star F _ 2 \star \cdots \star F _ n ) . \end{align*}


\begin{align*} \lambda _ y = ( \lambda _ { y , \gamma } ) _ { \gamma = 1 , 2 } , \lambda _ { y , \gamma } : = \lambda _ { y , \gamma } ( 0 ) = \frac { \chi _ { \Lambda } ( k ) } { 2 \pi | k | ^ { 1 / 2 } } \left ( \begin{array} { c } \mathbf { e } ^ { ( 1 ) } _ { \gamma } ( k ) 2 \cos ( k _ 1 y ) \\ - \mathbf { e } ^ { ( 2 ) } _ { \gamma } ( k ) 2 i \sin ( k _ 1 y ) \\ - \mathbf { e } ^ { ( 3 ) } _ { \gamma } ( k ) 2 i \sin ( k _ 1 y ) \end{array} \right ) . \end{align*}


\begin{align*} \mathcal { X } _ { n } ( x _ 1 ) = n ! \ , . \end{align*}


\begin{align*} \frac { \nabla _ F S _ F } { 2 } + f d i v _ F \nabla _ F ^ 2 h + f | \nabla _ B f | ^ 2 \nabla _ F h = \rho \nabla _ F S _ F . \end{align*}


\begin{align*} \min _ { \tilde { \beta } _ { t , n } , \tilde { \beta } _ { r , n } } & \mathrm { R e } ( \breve { \vartheta } _ { t , n } ^ * \tilde { \beta } _ { t , n } ) + \mathrm { R e } ( \breve { \vartheta } _ { r , n } ^ * \tilde { \beta } _ { r , n } ) \\ [ - 0 . 5 e m ] \mathrm { s . t . } & \tilde { \beta } _ { t , n } ^ 2 + \tilde { \beta } _ { r , n } ^ 2 = 1 , 0 \le \tilde { \beta } _ { t , n } , \tilde { \beta } _ { r , n } \le 1 . \end{align*}


\begin{align*} V _ r ( G ) ( k ) = \bigcup _ { E \leq G } V _ r ( E ) ( k ) , \end{align*}


\begin{align*} d ^ 1 _ { r , s } ( 0 , k ) = 0 ( r , s ) k \ne - \frac { 3 q } 2 . \end{align*}


\begin{align*} \frac { 3 n + 1 } { 4 } = 3 b 2 ^ { v - 1 } + 1 \equiv 1 \pmod { 2 } \end{align*}


\begin{align*} \mu ( z ) = - \frac { 1 } { 2 } ( 2 { \rm R e } z ) ^ 2 S h ( z ^ * ) . \end{align*}


\begin{align*} \mathcal { L } g - \mathcal { Q } ( g ) - \mathcal { C } ( g ) = \phi , \end{align*}


\begin{align*} V _ 1 ^ \lambda ( y _ 0 ) = - \Delta V ^ \lambda ( y _ 0 ) \leq \frac { V ^ \lambda ( y _ 0 ) } { \delta ^ 2 } < 0 . \end{align*}


\begin{align*} d ' ( x , y ) : = \begin{cases} \bar { d } ( x , y ) , & \mbox { i f } \bar { d } ( x , y ) < 1 \ , , \\ 1 , & \mbox { o t h e r w i s e } . \end{cases} \end{align*}


\begin{align*} \langle \omega _ \alpha f _ s f _ { r - s } \omega _ \beta \rangle = \sum _ { \lambda \vdash d } \frac { ( \dim \mathsf { V } ^ \lambda ) ^ 2 } { d ! } \omega _ \alpha ( \lambda ) f _ s ( \lambda ) f _ { r - s } ( \lambda ) \omega _ \beta ( \lambda ) \end{align*}


\begin{align*} X _ k = h _ k ( S _ 1 , S _ 2 , \ldots , S _ k , Q ) \forall k \in \{ 1 , 2 , 3 , \ldots \} \end{align*}


\begin{align*} x ( s ) = x ( \gamma ) + \int _ { \gamma } ^ { s } D F ( x ( \tau ) , t ) , s \in J . \end{align*}


\begin{align*} \div _ { \gamma _ n } b ( v ) = \div b ( v ) - v \cdot b ( v ) , \end{align*}


\begin{align*} Q ( i ) \approx \binom n 3 p ^ 3 \approx \frac 1 6 n ^ 3 p ^ 3 = n ^ 3 q ( t ) Y _ { u u ' } ( i ) \approx n p ^ 2 = n y ( t ) , \end{align*}


\begin{align*} \hat { E } _ n ^ { \epsilon , \Delta t } \le C ( T ) \Delta t \sum _ { k = 1 } ^ { n - 1 } \hat { E } _ k ^ { \epsilon , \Delta t } + C ( T ) ( 1 + | q _ 0 ^ \epsilon | ^ 2 + | p _ 0 ^ \epsilon | ^ 2 ) \Delta t , \end{align*}


\begin{align*} a _ { l 1 } = - \sigma _ 1 , \ \ a _ { l 2 } = - \sigma _ 2 , \ \ l \in L . \end{align*}


\begin{align*} 0 < \delta < 1 \kappa = \frac { 2 \Gamma \left ( \frac { 1 + \delta } { 2 } \right ) \Gamma \left ( \frac { 3 - \delta } { 2 } \right ) } { \Gamma \left ( \frac { \delta } { 2 } \right ) \Gamma \left ( 1 - \frac { \delta } { 2 } \right ) } = ( 1 - \delta ) \tan \frac { \pi \delta } { 2 } . \end{align*}


\begin{align*} c : = \gamma + \int _ { ( 0 , 1 ) } z \upsilon ( d z ) , \end{align*}


\begin{align*} \pi L ^ { - 1 } _ { \widetilde { k } } L _ { \widetilde { i + k } } = L ^ { - 1 } _ { \widetilde { k + 1 } } L _ { \widetilde { i + k + 1 } } \pi . \end{align*}


\begin{align*} a _ { j , m } = \frac { 1 } { m \left ( 1 - \exp ( - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } ) + z / m \right ) } b _ m = e ^ { 2 \pi i \lfloor s m \rfloor / m } . \end{align*}


\begin{align*} \begin{cases} - \Delta u ( x ) = u _ 1 ( x ) , & x \in \Omega , \\ ( - \Delta ) ^ { s _ 1 } u _ 1 ( x ) = v ^ p ( x ) , & x \in \Omega , \\ - \Delta v ( x ) = v _ 1 ( x ) , & x \in \Omega , \\ ( - \Delta ) ^ { t _ 1 } v _ 1 ( x ) = u ^ q ( x ) , & x \in \Omega , \\ u ( x ) = v ( x ) = u _ 1 ( x ) = v _ 1 ( x ) = 0 , & x \in \mathbb { R } ^ n \backslash \Omega , \end{cases} \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u & = \lambda a ( x ) f ( u ) \ ; \ ; & & \mbox { i n } \ ; \ ; \R ^ N , \\ u & > 0 & & \mbox { i n } \ ; \ ; \R ^ N , \\ u ( x ) & \to 0 & & \mbox { a s } \ ; | x | \to \infty , \end{alignedat} \right . \end{align*}


\begin{align*} T ( i y _ 0 ) = \frac { 1 6 \pi } { \cos ^ 2 ( 2 y _ 0 ) } . \end{align*}


\begin{align*} \left \{ e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } : 0 \leq j \leq m \right \} = \left \{ - e ^ { - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } } : - m / 2 + 1 \leq j \leq m / 2 \right \} . \end{align*}


\begin{align*} \partial _ t h = \frac { 1 } { 2 } \partial _ x ^ 2 h + \sqrt { 2 \mu _ 2 } \xi , \ \ \ h ( 0 , \cdot ) \equiv 0 . \end{align*}


\begin{align*} D _ { T ^ * } = ( 0 , T ^ * ] \times ( 0 , X ] \times ( 0 , 1 ) . \end{align*}


\begin{align*} G ^ { ( 4 ) } _ { \mathcal R } ( x , y ) = - { i e ^ { i k _ { + } \vert x \vert } } { ( 2 \pi ) ^ { - 1 } } \int _ { \mathcal { I } _ { s _ b , s _ b + \infty } } \sqrt { s - s _ b } f ( s ) e ^ { - \vert x \vert s ^ 2 } d s , \end{align*}


\begin{align*} \widetilde { Q } _ i : = \frac { 1 } { \ell ^ { 2 e + 1 } } Q _ i ( \ell ^ { e + 1 } X _ 1 , \ldots , \ell ^ { e + 1 } X _ n ) \in \Z _ \ell [ X _ 1 , \ldots , X _ n ] . \end{align*}


\begin{align*} \frac { \int _ { \Omega _ t } | f | ^ p } { \int _ \Omega | f | ^ p } = \frac { \int _ { \Omega _ { t _ 0 } } | f | ^ p + \int _ { \Omega _ t \setminus \Omega _ { t _ 0 } } | f | ^ p } { \int _ \Omega | f | ^ p } \leq { s _ p ( \Omega _ { t _ 0 } , \Omega ) } + \frac { \int _ { \Omega _ t \setminus \Omega _ { t _ 0 } } | f | ^ p } { \int _ \Omega | f | ^ p } . \end{align*}


\begin{align*} \lim _ { T \rightarrow + \infty } \frac { 1 } { T } \int _ { 0 } ^ { T } P _ t \phi \dd t = \int _ { H ^ 1 ( \mathbb { S } ^ 2 ) } \phi ( v ) \dd \mu ( v ) \ , . \end{align*}


\begin{align*} \dfrac { d x } { d \tau } = D [ A ( t ) x + g ( t ) ] . \end{align*}


\begin{align*} w ( x , t , k ) = \overline { w ( x , t , - \bar { k } ) } . \end{align*}


\begin{align*} h _ i = \binom { a _ i } { i } + \binom { a _ { i - 1 } } { i - 1 } + \cdots \mbox { a n d } h _ i ^ { \langle i \rangle } = \binom { a _ i + 1 } { i + 1 } + \binom { a _ { i - 1 } + 1 } { i } + \cdots , \end{align*}


\begin{align*} X & = \phi ^ { - 1 } \left ( ( X _ n ) _ { n = 1 } ^ { \infty } \right ) \\ B & = \phi ^ { - 1 } \left ( \times _ { n = 1 } ^ { \infty } B _ n ^ c \right ) \end{align*}


\begin{align*} I _ 1 = { \rm e x p } \left \{ \frac { 1 } { 2 \pi i } { \rm p . v . } \int _ { - \infty } ^ { \infty } \frac { { \rm l o g } \left ( 1 - b ^ 2 ( s ) \right ) } { s } d s \right \} , I _ 2 = \exp \left \{ \frac { 1 } { 2 } { \rm l o g } \left ( 1 - b ^ 2 \left ( 0 \right ) \right ) \right \} . \end{align*}


\begin{align*} \sigma : = \begin{cases} \min \left \{ \frac { 1 } { 4 } , \left ( \frac { c _ 0 } { 2 K _ 1 } \right ) ^ { \frac { 1 } { n } } \right \} , & \mbox { i f } \ , \ , n \ , \ , \mbox { i s o d d } ; \\ \min \left \{ \frac { 1 } { 4 } , \left ( \frac { 1 } { 2 K _ 2 } \right ) ^ { \frac { 1 } { n } } \right \} , & \mbox { i f } \ , \ , n \ , \ , \mbox { i s e v e n } ; \end{cases} \end{align*}


\begin{align*} | \widehat { Q _ { i j } ( u , v ) } | ( \zeta ) \lesssim \int _ { \zeta = \xi + \eta } \angle ( \xi , \eta ) | \xi | | \eta | \widehat u ( \xi ) \widehat v ( \eta ) \ , d \xi d \eta . \end{align*}


\begin{align*} \partial _ y ^ 3 u = \partial _ y \left ( \frac { 1 } { A h ^ 2 } \partial _ \tau u + \frac { 1 } { A h ^ 2 } u \partial _ x u - \frac { 1 } { h } \partial _ x \tilde { h } + \frac { 1 - A } { A h } \partial _ y \tilde { h } \partial _ y u \right ) . \end{align*}


\begin{align*} \begin{vmatrix} 2 n & - ( n + r ) \\ n + r & - 2 n \end{vmatrix} = ( n + r ) ^ 2 - 4 n ^ 2 = ( r - n ) ( r + 3 n ) \ne 0 . \end{align*}


\begin{align*} [ k ] _ n = t [ k - 1 ] _ { n - 1 } . \end{align*}


\begin{align*} \lim _ { m \to \infty } \frac { 1 } { m } \sum _ { j = - m / 2 + 1 } ^ { m / 2 } \frac { e ^ { 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \lfloor s m \rfloor } } { 1 - \exp \left \{ - 2 \pi i \frac { j + \theta _ 1 / 2 } { m } \right \} + \frac { z } { m } } = \frac { e ^ { - z s } } { 1 + ( - 1 ) ^ { \theta _ 1 - 1 } e ^ { - z } } . \end{align*}


\begin{align*} ( T _ t \mu ) \oplus ( T _ s \mu ) : = T _ { t + s } \mu \ , . \end{align*}


\begin{align*} \tau \sum _ { \ell = 1 } ^ { \infty } e ^ { - 2 \tau \ell } = \frac { \tau } { e ^ { 2 \tau } - 1 } \le C < \infty \end{align*}


\begin{align*} H _ y : = h _ { \alpha } + H _ f ^ + + \alpha V _ { y } - 2 \alpha ^ { 1 / 2 } \mathrm { R e } P A _ { y } ( x ) + \alpha \| \lambda _ { y } \| ^ 2 + 2 \alpha A _ { y } ^ { + } A _ { y } ^ - , \end{align*}


\begin{align*} ( H _ { b _ 0 + \epsilon } - z ) ^ { - 1 } = S _ \epsilon ( z ) - ( H _ { b _ 0 + \epsilon } - z ) ^ { - 1 } T _ \epsilon ( z ) . \end{align*}


\begin{align*} h ^ 0 x = \sum \limits _ { i = 1 } ^ s h ( c _ i ) I _ { A _ i } \end{align*}


\begin{align*} R _ { 2 ^ j } \stackrel { d } { = } R _ n + \hat { R } _ { 2 ^ j - n } - V _ { 0 , n , 2 ^ j } . \end{align*}


\begin{align*} \lim _ { t \to \infty } \frac { t ^ 2 - n ( t ) } { t \cdot \log ^ b t ^ 2 } = \lim _ { n \to \infty } \frac { ( \lambda _ n + o ( 1 ) ) ^ 2 - n } { ( \sqrt n + o ( \sqrt n ) ) \cdot \log ^ b ( \sqrt n + o ( \sqrt n ) ) ^ 2 } \\ = \lim _ { n \to \infty } \frac { a \sqrt n \cdot \log ^ b n + o ( \sqrt n ) } { ( \sqrt n + o ( \sqrt n ) ) \cdot \log ^ b ( \sqrt n + o ( \sqrt n ) ) ^ 2 } = a . \end{align*}


\begin{align*} \max \Big \{ \max ( u ) - \sum _ { j = 1 } ^ { \deg ( u ) - 1 } t _ j : u \in G ( I ) \Big \} = \max \big \{ \min ( u ) : u \in G ( I ) \big \} . \end{align*}


\begin{align*} \partial _ r \eta _ 0 ( r ) & = 0 , \\ \partial _ r \eta _ 1 ( r ) + k _ d ^ { - 1 } V _ \Gamma \eta _ 0 ( r ) & = 0 , \end{align*}


\begin{align*} & g ( q , u _ 0 , v ) \\ & : = \left | \left ( e ^ { - 2 i \vartheta } \left ( u _ 0 - q \right ) ^ 2 - E - i \eta \right ) ^ { - 1 } \right | ^ 2 + \left | \left ( e ^ { - 2 i \vartheta } \left ( u _ 0 - q - [ M _ A v ] _ 1 \right ) ^ 2 - E - i \eta \right ) ^ { - 1 } \right | ^ 2 . \end{align*}


\begin{gather*} - \rho ' \left ( \alpha ( x ) \right ) \rho ' ( y ) Z - \rho ' \left ( \alpha ( y ) \right ) \rho ' ( x ) Z = - Z \big ( \left [ y , [ \alpha ( x ) , \cdot ] \right ] \big ) - Z \big ( \left [ x , [ \alpha ( y ) , \cdot ] \right ] \big ) . \end{gather*}


\begin{align*} \begin{cases} \big \| \Theta [ s ] \big \| _ \mathrm { H S } \le C N ^ { 3 \beta } , \\ \mu [ s ] \le C , \\ H [ s ] ( e , \overline { e } ) - | \Theta [ s ] ( \overline { e } , \overline { e } ) | \ge c \| e \| ^ 2 , e \in ( \phi _ s ) _ \perp , \end{cases} \end{align*}


\begin{align*} A ( z ) = e ^ { z L _ 1 } z ^ { - 2 L _ 0 } g , \end{align*}


\begin{align*} C ( \xi ) = \int _ { F ^ { - 1 } ( \xi ) } | z | ^ { 2 n - 2 } d \sigma ( z ) . \end{align*}


\begin{align*} \| ( \Theta _ x , \theta ^ * _ t , \theta ^ * _ x , P _ t , P _ x , \Theta - \theta ^ * ) \| _ { H ^ 3 ( \mathbb { T } _ x ) } \le f ( t ) : = \varepsilon ^ { 1 + \sigma } g ( t ) . \end{align*}


\begin{align*} \begin{aligned} & C ( \mathbf { a } , \mathbf { d } ) ( \tau ) + C ( \mathbf { b } , \mathbf { c } ) ( \tau ) = \omega ^ { q / 2 } + \omega ^ 0 = 0 . \end{aligned} \end{align*}


\begin{align*} \lim _ { s \to r / 2 } \left ( Z _ r ( Q , s ) - \frac { \pi ^ { r / 2 } } { \sqrt { | Q | } \Gamma \left ( \frac { r } { 2 } \right ) ( s - \frac { r } { 2 } ) } \right ) = 2 Z ^ * _ { r / 2 } ( Q ) , \end{align*}


\begin{align*} \beta _ { i , i + j } ( I ) = \sum _ { u \in G ( I ) _ j } \binom { | \textup { s e t } ( u ) | } { i } , \mbox { f o r a l l $ i , j \ge 0 $ } . \end{align*}


\begin{align*} \Omega _ { \frac { q } { N } } ^ { - 1 } ( \lambda ) = \prod _ { \Box \in \lambda } \frac { 1 } { 1 + \frac { q c ( \Box ) } { N } } \end{align*}


\begin{align*} \ge \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) ^ { 2 k } \int _ { \gamma _ d - 2 \pi k / \log T } ^ { \gamma _ d } 1 \ , d t \ = \ \frac { 2 \pi k } { \log T } \sum _ { 0 < \gamma _ d \le T } m ( \gamma _ d ) ^ { 2 k } . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty \sum _ { k = 0 } ^ n A ( n , k ) \rightarrow \sum _ { n , k = 0 } ^ \infty A ( n + k , k ) \end{align*}


\begin{align*} \int _ { \Omega } \left | \nabla v _ { \epsilon } - a _ { \epsilon , A ( y ) } \nabla \phi _ { \epsilon } \right | ^ 2 d x = 0 . \end{align*}


\begin{align*} \alpha \int _ { \mathbb { R } ^ 3 } \mathrm { d } k \ ; f _ y ( k ) \ , \bigg \langle x u _ { \alpha } \ , \bigg | \ , \frac { ( h _ { \alpha } - e _ { \alpha } ) ^ 2 } { ( h _ { \alpha } - e _ { \alpha } + | k | ) } \ , \bigg | \ , x u _ { \alpha } \bigg \rangle = O \Big ( \frac { \alpha ^ 4 } { L } \Big ) . \end{align*}


\begin{align*} u ( t ) = \frac { ^ { \rho } I ^ { 1 - \gamma } u ( 0 ) } { \Gamma ( \gamma ) } \left ( \frac { t ^ \rho } { \rho } \right ) ^ { \gamma - 1 } - ^ { \rho } I ^ { \alpha } p ( t ) u ( t ) + \frac { 1 } { \Gamma ( \alpha ) } \int _ { 0 } ^ { t } \left ( \frac { t ^ { \rho } - s ^ { \rho } } { \rho } \right ) ^ { \alpha - 1 } s ^ { \rho - 1 } f ( s , u ( s ) , ^ \rho D ^ { \alpha , \beta } u ( s ) ) d s . \end{align*}


\begin{align*} \theta _ t = x _ 0 + \int _ 0 ^ t f ( s , \theta _ s - \omega _ s ) \ , d s , t \in [ a , b ] . \end{align*}


\begin{align*} z ( t ) = & S ( t ) z _ 0 - \int _ 0 ^ t S ( t - s ) ( i \nu z ( s ) + \epsilon ( \gamma z ( s ) - \mu \overline { z } ( s ) ) ) \d s - \tfrac { 1 } { 2 } \int _ 0 ^ t S ( t - s ) ( z ( s ) F _ \Phi ) \d s \\ & + i \kappa \int _ 0 ^ t S ( t - s ) ( \theta _ R ( | z | _ { X ^ { \mathfrak { s } } _ s } ) | z ( s ) | ^ 2 z ( s ) ) \d s - i \int _ 0 ^ t S ( t - s ) ( z ( s ) \d W ( s ) ) \end{align*}


\begin{align*} \| \mu \| _ { T V } = \int _ M h d x \leq \| h \| _ { 0 , q } . \end{align*}


\begin{align*} R ( g , T , \hat y , \hat \eta ) = \int \limits _ 0 ^ T \sqrt { \sum _ { j , k = 0 } ^ n g _ { j k } ( x ) \frac { d x _ j } { d t } \frac { d x _ k } { d t } } d t . \end{align*}


\begin{align*} \partial ^ \alpha ( f \circ g ) ( x ) = \sum _ { n = 1 } ^ { | \alpha | } \left [ f ^ { ( n ) } ( g ( x ) ) \cdot \sum _ { \gamma \in \Gamma _ { \alpha , n } } \bigg ( C _ { \gamma } \cdot \prod _ { j = 1 } ^ n ( \partial ^ { \gamma _ j } g ) ( x ) \bigg ) \right ] \forall \ , x \in U \ , , \end{align*}


\begin{align*} V _ \varepsilon ( \Phi _ \varepsilon ^ i ( X , t ) , t ) = \nu _ \varepsilon ( \Phi _ \varepsilon ^ i ( X , t ) , t ) \cdot \partial _ t \Phi _ \varepsilon ^ i ( X , t ) , ( X , t ) \in \Gamma _ \varepsilon ( 0 ) \times [ 0 , T ] . \end{align*}


\begin{align*} a _ n & = \operatorname { m e x } \{ a _ k , b _ k \ ; | \ ; k < n \} \\ b _ n & = f ( a _ n ) + b _ { n - 1 } + a _ n - a _ { n - 1 } \end{align*}


\begin{align*} \lambda _ n ^ 2 = n + a \sqrt n \cdot \log ^ b n + o ( 1 ) , \ ; \ ; \ ; n \to \infty , \end{align*}


\begin{align*} B = \{ x _ 3 Q _ 1 + Q _ 2 ^ 2 \} \subset \mathbb P ^ 3 \end{align*}


\begin{align*} \left | 1 - \frac { w ( \Phi ( \omega ) ) } { w ( \Phi ( \eta ) ) } \right | \leq | \tau | \cdot d ^ { 3 / 2 } \cdot v _ 0 ( 0 ) \cdot \max _ { r \in [ 0 , 1 ] } \frac { w ( \Phi ( \eta ) + r \tau ) } { w ( \Phi ( \eta ) ) } \leq 2 \delta \cdot d ^ { 3 / 2 } \cdot v _ 0 ^ d ( 2 \delta ) \cdot v _ 0 ( 0 ) = : C ^ \delta < \infty . \end{align*}


\begin{align*} { e } _ { V , \theta } : = \frac { 1 } { \theta } { \rm e n t } [ \mu ^ { * } ] + \frac { 1 - \lambda } { 2 } \int _ { M } V d \mu ^ { * } . \end{align*}


\begin{align*} | \overline { N } ( \phi ( \jmath ) , S ) ^ F | = \frac { | W _ G ( S ) ^ F | } { | W _ { G _ \jmath } ( S ) ^ F | } = { \mu \choose \nu } { \mu ' \choose \nu ' } \prod _ j \nu _ j ! \nu _ j ' ! ( \kappa _ G \cdot j ) ^ { \nu _ j + \nu _ j ' } . \end{align*}


\begin{align*} F ^ { \varepsilon } ( t , x , y , r , p , q , M , N , Z ) : = H ^ { \varepsilon } ( t , x , y , p , M , Z ) - \mathcal { L } ( x , y , q , N ) + \lambda r , \end{align*}


\begin{align*} \mathbf { \Pi } ^ { \lambda } \left \{ | C | ( B ) = n \right \} = \frac { ( \lambda | B | ) ^ { n } } { n ! } \exp \left ( - \lambda | B | \right ) . \end{align*}


\begin{align*} \partial _ t d ( \pi ( x , t ) , t ) + \partial _ t \pi ( x , t ) \cdot \nabla d ( \pi ( x , t ) , t ) = 0 . \end{align*}


\begin{align*} u _ { n } \left ( z \right ) = \mu _ { 2 n } \left ( z \right ) . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { N } \delta _ { N ^ { \frac { 1 } { d } } ( x _ { i } - x ) } . \end{align*}


\begin{align*} \mathbb { E } \left [ f \left ( \overline { \hat { X } ( t ) } \right ) \right ] - m i n ( f ) = \mathcal { O } \left ( \frac { 1 } { t } \right ) + \mathcal { O } ( h ^ { 1 / 2 } ) + \frac { \sigma _ * ^ 2 d } { 2 } . \end{align*}


\begin{align*} p _ { 2 1 } & = \widetilde { p } _ { 2 1 } p _ { 1 3 } = \widetilde { p } _ { 1 3 } \Omega , \\ q _ { 2 1 } & = \widetilde { q } _ { 2 1 } q _ { 1 3 } = \widetilde { q } _ { 1 3 } \Gamma . \end{align*}


\begin{align*} \left | B _ { n + 1 , \mu , 0 } ( 1 ) \Lambda _ p \right | _ p > \left | \sum _ { j = 1 } ^ k \lambda _ j S _ { n + 1 , \mu , j } ( 1 ) \right | _ p . \end{align*}


\begin{align*} Q _ { } ^ { \mathsf { c } } ( \varepsilon , q ) \cap Q _ { } ( \varepsilon _ { 1 } , n , q ) \subseteq \bigcup _ { \substack { I \cap I ' = \varnothing ; \\ 1 \leq i \leq 2 M + 1 } } Q _ { I ' } ( E _ { i , I } ) . \end{align*}


\begin{align*} q _ { \kappa , \delta } ( x ) = \left ( v _ \kappa , p _ 3 \left ( \phi ' _ \kappa + r _ 0 \kappa R ' _ 0 u _ \delta + v _ \kappa \right ) - p _ 3 \left ( \phi ' _ \kappa \right ) \right ) . \end{align*}


\begin{align*} v ^ 2 \frac { \partial } { \partial \tau } Q _ { - \overline { \tau } } = \frac { i } { 2 } Q ( - \overline { \tau } , 1 ) , v ^ 2 \frac { \partial } { \partial \tau } Q _ { \tau } = \frac { i } { 2 } Q ( \overline { \tau } , 1 ) , v ^ 2 \frac { \partial } { \partial \tau } \frac { Q ( \tau , 1 ) } { v ^ 2 } = i Q _ { \tau } . \end{align*}


\begin{align*} \ell _ n ( u ) = q d e t L ^ { - } ( u ) ( q d e t L ^ { + } ( u + \frac { 1 } { 2 } h n ) ) ^ { - 1 } . \end{align*}


\begin{align*} M ( A ) _ M ( i , j ) = A _ M ( i , j ) \ ; . \end{align*}


\begin{align*} m ^ \epsilon ( t ) = e ^ { - \frac { t } { \epsilon ^ 2 } } m _ 0 ^ \epsilon + \frac { 1 } { \epsilon } \int _ { 0 } ^ { t } e ^ { - \frac { t - s } { \epsilon ^ 2 } } d \beta ( s ) . \end{align*}


\begin{align*} \lambda _ { \mathcal { E } } ( x ) = r ! \ , \sum _ { E \in \mathcal { E } } \ ; \prod _ { i = 1 } ^ m \ ; \frac { x _ i ^ { E ( i ) } } { E ( i ) ! } = 1 - \sum _ { i = 1 } ^ m x _ i ^ r . \end{align*}


\begin{align*} x \cdot ( y + z ) = x \cdot y + x \cdot z \ ; . \end{align*}


\begin{align*} \int _ { \partial \mathcal { O } _ { ( x , 0 ) , \rho } \cap [ - 1 , 1 ] ^ 2 } | x _ 2 | ^ \gamma d \sigma & \le \Delta u ( \overline { \mathcal { O } _ { ( x , 0 ) , \rho } } \cap [ - 1 , 1 ] ^ 2 ) \\ & = \int _ { V } \nabla u \cdot \vec { \eta } d \sigma \\ & \le \int _ { V } C | x _ 2 | ^ \gamma d \sigma . \end{align*}


\begin{align*} D : = \sum _ j \alpha _ P ^ j D _ P ^ j , \end{align*}


\begin{align*} \zeta \cdot ( x _ 0 , \ldots , x _ N ) = ( \zeta ^ { \lambda _ 0 } x _ 0 , \ldots , \zeta ^ { \lambda _ N } x _ N ) \ , . \end{align*}


\begin{align*} \frac { d } { d x } \sum _ { n = 0 } ^ \infty B ( n ) \frac { x ^ n } { n ! } \end{align*}


\begin{align*} \{ x ^ { \odot \star } \in X ^ { \odot \star } \ : \ \lim _ { h \downarrow 0 } \| U ^ { \odot \star } ( t + h , t ) x ^ { \odot \star } - x ^ { \odot \star } \| = 0 \} & = \overline { \mathcal { D } ( A ^ { \odot \star } ( t ) ) } \\ & = \overline { \mathcal { D } ( A _ 0 ^ { \odot \star } ) } = X ^ { \odot \odot } = j ( X ) , \end{align*}


\begin{align*} s _ \lambda ( x _ 1 , \dots , x _ N , 0 , 0 , \dots , 0 ) = s _ \lambda ( x _ 1 , \dots , x _ N ) , \end{align*}


\begin{align*} [ X + \xi , Y + \eta ] _ H : = [ X , Y ] + \mathcal { L } _ X \eta - \mathcal { L } _ Y \xi - \frac { 1 } { 2 } d \left ( \eta ( X ) - \xi ( Y ) \right ) + \iota _ Y \iota _ X H . \end{align*}


\begin{align*} J ( K f ) = \lim _ n J ( K f _ n ) . \end{align*}


\begin{align*} \sigma _ n = \frac { r _ n ^ { \gamma - 2 } } { M _ n ^ { \gamma - 1 } } , g _ n ( y , s ) = \frac { r _ n ^ { \gamma } } { M _ n ^ { \gamma } } f _ n \left ( \bar x _ n + r _ n y , \bar t _ n + \frac { r _ n ^ \gamma } { M _ n ^ { \gamma - 1 } } s \right ) \end{align*}


\begin{align*} B _ { i , j } = \sum _ { k - m - 2 } ^ { m + 3 } L _ { i j k } \delta _ k \end{align*}


\begin{align*} R ( g , T , \hat y , \hat \eta ) = \int \limits _ 0 ^ T \sqrt { \sum _ { j , k = 0 } ^ n g ^ { j k } ( y ) \eta _ j \eta _ k } \ , d t = \sqrt { 2 H ( y , \eta _ 0 , \eta ) } T , \end{align*}


\begin{align*} \left \{ \begin{array} { l } ( - i \partial _ t + \theta \langle \nabla \rangle _ m ) \psi _ \theta = \Pi _ \theta [ V _ b * ( \psi ^ \dagger \psi ) \psi ] , \\ \psi _ \theta | _ { t = 0 } = \psi _ { 0 , \theta } , \end{array} \right . \end{align*}


\begin{align*} \log ( n + 1 ) ( n + 1 ) < \frac { \log H } { \varepsilon \log \log H } \log \left ( \frac { \log H } { \varepsilon \log \log H } \right ) = \frac { \log H } { \varepsilon } \left ( 1 - \frac { \log \left ( \varepsilon \log \log H \right ) } { \log \log H } \right ) . \end{align*}


\begin{align*} t = 2 \arctan ( e ^ x ) , h ( x ) = r ( t ) - \frac { \pi } { 2 } . \end{align*}


\begin{align*} \int \limits _ 0 ^ T \varphi ( s ) \int \limits _ { \R ^ d } B _ n ( s , x ) \nabla u _ n ( s , x ) \cdot \overline { \nabla g ( x ) } \d x \d s = \int \limits _ 0 ^ T \varphi ( s ) \int \limits _ { \R ^ d } \nabla u _ n ( s , x ) \cdot \overline { B _ n ( s , x ) ^ * \nabla g ( x ) } \d x \d s . \end{align*}


\begin{align*} \kappa _ \eta \dot \eta ^ \perp = \nabla _ { \dot \eta } \dot \eta , \end{align*}


\begin{align*} \begin{cases} \Delta ^ 2 v _ h = - \theta _ h & \textrm { i n } B _ R ( 0 ) \\ v _ h = \partial _ n v _ h = 0 & \textrm { o n } \partial B _ R ( 0 ) \ , . \end{cases} \end{align*}


\begin{align*} [ a ^ { 2 r } , b ] = [ a ^ r , b ] ^ { a ^ r } [ a ^ r , b ] . \end{align*}


\begin{align*} ( ( \phi _ { x _ 0 } ^ X ) ^ * d v _ X ) ( Z ) = \kappa _ { \phi , x _ 0 } ^ { X } d Z _ 1 \wedge \cdots \wedge d Z _ { 2 n } . \end{align*}


\begin{align*} \alpha ( A _ { \lambda 0 } , B ) = \dfrac { 1 } { q ^ 4 } \alpha ( A _ { \lambda 2 } , B ) . \end{align*}


\begin{align*} \left \{ \begin{alignedat} { 2 } - \Delta _ p u = K _ 1 ( x ) u ^ { \alpha _ 1 } \ , v ^ { \beta _ 1 } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; u > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; u \lfloor _ { \partial \Omega } = 0 , \\ - \Delta _ q v = K _ 2 ( x ) v ^ { \alpha _ 2 } \ , u ^ { \beta _ 2 } \ ; \ ; & \mbox { i n $ \Omega $ , } \ ; \ ; v > 0 \ ; \ ; & & \mbox { i n $ \Omega $ , } \ ; \ ; v \lfloor _ { \partial \Omega } = 0 , \end{alignedat} \right . \end{align*}


\begin{align*} v ( x , t ) = C \left ( \left ( \frac { 1 } { t + r ^ 2 } e ^ { - \frac { | x - r e _ n | ^ 2 } { t + r ^ 2 } } \right ) ^ { \beta } - \frac { e ^ { - \beta } } { r ^ { 2 \beta } } \right ) . \end{align*}


\begin{align*} S ' _ { \mathbf { t } } = \left \{ \Psi _ 7 \left ( G _ d + \frac { q } { 2 } x _ { m - 2 } + \frac { q } { 2 } \mathbf { t } \cdot \mathbf { y } ) \right ) : d \in \{ 1 , 2 \} , \mathbf { y } \in \mathbb { Z } _ { 2 } ^ { 2 } \right \} . \end{align*}


\begin{align*} \partial _ t f _ \epsilon = \partial _ x ^ 2 f _ \epsilon + F ( \partial _ x f _ \epsilon ) + \sqrt { \epsilon } \xi , \end{align*}


\begin{align*} \liminf _ { N \to \infty } \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } ( X _ { N } ) - \nu ) & \geq \liminf _ { N \to \infty } \min _ { Y _ { N } \in \mathbb { R } ^ { d \times N } } \mathcal { E } ^ { \neq } ( { \rm e m p } _ { N } ( Y _ { N } ) - \nu ) \\ & \geq \min _ { \rho \in \mathcal { P } ( \mathbb { R } ^ { d } ) } \mathcal { E } ( \rho - \nu ) \\ & = 0 \\ & = \mathcal { E } ( \mu - \nu ) . \end{align*}


\begin{align*} f ^ \epsilon ( \ell _ 1 , \ell _ 2 ) : = \omega \cdot \ell _ 2 - | \pi ( \ell _ 2 ) | ^ 2 - 2 \epsilon \pi ( \ell _ 1 ) \cdot \pi ( \ell _ 2 ) \ , , \epsilon \in \{ 0 , 1 \} \ , , \end{align*}


\begin{align*} \mathcal Z = \mathcal X = ( X _ t , p ^ { u t } , T ) . \end{align*}


\begin{align*} H ( t , x , y , p , M , Z ) : = \min \limits _ { u \in U } \left \{ - ( \sigma \sigma ^ { \top } M ) - f \cdot p - 2 ( \sigma \varrho ^ { \top } Z ^ { \top } ) - \ell \right \} . \end{align*}


\begin{align*} \tilde { l } ' : \ \eta _ i ^ { ' - 1 } ( D ) \to \mathbb { R } , \tilde { l } ' ( l ) : = \Re \frac { \eta ' _ i ( l ) - z _ 0 } { \partial _ { \gamma } \eta ' _ i ( l _ 0 ) } \end{align*}


\begin{align*} \frac { \mu \frac { 1 } { \sqrt { 1 - z } } } { ( 1 - \mu + \mu \sqrt { 1 - z } ) ^ 2 } = \frac { d } { d z } \biggl ( \frac { 2 } { 1 - \mu + \mu \sqrt { 1 - z } } \biggr ) . \end{align*}


\begin{align*} 0 = & \int _ { \Omega } ( | \nabla u | ^ 2 ( \phi ) - 2 \nabla u D \phi \nabla u + x _ n \chi _ { \{ u > 0 \} } ( \phi ) + \chi _ { \{ u > 0 \} } ) \langle \phi , x _ n \rangle d x \\ = & \frac { d } { d t } J ( u + t \phi ) | _ { t = 0 } , \end{align*}


\begin{gather*} \underbrace { s \left ( [ x , y ] \right ) + s ' \left ( \theta ( x , y ) \right ) } _ { \in J } + \underbrace { i \left ( [ x , y ] \right ) + i ' \left ( \theta ( x , y ) \right ) } _ { \in V } = \underbrace { \left [ s ( x ) , s ( y ) \right ] } _ { \in J } \\ + \underbrace { \rho \left ( s ( x ) \right ) i ( y ) + \rho \left ( s ( y ) \right ) i ( x ) + \theta ' \left ( s ( x ) , s ( y ) \right ) } _ { \in V } . \end{gather*}


\begin{align*} Z _ k = f ^ { ( k ) } \left ( ( Y _ j ^ { ( k ) } ) _ { j \in \tilde { J } _ k } \right ) \end{align*}


\begin{align*} d _ { \Gamma _ t } ( x ) \ ; = \ ; \inf \{ | x - y | \ , ; \ , y \in \Gamma _ t \} , \end{align*}


\begin{align*} f ( A _ 2 , \theta _ 2 , \omega _ 2 , S ) = d ^ { \frac { g \cdot \deg ( w ) } { 2 } } \cdot \det ( [ u ] ) ^ { - w } \cdot f ( A _ 1 , \theta _ 1 , \omega _ 1 , S ) , \end{align*}


\begin{align*} \phi ^ \pi = \exp ( - \pi \vee ) \phi \exp ( \pi \vee ) \end{align*}


\begin{align*} \lambda _ a ( 1 ) = a + ( 2 ^ { m - 1 } - 1 ) ^ a - a = \begin{cases} 1 , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; e v e n , } \\ 2 ^ { m - 1 } - 1 , \quad { \rm i f } \ ; a \ ; { \rm i s \ ; o d d . } \end{cases} \end{align*}


\begin{align*} { ( v \to \varphi ( v \cdot t g ) ) = v \to { t ^ { - r } } ( g ) ^ { - 1 } \varphi \left ( v \cdot \frac { ( g ^ \top ) ^ { - 1 } } { t } \right ) , } \end{align*}


\begin{align*} \mathbf { P r } \left ( \bigg \| \frac { d } { n \epsilon } \sum _ { i = 1 } ^ { n } J ( u _ 0 + \epsilon v _ { 0 , i } ) v _ { 0 , i } - \nabla \hat { J } ( u _ 0 ) \bigg \| \geq t \right ) \leq 2 \exp \left ( - \frac { c n t ^ 2 } { K ^ 2 } \right ) . \end{align*}


\begin{align*} u & \in \L _ t \big ( x _ 1 \big ( \textstyle \prod _ { s = \ell } ^ { d - 2 } x _ { n - s t - 1 } \big ) \big ( \prod _ { s = 0 } ^ { \ell - 1 } x _ { n - s t } \big ) , x _ 1 \big ( \prod _ { s = 0 } ^ { d - 2 } x _ { n - s t } \big ) \big ) , \\ v & = \big ( \textstyle \prod _ { s = \ell } ^ { d - 1 } x _ { n - s t - 1 } \big ) \big ( \prod _ { s = 0 } ^ { \ell - 1 } x _ { n - s t } \big ) . \end{align*}


\begin{align*} G ( \sigma , t ) = \frac { \sigma } { \sigma + i t } . \end{align*}


\begin{align*} \rho _ t = \ , \frac { \bar \rho } { J _ { X ( t , 0 , \cdot ) } } \circ X ( 0 , t , \cdot ) \ , , \end{align*}


\begin{align*} \sup _ { z \in D } | P g ( z ) | \rho _ D ^ { - 1 } ( z ) = : \lambda & \Longrightarrow \sup _ { z \in D } | S g ( z ) | \rho _ D ^ { - 2 } ( z ) \leq 6 4 \lambda + \lambda ^ 2 / 2 ; \\ \lim _ { z \to \partial D } | P g ( z ) | \rho _ D ^ { - 1 } ( z ) = 0 & \Longrightarrow \lim _ { z \to \partial D } | S g ( z ) | \rho _ D ^ { - 2 } ( z ) = 0 . \end{align*}


\begin{align*} \pi _ { 0 ; \varepsilon , \delta } ( \eta h _ a ) = \det ( \eta ) ^ \delta \chi _ { \varepsilon , 0 } ( h _ a ) ( a \in \R ^ \times , \ ; \eta \in \{ 1 , s \} ) . \end{align*}


\begin{align*} g = w - a ^ * ( 1 - \eta ) + \varepsilon \tau + \varepsilon \end{align*}


\begin{align*} \Psi ( x , t ) \leq t ^ { p ^ * ( x ) } + 1 + \mu ( x ) ^ { \frac { q ^ * ( x ) } { q ( x ) } } t ^ { q ^ * ( x ) } + 1 = \mathcal { G } ^ * ( x , t ) + 2 \quad ( x , t ) \in \overline { \Omega } \times [ 0 , \infty ) . \end{align*}


\begin{align*} A ( Z , Z ' _ Y ) = \sum _ { \alpha } Z _ N ^ { \alpha } \cdot A ^ { \alpha } ( Z _ Y , Z ' _ Y ) , C ( Z _ Y , Z ' ) = \sum _ { \alpha ' } C ^ { \alpha ' } ( Z _ Y , Z ' _ Y ) Z ' _ N { } ^ { \alpha ' } , \end{align*}


\begin{align*} \hat { \mathbf { R } } _ { D , b } = \frac { 1 } { | \mathbb { S } _ { D , b } | } \sum _ { ( m , n ) \in \mathbb { S } _ { D , b } } \mathbf { \hat { v } } _ { m , n } \mathbf { \hat { v } } _ { m , n } ^ * , \end{align*}


\begin{align*} \int _ 1 ^ \infty \frac { \Theta ' ( s ) } { s \Theta ( s ) } \dd s \ge - \frac 1 2 \log ( \Theta ( 1 ) ) + \sum _ { k = 2 } ^ \infty \frac { 1 } { k ^ 2 + k } \log ( \Theta ( k ) ) . \end{align*}


\begin{align*} \| \mathbf { X } \| _ { p - v a r } : = \sup _ { \mathcal { D } \subset [ 0 , 1 ] } \left ( \sum _ { t _ i \in \mathcal { D } } d _ { C C } ( \mathbf { X } _ { t _ i } , \mathbf { X } _ { t _ { i + 1 } } ) ^ p \right ) ^ { \frac { 1 } { p } } = \sup _ { \mathcal { D } \subset [ 0 , 1 ] } \left ( \sum _ { t _ i \in \mathcal { D } } \| \mathbf { X } _ { t _ i , t _ { i + 1 } } \| _ { C C } ^ p \right ) ^ { \frac { 1 } { p } } . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ s m _ i n _ i = 0 \end{align*}


\begin{align*} \mathcal { C } : = \{ x \in \Re ^ { m n } | x _ { i } = x _ { j } , 1 \leq i , j \leq m , x _ { i } \in \Re ^ { n } \} \end{align*}


\begin{align*} K _ n ^ { \lambda , \theta , T } ( y , y ' ) : = - \frac { 1 } { n } \sum _ { z ^ n = ( - 1 ) ^ { \theta _ 2 } } \frac { z ^ { h - h ' } e ^ { - ( \lambda + \theta _ 1 \pi i + T z ) ( [ t ' - t ] + \mathrm { 1 } _ { t ' = t } ) } } { 1 - e ^ { - ( \lambda + \theta _ 1 \pi i + T z ) } } . \end{align*}


\begin{gather*} ( \bar { w } , z _ \Gamma ) _ { H _ \Gamma } = \langle \partial _ { \boldsymbol { \nu } } \bar { u } + \bar { \eta } , z _ \Gamma \rangle _ { Z _ \Gamma ^ * , Z _ \Gamma } + \bigl ( \pi _ \Gamma ( v _ 1 ) - \pi ( v _ 2 ) - \bar { g } , z _ \Gamma \bigr ) _ { \ ! H _ \Gamma } , \end{gather*}


\begin{align*} m ( - R ^ G _ { T , s } , \sigma ) = m ( - R ^ G _ { T , s } , I ( \tau , \sigma ) ) . \end{align*}


\begin{align*} \begin{aligned} p ( P ) = & \frac { 1 } { \pi R ^ { 2 } \sin ^ { 2 } \theta } \int _ { 0 } ^ { R \sin \theta } \int _ { 0 } ^ { 2 \pi } p ( r ' , \theta ' ) r ' d r ' d \theta ' \\ & + \frac { 2 } { \pi R ^ { 2 } \sin ^ { 2 } \theta } \int _ { 0 } ^ { R \sin \theta } \int _ { 0 } ^ { r ' } \int _ { 0 } ^ { 2 \pi } \frac { ( \hat { u } - u ) v _ { \theta ' } } { \rho } r ' d \theta ' d \rho d r ' \doteq & I _ { 1 } ' + I _ { 2 } ' . \end{aligned} \end{align*}


\begin{align*} \frac { \partial _ i \tilde { \rho } } { \rho } = \frac { \partial _ i p } { \gamma p } , i = t , x , y , \end{align*}


\begin{align*} \iint _ \Omega | \nabla v | ^ 2 \Psi \ , \frac { d t } { | t | ^ { n - d - 2 } } \ , d x \approx \| S ( v | \Psi ) \| ^ 2 _ { L ^ 2 ( \R ^ d ) } . \end{align*}


\begin{align*} \frac { T _ { 2 n } } { n } \stackrel { } \to \frac { 1 } { \lambda _ + } + \frac { 1 } { \lambda _ - } : = c _ 2 \end{align*}


\begin{align*} \int _ E | f _ E | ^ p = \int _ { \Omega \setminus { E } } | f _ E | ^ p = : I . \end{align*}


\begin{align*} \langle \mathcal { F } _ 2 ( f \otimes g ) , \Phi \rangle = \langle f \otimes g , \mathcal { F } _ 2 ^ { - 1 } \Phi \rangle , \ \ \ \ \ \ \Phi \in \mathcal { S } ( \mathbb { R } ^ { 2 d } ) . \end{align*}


\begin{align*} \partial _ a E _ { m , n } ^ { a , b } [ g ( t _ 1 ) ] = \mathrm { C o v } ^ { a , b } _ { m , n } \left ( g ( t _ 1 ) , \sum _ { i = 1 } ^ { t _ 1 } L _ i ( a ) \right ) = \sum _ { i = 1 } ^ m L _ i ( a ) \cdot \mathrm { C o v } ^ { a , b } _ { m , n } \left ( g ( t _ 1 ) , 1 _ { \{ t _ 1 \ge i \} } \right ) \ge 0 . \end{align*}


\begin{align*} Q _ { j } ( \rho ) = \big \{ \omega \in \Omega : \exists \psi \in \mathbb { C } ^ { [ - L , L ] } , ( \psi ) \subset \widetilde { I } _ { j } , \| \psi \| = 1 , \big \| ( H _ { \omega } \upharpoonright \widetilde { I } _ { j } - E ) \psi \upharpoonright { \widetilde { I } _ { j } } \big \| , \big | \psi \upharpoonright { \partial \widetilde { I } _ { j } } \big | < \rho \big \} . \end{align*}


\begin{align*} ( \tilde \Delta - \lambda _ s ) ( E _ s - h _ s ) = - ( \Delta - \lambda _ s ) h _ s = - H _ s , \end{align*}


\begin{align*} X = Y \cup f ^ r ( Y ) \cup f ^ { 2 r } ( Y ) \cup \dots \cup f ^ { n _ 0 r } ( Y ) , \end{align*}


\begin{align*} \delta v _ { s , t } = \mathcal { D } ( v ) _ { s , t } + \int _ { s } ^ { t } h _ 2 v _ r \times \circ \dd B _ r \ , . \end{align*}


\begin{align*} \begin{cases} i \displaystyle \frac { \partial u } { \partial t } + H u = 0 \\ u ( 0 , x ) = u _ 0 ( x ) , \end{cases} \end{align*}


\begin{align*} \frac { \partial v _ \epsilon } { \partial t } \ ; = \ ; \frac { \dot { q } } { \epsilon ^ { 1 - \kappa } } \left ( \frac { \partial r _ \epsilon } { \partial t } \ , - \ , | \log \epsilon | \dot { \kappa } r _ \epsilon \right ) , \end{align*}


\begin{align*} x _ 1 ^ n + \cdots + x _ n ^ n - n \lambda x _ 1 \cdots x _ n = 0 \end{align*}


\begin{align*} \vartheta _ { T } ^ { ( \nu ) } : = \vartheta _ T \circ N ^ T _ \nu , \end{align*}


\begin{align*} \P \left \{ \adjustlimits \lim _ { \varepsilon \to 0 ^ + } \sup _ { s \in ( 0 , 1 ) } \frac { | H ( s + \varepsilon \ , , 0 ) - H ( s \ , , 0 ) | } { \varepsilon ^ { \theta } } = 0 \right \} = 1 \qquad . \end{align*}


\begin{align*} \left \{ \begin{aligned} \ ; \delta u ( y ) - \mathcal { L } u ( y ) & = - h ( y ) , \ ; \ ; D _ { n } , & \\ u ( y ) & = 0 , \ ; \partial D _ { n } . & \end{aligned} \right . \end{align*}


\begin{align*} x ( b ) - x ( a ) = \int _ { a } ^ { b } D [ A ( t ) x ( \tau ) + g ( t ) ] , \end{align*}


\begin{align*} \left \lbrace \begin{aligned} \partial _ t u ^ \epsilon ( t , q , p ) & = \frac { 1 } { \epsilon } \Bigl ( \nabla _ q u ^ \epsilon ( t , q , p ) \cdot p + \nabla _ p u ^ \epsilon ( t , q , p ) \cdot f ( q ) \Bigr ) \\ & + \frac { 1 } { \epsilon ^ 2 } \Bigl ( - \nabla _ p u ^ \epsilon ( t , q , p ) \cdot p + \nabla _ { p } ^ 2 u ^ \epsilon ( t , q , p ) : a ( q ) \Bigr ) , \\ u ^ \epsilon ( 0 , q , p ) & = \varphi ( q ) , \end{aligned} \right . \end{align*}


\begin{align*} [ \alpha , a \beta ] & = \alpha ( a ) \beta + a [ \alpha , \beta ] , \ a \in A , \ \alpha , \beta \in L , \\ ( a \alpha ) b & = a ( \alpha ( b ) ) , \ \ a , b \in A , \ \alpha \in L . \end{align*}


\begin{align*} V _ C = \begin{pmatrix} I _ { d \times d } & 0 _ { d \times d } \\ C & I _ { d \times d } \end{pmatrix} ; \end{align*}


\begin{align*} x _ { k + 1 } = x _ k - \gamma _ k \nabla f ( x _ k ) . \end{align*}


\begin{align*} \pi _ Y \circ \vartheta ( U ( \R ) ) = [ \lambda _ 1 , \lambda _ 2 ] \cap \left ( \bigsqcup _ { i = 1 } ^ { r } [ \varepsilon _ { 2 i - 1 } , \varepsilon _ { 2 i } ] \right ) . \end{align*}


\begin{align*} q _ n ^ { \epsilon , \Delta t } - q _ n ^ { 0 , \Delta t } & = Q _ n ^ { \epsilon , \Delta t } - q _ n ^ { 0 , \Delta t } - P _ n ^ { \epsilon , \Delta t } \\ & = q _ 0 ^ \epsilon - q _ 0 ^ 0 + \Delta t \sum _ { k = 0 } ^ { n - 1 } \bigl ( f ( q _ k ^ { \epsilon , \Delta t } ) - f ( q _ k ^ { 0 , \Delta t } ) + \sum _ { k = 0 } ^ { n - 1 } \bigl ( \sigma ( q _ k ^ { \epsilon , \Delta t } ) - \sigma ( q _ k ^ { 0 , \Delta t } ) \bigr ) - \epsilon p _ n ^ { \epsilon , \Delta t } . \end{align*}


\begin{align*} \sinh ( t ) V _ H ( t ) = Y ( t ) = \frac { \sinh ( \sqrt { - \kappa } t ) } { \sqrt { - \kappa } } V ( t ) , \end{align*}


\begin{align*} \pi _ k \xi = \frac { 1 } { 2 \pi i } \int \limits _ { \Gamma } \eta _ k \frac { \partial _ { \gamma } \eta _ i } { \eta _ i - z } d l . \end{align*}


\begin{align*} ^ { \rho } D ^ { \gamma } _ { 0 _ + } u ( t ) = ^ { \rho } D ^ { \beta ( 1 - \alpha ) } _ { 0 _ + } \Big ( f ( t , u ( t ) , ^ \rho D ^ { \alpha , \beta } u ( t ) ) - p ( t ) u ( t ) \Big ) . \end{align*}