\begin{align*} ( A + F ) y = ( b + f ) . \end{align*}


\begin{align*} \Upsilon : = K _ M \bigcap \bigcap _ { k = 1 } ^ { \infty } \Gamma _ k \end{align*}


\begin{align*} x = q _ i + p _ i \end{align*}


\begin{align*} \varphi _ { \lambda , u } = x _ { 0 } ^ { \lambda _ { 0 } } \cdots x _ { n } ^ { \lambda _ { n } } f _ 1 ^ { u _ 1 } \cdots f _ { l } ^ { u _ l } \end{align*}


\begin{align*} N ( \tau ) & = \frac { \operatorname { v o l } ( \mathbb { S } ^ { n - 2 } ) \operatorname { v o l } ( \partial \Omega ) } { n - 1 } \biggl [ \biggl ( \frac { \lambda + 3 \mu } { 2 \mu ( \lambda + \mu ) } \biggr ) ^ { n - 1 } + \frac { 1 } { ( 2 \mu ) ^ { n - 1 } } + \frac { n - 2 } { \mu ^ { n - 1 } } + \frac { 1 } { \alpha ^ { n - 1 } } \biggr ] \tau ^ { n - 1 } \\ & + o ( \tau ^ { n - 1 } ) \ \tau \to + \infty . \end{align*}


\begin{align*} u _ m - \beta ^ r u _ { m - r } = a ' \alpha ^ { m - r } t _ r . \end{align*}


\begin{align*} & \mathcal { B } ^ { \rm n r e } _ { h , r } ( \sigma ) = \Big \{ F \in \mathcal { B } _ { h , r } : F ( x ) = \mathcal { T } _ { N } F ( x ) \Big \} , \\ & \mathcal { B } ^ { \rm r e } _ { h , r } ( \sigma ) = \Big \{ F \in \mathcal { B } _ { h , r } : F ( x ) = \mathcal { R } _ { N } F ( x ) \Big \} . \end{align*}


\begin{align*} \sum _ { r = 1 } ^ n { T _ { i j r } T _ { k \ell r } } = 0 \forall i , j , k , \ell \in \{ 1 , \ldots , n \} . \end{align*}


\begin{align*} \P _ x ( Z _ d ( n _ 1 ) \leq \xi _ 1 , \ldots , Z _ d ( n _ m ) \leq \xi _ m ) = \det ( I - \bar { \chi } _ \xi K \bar { \chi } _ { \xi } ) _ { l ^ 2 ( \{ n _ 1 , \ldots n _ k \} \times \mathbb { N } } \end{align*}


\begin{align*} \dot { V } ( \bar { x } _ t ) = - \norm { \bar { x } ( t - h ) } ^ 2 . \end{align*}


\begin{align*} \Phi ^ \dagger ( v , p ) = 0 , \end{align*}


\begin{align*} \tilde Z _ n ^ N ( i ) : = \Delta _ n ^ { - \frac 1 2 } \left ( ( p _ N \tilde { \Delta } _ i ^ n Y ) ^ { \otimes 2 } - \int _ { t _ { i - 1 } } ^ { t _ i } p _ N \mathcal S ( t _ i - s ) \Sigma _ s \mathcal S ( t _ i - s ) ^ * p _ N d s \right ) . \end{align*}


\begin{align*} \sum _ { f \in \mathcal F _ { N , k } } a _ f ( n ) = \begin{cases} \frac { | \mathcal F _ { N , k } | } { n ^ { 1 / 2 } } + \O \left ( n \sigma _ 0 ( n ) 4 ^ { \nu ( N ) } \right ) , & n \\ \O \left ( n \sigma _ 0 ( n ) 4 ^ { \nu ( N ) } \right ) , & . \end{cases} \end{align*}


\begin{align*} \widehat { \Delta \mathbf { s } } = \mathbf { T } _ x \Delta \mathbf { s } = \mathbf { T } _ x \mathbf { X } ^ { - 1 } ( \nu \mu \mathbf { e } - \mathbf { X } \mathbf { s } - \mathbf { S } \Delta \mathbf { x } ) = \nu \mu \mathbf { e } - \mathbf { w } _ { x s } - \mathbf { R } _ { x s } \widehat { \Delta \mathbf { x } } . \end{align*}


\begin{align*} \log \int _ { \R ^ n } e ^ { f } \ , d \gamma _ t - \left ( \int _ { \R ^ n } f \ , d \gamma _ { y , t } - H ( \gamma _ { y , t } \ , | \ , \gamma _ t ) \right ) = H ( \gamma _ { y , t } \ , | \ , P ) , \end{align*}


\begin{align*} P ( x _ 1 , \ldots , x _ n ) = x _ 2 ^ { q + 1 } - x _ 1 ^ q x _ 3 . \end{align*}


\begin{align*} \langle e _ { i } , A f _ { j } \rangle = \langle A e _ { i } , A f _ { j } \rangle = \langle e _ { i } , f _ { j } \rangle = \delta _ { i j } . \end{align*}


\begin{align*} \dot { s } \varepsilon ( x ) \dot { s } = \displaystyle \varepsilon ( - x ^ { - 1 } ) \dot { s } h ( - x ) \varepsilon ( - x ^ { - 1 } ) . \end{align*}


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


\begin{align*} \lambda _ g ' \circ \bar \phi \circ \eta _ X & = \lambda _ g ' \circ \phi & \\ & = \phi \circ \lambda _ g & \\ & = \bar \phi \circ \eta _ X \circ \lambda _ g & \\ & = \bar \phi \circ \Q _ \tau \lambda _ g \circ \eta _ X & \end{align*}


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


\begin{align*} \lim _ { n \rightarrow \infty } \int _ { \mathbb { T } ^ d } \langle b ( u _ n ( x ) ) , \nabla u _ n ( x ) - \nabla u ( x ) \rangle d x = 0 . \end{align*}


\begin{align*} A _ { \alpha , \beta , u , v } = \begin{pmatrix} \alpha & u \alpha \\ \beta & v \beta \end{pmatrix} \end{align*}


\begin{align*} \frac { \partial \Phi _ { i , j } ^ l } { \partial x _ k } = 0 \textrm { f o r a l l $ p + 1 \le l \le n $ , a n d $ 1 \le k \le p $ } \end{align*}


\begin{align*} \sum _ s \phi _ { i s } ( \partial _ r ) \varphi _ s & = \phi _ { i i } ( \partial _ r ) \varphi _ i + \phi _ { i r } ( \partial _ r ) \varphi _ r = \frac { 1 } { 2 } \partial _ r ( \varphi ^ { - 1 } _ i ) \varphi _ i + \Gamma _ { r i } ^ i = 0 , \end{align*}


\begin{align*} 0 & \le \sum _ { l \in \Omega _ 1 } ( | l \cap W | - 1 ) ( | l \cap W | - 2 ) \\ & = \sum _ { l \in \Omega _ 1 } | l \cap W | ( | l \cap W | - 1 ) - 2 \sum _ { l \in \Omega _ 1 } | l \cap W | + 2 | \Omega _ 1 | \\ & \le | W | ( | W | - 1 ) - 2 \sum _ { l \in \Omega _ 1 } | l \cap W | + 2 | \Omega _ 1 | , \end{align*}


\begin{align*} M _ { \pm } ( x ; z ) = I + \mathcal { P } ^ { \pm } \left ( M _ { - } ( x ; \cdot ) R ( x ; \cdot ) \right ) , z \in \mathbb { R } , \end{align*}


\begin{align*} \alpha \circ T = T \circ \beta , [ T ( a ) , T ( b ) ] = T \big ( \varrho ( T ( a ) ) b - \varrho ( T ( b ) ) a \big ) . \end{align*}


\begin{align*} \tilde { P } _ { 2 q _ n - 1 } ( \theta ) = \sum _ { x _ 1 \in I _ 0 \cup I _ { \ell } } \tilde { P } _ { 2 q _ n - 1 } ( \theta _ { x _ 1 } ) \prod _ { \substack { j \in I _ 0 \cup I _ { \ell } \\ j \neq x _ 1 } } \frac { \sin \pi ( \theta - \theta _ j ) } { \sin \pi ( \theta _ { x _ 1 } - \theta _ j ) } . \end{align*}


\begin{align*} \mathcal D _ \pi = \sum _ { \widetilde { K } \ \ \widetilde { L } } ( e _ { \widetilde { K } } - 1 ) [ \widetilde { K } ] \in H _ 1 ( \widetilde M , \mathbb Z / 2 \mathbb Z ) \end{align*}


\begin{align*} e _ c \left [ i \right ] = a _ c { \left [ i \right ] } - a _ c ^ { \left ( o \right ) } . \end{align*}


\begin{align*} W _ { U _ { \tau _ k , R } } ( x ) = W \circ R _ { \tau _ k } ( x ) \end{align*}


\begin{align*} C _ 1 & { } = \left \{ ( f , g ) \in \mathrm { L } ^ 2 ( X , m ; \mathbb { R } ^ 2 ) : f \leq g \right \} , \\ C _ { 2 , \alpha } & { } = \left \{ ( f , g ) \in \mathrm { L } ^ 2 ( X , m ; \mathbb { R } ^ 2 ) : | f - g | \leq \alpha \right \} . \end{align*}


\begin{align*} \begin{aligned} & \lim _ { | k | \rightarrow \infty } ( z \Psi ^ - ) _ { 2 1 } ( x ; k ) = - \frac { 1 } { 2 i } \lim _ { | z | \rightarrow \infty } \nu ( x ; z ) \\ & = - \frac { 1 } { 4 } \left ( 2 i \bar { u } _ { x x } ( x ) + | u _ x ( x ) | ^ { 2 } \bar { u } _ x ( x ) \right ) e ^ { i c _ \pm } ( x ) = \widehat { \Psi } ^ - _ { 2 1 } ( x ) , \end{aligned} \end{align*}


\begin{align*} \mathfrak X ^ 0 ( Z ) = \{ ( x , w ) \in Z _ \mathrm { r e g } \times \P ^ { n } \ \mid \ x \ , \ , \ , \ , \alpha _ w | _ { Z _ \mathrm { r e g } } \} \end{align*}


\begin{align*} \varphi ( x , t , k ) = \phi ( x , t ; k ) e ^ { i ( k ^ 2 x + \eta ^ 2 t ) \sigma _ 3 } , \end{align*}


\begin{align*} \begin{aligned} \int _ B \varphi _ 0 \Delta u _ 1 \wedge \dots \wedge \Delta u _ m \wedge \beta _ n ^ { n - m } \leq \gamma \left ( \| \varphi _ 0 \| _ S + \int _ B \varphi _ 0 ( \Delta \varphi _ 0 ^ { + } ) ^ m \wedge \beta _ n ^ { n - m } \right ) ^ \kappa , \end{aligned} \end{align*}


\begin{align*} I _ f ( \theta ) : = \limsup _ { s \rightarrow + \infty } \frac { \ln \left | f \left ( s e ^ { i \theta } \right ) \right | } { s } , \end{align*}


\begin{align*} \theta _ i ^ { ( 2 ) } ( z _ j ) = \delta _ { i , j } \mbox { a n d } \theta _ { i } ^ { ( 1 ) } ( z _ j ) = a _ { i , j } . \end{align*}


\begin{align*} z = \frac { 1 } { \sqrt { 1 + a } } \cos ( t ) + \frac { i } { \sqrt { 1 - a } } \sin ( t ) , w = 0 . \end{align*}


\begin{align*} F ( x ^ { t + 1 } ) - F ( x ) + \frac { \alpha _ t ^ 2 } { 2 \gamma _ t } \| x - z ^ { t + 1 } \| ^ 2 \overset { \eqref { F - i n e q } } { \leq } \ & ( 1 - \alpha _ t ) \left ( F ( x ^ t ) - F ( x ) \right ) + \frac { \alpha _ { t } ^ 2 ( 1 - \beta _ t ) } { 2 \gamma _ { t } } \| x - z ^ t \| ^ 2 \\ \overset { \eqref { a l p h a - e q n } } { = } \ & ( 1 - \alpha _ t ) \left ( F ( x ^ t ) - F ( x ) + \frac { \alpha _ { t - 1 } ^ 2 } { 2 \gamma _ { t - 1 } } \| x - z ^ t \| ^ 2 \right ) . \end{align*}


\begin{align*} x x ' + x y ' + y y ' = n \ , . \end{align*}


\begin{align*} \ell ( s _ \gamma w s _ \alpha ) = & \ , \ell ( w s _ \alpha ) + 1 - \langle - ( w s _ \alpha ) ^ { - 1 } \gamma ^ \vee , 2 \rho \rangle \\ = & \ , \ell ( w ) + 2 - \langle \alpha ^ \vee , 2 \rho \rangle - \langle - w ^ { - 1 } \gamma ^ \vee - \langle - w ^ { - 1 } \gamma ^ \vee , \alpha \rangle \alpha ^ \vee , 2 \rho \rangle \\ = & \ , \ell ( s _ \gamma w ) + 1 + ( \langle - w ^ { - 1 } \gamma ^ \vee , \alpha \rangle - 1 ) \langle \alpha ^ \vee , 2 \rho \rangle . \end{align*}


\begin{align*} u & = x + y = x + \psi ( x ) , \\ \psi ( x ) & = u - x . \end{align*}


\begin{align*} \int _ { 0 } ^ { \lambda } \mu \big [ \xi ( \mu , t ) - \xi ( \mu , 0 ) ] \\ b i g d \mu = - & \int _ { 0 } ^ { t } \int _ { \lambda } ^ { \infty } \int _ { 0 } ^ { \lambda } \nu \Lambda ( \mu , \nu ) \zeta ( \mu , s ) \xi ( \nu , s ) \ d \nu d \mu d s . \end{align*}


\begin{align*} \mathcal { P } ( \delta ) : = \begin{cases} ( \delta , 0 ) \quad - \delta \\ \{ ( \beta _ 1 , \beta _ 2 ) \in ( R ^ - \cup \{ 0 \} ) ^ 2 \mid \beta _ 1 + \beta _ 2 = \delta \} \ , \ , . \end{cases} \end{align*}


\begin{align*} E [ M _ v ( t ) ^ 2 ] = E \big [ [ M _ v ] ( t ) \big ] . \end{align*}


\begin{align*} b ( 0 , x ) \equiv b _ 0 , \xi ( 0 , x ) = \xi _ 0 , S ( 0 , x ) = x \cdot \xi _ 0 . \end{align*}


\begin{align*} \| \mu _ 1 - \mu _ 2 \| _ { L , M } ^ * : = \sup _ { f \in C _ b ( M ) , \ , | f | _ L \leqslant 1 } \Big | \langle f , \mu _ 1 \rangle - \langle f , \mu _ 2 \rangle \Big | \le 2 , \end{align*}


\begin{align*} L ( x , \gamma ) : = J ( x ) + \sum \limits _ { k = 1 } ^ m \gamma _ k f _ k ( x ) = x ^ T A ( \gamma ) x + 2 b ( \gamma ) ^ T x + c ( \gamma ) \end{align*}


\begin{align*} g _ { \Delta } ( \omega ) : = \sum _ { \zeta _ { \Lambda } } \sum _ { \eta _ { \Lambda } } \mathbf { 1 } _ { \zeta _ { \Lambda } } ( \omega ) h _ { \Delta } ( \eta _ { \Lambda } \omega _ { \Lambda ^ c } ) \frac { \gamma _ { \Lambda } ( \eta _ { \Lambda } | \omega _ { \Lambda ^ c } ) } { \gamma _ { \Lambda } ( \zeta _ { \Lambda } | \omega _ { \Lambda ^ c } ) } , \omega \in \Omega , \end{align*}


\begin{align*} N : = \mathcal N ( F ^ q _ v ( 0 , p _ 0 ) ) = \operatorname { s p a n } \{ u _ \ell , w _ \ell \} . \end{align*}


\begin{align*} \left | G ^ { D _ { N } } ( x , y ) - \gamma \log \left ( 2 + N \frac { \max ( d ( x / N ) , d ( y / N ) ) } { 1 + | x - y | } \right ) \right | = O ( 1 ) . \end{align*}


\begin{align*} \mu \Big ( \bigcap _ { j = 0 } ^ { t _ n } T ^ { j h _ n } J \Big ) \geq ( 1 - \epsilon _ n ) \mu ( J ) . \end{align*}


\begin{align*} { \cal I } = 2 A _ - A _ + \left ( \frac { 1 } { 2 \pi i } \int _ { C } \frac { 1 } { \zeta - \zeta _ + } \cdot \frac { \dd \zeta } { \zeta - \zeta _ - } \right ) = \frac { 2 A _ - A _ + } { \zeta _ - - \zeta _ + } = \frac { 2 ( \zeta _ - + 1 ) ( \zeta _ + + 1 ) } { ( \zeta _ + - \zeta _ - ) ^ 3 } \end{align*}


\begin{align*} d ( p , g ) = \frac { d ( p , g ) } { d ( p , q ) } d ( p , q ) \leq \eta \left ( \frac { d ( g , \pi ^ { - 1 } ( y ) ) } { d ( q , \pi ^ { - 1 } ( y ) ) } \right ) r \leq \ell _ \eta r , \end{align*}


\begin{align*} \chi ( S ' _ b ) + \chi ( F ' _ b ) + \chi ( h ( F ' _ b ) ) & \ge \chi ( S ' _ b ) + \chi ( F ' _ b ) + \chi ( h ( F ' _ b ) ) + \chi ( \Sigma ) \\ & = \chi ( S ) + \chi ( F ) + \chi ( h ( F ) ) . \end{align*}


\begin{align*} \epsilon ^ { 1 } \wedge \dots \wedge \epsilon ^ { m } \left ( \xi \right ) = C ^ { 2 } \ , e ^ { 1 } \wedge \dots \wedge e ^ { m } \left ( \xi \right ) \end{align*}


\begin{align*} \bigg | \frac { \partial h _ { z _ i } ( T _ { z _ i } z _ i , T _ { z _ i } z _ j ) } { \partial z _ { i , \hbar } } \bigg | \leq C , \ \ 1 \leq i , j \leq m , \hbar = 1 , 2 , \end{align*}


\begin{align*} \aligned \frac d { d t } \mathcal { Y M } _ e ^ 0 ( \nabla ^ t ) _ { \big { | } _ { s = 0 } } = \frac d { d t } \mathcal { Y M } _ e ^ 0 ( \nabla ^ t ) _ { \big { | } _ { s = 0 } } = \int _ M \langle \delta ^ \nabla \big ( \exp ( \frac 1 2 | | R ^ \nabla | | ^ 2 ) R ^ \nabla \big ) , A \rangle \ , d v \ , . \endaligned \end{align*}


\begin{align*} y = \exp { \left ( \frac { \lambda } { \sigma - \frac { 1 } { 2 } } \right ) } , x = y ^ { - 1 } \log ^ { 2 } { q } , \end{align*}


\begin{align*} A ( q ) = A ( q , s ) = \left ( \prod _ { r = 1 } ^ { q } D _ { \ell _ { r } - 1 } \right ) \left ( \prod _ { r = q + 1 } ^ { s } D _ { \ell _ { r - 1 } } \right ) \end{align*}


\begin{align*} \beta _ i = v s _ { \alpha _ 1 } \cdots s _ { \alpha _ { i - 1 } } ( \alpha _ i ) \in \Phi , i = 1 , \dotsc , k . \end{align*}


\begin{align*} Z ( v , s ) = \Psi ^ q + \mathcal O ( \norm { ( v , s ) } ) . \end{align*}


\begin{align*} \exists j \leq J , \ ; \Pi _ { \mathrm { k e r } A ^ { \perp } } \frac { d ^ j } { d t ^ j } X _ t | _ { t = 0 } \neq 0 . \end{align*}


\begin{align*} 0 = c ( 1 , \gamma ^ - ) = c ( \gamma ^ { - 1 } \gamma , \gamma ^ - ) = c ( \gamma ^ { - 1 } , \gamma \gamma ^ - ) + c ( \gamma , \gamma ^ - ) = c ( \gamma ^ { - 1 } , \gamma ^ - ) + c ( \gamma , \gamma ^ - ) , \end{align*}


\begin{align*} o _ { k } ( n ) = d _ { k } ( n ) . \end{align*}


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


\begin{align*} q _ d ^ * = - ( d - 1 ) + O ( d ) \left ( - \frac { 1 - \log 2 } { 2 } d \right ) , d \to \infty . \end{align*}


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


\begin{align*} g ^ { q p } u _ { , p q i } = - v _ { i } + u _ { , i q } \xi ^ q \end{align*}


\begin{align*} \begin{cases} - \cdot ( K _ H ( x ) \nabla \varPhi ) = \frac { 1 } { \varepsilon ^ 2 } \left ( \varPhi - \left ( \frac { \alpha } { 2 } | x | ^ 2 + \beta \right ) | \ln \varepsilon | \right ) ^ p _ + , \ & x \in B _ { R ^ * } ( 0 ) , \\ \varPhi ( x ) = 0 , \ & x \in \partial B _ { R ^ * } ( 0 ) , \end{cases} \end{align*}


\begin{align*} \tilde c ( t ) : = \begin{cases} c ( 0 ) & ( t = 0 ) \\ c ( 1 - t ) & ( 0 < t < 1 ) . \end{cases} \end{align*}


\begin{align*} H = \mathbb { P } \end{align*}


\begin{align*} \left \{ \begin{array} { l l } i . ~ ~ \phi = 2 \pi N , ~ N ~ i s ~ i n t e g e r \\ i i . ~ ~ \phi = \frac { 4 \pi Z } { N } , \frac { 2 Z } { N } ~ i s ~ n o t ~ i n t e g e r . \\ \end{array} \right . \end{align*}


\begin{align*} \ell ( x _ \infty \cdots \prescript { \sigma ^ { n - 1 } } { } x _ \infty ) = n \ell ( x _ \infty ) . \end{align*}


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


\begin{align*} \langle T f , g \rangle _ \lambda = \langle T ^ { ( \lambda ) } _ a f , g \rangle _ \lambda \end{align*}


\begin{align*} \begin{aligned} & \int _ { \Omega } \nabla n \cdot ( S ( x , n , c ) \cdot \nabla c ) \ , \varphi ^ { 3 } \\ & \le \| S _ { 0 } \| _ { \mathcal { C } ( [ 0 , \gamma ] ) } C _ { 2 } ^ { \frac { 1 } { 4 } } C _ { 3 } ^ { \frac { 1 } { 4 } } \| \nabla c \| _ { L ^ { 2 } ( \Omega \cap B _ { \delta } ) } ^ { \frac { 1 } { 4 } } \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*} & \sup _ { x \in \left ( - \infty , x _ { 0 } \right ) } \left \| \langle x \rangle \mathcal { P } ^ { + } \left ( z ^ { - i } \bar { r } _ 1 ( z ) e ^ { 2 i z x } \right ) \right \| _ { L _ z ^ 2 } \leq \left \| z ^ { - i } \bar r _ 1 ( z ) \right \| _ { H ^ 1 } , i = 0 , 1 , \\ & \sup _ { x \in \left ( - \infty , x _ { 0 } \right ) } \left \| \langle x \rangle \mathcal { P } ^ - \left ( r _ 2 ( z ) e ^ { - 2 i z x } \right ) \right \| _ { L _ { z } ^ 2 } \leq \left \| r _ 2 ( z ) \right \| _ { H ^ 1 } . \end{align*}


\begin{align*} \hat h \left ( A _ \phi \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) = \hat h \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) \end{align*}


\begin{align*} { \frak G } _ x { \bf T } ( t ) = \sum _ { y = 0 } ^ n \int _ 0 ^ { \theta } { { \frak G } _ { x , y } ( s ) } T _ y \left ( t - s \right ) \dd s . \end{align*}


\begin{align*} y \left [ i \right ] = \sum \limits _ { \left ( { l , l ' } \right ) \in { \cal S } } { { \bf { h } } _ l ^ H { { \bf { F } } _ { l ' } } { \bf { d } } \left [ { i - { n _ l } - { \kappa _ { l ' } } } \right ] } + z \left [ { i } \right ] . \end{align*}


\begin{align*} \mathrm { v a r } [ S _ { m } ( y ) - S _ { k } ( x ) ] & = \mathrm { v a r } [ \varphi ^ { D _ { N } , \Delta ^ { k } ( x ) } ( y ) - \varphi ^ { D _ { N } , \Delta ^ { k } ( x ) } ( x ) + \varphi ^ { \Delta ^ { k } ( x ) , \Delta ^ { m } ( x ) } ( y ) ] \\ & \geq \mathrm { v a r } [ \varphi ^ { \Delta ^ { k } ( x ) , \Delta ^ { m } ( x ) } ( y ) ] \geq \mathrm { v a r } [ \varphi ^ { \Delta ^ { m + 1 } ( x ) , \Delta ^ { m } ( x ) } ( y ) ] \geq \gamma + o ( 1 ) . \end{align*}


\begin{align*} D ( X ) = \sum _ { 0 \le i \ne j \le n - 1 } B _ { i j } X ^ { 2 ^ i + 2 ^ j } . \end{align*}


\begin{align*} \varrho ^ \star ( x ) ( \xi ) = \varrho ^ * ( \alpha ( x ) ) \big ( ( \beta ^ { - 2 } ) ^ * ( \xi ) \big ) , \quad \forall \ x \in A , \xi \in V ^ * . \end{align*}


\begin{align*} \begin{array} { l } { \rm S I N R } _ { v , j } ^ q = 2 y _ { v , j } ^ q \sqrt { p _ { v , j } ^ { q } { \left ( { { R _ { \rm { P D } } } { G _ { v , j } ^ { q } } } \right ) } ^ 2 } - y _ { v , j } ^ q \left ( { \sum \limits _ { j ' \ne j } { \sum \limits _ { v ' \ne v } p _ { v ' , j ' } ^ { q } { \left ( { { R _ { \rm { P D } } } { G _ { v ' , j } ^ { q } } } \right ) } ^ 2 } + N _ { \rm V L C } B _ { \rm V L C } } \right ) . \end{array} \end{align*}


\begin{align*} U = U [ C , \beta , \Delta , p ] = \stackrel { d e f } { = } \sup _ { p > \Delta } \ \sup _ { \xi \in L _ p } \ \sup _ { X \in L _ p } Y [ C , \beta , \Delta ] ( \xi , X , p ) , \end{align*}


\begin{align*} \ell : \mathbb { R } \rightarrow \mathbb { R } \ell \left ( r \right ) = c \left ( r - \sin r \right ) ^ { 1 / 3 } \end{align*}


\begin{align*} \frac { \partial ^ 2 \mathbb { E } \left [ \varepsilon | \hat { \mathbf { H } } \right ] } { \partial a _ c \partial a _ c } > 0 ~ { \rm a n d } ~ \frac { \partial ^ 2 \mathbb { E } \left [ \varepsilon | \hat { \mathbf { H } } \right ] } { \partial a _ i \partial a _ i } > 0 , ~ i = 1 , 2 , \ldots , K \end{align*}


\begin{align*} T ^ { ( \lambda ) } _ a | _ W = T = T ^ { ( \lambda ) } _ b | _ W . \end{align*}


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


\begin{align*} Q ^ 2 - B Q - \Bigl [ I _ { n + 1 } \frac { \partial } { \partial x _ n } , Q \Bigr ] + C = 0 , \end{align*}


\begin{align*} \int _ 1 ^ \infty \frac { d s } { \sqrt { F ( s ) } } < \infty \quad \mbox { w h e r e } \ ; F ( s ) = \int _ 0 ^ s f ( t ) d t . \end{align*}


\begin{align*} | A | ^ { m - 1 } \left | P _ { [ u _ 1 , \dots , u _ m ] ^ \bot } A \right | _ { n - m } \le \prod _ { i = 1 } ^ m \left | P _ { u _ i ^ \bot } A \right | _ { n - 1 } . \end{align*}


\begin{align*} F _ { q } ( s ) = \frac { \Gamma ( m ) } { Q _ { m } ( 2 - q ) } \frac { 1 } { s ^ { m } } \ ; , \mbox { f o r } \ ; m \geq 2 . \end{align*}


\begin{align*} f ( z , t ) = ( 2 \pi ) ^ { - n - 1 } \ , \int _ { - \infty } ^ \infty e ^ { i \lambda t } \ , \big ( \sum _ { k = 0 } ^ \infty \rho _ k ^ \lambda ( f ) e ^ { n - 1 } _ { k , \lambda } ( z , 0 ) \big ) | \lambda | ^ n d \lambda . \end{align*}


\begin{align*} K ( c _ 1 , \dots , c _ n , 2 m ) = & p _ j \circ H ( c _ 1 , \dots , c _ n , m ) \\ = & p _ j \circ i _ { U _ k } ( c 1 , \dots , c _ n ) = p _ j ( c _ 1 , \dots , c _ n ) . \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty z ^ i \ , \phi _ s ( z ) \ , [ \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \ , \mathrm { d } z = \sum _ { k = 0 } ^ \infty \left [ v _ k ( j + \alpha - 1 ) + ( - 1 ) ^ { i + k } \ , \binom { j + \alpha - 1 } { k } \right ] \ , J _ { i , k } ^ { ( s ) } . \end{align*}


\begin{align*} \varphi _ 0 \eta _ 0 + \ldots + \varphi _ p \eta _ p = 0 , \end{align*}


\begin{align*} \Phi : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 } \rtimes S O _ { 3 } \Phi \left ( r u \right ) = \left ( \ell \left ( r \right ) u , \exp \left ( C _ { r u } \right ) \right ) \end{align*}


\begin{align*} f ( x - T ( x ) ) = - f ( - x - T ( x ) ) , \end{align*}


\begin{align*} u ^ { \prime \prime } + B u ^ \prime ( t ) + A ( u ) = f ( t ) \end{align*}


\begin{align*} \zeta _ k = \frac { 1 } { k } \sum _ { j = 0 } ^ { k - 1 } \delta _ { u _ j } , \end{align*}


\begin{align*} \hat \Phi \left ( \begin{pmatrix} a \\ b \end{pmatrix} , s \right ) = \hat h ( ( a , b ) ^ \top , s ) \begin{pmatrix} a \\ b \end{pmatrix} , \end{align*}


\begin{align*} \sup _ { Q \in \P ( \R ^ n ) } \left ( \int _ { \R ^ n } f \ , d Q - H ( Q ) \right ) = \log Z \in ( - \infty , \infty ) , \end{align*}


\begin{align*} d _ { \mathcal { F } } ( g _ 1 , g _ 2 ) : = \frac 1 2 \left ( d ( g _ 1 , \mathcal { F } _ { g _ 2 } ) + d ( g _ 2 , \mathcal { F } _ { g _ 1 } ) \right ) , \mbox { f o r a l l } g _ 1 , g _ 2 \in X , \end{align*}


\begin{align*} I : = \{ \beta \neq \gamma \in \Phi ^ + _ J \mid s _ \beta ( \gamma ) \notin \Phi ^ + _ J \} . \end{align*}


\begin{align*} 2 ^ { \star } : = \frac { n - 1 } { n - 2 } 2 \end{align*}


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


\begin{align*} 0 = R _ { 4 1 } ^ \delta - R _ { 6 3 } ^ \delta = h ^ 2 + \varepsilon _ 2 \delta _ 2 \left ( \lambda - \rho \right ) . \end{align*}


\begin{align*} \hat { c } _ { \Delta } ( \eta , \xi _ { \Delta } ) : = c _ { \Delta } ( \xi _ { \Delta } \eta _ { \Delta ^ c } , \eta _ { \Delta } ) \frac { \gamma _ { \Delta } ( \xi _ { \Delta } | \eta _ { \Delta ^ c } ) } { \gamma _ { \Delta } ( \eta _ { \Delta } | \eta _ { \Delta ^ c } ) } . \end{align*}


\begin{align*} X ^ \Delta ( t _ { n + 1 } ) = X ^ \Delta ( t _ { n } ) + F ( X ^ \Delta ( t _ { n } ) ) \Delta + \Delta ^ { 1 / 2 } G ( X ^ \Delta ( t _ { n } ) ) \xi _ { n + 1 } . \end{align*}


\begin{align*} \left \{ \begin{array} { l r } \partial _ u \tau = 2 \Gamma _ { u v } ^ v \tau ( 1 - \tau ) \\ \partial _ v \tau = 2 \Gamma _ { v u } ^ u ( 1 - \tau ) , \end{array} \right . \end{align*}


\begin{align*} \theta ^ h = 2 ~ { \rm a n d } ~ q \ge h \log _ 2 ( 1 + d 2 ^ { b _ 0 + 1 } ) . \end{align*}


\begin{align*} G ( x ) = - \frac { 2 ( a x ^ 2 + b x + c ) } { 2 a x + b } , x \ne - b / ( 2 a ) , \end{align*}


\begin{align*} V \widehat { \otimes } _ K W = \varprojlim _ { n , m } V _ { n } \widehat { \otimes } _ K W _ m . \end{align*}


\begin{align*} \begin{aligned} & \sum _ { 0 { \le } m _ 1 { \le } \cdots { \le } m _ n < \infty } \frac { ( \alpha ) _ { m _ 1 } } { { m _ 1 } ! } \frac { ( \beta ) _ { m _ 1 } } { ( \gamma ) _ { m _ 1 } } \frac { { m _ n } ! } { ( \alpha ) _ { m _ n } } \frac { ( \gamma ) _ { m _ { n } } } { ( \beta ) _ { m _ n } } \\ & \times \left \{ \prod _ { i = 1 } ^ { n } \frac { 1 } { ( m _ i + \alpha ) ^ { a _ i } ( m _ i + \beta ) ^ { b _ i } ( m _ i + \gamma ) ^ { c _ i } } \right \} , \end{aligned} \end{align*}


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


\begin{align*} { S ^ { \rm { O } } } = \frac { { { W _ s } I } } { { K L } } { \widetilde \varepsilon _ K } \left ( { { \gamma ^ { \rm { O } } } } \right ) . \end{align*}


\begin{align*} \begin{cases} \frac { d } { d t } X _ t = B ( X _ t , X _ t ) \\ X _ t | _ { t = 0 } = x _ 0 \end{cases} \end{align*}


\begin{align*} \Phi _ + = \{ e _ j - e _ k \mid 1 \leq j < k \leq n \} . \end{align*}


\begin{align*} \left | \Pi _ V e ^ { | z | J ^ \perp t } \tilde { Y } _ 0 \right | ^ 2 & = \left | e ^ { | z | J ^ \perp t } \Pi _ { V } \tilde { Y } _ 0 \right | ^ 2 = e ^ { 2 | z | \lambda _ R t } \left | \Pi _ { V } \tilde { Y } _ 0 \right | ^ 2 . \end{align*}


\begin{align*} \partial _ s \mathcal P ( x , y , s ) = - \frac { \mathcal { Q } ( x , y , \cos s ) } { 2 \sin ^ 2 s } , \end{align*}


\begin{align*} \Delta ^ 2 u = f \qquad & D , \\ [ 1 m m ] u = \frac { \partial u } { \partial \nu } = 0 & \partial D . \end{align*}


\begin{align*} \begin{aligned} \abs { \omega ( f ( a ) ) g ( a ) - \omega ( f ( b ) ) g ( b ) } & \leq \abs { \omega ( f ( a ) ) - \omega ( f ( b ) ) } \abs { g ( a ) } + \abs { \omega ( f ( b ) ) } \abs { g ( a ) - g ( b ) } \\ & \leq \norm { \omega \circ f } _ L d ( a , b ) \norm { g } _ L d ( 0 , a ) + 1 \cdot \norm { g } _ L d ( a , b ) \\ & \leq C \norm { g } _ L d ( a , b ) \end{aligned} \end{align*}


\begin{gather*} \mathcal { U } = \bigl \{ Q \in \mathbb { P } : Q \ll P ^ * ( \pi _ 1 , \ldots , \pi _ n ) Q \bigr \} . \end{gather*}


\begin{align*} \mathcal { R } _ j : = \widetilde { u } _ { \mathrm { a p } , j } - u _ { \mathrm { a p } , j } . \end{align*}


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


\begin{align*} M ( \lambda _ 1 / \lambda _ 2 ) = M ( \lambda _ 2 / \lambda _ 1 ) = | \lambda _ 1 \lambda _ 2 | \max ( 1 , | \lambda _ 1 / \lambda _ 2 | ) \max ( 1 , | \lambda _ 2 / \lambda _ 1 | ) = | \lambda _ 1 | ^ 2 . \end{align*}


\begin{align*} \int _ \Omega \Phi ( x ) \ , | \operatorname { g r a d } f ( x ) | \ , d V ( x ) = \int _ 0 ^ \infty d s \int _ { \Omega \cap \{ | f | = s \} } \Phi ( x ) \ , d S ( x ) , \end{align*}


\begin{align*} ( r _ m \eta ) _ x = \begin{cases} 1 , & \abs { x } _ { \infty } < m , \\ \ \eta _ x & \abs { x } _ { \infty } \geq m . \end{cases} \end{align*}


\begin{align*} \begin{aligned} \varphi & = S ( t ) \varphi _ 0 + \int _ { 0 } ^ { t } S ( t - s ) \mathbb { P } \nabla \cdot F ( s ) \dd s . \end{aligned} \end{align*}


\begin{align*} & \partial _ t Q ( t ) ^ { - 1 } P ^ { - 1 } \begin{pmatrix} w \\ \overline { w } \end{pmatrix} = Q ( t ) ^ { - 1 } P ^ { - 1 } \begin{pmatrix} S _ 2 ( t ) + e ^ { 3 i \lambda _ 1 ( \Phi _ 1 - \theta _ 1 ) } R _ 2 ( t ) \\ \overline { S _ 2 ( t ) + e ^ { 3 i \lambda _ 1 ( \Phi _ 1 - \theta _ 1 ) } R _ 2 ( t ) } \end{pmatrix} . \end{align*}


\begin{align*} \int _ 0 ^ \infty K _ \mu ( \alpha x ) K _ \nu ( \beta x ) x ^ { s - 1 } d x & = 2 ^ { s - 3 } \alpha ^ { - s - \nu } \beta ^ \nu \Gamma \left ( \frac { 1 } { 2 } ( s + \mu + \nu ) \right ) \Gamma \left ( \frac { 1 } { 2 } ( s - \mu - \nu ) \right ) \\ & \int _ 0 ^ 1 t ^ { \frac { s - \mu + \nu } { 2 } - 1 } ( 1 - t ) ^ { \frac { s + \mu - \nu } { 2 } - 1 } \left ( 1 - \left ( 1 - \frac { \beta ^ 2 } { \alpha ^ 2 } \right ) t \right ) ^ { - \frac { s + \mu + \nu } { 2 } } d t . \end{align*}


\begin{align*} V ( L , K [ n - 1 ] ) = \frac { 1 } { n } \int _ { S ^ { n - 1 } } h _ L ( u ) d S _ K ( u ) , \end{align*}


\begin{align*} V _ x ( t ) = \sum _ { x ' = 1 } ^ n \int _ 0 ^ \theta { \frak g } _ { x , x ' } ( s ) V _ { x ' } ( t - s ) \dd s + v _ x ( t ) . \end{align*}


\begin{align*} \tilde { L } _ { \lambda } ( E ) = L _ { \lambda } ( E ) - \ln { 2 } . \end{align*}


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


\begin{align*} | | \mathcal { A } _ q ^ p | | ^ 2 _ { L ^ 2 ( \mathbb R , \mathbb { H } ) } = E _ q \left ( | p | ^ 2 \right ) . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ { 3 } } A _ { t } \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x = - \mu \int _ { \mathbb { R } ^ { 3 } } \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x . \end{align*}


\begin{align*} Z ( v , s ) & = \Psi ^ q + v + \psi _ v ( 0 , p _ 0 ) v + s \psi _ p ( 0 , p _ 0 ) p ' ( 0 ) + \mathcal O ( \norm { ( v , s ) } ^ 2 ) \\ & = \Psi ^ q + v + \mathcal O ( \norm { ( v , s ) } ^ 2 ) , \end{align*}


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


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


\begin{align*} \psi _ { \Delta , \beta } ( p ) = \psi _ { \Delta , \ \beta } [ \psi ] ( p ) \stackrel { d e f } { = } \frac { p } { p - \Delta } \ \beta ^ p \ \psi ( p ) , \ \Delta < p \le b , \ \beta > 1 . \end{align*}


\begin{align*} F _ 1 = \frac { 1 } { \sqrt { | c _ 1 | } } \ \langle \langle \nu _ 1 \cdot \varphi , \varphi \rangle \rangle . \end{align*}


\begin{align*} \Gamma \left ( X , Y , Z \right ) = C \left ( X , J Y , Z \right ) + g \left ( X , \left ( \nabla _ { Z } J \right ) Y \right ) . \end{align*}


\begin{align*} \forall \Delta \Subset \Z ^ d \ \forall \xi _ { \Delta } : \gamma _ { \Delta } ( \xi _ { \Delta } \lvert \eta _ { \Delta ^ c } ) = \nu ( \xi _ { \Delta } \lvert \eta _ { \Delta ^ c } ) \eta \in \Omega , \end{align*}


\begin{align*} d _ 1 K _ { m , \epsilon } = \frac { \kappa _ m d _ 1 | q | ^ 2 } { ( | q | ^ 2 + \epsilon ) ^ { \kappa _ m + 1 } } , \end{align*}


\begin{align*} Q _ { j , - } ( x ; k ) = \mathcal { P } ^ - \left ( Q _ { j , - } ( x ; k ) J + D _ j \right ) ( z ) , z \in \mathbb { R } , \end{align*}


\begin{align*} f ( t , q ^ { \prime } ) = \displaystyle \sum _ { n = 0 } ^ { \infty } \left ( - \frac { q ^ { \prime } } { 1 - q ^ { \prime } } \right ) _ { 2 n } ( 1 - q ^ { \prime } ) ^ { 2 n } \frac { ( \alpha t ) ^ { 2 n + 1 } } { ( 2 n + 1 ) ! } = \sinh _ { q ^ { \prime } } \left ( \alpha t \right ) . \end{align*}


\begin{align*} \left ( \rho ^ \star \right ) ^ { s y m } - \left ( R i c ^ g \right ) ^ { J , + } = \frac { s ^ \star - s ^ g } { 4 } g . \end{align*}


\begin{align*} 2 \hat \beta ( l _ n , \xi ) - 2 \hat \beta ( l _ m , \xi ) = h ( l _ n \xi , l _ n \eta , l _ n \omega ) - h ( l _ m \xi , l _ m \eta , l _ m \omega ) . \end{align*}


\begin{align*} \left \langle Z _ { \kappa } \left ( x , \xi \right ) , Z _ { \kappa } \left ( y , \eta \right ) \right \rangle = \tfrac { 1 } { 4 } \left ( \left \langle x , \eta \right \rangle + \left \langle y , \xi \right \rangle \right ) \end{align*}


\begin{align*} \Pi _ { \mathrm { k e r } A } ( B ( y , z ) + B ( z , y ) ) = 0 \forall z \in \mathrm { k e r } A , y \in \mathrm { k e r } A ^ \perp \end{align*}


\begin{align*} \mu \left ( X \backslash \bigcup _ { k = 1 } ^ n \mathcal { R } _ k \right ) = 0 . \end{align*}


\begin{align*} \frac { d \psi _ i } { d t } = \Psi ^ { n } _ i ( \psi ) , \end{align*}


\begin{align*} \sup \limits _ i \int _ K | \tilde { F } _ i - ( 1 - b _ { t _ 0 , B } ( \Psi ) ) f F ^ { 1 + \delta } | ^ 2 _ h < + \infty . \end{align*}


\begin{align*} M = \sup \{ N _ { n } : n = 1 , 2 , 3 , \ldots \} < \infty . \end{align*}


\begin{align*} g ( x ) = \lim _ { n \rightarrow \infty } 2 ^ { n } f ( 2 ^ { n } x ) \end{align*}


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


\begin{align*} H ( \mu ) = \int _ { \R ^ m } h ( y ) \mu ( d y ) , \end{align*}


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


\begin{align*} \sum _ { k = 1 } ^ { \infty } e ^ { - t \tau _ k } = \int _ { \partial \Omega } \bigg [ \frac { 1 } { ( 2 \pi ) ^ { n - 1 } } \int _ { \mathbb { R } ^ { n - 1 } } \bigg ( \frac { i } { 2 \pi } \int _ { \mathcal { C } } e ^ { - t \tau } \sum _ { j \leqslant - 1 } \operatorname { T r } \phi _ { j } \ , d \tau \bigg ) \ , d \xi \bigg ] \ , d S . \end{align*}


\begin{align*} \delta _ { m , n } & = \frac { m } { n } \bigg ( 1 + \sqrt { n - 1 } \sqrt { \frac { n - m } { m } } \bigg ) < \frac { m } { n } \sqrt { 1 + n - 1 } \sqrt { 1 + \frac { n - m } { m } } = \sqrt { m } . \end{align*}


\begin{align*} \left \{ \begin{aligned} \Delta u & = v & & \quad \mbox { i n } \Omega , \\ \Delta v & = | \nabla u | ^ 2 & & \quad \mbox { i n } \Omega , \end{aligned} \right . \end{align*}


\begin{align*} \int _ { W ^ d } ( f _ i ( x _ j ) ) _ { i , j = 1 } ^ d ( g _ i ( x _ j ) ) _ { i , j = 1 } ^ d \prod _ { i = 1 } ^ d \nu ( d x _ i ) = \left ( \int _ { \mathbb { R } } f _ i ( x ) g _ j ( x ) \nu ( d x ) \right ) _ { i , j = 1 } ^ d . \end{align*}


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


\begin{align*} ( \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*} & ( r + \nu ) v ( s , i ) - H \left ( s , i , \partial _ s v ( s , i ) , \partial _ i v ( s , i ) \right ) = 0 , \\ - & ( r + \nu ) v ( s , i ) + H \left ( s , i , \partial _ s v ( s , i ) , \partial _ i v ( s , i ) \right ) = 0 , \end{align*}


\begin{align*} \hat { p } ^ i _ { j k } = \frac { \hat { p } ^ i _ j ( B ) \hat { p } ^ i _ k ( C ) } { \hat { p } ^ i ( D ) } . \end{align*}


\begin{align*} \langle v , w \rangle : = \sum _ { i \in Q _ 0 } v _ i w _ i - \sum _ { h \in Q _ 1 } v _ { h ' } w _ { h '' } \ , \ , v , w \in \mathbb { Z } ^ n . \end{align*}


\begin{align*} \frac { 1 } { t } \int _ 0 ^ t \bigg | \int _ { \mathbb { R } } e ^ { - 2 \pi i s x } d \mu _ { \delta _ 0 } ^ { \triangle } ( x ) \bigg | ^ 2 d s = O ( \log ( t ) / t ) ; \end{align*}


\begin{align*} V _ { \mathrm { o r i g } } = \sup _ { Q \in \P ( \R ^ n ) } \left ( \int _ { \R ^ n } g \ , d Q - \frac { 1 } { n } H ( Q \ , | \ , \gamma _ T ) \right ) = \frac { 1 } { n } \log \int _ { \R ^ n } e ^ { n g } \ , d \gamma _ T , \end{align*}


\begin{align*} \sup _ { y \in \Z ^ d } \sum _ { \Delta \ni y } \sum _ { z \neq y } \sum _ { \xi _ { \Delta } } \sum _ { i = 1 } ^ q \norm { \nabla ^ i _ z \hat { c } _ \Delta ( \cdot , \xi _ \Delta ) } _ \infty < \infty . \end{align*}


\begin{align*} F ^ q ( v , p ) = 0 \end{align*}


\begin{align*} ( \delta ^ \nabla \rho ) ( X _ 1 ) = - \sum _ i ( \nabla _ { e _ i } \rho ) ( e _ i , X _ 1 ) \ , . \end{align*}


\begin{align*} \begin{cases} - \Delta h _ { \hat { x } } ( x , y ) = 0 , \ \ & x \in T _ { \hat { x } } ( \Omega ) , \\ h _ { \hat { x } } ( x , y ) = \frac { 1 } { 2 \pi } \ln \frac { 1 } { | x - y | } , \ \ & x \in \partial T _ { \hat { x } } ( \Omega ) . \end{cases} \end{align*}


\begin{align*} h ( l \xi , l \eta , l \omega ) - h ( \xi , \eta , \omega ) = 2 \hat \beta ( l , \xi ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } ( \eta \gamma ^ n ) ^ + = \eta \gamma ^ + . \end{align*}


\begin{align*} 0 \rightarrow E \rightarrow \bigoplus _ { j = 1 } ^ n E _ j \rightarrow \mathcal { T } _ E \rightarrow 0 . \end{align*}


\begin{align*} \frac { 1 } { ( M ) } \mathcal { Y M } _ e ^ 0 ( { \nabla } ) = \exp ( \frac { | | R ^ \nabla | | ^ 2 } { 2 } ) - 1 = \exp \bigg ( \frac { 1 } { ( M ) } \mathcal { Y M } ( \nabla ) \bigg ) - 1 \end{align*}


\begin{align*} P _ { \min } ^ { \rm { N } } \left ( { \overline S , \overline \varepsilon , \overline R } \right ) = { p _ K } \left ( { \overline S , \overline \varepsilon , W } \right ) + \frac { { { p _ K } \left ( { \overline S , \overline \varepsilon , W } \right ) { { \left | { { { \widetilde h } _ b } } \right | } ^ 2 } + W { N _ 0 } } } { { { { \left | { { { \widetilde h } _ b } } \right | } ^ 2 } } } \left ( { { 2 ^ { \frac { { \overline R } } { W } } } - 1 } \right ) . \end{align*}


\begin{align*} \ell ( x ) = \sum _ { \alpha \in \Phi } \max ( 0 , \ell ( x , \alpha ) ) . \end{align*}


\begin{align*} \theta _ { i , ( 1 ) } ( v _ { \alpha , k } ) = - \sum _ { j = 1 } ^ d a _ { i , j } v _ { \alpha + 1 _ j , k } , \end{align*}


\begin{align*} [ x , y ] = x \cdot D ( y ) - \varepsilon ( x , y ) y \cdot D ( x ) . \end{align*}


\begin{align*} f ( A ) = \begin{cases} 1 , & s < - \frac d 2 , \\ ( \log A ) ^ \frac 1 2 , & s = - \frac d 2 , \\ A ^ { \frac { d } { 2 } + s } , & s > - \frac d 2 . \end{cases} \end{align*}


\begin{align*} \Phi ( v , p ) = 0 , \end{align*}


\begin{align*} \partial ^ \square H = \Delta _ \Gamma V + V \big | \nabla _ \Gamma \nu \big | ^ 2 \end{align*}


\begin{align*} ( p _ n ^ { - 1 } \cdot w _ n ) _ h = \left ( \frac { 1 } { t } - 1 \right ) ( x _ { p _ n } , y _ { p _ n } ) , \end{align*}


\begin{align*} X _ i = 0 , \omega _ { i j } V _ i V _ j = \tfrac { 1 } { 2 } h _ { i j } V _ i V _ j , \omega _ { i a } V _ i = h _ { i a } V _ i , \omega _ { a b } = 0 \end{align*}


\begin{align*} \sum _ { m _ 1 , \ldots , m _ n = 0 } ^ { \infty } \lambda _ { m _ 1 , \ldots , m _ n } a _ 1 ^ { m _ 1 } \cdots a _ n ^ { m _ n } \end{align*}


\begin{align*} I _ \varepsilon ( u _ \varepsilon ) = c _ \varepsilon , \ \ I ' _ \varepsilon ( u _ \varepsilon ) = 0 . \end{align*}


\begin{align*} d _ H ( \xi , w _ n ) = | \xi ^ { - 1 } \cdot w _ n | _ G \to 0 , \end{align*}


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


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


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


\begin{align*} \left \{ \begin{array} { l } g ( 0 ) = e \\ \frac { d g ( t ) } { d t } g ( t ) ^ { - 1 } = v ( t ) \end{array} \right . \end{align*}


\begin{align*} \begin{aligned} { \Gamma } _ m = \{ \mathcal { H } _ 1 ( A ) > 0 \} \cap \cdots \{ \mathcal { H } _ m ( A ) > 0 \} . \end{aligned} \end{align*}


\begin{align*} k ^ + = k _ { \mathcal { W } , g } ^ + & = \min \{ k \mid 0 \le k \le l _ S ( g ) \sigma ( k ) = \sigma ^ + \} , \\ k ^ - = k _ { \mathcal { W } , g } ^ - & = \min \{ k \mid 0 \le k \le l _ S ( g ) \sigma ( k ) = \sigma ^ - \} \end{align*}


\begin{align*} ( \mathbf q , \mathbf p ) = ( q _ 0 , \dots , q _ n , p _ 0 , \dots , p _ n ) \in \R ^ { n + 1 } \times \R ^ { n + 1 } . \end{align*}


\begin{align*} & m = \int _ { - \infty } ^ x e ^ { 2 i z ( x - y ) } w ( y ) d y , w ( x ) = - \partial _ x ( u _ x e ^ { i c _ - ( x ) } ) , \\ & n = \int _ { - \infty } ^ x e ^ { - 2 i z ( x - y ) } u _ y ( y ) e ^ { i c _ - ( x ) } d y . \end{align*}


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


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


\begin{align*} X ( t ) & = x + \int _ 0 ^ t A ( s , \overline { X } ( s ) ) d s + \int _ 0 ^ t B ( s , \overline { X } ( s ) ) d W ( s ) \\ & \quad \ + \int _ 0 ^ t \int _ { \{ | z | < 1 \} } H ( s , \overline { X } ( s ) , z ) \widetilde { N } ( d s , d z ) \\ & \quad \ + \int _ 0 ^ t \int _ { \{ | z | \ge 1 \} } J ( s , \overline { X } ( s ) , z ) N ( d s , d z ) . \end{align*}


\begin{align*} R \cdot R ( x , J x , J x , x ; u , J u ) & = - R \cdot R ( J x , x , u , J u ; x , J x ) - R \cdot R ( u , J u , x , J x ; J x , x ) \\ & = 2 \ , R \cdot R ( u , J u , x , J x ; x , J x ) . \end{align*}


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


\begin{align*} x x ' + x y ' + y y ' = 0 \ , . \end{align*}


\begin{align*} l _ { x y } ( i ) = c _ { x y } \lambda _ i \prod _ { z \in \partial x \setminus \{ y \} } \sum _ { j \in \Z } a _ { i j } l _ { z x } ( j ) \end{align*}


\begin{align*} R M S E = \sqrt { \frac { 1 } { L } \sum \limits _ { l = 1 } ^ { L } ( { \hat \theta _ { l } - \theta _ 0 } ) ^ 2 } \end{align*}


\begin{align*} S \textbf { \textit { u } } : = \nabla \textbf { \textit { u } } + \nabla \textbf { \textit { u } } ^ t , \textbf { \textit { u } } \in ( C ^ 1 ( \Omega ) ) ^ n , \end{align*}


\begin{align*} \left \{ \begin{aligned} \mathcal P _ X ^ D ( \alpha _ 1 , \dots , \alpha _ r ) _ x & = \mathcal P _ { X , x } & \\ \mathcal P _ X ^ D ( \alpha _ 1 , \dots , \alpha _ r ) _ x & = \alpha _ i & \end{aligned} \right . \end{align*}


\begin{align*} r _ { \ell - c } = s _ i \end{align*}


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


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


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


\begin{align*} [ \beta ^ t _ { a , b , n } \ , , \ , \gamma ^ t _ { a , b , m } ] = \delta _ { n + m , 0 } \end{align*}


\begin{align*} a s ^ { l } _ A ( x , y , z ) = ( x \cdot y ) \diamond \alpha ( z ) - \alpha ( x ) \diamond ( y \cdot z ) = 0 . \end{align*}


\begin{align*} J ( q ) = \int _ { 0 } ^ { I _ { \Phi \left ( \frac { q - \mu } { \sigma } \right ) } ( \alpha , \beta ) } \ , Q ( u ) \mathrm { d } u = \sum _ { i = 0 } ^ { \infty } \frac { f _ i \ , \ , \ , I _ { \Phi \left ( \frac { q - \mu } { \sigma } \right ) } ( \alpha , \beta ) ^ { ( i / \alpha + 1 ) } } { i / \alpha + 1 } . \end{align*}


\begin{align*} \pi \circ \phi = \mbox { i d } _ Y , \end{align*}


\begin{align*} B ( s , 2 ^ { - \beta } | x | ^ \beta ) \cap B ( s ' , 2 ^ { - \beta } | x | ^ \beta ) = \emptyset \quad \ s , s ' \in S \cap B ( x , r ) . \end{align*}


\begin{align*} f _ n ( t , z , x ) : = \sup _ { \| q \| \le n } \Big ( q \cdot z - g ( t , q , x ) \Big ) . \end{align*}


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


\begin{align*} A _ { i j } = \int _ { Y ^ { \ast } } \big ( \nabla \chi _ i + e _ i \big ) \cdot \big ( \nabla \chi _ j + e _ j \big ) d y , \end{align*}


\begin{align*} \psi ^ 0 ( t ) + \sum _ { j \ge 4 } \big ( \psi ( 2 ^ j t ) + \psi ( - 2 ^ j t ) + \psi ( 2 ^ j ( t + \pi ) ) + \psi ( 2 ^ j ( \pi - t ) ) \big ) = 1 \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*} Z _ { q } ( \beta ) = \frac { 1 } { \left ( \hbar \omega \right ) ^ { D N } } \left ( \frac { \Gamma \left ( \frac { 1 } { 1 - q } + 1 \right ) } { ( 1 - q ) ^ { D N } \ ; \Gamma \left ( \frac { 1 } { 1 - q } + D N + 1 \right ) } \right ) \frac { 1 } { \beta ^ { D N } } . \end{align*}


\begin{align*} d _ e = \sup _ { x \in M } \frac { \exp \big ( \frac { | | R ^ \nabla | | ^ 2 } 2 ( x ) \big ) } { \exp \big ( \frac { | | R ^ \nabla | | ^ 2 } 2 ( x ) \big ) - 1 } \ , . \end{align*}


\begin{align*} \Upsilon ^ { ( i - 1 ) } = \Delta ^ { ( i - 1 ) } , b _ { i - 1 } ( \Upsilon ) = b _ { i - 1 } ( \Delta ) , b _ { i } ( \Upsilon ) = 0 . \end{align*}


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


\begin{align*} | \delta ( \zeta , \gamma ^ + ) | = | \delta ( \zeta \theta ^ { - 1 } \theta , \gamma ^ + ) | = | \delta ( \zeta \theta ^ { - 1 } , \theta \gamma ^ + ) + \delta ( \theta , \gamma ^ + ) | \leq 4 M . \end{align*}


\begin{align*} & \mathcal { P } ^ + \left ( f ( z ) e ^ { - 2 i z x } \right ) = \int _ { 2 x } ^ { + \infty } \widehat { f } ( \xi ) e ^ { i z ( \xi - 2 x ) } \mathrm { d } \xi . \end{align*}


\begin{align*} \lambda _ x | _ { E _ { 2 , m } } = \lambda _ y , \lambda _ x | _ { E _ { 2 , m + e _ i - e _ j } } = \lambda ^ j \lambda _ x | _ { E _ { 2 , [ m - e _ j , m + e _ i ] } } = \phi ^ j \end{align*}


\begin{align*} g \big ( c ( t , x ) \big ) \begin{cases} \leq g \big ( c _ 0 ( x ) \big ) + \tilde { \varepsilon } , \\ \geq g \big ( c _ 0 ( x ) \big ) - \tilde { \varepsilon } \end{cases} \end{align*}


\begin{align*} \sigma _ 0 = \min \{ \frac { p _ 0 } { \nu _ 0 } , q _ 0 \} . \end{align*}


\begin{align*} \textsf { E } _ i s ^ { Z ( n ) } : = \sum _ { j \in { \mathcal { S } } } { P _ { i j } ( n ) s ^ j } = \bigl [ { f _ n ( s ) } \bigr ] ^ i , \end{align*}


\begin{align*} \tau ( x ) = \dfrac { \gamma } { 2 } \int _ { - \infty } ^ 0 e ^ { \gamma \tau } a _ { 0 , \delta } ( \hat { x } _ a ( \tau , x ) ) \ , d \tau . \end{align*}


\begin{align*} \tau _ { n , H } ^ M : = \inf \{ t \geq 0 : \Vert Y _ n ( t ) \Vert _ H > M \} \wedge T . \end{align*}


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


\begin{align*} \begin{aligned} \partial _ x \left ( e ^ { 2 i ( c _ - ( x ) + c ) } u ( x ) \right ) & = I _ 1 ' ( x ) + I _ 2 ' ( x ) . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} & | h ^ { \epsilon } ( t ) | \leq B \left ( \rho ^ { 2 } + | Y ^ { \epsilon } ( t - ) | ^ { 2 } \right ) ^ { \frac { 1 } { 2 } } , \\ & | \gamma ^ { \epsilon } ( t ) | \leq M \left ( \rho ^ { 2 } + | Y ^ { \epsilon } ( t - ) | ^ { 2 } \right ) ^ { \frac { 1 } { 2 } } . \end{aligned} \end{align*}


\begin{align*} \widetilde { W } & = ( \beta _ 1 , \dots , \beta _ N ) , \\ \Phi ( t ) ( v ) & = \sum _ { j = 1 } ^ N g ( y ) \mu _ j e _ j \langle v , f _ j \rangle , \ v \in \mathbb { R } ^ N , \\ ( f _ j ) _ { 1 \le j \le N } & : \ \mathbb { R } ^ N . \end{align*}


\begin{align*} \left ( \nabla _ { x } b \right ) _ { v } = C _ { x ^ { b } } \circ b \left ( v \right ) \end{align*}


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


\begin{align*} \langle \lambda , \lambda ' \rangle : = \sum _ { \alpha = 1 } ^ N \lambda _ { \alpha } \lambda ' _ { \alpha } , \quad \ , \lambda = ( \lambda _ { \alpha } ) _ { 1 \le \alpha \le N } , \ , \ , \ , \lambda ' = ( \lambda ' _ { \alpha } ) _ { 1 \le \alpha \le N } \in \mathbb { C } ^ N . \end{align*}


\begin{align*} \mu ' ( d x ) = e ^ { - \lambda } \delta _ \gamma ( d x ) + ( 1 - e ^ { - \lambda } ) f ( x - \gamma ) d x , \gamma \in \R , \end{align*}


\begin{align*} | r ^ { N , i } ( t , \boldsymbol { x } ) | \leq \frac { C } { N } \Big ( 1 + \frac { 1 } { N } \underset { j = 1 } { \overset { N } \sum } | x ^ i - x ^ j | \Big ) . \end{align*}


\begin{align*} | t _ n | _ p = 1 . \end{align*}


\begin{align*} w s _ \alpha ( - w ^ { - 1 } \gamma ) = - ( w s _ \alpha w ^ { - 1 } ) \gamma = - s _ { w \alpha } ( \gamma ) = - ( \gamma + w \alpha ) \in \Phi ^ - . \end{align*}


\begin{align*} a = \int _ { 0 } ^ 1 ( e ^ { \gamma x } - 1 ) x g ( x ) d x , b = 0 \nu ( d x ) = { \bf 1 } _ { \{ 0 < x \le c _ 1 \} } e ^ { \gamma x } g ( x ) d x . \end{align*}


\begin{align*} \mathbf { Q } _ { 0 , n } f & = \frac { q t } { q t - 1 } h _ n [ ( 1 - q t ) X / ( q t ) ] \cdot f \\ \mathbf { Q } _ { 1 , n } f & = \mathbf { D } _ n f . \end{align*}


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


\begin{align*} l \sum h _ { k } \nabla _ { k } \chi & = l \sum h _ { k } \left [ \frac { - f _ { k } } { f } + \frac { h _ { k } } { h } + \frac { \varphi ^ { ' } } { \varphi } h _ { k } + \frac { ( \nabla G \cdot e _ { i } ) \cdot w _ { k i } } { G } \right ] \\ & \leq C _ { 1 } l + l \frac { \nabla G \cdot e _ { k } } { G } h _ { k } w _ { k k } = C _ { 1 } l + l \frac { \nabla G \cdot e _ { k } } { G } \rho \rho _ { k } \leq C _ { 2 } l , \end{align*}


\begin{align*} k < & \Big ( \Pi _ { i = j } ^ { n - 1 } \big ( r _ i + 1 \big ) \Big ) h _ j , \end{align*}


\begin{align*} Q _ { j , - } - \mathcal { P } ^ - \left ( Q _ { j , - } J \right ) = \mathcal { P } ^ - ( D _ j ) , Q _ { j , + } - \mathcal { P } ^ - \left ( Q _ { j , - } J \right ) = \mathcal { P } ^ + ( D _ j ) . \end{align*}


\begin{align*} u _ \ast ( \eta _ 0 ) \leq \varphi ( p _ 0 ) = u _ \ast ( p _ 0 ) , \end{align*}


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


\begin{align*} c _ k = \frac 1 2 \sum _ { \substack { i + j = k \\ 0 \leq i , j \leq N } } ( i - j ) ^ 2 . \end{align*}


\begin{align*} \abs { a ^ \textrm { a p p } ( s ( \tau ) ) - a ( \tau ) } = \abs { \sqrt { \frac { - \gamma _ 2 s ( \tau ) } { \gamma _ 1 } } - \tau } = \mathcal O ( \tau ^ 3 ) = \mathcal O ( s ^ \frac 3 2 ) . \end{align*}


\begin{align*} \frac \varpi \pi x ' = \varphi _ M ^ { - 1 } ( \frac { \varpi ^ p } \pi x ' ) . \end{align*}


\begin{align*} \lim _ { n \to \infty } \mu ( T ^ { m _ n + j } A \cap B ) = \lim _ { n \to \infty } \mu ( T ^ { m _ n } A \cap T ^ { - j } B ) = \mu ( A ) \mu ( T ^ { - j } B ) = \mu ( A ) \mu ( B ) . \end{align*}


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


\begin{align*} J \nabla _ g f = X _ v \end{align*}


\begin{align*} & < v _ t , \xi > + \int _ { \mathbb { R } ^ { 3 } } ( ( v + u _ { \infty } ) \cdot \nabla v - ( b + B _ { \infty } ) \cdot \nabla b ) \cdot \xi \dd x + \nu \int _ { \mathbb { R } ^ { 3 } } \nabla v : \nabla \xi \dd x = 0 , \\ & < b _ t , \zeta > + \int _ { \Omega } ( b \times v + B _ { \infty } \times v + b \times u _ { \infty } ) \cdot \nabla \times \zeta \dd x + \mu \int _ { \Omega } \nabla \times b \cdot \nabla \times \zeta \dd x = 0 , \end{align*}


\begin{align*} M _ { \hat I } = \varprojlim _ n M / ( x _ 1 ^ n , \dots , x _ k ^ n ) . \end{align*}


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


\begin{align*} \begin{aligned} \int _ { \Omega } & \delta n \xi ( \cdot , t ) - \int _ { 0 } ^ { t } \int _ { \Omega } \delta n \xi _ { t } + \int _ { 0 } ^ { t } \int _ { \Omega } \nabla \delta n \cdot \nabla \xi \\ & = \int _ { 0 } ^ { t } \int _ { \Omega } ( \delta n S ( x , \tilde { n } _ { 1 } , c _ { 1 } ) \cdot \nabla c _ { 1 } + n _ { 2 } Z \cdot \nabla c _ { 1 } + n _ { 2 } S ( x , \tilde { n } _ { 2 } , c _ { 2 } ) \cdot \nabla \delta c ) \cdot \nabla \xi , \end{aligned} \end{align*}


\begin{align*} H _ N ( t ) - \widetilde H _ N ( t ) = \int _ { \Omega _ N } g _ { N , t } \ , d \mu _ { N , t } , g _ { N , t } : = \log \left ( \frac { d \tilde \nu _ { N , t } } { d \nu _ { N , t } } \right ) . \end{align*}


\begin{align*} \sum _ i \mu _ i \left ( \sum _ j \nu _ j \gamma _ { i , j } \right ) = \sum _ { i , j } ( \mu _ i \nu _ { i , j } ) \gamma _ { i , j } , \end{align*}


\begin{align*} \nabla X ^ { \flat } ( Y , Z ) = \nabla _ Y X ^ { \flat } Z = \langle \nabla _ Y X , Z \rangle \ , , \forall \ , X , Y \in \Gamma ( M ) \ , . \end{align*}


\begin{align*} ( \pi \otimes ( \psi \circ \det ) , ( \psi ( \varpi _ v ) \eta _ v ) _ v ) = ( \pi , ( \psi ( \varpi _ v ) \eta _ v ) _ v ) \end{align*}


\begin{align*} s _ \Sigma ( \bar x ) & \le s _ \Sigma ( x ) + \frac { 1 } { \lambda } d ( x , \bar x ) \le r _ m + \frac { \bar K } \lambda r _ { m ' } < \left ( \frac 1 { \bar K ^ 2 } + \frac 1 { \lambda \bar K } \right ) r _ { m ' - 2 } \le \frac { r _ { m ' - 2 } } { \bar K } \le \frac { r _ i } { \bar K } . \end{align*}


\begin{align*} \mathrm { E } ( X ^ n ) & = \frac { 1 } { \sigma \mathrm { B } ( \alpha , \beta ) } \sum _ { j = 0 } ^ { \infty } ( - 1 ) ^ j \binom { \beta - 1 } { j } \int _ { - \infty } ^ \infty x ^ n \left [ \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { j + \alpha - 1 } \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \mathrm { d } x . \end{align*}


\begin{align*} a \cdot b = \varrho ( T ( a ) ) b , \ \forall \ a , \ b \in V . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } z _ i = \lambda _ i \cdot \left ( { 1 \over 1 + \sum _ { j \in \mathbb Z _ 0 } \widetilde { z } _ j } \right ) ^ k , \ \ i \in \mathbb Z _ 0 , \\ \widetilde { z } _ i = \lambda _ i \cdot \left ( { 1 \over 1 + \sum _ { j \in \mathbb Z _ 0 } z _ j } \right ) ^ k , \ \ i \in \mathbb Z _ 0 . \end{array} \right . \end{align*}


\begin{align*} \big | \big ( \Gamma ( g , h ) , \ \omega \big ) _ { L _ v ^ 2 } \big | : = \Big | \int _ { \mathbb R ^ 3 } \Gamma ( g , h ) \ , \omega d v \Big | \leq C _ 1 \norm { g } _ { L _ v ^ 2 } \norm { \psi ( v , D _ v ) h } _ { L _ v ^ 2 } \norm { \psi ( v , D _ v ) \omega } _ { L _ v ^ 2 } , \end{align*}


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


\begin{align*} \Pi _ S ( X ) \rightarrow \oplus _ { i \in I } \Pi _ S ( D _ i , N _ i ) \rightarrow \cdots \rightarrow \oplus _ { J \subset I , \sharp J = n } \Pi _ S ( D _ J , N _ J ) \rightarrow \cdots \end{align*}


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


\begin{align*} \alpha _ { 0 } + \alpha _ { 1 } + \ldots + \alpha _ { t - 2 } = t \alpha _ { t } . \end{align*}


\begin{align*} \dot { x } ( t ) = A x ( t ) + A _ d x ( t - h ) , \forall t \geq 0 , \end{align*}


\begin{align*} \begin{cases} A ^ \top \nabla ^ { ( 1 , 0 ) } \eta ( A z _ \star , y _ \star ) - y _ \star ^ { ( 1 ) } \cdot e _ d ^ { ( 1 ) } = \textbf { 0 } _ d , \\ \nabla ^ { ( 0 , 1 ) } \eta ( A z _ \star , y _ \star ) - \tfrac { 1 } { p } \sum _ { i = 2 } ^ { 2 T } ( y _ \star ^ { ( i ) } ) ^ p \cdot e _ d ^ { ( i ) } - ( z _ \star ^ { ( 1 ) } - 2 T + \tfrac { 1 } { p } ) \cdot e _ d ^ { ( 1 ) } = \textbf { 0 } _ d . \end{cases} \end{align*}


\begin{align*} x _ 2 \partial _ { x _ 1 } v _ 3 - x _ 1 \partial _ { x _ 2 } v _ 3 + k \partial _ { x _ 3 } v _ 3 = 0 . \end{align*}


\begin{align*} \int _ 0 ^ { \infty } x ^ { \alpha } e ^ { - x } L _ n ^ { ( \alpha ) } ( x ) L _ m ^ { ( \alpha ) } ( x ) d x = \frac { \Gamma ( n + \alpha + 1 ) } { n ! } \delta _ { n m } . \end{align*}


\begin{align*} \ell ( x , v ( \beta _ i + \beta _ \ell ) ) \geq 0 \langle \mu , v \beta _ i + v \beta _ \ell \rangle = - 2 . \end{align*}


\begin{align*} \tilde { D } _ 1 & : = [ \gamma / 2 , \gamma ] \times [ - \sqrt { 1 + \alpha } ~ \gamma ^ 2 , \sqrt { 1 + \alpha } ~ \gamma ^ 2 ] , \\ \tilde { D } _ 2 & : = [ N , N + \gamma ] \times [ \sqrt { 1 + \alpha } ~ N ^ { \frac { \alpha + 2 } { 2 } } , \sqrt { 1 + \alpha } ~ N ^ { \frac { \alpha + 2 } { 2 } } + \gamma ^ 2 ] . \end{align*}


\begin{align*} \| x ( t ) - x ^ j ( t ) \| = \| w ( t - \tau ) - w ^ j ( t - \tau ) \| , t \in [ \tau - h , \tau ) , \end{align*}


\begin{align*} \mathbf { f } = & \log 3 \mathbf { e } _ 1 + \log 3 \mathbf { e } _ 2 + \log 3 \mathbf { e } _ 3 + \log \frac { 4 } { 3 } \mathbf { e } _ { 1 2 3 ' } \\ = & [ \log 4 , \log 4 , \log 4 , \log 1 6 , \log 1 6 , \log 1 6 , \log 4 8 ] ^ \intercal \end{align*}


\begin{align*} E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \biggr ) \biggr ] = E _ { \xi _ 1 ^ \theta ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \biggr ) \biggr ] . \end{align*}


\begin{align*} w _ n ( s ) = s { \frac { f ' _ n ( q s ) } { \beta ^ n } } n \in { \mathbb { N } } . \end{align*}


\begin{align*} X _ { i ; j k } - X _ { i ; k j } & = R _ { k j \ell i } X ^ \ell \\ X _ { j ; k i } - X _ { j ; i k } & = R _ { i k \ell j } X ^ \ell \\ X _ { k ; i j } - X _ { k ; j i } & = R _ { j i \ell k } X ^ \ell . \end{align*}


\begin{align*} \aligned & \sum _ { i = 1 } ^ { n - 1 } \langle i _ { e _ i } R ^ { \nabla } , i _ { e _ i } R ^ { \nabla } \rangle + \langle i _ { \frac \partial { \partial r } } R ^ { \nabla } , i _ { \frac \partial { \partial r } } R ^ { \nabla } \rangle \\ & = \sum _ { 1 \le j _ 1 \le n } \sum _ { i = 1 } ^ { n } \langle R ^ \nabla ( e _ i , e _ { j _ 1 } ) , R ^ \nabla ( e _ i , e _ { j _ 1 } ) \rangle \\ & = 2 | | R ^ \nabla | | ^ 2 \ , , \endaligned \end{align*}


\begin{align*} a ( u , v ) : = ( \triangle u , \Delta v ) . \end{align*}


\begin{align*} { S _ { \min } ^ { \rm { N } } } = \frac { { W I } } { { K L } } \overline \varepsilon . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { \infty } \epsilon _ i < \infty . \end{align*}


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


\begin{align*} \mathcal { G } ( t ) = \lambda ( t ) \mathcal { E } ( t ) + \mu \alpha ( t ) \mathcal { E } ^ r ( t ) \langle P ( u _ t ) , u \rangle + \nu \alpha ( t ) \delta ^ { \frac { 1 } { \ell } } ( t ) \mathcal { E } ^ { r + \frac { 1 } { q } + \frac { 1 } { \ell ^ { \prime } } } ( t ) . \end{align*}


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


\begin{align*} I _ { n + 1 } \frac { \partial ^ 2 } { \partial x _ n ^ 2 } + B \frac { \partial } { \partial x _ n } + C = \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + B - Q \Bigr ) \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } + Q \Bigr ) . \end{align*}


\begin{align*} R \cdot R ( x , J x , J x , x ; u , J u ) = \frac { 1 } { 2 } \ , L ( p , \bar { \pi } ^ h ) \ , Q ^ c ( g , R ) ( x , J x , J x , x ; u , J u ) . \end{align*}


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


\begin{align*} \hat { L } _ 0 = \{ 0 \} \sqcup \bigsqcup _ { m = 1 } ^ \infty \bigsqcup _ { \gamma \in \Gamma / ( \Gamma \cap N ) } \{ \gamma \cdot ( 0 , 0 , 0 , m ) \} \sqcup \bigsqcup _ { m = 1 } ^ \infty \bigsqcup _ { n = 0 } ^ { 3 m - 1 } \bigsqcup _ { \gamma \in \Gamma } \{ \gamma \cdot ( 0 , 0 , 3 m , n ) \} . \end{align*}


\begin{align*} \mathcal { H } ^ 2 ( t ; c , f ) : = \Bigg \{ \tilde { f } : \int _ { \{ \Psi < - t \} } | \tilde { f } | ^ 2 _ h c ( - \Psi ) < + \infty , \ \tilde { f } \in H ^ 0 ( \{ \Psi < - t \} , \mathcal { O } ( K _ M \otimes E ) ) \\ \& ( \tilde { f } - f ) _ { z _ 0 } \in \mathcal { O } ( K _ M ) _ { z _ 0 } \otimes J _ { z _ 0 } , z _ 0 \in Z _ 0 \Bigg \} , \end{align*}


\begin{align*} L u : = \big ( a _ { j k } ^ { \alpha \beta } \partial _ j \partial _ k u _ \beta \big ) _ { 1 \leq \alpha \leq N } , \end{align*}


\begin{align*} 0 \le V _ { \mathrm { o r i g } } - V _ { \mathrm { d e t } } \le \frac { n T ^ 2 } { 2 } \sum _ { i , j = 1 } ^ n \int _ { \R ^ n } | \partial _ { i j } g | ^ 2 \ , d \gamma _ { y ^ * , T } , \end{align*}


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


\begin{align*} \begin{cases} \nabla w \cdot \nabla ^ \perp \varphi = 0 , \\ w = \mathcal { L } _ H \varphi , \\ \varphi | _ { \partial \Omega } = 0 . \end{cases} \end{align*}


\begin{align*} W ^ J = \{ w \in W \mid w ( J ) \subseteq \Phi ^ + \} . \end{align*}


\begin{align*} d ( p , g ) \leq L d ( p , \mathcal { F } _ g ) = L d ( p , \mathcal { F } _ q ) \leq L d ( p , q ) < L \frac r L = r , \end{align*}


\begin{align*} \Re \mathcal { P } [ \varrho ] = - \frac { 5 } { 2 } a ^ 3 | w | ^ 4 \left ( - 2 a | w | ^ 2 + \sqrt { 5 } | w | ^ 2 - 2 \Re ( \varrho _ z ^ 2 ) \right ) , \end{align*}


\begin{align*} & \ell ( x , w ' \alpha ) + \ell ( x ' , \alpha ) \\ = & \ , \langle \mu , w ' \alpha \rangle + \langle \mu ' , \alpha \rangle - \Phi ^ + ( w w ' \alpha ) + \Phi ^ + ( w ' \alpha ) - \Phi ^ + ( w ' \alpha ) + \Phi ^ + ( \alpha ) \\ = & \ , \langle ( w ' ) ^ { - 1 } \mu + \mu ' , \alpha \rangle - \Phi ^ + ( w w ' \alpha ) + \Phi ^ + ( \alpha ) = \ell ( x x ' , \alpha ) . \end{align*}


\begin{align*} \frac { d ^ j } { d t ^ j } \tilde { X } _ t | _ { t = 0 } \end{align*}


\begin{align*} \begin{array} { l l } \begin{pmatrix} \lambda & 0 \\ 0 & \lambda ^ { - 1 } \end{pmatrix} , & \lambda > 1 , \\ \begin{pmatrix} a _ 1 & b _ 1 \\ c _ 1 & d _ 1 \end{pmatrix} , & a _ 1 d _ 1 - b _ 1 c _ 1 = 1 , a _ 1 + b _ 1 = c _ 1 + d _ 1 > 0 \end{array} \end{align*}


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


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


\begin{align*} \psi ( p ) = \psi _ { p _ 0 } [ \eta ] ( p ) \stackrel { d e f } { = } \inf _ { r \in R ( p ) } g ( p _ 0 , r , | \eta | _ r ) , \ p \le p _ 0 . \end{align*}


\begin{gather*} \int Q \bigl ( \oplus _ { i = 1 } ^ n f _ i \bigr ) \ , P _ \mathcal { U } ( d Q ) = \sum _ { i = 1 } ^ n \mu _ i ( f _ i ) \quad f _ i \in B ( \Omega _ i , \mathcal { A } _ i ) i \in I ; \end{gather*}


\begin{align*} L ( 1 , \chi _ K ) = \frac { 2 \pi h _ K } { w _ K \sqrt { \Delta _ K } } . \end{align*}


\begin{align*} ( I - F ) \Psi ^ - = I , \end{align*}


\begin{align*} \rho ^ \star ( \theta ^ \sharp , X ) & = D ^ g \theta ( J \theta ^ \sharp , J X ) + \frac { 1 } { 2 } g ( D ^ g _ { J X } \theta , J \theta ) + \frac { 1 } { 2 } g ( D ^ g _ X \theta , \theta ) + g ( d \theta , N _ X ) , \\ & = \frac { 1 } { 2 } D ^ g \theta ( J \theta ^ \sharp , J X ) + \frac { 1 } { 2 } g ( D ^ g _ X \theta , \theta ) + g ( d \theta , N _ X ) , \\ & = - \frac { 1 } { 2 } ( d \theta ) ^ { J , - } ( \theta ^ \sharp , X ) + g ( d \theta , N _ X ) . \end{align*}


\begin{align*} \begin{array} { l l l } \mathsf { g } _ { \mathbb { D } , \tilde { \mathbb { F } } ( \mathsf { S } ) } ( \tilde { F } ( L ) ) & = & \tilde { F } ( \mathsf { g } _ { \mathbb { C } , \mathsf { S } } ( L ) ) \circ \Phi _ { S , S } \\ & = & \tilde { F } ( \mathsf { d } _ { \mathbb { C } , \mathsf { S } } ( R ) ) \circ \Phi _ { S , S } \\ & = & \mathsf { d } _ { \mathbb { D } , \tilde { \mathbb { F } } ( \mathsf { S } ) } ( \tilde { F } ( R ) ) . \end{array} \end{align*}


\begin{align*} ( B \Psi - \Phi ) _ n = - B + 2 B K x _ n - \Phi _ n = 0 , \end{align*}


\begin{align*} | ( \mathcal { R } _ \lambda ( \sigma ) v _ T , v _ T ) | = \prod _ { j } \frac { 1 } { | a _ { i _ j + 1 } - a _ { i _ j } | } , \end{align*}


\begin{align*} \nabla _ { z , r } \eta ( z , r ) = \int _ { 0 } ^ { \infty } \nabla _ { z , r } { \mathcal { G } } * _ 1 G ( \cdot , r ' ) \frac { 1 } { r ' } \dd r ' . \end{align*}


\begin{align*} \partial _ t \begin{pmatrix} w \\ \overline { w } \end{pmatrix} & = i t ^ { - 1 } | W _ 1 | ^ 2 P \begin{pmatrix} i \mu & 0 \\ 0 & - i \mu \end{pmatrix} P ^ { - 1 } \begin{pmatrix} w \\ \overline { w } \end{pmatrix} + \begin{pmatrix} S _ 2 + e ^ { 3 i \lambda _ 1 ( \Phi _ 1 - \theta _ 1 ) } R _ 2 \\ \overline { S _ 2 + e ^ { 3 i \lambda _ 1 ( \Phi _ 1 - \theta _ 1 ) } R _ 2 } \end{pmatrix} . \end{align*}


\begin{align*} f _ { + a } ( x ) = e ^ { - \lambda _ s } f _ r ( x ) + ( 1 - e ^ { - \lambda _ s } ) f _ r \ast f _ s ( x ) . \end{align*}


\begin{align*} Z _ { i , j } = X _ { i , j } \ , { \rm c h } _ { \rm r e f } [ B ( m _ i , m _ j , s _ j - s _ i ) ] . \end{align*}


\begin{align*} v _ 0 = \sum Y _ { \ 1 } ^ { - } v _ k . \end{align*}


\begin{align*} a \cdot b a & = b , \\ a b \cdot c & = a \cdot ( b \cdot c ) , \\ ( a \cdot b ) a & = ( b \cdot a ) b , \end{align*}


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


\begin{align*} \int _ { \{ \Psi < - t \} } | F _ t + \alpha q | ^ 2 _ h c ( - \Psi ) - \int _ { \{ \psi < - t \} } | F _ t | ^ 2 _ h c ( - \Psi ) \geq 0 \end{align*}


\begin{align*} \operatorname { E } ( X ) = \frac { \sqrt { \pi \theta } } { \operatorname { B } ( a , b ) } \sum _ { n = 0 } ^ b \binom { b - 1 } { n } \frac { 1 } { \sqrt { a + n } } . \end{align*}


\begin{align*} F ^ 1 ( \phi _ i x ) = F ^ i x F _ 1 ( \phi _ i x ) = F _ i x \end{align*}


\begin{align*} c _ k = \sum _ { i = 1 } ^ { t - 1 } c _ i a _ i + c _ { k } \sum _ { i \in \Omega _ t ^ o } a _ i + c _ { k + 1 } \sum _ { i \in \Omega _ t ^ e } a _ i . \end{align*}


\begin{align*} \begin{cases} 0 & { i = j } \\ d _ { i j } & { \mbox { e l s e } } . \\ \end{cases} \end{align*}


\begin{align*} g \rightarrow g ^ { \sigma } : = \sigma ^ { - 1 } ( g ^ { - 1 } ) ^ { \vee } \sigma \end{align*}


\begin{align*} \Omega = \{ x \in \R ^ 3 \colon x _ 3 > | x _ 1 | + | x _ 2 | \} \end{align*}


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


\begin{align*} S : = F \cup \{ g \in G : \phi _ { g , 1 } \neq 0 \} \cup \{ g \in G : \psi _ { g , 1 } \neq 0 \} , \end{align*}


\begin{align*} \mathcal { Q } = \bigoplus _ { i = 1 } ^ n \mathcal { Q } _ i \mathcal { R } = \bigoplus _ { i = 1 } ^ { n - 1 } J _ { p _ i * } ( J _ { p _ i } ^ * ( \mathcal { Q } _ { i + 1 } ) ) . \end{align*}


\begin{align*} | \rho ( x ) | = ( x , \Sigma ) : = \inf \{ d ( x , y ) \mid y \in \Sigma \ , \} . \end{align*}


\begin{align*} \begin{cases} | \mathcal { K } ( c ' , x ) | \leqslant ( c ' + 1 ) C , & | x | < 1 , \\ \mathcal { K } ( c ' , x ) \leqslant ( - \alpha + \frac { \beta } { | x | } + \frac { C } { | x | } + c ' C ) , & | x | \geqslant 1 , \end{cases} \end{align*}


\begin{align*} & \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \Phi _ { j , + } ^ { 2 } ( \partial _ t \varphi + ( v _ j + u _ { \infty } ) \cdot \nabla \varphi ) \dd x \dd s + \int _ { \mathbb { R } ^ { 3 } } \Phi _ { 0 , + } ^ { 2 } \varphi _ 0 \dd x \\ & = - \mu _ j \int _ 0 ^ { T } \int _ { \mathbb { R } ^ { 3 } } \left ( \Phi _ { j , + } ^ { 2 } \Delta \varphi + \frac { 2 } { r } \partial _ r \Phi _ { j , + } ^ { 2 } \varphi - 2 | \nabla \Phi _ { j , + } | ^ { 2 } \varphi \right ) \dd x \dd s . \end{align*}


\begin{align*} \psi ^ \pm ( x , t ; k ) = e ^ { - i c _ - ( x ) \widehat { \sigma } _ 3 } \mu ^ \pm ( x , t ; k ) e ^ { i c _ + ( x ) \sigma _ 3 } , \end{align*}


\begin{align*} K _ { \lambda } = \frac { 1 } { ( 2 \pi ) ^ { d / 2 } \prod _ { j = 1 } ^ { d - 1 } j ! } \int _ { W ^ d } e ^ { - \sum _ { j = 1 } ^ d ( \lambda _ j - \bar { \lambda } ) z _ j } \hat { h } ^ { ( \bar { \lambda } , \ldots , \bar { \lambda } ) } ( z ) d z _ 1 \ldots d z _ d . \end{align*}


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


\begin{align*} \int _ 0 ^ T \left | \frac { \zeta ' } { \zeta } ( \sigma + i t ) \right | ^ 2 d t = O \left ( \frac { T } { ( \sigma - 1 / 2 ) ^ 2 } \right ) . \end{align*}


\begin{align*} B _ { 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*} \begin{aligned} \left | e ^ { - i c _ - ( x ) } - e ^ { - i \tilde { c } _ - ( x ) } \right | & = \left | e ^ { \frac { 1 } { 2 i } \int _ { - \infty } ^ x \left ( | u _ y ( y ) | ^ 2 - | \tilde { u } _ y ( y ) | ^ 2 \right ) d y } - 1 \right | \leq 2 \delta c _ 1 \| u _ y - \tilde { u } _ y \| _ { L ^ 2 } . \end{aligned} \end{align*}


\begin{align*} \frac { \partial p } { \partial x } & = h ^ x ( x , 0 ) + \int _ 0 ^ y \frac { \partial h ^ y } { \partial x } ( x , s ) \ , d s \\ & = h ^ x ( x , 0 ) + \int _ 0 ^ y g ^ { x y } ( x , s ) \ , d s \end{align*}


\begin{align*} \norm { \nabla ^ i _ z c _ { \Delta } ( \cdot , \xi _ { \Delta } ) } _ \infty = \norm { \nabla ^ i _ { 0 } c _ { \Delta - z } ( \cdot , \xi _ { \Delta - z } ) } _ \infty \norm { \nabla ^ i _ z \hat { c } _ { \Delta } ( \cdot , \xi _ { \Delta } ) } _ \infty = \norm { \nabla ^ i _ { 0 } \hat { c } _ { \Delta - z } ( \cdot , \xi _ { \Delta - z } ) } _ \infty . \end{align*}


\begin{align*} L = \bigoplus _ { i = 1 } ^ \ell \mathfrak a _ i b _ i . \end{align*}


\begin{align*} \Theta _ { \iota } = ( f ^ { - 1 } ) _ { \iota } \varphi h G + f ^ { - 1 } G \varphi h _ { \iota } \left ( 1 + \frac { \varphi ^ { ' } } { \varphi } h \right ) + f ^ { - 1 } \varphi h G _ { \iota } . \end{align*}


\begin{align*} C _ { \mu } = \{ x \in H : \mathcal { I } - d _ { - } ( x , H ) < 1 - \mu \} . \end{align*}


\begin{align*} g _ n ( t , q , x , \mu ) : = g \big ( t , \frac { q } { \sqrt { n } } , x , \mu \big ) \end{align*}


\begin{align*} - B _ p n ^ { p - p } \binom { p } { p } \sum _ { K = 0 } ^ { 0 } ( - 1 ) ^ { K + p } \binom { p - p } { K } , \end{align*}


\begin{align*} m _ k ( x _ k + s ) = \phi ( x _ k ) + \langle g _ k , s \rangle + \frac { 1 } { 2 } \langle H _ k s , s \rangle \end{align*}


\begin{align*} \aligned c + 1 + \frac { 1 } { 2 } \| u _ n \| _ { E _ a } \geq \Phi _ a ( u _ n ) - \frac { 1 } { 2 } \langle \Phi ' _ { j _ n } ( u _ n ) , u _ n \rangle = \left ( \frac { 1 } { 2 } - \frac { 1 } { 2 p } \right ) | u _ n | _ { H L } ^ { 2 p } \endaligned \end{align*}


\begin{align*} \phi _ { i s } ( \partial _ i ) = - \frac { \Gamma _ { i s } ^ s } { \varphi _ i } , \quad \forall s \neq i , \end{align*}


\begin{align*} P _ n P _ { n ' } = \begin{cases} P _ n , & n ' = n , \\ 0 , & n ' \ne n . \end{cases} \end{align*}


\begin{align*} \widetilde { \lambda } ( m ) \ r _ n ( m ) = E _ { n } \ \widetilde { r } _ { n + 1 } ( m ) + F _ n \ \widetilde { r } _ { n } ( m ) + G _ n \ \widetilde { r } _ { n - 1 } ( m ) \ , , \end{align*}


\begin{align*} a _ { P _ T } ( T ) = \delta ( T , T ) + O _ { \epsilon } ( k ^ { - 2 / 3 } \det ( T ) ^ { 1 + \epsilon } ) . \end{align*}


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


\begin{align*} \int | f | ^ 4 \lesssim \sum _ { \substack { R ^ { \frac { 1 } { 2 } } \le W \le R \\ W \ , } } \sum _ { \ell ( \tau ) = \frac { W } { R } } \sum _ { U \| U _ { \tau , R } } | U | ^ { - 1 } \Big ( \int \sum _ { \theta \subset \tau } | f _ \theta | ^ 2 W _ U \Big ) ^ 2 . \end{align*}


\begin{align*} \vert S \cap [ 1 , n ] \vert \leq \sum _ { j = 1 } ^ k \vert S _ j \vert \ll \sum _ { j = 1 } ^ k \frac { 2 ^ j \log c _ j } { j } \ll \frac { 2 ^ k \log c _ k } { k } = o ( \alpha ( n ) \cdot n / \log n ) \end{align*}


\begin{align*} \theta _ { S , \mu } \left ( \gamma ( \tau , z ) \right ) ( c \tau + d ) ^ { - \frac { g } { 2 } } e \left ( - c S [ z ] ( c \tau + d ) ^ { - 1 } \right ) = \sum _ { \eta \in ( 2 S ) ^ { - 1 } \mathbb Z ^ g / \mathbb Z ^ g } \varepsilon _ { S } ( \eta , \mu ; \gamma ) \theta _ { S , \eta } ( \tau , z ) , \end{align*}


\begin{align*} a _ { n - i } ( \gamma ^ { ( q ^ { n - i } - q ^ n ) } - 1 ) = 0 \end{align*}


\begin{align*} F ( r ) = \sigma _ k ^ { 1 / k } ( \lambda ( r ) ) \mbox { w i t h } \lambda ( r ) \in \Gamma _ k \end{align*}


\begin{align*} u ( \eta ) - \frac { \tilde { d } _ H ( \xi , \eta ) ^ \beta } { \delta ^ \beta } \leq u ^ \delta _ { \beta } ( \xi ) < u ^ \delta _ \beta ( p _ 0 ) = u ( q _ { \delta , \beta } ) - \frac { \tilde { d } _ H ( p _ 0 , q _ { \delta , \beta } ) ^ \beta } { \delta ^ \beta } \end{align*}


\begin{align*} H = \frac { 1 } { 2 m } \left ( \sum _ i p _ i ^ 2 + \lambda \left ( \sum _ i x _ i p _ i \right ) ^ 2 \right ) + { \cal V } ( r ) = \frac { 1 } { 2 m } \left ( ( 1 + \lambda r ^ 2 ) \sum _ i p _ i ^ 2 - \lambda \sum _ { i < j } J _ { i j } ^ 2 \right ) + { \cal V } ( r ) , \end{align*}


\begin{align*} f _ \alpha = \frac 1 2 | d ( ( h _ \alpha ) _ i + ( h _ \alpha ) _ j ) | , & \ g _ \alpha = \frac 1 2 | d ( ( h _ \alpha ) _ i - ( h _ \alpha ) _ j ) | , \\ f = \frac 1 2 | d ( h _ i + h _ j ) | , & g = \frac 1 2 | d ( h _ i - h _ j ) | . \end{align*}


\begin{align*} \lim _ { \epsilon \to 0 } \epsilon ^ { \frac { 1 } { p } } \| u _ { \epsilon } \| _ { L ^ p ( ( 0 , T ) \times \Gamma _ { \epsilon } ) } = \| u _ 0 \| _ { L ^ p ( ( 0 , T ) \times \Omega \times \Gamma ) } . \end{align*}


\begin{align*} [ \kappa ] _ n ^ 2 \stackrel { d e f } { = } \kappa _ 1 ^ 2 + \sum _ { i = 2 } ^ n ( \kappa _ i - \kappa _ { i - 1 } ) ^ 2 , \ n \ge 2 . \end{align*}


\begin{align*} ( \mathbf { u } - \mathbf { v } ) ( \underbrace { G _ 2 , G _ 3 , \cdots , G _ { 2 k } } _ { \bar { G } } ) = ( 0 , 0 , \cdots , 0 ) . \end{align*}


\begin{align*} [ \omega ] = 2 [ H ] - [ E ] , [ \alpha ] = 6 [ H ] + [ E ] . \end{align*}


\begin{align*} \abs { \omega _ n ( t _ 1 ) - \omega _ n ( t _ 2 ) } = \omega _ n ( t _ 2 ) - \omega _ n ( t _ 1 ) \leq C _ 1 \omega _ n ( t _ 2 - t _ 1 ) \leq C _ 1 \omega ( t _ 2 - t _ 1 ) . \end{align*}


\begin{align*} X = \begin{pmatrix} - 1 & 0 \\ 0 & 1 \end{pmatrix} . \end{align*}


\begin{align*} T _ \sigma ( f , g ) ( x ) = \langle K _ \sigma ( x , y , z ) , ( f \otimes g ) ( y , z ) \rangle , \end{align*}


\begin{gather*} \mathcal { A } _ { j m } ( x , \xi ) = \int _ { B _ 1 ( 0 ) } \phi ( y ) \mathcal { A } _ j ( x + m y , \xi ) d y , \end{gather*}


\begin{align*} & \sigma _ 3 = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} , \ \ \ \ P = \begin{pmatrix} 0 & u \\ - \bar { u } & 0 \end{pmatrix} , \\ & \eta = \sqrt { \alpha } \left ( k - \frac { \beta } { 2 k } \right ) , H = \alpha k U _ x + \frac { i \alpha \beta ^ 2 } { 2 } \sigma _ 3 \left ( \frac { 1 } { k } P - P ^ 2 \right ) . \end{align*}


\begin{align*} u _ n = P _ 1 ( n ) \alpha _ 1 ^ n + \cdots + P _ s ( n ) \alpha _ s ^ n \end{align*}


\begin{align*} \limsup _ { n \to \infty } \left | \Delta _ n ^ { - \frac 1 2 } \langle f _ 1 , f _ 2 \rangle \sum _ { i = 1 } ^ n \int _ { ( i - 1 ) \Delta _ n } ^ { i \Delta _ n } \langle ( \mathcal S ( i \Delta _ n - s ) - I ) f _ 1 , f _ 2 \rangle d s \right | = \infty . \end{align*}


\begin{align*} \int _ { B _ R } \dfrac { | \tau _ h D v | ^ p } { | h | ^ { p \frac { 2 \gamma } { p } } } = \int _ { B _ R } \dfrac { | \tau _ h D v | ^ p } { | h | ^ { 2 \gamma } } d x \le c \int _ { B _ R } \dfrac { | \tau _ h V _ p ( D v ) | ^ 2 } { | h | ^ { 2 \gamma } } d x h . \end{align*}


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


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } ( f _ { k } ( x ) + C _ { k } ) \ge 0 \ \ \forall \ x \in \mathbb { R } ^ { n } , \ \ C _ k \in \mathbb { R } _ + . \end{align*}


\begin{align*} \mathcal { P } ^ { \pm } ( h ) ( z ) = \pm \frac { 1 } { 2 } h ( z ) - \frac { i } { 2 } \mathcal { H } ( h ) ( z ) z \in \mathbb { R } . \end{align*}


\begin{align*} \mu ^ { \pm } ( x , t ; k ) = I + \int _ { \pm \infty } ^ { x } \mathrm { e } ^ { - \mathrm { i } k ^ 2 \left ( x - y \right ) \widehat { \sigma } _ 3 } \left ( P _ 1 \mu ^ { \pm } \right ) \left ( y , t ; k \right ) d y , \end{align*}


\begin{align*} \begin{aligned} \Delta u _ 1 \wedge \dots \wedge \Delta u _ k \wedge \beta _ n ^ { n - m } \wedge \omega \geq 0 . \end{aligned} \end{align*}


\begin{align*} d g _ 1 '' = ( \log \sqrt { \tau _ 0 } ) _ z d z \ g '' _ 1 + e ^ { 2 i \theta } \frac { 1 } { \sqrt { \tau _ 0 } } \left ( Q _ 0 d z + \frac { \tau _ 0 } { 4 } d \overline { z } \right ) \ J \widehat { g _ 1 '' } I ; \end{align*}


\begin{align*} \sin S _ c ( x , y ) \partial _ x S _ c ( x , y ) = \mathrm F ( x , y ) x + \mathrm G ( x , y ) y , \end{align*}


\begin{align*} & \langle \mu , \alpha \rangle + \chi _ R ( v ' \alpha ) + \chi _ L ( - w ' v ' \alpha ) \\ \leq & \langle \mu , \alpha \rangle + 1 - \chi _ R ( - v ' \alpha ) + \chi _ L ( - w ' v ' \alpha ) \\ = & 1 - \prescript L { } { } \ell { } ^ R ( x ' , - \alpha ) \leq 0 . \end{align*}


\begin{align*} f ( x ) = \sum _ { | I | = 0 , 1 , \dots , } \frac { 1 } { | I | ! } \partial ^ I f ( p ) ( x - p ) ^ I \mbox { f o r a l l } x \in U \end{align*}


\begin{align*} { \cal I } : = \frac { 1 } { 2 \pi i } \int _ { C } f ( \zeta ) \dd \zeta \end{align*}


\begin{align*} \beta _ n : = ( n + 1 ) h _ A + n ( \widetilde b + a ) + a \end{align*}


\begin{align*} \epsilon _ s P ( x _ 1 , \ldots , x _ n ) = P ( v _ 1 , \ldots , v _ n ) = 0 . \end{align*}


\begin{align*} \theta _ { r + 1 } ( \hat { f } ) ( \gamma _ 1 , \dots , \gamma _ r , \gamma _ { r + 1 } ) = \theta _ r ( f ) ( \gamma _ 1 , \dots , \gamma _ r \gamma _ { r + 1 } ) , \end{align*}


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


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


\begin{align*} & \mathbb { E } \int _ 0 ^ T \psi ( t ) \Big [ \Vert X ( t \wedge \tau _ u ^ M ) \Vert _ H ^ 2 - \Vert x \Vert _ H ^ 2 \Big ] d t \\ = & \mathbb { E } \int _ 0 ^ T \psi ( t ) \int _ 0 ^ { t \wedge \tau _ { u } ^ M } \Big [ 2 \langle \mathcal { A } ( s ) , X ( s ) \rangle + \Vert \mathcal { B } ( s ) \Vert _ { L _ 2 } ^ 2 \Big ] d s d t . \end{align*}


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


\begin{align*} R \cdot R = f \ , \ , Q ^ c ( g , R ) , \end{align*}


\begin{align*} M _ { N C } ( n , d ) \le H _ t ( n , d ) 2 ^ d + \left ( \binom { n } { d } - H _ t ( n , d ) \right ) \frac { 2 ^ { d + 1 } } { t + 1 } = \left ( h _ t ( n , d ) + ( 1 - h _ t ( n , d ) ) \frac { 2 } { t + 1 } \right ) \cdot 2 ^ d \binom { n } { d } \enspace . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \int _ { \R ^ { 2 } } ( | \nabla u _ { n } | ^ { 2 } + u _ { n } ^ { 2 } ) d x = 0 , \end{align*}


\begin{align*} \nabla _ i u : = \nabla _ { e _ i } u = e _ i ^ j \partial _ j u = e _ i ^ j u _ j \end{align*}


\begin{align*} c ^ { - 1 } \left ( \bar A \right ) \cap c ^ { - 1 } \left ( X \backslash A \right ) = c ^ { - 1 } \left ( \bar A \cap ( X \backslash A ) \right ) \neq \emptyset . \end{align*}


\begin{align*} \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w ' , \cdot ) | O ( \chi _ { n , m } ' ( w ' , M , \cdot ) ) = O ( \chi _ { n , m } ' ( w , M , \cdot ) ) ] \\ = \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w ' , \cdot ) | w , M ] \end{align*}


\begin{align*} \lim _ { M \rightarrow \infty } \lim _ { n \rightarrow \infty } I \leq \lim _ { n \rightarrow \infty } \mathbb { E } \int _ 0 ^ { T } f ( s ) \Vert X _ n ( s ) - X ( s ) \Vert _ H ^ 2 d s = 0 . \end{align*}


\begin{align*} \omega _ j = ( e _ 1 + \dots + e _ j ) | _ { \C ^ n _ 0 } \end{align*}


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


\begin{align*} & a _ { m } ( x ) + \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n + m + 1 } } { 2 \pi } a _ { n } ( x ) \int _ { - \infty } ^ { \infty } \frac { \varphi \left ( \tau \right ) z ^ { n + m + 1 } ( \tau ) d \tau } { \left ( \frac { 1 } { 2 } - i \tau \right ) \left ( \frac { 1 } { 2 } + i \tau \right ) } \\ & = \frac { ( - 1 ) ^ { m } } { 2 \pi } \int _ { - \infty } ^ { \infty } \frac { \varphi \left ( \tau \right ) z ^ { m + 1 } ( \tau ) d \tau } { \frac { 1 } { 2 } + i \tau } , m = 0 , 1 , \ldots . \end{align*}


\begin{align*} \mathcal { F } _ { \geq p } t e l ( \mathcal { C } ) : = \lbrace x \in t e l ( \mathcal { C } ) \mid \mathcal { A } ( x ) \geq p \rbrace . \end{align*}


\begin{align*} ( f _ 1 ( \mathbf { x } ) , f _ 2 ( \mathbf { x } ) , \cdots , f _ { n - 1 } ( \mathbf { x } ) ) = ( f _ 2 ( \mathbf { y } ) , f _ 3 ( \mathbf { y } ) , \cdots , f _ n ( \mathbf { y } ) ) = ( f _ 2 ( \mathbf { x } ) , f _ 3 ( \mathbf { x } ) , \cdots , f _ n ( \mathbf { x } ) ) . \end{align*}


\begin{align*} \begin{aligned} \int _ S g ( x ) p _ { S } ( x ) \ , d U ( x ) & = \int _ S \int _ S g ( x ) 1 _ { B _ y } ( x ) \ , d U ( y ) \ , d U ( x ) \\ & = \int _ S \left ( \frac { 1 } { U ( B _ y ) } \int _ { B _ y } g ( x ) \ , d x \right ) \ , d U ( y ) > \int _ S g ( y ) d U ( y ) = 0 , \end{aligned} \end{align*}


\begin{align*} \Phi ( t , c ( q ) ) = \lim _ { n \to \infty } \Phi \left ( t , c \left ( q - \frac { 1 } { n } \right ) \right ) \in \overline { X \backslash \bar A } \subset X \backslash A , \end{align*}


\begin{align*} \frac { 1 } { N } \underset { i = 1 } { \overset { N } { \sum } } \mathbb { E } \Big [ \sup _ { 0 \leq s \leq t } | x _ s ^ i - x _ s ^ { * , i } | ^ 2 \Big ] \leq \frac { C } { N ^ 2 } \Big ( 1 + \frac { 1 } { N ^ 2 } \sum _ { i , j = 1 } ^ N \mathbb E \big [ | \xi ^ i - \xi ^ j | ^ 2 \big ] \Big ) . \end{align*}


\begin{align*} \int _ 0 ^ \infty \frac { \exp [ - ( a + n ) y ] } { y } \mathrm { d } y = \operatorname { E } _ 1 ( 0 ) , \end{align*}


\begin{align*} \mu ^ n _ g ( t ) : = \sum _ { i = 1 } ^ n g _ i \psi _ i ( t ) , \end{align*}


\begin{align*} 0 \le c _ { r } = r ^ { 1 - d } \int _ { 0 } ^ { r } \rho ^ { d - 1 } n c \ , d \rho \le \frac { \gamma M _ { 0 } } { \sigma _ { d } } r \quad \mbox { f o r } r \in ( 0 , R ] . \end{align*}


\begin{align*} x _ n ( t ) : = x _ a ( t - n { \bar { t } } / { \gamma } , x ( n { \bar { t } } / { \gamma } ) ) \forall \ , t \in I _ n . \end{align*}


\begin{align*} \| \langle x \rangle I _ 1 ^ { \prime } ( x ) \| _ { L ^ 2 ( \mathbb { R } ^ + ) } = 2 \pi ^ { - 1 } \| \langle x \rangle \widehat { \bar { r } _ 1 ( z ) } ( 2 x ) \| _ { L ^ 2 ( \mathbb { R } ) } \leq c \| r _ 1 ( z ) \| _ { H ^ 1 ( \mathbb { R } ) } , \end{align*}


\begin{align*} \widetilde { \mu } \ , ' ( n ) \ r _ n ( m ) = H ' _ { m } \ \widetilde { r } \ , ' _ { n } ( m + 1 ) + I ' _ m \ \widetilde { r } \ , ' _ { n } ( m ) + J ' _ m \ \widetilde { r } \ , ' _ { n } ( m - 1 ) \ , , \end{align*}


\begin{align*} h ( x , u ) = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 2 } ^ { n } Z _ i ^ { \theta [ u ] } \Bigr ) \right ] = E _ { \xi _ 1 ^ \theta ( u ) } \left [ \varphi \Bigl ( x + u + \sum _ { i = 1 } ^ { n - 1 } Z _ i ^ { \rho } \Bigr ) \right ] . \end{align*}


\begin{align*} \phi ( g _ 2 H ) H = \pi _ N ( g _ 2 ) \psi ( \pi _ N ( g _ 2 ) ) H = g _ 2 H . \end{align*}


\begin{align*} \d : C ^ 0 _ c ( G , V ) ^ G \to C ^ 1 _ c ( G , V ) ^ G , ( \d v ) ( x ) = x \cdot v - v . \end{align*}


\begin{align*} ( M - X ^ m ( X ^ k - 1 ) I _ p ) \begin{pmatrix} X _ 1 X _ 2 \cdots X _ { p - 1 } \\ X _ 2 \cdots X _ { p - 1 } \\ \vdots \\ X _ { p - 1 } \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix} \end{align*}


\begin{align*} t \alpha = \Lambda ^ 0 < \dots < \Lambda ^ L = t \beta . \end{align*}


\begin{align*} ( w \varepsilon ^ \mu ) ( \alpha , k ) = ( w \alpha , k - \langle \mu , \alpha \rangle ) . \end{align*}


\begin{align*} \begin{aligned} ( \Pi _ V \tilde { Y } _ { \tau / 2 } ) & = \int _ 0 ^ { \tau / 2 } \left \| \Pi _ V e ^ { | z | J ^ \perp ( t - s ) } P ^ { - 1 } \Pi _ { \mathrm { k e r } A ^ \perp } \sigma \right \| _ F ^ 2 d s \ge c _ 1 \tau \end{aligned} \end{align*}


\begin{align*} A ( t ) U ( t , 0 ) x = & - i ( \Delta + b ( t , \xi ) \cdot \nabla + c ( t , \xi ) ) U ( t , 0 ) x \\ = & - i \Delta U ( t , 0 ) x - i b ( t , \xi ) \cdot \nabla U ( t , 0 ) x - i c ( t , \xi ) U ( t , 0 ) x \\ = & - i \Delta U ( t , 0 ) x - 2 i ( \nabla W ) ( t , \xi ) \cdot \nabla U ( t , 0 ) x - i \sum _ { j = 1 } ^ d ( \partial _ j W ( t , \xi ) ) ^ 2 U ( t , 0 ) x \\ & - i ( \Delta W ) ( t , \xi ) U ( t , 0 ) x - \mu ( \xi ) U ( t , 0 ) x - \widetilde { \mu } ( \xi ) U ( t , 0 ) x . \end{align*}


\begin{align*} d u ( t ) = \big [ - \Delta ^ 2 u + \Delta \varphi ( u ) \big ] d t + B ( t , u ) d W ( t ) , \end{align*}


\begin{align*} \epsilon _ 2 ( \eta _ a , t ) = \int _ N ^ t h ( x _ a + \delta u ( \tau ) ) - \dfrac { a _ { 0 , \delta } ( x _ a ) } { 2 } \ , d \tau \end{align*}


\begin{align*} \Gamma _ k : = \bigg \{ f \in L ^ p ( [ 0 , T ] , H ) : \int _ 0 ^ { T - \delta _ k } \Vert f ( t + \delta _ k ) - f ( t ) \Vert _ Y ^ p d t \leq \frac { 1 } { k } \bigg \} . \end{align*}


\begin{align*} \gamma ( s , \tau ) = \left ( r ^ * \cos \left ( \frac { s } { r ^ * } \right ) , r ^ * \sin \left ( \frac { s } { r ^ * } \right ) , \frac { c } { 4 \pi r ^ * } \tau \right ) ^ t , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ M \mathbb { E } \left [ y _ i ^ * y _ i | \right . \left . \hat { \mathbf { H } } \right ] = & \sum _ { j = 1 } ^ { M } a _ j ^ 2 \left ( \sum _ { l = 1 } ^ M \lvert \hat { \phi } ^ { \left ( l , j \right ) } \rvert ^ 2 + M \sigma _ { e _ i } ^ 2 \lVert \mathbf { p } _ j \rVert ^ 2 \right ) + a _ c ^ 2 \left ( \sum _ { i = 1 } ^ M \lvert \hat { \phi } ^ { \left ( i , c \right ) } \rvert ^ 2 + M \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ c \rVert ^ 2 \right ) + M \sigma _ n ^ 2 . \end{align*}


\begin{align*} [ E : F ] = [ F ( \alpha , \gamma ) : F ] = [ F ( \alpha , \gamma ) : F ( \gamma ) ] [ F ( \gamma ) : F ] \leq m ( q - 1 ) \end{align*}


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


\begin{align*} M ( r , f ) = \max _ { | x | = r } | f ( x ) | . \end{align*}


\begin{align*} A _ t \cdot r \nabla \theta & = \frac { 1 } { r } \Phi _ t , \\ ( B \times u ) \cdot r \nabla \theta & = \frac { 1 } { r } u \cdot \nabla \Phi , \\ \nabla \times B \cdot r \nabla \theta & = - \frac { 1 } { r } \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi . \end{align*}


\begin{align*} & \frac { 1 } { | \mathcal F _ { N , k } | } \sum _ { f \in \mathcal F _ { N , k } } R _ 2 ( g , \rho ) ( f ) = \frac { T ( g , \rho ) } { 4 L } \\ & + \O \left ( \frac { 1 } { L } \right ) + \O \left ( \frac { L ( \log \log x ) ^ 2 } { \pi _ N ( x ) } \right ) + \O \left ( \frac { x ^ { \pi _ N ( x ) c } 8 ^ { \nu ( N ) } } { k N } \right ) , \end{align*}


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


\begin{align*} \overbrace { [ x + a , y + b ] } = & ( x + a ) \star ( y + b ) - ( y + b ) \star ( x + a ) \\ = & x \ast y + \mathfrak { l } ( x ) b + \mathfrak { r } ( y ) a - y \ast x - \mathfrak { l } ( y ) a - \mathfrak { r } ( x ) b \\ = & [ x , y ] + ( \mathfrak { l } - \mathfrak { r } ) ( x ) b - ( \mathfrak { l } - \mathfrak { r } ) ( y ) a . \end{align*}


\begin{align*} \dim ( K _ i \cap K _ j ) = 1 \quad \quad K _ 1 \cap K _ 2 \cap K _ 3 = 0 \quad . \end{align*}


\begin{align*} \mathbf { x } ^ { \left ( \right ) } = & \mathbf { P } ^ { \left ( \right ) } \mathbf { A } ^ { \left ( \right ) } \mathbf { s } ^ { \left ( \right ) } = a _ c s _ c \mathbf { p } _ c + \sum _ { m = 1 } ^ { M } a _ m s _ m \mathbf { p } _ m , \end{align*}


\begin{align*} g _ \lambda ( z ) = \begin{cases} 0 & z = \lambda , \\ ( z - \lambda ) ^ { - 1 } & z \neq \lambda , \end{cases} \end{align*}


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


\begin{align*} \lim _ { n \to \infty } T _ { W , \psi } ( \mu _ n ( Q ^ n ) ) = T _ { W , \psi } ( \mu ) , I ( \mu _ n ( Q ^ n ) ) \le I ( \mu ) , \ \forall n . \end{align*}


\begin{align*} \beta ( \alpha ) = \limsup _ { n \to \infty } \frac { \ln q _ { n + 1 } } { q _ n } = \limsup _ { n \to \infty } \frac { - \ln \| n \alpha \| } { n } , \end{align*}


\begin{align*} \Psi ( x ) = \lim _ { n \rightarrow \infty } \max \{ | 2 | ^ { k } \mu ( 0 , 2 ^ { k + 1 } x ) ; 0 \leq k < n \} \end{align*}


\begin{align*} \mathbf { y } _ k = & a _ c s _ c \mathbf { H } _ k \mathbf { p } _ c + \mathbf { H } _ k \sum _ { i \in \mathcal { M } _ k } a _ i s _ i \mathbf { p } _ i + \mathbf { H } _ k \sum \limits _ { \substack { l = 1 \\ l \neq k } } ^ { K } \sum \limits _ { j \in \mathcal { M } _ l } a _ j s _ j \mathbf { p } _ j + \mathbf { n } _ k . \end{align*}


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


\begin{align*} M _ \omega ( f ) ( x ) = \begin{cases} f ( x ) \omega ( x ) & x \in K \\ 0 & x \notin K . \end{cases} \ , \end{align*}


\begin{align*} h _ m ( V _ m ( x ) ) = K ( t _ m ) , \end{align*}


\begin{align*} d u = \Delta _ p u \ , d t + \kappa \Delta u \ , d t + \sum _ { k \in \Z ^ d } \sum _ { i = 1 } ^ { d - 1 } \xi _ { k , i } ( x , t ) \cdot \nabla u \ , d B _ t ^ { k , i } , \end{align*}


\begin{align*} & \hphantom { { } = { } } \phi \circ ( \varphi \circ f ) ( x ) \\ * & = \widehat { \varphi } ( f ( o ) ) + D \phi ( p ) ^ i { } _ \alpha a ^ \alpha { } _ \beta x ^ \beta \\ * & \hphantom { { } = { } } + \frac 1 2 \left ( H \phi ( p ) ^ i { } _ { \alpha \beta } a ^ \alpha { } _ j a ^ \beta { } _ k + D \phi ( p ) ^ i { } _ \alpha b ^ \alpha { } _ { j k } \right ) x ^ j x ^ k + ( ) . \end{align*}


\begin{align*} \sum \limits _ { j = l } ^ { n - 1 } \chi _ j \leqslant ( \sum \limits _ { j = l } ^ { n } ) \end{align*}


\begin{align*} a = \frac { 1 } { 1 - q ^ { \prime } } ; b = \frac { 1 } { 1 - q } + 2 . \end{align*}


\begin{align*} P _ { \min } ^ { \rm { O } } \left ( { \overline S , \overline \varepsilon , \overline R | { W _ s } } \right ) = \frac { { \left ( { W - { W _ s } } \right ) { N _ 0 } } } { { { { \left | { { h _ b } } \right | } ^ 2 } } } \left ( { { 2 ^ { \frac { { \overline R } } { { \left ( { W - { W _ s } } \right ) } } } } - 1 } \right ) + { p _ K } \left ( { \overline S , { W _ s } } \right ) , \end{align*}


\begin{align*} \left \{ \begin{aligned} S ^ \prime _ t & = - \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t , & S _ 0 & = s _ 0 , \\ I ^ \prime _ t & = \beta ( 1 - \theta L _ t ) S _ t ( 1 - \theta L _ t ) I _ t - \gamma I _ t - I _ t \phi \left ( I _ t \right ) , & I _ 0 & = i _ 0 , \\ R ^ \prime _ t & = \gamma I _ t , & R _ 0 & = r _ 0 , \\ D ^ \prime _ t & = I _ t \phi \left ( I _ t \right ) , & D _ 0 & = d _ 0 . \\ \end{aligned} \right . \end{align*}


\begin{align*} f _ \alpha f _ \beta \cdot v _ { \omega _ i } + \sum _ { \mathbf { \gamma } } c _ { \mathbf { \gamma } } f _ { \gamma _ 1 } f _ { \gamma _ 2 } \cdot v _ { \omega _ i } = 0 . \end{align*}


\begin{align*} G _ { \iota } = ( \nabla G \cdot e _ { \iota } ) w _ { \iota \iota } , \end{align*}


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


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


\begin{align*} \phi _ x ^ \ell : = \frac 1 \ell \sum _ { y = 0 } ^ { \ell - 1 } \phi _ { x + y } , \bar \phi _ x ^ \ell : = E _ { \tilde \nu _ { N , t } } \left [ \phi _ x ^ \ell ~ \Big | ~ \sum _ { y = 0 } ^ { \ell - 1 } r _ { x + y } \right ] . \end{align*}


\begin{align*} T _ x T _ y = & T _ { x y } \qquad \qquad \qquad \ell ( x y ) = \ell ( x ) + \ell ( y ) , \\ T _ s ^ 2 = & ( v - v ^ { - 1 } ) T _ s + 1 ~ ~ s \in \widetilde W . \end{align*}


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


\begin{align*} \begin{aligned} \mathcal { H } _ m ( K _ m ) = C _ { m , n } \delta _ a \end{aligned} \end{align*}


\begin{align*} \sum _ { r = 1 } ^ p \sum _ { s = 1 } ^ q \alpha _ { r s } ( ( X ^ * X - \| X \| ^ 2 ) e _ r \otimes f _ s ) = 0 . \end{align*}


\begin{align*} \left \{ \begin{array} { l r } \frac { \partial X ( x , t ) } { \partial t } = \frac { 1 } { f ( v ) } \sigma _ { k } ( x , t ) \varphi ( X \cdot v ) ( X \cdot v ) G ( X ) v - X ( x , t ) ; \\ X ( x , 0 ) = X _ { 0 } ( x ) , \end{array} \right . \end{align*}


\begin{align*} Z _ { q } ( \beta ) = \frac { 1 } { h ^ { D N } N ! } \displaystyle \int _ { 0 } ^ { \infty } \ ; d \bar { x } \ ; d \bar { p } \ ; \exp _ { q } \left ( - \beta \displaystyle \sum _ { i = 1 } ^ { D N } \frac { p _ { i } ^ { 2 } } { 2 m } \right ) , \end{align*}


\begin{align*} ( A ^ { \tilde \Psi } [ \eta _ 1 , \dots , \eta _ n ] ) ( x ) = f ( x ) - g ( x ) . \end{align*}


\begin{align*} \nabla _ { e _ i } X = r \nabla _ { e _ i } \frac \partial { \partial r } = r H e s s ( r ) ( e _ i , e _ j ) e _ j \ , . \end{align*}


\begin{align*} \bar \phi : = ( \bar \phi _ { v , v ' } ) _ { V \times V ' } \colon X ^ { ( V ' ) } \to X ^ { ( V ) } \qquad \qquad \bar \psi : = ( \bar \psi _ { v ' , v '' } ) _ { V ' \times V '' } \colon X ^ { ( V '' ) } \to X ^ { ( V ' ) } , \quad \end{align*}


\begin{align*} { S ^ { \rm { S - N } } } = \frac { { { W _ m } I } } { { K L } } { \widetilde \varepsilon _ K } \left ( { { \gamma ^ { \rm { S - N } } } } \right ) , \end{align*}


\begin{align*} u ( x ' , t ) : = P _ t ^ L * f ( x ' ) , ( x ' , t ) \in \R ^ n _ + . \end{align*}


\begin{align*} { S } ^ { \rm P R } ( { \cal A } ) = \kappa ( \rho ) W | _ { \mathcal D } \in \R ^ { N L \times N L } \ , , \end{align*}


\begin{align*} J ( \tau , w ( \cdot ) , u ( \cdot ) ) = \sigma ( x ( \vartheta ) , x _ \vartheta ( \cdot ) ) + \int _ \tau ^ \vartheta f ^ 0 ( t , x ( t ) , x ( t - h ) , u ( t ) ) \mathrm { d } t , \end{align*}


\begin{align*} 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} , \ i \mapsto \begin{pmatrix} \sqrt { a } & 0 \\ 0 & - \sqrt { a } \end{pmatrix} , \ j \mapsto \begin{pmatrix} 0 & \sqrt { b } \\ \sqrt { b } & 0 \end{pmatrix} , \ i j \mapsto \begin{pmatrix} 0 & \sqrt { a b } \\ - \sqrt { a b } & 0 \end{pmatrix} . \end{align*}


\begin{align*} \begin{aligned} \| x ( ( n + 1 ) \bar { t } / \gamma ) \| _ { \mathcal { A } } & \le \beta ( \| x ( n \bar { t } / \gamma ) \| _ { \mathcal { A } } , \bar { t } ) + \tilde { d } & & \forall \ , n \in \mathbb { N } . \end{aligned} \end{align*}


\begin{align*} T \in Z ( V ) \iff \mu _ T ( x , \mu _ T ( y , z ) ) = 0 \forall x , y , z \in V . \end{align*}


\begin{align*} \prescript L { } { } \widetilde W { } ^ R \cap \left ( \widetilde W _ L x \widetilde W _ R \right ) = & \left \{ \prescript L { } { } x { } ^ R \right \} , \\ \prescript { - L } { } { } \widetilde W { } ^ { - R } \cap \left ( \widetilde W _ L x \widetilde W _ R \right ) = & \left \{ \prescript { - L } { } { } x { } ^ { - R } \right \} . \end{align*}


\begin{align*} \beta = \frac { 3 - \rho } { 2 } \end{align*}


\begin{align*} & \lim _ { n \to \infty } \mathbb P \left [ \sup _ { t \in [ 0 , T ] } | ( 1 ) _ n | > \epsilon \right ] \\ \leq & \lim _ { n \to \infty } \mathbb P \left [ \sup _ { t \in [ 0 , T ] } | ( 1 . 1 ) _ { n , N } | > \epsilon \right ] + \sup _ { n \in \mathbb N } \mathbb P \left [ \sup _ { t \in [ 0 , T ] } | ( 1 . 2 ) _ { n , N } | > \epsilon \right ] \\ = & 0 + \frac 1 { \epsilon } \sup _ { n \in \mathbb N } \mathbb E \left [ \sup _ { t \in [ 0 , T ] } | ( 1 . 2 ) _ { n , N } | \right ] \leq \delta . \end{align*}


\begin{align*} 0 = & p s _ { \delta _ N , i } \left ( \frac { \delta _ N } { s _ { \delta _ N , i } } \right ) ^ { \frac { 2 p } { p - 1 } } \int _ { \mathbb { R } ^ 2 } \phi ( T _ { z _ { N , i } } y ) ^ { p - 1 } _ + \phi ' ( T _ { z _ { N , i } } y ) \frac { ( T _ { z _ { N , i } } ) _ h ^ t \cdot T _ { z _ { N , i } } y } { | T _ { z _ { N , i } } y | } \tilde { u } _ { N , i } ( y ) d y + O \left ( \frac { \varepsilon _ N ^ 2 } { | \ln \varepsilon _ N | ^ { p - 1 } } \right ) . \end{align*}