\begin{align*} r = \mathrm { a d } ^ 0 \rho _ \pi : G _ { F , S } \to H ( E ) . \end{align*}


\begin{align*} h ( x ) = \prod _ { j = 1 } ^ d \lambda _ j \int _ { x _ d } ^ \infty d b _ d \int _ { x _ { d - 1 } } ^ { b _ { d } } d b _ { d - 1 } \cdots \int _ { x _ 1 } ^ { b _ { 2 } } d b _ 1 e ^ { \sum _ { i = 1 } ^ d \lambda _ i ( x _ i - b _ i ) } h ( b _ 1 , \ldots , b _ d ) . \end{align*}


\begin{align*} \left \{ \begin{array} { l l } \aligned & - \Delta u + V ( x ) u = \left ( I _ \alpha \ast | u | ^ { p } \right ) | u | ^ { p - 2 } u + \lambda u ~ \R ^ N , \\ & u \in H ^ { 1 } \left ( \mathbb { R } ^ { N } \right ) \endaligned \end{array} \right . \end{align*}


\begin{align*} \psi ( t ) : = \frac { 2 \sqrt { n } } { n + 1 } \Bigl ( t ( n + 1 - t ) \Bigr ) ^ { 1 / 2 } + \left | 1 - \frac { 2 t } { n + 1 } \right | . \end{align*}


\begin{align*} A ( t ) = 2 { \rm R e } ( u , u _ t ) + b | | u | | ^ 2 _ { L ^ 2 ( \mathbb { H } ^ n ) } . \end{align*}


\begin{align*} b ( t , \xi ) = 2 \nabla W ( t , \xi ) , \end{align*}


\begin{align*} \lim _ { n \to \infty } \frac { \left | R _ q ( n ) \right | } { | B _ S ( n ) | } = 0 \end{align*}


\begin{align*} u ( z ) = \Re f ( z ) , z \in D . \end{align*}


\begin{align*} \mathcal { H } _ m ( A ) : = S _ m ( \lambda ( A ) ) , \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \sup _ { t \leq T } \Vert X ( t , x _ n ) - X ( t , x ) \Vert _ H = 0 , \ \mathbb { P } . \end{align*}


\begin{align*} \tilde { \mathbf { W } } = ( \mathbf { G } ^ H \mathbf { \Phi } ^ H \mathbf { H } \mathbf { H } ^ H \mathbf { \Phi } \mathbf { G } + \sigma ^ 2 \mathbf { I } _ N ) ^ { - 1 } \mathbf { G } ^ H \mathbf { \Phi } ^ H \mathbf { H } , \end{align*}


\begin{align*} \tilde { y } ^ { k } = \Pi _ { \mathcal { N } } \{ y ^ k - \frac { 1 } { \beta } [ \lambda ^ k - \beta ( \tilde { x } ^ { k } - y ^ k ) ] \} . \end{align*}


\begin{align*} \overline { \Omega } = \bigcup _ { { \bf k } \in K _ { \epsilon } } \epsilon \big ( \overline { Y } + { \bf k } \big ) . \end{align*}


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


\begin{align*} \Delta x _ k + \bar u ^ { - 1 } \nabla \bar u \cdot \nabla x _ k = 0 \mbox { f o r a l l } k = 1 , \dots , n . \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty b _ k ( h _ m ( n ) ) q ^ n \end{align*}


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


\begin{align*} \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x = \int _ { \Omega } q ^ 2 \left ( K _ H ( x ) \nabla \left ( \frac { u } { q } \right ) | \nabla \left ( \frac { u } { q } \right ) \right ) d x . \end{align*}


\begin{align*} \dot { x } ( t ) & = - A ^ { \circ } x ( t ) \quad ( t > 0 ) , \\ y ( t ) & = B ^ { \circ } x ( t ) \quad ( t \geq 0 ) , \\ x ( 0 ) & = x _ 0 \in X ^ \circ . \end{align*}


\begin{align*} e ^ { 2 w ( x ) } : = \left ( \int _ { \Bar { \mathcal F } _ x } \dd \mu _ x \right ) ^ { - 1 } \left ( \int _ { \Bar { \mathcal F } _ x } e ^ { 2 { \bar f } _ x } \dd \mu _ x \right ) , \end{align*}


\begin{align*} ( \varinjlim _ { n } A _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) ) ^ { R G - l a } = \varinjlim _ { n } A _ { H ' , n } \widehat { \otimes } _ { A _ { H ' , m } } S _ { H ' , m } ( V ) . \end{align*}


\begin{align*} \begin{aligned} h \left ( \mathrm { d o m } \ , E ^ \Phi _ A \right ) & = \mathrm { d o m } \ , E ^ \Psi _ B \\ h \left ( \mathrm { d o m } \ , P ^ \Phi _ { \bar A } \right ) & = \mathrm { d o m } \ , P ^ \Psi _ { \bar B } \\ \end{aligned} \end{align*}


\begin{align*} I : = \{ i \in S \mid \} . \end{align*}


\begin{align*} h _ { p } ^ { \left ( 0 \right ) } = \frac { 1 } { 2 \pi i } \int _ { \mathbb { T } } \widetilde { \varphi } _ { 0 } ^ { - } \left ( u \right ) u ^ { p } d u , \end{align*}


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


\begin{align*} y ( q ) = \psi ( q ^ 3 ) \left ( \frac { \varphi _ 6 \varphi _ 9 ^ 2 } { \varphi _ 3 \varphi _ { 1 8 } } - 2 q \psi ( q ^ 9 ) \right ) = \frac { \varphi _ 6 ^ 2 } { \varphi _ 3 } \frac { \varphi _ 6 \varphi _ 9 ^ 2 } { \varphi _ 3 \varphi _ { 1 8 } } - 2 q \psi ( q ^ 3 ) \psi ( q ^ 9 ) = z ( q ^ 3 ) - 2 q \psi ( q ^ 3 ) \psi ( q ^ 9 ) \end{align*}


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


\begin{align*} x '' : = w '' \varepsilon ^ { \mu '' } : = x ' r _ { \mathbf a } = w ' s _ \alpha \varepsilon ^ { \mu ' - ( 1 + \langle \mu ' , \alpha \rangle ) \alpha ^ \vee } < x ' . \end{align*}


\begin{align*} \begin{aligned} E _ { \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 2 } ^ { k + 1 } Z _ i ^ { \mathsf { L } ^ { k + 1 } [ u ] } \biggr ) \biggr ] & = E _ { \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) } \biggl [ \varphi \biggl ( x + u + \sum _ { i = 1 } ^ { k } Z _ i ^ { \mathsf { M } ^ { k } } \biggr ) \biggr ] \\ & = W ( \xi _ 1 ^ { \mathsf { L } ^ { k + 1 } } ( u ) , k , x + u , \mathsf { M } ^ { k } ) . \end{aligned} \end{align*}


\begin{align*} \mathcal { Q } = \int _ { 0 } ^ t v _ 1 ( s , x - \lambda t + \lambda s ) \dd s - \int _ { t _ { 1 , e n } ( x , t ) } ^ { t _ { 1 , e x } ( x , t ) } v _ 1 ( s , x - \lambda t + \lambda s ) \dd s \end{align*}


\begin{align*} \Phi ( n ) : = n \cdot \phi ( n ) , \forall n \in E . \end{align*}


\begin{align*} P _ { n } f ( x ) = \sum _ { | \nu | = n } \widehat { f } ( \nu ) \Phi _ { \nu } ( x ) . \end{align*}


\begin{align*} J _ { \alpha } ( \alpha ( x ) , \alpha ( y ) , [ x , z ] ) = [ J _ { \alpha } ( x , y , z ) , \alpha ^ { 2 } ( x ) ] , \end{align*}


\begin{align*} d ( \phi \wedge \beta ) = d \phi \wedge \beta - \phi \wedge d \beta = d \phi \wedge \beta = \alpha \wedge \beta . \end{align*}


\begin{align*} \mathcal { E } _ { | _ { x \times C } } = \varphi ( x ) , \end{align*}


\begin{align*} \gamma ^ k = \max _ { x _ n ^ D \in \mathcal { N } _ D } 5 0 0 \cdot \frac { | \Gamma _ D | \mathrm { N } _ I \lambda _ d ^ 2 } { | \Omega | \mathrm { N } _ D \lambda _ 1 } \frac { | \nabla w ( x _ n ^ D ) | ^ 2 } { | \nabla w ( y _ n ) | ^ 2 } . \end{align*}


\begin{align*} n = \frac { \beta ^ { 3 } - 3 } { 6 } \end{align*}


\begin{align*} \lambda _ 0 = 1 , \lambda _ t = \prod _ { i = 1 } ^ { t } ( 1 - \alpha _ i ) . \end{align*}


\begin{align*} G ( U , x , t ) = ( \nabla _ V \tilde { \eta } ) ( A ( U , x , t ) , x , t ) \end{align*}


\begin{align*} \begin{aligned} & \frac 1 { \rho _ 1 ^ { n - 2 d _ e \| R ^ \nabla \| ^ 2 _ \infty } } \int _ { B _ { \rho _ 1 } ( x _ 0 ) } \big ( \exp ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) - 1 \big ) \ , d v \\ & \leq \frac 1 { \rho _ 2 ^ { n - 2 d _ e \| R ^ \nabla \| ^ 2 _ \infty } } \int _ { B _ { \rho _ 2 } ( x _ 0 ) } \big ( \exp ( \frac { | | R ^ \nabla | | ^ 2 } 2 ) - 1 \big ) \ , d v \ , , \end{aligned} \end{align*}


\begin{align*} \theta ' _ i = \pi _ i ( X _ 2 , Y _ 2 , \dots , X _ { i - 1 } , Y _ { i - 1 } ) , i \ge 2 . \end{align*}


\begin{align*} S ^ { i j } _ k = \frac { \partial S _ k [ D ^ 2 u ] } { \partial r _ { i j } } . \end{align*}


\begin{align*} d Y _ t = ( \mathcal A Y _ t + \alpha _ t ) d t + \sigma _ t d W _ t , t \in [ 0 , T ] , \end{align*}


\begin{align*} \tau ( a ) = \int _ { X _ A } \hat a \ , d \mu \quad a \in A _ { s a } . \end{align*}


\begin{align*} \bar a _ { i , j } ( f ) = a _ { i , j } * f = \int _ { \mathbb R ^ 3 } a _ { i , j } ( v - v _ * ) f ( v _ * ) d v _ * . \end{align*}


\begin{align*} \prod _ { 1 \leq j \leq N , n _ j \mid n _ 1 } \vert \eta _ { j , ( 1 ) } \vert _ F = \frac { 1 } { p ^ { D } } \end{align*}


\begin{align*} S \left ( \rho , x \right ) = 8 t \rho ^ { 3 } + 2 x \rho . \end{align*}


\begin{align*} \frac D 2 = \{ q _ 1 \mid q _ 1 \overline { q p } q \in D \} . \end{align*}


\begin{align*} \phi ( s , t ) = \gamma ( s ) + \sum _ { i = 1 } ^ { n - 1 } t _ i w _ i , t = ( t _ 1 , \ldots , t _ { n - 1 } ) , \end{align*}


\begin{align*} \left [ \mathbf { F } \right ] _ { i , j } = \frac { 2 } { \sigma _ { m } ^ { 2 } } \mathrm { R e } \Big \{ \sum _ { l = 1 } ^ { L } \frac { \partial \mbox { \boldmath $ \mu $ } \left [ l \right ] ^ { H } } { \partial \xi _ { i } } \frac { \partial \mbox { \boldmath $ \mu $ } \left [ l \right ] } { \partial \xi _ { j } } \Big \} , \ i , j = 1 , \cdots , 4 , \end{align*}


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


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


\begin{align*} | W | q ^ { d ^ 2 - 1 } \theta _ d \le \sum _ { i = 0 } ^ 3 \left | \bigcup _ { F \in C _ i } F \cap M \right | . \end{align*}


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


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


\begin{align*} \lambda ( q ) = \sum \limits _ { 1 \leq n \leq q \atop { n ^ 2 + n + 1 \equiv 0 \ , ( q ) } } 1 \ , , \end{align*}


\begin{align*} \left \{ \begin{aligned} - d y _ t = & [ - k _ 3 p _ t + f _ 0 ( t ) ] d t + [ - k _ 3 q _ t + g _ 0 ( t ) ] d \overleftarrow B _ t - z _ t d W _ t , \\ d p _ t = & [ k _ 2 y _ t + F _ 0 ( t ) ] d t + [ k _ 2 z _ t + G _ 0 ( t ) ] d W _ t - q _ t d \overleftarrow B _ t , \\ y _ T = & \xi , \ , \ , p _ 0 = y _ 0 + \Psi _ 0 . \end{aligned} \right . \end{align*}


\begin{align*} z = \pm \sqrt { \frac { s _ 0 } { ( 1 + a ) ( 1 + a + s _ 0 ) } } , \ w = \sqrt { \frac { 1 + a } { 1 + a + s _ 0 } } \left ( \cos ( t ) + i \sin ( t ) \right ) , \end{align*}


\begin{align*} X = \sqrt { - \frac { \theta } { \log ( Y ) } } \end{align*}


\begin{align*} y \left ( \rho \right ) : = y \left ( \rho , x \right ) = \frac { 1 } { \frac { 1 } { 2 } - i \rho } \sum _ { j = 0 } ^ { \infty } a _ { j } ( x ) z ^ { n } \left ( \rho \right ) , \quad \sum _ { j = 0 } ^ { \infty } \left \vert a _ { j } ( x ) \right \vert ^ { 2 } < \infty . \end{align*}


\begin{align*} \mathbb { T } _ { \rho } ( \mathbb { D } ^ 2 , \{ \mathbf { h } _ 0 ^ { \mathbb { D } ^ 2 } \} ) = \left [ \ell _ 0 ( \mathbf { h } _ 0 ^ { \mathbb { D } ^ 2 } ) , \mathbf { c } _ 0 \right ] . \end{align*}


\begin{align*} \gamma ( \| \cdot \| _ { \infty } , \tau _ { \operatorname { c o } } ) = \beta _ { 0 } = \mu ( \mathrm { C } _ { \operatorname { b } } ( \Omega ) , \mathrm { M } _ { \operatorname { t } } ( \Omega ) ) \end{align*}


\begin{align*} d _ 2 ( \mathbf { x } ( 1 ) , \mathbf { s } ( 1 ) ) \le \mathbf { z } ( 1 ) \le \nu \tilde { \Gamma } \mu = \tilde { \Gamma } \mu ( 1 ) . \end{align*}


\begin{align*} \Vert Y _ n ( t ) \Vert _ H ^ 2 = & \Vert P _ n x \Vert _ H ^ 2 + \int _ 0 ^ t \Big [ 2 \langle A ( s , Y _ n ( s ) ) , Y _ n ( s ) \rangle + \Vert P _ n B ( s , Y _ n ( s ) ) Q _ n \Vert _ { L _ 2 } ^ 2 \Big ] d s \\ & + 2 \int _ 0 ^ t \Big ( Y _ n ( s ) , B ( s , Y _ n ( s ) ) Q _ n d W ( s ) \Big ) . \end{align*}


\begin{align*} R ( \lambda , M _ { q } ) f = \frac { 1 } { \lambda - q } f , f \in \mathrm { C } _ { \operatorname { b } } ( \Omega ) , \end{align*}


\begin{align*} R _ 2 ( t ) & = - i \lambda _ 6 t ^ { - 1 } \mathcal { F } M ( t ) ^ { - 1 } \mathcal { F } ^ { - 1 } \mathcal { N } _ 2 ( \mathcal { F } M { { \mathcal F } } ^ { - 1 } v _ 1 , \mathcal { F } M { { \mathcal F } } ^ { - 1 } v _ 2 ) ( t ) \\ & + i \lambda _ 6 t ^ { - 1 } \mathcal { N } _ 2 ( v _ 1 , v _ 2 ) ( t ) , \end{align*}


\begin{align*} \mathbb { E } _ { \mathbb { P } } \left [ \left ( \int _ { T _ { 0 } } ^ { \left ( T _ { 0 } + \delta \right ) \wedge T } \left | \triangle b _ { t } ^ { i , n , \infty } \right | ^ { 2 } d t \right ) ^ { p } \right ] \leq \frac { p ! \beta ^ { p } \delta ^ { p } } { n ^ { p } } , i = 1 , \cdots , n . \end{align*}


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


\begin{align*} \bar u _ s = \left \{ \begin{array} { l } \alpha _ s , \ \ \ s \in [ t , t + \varepsilon ] , \\ u _ s , \ \ \ , \end{array} \right . \end{align*}


\begin{align*} \dot x = & \ , - \gamma \left ( y _ \delta ( { x } , t ) - \bar { y } \right ) \ , u ( t ) & & { x } ( 0 ) = { x } _ 0 \\ \dot { \bar { y } } = & \ , \gamma \left ( y _ \delta ( { x } , t ) - \bar { y } \right ) & & \bar { y } ( 0 ) = \bar { y } _ 0 \end{align*}


\begin{align*} \left \{ \begin{array} { l } \displaystyle \mathcal { A } ^ * \varphi + \frac { \partial f } { \partial y } ( x , y _ u ) \varphi = \frac { \partial L } { \partial y } ( x , y _ u , u ) \Omega , \\ \varphi = 0 \Gamma . \end{array} \right . \end{align*}


\begin{align*} \frac { 2 } { m ( m - 1 ) } \sum _ { 1 \le i < j \le m } | P _ { u _ i ^ \bot } Z | | P _ { u _ j ^ \bot } Z | \le \left ( \frac { 1 } { m } \sum _ { i = 1 } ^ m | P _ { u _ i ^ \bot } Z | \right ) ^ 2 . \end{align*}


\begin{align*} \frac { m ( I _ k \cap H ) } { m ( I _ k ) } & = \frac { m ( I _ k \cap P ) } { m ( I _ k ) } + \frac { m ( I _ k \cap ( H \setminus P ) ) } { m ( I _ k ) } \\ & < \frac { m ( I _ k \cap P ) } { m ( I _ k ) } + \frac { n _ k + 1 } { 2 ^ { n _ k } } \end{align*}


\begin{align*} { { \bf { h } } _ l } = { \alpha _ l } { \bf { a } } \left ( { { \theta _ l } } \right ) = { \alpha _ l } { \left [ { 1 , { e ^ { - j \pi \cos { \theta _ l } } } , \cdots , { e ^ { - j \pi \left ( { { M _ t } - 1 } \right ) \cos { \theta _ l } } } } \right ] ^ T } . \end{align*}


\begin{align*} d ( \theta ^ k , \tilde { \theta } ^ { k } , \zeta ^ k ) : = \theta ^ k - \tilde { \theta } ^ k - G ^ { - 1 } \zeta ^ k . \end{align*}


\begin{align*} \Phi _ { t } + u \cdot \nabla \Phi & = \mu \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi . \end{align*}


\begin{align*} J = \{ j \in \{ 1 , \dotsc , m \} \colon u _ j \in B _ i v _ j \in B _ i \} \end{align*}


\begin{align*} X = \{ z = ( z _ k ) : | | z | | < \infty \} , \end{align*}


\begin{align*} \widetilde \omega _ a = \begin{cases} 0 , & k \neq 0 , \\ \frac { 1 } { \langle \omega _ a , \theta \rangle } \omega _ a , & k = 0 . \end{cases} \end{align*}


\begin{align*} \operatorname { V o l u m e } \big ( B _ { \rho } ( x _ 0 ) \big ) = o ( { \rho } ^ { \lambda } ) \rho \rightarrow \infty \ , , \end{align*}


\begin{align*} \overline { h } \left ( r \right ) = \int _ { \left \vert x \right \vert > r } \nu _ { \alpha } ^ { \prime } \left ( d x \right ) + r ^ { - 2 } \int _ { \left \vert x \right \vert \leq r } \left \vert x \right \vert ^ { 2 } \nu _ { \alpha } ^ { \prime } \left ( d x \right ) + r ^ { - 1 } \int _ { 1 < \left \vert x \right \vert \leq r } x \nu _ { \alpha } ^ { \prime } \left ( d x \right ) \end{align*}


\begin{align*} \Omega ^ 1 _ A \cong \bigoplus _ { i = 1 } ^ n A . d x _ i , \end{align*}


\begin{align*} \mathcal G _ t = \mathcal A _ t + \gamma S _ { } + 2 \gamma S _ - , \end{align*}


\begin{align*} \Phi ( \omega ) = \{ x \in C ( \omega ) \mid \langle \mathcal { F } ( x ^ * ) ( \omega ) , x - x ^ * ( \omega ) \rangle < 0 \} \end{align*}


\begin{align*} H _ { j , n } = ( - 1 ) ^ { n } \binom { j } { n } ( \beta ) ^ { ( j ) } \sqrt { \frac { n ! } { \left ( \beta \right ) ^ { \left ( n \right ) } } } . \end{align*}


\begin{align*} \lim _ { m \to \infty } T ^ { c _ 2 ( m ) } x _ { 0 0 } = x _ { 0 1 } \quad \textup { a n d } \lim _ { n \to \infty } T ^ { c _ 1 ( n ) } x _ { 0 1 } = x _ { 1 1 } , \end{align*}


\begin{align*} \mu \left ( \left \{ \tau \in \partial _ e ( T ( A ) ) \mid \left | \frac { 1 } { n } \sum _ { i = 0 } ^ { n - 1 } \tau ( \alpha ^ i ( a ) ) - \tau _ \mu ( a ) \right | > \varepsilon \right \} \right ) \le c _ 1 e ^ { - c _ 2 n \varepsilon ^ 2 } . \end{align*}


\begin{align*} \norm { u } _ { L ^ 1 _ m L ^ 2 _ v } : = \sum _ { m \in \mathbb Z ^ 3 } \norm { \mathcal F _ x u ( m , \cdot ) } _ { L ^ 2 _ v } , \end{align*}


\begin{align*} J _ \nu ( t ) = \sum _ { k = 0 } ^ \infty \frac { ( - 1 ) ^ k } { k ! \Gamma ( k + \nu + 1 ) } \left ( \frac { t } { 2 } \right ) ^ { 2 k + \nu } \end{align*}


\begin{align*} w _ { \tau } ( x ) : = \begin{cases} \dfrac { M ( a + b ) } { 2 } & a + \tau < x < b - \tau , \\ [ 1 e x ] w _ { 0 } ( x ) & x \in ( a , b ) \setminus ( a + \tau , b - \tau ) . \end{cases} \end{align*}


\begin{align*} G _ { m , \epsilon } ( t ) & = \int _ 0 ^ 1 K _ m ( t , r ) d \Theta _ { \epsilon } ( r ) \\ & = \Theta _ { \epsilon } ( 1 ) K _ m ( t , 1 ) - \int ^ { 1 } _ { 0 } \Theta _ { \epsilon } ( r ) \frac { \partial K _ m ( t , r ) } { \partial r } d r , \end{align*}


\begin{align*} b ( v , w ) = - ( - ) ^ { \deg v \deg w } b ( w , v ) , \end{align*}


\begin{align*} [ V ] & = \frac { [ \mathbb C _ x ] } { D } \\ D & = \prod _ { n = 1 } ^ \infty ( 1 - e ^ { - n \delta } ) ^ { r } \prod _ { n = 1 } ^ \infty \prod _ { \alpha \in \Delta _ + ^ 0 } ( 1 - e ^ { - \alpha } e ^ { ( 1 - n ) \delta } ) ( 1 - e ^ { \alpha } e ^ { - n \delta } ) . \end{align*}


\begin{align*} \widehat { f } ( \lambda , \omega ) = c _ { n , m } \ , P _ m ( \lambda \omega ) \ , \int _ 0 ^ \infty g ( r ) \ , \frac { J _ { n / 2 + m - 1 } ( \lambda r ) } { ( \lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r . \end{align*}


\begin{align*} \sum _ { \substack { p \leq x \\ ( p , N ) = 1 } } a _ f \left ( p ^ { 2 l } \right ) \ll ( l \log l ) \sqrt { x } \log x \log ( N ( k - 1 ) ) . \end{align*}


\begin{align*} \begin{aligned} v _ t + ( w + u _ { \infty } ) \cdot \nabla v + \nabla p & = ( d + B _ { \infty } ) \cdot \nabla b + \nu \Delta v , \\ b _ t + ( w + u _ { \infty } ) \cdot \nabla b + \nabla q & = ( d + B _ { \infty } ) \cdot \nabla v + \mu \Delta b , \\ \nabla \cdot v = \nabla \cdot b & = 0 , \\ w = \chi _ { \varepsilon } * v , \ d & = \chi _ { \varepsilon } * b . \end{aligned} \end{align*}


\begin{align*} N e w t o n ( F ) \cap \mathbb { Z } ^ m = S u p p ( F ) = \bigcup \limits _ { i = 0 } ^ l S u p p ( s _ { \lambda ^ i } ) = \bigcup \limits _ { i = 0 } ^ l N e w t o n ( s _ { \lambda ^ i } ) \cap \mathbb { Z } ^ m . \end{align*}


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


\begin{align*} D = 3 \left ( \left ( a - b \right ) ^ { 4 } - \left ( a ^ { 4 } + b ^ { 4 } + \left ( a - 1 \right ) ^ { 4 } + \left ( b - 1 \right ) ^ { 4 } \right ) + 1 \right ) \end{align*}


\begin{align*} \tilde { \mathbf { H } } = \mathbf { J } _ { 1 } \mathbf { \Sigma } _ { 1 } \mathbf { J } _ { 1 } ^ { \mathbf { H } } . \end{align*}


\begin{align*} \tau = \frac { \varrho _ z } { | w | } . \end{align*}


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


\begin{align*} G _ n ( x , z ) = \prod _ { j = 1 } ^ d \lambda _ j ^ n e ^ { - \sum _ { j = 1 } ^ d \lambda _ j ( z _ j - x _ j ) } \det ( q _ { n + i - j } ( z _ j - x _ i ) ) _ { i , j = 1 } ^ d . \end{align*}


\begin{align*} \lim \limits _ { | x | \nearrow R } u ( x ) = \lim \limits _ { | x | \nearrow R } v ( x ) = \infty \end{align*}


\begin{align*} \phi ( p ) = ( 1 + | p | ^ { \gamma } ) ^ { \alpha } \quad \gamma > 1 \alpha \in ( 0 , 1 / \gamma ) . \end{align*}


\begin{align*} \mathrm { E } ( X ^ n ) = & \frac { \mu ^ n } { \mathrm { B } ( \alpha , \beta ) } \sum _ { j = 0 } ^ { \infty } ( - 1 ) ^ j \binom { \beta - 1 } { j } \sum _ { i = 0 } ^ n \binom { n } { i } \left ( \frac { \sigma } { \mu } \right ) ^ i \\ & \times \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*} \frac { f ' ( x ) } { f ( x ) } ~ { \rm f o r } ~ f ( x ) = \frac { p ( x ) } { g ( x ) } , ~ g ( x ) = \prod _ { j = 1 } ^ k ( x - z _ j ) , \end{align*}


\begin{align*} q _ 1 ( z ) : = \mathcal { C } \left ( Q _ { j , - } J \right ) ( z ) q _ 2 ( z ) : = \mathcal { C } \left ( Q _ { j , - } J + D _ j \right ) ^ H ( z ) \end{align*}


\begin{align*} \phi ( \sigma , \tau ) = [ \widetilde \sigma , \widetilde \tau ] = \widetilde \sigma \widetilde \tau { \widetilde \sigma } ^ { - 1 } { \widetilde \tau } ^ { - 1 } \in \langle \epsilon \rangle \cong \mathbb Z / 2 \mathbb Z . \end{align*}


\begin{align*} \alpha | z | ^ 2 + \beta | w | ^ 2 + \Re \left ( a z ^ 2 + b w ^ 2 \right ) - 1 = 0 , \end{align*}


\begin{align*} \sum _ { j = 1 } ^ { n } k _ { i j } c _ { r j k } = \sum _ { j = 1 } ^ { n } q _ { r i j } k _ { j k } . \end{align*}


\begin{align*} R _ { D } ^ \pm = R _ { D ^ 0 } ^ \pm + d ^ { D ^ 0 } S | _ { E _ \pm \otimes E _ { \mp } \otimes E _ \pm } , \end{align*}


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


\begin{align*} \frac { \| \hat x - x \| _ 2 } { \| \hat x \| _ 2 } = O \big ( n ^ 4 \ , { \bf u } \ , \kappa ( A ) \ , { \bf g _ { \rm G E P P } } ( A ) \big ) , \end{align*}


\begin{align*} \begin{aligned} | \mathcal { R } _ { 3 , \epsilon } ( t ) | & \leq \vartheta \lambda ^ { 2 } ( \epsilon ) \int _ 0 ^ t 1 + | \varphi ( s ) | + | { X } ^ { \epsilon } ( s ) | d s , \end{aligned} \end{align*}


\begin{align*} r _ { \ell _ 0 } \leq e ^ { ( \delta _ n - L + 5 0 \varepsilon ) q _ n } \max _ { \ell _ 1 = \ell _ 0 \pm 1 } r _ { \ell _ 1 } . \end{align*}


\begin{align*} \max _ { i , j } \left \| \lambda _ i ^ j ( \mathbf { w } _ { \bar { x } \bar { s } } ) - \nu \right \| = \max _ { i , j } \left \| \lambda _ i ^ j ( \mathbf { w } _ { x s } ) - \nu \right \| \le \gamma . \end{align*}


\begin{align*} h ( T ' ) & = y q \log ( y q ) + r \Delta ( q + 1 ) + f ( r ) \\ & = q y \log ( q ) + q y \log ( y ) + r \log ( q ) + r + f ( r ) + o ( 1 ) \\ & = m \log ( m - r ) - r \log ( y ) + r + r \log ( r ) + o ( 1 ) \\ & = m \log ( m ) - r \log \left ( \frac y r \right ) + o ( 1 ) , \end{align*}


\begin{align*} \phi \left ( \begin{pmatrix} 1 & u \\ 0 & 1 \end{pmatrix} \begin{pmatrix} t & 0 \\ 0 & \frac { 1 } { t } \end{pmatrix} \right ) = 2 t \sum _ { n = 1 } ^ \infty \rho _ \phi ( n ) K _ { \nu } ( 2 \pi n t ^ 2 ) \cos ( 2 \pi n u ) . \end{align*}


\begin{align*} G = H \wr \langle t \rangle = A \rtimes \langle t \rangle A = \bigoplus \nolimits _ { i \in \mathbb { Z } } H ^ { t ^ i } \end{align*}


\begin{align*} \lim \limits _ { \alpha \to \infty } \sup _ { \Vert f \Vert _ { L ^ 1 ( \nu ) } \leq 1 } \psi _ \alpha ( K ^ { * } f ) = 0 \iff \lim \limits _ { \alpha \to \infty } \sup _ { \omega = \frac { 1 } { H } \sum _ { h = 1 } ^ { H } \delta _ { a _ h } } \psi _ { \alpha } ( K ^ * \omega ) = 0 \end{align*}


\begin{align*} \mu ( B ( x , r ) ) = \mu ( B ( x , r ) \cap V _ m ( x ) ) . \end{align*}


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


\begin{align*} \tilde { B } = \{ z \in \R ^ n : | \Pi _ { \mathrm { k e r } A ^ \perp } z | ^ 2 \le \delta | \Pi _ { \mathrm { k e r } A } z | ^ { 2 r } \} , \end{align*}


\begin{align*} \Lambda = \bigcup _ { n = 0 } ^ { \infty } \Lambda _ n = \sum _ { k = 0 } ^ { \infty } ( R ^ { t } ) ^ { k } L , \end{align*}


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


\begin{align*} S _ { H ' } ( C ^ \bullet ) \widehat { \otimes } _ { A _ { H ' , \infty } } A : = \varinjlim _ n S _ { H ' , n } ( C ^ \bullet ) \widehat { \otimes } _ { A _ { H ' , n } } A = C ^ \bullet . \end{align*}


\begin{align*} \rho _ k ^ { - 1 } L _ k \overset { \eqref { c o n i c - L } } { = } C + \rho _ k ^ { - 1 } B + \rho _ k ^ { - 1 } L _ { \nabla g } \sum _ { i = 0 } ^ { k - 1 } \rho _ i \eta _ i + \rho _ k ^ { - 2 } \overset { \eqref { d e f 1 } } { \leq } C + \rho _ 0 ^ { - 1 } B + \rho _ 0 ^ { - 1 } L _ { \nabla g } \theta + \rho _ 0 ^ { - 2 } \overset { \eqref { d e f 2 } } { = } L . \end{align*}


\begin{align*} \omega _ { \lambda , z } = \pi ^ * ( d \log \varphi _ { \lambda , z } ) \end{align*}


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


\begin{align*} f ^ \lambda ( z ) = ( 2 \pi ) ^ { - n } \ , | \lambda | ^ n \ , \sum _ { k = 0 } ^ \infty R _ k ^ { n - 1 } ( \lambda , f ) \ , \varphi _ { k , \lambda } ^ { n - 1 } ( z ) . \end{align*}


\begin{align*} L _ { k } ^ { \left ( 1 \right ) } \left ( s , \rho \right ) = \int _ { - \infty } ^ { \infty } \frac { e ^ { i S ( \tau , x - s ) } \tau ^ { k - 1 } } { \tau - \rho } d \tau \end{align*}


\begin{align*} M ( t ) = \sum _ { \substack { j , h : \\ j + h \in \{ q - 1 , q \} } } h \cdot C _ { j , h } ( t ) . \end{align*}


\begin{align*} U _ r ( p ) = \begin{cases} \max \{ \varphi ( p ) + \delta , u ( p ) \} & \\ u ( p ) & \end{cases} \end{align*}


\begin{align*} \mathbb { E } _ { n , m } ' [ ( \alpha + \beta ) ( \gamma + \delta ) ] \\ = \mathbb { E } _ { n , m } [ \chi _ { n , m } ( w , x _ 1 ) \chi _ { n , m } ( w , x _ 2 ) ] \\ \\ = \mathcal { K } _ { n , m } ( x _ 1 , x _ 2 ) \end{align*}


\begin{align*} G '' ( c ) = - s c ^ { s - 2 } \phantom { x x } \phantom { x x } g ( c ) = G ( c ) - G ' ( c ) c = c ^ s \end{align*}


\begin{align*} \gamma ^ { - 1 } S _ { \gamma _ N x } \gamma & = \lbrace \gamma ^ { - 1 } \gamma ' \gamma , \gamma ' \in S _ { \gamma _ N x } \rbrace \\ & = \lbrace \gamma ^ { - 1 } \gamma ' \gamma , \gamma ' _ N \gamma _ N \cdot x \in \Bar { P } ^ 0 \gamma _ N x \rbrace \\ & = \lbrace \gamma ^ { - 1 } \gamma ' \gamma , \gamma _ N ^ { - 1 } \gamma ' _ N \gamma _ N \cdot x \in \Bar { P } ^ 0 x \rbrace \\ & \subset S _ x , \end{align*}


\begin{align*} - K _ X = ( n - 1 ) A _ X , \end{align*}


\begin{align*} \Delta _ \Gamma V + V \big | \nabla _ \Gamma \nu \big | ^ 2 = A \colon D _ \Gamma ^ 2 H + B \cdot \nabla _ \Gamma H + C H [ 0 , T ] \times M \end{align*}


\begin{align*} \Gamma ( G , M ( S ) ) = [ H ^ 0 ( G , r ( M ) ( S ) ) \to H ^ 0 ( G , r ^ 2 ( M ) ( S ) ) \to \dots ] , \end{align*}


\begin{align*} \frac { d \mu _ 1 ^ { n } ( t ) } { d t } = 0 , \end{align*}


\begin{align*} u _ t & { } = ( 1 - t ) u + t ( v - \alpha ) , & v _ s & { } = ( 1 - s ) v + s ( u + \alpha ) . \end{align*}


\begin{align*} [ x , y ] = x \diamond y - \varepsilon ( x , y ) y \diamond x , \ ; \ ; \forall x , y \in \mathcal { H } ( A ) . \end{align*}


\begin{align*} { \hat H } & = - \left ( ( 1 + \lambda r ^ 2 ) \hat { \Delta } + \lambda r \frac { \partial } { \partial r } + \lambda { \hat J } ^ 2 \right ) + { \cal V } ( r ) \\ & = - \left ( ( 1 + \lambda r ^ 2 ) \frac { \partial ^ 2 } { \partial r ^ 2 } + ( d - 1 + d \lambda r ^ 2 ) \frac { \partial } { \partial r } - \frac { { \hat J } ^ 2 } { r ^ 2 } \right ) + { \cal V } ( r ) , . \end{align*}


\begin{align*} \Gamma ^ k _ { i j } = 0 \quad \forall i \neq j \neq k \neq i , \end{align*}


\begin{align*} \delta _ 1 = 2 \big [ \nu F ( \nu ) - J ( q ) \big ] \quad \ , \ , \ , \quad \ , \ , \ , \delta _ 2 = \nu - 2 \ , J ( m ) , \end{align*}


\begin{align*} f ( x ) = \sum _ { \nu } \widehat { f } ( \nu ) \Phi _ { \nu } ( x ) = \sum _ { n = 0 } ^ { \infty } P _ { n } f ( x ) , \end{align*}


\begin{align*} \begin{aligned} | F ^ { i j } ( \nabla _ \beta u ) _ { i j } | \leq \ , & | \nabla _ \beta f ^ { 1 / k } | + C \left ( f ^ { 1 / k } + b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \right ) \\ \leq \ , & C m ^ { 1 / k } m ^ { - 1 / 2 ( k - 1 ) } + C b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \\ \leq \ , & C b _ \alpha ^ { - 1 / 2 } m ^ { 1 / k } + C b _ \alpha \sum _ { i = 1 } ^ n F ^ { i i } \end{aligned} \end{align*}


\begin{align*} G _ n ( x , z ) = \mathrm { d e t } \left ( f _ { n + i - j } ( z _ j - x _ i ) \right ) _ { i , j = 1 } ^ d , \end{align*}


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


\begin{align*} \begin{aligned} u ( x _ 0 ) & = \varphi ( x _ 0 ) = \frac { 1 } { 2 } \sum _ { \beta = \alpha + 1 } ^ { n - 1 } b _ \beta x _ \beta ^ 2 + \frac { 1 } { 6 } \sum _ { \xi , \beta , \gamma \geq \alpha + 1 } \varphi _ { \xi \beta \gamma } ( 0 ) x _ \xi x _ \beta x _ \gamma + O ( | \tilde { x } | ^ 4 ) \\ & \geq \frac { 1 } { 2 } \delta ^ 2 b _ \alpha ^ 2 - C \delta ^ 3 b _ \alpha ^ 2 \geq \frac { 7 } { 1 6 } \delta ^ 2 b _ \alpha ^ 2 \end{aligned} \end{align*}


\begin{align*} \int _ { 0 } ^ { \infty } \omega ( \mu ) \big [ \xi ( \mu , t ) - \xi ( \mu , 0 ) \big ] \ d \mu = \int _ { 0 } ^ { t } \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \mu } \varpi _ 1 ( \mu , \nu ) \xi ( \mu , s ) \xi ( \nu , s ) \Lambda ( \mu , \nu ) \ d \nu d \mu d s , \end{align*}


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


\begin{align*} \int _ { a x } ^ { x } | \theta ( t + \delta t ) - \theta ( t ) - \delta t | ^ 2 d t & > \int _ I | \theta ( t + \delta t ) - \theta ( t ) - \delta t | ^ 2 d t = \delta ^ 2 \int _ I t ^ 2 d t \\ & > \delta ^ 2 \int _ { a x } ^ { x / 2 } t ^ 2 d t = \frac { \delta ^ 2 x ^ 3 } { 3 } \left ( \frac { 1 } { 8 } - a ^ 3 \right ) . \end{align*}


\begin{align*} ( A , B ) \cdot a = a \end{align*}


\begin{align*} W _ r ^ M ( x ) = a _ i i \in \left [ \frac { i - 1 } M , \frac i M \right ) , \end{align*}


\begin{align*} \left | \int _ { \mathbb { R } ^ { 3 } } \nabla \times B \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x \right | = \left | \int _ { \mathbb { R } ^ { 3 } } \nabla \times b \cdot B 1 _ { ( 0 , \infty ) } ( \Phi ) \dd x \right | \lesssim | | \nabla \times b | | _ { L ^ { 2 } } | | b | | _ { L ^ { 2 } } . \end{align*}


\begin{gather*} g ( t ) \allowbreak = \allowbreak \frac { 1 } { ( 1 - t ^ { 2 } ) ^ { \beta } } \left ( 1 + \frac { \rho t ^ { 2 } } { 1 - t ^ { 2 } } \right ) ^ { - \beta } \allowbreak = \\ = \allowbreak \frac { 1 } { ( 1 - t ^ { 2 } ( 1 - \rho ) ) ^ { \beta } } = \sum _ { n \geq 0 } t ^ { 2 n } ( 1 - \rho ) ^ { n } \left ( \beta \right ) ^ { \left ( n \right ) } / n ! . \end{gather*}


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


\begin{align*} \mathcal { U } = \mathrm { k e r } A \cap \S ^ { n - 1 } . \end{align*}


\begin{align*} \left ( d J \theta \right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } = - \left ( d \theta \right ) ^ { ( 2 , 0 ) + ( 0 , 2 ) } _ { J \cdot , \cdot } . \end{align*}


\begin{align*} \gamma _ { \Theta } ^ s ( E ) = \sup \{ | \langle \nu , 1 \rangle | \} , \end{align*}


\begin{align*} x = \epsilon t ^ 2 + 1 , y = 2 t , z = 1 - \epsilon t ^ 2 , t \in F . \end{align*}


\begin{align*} \cdot : G \times G / K \to G / K , x \cdot y K : = x y K \end{align*}


\begin{align*} \partial _ t \Phi ^ { 2 } _ { + } + u \cdot \nabla \Phi ^ { 2 } _ { + } & = 0 \textrm { i n } \ \mathbb { R } ^ { 3 } \times ( 0 , T ) , \end{align*}


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


\begin{align*} \sin \theta \sin ( n \theta ) = \frac { 1 } { 2 } \left [ \cos ( ( n - 1 ) \theta ) - \cos ( ( n + 1 ) \theta ) \right ] = \end{align*}


\begin{align*} \begin{aligned} & \mathbf { T } _ { u } \triangleq \begin{pmatrix} & \mathbf { u } _ { 1 } & \mathbf { u } _ { 2 : n } ^ T \\ & \mathbf { u } _ { 2 : n } & \beta _ { u } \mathbf { I } + \frac { \mathbf { u } _ { 2 : n } \mathbf { u } _ { 2 : n } ^ T } { \beta _ { u } + \mathbf { u } _ { 1 } } \end{pmatrix} , \beta _ { u } = \sqrt { \mathbf { u } _ { 1 } ^ 2 - \| \mathbf { u } _ { 2 : n } \| ^ 2 } \end{aligned} \end{align*}


\begin{align*} 4 \beta \int \rho ^ { - Q + 2 } Z v \ ; \Delta _ { H } v = 2 \beta \int \operatorname { d i v } ( \rho ^ { - Q + 2 } Z ) | \nabla _ { H } v | ^ 2 - 4 \beta \sum _ { i = 1 } ^ { m } \int X _ i v [ X _ i , \rho ^ { - Q + 2 } Z ] v . \end{align*}


\begin{align*} \sum _ { v \in G _ n } \deg _ { G _ n } ( v ) = \sum _ { v \in G _ n } ( X _ v + \bar X _ { v } ) \in ( ( 1 - \delta ) n d , ( 1 + 2 \delta ) n d ) \subset ( n d / 2 , 2 n d ) . \end{align*}


\begin{align*} \mu _ { \rho , \{ \mathcal { D } _ { n } \} } = \delta _ { \rho \mathcal { D } _ { 1 } } * \delta _ { \rho ^ { 2 } \mathcal { D } _ { 2 } } * \delta _ { \rho ^ { 3 } \mathcal { D } _ { 3 } } * \cdots . \end{align*}


\begin{align*} \int _ { \mathfrak X } | \omega | & = { \pi _ s } _ ! \int _ { \mathfrak U } | \omega | = { \pi _ s } _ ! \Big ( { \psi _ s } _ ! \int _ { \mathfrak Y } | \omega | \Big ) = { \phi _ s } _ ! \int _ { \mathfrak Y } | \omega | , \end{align*}


\begin{align*} g ( x ) = \int _ \S W _ r ( y ) \prod _ { i = 1 } ^ n \eta _ i ( y ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) ) \ \d y . \end{align*}


\begin{align*} \displaystyle G ^ { a b } \simeq \Z _ { p } ^ { r } \times \prod _ { s = 1 } ^ { n } \Z / p ^ { i _ { s } } \Z \end{align*}


\begin{align*} q = k \eta ( k ) + \exp ( k \eta ( k ) ) \geq 2 0 ( | s _ 1 | + | s _ 2 | ) \end{align*}


\begin{align*} \begin{aligned} & \ \ \ \ \eta ^ { a m + 1 } ( 3 z ) \eta ^ { b m - 1 } ( z ) \ | \ U ( m ) \\ & \equiv _ m \left ( \sum _ { n = 0 } ^ \infty b _ 3 ( n ) q ^ { n + \frac { m ( 3 a + b ) + 2 } { 2 4 } } \ | \ U ( m ) \right ) \cdot \prod _ { n = 1 } ^ \infty ( 1 - q ^ { 3 n } ) ^ a ( 1 - q ^ { n } ) ^ b . \end{aligned} \end{align*}


\begin{align*} \sum _ x | \tilde p _ x ( \ell ) | ^ 2 = \left ( \frac { 2 \pi \ell } { \theta _ n } \right ) ^ 2 \sum _ x | \tilde q _ x ( \ell ) | ^ 2 . \end{align*}


\begin{align*} F ( u , q ) = 0 \end{align*}


\begin{align*} \lim _ { t \to 0 } j ( t ) = \int _ { \Omega } ( K _ H ( x ) \nabla u | \nabla u ) d x > 0 . \end{align*}


\begin{align*} u _ { \varepsilon } ( x ' , t ) : = u ( x ' , t + \varepsilon ) \quad f _ { \varepsilon } ( x ' ) : = u ( x ' , \varepsilon ) , x ' \in \R ^ n , \ , t > 0 . \end{align*}


\begin{align*} \liminf _ { N \to \infty } \frac { 1 } { N } \sum _ { n = 0 } ^ { N - 1 } \max _ { 1 \leq i < j \leq k } \widetilde { d } ( T _ { \vec { v } } ^ n ( x _ i , t _ i ) , T _ { \vec { v } } ^ n ( x _ j , t _ j ) ) = 0 \end{align*}


\begin{align*} \tilde S ^ { ( q ) } _ { j , j ' } = \frac { 2 \Theta ( \mu _ j , \mu _ { j ' } ) } { \mu _ { j } + \mu _ { j ' } } \tilde F _ { j , j ' } . \end{align*}


\begin{align*} \sigma : = | u _ { , i j k } | ^ 2 \end{align*}


\begin{align*} H _ { j , n } = E X ^ { j } h _ { n } ( X ) = \frac { 1 } { \sqrt { n ! } } E X ^ { j } H _ { n } ( X ) = \left \{ \begin{array} { c c c } 0 & & n > j j - n \\ \frac { j ! } { 2 ^ { ( j - n ) / 2 } ( ( j - n ) / 2 ) ! \sqrt { n ! } } & & j - n \end{array} \right . \end{align*}


\begin{align*} & | X ^ \beta ( 0 ; t , x ) - X ^ \beta ( 0 ; t , y ) | \le | x - y | + \int _ 0 ^ t \left | \frac { d } { d s } X ^ \beta ( s ; t , x ) - \frac { d } { d s } X ^ \beta ( s ; t , y ) \right | \dd s \\ & = | x - y | + \int _ 0 ^ t | u ( X ^ \beta ( s ; t , x ) , s ) - u ( X ^ \beta ( s ; t , y ) , s ) | \dd s \\ & \le | x - y | + \int _ 0 ^ t \varphi _ \Theta ( | X ^ \beta ( s ; t , x ) - X ^ \beta ( s ; t , y ) | ) B \dd s . \end{align*}


\begin{align*} \theta ^ { k + 1 } = \theta ^ { k } - \alpha d ( \theta ^ { k } , \tilde { \theta } ^ { k } , \zeta ^ k ) , \end{align*}


\begin{align*} \begin{aligned} \int _ { \mathbb { R } } s _ 1 ( z ) s _ 2 ( z ) d z = \int _ { \mathbb { R } } \left [ \mathcal { P } ^ { - } ( s J ) + s J \right ] \left [ \mathcal { P } ^ { - } ( s J ) \right ] ^ { H } \mathrm { d } z = 0 . \end{aligned} \end{align*}


\begin{align*} \lim _ { x \to \infty } f _ s ( x ) / f ( x ) = ( 1 - e ^ { - \lambda _ s } ) ^ { - 1 } . \end{align*}


\begin{align*} a ( Z ) = a \big ( ( Z ^ * Z ) ^ \frac { 1 } { 2 } \big ) \end{align*}


\begin{align*} \mathbf { U } _ V = \mathbf { I } - \mathbf { u } _ { \lambda _ 1 } ( \mathbf { u } _ { \lambda _ 1 } ^ { H } \mathbf { u } _ { \lambda _ 1 } ) ^ { - 1 } \mathbf { u } _ { \lambda _ 1 } ^ { H } , \end{align*}


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


\begin{align*} \mathbb { P } ( \hat { \eta } ^ { D , M } ( D \times [ b , \infty ) ) < \infty ) = 1 , \end{align*}


\begin{align*} \rho ( \xi , Q _ n ) _ p = \max _ { z \in Q _ n ^ \perp } \frac { \langle \xi , z \rangle } { \| z \| _ { p ' } } \ge \frac { \langle \xi , P \xi \rangle } { \| P \xi \| _ { p ' } } . \end{align*}


\begin{align*} k _ 3 ( \langle H \rangle , a ) = \frac { 1 } { 2 } \times \mathop { \sum \sum } _ { \chi _ 4 ( x - y ) \in \{ 1 , \chi _ 4 ( g ) \} } 1 , \end{align*}


\begin{align*} Z = \prod _ { k \in \N } Z _ k , \end{align*}


\begin{align*} \phi _ 1 ' = \phi _ 3 , \end{align*}


\begin{align*} C F ( F ) = \widehat { t e l } ( C F ( F ) , i d ) \rightarrow C ^ { L , k } = \varinjlim _ { p } \varprojlim _ { q } \varinjlim _ { n _ { i } } C F ^ { L , k } _ { [ p , q ] } ( H _ { n _ { i } , t } ) . \end{align*}


\begin{align*} \widehat { \iota } ( D ) = \frac { \beta - e } { d } \end{align*}


\begin{align*} \theta _ { i , ( 1 ) } ( f ) = \sum _ { j } - a _ { i , j } \theta _ { j , ( 2 ) } ( f ) . \end{align*}


\begin{align*} \lambda ^ { - m } \ , \int _ { S ^ { n - 1 } } \widehat { f } ( \lambda , \omega ) \ , Y _ m ( \omega ) \ , d \omega = c _ { n , m } \int _ 0 ^ \infty r ^ { - m } f _ m ( r ) \ , \frac { J _ { n / 2 + m - 1 } ( \lambda r ) } { ( \lambda r ) ^ { n / 2 + m - 1 } } r ^ { n + 2 m - 1 } d r \end{align*}


\begin{align*} \int _ { - \infty } ^ \infty z ^ i \ , \phi _ s ( z ) \ , [ \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \mathrm { d } z = & \int _ 0 ^ \infty z ^ i \ , \phi _ s ( z ) \ , [ \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \mathrm { d } z \\ & + \int _ 0 ^ \infty ( - 1 ) ^ i \ , z ^ i \ , \phi _ s ( z ) [ 1 - \Phi _ s ( z ) ] ^ { j + \alpha - 1 } \mathrm { d } z . \end{align*}


\begin{align*} y _ i = a _ c s _ c \left ( \hat { \mathbf { h } } _ { i , * } + \tilde { \mathbf { h } } _ { i , * } \right ) \mathbf { p } _ c + \sum _ { j = 1 } ^ { M } a _ j s _ j \left ( \hat { \mathbf { h } } _ { i , * } + \tilde { \mathbf { h } } _ { i , * } \right ) \mathbf { p } _ j + n _ i . \end{align*}


\begin{align*} \ell ( x s _ \alpha , s _ \alpha v \beta ) = & \ell ( x , v \beta ) + \ell ( s _ \alpha , s _ \alpha v \beta ) \\ = ~ \ , & \ell ( x , v \beta ) + \begin{cases} 1 , & v \beta = - \alpha , \\ - 1 , & v \beta = \alpha , \\ 0 , & v \beta \neq \pm \alpha . \end{cases} \end{align*}


\begin{align*} E = \bigcap _ { k = 0 } ^ \infty E _ k . \end{align*}


\begin{align*} d _ n ( K , \ell _ \infty ^ { N ' } ) \le \delta , K : = \{ \xi ' ( \omega ) \colon \Lambda ( \omega ) = \Lambda ^ \circ , \ ; \Gamma ( \omega ) = \Gamma ^ \circ \} . \end{align*}


\begin{align*} \det ( Y ) ^ { k / 2 } | F ( Z ) | \ll _ { n , \epsilon } \begin{cases} k ^ { \frac { 3 n ( n + 1 ) } { 8 } } \det ( Y ) ^ { \frac { n + 1 } { 4 } + \epsilon } \\ k ^ { \frac { n ( n + 1 ) } { 4 } } \det ( Y ) ^ { \frac { 3 ( n + 1 ) } { 8 } + \epsilon } . \end{cases} \end{align*}


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


\begin{align*} 0 = D f | _ { \Gamma _ 0 } ( \Delta ) = \phi ( C \Delta + \Delta ^ T C ^ T + \Delta ^ T D \Gamma _ 0 + \Gamma _ 0 ^ T D \Delta ) . \end{align*}


\begin{align*} { \tt P h i 1 = p l g c i r m a p ( v e r , i n f ) } \end{align*}


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


\begin{align*} h _ { 1 1 i i } = h _ { i i 1 1 } + \kappa _ 1 ^ 2 \kappa _ i - \kappa _ 1 \kappa _ i ^ 2 - \kappa _ 1 + \kappa _ i . \end{align*}


\begin{align*} Z _ { V } ( [ \alpha ] ) = - \int _ { V } \alpha + \sqrt { - 1 } \int _ { V } \omega , \end{align*}


\begin{align*} \lim _ { n \to \infty } \| T _ n p _ n f - p _ n T f \| = 0 f \in X . \end{align*}


\begin{align*} \left \{ \begin{aligned} g ( x ) & : = { \rm s g n } ( x ) f ( | x | ) , \\ \tilde V _ w ^ { \varepsilon } ( x , t ) & : = \frac { 1 } { 2 } ( h ( | x | ) ) ^ 2 - \dot y ( t ) h ( | x | ) + \varepsilon f ' ( | x | ) - \frac { \varepsilon } { 2 } y ( t ) h '' ( | x | ) . \end{aligned} \right . \end{align*}


\begin{align*} M ^ d \times ( - \infty , 0 ) \ni ( x , z ) \mapsto \mathcal { S } ( x , z ) = ( x , \varrho ( x , z ) ) \in \Omega , \end{align*}


\begin{align*} \vert S \cap [ 1 , n ] \vert \leq \sum _ { \ell = 1 } ^ U \vert S _ { \ell } \vert = O ( U ^ 2 ) = O ( ( \log n ) ^ 2 ) = o ( n / \log n ) \end{align*}


\begin{align*} \sum _ { n = 0 } ^ \infty b _ 5 \left ( \frac { m n - 1 } { 6 } \right ) q ^ n \in S _ { 2 m - 2 } ( \Gamma _ 0 ( 1 8 0 ) , \chi _ 5 ) _ m , \end{align*}


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


\begin{align*} w ( x ' , t ) = W ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x ' ) ) , \ \ \varphi ( x ' , t ) = \varPhi ( \bar { R } _ { - \alpha | \ln \varepsilon | t } ( x ' ) ) , \end{align*}


\begin{align*} E _ { \alpha , \beta } ( z ) = \sum _ { k = 0 } ^ { + \infty } \frac { z ^ k } { \Gamma ( \alpha \ , k + \beta ) } , \ \alpha > 0 , \ \beta , z \in \C . \end{align*}


\begin{align*} h : = b \vert _ { E _ 1 } + \sum \limits _ { k = 2 } ^ \delta e ^ { 2 f _ k } b \vert _ { E _ k } + g , \end{align*}


\begin{align*} \phi = \frac { 1 } { 2 } ( u _ { , p } \xi ^ p + v _ q x ^ q ) . \end{align*}


\begin{align*} \delta _ n ' = \frac { \ln q _ { n + 1 } + \ln | \sin ( \pi ( \theta - \frac { 1 } { 2 } + m _ n \alpha ) ) | } { q _ n } \leq \frac { \ln q _ { n + 1 } + \ln ( \pi \| q _ n \alpha \| ) } { q _ n } < \frac { \ln \pi } { q _ n } . \end{align*}


\begin{align*} \varphi _ { 0 } ^ { - } \left ( \rho \right ) : = \frac { 1 } { 2 \pi i } \int _ { - \infty } ^ { \infty } \frac { \varphi _ { 0 } \left ( \tau \right ) } { \tau - \rho } d \tau , \quad \rho \in \mathbb { R } , \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 ) \\ & = \frac { 1 } { e ^ { - \psi } } \left ( d \phi ( x ) \int _ { y } ^ \infty e ^ { - { \psi ( x , v ) } } d v - \int _ { y } ^ \infty e ^ { - { \psi ( x , v ) } } d _ x { \psi } ( x , v ) d v \right ) . \end{align*}


\begin{align*} k ( t , x , y ) : = \rho ( - R ^ { - 1 } B P _ t x - R ^ { - 1 } B y ) \quad \end{align*}


\begin{align*} f _ * \mathcal L ^ \vee ( d ) [ 2 d ] = ( f _ * \mathcal L ) ^ \vee \end{align*}


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


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


\begin{align*} R \sqrt { R } \geq 8 R \geq 8 \max ( 4 , 8 \| g \| ) > C = 4 \max ( 1 , \sqrt { \| g \| } ) , \end{align*}


\begin{align*} F _ { q } ( s - s _ { 0 } ) = L _ { q } \left [ f ( t ) \exp _ { q } \left [ \frac { s _ { 0 } t } { 1 - ( 1 - q ) s t } \right ] \right ] . \end{align*}


\begin{align*} \tilde { u } ( x , t ) = \begin{cases} u ( x , t ) , & t \in [ 0 , T _ ] \\ u _ 1 ( x , t ) , & t \in [ T _ , T _ + T _ 1 ] , \end{cases} \end{align*}


\begin{align*} \mathbf { R } _ { x s } = \left ( \mathbf { R } _ { x s } ^ i : i = 1 , \cdots , k \right ) , \end{align*}


\begin{align*} H _ { x , x ' } ^ { ( n ) } = \frac { 1 } { ( n + 1 ) ^ 2 } \sum _ { j , j ' = 0 } ^ n \Phi \left ( \frac { \theta j } { n + 1 } , \frac { \theta ' j ' } { n + 1 } \right ) \exp \left \{ \frac { 2 i \pi j x } { n + 1 } \right \} \exp \left \{ \frac { 2 i \pi j ' x ' } { n + 1 } \right \} \end{align*}


\begin{align*} \frac { \partial \textbf { \textit { W } } } { \partial t } - ( B - Q ) \textbf { \textit { W } } = - \textbf { \textit { Y } } . \end{align*}


\begin{align*} \begin{cases} \partial _ t u = \Delta u - ( u \cdot \nabla ) u - \nabla p - \nabla \cdot ( \nabla n \otimes \nabla n ) , \\ \nabla \cdot u = 0 , \\ \partial _ t n = \Delta n - ( u \cdot \nabla ) n - \Phi ( n ) , \\ u = 0 \quad \frac { \partial n } { \partial \nu } = 0 \quad \partial \mathcal { O } , \\ u ( 0 ) = u _ 0 , n ( 0 ) = n _ 0 , \end{cases} \end{align*}


\begin{align*} \widetilde { X } _ n ( t ) = & P _ n x + \int _ 0 ^ t P _ n A ( s , \widetilde { X } _ n ( s ) ) d s + \int _ 0 ^ t P _ n B ( s , \widetilde { X } _ n ( s ) ) Q _ n d \widetilde { W } ( s ) , t \in [ 0 , T ] . \end{align*}


\begin{align*} \prescript J { } \pi ( s _ \beta w ) = ( w ' , \mu + \Phi ^ + ( - \beta ) w ^ { - 1 } \beta ^ \vee ) \end{align*}


\begin{align*} \left \{ \begin{array} { l l } - \Delta v + u v = f , & x \in \Omega , \\ \nabla v \cdot \nu + v = \eta , & x \in \partial \Omega \end{array} \right . \end{align*}


\begin{align*} f ( x ) = \sum _ { 1 \le i < j \le M } \left | \det \begin{pmatrix} x _ i & x _ j \\ z _ i & z _ j \end{pmatrix} \right | \quad \hbox { a n d } g ( x ) = \sum _ { 1 \le i < j < k \le M } \left | \det \begin{pmatrix} x _ i & x _ j & x _ k \\ y _ i & y _ j & y _ k \\ z _ i & z _ j & z _ k \end{pmatrix} \right | . \end{align*}


\begin{align*} x = \sum _ { n \geq 1 } \mu _ d ( n ) y ^ n . \end{align*}


\begin{align*} D '' u _ { m , m ' , \epsilon , j } + P _ { m , m ' } \big ( \sqrt { 2 \pi b N _ 1 \delta \tilde \delta _ m + N _ 1 \tilde { \lambda } _ { m ' } } h _ { m , m ' , \epsilon , j } \big ) = \lambda \end{align*}


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


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


\begin{align*} \frac { \partial f } { \partial y } ( t z ) = - \frac { \psi _ 0 } { \psi _ 1 } \frac { \partial f } { \partial x } ( t z ) . \end{align*}


\begin{align*} p : v = v _ 1 \xrightarrow { \alpha _ 1 } v _ 2 \xrightarrow { \alpha _ 2 } \cdots \xrightarrow { \alpha _ { n - 1 } } v _ n = v ' \end{align*}


\begin{align*} A _ i = \{ t \in [ 0 , 1 ] \colon d ( \gamma ( t ) , p _ i ) = d ( p , q ) / n \} . \end{align*}


\begin{align*} \frac { 1 } { r } \left ( \Phi _ t + u \cdot \nabla \Phi - \mu \left ( \Delta - \frac { 2 } { r } \partial _ r \right ) \Phi \right ) = 0 \textrm { o n } \ L ^ { 2 } ( \mathbb { R } ^ { 3 } ) . \end{align*}


\begin{align*} g x _ i = a _ { i 1 } x _ 1 + \cdots + a _ { i n } x _ n \end{align*}


\begin{align*} | s _ { h , q } - s _ h | \le \epsilon _ 0 = 1 / 2 ^ { b _ 0 } \end{align*}


\begin{align*} \mathbf { y } = \sum _ { k \in \mathcal { K } } \mathbf { h } _ { k } x _ { k } + \mathbf { n } _ { \textrm { u l } } , \end{align*}


\begin{align*} f _ { N } ( x , y | \rho ) = f _ { N } ( x ) f _ { N } ( x ) \sum _ { n \geq 0 } \rho ^ { n } h _ { n } ( x ) h _ { n } ( y ) , \end{align*}


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


\begin{align*} D '' u _ { \epsilon , j } = D '' \left ( ( 1 - v ' _ { \epsilon } ( \Psi ) ) f F ^ { 1 + \delta } \right ) . \end{align*}


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


\begin{align*} a _ 0 b _ 0 = 1 , \ \sum _ { k = 0 } ^ n \Gamma ( k + 1 - \alpha ) \Gamma ( \alpha + n - k ) a _ { n - k } b _ k = 0 , \ n \ge 1 . \end{align*}


\begin{align*} \Gamma _ { r } = \mathbb { Z } ^ { d } _ { * } \cap \bigg \{ \mathbf { k } : \sum _ { j } \langle j \rangle ^ { \eta } { \bf k } _ { j } w _ { j } \geq r | { \bf k } | _ { \eta } , \sum _ { j } w _ { j } = 1 , w _ j > 0 \bigg \} , \end{align*}


\begin{align*} c _ { \psi } = \sin ^ 2 ( \alpha ) \int _ 0 ^ { 2 \pi } \frac { \psi _ 0 \cos \theta + \psi _ 1 \sin \theta } { \left ( \dfrac { \cos \theta } { r _ 0 ^ 2 } - \dfrac { \sin \theta \cos \alpha } { r _ 0 r _ 1 } \right ) \psi _ 0 + \left ( \dfrac { \sin \theta } { r ^ 2 _ 1 } - \dfrac { \cos \theta \cos \alpha } { r _ 0 r _ 1 } \right ) \psi _ 1 } d \theta \end{align*}


\begin{align*} ( [ K \delta K ] ( \chi \cdot \phi ) ) ( \sigma , g ) & = \sum _ i \delta _ i \cdot ( \chi ( \phi ) ) ( \sigma , g \delta _ i ) \\ & = \sum _ i \chi ( \det ( g \delta _ i ) ) \delta _ i \cdot \phi ( \sigma , g \delta _ i ) \\ & = \chi ( \det \delta ) \chi ( \det g ) \sum _ i \delta _ i \cdot \phi ( \sigma , g \delta _ i ) . \end{align*}


\begin{align*} \mathcal { F } F C l _ { D i f f ( S ^ 1 ) } ^ { * } ( S ^ 1 , V ) = \mathcal { F } C l ^ { * } ( S ^ 1 , V ) \rtimes D i f f ( S ^ 1 ) . \end{align*}


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


\begin{align*} \left | \sum _ { j = 0 } ^ { n - 1 } A ( \sigma ^ j ( x ) ) - \sum _ { j = 0 } ^ { n - 1 } A ( \sigma ^ j ( y ) ) \right | < \epsilon \ , \end{align*}


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


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


\begin{align*} \varepsilon ' = \frac { \varepsilon d ( p _ { n + 1 } , q _ { n + 1 } ) } { C _ { n + 1 } ( 1 + d ( p _ { n + 1 } , q _ { n + 1 } ) ) } \end{align*}


\begin{align*} \Delta { { \mathbf { y } } ^ { T } } \mathbf { A x } - \Delta { { \mathbf { x } } ^ { T } } \left ( { { \mathbf { A } } ^ { T } } \mathbf { y } + \mathbf { C } \mathbf { x } + \mathbf { s } \right ) = 0 . \end{align*}


\begin{align*} h ( \nu | \mu ) : = \limsup _ { n \to \infty } \frac { 1 } { \abs { \Lambda _ n } } h _ { \Lambda _ n } ( \nu | \mu ) , \end{align*}


\begin{align*} \sum _ { 1 \leq i , j \leq n } \Psi ( v _ i v _ i ^ * v ' _ j ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi \left ( \sum _ { i = 1 } ^ n v _ i v _ i ^ * v ' _ j \right ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi \left ( p v ' _ j \right ) { v ' _ j } ^ * = \sum _ { j = 1 } ^ n \Psi ( v ' _ j ) { v ' _ j } ^ * . \end{align*}


\begin{align*} \forall z \in \Z ^ d \ \forall i = 1 , \dots , q \ \forall \eta \in \Omega : \sum _ { \Delta \Subset \Z ^ d } \sum _ { \xi _ { \Delta } } \nabla _ z ^ i \left ( c _ { \Delta } ( \cdot , \xi _ { \Delta } ) - \hat { c } ( \cdot , \xi _ { \Delta } ) \right ) ( \eta ) = 0 . \end{align*}


\begin{align*} \begin{aligned} \eta ( U | \bar U ; x , t ) & \doteq \eta ( U , x , t ) - \eta ( \bar U , x , t ) - G ( \bar U , x , t ) \cdot ( A ( U , x , t ) - A ( \bar U , x , t ) ) \\ & = \tilde { \eta } ( V , x , t ) - \tilde { \eta } ( \bar V , x , t ) - ( \nabla _ V \tilde { \eta } ) ( \bar V , x , t ) \ ; ( A ( U , x , t ) - A ( \bar U , x , t ) ) \ , \\ & \doteq \tilde { \eta } \big ( V | \bar V ; x , t \big ) \ , , \end{aligned} \end{align*}


\begin{align*} F ( X ) = \sum _ { i = 1 } ^ n f _ i ( x _ i ) , ~ G ( Y ) = \sum _ { i = 1 } ^ n g _ i ( y _ i ) . \end{align*}


\begin{align*} \left ( s ^ H + \| N \| ^ 2 - 2 \| \theta \| ^ 2 \right ) \| \theta \| ^ 2 = 0 . \end{align*}


\begin{align*} K _ { \lambda , \varrho ( I ) } = \sum _ { J \subset I } \kappa _ { \lambda , J } , \end{align*}


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


\begin{align*} \# \{ \beta \in o \mid \varphi ( v ' \beta ) < 0 \} = & \# \{ \beta \in - s _ { \alpha } ( o ) \mid \varphi ( v ' \beta ) < 0 \} \\ = & \# \{ \beta \in o \mid \varphi ( - v \beta ) < 0 \} \\ \leq & \# \{ \beta \in o \mid \varphi ( v \beta ) \geq 0 \} \\ = & 2 - \# \{ \beta \in o \mid \varphi ( v \beta ) < 0 \} . \end{align*}


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


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


\begin{align*} d ^ \nabla R ^ \nabla = 0 \ , . \end{align*}


\begin{align*} Y = \left ( \frac { \eta } { m + 1 } x _ { n + 1 } ^ { m + 1 } + \lambda x _ { n + 1 } \right ) e _ { n + 1 } . \end{align*}


\begin{align*} 2 \widetilde y _ n & = 2 e ^ { r n ( A + a ) + r A } \\ & > ( e ^ { - r A } + e ^ { r a } ) e ^ { r n ( A + a ) + r A } \\ & = e ^ { r n ( A + a ) } + e ^ { r ( n + 1 ) ( A + a ) } \\ & = \widetilde x _ n + \widetilde x _ { n + 1 } . \end{align*}


\begin{align*} \tau _ { k } ^ \star = \frac { \sqrt { 1 + \iota _ { k } } \tilde { \mathbf { h } } _ { k } ^ H \mathbf { w } _ { k } } { \sum _ { p \in \mathcal { K } } | \tilde { \mathbf { h } } _ { k } ^ H \mathbf { w } _ { p } | ^ 2 + \sigma _ { k } ^ 2 } , \forall k \in \mathcal { K } . \end{align*}


\begin{align*} H ( v , w , t ^ { - 1 } ) = t ^ { ( _ { \bullet } v - w ) \cdot ( _ { \bullet } v - w ) - _ { \bullet } v \cdot _ { \bullet } v } \frac { \mathit { g l } ( _ { \bullet } v , t ) } { \mathit { g l } ( _ { \bullet } v - w , t ) } . \end{align*}


\begin{align*} \tilde { h } _ d ( x ) = x \log d + \log \left ( \frac { \Gamma ( x ) } { \Gamma \left ( x + \frac { 1 } { 2 } \right ) \Gamma \left ( x + \frac { d - 1 } { 2 } \right ) } \right ) + \log \left ( \frac { 2 ^ { d - 2 } \Gamma \left ( \frac { d } { 2 } \right ) ^ 2 } { \sqrt { \pi } d ^ { \frac { d - 1 } { 2 } } } \right ) . \end{align*}


\begin{align*} Q _ h : = { P } _ { k - 1 } ^ { \mathrm { d i s c } } ( \mathcal { T } ) \cap Q . \end{align*}


\begin{align*} q _ \pi ( G _ 0 ) = q _ \pi ( V \times K ) = q _ \pi ( V ) q _ \pi ( K ) . \end{align*}


\begin{align*} P _ { \min } ^ { { \rm { S - N } } } \left ( { \overline S , \overline \varepsilon , \overline R } \right ) = \bigcap { \left \{ { P \in { { \mathbb { R } } ^ + } : \left ( { \overline S , \overline R } \right ) \in { \mathcal { R } } _ { { \rm { S v B } } } ^ { { \rm { S - N } } } \left ( { W , P , K , \overline \varepsilon } \right ) } \right \} } . \end{align*}


\begin{align*} \begin{aligned} { \bf { d } } \left [ { i - { \kappa _ l } } \right ] & = { { { \bf { \bar x } } } _ { \rm { t } } } \Big [ { \left \lfloor { \frac { { i - { \kappa _ l } + { N _ { \rm { C P } } } } } { { K + { N _ { \rm { C P } } } } } } \right \rfloor , } \Big . \\ & \ \ \ \ \Big . { { \rm { m o d } } \left ( { i - { \kappa _ l } + { N _ { \rm { C P } } } , K + { N _ { \rm { C P } } } } \right ) - { N _ { \rm { C P } } } } \Big ] . \end{aligned} \end{align*}


\begin{align*} \begin{aligned} \dfrac { \partial h ( s ) } { \partial s } & = \left . \dfrac { \partial h ( s ) } { \partial s } \right | _ x + \left . \dfrac { \partial ^ 2 h ( s ) } { \partial s ^ 2 } \right | _ x ( s - x ) + O ( ( s - x ) ^ 2 ) . \\ \end{aligned} \end{align*}


\begin{align*} \frac { 1 } { A r } = h ^ q , \end{align*}


\begin{align*} \begin{aligned} X ^ { \epsilon } ( t ) & = x _ { 0 } + \int ^ { t } _ { 0 } b ( X ^ { \epsilon } ( s ) ) d s - \lambda ( \epsilon ) \epsilon ^ { 2 } \int ^ { t } _ { 0 } \sigma ( X ^ { \epsilon } ( s - ) ) \theta _ { \epsilon } ( s - ) \tilde { N } ^ { \epsilon ^ { - 2 } } ( d s ) \\ & \ \ \ \ - \frac { \lambda ( \epsilon ) \epsilon ^ { 2 } } { 2 } \int _ 0 ^ t \sigma ( X ^ { \epsilon } ( s - ) ) d \theta _ { \epsilon } ( s ) . \end{aligned} \end{align*}


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


\begin{align*} \ell ( a , b , \theta ) = & n [ \log ( 2 \theta ) - \log \{ \operatorname { B } ( a , b ) \} ] - 3 \sum ^ n _ { i = 1 } \log ( x _ { i } ) - a \theta \sum ^ n _ { i = 1 } \frac { 1 } { x ^ 2 _ i } \\ & + ( b - 1 ) \sum ^ { n } _ { i = 1 } \log \left \{ 1 - \exp \left ( - \frac { \theta } { x _ i ^ 2 } \right ) \right \} . \end{align*}


\begin{align*} \lim _ { k \to + \infty } \int _ { \tau } ^ { \tau + T } \widehat { Q } _ 1 ^ k ( t ) \d t = 0 . \end{align*}


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


\begin{align*} \prescript J { } \pi ( w _ 1 ) = ( w _ 1 ' , \mu _ 1 ) , \prescript J { } \pi ( w _ 2 ) = ( w _ 2 ' , \mu _ 2 ) . \end{align*}


\begin{align*} \prod _ { i = 1 } ^ k | V _ i \setminus X _ i | - \frac { m ^ k } { 4 } \geq m ^ k \left ( \left ( 1 - \frac { 1 } { 2 k ^ 2 } \right ) ^ k - \frac { 1 } { 4 } \right ) \geq m ^ k \left ( 1 - \frac { 1 } { 2 k } - \frac { 1 } { 4 } \right ) \geq \frac { m ^ k } { 2 } \end{align*}


\begin{align*} x _ \Upsilon = \prod _ { \tau \in \Upsilon _ d } x _ \tau . \end{align*}


\begin{align*} B _ { a } : = \begin{pmatrix} 1 \\ & a \\ & & \ddots \\ & & & 1 \\ & & & & a \end{pmatrix} \end{align*}


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


\begin{align*} x ^ 2 + x y + y ^ 2 = n \ , , ( x , y ) = 1 \ , , x \in \mathbb { N } \ , , y \in \mathbb { Z } \setminus \{ 0 \} \end{align*}


\begin{align*} \dot x ( J ) = \dot x _ { \mathcal { W } , g } ( J , u ) , \dot y ( J ) = \dot y _ { \mathcal { W } , g } ( J , u ) , \dot z ( J ) = \dot z _ { \mathcal { W } , g } ( J , u ) \end{align*}


\begin{align*} C _ { R P I - R I } = O \{ \beta ( 2 N - 2 ) ^ 2 + 2 N - 3 \} \end{align*}


\begin{align*} \lim _ { k \to \infty } \sup _ { z _ 0 \in \mathbb { R } } \int _ { B ( z _ 0 e _ z , R ) } | b _ { n _ { k } } | ^ { 2 } \dd x = 0 . \end{align*}


\begin{align*} \Phi _ { X _ 0 } ( X ) = \frac { d } { d t } \big \vert _ { t = 0 } ( \varphi _ { X _ 0 } \circ \exp ) ( t X ) . \end{align*}


\begin{align*} L ( x ) = x ^ { 2 ^ r } + t x . \end{align*}


\begin{align*} \mu _ r ( \psi ( \cdot ) ) = \mu _ r ( \cdot ) : = \sum _ { i = 1 } ^ { \infty } i ^ r \psi _ i ( \cdot ) . \end{align*}


\begin{align*} \sigma _ { n - 1 } ( \kappa ) = \sigma , \partial \Sigma = \Gamma , \end{align*}


\begin{align*} \dot g ( J ) \in A t ^ { \rho ( \dot g ( J ) ) } , \end{align*}


\begin{align*} t \cdot ( U , V ) = ( U D ( \overline { t } ) , D ( t ) V ) , \end{align*}


\begin{align*} \lambda _ S = \langle \alpha , \beta \rangle \alpha - \langle \alpha , \alpha \rangle \beta . \end{align*}


\begin{align*} \sum _ { K \ \ L } \ \sum _ { \widetilde { K } \ \ \pi ^ { - 1 } ( K ) } ( e _ { \widetilde { K } } - 1 ) \langle [ \widetilde { K } ] , \alpha \rangle = 0 . \end{align*}


\begin{align*} \mathcal { L } f = - \Gamma ( \sqrt \mu , f ) - \Gamma ( f , \sqrt \mu ) , \Gamma ( g , h ) = \mu ^ { - { 1 \over 2 } } Q _ L ( \sqrt { \mu } g , \sqrt { \mu } h ) . \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 ( x , 0 ) = X _ { 0 } ( x ) , \end{array} \right . \end{align*}


\begin{align*} & \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | F _ { t _ 0 , \tilde { c } } | ^ 2 _ h c ( - \Psi ) \\ = & \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | \tilde F | ^ 2 _ h c ( - \Psi ) + \int _ { \{ z \in M : - \Psi ( z ) \in S _ { i , n } \backslash U _ k \} } | F _ { t _ 0 , \tilde { c } } - \tilde F | ^ 2 _ h c ( - \Psi ) . \end{align*}


\begin{align*} ( a / b ) ^ k = ( \alpha / \beta ) ^ l . \end{align*}


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


\begin{align*} & \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ { n } \cdot B _ { n } \dd x = \int _ { \Omega } \textrm { c u r l } ^ { - 1 } B _ { 0 , n } \cdot B _ { 0 , n } \dd x = h _ n , \\ & \mathcal { I } _ { h _ n } \leq \frac { 1 } { 2 } \int _ { \Omega } \left ( | u _ { n } | ^ { 2 } + | B _ { n } | ^ { 2 } \right ) \dd x \leq \frac { 1 } { 2 } \int _ { \Omega } \left ( | u _ { 0 , n } | ^ { 2 } + | B _ { 0 , n } | ^ { 2 } \right ) \dd x , \end{align*}


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


\begin{align*} \sigma \left ( \sum a _ n \varepsilon ^ n \right ) = \sum a _ n ^ q \varepsilon ^ n . \end{align*}


\begin{align*} f ( x ) = a x ^ { q + 1 } + b x ^ q + c x + d \end{align*}


\begin{align*} z ^ { - 1 } k ^ { - 1 } b ( k ) = - \frac { 1 } { 2 } \widehat { \bar { u } _ x K \psi ^ - _ { 1 1 } } ( z ) + i \widehat { \bar { u } } ( z ) . \end{align*}


\begin{align*} \epsilon _ { f \otimes \chi _ D } = \chi _ D ( - r ) \mu ( q ' ) q '^ { 1 / 2 } \lambda _ f ( q ' ) \epsilon _ f . \end{align*}


\begin{align*} Y _ n ( t ) = P _ n x + \int _ 0 ^ t P _ n A ( s , Y _ n ( s ) ) d s + \int _ 0 ^ t P _ n B ( s , Y _ n ( s ) ) Q _ n d W ( s ) , \end{align*}


\begin{align*} { \frac { 1 } { \mathcal { A } _ { \gamma } ( s ) } } = { \frac { 1 } { q - s } } + { \gamma } , \end{align*}


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


\begin{align*} \mathrm { E } ( Z ^ s ) = \sum _ { r = 0 } ^ \infty w _ { r } ( \alpha , \beta ) \ , \tau _ { s , r + a - 1 } \mathrm { E } ( Z ^ s ) = \sum _ { i , j = 0 } ^ \infty \sum _ { r = 0 } ^ j w _ { i , j , r } ( \alpha , \beta ) \ , \tau _ { s , r } , \end{align*}


\begin{align*} \langle C , C \rangle = \sum _ { i = 1 } ^ s | c _ { i } | ^ { 2 } \langle g _ { i } , g _ { i } \rangle . \end{align*}


\begin{align*} ( g _ t , u _ t ) = \psi _ t ^ * \big ( \left . \big ( \bar g , \bar u \big ) \right | _ { M \setminus \Omega _ t } \big ) . \end{align*}


\begin{align*} V _ { \lambda / \mu } : = L _ T , \end{align*}


\begin{align*} & \| y _ { \bar u + \theta ( u - \bar u ) } - \bar y \| _ { C ( \bar \Omega ) } = \| \phi + \theta ( y _ u - \bar y ) \| _ { C ( \bar \Omega ) } \\ & \le ( C _ 2 C _ { f , K _ U } \sqrt { | \Omega | } \| y _ u - \bar y \| _ { C ( \bar \Omega ) } + 1 ) \| y _ u - \bar y \| _ { C ( \bar \Omega ) } . \end{align*}


\begin{align*} \Delta _ + ^ 0 = \{ \alpha _ i - \alpha _ j | 1 \leq 1 < j \leq r \} \end{align*}


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


\begin{align*} N ( x ) \cdot N ( y ) = N \big ( N ( x ) \cdot y + x \cdot N ( y ) - N ( x \cdot y ) \big ) , \quad \forall \ x , y \in A . \end{align*}


\begin{align*} \prod _ { j = 1 } ^ d q _ j ^ { - x _ j } \left ( \left ( \frac { q _ i } { 1 - q _ i } \right ) ^ { j - 1 } q _ i ^ { x _ j } \right ) _ { i , j = 1 } ^ d . \end{align*}


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


\begin{align*} \psi _ \epsilon = \sup _ { \| x - x ' \| < \delta , \| y - y ' \| \le \delta } u ( x ' , y ' ) \end{align*}


\begin{align*} ( \tilde h , \tilde v ) : = ( \hat h , \hat v ) + \big ( L _ Y \bar g , Y ( \bar u ) \big ) \end{align*}


\begin{align*} \lim _ { | z | \rightarrow \infty } \Psi ^ - _ { 1 1 } ( x ; z ) = e ^ { - i c _ - ( x ) } . \end{align*}


\begin{align*} \lim _ { \delta \to 0 } \lim _ { \epsilon \to 0 } \lambda ^ 2 ( \epsilon ) \log \mathbb P \Big ( \sup _ { t \in [ 0 , 1 ] } | X ^ { \epsilon } ( t ) - \varphi ( t ) | < \delta , \tilde { \chi } _ 1 < 1 \Big ) = - \infty . \end{align*}


\begin{align*} d X _ t = \left ( b _ 0 ( t , X _ t ) + \langle b ( t , X _ t , \cdot ) , \mu \rangle \right ) d t + d B ^ H _ t , \quad \operatorname { L a w } ( X ) = \mu , \quad \operatorname { L a w } ( X _ 0 ) = \mu _ 0 . \end{align*}


\begin{align*} \lim _ { \substack { n \rightarrow \infty \\ M \rightarrow \infty } } \Vert X _ n ( t \wedge \tau _ u ^ M ) - X ( t \wedge \tau _ u ^ M ) \Vert _ H = 0 , \mathbb { P } \otimes d t . \end{align*}


\begin{align*} H ( \widetilde { Q } ) \le \sum _ { i = 1 } ^ { n } \tilde { t } _ i H ( Q _ i ) , \end{align*}


\begin{align*} \forall \eta _ \Lambda \in \Omega _ \Lambda : \sum _ { \Delta \subset \Lambda } \sum _ { \xi _ \Delta } \mu ( \eta _ \Lambda ) c _ \Delta ( \eta _ \Lambda , \xi _ \Delta ) = \sum _ { \Delta \subset \Lambda } \sum _ { \xi _ \Delta } \mu ( \xi _ \Delta \eta _ { \Lambda \setminus \Delta } ) c _ \Delta ( \xi _ \Delta \eta _ { \Lambda \setminus \Delta } , \eta _ \Delta ) , \end{align*}


\begin{align*} \mathcal { I } - d ^ { - } ( p , H ) & = \mathcal { I } - d _ { - } ( p , H ) = \mathcal { I } - \liminf x _ n = \mathcal { I } - \liminf ( 1 - y _ n ) \\ & = 1 - \mathcal { I } - \limsup y _ n = 1 - \mathcal { I } - d ^ { - } ( p , H ^ { c } ) \end{align*}


\begin{align*} R \cdot R = L \ , \ , Q ( g , R ) , \end{align*}


\begin{align*} \int _ { A _ { n , j } \cap A _ n } & \int _ { \eta _ j ( K ) } ^ { \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) } D ^ p ( x _ { \tau _ n + t } ( \omega ) ) d t d \P \\ & \ge c \delta ^ p K ^ { 2 r p } \int _ { A _ { n , j } \cap A _ n \cap \{ \bar { \tau } \ge T \} } ( \tau _ { n + 1 } ( \omega ) - \tau _ n ( \omega ) ) d \P - c \eta _ j ( K ) \delta ^ p K ^ { 2 r p } \P ( A _ { n , j } \cap A _ n ) . \end{align*}


\begin{align*} \mathbf M = \frac 1 { 2 | x - y | ^ 2 } \begin{pmatrix} 2 h ^ 2 & \ ( 1 + \cos S _ c ) \rho h - 2 r h \\ 2 \rho h - ( 1 + \cos S _ c ) r h & \ 2 ( r - \rho ) ^ 2 \end{pmatrix} . \end{align*}


\begin{align*} \sigma _ { n - 1 } ( \kappa ) = \sigma , \partial \Sigma = \Gamma . \end{align*}


\begin{align*} \bar { v } _ { i } : = \begin{pmatrix} 0 \\ 0 \\ 0 \\ 0 \\ v _ { i } \end{pmatrix} \in R ^ { 2 n } . \end{align*}


\begin{align*} u _ j ( x ) & = \int _ { x - r } ^ { x + r } \prod _ { \stackrel { i = 1 } { i \neq j } } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) ( - \partial \eta _ j ( x ) ) \sin ^ { [ n ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) \ \d y , \\ q ( x ) & = \int _ { x - r } ^ { x + r } \prod _ { i = 1 } ^ n ( \eta _ i ( y ) - \eta _ i ( x ) ) \sin ^ { [ n + 1 ] } ( \tilde \Psi ( y ) - \tilde \Psi ( x ) ) ( - \partial \tilde \Psi ( x ) ) \ \d y \end{align*}


\begin{align*} \left ( \lambda _ 1 I - \frac { \partial } { \partial x _ 1 } \right ) \cdots \left ( \lambda _ d I - \frac { \partial } { \partial x _ d } \right ) h ( x ) = \prod _ { j = 1 } ^ d \lambda _ j h ( x ) \end{align*}


\begin{align*} \sum _ { k = 0 } ^ { m } \gamma _ { k } y _ { k } \ge 0 \ \ \forall \ y \in \Omega _ { 0 } \end{align*}


\begin{align*} & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { 1 } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } c ( s ) e ^ { - s } d s \\ \le & \limsup _ { n \to + \infty } \sum _ { i = 0 } ^ { n - 1 } \frac { \sup _ { I _ { i , n } \backslash U _ k } c ( t ) } { \inf _ { I _ { i , n } \backslash U _ k } c ( t ) } \int _ { I _ { i , n } \backslash U _ k } e ^ { - s } d s \\ = & \int _ { ( t ' _ 2 , t ' _ 1 ] \backslash U _ k } e ^ { - s } d s . \end{align*}


\begin{align*} \hat { \varphi } ( \tau , w ( \cdot ) ) = \inf \limits _ { u ( \cdot ) } J ( \tau , w ( \cdot ) , u ( \cdot ) ) . \end{align*}


\begin{align*} \| u \| _ { \mathcal Q ^ { \alpha , p } } : = \left ( \int _ { \mathbb R ^ N } \left | I _ \frac \alpha 2 \ast | u | ^ { p } \right | ^ 2 d x \right ) ^ { \frac 1 { 2 p } } \end{align*}


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


\begin{align*} V ( r ) = \frac { L ( L + 1 ) } { r ^ 2 } - \frac { Q } { r } f + \kappa B _ 1 \frac { r } { f } + \kappa B _ 2 \frac { 1 } { f ^ 2 } + \kappa B _ 3 \frac { r } { f ^ 3 } + \kappa B _ 4 \frac { 1 } { f ^ 4 } + \kappa B _ 5 \frac { r } { f ^ 5 } + \kappa B _ 6 \frac { 1 } { f ^ 6 } . \end{align*}


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


\begin{align*} \iota ( A ) = e _ { r } ^ { ( \pi s _ r ) ( \lambda ) _ r - \pi ( \lambda ) _ r } ( A ) . \end{align*}


\begin{align*} \tilde { x } ^ { k } = \Pi _ { \mathcal { C } } \{ x ^ k - \frac { 1 } { \beta r } [ F ( x ^ { k } ) - \lambda ^ k + \beta ( x ^ k - y ^ k ) ] \} . \end{align*}


\begin{align*} \alpha \circ T = T \circ \beta ~ ~ ~ ~ T ( a ) \ast T ( b ) = T \big ( \mathfrak { l } ( T ( a ) ) b + \mathfrak { r } ( T ( b ) ) a \big ) . \end{align*}


\begin{align*} & \psi ^ \pm ( x , t ; k ) = e ^ { - i c _ - \sigma _ { 3 } } \mu ^ \pm ( x , t ; k ) e ^ { i c \sigma _ 3 } , \\ & \mu ^ \pm ( x , t ; k ) = e ^ { - i c _ - \sigma _ { 3 } } \varphi ^ \pm ( x , t ; k ) e ^ { - i \left ( k ^ 2 x + \eta ^ 2 t \right ) \sigma _ 3 } C ^ \pm ( k ) e ^ { i \left ( k ^ 2 x + \eta ^ 2 t \right ) \sigma _ 3 } e ^ { i c \sigma _ 3 } . \end{align*}


\begin{align*} \nabla \mathcal { N } _ { \epsilon } ( x ) = W _ { L } \cdot \sigma _ { \epsilon } ' ( N _ { \epsilon } ^ { ( L - 1 ) } ( x ) ) \cdot W _ { L - 1 } \cdot \ldots \cdot \sigma _ { \epsilon } ' ( N _ { \epsilon } ^ { ( 1 ) } ( x ) ) \cdot W _ { 1 } . \end{align*}


\begin{align*} c _ 1 ( q , k , p ) : = \frac { 1 } { 4 } \Big ( \hat W _ r ( q - k ) + \hat W _ r ( q + k ) - 2 \hat W _ r ( q ) - ( 4 \lambda + 2 \mu ) \hat W _ r ( q ) \Big ) , \end{align*}


\begin{align*} u ( x , 0 ) = q ( x ) , x \in \left ( - \infty , \infty \right ) , \end{align*}


\begin{align*} v _ a ( u ) = \lim _ { r \to 0 } { \frac { \sigma ( a , r ) } { { r } ^ { \frac { 4 n ( m - 1 ) } { m } } } } \end{align*}


\begin{align*} \lim \nolimits _ { k \to \infty } C ( l ' , r ' ; P _ { k , S } ^ { l , r } ) = \delta _ S ( l , r ; \ , l ' , r ' ) , \end{align*}


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


\begin{align*} \mathbf { u } _ { \lambda _ { 1 } } = \left [ \begin{array} { l } \mathbf { u } _ { \lambda _ { 1 1 } } \\ \mathbf { u } _ { \lambda _ { 1 2 } } \end{array} \right ] \end{align*}


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


\begin{align*} \boldsymbol { A ^ * A } x = \| \boldsymbol { A } \| ^ 2 x . \end{align*}


\begin{align*} f ( \kappa ) = \sigma , \partial \Sigma = \Gamma , \end{align*}


\begin{align*} | k _ i ^ N ( t , \nu ^ { N , 1 } _ { \boldsymbol { x } } , \ldots , \nu ^ { N , N } _ { \boldsymbol { x } } , \lambda ^ { N , 1 } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } , \ldots , \lambda ^ { N , N } _ { \boldsymbol { \phi } ( t , \boldsymbol { x } ) } ) - k \big ( t , \nu ^ { N , i } _ { \boldsymbol { x } } , \Phi ( t , \nu ^ { N , i } _ { \boldsymbol { x } } ) \big ) | = O \Big ( \frac { 1 } { N } \Big ( 1 + \frac { 1 } { N } \underset { j \neq i } \sum | x ^ i - x ^ j | \Big ) \Big ) ; \end{align*}


\begin{align*} \sigma ( \alpha + \beta ) = \sigma ( \alpha ) + \sigma ( \beta ) \end{align*}


\begin{align*} \partial \Omega _ \varepsilon = \{ x + \varepsilon \rho ( x ) \mathbf { n } _ x \mid x \in \partial \Omega \} , \end{align*}


\begin{align*} I ( u ) = m | | u | | ^ 2 _ { L ^ 2 ( \mathbb { H } ^ n ) } + | | \nabla _ { H } u | | ^ 2 _ { L ^ 2 ( \mathbb { H } ^ n ) } - { \rm R e } \int _ { \mathbb { H } ^ n } \overline { u } f ( u ) d x . \end{align*}


\begin{align*} \frac { \partial \mathbb { E } \left [ \varepsilon | \hat { \mathbf { H } } \right ] } { \partial a _ i } = & 4 a _ i \sum _ { l = 1 } ^ { M - 1 } \sum _ { q = l + 1 } ^ { M } \Re \left [ \hat { \phi } ^ { \left ( l , i \right ) ^ * } \hat { \phi } ^ { \left ( q , i \right ) } \right ] - 2 \Re { \left \{ \hat { \phi } ^ { \left ( i , i \right ) } \right \} } + 4 a _ i \left ( \sum _ { l = 1 } ^ M \lvert \hat { \phi } ^ { \left ( l , i \right ) } \rvert ^ 2 + M \sigma _ { e , i } ^ 2 \lVert \mathbf { p } _ i \rVert ^ 2 \right ) . \end{align*}


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


\begin{align*} T _ n ^ v \cong \bigoplus _ { s = q + 1 } ^ { p } R / ( v ( \kappa ^ n _ s ) / v ( \mu ^ { n + 1 } _ { \alpha ( s ) } ) ) , \end{align*}


\begin{align*} [ X + \xi , Y + \eta ] _ H = \mathcal { L } _ X ( Y + \eta ) - \iota _ Y d \xi + H ( X , Y , \cdot ) , \end{align*}


\begin{align*} n - m + f = 2 , \end{align*}


\begin{align*} & \frac { ( n - 2 ) ( 1 0 - n ) } { 4 } \int _ { B _ { \rho } } r ^ { 2 - n } u _ r ^ 2 \d x \\ & \leq C \int _ { B _ { 2 \rho } \setminus B _ { \rho } } r ^ { 2 - n } | \nabla u | ^ 2 \d x + C \varepsilon \int _ { B _ { 4 \rho } } r ^ { 3 - n } | \nabla u | ^ 2 \d x , \rho \leq 1 / 4 \varepsilon \leq \varepsilon _ 0 , \end{align*}


\begin{align*} \partial _ \theta \Phi = \partial _ \lambda \Phi = \partial _ t \Phi = 0 \end{align*}


\begin{align*} \psi _ i = \left \{ \begin{aligned} & \sigma ^ 0 & & ( i \in S ) \\ & \tau & & ( i \notin S ) . \end{aligned} \right . \end{align*}


\begin{align*} c _ s ^ 2 \partial _ { z 1 } T = - \frac { 1 } { \Delta t } { \varsigma } _ 3 { m } _ 3 ^ { ( 1 ) } , \end{align*}


\begin{align*} v ^ { - 1 } \sum _ { k = 1 } ^ N ( \sigma \circ w _ \infty ) ^ k \mu _ \infty \in X _ \ast \otimes \mathbb Q \end{align*}


\begin{align*} \varphi ( y ) \geq \inf \left \{ \varphi ( y ) \ | \ | y | = R \right \} , | y | \leq R . \end{align*}


\begin{align*} & S ' ( h , v ) = 0 \mbox { i n a c o l l a r n e i g h b o r h o o d $ U $ o f $ \hat \Sigma $ i n } M \setminus \Omega \\ & \left \{ \begin{array} { l } h ^ \intercal = 0 \\ H ' ( h ) = 0 \end{array} \right . \mbox { o n } \hat \Sigma . \end{align*}


\begin{align*} \sum _ { \ell \in L _ { i } } c _ { \ell } ( 0 ) \cdot u _ { i } ^ { \gamma _ { \ell } } = 0 \end{align*}


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


\begin{align*} A _ 2 ^ 2 \le \frac { 1 } { n } \sum _ { i = 1 } ^ n \mathcal { W } _ 1 ^ 2 ( P _ i , Q ^ * _ i ) \le \frac { 1 } { n } \sum _ { i = 1 } ^ n \mathcal { W } _ 2 ^ 2 ( P _ i , Q ^ * _ i ) \le \frac { 2 R _ f } { \kappa n } . \end{align*}


\begin{align*} ( r , q ) = 1 \ , . \end{align*}


\begin{align*} \alpha ^ { \tau } \oplus \beta ^ { \tau } = \{ \gamma _ 1 \oplus \gamma _ 2 \mid \gamma _ 1 \in \alpha ^ { \tau } , \gamma _ 2 \in \beta ^ { \tau } \} ^ { \tau } . \end{align*}


\begin{align*} g = D ^ 2 u , u \in C ^ \infty ( N , \R ) . \end{align*}


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


\begin{align*} { \bf { x } } \left [ n \right ] = \sum \limits _ { l = 1 } ^ L { { { \bf { f } } _ l } s \left [ { n - { \kappa _ l } } \right ] } , \end{align*}


\begin{align*} a ( D ( x ) V ) = a ( D ( x ) D ( t ) V _ 1 ) = a ( D ( t ) D ( x ) V _ 1 ) = a ( D ( x ) V _ 1 ) , \end{align*}


\begin{align*} J _ 1 + J _ 3 = 0 . \end{align*}


\begin{align*} f ( z ) = C \int _ 0 ^ z \frac { \zeta ^ \alpha ( \zeta - 1 ) ^ \beta ( \zeta - t ) ^ \gamma } { ( \zeta - z _ 0 ) ^ 2 ( \zeta - \overline { z } _ 0 ) ^ 2 } \ , d \zeta + z _ 1 , \end{align*}


\begin{align*} \delta z = - \frac { 1 } { 8 } { F _ z } . \end{align*}


\begin{align*} \begin{aligned} G _ { \epsilon } ( t _ 1 , \cdots , t _ k ) & = \Big ( \int ^ { 1 } _ { 0 } K ( t _ 1 , r ) \theta _ { \epsilon } ( r ) d r , \cdots , \int ^ { 1 } _ { 0 } K ( t _ k , r ) \theta _ { \epsilon } ( r ) d r \Big ) ^ { \tau } \\ & = \Big ( G _ { \epsilon } ( t _ 1 ) , \cdots , G _ { \epsilon } ( t _ k ) \Big ) \end{aligned} \end{align*}


\begin{align*} \frac { \left ( 6 n - \beta ^ { 3 } + 3 \right ) \left ( n ^ { 2 } + n + 1 \right ) } { 3 } = 0 \end{align*}


\begin{align*} \operatorname { S p } ( \mathcal { L } _ { 1 , \alpha } \otimes I _ { H ^ k ( \mathbb { R } ^ { d - 1 } ) } + I _ { H ^ k _ { \alpha } ( \mathbb { R } ) } \otimes \Delta _ y ) = \operatorname { S p } ( \mathcal { L } _ { 1 , \alpha } ) + \operatorname { S p } ( \Delta _ y ) . \end{align*}


\begin{align*} F ( u ) = \int _ { a } ^ { b } f ( x ) \ , K ( u , x ) d x . \end{align*}


\begin{align*} \mu _ d ( n ) = \sum _ { \substack { n _ 1 , \ldots , n _ d \geq 1 \\ \prod _ { i = 1 } ^ d n _ i = n } } \mu ( n _ i ) \end{align*}


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


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


\begin{align*} \bar { \mathbf { A } } = \mathbf { A } \mathbf { D } ^ T , \bar { \mathbf { C } } = \mathbf { D } \mathbf { C } \mathbf { D } ^ T . \end{align*}


\begin{align*} d \hat { \sigma } _ i ( T , X _ j ) = - T ( r ^ j _ i ) + X _ j ( \lambda _ i ( T ) ) - \lambda _ i ( \nabla ^ v _ { X _ j } T ) + \lambda _ j ( T ) r ^ j _ i = 0 , \quad \forall j \neq i . \end{align*}


\begin{align*} \det D ^ { 2 } u = \mu , \end{align*}


\begin{align*} [ ( \lambda _ 1 , \mu _ 1 ) , \kappa _ 1 ] \cdot [ ( \lambda _ 2 , \mu _ 2 ) , \kappa _ 2 ] : = [ ( \lambda _ 1 + \lambda _ 2 , \mu _ 1 + \mu _ 2 ) , \kappa _ 1 + \kappa _ 2 + \lambda _ 1 \mu _ 2 ^ t - \mu _ 1 \lambda _ 2 ^ t ] . \end{align*}


\begin{align*} p = 1 \oplus 0 \oplus 1 \oplus 0 \oplus \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} , q = 0 \oplus 1 \oplus 1 \oplus 0 \oplus \begin{pmatrix} a ^ 2 & a b \\ a b & b ^ 2 \end{pmatrix} \end{align*}


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


\begin{align*} \mathbb { E } e ^ { - \langle Z _ { \lambda } ^ { S _ { n } } , f \rangle } = \mathbb { E } e ^ { - \langle Y _ { \infty } ^ { S _ { n } } , f \rangle } , \end{align*}


\begin{align*} s _ 3 = \frac { ( \lambda + \mu ) ( \lambda + 2 \mu ) } { \lambda + 3 \mu } , s _ 4 = \frac { 2 \mu ( \lambda + 2 \mu ) } { \lambda + 3 \mu } , t _ 1 = \frac { \mu } { 2 | \xi | } \biggl ( \frac { 1 - 2 s _ 1 } { \tau - 2 \mu s _ 1 | \xi | } - \frac { 1 } { \tau - 2 \mu | \xi | } \biggr ) . \end{align*}


\begin{align*} Y = m ( X ) + \eta \end{align*}


\begin{align*} \lambda _ k = \frac 1 4 \hat W _ r ( q ) \Big [ \hat W _ r ( q + k ) + \hat W _ r ( q - k ) - 2 \hat W _ r ( q ) \Big ] \end{align*}


\begin{align*} { \cal P } = [ O _ N , \dots , O _ N , P _ { i j } ^ { ( \ell ) } , O _ N , \dots , O _ N ] \in \R ^ { N \times N L } \quad \mathrm { w i t h } P _ { i j } ^ { ( \ell ) } = \mathbf { e } _ i \mathbf { e } _ j ^ T \in \R ^ { N \times N } . \end{align*}


\begin{align*} & \partial _ t \left ( \int _ 0 ^ t U ( t , s ) e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) d s \right ) \\ = & \int _ 0 ^ t \partial _ t U ( t , s ) e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) d s + U ( t , t ) e ^ { ( \alpha - 1 ) W ( t ) } g ( y ( t ) ) \\ = & \int _ 0 ^ t A ( t ) U ( t , s ) e ^ { ( \alpha - 1 ) W ( s ) } g ( y ( s ) ) d s + e ^ { ( \alpha - 1 ) W ( t ) } g ( y ( t ) ) . \end{align*}


\begin{align*} ( 2 \pi ) ^ { - n } | \lambda | ^ n \ , \int _ { \C ^ n } \widehat { f } ( a , w ) \varphi _ { k , \lambda } ^ { n - 1 } ( z - w ) e ^ { \frac { i } { 2 } \lambda \Im ( z \cdot \bar { w } ) } d w = f ^ { - \lambda } \ast _ { - \lambda } \varphi _ { k , \lambda } ^ { n - 1 } ( z ) = \widehat { f } ( a , z ) . \end{align*}


\begin{align*} \omega : = - p _ 1 ^ * ( \log \sqrt { \tau _ 0 } ) _ z d z + p _ 2 ^ * \omega _ 0 . \end{align*}


\begin{align*} \langle \alpha ^ \vee , 2 \rho \rangle = \frac 1 2 \left ( \langle \alpha ^ \vee , 2 \rho \rangle + \langle s _ \alpha ( \alpha ^ \vee ) , s _ \alpha ( 2 \rho ) \rangle \right ) = \frac 1 2 \langle \alpha ^ \vee , 2 \rho - s _ \alpha ( 2 \rho ) \rangle . \end{align*}


\begin{align*} b ^ n _ l = \left ( l + \frac { 1 } { 2 } \right ) I _ { n , l } ~ , ~ a ^ n _ l = - \frac { 1 } { n } \left ( l + \frac { 1 } { 2 } \right ) I _ { n , l } \end{align*}


\begin{align*} F ( m ; z ) = \sum _ { n = 0 } ^ \infty b _ 3 \left ( \frac { m n - 1 } { 1 2 } \right ) q ^ n , \end{align*}


\begin{align*} \begin{array} { l } R _ { \max } = \mathop { \max } \limits _ { \bf p } { R _ T } \\ { \rm { s } } { \rm { . t } } { \rm { . } } \\ C 2 , \ , C 3 , \ , C 8 , \ , { \rm a n d } \ , C 9 . \end{array} \end{align*}


\begin{align*} \begin{aligned} \Big | \langle \boldsymbol { \nu } , A _ t \rangle - \bar A _ t + \langle \boldsymbol { \nu } , f _ { \alpha , x _ \alpha } \rangle - \bar f _ { \alpha , x _ \alpha } ) \Big | & \le c _ 3 \langle \boldsymbol { \nu } , \eta ( \lambda | \bar U ; x , t ) \rangle = c _ 3 \mathcal { H } ( \boldsymbol { \nu } , U , \bar { U } ; x , t ) \ ; . \end{aligned} \end{align*}


\begin{align*} & \Gamma _ 1 ( X ) = \sum \limits _ { 1 \leq d \leq z } \mu ( d ) \Sigma \big ( X , d ^ 2 \big ) \ , , \\ & \Gamma _ 2 ( X ) = \sum \limits _ { d > z } \mu ( d ) \Sigma \big ( X , d ^ 2 \big ) \ , , \\ & \Sigma \big ( X , d ^ 2 \big ) = \sum \limits _ { 1 \leq n \leq X \atop { n ^ 2 + n + 1 \equiv 0 \ , ( d ^ 2 ) } } 1 \ , , \\ & \sqrt { X } \leq z < X \ , , \end{align*}


\begin{align*} \lim _ { p \to \infty } \frac { \psi ( p ) } { p } = 0 . \end{align*}


\begin{align*} g _ i ^ { ( e q ) } = { { w } _ i } T , \end{align*}


\begin{align*} \Bigl [ I _ { n + 1 } \frac { \partial } { \partial x _ n } , Q \Bigr ] f = I _ { n + 1 } \frac { \partial } { \partial x _ n } ( Q f ) - Q \Bigl ( I _ { n + 1 } \frac { \partial } { \partial x _ n } \Bigr ) f = \frac { \partial Q } { \partial x _ n } f , f \in C ^ { \infty } ( \Omega ) . \end{align*}


\begin{align*} \mathcal S ( f ) = \left ( \frac { f '' } { f ' } \right ) - \frac 1 2 \left ( \frac { f '' } { f ' } \right ) ^ 2 \end{align*}


\begin{align*} \sum _ { i = 1 } ^ m f _ i ( u ) = 0 , \end{align*}


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


\begin{align*} H _ \epsilon = \bigcap _ { b \in B _ 1 \times \cdots \times B _ k } T ^ { - \overline { \eta ( \epsilon ) } \cdot b } E \end{align*}


\begin{align*} \frac { 2 d + 1 } { 2 d } \mathbb { E } [ h ^ { V } | \sigma ( h _ { z } ^ { V } : z \in V \backslash U ) ] & = \mathbb { E } [ \mathbb { E } [ h _ { x } ^ { V } | \sigma ( h _ { z } ^ { V } : z \in V \backslash \{ x \} ) ] | \sigma ( h _ { z } ^ { V } : z \in V \backslash U ) ] \\ & = \mathbb { E } \Big [ \frac { 1 } { 2 d } \sum _ { i } h _ { x \pm e _ { i } } - \frac { 1 } { 2 d ^ { 2 } } \sum _ { i < j } h _ { x \pm e _ { i } \pm e _ { j } } - \frac { 1 } { 4 d ^ { 2 } } \sum _ { i } h _ { x \pm 2 e _ { i } } \Big | \sigma ( h _ { z } ^ { V } : z \in V \backslash U ) \Big ] . \end{align*}


\begin{align*} C ( z ) = \sum _ { i } c _ { i } g _ { i } ( z ) \end{align*}


\begin{align*} f ( x ) = \frac { 1 } { \sigma \mathrm { B } ( \alpha , \beta ) } \ , \left [ \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \alpha - 1 } \left [ 1 - \Phi _ s \left ( \frac { x - \mu } { \sigma } \right ) \right ] ^ { \beta - 1 } \phi _ s \left ( \frac { x - \mu } { \sigma } \right ) . \end{align*}


\begin{align*} \lim _ { n \rightarrow \infty } \inf _ { k \in { \mathbb Z } } \left ( \frac { \mu ( f ^ { k - n } ( W ) ) } { \mu ( f ^ { k } ( W ) ) } \right ) = \infty \tag * { $ \mathcal { U E } 2 $ } \end{align*}


\begin{align*} \rho ( F ) = \{ \eta \in \Omega : F ( \eta ) ^ { - 1 } \} \end{align*}


\begin{align*} \tilde { A } _ n ( \ell ) a _ { \ell } = \tilde { Q } _ { n , 1 } ( \ell ) \tilde { \beta } _ { n , 1 } ^ { \ell } + \cdots + \tilde { Q } _ { n , \tilde { s } ( n ) } ( \ell ) \tilde { \beta } _ { n , \tilde { s } ( n ) } ^ { \ell } \end{align*}


\begin{align*} \tilde { b } = \nabla \times ( \phi \nabla \theta ) + \frac { 1 } { \tau } G \nabla \theta , \end{align*}


\begin{align*} \lim _ { \Delta \uparrow \Z ^ d } \sup _ { \eta , \xi \in \Omega } \abs { \gamma _ { \Lambda } ( \eta _ \Lambda | \eta _ { \Delta \setminus \Lambda } \xi _ { \Delta ^ c } ) - \gamma _ \Lambda ( \eta _ { \Lambda } | \eta _ { \Lambda ^ c } ) } = 0 . \end{align*}


\begin{align*} [ V _ \lambda ] = q ^ { - \frac { c } { 2 4 } } q ^ { \frac { ( \lambda | \lambda + 2 \rho ) } { 2 ( k + h ^ \vee ) } - \lambda ( x ) } \ \prod _ { n = 1 } ^ \infty \prod _ { \alpha \in \Delta _ { } ^ { 0 } } ( 1 - q ^ { n - \epsilon ( \alpha ) } ) ^ { - 1 } \prod _ { \alpha \in \Delta _ { } ^ { + , \sharp } } ( 1 - e ^ { \alpha ( h ) } q ^ { n - \epsilon ( \alpha ) } ) ^ { - 1 } ( 1 - e ^ { - \alpha ( h ) } q ^ { n + \epsilon ( \alpha ) - 1 } ) ^ { - 1 } , \end{align*}


\begin{align*} \partial _ t \rho _ n = - \sum _ { i = 1 } ^ N \nabla _ { x _ i } \cdot \left ( \left ( \frac { 1 } { n - 1 } \sum _ { j = 1 } ^ n K \left ( x _ i - x _ j \right ) \right ) \rho _ n \right ) + \sum _ { i = 1 } ^ n \Delta _ { x _ i } \rho _ n . \end{align*}


\begin{align*} \eta ( K ) = 4 K ^ { \frac { 2 r - 2 } { 3 } } . \end{align*}


\begin{align*} & S ( k ) = I + \mathcal { O } ( k ^ { - 1 } ) , k \rightarrow \infty . \end{align*}


\begin{align*} \max _ { \theta \in [ 0 , 1 ] } \ \max _ { j = 1 , . . . , k + 1 } \prod _ { \ell \neq j } \ \frac { | \sin \pi ( \theta - \theta _ { \ell } ) | } { | \sin \pi ( \theta _ j - \theta _ { \ell } ) | } < e ^ { \gamma k } . \end{align*}


\begin{align*} \varrho = 0 , \varrho _ { Z Z } ( L , L ) \end{align*}


\begin{align*} \alpha ^ * & = 1 + \lim _ { z \rightarrow 0 } z \frac { \Phi ( z ) } { \phi ( z ) } \frac { 1 - \Phi ( z ) } { 1 - 2 \Phi ( z ) } = 1 + \frac { \sqrt { 2 \pi } } { 4 } \lim _ { z \rightarrow 0 } \frac { z } { 1 - 2 \Phi ( z ) } \\ & = 1 + \frac { \sqrt { 2 \pi } } { 4 } \lim _ { z \rightarrow 0 } \frac { 1 } { - 2 \phi ( z ) } = 1 - \frac { \pi } { 4 } \approx 0 . 2 1 4 6 . \end{align*}


\begin{align*} A \widehat { \otimes } _ { A _ { H ' , n } } S _ { H ' , n } ( V ) = V . \end{align*}


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


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


\begin{align*} d X _ t ^ i = \left ( b _ 0 ( t , X ^ i ) + \frac { 1 } { n - 1 } \sum _ { j = 1 , j \neq i } ^ n b \left ( t , X ^ i , X ^ j \right ) \right ) d t + d B ^ { H , i } _ t , i = 1 , \cdots , n , \end{align*}


\begin{align*} \rho ^ { \sigma } ( g ) : = \rho ( g ^ { \sigma } ) , \ \ g \in G . \end{align*}


\begin{align*} \begin{cases} \forall s \mbox { a . e . } \in [ 0 , t ] , \ : \forall q \geq 0 , \ : ( \dot z _ \omega ^ { x , t } ( s ) - \omega ( s ) ) ( q - z _ \omega ^ { x , t } ( s ) ) \leq 0 \\ z _ { \omega } ^ { x , t } ( t ) = x , \end{cases} \end{align*}


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


\begin{align*} \begin{aligned} ( T \backslash G ) \times G & \to G \\ ( g , \gamma ) & \mapsto g ^ { - 1 } \gamma g , \end{aligned} \end{align*}


\begin{align*} \widehat { A } ^ \Gamma = \{ \chi \in \widehat { A } \ : : \ : \gamma \chi = \chi \ , \forall \gamma \in \Gamma \} \end{align*}


\begin{align*} \langle \nu ( b ) , 2 \rho \rangle \leq \ell ( y ) \leq \ell ( y ' ) = \langle \nu ( b ) , 2 \rho \rangle . \end{align*}


\begin{align*} 0 = \nabla _ { i } Q = \frac { \nabla _ { i } P } { P } - A \nabla _ { i } \left ( \frac { \rho ^ { 2 } } { 2 } \right ) , \end{align*}


\begin{align*} ( \tilde { \kappa } _ 1 ) _ i = \tilde { h } _ { 1 1 i } = h _ { 1 1 i } , \end{align*}


\begin{align*} \sigma ( \lambda \alpha ) = \sigma ( \lambda ) \sigma ( \alpha ) = \lambda \sigma ( \alpha ) . \end{align*}


\begin{align*} \mathfrak P _ l f ( x ) = 2 ^ { \frac d 2 j } 2 ^ { - l } \int \int e ^ { i \lambda \Phi _ s ( x , y ) } \widetilde A _ s ( x , y ) d s \ , f ( y ) d y , \end{align*}


\begin{align*} x x ' + x y ' + y y ' = - 2 n \ , . \end{align*}


\begin{align*} \eta ( 1 ) = \gamma ^ { - 1 } \mbox { a n d } \eta ( g ) = 1 . \end{align*}


\begin{align*} V [ e _ 1 ] = \{ v \in V \setminus V ' \mid ( v , e _ 1 ) \in E ( v , f _ 1 ) \notin E \} \end{align*}


\begin{align*} n J _ n ^ { a , b } = J ^ { a , b } + o ( 1 ) , \quad \mbox { a s $ n \to + \infty $ } , \end{align*}


\begin{align*} x = Q _ { \mathrm { B } } ( u ) = I _ u ^ { - 1 } ( \alpha , \beta ) = \sum _ { i = 1 } ^ { \infty } d _ i \ , u ^ { i / \alpha } . \end{align*}


\begin{align*} \mathbf { H } ( \varphi ) = \mathbf { H } ( \varphi _ { 0 } ) T ( T _ { + } ) , \end{align*}


\begin{align*} \left \vert \widetilde { \varphi } _ { 0 } ^ { - } \left ( z ( \rho ) \right ) \right \vert = c _ { l } \left \vert \frac { 1 } { 2 } - i \rho \right \vert ^ { l + 1 } \left \vert J _ { l } \left ( \rho \right ) \right \vert \leq \operatorname * { C o n s t } \left ( \left ( 1 + \left \vert \rho \right \vert \right ) ^ { \frac { 5 } { 2 } l - \frac { 1 } { 2 } - 2 \alpha } \mathbf { L } ( \rho ) + 1 \right ) \end{align*}


\begin{align*} \sum _ { i = 1 } ^ { n } w _ i \frac { d y _ i } { y _ i } + w _ { n + 1 } \frac { d \varphi } { \varphi } = 0 \end{align*}


\begin{align*} \prescript J { } \pi ( w _ 1 ) = ( w _ 1 ' , \mu _ 1 ) , \prescript J { } \pi ( w _ 2 ) = ( w _ 2 ' , \mu _ 2 ) . \end{align*}


\begin{align*} t _ \theta : = \begin{bmatrix} \cos \theta \dfrac { 1 } { \sqrt { k } } \mathbb { 1 } _ k \\ \hline \sin \theta \dfrac { 1 } { \sqrt { l } } \mathbb { 1 } _ l \end{bmatrix} \mbox { a n d } U _ \theta : = \begin{bmatrix} \ ; \cos \theta \sqrt { \dfrac { m } { k } } V & \vline & \sin \theta \sqrt { \dfrac { m } { l } } W \ ; \end{bmatrix} , \end{align*}


\begin{align*} \overline { \mathfrak { D } _ { q ^ n } } \ \overline { U _ { q ^ n } } \dot { s } \varepsilon ( v ) \zeta = ( A + B + C ) \overline { U _ { q ^ n } } \dot { s } \overline { U ^ * _ { q ^ n } } { \bf 1 } _ { - } \in \Bbbk { \bf G } \zeta . \end{align*}


\begin{align*} \Lambda ( \alpha ) = \lim _ { \epsilon \to 0 } \Lambda _ { \epsilon ^ 2 } ( \alpha ) . \end{align*}


\begin{align*} & \sup \{ | \varphi _ { 2 k j _ 0 } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j _ 0 } ( z ) | \leq r \} \\ & = \sup \{ | \varphi _ { k j _ 0 } \circ \varphi _ { k j _ 0 } ( z ) - \rho \circ \varphi _ { k j _ 0 } ( z ) | ; \ z \in \mathbb { B } _ n , \ | \varphi _ { k j } ( z ) | \leq r \} \\ & \leq \sup \{ | \varphi _ { k j _ 0 } ( z ) - \rho ( z ) | ; \ z \in \mathbb { B } _ n , \ | z | \leq r \} \leq \varepsilon \end{align*}


\begin{align*} \rho ( x ) = \sup _ { r > 0 } \Big \{ r : \frac { 1 } { r ^ { n - 2 } } \int _ { B ( x , r ) } V ( y ) d y \leq 1 \Big \} . \end{align*}


\begin{align*} X _ x ( b ) ( \overline { \mathbb F _ q } ) = \{ g \in G ( \breve F ) / I \mid g ^ { - 1 } b \sigma ( g ) \in I x I \} . \end{align*}


\begin{align*} \left [ c _ { \chi _ 1 \chi ^ { - 1 } } \right ] = \left [ \beta _ \eta \right ] = 0 \in \mathrm { H } ^ 2 ( Q , S ^ 1 ) . \end{align*}


\begin{align*} G _ - ( x ; k ) = \mathcal { P } ^ { - } \left ( G _ { - } J \right ) ( z ) + \mathcal { P } ^ { - } \left ( J \right ) , \end{align*}


\begin{align*} \prod _ { k = 1 } ^ n a _ k + \sum _ { j = 1 } ^ n \left ( \prod _ { k = 1 } ^ { j - 1 } ( a _ k + b _ k ) \right ) b _ j \left ( \prod _ { k = j + 1 } ^ n a _ k \right ) = \prod _ { i = 1 } ^ n ( a _ i + b _ i ) , \end{align*}


\begin{align*} \lim _ { x \to \infty } e ^ { \gamma x } f _ 2 ( x ) = \lim _ { x \to \infty } c f _ e ( x ) = c \lim _ { x \to \infty } \sum _ { n c _ 1 \ge x } ^ \infty ( ( \lambda \Lambda _ e ) ^ n / n ! ) \ , g _ { e } ^ { \ast n } ( x ) = 0 . \end{align*}


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


\begin{align*} \lim _ { \Lambda \uparrow \Z ^ d } \frac { 1 } { \mu ( \eta _ { \Lambda } ) } \int \mathbf { 1 } _ { \eta _ { \Lambda } } ( \xi ) f ( \xi ) \mu ( d \xi ) = f ( \eta ) . \end{align*}


\begin{align*} \langle \nu ( x ) , 2 \rho \rangle = \sum _ { \alpha \in \Phi ^ + } \frac 1 N \sum _ { k = 1 } ^ N \ell ( x , ( \sigma \circ w ) ^ k v \alpha ) \geq \ell ( x ) . \end{align*}


\begin{align*} \langle \mathcal { G } ^ { \mathrm { e n d } } \cdot , \cdot \rangle = \mathcal { G } \end{align*}


\begin{align*} R = \left ( \prod _ { m = i + 1 } ^ { j - 1 } M _ m ^ T \bar { Q } _ m \right ) \times M _ j ^ T , \end{align*}


\begin{align*} \frac { \partial f ( 0 ) } { \partial z } = \frac { \partial ^ 2 f ( 0 ) } { \partial z ^ 2 } = \frac { \partial ^ 3 f ( 0 ) } { \partial z ^ 3 } = 0 \frac { \partial ^ 4 f ( 0 ) } { \partial z ^ 4 } & = \frac { 4 \sqrt { 2 } } { \pi } \frac { \Gamma ( 3 / 2 - \pi / 4 ) } { \Gamma ( 1 - \pi / 4 ) } \left ( \frac { 3 } { \pi } - 1 \right ) < 0 . \end{align*}


\begin{align*} a ( x _ a ; x _ c ; \beta ) - a ( x _ a ; x _ b ; \alpha ) - a ( x _ a ; x _ d ; { \beta - \alpha } ) = 0 , \\ c ( x _ a ; x _ c ; \beta ) - c ( x _ a ; x _ b ; \alpha ) - a ( x _ a ; x _ d ; { \beta - \alpha } ) = 0 , \end{align*}


\begin{align*} H = Z \circ \varphi . \end{align*}


\begin{align*} f _ 1 ( x ) = ( e ^ { \lambda \Lambda _ 1 } - 1 ) ^ { - 1 } \sum _ { n = 1 } ^ { \infty } ( ( \lambda \Lambda _ 1 ) ^ n / n ! ) g _ 1 ^ { \ast n } ( x ) \end{align*}


\begin{align*} \rho { u _ z } = \sum \limits _ { i = 0 } ^ { 1 8 } { { c _ { i z } } { f _ i } + \frac { { \Delta t } } { 2 } { F _ z } } . \end{align*}


\begin{align*} ( ( G _ \alpha ( \epsilon _ N ) \circ G _ \alpha T _ \alpha ( \iota ) \circ \psi _ L ) ( l ) ) ( s ) & = \epsilon _ N ( ( G _ \alpha T _ \alpha ( \iota ) \circ \psi _ L ) ( l ) ( s ) ) \\ & = \epsilon _ N ( T _ \alpha ( \iota ) \circ \psi _ L ( l ) ( s ) ) \\ & = \epsilon _ N ( T _ \alpha ( \iota ) ( s \otimes l ) ) \\ & = \epsilon _ N ( s \otimes \iota ( l ) ) \\ & = \iota ( l ) ( s ) \end{align*}


\begin{align*} A _ k ( \theta , E ) = \left ( \begin{array} { c c } P _ k ( \theta , E ) & - P _ { k - 1 } ( \theta + \alpha , E ) \\ P _ { k - 1 } ( \theta , E ) & - P _ { k - 2 } ( \theta + \alpha , E ) \end{array} \right ) , \end{align*}


\begin{align*} Q ( u ) = \sum _ { i = 0 } ^ { \infty } f _ i \ , \ , u ^ { i / \alpha } , \end{align*}


\begin{align*} \nu ^ { n } _ m ( t ) : = \sum _ { i = m } ^ { n } i \psi ^ { n } _ i ( t ) , \end{align*}


\begin{align*} f ( x ) = \left \{ \begin{array} { l l l } x & & x \geq 0 \\ a ( e ^ { x } - 1 ) & & x < 0 \end{array} \right . a > 0 . \end{align*}


\begin{align*} \beta = 3 \left ( 2 i - 1 \right ) \end{align*}


\begin{gather*} \Gamma _ 0 = \bigl \{ P \in \Gamma : P ( \abs { f } ) < \infty P ( f ) = 0 f \in F \bigr \} \end{gather*}


\begin{gather*} \lim _ k \sup _ { Q \in \mathcal { U } } Q \bigl \{ \abs { f } \ , 1 ( \abs { f } > k ) \bigr \} = 0 \quad f \in F . \end{gather*}


\begin{align*} \tilde { m } _ n = \begin{cases} m _ n - q _ { n - 1 } , \ell _ n = a _ { n + 1 } \\ m _ n + q _ { n - 1 } , \ell _ n = - a _ { n + 1 } \end{cases} . \end{align*}


\begin{align*} \widehat { \tau } : x \mapsto \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \tau _ { 1 } ( x _ { i i } ) \forall x = ( x _ { i j } ) \in \mathcal { M } _ { n } ( M _ { 1 } ) . \end{align*}


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


\begin{align*} s _ { D } ( x ) : = a ^ { D } ( x , x ) . \end{align*}


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


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


\begin{align*} 1 = \lim _ { x \to \infty } \frac { F ( x + y + \Delta ) } { F ( x - c + \Delta ) } \le \lim _ { x \to \infty } \frac { f ( x + y ) } { f ( x ) } \le \lim _ { x \to \infty } \frac { F ( x + y - c + \Delta ) } { F ( x + \Delta ) } = 1 \end{align*}