\begin{align*} U ( \vec { a } , 1 ) \left ( S , f _ \alpha \otimes \rho ( E ^ \alpha ) \right ) : = & e ^ { - i [ \hat { f } _ 0 \cdot \hat { H } ( \vec { a } , \rho ) + \hat { f } _ 1 \cdot \hat { P } ( \vec { a } , \rho ) ] } \left ( S + \vec { a } , f _ \alpha ( \cdot - \vec { a } ) \otimes \rho ( E ^ \alpha ) \right ) \\ = & e ^ { i [ a ^ 0 \hat { H } ( \rho ) - a ^ 1 \hat { P } ( \rho ) ] } \left ( S + \vec { a } , f _ \alpha ( \cdot - \vec { a } ) \otimes \rho ( E ^ \alpha ) \right ) . \end{align*}


\begin{align*} \pi _ { \psi } [ ( x ^ { \ast } , 0 ) + ( x , 0 ) + ( 0 , k ) ] g ( y ^ { \ast } ) = \psi ( k + \langle x ^ { \ast } + y ^ { \ast } , x \rangle ) g ( x ^ { \ast } + y ^ { \ast } ) , \end{align*}


\begin{align*} X ' : = \{ g \in L ^ 0 ( \Omega ) : f g \in L ^ 1 ( \Omega ) \} . \end{align*}


\begin{align*} & [ \alpha _ { \mathcal { L } } ( a ) , \alpha _ { \mathcal { L } } ( b ) , [ x , y , z ] _ { \mathcal { L } } ] _ { \mathcal { L } } \\ & = [ [ a , b , x ] _ { \mathcal { L } } , \alpha _ { \mathcal { L } } ( y ) , \alpha _ { \mathcal { L } } ( z ) ] _ { \mathcal { L } } + [ \alpha _ { \mathcal { L } } ( x ) , [ a , b , y ] _ { \mathcal { L } } , \alpha _ { \mathcal { L } } ( z ) ] _ { \mathcal { L } } + [ \alpha _ { \mathcal { L } } ( x ) , \alpha _ { \mathcal { L } } ( y ) , [ a , b , z ] _ { \mathcal { L } } ] _ { \mathcal { L } } , \end{align*}


\begin{align*} P l ( k , n ) = \sqcup _ { i = 0 } ^ { m } \sigma ( \widetilde { E } ( \lambda ^ i ) ) . \end{align*}


\begin{align*} \phi ( A _ 1 \otimes \cdots \otimes A _ m ) = U ( \varphi _ 1 ( A _ 1 ) \otimes \cdots \otimes \varphi _ m ( A _ m ) ) V \end{align*}


\begin{align*} V _ k ^ 2 ( t ) & = 2 \int _ 0 ^ t \int _ { 0 } ^ { s } R _ k ( s , \varphi , \varphi , R ( s ' , \varphi , \varphi , \varphi ) ) \dd s ' \dd s + \int _ 0 ^ t \int _ { 0 } ^ { s } R _ k ( s , \varphi , R ( s ' , \varphi , \varphi , \varphi ) , \varphi ) \dd s ' \dd s \\ & = : \int _ 0 ^ t \int _ 0 ^ s ( 2 A + B ) ( s , s ' ) \dd s ' \dd s . \end{align*}


\begin{align*} h _ i \ , \xi _ j = 0 . \end{align*}


\begin{align*} A ( s ) = ( 1 + s ) \log ( 1 + s ) - s \ , \ , \tilde { A } ( s ) = \exp ( s ) - s - 1 . \end{align*}


\begin{align*} \begin{cases} - \Delta u = g ( u ) , \ \ u \ge 0 , & \ \ B _ R , \\ u = 0 , \ \ & \ \ \partial B _ R , \end{cases} \end{align*}


\begin{align*} \aligned & \Bigl ( \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | \eta _ \epsilon | ^ { 2 ^ * _ \mu } ) | \eta _ \epsilon | ^ { 2 ^ * _ { \mu } } \Bigr ) ^ { \frac 1 { 2 ^ * _ \mu } } \leq C ( N , \mu ) ^ { \frac 1 { 2 ^ * _ { \mu } } } | \eta _ \epsilon | ^ 2 _ { 2 ^ * } = C ( N , \mu ) ^ { \frac { N } { 2 } \cdot \frac { 1 } { 2 ^ * _ \mu } } S ^ { \frac { N } { 2 } } _ { H , L } + O ( \epsilon ^ { N - 2 } ) . \endaligned \end{align*}


\begin{align*} \gamma _ 1 ( a ) : = \sup \{ \gamma \geq 0 : ~ \exists u \in C ^ 2 ( \Omega ) , ~ ~ u < 0 ~ ~ \textnormal { a n d } ~ ~ L _ a u \geq - \gamma u f \} . \end{align*}


\begin{align*} \Vert d _ { 0 , N } \Vert _ 1 \leq \Vert d _ { 0 , N } \Vert _ 2 \leq \limsup _ { n \rightarrow \infty } n ^ { - 1 / 2 } \big \Vert \sum _ { i = 1 } ^ n d _ { i , N } \big \Vert _ 2 \ , . \end{align*}


\begin{align*} U _ k = \prod _ { p \in { \cal P } _ 1 } p ^ { u _ { k p } } , V _ k = \prod _ { p \in { \cal P } _ 2 } p ^ { u _ { k p } } , W _ k = \prod _ { p \in { \cal P } _ 3 } p ^ { u _ { k p } } . \end{align*}


\begin{align*} \left ( \frac { 1 } { \sqrt { 2 \pi } } \right ) ^ 2 & \int _ { S _ 0 } e ^ { - i [ q ^ 2 x ^ 2 + q ^ 3 x ^ 3 ] } F _ \alpha \left ( \hat { H } ( \rho ) , \hat { P } ( \rho ) , x ^ 2 , x ^ 3 \right ) \ d x ^ 2 d x ^ 3 \otimes \rho ( E ^ \alpha ) \\ = & \hat { F } _ \alpha \left ( \hat { H } ( \rho ) , \hat { P } ( \rho ) , q ^ 2 , q ^ 3 \right ) \otimes \rho ( E ^ \alpha ) . \end{align*}


\begin{align*} \partial \Phi / \partial U = \Phi _ { U } = U ^ T P , \Phi _ { U } \lbrack P ^ { - 1 } ( ( A _ i U ) _ { x _ i } + A ^ T _ i U _ { x _ i } + C U ) \rbrack = ( U ^ T A _ i U ) _ { x _ i } = ( \Psi _ i ) _ { x _ i } . \end{align*}


\begin{align*} \Phi _ t + ( \Psi _ i ) _ { x _ i } = 0 , \end{align*}


\begin{align*} \left ( y ^ 2 + f y + r > 1 > y \right ) \left ( \wedge 1 = \frac { 1 } { | N ( 1 ) | } > \frac { y } { | N ( y ) | } = y \right ) \wedge \end{align*}


\begin{align*} I _ 2 \leq & C ( t - s ) \int _ { - R } ^ R \int _ 0 ^ t \left ( \int _ { \R } G _ { t - r } ( x - z ) d z \right ) d r d x = C ( t - s ) R \int _ 0 ^ t ( t - r ) d r \leq C ( t - s ) R . \end{align*}


\begin{align*} P ^ \omega ( ( C _ t \cap D _ t ) ^ c \mid S _ t ) & = 1 - P ^ \omega ( C _ t \cap D _ t \mid S _ t ) \\ & \leq 1 - P ^ \omega ( D _ t \mid C _ t , S _ t ) P ^ \omega ( C _ t \mid S _ t ) \\ & = 1 - ( 1 - P ^ \omega ( D _ t ^ c \mid C _ t , S _ t ) ) ( 1 - P ^ \omega ( C _ t ^ c \mid S _ t ) ) \\ & \leq P ^ \omega ( D _ t ^ c \mid C _ t , S _ t ) + P ^ \omega ( C _ t ^ c \mid S _ t ) . \end{align*}


\begin{align*} \lim _ { N \to \infty } N ^ { - 1 } M _ j ( \tau _ N ) & = \frac { \nu q } { \lambda } \int _ 0 ^ 1 ( 1 - p y ) ^ { - 1 } y ^ { j - 1 } d y . \end{align*}


\begin{align*} \phi \left ( \bigotimes \limits _ { i = 1 } ^ { m - 1 } X _ i E _ { j _ i j _ i } X _ i ^ * \otimes B \right ) = W _ { X } \left ( \bigotimes \limits _ { i = 1 } ^ { m - 1 } E _ { j _ i j _ i } \otimes \varphi _ { j _ 1 , \ldots , j _ { m - 1 } , X } ( B ) \right ) W _ { X } ^ * \end{align*}


\begin{align*} \omega ( a ^ * a ) = 1 \end{align*}


\begin{align*} \frac { \partial f } { \partial t } + \nabla \cdot ( u f ) - \nu \Delta f = 0 , \end{align*}


\begin{align*} \int _ { \mathbb { S } ^ n } u x _ l \mathrm { d } A = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } , \ \ l = 1 , 2 , \cdots , n + 1 . \end{align*}


\begin{align*} ( t , w ) \longmapsto \left \{ \begin{array} { r l } \widetilde { \rho } _ w ( t , w ) = \rho _ w ( t , t ^ s w ) ; & t \ge 0 , w \in \C \\ \frac { 1 } { 2 i } e ^ { i \pi ( 1 - \beta _ 1 ) } ; & t \le 0 , w \in \C \end{array} \right . \end{align*}


\begin{align*} \sum \limits _ { j \in I _ k } s _ j = 0 , k = 1 , \ldots , m . \end{align*}


\begin{align*} \varphi : = \varphi _ W \underline { \varphi } : = \varphi _ { \underline { W } } ; \end{align*}


\begin{align*} \Sigma ( \delta ) = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \boldsymbol { K } _ 1 ( \mu _ { 1 3 } , \gamma _ { 1 3 } ) \le \delta _ 1 , \boldsymbol { K } _ 2 ( \mu _ { 2 3 } , \gamma _ { 2 3 } ) \le \delta _ 2 \right \} , \end{align*}


\begin{align*} n ^ { - 1 } g \left | \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p \bar j } \right | ^ 2 - g \sum _ { p , q = 1 } ^ n u ^ { p \bar p } u ^ { q \bar q } | u _ { p \bar q j } | ^ 2 + g \sum _ { p = 1 } ^ n u ^ { p \bar p } u _ { p \bar p j \bar j } = n ( 2 R e ( g _ { v j } v _ { \bar j } ) + g _ v v _ { j \bar j } + g _ { v v } | v _ j | ^ 2 + g _ { j \bar j } ) . \end{align*}


\begin{align*} \mathcal { G } ( \mathbb { R } ) = \frac { C ^ { \infty } } { C ^ { \infty } } . \end{align*}


\begin{align*} L ( u , \chi ) = \prod _ { i = 1 } ^ { d ( \chi ) } ( 1 - \gamma _ i u ) \end{align*}


\begin{align*} \norm { ( v _ 0 ^ { \alpha , \epsilon } , v _ 1 ^ { \alpha , \epsilon } ) } _ { \dot { H } ^ 1 \times L ^ 2 } ^ 2 = ( \alpha / \epsilon ) ^ { - 6 } \alpha ^ 2 \norm { ( v _ 0 , v _ 1 ) } _ { \dot { H } ^ 1 \times L ^ 2 } ^ 2 \ , = 1 6 \pi \epsilon ^ 6 / 3 \alpha ^ 4 . \end{align*}


\begin{align*} \lim _ { k \to \infty } I ( k ) = + \infty . \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( V ^ { \lambda } ) & = L ( V ^ { \lambda } ) + N ( V ^ { \lambda } ) + \omega M ( V ^ { \lambda } ) + \mathbf { c } \cdot \mathbf { P } ( V ^ { \lambda } ) \\ & = \lambda ^ 2 L ( V ) + \lambda ^ { \frac { d } { 2 } + 1 } N ( V ) + \omega M ( V ) + \lambda \mathbf { c } \cdot \mathbf { P } ( V ) , \end{align*}


\begin{align*} \sup _ { | z | = 1 } \big ( \log | { \rm d e t } ( z - U _ n ) | - \log n + \frac { 3 } { 4 } \log \log n \big ) , \end{align*}


\begin{align*} \mathcal { Z } _ t = \exp \bigg \{ \int _ 0 ^ t e ^ { - \frac { \beta } { 2 } s } h ( x _ s ^ u , u _ s ) d \xi _ s - \frac { 1 } { 2 } \int _ 0 ^ t e ^ { - \beta s } | h ( x _ s ^ u , u _ s ) | ^ 2 d s \bigg \} \end{align*}


\begin{align*} \alpha = 1 - \frac { 3 } { 4 } \frac { \log n } { n } , \end{align*}


\begin{align*} \frac { u _ 0 ( r ) } { u _ 0 ' ( r ) } \cdot \mathcal U ' ( r ) + \mathcal U ( r ) = \frac { u ' ( r ) u _ 0 ( r ) - u _ 0 ' ( r ) u ( r ) } { u _ 0 ( r ) u _ 0 ' ( r ) } + \frac { u ( r ) } { u _ 0 ( r ) } = \mathcal W ( r ) . \end{align*}


\begin{align*} \mathcal { M } _ { q , \alpha ^ 1 } f = \mathcal { M } _ { q , \alpha ^ 2 } f , f \in H ^ { s _ M } ( \Omega ^ c ) \end{align*}


\begin{align*} \eta ^ { - 1 } S \eta = \iota . \end{align*}


\begin{align*} 0 < \epsilon : = r _ * - \underline { r } \leq \overline C _ 3 \alpha ^ 2 . \end{align*}


\begin{align*} C ^ n ( G , M ) & : = F ^ n ( G , M ) ^ G \\ d ^ n & : = d ^ n \mid _ { C ^ n ( G , M ) } . \end{align*}


\begin{align*} | W _ t \ ! - \ ! W _ \tau | ^ 2 = | W _ t | ^ 2 + | W _ \tau | ^ 2 \pm 2 | W _ t | | W _ \tau | \cos \overline \beta = \varepsilon _ 1 ^ 2 \sin ^ 2 \overline \beta + ( | W _ t | \ ! \pm \ ! \varepsilon _ 1 \cos \overline \beta ) ^ 2 \ge \varepsilon _ 1 ^ 2 \sin ^ 2 \overline \beta \ge \frac { 4 } { \pi ^ 2 } \varepsilon _ 1 ^ 2 \overline \beta ^ 2 \end{align*}


\begin{align*} c ( x ) \triangleq \left \{ \begin{array} { l r } 0 & x = 0 , \\ K + v x & x > 0 . \end{array} \right . \end{align*}


\begin{align*} \mathrm { S p a n } \{ \tilde { \lambda } \} + \mathrm { K e r } ( S ^ * ) = \mathrm { S p a n } \{ \lambda ^ * \} + \mathrm { K e r } ( S ^ * ) \end{align*}


\begin{align*} \delta _ t ( x + \lambda y ) = \delta _ { x , 0 } \delta _ t ( \lambda y ) + ( 1 - \delta _ { x , 0 } ) \delta _ t ( x + \lambda y ) \ . \end{align*}


\begin{align*} \| f ^ { = d } \| _ 2 ^ 2 \le \left ( \frac { e \cdot 1 2 r } { \theta } \right ) ^ { d } \gamma _ 1 \cdot \gamma \leq \left ( \frac { 3 3 r } { d } \right ) ^ d \gamma _ 1 \cdot \gamma \cdot \log ^ { d } \left ( \frac { \gamma _ 2 } { \gamma _ 1 } \right ) , \end{align*}


\begin{align*} \binom { m } { n } : = \frac { m ( m - 1 ) \cdots ( m - n + 1 ) } { n ! } . \end{align*}


\begin{align*} \frac { 1 } { \sum _ { j \in \N } f ( j ) ^ 2 } \sum _ { v \not \in \mathcal { L } } f ( v ) ^ 2 = o ( 1 ) \textrm { a s } N \to \infty . \end{align*}


\begin{align*} \int _ 0 ^ T \int _ \mathbb { R } \mathrm { s g n } ( F _ { N } - \alpha ) \chi \mathbf { S } [ F _ { N } ] ( t , x ) d x d t & = \int _ 0 ^ T \int _ \mathbb { R } \mathbf { 1 } _ { \left \{ F ( t , x ) \neq \alpha \right \} } \mathrm { s g n } ( F _ { N } - \alpha ) \chi \mathbf { S } [ F _ { N } ] ( t , x ) d x d t \\ & \underset { N \rightarrow \infty } { \rightarrow } \int _ 0 ^ T \int _ \mathbb { R } \mathrm { s g n } ( F - \alpha ) \chi \mathbf { S } [ F ] ( t , x ) d x d t \end{align*}


\begin{align*} \widehat \psi _ r ( C _ r ) = & \inf _ { x \in [ 0 , 1 ] } \left [ J _ r ( x , d _ r ( x ) ) + ( C _ r - d _ r ( x ) ) \sup _ { \beta \in [ d _ r ( x ) , C _ r ] } \theta ( x , \beta ) \right ] \\ \leq & \ , \inf _ { x \in [ 0 , 1 ] } ( C _ r - d _ r ( x ) ) \sup _ { \beta \in [ d _ r ( x ) , C _ r ] } \theta ( x , \beta ) = 0 , \end{align*}


\begin{align*} y \Bigl ( \bigl \{ T _ { 1 } ( Z ) , \cdots , T _ { q - 1 } ( Z ) \bigr \} , \ , T _ { q } ( Z ) \Bigr ) = T _ { q } ( N ) , \ \forall q \ge 1 . \end{align*}


\begin{align*} R ^ i ( f \times f ) _ * ( W ) = \R ^ i ( f \times f ) _ * ( \pi _ 1 ^ * ( V ) \otimes \pi _ 2 ^ * ( V ) ) \cong \bigoplus _ { t } \R ^ t f _ * ( V ) \otimes \R ^ { i - t } f _ * ( V ) . \end{align*}


\begin{align*} u ( x ) = \left ( \frac { 1 } { N } \sum _ { i \in [ N ] \setminus j } A _ N ( i , j ) - \lambda \right ) ^ { - 1 } \frac { 1 } { N } \sum _ { i \in [ N ] \setminus j } A _ N ( i , j ) \overline { u } _ i . \end{align*}


\begin{align*} \Big \| \sum _ { j = 1 } ^ m \alpha _ j \nabla u _ j \Big \| _ { p , q , \mu } \leq \Big ( \sum _ { j = 1 } ^ \infty | a _ j | ^ q \Big ) ^ \frac 1 { q } \end{align*}


\begin{align*} \widetilde { \mu } _ { \omega , \mathbf { c } } : = \inf \{ S _ { \omega , \mathbf { c } } ( \Psi ) | \ \Psi \in \mathcal { H } ^ 1 \backslash \{ ( \mathbf { 0 } , \mathbf { 0 } , \mathbf { 0 } ) \} , \ I _ { \omega , \mathbf { c } } ( \Psi ) = 0 , \ \mathbf { c } \cdot \mathbf { P } ( \Psi ) \ge 0 \} . \end{align*}


\begin{align*} \mu ( \Gamma ) = \delta _ { x _ 0 } ( \Gamma ) + \int _ { \textbf { X } } \int _ { \textbf { A } } p ( \Gamma | y , a ) \sigma ^ s ( d a | y ) \mu ( d y ) , ~ \Gamma \in { \cal B } ( \textbf { X } ) . \end{align*}


\begin{align*} [ x ] = [ y ] \Leftrightarrow \prod _ { i \neq j } x _ i y _ j = y _ i x _ j \rlap { . } \end{align*}


\begin{align*} \eta ( B ^ { \prime } ) = \eta ( B ) + 1 \ge T + \max _ { 1 \le i \le s } a _ i . \end{align*}


\begin{align*} S _ { \mathrm { m a i n } } = \sum _ { d _ 1 , d _ 2 < D } \frac { 1 } { \sqrt { d _ 1 d _ 2 } } \sum _ { I d _ 1 , J d _ 2 < D } \alpha _ { I d _ 1 } \bar { \alpha } _ { J d _ 2 } & \sum _ { I L = J M } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } . \end{align*}


\begin{align*} \frac { 1 } { ( q ; q ) _ \infty ( x ^ \alpha ; q ) _ \infty ( q x ^ { - \alpha } ; q ) _ \infty } & = E _ 0 + E _ \alpha + E _ { 2 \alpha } + \dotsm + q ( E _ { - \alpha } + E _ { \alpha } + 2 E _ { 2 \alpha } + \dotsm ) + q ^ 2 \dotsm , \end{align*}


\begin{align*} A _ k = \frac { 2 ^ { 2 s } } { b _ k } \quad \mbox { w i t h } b _ k = \begin{cases} l & \mbox { i f } A _ { 0 } A _ { 1 } \cdots A _ { k - 1 } \in \left [ \frac { 1 } { 2 } , 1 \right ) , \\ l + 1 & \mbox { i f } A _ { 0 } A _ { 1 } \cdots A _ { k - 1 } \in [ 1 , 2 ] . \end{cases} \end{align*}


\begin{align*} h = h _ 1 * \dots * h _ p \hbox { a n d } \mathcal { L } _ \mathcal { A } ( h ) = \mathcal { L } _ \mathcal { A } ( h _ 1 ) * \dots * \mathcal { L } _ \mathcal { A } ( h _ p ) . \end{align*}


\begin{align*} \mathcal { P } ( X _ \eta ) = \mathcal { P } ( X _ \varphi ) = \{ N _ \ast ( \nu _ { \eta * } \vee \nu _ \eta \vee \kappa ) : \kappa \in \mathcal { P } ( \{ 0 , 1 \} ^ { \Z } ) \} , \end{align*}


\begin{align*} & ( \sqrt { | \Lambda ^ - | } W ^ - - R \sqrt { \Lambda ^ + } W ^ + - S G ) ^ T ( \sqrt { | \Lambda ^ - | } W ^ - - R \sqrt { \Lambda ^ + } W ^ + - S G ) + \\ & \begin{bmatrix} \sqrt { \Lambda ^ + } W ^ + \\ G \end{bmatrix} ^ T \begin{bmatrix} I - R ^ T R & - R ^ T S \\ - S ^ T R & I - S ^ T S \end{bmatrix} \begin{bmatrix} \sqrt { \Lambda ^ + } W ^ + \\ G \end{bmatrix} - G ^ T G \end{align*}


\begin{align*} \mathcal { H } _ { \partial H } + 2 \lambda f ' \big ( | E \Delta H | \big ) \left ( \mathbf { 1 } _ { \overline H } - \mathbf 1 _ { \overline E } \right ) = \mu \partial H , \end{align*}


\begin{align*} \Delta _ C = \Delta - \tau \circ \Delta . \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } L _ a \phi _ 1 = - \gamma _ 1 \phi _ 1 f & \textnormal { i n } & \Omega , \\ \phi _ 1 = 0 & \textnormal { o n } & \partial \Omega \\ \phi _ 1 < 0 & \textnormal { i n } & \Omega . \end{array} \right . \end{align*}


\begin{align*} \big | \overline { \rho } ( \alpha _ t ( f ) ) ( z ) - \beta _ t ( \overline { \rho } ( f ) ) ( z ) \big | & = \big | \alpha _ t ( f ) ( \rho ( z ) ) - \overline { \rho } ( f ) ( \eta _ { t ^ { - 1 } } ( z ) ) \big | \\ & = \big | f ( \theta _ { t ^ { - 1 } } ( \rho ( z ) ) ) - f ( \rho ( \eta _ { t ^ { - 1 } } ( z ) ) ) \big | < \varepsilon . \end{align*}


\begin{align*} \cosh ( b ) & = \cosh ( a ) \cosh ( r ) - \sinh ( a ) \sinh ( r ) \cos ( \theta ) \\ & = \cosh ( a ) \cosh ( r ) - \sinh ( a ) \cosh ( r ) \tanh ( a ) \\ & = \frac { \cosh ( r ) } { \cosh ( a ) } [ \cosh ( a ) ^ 2 - \sinh ^ 2 ( a ) ] = \frac { \cosh ( r ) } { \cosh ( a ) } . \end{align*}


\begin{align*} f ( \omega ) : = \mu _ { \omega , \mathbf { c } } - S _ { \omega , \mathbf { c } } ( U _ 0 ) = \mu _ { \omega , \sqrt { \omega } \ \mathbf { c } _ 0 } - S _ { \omega , \sqrt { \omega } \ \mathbf { c } _ 0 } ( U _ 0 ) . \end{align*}


\begin{align*} \mathrm { R e d } = \prod _ { \ell \in \mathcal { S } } \mathrm { R e d } _ { \mathcal { T } ( \ell ) } , , \Pi = \prod _ { \ell \in \mathcal { S } } \Pi _ { \mathcal { T } ( \ell ) } , \end{align*}


\begin{align*} f = \sum _ { k = 0 } ^ \infty p _ k . \ ; \ ; \ ; \phantom { ( k \geq 0 ) } \end{align*}


\begin{align*} a _ 0 ^ 2 = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 , \ a _ 1 ^ 2 = O ( \varepsilon ) \| u \| _ { L ^ 2 ( \mathbb { S } ^ n ) } ^ 2 , \end{align*}


\begin{align*} \tau _ 2 ( \rho S _ t + \rho u \cdot S _ x ) + S = \mu u _ x . \end{align*}


\begin{align*} x ( y z ) = \lambda _ 1 ( x y ) z & + \lambda _ 2 ( y x ) z + \lambda _ 3 z ( x y ) + \lambda _ 4 z ( y x ) \\ & + \lambda _ 5 ( x z ) y + \lambda _ 6 ( z x ) y + \lambda _ 7 y ( x z ) + \lambda _ 8 y ( z x ) \intertext { a n d } ( y z ) x = \mu _ 1 ( x y ) z & + \mu _ 2 ( y x ) z + \mu _ 3 z ( x y ) + \mu _ 4 z ( y x ) \\ & + \mu _ 5 ( x z ) y + \mu _ 6 ( z x ) y + \mu _ 7 y ( x z ) + \mu _ 8 y ( z x ) \end{align*}


\begin{align*} \Sigma _ { \mathrm { D } } ( \delta ) : = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } _ 1 \times \mathcal { S } _ 2 ) : \boldsymbol { W } _ { p _ 1 } ( \gamma _ 1 , \mu _ 1 ) \leq \delta _ 1 ^ { 1 / p _ 1 } , \ \boldsymbol { W } _ { p _ 2 } ( \gamma _ 2 , \mu _ 2 ) \leq \delta _ 2 ^ { 1 / p _ 2 } \right \} . \end{align*}


\begin{align*} P _ { \mathcal { H } _ \chi ^ { \rm s t i f f ( s o f t ) } } \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } = \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \widehat { P } _ \chi , \bigl ( \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) ^ * P _ { \mathcal { H } _ \chi ^ { \rm s t i f f ( s o f t ) } } = \widehat { P } _ \chi \bigl ( \Pi _ \chi ^ { \rm s t i f f ( s o f t ) } \bigr ) ^ * , \end{align*}


\begin{align*} \Pi _ \chi ^ { \rm s t i f f } = \Pi _ 0 ^ { \rm s t i f f } + \sum _ { n = 1 } ^ \infty \chi ^ n \Pi _ { n } , \mbox { i n $ H ^ 1 ( Y ; \C ^ 3 ) $ , } \end{align*}


\begin{align*} \Lambda _ n : = X _ { \bullet \bullet \cdots \bullet } \quad S _ n : = \sum _ { | F | = n } X _ F \end{align*}


\begin{align*} Y _ { T } ( r ) & = N \left ( y _ { T } ( Y _ { T } , r ) \right ) \\ & = N \left ( y _ { S } ( Y _ { T } , r ) \right ) . \end{align*}


\begin{align*} f _ N ( x ) = g ( T x ) x \in A ^ k . \end{align*}


\begin{align*} \mathcal { K } _ t \varphi = \varphi ( \Phi ( t ; x ) ) , \forall x \in \mathbb { R } ^ d . \end{align*}


\begin{align*} \varphi ( u ( t ) ) & = \varphi ( u _ 0 ) + \int _ 0 ^ t \varphi ' ( u ( s ) ) b ( s ) \ , d s + \int _ 0 ^ t \varphi ' ( u ( s ) ) G ( s ) \ , d W ( s ) \\ & + \frac 1 2 \int _ 0 ^ t \operatorname { T r } _ { G ( s ) } \varphi '' ( u ( s ) ) \ , d s . \end{align*}


\begin{align*} \frac { V _ { i j } } { V } = e _ i e _ j + \delta _ { i j } \bigg ( \frac { 1 } { 1 + \lambda _ i ^ 2 } - 2 e _ i ^ 2 \bigg ) \geq \frac { 1 - \lambda _ i ^ 2 } { ( 1 + \lambda _ i ^ 2 ) ^ 2 } \geq C ( \eta ) \end{align*}


\begin{align*} z ( s ) & = h ^ 2 \beta _ h ( s ) + \frac { h ^ 2 } { 2 } \overline { \beta _ h ( s ) } + \frac { \sigma ^ 2 } { 2 } \overline { \beta _ \sigma ( s ) } , \\ \zeta ( s ) & = i \beta _ h ( s ) ( K _ 1 + K _ 3 ) - i \overline { \beta _ h ( s ) } K _ 2 - i \overline { \beta _ \sigma ( s ) } K , \\ \widetilde \gamma ( s ) & = - i s ( \beta _ h ( s ) - 1 ) ( | K _ 1 | ^ 2 + | K _ 3 | ^ 2 ) + i s ( \overline { \beta _ h ( s ) } - 1 ) | K _ 2 | ^ 2 + i s ( \overline { \beta _ \sigma ( s ) } - 1 ) | K | ^ 2 . \end{align*}


\begin{align*} b _ i = x _ i + \sum _ { j > i } s _ { i j } x _ j = \displaystyle { x _ i + \sum _ { \substack { j : f ( j ) < f ( i ) \\ g ( j ) \ne g ( i ) } } s _ { i j } x _ j } . \end{align*}


\begin{align*} \mu _ 1 : = \sup \{ \lambda \geq 0 : ~ \exists u \in \mathcal P _ 0 ( \Omega ) , ~ \textnormal { s o l u t i o n o f } ~ \eqref { e q 3 1 9 } \} , \end{align*}


\begin{align*} T _ { \rho } ^ { ( i ) } = ( I \otimes \cdots \otimes \underbrace { T _ \rho } _ i \otimes \cdots \otimes I ) . \end{align*}


\begin{align*} T ^ { ( k ) } ( f _ 1 , f _ 2 ) = \sum _ { | l | > \Gamma } T _ l \left ( \Delta _ { \mathfrak { r } _ 1 l + k _ 1 } ^ { ( 1 ) } f _ 1 , \Delta _ { \mathfrak { r } _ 2 l + k _ 2 } ^ { ( 2 ) } f _ 2 \right ) . \end{align*}


\begin{align*} D _ { q } & \equiv \sum _ { n = 0 } ^ { \lfloor q / s \rfloor } C _ { q - n s , n } \Big ( \frac { \Delta t } { \Delta x ^ s } \Big ) ^ { n } . \end{align*}


\begin{align*} N ( x ) = F ( x ) + O ( x ^ { \theta } ) = A x + O ( x ^ { \theta } + x ^ { 1 - \theta } ) , \end{align*}


\begin{align*} y ( q x ) = \Gamma ( x ) y ( x ) . \end{align*}


\begin{align*} \abs { \frac { 1 } { n } \sum _ { j = 0 } ^ { n - 1 } U ^ j ( \mu ) ( x ) } \leq \frac { \epsilon } { \chi _ d } \rho ( x ) x \in ( e M e ) _ + n \in \{ 1 , \ldots , N _ d \} . \end{align*}


\begin{align*} \left ( T _ \lambda ^ { \vec { v } } \right ) _ { i , j } = \sum _ { m = 1 } ^ { n - j } - x _ { j + m } \alpha _ { n - i - m + 1 } \end{align*}


\begin{align*} G ( z ) : = \log \beta ( z ) + \frac { u ^ 2 } { 2 } + B \rho \end{align*}


\begin{align*} \langle u _ { p \alpha + k } ^ { ( i ) } ( x _ 1 , \ , \cdot \ , , t ) \rangle _ { \Upsilon _ i } : = \int _ { \Upsilon _ i } u _ { p \alpha + k } ^ { ( i ) } \big ( x _ i , \bar { \xi } _ i , t \big ) \ , d \bar { \xi } _ i = 0 . \end{align*}


\begin{align*} \mathrm { R W } ( d ) & = \sup _ { \lambda \ge 1 } \left \{ \inf _ { \pi \in \Pi ( \mu _ { 1 , L + 1 } , \dotsc , \mu _ { L , L + 1 } ) } \int _ { \mathcal { V } } \min _ { \ell \in [ L ] } \{ y _ \ell + \phi _ { \lambda , \ell } ( x _ 1 , \dotsc , x _ L ) \} d \pi ( s ) - \langle \lambda , \delta \rangle \right \} , \end{align*}


\begin{align*} | \Im \xi _ j | = | \eta _ 2 | > 2 \varpi , \xi _ j \cdot \xi _ j = \lambda , j = 1 , 2 , \xi _ 1 + \xi _ 2 = \eta . \end{align*}


\begin{align*} g ( s ) = 0 , g ( s + 1 ) = 0 , h ( s ) = 0 \end{align*}


\begin{align*} L _ { c l } = 3 M _ { c } K _ { c } \end{align*}


\begin{align*} S _ D ( n , k ) = \frac { 1 } { 2 ^ k k ! } \left [ \sum _ { \ell = 0 } ^ k D ( n , \ell ) \binom { n - \ell } { k - \ell } + n \cdot 2 ^ { n - 1 } ( k - 1 ) ! S ( n - 1 , k - 1 ) \right ] , \end{align*}


\begin{align*} \hat { b } _ { n + 1 } - \tfrac 1 2 = \tfrac { 1 } { 2 } b _ { n } ^ { \frac { 1 } { \beta } } . \end{align*}


\begin{align*} \tilde { \alpha } _ i = ( a _ l , \dots , a _ { j + 1 } , x , a _ j , \dots , a _ 1 ) . \end{align*}


\begin{align*} w ( x ) = \frac { \delta ( \gamma - 1 ) } { 2 \gamma } \left ( 1 - | x | ^ 2 \right ) , \ \ x \in B _ 1 ( 0 ) , \end{align*}


\begin{align*} 0 = \frac { 2 } { m } \Delta \left ( | X | ^ 2 \right ) + \frac { 2 } { m } \nabla _ X \left ( | X | ^ 2 \right ) . \end{align*}


\begin{align*} \frac { 1 } { 2 \pi i } \ , \intop _ { \frac { 1 } { 2 } - \mu - i \infty } ^ { \frac { 1 } { 2 } - \mu + i \infty } \Gamma ( z ) \ , \Gamma ( s - z ) \ , \zeta _ { p } ( 2 z ) \ , x ^ { 2 z } d z = \sqrt { \pi } \ , x \sum _ { m = 1 } ^ { \infty } \ , \frac { 1 } { 2 \pi i } \ , \intop _ { \mu - i \infty } ^ { \mu + i \infty } \ , \Gamma \left ( z \right ) \ , \Gamma \left ( s + z - \frac { 1 } { 2 } \right ) \ , \left ( 2 z , 2 \pi p m \right ) _ { m } \ , ( \pi x m ) ^ { - 2 z } \ , d z , \end{align*}


\begin{align*} \# { \rm R e s } ( P ) \cap [ 1 - C _ 1 h , 1 + C _ 1 h ] + i [ - C _ 2 h , \infty ) = \mathcal { O } ( h ^ { - n } ) . \end{align*}


\begin{align*} \begin{aligned} \int _ { \R ^ N } | v ( x ) | \ , | \left \langle \nabla \zeta ( x ) , \nabla \gamma _ R ( x ) \right \rangle | \ , d x & \le \frac { \bar C } { R } \ , \int _ { B _ R \setminus B _ { R / 2 } } | v ( x ) | \ , | \nabla \zeta ( x ) | \ , d x \\ & \le \frac { \bar C C } { R } \ , \int _ { B _ R \setminus B _ { R / 2 } } | v ( x ) | \psi ( x ) \ , d x . \end{aligned} \end{align*}


\begin{align*} \sup _ { \substack { 0 \leq r \leq t \\ x ' \in \R } } \int _ { \R ^ 3 } \big \| \Delta _ { h } ( r , z , t , x ) \big \| _ p \big \| \Delta _ { h } ( r , z , t , x ' ) \big \| _ p | h | ^ { 2 H - 2 } d h d z d x = K _ 1 + K _ 2 + K _ 3 + K _ 4 , \end{align*}


\begin{align*} f ( k , t , \mathbf { p } _ { n + 1 } ) = \overline { F } ( k - 1 , t , \mathbf { p } _ { n + 1 } ) - \overline { F } ( k , t , \mathbf { p } _ { n + 1 } ) . \end{align*}


\begin{align*} T ^ \varphi _ { j i , i } = R ^ \varphi _ { j i , i } = \frac { 1 } { 2 } ( S ^ \varphi ) _ j - \alpha \varphi ^ a _ { t t } \varphi ^ a _ j = 0 . \end{align*}


\begin{align*} b _ T ( \ell _ j ) = b _ T ( ( \ell _ j ) _ { \sf p } ) + b _ T ( ( \ell _ j ) _ { \sf q } ) . \end{align*}


\begin{align*} \widetilde { T } _ { i } : = \partial f _ { i } \cap \left ( - \sum _ { j \neq i } \partial f _ { j } \right ) , i = 1 , \ldots , m , \end{align*}


\begin{align*} \pi _ { \mu } ( f ) v = \begin{pmatrix} f ( T ^ t ) & 0 \\ \star & \star \end{pmatrix} \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} = \begin{pmatrix} v _ 1 \\ v _ 2 \end{pmatrix} . \end{align*}


\begin{align*} \sup _ { h \in \N } \mathcal { G } ( u _ h , \beta _ h , B _ r ) < + \infty \mbox { a n d } \lim _ { h \to + \infty } \Psi ( u _ h , \beta _ h , B _ r ) = \lim _ { h \to + \infty } \mathcal { H } ^ 1 ( J _ { u _ h } ) = 0 . \end{align*}


\begin{align*} \partial _ t \partial _ x w ( t , x ) = \frac 1 2 \partial _ { x x } \partial _ x w ( t , x ) + \frac 1 x \partial _ x \partial _ x w ( t , x ) - \frac { 1 } { x ^ 2 } \partial _ x w ( t , x ) . \end{align*}


\begin{align*} f _ { r _ 1 | \Phi ( \mathcal { A } ) > 0 } ( r ) = \upsilon ( \lambda , R _ S ) r e ^ { - \lambda \pi \frac { R _ S } { R _ E } r ^ 2 } , R _ { m i n } \leq r \leq R _ { m a x } , \end{align*}


\begin{align*} W ^ * \pi ( \lambda ) g _ { q p t } & = W ^ * \left ( \pi ( \lambda ) g + e ^ { \frac { - 2 \pi i t } { 3 } } e ^ { \frac { 2 \pi i k p } { M } } \pi ( \lambda ' ) g \right ) \\ & = W ^ * \pi ( \lambda ) g + e ^ { \frac { - 2 \pi i t } { 3 } } e ^ { \frac { 2 \pi i k p } { M } } W ^ * \pi ( \lambda ' ) g . \end{align*}


\begin{align*} k + r = k ' + r ' . \end{align*}


\begin{align*} \sum _ { i = 0 } ^ { N - 1 } D \varphi ( T ^ i x ) = \sum _ { l = 0 } ^ { k } \sum _ { 0 \leq i < q ( l ) } S ( l ) D \varphi ( ( T ^ { ( l ) } ) ^ i x ( l ) ) \end{align*}


\begin{align*} \sup _ { x ^ \prime \in \mathbb { R } ^ d } \left [ \varphi _ 1 ( x ^ \prime , x _ 1 ) + \varphi _ 2 ( x ^ \prime , x _ 2 ) \right ] = b + B ( x _ 1 - x _ 2 ) - ( x _ 1 - x _ 2 ) ^ \top W ( x _ 1 - x _ 2 ) . \end{align*}


\begin{align*} g _ { n } ( x ) = \left \{ \begin{tabular} { l } $ f _ { n } ( x ) $ \ i f \ $ x \in E _ { n } $ \\ $ f _ { n + m } ( x ) $ i f $ x \in E _ { n + m } \backslash E _ { n + m - 1 } , \ m = 1 , 2 , 3 , . . . . $ \end{tabular} \right . \end{align*}


\begin{align*} \theta _ f : \mathcal { X } \ni x \mapsto ( f _ j ( x ) ) _ { j = 1 } ^ n \in \ell ^ p ( [ n ] ) ; \theta _ g : \mathcal { X } \ni x \mapsto ( g _ k ( x ) ) _ { k = 1 } ^ n \in \ell ^ p ( [ n ] ) \end{align*}


\begin{align*} | D ( x , y ) | & < ( 1 - m ) \left ( 1 - y ^ { - 1 } \right ) \frac { x } { \log y } + M \sqrt { x } \log x + 1 \\ & \le \left ( ( 1 - m ) \left ( 1 - y ^ { - 1 } \right ) + M \frac { \log ^ 2 y } { \sqrt { y } } + \frac { \log y } { y } \right ) \frac { x } { \log y } , \end{align*}


\begin{align*} f _ { i j } = f _ { j k 0 } + f _ { j k 1 } Q _ k + \ldots + f _ { j k s } Q _ k ^ s \end{align*}


\begin{align*} \Phi ( x , y ) & = \# \{ n \le x \colon P ^ - ( n ) > y \Omega ( n ) \le 2 \} \\ & = \pi ( x ) - \pi ( y ) + 1 + \sum _ { y < p \leq \sqrt { x } } \sum _ { p \le q \le x / p } 1 \\ & = \pi ( x ) - \pi ( y ) + 1 + \sum _ { y < p \leq \sqrt { x } } ( \pi ( x / p ) - \pi ( p ) + 1 ) , \end{align*}


\begin{align*} \mathcal U _ \infty = \lim _ { r \to \infty } \frac { u ( r ) } { u _ 0 ( r ) } = \lim _ { r \to \infty } \frac { u ' ( r ) } { u _ 0 ' ( r ) } = \mathcal W _ \infty \end{align*}


\begin{align*} Q ( t _ 1 , t _ 2 , t _ 3 ) = & \left ( \widetilde { P } _ 1 ' \widetilde { P } _ 2 ' \right ) ( t _ 2 ) \left ( \widetilde { P } _ 1 ' \widetilde { P } _ 2 ' \right ) ( t _ 3 ) \left ( \widetilde { P } _ 1 ' ( t _ 2 ) \widetilde { P } _ 2 ' ( t _ 3 ) - \widetilde { P } _ 1 ' ( t _ 3 ) \widetilde { P } _ 2 ' ( t _ 2 ) \right ) \cdot \\ & \left ( \widetilde { P } _ 2 '' \widetilde { P } _ 1 ' ( t _ 1 ) - \widetilde { P } _ 1 '' \widetilde { P } _ 2 ' ( t _ 1 ) \right ) \end{align*}


\begin{align*} ( \rho \otimes \varphi ) ^ { \sigma ' , s ' } _ { \sigma , s } ( g ) : = \rho _ { g \sigma } ^ { \sigma ' } \cdot \varphi _ s ^ { s ' } ( g ) . \end{align*}


\begin{align*} x ^ { r + 1 } - z = 0 \end{align*}


\begin{align*} w _ b = \sum _ { \ell } c _ \ell a _ \ell + \Psi _ b , \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \alpha _ i z _ { i p } = \sum _ { i = 1 } ^ n \beta _ i z _ { i p } \leq \sum _ { i = 1 } ^ n \gamma _ i z _ { i p } , \end{align*}


\begin{align*} B ( T ) = - \frac { \pi ^ { 3 / 2 } \sigma ^ { 3 } } { \sqrt { 3 } } \sqrt { \frac { k T } { \varepsilon } } \exp \left ( \frac { \varepsilon } { k T } \right ) \left ( 1 + \frac { 3 5 } { 3 6 } \frac { k T } { \varepsilon } + \frac { 5 0 0 5 } { 2 5 9 2 } \left ( \frac { k T } { \varepsilon } \right ) ^ { 2 } + \mathcal { O } ( T ^ { 3 } ) \right ) . \end{align*}


\begin{align*} \sum _ { k = 1 } ^ g A _ k \otimes X _ k \leq I . \end{align*}


\begin{align*} \int _ { [ 0 , 1 ] } \d y \ , \widehat { r } _ \beta ( x , y ) = \beta \forall \ , x \in [ 0 , 1 ] , \end{align*}


\begin{align*} \dfrac { \partial u ^ { j \bar k } } { \partial u _ { p \bar q } } = - u ^ { j \bar q } u ^ { p \bar k } . \end{align*}


\begin{align*} \alpha ( \mathcal { K } ( n , k ) ) \leq \frac { 4 \binom { n } { 4 } } { 1 - \frac { 2 \binom { n - k } { k } } { - 2 \binom { n - k - 1 } { k - 1 } } } = 4 \binom { n - 1 } { k - 1 } . \end{align*}


\begin{align*} a _ i \ = \ a _ { m - i } \ = \ 9 , . \end{align*}


\begin{align*} \delta = \exp \left ( - \exp \left ( c \varepsilon ^ { - 6 } \right ) \right ) \end{align*}


\begin{align*} \mathsf { d } _ { c c } ( Z ) = \mathsf { d } _ { c c } \Bigg ( \prod _ { j = 1 } ^ k y \big ( \bigl \{ X _ { n i } ^ { ( j ) } \bigr \} \big ) \Bigg ) \leq 2 ^ { k - 1 } \mathsf { d } _ { \mathrm { c o m } } ^ { ( k ) } \big ( \bigl \{ X _ { n i } ^ { ( 1 ) } \bigr \} , \dots , \bigl \{ X _ { n i } ^ { ( k ) } \bigr \} \big ) \leq 1 . \end{align*}


\begin{align*} \sum _ { a , b } M ( a , b ) f ^ g ( b ) = \sum _ { a , b } M ( a , b ) f ( g b ) = \sum _ { a , b } M ( g a , g b ) f ( g b ) = \mu f ( g a ) = \mu f ^ g ( a ) \ , , \end{align*}


\begin{align*} S _ { \varepsilon } ( t ) = \left ( \begin{array} { c c } a _ { \varepsilon } ( t ) & 0 \\ 0 & 1 \end{array} \right ) , \end{align*}


\begin{align*} \lim _ { k \to \infty } \frac { \nu _ k } { \sigma ^ 2 _ k } = 0 ; \end{align*}


\begin{align*} w ( t ) = w _ 0 ( t ) + \int _ 0 ^ t S ( t - s ) F ( s ) w ( s ) \ , d s + \int _ 0 ^ t S ( t - s ) \Sigma ( s ) w ( s ) C \ , d W ( s ) \end{align*}


\begin{align*} X _ \eta = X _ \varphi = \overline { [ \underline { \varphi } , \varphi ] } . \end{align*}


\begin{align*} p > 2 : \lim _ { N \to \infty } \sum _ { i = 1 } ^ { M _ N } \mathbb E \Big [ \big | X _ { N , M _ N } ^ { ( i ) } \big | ^ p \Big ] = 0 \ , . \end{align*}


\begin{align*} \frac { 1 } { \sum _ { j \in \N } f ( j ) ^ 2 } \sum _ { v \in L } f ( v ) ^ 2 = o ( 1 ) \textrm { a s } N \to \infty . \end{align*}


\begin{align*} Q = L _ 1 \circ L _ 2 \circ L _ 3 . \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } p ' y ^ 2 = z \cdots \mbox { ( 6 $ - $ 1 ) } \\ x ^ 2 = w \cdots \mbox { ( 6 $ - $ 2 ) } \\ p ' w ^ 2 = p x + q z \cdots \mbox { ( 6 $ - $ 3 ) } \\ z ^ 2 = p y + q w \cdots \mbox { ( 6 $ - $ 4 ) } \\ p ' y w = a x + b z \cdots \mbox { ( 6 $ - $ 5 ) } \\ x z = a y + b w \cdots \mbox { ( 6 $ - $ 6 ) } \\ p ' y w = c x + d z \cdots \mbox { ( 6 $ - $ 7 ) } \\ x z = c y + d w \cdots \mbox { ( 6 $ - $ 8 ) } . \end{array} \right . \end{align*}


\begin{align*} \left ( { \nabla } _ { X } C \right ) \xi = - \omega A _ { \xi } X - h ( X , B \xi ) \end{align*}


\begin{align*} S _ i ^ * T T ^ * = S _ i ^ * T T ^ * \left ( \sum _ { j = 1 } ^ d S _ j S _ j ^ * \right ) = T T ^ * S _ i ^ * . \end{align*}


\begin{align*} u _ j = \sum _ { s _ 1 , \dots , s _ j = 1 } ^ { d _ 1 } \alpha _ { s _ 1 \dots s _ j } X _ { s _ 1 } \otimes \cdots \otimes X _ { s _ j } & = \sum _ { s _ 1 , \dots , s _ j = 1 } ^ { d _ 1 } \big ( \alpha _ { s _ 1 \dots s _ j } ^ 1 X _ { s _ 1 } \big ) \otimes \cdots \otimes \big ( \alpha _ { s _ 1 \dots s _ j } ^ j X _ { s _ j } \big ) \\ & = : \sum _ { n = 1 } ^ { d _ 1 ^ j } X _ { n 1 } \otimes \cdots \otimes X _ { n j } , \end{align*}


\begin{align*} Q = \left ( \begin{array} { c c } 3 & 0 \\ 0 & 3 \end{array} \right ) , A = \left ( \begin{array} { c c c } 1 & 2 \\ 2 & 1 \end{array} \right ) , L = \left ( \begin{array} { c c c } 2 & - 2 \\ - 2 & 2 \end{array} \right ) . \end{align*}


\begin{align*} \Sigma ( \delta ) = \left \{ \gamma \in \mathcal { P } ( \mathcal { S } ) : \boldsymbol { W } _ { p _ 1 } ( \gamma _ { 1 } , \mu _ 1 ) \leq \delta _ 1 ^ { 1 / p _ 1 } , \ \boldsymbol { W } _ { p _ 2 } ( \gamma _ { 2 } , \mu _ 2 ) \leq \delta _ 2 ^ { 1 / p _ 2 } \right \} . \end{align*}


\begin{align*} D _ I = \left \{ \left ( \begin{array} { c c } e ^ { - t _ 1 } \mathrm { I } _ 2 & 0 \\ 0 & e ^ { - t _ 2 } \mathrm { I } _ 3 \\ \end{array} \right ) , t _ 1 > t _ 2 \right \} , \textrm { w h e r e } \mathrm { I } _ k \textrm { i s t h e $ k \times k $ i d e n t i t y m a t r i x . } \end{align*}


\begin{align*} J \left ( t , \xi ; u \right ) \triangleq \left . Y _ { s } ^ { t , \xi ; u } \right \vert _ { s = t } \end{align*}


\begin{align*} \alpha A \circ \beta B = \alpha \beta ( A \circ B ) . \end{align*}


\begin{align*} \pi ^ 2 = 3 6 L i _ 2 \left ( \frac { 1 } { 2 } \right ) - 3 6 L i _ 2 \left ( \frac { 1 } { 4 } \right ) - 1 2 L i _ 2 \left ( \frac { 1 } { 8 } \right ) + 6 L i _ 2 \left ( \frac { 1 } { 6 4 } \right ) . \end{align*}


\begin{align*} a _ \varepsilon ( x _ n ) : = \sup _ { x ^ \prime \in \mathbb { R } ^ { n - 1 } } \int _ { B _ { \delta } ( \hat { x } ^ \prime ) \times ( 0 , \delta ) } \rho _ \varepsilon ( | A _ \gamma ( \hat { x } , \hat { x } ) ( \hat { x } - \bar { y } ) | ) \ ; y . \end{align*}


\begin{align*} \left ( { D } ^ { \alpha } _ { N } U ^ { n - \sigma } , U ^ { n , \sigma } \right ) \ , \le & \ , \| f ^ { n - \sigma } \| \ , \| U ^ { n , \sigma } \| . \end{align*}


\begin{align*} \theta _ { X ^ { \ast } , L } ( f ) ( \epsilon , y ) & = \int _ { X ^ { \ast } / X ^ { \ast } \cap L } f ( [ \dot { y } ^ { \ast } , 0 ] + [ ( \epsilon , y ) , 0 ] ) d \dot { y } ^ { \ast } \\ & = \int _ { X ^ { \ast } / X ^ { \ast } \cap L } \psi ( \epsilon \tfrac { \langle \dot { y } ^ { \ast } , y \rangle } { 2 } ) f ( \epsilon , y + \epsilon \dot { y } ^ { \ast } ) d \dot { y } ^ { \ast } , \end{align*}


\begin{align*} i \frac { \partial \psi } { \partial t } + \triangle \psi - V ( x ) \psi + h ( | \psi | ^ { 2 } ) \psi = 0 \hbox { i n } \mathbb { R } ^ N \end{align*}


\begin{align*} \overline { \overline { C } } _ { X ^ { \ast } } ( k _ 1 , k _ 2 ) & = \widetilde { C } _ { X ^ { \ast } } ( k _ 1 , k _ 2 ) \widetilde { s } ( k _ 1 ) \widetilde { s } ( k _ 2 ) \widetilde { s } ( k _ 1 k _ 2 ) ^ { - 1 } = 1 ; \end{align*}


\begin{align*} \textsf { E } \exp ( C _ { 0 } \lambda \Vert B _ { i } \Vert _ { 2 } ^ { 2 } ( \max _ { j \le n } \eta _ { j } - \textsf { E } \max _ { j \le n } \eta _ { j } ) ) = \prod _ { l = 1 } ^ { n } \textsf { E } \exp ( C _ { 0 } \lambda \Vert B _ { i } \Vert _ { 2 } ^ { 2 } \frac { T _ { l } - 2 } { 2 ( n - l + 1 ) } ) , \end{align*}


\begin{align*} \mathrm { R W } _ 0 ( d ) = \sup _ { \eta \ge 1 } \left \{ \mathbb { E } _ { \mu } \left [ \max \{ Y _ 2 + \eta h _ 1 ( X ) , Y _ 1 + \eta h _ 0 ( X ) \} \right ] - \eta \delta _ 0 \right \} , \end{align*}


\begin{align*} \psi ^ 0 _ { \mu , \xi } ( x ) = \mu \frac { \partial U _ { \mu , \xi } } { \partial \mu } , \psi ^ h _ { \mu , \xi } ( x ) = \mu \frac { \partial U _ { \mu , \xi } } { \partial \xi _ { h } } . \end{align*}


\begin{align*} \widetilde { \Delta } \circ \widetilde { \varphi } ( x , m ) & = \widetilde { \Delta } ( \varphi _ L ( x ) , \varphi _ M ( m ) ) \\ & = \Delta ( \varphi _ L ( x ) ) + \rho _ { \beta } ( \varphi _ M ( m ) ) - \tau \circ ( \rho _ { \beta } ( \varphi _ M ( m ) ) ) . \end{align*}


\begin{align*} C _ { \gamma } ^ k [ 0 , 1 ] = \big \{ x \in \mathbb { R } ^ k : 0 < x _ 1 < \dots < x _ k < 1 \big \} . \end{align*}


\begin{align*} \left . \begin{array} { l l } \mathcal { S } ( \xi , - \eta ) = \min \limits _ { ( u , v ) \in \mathcal { M } } \mathcal { S } ( u , v ) . \end{array} \right . \end{align*}


\begin{align*} \left \| \partial _ { x _ j } ^ k f _ j \right \| _ \infty \lesssim \lambda ^ k \| f _ j \| _ \infty , \textrm { f o r $ j = 1 , 2 $ } . \end{align*}


\begin{align*} d _ { w } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \right ) & = \int _ { - \infty } ^ { \infty } \left | \mathbb { P } \left ( \frac { \log Z _ { n } - n \mu } { \sigma \sqrt { n } } \leq x \right ) - \Phi ( x ) \right | \mbox { d } x \\ & \leq \frac { C } { n ^ { \delta / 2 } } \int _ { - \infty } ^ { \infty } \frac { 1 } { 1 + | x | ^ { 1 + \delta ^ { ' } } } \mbox { d } x \\ & \leq \frac { C } { n ^ { \delta / 2 } } , \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow \infty } w ^ { k } = w ^ { * } \in W ^ { * } . \end{align*}


\begin{align*} & \alpha \leq \overline { C } _ 1 \max _ { q \in S ^ 2 : \ , q \cdot v = 0 } \ ; \max _ { 0 \le s \le t } \ ; ( q \cdot W _ s ) \leq \overline { C } _ 1 \varepsilon , \quad \widehat { \alpha } \leq \overline { C } _ 2 \varepsilon , \epsilon \leq \overline { C } _ 3 \ , \alpha ^ 2 . \end{align*}


\begin{align*} \varphi _ t = \sum _ { N _ t } \tau _ j ^ { - 1 } a _ j \psi _ j , \end{align*}


\begin{align*} \Delta _ \varepsilon \left [ f ( x ) \right ] _ { x = \alpha } \overset { } { = } f ( \alpha + \varepsilon ) - f ( \alpha ) . \end{align*}


\begin{align*} \Pi _ { \psi } ( g ) f ( [ \epsilon , w ] ) & = \widetilde { C } _ { X ^ { \ast } } ( g , h _ { - \epsilon } ) \widetilde { \beta } ( h _ { - 1 } g ^ { s ( - \epsilon ) } ) ^ { - 1 } f ( [ - \epsilon , w g ^ { s ( - \epsilon ) } ] ) \\ & = e ^ { \tfrac { \pi i ( 1 - \epsilon ) } { 4 } } ( - c , - \epsilon ) _ { \R } \beta ( d , - \epsilon c ) ^ { - 1 } f ( [ - \epsilon , w g ^ { s ( - \epsilon ) } ] ) . \end{align*}


\begin{align*} \begin{aligned} \Pr ( X _ t \le k \ , \vert \ , X _ 0 = r ) & = \frac { n - k } { n ^ t } \binom { n - r } { n - k } \sum _ { j = 0 } ^ { k - r } ( - 1 ) ^ { k - r - j } \binom { k - r } { j } \frac { ( r + j ) ^ t } { n - r - j } \\ \end{aligned} \end{align*}


\begin{align*} \nu _ i \left ( j f _ j Q _ i ' Q _ i ^ { j - 1 } \right ) = v ( j ) + \nu \left ( f _ j Q _ i ^ j \right ) + \nu \left ( Q ' _ i \right ) - \nu ( Q _ i ) \geq \nu _ i ( f ) + \alpha _ i . \end{align*}


\begin{align*} \partial _ j \Phi & = \lambda \partial _ j \Psi e ^ { \lambda \Psi } , \partial _ { j , k } \Phi = \lambda \partial _ { j , k } \Psi e ^ { \lambda \Psi } + \lambda ^ 2 ( \partial _ j \Psi ) ( \partial _ k \Psi ) e ^ { \lambda \Psi } , \end{align*}


\begin{align*} m ^ { 1 - r } \cdot \tilde { B } _ r ( x ) = \sum _ { y m = x } \tilde { B } _ r ( y ) . \end{align*}


\begin{align*} \sigma \left ( ( A ^ { * _ 1 } , \underline { A } ^ { * _ 2 } ) \right ) : = \left ( \sigma ( A ^ { * _ 1 } ) , \sigma ( \underline { A } ^ { * _ 2 } ) \right ) . \end{align*}


\begin{align*} \mathcal { F } _ { X ^ { \ast } X } : & \mathcal { H } ( X ) \longrightarrow \mathcal { H } ( X ^ { \ast } ) ; \\ & f \longmapsto \mathcal { F } _ { X ^ { \ast } X } ( f ) ( h ) = \int _ { X ^ { \ast } } f ( [ x ^ { \ast } , 0 ] h ) d x ^ { \ast } . \end{align*}


\begin{align*} C _ { \omega , \mathbf { c } } : = \left \{ U \in \mathcal { H } ^ 1 \backslash \{ ( \mathbf { 0 } , \mathbf { 0 } , \mathbf { 0 } ) \} \ | \ S _ { \omega , \mathbf { c } } ( U ) < \widetilde { \mu } _ { \omega , \mathbf { c } } , \ I _ { \omega , \mathbf { c } } ( U ) < 0 , \ \mathbf { c } \cdot \mathbf { P } ( U ) \ge 0 \right \} . \end{align*}


\begin{align*} \mathbb { P } ( { E } _ { i , 2 } ) & = \underset { \substack { k _ 1 + \cdots + k _ K \\ = m - i } } { \sum } \frac { ( m - i ) ! } { k _ 1 ! \cdots k _ K ! } \mathbb { P } _ K ^ { k _ 1 } \mathbb { P } _ { e , 1 } ^ { k _ 1 } \times \cdots \times \mathbb { P } _ K ^ { k _ K } \mathbb { P } _ { e , K } ^ { k _ K } , \end{align*}


\begin{align*} \mathcal { J } _ { p } ( x , s ) : = \intop _ { 0 } ^ { \infty } \ , \frac { y ^ { s - \frac { 1 } { 2 } } J _ { s - \frac { 1 } { 2 } } ( 2 \pi x y ) } { \sigma \left ( y \right ) e ^ { 2 \pi y } - 1 } \ , d y \end{align*}


\begin{align*} \sigma ( c _ { i j k } ) = c _ { \sigma i \sigma j \sigma k } \end{align*}


\begin{align*} \begin{aligned} L ( K , u ^ * ) & = \{ t v \in \mathcal { U } : \ G ' ( u ^ * ) u ^ * + G ' ( u ^ * ) v \in K , \ \forall \ t > 0 \} \\ & = \{ t v \in \mathcal { U } : \ G ' ( u ^ * ) u ^ * + G ' ( u ^ * ) v \in \mathcal { K } - G ( u ^ * ) + G ' ( u ^ * ) u ^ * , \ \forall \ t > 0 \} \\ & = \{ t v \in \mathcal { U } : \ G ( u ^ * ) + G ' ( u ^ * ) v \in \mathcal { K } , \ \forall \ t > 0 \} \\ & = L ( \mathcal { K } , u ^ * ) . \end{aligned} \end{align*}


\begin{align*} Q _ { 3 , \vec { v } _ { s , t , f , 0 } ^ { ( 1 ) } } = \left ( \left ( \begin{array} { r r r } - s & 0 & 1 \\ 1 & 0 & 0 \\ - t & 1 & 0 \end{array} \right ) , \left ( \begin{array} { r r r } 0 & 1 & 0 \\ 0 & t & 1 \\ 1 & s & 0 \end{array} \right ) , \left ( \begin{array} { r r r } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right ) \right ) . \end{align*}


\begin{align*} \Gamma _ { B _ + ( T _ 1 \cdots T _ k ) } ( A ) = B ( \Gamma _ { T _ 1 } \cdots \Gamma _ { T _ k } ) , \end{align*}


\begin{align*} \left \{ \begin{array} { l } \varphi ^ + ( M _ I ) = P _ + ( \varphi ^ - ( M _ { I ' } ) a ^ { i _ p } ) \\ \varphi ^ - ( M _ I ) = - P _ - ( \varphi ^ - ( M _ { I ' } ) a ^ { i _ p } ) . \end{array} \right . \end{align*}


\begin{align*} \Gamma _ { s } ^ { N , \phi } ( \pi ^ N ) & = \frac { 2 \kappa } { N ^ 2 } \sum _ { ( x , \sigma ) \in V } \eta ^ N _ s ( x , \sigma ) ( \partial _ x \phi ( \tfrac { x } { N } , \sigma ) ) ^ 2 + \frac { 1 } { N ^ 2 } \sum _ { ( x , \sigma ) \in V } \sum _ { \sigma ' \in S } c ( \sigma , \sigma ' ) \eta ^ N _ s ( x , \sigma ) ( \phi ( \tfrac { x } { N } , \sigma ' ) - \phi ( \tfrac { x } { N } , \sigma ) ) ^ 2 \\ & \ \ \ \ \ \ \ \ + R _ 4 ( \phi , s , N , \sigma ) , \end{align*}


\begin{align*} H _ \delta = H \cap \left ( \bigcap _ { \substack { g \in G \\ n _ g \neq 0 } } g H g ^ { - 1 } \right ) . \end{align*}


\begin{align*} E _ t ( \phi , \omega , r ) : = G ( \phi , \omega , r ; t ) = \big ( x ( \phi , \omega , r ; t ) , y ( \phi , \omega , r ; t ) , z ( \phi , \omega , r ; t ) \big ) , \end{align*}


\begin{align*} F _ { f } ( z ) : = \log \dfrac { f ( z ) } { z } = 2 \sum _ { n = 1 } ^ { \infty } \gamma _ { n } ( f ) z ^ n , \ ; \ ; z \in \mathbb { D } , \ ; \ ; \log 1 : = 0 , \end{align*}


\begin{align*} \begin{pmatrix} \Phi _ t & \Phi _ { t + 1 } \\ \Phi _ { t + 1 } & \Phi _ { t + 2 } \end{pmatrix} \end{align*}


\begin{align*} \Phi ( x , y ) = \Phi ( x , z ) + \sum _ { y < p \le z } \Phi ( x / p , p ^ - ) , \end{align*}


\begin{align*} \| w _ t \| _ { H ^ 1 ( \Gamma ) } = \| \mathcal { T } ^ \ast v _ t \| _ { H _ \Gamma ^ 1 ( \partial \mathrm { M } ) } \le t \| v _ t \| _ { L ^ 2 ( \mathrm { M } _ 0 ) } . \end{align*}


\begin{align*} & x ^ { \alpha } y ^ { \beta } z ^ { \gamma } + x ^ { \beta } y ^ { \gamma } z ^ { \alpha } + x ^ { \gamma } y ^ { \alpha } z ^ { \beta } - x ^ { \beta } y ^ { \alpha } z ^ { \gamma } - x ^ { \gamma } y ^ { \beta } z ^ { \alpha } - x ^ { \alpha } y ^ { \gamma } z ^ { \beta } \\ = & ( x - y ) ( x - z ) ( y - z ) \sum _ { i = 0 } ^ { \beta - \gamma - 1 } \sum _ { j = 0 } ^ { \alpha - \beta - 1 } \sum _ { k = 0 } ^ { \alpha - \beta - 1 - j + i } x ^ { \gamma + i + j } y ^ { \beta - 1 - i + k } z ^ { \alpha - 2 - j - k } . \end{align*}


\begin{align*} \overline { \overline { C } } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) & = \widetilde { C } _ { X ^ { \ast } } ( g _ 1 , g _ 2 ) \widetilde { s } ( g _ 1 ) \widetilde { s } ( g _ 2 ) \widetilde { s } ( g _ 1 g _ 2 ) ^ { - 1 } . \end{align*}


\begin{align*} C _ { \mathrm { U B F } } = \Big ( \frac { e } { \theta } \Big ) ^ { \lceil \frac { 2 \eta } { b } \rceil } = \Big ( \frac { 5 e ^ 2 \Delta } { b ^ 2 } \Big ) ^ { \lceil \frac { 2 \eta } { b } \rceil } , \end{align*}


\begin{align*} \| u _ n - u _ m \| _ p ^ p = \sum _ { x \in V } \mu ( x ) | u _ n ( x ) - u _ m ( x ) | ^ p < \epsilon , \quad \forall n , m \geq N . \end{align*}


\begin{align*} p _ k ( \lambda , \mu ) = \sum \limits _ { j = 0 } ^ { 2 n + 1 } ( - 1 ) ^ j \lambda ^ { - j - 1 } \mu ^ j \end{align*}


\begin{align*} A ( w , D _ \tau w ) : D ^ 2 _ { \tau } w + b ( w , D _ \tau w ) \cdot D _ \tau w + \mu w = ( n - 1 ) - \mu - ( 1 + w ) R , \end{align*}


\begin{align*} A \circ B & = ( \mathcal { R } A + i \mathcal { I } A ) \circ ( \mathcal { R } B + i \mathcal { I } B ) \\ & = ( \mathcal { R } A \circ \mathcal { R } B - \mathcal { I } A \circ \mathcal { I } B ) + i ( \mathcal { R } A \circ \mathcal { I } B + \mathcal { I } A \circ \mathcal { R } B ) . \end{align*}


\begin{align*} \hat { m } ( x , t ) = \begin{cases} \bar { m } ( x , t ) & \mbox { i f } \ , x \in \Omega \ , \mbox { a n d } \ , t \in ( 0 , T ) , \\ 0 & \mbox { i f } \ , x \in \Omega \ , \mbox { a n d } \ , t \geq T , \end{cases} \end{align*}


\begin{align*} \theta ( a ^ * a ) \geq \theta ( a ) ^ * \theta ( a ) = \sigma ( a ) ^ * \sigma ( a ) = \sigma ( a ^ * a ) . \end{align*}


\begin{align*} ( \Delta + \lambda - q _ 1 ) \tilde { v } _ 2 = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , \tilde { v } _ 2 { _ { | \partial M } } = v _ 2 { _ { | \partial M } } , \end{align*}


\begin{align*} { \rm E n t S p } \left ( G , \mu \right ) : = \left \{ h _ { \mu } ( X , \nu ) : ( X , \nu ) \mbox { i s a n e r g o d i c } ( G , \mu ) \mbox { - s t a t i o n a r y s y s t e m } \right \} . \end{align*}


\begin{align*} x ^ { - 2 s } = x ^ { - 1 } \left ( 1 - 2 \log ( x ) \ , \left ( s - \frac { 1 } { 2 } \right ) + O \left ( \left ( s - \frac { 1 } { 2 } \right ) ^ { 2 } \right ) \right ) , \end{align*}


\begin{align*} \underline { \varphi } ( h ) = \varphi ^ * ( \Gamma _ { H , H ^ * } ( h ) ) . \end{align*}


\begin{align*} \mathbf { R } \in L ^ { \infty } ( \mathcal { D } _ { a T } , \mathbb { R } ^ { 3 \times 3 } ) . \end{align*}


\begin{align*} X _ j = \frac { \partial } { \partial x _ j } , \enspace j = 1 , \ldots , k , X _ { k + j } = | x | ^ { 2 \alpha } \frac { \partial } { \partial x _ { k + j } } , \enspace j = 1 , \ldots , N - k \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle D _ u \theta ( u ^ * ) , v \rangle _ \mathcal { U } + \langle \tilde { \lambda } ^ * , S v \rangle _ { * } = 0 \forall \ v \in \mathcal { U } ; \\ & - \langle \tilde { \lambda } ^ * , \zeta - S u ^ * \rangle _ * \geq 0 \forall \ \zeta \in \overline { K \cap R ( S ) } ^ { \| \cdot \| _ * } , \\ \end{aligned} \right . \end{align*}


\begin{align*} | c ( \varphi ) | \norm { S ( k ) ( \bar { \xi } - \bar { \xi } \circ T ) } _ { \sup } & = \| S ( k ) \chi \| _ { \sup } \leq \norm { S ( k ) \varphi } _ { \sup } + \norm { S ( k ) ( \widetilde { v } _ 0 \circ T - \widetilde { v } _ 0 ) } _ { \sup } \\ & \leq ( \| v _ 0 \| _ { \sup } + o ( 1 ) ) \norm { S ( k ) ( \bar { \xi } - \bar { \xi } \circ T ) } _ { \sup } . \end{align*}


\begin{align*} & \quad \Pr \left ( > \theta | \Phi , \Phi ( \mathcal { A } ) > 0 \right ) \\ & \approx \sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } \prod \limits _ { x _ { i } \in \Phi \backslash \{ x _ 1 \} \cap { \mathcal { A } } } \frac { 1 } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M r _ i ^ { \alpha } } \right ) ^ M } . \end{align*}


\begin{align*} \begin{cases} P ( ( - \Delta ) ^ s ) u ^ { ( 2 ) } ( x ) + F ^ { ( 2 ) } ( x ) u ^ { ( 2 ) } ( x ) = 0 & \Omega , \\ u ^ { ( 2 ) } = f ^ 2 & \Omega ^ c : = \mathbb { R } ^ n \setminus \overline { \Omega } . \end{cases} \end{align*}


\begin{align*} \mathrm { R W } _ { C } ( d ) & = \sum _ { \ell = 1 } ^ L \mathbb { E } _ { ( Y _ \ell , X ) \sim \mu _ { \ell , L + 1 } } \left [ ( Y _ \ell - \delta _ 0 ) I ( D ( X ) = \ell ) \right ] \\ & = \mathbb { E } _ { X } \left [ \sum _ { \ell = 1 } ^ L \left ( \mathbb { E } [ Y _ \ell \mid X ] - \delta _ 0 \right ) I ( D ( X ) = \ell ) \right ] . \end{align*}


\begin{align*} y ^ m = P ( z ) , \end{align*}


\begin{align*} \overline { \mathrm { u m d i m } } _ { Y ^ \mathbb { N } \times I _ { \varphi } ( G ) } ( F , D ) = \overline { \mathrm { u m d i m } } _ { I _ { \varphi } ( G ) } ( G , d , \mathbb { P } ) . \end{align*}


\begin{align*} \mathcal { T } _ D ( f ) ( z ) = - \frac { 1 } { 2 \pi } \iint _ D \left ( \frac { f ( \zeta ) } { \zeta } \frac { \zeta + z } { \zeta - z } + \frac { \overline { f ( \zeta ) } } { \overline { \zeta } } \frac { 1 + z \overline { \zeta } } { 1 - z \overline { \zeta } } \right ) \ , d \xi \ , d \eta , \end{align*}


\begin{align*} M \vec x ^ 2 & = c _ 1 \odot M \vec x ^ 1 , \\ \vec x ^ 1 - \vec z ^ { \vec x ^ 1 } & = c _ 2 \odot ( \vec x ^ 2 - \vec z ^ { \vec x ^ 2 } ) . \end{align*}


\begin{align*} \mathcal { T } _ D ^ k ( \langle h _ k , P _ r ( \theta - \cdot ) + i Q _ r ( \theta - \cdot ) \rangle ) = \langle h _ k , ( P _ r ( \theta - \cdot ) + i Q _ r ( \theta - \cdot ) ) ( e ^ { i ( \cdot ) } - r e ^ { i \theta } + \overline { e ^ { i ( \cdot ) } - r e ^ { i \theta } } ) ^ k \rangle , \end{align*}


\begin{align*} \left \{ \begin{array} { l } \partial _ { t } U ( t , \xi ) = i \langle \xi \rangle A ( t ) U ( t , \xi ) + i \langle \xi \rangle ^ { - 1 } Q ( t ) U ( t , \xi ) + F ( t , \xi ) , \quad ( t , \xi ) \in ( 0 , T ] \times \mathcal { I } _ { \hbar } , \\ U ( 0 , \xi ) = U _ { 0 } ( \xi ) , \quad \xi \in \mathcal { I } _ { \hbar } . \end{array} \right . \end{align*}


\begin{align*} s _ k ( A _ { s } ( \mathrm { i , j } ) ) = 0 { \rm f o r \quad } s = 1 , \ldots , m . \end{align*}


\begin{align*} \mathcal { P } _ 0 ( \Omega ) : = \{ u \in P S H ( \Omega ) \cap C ^ { 2 } ( \Omega ) \cap C ^ 0 ( \bar \Omega ) \ , ; \ , u _ { | _ { \partial \Omega } } = 0 \} . \end{align*}


\begin{align*} L ^ 1 * M ^ { p , q } & = M ^ { p , q } , \end{align*}


\begin{align*} - X '' + R ( \dot { \gamma } , X ) \dot { \gamma } = 0 , \end{align*}


\begin{align*} \left \lbrace \begin{array} { l c r } L v = - \lambda _ 1 u G & \textnormal { o n } & \Omega \\ v = 0 & \textnormal { i n } & \partial \Omega , \\ \end{array} \right . \end{align*}


\begin{align*} \omega ^ + = \tilde { \omega } ^ + + c _ 1 , \ , \ , \omega ^ - = \tilde { \omega } ^ - + c _ 2 , \ , \ , \omega ^ { ( 0 ) } = \tilde { \omega } ^ { ( 0 ) } + c _ 0 \end{align*}


\begin{align*} \| F \| _ { [ \mathcal { X } ] ' } = \| f \| . \end{align*}


\begin{align*} \Big \Vert \sup _ { k \leq 2 ^ { L - m ( L ) } } \Big | \sum _ { \ell = 1 } ^ k ( \widetilde U _ { \ell , L } - U _ { \ell , L } ) \Big | \Big \Vert _ 2 ^ 2 \ll 2 ^ { L - m ( L ) } \ , . \end{align*}


\begin{align*} \prod _ { \substack { l , m , n \geq 1 \\ l , m \leq n ; \ , \gcd ( l , m , n ) = 1 } } \left ( 1 - z ^ n \right ) ^ { \frac { l m ^ 2 } { n ^ 4 } } = \sqrt [ 3 ] { \left ( 1 - z \right ) } \ ; \exp \left \{ - \frac { 1 } { 1 2 } \left ( L i _ 2 ( z ) + \frac { z ( 7 - 5 z ) } { ( 1 - z ) ^ 2 } \right ) \right \} . \end{align*}


\begin{align*} h & = \psi ( g ^ { 1 } , \ldots , g ^ { k } ) \end{align*}


\begin{align*} \abs { \nabla \left ( \mathcal { P } _ a \circ w \right ) } = \abs { \nabla \left ( \mathcal { P } \circ ( w ( x ) - a ) \right ) } = \abs { ( \nabla \mathcal { P } ) ( w ( x ) - a ) } \abs { \nabla w ( x ) } . \end{align*}


\begin{align*} \Omega ( m ^ { 0 . 5 } \cdot s ^ { \omega } \cdot m ^ 2 ) = \Omega ( s ^ { \omega } \cdot m ^ { 2 . 5 } ) , \end{align*}


\begin{align*} \mathrm { I } ( X , \mathcal { F } ) = \int _ { \Omega } \int _ { S } \log \frac { d P \left ( X | \mathcal { F } \right ) } { d P ( X ) } d P \left ( X | \mathcal { F } \right ) d \mathbb { P } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle D _ u \theta ( u ^ * ) , v \rangle _ \mathcal { U } + \langle \lambda ^ { * } , S v \rangle = 0 \forall \ v \in \mathcal { U } ; \\ & - \langle \lambda ^ { * } , \zeta - S u ^ * \rangle \geq 0 \forall \ \zeta \in \overline { K \cap R ( S ) } , \\ \end{aligned} \right . \end{align*}


\begin{align*} \mathbb { V } ( \overline { S } _ n ) = \mathbb { E } ( \overline { S } _ n ^ 2 ) - \mathbb { E } ( \overline { S } _ n ) ^ 2 . \end{align*}


\begin{align*} v ( t ) = v _ 0 ( t ) + \int _ 0 ^ t S ( t - s ) f ' ( u ( s ) ) v ( s ) \ , d s + \int _ 0 ^ t S ( t - s ) \sigma ' ( u ( s ) ) v ( s ) B \ , d W ( s ) , \end{align*}


\begin{align*} f ( t ) = t ^ { s } \kappa ( t ) , \end{align*}


\begin{align*} Z _ q ( G ) \le T + \max _ { 1 \le i _ 1 < \ldots < i _ q \le s } \sum _ { j = 1 } ^ q a _ { i _ j } . \end{align*}


\begin{align*} \mathcal { I } _ { \mathrm { D M R } } ( \delta ) : = \sup _ { \gamma \in \Sigma _ { \mathrm { D M R } } ( \delta ) } \int _ { \mathcal { X } } f \ , d \gamma , \mbox { } \delta \ge 0 , \end{align*}


\begin{align*} c l V _ n \cap c l V _ m = \emptyset , m \neq n , \ ; \ ; \min _ { c l V _ n } | f _ n | \geq 1 - 1 / n . \end{align*}


\begin{align*} \rho _ { \beta } \circ \varphi _ M ( m ) = ( \varphi _ M ( m ) \otimes \alpha ) \rho _ { \beta } ( m ) + ( \beta \otimes \varphi _ L ) \rho _ { \beta } ( m ) , m \in M . \end{align*}


\begin{align*} \frac { u _ 1 } { c ( v _ 1 ) } + \cdots + \frac { u _ n } { c ( v _ n ) } = 0 . \end{align*}


\begin{align*} g _ 2 ( x ) = \frac { | x | ^ \alpha } \alpha - \frac { | x | ^ \beta } \beta , \qquad \alpha > \beta > 0 \ , . \end{align*}


\begin{align*} \theta _ j ^ y = X ^ { \mathcal { X } _ j ^ y , \mathcal { N } _ j ^ y } _ { T _ c ( X ^ { \mathcal { X } _ j ^ y , \mathcal { N } _ j ^ y } ) } \end{align*}


\begin{align*} \sum _ { \substack { \varepsilon \in F ' / K \\ \pi _ { K _ 1 } ( \varepsilon ) = \lambda } } \sum _ { \substack { \mu \in L ' / K \\ p _ K ( \mu ) = \varepsilon } } c ( m + q ( \mu _ { K _ \R ^ \perp } ) , \mu ) = \sum _ { \substack { \gamma \in L _ 1 ' \\ \gamma | _ { K _ 1 } = \lambda } } \sum _ { \substack { \delta \in M ' / L \\ \pi ( \delta ) = \gamma } } c ( m + q ( \gamma _ { ( K _ 1 ) _ \R ^ \perp } ) , \delta ) \end{align*}


\begin{align*} \Phi [ u ] ( t ) = e ^ { - t H ^ \beta } u _ 0 + \int _ 0 ^ t e ^ { - ( t - \tau ) H ^ \beta } \left ( f ( u ( \tau ) ) \right ) \ , d \tau . \end{align*}


\begin{align*} r '' : = \sqrt { \frac { d - 1 } { d } } r ' \le r ' . \end{align*}


\begin{align*} 0 = \int _ U \nabla u \cdot \nabla ( w ^ t - w ^ 0 ) \d x - \int _ { U } f ( u ) ( w ^ t - w ^ 0 ) \d x = : I _ 1 ( t ) - I _ 2 ( t ) , \ \ \ t \in [ 0 , 1 / \kappa ) . \end{align*}


\begin{align*} & f \in C ^ 0 ( [ - 1 , 1 ] ) \cap C ^ 1 ( - 1 , 1 ) , \\ & \lim \limits _ { s \rightarrow - 1 } f ^ \prime ( s ) = - \infty , \ ; \ ; \ ; \ ; \lim \limits _ { s \rightarrow 1 } f ^ \prime ( s ) = + \infty , \ ; \ ; \ ; \ ; f ^ { \prime \prime } ( s ) \geq \alpha > 0 . \end{align*}


\begin{align*} | \Omega ^ { * } | & \le \sum _ { j = 1 } ^ { m } | \Omega _ { j } ^ { * } | \le C \sum _ { j = 1 } ^ { m } \sum _ { k _ { j } } | Q _ { j , k _ { j } } | \le C ( \lambda \gamma ) ^ { \frac { - n } { m n - \alpha } } . \end{align*}


\begin{align*} y ^ { i } \left ( t \right ) = \xi ^ { i } + \int _ { t } ^ { T } f ^ { i } \left ( s , y _ { s } ^ { i } , z _ { s } ^ { i } \right ) \mathrm { d } s - \int _ { t } ^ { T } z _ { s } ^ { i } \mathrm { d } W _ { s } , \end{align*}


\begin{align*} \left \| \frac { 1 } { m + 1 } \sum _ { i = 0 } ^ m w _ { n + i } - w \right \| & < C \left [ \left \| \frac { 1 } { m + 1 } \sum _ { i = 0 } ^ m x _ { n + i } - x \right \| + \left \| \frac { 1 } { m + 1 } \sum _ { i = 0 } ^ m y _ { n + i } - y \right \| \right . \\ & + \left . \left \| \frac { 1 } { m + 1 } \sum _ { i = 0 } ^ m z _ { n + i } - z \right \| \right ] . \end{align*}


\begin{align*} \mathbb { E } ^ 2 [ f ] \left ( \frac { C r ^ 2 \log ( 1 / \mathbb { E } [ f ] ) } { t } \right ) ^ t = p ^ { 2 t } \left ( C ' p ^ { - 2 } \log ( 1 / p ) \right ) ^ t = \left ( C ' \log ( 1 / p ) \right ) ^ t , \end{align*}


\begin{align*} W = h \ , \tfrac { 1 } { 2 } \nabla _ \xi H ^ 2 ( \nabla v ) \end{align*}


\begin{align*} \| f \| _ { \alpha } = \max _ { z \in G } | f ( z ) | + \sup _ { z , w \in G , z \ne w } \frac { | f ( z ) - f ( w ) | } { | z - w | ^ { \alpha } } \end{align*}


\begin{align*} \frac { \partial \mathrm { R } _ { \mathrm { D } } ( \lambda _ { \mathrm { D } } ^ \star , \delta ) } { \partial \lambda _ \ell } = 0 . \end{align*}


\begin{align*} \lambda _ { m } = \frac { \langle g , M g \rangle } { \langle g , g \rangle } = \lambda _ { k } . \end{align*}


\begin{align*} \overline { a } = r e ^ { h } , \end{align*}


\begin{align*} \epsilon ( f ) : = \max \{ \overline \nu ( x - a ) \mid a \mbox { i s a r o o t o f } f \} . \end{align*}


\begin{align*} b : = \begin{cases} \frac { b _ 1 } { b _ 2 } , & \frac { d } { 4 } + \frac { 1 } { 2 } \leq \alpha < \frac { d } { 2 } + 1 , \\ 1 , & \alpha \geq \frac { d } { 2 } + 1 . \end{cases} . \end{align*}


\begin{align*} \prod _ { x : X } f ( x ) ^ { k } ( m _ g ( x ) f ( x ) ^ { k _ h } - m _ h ( x ) f ( x ) ^ { k _ g } ) = 0 \end{align*}


\begin{align*} \lim _ { s \to \infty } \frac { g _ j ( s ) } { s ^ { k _ j } } = 1 \end{align*}


\begin{align*} \varphi _ { \lambda + \mu } ( [ x _ \mu y ] , z ) = [ x _ \mu \varphi _ { \lambda } ( y , z ) ] - ( - 1 ) ^ { | x | | y | } [ y _ { \lambda } \varphi _ { \mu } ( x , z ) ] , \end{align*}


\begin{align*} \lim _ { n \to \infty } \norm { S _ n ' ( y ' - \sigma _ k ' ( y ' ) ) } = 0 . \end{align*}


\begin{align*} \sharp \{ p \leq r \leq q \ , | \ , D _ { \lambda } ( r ) = \times \} \geq \sharp \{ p \leq r \leq q \ , | \ , D _ { \lambda } ( r ) = \circ \} \end{align*}


\begin{align*} \theta ( ( F ( t _ i ) - F ( t _ { i - 1 } ) ) p _ i ) = \pm \theta ( ( F ( t _ 2 ) - F ( t _ { 1 } ) ) ( \sigma _ i \ast p _ i ) ) \ . \end{align*}


\begin{align*} T _ i ( u ) = \int _ 0 ^ u e ^ { 2 \rho _ i ( v ) } d v \xrightarrow [ u \to \infty ] { a . s . } T _ i ^ { \infty } , \end{align*}


\begin{align*} \mathcal { F } ' _ { X X ^ { \ast } } : & L ^ 2 ( X ) \longrightarrow L ^ 2 ( X ^ { \ast } ) ; \\ & f \longmapsto \mathcal { F } ' _ { X X ^ { \ast } } ( f ) ( x ^ { \ast } ) = \int _ { X } \psi ( \langle x , x ^ { \ast } \rangle ) f ( x ) d x , \end{align*}


\begin{align*} \mathcal I _ 1 = \left ( \left . b _ { \ell i } \left ( X _ \ell - \sum \limits _ { j = 1 } ^ { s _ { \ell i } } b _ { \ell i j } { \textbf { X } } ^ { \lambda _ j } \right ) \ \right | \ i , \ell \in I \setminus \{ i _ { } \} , i < \ell \right ) . \end{align*}


\begin{gather*} \theta : \Omega \times [ 0 , 3 \cdot 2 ^ { - k } ] \to \mathbb { R } , \\ \theta ( ( x _ 1 , x _ 2 ) , t ) = 2 \left \| f _ 0 \right \| _ { L ^ \infty } C _ 0 ^ { - 1 } \left ( \mathrm { e r f } \left ( \frac { - x _ 1 } { \sqrt { 4 \nu t } } \right ) + \mathrm { e r f } \left ( \frac { x _ 1 - a } { \sqrt { 4 \nu t } } \right ) \right ) . \end{gather*}


\begin{align*} \begin{cases} \displaystyle \dot \lambda _ t = \vec h _ { \bar u ( t ) } ^ \nu ( \lambda _ t ) & t \in [ 0 , 1 ] , \\ \displaystyle h _ { \bar u ( t ) } ^ \nu ( \lambda _ t ) = \max _ { v \in \R ^ k } h _ { v } ^ \nu ( \lambda _ t ) & t \in [ 0 , 1 ] , \\ \nu \leq 0 . & \end{cases} \end{align*}


\begin{align*} v ^ m : = \prod _ { i \in \mathsf { I } } v ( i ) ^ { m ( i ) } . \end{align*}


\begin{align*} \beta _ { i , k } : = \begin{cases} 0 & \mbox { i f } d _ { i , k } - x _ i \leq 0 \\ c \P ( \omega _ k ) \frac { x _ i - d _ { i , k } } { \bar { y } } & \mbox { i f } 0 \leq d _ { i , k } - x _ i \leq \bar { y } \\ - c \P ( \omega _ k ) & \mbox { i f } d _ { i , k } - x _ i \geq \bar { y } . \end{cases} \end{align*}


\begin{align*} \mathbf { n } _ \lambda = \lambda ^ { 7 / 2 } \mathbf { e } _ \lambda . \end{align*}


\begin{align*} W ^ T \Lambda W = \begin{bmatrix} \sqrt { \Lambda ^ + } W ^ + \\ G \end{bmatrix} ^ T \begin{bmatrix} I - R ^ T R & - R ^ T S \\ - S ^ T R & I - S ^ T S \end{bmatrix} \begin{bmatrix} \sqrt { \Lambda ^ + } W ^ + \\ G \end{bmatrix} - G ^ T G . \end{align*}


\begin{align*} f ^ { - } ( z ) : = \sum _ { n < 0 } C _ { h } ^ { - } ( n ) \Gamma ( k - 1 , 4 \pi \left | n \right | y ) q ^ { n } . \end{align*}


\begin{align*} \Gamma _ k : = \begin{cases} 1 , & k = 1 , \\ ( 1 - \alpha _ k ) \Gamma _ { k - 1 } , & k \ge 2 . \end{cases} \end{align*}


\begin{align*} p _ { h } = 2 \left ( 1 + \frac { s } { n } \right ) ^ { h } , \quad \mbox { f o r } h = 0 , 1 , 2 , \ldots . \end{align*}


\begin{align*} T _ g ( u ) & : = \int _ \Omega \langle \nabla g , \nabla u \rangle \ , d x - \frac { C _ { N , s } } { 2 } \iint _ { \R ^ { 2 N } } \frac { ( g ( x ) - g ( y ) ) ( \hat { u } ( x ) - \hat { u } ( y ) ) } { | x - y | ^ { N + 2 s } } \ , d x \ , d y \\ & \qquad - \int _ \Omega V ( x ) g u \ , d x - \int _ \Omega f u \ , d x , \end{align*}


\begin{align*} q ( r ' ) ^ { - 2 / q } = \sqrt { q } \sqrt { q \left ( \frac { 1 } { ( r ' ) ^ 2 } \right ) ^ { 2 / q } } \le \sqrt { 2 } \sqrt { q } \sqrt { \log ( 1 / \gamma ) } . \end{align*}


\begin{align*} & A = \pm \prod _ { p \in { \cal P } _ 3 } p ^ { a _ p - b _ p + \sum _ { i = 1 } ^ n ( \alpha _ i - \beta _ i ) z _ { i p } ^ 0 } , \\ & B = \pm \prod _ { p \in { \cal P } _ 2 } p ^ { b _ p - c _ p + \sum _ { i = 1 } ^ n ( \beta _ i - \gamma _ i ) z _ { i p } ^ 0 } , \\ & C = \pm \prod _ { p \in { \cal P } _ 1 } p ^ { c _ p - a _ p + \sum _ { i = 1 } ^ n ( \gamma _ i - \alpha _ i ) z _ { i p } ^ 0 } . \end{align*}


\begin{align*} U L ( x ) & = \begin{bmatrix} x & 0 \\ 0 & 1 \end{bmatrix} & L R ( x ) & = \begin{bmatrix} 1 & 0 \\ 0 & x \end{bmatrix} & D ( x ) & = \begin{bmatrix} x & 0 \\ 0 & x \end{bmatrix} . \end{align*}


\begin{align*} R ^ \times = \mathfrak n \times K ^ \times . \end{align*}


\begin{align*} \sum _ { m , v \in \mathcal { M } ' } \sum _ { p \in P } \frac { r ( m ) r ( v ) } { p ^ { \sigma } } \Phi \left ( T \log \frac { m p } { v } \right ) \ge \sum _ { p \in P } \sum _ { \substack { m , v \in \mathcal { M } \\ m p = v } } \frac { f ( m ) f ( v ) } { p ^ { \sigma } } = \sum _ { v \in \mathcal { M } } f ( v ) ^ 2 \sum _ { p | v } \frac { 1 } { f ( p ) p ^ { \sigma } } . \end{align*}


\begin{align*} T _ 6 = - \frac { s } { 2 } \iint _ Q \sigma _ { t t } | u | ^ 2 d x d t \geq - C s \iint _ Q \xi ^ { 3 / 2 } | u | ^ 2 d x d t . \end{align*}


\begin{align*} \sup _ { \pi \in \mathcal { G } _ { \mathrm { D } , \lambda } } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ \lambda d \pi = \sup _ { \pi \in \Gamma \left ( \Pi ( \mu _ 1 , \mu _ 2 ) , \varphi _ \lambda \right ) } \int _ { \mathcal { V } \times \mathcal { V } } \varphi _ { \lambda } d \pi . \end{align*}


\begin{align*} r ( p ) \ = \ r ( 4 \underbrace { 9 \cdots 9 } _ { m } ) \ = \ \underbrace { 9 \cdots 9 } _ { m } 4 \ = \ 2 \cdot 4 \underbrace { 9 \cdots 9 } _ { m - 1 } 7 \ = \ 2 ( p - 2 ) . \end{align*}


\begin{align*} \Phi ^ { \lambda } ( x ) : = \lambda ^ { \frac { d } { 2 } } \Phi ( \lambda x ) . \end{align*}


\begin{align*} \int _ 0 ^ { \infty } \sqrt { f ( \chi ^ { - 1 } ( u - 1 ) ) } \ , d u = \int _ 0 ^ { \infty } \sqrt { f ( y ) } \chi ' ( y ) \ , d y \\ < \int _ 0 ^ { \infty } f ( y - 1 ) \ , d y \le V + 1 . \end{align*}


\begin{align*} \check { \psi } \left ( x \right ) = \frac { 1 } { ( 2 \pi \lambda ) ^ d } \sum _ { k \in \mathbb Z _ \lambda ^ d } \psi ( k ) e ^ { i k \cdot x } . \end{align*}


\begin{align*} \left \{ \hat { X } _ { n + 1 } ^ { y , \Gamma } = \hat { X } _ n ^ { y , \Gamma } + 1 \right \} \subseteq \left \{ X _ { n + 1 } ^ { y , \Gamma } = X _ n ^ { y , \Gamma } + e _ 2 \right \} . \end{align*}


\begin{align*} \hat { g } = & \ , \frac { x ^ { n - 1 } d x ^ 2 } { 2 P ( x ) } + \frac { 2 P ( x ) } { x ^ { n - 1 } } ( d \phi + \bar { \sigma } ) ^ 2 + 2 x \bar { g } , \\ \hat { J } = & \ , d \left [ x ( d \phi + \bar \sigma ) \right ] . \end{align*}


\begin{align*} \| \Phi _ { \omega , \mathbf { c } } \| _ { \mathcal { H } ^ 1 } ^ 2 \le C L _ { \omega , \mathbf { c } } ( \Phi _ { \omega , \mathbf { c } } ) = 6 C \mu _ { \omega , \mathbf { c } } . \end{align*}


\begin{align*} \mathcal { I } : = \left \{ A \in \mathcal { B } ( W _ { \Omega } ) : \mathcal { S } ^ { - 1 } ( A ) = A \right \} . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ n \alpha _ i z _ i = \sum _ { i = 1 } ^ n \beta _ i z _ i = \sum _ { i = 1 } ^ n \gamma _ i z _ i - 1 , z _ i \geq 0 , i = 1 , \dots , n , \end{align*}


\begin{align*} \rho ^ f _ N ( t ) = \Pi _ 0 ^ { i / N } \exp f ( s ) \ , d s , \mbox { i f } \frac i N \leq t < \frac { i + 1 } N . \end{align*}


\begin{align*} \bar f ( X , \sum a ^ i \partial _ i ) = ( f X - \sum a ^ i \xi _ i , \sum \eta ^ j ( X ) \partial _ j ) , \bar Q ( X , \sum a ^ i \partial _ i ) = ( Q X , \sum a ^ i \partial _ i ) . \end{align*}


\begin{align*} G _ \alpha ( r ) : = ( r ^ 2 + \alpha ) ^ { \frac p 2 } \textrm { f o r a l l } \ ; \ ; r \in \R \ , . \end{align*}


\begin{align*} \limsup _ \lambda \| e _ \lambda ^ * ( b - \omega ( b ) I ) e _ \lambda \| & = \limsup _ \lambda \| e _ \lambda ^ * ( s ^ * + t ) e _ \lambda \| \\ & \leq \limsup _ \lambda ( \| e _ \lambda s ^ * \| + \| t e _ \lambda \| ) \\ & = \limsup _ \lambda ( \| s e _ \lambda \| + \| t e _ \lambda \| ) \\ & = 0 . \end{align*}


\begin{align*} c & | \log ( y - l _ \alpha ) - \log ( x - l _ \alpha ) | = \int _ x ^ { y } \frac { c } { s - l _ \alpha } d s \leq \Big | \int _ x ^ { y } D ^ { n + 1 } \varphi ( s ) d s \Big | \\ & \leq | D ^ { n } \varphi ( x ) - D ^ { n } \varphi ( y ) | \leq \| D ^ { n } \varphi \| _ { C ^ { \tau } } | ( y - l _ \alpha ) - ( x - l _ \alpha ) | ^ { \tau } . \end{align*}


\begin{align*} g _ n ( x ) = \left \{ \begin{array} { c l } x & \mbox { i f } x \in I ^ n \\ x _ 0 & \mbox { i f } x \notin I ^ n \end{array} , \right . \end{align*}


\begin{gather*} A \circ B = ( A \circ f ) \circ ( f ^ { - 1 } \circ B ) , \ f \in \mathbb { C } ( x ) , \ f \not = 0 , \\ \partial ^ 2 = \left ( \partial + \frac 1 { x - c } \right ) \circ \left ( \partial - \frac 1 { x - c } \right ) , \ c \in \mathbb { C } . \end{gather*}


\begin{align*} \theta ( a _ 1 \cdots a _ k ) = \theta ( a _ 1 ) \cdots \theta ( a _ k ) , \end{align*}


\begin{align*} \langle A h , g \rangle _ { \mu _ p } = 0 . \end{align*}


\begin{align*} A \tilde { S } = J _ { 2 n } \tilde { S } J _ { 2 n } ^ T A + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} A _ t ( x ) = x - \gamma ( x , t ) - \frac { 1 } { N } \sigma ( t ) , \end{align*}


\begin{align*} ( 1 - \rho ) f _ 0 ( t ) + \rho \sum _ { r = 1 } ^ \infty \zeta _ { d , r } f _ r ( t ) \end{align*}


\begin{align*} 1 = \langle D ^ { k - 1 } ( f ) , g \rangle = \langle c F , g \rangle = c \langle F , g \rangle = c . \end{align*}


\begin{align*} n ! ! = \begin{cases} n \times ( n - 2 ) \times \ldots \times 3 \times 1 & \mbox { i f $ n $ i s o d d } \\ n \times ( n - 2 ) \times \ldots \times 4 \times 2 & \mbox { o t h e r w i s e . } \end{cases} \end{align*}


\begin{align*} \int _ \mathrm { M } ( q _ 2 - q _ 1 ) v _ 1 v _ 2 d x = - \int _ \mathrm { M } [ \Delta , \psi ] v v _ 1 d x . \end{align*}


\begin{align*} D _ { S , x } T _ \rho f = \sum _ { T \supseteq S } \rho ^ { | S | } T _ \rho D _ { S , x } [ f ^ { = T } ] = \rho ^ { | S | } \sum _ { T \subseteq [ n ] } T _ \rho D _ { S , x } [ f ^ { = T } ] = \rho ^ { | S | } T _ \rho D _ { S , x } f \end{align*}


\begin{align*} \sum _ { i = \ell + 1 } ^ { k } s _ i ^ p ( A ) = 0 \hbox { a n d } \displaystyle \sum _ { i = k - \ell + 1 } ^ { k } s _ k ^ p ( B ) = 0 , \end{align*}


\begin{align*} \Omega ( \ell _ 1 ) \left ( Z ( \mathcal { G } ( \ell _ 1 ) ) ( \mathbb { Q } _ p ) \mathcal { G } ( \ell _ 1 ) ( \mathbb { Q } ) \right ) = \mathbb { Q } _ p ^ \times \Omega ( \ell _ 1 ) \left ( \mathcal { G } ( \ell _ 1 ) ( \mathbb { Q } ) \right ) \end{align*}


\begin{align*} \lim _ { N \to \infty } \| r _ N - r \| _ \infty = 0 \end{align*}


\begin{align*} S ( x , \omega ) = S \left ( x , \zeta _ { + } ( x , \omega ) , \zeta _ { - } ( x , \omega ) \right ) \subseteq L _ { x } \backslash G \end{align*}


\begin{align*} \left \{ \begin{array} { @ { \ , } l l l } 0 = - q c ( b - d ) \\ 0 = - c ^ 2 \\ q ^ 2 = q b ^ 2 \\ q ( d - b ) = - c d . \\ \end{array} \right . \end{align*}


\begin{align*} \mathcal { F } _ k = F _ { H _ { k + 1 } } \cap \bigcap _ { y \in I _ { H _ { k + 1 } } } \bigcap _ { j = 0 } ^ { l _ k - 1 } \ ; D _ { H _ k } ^ { \mathcal { X } _ j ^ y , \mathcal { N } _ j ^ y } . \end{align*}


\begin{align*} \mathbb { E } \bigg ( \int _ t ^ \infty z _ n d W _ s - \int _ t ^ \infty z d W _ s \bigg ) ^ 2 = \mathbb { E } \int _ t ^ \infty | z _ n - z | ^ 2 d s \leq \mathbb { E } \int _ t ^ \infty e ^ { - \beta s } | z _ n - z | ^ 2 d s \rightarrow 0 , \ \ n \rightarrow \infty . \end{align*}


\begin{align*} w _ { i j , k } - w \left ( R ^ \varphi _ { i j , k } - \frac { S ^ \varphi _ k } { m - 1 } \delta _ { i j } \right ) - \left ( R ^ \varphi _ { i j } - \frac { S ^ \varphi } { m - 1 } \delta _ { i j } \right ) w _ k - R ^ \varphi _ { i j , k } + \frac { S ^ \varphi _ k } { m } \delta _ { i j } = 0 \ , . \end{align*}


\begin{align*} E _ \alpha = \inf \Big \{ J ( u ) = \frac 1 2 \int _ { \R ^ N } | \nabla u | ^ 2 - \int _ { \R ^ N } F ( u ) \ \Big | \ { u \in \mathcal M _ \alpha } \Big \} , \end{align*}


\begin{align*} Z _ 4 ( H ) = \overline { \{ ( p , q ) \in X \times X \ , | \ , \exists \ , C ' \in | H | \mbox { s m o o t h s . t . } p , q \in C ' , \ , 4 [ p - q ] = 0 \in J C ' \} } \subseteq X \times X \end{align*}


\begin{align*} \lim _ { j \to + \infty } \int _ \Omega ( - w _ j ) ^ { n + 1 } d V _ g = \int _ \Omega ( - w ) ^ { n + 1 } d V _ g . \end{align*}


\begin{align*} T _ w ^ \circ ( M , u ^ * ) = L ^ { \circ } ( \mathcal { K } , u ^ * ) . \end{align*}


\begin{align*} \tilde { S } _ { \gamma _ i \gamma _ j } & = - J _ { 2 | \alpha _ i | } \tilde { S } _ { \gamma _ i \gamma _ j } J ^ T _ { 2 | \alpha _ j | } + \mathcal { O } ( \| H \| ) . \end{align*}


\begin{align*} h _ { \bar u ( t ) } ^ \nu ( \lambda _ t ) = \max _ { v \in \R ^ k } h _ v ^ \nu ( \lambda _ t ) , \qquad t \in [ 0 , 1 ] , \end{align*}


\begin{align*} & \int _ 0 ^ { \infty } e ^ { - \lambda s } ( 1 - p _ 0 ( s ) ) d s = \frac 1 { \lambda } \int _ 0 ^ { \infty } \frac { q } { 1 - p e ^ { - \lambda s } } \cdot \lambda e ^ { - \lambda s } d s . \end{align*}


\begin{align*} \left \langle \mathcal { M } _ { q , \alpha } f , g \right \rangle : = A _ q ( u _ f , g ) , \end{align*}


\begin{align*} ( s \kappa ^ { p } + n \sqrt { - m } ) = \varepsilon \eta ^ { p } , \end{align*}


\begin{align*} \lim _ { j \to + \infty } \int _ \Omega ( - u _ j ) ^ { n + 1 } g d V = \int _ \Omega ( - u ) ^ { n + 1 } g d V . \end{align*}


\begin{align*} \begin{cases} \partial _ t \nu _ t ( x ) = - \frac { \delta V ^ { \sigma } } { \delta \nu } ( \nu _ t , \mu _ t , x ) \nu _ t ( x ) , \\ \partial _ t \mu _ t ( y ) = \frac { \delta V ^ { \sigma } } { \delta \mu } ( \nu _ t , \mu _ t , y ) \mu _ t ( y ) , \end{cases} \end{align*}


\begin{align*} \| T ^ \ast \varphi _ j \| _ { H _ \Gamma ^ { 3 / 2 } ( \partial \mathrm { M } ) } = \tau _ j . \end{align*}


\begin{align*} \bigcup _ { i = 1 } ^ n \{ h _ 1 g _ i , \ldots , h _ { \frac { d } { n } } g _ i \} . \end{align*}


\begin{align*} \left . \frac { d } { d \alpha } \right | _ { \alpha = 0 } \chi _ F ( \alpha ) = ( - 1 ) ^ { n - 1 } ( Y _ F , \varphi _ n ) \end{align*}


\begin{align*} \int _ { \R ^ 2 } | w _ N ( x ) | ^ \alpha \d x = N ^ { 2 \beta ( \alpha - 1 ) } \int | w ( x ) | ^ \alpha \d x , \forall \alpha > 0 . \end{align*}


\begin{align*} \nu _ { y ' } ^ g ( x ) = \langle y ' T _ { g } ( x ) \Omega _ \rho , \Omega _ \rho \rangle _ { \rho } . \end{align*}


\begin{align*} H p ( x ) = \sum _ { j = 0 } ^ { \left [ \frac { k } { 2 } \right ] } \frac { ( - 1 ) ^ j \Gamma \left ( \frac { n } { 2 } + k - j - 1 \right ) } { 4 ^ j j ! \Gamma \left ( \frac { n } { 2 } + k - 1 \right ) } \| x \| ^ { 2 j } \Delta _ n ^ j p ( x ) . \end{align*}


\begin{align*} { } _ p F _ q ( a _ 1 , \ldots , a _ p ; b _ 1 , \ldots , b _ q ; x ) = \sum _ { j = 0 } ^ \infty \frac { ( a _ 1 ) _ j \cdots ( a _ p ) _ j } { ( b _ 1 ) _ j \cdots ( b _ q ) _ j } \frac { x ^ j } { j ! } , \end{align*}


\begin{align*} | K | _ { n a t } = \int ^ { [ n ] \in \mathcal { A } } K _ n . | \square _ S [ n ] | _ { n a t } \end{align*}


\begin{align*} \begin{array} { l l l } V _ 1 ( x ) & = [ K + \min _ { x \leq y \leq x + B } \{ v ( y - x ) + L _ 1 ( y ) \} - L _ 1 ( x ) ] ^ - ; \end{array} \end{align*}


\begin{align*} ( \Delta + \lambda - q _ 1 ) \tilde { u } _ 2 = 0 \ ; \mathrm { i n } \ ; \mathrm { M } , \tilde { u } _ 2 { _ { | \partial M } } = u _ 2 { _ { | \partial M } } , \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) = n - | E ( T ) | . \end{align*}


\begin{align*} \sum _ { i = 1 } ^ D \sum _ { j = 1 } ^ D \alpha _ i \bar { \alpha } _ j \sum _ { f \in \mathcal { B } _ { 2 k } ( p ) } \omega _ f \lambda _ f ( i ) \lambda _ f ( j ) \big | L ( f , n + 1 ) \big | ^ 2 = 0 \end{align*}


\begin{align*} ( u _ i ^ { * } ( C + D ) u _ i ) ^ { \gamma } + ( u _ i ^ { * } ( C - D ) u _ i ) ^ { \gamma } \geq 2 ( u _ i ^ { * } C u _ i ) ^ { \gamma } \hbox { f o r $ i = 1 , \ldots , n . $ } \end{align*}


\begin{align*} \overline { \beta ( a , b , c ) } = \sigma _ 1 \sigma _ 2 ^ { a + 1 } \sigma _ 1 \sigma _ 2 ^ { b + 1 } \sigma _ 1 \sigma _ 2 ^ { c + 1 } . \end{align*}


\begin{align*} S _ { \omega , \mathbf { c } } ( V ) = \frac { 1 } { 3 } K _ { \omega , \mathbf { c } } ( V ) + \frac { 1 } { 6 } L _ { \omega , \mathbf { c } } ( V ) = \mu _ { \omega , \mathbf { c } } \end{align*}


\begin{align*} a ^ + = n - z - \mathfrak { S } ( f _ k ) , \end{align*}


\begin{align*} u \mapsto ( u _ j : = \Pi _ { j , \lambda } \chi _ j u ) . \end{align*}


\begin{align*} L _ { \mu _ m , \xi _ m } ( \phi _ m ) = h _ m , \| \phi _ m \| = 1 \quad \mbox { a n d } \| h _ m \| \rightarrow 0 . \end{align*}


\begin{align*} \widetilde { \mathcal { F } } _ { \gamma } : = f ^ { + } - \gamma E _ { g | V ( p ) } = \sum _ { n \gg - \infty } \left ( C _ { f } ^ { + } ( n ) - n ^ { 1 - k } C _ { g } ( n / p ) \right ) q ^ { n } = \sum _ { n \gg - \infty } n ^ { 1 - k } d _ { \gamma } ( n ) q ^ { n } . \end{align*}


\begin{align*} \textsf { A d } g ( X ) : = \bar { g } X { \bar { g } } ^ { - 1 } \end{align*}


\begin{align*} \lim _ { l \to \infty } \sup _ { x \in M _ + , x \neq 0 } \abs { ( B _ l ( \nu _ k ) - \bar { \nu } _ k ) ( x ) } / \rho ( x ) & = \lim _ { l \to \infty } \sup _ { x \in M _ + , x \neq 0 } \abs { ( B _ l ( \nu _ k ) - \bar { \nu } ) ( x ) } / \rho ( x ) = 0 . \end{align*}


\begin{align*} F _ 1 = [ & - \det \Delta \cdot ( \Delta ^ { - 1 } ( \partial F _ i / \partial x _ 1 ) ) \\ & - \det \Delta \cdot ( \Delta ^ { - 1 } ( \partial F _ i / \partial x _ 0 ) ) ] . \end{align*}


\begin{align*} M _ a ( \cdot ) : = \begin{cases} \frac { 1 } { a ^ d } \sum _ { 0 \leq i _ 1 < a } \cdots \sum _ { 0 \leq i _ d < a } T _ 1 ^ { i _ 1 } \cdots T _ d ^ { i _ d } ( \cdot ) & G = \Z _ + ^ d , ~ a \in \N , \\ \\ \frac { 1 } { a ^ d } \int _ { Q _ a } T _ t ( \cdot ) d t & G = \R _ + ^ d , ~ a \in \R _ + . \end{cases} \end{align*}


\begin{align*} D _ { D } [ \partial _ { t } \mu _ { S } ] ( e ) + A _ { B } ^ * [ \partial _ { t } \mu _ { B } ] ( e ) = G _ { D , 2 } ( e ) \end{align*}


\begin{align*} c l ( v ) : = \frac { 1 } { \sqrt { 2 } } ( v - i J v ) \wedge ( \cdot ) - \frac { 1 } { \sqrt { 2 } } ( v + i J v ) \lrcorner ( \cdot ) , \end{align*}


\begin{align*} \begin{aligned} \Delta \left ( \frac { x _ i - x _ j + m } { \ell } \right ) & = \int _ \mathbb { R } \chi \left ( \frac { x _ i + \ell x _ 0 - x _ j + m } { \ell } \right ) \chi ( x _ 0 ) \ , d x _ 0 \\ & = \frac { 1 } { \ell } \int _ \mathbb { R } \chi \left ( \frac { x _ i - x _ 0 ' + m } { \ell } \right ) \chi \left ( \frac { x _ j - x _ 0 ' } { \ell } \right ) \ , d x _ 0 ' \end{aligned} \end{align*}


\begin{align*} u ^ \bigstar _ \mu ( x ) = \int _ { C _ D | x | ^ D } ^ \infty ( - u ^ * _ \mu ) ' ( t ) \dd t , \end{align*}


\begin{align*} f ^ { ( 1 ) } ( \boldsymbol \alpha ) \triangleq & - \log p ^ { ( 1 ) } ( \mathbf R ; \boldsymbol \alpha ) - L \log \pi \\ = & \log | \boldsymbol \Sigma ^ { ( 1 ) } _ { \boldsymbol \alpha } | + \left ( \boldsymbol \Sigma _ { \boldsymbol \alpha } ^ { ( 1 ) - 1 } \widehat { \mathbf \Sigma } _ { \mathbf R } \right ) . \end{align*}


\begin{align*} \begin{cases} ( - \Delta ) ^ s u = V ( x ) u & , \\ u = 0 & , \end{cases} u > 0 , \end{align*}


\begin{align*} [ b _ k , D ^ * _ k ] = 0 \forall k \in \Lambda ^ * \ . \end{align*}


\begin{align*} \left \vert f \left ( z \right ) \right \vert \geq \left \vert g \left ( z \right ) \right \vert - \left \vert g \left ( z \right ) - f \left ( z \right ) \right \vert > s - \left ( s - n \right ) = n \end{align*}


\begin{align*} m _ \nu ( a , b ) = \inf _ { \mathcal { T } ( a , b ) } \max _ { s > 0 } J _ \nu ( s \circ ( u , v ) ) . \end{align*}


\begin{align*} p - \sum _ { i = 1 } ^ { n } \Big ( m _ i ( p - x _ { i } ) \ss _ i ^ { 3 } \Big ) = 0 . \end{align*}


\begin{align*} \bar { \Delta } = \underset { W \rightarrow \infty } { \lim } \frac { 1 } { W } \int ^ { W } _ { 0 } \Delta ( t ) d t , \end{align*}


\begin{align*} Y = \{ j \in [ n ] \ : \ \mu _ { C _ j } \} \end{align*}


\begin{align*} ( \nabla ( u _ { \kappa } ) _ \mu ^ \bigstar ) _ \mu ^ * ( t ) = \kappa ( \nabla u _ \mu ^ \bigstar ) _ \mu ^ * ( \kappa ^ D t ) . \end{align*}


\begin{align*} \Phi _ { k , i } ( \alpha _ 1 , \ldots , \alpha _ k ) = 0 , \forall i = 1 , \ldots , n , \end{align*}


\begin{align*} \left | \frac { A _ 1 } { A _ 2 } \right | = \left | \frac { B _ 1 } { B _ 2 } \right | = \left | \frac { C _ 1 } { C _ 2 } \right | = 1 , \end{align*}


\begin{align*} \left . u _ 1 \right | _ W = f _ 1 = f _ 2 = \left . u _ 2 \right | _ W W . \end{align*}


\begin{align*} G _ k ( t - s ) = \int _ { \Lambda ^ * } \chi ( r ) \chi ^ \perp ( r + k ) e ^ { - i ( t - s ) ( E _ r + E _ { r + k } ) } \d r \ . \end{align*}


\begin{align*} \phi = \phi _ { e } + \sum _ { m = 1 } ^ { \infty } C _ { q _ m } h ^ { q _ m } . \end{align*}


\begin{align*} E ^ 2 ( \lambda ) & = \left ( { \mathrm { s o t } } \lim _ { n \to \infty } f _ { \lambda , n } ( A ) \right ) ^ 2 \\ & = { \mathrm { s o t } } \lim _ { n \to \infty } f ^ 2 _ { \lambda , n } ( A ) \\ & \ge { \mathrm { s o t } } \lim _ { m \to \infty } f _ { \lambda , m } ( A ) = E ( \lambda ) \end{align*}


\begin{align*} [ 0 , 1 ) \times V \ni ( t , z ) \mapsto d _ z \phi _ t ( p , \cdot ) = d _ z \left ( \exp _ z ( ( t - 1 ) d _ z \mathfrak f _ p ) \right ) \end{align*}


\begin{align*} \partial _ t c = \Delta c - f ^ \prime ( c ) \ ; \ ; \ ; \Omega _ T \end{align*}


\begin{align*} C _ { L , K } ( m , \delta ) = \sum _ { \substack { \mu \in L ' / K \\ p _ K ( \mu ) = \delta } } c ( m + q ( \mu _ { K ^ \perp } ) , \mu + L ) \end{align*}


\begin{align*} g ( p , u ) = \left \{ \begin{array} { l l } + e _ 1 & \\ - e _ 1 & \\ - e _ 2 & \\ + e _ 2 & \end{array} \right . \end{align*}


\begin{align*} { \rm U } ( L ) = \bigoplus _ { \mathbf { m } \in \N _ 0 ^ { d } } { \rm U } _ \mathbf { m } . \end{align*}


\begin{align*} \mathcal { M } _ { L , x _ 0 } = \bigcap _ { E \in \sigma ^ { ( \mathcal { I } , r e d ) } ( H _ { \omega , \Lambda _ { L } ( x _ 0 ) } ) } \mathcal { M } _ { L , x _ 0 } ^ { ( E ) } . \end{align*}


\begin{align*} \lim \limits _ { k \rightarrow \infty } \| w ^ { k } - w ^ { * } \| = \sigma \geq 0 \end{align*}


\begin{align*} \varphi _ t = \sum _ { N _ t } \tau _ j ^ { - 1 } a _ j \psi _ j . \end{align*}


\begin{align*} G _ { m , n } ( x ) = \mathbb { P } \left ( \sum _ { i = m + 1 } ^ { n } \frac { X _ { i } - \mu } { \sigma \sqrt { n } } \leq x \right ) = G _ { n - m } \left ( \frac { x \sqrt { n } } { \sqrt { n - m } } \right ) = G _ { n - m } \Big ( x \left ( 1 + R _ { n } \right ) \Big ) , \end{align*}


\begin{align*} h \left ( G / P _ { \mathsf { f } ( I ) } , \bar { \nu } _ { 0 } \right ) = h \left ( G \times _ { P _ { \mathsf { f } ( I ) } } X _ { 0 } , \bar { \nu } _ { 0 } \times \lambda \right ) \ge h ( X , \nu ) . \end{align*}


\begin{align*} \mathcal { I } ( \delta ) = \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \left \{ \langle \lambda , \delta \rangle + \mathcal { I } ^ { \star } ( \lambda ) \right \} = \inf _ { \lambda \in \mathbb { R } ^ 2 _ + } \left \{ \langle \lambda , \delta \rangle + \sup _ { \gamma \in \Pi \left ( \mu _ 1 , \mu _ 2 \right ) } \int _ { \mathcal { V } } f _ \lambda ( v ) \ , d \gamma ( v ) \right \} , \end{align*}


\begin{align*} \psi _ { \mu _ N } = \frac 1 { N + 1 } \ , \sum _ { i = 1 } ^ { N + 1 } \psi _ { \delta _ { x _ i } } \end{align*}


\begin{align*} S & \ll \widehat { Y } ^ { ( n + 1 ) / 2 + \varepsilon } | d | \left ( \widehat { V } ^ n \widehat { Z } ^ { - n / 2 } + \widehat { Y } ^ { n / 3 } \right ) \sum _ { \substack { | r | = \widehat { Y } \\ d \mid r } } | b _ 2 r _ 3 ' | ^ { 1 / 2 } | b _ 1 ' b _ 2 ' r _ 3 '' | \\ & = \widehat { Y } ^ { ( n + 3 ) / 2 + \varepsilon } | d | \left ( \widehat { V } ^ n \widehat { Z } ^ { - n / 2 } + \widehat { Y } ^ { n / 3 } \right ) \sum _ { \substack { | r | = \widehat { Y } \\ d \mid r } } | b _ 1 | ^ { - 1 } | b _ 2 r ' _ 3 | ^ { - 1 / 2 } , \end{align*}


\begin{align*} A _ R ( a , b ) : = \bigl \{ ( u , v ) \in \mathcal { T } ( a , b ) : | \nabla u | ^ 2 _ 2 + | \nabla v | ^ 2 _ 2 < R ^ 2 \bigr \} . \end{align*}


\begin{align*} A ^ h : = \begin{pmatrix} y ^ { \alpha _ y } & 0 \\ 0 & ( x + h ) ^ { \alpha _ x } \end{pmatrix} , \end{align*}


\begin{align*} \tau = \inf \{ t \ge 0 , \ B ( t ) + \theta - \eta t = 0 \} , \end{align*}


\begin{align*} \int \omega ( x ) , \omega ( x ) : = \prod _ { i = 1 , 2 } t _ i ^ a ( t _ i - 1 ) ^ b ( t _ i - x ) ^ c \cdot ( t _ 1 - t _ 2 ) ^ g \ d t _ 1 \wedge d t _ 2 \end{align*}


\begin{align*} p _ { \Phi , f u l l } ( x , \xi , \tau ) & = | \xi | ^ 2 - \tau ^ 2 | \nabla \Phi ( x ) | ^ 2 + 2 i \tau \xi \cdot \nabla \Phi ( x ) + \tau \Delta \Phi ( x ) \\ p _ { \Phi } ( x , \xi , \tau ) & = | \xi | ^ 2 - \tau ^ 2 | \nabla \Phi ( x ) | ^ 2 + 2 i \tau \xi \cdot \nabla \Phi ( x ) \end{align*}


\begin{align*} \begin{aligned} J _ 3 \ge & - C s ^ 2 \lambda ^ 2 \iint _ { Q } \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 u ^ 2 d x d y d t - C s ^ 2 \lambda ^ 2 \int _ { 0 } ^ { T } \int _ { \omega _ 0 } \xi ^ 3 \left | A \nabla \eta \cdot \nabla \eta \right | ^ 2 u ^ 2 d x d y d t , \end{aligned} \end{align*}


\begin{align*} f ( z ) = \frac { z - a } { 1 - \overline { a } z } \ , \frac { z - \overline { a } } { 1 - a z } \ , \widetilde { f } ( z ) , \end{align*}


\begin{align*} w ( z ) = i c - I + \frac { 1 } { 2 \pi } \langle h _ b , P _ r ( \theta - \cdot ) \rangle - \frac { 1 } { 2 \pi } \iint _ { | \zeta | < 1 } \left ( \frac { f ( \zeta ) } { \zeta } \ , \frac { \zeta + z } { \zeta - z } + \frac { \overline { f ( \zeta ) } } { \overline { \zeta } } \ , \frac { 1 + z \overline { \zeta } } { 1 - z \overline { \zeta } } \right ) \ , d \xi \ , d \eta , \end{align*}


\begin{align*} \nu ( f _ 0 + f _ 1 g + \ldots + f _ n g ^ n ) : = v ( f _ 0 ( \eta ) ) . \end{align*}


\begin{align*} \eta _ { \omega } = \lim _ { n \to \infty } \omega _ { n } . \eta \mbox { e x i s t s } . \end{align*}


\begin{align*} \lim _ { j \to + \infty } \big \| \exp ( A ) ^ j \exp ( B ) ^ { - \lfloor \varepsilon j \rfloor } \big \| = \infty , \mbox { o r } \lim _ { j \to - \infty } \big \| \exp ( A ) ^ j \exp ( B ) ^ { - \lfloor \varepsilon j \rfloor } \big \| = \infty . \end{align*}


\begin{align*} \tilde { p } = \begin{cases} p _ { h + 1 } & h < h _ { q } - 1 , \\ q & h = h _ { q } - 1 . \end{cases} \end{align*}


\begin{align*} \langle \pi _ { \zeta } ( B _ j B _ k ) \xi , \pi _ { \zeta } ( B _ { \ell } ) \xi \rangle = \delta _ { j , \ell } \langle \pi _ { \zeta } ( B _ k ) \xi , \xi \rangle = 0 . \end{align*}


\begin{align*} I & = \left ( \sum _ { A \in \mathcal { P } } \left ( 1 - \frac { \alpha ' ( A ) } { \alpha ( A ) } \right ) \alpha ( A ) \log \frac { \alpha ( A ) } { \beta ( A ) } \right ) - H _ { \alpha ' \parallel \alpha } \left ( \mathcal { P } \right ) \\ & \le \sum _ { A \in \mathcal { P } } \left ( 1 - \frac { \alpha ' ( A ) } { \alpha ( A ) } \right ) \alpha ( A ) \log \frac { \alpha ( A ) } { \beta ( A ) } . \end{align*}


\begin{align*} Y ( A , B , C ) : = \max \{ | A + B z + C z ^ 2 | + 1 - | z | ^ 2 : z \in \overline { \mathbb { D } } \} . \end{align*}


\begin{align*} f ( z ) = z + \sum _ { n = 2 } ^ { \infty } a _ n z ^ n , \ ; \mbox { f o r } \ ; z \in \mathbb { D } : = \{ z : \in \mathbb { C } : | z | < 1 \} \end{align*}


\begin{align*} X _ { \mathbf { K } } ( \mathbb { C } ) = \mathbf { G } ( \mathbb { Q } ) \backslash \left ( \mathbf { X } \times \mathbf { G } ( \mathbb { A } _ f ) / \mathbf { K } \right ) \end{align*}


\begin{align*} h ^ { G _ N } ( x , y ) = \left \{ \begin{array} { l l } 1 , & , \\ 0 , & . \end{array} \right . \end{align*}


\begin{align*} \theta _ { 1 / 2 } ( \tfrac { a z + b } { c z + d } , \epsilon ) = \lambda ^ { \pm } ( \gamma , \epsilon ) \sqrt { \det \gamma ( c z + d ) } \sum _ { n \in \Z } e ^ { i ( \det \gamma ) \epsilon \pi n ^ 2 z } , \end{align*}


\begin{align*} < \beta \circ \rho _ { \beta } ^ * ( f , \gamma ) , m > & = < ( f , \gamma ) , \rho _ { \beta } \circ \beta ( m ) > \\ & = < ( f , \gamma ) , ( \alpha \otimes \beta ) \circ \rho _ { \beta } ( m ) > \\ & = < \rho _ { \beta } ^ * \circ ( \alpha \otimes \beta ) ( f , \gamma ) , m > \\ & = < \rho _ { \beta } ^ * ( \alpha ( f ) , \beta ( \gamma ) ) , m > \end{align*}


\begin{align*} d ( J _ { \mathrm { s e c t } ( \delta / 2 ) } ( t ) ) = J _ { \mathrm { s e c t } ( \delta / 2 ) } ( t ^ { [ 1 ] } ) \cdot \omega ( t ^ { [ 2 ] } ) , \end{align*}


\begin{align*} C = Y _ 0 \subset Y _ 1 \subset \dots \subset Y _ k = Y \end{align*}


\begin{align*} \| f ^ { = d } \| _ q \leq ( 1 / \rho ) ^ d \gamma ' = \gamma \left ( 9 9 \cdot 2 \ , r \ , r ' q \ , ( r ' ) ^ { - 2 / q } \sqrt { \frac { \log ( 1 / \gamma ) } { d } } \right ) ^ d = \gamma \left ( 9 9 \cdot 2 \cdot \sqrt { 2 } \ , r \ , q \ , ( r ' ) ^ { - 2 / q } \right ) ^ d \end{align*}


\begin{align*} = \left ( 1 - x \right ) ^ { \frac { z ( 1 + z ) L i _ 2 ( y ) } { ( 1 - z ) ^ 3 } } , \end{align*}


\begin{align*} \chi _ K R ( \lambda ) \chi _ K = \chi _ K R _ K ( \lambda ) \chi _ K : C ^ \infty ( M ) \to \mathcal { D } ' ( M ) \end{align*}


\begin{align*} D _ \Theta X _ T ^ { ( \ast , n ) } \le \left ( D _ \Theta X ^ { ( n ) } \right ) _ T ^ \ast : = \sup _ { s \in [ 0 , T ] } D _ \Theta X _ s ^ { ( n ) } . \end{align*}


\begin{align*} \theta = \sum _ { j = 1 } ^ { \ell } f _ j d \log z _ j + \sum _ { k = \ell + 1 } ^ { n } f _ k d z _ k . \end{align*}


\begin{align*} c _ k ( D _ n ) = c _ k ( \lambda E ( 1 , n ) ) , \end{align*}


\begin{align*} \mathrm { R W } _ 0 ( d ) & : = \inf _ { \gamma \in \Sigma _ 0 ( \delta ) } \mathbb { E } [ Y _ 1 ( 1 - d ( X ) ) + Y _ 2 d ( X ) ] \\ \mathrm { R W } ( d ) & : = \inf _ { \gamma \in \Sigma ( \delta ) } \mathbb { E } [ Y _ 1 ( 1 - d ( X ) ) + Y _ 2 d ( X ) ] , \end{align*}


\begin{align*} \begin{aligned} T _ 4 = & s \iint _ { Q } ( A \nabla \sigma ) _ i \frac { \partial A } { \partial x _ i } \nabla u \cdot \nabla u d x d t \ge - C s \lambda \iint _ Q \xi A \nabla u \cdot \nabla u d x d t . \end{aligned} \end{align*}


\begin{align*} T ( f ) ( z ) : = - \frac { 1 } { \pi } \iint _ D \frac { f ( \zeta ) } { \zeta - z } \ , d \xi \ , d \eta . \end{align*}


\begin{align*} \gamma _ { \xi } = \overline { \gamma _ { \overline { \xi } } } \end{align*}


\begin{align*} \gamma = \left \{ w \in \C ; | w | = R \left ( \frac { w } { | w | } \right ) \right \} \end{align*}


\begin{align*} \langle z ^ { r } , & z ^ { r ' } \rangle = \frac { 1 } { \pi } \int _ { \mathbb { C } } z ^ { r } \cdot \overline { z ^ { r ' } } e ^ { - | z ^ 2 | } \ d x \ d p , \ z = x + \sqrt { - 1 } p . \end{align*}


\begin{align*} \Delta \chi _ T ( t ) & = \ < Y _ T , \lambda _ 1 ^ t ( \lambda _ 1 - 1 ) \ > = \ < \Delta Y _ T , \lambda _ 1 ^ t \otimes ( \lambda _ 1 - 1 ) \ > \\ & = \sum _ { ( T ) } \ < Y _ { T ( 1 ) } \otimes Y _ { T ( 2 ) } , \lambda _ 1 ^ t \otimes ( X _ \bullet + X _ { \bullet \bullet } + \cdots ) \ > \\ & = \ < Y _ { T _ 1 } \cdots Y _ { T _ k } , \lambda _ 1 ^ t \ > \quad \\ & = \chi _ { T _ 1 } ( t ) \cdots \chi _ { T _ k } ( t ) . \end{align*}


\begin{align*} \prod _ { \substack { ( a , b , c , d ) = 1 \\ a , b , c , d \geq 1 } } \left ( \frac { 1 } { 1 - w ^ a x ^ b y ^ c z ^ d } \right ) ^ { \frac { 1 } { a ^ r b ^ s c ^ t d ^ u } } = \exp \left \{ \left ( \sum _ { h = 1 } ^ { \infty } \frac { w ^ h } { h ^ r } \right ) \left ( \sum _ { i = 1 } ^ { \infty } \frac { x ^ i } { i ^ s } \right ) \left ( \sum _ { j = 1 } ^ { \infty } \frac { y ^ j } { j ^ t } \right ) \left ( \sum _ { k = 1 } ^ { \infty } \frac { z ^ k } { k ^ u } \right ) \right \} . \end{align*}


\begin{align*} f _ 1 ( x _ i ) = \left \{ \begin{array} { l l } f ( x _ i ) , & \hbox { $ i = 1 , 2 , \ldots , n $ ; } \\ 0 , & \hbox { $ i = n + 1 $ . } \end{array} \right . \end{align*}


\begin{align*} \int _ { \Omega } a ^ { i j , k l } ( D ^ { 2 } u ) u _ { i j } \eta _ { k l } d x = 0 , \forall \eta \in C _ { 0 } ^ { \infty } ( \Omega ) , \end{align*}


\begin{align*} A + B = \{ a + b : a \in A , \ , b \in B \} , A \cdot B = \{ a b : a \in A , \ , b \in B \} \ , . \end{align*}


\begin{align*} \Delta ( f ) _ { \mid p } \coloneqq \sum _ { i = 0 } ^ { \infty } ( a ( i , \ldots , i ) \bmod p ) t ^ i \in \mathbb F _ p [ [ t ] ] \ , . \end{align*}


\begin{align*} \mathcal { O } _ i = \left \{ ( A , B ) \in \binom { [ n ] } { k } \times \binom { [ n ] } { k } : \ | A \cap B | = k - i \right \} . \end{align*}


\begin{align*} f ( q ) V \ : = \ \oplus _ { i \in \Z } V ^ { a _ i } \{ i \} , \ \ f ( q ) = \sum _ i a _ i q ^ i , \ a _ i \in \mathbb { N } , \end{align*}


\begin{align*} y _ { q s } ^ { k + \ell + 1 } = \frac { y _ { q p } ^ { k + \ell } } { y _ { s p } ^ { k + \ell } } . \end{align*}


\begin{align*} f : \square [ \underline { n } ] = ( \square [ n _ 1 ] * \dots * \square [ n _ p ] ) * \square [ n _ { p + 1 } ] \longrightarrow \square [ n ] \end{align*}


\begin{align*} \lambda _ { j _ 1 } = \lambda _ { j _ 1 ^ * } = \lambda _ { j _ 2 } = \lambda _ { j _ 2 ^ * } = \lambda _ t \end{align*}


\begin{align*} \tilde { \Lambda } ( z ) = \ln ' ( \Lambda ( z ) ) \pi ( z ) , \end{align*}


\begin{align*} \mathfrak { S } ( f _ k ) + \overline { \mathfrak { S } } ( f _ k ) \geq ( v _ l - z _ l - z _ r ) + 2 z _ r = v _ l - z _ l + z _ r . \end{align*}


\begin{align*} \left \{ \begin{aligned} & \langle D _ u f ( u ^ * ) , v \rangle _ { \mathcal { U } } + \langle \bar { \lambda } , G ' ( u ^ * ) v \rangle = 0 \forall \ v \in \mathcal { U } ; \\ & - \langle \bar { \lambda } , \zeta - G ' ( u ^ * ) u ^ * \rangle \geq 0 \forall \ \zeta \in K , \\ \end{aligned} \right . \end{align*}


\begin{align*} x _ { 1 } = \exp ( X ) . \end{align*}


\begin{align*} f ( \cdot , t ) = f _ 0 \circ \big ( y ^ { ( i , k , n ; L ) } _ t \big ) ^ { - 1 } . \end{align*}


\begin{align*} \int _ { \mathbb { R } + i B } \hat { \psi } ( \lambda ) \frac { d } { d \lambda } \log \zeta _ 1 ( \lambda ) d \lambda = - 2 \pi i \left \langle \sum \limits _ \gamma \frac { T _ \gamma ^ \# \delta ( t - T _ \gamma ) } { | \det ( I - \mathcal { P } _ \gamma ) | } , \psi \right \rangle \end{align*}


\begin{align*} \mathfrak { h } _ j ( \varphi ) & = \lim _ { k \to \infty } Q ( 0 , k ) ^ { - 1 } \circ P _ { U ^ { ( k ) } _ j } \circ \mathcal { M } ^ { ( k ) } \circ S ( k ) ( \varphi ) \\ & = \sum _ { l \geq 0 } Q ( 0 , l ) ^ { - 1 } \circ P _ { U ^ { ( l ) } _ j } \circ \big ( \mathcal { M } ^ { ( l ) } \circ S ( l ) - Z ( l ) \circ \mathcal { M } ^ { ( l - 1 ) } \circ S ( l - 1 ) \big ) ( \varphi ) \end{align*}


\begin{align*} f ( k , t , n ) = F ( k , t , n ) - F ( k - 1 , t , n ) . \end{align*}


\begin{align*} u _ 0 ( r ) = C _ \lambda r ^ \lambda = C _ \lambda | x | ^ \lambda , v _ 0 ( r ) = C _ \mu r ^ \mu = C _ \mu | x | ^ \mu \end{align*}


\begin{align*} n = 1 + \ln \sigma ^ { - \frac { 1 } { \ln 2 } } = 1 + \log _ { \frac { 1 } { \eta ' } } \sigma ^ { \frac { \ln \eta ' } { \ln 2 } } . \end{align*}


\begin{align*} f ( Z ) = ( I \otimes x ^ * ) \left ( I - \sum _ { j = 1 } ^ d Z _ j \otimes T _ { 0 , j } ^ * \right ) ^ { - 1 } \left ( \sum _ { j = 1 } ^ d Z _ j \otimes ( T _ j ^ * x ) \right ) . \end{align*}


\begin{align*} w ( z ) = \Phi _ { 0 } ( z ) + \sum _ { k = 1 } ^ { n - 1 } \left ( T ^ k ( \Phi _ k ) ( z ) \right ) + T ^ n ( f ) ( z ) , \end{align*}


\begin{align*} M _ 1 ( f ) = \int _ { Q _ { 0 , \ldots , 0 } } T _ t ( f ) d t = \int _ { [ 0 , 1 ) ^ d } T _ t ( f ) d t . \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } ( I _ \mu * | u | ^ { p } ) | v | ^ { q } \leq D _ 0 ( | \nabla u | ^ 2 _ 2 + | \nabla v | ^ 2 _ 2 ) ^ { \frac { \gamma _ p + \gamma _ q } { 2 } } , \ D _ 0 : = \frac { 1 } { \gamma _ p + \gamma _ q } \Bigl ( \frac { \gamma _ p + \gamma _ q - 2 } { 2 ( \gamma _ p + \gamma _ q ) l ( a , b ) } \Bigr ) ^ { \frac { \gamma _ p + \gamma _ q - 2 } { 2 } } , \end{align*}


\begin{align*} \lambda _ { j _ 1 } = \lambda _ { j _ 1 ^ * } = \dots = \lambda _ { j _ r } = \lambda _ { j _ r ^ * } \end{align*}


\begin{align*} I ' = \left \{ i \in I ^ * \ \left | \ Q _ i \in \textbf { Q } _ { n _ 0 } \right . \right \} . \end{align*}


\begin{align*} \| T u \| _ { ( L ^ p ( V ) ) ^ \ast } = \sup _ { \| f \| _ p = 1 } \langle T u , f \rangle \leq \| u \| _ q . \end{align*}


\begin{align*} \frac { B ( r ) } { B ' ( r ) } \cdot \left [ \mathcal Y ( r ) ^ { p - 1 } \right ] ' + \mathcal Y ( r ) ^ { p - 1 } = Q _ 2 ( \mathcal V ( r ) , v _ 0 ( r ) ) \cdot Q _ 3 ( \mathcal W ( r ) , u _ 0 ' ( r ) ) , \end{align*}


\begin{align*} Q _ { 1 , \vec { v } _ { s , t , f , 0 } } = \end{align*}


\begin{align*} \langle b \delta _ t , \lambda ( x ) b \delta _ t \rangle = \big \langle b \delta _ t , \sum _ { s \in G } \lambda _ s ( b _ s ) b \delta _ t \big \rangle = \sum _ { s \in G } \langle b \delta _ t , b _ s b \delta _ { s t } \rangle = b ^ * b _ e b = b ^ * \mathbb { E } ( x ) b . \end{align*}


\begin{align*} f ' = a _ 1 + 2 a _ 2 x + \ldots + r a _ r x ^ { r - 1 } . \end{align*}


\begin{align*} \mu ^ \delta _ { K S } ( f ) ( \tau ) = \left ( \frac { \eta ^ \delta ( \tau ) } { \eta ^ \delta ( p \tau ) } \right ) ^ { 2 ( c ^ 2 - 1 ) } = \left ( \frac { \Delta ^ \delta ( \tau ) } { \Delta ^ \delta ( p \tau ) . } \right ) ^ { ( c ^ 2 - 1 ) / 1 2 } \end{align*}


\begin{align*} \mu ( f ) = \gamma _ i + v \left ( \tilde { b } \right ) . \end{align*}


\begin{align*} \widetilde { R } ( h , x ) = \begin{cases} ( R h , x ) & \underline { \varphi } ( h ) ( 0 ) = \varphi ( h ) ( 0 ) , \\ ( R h , \sigma x ) & 0 = \underline { \varphi } ( h ) ( 0 ) < \varphi ( h ) ( 0 ) = 1 . \end{cases} \end{align*}


\begin{align*} \mathcal { K } _ { c l r \left ( i \right ) } = \mathcal { K } _ { c l r \left ( i - 1 \right ) } \setminus k _ { n e w \left ( i - 1 \right ) } \end{align*}


\begin{align*} S _ { \mathrm { o f f } } ^ { c < p ^ 2 } : = \sum _ { d _ 1 , \ , d _ 2 < D } \ , \ , \ , \ & \sum _ { I d _ 1 , \ , J d _ 2 < D } x _ { I d _ 1 } \bar { x } _ { J d _ 2 } \sqrt { I J } \sum _ { L , M \geq 1 } \frac { G _ k ( L M d _ 1 d _ 2 \slash p ) } { \sqrt { L M } } \\ & \times \left ( 2 \pi i ^ { 2 k } \sum _ { \substack { p | c \\ c < p ^ 2 } } \frac { \mathcal { S } ( I L , J M , c ) } { c } \ , J _ { 2 k - 1 } \left ( \frac { 4 \pi \sqrt { I L J M } } { c } \right ) \right ) \end{align*}


\begin{align*} \lambda = 1 / N ^ { \frac { 5 } { 3 } + \alpha } 1 \leq n \leq N ^ { 1 / 3 } \ , \epsilon = 1 / N ^ \beta \end{align*}


\begin{align*} - X '' + R X = \lambda X , \end{align*}


\begin{align*} \mathbf { P G } ^ 1 ( \mathbb { Q } _ p ) \rightarrow \prod _ { \ell \in \mathcal { S } , \sigma \in \mathcal { T } ( \ell ) } \mathbf { P G } ^ 1 [ \ell , \sigma ] \backslash \mathbf { P G } ^ 1 ( \mathbb { Q } _ p ) = \Gamma \backslash \prod _ { \ell \in \mathcal { S } , \sigma \in \mathcal { T } ( \ell ) } \mathbf { P G } ^ 1 ( \mathbb { Q } _ p ) \end{align*}


\begin{align*} z \bigl ( Y , \ , t \bigr ) = y ^ { * } ( N , \ , t ) , \ \forall t , \ \pi \end{align*}


\begin{align*} | f ( u ) - f ( v ) | \leq C | u - v | \sum _ { k = 0 } ^ { \infty } \frac { \lambda ^ k } { k ! } \left ( | u | ^ { p k + m - 1 } + | v | ^ { p k + m - 1 } \right ) . \end{align*}


\begin{align*} \left . \begin{array} { l l } \displaystyle G ( t ) = \int _ { \mathbb { R } ^ N } | x | ^ 2 \left ( a _ 2 | \phi | ^ 2 + a _ 1 | \psi | ^ 2 \right ) d x . \end{array} \right . \end{align*}


\begin{align*} & \quad \Pr \left ( > \theta | \Phi , \Phi ( \mathcal { A } ) > 0 \right ) \\ & \approx \sum _ { m = 1 } ^ { M } C _ { M } ^ m ( - 1 ) ^ { m + 1 } e ^ { - m \eta \theta r _ 1 ^ { \alpha } / \rho } \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \ ! \prod \limits _ { x _ { i } \in \Phi \backslash \{ x _ 1 \} \cap { \mathcal { A } } } \frac { 1 } { \left ( 1 + \frac { m \eta \theta r _ 1 ^ { \alpha } } { M r _ i ^ { \alpha } } \right ) ^ M } . \end{align*}


\begin{align*} \left \{ \begin{aligned} & - \epsilon ^ 2 \Delta u + V ( x ) u = \lambda u + u \log u ^ 2 , \ , \ , \ , \ , \hbox { i n } \mathbb { R } ^ N , \\ & \int _ { \mathbb { R } ^ { N } } | u | ^ { 2 } d x = a ^ { 2 } \epsilon ^ N , \end{aligned} \right . \end{align*}


\begin{align*} a _ p + \sum _ { i = 1 } ^ n \alpha _ i z _ { i p } = c _ p + \sum _ { i = 1 } ^ n \gamma _ i z _ { i p } < b _ p + \sum _ { i = 1 } ^ n \beta _ i z _ { i p } , \end{align*}


\begin{align*} x y - z t = 1 \end{align*}


\begin{align*} W _ { \vec { k } , \vec { l } } ( \vec { y } , \vec { x } ) & = \overline { W _ { \vec { k } , \vec { l } } } ( \vec { x } , \vec { y } ) , \tilde { W } _ { \vec { k } , \vec { l } } ( \vec { y } , \vec { x } ) = \overline { \tilde { W } _ { \vec { k } , \vec { l } } } ( \vec { x } , \vec { y } ) , \check { W } _ { \vec { k } , \vec { l } } ( \vec { y } , \vec { x } ) = \overline { \check { W } _ { \vec { k } , \vec { l } } } ( \vec { x } , \vec { y } ) , \end{align*}


\begin{align*} \int _ { \mathbb { R } ^ N } ( I _ \mu \ast | u _ n | ^ { 2 ^ * _ \mu } ) | u _ n | ^ { 2 ^ * _ \mu } = 1 , | \nabla u _ n | ^ 2 _ 2 \rightarrow S _ { H , L } , \quad \ n \rightarrow \infty . \end{align*}


\begin{align*} S ( p , q , a , b , c , d ) = \begin{pmatrix} 0 & 1 \\ p & q \\ a & b \\ c & d \end{pmatrix} . \end{align*}


\begin{align*} \alpha _ { g } = \displaystyle \lim _ { m \rightarrow \infty } \dfrac { C _ { D ^ { k - 1 } ( f ) } ( p ^ { 2 m + 1 } ) } { \beta ^ { 2 m } } . \end{align*}


\begin{align*} \chi _ { k , \psi } = \iota _ { \xi _ { k , \psi } } = \iota _ { \xi _ { k , \psi } \cdot \chi _ { 0 } } . \end{align*}


\begin{align*} \rho _ 1 = \frac 1 2 \min \{ { C _ { t } ^ { - 1 } } , \ell ^ { - 1 } \alpha \} \end{align*}


\begin{align*} \frac { \partial } { \partial h } E _ 1 ( h ; y , u ) = - h ^ { - 1 } \cdot \frac { \partial } { \partial h } \lambda ( y ^ u , y ^ h ) + h ^ { - 2 } \lambda ( y ^ { u } , y ) - h ^ { - 2 } \lambda ( y ^ { u - h } , y ^ h ) . \end{align*}